[    {        "title": "0705.0474v1.Domain_imaging_and_domain_wall_propagation_in__Ga_Mn_As_thin_films_with_tensile_strain.pdf",        "content": " 1Domain imaging and domain wall propagation in (Ga,Mn)As thin \nfilms with tensile strain \nK. Y. Wang \n Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom \n A. W. Rushforth, V. A. Grant,  R. P. Campion, K. W. Edmonds, C. R. Staddon, \nC. T. Foxon, and B. L. Gallagher \nSchool of Physics and Astronomy, Un iversity of Nottingham, Nottingham NG7 \n2RD, United Kingdom \n                                        J. Wunderlich, D. A. Williams \n           Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom \n2RD, United Kingdom \n \nAbstract \n \nWe have performed spatially resolved Pola r Magneto-Optical Kerr Effect Microscopy \nmeasurements on as-grown and annealed Ga 0.95Mn 0.05As thin films with tensile strain. \nWe find that the films exhibit very strong perpendicular magnetic anisotropy which is \nincreased upon annealing. During magnetic reversal, the domain walls propagate \nalong the direction of surface ripples for the as-grown sample at low temperatures and along the [110] direction for the annealed sample. This indicates that the magnetic \ndomain pattern during reversal is determin ed by a combination of magnetocrystalline \nanisotropy and a distribution of pinning si tes along the surface ripples that can be \naltered by annealing. These mechanisms c ould lead to an effective method of \ncontrolling domain wall propagation. \n \n \n  2 \n \nThe (III,Mn)V ferromagnetic semiconductor family has attracted much attention for \nits potential applications in spintronics, in which, for examples, logic and memory operations may be integrated on a singl e device [1,2]. (Ga,Mn )As, with Curie \ntemperature as high as 173 K [3 ], is one of the most pr omising candidates for such \napplications. Understanding and control ling the magnetic properties, including \ndomain structure and domain wall propagation in these materials is important for \nrealizing and manipulating spintronics devices.  \n (Ga,Mn)As epilayers grown on a relaxed ( 001) (In,Ga)As buffer layer experience a \ntensile strain due to the difference in the lattice constant in each layer. Under these \nconditions the magnetic easy axis is known to  be perpendicular to the plane [4] in \nagreement with theoretical predictions [1 ]. Magnetic domain patterns in (Ga,Mn)As \nwith perpendicular magnetic anisotropy have been studied previously using Scanning \nHall Probe Microscopy [5] and Polar Ma gneto-Optical Kerr Effect Microscopy \n(PMOKM) [6-8].Ref. 5 showed that domains formed a stripe pattern with a typical \nwidth of a few microns at lo w temperatures. The stripe dom ains were aligned nearly \nparallel to the <110> directions. This was attributed to th e magnetocrystalline \nanisotropy of the films. In this paper we employ PMOKM to study magnetic domains \nover an area (150 µm ×150 µm ) much larger than reported  in Ref. [5] (7.3 µm × 7.3 \nµm), and show a clear correlation between magnetic and topographic features. These \nimages reveal that, at the lowest temperatures the magnetic stripe domains are aligned along the topological crosshatch  pattern, known to form along the <110>  directions \ndue to mismatch dislocations in the buffe r layer [ 9, 10]. This indicates that an \nadditional mechanism, besides the magnetocr ystalline anisotropy, is responsible for  3determining the size and direction of  the magnetic stripe domains during \nmagnetization reversal at low temperatures.  The epilayer structure was grown by Molecular Beam Epitaxy using a modified \nVarian GEN-II system [11]. The growth sequence consisted of a semi-insulating \nGaAs(001) substrate onto which was deposit ed a high temperature GaAs buffer layer \nat 580°C. This was followed by a 580nm In\n0.15Ga0.85As layer grown at 500°C and a \n25nm Ga 0.95Mn 0.05As layer grown at 255°C. Post-gro wth annealing was performed in \nair at 190 0C for 120 hours, which is an establis hed procedure for increasing the T C of \n(Ga,Mn)As thin films [12]. X-ray diffracti on measurements were performed using a \nPhilips X’Pert Materials Research Diffract ometer. By analysing both symmetric and \nasymmetric reciprocal space maps taken in the two <110> directions [13], the degree \nof relaxation of the (In,Ga)As layer was estimated to be 70% in both directions, \nconsistent with previous work [14] which showed that above a certain thickness, the \n(In,Ga)As layer can support it self without further relaxa tion. The wafer was cleaved \ninto separate pieces for PMOKM meas urements and SQUID magnetometry. The \nPMOKM images were obtained using a comme rcial system with a high pressure Hg \nlamp and a high resolution CCD camera givi ng a spatial resolution of 1 µm. The  \nSQUID measurements were carried out us ing a commercial Quantum Design system. \n \nFigures 1(a) and (c) show magnetic hysteresis loops measured using PMOKM for the \nas-grown and annealed film respectively. The Kerr rotation angle, averaged over the \nimage area, is proportiona l to the component of the magnetization pointing \nperpendicular to the plane of the film. Both samples show nearly  square hysteresis \nloops, indicating that the magnetic easy axis  is perpendicular to  the plane over the \nwhole temperature range below the Curie temperature (T C). This is also confirmed by  4the temperature dependence of the remnant magnetization (Figs1 (b) and (d)), which \nis fit well by a Brillouin function with S= 5/2. Low temperature annealing increases T C \nfrom 66K to 137K and increases the magneti zation, as observed by the increase in the \nKerr rotation angle and confirmed by SQUID magnetometry. This is expected since \nthe annealing process remove s interstitial Mn ions [15]  which act as double donors \nand couple antiferromagnetically  to the substituti onal Mn ions. The removal of these \nimpurities also leads to increase of the tensile strain and increase of hole density [16], thus resulting in a stronger perpendicular magnetic anisotropy [17]. Evidence for this is the fact that, for the annealed sample, the hysteresis loops are more square. The \ncoercive field is also reduced, possibly due  to a reduction in the number of pinning \nsites or the pinning energy, as will be discus sed later. We note that, for the as–grown \nsample, the T\nC measured by SQUID is slightly smaller than that measured by \nPMOKM, probably due to small variations be tween the two pieces of the wafer used \nfor each measurement. There is also a sm all in-plane component of the remnant \nmagnetization measured in the as-grown samp le. This is not observed in the annealed \nsample.  Figures 2(a) and (b) show Atomic Force Microscopy (AFM) images of the as-grown \nsample. These images reveal ripples in th e surface running along the <110> directions \nto form a crosshatch pattern. The ridges r unning along the [110] directions have an \naverage separation of 1 µm and those runni ng along the [1-10] direction have an \naverage separation of 0.5 µm. The ridges are typically ~10nm in height. These ridges are known to occur during the growth of the (In\n,Ga)As buffer layer, due to the \nformation of strain-relieving misfit dislocatio ns [8]. PMOKM images at 6K of an as-\ngrown sample from the same wafer are shown in Figs. 2(c) and (d). Initially, the film  5is saturated with negative magnetic field of -300 Oe, which is much larger than the \ncoercive field. The field is then swept to 145 Oe, just less than the coercive field. \nPMOKM images are then captured at a rate  of 15 frames per second. The domain \nimages captured at 0s and 1.3s are shown in Figures 2(c) and (d), respectively. These images reveal that the magnetization re versal proceeds through the nucleation and \npropagation of magnetic domains  (shown by dark features in  Figs 2(c) and (d)). The \ndomains propagate along the ripples, predom inantly in the [110] direction, increasing \nin length, but not in width. After 12s a stable  state is obtained indicating that thermal \nexcitation and the small applied field do not provide enough energy to overcome the local barriers to domain wall motion. The magnetic domains form stripes of typical width 2-3 µm, a similar length scale to th e separation of the ridges. At higher \ntemperatures, the magnetic reversal has two main differences to that at low \ntemperatures. Firstly, full reversal occurs  in a time period of tens of seconds. \nSecondly, the domain walls propagate along ra ndom directions forming a cauliflower-\nlike domain pattern during the magnetic reversal and the average domain size increases with increasing temperature. This  is due to a lowering of the in-plane \nmagnetic anisotropy barriers and a more e ffective thermal activation of domain wall \nmotion. This behaviour at high temperatures is similar to that observed in a previous \nPMOKM study of tensile strain ed (Ga,Mn)As films [6]. \n Figure 3 shows successive snapshots of magnetic domain reversal at T =90 K  in the \n(Ga,Mn)As film after annealing. Simila r images are observed over the whole \ntemperature range for the annealed sample. They differ from the images obtained for \nthe as-grown sample in that the reversed domains are typically much larger (several tens of microns) during the reversal. Th e domain walls align along the [1-10]  6direction and propagate rapidl y along the [110] axis betw een pinning sites, until the \nmagnetization is almost fully reversed with only a few unreversed stripe domains remaining. The width of the unreversed stri pe domains is a few microns, which is \nmuch wider than the typical domain wall width (~15 nm) in (Ga,Mn)As [18]. At lower temperatures similar behaviour is observed.   A previous study [5] of the magnetic do main structure of (Ga,Mn)As with \nperpendicular magnetic anisotropy found that  the magnetic domains formed a stripe \npattern and the width of th e stripe domains did not de pend strongly on temperature \nuntil temperatures close to T\nC. Also the orientation of the stripes rotated from the \n<110> directions to the <100>  directions as temperature increased. For our as-grown \nsample stripe domains are only observed at  T<20K and are always oriented along the \n<110> directions. This suggests that the fo rmation of stripe domains in our sample \narises because the domain walls are pinned at  features which are themselves aligned \nalong the <110> directions. Above about 20K , thermal excitation is sufficient to \novercome the energy barriers at  these pinning sites and th e stripe domain pattern is \nnot observed. Recently, it has been proposed [10] that it may be energetically favourable for the interstitial  Mn atoms to be concentrated at the ridges of the \ncrosshatch pattern. Alternatively, the misfit dislocations may provide more favourable \nsites for Mn interstitials in the trough regi ons. In either case, a distribution of the \ndensity of Mn interstitials along the direction of the su rface ripples may provide a \nnetwork of pinning sites that cause the domain  walls to be preferen tially aligned along \nthe <110> directions. Annealing removes the interstitials, thereby reducing the \ndensity of such pinning sites and the stri pe domain pattern is no longer observed at \nany temperature. However, the fact that the domain walls are always  aligned with the  7[110] direction after annealin g may be due to residual pinni ng sites or may be due to \nthe weak magnetocrysta lline anisotropy [5]. \n We have studied the magnetic prope rties of a tensile strained Ga\n0.95Mn 0.05As thin film \nby using SQUID and PMOKM. Both the as-gro wn and annealed samples show very \nstrong uniaxial anisotropy with easy axis  along [001] direc tion and the Curie \ntemperature up to 137 ± 2K for the annealed sample. During the magnetic reversal, for the as-grown sample the domain walls te nd to propagate along th e direction of the \nsurface ripples at low temperatures, which i ndicates that an additional mechanism to \nthe magnetocrystalline anisotropy is respons ible for the magnetic domain pattern and \nthe propagation of magnetic domain walls. Th is may be the presence of pinning sites \ncaused by Mn interstitials distributed along the crosshatch pattern. This mechanism \nand the strongly anisotropic domain wall  motion found after annealing have \nimplications for studies of current-dri ven domain wall motion in ferromagnetic \nsemiconductor devices and may provide an effective method for control of the formation of magnetic domains a nd propagation of domain walls. \nACKNOWLEDGEMENT \nThis project was supported by EC si xth framework grant: FP6-IST-015728 and \nEPSRC (GR/S81407/01). \n \nREFERENCES: \n \n1 T. Dietl, H. Ohno, F. Matsukura, J. Cibèrt, and D. Ferrand, Science 287, 1019 (2000). \n2 S. A. Wolf, D. D. Awschalorn, R.  A. Buhrman, J. M. Daughton, S. Von Molnár, M. L. Roukers, A. Y. \nChtchlkanova, and D. M. Treger, Science, 294, 1488 (2001). \n3 T. Jungwirth, K. Y. Wang, J. Mašek, K. W. Edmonds,  J. König, J. Sinova, M. Polini, N. A. Goncharuk, \nA. H. MacDonald, M. Sawicki, A. W. Rushforth, R. P. Campion, L. X. Zhao, C. T. Foxon, and B. L.  8Gallagher, Phys. Rev. B 72, 165204 (2005); K. Y. Wang, R. P. Campion, K. W. Edmonds, M. Sawicki, \nT. Dietl, C. T. Foxon, and B. L. Gallagher, Proc. 27th. Conf. on Phys. Of Semicon., edited by J. \nMenéndez and C. Van de Walle, Flagstaff, AZ, July 2004 (Melville,  New York, 2005), p. 333; e-print: \ncond-mat/0411475. 4A. Shen, H. Ohno, F. Matsukura, Y. Sugawara, N. Akiba, T. Kuroiwa, A. Oiwa, A. Endo, S. \nKatsumoto, and Y. Iy , J. Crystal Growth 175/176 , 1069 (1997) . \n5 T. Shono, T. hasegawa, T.Fukumura, F. Matsukura and H. Ohno, Appl. Phys. Lett.  77, 1363 (2000). \n6 L. Thevenard, L. Largeau, O. Mauguin, G. Patriarche, A. Lemaitre, N. Vernier, and J. Ferre. Phys. Rev. B, 73, 195331 (2006). \n7 M. Yamanouchi, D. Chiba, F. Matsukura, T. Dietl, and H. Ohno, Phys. Rev. Lett. 96, 096601 (2006). \n8 D. Chiba, F. Matsukura and H. Ohno, Appl. Phys. Lett. 89,162505 (2006).  \n9 O. Yastrubchak, T. Wosinski, T. Figielski, E.  Lusakowska, B. Pecz an d A. L. Toth, Physica E 17, 561 \n(2003). \n10 O. Maksimov, B. L. Sheu, G. Xiang, N. Keim , P. Schiffer, and N. Sam arth, J. Cryst. Growth, 269, \n298 (2004). \n11 R. P. Campion, K. W. Edmonds, L. X. Zhao, K. Y. Wang, C. T. Foxon, B. L. Gallagher, and C. R. \nStaddon,  J. Cryst. Growth  247, 42 (2003). \n12 K. W. Edmonds, K. Y. Wang, R. P. Campion, A. C. Neumann, N. R. S. Farley, B. L. Gallagher, and C. T. Foxon, Appl. Phys. Lett. 81, 4991 (2002). \n13 P. F. Fewster and N. L. Andrew, “Strain a nalysis by x-ray diffraction”, Thin Solid Films, 319, 1 \n(1998). 14 L. K. Howard, P. Kidd and R. H. Dixon, J. Cryst. Growth 125, 281 (1992). \n15 K.W. Edmonds, P. Bogusÿawski, K.Y. Wang, R.P. Campion, S.N. Novikov, N.R.S. Farley, B.L. \nGallagher, C.T. Foxon, M. Sawicki, T. Dietl, M. Buon giorno Nardelli, and J. Bernholc, Phys. Rev. Lett. \n92, 037201 (2004). \n16 L. X. Zhao, C. R. Staddon, K. Y. Wang, K. W. Edmonds, R. P. Campion,  B. L Gallagher, C. T. \nFoxon, Appl. Phys. Lett. 86, 071902 (2005) . \n17 T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B, 63,195205(2001). \n18 T. Dietl, J. König, and A. H. MacDonald, Phys. Rev. B, 64, 241201(R) (2001). \n \n  9 \nFigure Captions: \n  \nFig.1 (a) and (c): Magneto-optical Kerr rota tion hysteresis loops obtained from \nPMOKM for (a) as-grown and (c) annealed Ga 0.95Mn 0.05As thin film at different \ntemperatures. (b) and (d) Temperature depe ndent of normalized magnetization from \nPMOKM (solid symbols) and SQUID (open symbols) for (b) the as-grown and (d) \nannealed Ga 0.95Mn 0.05As thin film. The line is the Brillouin function with S=5/2, using \nthe Tc obtained from the PMOKM measurements.   Fig. 2 (a) Topology and (b ) derivative AFM image for the as-grown  Ga\n0.95Mn 0.05As \nthin film; successive snapshots of the magne tization reversal at 6K for the as-grown \nGa0.95Mn 0.05As thin film at H = 145 Oe, (c) 0 s; (d) 1.3 s  respectively.  \n \n \n \nFig.3 Successive PMOKM snapshots of the magnetic domain pattern during the \nmagnetization reversal at 90K for the annealed  Ga 0.95Mn 0.05As thin film at H = 23.7 \nOe, (a) 5 s; (b) 5.6 s; (c)7.3  s; (d) 9.23 s; and (e) 9.37s,  respectively. Arrow in (e) \nindicates a persistent residual stripe domain. \n \n \n \n  \n \n \n \n  \n \n   10 \n0.00.20.40.60.81.0\n0 30 60 90 120 1500.00.20.40.60.81.0\n-300 -200 -100 0 100 200 300-0.6-0.4-0.20.00.20.40.6-0.3-0.2-0.10.00.10.20.3(b)\n As-Grown\n [001]\n [001]\n [100]\n [001]\n  M/Ms\n(d)\n  Annealed\n [001]\n [001]\n [001]\n  M/MS\nT(K)(c)\n  θκ(deg.)\nH(Oe) 9K\n 70K\n 100K\n 135K(a)\n  θκ(deg.) 8K\n 35K\n 60K\n \n \n \n  \nFig.1 K. Y. Wang et al. \n   \n \n \n \n  \n \n  \n \n  \n \n  \n \n  \n \n  \n \n \n \n  \n \n  \n \n  11 \n \n \n  \n[110]\n[1-10](a)\n(d) (c)(b)\n[110]\n[1-10](a)\n(d) (c)(b)\n \n \n  \n \n \nFigure 2 K. Y. Wang et al. \n   \n \n  \n \n  \n \n  \n \n  \n  12 \n \n \n  \n[110]\n[1-10] (c)(b) (a)\n(d)\n(e)\n[110]\n[1-10] (c)(b) (a)\n(d)\n(e)\n[110]\n[1-10] (c)(b) (a)\n(d)\n(e)\n \n \n \n \nFigure 3 K. Y. Wang et al.  \n \n \n \n "    },    {        "title": "0705.2870v1.Anisotropy_and_magnetization_reversal_with_chains_of_submicron_sized_Co_hollow_spheres.pdf",        "content": " 1Anisotropy and magnetization reversal with chains of \nsubmicron-sized Co hollow spheres \nLin He and Chinping Chena \n1Department of Physics, Peking Univer sity, Beijing 100871, People’s Republic of \nChina a) E-mail : cpchen@pku.edu.cn\n, Phone : +86-10-62751751  \nFang Liang and Lin Guob \n2School of Materials Science and E ngineering, Beijing University of \nAeronautics and Astronautics, Beijin g 100083, People’s Republic of China   \nb) E-mail: guolin@buaa.edu.cn  \nPACS : 75.30.Gw, 75.60.Jk, 75.50.Tt, 75.75.+a \nKeywords: chain of nano-sized Co hollow sp heres, coercivity, Néel-Brown analysis, \nmagnetic anisotropy, blocking temperature.  \n Abstract  \nMagnetic properties with chains of hcp Co  hollow spheres have been studied. The \ndiameter of the spheres ranges from 500 to  800 nm, with a typical shell thickness of \nabout 60 nm. The shell is poly crystalline with an average crystallite size  of 20 to 35 \nnm. The blocking temperature dete rmined by the zero-field-cooling M\nZFC(T) \nmeasurement at H = 90 Oe is about 325 K. The corre sponding effective anisotropy is \ndetermined as,  Keff ~ 4.6 ×104 J/m3. In addition, the blocking temperature and the  2effective anisotropy determined by the analysis on HC(T) are 395 K and 5.7 ×104 J/m3, \nrespectively. The experimentally determined anisotropy is smaller by one order of magnitude than the magnetocrystalline anisot ropy of the bulk hcp Co, which is about \n3 to 5 ×10\n5 J/m3. A further analysis on HC(T) shows that the ma gnetization reversal \nfollows a nucleation rotational mode with an effective switching volume, V* ~ 2.3 \n×103 nm3. The corresponding effective diameter is calculated as 16.4 nm . It is slightly \nlarger than the coherence length of Co, a bout 15 nm. The possible reason for the much \nreduced magnetic anisotropy is discussed briefly.                 31. Introduction \nAs the synthesis technique in material sc ience makes progress to reach the realms \nof nanometer, the emergence of nano-scaled magnetic material brings about a series of new applications due to its unique prope rties. Recently, much attention has been \nfocused on the fabrication of nano- or s ubmicron-sized hollow spheres because of \ntheir potential applications in  catalysts, artificial cells,  coatings, medical delivery \nvehicle systems for inks, and dyes [1-4], etc. The magnetic properties of the hollow \nsphere structure have been reported al so. For example, the magnetization ground \nstates of magnetic hollow sphere are studi ed by analytical cons ideration by D. Goll et \nal. [5]. Two stable ground stat es, the single domain and the vortex curling, have been \nobtained in theory for this special struct ure. Experimentally, there are many works \nconcerning the synthesis and magnetic propert ies of Ni or Co hollow spheres [6-8]. \nHowever, these works mainly concentrat e on the characteriz ation of magnetic \nproperties only. Therefore, an in-depth in vestigation is of great interest and \nimportance.      In our previous works, we have repo rted the synthesis and characterization of \nuniform chains of Co hollow spheres [9] or Co hollow spheres with each cavity void enclosing a solid nanoparticle [10]. In th e present paper, we report a study on the \nmagnetic anisotropy properties for the ch ains of hcp Co hollow structure. The \nexperimental details are presented in [9].  The blocking temperature obtained from the \nM\nZFC(T) measurement is, TBZFC(90Oe) ~ 325 K, whereas it is TBNB ~ 395 K \ndetermined by the Néel–Brown analysis on HC(T). The corresponding effective  4anisotropy constant is derived as Keff ~ 4.6 ×104 J/m3 and 5.7 ×104 J/m3, respectively. \nThese values are one order smaller than the magnetocrystalline anisotropy, K1, of the \nbulk hcp Co, which is about 3 to 5 ×105 J/m3 [11-13].  \n 2. Sample preparation and characterization  \nThe preparation procedure and mechanism of synthesis are reported in [9]. In \nbrief, the sample was synthesized by the reduction of CoCl\n2·6H 2O with hydrazine \nmonohydrate in the presence of poly(vinylpyrrolidone) (PVP, Mw 58,000). First, a \nsolution was prepared by dissolving cobalt sa lt and PVP in ethylene glycol (EG). This \nwas followed by a dropwise addition of hydr azine monohydrate at room temperature \nwith simultaneous vigorous agitation. S econd, the homogeneous mixture was heated \nto the boiling point of  EG for refluxing (197 oC). After the mixture was refluxed for 4 \nh, a dark precipitate was obt ained, which was separated by centrifugation and washed \nwith absolute ethanol.  \nThe sample has been characterized and pres ented in the previous  report [9]. The \nmajor results are described as follows. Th e XRD spectrum is in agreement with the \nhcp Co (JCPDS 05-0727). No impurity phase su ch as the cobalt oxide or the precursor \ncompounds have been detected. The structural analysis by high-resolution transmission electron microscopy (HRTEM ) has also confirmed the hcp crystal \nstructure [9]. Scanning electron microscopy (SEM) images of the sample are obtained by a JSM-5800, as shown in Fig. 1. The sample  consists of cobalt  hollow spheres with \nsize ranging from 500 to 800 nm. The sphere s are connecting with  each other in  5structure, forming branched necklace-like chai ns with a length of tens of micrometers. \nThe surface of these hollow spheres is r ough. It reflects a polycrystalline structure \nformed of Co nanocrystallites. According to the detailed investigation by SEM and \nTEM, as presented in [9], there is a wide distribution in the size of crystallites. The \naverage size is about 20-35 nm in diameter . The shell thickness is estimated to be \nabout 60 nm.  PVP plays a crucial role in the form ation of chains of hollow sphere \nstructure. It serves not only as the soft template, but also as the surface reagent. Without PVP, only Co solid spheres with ir regular shape were obtained. At the end of \nthe experiments, the residual PVP was washed away thoroughly.   \n \n3. Magnetic measurements and analysis \nThe temperature and field depe ndence of the magnetization, M(T) and M(H), was \ninvestigated by a Quantum Design SQ UID magnetometer. In particular, M(T) was \nstudied by zero-field-cooling (ZFC) and field-cooling (FC)  modes from 5 K to 380 K. \nFor the M\nZFC(T) curve, the sample was first cooled down to T = 5 K under zero \napplied field, and then the magnetization signal was recorded in an applied field, Happ \n= 90 Oe, in the warming process. For the MFC(T) curve, the procedure was the same, \nexcept that the sample was cooled under a cooling magnetic field, HCOOL = 20 kOe.  \nIn Fig. 2, the curves for MZFC(T) and MFC(T) are presented. These two separate \nwidely from each other. In addition, a maximum shows up on the MZFC(T) curve, as \nshown in the inset of Fig. 2. The position of  the maximum is defined as the blocking \ntemperature, TBZFC(90 Oe) ~ 325 K. These properties are attributed to the presence of  6a magnetic anisotropy barrier, EA. Above TBZFC, MZFC(T) and MFC(T) still separate \nslightly from each other. This is attribut able to the size distribution of switching \nvolume, which in turn is dictated by the si ze of nanocrystallites forming the shell of \nsphere. The size of the switching vol ume in comparison with that of the \nnanocrystallites and the coherence length of Co will be elaborated later.  \nM(H) curves at T = 50, 70, 100, 150, 200, 250, 300, and 380 K were measured by \nthe sweeping field within the limit from  -50 kOe to 50 kOe. Fig. 3 shows three \nrepresentative hysteresis loops in the low field region measured at T = 50, 150 and \n380 K. We see that the remane nt magnetization and the coer civity decrease with the \nincreasing temperature. In the high field region, however, the saturation magnetization \nMS(T) stays almost a constant at these temp eratures. The inset shows the saturation \nbehavior of M(H) curve in the high field at T = 300 K. The saturation value is \ndetermined as 168 emu/g, which is equal to the bulk value. This reveals a different \nfeature from that of the recently reported nano-sized Ni hollow spheres with a much \nreduced saturation magnetizati on, ~ 24% [6] or ~ 38% [7] of the bulk value at 300 K.  \nFig. 4 shows the temperature dependence of  the reduced saturation magnetization, \nMS(T)/MS(50K), and the remanent ratio, MR(T)/ MS(T). The saturation magnetization \nalmost remains constant in the investigated  temperature range, whereas the magnitude \nof MR/MS decreases almost linearly with increa sing temperature. The remanent ratio \nof the sample is about 0.12 at 50 K, and it goes down to about 0.02 at 380 K. This is much smaller than the theoretically analyzed value, i.e. 0.5, for a system of randomly \noriented Stoner-Wohlfarth (SW) particles [14]. Hence, it is  a sign that the  7magnetization reversal of the present sample  is not by the coherent rotational mode.  \n \n4. Anisotropy and blocking temperature    \nThe magnetic anisotropy and the magnetizat ion reversal behavior are analyzed \nfrom the results of M(T) and M(H) measurements. In Fig. 5, the coercivity, HC(T), \ndetermined from the hysteresi s loops is plotted along with ΔM(T) = MFC – MZFC. \nThese two exhibit a proportio nal relation. The temperature dependent coercivity can \nbe analyzed by the Néel–Brown model [15,16] . By this analysis, the field dependent \nanisotropy barrier is written as EA(H) = E0(1-H/H0)α, where E0 is the energy barrier at \nzero field and H0 is the switching field at zero temp erature. By taking into account the \nthermal activation effect over this ba rrier, it is then expressed as EA(H) = \nkBTln[t(T)/t0], in which t is the time necessary to jump over the energy barrier at \ntemperature T and t0 is a constant typically  of the order from 10-9 to 10-11 s. Usually, \nln[t(T)/t0] equals to 25. The temperature depende nt property of the coercivity then \nfollows,  \n⎪⎭⎪⎬⎫\n⎪⎩⎪⎨⎧\n⎥⎦⎤\n⎢⎣⎡− =α/1\n0\n00 )/ln( 1 )( ttETkH THB\nC .         ( 1 )  \nFor particles with size larger than the magnetic coherence length, by taking into \naccount the intraparticle interaction in cluding the exchange and the dipolar \ninteractions between different magnetic domains and neglecting the interparticle \ninteraction, the exponent α has been shown theoretically ~ 3/2 by R. H. Victora [17].   \nCritical length scales for the magnetic localization and the related magnetization \nreversal property have been discussed in detail by Skomski et al. for transition-metal  8nanowires, including single crys tal and polycrystalli ne [18]. The crysta llite size of the \npresent polycrystalline sample is about 20 ~ 35 nm on average. It is larger than the \ncoherence length of Co, ~ 15 nm [19]. This  would put the sample in the ordinary \nweak coupling region in the phase diagram proposed by Skomski et al.  for the \npolycrystalline nanowire, see Fig. 2 in [ 18], despite the much more complicated \nstructure with the present sample. Therefore, HC(T) is expected to follow Eq. (1) with \nα = 3/2 for the magnetization reversal by nu cleation rotational mode. The dashed line \nin Fig. 5 is the fitting result of Eq. (1) by fixing α = 3/2. It describes reasonably well \nthe behavior of HC(T) and ΔM(T), as shown in Fig. 5. The values of the fitting \nparameters determined by Eq. (1) are H0 = 415 Oe and E0/[kBln(t/t0)] = 395 K. Thus, \nthe blocking temperature obtaine d by the Néel–Brown analysis on HC(T) is TBNB = \n395 K, which is larger than TBZFC = 325 K. It reflects the blocking property \ncorresponding to a switching volume size in the large end of the distribution. On the \nother hand, TBZFC describes the blocking property for the average switching volume \nsize.  \nThe effective switching volume is estimated as, V* ~ 2.3 ×103 nm3, according to \nthe expression V* = E0/(μ0MSH0) with E0 and H0 obtained from the fitting by Eq. (1) \n[20]. The corresponding effective diameter is about 16.4 nm. This is slightly smaller \nthan the average diameter of the nanocryst allites, 20 ~ 35 nm [9]. Importantly, this \nvalue is consistent with the coherence leng th of Co, ~ 15 nm. This  further supports the \npicture of magnetization reversal by nuc leation rotation. With the knowledge of TB \nand V*, the effective anisotropy, Keff, can be calculated by KeffV* = 25 kBTB. It is  9about 4.6 ×104 J/m3 derived from TBZFC or 5.7 ×104 J/m3 from TBNB. This is one order \nsmaller than the magnitude of magnetocr ystalline anisotropy of the bulk hcp Co, K1 ~ \n3 to 5 ×105 J/m3 [11-13]. For large particles form ed of nanocrystallites which are \nsmall with randomly oriented axes of anis otropy and in the strongly exchange coupled \nregion, the average magnetocrys talline anisotropy becomes complicated [21]. In the \npresent work, however, the shell thickness of about 60 nm is larger than the \nnanocrystallite size of 20 to 35 nm, which in turn is larg er than the switching volume \nsize of about 16.4 nm. The magne tization reversal for an in dividual grain proceeds by \nthe nucleation rotational mode. As shown by Fig. 3 in [18], a localization of the \nnucleation mode would lead to a softening of magnetism in a si ngle crystal nanowire. \nIn the present work, we consider spherical grains rather than elongated ones with the \nnucleation rotational mode. Perhaps, this is the reason leading to the significantly reduced anisotropy.   \n \n5. Conclusion  \nIn summary, the magnetic properties of ch ains of submicron-sized hcp Co hollow \nspheres have been investigated. The eff ective anisotropy is found to be one order \nsmaller than the magnetocrys talline anisotropy of the bul k hcp Co. According to the \nNéel–Brown analysis on H\nC(T), the magnetization reversal is by nucleation rotational \nmode. The blocking temperature dete rmined from the maximum of the MZFC(T) curve, \nmeasured in the applied field Happ = 90 Oe, is TBZFC ~ 325 K. This is lower than the \none derived from the behavior of HC(T), TBNB ~ 395 K. The difference between these  10two values is attributable to the size di stribution of the switching volume, which in \nturn is affected by the size of the nanocrys tallites forming the shell of hollow spheres.  \n Acknowledgement  \nAuthors acknowledge the support from the National Natural Science Foundation of \nChina (No. 20673009), the program for New Cent ury Excellent Talents in University \n(NCET-04-0164) and the Specialized Resear ch Fund for the Doctral Program of \nHigher Education (SRFDP-2006006005).  \n \n               11References \n[1] D. Zhang, L. Qi, J. Ma  and H. Cheng, Adv. Mater 14,1499  (2002). \n[2] A. B. Bourlinos, M. A. Karakassides and D.  Petridis, Chem. Commun. \n1518 (2001). \n[3] Z. Zhong, Y. Yin, B. Gates and Y. Yia, Adv. Mater. 12, 206 (2000). \n[4] F. Caruso, R. A. Caruso and H. Mohwald, Chem. Mater. 11, 3309 (1999).  \n[5] D. Goll, A. E. Berkowitz and H. N. Bertram, Phys. Rev.B 70, 184432 (2004). \n[6] Q. Li, H. Liu, M. Han, J. Zhu, Y . Liang, Z. Xu and Y . Song, Adv. Mater. 17, 1995   \n(2005). \n[7] J. C. Bao, Y . Y . Liang, Z. Xu and L. Si, Adv. Mater. 15, 1832 (2003). \n[8] H. Yoshikawa, K. Hayashida, Y. Ko zuka, A. Horiguchi, K. Awaga, S. Bandow \nand S. Iijima, Appl. Phys. Lett. 85, 5287 (2004). \n[9] L. Guo, F. Liang, X. G. Wen,  S. H. Yang, W. Z. Zheng, L.  He, C. P. Chen and Q. \nP. Zhong, Adv. Fun. Mater. 17, 425 (2007).  \n[10] F. Liang, L. Guo, Q. P. Zhong, X. G. Wen, C. P. Chen, N. N. Zhang and W. G. \nChu, Appl. Phys. Lett. 89, 103105 (2006). \n[11] R. Skomski, J. Phys.: Condens. Matter 15, R841 (2003). \n[12] D. Weller, G. R. Harp, R. F. C. Farro w, A. Cebollada and J. Sticht, Phys. Rev. \nLett. 72, 2097 (1994). \n[13] X. Liu, M. M. Steiner, R. Sooryakumar , G. A. Prinz, R. F. C. Farrow and G. \nHarp, Phys. Rev. B 53, 12166 (1996). \n[14] Sōshin Chikazumi, 1997, Physics of Ferromagnetism  2nd (Oxford: Oxford  12university press) P472. \n[15] L. Néel, Ann. Geophys. 5, 99 (1949). \n[16] W. F. Brown, Phys. Rev. 130, 1677 (1963). \n[17] R. H. Victora, Phys. Rev. Lett. 63, 457 (1989). \n[18] R. Skomski, H. Zeng, M. Zheng, and D. J. Sellmyer, Phys. Rev. B 62, 3900 \n(2000). \n[19] D. J. Sellmyer, M. Zheng and R. Skomski, J. Phys.: Condens. Matter 13, R433    \n(2001).    \n[20] W. Wernsdorfer, B. Doudi n, D. Mailly, K. Hasselbach, A.  Benoit, J. Meier, J.-Ph. \nAnsermet, and B. Barbara, Phys. Rev. Lett. 77, 1873 (1996). \n[21] R. M. H. New, R. F. W. P ease and R. L. White, IEEE Trans. Mag. 31, 3805      \n(1995).   \n           13F i g u r e s          \n       Figure 1: SEM image for the submicron-si zed chains of Co hollow spheres.  \n         Figure 2: M\nZFC(T) and MFC(T) curves measured in an applied field, Happ = 90 Oe \nbetween 5 K and 380 K. The inset sh ows an enlarged view of the MZFC(T) curve.  \n \n \n  14 \n        Figure 3: Three hysteresis loops measured at T = 50 K, 150 K, and 380 K. The inset \nshows the saturation magnetiza tion measured at 300 K. \n         Figure 4: Reduced remanent ratio, M\nR(T)/MS(T), and reduced saturation \nmagnetization, MS(T)/MS(50K), versus temperature.   15 \n          \nFigure 5: Temperature variation of H\nC(T) and ΔM(T) = MFC – MZFC. The dashed curve \nis for the fitting result by Eq. (1). The solid curve is for ΔM. \n  "    },    {        "title": "0707.2344v1.Magnetodipolar_interlayer_interaction_effect_on_the_magnetization_dynamics_of_a_trilayer_square_element_with_the_Landau_domain_structure.pdf",        "content": "1Magnetodipolar interlayer interaction effect on the magnetization dynamics\nof a trilayer square element with the Landau domain structure\nD.V. Berkov, N.L. Gorn\nInnovent Technology Development, Prüssingstr. 27b, D-07745 Jena, Germany\nABSTRACT\nWe present a detailed numerical simulation study of the effects caused by the magnetodipolar\ninteraction between ferromagnetic (FM) layers of a trilayer magnetic nanoelement on its mag-\nnetization dynamics. As an example we use a Co/Cu/Ni80Fe20 element with a square lateral\nshape where the magnetization of FM layers forms a closed Landau-like domain pattern. First\nwe show that when the thickness of the non-magnetic (NM) spacer is in the technology\nrelevant region h ~ 10 nm, magnetodipolar interaction between 90o Neel domain walls in FM\nlayers qualitatively changes the equilibrium magnetization state of these layers. In the main of\nthe paper we compare the magnetization dynamics induced by a sub-nsec field pulse in a\nsingle-layer Ni80Fe20 (Py) element and in the Co/Cu/Py trilayer element. Here we show that\n(i) due to the spontaneous symmetry breaking of the Landau state in the FM/NM/FM trilayer\nits domains and domain walls oscillate with different frequencies and have different spatial\noscillation patterns; (ii) magnetization oscillations of the trilayer domains are strongly supp-\nressed due to different oscillation frequencies of domains in Co and Py; (iii) magnetization\ndynamics qualitatively depends on the relative rotation sense of magnetization states in Co\nand Py layers and on the magnetocrystalline anisotropy kind of Co crystallites. Finally we\ndiscuss the relation of our findings with experimental observations of magnetization dynamics\nin magnetic trilayers, performed using the element-specific time-resolved X-ray microscopy.\nKeywords: thin magnetic films; magnetic nanoelements; magnetization dynamics; numerical\nsimulations\nPACS numbers: 75.40.Mg; 75.40.Gb; 75.60.Ch; 75.75.+a;2I. INTRODUCTION\nStudies of nanoelements patterned out of magnetic multilayers constitute now a rapidly gro-\nwing research area due to their already existing and very promising future applications. Using\nmodern patterning technologies it is possible to produce arrays of nanoelements with lateral\nsizes of  ~ 102 - 103 nm and nearly arbitrary shapes. Such elements and their arrays can be\nemployed in magnetic random access memory (MRAM) cells, miniaturized magnetoresistan-\nce sensors (in read/write hard disk heads), advanced high-density storage media, spintronic\ndevices [1] etc.\nSmall lateral sizes of these single- and multilayered structures lead to qualitatively new featu-\nres of their magnetization dynamics, with the quantization of their spin wave eigenmodes\nbeing the most famous example (see, e.g., [2, 3, 4, 5]). Thorough understanding of this novel\nfeatures is crucially important both for the progress of the fundamental research in this area\nand for the development of reliable high-technology products based on such systems.\nIn the last decade extensive experimental and theoretical effort was dedicated to the studies of\nmagnetization dynamics of single-layer nanoelements. Among them the nanodisks possessing\nclosed magnetization configuration with the central vortex (for continuous disks) or without it\n(for rings with the hole in the middle) represent the simplest non-trivial example due to their\ncircular form and hence - axially symmetric magnetization configuration. Magnetization\ndynamics of these nanodisks has been extensively studied using advanced experimental\ntechniques, analytical theories and numerical simulations [6] and is satisfactory understood.\nThe next complicated case is a square or rectangular single-layer nanoelement with either\nsaturated magnetization state or closed Landau domain structure. In the state close to satura-\ntion the main non-trivial effect is due to the strong demagnetizing field near the element edges\nperpendicular to the field and magnetization direction; corresponding dynamics could also be\nunderstood quantitatively combining experimental and theoretical methods (see, e.g., [2, 4]\nand references therein). The closed Landau magnetization pattern is much more demanding at\nleast from the theoretical point of view, because magnetization dynamics of this structure\nexhibits both highly localized (oscillation of the central core and domain walls) and extended\n(oscillation of domain areas) modes. But many important features of this dynamics could be\nalso understood very recently using such advanced experimental methods as time-resolved\nKerr microscopy and space resolved quasielastic Brilloin light scattering techniques,\nsupported by detailed numerical simulations [7, 8, 9, 10, 11].\nHowever, single-layer elements are not very interesting from the point of view of potential\napplications, because almost any technical device based on magnetic nanolayers employs - for\nvarious, but fundamental reasons - mainly multilayer structures. An additional layer (or seve-\nral such layers) is required, e.g., as a reference layer with 'fixed' magnetization to detect via\nsome MR effect the resistance change when the 'free' layer changes its magnetization direc-\ntion, or as an electron spin polarizer in spintronic devices, etc. For this reason the magnetiza-\ntion dynamics of multilayered structures is of the major interest.\nIn such structures the interlayer interaction effects play often a very important role. Even if\nwe leave aside a strong exchange (RKKY) coupling present in structures with very thin non-\nmagnetic spacers consisting of some specific materials like Ru, we are still left with the un-\navoidable magnetodipolar coupling between the layers. This coupling is identically zero only\nfor a multilayer structure with infinitely extended and homogeneously magnetized layers - the\nsituation which virtually never is encountered in practice. For this reason understanding of the\nmagnetodipolar interlayer interaction influence is absolutely necessary for further progress.3This interaction is especially strong in situations, where the layer magnetization is - at least in\nsome regions - perpendicular to the free layer surface, thus inducing very large \"surface mag-\nnetic charges\" and consequently - high stray fields. Typical example of such systems are mul-\ntilayers with a perpendicular magnetic anisotropy and nanoelements where the magnetization\nlies in the layer plane, but is nearly saturated, so that large stray fields emerge near the side\nedges of a multilayer stack. The influence of the interlayer interaction in such systems has\nbeen extensively investigated in the past for the quasistatic magnetization structures (see, e.g.,\nthe review [12] and references therein) and very recently - by studies of the magnetization\ndynamics [13].\nTo avoid the strong interaction caused by the magnetization directed normally to the free sur-\nface, a commonly used idea is to employ multilayer nanoelements with closed magnetization\nstructures. In nanodisks such a structure is represented by an in-plane rotating magnetization,\ncontaining a central vortex as the only element producing strong demagnetizing field. Magne-\ntodipolar interaction between the cores of such vortices in such multilayered circular nanodots\nhas been investigated recently in [13].\nFor the next commonly used nanoelement shape - magnetic rectangle - the closed magnetiza-\ntion structure is achieved by the famous Landau pattern with four homogeneously magnetized\ndomains and four 90o domain walls (in case of a square element). For sufficiently thin films\ncommonly used in technologically relevant systems these walls are the so called Neel walls\n[14] with the magnetization lying almost in the element plane. Such a magnetization\nconfiguration contains only volume 'magnetic charges' (no free poles on the element surface,\nand hence - no surface charges), which are usually weaker than surface charges. For this\nreason the interlayer interaction mediated by the domain walls is multilayer elements with the\nclosed magnetization structure is expected to be weaker than in multilayer stacks with\nsaturated  magnetization.\nHowever, recently several research groups [15, 16, 17, 18] have demonstrated that the stray\nfield caused by the magnetization of vortex and/or Neel walls in one layer of a multilayered\nsystem can still strongly affect the magnetic state of other layers. Most of these studies use\nsimplified DW models which enable a semianalytical treatment of the problem (see, e.g., [15,\n16]), but rigorous micromagnetic simulations have confirmed that the stray field of a vortex\nwall with the Neel cap [17] or of a purely Neel wall [18] can strongly affect the magnetic state\nof other layers even if the interlayer separation is as large as ~ 100 nm [18].\nIn this study we consider multilayer square elements with lateral sizes ~ 1 mkm and thicknes-\nses of magnetic layers and non-magnetic spacers ~ 10 nm (i.e., geometry typical for numerous\napplications). We shall demonstrate that in such systems the magnetodipolar interlayer inter-\naction due to the Neel domain walls of the closed Landau structure is strong enough to change\nqualitatively both the quasistatic magnetization structure and magnetization dynamics of a\nsystem. The paper is organized as follows. After the brief description of our simulation me-\nthodology (Sec. II.A) we show reference results for a square single-layer element which will\nserve for comparison with multilayered systems. Afterwards we analyze in detail the effect of\nthe interlayer interaction on the quasistatic magnetization structure and magnetization dyna-\nmics for a square FM/NM/FM trilayer (Sec. II.C), considering both the influence of the initial\nmagnetization state - compare Sec. II.C and D, and the effect of the random polycrystalline\ngrain structure - compare Sec. II.D and E. In Sec. III we compare our results with (unfor-\ntunately very few) available experimental and numerical studies of similar systems, and\ndiscuss the possibility of an experimental verification of our simulation predictions.4II. NUMERICAL SIMULATION RESULTS\nA. Numerical simulations setup\nIn this study we have simulated the trilayer element Co/Cu/Py with the following geometry\nshown in Fig. 1 (when not stated otherwise): lateral sizes 1 x 1 mkm2, Co and Py layer thick-\nnesses hCo = hPy = 25 nm, Cu interlayer thickness (non-magnet spacer thickness between Co\nand Py layers) hsp = 10 nm. Both magnetic layers were discretized into Nx x Nz x Ny = 200 x\n200 x 4 rectangular prismatic cells. We have checked that the discretization into at least 4 in-\nplane sublayers was necessary to reproduce correctly the 3D magnetization structure of 90o\ndomain walls (DW) present in the equilibrium Landau magnetization state of square\nnanoelements.\nThe following magnetic parameters have been used: for Py - saturation magnetization\nPy\nS 860 M = G, exchange stiffness constant APy = 1\u000110-6 erg/cm and cubic magnetocrystalline\ngrain anisotropy Py 3\ncub510 K =× erg/cm3; for Co - Co\nS 1400 M = G, ACo = 3\u000110-6 erg/cm and\nmagnetocrystalline grain anisotropies Co 5\ncub610 K =× erg/cm3 for the cubic fcc modification of\nCo and Co 6\nun410 K =× erg/cm3 for its uniaxial hcp modification (see [19] for discussion of all\nvalues for Co). The average grain (crystallite) size 10 D\u0001\u0002= nm (in all directions) with 3D\nrandomly oriented anisotropy axes of various crystallites was used for both magnetic\nmaterials. There was no correlation between the crystallites in Py and Co layers.\nSimulations of both the equilibrium magnetization structure and magnetization dynamics\nwere performed using our commercially available MicroMagus package (see [20] for\nimplementation details). For simulations of the magnetization dynamics the package employs\nthe optimized Bulirsch-Stoer method with the adaptive step-size control to integrate the\nLandau-Lifshitz-Gilbert equation for the magnetization motion with the standard linear\nGilbert damping (damping constant was set to l = 0.01 throughout our simulations). Due to\nthe small amplitude of magnetization oscillations studied here we believe that this simplest\ndamping form adequately describes the energy dissipation in our system. We also did not take\ninto studies additional damping caused by the spin pumping effect in magnetic multilayers\n(see, e.g., [21]), because this is beyond the scope of this paper.\nWe have studied magnetization dynamics of our system in a pulsed magnetic field applied\nperpendicularly to the element plane. To obtain magnetization excitation eigenmodes of an\nequilibrium magnetization state we have applied a short field pulse in the out-of-plane\ndirection with the maximal field value Hmax = 100 Oe and the trapezoidal time dependence\nwith rise and fall times tr = tf = 100 ps and the plateau duration tpl = 300 ps. To obtain the\neigenmode spectrum, we have set the dissipation constant to zero (l = 0) and recorded\nmagnetization trajectories of each cell during the pulse and for Dt = 10 ns after the pulse was\nover. Spatial profiles of the eigenmodes (spatial maps of the oscillation power distribution in\nthe element plane) were then obtained in the meanwhile standard way [2, 22] using the\nFourier analysis of these magnetization trajectories after the pulse decay. Because the applied\nfield pulse was spatially homogeneous, we could observe only eigenmodes which symmetry\nwas not lower than the symmetry of the equilibrium magnetization state of the studied system.\nIn principle, the analysis of the eigenvalues and eigenvectors of the energy Hessian matrix\n[23] allows to obtain all eigenmodes, but our method can be used for much larger systems5because it does not require the explicit search of eigenvalues for large matrices with sizes Z x\nZ proportional to total number of discretization cells Z  ~ Nx x Nz x Ny. For the qualitative\nanalysis of the influence of various physical factors on the magnetization dynamics aimed in\nthis paper our method provides enough information.\nTo study the transient magnetization dynamics which could be compared to real experiments\nwe have applied the same field pulse as described above and recorded magnetization time\ndependencies for each discretization cell during the pulse and for Dt = 3 ns after the pulse.\nHere the dissipation constant was set to l = 0.01 - the value commonly reported in literature\nfor thin Py films; the same constant was used for Co layer. We have checked that increasing\nthis value up to l = 0.05 led, as expected, to faster overall oscillation decay, but did not\nproduce any qualitative changes in the magnetization dynamics.\nB. Single layer Py element as the reference system\nKeeping in mind that we are going to study interaction effects in a trilayer system, we first\npresent reference results for the single Py nanoelement with the same parameters as the Py\nlayer of the complete trilayer (the square element 1 x 1 mkm2, with the thickness hPy = 25 nm,\nPy\nS 860 M = G, APy = 1\u000110-6 erg/cm, Py 3\ncub510 K =× erg/cm3). Fig. 2a shows the equilibrium\nmagnetization structure of such a nanoelement obtained starting from the initial state\nconsisting of four homogeneously magnetized domains in corresponding triangles (as shown,\ne.g., in Fig. 4a), whereby magnetic moments of four central cells were oriented perpendicular\nto the layer plane (along the y-axis). As expected, the very small random grain anisotropy of\nPermalloy has virtually no influence on the magnetization state, so that the equilibrium\nmagnetization forms a nearly perfect closed Landau magnetization pattern with four 90o Neel\ndomain walls between the domains and the central vortex showing upwards.\nSpectrum of magnetization excitations for this Landau pattern is shown in Fig. 2b together\nwith spatial maps of the oscillation power of the out-of-plane magnetization component for\neach significant spectral peak. We remind (see Sec. IIA) that the field pulse used to draw the\nmagnetization out of its equilibrium state was spatially homogeneous, and hence only modes\nwith the corresponding symmetry could be excited. Excitation modes of the Landau domain\npattern have been recently studied in detail in [10, 11], so here we will only briefly mention\nseveral issues important for further comparison with the trilayer system.\nThe lowest peak in the excitation spectrum in Fig. 2b corresponds, in a qualitative agreement\nwith the results from [10], to the domain wall oscillations, whereby due to the spatial\nsymmetry of the exciting field pulse we observe only the oscillation mode where all domain\nwalls oscillate in-phase. We are not aware of any analytical theory which would allow to cal-\nculate the frequency of a 90o domain wall oscillations and thus could be compared to our\nsimulations. From the qualitative point of view, DW oscillations are exchange-dominated and\ntheir frequency is the lowest one among other exchange-dominated magnetization excitations,\nbecause the equilibrium magnetization configuration inside a DW is inhomogeneous and thus\nits stiffness with respect to small deviations from the equilibrium is smaller than for a colline-\nar magnetization state.\nPeaks with higher frequencies correspond to the oscillations within four triangular domains of\nthe Landau structure; again, only symmetric in-phase oscillations have been observed. Accor-\nding to the analysis performed in [10, 11] domain excitations can be classified into the\nfollowing types. First, there exist modes which power distribution has nodes (between the\npeaks) in the radial direction, i.e., from the square center to its edges. Corresponding wave6vector is perpendicular to the magnetization direction in the domains (^kM). Such modes\nare similar to Damon-Eshbach modes in extended thin films and are called radial (wave vec-\ntor in the radial direction) [10] or transverse (because ^kM) [11]. Second, there exist modes\nwith power distribution nodes along the contour around the square center. In this case regions\nwith high power form elongated bands from the center to the edges of the square. For these\nmodes the wave vector of their spatial power distribution is roughly parallel to the local\nmagnetization direction in each domain (kM\u0001); they behaviour is similar to the backward\nvolume modes in extended thin films. For obvious reasons this second type is called azi-\nmuthal [10] or longitudinal [11] modes.\nAs it can be seen from Fig. 2b, our field pulse excites mainly an azimuthal mode with the\nfrequency f \u0002 3.2 GHz and several modes which can be classified as mixed radial-azimuthal\nmodes, because their spatial power distribution has nodes along both the radial direction and\nthe contours around the square center. These our results can be compared to simulations from\n[11], where the Py element with the same lateral sizes 1000 x 1000 nm2, but with the smaller\nthickness h = 16 nm was studied. Qualitatively our power maps are very similar to those\nshown in [11], but there are some important discrepancies. First of all, our overall excitation\nspectrum is very different from that presented in [11] (compare our Fig. 2b with Fig. 1d from\n[11]), although several peak positions are very close. Our power maps for specific modes also\nhave some qualitative similarities to several maps presented in [11], but detailed comparison\ndoes not make much sense due to the different total power spectra as mentioned above. All\nthese difference may arise because the simulated nanoelement in [11] was not discretized in\nthe layer plane,  but we believe that the major reasons are (i) the much shorter excitation pulse\n(td = 2.5 ps pulse length) used in [11] compared to our (300 ps) and (ii) the presence of the\nfinite damping in simulations from [11]. This problem requires further investigation, but is\nbeyond the scope of this paper.\nTransient magnetization dynamics for the single-layer Py element after the application of the\nsame field pulse as used for the studies of the excitation spectrum is shown in Fig. 3. We\nremind that for these simulations we have used the non-zero damping l = 0.01 typical for Py\nfilms. Fig 3 shows the time dependence of the angle between the average layer magnetization\nand the element plane () () tmt Y^\u0002  (panel (a)), spatial maps of the out-of-plane magnetizati-\non projection during the pulse (b) and after the pulse (c). By displaying the out-of-plane mag-\nnetization projection we have subtracted the equilibrium magnetization eq() m r, so that maps\nin Fig. 3 (and all other figures where the transient magnetization dynamics is shown) repre-\nsent the difference eq(,) () m m tm D^ ^ ^ = -r r. Homogeneous grey background around the mag-\nnetic element shows the reference grey intensity for 0 mD^=.\nFirst of all, we emphasize that even the small damping l = 0.01 used here leads to relatively\nfast oscillation decay (within ~ 3 ns after the pulse). After the initial increase of the perpendi-\ncular magnetization projection due to the field pulse (see the bright contrast across the whole\nsquare in Fig. 3b) is over, the time dependence of the average magnetization is dominated by\nrelatively fast oscillations of the domains, slightly modulated by oscillations of a lower frequ-\nency due to the domain wall motion. Corresponding patterns can be seen in Fig. 3c, where\none can directly recognize that domain walls and domains themselves oscillate with very\ndifferent frequencies. Further, comparison of the time-dependent maps from Fig. 3c with the\nspatial power maps in Fig. 2b shows the qualitative relation between the eigenmodes and the\ntransient dynamics of the Py square in this case: not only the contrast due to the DW oscilla-7tions, but also characteristic wave patterns inside the domains and near the outer regions of\ndomain walls agree qualitatively with the eigenmode power distributions shown in Fig. 2b.\nAnalogous simulations (with qualitatively similar results) have been carried out in [24] in\norder to explain the magnetization dynamics observed there using the time-resolved X-ray\nmicroscopy. We shall return to the analysis of these results by comparing our simulation with\nexperimental data in Sec. III.\nC. Trilayer Co/Cu/Py element: Landau structures with the same rotation sense in both\nmagnetic layers\n1. Deformation of the quasistatic magnetization structure\nIn this section we consider the trilayer element Co(25nm)/Cu(10nm)/Py(25nm), 1 x 1 mkm2\nin-plane size, with magnetic parameters given in Sec. II.B and cubic random anisotropy of Co\ngrains with Co 5\ncub610 K =× erg/cm3. In order to determine the equilibrium magnetization state\nof any system by minimizing its magnetic free energy we have to choose the initial (starting)\nmagnetization state.  As such a state we take in this section for both Co and Py layers the\nclosed in-plane magnetization configuration with sharply formed four triangular domains and\nfour magnetic moments in the middle of each layer pointing in the same out-of-plane direc-\ntion - along the +y-axis (to fix the orientation of the central vortex). An important point is that\nthe rotation sense of the starting magnetization state is the same for both layers. This initial\nstate is shown schematically in Fig. 4a. The situation, when the initial state consists of two\nclosed magnetization configurations with opposite rotation senses in Co and Py layers, is\nconsidered in the next subsection.\nThe corresponding equilibrium state which comes out as the result of the energy minimization\nis shown in Fig. 4c. The most striking feature of this state is the strong deformation of a 'nor-\nmal' Landau pattern (see Fig. 2a). Namely, central vortices in Co and Py layers are signifi-\ncantly displaced in opposite directions and domain walls are bended - also in opposite direc-\ntions for Co and Py. Equilibrium domains in both layers do not have anymore a shape of iso-\nsceles triangles, but rather form 'triangles' with slightly bended sides of different lengths. The\ndegree of the deformation described above depends both on the Cu spacer thickness and the\nlateral size of the squared trilayer structure (results not shown).\nThe reason for this unusual deformation can be understood by analyzing the intermediate\nmagnetization configurations arising during the energy minimization. At the first stage of this\nprocess the 'normal' Landau domain configuration inside each layer is formed. It is well\nknown that the 90o Neel domain walls of this magnetization configuration possess both\nvolume and surface 'magnetic charges' along them. We consider in detail the configuration of\nsurface charges, because the density of these 'charges' is simply proportional to the out-of-\nplane magnetization component () m^r and is thus easier to visualize. The corresponding map\nof the out-of-plane magnetization () m^r for the equilibrium Landau state of a 25 nm thick\nsquare Py nanoelement is shown in Fig. 4b. One can clearly see significant enhancement of\nthe () m^r-magnitude along all four domain walls. Thus two lines of the opposite 'surface\nmagnetic' charges are formed along each wall, building a kind of a 'linear dipole' (shown\nschematically in Fig. 4b with arrows and + and – signs). Now, it is important to realize that\nthe orientation of these dipoles for the given domain wall is the same on both upper and lower\nsurfaces of the nanoelement. For this reason, for the geometry shown in Fig. 1 and initial\nmagnetization states with the same rotation senses (as shown in Fig. 4a) we have on the upper8Co surface and lower Py surface linear dipoles with the same (parallel) orientations along all\nfour walls in each layer. The volume 'charges' formed due to the non-zero magnetization\ndivergence in the nanoelement volume have a qualitatively similar distribution. They also\ncontribute to the effect described below; with the decrease of the film thickness, when the\nNeel wall tends to a perfectly 'in-plane' magnetization structure, the contribution from these\nvolume charges becomes dominating.\nThe linear dipoles described above dipoles obviously repel each other, and due to the small\ninterlayer distance (which is in this case significantly smaller than the wall width) this\nrepulsion is very strong. With other words, corresponding 90o Neel domain walls in Co and\nPy layers 'feel' a strong mutual repulsion, so that they start to move away from each other. As\nthe result, domain walls in Co and Py layers shift in opposite directions, forming the final\nequilibrium structure displayed in Fig. 4c. We note in passing that for the starting state used in\nthese simulations, the central vortices in Co and Py layer have the same orientation and thus\nattract each other. However, due to the small vortex area this attraction can not compensate\nfor the strong repulsion of all domain walls, although the surface density of magnetic charges\nwithin the vortices is much higher than along the walls to due a large values of () m^r within\nthe vortex.\nFor further consideration it also important to note that the equilibrium magnetization state of\nCo is disturbed by its random grain anisotropy more than for the Py layer, for which the influ-\nence of this anisotropy is very small. Corresponding disturbance can be seen on the spatial\nmap of () m^r for Co (Fig. 4c, middle panel of the upper row), but for the moderate cubic ani-\nsotropy of Co Co 5\ncub610 K =× erg/cm3 and the small average crystallite size 10 D\u0001\u0002= nm, this\ndisturbance is still rather weak.\n2. Eigenmodes and transient magnetization dynamics\nStrong deformation of the equilibrium domain structure discussed in the previous subsection\nhas a qualitative impact on magnetization dynamics in the trilayer as compared to a single-\nlayer systems.\nFirst of all, spectrum of eigenmodes of the Py layer from Co/Cu/Py trilayer (shown in Fig. 5,\nupper panel) is qualitatively different from the corresponding single-layer Py square (Fig. 2b).\nIrregular domain structure results in a quasi-continuos (for our resolution) oscillation power\nspectrum, because peaks corresponding to the oscillations of each domain and domain wall\nare located at different positions. In particular, all domain walls oscillate with various frequ-\nencies, as shown in Fig. 5 by the peaks of the 1st group (a, b, c). These peaks can be attributed\nto oscillations of different domain walls as displayed on the power spatial maps in the upper\nrow of these maps. Eigenmode frequencies for oscillations within the domains also differ\nsignificantly for different Py domains, as shown by the peak positions of the 2nd group in the\nspectrum. In addition, the power distribution patterns within each domain become highly\nirregular, as shown by corresponding maps in the second map row in the same figure. For\nhigher frequencies (group 3), the power distribution is even more complicated, although some\ntypical attributes of longitudinal and transverse modes can still be recognized (3rd map row).\nTransient magnetization dynamics of the same trilayer system with the finite damping (it was\nset to l = 0.01 for both Co and Py layers) also strongly differs from the monolayer case. Cor-\nresponding time dependencies for the out-of-plane angles () () tmt Y^\u0002 of the average magne-9tization are shown in Fig. 6a for the Co layer (thin solid line), Py layer (dashed line) and the\ntotal system (thick solid line).\nThe out-of-plane magnetization deviation during the field pulse is smaller for the Co layer\nthan for the Py layer, due to the higher saturation magnetization of Co which leads to larger\ndemagnetizing field caused by the out-of-plane excursion of Co magnetization. Due to the\nsame larger saturation magnetization of Co (and equal Co and Py layer thicknesses), the basic\noscillation frequency is now close to the oscillation frequency of domains for the single-layer\nCo nanosquare: Co layer in the trilayer element determines the overall oscillation frequency,\n'locking' (capturing) the frequency of the Py layer domains also. Because the eigenfrequencies\nof Co and Py domains do not coincide, this phenomenon leads to a much faster decay of the\ndomain oscillations in the Py layer compared to the case of the single-layer Py element (com-\npare the dashed lines in Fig. 6a to Fig. 3a). In fact, shortly after the pulse is over (t > 0.6 ns),\nPy domain oscillations are barely visible both in the average magnetization time-dependence\n(Fig. 6a) and spatial maps of the out-of-plane magnetization component (Fig. 6c). Low-frequ-\nency oscillations of the average magnetization of Py are entirely determined by the oscillati-\nons of bended DWs. Note that oscillations of different walls are out of phase due to the diffe-\nrent eigenfrequencies of the four DWs in the disturbed Landau structure (see Fig. 5), so that\nfor the given time moment different walls (and even different regions of one and the same\nwall) can exhibit opposite magnetization contrasts as displayed on the last images in both map\nrows in Fig. 6b and 6c. We shall return to this important circumstance by comparing our\nsimulations to experimental data in Sec. III.\nD. Trilayer Co/Cu/Py element: Landau structures with opposite rotation senses in Co\nand Py magnetic layers\nEquilibrium magnetization configuration. It is well known that the initial magnetization state\nused to start the energy minimization in micromagnetics can have a decisive influence on the\nequilibrium configuration resulting from this minimization, because any realistic ferromagne-\ntic system possesses many energy minima due to several competing interactions present in\nferromagnets. For this reason we have studied the influence of the starting configuration on\nthe equilibrium magnetization and dynamical properties of our trilayer system, choosing as an\nalternative starting state the same Landau-like domains structure as described at the beginning\nof subsection II.C.1, but with opposite rotation senses for Co and Py layers (see Fig. 7a).\nIn this case Landau patterns are also formed at the initial energy minimization stage in both\nmagnetic layers. However, closed magnetization states in Co and Py layers have now opposite\nrotation senses. For this reason linear magnetic dipoles appearing along each domain wall at\nthe upper Co and lower Py surfaces as described in subsection II.C.1 above, are oriented anti-\nparallel. Hence the domain walls (which form these dipoles) attract each other, so that these\nwalls become wider and do not move across the layers. This naturally leads to a nearly sym-\nmetrical magnetization configurations in both Co and Py. This configuration is qualitatively\nsimilar to a 'normal' Landau pattern in a single square nanoelement, but domain walls are\nmuch broader (compare Fig. 7b to Fig. 2a).\nTable 1. Energies of equilibrium magnetization states shown in Fig. 4c and 7b.10Initial magn. state Total energy\n(nanoerg)Anisotropy\nenergyExchange\nenergyDemag.\nenergy\nThe same rotation\nsenses in Co and Py4.193 2.851 0.715 0.627\nOpposite rotation\nsenses in Co and Py3.602 2.838 0.510 0.254\nIt is instructive to compare the energies of equilibrium magnetization configurations obtained\nfrom the two different starting states as explained above. From Table 1 one can see that the\ntotal energy of the configuration with opposite rotation senses of closed magnetization states\nin Co and Py layers (Fig. 7b) is lower than the energy of the state with the same rotation sen-\nses in both layers (Fig. 4c). The total energy decrease is mainly due to the smaller exchange\nenergy (wider domain walls) and demagnetizing energy (attraction of domain wall dipoles) in\nthe 'opposite' state. However, due to the dominant contribution of the magnetocrystalline\nanisotropy energy (which is nearly equal in both cases) the total energy difference is not very\nlarge, so in experimental realizations both states can be expected.\nMagnetization dynamics: eigenmodes. The almost symmetrical equilibrium magnetization\nstate results in the excitation spectrum with much narrower peaks than for the strongly\ndisturbed asymmetrical state considered in the previous subsection. Corresponding spectrum\nof eigenmodes for the Py layer (from the Co/Cu/Py trilayer) is shown in Fig. 8 together with\nspatial maps of the oscillation power. All domain walls have now nearly the same oscillation\nfrequency (similar to the situation for the single-layer Py element), but due to the increased\nwidth of the domain walls corresponding oscillation regions are also much wider - compare\nthe 1st map on Fig. 8 with the 1st map on Fig. 2b. Broadening of domain walls manifests itself\nalso in the significant decrease of the corresponding oscillation frequency (\u0002 2.2 GHz for the\nPy layer within the trilayer vs \u0002 3.2 GHz for the single-layer Py element).\nSpectral peaks corresponding to the oscillations of domains themselves are also much narro-\nwer than for the highly asymmetrical state discussed above, so that several modes can be well\nresolved (Fig. 8). Although the oscillation frequencies for various domains coincides (within\nour resolution \u0003f ~ 0.1 GHz) and oscillation power patterns for various domains are very\nsimilar (at least for modes 2 and 3 shown in Fig. 8), the absolute values of the spatial power\nsignificantly differs from domain to domain. We attribute this effect to the random anisotropy\nfluctuations of the Co layer. It is well known that due to the small average crystallite size\nthese fluctuations are largely 'averaged out' [25]. However, remaining small fluctuations of\nthe out-of-plane magnetization in the Co layer on a large spatial scale have a significant\ninfluence on the Py layer eigenmodes due to the large saturation magnetization of Co and\nsmall spacer thickness. In particular, these fluctuations may lead to the redistribution of the\noscillation power between the domains as can be seen on the spatial power maps in Fig. 8.\nMagnetization dynamics: transient behaviour. Due to the qualitatively different equilibrium\nmagnetization state for the trilayer with oppositely oriented Landau structures in Co and Py\nlayers its transient magnetization oscillations (after the field pulse) for the finite damping case\nare also very different from both the single-layer square and the trilayer possessing Landau\nstructures with the same rotation senses in both Co and Py layers. Corresponding simulation\nresults are shown in Fig. 9 in the same format as in Fig. 6.11First of all, due to the largely restored symmetry of the equilibrium magnetization configura-\ntion, magnetization oscillations of different domain walls and different domains are now 'in-\nphase'. Due the much 'softer' magnetization configurations of the domains their oscillations\nhave now a much higher amplitude than for the trilayer with 'parallel' Landau structures\n(compare after-pulse oscillations and magnetization maps in Fig. 6 and Fig. 9). For this reason\nthe relative contribution of domain wall oscillations to the time dependence of the average\nmagnetization is almost negligible. It can be seen that domain oscillations are dominated by\nthe propagating spin wave which is excited at the square center (core of the Landau structure).\nIts wavefront has initially a nearly squared form, but when the wave propagates towards the\nelement edges, its front becomes circular. Several nodes appear along this wave front for the\nsufficiently long propagation time (see several last maps in Fig. 9) in accordance with the\nspatial power distribution of the system eigenmodes. However, the propagating time shown in\nFig. 9 is too short to establish the nodal structure with as many nodes as shown in Fig. 6.\nE. Trilayer Co/Cu/Py element: influence of the Co anisotropy type\nIt is well known that thin polycrystalline Co films may possess two kinds of the magnetocrys-\ntalline grain anisotropy, according to the two possible grain types: fcc grains have the cubic\nanisotropy Co 5\ncub610 K =× erg/cm3 (the case which was analyzed above) and hcp grains have\nthe much stronger uniaxial anisotropy Co 6\nun410 K =× erg/cm3 (see [19] and original\nexperiments in [26] for the corresponding discussion). Films with mixed fcc-hcp structure are\nalso possible. For this reasons we have studied the effect of the Co anisotropy type on the\nequilibrium magnetization structure and magnetization dynamics of our trilayer simulating the\nsystem with all parameters as given above, and with the starting Landau states with the same\nrotation sense in both magnetic layers, but with the uniaxial anisotropy of Co grains\nCo 6\nun410 K =× erg/cm3. Grain anisotropy axes were again distributed randomly in 3D.\nCorresponding results are presented in Fig. 10 and 11.\nThe major effect of such a large magnetocrystalline grain anisotropy is the strong disturbance\nof the equilibrium magnetization structure, as it can be seen from Fig. 10. Despite the small\naverage grain size 10 D\u0001\u0002= nm, the anisotropy fluctuations in Co even after their 'averaging-\nout' [25] are sufficiently strong to induce large deviations from the ideal Landau structure and\nto enforce significant randomly varying out-of-plane magnetization component on the lateral\nCo surfaces. The stray field induced on the Py layer by this Co() m^r-component nearly\ndestroys the original Landau magnetization structure of this layer, so that only the overall\nmagnetization rotation sense is preserved. The initially triangular domains of the Landau\nstructure now have a highly irregular form and only small pieces of domain walls (mainly\nnear the Py square corners) can be recognized (Fig. 10b).\nCorrespondingly, the magnetization dynamics of such a trilayer element again differs quali-\ntatively from all cases studied above (Fig. 11). Oscillations of the domain walls are almost\ninvisible. Average magnetization time dependence is entirely dominated by the circular wave\nemitted from the central vortex as shown in Fig. 11b and 11c. Due to the strongly disturbed\ndomain structure and absence of well defined domain walls (at least in the middle of the Py\nsquare) the wave front is roughly circular from the very beginning. However, irregularities of\nthe equilibrium magnetization structure within the domains lead to large modulations of the\noscillation amplitude along the wave front, as it can be recognized already for the initial stage\nof the wave propagation (Fig. 11b).12III. VERIFICATION OF SIMULATION PREDICTIONS AND COMPARISON WITH\nEXPERIMENTAL OBSERVATIONS\nAlthough, as already mentioned in the Introduction, both static magnetization structures and\nmagnetization dynamics in multilayer nanoelements have been extensively studied in the last\nseveral years, we are not aware of any experiments which could be used for direct confirma-\ntion or disprove of our simulation results.\nIn principle, our predictions concerning the equilibrium magnetization structure in square\nmkm-sized multilayer elements  (Fig. 4, 8, 10, 12) can be verified quite easily. Fabrication of\nmkm- and sub-mkm patterned multilayer elements of corresponding sizes is possible using\nseveral experimental techniques. Deformation of the 'normal' Landau structure predicted by us\nis strong enough for both the same and opposite magnetization rotation senses in individual\nlayers of the magnetic element. Hence it should be possible to detect this deformation with the\nstate-of-the-art methods for the observation of magnetization structures in nm-thick layers,\ne.g., using the meanwhile standard high-resolution MFM-facilities. We believe, that in an\narray of square nanoelements composed as described in this paper both types of the equilibri-\num magnetization states (depending on random initial fluctuations of the magnetization) will\nbe formed, so that structures shown both in Fig. 4 and Fig. 7 can be found.\nMagnetization dynamics of thin film systems can be measured nowadays not only with a very\nhigh lateral and temporal resolution, but also element-specific using the synchrotron X-ray\nradiation [24, 27, 28] with the potential resolution of several tens of nanometers. Layer-\nselective measurements are also possible using the Kerr microscopy technique [29], whereby\nthe resolution lies in the sub-mkm region (see, e.g., the recent detailed study of the magnetiza-\ntion dynamics of the Landau state for Py squares with sizes ~ 10 - 40 mkm in [32]). As\nalready mentioned in the Introduction, most papers on this topic are devoted to the\nmagnetization dynamics of mkm-sized single-layer elements. Excitations for trilayered\ncircular Py/Cu/Py nanodots at different external fields were studied in thermodynamical\nequilibrium in [30];  mainly the interlayer interaction effects due to the magnetic poles on the\nedges of nearly saturated layers have been described. There are also a few papers where the\nmagnetization switching of rectangular magnetic trilayers is studied (see, e.g., [31]), where\nthe major effect is also due to the strong stray fields induced near the edges of a nanoelement\nin a magnetically saturated state.\nWe are aware of only two experimental studies which results can be more or less directly rela-\nted to the subject of this paper, namely, the interlayer dipolar interaction dominated by the\nnearly in-plane domain walls of the closed (Landau-like) magnetization configuration. In both\ncases [24, 27] the magnetization dynamics of Co/Cu/Py square trilayers was studied using the\nelement-specific time-dependent synchrotron X-ray microscopy.\nIn the pioneering paper [24] the transient magnetization dynamics of a relatively large 4 x 4\nmkm2 trilayer Co(50nm)/Cu(2nm)/Py(50nm) was studied by the pump-and-probe X-ray\nmagnetic circular dichroism (XMCD) microscopy in the field pulse perpendicular to the\nsample plane. This technique has allowed to investigate the magnetization dynamics with the\ntemporal resolution ~ 50 ps and potential spatial resolution ~ 20 nm (however, the actual reso-\nlution achieved in [24] is much poorer and is difficult to estimate due to a significant shot\nnoise). Stoll et al. [24] did not study the equilibrium magnetization state of their system and\nhave presented spatial maps of the out-of-plane magnetization component of the Py layer only\nfor various time moments during and after the field pulse. Magnetization maps shown in [24]\nare effectively differential images between the excited and equilibrium magnetic states, so13that the contrasts due to the static domain walls and central vortex of the closed magnetization\nstructure are excluded.\nThe main qualitative features of experimental images presented in [24] are the following: (i)\nbright and relatively narrow bands along the diagonals of the squared magnetic element, i.e.,\nwhere the domain walls of the 'normal' equilibrium Landau pattern are located; (ii) at the\nsame time different domain walls exhibit contrast of different brightness and even of different\nsigns, indicating that oscillations phases and/or frequencies for different walls are different;\n(iii) after the decay of the field pulse virtually no contrast within the domains themselves can\nbe seen, so that the average out-of-plane magnetization component after the field pulse is zero\n(Py()0 m^\u0001\u0002=r ) within the experimental resolution.\nThe authors of [24] attributed the narrow contrast bands mentioned above to the domain walls\noscillations. To support their experimental findings, Stoll et al. have performed dynamic mic-\nromagnetic simulations, where they have included the Py layer only and discretized this layer\nonly in the lateral plane. Simulated out-of-plane magnetization images shown in [24]\ndemonstrate, of course, the time-dependent contrast between the oscillations of domain walls\nand domains themselves, but clearly fail to reproduce all other qualitative features of their\nexperimental images listed above.\nWe have also performed simulations of the Py single layer element with the sizes used in\n[24], discretizing it into 400 x 400 x 4 (totally 6.4\u0001105) cells. We note that due to the low\nanisotropy and relatively low saturation magnetization of  Py the size of our discretization\ncells 10 x 10 x 12.5 nm3 was small enough to reproduce main features of the Py dynamics.\nProper simulation of the magnetic trilayer with the same lateral sizes including the 50 nm\nthick Co layer would require to halve the cell size in each dimension, so that the overall cell\nnumber would be prohibitively large for the state-of-the-art micromagnetic simulations. Our\nPy() m^r-images (Fig. 12) qualitatively agree with simulation data from [24], demonstrating\nonce more that when the interaction with the Co layer is neglected, in-phase oscillations of all\nfour DWs of the Landau pattern should be observed. In addition, the strong contrast within\nthe domains after the field pulse is clearly seen in Fig. 12c, manifesting itself also in strong\nafter-pulse oscillations of the average out-of-plane magnetization Py() m t^\u0001\u0002 as shown in Fig.\n12a. The amplitude of these after-pulse oscillations is comparable with the maximal value of\nPym^\u0001\u0002 achieved during the pulse. Taking into account that the domain contrast during the\npulse is clearly seen in the experimental images presented in [24], it is unlikely that\napproximately the same contrast after the pulse would be completely overlooked. All in one,\nexperimental findings from [24] can not be explained satisfactory when the dynamics of Co\nlayer and the interlayer interaction in the trilayer Co/Cu/Py is neglected.\nTaking into account that we could not simulate (at least not with proper resolution) the comp-\nlete system studied in [24], we can compare the results of Stoll et al. with our simulation data\nonly qualitatively. First of all, we note that the straight lines corresponding to the domain wall\noscillations indicate that Co and Py layers in this experiment possess Landau magnetization\nstates with opposite rotation senses, because for the trilayer with the same magnetization\nrotation senses in both magnetic layers domain walls should be strongly bended (see Fig. 4\nand 6 above).\nFrom the remaining possibilities, dynamic magnetization images of the trilayer with 'opposite'\nLandau patterns and fcc Co crystallites (Fig. 9) demonstrate well pronounced straight lines14corresponding to the domain wall oscillations similar to those observed in [24]. However, due\nto the high symmetry of the equilibrium state, all domain walls oscillate in-phase and with the\nsame amplitude, in contrast with strongly out-of-phase DW-oscillations with different\namplitudes seen in Fig. 2 from [24]. For the same trilayer with hcp-Co magnetization\ndynamics images are asymmetric (Fig. 11), but due to the very blurred boundaries between Py\ndomains virtually no contrast is observed along the square diagonals (which would\ncorrespond to DW-oscillations).\nAt this point it should be noted that the thickness of magnetic layers studied in [24] (hCo = hPy\n= 50 nm) is twice as large as in our simulated system (hCo = hPy = 25 nm). In a system with\nsuch thick layers, domain wall structure disturbed due to the interlayer interaction, could be\npartially recovered due to thicker magnetic layers. To check this idea, we have  simulated the\nCo/Cu/Py trilayer with the same parameters and initial magnetization structure, as for the sys-\ntem shown in Fig. 10 and 11, but with the Py thickness hPy = 50 nm; here the Py layer was\ndiscretized in 8 in-plane sublayers, so that the size of the discretization cell was preserved.\nCorresponding simulation results are shown in Fig. 13 (equilibrium state) and 14\n(magnetization dynamics). One can see, that for such increased Py thickness domain walls in\nthe equilibrium magnetization state are, indeed, partially recovered (see Fig. 13), so that their\noscillations are clearly visible in the dynamic patterns (Fig. 14). Due to the remaining asym-\nmetry of the magnetization structure, oscillations of different DWs have different spatial\npatterns, amplitudes and frequencies, in a qualitative agreement with the images displayed in\n[24]. The average out-of-plane magnetization projection exhibits only very weak oscillations\nafter the field pulse, which is also in agreement with [24]. However, we observe significant\nmagnetization contrast near the square center which is due to the wave emitted by the vortex\ncore; this contrast was not found experimentally [24].\nIn the second paper mentioned above [27] the magnetization dynamics of 1 x 1 mkm2\nCo(20nm)/Cu(10nm)/Py(20nm) trilayer was studied in the in-plane pulsed magnetic field, so\nthat mainly the central vortex motion could be seen both in Co and Py layers. The authors\ndisplay also the equilibrium magnetization structures of both magnetic layers (see Fig. 1 in\n[27]), also obtained using XMCD-microscopy. Unfortunately, the resolution of these images\nis still not good enough to make any quantitative statements, so one can only say that both Co\nand Py layers possess closed magnetization structures with the same rotation senses and that\nthese structures are somewhat disturbed (compared to 'ideal' Landau patterns). However, no\nmeaningful quantitative comparison to our results presented in Fig. 4 is possible.\nIV. CONCLUSION\nIn this paper we have studied the effects of magnetodipolar interlayer interaction in trilayer\nelements with lateral sizes in sub-mkm region and magnetic layers and spacer thicknesses of\nseveral nanometers. We have shown that due to such a small interlayer distance even relative-\nly weak stray field induced by the 90o Neel domain walls of the closed magnetization state\n(Landau-like pattern) causes qualitative changes of both the equilibrium magnetization struc-\nture and magnetization dynamics in these systems. We have also demonstrated that the effect\nof such an interaction may be very different, depending not only on the initial magnetization\nstate used to find the equilibrium magnetization pattern of a system, but also on the crystallo-\ngraphic structure of magnetic layers. This random crystal grain structure significantly affects\nthe magnetization dynamics also for very small crystallite size, where the random magneto-\ncrystalline anisotropy of the grains is largely 'averaged-out'. The statement about this anisotro-\npy averaging is often used to justify the neglect of this random anisotropy when simulating15the corresponding magnetization dynamics; our results reveal that in many important cases\nsuch a neglect may be the prohibitive oversimplification of a problem.\nOur simulations clearly demonstrate that for the qualitative and especially quantitative under-\nstanding of magnetization dynamics in multilayers, magnetodipolar interlayer interaction\neffects must be included into consideration, even when the equilibrium magnetization struc-\nture forms a closed flux state and thus its stray field is believed to be relatively weak.\nAlthough we are not aware of any experimental studies which results could be directly com-\npared to our simulation data, our main predictions can be relatively easily verified with avail-\nable experimental techniques, as discussed in detail in Sec. III.. However, the accurate sample\ncharacterization both from crystallographic and magnetic points of view is required to enable\na meaningful comparison with experimental results.\nACKNOWLEDGEMENTS. The authors acknowledge fruitful discussions with P. Fischer\nand A. Slavin. This research was partly supported by the Deutsche Forschungsgemeinschaft\n(DFG grant BE 2464/4-1).\nReferences:\n[1] U. Hartmann (Ed.), Magnetic Multilayers and Giant Magnetoresistance: Fundamentals and Industrial\nApplications, Springer Series in Surface Sciences, Vol. 37, 2000; H. Hopster and H.P. Oepen (Eds.), Magnetic\nMicroscopy of Nanostructures, Series: NanoScience and Technology, 2005; H. Zabel and S.D. Bader (Eds.),\nMagnetic Heterostructures: Advances and Perspectives in Spinstructures and Spintransport, Springer Tracts in\nModern Physics, Vol. 227, 2007\n[2] C. Bayer, J. Jorzick, B. Hillebrands, S.O. Demokritov, R. Kouba, R. Bozinoski, A.N. Slavin, K. Guslienko,\nD.V. Berkov, N.L. Gorn, M. P. 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Schultz, Small-amplitude magnetization dynamics in\npermalloy elements investigated by time-resolved wide-field Kerr microscopy, Phys. Rev., B71 (2005) 13440518Figure captions:\nFig. 1. Geometry of the simulated system, co-ordinate axes and the pulsed field direction used\nin simulations.\nFig. 2. Magnetic properties of the single-layer squared Permalloy element (1000 x 1000 x 25\nnm3): (a) - equilibrium Landau magnetization structure in zero external field shown as grey\nscale maps of magnetization projections; (b) - spectrum of eigenmodes excited by the small\nout-of-plane pulsed homogeneous field for the state shown in (a). Grey-scale maps below the\nspectrum show the spatial distribution of the oscillation power for corresponding peaks\n(bright areas correspond to a large oscillation power)\nFig. 3. Magnetization dynamics of a single-layer Permalloy element with the same sizes as in\nFig. 2 in the pulsed out-of-plane field: (a) - time dependence of the angle \u0001perp between the\naverage layer magnetization and the element plane (perp~ ()ym Y\u0001\u0002r, see Fig. 1); the trapezoi-\ndal pulse form is shown at the same panel as thin solid line; (b) - grey scale maps of my(r)\n(out-of-plane magnetization projection) during the pulse; (c) grey scale maps of my(r) after\nthe pulse. Oscillations of both domain walls and domains themselves are clearly seen.\nFig. 4. To the formation of a static equilibrium magnetization structure in zero external field\nfor the trilayer Co/Cu/Py element with the lateral sizes 1000 x 1000 nm2, Co and Py thicknes-\nses hCo = hPy = 25 nm and spacer thickness hCu = 10 nm. The Co layer possesses a fcc poly-\ncrystalline structure with the average crystallite size 10 D\u0001\u0002= nm and cubic grain anisotropy\nKcub = 6\u0001105 erg/cm3. (a) - initial magnetization structure used as the starting state by the\ncalculation of the equilibrium structure shown in (c) as grey scale maps of magnetization\nprojections; (b) initial distribution of the surface charges responsible for the repulsion of 90o\ndomain walls initially located along the main diagonals of the square in Co and Py layers.\nFig. 5. Eigenmodes spectrum for a Permalloy layer of the trilayer element with the static\nmagnetization structure shown in Fig. 4c. Due to the symmetry breaking of the underlying\nmagnetization state spectral lines corresponding to the oscillations of different domain walls\nare positioned at different frequencies (spectral group 1). Spectral peaks corresponding to the\ndomain oscillations form two quasi-continuous groups (groups 2 and 3), whereby each line\nwithin a group corresponds to magnetization oscillations within a specific domain as shown\nby grey-scale maps of the oscillation power distributions below.\nFig. 6. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstructure shown in Fig. 4c in the pulsed out-of-plane field: (a) - time dependence of the angle\n\u0001perp between the average magnetization and the element plane for the magnetization of the\ntotal element (thick green line), Co layer (thin blue line) and Py layer (thick dashed line).\nGrey scale maps of the out-of-plane magnetization projection of the Py layer during the pulse\n(b) and after the pulse (c). Due to different eigenfrequencies oscillations of different domain\nwalls are out-of-phase here and oscillation of domains themselves are strongly suppressed\ncompared to the case of a single-layered Py element (Fig. 3c).\nFig. 7. Static equilibrium magnetization structure in zero external field for the same Co/Cu/Py\nelement as shown in Fig. 4, but starting from Landau magnetization states with opposite\nrotation senses in Co and Py layers (a). Resulting equilibrium state is shown at the panel (b)19as grey scale maps of magnetization projections. It can be seen that due to the attraction of\ndomain walls for the starting magnetization states the symmetry of the final equilibrium\nmagnetization structure is nearly preserved.\nFig. 8. Eigenmodes spectrum for a Permalloy layer of the trilayer element with the static\nmagnetization structure shown in Fig. 7b. Due to the largely preserved symmetry of domain\nwalls their oscillations have nearly the same frequency (spectral line 1 and grey-scale map 1).\nOscillation power distribution in domain regions (maps 2-4) is still asymmetric due to magne-\ntodipolar interaction with the Co layer which magnetization has a noticeable and spatially\nvarying out-of-plane component due to a significant magnetocrystalline anisotropy (fcc Co).\nFig. 9. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstructure shown in Fig. 7b in the pulsed out-of-plane field presented in the same way as in\nFig. 6. It can be seen that the magnetization dynamics is largely dominated by the propagation\nof the spin wave excited at the central vortex; its wave front has initially the square shape\nwhich transforms during the propagation into a nearly circular one.\nFig. 10. Static equilibrium magnetization state for Hext = 0 for the same Co/Cu/Py element as\nshown in Fig. 7 (starting from Landau magnetization states with opposite rotation senses in\nCo and Py layers - see (a)) but with the Co layer having a hcp polycrystalline structure with\nthe uniaxial grain anisotropy Kun = 4\u0001106 erg/cm3. Due to such a large random anisotropy\nvalue the symmetry of the final magnetization state is strongly disturbed (b) and boundary\nregions between the domains are very wide.\nFig. 11. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstructure shown in Fig. 10b in the pulsed field presented in the same way as in Fig. 9. Due to\nthe strong disturbance of the static magnetization state the oscillations of domain walls are\nnearly invisible. Although the front of the dominating spin wave remains approximately\ncircular, the wave amplitude shows significant inhomogeneities along this wave front.\nFig. 12. Simulated transient magnetization dynamics for a single-layer Py element with lateral\nsizes 4 x 4 mkm2 and thickness hPy = 50 nm (after the same field pulse and presented in the\nsame way as in Fig. 3). Strong magnetization oscillations within the domain regions after the\nfield pulse can be seen.\nFig. 13. Static equilibrium magnetization state for Hext = 0 for the same Co/Cu/Py element as\nshown in Fig. 10 (starting from Landau states with opposite rotation senses in Co and Py\nlayers - see (a)) but with the thicker Py layer: hPy = 50 nm. Due to the increased Py layer\nthickness domain walls within this layer are partially recovered (see the grey-scale map of\nmy(r) for Py in Fig. 13b).\nFig. 14. Magnetization dynamics of a trilayer Co/Cu/Py element with the static magnetization\nstate from Fig. 13b presented in the same way as in Fig. 11. In contrast to the case shown in\nFig. 11, oscillations of domain walls can be clearly seen. However, these oscillations remain\nstrongly asymmetric, what can be seen especially well on the magnetization maps after the\nfield pulse (c).20xy\nzCoPy\nhCohPy\ndHext\nFig. 1\nf, GHz\n0 2 4 6 8 10 12 14Pav(my(r))\n010x10-620x10-630x10-640x10-6\n(a)\n(b)\nFig. 2\n21t, ns\n0.0 0.5 1.0 1.5 2.0 2.5 3.0Yperp\n0.0000.0050.010\n(a)\n(b)\n(c)\nFig. 322Co Py\n(a)\n(c)\n(b)\nFig. 423\nf, GHz\n0 2 4 6 8 10 12 14<P(my(r))>\n05x10-610x10-615x10-6\n   1\na b c     2\na  b  c\n     3\na  b  c\n1:\n2:\n3:a b c\na b\nbc a\nFig. 524t, ns\n0.0 0.5 1.0 1.5 2.0Yperp\n0.0000.0050.010 Total magn.\nCo layer\nPy layer(a)\n(c)\n(b)\nFig. 625(a)\n(b)Co Py\nFig. 7\nf, GHz\n0 2 4 6 8 10 12<P(my(r))>\n050x10-6100x10-6\n1\n23 4\n1 2 3 4\nlog(P)\nFig. 826\nt, ns\n0.0 0.5 1.0 1.5 2.0Yperp\n-0.0050.0000.0050.0100.015\nTotal element \nCo layer\nPy layer\n(a)\n(c)(b)\nFig. 927\n(a)\nCo Py\n(b)\nFig. 1028t, ns0.0 0.5 1.0 1.5 2.0Yperp\n-0.0050.0000.0050.0100.015\nTotal element \nCo layer\nPy layer\n(a)\n(c)(b)\nFig. 1129t, ns\n0.0 0.5 1.0 1.5 2.0 2.5 3.0Yperp\n-0.0050.0000.0050.0100.015\n(a)\n(c)\n(b)\nFig. 1230(a)\nCo Py\n(b)\nFig. 1331t, ns\n0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Yperp\n0.0000.0050.010Total element \nCo layer\nPy layer\n(a)\n(c)(b)\nFig. 14"    },    {        "title": "0707.3329v1.Magnetocrystalline_anisotropy_controlled_local_magnetic_configurations_in__Ga_Mn_As_spin_transfer_torque_microdevices.pdf",        "content": "Local control of magnetocrystalline anisotropy in (Ga,Mn)As microdevices:\nDemonstration in current induced switching\nJ. Wunderlich,1A. C. Irvine,2J. Zemen,3V. Hol\u0013 y,4A. W. Rushforth,5E. De Ranieri,2, 1U. Rana,2, 1K.\nV\u0013 yborn\u0013 y,3Jairo Sinova,6C. T. Foxon,5R. P. Campion,5D. A. Williams,1B. L. Gallagher,5and T. Jungwirth3, 5\n1Hitachi Cambridge Laboratory, Cambridge CB3 0HE, UK\n2Microelectronics Research Centre, Cavendish Laboratory, University of Cambridge, CB3 0HE, UK\n3Institute of Physics ASCR, Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\n4Charles University, Faculty of Mathematics and Physics,\nDepartment of Electronic Structures, Ke Karlovu 5, 121 16 Prague 2, Czech Republic\n5School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK\n6Department of Physics, Texas A&M University, College Station, TX 77843-4242, USA\n(Dated: November 8, 2021)\nThe large saturation magnetization in conventional dense moment ferromagnets o\u000bers \rexible\nmeans of manipulating the ordered state through demagnetizing shape anisotropy \felds but these\ndipolar \felds, in turn, limit the integrability of magnetic elements in information storage devices. We\nshow that in a (Ga,Mn)As dilute moment ferromagnet, with comparatively weaker magnetic dipole\ninteractions, locally tunable magnetocrystalline anisotropy can take the role of the internal \feld\nwhich determines the magnetic con\fguration. Experiments and theoretical modeling are presented\nfor lithographically patterned microchannels and the phenomenon is attributed to lattice relaxations\nacross the channels. The utility of locally controlled magnetic anisotropies is demonstrated in\ncurrent induced switching experiments. We report structure sensitive, current induced in-plane\nmagnetization switchings well below the Curie temperature at critical current densities \u0018105Acm\u00002.\nThe observed phenomenology shows signatures of a contribution from domain-wall spin-transfer-\ntorque e\u000bects.\nPACS numbers: 75.50.Pp, 75.60.Jk, 85.75.-d\nI. INTRODUCTION\n(Ga,Mn)As and related ferromagnetic semiconductors\nare unique due to their dilute moment nature and the\nstrong spin-orbit coupling.1,2Doped with only \u00181-10%\nof Mn magnetic moments, the saturation magnetization,\nMs, and the magnetic dipole interaction \felds are \u0018100-\n10 times weaker in these materials than in conventional\nferromagnets. This could make possible dense integra-\ntion of ferromagnetic semiconductor microelements with\nminimal dipolar cross-links. Despite the low Msthe mag-\nnetic anisotropy \felds, Ha, routinely reach \u001810 mT due\nto the large, spin-orbit coupling induced magnetocrys-\ntalline terms.3,4The magnetocrystalline anisotropy can,\ntherefore, take the role normally played by dipolar shape\nanisotropy \felds in the conventional systems. The com-\nbination of appreciable and tunable Haand lowMsleads\nto outstanding micromagnetic characteristics. One par-\nticularly important example is the orders of magnitude\nlower critical current in the spin-transfer-torque magneti-\nzation switching5,6than observed for dense moment con-\nventional ferromagnets, which follows from the approx-\nimate scaling of jc\u0018HaMs. Critical currents for do-\nmain wall switching of the order 105Acm\u00002have been\nreported and the e\u000bect thoroughly explored in perpendic-\nularly magnetized (Ga,Mn)As thin \flm devices at tem-\nperatures close to the Curie temperature.7,8,9\nHere we demonstrate that it is possible to locally tune\nand control spin-orbit coupling induced magnetocrys-\ntalline anisotropies in (Ga,Mn)As, which is achievedin our devices by lithographically producing strain\nrelaxation. This is the central result of our work and it\nrepresents the necessary prerequisite for future highly\nintegrated microdevices fabricated in the dilute-moment\nferromagnets. It also makes possible a range of new stud-\nies of extraordinary magnetotransport and magnetization\ndynamics e\u000bects in such systems. As a demonstration\nwe link the achieved local control of magnetocrystalline\nanisotropy with a study of current induced domain\nwall switching which is currently one of the most hotly\ndebated areas of theoretical and experimental spintronics\nresearch.7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25We\nreport in-plane domain-wall switchings well below the\nCurie temperature at jc\u0018105Acm\u00002whose character-\nistics strongly depend on the locally induced changes\nof magnetic anisotropy. The phenomenology of the\ncurrent induced switching we observe shows signatures\nof domain wall spin-transfer-torque e\u000bects.\nThe paper is organized as follows: In Section II A\nwe introduce the studied (Ga,Mn)As microstructures\nand the anisotropic magnetoresistance (AMR) technique\nfor detecting local magnetization orientation along the\nchannels.19This technique is particularly useful in di-\nlute moment ferromagnets where direct imaging meth-\nods, such as the magneto-optical Kerr e\u000bect, lack the re-\nquired sensitivity due to the low Ms. Numerical simula-\ntions of the lattice relaxation in the microbars and micro-\nscopic calculations of the corresponding changes of mag-\nnetocrystalline anisotropies are discussed in Section II B.\nCurrent induced switching experiments in our structuresarXiv:0707.3329v1  [cond-mat.mes-hall]  23 Jul 20072\nwith locally controlled anisotropies are presented in Sec-\ntion III. A brief summary of the main results is given in\nSection IV.\nII. LATTICE RELAXATION AND LOCAL\nCONTROL OF MAGNETIC ANISOTROPY\nA. Experiment\nFig. 1 shows scanning electron micrographs of one of\nthe devices studied. The structure consists of a macro-\nscopic Van der Pauw device and an L-shaped channel pat-\nterned on the same wafer, the arms of which are Hall-bars\naligned along the [1 10] and [110] directions. The trench-\nisolation patterning was done by e-beam lithography and\nreactive ion etching in a 25 nm thick Ga 0:95Mn0:05As\nepilayer, which was grown along the [001] crystal axis\non a GaAs substrate. Results for two samples are re-\nported: device A(B) has 4(1) \u0016m wide, 80(20) \u0016m\nlong Hall bars. Isolated magnetic elements with the\ndimensions of these Hall bars and Ms\u001850 mT of the\nGa0:95Mn0:05As would have in-plane shape anisotropy\n\felds below\u00181 mT, which is an order of magnitude lower\nthan the magnetocrystalline anisotropy \felds. In-plane\nshape anisotropies are further reduced in our devices as\nthey are de\fned by narrow (200nm) trenches with the\nremaining magnetic epilayer left in place. The Curie\ntemperature of 100 K was obtained from Arrot plots of\nanomalous Hall data. Hole density of 5 \u00021020cm\u00003was\nestimated from high-\feld Hall measurements. At this\ndoping the compressive strain in the Ga 0:95Mn0:05As epi-\nlayer grown on the GaAs substrate produces a strong\nmagnetocrystalline anisotropy which forces the magneti-\nzation vector to align parallel with the plane of the mag-\nnetic epilayer.3,4\nMagnetization orientations in the individual micro-\nbars are monitored locally by measuring longitudinal and\ntransverse components of the AMR at in-plane magnetic\n\felds. The magnetization rotation experiments at sat-\nuration magnetic \feld measured on device B and on\nthe macroscopic Van der Pauw device are presented in\nFigs. 2(a) and (b). (For the detailed discussion of the\norigins of the AMR and microscopic modeling of this ex-\ntraordinary magnetoresistance coe\u000ecient in (Ga,Mn)As\nsee Ref. 26.) Examples of magnetoresistance measure-\nments for external magnetic \feld sweeps in which the\n\feld angle\u0012, measured from the [1 10] axis, is constant are\nshown in Figs. 2(c) and (d). The strongly \u0012-dependent\nlow-\fled magnetoresistance is attributed to magnetiza-\ntion rotations. At high \felds, the magnetoresistance be-\ncomes purely isotropic, i.e., the di\u000berences between resis-\ntances for di\u000berent angles \u0012become independent of the\nmagnitude of the external \feld. This property and the\nmuch smaller magnitude of the isotropic magnetoresis-\ntance compared to the low-\feld anisotropic magnetore-\nsistance allows us to use the high-\feld measurements in\nFigs. 2(a),(b) for determining the one to one correspon-\nFIG. 1: (a) Scanning electron micrograph of the L-shaped\nmicrodevice B and the macroscopic Van der Pauw device.\n(b) Detail of the L-shaped microdevice with the longitudinal\n(L) and transverse (T) resistance contacts in the bars and the\ncorner (C) resistance contacts. Positive hole current in the\np-type (Ga,Mn)As is de\fned to propagate from the [1 10]-bar\nto the [110]-bar.\ndence between a change in the low-\feld resistance and a\nchange in magnetization orientation. Note that the 45\u000e\nphase shift between the longitudinal and transverse AMR\ntraces (see Figs. 2(a),(b)) allows us to determine unam-\nbiguously the change in the magnetization angle if both\nresistance components are measured simultaneously. The\ntechnique of detecting magnetization rotations via AMR\nmeasurements is exploited in Section III where we com-\npare \feld induced and current induced magnetization\nswitchings. Importantly, the multiterminal design of our\nL-shaped microbars also allows to apply this electrical\nmeasurement of magnetization angle locally at the cor-\nner and at di\u000berent parts of the L-shaped Hall bars and,\ntherefore, to track the propagation of domain walls if\npresent in the system.\nIn this section we use the \fxed- \u0012magnetoresis-\ntance measurements to \frst determine local magnetic\nanisotropies in the individual microbars. Values of \u0012cor-\nresponding to easy-axis directions have the smallest low-\n\feld magnetoresistance. For values of \u0012not correspond-\ning to easy-axis directions the magnetization undergoes\na (partially) continuous rotation at low \felds resulting in\ndi\u000berent orientations, and hence di\u000berent measured re-\nsistances, at saturation and remanence. We \fnd that the\ntechnique can be used to determine the easy-axis direc-\ntions within\u00061\u000e.\nThe e\u000bect of microfabrication on the magnetic\nanisotropy is apparent in Fig. 3. In the bulk, magne-\ntization angle 30\u000ecorresponds to an easy-axis while 7\u000e\nand 55\u000eare signi\fcantly harder. For device B, 7\u000eis an\neasy-axes in the [1 10]-bar and 55\u000eis an easy-axis in the\n[110]-bar. All easy-axes found in devices A and B and in\nthe bulk are summarized in Tab.I. The bulk material has\nthe cubic anisotropy of the underlying zincblende struc-\nture plus an additional uniaxial [1 10] anisotropy as is typ-\nical (Ga,Mn)As epilayers.27This results in two easy-axes\ntilted by 15\u000efrom the [100] and [010] cube edges towards3\nthe [1 10] direction. In the microdevices, the easy-axes\nare rotated from their bulk positions towards the direc-\ntion of the respective bar and the e\u000bect increases with\ndecreasing bar width.\n-0000\n45\n90\n135\n180225270315\n-0 0 090 -606\n-606 Δρ Δρ Δρ ΔρL [ 10-5Ω cm ]\n-0000\n45\n90\n135\n180225270315\n-0 0 090 -606\n-606 Δρ Δρ Δρ ΔρL [ 10-5Ω cm ]\n-0000\n45\n90\n135\n180225270315\n-0 0 0-606\n-606 Δρ Δρ Δρ ΔρT [ 10-5Ω cm ]\n18090-0000\n45\n90\n135\n180225270315\n-0 0 0-606\n-606 Δρ Δρ Δρ ΔρT [ 10-5Ω cm ]\n18090\n-4 -2 0 2 4-100010067.5°90°\n45°\n22.5°\n0°ΔΔΔΔRL [Ω]\nBθθθθ  [T]-4 -2 0 2 4-1000100\n67.5°45°\n90°0°\n22.5°ΔΔΔΔRL [Ω]\nBθθθθ  [T]\nBAR(a) (b)\n(c) (d)\nFIG. 2: Device B longitudinal (a) and transverse (b) AMRs\nmeasured at 4.2 K in a rotating 4 T in-plane \feld with the\n\feld angle measured from the [1 10] axis, and bulk transverse\nAMR measured in the Van der Pauw device with current lines\noriented along the [010] axis. (\u0001 \u001a\u0011\u001a\u0000\u001awhere\u001ais the av-\nerage value over all angles.) In-plane, \fxed-angle \feld sweep\nmeasurements of the longitudinal magnetoresistances of the\n(c) [1 10]-bar and (d) [110]-bar bar of device B. (Same average\nresistances as in (a) and (b) are subtracted to obtain \u0001 R)\nsample bulk A [110]A [110] B [110] B [110]\neasy-axis angle \u000630\u000e\u000615\u000e\u000636\u000e+7\u000e,\u00008\u000e+55\u000e,\u000063\u000e\nTABLE I: Easy-axes angles, measured from the [1 10] crystal\ndirection, determined by magnetoresistance measurements in\nthe macroscopic Van der Pauw device (bulk) and in the [1 10]\nand [110]-bars of the L-shaped devices A and B.\nB. Theory\nThe local changes in the magnetocrystalline anisotropy\ncan be understood in the following way. Ga 0:95Mn0:05As\nepilayers grown on GaAs substrate are compressively\nstrained in the (001) plane with the typical value of\nthe strain parameter f\u0011(a\u0003\nGaMnAs\u0000a\u0003\nGaAs )=a\u0003\nGaAs\u0019\n0:2\u00000:3%, wherea\u0003\nGaAs anda\u0003\nGaMnAs are the lattice pa-\nrameters of the cubic fully relaxed GaAs and (Ga,Mn)As\n\flm, respectively. With the (Ga,Mn)As material re-\nmoved in the trenches along the bars, the lattice can\nrelax in the transverse direction and the corresponding\nextension can be roughly estimated as ft=w\u00180:01%,\nDevice B\n[110]-bar-[110]-barBulkFIG. 3: Comparison of the low-\fled measurements at 4.2 K\nof the transverse resistance in the bulk Van de Pauw device\n(upper panel) and of the longitudinal resistance of the [1 10]\nand [110]-bar in device B (lower panels).\nwheret= 25 nm is the thickness of the (Ga,Mn)As \flm\nandwis the bar width.\nOn a quantitative level, the strength of the lattice re-\nlaxation in the microbars is obtained from numerical elas-\ntic theory simulations for the realistic sample geometry.\n(GaAs values of the elastic constants are considered for\nthe whole wafer including the Ga 0:95Mn0:05As epilayer.)\nResults of such calculations are illustrated in Fig. 4 for\nthe [1 10]-bar of device B. In panel (a) we show the\nstrain component along the growth-direction [001]-axis\nwith respect to the lattice parameter of a fully relaxed\ncubic GaAs, e[001] = (a[001]\u0000a\u0003\nGaAs )=a\u0003\nGaAs . Since all\nstrain components scale linearly with fwe plote[001]=f.\nThe \fgure highlights the growth induced lattice match-\ning strain; because of the in-plane compression of the\n(Ga,Mn)As lattice the elastic medium reacts by expand-\ning the lattice parameter in the growth direction, as com-\npared toa\u0003\nGaMnAs , i.e.,e[001]=f > 1.\nWithin the plane, the lattice can relax only in the di-\nrection perpendicular to the microbar orientation. The\ncorresponding strain component, calculated again with\nrespect to the GaAs, is plotted in Fig. 4(b) over the en-\ntire cross-section of device B and, in Figs. 4(c) and (d),\nalong various cuts through the [001]-[110] plane. While\nin the center of the bar the in-plane relaxation is rela-\ntively weak, i.e. the lattice parameter remains similar to\nthat of the GaAs substrate, the lattice is strongly relaxed\nnear the edges of the bar. Averaged over the entire cross-4\nsection of the (Ga,Mn)As bar we obtain relative in-plane\nlattice relaxation of several hundredths of a per cent, i.e.,\nof the same order as estimated by the ft=w expression.\nThe microscopic magnetocrystalline energy calculations\ndiscussed in the following paragraphs con\frm that these\nseemingly small lattice distortions can fully account for\nthe observed easy-axis rotations in the strongly spin-orbit\ncoupled (Ga,Mn)As.\nOur microscopic calculations of the magnetization an-\ngle dependent total energies are based on combining the\nsix-band k\u0001pdescription of the GaAs host valence band\nwith kinetic-exchange model of the coupling to the local\nMnGad5-moments.3,4The theory is well suited for the\ndescription of spin-orbit coupling phenomena in the top\nof the valence band whose spectral composition and re-\nlated symmetries are dominated, as in the familiar GaAs\nhost, by the p-orbitals of the As sublattice. The k\u0001pmod-\neling also provides straightforward means of accounting\nfor the e\u000bects of lattice strains on the (Ga,Mn)As band\nstructure.3,4(As in the above macroscopic simulations\nwe assume that the elastic constants in (Ga,Mn)As have\nthe same values as in GaAs.) This theory, which uses no\nadjustable free parameters, describes accurately the sign\nand magnitude of the AMR data in Fig. 2.26It has also\nexplained the previously observed transitions between in-\nplane and out-of-plane easy magnetization orientations\nin similar (Ga,Mn)As epilayers grown under compressive\nand tensile strains and provided a consistent account of\nthe signs and magnitudes of corresponding AMR e\u000bects.2\nFor the modeling of the magnetocrystalline energy\nof the microbars we assume homogeneous strain in the\n(Ga,Mn)As layer corresponding to the average value of\ne[110] obtained in the macroscopic elastic theory sim-\nulations. The input parameters of the microscopic\ncalculations3,4are then strain components, related to the\nfully relaxed cubic (Ga,Mn)As lattice, in the [100]-[010]-\n[001] (x\u0000y\u0000z) coordinate system which are given by:\neij=0\n@exxexy0\neyxeyy0\n0 0ezz1\nA\n=0\n@e[110]\n2\u0000f\u0006e[110]\n20\n\u0006e[110]\n2e[110]\n2\u0000f 0\n0 0 e[001]\u0000f1\nA;(1)\nwhere\u0006corresponds to the [1 10]-bar and [110]-bar re-\nspectively.\nIn Fig. 5(b) we plot calculated magnetocrystalline en-\nergies as a function of the in-plane magnetization angle\nforf= 0:3% andexyranging from zero (no in-plane lat-\ntice relaxation) to typical values expected for the [1 10]-\nbar (exy>0) and for the [110]-bar ( exy<0). Consistent\nwith the experiment, the minima at [100] and [010] for\nexy= 0 move towards the [1 10] direction for lattice ex-\npansion along [110] direction ( exy>0) and towards the\n[110] direction for lattice expansion along [1 10] direction\n(exy<0). Note that the asymmetry between experi-\nmental easy-axes rotations in the two bars is due to thea [110]-uniaxial component present already in the bulk\nmaterial whose microscopic origin is not known but can\nbe modeled27by an intrinsic (not induced by micropat-\nterning) strain ebulk\nxy\u0018+0:01%.\nFIG. 4: Numerical simulations of lattice parameters in the\n1\u0016m wide [1 10]-bar of device B de\fned by 200 nm wide\nand 75 nm deep trenches in the 25 nm thick (Ga,Mn)As\n\flm on a GaAs substrate. (a) Strain component along the\n[001]-axis with respect to the lattice parameter of a fully\nrelaxed cubic GaAs, e[001] = (a[001]\u0000a\u0003\nGaAs )=a\u0003\nGaAs . The\nepitaxial growth induced strain parameter fis de\fned as,\nf= (a\u0003\nGaMnAs \u0000a\u0003\nGaAs )=a\u0003\nGaAs wherea\u0003\nGaMnAs> a\u0003\nGaAs is\nthe lattice parameter of the cubic fully relaxed (Ga,Mn)As\n\flm. (b) Same as (a) for in-plane strain component e[110]in\nthe direction perpendicular to the bar orientation. (c) and\n(d) Strain components e[110]along di\u000berent cuts through the\n[001]-[110] plane. The cuts and the corresponding e[110]=f\ncurves are highlighted by colored arrows in (b) and the cor-\nresponding color coding of curves in (c) and (d).\nIII. DEMONSTRATION IN CURRENT\nINDUCED SWITCHING\nThe L-shaped geometry of our devices is well suited\nfor a systematic study of the link between the locally\nadjusted magnetic anisotropies in the individual micro-\nbars and their current induced switching characteristics.\nApart from the distinct magnetocrystalline anisotropy\n\felds, the two bars in each device have identical material\nparameters and lithographical dimensions. They can also\nbe expected to share a common domain-wall nucleation\ncenter at the corner of the L-shaped channel since in this\nregion the lattice relaxation e\u000bects and the correspond-\ning enhancement of the magnetocrystalline anisotropies\nare less pronounced. Apart from this e\u000bect, the domain\nwall nucleation at the corner can be expected to be sup-5\nFIG. 5: (a) Schematics of the easy-axes orientations in the\n[110] and [110]-bars of the L-shaped devices A and B. Arrows\nindicate the direction and strength of the patterning induced\nlattice relaxation. (b) Theoretical magnetocrystalline ener-\ngies as a function of the in-plane magnetization angle for zero\nshear strain (black line), for exy= 0:004;::;0:02% (red lines)\ncorresponding to lattice extension along [110] axis, and for\nexy=\u00000:004;::;\u00000:02% (blue lines) corresponding to lat-\ntice extension along [1 10] axis. The magnetic easy-axes at\nexy= 0, 0.02% and -0.02% are highlighted by black, red, and\nblue arrows, resp. Lattice deformations breaking the [1 10]-\n[110] symmetry of the microscopic magnetocrystalline energy\npro\fle are illustrated by the diamond-like unit cells extended\nalong [110] axis for the [1 10]-bar (red diamond) and along the\n[110] axis for the [110]-bar (blue diamond).\nported by an enhanced current induced heating in this\npart of the device.\nThe basic phenomenology of current induced switch-\nings that we observe in all our L-shaped microbars is\nillustrated in Figs. 6 and 7. The particular \feld-assisted\nswitching data plotted in the \fgures were measured in\nthe [110]-bar of device A at \u0012= 7\u000e. At this o\u000b-easy-\naxis angle the current induced switching can be easily\ninduced and detected due to the hysteretic bistable char-\nacter of the low \feld magnetization and the clear AMR\nsignal upon reversal (see Fig. 7(a)). We start with assess-\ning the role of heating in the current induced switching\nexperiments. Figs. 6(a) and (b) compare the temper-\nature dependence of the longitudinal resistance at lowcurrent density (103Acm\u00002) with the dependence on\ncurrent density measured in liquid helium. As seen from\nthe plots, the maximum current density of 1 \u0002106Acm\u00002\nused in the experiments corresponds to heating the sam-\nple by approximately 20 K, which is well below the Curie\ntemperature of 100 K. Nevertheless, a suppression due to\nheating of the e\u000bective barrier between metastable and\nstable states and thermally induced reversals are possi-\nble near the switching \felds and these e\u000bects have to be\nconsidered when analyzing the current induced switching\nexperiments below.\nThe measurements presented in Figs. 7(b)-(f) were per-\nformed by \frst applying a saturation \feld and then re-\nversing the \feld and setting it to a value close to but\nbelow the switching \feld in the \feld-sweep experiment\n(see Fig. 7(a)). Then, the \frst current ramp was applied\nwhich triggered the reversal, followed by subsequent con-\ntrol current ramps of the same polarity which showed\nno further changes in the magnetization. Constant cur-\nrent sweep rate of 5 \u0002104Acm\u00002s\u00001was used in all\nexperiments. In Figs. 7(b)-(f) we plot the di\u000berence, \u000eR,\nbetween resistances of the \frst and the subsequent cur-\nrent ramps. We note that no switchings were observed\nin these experiments up to the highest applied currents\nin the [1 10]-bar. In this bar with the stronger magne-\ntocrystalline anistropy, the magnitude of the low current\n(103Acm\u00002) switching \feld at \u0012= 7\u000eis\u00198 mT, as com-\npared to the\u00195:5 mT switching \feld in the [110]-bar.\nFirst we discuss data in Fig. 7(b) and (c) taken at -\n4 mT external \feld and negative current ramps. The\ntwo independent experiments (panels (b) and (c) re-\nspectively) performed at nominally identical conditions\ndemonstrate the high degree of reproducibility achieved\nin our devices. This includes the step-like features\nwhich we associate with domain wall depinning/pinning\nevents preceding full reversal. To understand this process\nin more detail we complement the longitudinal (black\ncurve) and transverse (red curve) resistance measure-\nments in the [110]-bar with the resistance measurements\nat the corner (blue curve) of the L-shaped channel. The\nschematic plot of the respective voltage probes is shown\nin the inset. The \frst magnetization switching event at\nj\u0019\u00005\u0002105Acm\u00002is detected by the step in the \u000eRC\nsignal, i.e., occurs in the corner region between the RC\ncontacts. For current densities in the range between j\u0019\n\u00005\u0002105Acm\u00002andj\u0019\u00006\u0002105Acm\u00002the domain wall\nremains pinned in the corner region. The next domain\nwall propagation and pinning event in \u000eRCis observed\nbetweenj\u0019\u00006\u0002105Acm\u00002andj\u0019\u00007\u0002105Acm\u00002\nand forjjj>7\u0002105Acm\u00002the region between the RC\ncontacts is completely reversed. The depinning events\natj\u0019 \u0000 5\u0002105Acm\u00002andj\u0019 \u0000 6\u0002105Acm\u00002are\nalso registered by the RLandRTcontacts through noise\nspikes in the respective \u000eRLand\u000eRTsignals. However,\nbeyond these spikes, \u000eRLand\u000eRTremain constant for\njjj<7\u0002105Acm\u00002indicating that the domain wall has\nnot reached the section of the [110]-bar between the RL\ncontacts at these current densities. Constant \u000eRCand6\nstep-like changes in \u000eRLand\u000eRTatjjj>7\u0002105Acm\u00002\nare signatures of the domain wall leaving the corner sec-\ntion and entering the part of the [110]-bar between the\nRLcontacts. The reversal of this part is completed at\nj\u0019\u00008\u0002105Acm\u00002. Note that both the \u000eRL, averaging\nover the whole bar between the longitudinal contacts, and\nthe\u000eRT, re\recting the local structure near the respective\ntransverse contacts, show switching at the same current\nand the sense and magnitude of the overall change in \u000eRL\nand\u000eRTare consistent with those observed in the \feld\nsweep measurement (see Fig. 7(a)). This indicates that\nthe contacts have a negligible e\u000bect on the anisotropy in\nthis bar and allows us to unambiguously determine the\nmagnetization angles of the initial state, 39 \u00061\u000e, and of\nthe \fnal state, 211 \u00061\u000e. This -4 mT \feld assisted cur-\nrent induced switching is not observed at positive current\nramps up to the highest experimental current density of\nj= 1\u0002106Acm\u00002which indicates that spin-transfer-\ntorque e\u000bects can be contributing to the reversal. Note\nalso that the domain wall propagates in the direction\nopposite to the applied hole current, in agreement with\nprevious spin-transfer-torque studies of perpendicularly\nmagnetized (Ga,Mn)As \flms.8(The anomalous direction\nof the domain wall propagation is assigned to the anti-\nferromagnetic alignment of hole spins with respect to the\ntotal moment in (Ga,Mn)As.7,8,9)\nFIG. 6: (a) Temperature dependence of relTRL\u0011[RL(T)\u0000\nRL(4:2)]=RL(4:2) at current density 103Acm\u00002. For com-\npleteness, relTRLover a wide range of temperatures below\nand above the Curie temperature is shown in the inset. (b)\nFirst (solid line) and second (dashed line) current ramps at\n-4 mT \feld applied along \u0012= 7\u000e; relative resistances are\nplotted with respect to the zero-current resistance in the \frst\nramp. Switching at jc\u0019 \u00007:5\u0002105Acm\u00002is marked.\nA suppression of the role of the spin-transfer-torque\nrelative to the thermally assisted switching mechanism\n510RCRT\nRL-10-50-20-100\nj[105A/cm2](b)\n-20-100δδδδR[Ω]-4mT\n-20-100δδδδR[Ω](e)\n-5mT\n510 -10-50\nj[105A/cm2]-20-1001020-125-100-75-50ΔΔΔΔRL[Ω]\nBθθθθ[mT]-10-50510ΔΔΔΔRT[Ω]\n-20-1001020-125-100-75-50[\nBθθθθ[mT]-10-50510\nθ= 7°(a)\nExp. #1\nExp. #2Device A [110]-bar\n(c) (d)\n(f)+jδδδδRLδδδδRT\nδδδδRCFIG. 7: (a) Field-sweep measurements at \u0012= 7\u000ein the [110]-\nbar of device A. (b) Di\u000berences between the \frst and second\nnegative current ramps for the longitudinal (black lines) and\ntransverse (red lines) resistance in the [110]-bar and in the\ncorner (blue lines) of device A at -4 mT external \feld applied\nalong\u0012= 7\u000e. Arrows indicate the current ramp direction.\n(c) Same as (b) for the second independent experiment. (d)\nSame as (b) and (c) for positive current ramps. The inset\nshows contacts used for measurements of RL,RT, andRC\nin all panels. (e),(f) -5 mT \feld assisted current induced\nswitching experiments.\nis expected at \felds closer to the coercive \feld. The\ndata taken at -5 mT \feld shown in Fig. 7(e) and (f) are\nfully consistent with this expectation. Current induced\nswitchings are observed here at lower critical currents and\nfor both current polarities. Nevertheless, the asymmetry\nbetween the negative and positive critical currents is still\napparent and consistent with a picture of cooperative\ne\u000bects of heating and spin-transfer-torque for negative\ncurrents and competing e\u000bects of the two mechanisms\nfor positive currents.\nThe distinct current induced switching characteristics\nachieved by patterning one bar along the [110] direction\nand the other bar along the [1 10] direction are illustrated\nin Figs. 8 and 9 on a set of experiments in device B. The\nmeasurements shown in Figs. 8(b)-(d) were taken on the7\n[110]-bar in an external \feld of a magnitude of -9 mT\napplied along \u0012= 0\u000e(see corresponding \feld sweep mea-\nsurements in Fig. 8(a)). Up to the highest experimental\ncurrent densities, the switching (from magnetization an-\ngle 9\u000eto 180\u000e) is observed only for the positive current\npolarity. A less detailed tracking of the domain wall is\npossible in this experiment compared to the data in Fig. 7\ndue to the larger magnitude of the external \feld (larger\ncoercive \feld of device B) and smaller separation of the\ncontacts used to monitor RCin this device. Neverthe-\nless, the -9 mT \feld assisted reversal process shown in\nFig. 8 is clearly initiated in the corner and, again, the\ndomain wall propagates in the direction opposite to the\napplied hole current. Since for the opposite magnetic\n\feld sweep we observe the current induced switching at\n+9 mT also at positive currents (compare Figs. 8(b) and\n(d)), the Oersted \felds are unlikely to be the dominant\nswitching mechanism. Note also that the Oersted \felds\ngenerated by our experimental currents are estimated to\nbe two orders of magnitude weaker than the anisotropy\n\felds.9\nδR[Ω]\nj[105A/cm2](b)-505\n-9mT\n0 510-10-50 + 9mTRL\nRT\nRL\n-10-5±9mT\nRCRT-505ΔΔΔΔRT[Ω]\n-20-1001020-110-100-90-80ΔΔΔΔRL[Ω]\nB[mT]-505RT[\n-20-1001020-110-100-90-80RL[\nB[mT]θ= 0°\nθθθθθθθθ(a)Device B [110]-bar-\n-j(c)\n(d)-δδδδRT\n-δδδδRTδδδδRc\nδδδδRcδδδδRLδδδδRL\nFIG. 8: (a) Field-sweep measurements at \u0012= 0\u000ein the [1 10]-\nbar of device B. (b) Di\u000berence between the \frst and second\npositive current ramps in the [1 10]-bar of device B at -9 mT\n\feld applied along \u0012= 0\u000e. Note that \u0000\u000eRTis plotted for\nclarity. (c) Same as (b) at negative current ramps at \u00069 mT.\nThe inset shows contacts used for measurements of RLand\nRTin all panels.(d) Same as (b) at +9 mT \feld.\nThe character of the current induced switching in de-\nvice B at -9 mT is completely di\u000berent in the [110]-bar\ncompared to the [1 10]-bar, as shown in Figs. 9(c) and(d). The switching occurs at much lower current den-\nsities due to the lower coercive \feld of the [110]-bar at\n\u0012= 0\u000e(compare Figs. 8(a) and 9(a)), and the asym-\nmetry between the positive and negative switching cur-\nrents is small, suggesting that heating plays an important\nrole in this experiment. Although we see clear jumps in\n\u000eRL, which are consistent with the \feld-sweep data in\nFig. 9(a), the absence of the \u000eRTswitching signal in the\n[110]-bar hinders the unambiguous determination of the\nswitching angles. This feature is ascribed to a fabrica-\ntion induced strong pinning at the RTcontacts; indeed\nthe \feld-sweep measurements for the [110]-bar show an\nincomplete switching at 10 mT in the longitudinal resis-\ntance and no clear signature of switching for the trans-\nverse resistance contacts at this \feld. (Full saturation of\nthe entire bar including the transverse contacts region is\nachieved at 100 mT.)\n(a)ΔΔΔΔRT[Ω]\n-20-1001020ΔΔΔΔRL[Ω]\nBθθθθ[mT]-10-50\n-100-80-60-40-200\n10ΩΩΩΩ\nθ= 0°\n-10-50-20-100\nj[105A/cm2]δR[Ω]-9mT\n(c)\n510Device B [110] bar\n(d)\nRT\n+jRLRC0 mT\n(b)\n-10-50-10-50510\nj[105A/cm2]δR[Ω]δδδδRL\nδδδδRT\nδδδδRc\nδδδδRLδδδδRT\nFIG. 9: (a) Field-sweep measurements at \u0012= 0\u000ein the [110]-\nbar of device B. (b) Di\u000berence between the \frst and second\nnegative current ramps in the [110]-bar of device B at zero\n\feld. (c) Di\u000berence between the \frst and second negative\ncurrent ramps at -9 mT \feld applied along \u0012= 0\u000e. (d) Same\nas (c) at for positive current ramps. The inset shows contacts\nused for measurements of RLandRTin all panels.\nIn Fig. 9(b) we exploit the pinning at the RTcontacts\nto study current induced switching at zero magnetic \feld.\nNote that if the switching of the whole bar was com-\nplete the zero-\feld 180\u000erotation from negative to pos-\nitive easy-axis directions would be undetectable by the8\nAMR measurement. We again see no switching signal in\n\u000eRTbut a clear step in \u000eRL. As for all \feld-assisted ex-\nperiments, the sense and magnitude of the jump in \u000eRL\nfor zero \feld correlates well with the \feld sweep mea-\nsurements (see the dashed line in Fig. 9(a)). Also consis-\ntent with the trends in the \feld-assisted experiments, the\nswitching occurs at larger current than in the -9 mT \feld\nassisted switching. Up to the highest experimental cur-\nrent density, the zero-\feld switching is observed only in\nthe negative current ramp, as we would expect for the do-\nmain wall propagation from the corner (see the \u000eRcsignal\nin Fig. 9(b)) to the [110]-bar due to spin-transfer-torque.\nWe emphasize however that a detailed understanding of\nthe origin of the observed current induced switchings in\nour L-shaped devices is beyond the scope of this work.\nOur main aim was to demonstrate that the local control\nof the magnetocrystalline anisotropy we achieved in these\ndilute moment ferromagnetic structures is a new powerful\ntool for investigating spin dynamics phenomena.\nIV. SUMMARY\nIn summary, (Ga,Mn)As microchannels with locally\ncontrolled magnetocrystalline anisotropies and inher-\nently weak dipolar \felds represent a new favorable class\nof systems for exploring magneto-electronic e\u000bects at mi-\ncroscale. We have observed easy-axes rotations which\ndepend on the width and crystal orientation of the mi-\ncrochannel. Based on numerical simulations of strain\ndistribution for the experimental geometry and micro-\nscopic calculations of the corresponding spin-orbit cou-\npled band structures we have explained the e\u000bect in\nterms of lattice relaxation induced changes in the mag-\nnetocrystalline anisotropy. The observation and expla-\nnation of micropatterning controlled magnetocrystallineanisotropy of the (Ga,Mn)As dilute moment ferromagnet\nrepresents the central result of our paper. In addition to\nthat we have demonstrated that the structures are well\nsuited for a systematic study of current induced switch-\ning phenomena well bellow Curie temperature at rela-\ntively low current densities. We have found indications\nthat domain-wall spin-transfer-torque e\u000bects contribute\nstrongly to the observed switchings. This suggests that\nour structures represent a new favorable system for ex-\nploring these technologically important yet still physi-\ncally controversial spin dynamics phenomena.\nNote added : After the completion of our work, inde-\npendent and simultaneous studies of the lattice relax-\nation induced changes of magnetocrystalline anisotropies\nin (Ga,Mn)As have been posted on the Los Alamos\nArchives and some of them published during the process-\ning of our manuscript.28,29,30The crystal orientations and\nwidths of the nanochannels considered in these works are\ndi\u000berent than in our study. Nevertheless, the reported\ne\u000bects are of the same origin and our works provide a\nmutual con\frmation that the seemingly tiny changes in\nthe lattice constant can completely overwrite the mag-\nnetocrystalline energy landscape of the host (Ga,Mn)As\nepilayer.\nAcknowledgment\nWe acknowledge discussions with A. H. MacDonald,\nV. Nov\u0013 ak, and support from EU Grant IST-015728,\nfrom EPSRC Grant GR/S81407/01, from GACR and\nAVCR Grants 202/05/0575, 202/06/0025, 202/04/1519,\nFON/06/E002, AV0Z1010052, LC510, from MSM Grant\n0021620834, from NSF Grant DMR-0547875, and from\nONR Grant N000140610122.\n1F. Matsukura, H. Ohno, and T. Dietl, in Handbook of Mag-\nnetic Materials , edited by K. H. J. Buschow (Elsevier, Am-\nsterdam, 2002), vol. 14, p. 1, From Ohno Lab Homepage.\n2T. Jungwirth, J. Sinova, J. Ma\u0014 sek, J. Ku\u0014 cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006), cond-\nmat/0603380.\n3T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B 63 ,\n195205 (2001), cond-mat/0007190.\n4M. Abolfath, T. Jungwirth, J. Brum, and A. H. MacDon-\nald, Phys. Rev. B 63 , 054418 (2001), cond-mat/0006093.\n5J. Sinova, T. Jungwirth, X. Liu, Y. Sasaki, J. K. Furdyna,\nW. A. Atkinson, and A. H. MacDonald, Phys. Rev. B 69 ,\n085209 (2004), cond-mat/0308386.\n6D. Chiba, Y. Sato, T. Kita, F. Matsukura, and H. Ohno,\nPhys. Rev. Lett. 93, 216602 (2004), cond-mat/0403500.\n7M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno,\nNature 428, 539 (2004).\n8D. Chiba, M. Yamanouchi, F. Matsukura, T. Dietl, and\nH. Ohno, Phys. Rev. Lett. 96, 096602 (2006), cond-\nmat/0601464.9M. Yamanouchi, D. Chiba, F. Matsukura, T. Dietl, and\nH. Ohno, Phys. Rev. Lett. 96, 096601 (2006), cond-\nmat/0601515.\n10P. P. Freitas and L. Berger, J. Appl. Phys. 57, 1266 (1985).\n11A. Yamaguchi, T. Ono, S. Nasu, K. Miyake, K. Mibu, and\nT. Shinjo, Phys. Rev. Lett. 92, 077205 (2004).\n12E. Saitoh, H. Miyajima, T. Yamaoka, and G. Tatara, Na-\nture432, 203 (2004).\n13G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004), cond-mat/0308464.\n14Z. Li and S. Zhang, Phys. Rev. B 70 , 024417 (2004).\n15G. Tatara, N. Vernier, and J. Ferre, Appl. Phys. Lett. 86,\n252509 (2004), cond-mat/0411250.\n16S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95, 107204\n(2005).\n17A. Thiaville, Y. Nakatani, J. Miltat, and Y. Suzuki, Euro-\nphys. Lett. 69, 990 (2005).\n18C. Gould, K. Pappert, C. R uster, R. Giraud, T. Borzenko,\nG. M. Schott, K. Brunner, G. Schmidt, and L. W.\nMolenkamp, Jpn. J. Appl. Phys. 1 45 , 3860 (2006), cond-9\nmat/0602135.\n19M. Hayashi, L. Thomas, Y. B. Bazaliy, C. Rettner,\nR. Moriya, X. Jiang, and S. S. P. Parkin, Phys. Rev. Lett.\n96, 197207 (2006).\n20L. Thomas, M. Hayashi, X. Jiang, R. Moriya, C. Rettner,\nand S. S. P. Parkin, Nature 443, 197 (2006).\n21V. K. Dugaev, V. R. Vieira, P. D. Sacramento, J. Bar-\nnas, M. A. N. Ara\u0013 ujo, and J. Berakdar, Phys. Rev. B 74 ,\n054403 (2006).\n22J. Xiao, A. Zangwill, and M. D. Stiles, Phys. Rev. B 73 ,\n054428 (2006).\n23J. I. Ohe and B. Kramer, Phys. Rev. Lett. 96, 027204\n(2006).\n24M. A. N. Araujo, V. K. Dugaev, V. R. Vieira, J. Berakdar,\nand J. Barnas, Phys. Rev. B 74 , 224429 (2006), cond-\nmat/0610235.\n25R. A. Duine, A. S. N\u0013 u~ nez, and A. H. MacDonald, Phys.\nRev. Lett. 98, 056605 (2007), cond-mat/0607663.\n26A. W. Rushforth, K. V\u0013 yborn\u0013 y, C. S. King, K. W. Ed-monds, R. P. Campion, C. T. Foxon, J. Wunderlich,\nA. C. Irvine, P. Va\u0014 sek, V. Nov\u0013 ak, et al. (2007), cond-\nmat/0702357.\n27M. Sawicki, K.-Y. Wang, K. W. Edmonds, R. P. Cam-\npion, C. R. Staddon, N. R. S. Farley, C. T. Foxon, E. Pa-\npis, E. Kaminska, A. Piotrowska, et al., Phys. Rev. B 71 ,\n121302 (2005), cond-mat/0410544.\n28S. H umpfner, M. Sawicki, K. Pappert, J. Wenisch,\nK. Brunner, C. Gould, G. Schmidt, T. Dietl, and L. W.\nMolenkamp, Appl. Phys. Lett. 90, 102102 (2007), cond-\nmat/0612439.\n29K. Pappert, S. H umpfner, C. Gould, J. Wenisch, K. Brun-\nner, G. Schmidt, and L. W. Molenkamp (2007), cond-\nmat/0701478.\n30J. Wenisch, C. Gould, L. Ebel, J. Storz, K. Pappert, M. J.\nSchmidt, C. Kumpf, G. Schmidt, K. Brunner, and L. W.\nMolenkamp (2007), cond-mat/0701479."    },    {        "title": "0709.3243v2.Local_electronic_structure_and_magnetic_properties_of_LaMn0_5Co0_5O3_studied_by_x_ray_absorption_and_magnetic_circular_dichroism_spectroscopy.pdf",        "content": "arXiv:0709.3243v2  [cond-mat.str-el]  17 Mar 2008Local electronic structure and magnetic properties of LaMn 0.5Co0.5O3studied by x-ray\nabsorption and magnetic circular dichroism spectroscopy\nT. Burnus,1Z. Hu,1H. H. Hsieh,2V. L. J. Joly,3P. A. Joy,3M. W.\nHaverkort,1Hua Wu,1A. Tanaka,4H.-J. Lin,5C. T. Chen,5and L. H. Tjeng1\n1II. Physikalisches Institut, Universit¨ at zu K¨ oln, Z¨ ulp icher Straße 77, 50937 K¨ oln, Germany\n2Chung Cheng Institute of Technology, National Defense Univ ersity, Taoyuan 335, Taiwan\n3Physical and Materials Chemistry Division, National Chemi cal Laboratory, Pune 411008, India\n4Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan\n5National Synchrotron Radiation Research Center, 101 Hsin- Ann Road, Hsinchu 30076, Taiwan\n(Dated: June 26, 2018)\nWe have studied the local electronic structure of LaMn 0.5Co0.5O3using soft-x-ray absorption\nspectroscopy at the Co- L3,2and Mn- L3,2edges. We found a high-spin Co2+–Mn4+valence state\nfor samples with the optimal Curie temperature. We discover ed that samples with lower Curie\ntemperatures contain low-spin nonmagnetic Co3+ions. Using soft-x-ray magnetic circular dichroism\nwe established that the Co2+and Mn4+ions are ferromagnetically aligned. We revealed also that\nthe Co2+ions have a large orbital moment: morb/mspin≈0.47. Together with model calculations,\nthis suggests the presence of a large magnetocrystalline an isotropy in the material and predicts a\nnontrivial temperature dependence for the magnetic suscep tibility.\nPACS numbers: 71.27.+a, 78.70.Dm, 71.70.-d, 75.25.+z\nThemanganitescontinuetoattractconsiderableatten-\ntion from the solid state physics and chemistry commu-\nnityoverthelastfivedecadesbecauseoftheirspectacular\nmaterialproperties.1,2,3,4TheparentcompoundLaMnO 3\nisanA-typeantiferromagneticinsulatorwithorthorhom-\nbic perovskite crystal structure. Replacing La by Sr,\nCa or Ba results in multifarious electronic and magnetic\nproperties including the transformation into a ferromag-\nnetic state accompanied by a metal-insulator transition\nand the occurrenceofcolossalmagnetoresistance.5,6Sub-\nstitution of the magnetic Mn ions by Co also yields fer-\nromagnetism in the LaMn 1−xCoxO3series. The Curie\ntemperature reaches a maximum for x= 0.5 (TC= 220–\n240 K).7,8,9,10,11This should be contrasted with the\nend member of this series, namely the rhombohedral\nLaCoO 3, whichisanonmagneticinsulatoratlowtemper-\natures, showing yet the well-known spin-state transition\nat higher temperatures which by itself is subject of five\ndecades of intensive study.7,9,12\nExplaining the appearance of ferromagnetism in the\nmanganites by Co substitution is, however, not a triv-\nial issue. Assuming that ordering of the Co and Mn\nions had not been achieved for the x= 0.5 composi-\ntion, Goodenough et al.concluded early on that the\nferromagnetism is generated by Mn3+–O–Mn3+superex-\nchange interactions.7On the other hand, later magnetic\nsusceptibility and Mn NMR studies suggested that it is\nthe exchange interaction involving the ordering of Co2+–\nMn4+transition-metal ions which causes the ferromag-\nnetism in LaMn 0.5Co0.5O3.8,9,13,14,15,16,17\nOnly few high-energy spectroscopic studies are re-\nported for the Co substituted manganites. Using soft-x-\nray absorption spectroscopy (XAS), Park et al.found in\ntheir low Co compositions that the Co ions are divalent,\nfavoring a Mn3+–Mn4+double-exchange mechanism for\nthe ferromagnetism.18Extrapolating this Co divalent re-sult to the x= 0.5 composition would provide support to\nthe suggestion that the ferromagnetism therein is caused\nby the Co2+–Mn4+exchange interaction. However, no\nXAS data have been reported so far for this x= 0.5\ncomposition. Using K-edge XAS, Toulemonde et al.re-\nvealed that the Co ion is also divalent in their hole doped\nand Co substituted manganite.19Yet, these results for\nthe low Co limit have been questioned by van Elp, who\nclaimed that the Co ions should be in the intermediate-\nspin trivalent state rather than in the high-spin divalent\nstate.20\nFurther discussionis alsoraisedby the workofJoyand\ncoworkers,21,22who have synthesized two different sin-\ngle phases of LaMn 0.5Co0.5O3and inferred from a com-\nbination of magnetic susceptibility and x-ray photoelec-\ntron spectroscopy measurements that the phase with the\nhigherTCcontains high-spin Mn3+and low-spin Co3+\nions, while the lower TCphase has Co2+and Mn4+.\nVery recently, however, long-range charge ordering has\nbeen observed in neutron diffraction experiments by Bull\net al.23and Troyanchuk et al.24on the high- TCphase,\npointing towards the Co2+–Mn4+scenario. Also, the\nmost recent magnetic susceptibility and K-edge XAS\ndata by Kyˆ omen et al.favor the presence of essen-\ntially Co2+–Mn4+at low temperatures.25The issue of\nMn/Co ordering including the possible coexistence of or-\ndered and disordered regions remains one of the impor-\ntant topics.11,26,27Interesting is that the magnetization\nof polycrystalline samples of LaCo 0.5Mn0.5O3does not\nsaturate in magnetic fields up to 7 T,16and that there\nare indications for a large magnetic anisotropy.24\nOn the theoretical side, not much work has been car-\nriedoutsofar. Arelativelyearlyband-structurestudyby\nYanget al.on the LaMn 0.5Co0.5O3system predicted a\nhalf-metallicbehaviorwithamagneticmomentof3 .01µB\nfor Mn and 0 .54µBfor Co ions, suggesting Mn3+–Co3+2\nvalence states.28This study, however, was performed be-\nfore the existence of the charge-ordered crystal structure\nwas reported.23,24\nHere, we present our experimental study of the local\nelectronicstructureofLaMn 0.5Co0.5O3bothforthehigh-\nand low- TCphases using the element-specific XAS and\nx-ray magnetic circular dichroism (XMCD) at the Co-\nL2,3and Mn-L2,3edges, i.e., transitions from the 2 pcore\nto the 3dvalence orbitals. Our objective is not only to\nestablish the valence and spin states of the Co and Mn\nions but also to investigate the possible presence of an\norbital moment associated with a Co2+ion, in which\ncase the material should have a large magnetocrystalline\nanisotropy and a nontrivial temperature dependence of\nits magnetic susceptibility.\nIn XAS and XMCD we make use of the fact that the\nCoulomb interaction of the 2 pcore hole with the 3 delec-\ntrons is much larger than the 3 dband width, so that\nthe absorption process is strongly excitonic and there-\nfore well understood in terms of atomiclike transitions\nto multiplet-split final states.29,30,31Unique to soft-x-ray\nabsorption is that the dipole selection rules are very ef-\nfective in determining which of the 2 p53dn+1final states\ncan be reached and with what intensity, starting from a\nparticular 2 p63dninitial state ( n= 7 for Co2+,n= 6 for\nCo3+,n= 4 for Mn3+, andn= 3 for Mn4+). This makes\nthe technique an extremely sensitive local probe, ideal\nto study the valence32,33and spin12,34,35,36,37,38charac-\nter as well as the orbital contribution to the magnetic\nmoment39,40,41of the ground or initial state.\nThe two single-phase LaMn 0.5Co0.5O3polycrystalline\nsamples were synthesized as described previously21,22,42\nand the single phase nature of the two phases (low- TC\nphase and high- TCphase) were confirmed by temper-\nature dependent magnetization measurements. These\nmeasurements showed a single sharp magnetic transition\natTC= 225K(calledhigh- TCphase)forthe samplesyn-\nthesized at 700◦C and a sharp transition at TC= 150\nK (called low- TCphase) for the sample synthesized at\n1300◦C. On the other hand, more than one magnetic\ntransition or broad magnetic transitions were observed\nfor samples synthesized at other temperatures indicating\ntheir mixed phase behavior, as described in Ref. 42. The\nmagnetizationat5Kinafieldof5Tis50.4emu/gforthe\nhigh-TCphaseand42.4emu/gforthe low- TCphase. The\nCo- and Mn- L2,3XAS and XMCD spectra were recorded\nat the Dragon beamline of the National Synchrotron Ra-\ndiation Research Center (NSRRC) in Taiwan with an en-\nergy resolution of 0.25 eV. The sharp peak at 777.8 eV of\nthe Co-L3edge ofsingle crystalline CoOand at 640eVof\nthe Mn-L3of singlecrystalline MnO wereused for energy\ncalibration. The isotropic XAS spectra were measured at\nroom temperature, whereas the XMCD spectra at both\nthe Co-L2,3and the Mn- L2,3edges were measured at\n135 K in a 1 T magnetic field with approximately 80%\ncircularly polarized light. The magnetic field makes an\nangle of 30◦with respect to the Poynting vector of the\nsoft x-rays. The spectra were recorded using the total/s55/s55/s53 /s55/s56/s48 /s55/s56/s53 /s55/s57/s48 /s55/s57/s53 /s56/s48/s48/s67/s111/s79/s76/s97/s67/s111/s79\n/s51\n/s76/s97/s77/s110\n/s48/s46/s53/s67/s111\n/s48/s46/s53/s79\n/s51\n/s32/s32/s32 /s84\n/s67/s32/s61/s32/s49/s53/s48/s32/s75\n/s76/s97/s77/s110\n/s48/s46/s53/s67/s111\n/s48/s46/s53/s79\n/s51\n/s32/s32/s32 /s84\n/s67/s32/s61/s32/s50/s50/s53/s32/s75/s67/s111/s45 /s76\n/s50/s67/s111/s45 /s76\n/s51\n/s32/s32/s73/s110 /s116/s101/s110 /s115/s105/s116/s121 \n/s80/s104/s111/s116/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s97\n/s98\n/s99\n/s100\nFIG. 1: Co- L2,3XAS spectra of (a) LaCoO 3as a Co3+ref-\nerence, of the LaMn 0.5Co0.5O3samples with (b) TC= 150 K\nand (c)TC= 225 K, and (d) of CoO as a Co2+reference.\nelectron yield method (by measuring the sample drain\ncurrent) in a chamber with a base pressure of 2 ×10−10\nmbar. Clean sample areas were obtained by cleaving the\npolycrystals in situ.\nFigure 1 shows the Co- L2,3XAS spectra of\nLaMn0.5Co0.5O3for both the high- TC[curve (c)] and the\nlow-TCphase[curve(b)]. Thespectraweretakenatroom\ntemperature. For comparison, the spectrum of LaCoO 3\nin the low-temperature nonmagnetic state [curve (a)] is\nincluded as a low-spin trivalent Co reference and also of\nCoO (curve d) as a divalent Co reference. The spectra\nare dominated by the Co 2 pcore-hole spin-orbit coupling\nwhich splits the spectrum roughly in two parts, namely\ntheL3(hν≈777–780 eV) and L2(hν≈793–796 eV)\nwhite lines regions. The line shape of the spectrum de-\npends stronglyonthemultiplet structuregivenbythe Co\n3d–3dand 2p–3dCoulomb and exchange interactions, as\nwell as by the local crystal fields and the hybridization\nwith the O 2 pligands.\nImportant is that XAS spectra are highly sensitive to3\n/s55/s55/s53 /s55/s56/s48 /s55/s56/s53 /s55/s57/s48 /s55/s57/s53 /s56/s48/s48/s76 /s97/s77/s110\n/s48/s46/s53/s67/s111 \n/s48/s46/s53/s79\n/s51\n/s32/s84\n/s67/s32/s61/s32/s49/s53/s48/s32/s75\n/s32/s84\n/s67/s32/s61/s32/s50/s50/s53/s32/s75/s44/s32/s215/s32/s48/s46/s56\n/s32/s68/s105/s102/s102/s101/s114/s101/s110/s99/s101\n/s76 /s97/s67/s111 /s79\n/s51/s32/s215/s32/s48/s46/s50/s67/s111 /s45 /s76\n/s50/s67/s111 /s45 /s76\n/s51\n/s32/s32/s73 /s110/s116/s101/s110/s115/s105/s116/s121 \n/s80/s104/s111 /s116/s111 /s110/s32/s69/s110/s101/s114/s103 /s121 /s32/s40/s101/s86/s41/s97\n/s98\nFIG. 2: (Color online) Co- L2,3XAS spectra of (a) the\nLaMn 0.5Co0.5O3samples with TC= 225 K (solid black curve)\nandTC= 150 K (dashed red curve), their difference (dotted\nblue curve), and (b) of LaCoO 3as Co3+reference.\nthe valence state: an increase of the valence state of the\nmetal ion by one causes a shift of the XAS L2,3spec-\ntra by one or more eV toward higher energies.32,33This\nshift is due to a final state effect in the x-ray absorp-\ntion process. The energy difference between a 3 dn(3d7\nfor Co2+) and a 3 dn−1(3d6for Co3+) configuration is\n∆E=E(2p63dn−1→2p53dn)−E(2p63dn→2p53dn+1)≈\nUpd−Udd≈1–2 eV, where Uddis the Coulomb repulsion\nenergybetweentwo3 delectronsand Updthe onebetween\na 3delectron and the 2 pcore hole. In Fig. 1 we see a shift\nof the “center of gravity” of the L3white line to higher\nphoton energies by approximately 1.5 eV in going from\nCoO to LaCoO 3. The energy position and the spectral\nshape of the high- TCphase of LaMn 0.5Co0.5O3is very\nsimilar to that of CoO, indicating an essentially divalent\nstate of the Co ions.\nWhile the spectral features of the low- TCphase of\nLaMn0.5Co0.5O3are also very similar to those of CoO\nand the high- TCphase asfarasthe low-energysideofthe\nL3white line is concerned, this is no longer true for the\nhigh-energy side. The spectral weight at about 780 eV is\nincreased when one compares the high- TCwith the low-\nTCphase, and this increase is revealed more clearly by\ncurves(a) ofFig.2. It isnaturaltoassociatethis increase\nwith the presenceofCo3+speciessincethe LaCoO 3spec-\ntrum has its main peak also at 780 eV. In order to ver-\nify this in a more quantitative manner, we rescaled the\nspectrum of the high- TCphase with respect to that of\nthe low-TCphase and calculate their difference. We find\nthat a rescaling factor of about 0.8 results in a differ-\nence spectrum (dotted blue curve of Fig. 2) which resem-\nbles very much the spectrum of LaCoO 3. This in turn\nmay be taken as an indication that the low- TCphase has\nabout 20% of its Co ions in the low-spin trivalent state.\nThis result contradicts the reports in Refs. 21 and 22640 645 650 655 660eSrMnO3LaMn0.5Co0.5O3\nTc = 225 K\nMnOLaMnO3LaMn0.5Co0.5O3\nTc = 150 KMn-L2Mn-L3\n  Intensity\nPhoton Energy (eV)a\nb\nc\nd\nFIG. 3: Mn- L2,3XAS spectra of the LaMn 0.5Co0.5O3sample\nwith (a) TC= 150 K and (b) TC= 225 K together with (c)\nSrMnO 3(Mn4+, taken from Ref. 43), (d) LaMnO 3(Mn3+)\nand (e) MnO (Mn2+) for comparison.\nwhich suggested that it was the high- TCsample which\ncontained trivalent Co ions. The different result com-\ning from the x-ray photoemission (XPS) study22could\nbe due to the following reason: Unlike XAS in which the\nmultiplet structure of the Co- L2,3spectra is very charac-\nteristic for the Co valence, the XPS yields rather broad\nand featureless Co 2 pcore-level spectra with very little\ndistinction between Co2+and Co3+. To use XPS core-\nlevel shifts to determine the valence state of insulating\nmaterials is also not so straight forward due to the fact\nthat the chemical potential with respect to the valence or\nconduction band edges is not well defined. The present\nfinding of the presence of low-spin Co3+species natu-\nrally explains why the low- TCsample has less than the\noptimalTC: the nonmagnetic ions suppress strongly the\nspin-spin coupling between neighboring metal ions.\nFigure 3 shows the room temperature Mn- L2,3XAS\nspectra of the low- TCLaMn0.5Co0.5O3[curve (a)] and\nthe high- TCLaMn0.5Co0.5O3[curve (b)] together with\nLaMnO 3as a trivalent Mn reference [curve (c)] and MnO4\n/s54/s52/s48 /s54/s52/s53 /s54/s53/s48 /s54/s53/s53 /s54/s54/s48/s76 /s97/s77/s110\n/s48/s46/s53/s67/s111 \n/s48/s46/s53/s79\n/s51/s32/s84\n/s99/s32/s61/s32/s49/s53/s48/s32/s75\n/s32/s84\n/s99/s32/s61/s32/s50/s50/s53/s32/s75/s44/s32/s215/s32/s48/s46/s56\n/s32/s68/s105/s102/s102/s101/s114/s101/s110/s99/s101\n/s76/s97/s77/s110/s79\n/s51/s32/s215/s32/s48/s46/s50/s77/s110/s45 /s76\n/s50/s77/s110/s45 /s76\n/s51\n/s32/s32/s73/s110 /s116/s101/s110 /s115/s105/s116/s121 \n/s80/s104/s111/s116/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s97\n/s98\nFIG. 4: (Color online) Mn- L2,3XAS spectra of (a) the two\nLaMn 0.5Co0.5O3samples with TC= 150 K (dashed red\ncurve),TC= 225 K (black solid curve) and their difference\n(dotted blue curve), and (b) LaMnO 3(Mn3+) for comparison.\nas a divalent Mn reference [curve (d)]. Again we see a\ngradual shift of the center of gravity of the L3white line\nto higher energies from MnO to LaMnO 3and further to\nSrMnO 3, reflecting the increase of the Mn valence state\nfrom 2+ via 3+ to 4+. The Mn- L2,3spectrum of the\nhigh-TCLaMn0.5Co0.5O3samples is similar to that of\nSrMnO 3and LaMn 0.5Ni0.5O3,44in which a Ni2+/Mn4+\nvalence state was found. The Mn- L2,3XAS spectrum\nthus reveals an essentially Mn4+state in the high- TC\nLaMn0.5Co0.5O3, consistent with the observation of the\nCo2+valence in the Co- L2,3XAS spectra above, i.e. ful-\nfilling the charge balance requirement.\nTo investigate whether the presence of Co3+species\nin the low- TCLaMn0.5Co0.5O3is also accompanied by\nthe occurrence of Mn3+ions as charge compensation, we\nhave carried out a similar analysis as for the Co spectra.\nFigure 4 shows the low- TCspectrum (red dashed curve)\nandthehigh- TCone(blacksolidcurve)rescaledto80%of\nlowTC. Their difference spectrum is shown as the dotted\nblue curve. We find that the line shape resembles very\nmuch that of the high- TCsample itself, suggesting that\nmost ofthe Mn in the low- TCsamplearealsotetravalent.\nThis in turn would imply that the low- TCsample has to\nhave excess of oxygen to account for the presence of the\nCo3+species. Nevertheless, a closer look reveals that the\nenergy position of the difference spectrum lies between\nthat of the Mn4+and the Mn3+spectra, and that the\nvalley at 641–642 eV, at which energy a typical Mn3+\nsystem like LaMnO 3has its maximum, is not so deep.This suggests that in the low- TCsample, there are also\nsome Mn3+ions or strongly hybridized Mn3+and Mn4+\nions. Such a charge compensation for the Co3+could in-\ndicatethattheorderingoftheMnandCoionsislessthan\nperfect, so that the dislocated Co ions in the Mn4+posi-\ntions would have smaller metal-oxygen distances, leading\nto the stabilization of the low-spin trivalent state of the\nCo.\nHavingestablished the valences ofthe Coand Mn ions,\nwe now focus our attention on their magnetic properties.\nInthetoppanel(a)ofFig.5, wepresentthe XMCDspec-\ntra at the Co- L2,3edges of the high- TCLaMn0.5Co0.5O3\ntaken at 135 K. The spectra µ+(black solid curve) and\nµ−(red dashed curve) stand, respectively, for parallel\nand antiparallel alignments between the photon spin and\nthe magnetic field. One can clearly observe large differ-\nences between the two spectra with the different align-\nments. The difference spectrum, ∆ µ=µ+−µ−, i.e.\nthe XMCD spectrum, is also shown (blue dotted curve).\nIn the bottom panel (b) of Fig. 5 we show the XMCD\nspectra at the Mn- L2,3edges. Also here we can observe\na large XMCD signal. It is important to note that the\nXMCD is largely negative at both the Co and the Mn\nL3edges, indicating that the Co2+and Mn4+ions are\naligned ferromagnetically.\nVeryinterestingaboutthe XMCDatthe Co- L2,3edges\nis that it is almost zero at the L2while it is largely nega-\ntive at the L3. This is a direct indication that the orbital\ncontribution( Lz,morb)totheComagneticmomentmust\nbe large. In making this statement, we effectively used\nthe XMCD sum rule derived by Thole et al.,39in which\nthe ratio between the energy-integrated XMCD signal\nand the energy-integrated isotropic spectra gives a direct\nvalue for Lz. Nevertheless, for a quantitative analysis\nit is preferred to extract experimentally the Lz/Szratio\nby making use of an approximate XMCD sum rule de-\nveloped by Carra et al.45for the spin contribution (2 Sz,\nmspin) to the magnetic moment. This is more reliable\nthan extracting the individual values for LzandSzsince\none no longer needs to make corrections for an incom-\nplete magnetization, due to, for example, possible strong\nmagnetocrystalline anisotropy in a polycrystalline mate-\nrial. The sum rules of Thole et al.39and Carra et al.45\ngive for the morb/mspinorLz/2Sz,\nmorb\nmspin=Lz\n2Sz+7Tz\n=2\n3/integraltext\nL3∆µ(E)dE+/integraltext\nL2∆µ(E)dE/integraltext\nL3∆µ(E)dE−2/integraltext\nL2∆µ(E)dE,(1)\nwhereTzdenotes the magnetic dipole moment. This Tz\nfor ions in octahedral symmetry is a small number and\nnegligible compared to Sz.46,47Using this equation, we\nextractmorb/mspin= 0.47 out of our Co- L2,3XMCD\nspectrum. This is a large value and is in fact close to the\nvalue of 0.57 for CoO,49a compound well known for the\nimportant role of the spin-orbit interaction for its mag-\nnetic and structural properties.50,51,52,53,54,55,56,57,58,595\n/s55/s55/s53 /s55/s56/s48 /s55/s56/s53 /s55/s57/s48 /s55/s57/s53 /s56/s48/s48\n/s54/s52/s48 /s54/s52/s53 /s54/s53/s48 /s54/s53/s53 /s54/s54/s48/s67/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110/s67/s97/s108/s99/s117/s108/s97/s116/s105/s111/s110\n/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116/s69/s120/s112/s101/s114/s105/s109/s101/s110/s116\n/s32/s181\n/s32/s181/s43\n/s32/s181/s43\n/s181/s67/s111/s45 /s76\n/s50\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121 \n/s69/s110/s101/s114/s103 /s121 /s32/s40/s101/s86/s41/s32/s181\n/s32/s181/s43\n/s32/s181/s43\n/s181/s40/s97/s41/s67/s111/s45 /s76\n/s51\n/s77/s110/s45 /s76\n/s50/s77/s110/s45 /s76\n/s51\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121 \n/s69/s110/s101/s114/s103 /s121 /s32/s40/s101/s86/s41/s40/s98/s41\nFIG. 5: (Color online) Co- L2,3(a) and Mn- L2,3(b) spectra\nof LaMn 0.5Co0.5O3taken with circularly polarized x-rays at\n135 K. The photon spin was aligned parallel ( µ+, black solid)\nand antiparallel ( µ−, red dashed) to the 1 T magnetic field,\nrespectively; the difference spectra is shown in dotted blue .\nTop: measured spectra. Bottom: simulated spectra.The unquenched orbital moment is closely related to the\nopent2gshell of the 3 d7configuration.60,61\nApplying the sum rules for the Mn- L2,3XMCD spec-\ntra, we obtain morb/mspin= 0.09. This means that the\norbitalmoment forthe Mn4+ionsisnearlyquenched. In-\ndeed, for the 3 d3configuration in the Mn4+compounds,\nthe majority t2gshell is fully occupied and thus a prac-\ntically quenched orbital moment is to be expected.\nTo critically check our findings concerning the local\nelectronic structure of the Co and Mn ions, we will ex-\nplicitly simulate the experimental XMCD spectra using\nthe configuration interaction cluster model.29,30,31The\nmethod uses a CoO 6and MnO 6cluster, respectively,\nwhich includes the full atomic multiplet theory and the\nlocal effects of the solid. It accounts for the intra-atomic\n3d–3dand 2p–3dCoulomb interactions, the atomic 2 p\nand 3dspin-orbit couplings, the oxygen 2 p–3dhybridiza-\ntion, and local crystal field parameters. Parameters for\nthemultipolepartoftheCoulombinteractionsweregiven\nby the Hartree-Fock values,29while the monopole parts\n(Udd,Upd) as well as the oxygen 2 p–3dcharge trans-\nfer energies were determined from photoemission experi-\nmentsontypicalCo2+andMn4+compounds.62The one-\nelectron parameters such as the oxygen 2 p–3dand oxy-\ngen 2p–oxygen 2 ptransfer integrals were extracted from\nband-structurecalculations63withinthelocal-densityap-\nproximation (LDA) using the low-temperature crystal\nstructure of the high- TCphase.24The simulations have\nbeen carried out using the XTLS 8.3 program29with\nthe parameters given in Ref. 64.\nImportantforthelocalelectronicstructureoftheCo2+\nionisits local t2gcrystalfield scheme. Thistogetherwith\nthe spin-orbit interaction determines to a large extent its\nmagnetic properties. To extract the crystal field param-\neters needed as input for the cluster model, we have per-\nformed constrained LDA+U calculations63without the\nspin-orbit interaction. We find that the zx+xyorbital\nlieslowest,whilethe yzislocated22meVandthe zx−xy\n27 meV higher. Here, we made use of local coordinates\nin which the zdirection is along the long Co–O bond\n(2.078˚A), theyalong the second-longest bond (2.026\n˚A), and the xalong the short bond (1.997 ˚A). The clus-\nter model finds the easy axis of the magnetization to lie\nin theyzdirection with a single-ion anisotropy energy of\nabout 0.5–1.5 meV, i.e., larger than can be achieved by\nthe applied magnetic field. Since we are dealing with a\npolycrystallinesample, the sum of spectra taken with the\nlight coming from all directions has to be calculated; we\napproximated this by summing two calculated spectra:\none for light with the Poynting vector along the yzaxis\nand one with the Poynting vector perpendicular to this.\nThe exchange field direction is kept along the yzin both\ncases.\nThe results of the cluster model calculations are in-\ncluded in Fig.5, in the toppanel (a)for the Co L2,3edges\nand the bottom panel (b) for the Mn. One can see that\nthe line shapes of the experimental Co and Mn spectra\nare well explained by the simulations: all the characteris-6\n/s53 /s49/s48 /s49/s53 /s50/s48 /s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s53/s48 /s52/s48 /s51/s48 /s50/s48 /s49/s48 /s48/s45/s54/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48/s32\n/s32/s69/s110/s101/s114/s103 /s121 /s32/s40/s109/s101/s86/s41\n/s72\n/s101/s120/s32/s40/s109/s101/s86/s41\n/s32/s32\n/s116\n/s50 /s103/s32/s40/s109/s101/s86/s41\n/s32/s32/s32\n/s32/s72\n/s101/s120/s32/s40/s109/s101/s86/s41\n/s32\nFIG. 6: Energy level diagram of the Co2+ion (left panel)\nin a cubic field depending on the strength of the exchange\nfieldHex, (middle panel) the effect of lowering the symmetry\n[∆eg= 4∆t2g= 4(Ezx−xy,yz−Ezx+xy)], and (right panel)\nthe low-symmetry energy splitting depending on Hex.\nticfeaturesarereproduced. Wewouldliketoremarkthat\nthe experimentalXMCD spectra(blue dotted curves)are\nin general about 30% smaller than the simulated XMCD\nspectra (blue dotted curves). This is due to the fact that\nthe experimental spectra were notcorrected for the in-\ncomplete degree of circular polarization ( ≈80%) of the\nbeamline, nor for the fact that magnetic field makes an\nangle of 30◦with respect to the Poynting vector of the\nlight, nor for the reduction of the magnetization at 135 K\nat which the samplewasmeasured– comparedto the cal-\nculation which were done at 0 K. From these simulations\nwe thus can safely conclude that our interpretation for\nthe Co and Mn valences and magnetic moments is sound.\nOur finding of a large orbital contribution to the Co\nmagnetic moment has important implications for the in-\nterpretation of the magnetic susceptibility data. In most\nof the studies published so far, one tried to extract mag-\nnetic quantum numbers from the magnetic susceptibility\ndata using the Curie or Curie-Weiss law by finding a\ntemperature region in which the inverse of the magnetic\nsusceptibility is linear with temperature. One usually\ntakes the high temperature region. We will show below\nthat this standard procedure will notprovide the mag-\nnetic quantum numbers relevant for the ground state of\nthis material.\nThe fact that the 3 dspin-orbit interaction in this Co\nmaterial is “active” has as a consequence that the en-\nergy difference between the ground state and the first\nexcited state will be of the order of the spin-orbit split-\ntingζ, which is about 66 meV for the Co2+ion. We\nhave illustrated this in Fig. 6 which shows the energy\nlevel diagram of the Co2+ion, both in cubic symmetry(left panel) and in the low-temperature and ferromag-\nnetic state of the LaMn 0.5Co0.5O3system (right panel)\nwhere we have used the crystal field scheme as described\nabove.\nTo demonstrate the consequences of the presence of\nsuch a set of low lying excited states, we calculated the\nmagnetic susceptibility χof the Co2+in cubic symmetry\nforanapplied magneticfield of0.01T andwithout an ex-\nchangefield. The results arepresented in Fig. 7 where we\ndepict alsothe (apparent)effectivemagneticmoment µeff\n[µ2\neffis defined here as 3 kBdivided by the temperature\nderivative of 1 /χ(T)] and the (apparent) Weiss temper-\nature Θ [Θ is defined here as the intercept of the tan-\ngent to the 1 /χ(T) curve with the abscissa]. One can\nclearly observe that 1 /χ(T) is not linear with tempera-\nture for temperatures between TC= 225 K and roughly\n800K. Only for temperatures higher than 800 K, one can\nfind a Curie-Weiss-like behavior, but then the (apparent)\nWeiss temperature has nothing to do with magnetic cor-\nrelations since they were not included in this single ion\ncalculations. Instead, the (apparent) Weiss temperature\nmerely reflects the fact that the first excited states are\nthermally populated. This means in turn that one can-\nnot directly extract the relevant groundstate quantum\nnumbers from the high temperature region.\nIn principle, one could hope to find a Curie-Weiss be-\nhavior by focusing on the very low temperature region\nonly, e.g., below 50 K, but there one has to take into\naccount that there is a very large van Vleck contribution\nto the magnetic susceptibility due to the fact that the\nfirst excited states are lying very close, i.e., in the range\nof the spin-orbit splitting. The extrapolation to T= 0\nK would then give the real value for µeffof the ground\nstate. In the case of LaMn 0.5Co0.5O3, however, the pres-\nence of ferromagnetism, which already sets in at 225 K,\nwill completely dominate the magnetic susceptibility and\nthus hinder the determination of µeffof the ground state\nusing this procedure. Obviously, one can determine in\nprinciple the magnetic moments in a ferromagnet from\nthe saturation magnetization, but apparently this is the\nissue for LaMn 0.5Co0.5O3where one is debating about\nthe importance of Mn/Co disorder and its relationship\nto reduced magnetizations and less than optimal Curie\ntemperatures.\nAnother often used “magnetic”technique to determine\nthe moments in this ferromagnetic material is neutron\nscattering. Troyanchuk et al.found a mean value of\n2.5µBperformulaunit(LaCo 0.5Mn0.5O3).24Theauthors\nclaimed that this is in good agreement with the Co2+–\nMn4+scenario. Indeed, assuming spin-only moments as\nis generally done (but which is not correct as shown\nabove), one would already expect 3 µBfor a Co2+ion\nand 3µBfor a Mn4+ion, totaling to 6 µB, i.e. 3µB/f.u.,\nwhich is somewhat larger than the experimental finding\nand which can be understood consistently if one assumes\nthat the Co–Mn ordering in their sample is not perfect.\nIt is important to note that the low-spin Co3+–Mn3+\nscenario can be ruled out since this yields only 2 µB/f.u.,7\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48 /s49/s48/s48/s48 /s49/s50/s48/s48/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48/s48/s46/s50/s53/s48/s46/s51/s48/s48/s46/s51/s53/s48/s46/s52/s48/s48/s46/s52/s53\n/s48 /s52/s48/s48 /s56/s48/s48 /s49/s50/s48/s48/s45/s52/s48/s48/s52/s48/s48 /s52/s48/s48 /s56/s48/s48 /s49/s50/s48/s48/s51/s46/s53/s52/s46/s48/s52/s46/s53/s53/s46/s48/s53/s46/s53/s54/s46/s48\n/s32/s32/s49/s47 /s32/s40/s109/s111 /s108/s47/s109/s101/s109/s117/s41\n/s84 /s32/s40/s75/s41/s84 /s32/s40/s75/s41\n/s32/s32/s40/s75/s41/s84 /s40/s75/s41\n/s32/s32/s181\n/s101/s102/s102/s32/s40 /s181\n/s66/s41\nFIG. 7: Calculated inverse susceptibility for a single Co2+\nion in a cubic crystal field; (top inset) the (apparent) effec-\ntive moment µeffand (bottom inset) the (apparent) Weiss\ntemperature Θ as defined in the text.\ni.e., too low to explain the experiment. Nevertheless, a\nCo3+–Mn3+scenario in which the Co3+ion is in the in-\ntermediate ( S= 1) or high spin state ( S= 2) cannot be\nexcluded on the basis of the moments measured by the\nneutrons alone.12,38\nOur cluster model calculations based on the XAS and\nXMCD spectra reveal that the Co2+ion hasmspin=2.12µBandmorb= 0.99µBand that the Mn4+has\nmspin= 2.84µBandmorb= 0.02µB, totaling to\n2.99µB/f.u. This is not inconsistent with the magnetiza-\ntion results of Asai et al.,16if we make an extrapolation\nto higher magnetic fields as to estimate the saturated\ntotal moment. Our result is larger than the neutron re-\nsults, but also not inconsistent if one is willing to accept\nthat there is an appreciable amount of Co–Mn disorder\nin the neutron sample. Crucial is that our XAS and\nXMCD spectra rule out allthe Co3+–Mn3+scenarios:\n(1) our Co L2,3spectra give a positive match with those\nof Co2+compounds, while they do not fit those of low-\nspin Co3+and high-spin Co3+compounds;12,38(2) our\nMnL2,3spectra are very similar to those of Mn4+com-\npounds, and very dissimilar to those of Mn3+.\nTo summarize, we have utilized an element-specific\nspectroscopic technique, namely, soft-x-ray absorption\nand magnetic circular dichroism spectroscopy, to unravel\nthe local electronic structure of LaMn 0.5Co0.5O3system.\nWe have firmly established the high-spin Co2+–Mn4+\nscenario. 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Hartree-Fock results have been used\nfor the Slater integrals, which were reduced to ( F2\ndd) 90%,\n(F4\ndd) 100%, and ( F2\np,G1\npd,G3\npd) 0.95%. Parameters MnO 6\ncluster: U3d3d= 5.0 eV,U3d2p= 6.0 eV; ∆ = −3.0 eV;\nTpp= 0.7 eV; ∆ionic\nCF= 0.95 eV; Vpdσ=−1.6 eV;\nVpdπ= 0.74 eV;Hex6.5 meV;Beff\next= cos30◦×1 T. The\nSlater integrals were reduced to 70%."    },    {        "title": "0801.0886v2.Voltage_control_of_magnetocrystalline_anisotropy_in_ferromagnetic___semiconductor_piezoelectric_hybrid_structures.pdf",        "content": "arXiv:0801.0886v2  [cond-mat.mtrl-sci]  22 Jan 2008Voltage control of magnetocrystalline anisotropy\nin ferromagnetic – semiconductor/piezoelectric hybrid st ructures\nA. W. Rushforth,1,∗E. De Ranieri,2J. Zemen,3J. Wunderlich,2,3K. W. Edmonds,1C. S. King,1\nE. Ahmad,1R. P. Campion,1C. T. Foxon,1B. L. Gallagher,1K. V´ yborn´ y,3J. Kuˇ cera,3and T. Jungwirth3,1\n1School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom\n2Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Ki ngdom\n3Institute of Physics ASCR v.v.i., Cukrovarnick´ a 10, 162 53 Praha 6, Czech Republic\n(Dated: November 6, 2018)\nWe demonstrate dynamic voltage control of the magnetic anis otropy of a (Ga,Mn)As device\nbonded to a piezoelectric transducer. The application of a u niaxial strain leads to a large re-\norientation of the magnetic easy axis which is detected by me asuring longitudinal and transverse\nanisotropic magnetoresistance coefficients. Calculations based on the mean-field kinetic-exchange\nmodel of (Ga,Mn)As provide microscopic understanding of th e measured effect. Electrically induced\nmagnetization switching and detection of unconventional c rystalline components of the anisotropic\nmagnetoresistance arepresented, illustratingthegeneri cutilityofthepiezovoltagecontroltoprovide\nnew device functionalities and in the research of micromagn etic and magnetotransport phenomena\nin diluted magnetic semiconductors.\nPACS numbers: 75.47.-m, 75.50.Pp, 75.70.Ak\nThe control of magnetic properties of ferromagnetic\nmaterials by electrical means is an important prerequi-\nsite for successful implementation of spintronics in infor-\nmation processing technologies, and for major advance-\nments in sensor and transducer applications. The ma-\nterial class of (III,Mn)V dilute magnetic semiconductors\n(DMSs) plays a major role in this research because it\ncombines ferromagnetic and semiconducting properties\nin one physical system and because of the potential com-\npatibility with modern microelectronictechnologies. The\nintense researchinto DMS systems has led to several suc-\ncessful demonstrations of direct gate control of ferromag-\nnetism viacarrierredistribution [1, 2]. However, the high\ncarrier-doping levels ( >∼1%) necessary to achieve fer-\nromagnetism render the direct gating efficiency in these\nsemiconductor materials relatively low.\nWhile unfavorable for the direct gating, the large con-\ncentrations (Fermi energies) of the spin polarized holes\nthat mediate ferromagnetic coupling between the Mn lo-\ncalmomentsproducelargemagneticstiffness,resultingin\na mean-field like magnetization and macroscopic single-\ndomain characteristics of these dilute moment ferromag-\nnets. At the same time, magnetocrystalline anisotropies\nderived from spin-orbit coupling effects in the hole va-\nlence bands remain large, even at these high hole con-\ncentrations. This leads to the sensitivity of magnetic\neasy-axes orientations to strains as small as 10−4[3, 4].\nSo far the strain effects have been controlled by lattice\nparameter engineering during growth [5, 6] or through\npost growth lithography [3, 4, 7].\nIt has been demonstrated that sizeable strains can be\ninduced in GaAs structures using a piezoelectric trans-\nducer [8]. Here we utilize this technique to demon-\nstrate the dynamic voltage control via strain of the mag-\nnetic anisotropy in a (Ga,Mn)As device bonded to apiezo-transducer. We demonstrate that the favorablemi-\ncromagnetic characteristics of DMSs and the relatively\nsimple band structure allow for a microscopic descrip-\ntion of these effects on an unprecedented level of ac-\ncuracy compared to metal ferromagnet/piezoelectric de-\nvices [9, 10, 11, 12]. Finally we discuss the realization\nof electrically induced magnetization switching and of\nthe detection of unconventional crystalline components\nof the anisotropic magnetoresistance (AMR). These are\ntwo examples illustrating the generic utility of the piezo\nvoltage control to provide new device functionalities and\nin the research of micromagnetic and magnetotransport\nphenomena in DMSs.\nThe25nmthickGa 0.94Mn0.06Asepilayerwasgrownby\nlow-temperature molecular-beam-epitaxy on GaAs sub-\nstrate and buffer layers (see [13] for details). The mate-\nrial is under compressive in-plane strain of ∼3 x 10−3\n[14] due to the lattice mismatch with the GaAs. From\nSQUIDmagnetometryon an unstrained sample the mag-\nnetic easy axis is in-plane in a direction determined\nby competition between biaxial [100]/[010] and uniaxial\n[1¯10] anisotropies. The uniaxial term dominates above\n10 K. At 50 K the cubic and uniaxial anisotropy con-\nstants determined from hard axis magnetisation curves\nareKc= 85 Jm−3andKu= 261 Jm−3±20%.\nA (Ga,Mn)As Hall bar, fabricated by optical lithogra-\nphy, and orientated alongthe [1 ¯10]direction, wasbonded\nto the PZT piezo-transducer using a two-component\nepoxy after thinning the substrate to 150 ±10µm by\nchemical etching. The stressor was slightly misaligned so\nthat a positive/negative voltage produces a uniaxial ten-\nsile/compressive strain at ≈ −10◦to the [1¯10] direction.\nThe induced strain was measured by strain gauges,\naligned along the [1 ¯10] and [110] directions, mounted on\na second piece of 150 ±10µm thick wafer bonded to the2\n\u0001\u0002\u0003\u0004\u0001\u0002\u0003\u0005\u0001\u0002\u0003\u0006 \u0002\u0003\u0002 \u0002\u0003\u0006 \u0002\u0003\u0005 \u0002\u0003\u0004\u0004\u0007\b\u0002\b\u0006\b\u0005\b\u0004\t\n\u000b\n\u0001\b\f\u0002\r\n\u000e\u000e\u000f\u0010\u0011\u000e\t\u0001\u000b\n\u0012\u000e\t\u0013\u000b\u000e\u0006\f\u0002\n\u000e\u0007\f\u0002\n\u0001\u0002\u0003\u0004\u0001\u0002\u0003\u0005\u0001\u0002\u0003\u0006 \u0002\u0003\u0002 \u0002\u0003\u0006 \u0002\u0003\u0005 \u0002\u0003\u0004\u0005\u0004\u0007\b\u0002\b\u0006\b\u0005\b\u0004\b\u0007\u0006\u0002\t\u0014\u000b\n\u0015\b\f\u0002\r\n\u000e\u000e\u000f\u0010\u0011\u000e\t\u0001\u000b\n\u0012\u000e\t\u0013\u000b\u000e\u0006\f\u0002\n\u000e\u0007\f\u0002\u0001\u0002\u0003\u0004\u0001\u0002\u0003\u0005\u0001\u0002\u0003\u0006 \u0002\u0003\u0002 \u0002\u0003\u0006 \u0002\u0003\u0005 \u0002\u0003\u0004\b\u0006\u0003\u0016\f\b\u0017\u0003\u0002\u0002\b\u0017\u0003\u0002\f\b\u0017\u0003\b\u0002\n\t\u0018\u000b\n\u0001\b\f\u0002\r\n\u000e\u000e\u000f\u0010\u0010\u000e\t\u0019\u0001\u000b\n\u0012\u000e\t\u0013\u000b\u000e\u0017\f\u0002\n\u000e\u001a\f\u0002\n\u0001\u0002\u0003\u0004\u0001\u0002\u0003\u0005\u0001\u0002\u0003\u0006 \u0002\u0003\u0002 \u0002\u0003\u0006 \u0002\u0003\u0005 \u0002\u0003\u0004\b\u0017\u0003\u0002\f\b\u0017\u0003\b\u0002\b\u0017\u0003\b\f\n\u0015\b\f\u0002\r\t\u001b\u000b\n\u000e\u000e\u000f\u0010\u0010\u000e\t\u0019\u0001\u000b\n\u0012\u000e\t\u0013\u000b\u000e\u0017\f\u0002\n\u000e\u001a\f\u0002\u0002\u0002\u0002\n\u0017\u0002\u0002\n\u0002\u0002\u0002\n\u0007\u0002\u0002\nFIG. 1: The longitudinal resistances, R xx((a) and (b)) and\nthe transverse resistances R xy((c) and (d)) as a function of\nmagnetic field for angles close to the easy axes (30◦at -150\nV and 80◦at +150 V). The curves close to the easy axes in\neach case are relatively flat as a function of field, indicatin g\nsmall rotation of the angle of the magnetisation. T=50 K\npiezo-stressor. Differential thermal contraction of GaAs\nand PZT on cooling to 50 K produces a measured biax-\nial, in plane, tensile strain at zero bias of 10−3and a\nuniaxial strain estimated to be of the order of ∼10−4\n[15] which could not be accurately measured. At 50 K,\nthe magnitude of the additional strain for a piezo voltage\nof±150 V is approximately 2 ×10−4.\nIn this study the orientation of the in-plane mag-\nnetisation of the (Ga,Mn)As Hall bar was determined\nfrom the measured longitudinal and transverse AMR.\nTo a good approximation ( ≈10%), these are given by\n∆ρxx/ρav=Ccos2φandρxy/ρav=Csin2φ, whereφ\nis the angle between the magnetisation direction and the\nHall bar (current) direction [16].\nFigure 1 shows magnetoresistance measurements at\n50 K for external magnetic field sweeps at constant field\nangleθmeasured from the Hall bar direction. The\nstrongly θ-dependent low-field magnetoresistance, which\nsaturates at higher field is due to AMR, i.e., to mag-\nnetisation rotations. We have subtracted an isotropic,\nθ-independent magnetoresistance contribution, approxi-\nmated as linear, from the measured longitudinal resis-\ntances.\nWhen the external field is close to the magnetic easy\naxis, the measured resistances at saturation and rema-\nnence should be almost the same and a significant mag-\nnetoresistance due to rotation of the magnetisation can\nonly be present at very low applied fields. For external\nfields away from the easy axis, large magnetoresistances\ncorrespondingto largerotationsofthe magnetisationori-\nentation are present. This enables us to determine the\neasy axis directions within ±5◦.\nThe effect of the piezo-stressor is clearly apparent inFigure 1. At 50K, SQUID measurements show that the\nmagnetic easy axis is oriented along the [1 ¯10]- direc-\ntion for the as-grown (Ga,Mn)As wafer, consistent with\n|Kc|<|Ku|. Theeasyaxisforthe Hall barbondedto the\nstressorrotates to an angle φ= 65◦upon cooling to 50 K\ndue to a uniaxial strain induced by anisotropic thermal\ncontraction of the piezo stressor [15]. Application of a\nbias of +150 V to the stressor causes the easy axis to ro-\ntate further to φ= 80◦while for -150 V it rotates in the\nopposite sense to φ= 30◦. This directly demonstrates\nelectric field control of the magnetic anisotropy in our\n(Ga,Mn)As/PZT hybrid system.\nThe magnetic anisotropy for our system can be de-\nscribed phenomenologically by an energy functional\nE(ˆM) =−Kc/4sin22φ+Kusin2φ+K′\nusin2(φ+φ0),\nwhere the last term, with φ0≈10◦, is due to the mis-\naligned stressor. The observed behaviour is then con-\nsistent with the (Ga,Mn)As being in tensile strain along\nthe axis of the stressor on cool down and applied posi-\ntive(negative)voltageincreasing(decreasing)this strain.\nNote that the misalignment allows smooth rotation of a\nsingle easy axis in the experimentally accessible voltage\nrange.\nWe now calculate the expected magnetic anisotropy\ncharacteristicsofthe studied (Ga,Mn)As/PZTsystemby\ncombining the six-band k·pdescription of the GaAs\nhost valence band with the kinetic-exchange model of\nthe coupling to the local Mn Gad5-moments [5, 6]. This\napproach is well-suited to the description of spin-orbit\ncoupling phenomena near the top of the valence band\nwhose spectral composition and related symmetries are\ndominated bythe p-orbitalsofthe As sublattice, andalso\nprovides straightforward means of incorporating lattice\nstrains [3, 5, 6].\nDue to the presence of unintentional compensating de-\nfects in (Ga,Mn)As films, the concentrationsofferromag-\nnetically ordered Mn local moments and holes cannot be\naccurately controlled during growth or determined post\ngrowth[17]. Wethereforeconsiderinouranalysisuncom-\npensated Mn Gamoment concentrations within an inter-\nvalx= 3−5% which safely contain the expected value of\nx. The magnetocrystalline anisotropy constants depend\nstrongly on the local moment density and the hole com-\npensation ratio p/NMn, where p is the hole density and\nNMnis the concentration of Mn ions. For fixed pand\nNMn, the cubic term Kc, calculated without adjustable\nparameters, agrees with the measured 50 K value for\np/NMn= 0.6−0.4 forx= 3−5% in good agreement\nwith the estimated compensation ratio in our as-grown\nmaterial [17].\nThe origin of the uniaxial anisotropy term in bare\n(Ga,Mn)As wafers is not known, but it can be modelled\n[3, 18] by introducing a shear strain eintalong the [1 ¯10]\naxis. For p/NMn= 0.6−0.4 we obtain the experimen-\ntalT= 50 K value of Kufor compressive shear strain\neint= 3−2×10−4within the considered range of x’s.3\n\u0001 \u0002\u0001 \u0003\u0001\u0001 \u0003\u0002\u0001 \u0004\u0001\u0001 \u0004\u0002\u0001 \u0005\u0001\u0001 \u0005\u0002\u0001\u0006\u0001\u0007\u0005\u0006\u0001\u0007\u0004\u0006\u0001\u0007\u0003\u0001\u0007\u0001\u0001\u0007\u0003\n\b\u0001\t\n\u0002\t\u0005\u0001\t\u000b\f\r\n\u000e\u000e\u0006\u0003\u0002\u0001\u000f\n\u000e\u0010\u0003\u0002\u0001\u000f\n\u000e\u000e\u000e\u000e\u000e\u000e\u000e\u0001\u000f\n\u000e\u000e\u0011\u000e\u000b\u0012\u0013\u0014\u0015\u0005\r\n\u0001\u000e\u000b\u0016\u0017\u0018\u0019\u0017\u0017\u001a\r\n\u0001 \u0002\u0001 \u0003\u0001\u0001 \u0003\u0002\u0001 \u0004\u0001\u0001 \u0004\u0002\u0001 \u0005\u0001\u0001 \u0005\u0002\u0001\u0006\u0001\u0007\u001b\u0006\u0001\u0007\u0004\u0001\u0007\u0001\u0001\u0007\u0004\u0001\u0007\u001b \u000b\u001c\r\n\u000e\u000e\u0002\u0003\u001d\u001d\u0014\u0003\f\u001e\u000e\u000b\u001f\r\n\u0004\u000e\u000b\u0016\u0017\u0018\u0019\u0017\u0017\u001a\r\u000e\u000e\u0006\u0003\u0002\u0001\u000f\n\u000e\u0010\u0003\u0002\u0001\u000f\n\u000e\u000e\u000e\u000e\u000e\u000e\u000e\u0001\u000f\n\u0001 \u0002\u0001 \u0003\u0001\u0001 \u0003\u0002\u0001 \u0004\u0001\u0001 \u0004\u0002\u0001 \u0005\u0001\u0001 \u0005\u0002\u0001\u0006\u0001\u0007\b\u0006\u0001\u0007\n\u0006\u0001\u0007\u001b\u0006\u0001\u0007\u0004\u0001\u0007\u0001\u0001\u0007\u0004\u0001\u0007\u001b\u0001\u0007\n\u0001\u0007\b\n\u000b \r\n\u000e\u000e\u0002\u0003\u001d\u001d\u0014\u0003\f\u001e\u000e\u000b\u001f\r\n\u0004\u000e\u000b\u0016\u0017\u0018\u0019\u0017\u0017\u001a\r\u000e\u0006\u0003\u0002\u0001\u000f\n\u000e\u0010\u0003\u0002\u0001\u000f\n\u000e\u0001\u000f\nFIG. 2: (a) The microscopic E(ˆM) curves for the three piezo\nvoltages. φis the angle of the magnetisation with respect to\nthe Hall bar. (b) The longitudinal AMR from theory calcula-\ntions with a non saturating magnetic field of 20 mT rotated in\nthe plane of the film (c) The experimental AMR curves with\na field of 40 mT rotated in the plane of the film. ρavis the\nρxxaveraged over 360◦in the low field regime. θis the angle\nof the magnetic field with respect to the Hall bar. T=50 K\nThe calculations reproduce the measured 0 V easy axis\nfor a tensile strain of estr= 6−4×10−4, along the\nstressor axis and the experimental easy axes for ±150 V\nare obtained by increasing/decreasing the estrstrain by\n3−2×10−4. These changes in strain agree with the\nmeasured values for ±150 V, and the 0 V strain due to\ndifferential contraction is of the expected order. The re-\nsulting microscopic E(ˆM) curves for the three voltages\nare shown in Figure 2(a).\nThe magnetoresistance calculated microscopicallyfrom the same band structure model combined with\nBoltzmann transport theory [16] gives AMR at satura-\ntion of the same sign and comparable magnitude to the\nexperiment if we assume the above compensation ratios.\nThis allows us to microscopically simulate AMR mea-\nsurements assuming the single domain behaviour. In\nFigure 2(b) we show the results of simulations and in\nFigure 2(c) experimental data for the situation where a\nmagnetic field of magnitude smaller than the saturation\nfield is rotated in the plane of the (Ga,Mn)As epilayer.\nBoththeoryandexperiment showthatthese AMR traces\nare no longer sinusoidal since the magnetisation does not\ntrack the applied rotating field. Ranges of magnetic field\nanglesθfor which resistance is more slowly varying cor-\nrespond to angles close to the easy axis, where the mag-\nnetisation vector lags behind θ. On the other hand ro-\ntation around the hard axis is more abrupt, and in this\nregion the AMR can develop hysteretic features whose\nwidths increase with decreasing magnitude of the rotat-\ning field. At +150 V the hard axis is close to the Hall\nbar axis resulting in sharper minima than maxima in the\ncorresponding experimental and theoretical AMR traces,\nwhile the trend is clearly opposite for the -150 V bias\ndata, consistent with the easy axis directions obtained\nfrom the field sweep measurements.\n\u0001\u0002\u0003\u0002\u0004\u0001\u0002\u0003\u0002\u0005 \u0002\u0003\u0002\u0002 \u0002\u0003\u0002\u0005 \u0002\u0003\u0002\u0004\u0001\u0006\u0006\u0001\u0006\u0002\u0001\u0005\u0006\u0001\u0005\u0002\u0001\u0007\u0006\u0001\u0007\u0002\b\t\n\u000b\u000b\f\r\u000e\u000b\b\u0001\n\u000f\b\u0010\n\u0001\u0011\u0006 \u0002 \u0011\u0006 \u0006\u0002 \u0012\u0006 \u0013\u0002\u0002 \u0013\u0011\u0006 \u0013\u0006\u0002\u0001\u0005\u0006\u0001\u0005\u0002\u0001\u0007\u0006\u0001\u0007\u0002\u0007\n\u0011\n\u0013\b\u0014\n\u000b\u000b\f\r\u000e\u000b\b\u0001\n\u0015\u0016\u0017\u0018\u0019\u000b\u001a\u0019\u001b\u001c\t\u001d\u0017\u000b\b\u001a\nFIG. 3: (a) Low field magnetic hysteresis curve at +150 V.\nThe field is swept from saturating negative field at 165◦to the\nposition show by the black arrow. Then (b) the piezo voltage\nis swept inducing a rotation of the angle of the magnetisatio n,\nindicated by the red arrows. Numbered arrows represent the\norder and direction of the voltage sweeps. T=30 K.4\nHaving established a microscopic understanding\nof the control of the magnetic anisotropy in the\n(Ga,Mn)As/PZThybridsystemwenowproceedwith the\ndemonstration of an electrically induced magnetisation\nswitching. The bias-dependent hysteresis loops which al-\nlow for such a reversal process are shown in Figure 3(a).\nWith the piezo voltage at +150V, the initial magnetisa-\ntion state is prepared by sweeping the external magnetic\nfield fromnegativesaturatingfield at165◦to the position\nshown by the blackarrow. This causes the magnetisation\ntorotatefrom165◦to 260◦, atB=0T(i.e. alongtheeasy\naxis at +150 V), then to 275◦for the small positive field\nof approximately 18 mT (marked by the black arrow).\nThen, with the external magnetic field held constant the\npiezo voltage is swept (Figure 3(b)) and the magnetisa-\ntion rotates from 275◦to 25◦(i.e. close to the easy axis\nfor -150 V) resulting in a change of Rxyas shown by the\nred arrows. This sequence switches the magnetisation\nfrom the 4th to the 1st quadrant, where it remains for\nsubsequent voltage sweeps. The magnetisation can be\nswitched back again by reversing the sequence, with the\nmagnetic field set to the opposite polarity.\nFinally we report on the detection of an unconven-\ntional crystalline component of the AMR allowed by the\npiezo voltage control. The AMR in (Ga,Mn)As is known\nto consist of a non-crystalline component, reflecting the\nsymmetry breaking imposed by a preferred current direc-\ntion, and crystalline terms reflecting the underlying crys-\ntal symmetry. The crystalline terms typically represent\n10% of the total AMR in 25nm (Ga,Mn)As layers [16].\nFigure 4 shows the change in the longitudinal ∆ ρxx/ρav\nand transverse ∆ ρxy/ρavcomponents of the AMR for\npiezo voltages of ±150 V. ∆ ρxx=ρxx−ρav, andρavis\nthe average of ρxxover 360oin the plane. The distortion\nof the lattice by the piezo transducer leads to modifica-\ntion of the crystalline components of the AMR, shown in\nthe figure by subtracting the curves at piezo voltages of\n±150 V. This modification represents ≈10% of the total\nAMR and is comparable to the absolute magnitude of\nthe crystalline terms. The appearance of a fourth order\nterm in the transverse AMR is expected under a uniaxial\ndistortion [19], but this higher order term was not found\nto be significant in unstrained 25nm (Ga,Mn)As layers\n[16].\nTo conclude, we have demonstrated the voltage con-\ntrol of the magnetic anisotropy and non-volatile switch-\ning of the magnetisation direction in (Ga,Mn)As induced\nby strain applied with a piezoelectric transducer. Micro-\nscopic theory calculations capture the physics involved.\nThese techniques open up new avenues for exploring a\nvarietyofmicromagneticandmagnetotransportphenom-\nena in DMS systems.\nAcknowledgements We are grateful to J.\nChauhan and D. Taylor for sample fabrication.\nWe acknowledge support from EU Grant IST-\n015728, from UK Grant GR/S81407/01, from CR/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23/g39/g85 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/g85/g68/g89 /;#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/g57/;#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23 /g39/g85 /;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/g85/g68/g89 /;#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23#23/g57/;#23#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23#23 \n/g73/;#23#23#23/;#23#23#23#23#23#23#23#23#23#23#23/g71/g72/g74/g85/g72/g72/g86/;#23#23#23#23#23#23#23#23#23#23#23#23 /;#23#23#23/g91/g91 \n/;#23#23#23/g91/g92 \nFIG. 4: The change in (a) the longitudinal ∆ ρxx/ρavand (b)\nthe transverse ∆ ρxy/ρavcomponents of the AMR for piezo\nvoltages of ±150 V.\nGrants 202/05/0575, 202/04/1519, FON/06/E002,\nAV0Z1010052, KJB100100802 and LC510.\n∗Corresponding author; Electronic address:\nAndrew.Rushforth@Nottingham.ac.uk\n[1] D. Chiba, M. Yamanouchi, F. Matsukura, and H. Ohno,\nScience301, 943 (2003).\n[2] D. Chiba, F. Matsukura, and H. Ohno, Appl. Phys. Lett.\n89, 162505 (2006).\n[3] J. Wunderlich, et al., Phys. Rev. B 76, 054424 (2007).\n[4] J. Wenisch, et al., Phys. Rev. Lett. 99, 077201 (2007).\n[5] T. Dietl, H. Ohno, and F. Matsukura, Phys. Rev. B 63,\n195205 (2001).\n[6] M. Abolfath, T. Jungwirth, J. Brum, and A.H. MacDon-\nald, Phys. Rev. B 63, 054418 (2001).\n[7] S. H¨ umpfner, et al., Appl. Phys. Lett. 90, 102102 (2007).\n[8] For technical details of strain transfer in related non-\nmagnetic devices see M. Shayegan, et al., App. Phys.\nLett.83, 5235 (2003).\n[9] Sang-Koog Kim, et al.,J. Magn. Magn. Mater. 275, 127\n(2003).\n[10] Jeong-Won Lee, Sung-Chul Shin and Sang-Koog Kim,\nAppl. Phys. Lett. 82, 2458 (2003).\n[11] Bernhard Botters et al., Appl. Phys. Lett. 89, 242505\n(2006).\n[12] H. Boukari, C. Cavaco, W. Eyckmans, L. Lagae, and J.\nBorghs, J. Appl. Phys. 101, 054903 (2007).\n[13] R.P. Campion, et al., J. Cryst. Growth 251, 311-316\n(2003).\n[14] L.X. Zhao, et al., Appl. Phys. Lett. 86, 071902 (2005).\n[15] B. Habib, J. Shabani, E.P. De Poortere, M. Shayegan,\nand R. Winkler, Appl. Phys. Lett. 91, 012107 (2007).\n[16] A.W. Rushforth, et al., Phys. Rev. Lett. 99, 147207\n(2007).\n[17] T. Jungwirth, et al., Phys. Rev. B 73, 165205 (2006)\n[18] M. Sawicki, et al., Phys. Rev. B 71, 121302(R) (2005).\n[19] K. V´ yborn´ y, et al., to be published."    },    {        "title": "0801.4518v1.Anisotropic_magnetization_studies_of__R_2_Co_Ga_8___R___Gd__Tb__Dy__Ho__Er__Tm__Y_and_Lu__single_crystals.pdf",        "content": "arXiv:0801.4518v1  [cond-mat.str-el]  29 Jan 2008APS/123-QED\nAnisotropic magnetization studies of R 2CoGa 8\n(R = Gd, Tb, Dy, Ho, Er, Tm, Y and Lu) single crystals\nDevang A. Joshi, R. Nagalakshmi, S. K. Dhar and A. Thamizhavel\nDepartment of Condensed Matter Physics and Material Scienc es,\nTata Institute of Fundamental Research,\nHomi Bhaba Road, Colaba, Mumbai 400 005, India.\n(Dated: December 17, 2018)\nAbstract\nSingle crystals of R 2CoGa8series of compounds were grown, for the first time, by high tem pera-\nture solution growth (flux) method. These compounds crystal lize in a tetragonal crystal structure\nwiththespacegroup P4/mmm. IthasbeenfoundthatR 2CoGa8phaseformsonlywiththeheavier\nrare earths, starting from Gd with a relatively large c/aratio of≈2.6. The resultant anisotropic\nmagnetic properties of the compounds were investigated alo ng the two principal crystallographic\ndirections of the crystal viz., along [100] and [001]. The no nmagnetic compounds Y 2CoGa8and\nLu2CoGa8show diamagnetic behavior down to the lowest temperature (1 .8 K) pointing out the\nnon-magnetic nature of Co in these compounds and a relativel y low density of electronic states at\nthe Fermi level. Compounds with the magnetic rare earths ord er antiferromagnetically at tempera-\ntures lower than 30 K. The easy axis of magnetization for R 2CoGa8(R = Tb, Dy and Ho) is found\nto be along the [001] direction and it changes to [100] direct ion for Er 2CoGa8and Tm 2CoGa8.\nThe magnetization behavior is analyzed on the basis of cryst alline electric field (CEF) model.\nThe estimated crystal field parameters explains the magneto crystalline anisotropy in this series of\ncompounds.\nPACS numbers: 71.20.Eh, 71.27.+a, 71.70.Ch, 75.50.Gg, 75.50.Ee\nKeywords: R 2CoGa8, antiferromagnetism, crystalline electric field, metamagnetism.\n1I. INTRODUCTION\nRnTX3n+2(R = rare earths, T = Co, Rh and Ir and X = In and Ga) form a family of\ncompounds, mainly consisting of two groups with n = 1 and n = 2. Both t he groups of\ncompounds are structurally similar and exhibit a variety of interestin g physical phenomena,\nwhich include heavy fermions, superconductivity and their coexiste nce, pressure induced\nsuperconductivity, magnetic ordering etc. Compounds ofn=1gro uphave beeninvestigated\nextensively compared to n = 2. The latter group of compounds (R 2TX8) were first reported\nby Kalychak et al1, who reported the crystallographic details on these compounds. L ater\non, the interest in these compounds grew further due to the inter esting behavior shown by\nthe Ce compounds. Ce 2RhIn8orders antiferromagnetically with a N´ eel temperature of 2.8 K\nand it undergoes pressure induced superconductivity at 2 K under a pressure of 2.3 GPa3.\nCe2CoIn8is a Kondo lattice exhibiting heavy fermion superconductivity with a Tc= 0.4 K4,5\nat ambient pressure, while Ce 2IrIn8is a heavy fermion paramagnet2.\nThe magnetic properties of polycrystalline R 2CoIn8compounds have been reported by\nDevanget al5. These compounds crystallize in the tetragonal structure with th e space\ngroupP4/mmm . Some of their interesting features are: a crystal field split nonma gnetic\ndoublet ground state of Pr3+ions in Pr 2CoIn8, field induced ferromagnetic behavior at\nlow temperatures in Dy 2CoIn8and Ho 2CoIn8and an anomalously high magnetoresistance\n(∼2700 %) at 2 K in Tb 2CoIn8. Considering these interesting behaviors in the indium\nanalogs, and to the best of our knowledge there are no reports on the corresponding gallium\ncompounds, we decided to study the R 2CoGa8for various rare earths. Here we report on\nour detailed structural and magnetization studies in this series of s ingle crystals.\nII. EXPERIMENT\nSingle crystals of R 2CoGa8(R = rare earths) compounds were grown by the flux method.\nThe starting materials used for the preparation of R 2CoGa8single crystals were high purity\nmetals of rare-earths (99.95%), Co (99.9%) and Ga (99.999%). Owin g to the low melting\npoint of Ga, the single crystals were grown in Ga flux. Considering the great affinity of\nCo and Ga atoms to form CoGa 3, we decided to make R 2Co button by arc melting and\nuse it with excess Ga flux. From the binary phase diagram of R-Co we f ound that R 2Co\n2phase does not exist, so the melt formed will have arbitrary phases which are assumed\nto remain unstable in presence of Ga within the required temperatur e range. From the\nprevious studies4, it was observed that the R 2CoIn8crystals were grown in the temperature\nrange between 750 and 450◦C. We have also employed the same temperature range for\nthe growth of R 2CoGa8compounds. A button of R 2Co with excess of Ga (R:Co:Ga =\n2:1:27) was taken in an alumina crucible and then sealed in an evacuated quartz ampoule.\nThe ampoule was then heated up to 1050◦C over a period of 24 hours and held at this\ntemperature for 24 hours, so that the melt becomes homogeneou s. The furnace was then\ncooled very rapidly down to 750◦C to avoid the formation of any unwanted phase. From\n750 to 400◦C the furnace was cooled down at the rate of 1◦C/h, followed by a fast cooling\nto room temperature. The crystals were separated by centrifug ing and as well treating them\nin hot water. The crystals obtained were platelets of size roughly 5 ×5×1 mm3. In some\nof the cases the small platelets stick together to form a big crysta l (roughly 7 ×5×4\nmm3). An energy dispersive X-ray analysis (EDAX) was performed on all the obtained\nsingle crystals to identify their phase. The EDAX results confirmed t he crystals to be of\nthe composition 2:1:8. To check for the phase purity, powder x-ray diffraction pattern of\nall the compounds were recorded by powdering a few small pieces of single crystal. The\nR2CoGa8phase forms only for heavier rare earths (Gd, Tb, Dy, Ho, Er, Tm a nd Lu, Y).\nOur attempts to make the compound with lighter rare earths failed. We did not attempt to\nmake Yb and Eu compounds. It was found that in all the cases the cr ystallographic (001)\nplane was perpendicular to the flat plates of the crystal. The cryst als were oriented along\nthe crystallographic axis [100] and [001] using Laue X-ray diffractom eter, and cut along the\nprincipal directions for the purpose of magnetization measuremen ts by spark erosion cutting\nmachine. The magnetic measurements were performed using a supe rconducting quantum\ninterference device (SQUID - Quantum Design) and vibrating sample magnetometer (VSM\nOxford Instruments).\n3III. RESULTS\nA. Crystal structure\nThe R 2CoGa8series of compounds form in a tetragonal structure with a space g roup\nP4/mmm (# 123). One of the compounds of the series, Ho 2CoGa8is well cited for this\nparticular structure. In order to confirm the phase homogeneity of the compound with\nproper lattice and crystallographic parameters, a Rietveld analysis of the observed X-ray\npattern of all the compounds was done. The lattice parameters th us obtained are listed\nin Table I and a representative Rietveld refined plot of Ho 2CoGa8is shown in Fig. 1. The\ncrystallographic parameters for each of the constituent atoms ( at various crystallographic\nsites) in Ho 2CoGa8are presented in Table II. Both the lattice parameters aandcare\nsmaller than the corresponding In compounds5. One of the possible reasons is due to the\nsmaller metallic radii of Ga ( ≈1.3˚A) compared to that of In ( ≈1.6˚A). The unit cell\nvolumes of the R 2CoGa8compounds are plotted against their corresponding rare earths in\nFig 2. The unit cell volume and both the lattice parameters decrease as we move from Gd\nto Lu. This is attributed to the well known lanthanide contraction. T he(c/a)ratio remains\nconstant ( ≈2.6) for all the compounds but it is slightly less than that of the corre sponding\nIn compounds ( ≈2.63). The large (c/a)ratio indicates the significant structural anisotropy\nin these compounds. The crystal structure of R 2CoGa8is shown in Fig. 3. The central\n6\n4\n2\n0\n-2 Intensity (a.u.)\n8070605040302010\n 2θ (º)  Obs.\n  Cal.\n  Diff.\n  Bragg Pos.\n Ho2CoGa8\nFIG. 1: Powder X-ray diffraction pattern recorded for crushed single crystals of Ho 2CoGa8at room\ntemperature. The solid line through the experimental data p oints is the Rietveld refinement profile\ncalculated for the tetragonal Ho 2CoGa8.\n4Rare earth Lattice parameter Volume c/a\na(˚A) b(˚A) ( ˚A3)\nGd 4.265 11.099 201.89 2.602\nTb 4.243 11.043 198.80 2.602\nDy 4.231 11.027 197.39 2.606\nHo 4.219 10.994 195.69 2.605\nEr 4.210 10.964 194.32 2.604\nTm 4.199 10.938 192.85 2.604\nLu 4.181 10.903 190.93 2.607\nY 4.249 11.053 199.55 2.601\nTABLE I: Lattice parameters, unit cell volume and c/aratio for the R 2CoGa8series of compounds\nAtom Site x y z Ueq(˚A2) Occ.\nsymmetry\nHo 2g 0 0 0.306 0.588 2\nCo 1a 0 0 0 1.186 1\nGa1 2e 0 0.5 0.5 1.771 2\nGa2 2h 0.5 0.5 0.295 0.40 2\nGa3 4i 0 0.5 0.114 0.074 4\nTABLE II: Refined crystallographic parameters for Ho 2CoGa8\nportion of the unit cell between the rare earth planes (along the caxis) and including them\nis similar to the unit cell of RGa 3phase. In fact, the lattice constant aof R2CoGa8phase is\napproximately equal to that of the RGa 3(cubic) phase6and the remaining structure above\nand below it forms the CoGa 2layers. The unit cell of R 2CoGa8may be viewed as formed\nby the stacking of RGa 3units and CoGa 2layer alternately along the caxis. It is imperative\nto mention here one similarity between RGa 3and R 2CoGa8compounds; viz., in both cases\nthe compounds form only with heavy rare earths. Gd is on the borde rline of stability which\nforms the R 2CoGa8phase but not the RGa 3phase6. A similar comparison is also possible\n5201\n198\n195\n192 V (Å3)\nGd Tb Dy Ho Er Tm Lu YR2CoG a8\nFIG. 2: (Color online) Unit cell volume of R 2CoGa8compounds plotted against the corresponding\nrare earths.\nRCo\nGa\n[001]\n[100]\nFIG. 3: Tetragonal unit cell of R 2CoGa8compounds.\nfor indides (R 2CoIn87and RIn 38), where only Lu and La are on the borderline of stability\nand do not form R 2CoIn8phase, where as RIn 3forms for the entire rare earth series. Hence\nthe non formation of RGa 3phase with lighter rare earths is one of the possible reasons for\nthe non formation of the corresponding R 2CoGa8compounds. Therefore, RX 3(X = In and\nGa) can plausibly be considered as one of the basic building blocks for t he formation of\nR2CoX8(X = In and Ga) compounds.\n66\n4\n2\n0\n-2\n-4 χ (10-4emu/mole)\n300250200150100500\n Temperature (K) Y2CoGa8\n Lu2CoGa8\n Fit\n \nFIG. 4: (Color online) Temperature dependence of the magnet ic susceptibility in Y 2CoGa8and\nLu2CoGa8. The solid line through the data points implies the modified C urie-Weiss fitting.\nIV. MAGNETIC PROPERTIES\nA. Y 2CoGa 8, Lu2CoGa 8\nThese two compounds with the non-magnetic Y and Lu respectively, show diamagnetic\nbehavior. The susceptibility of both the compounds is shown in Fig. 4. It is negative and\nnearly temperature independent at high temperatures and exhibit s a weak upturn at low\ntemperatures crossing into the positive region below 10 K. Assuming a magnetic moment of\n1µB/impurity, a good fit of the modified Curie-Weiss expression\nχ=χ0+C\nT−θp, (1)\nto the data lead to paramagnetic impurity ion concentration of a few ppm and θpis nearly\nzero. The value of χ0is found to be -2.08 ×10−4emu/mol and -3.13 ×10−4emu/mol\nfor Y2CoGa8and Lu 2CoGa8, respectively. These data show conclusively the nature of non-\nmagnetic Co atoms in this series of compounds.\nThe diamagnetic behavior of Y 2CoGa8and Lu 2CoGa8is in contrast with the Pauli-\nparamagnetic behavior shown by the non magnetic indide Y 2CoIn8and indicates a low\ndensity of electronic density of states at the Fermi level in the gallid es, such that the dia-\nmagnetic contribution due to the filled electronic shells exceeds the P auli paramagnetic\ncontribution from the conduction electrons. Indeed, the co-effic ient of the electronic heat\ncapacity ( γ), which is proportional to the density of states at the Fermi level, of Y2CoGa8\n70.15\n0.10\n0.05 χ (emu/mol)Gd2CoGa8\nH // [001]\nH // [100]\n (a)5 kOe\n40\n20\n0 χ−1(mol/emu·Gd)\n300250200150100500\n Temperature (K)Gd2CoGa8\n H // [100]\n H // [001]\n Curie-Weiss Fit\n(b)0.160\n0.156 χ (emu/mol)\n3020100  K\nFIG. 5: (Color online) (a) Magnetic susceptibility of Gd 2CoGa8, inset shows the magnified view of\nlow temperature susceptibility, (b) inverse magnetic susc eptibility with a modified Curie-Weiss fit.\nis (2 mJ/K2·mol, comparably less than that of Y 2CoIn85(13 mJ/K2·mol). If the density\nof states at the Fermi level decreases on replacing In by Ga in thes e R2CoX8(X = In and\nGa) series of compounds, it should result in a weaker conduction elec tron mediated RKKY\nmagnetic interactions between the rare earth ions in R 2CoGa8compounds. This may be one\nof the reasons that the N´ eel temperatures of R 2CoGa8series of compounds are lower than\nthose of the corresponding indides R 2CoIn8.\nB. Gd 2CoGa 8\nWe next describe the magnetization of Gd 2CoGa8, as Gd is a Sstate ion in which the\ncrystal electric field induced anisotropy is zero in the first order. T he susceptibility of\nGd2CoGa8along [100] and [001] directions in an applied magnetic field of 5 kOe is sho wn\nin Fig. 5(a). The low temperature part is shown as an inset of Fig. 5(a ). The susceptibility\nshows a peak due to antiferromagnetic transition at TN= 20 K, followed by an upturn below\n≈15 K. Quantitatively similar behavior is seen with the field applied in [100] a nd [001]\n82.0\n1.5\n1.0\n0.5\n0 Magnetization ( µΒ / Gd)\n121086420\n Magnetic Field (T)Gd2CoGa8 \n  H // [001]      \n  H // [100]\n  T = 2 K\nFIG. 6: (Color online) Magnetic isotherm of Gd 2CoGa8at 2 K with the field along [100] and [001]\ndirections.\ndirections, respectively. The minor difference in the susceptibility alo ng the two axes may\nbe due to the second order anisotropy arising from the dipole-dipole interaction. The linear\nbehavior of the magnetic isotherms at 2 K (Fig. 6) for both the axes further corroborates the\nantiferromagnetic nature of the magnetically ordered state. In t he paramagnetic state the\nsusceptibility was fitted to the modified Curie-Weiss law as shown in Fig 5 (b). The obtained\neffective magnetic moments presented in Table III are close to that of the theoretically\nexpected one. The paramagnetic Curie temperatures θpare -69 K and -67 K respectively, for\n[100] and [001] directions. The relatively high value along both the dire ctions indicate strong\nantiferromagnetic interaction among the Gd3+moments. The upturn in the susceptibility at\nlow temperatures is often attributed to a canting of antiferromag netically aligned moments.\nA similar behavior has earlier been seen in GdCo 2Si211, which was later shown to be due\nto a non-collinear amplitude modulated structure12. So we assume that a similar or some\ncomplicated stable magnetic structure is present in Gd 2CoGa8.\nC. R 2CoGa 8(R = Tb, Dy and Ho)\nIn these three compounds the easy-axis of magnetization lies along the [001] direction.\nThe data clearly show the anisotropic behavior of the magnetization in both the paramag-\nnetic and anitferromagnetically ordered states arising due to the in fluence of crystal electric\nfields (CEF) ontheHund’s rulederived groundstatesofthefree R3+ions. Figure7(a)shows\n90.4\n0.3\n0.2\n0.1\n0χ (emu/mol)Tb2CoGa8\n5 kOe\n H // [001]\n H // [100]\n(a)\n30\n20\n10\n0(χ − χ0)-1(mol/emu·Tb)\n300250200150100500\n Temperature (K) H // [001]\n H // [100]\n CEF-Fit\n Tb2CoGa8\n(b)\nFIG. 7: (Coloronline) (a) Magnetic susceptibility ofTb 2CoGa8, (b)inversemagneticsusceptibility;\nsolid lines through the data point indicate the CEF fit.\nthe susceptibility of Tb 2CoGa8from 1.8 to 300 K in a magnetic field of 5 kOe along the two\ncrystallographic directions ([100] and [001]). The data show an antif erromagnetic transition\nat TN= 28 K. The anisotropic behavior of the susceptibility below the N´ eel temperature\nshows [001] direction as the easy axis of magnetization. In contras t, the susceptibility along\n[100] remains below that of [001] direction in the entire temperature range followed by a\nknee at the ordering temperature. The Curie-Weiss fits of the inve rse susceptibility in the\nparamagnetic state gives µeffandθpas 9.55µB/Tb and 9.66 µB/Tb and -58 K and -16 K\nfor field parallel to [100] and [001] directions, respectively. The obt ained effective moment\nfor both the axes is close to that of the theoretical value (9.72 µB/Tb). The polycrystalline\naverage of θpis -45.3 K, indicating the presence of strong antiferromagnetic inte ractions in\nthe compound.\nThe anisotropic magnetic behavior is further corroborated by the magnetic isotherms\n(measured at various temperatures along the [001] direction) of t he compound with field\napplied along the two crystallographic axes, respectively which is sho wn in Fig. 8(a). The\n105\n4\n3\n2\n1\n0Magnetization ( µΒ / Tb)\n121086420\n Magnetic Field (T)   2 K\n   5 K\n 10 K\n 15 K\n 20 K\n 25 K\n 30 K\n Tb2CoGa8\nH // [001]\nH // [100]\nT = 2 K\nFIG. 8: (Color online) Magnetic isotherms of Tb 2CoGa8at 2 K for the field along [100] and at\nvarious temperatures for [001] direction.\nmagnetization at T = 2 K undergoes multiple metamagnetic transitions at 3.5, 4.8, 6.6,\n9 and 10 T. The magnetic transition at 6.6 T is a predominant one where as the others\nrepresent minor reorientation of the moments with field. The satur ation magnetization\nobtained at 12 T at 2 K is about 4.2 µB/Tb, which is less than half of the saturation\nmoment of Tb3+ions (9µB/Tb). Further high magnetic field is required to obtain the full\nsaturation value of the Tb moments. As the temperature is increas ed the sharpness of the\nmetamagnetic transition decreases and it shifts towards lower field s due to the extra thermal\nenergy available for the reorientation of the moments. At 30 K the m agnetization is linear\nindicating the paramagnetic state of the compound. The magnetic is otherm at 2 K with the\nfield along the hard direction [100] (represented by triangles in Fig. 8 (a)) is a straight line\nwith a magnetization value of 2.6 µB/Tb at 12 T.\nThe thermal variation of the magnetic susceptibility of Dy 2CoGa8is similar to that of\nTb2CoGa8. An antiferromagnetic transition occurs at TN=18 K as shown in Fig. 9(a).\nIn the paramagnetic state the inverse susceptibility was fitted to C urie-Weiss law with\nµeff= 10.4µB/Dy and 10.5 µB/Dy and θp= -45 K and -6 K along the [100] and [001] axis,\nrespectively. The effective moments are close to the theoretical v alue (10.63 µB) and the\npolycrystalline average of θpis -32 K indicating an antiferromagnetic interaction. The mag-\nnetic isotherms with field along the two crystallographic directions ar e shown in Fig. 10(a).\nThe temperature variation of magnetic isotherms was measured on ly along the easy axis\n[001]. The magnetization along the [001] at 2 K undergoes two metama gnetic transitions\n1120\n10\n0 (χ − χ0)-1 (mol/emu·Dy)\n300250200150100500\n Temperature (K)Dy2CoGa8\n H // [001]\n H // [100]\n CEF-Fit\n(b)0.6\n0.4\n0.2\n0 χ (emu/mol)Dy2CoGa8\n5 kOe H // [001]\n H // [100]\n(a)\nFIG. 9: (Color online)(a) Magnetic susceptibility ofDy 2CoGa8, (b)inversemagneticsusceptibility;\nsolid lines through the data point indicate the CEF fit.\natHc1= 3.4 T and Hc2= 8.6 T. After the second metamagnetic transition the magnetiza-\ntion reaches about 7.1 µB/Dy at 12 T. This value is less than the ideal saturation value of\n10µB/Dy for Dy3+ion. A hysteresis was observed (not shown in the figure) between t he\ntwo metamagnetic transitions, which may be due to the anisotropic b ehavior of the reori-\nented moments. The temperature variation of the magnetic isothe rm is similar to that of\nTb2CoGa8, namely the decrease in the sharpness and shift towards lower mag netic fields of\nthe metamagnetic transitionwith temperature. In Tb 2CoGa8the magnetizationat 12 Twas\nfound to increase initially with temperature and then decreases nea r the N´ eel temperature\nof the compound, whereas in Dy 2CoGa8it decreases continuously with temperature. This\neffect is attributed to the strong antiferromagnetic coupling of th e Tb3+moments compared\ntothat ofDy3+moments. This isalso evident fromthepolycrystalline average θpofboththe\ncompounds. Because of the strong coupling the thermal energy a cts as a helping hand for\nthe reorientation of themoments, whereas inDy 2CoGa8the field energy is sufficient to break\nthe antiferromagnetic coupling. From the differential plots of the is othermal magnetization\n1210\n8\n6\n4\n2\n0Magnetic Field (T)\n20 15 10 5 0\nTemperature (K)Dy2CoGa8\nH // [001]\nHc1Hc26\n4\n2\n0 Magnetization (  / Dy)\n12 10 8 6 4 2 0\n Magnetic Field (T)Dy2CoGa8\n    2 K\n   6 K\n   9 K\n 12 K\n 15 K\n 20 K\n \n H // [001]\nH // [100]\nT = 2 K (a)\n(b)\nAF-IAF-IIFERRO\nFIG. 10: (Color online) Magnetic isotherms of Tb 2CoGa8at 2 K for the field along [100] and at\nvarious temperatures for [001] direction and (b) the magnet ic phase diagram of Tb 2CoGa8.\ncurves (not shown here), we have constructed the magnetic pha se diagram as depicted in\nFig. 10(b). Hc1at first increases with the increase in temperature and then decre ases above\n10 K and finally vanishes for temperature above 15 K , while the Hc2decreases continuously\nwith the increase in the temperature. At low temperatures and for fields less than 3.4 T,\nthe systems is in a purely antiferromagnetic state as indicated by (A F-I) in Fig. 10(b) and\nthen undergoes a complex magnetic structure (AF-II) for fields b etweenHc1andHc2and\nfinally enters into the field induced ferromagnetic state.\nThe magnetic susceptibility of Ho 2CoGa8in an applied magnetic field of 5 kOe along\n[100] and [001] axes is shown in Fig. 11(a). The compound orders ant iferromagnetically at\nTN= 6 K. Overall, the susceptibility along the [001] and [100] direction sho ws a similar\nbehavior as observed in Tb 2CoGa8and Dy 2CoGa8analogs. This indicates a less anisotropic\nbehavior of Ho 2CoGa8compared to the former two compounds. The Curie-Weiss fit of the\n131.5\n1.0\n0.5\n0 χ (emu/mol)Ho2CoGa8  H // [001]\n H // [100] 5 kOe\n20\n15\n10\n5\n0 (χ − χ0)-1(mol/emu·Ho)\n300250200150100500\n Temperature (K)Ho2CoGa8\n H // [001]\n H // [100]\n CEF-Fit\n \nFIG. 11: (Color online) (a) Magnetic susceptibility of Ho 2CoGa8, (b) inverse magnetic susceptibil-\nity; solid lines through the data point indicate the CEF fit.\ninverse susceptibility in the paramagnetic state of the compound giv esµeff= 10.48µB/Ho\nand 10.6 µB/Ho and θp= -18.6 K and -1.5 K along the [100] and [001] direction respectively.\nThe effective moments are close to the theoretically expected value (10.6µB/Ho) and the\npolycrystalline average of θPis -12.9 K, in accordance with the antiferromagnetic transition\nin the compound. A lower absolute value of θpcompared to those of compounds with R\n= Gd, Tb and Dy, indicates weaker interaction among the Ho3+moments. This is evident\nin the magnetic isotherm of the compound along the easy axis of magn etization (as shown\nin Fig. 12). The metamagnetic transition occurs at lower fields compa red to the former\ncompounds. The magnetic isotherm at 2 K (Fig. 12) along the [001] ax is undergoes two\nmetamagnetic transitions at Hc1= 1.55 T and at Hc2= 4.0 T. Both of them induce large\nchanges in the magnetization. The magnetization at 12 T and 2 K is 8.2 µB/Ho. This is\nclose to the saturation moment of 10 µBfor Ho3+moments. Thus the second metamagnetic\ntransition drives the compound to a field induced ferromagnetic sta te. With the increase\nin the temperature the metamagnetic transitions broaden and shif ts towards lower fields\n148\n6\n4\n2\n0 Magnetization ( µΒ/ Ho)\n121086420\n Magnetic Field (T)    2 K\n 3.5 K\n    5 K\n 6.5 K\n  10 K\n  15 K\n  25 KHo2CoGa8\n   H // [001]\nFIG. 12: (Color online) Magnetic isotherms of Ho 2CoGa8at 2 K for the field along [100] and at\nvarious temperatures for [001] direction.\nand their sharpness decreases. Above the ordering temperatur e of the compound (6 K),\nthe magnetic isotherms at 10 K and 15 K tend towards saturation at high fields and the\nmoment at 12 T is above 8.2 µB/Ho. At 25 K the magnetic isotherm is a straight line as\nexpected for a paramagnetic state. The magnetization with field alo ng the hard direction\n[100] is not a straight line as observed for other compounds, but th e moment tends towards\nsaturation with field; ≈7µB/Ho at 12 T. At high fields (above 8 T), the behavior of the\ncompound is similar in both the directions (except for the moment is litt le less along the\nhard direction). This is because the field energy is sufficient to overc ome the anisotropic\nenergy barrier existing within the compound. This also supports the early explanation of\nless anisotropy of Ho 2CoGa8compared to that of Tb 2CoGa8and Dy 2CoGa8. From the\ndifferential plots of the isothermal magnetization along [001] direct ion we have constructed\nthe magnetic phase diagram as shown in Fig 13(b). It is obvious from t he figure that both\nthe metamagnetic transitions shift towards lower fields with the incr ease in temperature and\nfinally merge with each other at the ordering temperature.\nD. R 2CoGa 8(R = Er and Tm)\nThough these compounds also order antiferromagnetically, the ea sy axis of magnetization\nis now along the [100] direction. Er 2CoGa8and Tm 2CoGa8order antiferromagnetically at\n2 and 3 K, respectively. The change in the direction of the easy axis o f magnetization\n155\n4\n3\n2\n1\n0Magnetic Field (T)\n8 6 4 2 0\n Temperature (K)Ho2CoGa8\nH // [001]\nAF-IPARAAF-IIFERRO\nHc1Hc2\n(b)8\n6\n4\n2\n0 Magnetization ( µΒ/ Ho)\n12 10 8 6 4 2 0\n Magnetic Field(T) H // [100]\n H // [001]Ho2CoGa8\nT = 2 K\n(a)\nFIG. 13: (Color online)(a) Magnetic isotherm of Ho 2CoGa8at 2 K for the field along [100] and\n[001] directions, (b) the magnetic phase diagram of Ho 2CoGa8.\nfollows the general trend observed in a number of tetragonal rar e-earth series of compounds\nRRh4B49, RAgSb 213and RRhIn 514. The obtained value of µeffandθpare presented in\nTable III. The effective moment for both the compounds along both the crystallographic\ndirections is close to their theoretically expected value (9.59 µB/Er and 7.57 µB/Tm). The\npolycrystallineaverageoftheparamagneticCurietemperaturefo rEr2CoGa8andTm 2CoGa8\nare -7.13 K and -4.2 K respectively. Both of them are negative and ar e lower compared to\nthe other R 2CoGa8compounds, indicating a gradual weakening of the antiferromagne tic\ninteractions among the rare earth moments, in this series.\nThe susceptibility of Er 2CoGa8along both the directions is shown in Fig. 14a. The\nsusceptibility along the [100] direction shows a peak at the antiferro magnetic transition\ntemperature, where as that along the [001] direction does not sho w any ordering down to\n163.0\n2.0\n1.0\n0 χ (emu/mol)\n25 20 15 105 0\nTemperature (K) Er2CoGa8 H // [100]\n H // [001]\n(a)5 kOe\n25\n20\n15\n10\n5(χ − χ0)−1(mol/emu·Er)\n300250200150100500\n Temperature (K) H // [001]\n H // [100]\n CEF-FitEr2CoGa8\n(b)\nFIG. 14: (Color online) (a) Magnetic susceptibility of Er 2CoGa8for the temperature range 1.8 to\n25 K, (b) inverse magnetic susceptibility; solid lines thro ugh the data point indicate the CEF fit.\n8\n6\n4\n2\n0 Magnetization ( µΒ/Er)\n121086420\n Magnetic Field (T)Er2CoGa8\nH // [001]\n H // [100]\nT = 2 K\nFIG. 15: (Color online) Magnetic isotherm of Er 2CoGa8at 2 K for the field along [100] and [001]\ndirections.\n171.8 K. The inverse susceptibility is plotted in Fig. 14(b). Both the plots are close to each\nother indicating a weak anisotropy of the compound. In addition, th ere is a crossover of\nthe susceptibility at about 164 K, indicating a change in easy axis of ma gnetization with\ntemperature. Similar behavior has been observed in Er 2PdSi315. Such a behavior may\npossibly arise due to the magnetic behavior of the compound lying on t he border of the\nanisotropic crossover along the crystallographic axis. This is also su pported by the crystal\nfield calculation on the compound discussed in the next section. The m agnetic isotherm\nalong both the direction is shown in Fig. 15. The magnetization along th e [100] direction\nundergoes a metamagnetic transition at 0.25 T and tends to satura te at higher fields. The\nmoment at 1.8 K and 12 T is 9 µB/Er. With increase in the temperature (at 4 K and\n10 K, not shown in the figure) the metamagnetic transition vanishes but the behavior of\nthe magnetization remains the same with the moment little less then th at at 2 K. The\nmagnetization behavior along the [001] direction is almost similar to tha t of [100] direction.\nFor field parallel to [100] the magnetization reaches a moment of 5.6 µB/Er at 1.8 K and\n12 T.\nIn case of Tm 2CoGa8the susceptibility along both the crystallographic direction show\nantiferromagnetic behavior at 50Oe(inset ofFig. 16a). When the fi eld isincreased to 5 kOe,\nantiferromagnetic transition along both the axis vanishes. This indic ates a weak interaction\namongthemomentssuch thattheantiferromagneticpeakshiftsw ithsmall appliedfieldsand\nvanishes even at 5 kOe. The magnetic isotherm at 1.8 K for both the c rystallographic axis is\nshown in Fig. 17. The behavior of magnetization along both the axis is q ualitatively similar.\nBut the magnetization along the [100] direction is higher than that of the [001] direction.\nAt approximately 11.2 T there is a cross over with the magnetization a long the [001] axis\nexceeding the magnetization along the [100] axis. High field magnetiza tion measurements\nare necessary to probe this behavior.\nV. DISCUSSION\nAn interesting magnetic behavior is exhibited by the R 2CoGa8series of compounds. The\nnonmagnetic compounds Y 2CoGa8and Lu 2CoGa8show diamagnetic behavior. Compounds\nwith magnetic rare earths (R = Gd, Tb, Dy, Ho, Er and Tm) order ant iferromagnetically at\nlow temperatures. The N´ eel temperatures are listed in Table III. For comparison the N´ eel\n184\n3\n2\n1\n0 χ (emu/mol)Tm2CoGa8\n5 kOe\n H // [100]\n H // [001]\n40\n20\n0 χ−1 (mol/emu·Tm)\n300250200150100500\nTemperature (K)Tm2CoGa8\n H // [100]\n H // [001]\n CEF-Fit\n 3.0\n2.0\n1.0 χ (emu/mol)\n151050H = 50 Oe\nFIG. 16: (Color online)(a) Magnetic susceptibility of Tm 2CoGa8, inset shows the magnified view\nof low temperature susceptibility, (b) inverse magnetic su sceptibility; solid lines through the data\npoint indicate the CEF fit.\n5\n4\n3\n2\n1\n0Magnetization ( µΒ/Tm)\n121086420\n Magnetic Field (T) H // [001]\n H // [100]Tm2CoGa8\nT = 2 K\nFIG. 17: (Color online) Magnetic isotherm of Tm 2CoGa8at 2 K for the field along [100] and [001]\ndirections.\n19R2CoIn8 R2CoGa8\nTN TN µeff(µB/R) θp(K) χ0(emu/mol)\nR (K) (K) [100] [001] [100] [001] [100] [001]\nGd 33.5 20 7.94 7.9 -69 -67 7.2 ×10−43.4×10−4\nTb 30 28 9.55 9.66 -58 -16 0 0\nDy 17.4 18 10.53 10.47 -45 -6 9.0 ×10−41.2×10−3\nHo 7.6 6 10.5 10.6 -18.6 -1.5 8.0 ×10−43.0×10−4\nEr N.A 3 9.5 9.59 -5.2 -11.2 0 0\nTm N.A 2 7.57 7.35 -1.7 -12.6 1.7 ×10−32.5×10−4\nLu N.A Dia - - - - - -\nY P.P Dia - - - - - -\nN.A : Data not available, P.P : Pauli paramagnetic, Dia: Diam agnetic\nTABLE III: Comparison of N´ eel temperatures of R 2CoIn8with R 2CoGa8compounds. Paramag-\nnetic Curie temperature ( θp), effective magnetic moment ( µeff) andχ0for R2CoGa8compounds.\n30\n20\n10\n0 TN (K)\nGdTbDyHoErTm\nRare EarthsR2CoGa8\n Observed\n de-Gennes\nFIG.18: (Coloronline) N´ eel temperatureoftheR 2CoGa8compoundscomparedwiththatexpected\nfrom de-Gennes scaling. The lines joining the data points ar e the guide to eyes.\ntemperatures of the corresponding indides are also listed. The tra nsition temperature is less\n20comparedtothatofthecorrespondingpolycrystallineindiumanalog sR2CoIn8. Accordingto\nthe de-Gennes scaling, in the mean field approximation the magnetic o rdering temperatures\nTMof the isostructural members of a rare earth series of compound s are proportional to\n(gJ−1)2J(J+ 1), where gJis the Land´ e gfactor and Jis the total angular momentum.\nThe N´ eel temperature of the R 2CoGa8series of compound is plotted in Fig. 18 along with\ntheir de-Gennes expected values, normalized to TNof Gd2CoGa8. A fairly large deviation\nof TNof the Tb and Dy compounds from the de-Gennes expected scaled v alue is noticeable.\nThe ordering temperature of Tb 2CoGa8is even higher than that of the corresponding Gd\ncompound. A similar behavior was also observed for RRh 4B49compounds and Noakes et\nal.10showed it to be due to the CEF effects. The magnetic susceptibility da ta presented\nabove therefore provide a good opportunity to attempt a crysta l electric field analysis on\nthis series of compounds. From the estimated crystal field parame ters we found that the\nenhancement in the ordering temperature of Tb and Dy can be attr ibuted to CEF effects,\nas described later in this section.\nThe rare-earth atom in R 2CoGa8occupies the 2 gWyckoff’s position with a tetragonal\nC4vpoint symmetry. The CEF Hamiltonian for a tetragonal symmetry is g iven by,\nHCEF=B0\n2O0\n2+B0\n4O0\n4+B4\n4O4\n4+B0\n6O0\n6+B4\n6O4\n6, (2)\nwhereBm\nℓandOm\nℓare the CEF parameters and the Stevens operators, respective ly16,17. The\nCEF susceptibility is defined as\nχCEFi=N(gJµB)21\nZ/parenleftBigg/summationdisplay\nm/negationslash=n| /angbracketleftm|Ji|n/angbracketright |21−e−β∆m,n\n∆m,ne−βEn+/summationdisplay\nn| /angbracketleftn|Ji|n/angbracketright |2βe−βEn/parenrightBigg\n,\n(3)\nwheregJis the Land´ e g-factor,Enand|n/angbracketrightare thenth eigenvalue and eigenfunction,\nrespectively. Ji(i=x,yandz) is a component of the angular momentum, and ∆ m,n=\nEn−Em,Z=/summationtext\nne−βEnandβ= 1/kBT. The magnetic susceptibility including the\nmolecular field contribution λiis given by\nχ−1\ni=χ−1\nCEFi−λi. (4)\nThe inverse susceptibility of the R 2CoGa8(R = Tb, Dy, Ho, Er and Tm) as shown\nin Fig. [7(b), 9(b), 11(b), 14(b) and 16(b)] respectively was fitte d to the above discussed\n21250\n200\n150\n100\n50\n0 Energy (K) Doubl et\n Singlet\n \nTb Dy Ho Er Tm\nFIG. 19: (Color online) CEF energy level splitting of the gro und state of the R3+ions in R 2CoGa8\ncompounds.\nRB0\n2B0\n4 B4\n4 B0\n6 B4\n6 λ[100]λ[001]J[100]\nex/kBJ[001]\nex/kB\n(K) (K) (K) (K) (K) (mol/emu) (mol/emu) (K) (K)\nTb -1.61 0.0049 -0.0515 -7.0 ×10−51.63×10−3-2.95 -4.7 -2.24 -4.94\nDy -0.7 0.005 0.001 9.0 ×10−81.0×10−4-2.5 -2.0 -1.28 -1.94\nHo -0.22 -1.45 ×10−3-0.0314 -2.0 ×10−5-2.0×10−5-1.2 -1.2 -0.51 -0.59\nEr 0.089 1.3 ×10−40 4.3 ×10−62.5×10−4-0.782 -0.274 -0.35 -0.32\nTm 0.35 1.72 ×10−60.0248 -8.46 ×10−69.1×10−40.723 0 -0.53 -0.075\nTABLE IV: CEF paramters for R 2CoGa8compounds obtained from the inverse susceptibility fit\nCEF model Eqn. (2-4). The values of the CEF parameters thus obt ained are presented in\nTable IV and the corresponding energy levels are shown in Fig. 19. Th e dominant crystal\nfield parameter B0\n2is negative for Tb, Dy and Ho compounds. This is consistent with the\nuniaxial ([001] as an easy axis) magnetic anisotropy present in thes e compounds. The sign\nof theB0\n2changes for Er and Tm compounds which is consistent with the chang e in the easy\naxis of magnetization. For Er 2CoGa8, the estimated B0\n2is 0.089 K, which is close to zero,\nindicating thatthecompoundisontheborderlineofthemagneticanis otropycrossover. The\ncurrent setofCEFparameterscouldevenexplainthecrossover in themagneticsusceptibility\nof Er2CoGa8. According to mean field theory, the CEF parameter B0\n2can be related to the\nexchange constant and paramagnetic Curie temperature by the r elation18\n22θ[001]\np=J(J+1)\n3kBJex[001]−(2J−1)(2J+3)\n5kBB0\n2, (5)\nθ[100]\np=J(J+1)\n3kBJex[100]+(2J−1)(2J+3)\n10kBB0\n2. (6)\nThe obtained value for Jexalong both the crystallographic directions is presented in\nTable IV. The negative value along both the principal directions implies that the anitferro-\nmagnetic interaction is dominant in this series of compounds. In the c ase of R 2CoGa8(R =\nTb, Dy and Ho) the exchange constant Jex, has a higher value along [001] compared to that\nof [100] direction. This implies that the antiferromagnetic exchange interaction is stronger\nalong the [001] direction, thus supporting our experimental obser vation of magnetic easy\naxis for these compounds. Whereas, for R 2CoGa8(R = Er and Tm) the Jexis dominant\nalong the [100] direction, which is again consistent with our magnetic s usceptibility data.\nThe magnetic transition temperature ( TM) of a compound in presence of crystalline elec-\ntric field is given by Noakes et al10as,\nTM= 2Jex(gJ−1)2/angbracketleftJ2\nz(TM)/angbracketrightCEF, (7)\nWhere/angbracketleftJ2\nz(TM)/angbracketrightCEFis the expectation value of J2\nzatTMunder the influence of crystalline\nelectric fieldalone. Jexistheexchange constant fortheRKKYexchange interactionbetw een\nthe rare earth atoms. Now for the compounds with tetragonal st ructure the dominating\ncrystal field term is B0\n2, so neglecting the higher order term and when the ordering is along\nthe [001] direction, the above equation can be rewritten as10\nTM= 2Jex(gJ−1)2/summationdisplay\nJzJ2\nzexp(−3J2\nzB0\n2/TM)\nexp(−3J2\nzB0\n2/TM). (8)\nThe transition temperature T Mwas calculated using the values of the exchange constant\nJexobtained from Eqn. 5 for the easy-axis of magnetization and with th e corresponding B0\n2\ncrystal field parameter. The magnetic transition temperature fo r Tb2CoGa8and Dy 2CoGa8\nthus estimated was found to be ( ≈82 K) and ( ≈22 K), respectively. This shows the\nenhancement in the transition temperature in presence of CEF com pared to that expected\nfrom de-Gennes scaling. Though the estimated TNin Tb2CoGa8is unexpectedly high, the\npreceding analysis shows that the enhancement of TNof Tb2CoGa8can be due to CEF\neffects.\n23VI. CONCLUSION\nTo conclude, we have successfully grown the single crystals of R 2CoGa8(R = Gd, Tb, Dy,\nHo, Er, Tm, Lu andY) forthefirst timeby using Gaasflux. This series ofcompounds forms\nonly with the higher rare earths. The phase purity of the crystals w as confirmed by means\nof powder X-ray diffraction. Y 2CoGa8and Lu 2CoGa8show diamagnetic behavior indicating\na low density of states at the Fermi level and a filled 3 dof Co band. Compounds with\nmagneticrareearthsorderantiferromagneticallyatlowtemperat ures. TheN´ eeltemperature\nof Tb2CoGa8and Dy 2CoGa8deviates appreciably from expected value of de-Gennes scaling.\nThe reason is attributed to crystal field effects. The easy axis of m agnetization for R 2CoGa8\n(R = Tb, Dy and Ho) is along the crystallographic [001] direction. It ch anges to (100) plane\nfor Er 2CoGa8and Tm 2CoGa8.\n1Kalychak Ya. M, Zeremba V. I, Baranyak V. M, Bruskov V. A and Za valij P. Yu, Izv. Acad,\nNauk SSSR Metally 1, 209 (1989).\n2J.D. Thompson, R. Movshovich, Z. Fisk, F. Bouquet, N.J. Curr o, R.A. Fisher, P.C. Hammel,\nH. Hegger, M.F. Hundley, M. Jaime, P.G. Pagliuso, C. Petrovi c, N.E. Phillips and J.L. Sarrao,\nJ. Magn. Magn. Mater. 226-230 , 5 (2001).\n3M. Nicklas, V. A. Shidorov, H. A. Borges, P. G. Pagliuso, C. Pe trovik, Z. Fisk, J.L. Sarrao and\nJ.D. Thompson, Phys. Rev. B 67, 020506 (2003).\n4G. Chen, S. Ohara, M. Hedo, Y. Uwatoko, K. Saito, M. Sorai and I . Sakamoto, J. Phys. Soc.\nJapan712836 (2002).\n5Devang A. Joshi, C.V. Tomy and S.K. Malik, J. Phys.: Condens. Matter19136216 (2007).\n6S. Cirafici and E. Franceschi, J. Less-Common Met. 77, 269 (1981).\n7Ya. M. Kalychak, J. Alloys Compounds 291 (1999) 80\n8K. H. J. Buschow, J. Chem. Phys., 50, 137 (1969)\n9B. D. Dunlap, L. N. Hall, F. Behroozi, G. W. Crabtree and D. G. N iarchos, Phys. Rev. B 29\n6244 (1984).\n10D. R. Noakes and G. K. Shenoy, Phys. Lett. 91A, 35 (1982).\n11R. Mallik and E. V. Sampathkumaran, Phys. Rev. B 58, 9178 (1998).\n2412M. Rotter, M. Loewenhaupt, M. Doerr, A. Lindbaum and H. Micho r, Phys. Rev. B 64014402\n(1998).\n13K. D. Myers, S. L. Budko, I. R. Fisher, Z. Islam, H. Kleinke, A. H. Lacerda and P. C. Canfield,\nJ. Magn. Magn. Mater. 205, 27 (1999).\n14N. V. Hieu, T. Takeuchi, H, shishido, C. Tonohiro, T. Yamada, H. Nakashima, K. Sugiyama,\nR. Settai, T. D. Matsuda, Y. Haga, M. Hagiwara, K. Kindo, S. Ar aki, Y. Nozue and Y. Onuki,\nJ. Phys. Soc. Japan 76, 064702 (2007).\n15M. Frontzek, A. Kreyssig, M. Doerr, M. Rotter, G. Behr, W. L¨ o ser, I. Mazilu and M. Loewen-\nhaupt, J. Magn. Magn. Mater., 301, 398 (2006).\n16K. W. H. Stevens, Proc. Phys. Soc., London, Sect. A 65, 209 (1952).\n17M. T. Hutchings, in Solid State Physics: Advances in Research and Applications , edited by F.\nSeitz and B. Turnbull (Academic, New York, 1965), Vol.16, p. 227.\n18J. Jensen and A. R. Mackintosh, Rare earth magnetism structu res and excitations, Calrendon\npress, Oxford Chap 2, p. 73 (1991).\n25"    },    {        "title": "0802.1574v1.Domain_walls_in__Ga_Mn_As_diluted_magnetic_semiconductor.pdf",        "content": "Domain walls in (Ga,Mn)As diluted magnetic semiconductor\nAkira Sugawara,1H. Kasai,2A. Tonomura,1, 2P. D. Brown,3R. P. Campion,4\nK. W. Edmonds,4B. L. Gallagher,4J. Zemen,5and T. Jungwirth5, 4\n1Initial Research Project, Okinawa Institute of Science and Technology,\nc/o Hitachi Advanced Research Laboratory, Akanuma 2520, Hatoyama, Saitama 350-0395, Japan\u0003\n2Advanced Research Laboratory, Hitachi Ltd., Akanuma 2520, Hatoyama, Saitama 350-0395, Japan\n3School of Mechanical, Materials and Manufacturing Engineering,\nUniversity of Nottingham, University Park, Nottingham NG7 2RD, United Kingdom\n4School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom\n5Institute of Physics ASCR, Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\n(Dated: April 25, 2022)\nWe report experimental and theoretical studies of magnetic domain walls in an in-plane magne-\ntized (Ga,Mn)As dilute moment ferromagnetic semiconductor. Our high-resolution electron holog-\nraphy technique provides direct images of domain wall magnetization pro\fles. The experiments\nare interpreted based on microscopic calculations of the micromagnetic parameters and Landau-\nLifshitz-Gilbert simulations. We \fnd that the competition of uniaxial and biaxial magnetocrys-\ntalline anisotropies in the \flm is directly re\rected in orientation dependent wall widths, ranging\nfrom approximately 40 nm to 120 nm. The domain walls are of the N\u0013 eel type and evolve from\nnear-90\u000ewalls at low-temperatures to large angle [1 \u001610]-oriented walls and small angle [110]-oriented\nwalls at higher temperatures.\nPACS numbers: 68.37.Lp, 75.50.Pp, 75.60.Ch\nMagnetic domain walls (DWs) are extensively explored\nfor their potential in integrated memory and logic devices\n[1, 2] and because of the number of open basic physics\nquestions related to their dynamics in external magnetic\nand electric \felds [3, 4, 5, 6, 7]. Dilute moment ferromag-\nnetic semiconductors, of which (Ga,Mn)As is archetyp-\nical, are playing an increasingly important role in this\nresearch area [8, 9, 10, 11]. p-type (Ga,Mn)As has a\nsaturation magnetization ( Ms) which is two orders of\nmagnitude lower than in conventional metal ferromag-\nnets, while the magnetocrystalline anisotropy energies\n(K) and spin sti\u000bness ( A) are comparable to the met-\nals [12, 13, 14, 15]. The low Msis due to the dilute Mn\nmoments while the holes in the spin-orbit-coupled va-\nlence bands, mediating the long-range Mn-Mn coupling,\nproduce the large KandA.\nAmong the immediate implications of these character-\nistics are weak dipolar stray \felds which would allow\nfor dense integration of (Ga,Mn)As micro-elements with-\nout unintentional cross-links, macroscopic-size domains,\nsquare hysteresis loops, and mean-\feld-like temperature\ndependent magnetization [16, 17]. The strong in-plane\nbiaxial and uniaxial magnetocrystalline anisotropies [16,\n17, 18] and weak dipole \felds lead also to the formation\nof unique segmented domain structures and spontaneous\ndomain reorganization with changing temperature [19].\nAn outstanding feature of DW dynamics in (Ga,Mn)As\nis the orders of magnitude lower critical current for DW\nswitching than observed for conventional ferromagnets\n[8, 9].\nThe basic tool for studying DWs is magnetic struc-\nture imaging but here the low saturation moment in\n(Ga,Mn)As is a problem, greatly reducing the sensitiv-ity of conventional magneto-optical and scanning Hall\nprobe microscopy techniques. For out-of-plane magne-\ntized (Ga,Mn)As \flms, grown under tensile strain on\n(In,Ga)As, the resolution achieved with these techniques\nis limited to\u00181\u0016m [8, 20, 21] and the sensitivity is\nfurther reduced for (Ga,Mn)As layers grown on GaAs\nunder compressive strain with in-plane magnetization\n[16, 22, 23]. Imaging internal DW con\fgurations, which\nare particularly intriguing in the in-plane materials, re-\nquires\u0018100\u000010 nm resolution and has therefore re-\nmained far beyond the reach of the conventional mag-\nnetic microscopy techniques.\nIn this paper we present the detailed study of in-plane\nDWs in (Ga,Mn)As. We obtain direct images of DW\nmagnetization pro\fles and determine the type and width\nof the DWs and their dependence on the wall orientation\nand temperature. The high sensitivity to in-plane mag-\nnetization is achieved by employing transmission elec-\ntron microscopy techniques [24] based on the Lorentz de-\n\rection of transmitted electrons by the in-plane compo-\nnent of the magnetic induction and on holographic elec-\ntron phase retrieval. The interpretation of experiments\nis based on kinetic-exchange-theory [25] calculations of\nmicromagnetic parameters and Landau-Lifshitz-Gilbert\n(LLG) simulations.\nA Ga 0:96Mn0:04As (500 nm)/GaAs (1 nm)/AlAs\n(50 nm)/bu\u000ber-GaAs (100 nm) multilayer was deposited\non a GaAs(001) substrate using molecular beam epitaxy.\nElectron transparent uniform (Ga,Mn)As foils with a\nwide \feld of view around 100 \u0016m were produced by se-\nlectively etching away the substrate using the AlAs as\na stop layer. The cubic anisotropy favours magnetiza-\ntion along the in-plane h100icrystalline axes, and thearXiv:0802.1574v1  [cond-mat.mtrl-sci]  12 Feb 20082\nuniaxial anisotropy favours magnetization along one of\ntheh110iaxes. We label the in-plane uniaxial easy axis\nas the [1 \u001610] direction, as this was found to be the easy\naxis in layers grown under similar conditions in the same\nsystem [28]. SQUID magnetization measurements on an\nunetched part of the wafer yield the cubic anisotropy con-\nstantKc=1.18(0.32) kJ/ m3and the uniaxial anisotropy\nconstantKu=0.18(0.11) kJ/ m3atT=10(30) K, and the\nCurie temperature Tc= 60 K.\nElectron holography measurements were performed us-\ning a 300 kV transmission electron microscope equipped\nwith a cold \feld-emission electron gun (Hitachi High-\nTechnologies, HF-3300X). The specimen was located at\na special stage in between the condenser and the weakly-\nexcited objective lens. In this con\fguration the inter-\nnal magnetic structures are expected to be undisturbed\nby the residual magnetic \feld, which is estimated to be\nsmaller than 10\u00004T and oriented perpendicular to the\nspecimen. Electron holography experiments were per-\nformed within a near-edge sample region, because of the\nneed for a reference electron wave traveling through vac-\nuum. The electrostatic phase gradient associated with\nslight thickness variations arising from inhomogeneous\nchemical etching of the sample near edges was com-\npensated for by subtracting a phase image of the sam-\nple in the high temperature paramagnetic state together\nwith a linear wedge correction. The sampling resolu-\ntion of the CCD camera used for image acquisition was\n1.6 nm, whilst the numerical phase reconstruction was\nperformed using a low-pass Fourier mask corresponding\nto the wavelength of 20 nm for holograms acquired with\n5 nm-spacing interference fringes.\nFig. 1 shows the DW phase images (right panels) ac-\nquired from a 3 \u0016m\u00021\u0016m corner region of the sample foil\nat 30.5 K, 25.4 K, and 9.8 K, respectively, together with\na larger area overviews (left panels) obtained by Fresnel\nmode Lorentz imaging [19]. The phase ( \u001e) is ampli\fed\nby a factor of 4 for clarity, and its cosine (i.e. cos(4\u001e)) is\ndislpayed in gray-scale. The magnitude and direction of\nthe magnetic induction Bare determined separately from\nthe relationship between the phase gradient and magnetic\ninduction,@\u001e=@r= 2\u0019et=h B(r), whereeis the electron\ncharge,tis the \flm thickness, and his the Planck con-\nstant. The local Bdirections are parallel to the tangent\nof the equiphase lines, as indicated by black arrows in\nFig. 1, and the magnitude of Bis inverse proportional to\nthe spacing of the equiphase lines.\nThe Fresnel-mode Lorentz and holographic phase re-\nconstructed images show consistently the location of sev-\neral DWs by bright contrast lines in the former case and\nby sharply bent equiphase lines in the latter case. The\nhigh resolution electron holography then provides the de-\ntailed information on the internal structure of the DWs.\nAs expected the magnitude of Bdecreases with increas-\ning temperature. The direction of Brotates gradually\nacross the wall boundary for all detected DWs imply-\nFIG. 1: Lorentz micrographs (left column) and phase images\nampli\fed by a factor of four (right column) acquired at (a)\n30.5 K, (b) 25.4, and (c) 9.8 K, respectively. Three di\u000berent\ntypes of DWs are observed, marked by white arrows: (i) a\nwall parallel to [1 \u001610] that is near-180\u000etype at high tempera-\nture and near-90\u000etype at low temperature; (ii) a pair of DW\nthat is near-180\u000ehead-on type at high temperature and near-\n90\u000etype at low temperature, and (iii) a near-90\u000etype wall\nparallel to [110] that appeared at low temperature. The local\nB directions are denoted by black arrows.\ning the N\u0013 eel type walls in the studied (Ga,Mn)As \flm.\nFor the DW denoted as (i) in Fig. 1, Brotates from\nthe near [100]/[010] directions towards the [1 \u001610] direc-\ntion with increasing temperature, i.e., we acquired direct\nimages of a transition from the near-90\u000ein-plane DW at\nlow-temperatures to a near-180\u000ewall at higher tempera-\ntures. As discussed below this is a demonstration in DW\nphysics of the competition between cubic and uniaxial\nanisotropies in the (Ga,Mn)As ferromagnet.\nFIG. 2: Magnitude ( jBj) and direction ( \u0012) ofBas a function\nof temperature ( T). Only results with subtraction of para-\nmagnetic phase images for thickness variation correction and\nfurther linear-wedge correction are displayed.\nFig. 2 summarizes the variation of the magnitude of\nBand the angle \u0012ofBmeasured from the [1 \u001610]-axis\nas a function of temperature for the left end of the DW\n(i). Filled symbols, corresponding to the phase images in\nFig. 1, include linear wedge correction for thickness vari-3\nation near the sample foil edge along the [1 \u001610]-direction.\nFor comparison we also plot the uncorrected data which\nshow similar behavior only the variation of \u0012appears\nsmaller. We also note that the Lorentz wall contrast\nobserved in the near-edge regions disappeared at tem-\nperatures typically 10 K lower than the far-edge region.\nThis suggests that the local temperature near the edges\nof the sample foil was higher than indicated by the ther-\nmal read-out from the liquid helium sample holder, due\nto restricted or insu\u000ecient heat transfer.\nFIG. 3: Phase image ampli\fed by a factor of 4, acquired at\n8 K (a) and 22 K (b). x-di\u000berentiated (c) and y-di\u000berentiated\n(d) images of (a). The DW positions are indicated by white\narrows. Projected pro\fle of the phase gradient across the\n[1\u001610] (x) -oriented DW (e) and [110] ( y) -oriented DW (f)\nalong lines indicated by white rectangles.\nPhase images in Fig. 3(a),(b) show vortex-like DWs\nwhich clearly demonstrate the dependence of the mag-\nnetization rotation angle and width of the DWs on the\ncrystallographic orientation of the wall and temperature.\nThe [110] and [1 \u001610]-oriented walls evolve from near-90\u000e\nwalls at low-temperatures to a large angle [1 \u001610]-oriented\nwall and a small angle [110]-oriented wall at higher tem-\nperatures. The width Wmof the walls is obtained by\ndi\u000berentiating the phase images with respect to the x\n([110]) and y([1\u001610]) directions (see Figs. 3(c)-(f)) and\nby \ftting the measured phase gradient pro\fle with a\nhyperbolic-tangent function [26]. In particular, the By\npro\fle along the x-axis for the narrow [1 \u001610]-oriented wall\nwas \ftted by By=By0tanh(2(x\u0000x0)=Wm) +C, where\nx0, andCare the central position of the wall and a com-\npensation term for the phase gradient, respectively. We\nobtainedWm= 45\u000610 nm for the measurement at 8 K\nand 54\u000617 nm at 22 K. At 30 K the width is, within the\nerror bar, identical to Wmat 22 K. Analogous \ftting pro-\ncedure for the [110]-oriented walls yield Wm= 85\u000615 nm\nfor the measurement at 8 K and 117 \u000635 nm at 22 K.\nAt 30 K the [110]-oriented walls have not been resolved.\nQuantitatively accurate theoretical modeling of the mi-\ncromagnetics in (Ga,Mn)As is inherently di\u000ecult due\nto the strong disorder and experimental uncertaintiesin Mn and hole densities. On a qualitative or semi-\nquantitative level, the physics underlying the observed\nphenomenology of in-plane DWs in (Ga,Mn)As can be\nconsistently described using the well established kinetic-\nexchange model, as we now discuss in detail. The descrip-\ntion is based on the canonical transformation which for\n(Ga,Mn)As replaces hybridization of Mn d-orbitals with\nAs and Ga sp-orbitals by an e\u000bective spin-spin interac-\ntion ofL= 0;S= 5=2 local-moments with host valence\nband states [25]. The spin-orbit coupling in the band\nstates produces the large magnetocrystalline anisotropies\nand, together with the mixed heavy-hole/light-hole char-\nacter, the large spin sti\u000bness [12, 13, 14, 15].\nFIG. 4: (a) Microscopic calculations of Ku=Kcfor the whole\nrange of doping parameters considered; inset shows the ef-\nfective barrier energies for the two walls and the speci\fed\nhole and local moment densities. (b) LLG simulations for\nthe low temperature micromagnetic parameters of the stud-\nied (Ga,Mn)As. Magnetization orientations in the individual\ndomains are highlighted by arrows.\nBased on previous detailed characterizations [27] of\nthe as-grown (Ga,Mn)As materials we assumed in our\ncalculations a range of relevant hole densities, p= 3\u0000\n4\u00021020cm\u00003and Mn local moment dopings, xMn=\n3\u00004:5% (NMn= 6\u000010\u00021020cm\u00003). The cor-\nresponding mean-\feld Curie temperatures are between\n50 and 100 K, consistent with experiment. First we\ninspected the theoretical dependence of in-plane mag-\nnetocrystalline anisotropies on growth-induced lattice-\nmatching strains. We found that typical strains in as-\ngrown materials have a negligible e\u000bect on the in-plane\nanisotropy energy pro\fles and, therefore, releasing the\nstrain during the preparation of the thinned, electron-\ntransparent (Ga,Mn)As foil should not a\u000bect signi\fcantly\nthe properties of in-plane DWs. The in-plane magne-\ntocrystalline anisotropy energy is accurately described by\nE(\u0012) =\u0000Kc=4 sin22\u0012+Kusin2\u0012, where\u0012is measured\nfrom the [1 \u001610] crystal axis. The microscopic origin of\nthe [110]-uniaxial anisotropy component present in most\n(Ga,Mn)As materials is not known but can be modeled\n[28] by introducing a shear strain exy\u00180:001\u00000:01%.\nFor the considered range of hole and local moment densi-\nties we obtained theoretical T= 0 values of Kcbetween\napproximately 0.5 and 1.5 kJm\u00003, consistent with the4\nlow-temperature SQUID measurement of Kc. The the-\noreticalKcvalues were found to be independent of exy,\nwhich is the only free parameter in the theory and whose\nmagnitude and sign was \fxed to match the experimen-\ntal low-temperature Ku=Kcratio. We then calculated\nthe temperature dependence of Ku=Kcand the corre-\nsponding easy-axis angle \u0012= 1=2 arccos(Ku=Kc) (the\nother easy-axis is placed symmetrically with respect to\nthe [1 \u001610]-direction.) The results, shown in Fig. 4(a),\nare fairly universal for all p's andNMn's considered in\nthe calculations and consistent with experimental data\nin Fig. 2.\nAn order-of-magnitude estimate of the theoretical DW\nwidth is given [14] by the length-scalep\nA=K eff, where\nKeffis the e\u000bective anisotropy energy barrier separat-\ning the bistable states on respective sides of the DW. For\nNMn\u00191021cm\u00003, the mean low-temperature value of\nKeff\u0019Kc=4\u00190:3 kJm\u00003(with 20% variation in the\nconsidered range of hole densities) and the spin-sti\u000bness\nA\u00190:4 pJm\u00001(nearly independent of p), yielding typ-\nical DW width Wm\u001840 nm, in agreement with experi-\nment. Despite the relatively small Ku=Kc\u00190:15 and the\ncorresponding small tilt by \u001910\u000eof the easy-axis from\nthe [100]/[010] directions at low temperatures, K[1\u001610]\neff\u0019\nKc=4 +Ku=2 for the larger-angle, [1 \u001610]-oriented DW is\nalready about twice as large as K[110]\neff\u0019Kc=4\u0000Ku=2\nfor the smaller-angle, [110]-oriented DW (see Fig. 4(a)),\nexplaining the sizable di\u000berence between the respective\nexperimental DW widths at 8 K. The observed tempera-\nture dependence of Wmcan be qualitatively understood\nby considering the approximate magnetization scaling of\nKc\u0018M4,Ku\u0018M2, andA\u0018M2[15, 17]. This im-\nplies for the [1 \u001610]-oriented DW that Wminitially increase\nwith temperature and then saturates at high Twhile the\n[110]-wall width steadily increases with T, becoming un-\nresolvable at Ku(T)=Kc(T)\u00191=2. (Note that a more\nquantitative discussion of the temperature dependence of\nWmis also hindered by the relatively large experimental\nerror bars for this quantity.)\nMagnetic dipolar \felds play a marginal role in the di-\nlute moment (Ga,Mn)As ferromagnets which may ex-\nplain, together with the <180\u000ewall-angle, the N\u0013 eel\ntype of the observed DWs despite their relatively small\nthickness. We further elaborate on this and on the\nabove qualitative arguments by performing the LLG sim-\nulations using micromagnetic parameters of the stud-\nied (Ga,Mn)As material. These calculations, shown in\nFig.4(b) for T= 8 K, con\frm the N\u0013 eel type of the DWs,\nthe evolution from near-90\u000ewalls at low-temperatures to\nlarger angle [1 \u001610]-oriented walls and smaller angle [110]-\noriented walls at higher temperatures, and the increasing\nanisotropy of the DW widths with increasing temper-\nature with Wmranging from 40 to 100 nm. This leads\nus to the conclusion that our combined experimental and\ntheoretical work represents an important extension to ourunderstanding of the micromagnetics of in-plane magne-\ntized (Ga,Mn)As down to the smallest relevant length-\nscale, the individual DW width. Since our \fndings are\nlikely una\u000bected by the constraints of the experimental\ntechnique on the lateral and vertical sample dimensions,\nour approach has a generic utility as a basis for DW stud-\nies in dilute moment ferromagnets.\nWe acknowledge valuable discussions with M.R. Sche-\ninfein and T. Yoshida and support from EU Grant\nIST-015728, from UK Grant GR/S81407/01, from\nCR Grants 202/05/0575, 202/04/1519, FON/06/E002,\nAV0Z1010052, and LC510.\n\u0003Present address: Advanced Research Laboratory, Hitachi\nLtd.; Electronic address: akira.sugawara.ne@hitachi.com\n[1] D. A. Allwood et al., Science 309, 1688 (2005).\n[2] L. Thomas, Nature 443, 197 (2006).\n[3] G. Tatara and H. Kohno, Phys. Rev. Lett. 92, 086601\n(2004).\n[4] S. Zhang and Z. Li, Phys. Rev. Lett. 93, 127204 (2004).\n[5] S. E. Barnes and S. Maekawa, Phys. Rev. Lett. 95,\n107204 (2005).\n[6] R. A. Duine, A. S. N\u0013 u~ nez, J. Sinova, and A. H. MacDon-\nald, Phys. Rev. B 75 , 214420 (2007).\n[7] M. D. Stiles, W. M. Saslow, M. J. Donahue, and A. Zang-\nwill, Phys. Rev. B 75 , 214423 (2007).\n[8] M. Yamanouchi, D. Chiba, F. Matsukura, and H. Ohno,\nNature 428, 539 (2004).\n[9] J. Wunderlich et al., et al., Phys. Rev. B 76, 054424\n(2007).\n[10] R. A. Duine, A. S. N\u0013 u~ nez, and A. H. MacDonald, Phys.\nRev. 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Phys. 101, 106101 (2007).\n[22] A. Pross et al., J. Appl. Phys. 95, 3225 (2004).\n[23] A. Pross et al., J. Appl. Phys. 95, 7399 (2004).\n[24] H. Hopster and H. P. Oepen, Magnetic Microscopy of\nNanostructures (Springer-Verlag Berlin, 2004).\n[25] T. Jungwirth, J. Sinova, J. Ma\u0014 sek, J. Ku\u0014 cera, and A. H.\nMacDonald, Rev. Mod. Phys. 78, 809 (2006).\n[26] A. Hubert and R. Sch afer, Magnetic Domains: The Anal-5\nysis of Magnetic Microstructures (Springer, 1998).\n[27] T. Jungwirth et al., Phys. Rev. B 72 , 165204 (2005).[28] M. Sawicki et al., Phys. Rev. B 71 , 121302 (2005)."    },    {        "title": "0802.3344v2.Lithographically_and_electrically_controlled_strain_effects_on_anisotropic_magnetoresistance_in__Ga_Mn_As.pdf",        "content": "arXiv:0802.3344v2  [cond-mat.mtrl-sci]  27 Feb 2008Lithographically and electrically controlled strain effec ts on anisotropic\nmagnetoresistance in (Ga,Mn)As\nE. De Ranieri,1,2A. W. Rushforth,3K. V´ yborn´ y,4U. Rana,1,2E. Ahmed,3R. P. Campion,3\nC. T. Foxon,3B. L. Gallagher,3A. C. Irvine,1J. Wunderlich,2,4and T. Jungwirth4,3\n1Microelectronics Research Centre, Cavendish Laboratory, University of Cambridge, CB3 0HE, UK\n2Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Ki ngdom\n3School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom\n4Institute of Physics ASCR v.v.i., Cukrovarnick´ a 10, 162 53 Praha 6, Czech Republic\n(Dated: October 28, 2018)\nIt has been demonstrated that magnetocrystalline anisotro pies in (Ga,Mn)As are sensitive to\nlattice strains as small as 10−4and that strain can be controlled by lattice parameter engin eering\nduring growth, through post growth lithography, and electr ically by bonding the (Ga,Mn)As sample\nto a piezoelectric transducer. In this work we show that anal ogous effects are observed in crystalline\ncomponents of the anisotropic magnetoresistance (AMR). Li thographically or electrically induced\nstrain variations can produce crystalline AMR components w hich are larger than the crystalline\nAMR and a significant fraction of the total AMR of the unproces sed (Ga,Mn)As material. In these\nexperiments we also observe new higher order terms in the phe nomenological AMR expressions and\nfindthat strain variation effects can play important role in t he micromagnetic and magnetotransport\ncharacteristics of (Ga,Mn)As lateral nanoconstrictions.\nPACS numbers: 75.47.-m, 75.50.Pp, 75.70.Ak\n1. INTRODUCTION\nGaAsdopedwith ∼1−10%ofthemagneticacceptorMnisauniquematerialforexploringsp in-orbitcouplingeffects\non micromagnetic and magnetotransport characteristics of ferr omagnetic spintronic devices. Spin polarized valence\nband holes that mediate ferromagnetic coupling between Mn local mo ments produce large magnetic stiffness, resulting\nin a mean-field like magnetization and macroscopic single-domain behav ior of these dilute moment ferromagnets. At\nthe same time, magnetocrystalline anisotropies derived from spin-o rbit coupling effects in the hole valence bands are\nlargeleading to the sensitivity of magnetic state to strains as small a s 10−4[1, 2, 3, 4, 5]. Experimentally, strain effects\ncan be controlled by lattice parameter engineering during growth [6 , 7], through post growth lithography [1, 2, 3], or\nelectrically by bonding the (Ga,Mn)As sample to a piezoelectric transd ucer [4, 5, 8]. Easy axis rotations from in-plane\nto out-of-plane directions have been demonstrated in these stud ies in (Ga,Mn)As films grown under compressive and\ntensile lattice matching strains, and the orientation of the in-plane e asy axis (axes) has been shown to respond to\nstrain relaxation in lateral microstructures or controlled dynamica lly by piezoelectric transducers.\nStrain control of magnetocrystalline effects on transport in (Ga,M n)As, we focus on in this paper, has so far been\nexplored less extensively. Our work in this direction is motivated by pr evious experimental and theoretical analyses of\nthe crystalline terms of the anisotropic magnetoresistance (AMR) . The studies have shown that these magnetotrans-\nport coefficients can be large and reflect the rich magnetocrystallin e anisotropies of the studied (Ga,Mn)As materials\n[9, 10, 11, 12, 13, 14, 15, 16, 17]. Furthermore, the possibility of d irectly controlling the crystalline AMR by strain\nhas been demonstrated in (Ga,Mn)As films grown under compressive and tensile lattice matching strains [10, 12, 18].\nHere we report and analyze AMR measurements in strain relaxed (Ga ,Mn)As micro and nanostructures and in a\n(Ga,Mn)As film bonded to a piezo-stressor. We show that post-gro wth induced lattice distortions can significantly\nmodify crystalline AMR terms and give rise to new, previously undetec ted components in phenomenological AMR\nexpansions.\nThe paper is organized as follows: In the 2nd section we derive a phen omenological description of the AMR relevant\nto the experimentally studied (Ga,Mn)As systems. 3rd and 4th sect ions report AMR data acquired in macroscopic\nand strain-relaxed microscopic Hall bars, in (Ga,Mn)As nanoconstr iction devices, and in voltage controlled piezoelec-\ntric/(Ga,Mn)As hybrid structures. A brief summary of the results is given in the 5th Section.2\n2. PHENOMENOLOGICAL DESCRIPTION OF THE AMR\nWe consider a thin film geometry and a magnetization vector, /vectorM/|/vectorM|= (cosψ,sinψ), in the plane of the film with\nits two components defined with respect to the orthogonal cryst allographic basis {[100],[010]}. The resistivity tensor,\nˆρ=/parenleftbiggρ11(cosψ,sinψ)ρ12(cosψ,sinψ)\nρ21(cosψ,sinψ)ρ22(cosψ,sinψ)/parenrightbigg\n, (1)\nwritten in the same basis describes the longitudinal and transverse resistivities of a pair of Hall bar devices oriented\nalong the [100]-direction ( ρ11andρ21) and the [010]-direction ( ρ22andρ12). Resistivities of a pair of orthogonal Hall\nbars tilted by an angle θfrom the [100]/[010] directions are given by R−θˆρRθ, whereRθ=/parenleftbiggcosθ−sinθ\nsinθcosθ/parenrightbigg\nis the\nrotation matrix. Written explicitly, the longitudinal ( ρL) and transverse ( ρT) resistivities for the Hall bar rotated by\nthe angleθfrom the [100]-direction read\nρL= (cosθ,sinθ)·ˆρ·/parenleftbiggcosθ\nsinθ/parenrightbigg\n,\nρT= (cosθ,sinθ)·ˆρ·/parenleftbigg−sinθ\ncosθ/parenrightbigg\n.(2)\nWe first derive expressions for the non-crystalline AMR component s [16, 19] which depend only on the angle ψ−θ\nbetween the current (Hall bar orientation) and the magnetization vector, and which account for the AMR in isotropic\n(polycrystalline) materials. We expand the elements of ˆ ρin Eq. (1) in series of cosnψand sinnψ, or equivalently\nof cosnψand sinnψ[19]. The form of Eq. (2) implies that corresponding expansions of ρLandρTin series of\ncos(nψ+mθ) and sin(nψ+mθ) contain only terms with m= 0,±2. Among those, the cos2( ψ−θ) and sin2( ψ−θ)\nare the only terms depending on ψ−θ. It explains why the non-crystalline AMR components, which are obt ained by\ntruncating Eq. (1) to\nˆρ=ρav/parenleftbigg\n1 0\n0 1/parenrightbigg\n+2ρavCI/parenleftbigg−1\n2+cos2ψsinψcosψ\nsinψcosψ−1\n2+sin2ψ/parenrightbigg\n, (3)\ntake the simple form ∆ ρL/ρav≡(ρL−ρav)/ρav=CIcos2(ψ−θ) andρT/ρav=CIsin2(ψ−θ) [16, 19]. Here ρavis\nthe average (with respect to ψ) longitudinal resistivity, and CIis the non-crystalline AMR amplitude.\nAll terms in the expansion of Eq. (2) which depend explicitly on the orie ntation of the magnetization vector with\nrespect to the crystallographic axes contribute to the crystalline AMR [16, 19]. Symmetry considerations can be\nused to find the form of ˆ ρ(ψ) in Eq. (1) specific to a particular crystal structure. Explicit expr essions for ˆ ρ(ψ) in\nunperturbed cubic crystals, and cubic crystals with uniaxial strain s along [110] and [100] axes are derived in the\nAppendix. Here we write the final expression for ρLandρTobtained from the particular form of ˆ ρ(ψ) and from\nEq. (2). For the cubic lattice, omitting terms with the periodicity in ψsmaller than 90◦, we obtain,\n∆ρL\nρav=CIcos2(ψ−θ)+CICcos(2ψ+2θ)+CCcos4ψ+... (4)\nρT\nρav=CIsin2(ψ−θ)−CICsin(2ψ+2θ)+... . (5)\nFor the higher order cubic terms see the Appendix.\nAdditional components emerge in ∆ ρL/ρavandρT/ρavfor the uniaxially strained lattice which we denote as ∆uni\nL\nand ∆uni\nT, respectively. Omitting terms with the periodicity in ψsmaller than 180◦we obtain,\n(±)∆uni\nL=Cs\nIUsin2θ+Cs\nUsin2ψ\n(±)∆uni\nT=Cs\nIUcos2θ(6)\nfor strain along the in-plane diagonal directions ( s= [110] corresponds to ”+” and [1 ¯10] to ”-”), and\n(±)∆uni\nL=Cs\nIUcos2θ+Cs\nUcos2ψ\n(±)∆uni\nT=−Cs\nIUsin2θ+Cs\nU,Tsin2ψ(7)\nfor strain along the in-plane cube edges ( s= [100] corresponds to ”+” and [010] to ”-”) For higher order uniax ial\nterms see again the Appendix.3\n3. EXPERIMENTS IN LITHOGRAPHICALLY PATTERNED (GA,MN)AS MI CRODEVICES\nWe now proceed with the discussion of AMR measurements in (Ga,Mn)A s microdevices in which strain effects\nare controlled by lithographically induced lattice relaxation [1, 2, 3]. Op tical micrograph of the first studied device\nis shown in Fig. 1(a). The structure consists of four 1 µm wide Hall bars and one 40 µm wide bar connected in\nseries. The wider bar is aligned along the [010] crystallographic direct ion, the micro-bars are oriented along the [110],\n[110], [100], and [010] axes. The Hall bars are defined by 500 nm wide tre nches patterned by e-beam lithography\nand reactive ion etching in a 25 nm thick Ga 0.95Mn0.05As epilayer, which was grown along the [001] crystal axis on a\nGaAs substrate. The Curie temperature of the as-grown (Ga,Mn) As is 60 K. A compressive strain in the (Ga,Mn)As\nepilayer grown on the GaAs substrate leads to a strong magnetocr ystalline anisotropy which forces the magnetization\nvector to align parallel with the plane of the magnetic epilayer [6, 7]. Th e growth strain is partly relaxed in the\nmicrobars, producing an additional, in-plane uniaxial tensile strain in t he transverse direction [1, 2, 3].\n05\n0\n5/g71/g85 L//g85av [%]\nj||[010]90 /g113[010]\n0/g113/g9245 /g113[110]135 /g113\n315 /g113\n270 /g113225 /g113\n(b)\nFIG. 1: (a) Micrograph of the first studied device with four mi croscopic bars in series (the macroscopic Hall bar is not sho wn).\nInset: enlarged view of one of the microchannels; black area s are the isolating trenches defining the channel. (b) Polar p lot of\nthe percentage change in resistivity (AMR) as a function of t he angle between the applied field and the [100] direction for the\nmicro- and macroscopic bars aligned along the [010] axis. St rain relaxation due to patterning leads to a reduction of the AMR\nmagnitude of about 30%. For better clarity we plot, instead o f ∆ρL/ρavdefined in the Section 2, δρL/ρav≡(ρL−ρL,min)/ρav.\nHereρL,minis the minimum (with respect to ψ) longitudinal resistivity.\nMagnetoresistance traces were measured with the saturation ma gnetic field applied in the plane of the device, i.e.,\nin the pure AMR geometry with zero (antisymmetric) Hall signal and w ith magnetization vector aligned with the\nexternal magnetic field. The sample was rotated by 360◦with 5◦steps. Longitudinal resistances of all five Hall-bars4\nwere measured simultaneously with lock-in amplifiers.\nIn Fig. 1(b) we show AMR data from magnetization rotation experime nts in the 40 µm and 1µm wide bars aligned\nalong the [010] direction. Both curves have a minimum for magnetizat ion oriented parallel to the Hall bar axis and\na maximum when magnetization is rotated by 90◦. Although this is a typical characteristic of the non-crystalline\nAMR term in (Ga,Mn)As the large difference between the AMR magnitud es in the two devices points to a strong\ncontribution of the crystalline AMR coefficient C[010]\nUin Eq. (7), originating from the strain induced by transverse\nlattice relaxation in the microbar. We find that the magnitude of the c oefficient,C[010]\nU= 0.77, amounts to about\n30% of the magnitude of the total AMR in the unrelaxed macroscopic bar.\nIn Figs. 2(a) and (b) we plot AMR traces for microbars patterned a long the [100]/[010] and [110]/[1 ¯10] crystallo-\ngraphicdirections. Strikingly, the overallmagnitudeofthe AMR tra cesforthe [110]/[1 ¯10]orientedHall barsis about a\nfactor of 3 smaller than for the [100]/[010]bars and appears to hav e a much stronger relative contribution of the cubic\ncrystalline term (the term proportional to Ccin Eq. (4)). However, by extracting the 90◦-periodic AMR components\nfor all microbars, as well as for the macroscopic Hall bar, we find a c onsistent value of Cc=−0.17±0.01%. This\nimplies that it is rather a suppression (enhancement) of the uniaxial AMR components for the [110]/[1 ¯10] ([100]/[010])\noriented bars which accounts for the difference in AMR traces in Figs . 2(a) and (b). Since θ=n×45◦for the Hall bars\nstudied in Figs. 1 and 2 and the lattice relaxation induced strains in the se microbars are in the transverse direction\nwe can rewrite the ψ-dependent uniaxial terms for the longitudinal AMR in Eqs. (6,7) in a c ompact form,\n∆uni\nL=Cs\nUcos2(ψ−θ). (8)\nThis expression, together with Eq. (4), implies that the amplitude of the total uniaxial (180◦-periodic) contribution\nto the AMR in the [110]/[1 ¯10] devices ( |CI+C[100]/[010]\nU +CIC|) can indeed differ from that of the [100]/[010] devices\n(|CI+C[110]/[1¯10]\nU −CIC|), provided that CICis non-zero and/or C[100]/[010]\nU /ne}ationslash=C[110]/[1¯10]\nU .\nAnother observationwe make is a broken [100]-[010]symmetry betw een the two AMR tracesin Fig. 2(a) and in each\nof the two traces in Fig. 2(b). While in the former case this behavior c an be captured by Eq. (8) taking C[100]\nU/ne}ationslash=C[010]\nU,\nthe shape of the AMR curves in Fig. 2(b) is inconsistent with the form of Eq. (8). We have attempted to model\nthe broken [100]-[010] symmetry by introducing a contribution to th eC[100]\nUcoefficient which is independent of the\nmicrobar orientation, i.e., assuming that its origin is distinct from transverse strains induced by the micropatterning.\nFrom the difference between the two AMR curves in Fig. 2(a) and fro m the [1¯10]-bar AMR in Fig. 2(b) we obtained\nthat this contribution is 0.3%, and from the [110]-bar AMR we obtained 0.1%. A bar independent contribution to\nC[100]\nUtherefore explains only part of the observed [100]-[010] broken sy mmetry effects; we attribute the remaining\npart to possible material inhomogeneities or non-uniformities and mis alignments in the micropatterning.\nImportantly, the above experimental uncertainties have no effec t on the main conclusion of our experiments that\nthe lattice relaxationinduced uniaxial AMR coefficient is larger than th e cubic crystalline component and a significant\nfractionofthetotalAMRoftheunpatternedmaterial. Bynormaliz ingthevalueofthetransversestrain-induced C[010]\nU\ncoefficient and the CCcoefficient to the respective values at 4 K, we can also compare their temperature dependencies\nwithin the measured range of temperatures of 4 to 70 K. Clearly the CCcoefficient decreases more rapidly with\nincreasing temperature. This recalls the behavior of the magnetoc rystalline anisotropy terms in magnetization, where\nthe uniaxial term decreases in a less pronounced way than the cubic one, since the former scales roughly with M2,\nwhile the latter with M4. As a result of this, the transverse strain-induced term becomes more dominant at higher\ntemperatures, changing from 31% of the total AMR at 4 K, to 38% a t 70 K (not shown). (Note that non-zero AMR\nis still observable at 70 K which is above the Tcof the as-grown material. This is presumably because of the increas e\nin the Curie temperature due to partial annealing during the device f abrication processes.)\nA detail analysis of the longitudinal resistance measurements in the microbars allows us to identify higher order\ncubic terms (see Eq. (12) in the Appendix). By subtracting the 2nd and 4th order terms from AMR data measured\non the [010] microbar we find a clear signature of an 8th order (45◦-periodic) cubic term with an amplitude of 0.04%,\nas shown in Figure 2(d). In Sections 4 we give another example of the unusual high order AMR terms (and explain\nin more detail how these are extracted from the data) which emerg e from post-growth induced lattice distortion\nexperiments.\nMeasurements in the Hall bars discussed above demonstrate that (sub)micrometer lithography of (Ga,Mn)As ma-\nterials grown under lattice matching strains inevitably produces str ain relaxation which may be large enough to\nsignificantly modify magnetotransport characteristics of the str ucture. Lateral micro and nanoconstrictions, utilized\nin magnetotransport studies of non-uniformly magnetized system s or as pinning centers for domain wall dynamics\nstudies, are an important class of devices for which these effects a re highly relevant. In Fig. 3 we show data measured\nin devices consisting of two 4 µm wide bars patterned from the same (Ga,Mn)As wafer as above alon g the [110]5\n0.0 1.5 \n0.0 \n1.5 90 /g113\n0/g11345 /g113135 /g113\n270 /g113225 /g113j|| [110] \nj|| [1-10] (b)\n0.0 1.5 3.0 4.5 \n0.0 \n1.5 \n3.0 \n4.5 /g71/g85 L//g85av [%] 90 /g113[010] \n0/g113[100] 45 /g113[110] 135 /g113\n270 /g113225 /g113j|| [100] \nj|| [010] (a)\n/g92\n0.00 0.04 \n0.00 \n0.04 90 /g113[010] \n0/g113[100] 45 /g113[110] 135 /g113\n270 /g113225 /g113j|| [010] (d)\n0 10 20 30 40 50 60 70 0.0 0.2 0.4 0.6 0.8 1.0 \nC0(4 K): \nCu[010] = 0.77% \nCu = -0.3% \nCc= -0.17% Cc\nCu[010] \nT [K] C/ C 0(c)\n/g92/g71/g85 L//g85av [%] /g71/g85 L//g85av [%] /g92\nFIG. 2: (a),(b) AMR curves for the microscopic bars aligned a long the in-plane cubic and diagonal directions respective ly.\nThe magnitude of the AMR is about a factor of 3 smaller in the mi crobars along the diagonal directions. (c) Temperature\ndependence of the crystalline AMR coefficients normalized to the respective values C0at 4 K (shown in the inset). (d) Polar\nplot showing the 8th order term found in the AMR of the microsc opic bars, with a magnitude of 0.04%.\n(or [100]) crystallographic direction and connected by a 150 nm wide a nd 500 nm long constriction. Magnetic field\nsweep experiments at a fixed field angle, plotted in panel (b), illustra te a marked increase in the constriction of the\nanisotropy field along the [100] bar direction at which magnetization r otates from saturation field orientations towards\nthe easy [100]-axis. Because of the dilute moment nature of the (Ga ,Mn)As ferromagnet, shape anisotropy plays only\na minor role here and the effect is ascribed to strain relaxation and co rresponding changes in the magnetocrystalline\nanisotropy in the constriction.\nAMR measurements in rotating B= 4 T field shown in Figs. 3(c) and (d) provide further indication of the\npresence of strong strain relaxation induced magnetocrystalline e ffects in devices with narrow constrictions. The\ncomparison between AMRs of the wider contacts and of the constr iction shows very similar phenomenology to that of\nthe macroscopicand strain-relaxedmicroscopic Hall bars discusse d in the first part of this section (compare Fig. 2 and\nFigs. 3(c) and (d). We again identify the uniaxial crystalline AMR term in the constriction due to microfabrication\nwhich is of the same sign and similar magnitude as observed in the micro H all bars. Consistency is also found when\ncomparing the character of the AMR curves for the micro Hall-bars and for the constriction devices patterned along\ndifferent crystallographic directions (see Fig. 2 and Figs. 3(c) and ( d)).6\n024\n0\n2\n402\n0\n2\n0/g113\n[100] 45 /g113[110] 90 /g113[010] \n45 /g11390 /g113\n0/g113/g71R [%] /g92j|| [1-10] \nRc \nR_4 /g80m2/g80m(a)\n135 /g113\n270 /g113315 /g113135 /g113246\n[100] [010] \n-200 -100 0 100 200 246 [010] \n[100] \nj|| [100] (b)\n/g71R [%] \nB [mT] R_4 /g80m\nRc \n/g92(c) (d)/g71R [%] \nFIG. 3: (a) Scanning electron micrographs of a 4 µm wide bar containing a 150 nm wide and 500 nm long constrictio n patterned\nfrom the same wafer material as in Figs. 1 and 2. (b) Resistanc e variations during in-plane magnetic field sweeps from nega tive\nto positive saturation fields applied along [100] (black) an d [010] (blue) directions, measured in the constriction dev ice patterned\nalong the [100] crystallographic axis. (c,d) AMR measureme nts in the wider contact (black) and across the constriction (red) in\na rotating saturation field of 4 T for devices patterned along the [100] direction (c) and along the [110] direction (d). Pe rcentage\nchange in resitances rather than resistivities are plotted for this non-uniform geometry device; the distinction is no t relevant\nfor the discussion of the relative changes in the longitudin al magnetoresistance.\n4. EXPERIMENTS IN (GA,MN)AS/PIEZO-TRANSDUCER HYBRID STRU CTURES\nLithographic patterning of micro and nanostructures in (Ga,Mn)As provides powerful means for engineering the\ncrystalline AMR components. In this section we show that further, dynamical control of these effects is achieved in\nhybrid piezoelectric/(Ga,Mn)As structures. A25 nm thick Ga 0.94Mn0.06As epilayerutilized in the study was grownby\nlow-temperature molecular-beam-epitaxy on GaAs substrate and buffer layers [4]. A macroscopic Hall bar, fabricated\nin the (Ga,Mn)As wafer by optical lithography, and orientated along the [1¯10] direction, was bonded to the PZT\npiezo-transducer using a two-component epoxy after thinning th e substrate to 150 ±10µm by chemical etching. The\nstressor was slightly misaligned so that a positive/negative voltage p roduces a uniaxial tensile/compressive strain at\n≈ −10◦to the [1 ¯10] direction. The induced strain was measured by strain gauges, a ligned along the [1 ¯10] and [110]7\ndirections, mounted on a second piece of 150 ±10µm thick wafer bonded to the piezo-stressor. Differential thermal\ncontraction of GaAs and PZT on cooling to 50 K produces a measured biaxial, in- plane tensile strain at zero bias\nof 10−3and a uniaxial strain estimated to be of the order of ∼10−4[20]. At 50 K, the magnitude of the additional\nstrain for a piezo voltage of ±150 V is approximately 2 ×10−4.\n012\n045 90 \n135 \n180 \n225 \n270 315 0\n1\n2[010] \n[110] \n[11 0] [100] \n/g73 (degrees) /g71R (%)  +150V \n -150V \n 0V \n0.0 0.1 0.2 \n045 90 \n135 \n180 \n225 \n270 315 0.0 \n0.1 \n0.2 [11 0] [010] \n[110] \n[100] \n/g73 (degree /g71R (%) /g71Rxx [+150V]- /g71Rxx [- \n/g71Rxy [+150V]- /g71Rxy [-\n0.00 0.05 0.10 \n045 90 \n135 \n180 \n225 \n270 315 0.00 \n0.05 \n0.10 /g73 (degrees) \n[11 0] [100] [110] [010] /g71R (%)  +150V \n -150V \n 0V /g71/g85 T//g85av [%] longitudinal \ntransversal \n/g92/g92/g71/g85 L/g18/g85 av ,/g71/g85 T//g85av [%] \n/g92(a) (b)\n(c)\n/g71/g85/g18/g85 av (+150V) - /g71/g85 //g85av (-150V) [%] \nFIG. 4: (a) The longitudinal (solid curves) and the transver se (dashed curves) AMRs for piezo voltages ±150V. (b) The\ndifferences between longitudinal and transverse AMRs for pi ezo voltages ±150V. (c) Fourth order components of the transverse\nAMR at piezo voltages ±150V and 0V (2nd order components were subtracted as describ ed in the text). In all cases T=50 K\nand the field of 1 T was rotated in the plane of the (Ga,Mn)As lay er. As in Fig.s 1-3 we plot better clarity δρL/ρav≡\n(ρL−ρL,min)/ρavandδρT/ρav≡(ρT−ρT,min)/ρav. HereρL(T),minis the minimum(with respect to ψ)longitudinal(transverse)\nresistivity.\nPrevious measurements [4] of the device identified large changes in t he magnetic easy axis orientation induced by\nthe piezoelectric stressor. Here we focus on the effects of the st ressor on the magnetotransport coefficients. The AMR\nmeasured at 50 K for ±150 V on the transducer is shown in Fig. 4(a). The modification of the AMR induced by\nthe strain can be extracted by subtracting curves at ±150 V (see Fig. 4(b)). It is expected that only the crystalline\nterms are modified; indeed the modification in the longitudinal resistiv ityρLis due to the second and fourth order\ncrystalline AMR terms. This is consistent with our previous analysis on unstressed Hall bars where we found that\nthere were second and fourth order crystalline terms represent ing approximately 10% of the total AMR. There is also\na modification of ρTof similar magnitude. This is predominantly due to the fourth order te rm.\nTo extract the absolute value of the fourth order term in ρTat each voltage we have performed the following\nanalysis: Starting with the raw ρTdata we subtract any offset due to mixing of ρLinto theρTsignal which may8\noccur as a consequence of small inaccuracies in the Hall bar geomet ry or small inhomogeneity in the wafer. This is\na correction of approx 0.4% of the ρLsignal which should have no significant effect on the subsequent ana lysis of\nfourth order terms. (The fourth order components in ρLare typically 0.1%, so the effect on ρTwould be 0.4% ×0.1%\n= 0.0004%, i.e., negligibly small.) We then remove any unintentional antisymmetric (Ha ll) component from ρTby\nshifting the data by 180◦and averaging. The second order terms are subsequently remove d fromρTby shifting the\ndata by 90◦and averaging. The result of this procedure is plotted in Fig. 4(c).\nAt 0 V the fourth order component is approximately 0.03% (peak to t rough). At +150V it is further enhanced\nto approximately 0.1% while at -150V the magnitude is reduced to appr oximately 0.01% which is a value similar to\nthe fourth order term observed after carefully reexamining a (Ga ,Mn)As wafer without the piezo-stressor attached to\nit [16]. For the present device, measurements of the magnetic aniso tropy indicate that the application of -150V to\nthe piezo transducer counteracts the uniaxial strain induced by d ifferential thermal contraction on cooling to return\nthe device close to the unstrained state [4]. The presence of a fou rth order term in the transverse AMR is allowed\nunder a uniaxial distortion, see Eq. (16), but is not expected if only cubic symmetry is present. The data presented in\nFig. 4(c) clearly demonstrates that the uniaxial strain produced b y the piezo transducer induces a significant fourth\norder term in the transverse AMR, which is usually considered to be o f insignificant magnitude in the unstrained\nwafer. The analysis demonstrates that by applying voltage on the p iezoelectric transducer one can significantly\nenhance crystalline AMR components, as compared to the bare (Ga ,Mn)As wafer, as well as efficiently compensate\nadditional strain effects induced by, e.g., different thermal expans ion coefficients in hybrid multilayer structures.\n5. CONCLUSIONS\nWe have demonstrated that beside the previously observed effect s on magnetic anisotropies, post-growth strain\nengineering can be also used to manipulate efficiently the AMR of (Ga,Mn )As. Since magnetic anisotropy is a\nproperty of the total energy of the system while AMR reflects qua siparticle scattering rate characteristics [16] there\nis no straightforward link between the two observations. Experime nts and phenomenological analysis of the data\nhave been presented for two distinct approaches to post-growt h strain control: We used the transverse in-plane\nrelaxation of the GaAs/(Ga,Mn)As lattice mismatch strain in lithograp hically patterned narrow Hall bars, and a\ndynamically controlled strain was induced using a piezo-transducer. Our main results include the observation of\nAMR changes due to strain which can be comparable in magnitude to th e strongest, non-crystalline AMR component\nin bare (Ga,Mn)As, and we have also reported previously undetecte d high-order crystalline AMR terms. Finally\nwe have demonstrated that strain-induced effects can play an impo rtant role in magnetoresistance characteristics of\n(Ga,Mn)As nanoconstrictions.\nAcknowledgements\nWe acknowledge support from EU Grant IST-015728, from UK Gran t GR/S81407/01, from Grant Agency and\nAcademy of Sciences and Ministry of Education of the Czech Republic Grants FON/06/E002, AV0Z10100521,\nKJB100100802, KAN400100652, and LC510.\nAppendix – derivation of phenomenological AMR expressions\nTo derive the appropriate AMR expansions for cubic and uniaxially dist orted crystals we consider the resistivity\ntensor in Eq. (1) with the two Hall bars and magnetization vector fix ed in space and perform the relevant symmetry\noperations to the underlying crystal. (Note that the values of ψandθmay change under the effect of the symmetry\noperations since the angles are defined with respect to the crysta llographic directions.) The relevant operations for\nthe cubic crystal are summarized in Tab. I; the last operation, the invariance under ψ→90◦−ψassuming the Hall\nbars and the crystal fixed, is derived from the microscopic theore tical expression for the AMR [18]. The general form\nof Eq. (1) constrained by these cubic symmetry considerations re ads:\nˆρ= ˆρcub=/parenleftBigg\nu(cos2ψ) cos ψsinψv(cos2ψ)+v(sin2ψ)\n2\ncosψsinψv(sin2ψ)+v(cos2ψ)\n2u(sin2ψ)/parenrightBigg\n. (9)9\nsymmetry operation implied conditions on ˆ ρ\nsymmetry along [010] ρ11(cosψ,sinψ) =ρ11(−cosψ,sinψ)\nsymmetry along [110] ρ11(cosψ,sinψ) =ρ22(sinψ,cosψ)\nρ12(cosψ,sinψ) =ρ21(sinψ,cosψ)\nsymmetry along [1 ¯10] ρ11(cosψ,sinψ) =ρ22(−sinψ,−cosψ)\nrotation by 90◦ρ12(cosψ,sinψ) =−ρ21(−sinψ,cosψ)\ninvariance under ψ→90◦−ψ ρ12(cosψ,sinψ) =ρ12(sinψ,cosψ)\n(fixed crystal)\nTABLE I: Symmetry operations used for a cubic crystal.\nsymmetry operation implied conditions on ˆ ρ\nsymmetry along [110] ρ11(cosψ,sinψ) =ρ22(sinψ,cosψ)\nρ12(cosψ,sinψ) =ρ21(sinψ,cosψ)\nsymmetry along [1 ¯10] ρ11(cosψ,sinψ) =ρ22(−sinψ,−cosψ)\nρ12(cosψ,sinψ) =ρ21(−sinψ,−cosψ)\ninvariance under ψ→90◦−ψ ρ12(cosψ,sinψ) =ρ12(sinψ,cosψ)\n(fixed crystal)\nTABLE II: Symmetry operations used for cubic crystal uniaxi ally strained along [110].\nFunctionsuandvcan be expanded in Taylor series of cosnψas done in the original work by D¨ oring [21] or,\nequivalently, in series of cos nψ. For example for uin Eq. (9) we obtain,\nu(cos2ψ) =a0+a2cos2ψ+a4cos4ψ+... (10)\nand\nu(sin2ψ) =a0−a2cos2ψ+a4cos4ψ−... . (11)\nEqs.(9)-(11) togetherwithEq.(2)yield, aftertransformingall productsofgoniometricfunctionsandrecollectingthem\ninto sines and cosines of sums of angles, the following structure of t he longitudinal and transverse AMR expressions:\n∆ρL\nρav=CCcos4ψ+CC8cos8ψ+... (12)\n+CIcos(2ψ−2θ)+CICcos(2ψ+2θ)+\n+CI6cos(6ψ−2θ)+CIC6cos(6ψ+2θ)+\n...,\nand\nρT\nρav= (13)\n+CIsin(2ψ−2θ)−CICsin(2ψ+2θ)+\n+CI6sin(6ψ−2θ)−CIC6sin(6ψ+2θ)+\n...\nEq. (4,5) in Section 2 are obtained by keeping all terms in (12,13) up to 4ψ. Note that there is a simple relation-\nship between the longitudinal and transverse AMRs, ρT/ρav=−1\n2(∂(∆ρL/ρav)/∂θ), which is a consequence of the\nsymmetry (ˆ ρ)ij= (ˆρ)jiin Eq. (9).\nAnalogous procedure can be applied to cubic crystals with uniaxial st rain along the [110]-direction; corresponding\nsymmetry operations are listed in Tab. II and ˆ ρin this case reads,\nˆρ= ˆρcub+/parenleftbigg\nt(cos2ψ)cosψsinψ1\n2[w(cos2ψ)+w(sin2ψ)]\n1\n2[w(cos2ψ)+w(sin2ψ)]t(sin2ψ)cosψsinψ/parenrightbigg\n. (14)10\nsymmetry operation implied conditions on ˆ ρ\nsymmetry along [100] ρ11(cosψ,sinψ) =ρ11(cosψ,−sinψ)\nρ22(cosψ,sinψ) =ρ22(cosψ,−sinψ)\nρ12(cosψ,sinψ) =−ρ12(cosψ,−sinψ)\nρ21(cosψ,sinψ) =−ρ21(cosψ,−sinψ)\nsymmetry along [010] ρ11(cosψ,sinψ) =ρ11(−cosψ,sinψ)\nρ22(cosψ,sinψ) =ρ22(−cosψ,sinψ)\nρ12(cosψ,sinψ) =−ρ12(−cosψ,sinψ)\nρ21(cosψ,sinψ) =−ρ21(−cosψ,sinψ)\nTABLE III: Symmetry operations used for a cubic crystal unia xially strained along [100].\nEq. (14) yields the following uniaxial AMR terms,\n∆ρL\nρav=C[110]\nUsin2ψ+C[110]\nU6sin6ψ+C[110]\nU10sin10ψ+... (15)\n+C[110]\nIUsin2θ+\n+C[110]\nIU4+sin(4ψ−2θ)+C[110]\nIU4−sin(4ψ+2θ)+\n+C[110]\nIU8+sin(8ψ−2θ)+C[110]\nIU8−sin(8ψ+2θ)+\n...\nand\nρT\nρav= (16)\n+C[110]\nIUcos2θ+\n−C[110]\nIU4+cos(4ψ−2θ)+C[110]\nIU4−cos(4ψ+2θ)+\n−C[110]\nIU8+cos(8ψ−2θ)+C[110]\nIU8−cos(8ψ+2θ)+\n....\nThe terms which contain at most 2 ψreproduce Eq. (6).\nCubic crystal with uniaxial strain along [100]-axis are described by (s ee Tab. III),\nˆρ=/parenleftbiggu(cos2ψ)+∆u(cos2ψ) sinψcosψ[v(cos2ψ)+∆v(cos2ψ)]\nsinψcosψ[v(sin2ψ)−∆v(sin2ψ)]u(sin2ψ)−∆u(sin2ψ)/parenrightbigg\n. (17)\nNote that (ˆ ρ)ij/ne}ationslash= (ˆρ)jiin this case. Eq. (17) yields the following uniaxial AMR terms,\n∆ρL\nρav=C[100]\nUcos2ψ+C[100]\nU6cos6ψ+C[100]\nU10cos10ψ+... (18)\n+C[100]\nIUcos2θ+\n+C[100]\nIU4+cos(4ψ−2θ)+C[100]\nIU4−cos(4ψ+2θ)+\n+C[100]\nIU8+cos(8ψ−2θ)+C[100]\nIU8−cos(8ψ+2θ)+\n... ,\nand\nρT\nρav= +C[100]\nU,Tsin2ψ+C[100]\nU4,Tsin4ψ+C[100]\nU6,Tsin6ψ+... (19)\n−C[100]\nIUsin2θ+\n+C[100]\nIU4+sin(4ψ−2θ)−C[100]\nIU4−sin(4ψ+2θ)+\n+C[100]\nIU8+sin(8ψ−2θ)−C[100]\nIU8−sin(8ψ+2θ)+\n... .11\nAgain the lowest order terms reproduce Eq. (7).\n[1] S. H¨ umpfner, M. Sawicki, K. Pappert, J. Wenisch, K. Brun ner, C. Gould, G. Schmidt, T. Dietl, and L. W. Molenkamp,\nAppl. Phys. Lett. 90, 102102 (2007), arXiv:cond-mat/0612439.\n[2] J. Wunderlich, A. C. Irvine, J. Zemen, V. Holy, A. W. Rushf orth, E. D. Ranieri, U. Rana, K. Vyborny, J. Sinova, C. T.\nFoxon, et al., Phys. Rev. B 76, 054424 (2007), arXiv:0707.3329.\n[3] J. Wenisch, C. Gould, L. Ebel, J. Storz, K. Pappert, M. J. S chmidt, C. Kumpf, G. Schmidt, K. Brunner, and L. W.\nMolenkamp, Phys. Rev. Lett. 99, 077201 (2007), arXiv:cond-mat/0701479.\n[4] A. W. Rushforth, E. D. Ranieri, J. Zemen, J. Wunderlich, K . W. Edmonds, C. S. King, E. Ahmad, R. P. Campion, C. T.\nFoxon, B. L. Gallagher, et al. (2008), arXiv:0801.0886.\n[5] M. Overby, A. Chernyshov, L. P. Rokhinson, X. Liu, and J. K . 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Goennenwein, S. Russo, A. F. Morpurgo, T. M. Klap wijk, W. Van Roy, and J. De Boeck, Phys. Rev. B 71,\n193306 (2005), arXiv:cond-mat/0412290.\n[14] K. Y. Wang, K. W. Edmonds, R. P. Campion, L. X. Zhao, C. T. F oxon, and B. L. Gallagher, Phys. Rev. B 72, 085201\n(2005), arXiv:cond-mat/0506250.\n[15] W. Limmer, M. Glunk, J. Daeubler, T. Hummel, W. Schoch, R . Sauer, C. Bihler, H. Huebl, M. S. Brandt, and S. T. B.\nGoennenwein, Phys. Rev. B 74, 205205 (2006), arXiv:cond-mat/0607679.\n[16] A. W. Rushforth, K. V´ yborn´ y, C. S. King, K. W. Edmonds, R. P. Campion, C. T. Foxon, J. Wunderlich, A. C. Irvine,\nP. Vaˇ sek, V. Nov´ ak, et al., Phys. Rev. Lett. 99, 147207 (2007), arXiv:cond-mat/0702357.\n[17] A. W. Rushforth, K. V´ yborn´ y, C. S. King, K. W. Edmonds, R. P. Campion, C. T. Foxon, J. Wunderlich, A. C. Irvine,\nV. Nov´ ak, K. Olejn´ ık, et al. (2007), arXiv:0712.2581.\n[18] T. Jungwirth, M. Abolfath, J. Sinova, J. Kuˇ cera, and A. H. MacDonald, Appl. Phys. Lett. 81, 4029 (2002), arXiv:cond-\nmat/0206416.\n[19] T. McGuire and R. Potter, IEEE Trans. Magn. 11, 1018 (1975).\n[20] B. Habib, J. Shabani, E. P. D. Poortere, M. Shayegan, and R. Winkler, Appl. Phys. Lett. 91, 012107 (2007),\narXiv:0706.0736.\n[21] W. D¨ oring, Ann. Phys. (Leipzig) 424, 259 (1938)."    },    {        "title": "0803.0293v1.X_ray_absorption_and_x_ray_magnetic_dichroism_study_on_Ca3CoRhO6_and_Ca3FeRhO6.pdf",        "content": "arXiv:0803.0293v1  [cond-mat.str-el]  3 Mar 2008X-ray absorption and x-ray magnetic dichroism study on Ca 3CoRhO 6and Ca 3FeRhO 6\nT. Burnus,1Z. Hu,1Hua Wu,1J. C. Cezar,2S. Niitaka,3,4H. Takagi,3,4,5C. F. Chang,1N. B.\nBrookes,2H.-J. Lin,6L. Y. Jang,6A. Tanaka,7K. S. Liang,6C. T. Chen,6and L. H. Tjeng1\n1II. Physikalisches Institut, Universit¨ at zu K¨ oln, Z¨ ulp icher Str. 77, 50937 K¨ oln, Germany\n2European Synchrotron Radiation Facility, BP 220, 38043, Gr enoble, France\n3RIKEN, Institute of Physical and Chemical Research, 2-1, Hi rosawa, Wako, Saitama 351-0198, Japan\n4CREST, Japan Science and Technology Agency (JST), Kawaguch i, Saitama 332-0012, Japan\n5Department of Advanced Materials Science, University of To kyo,\n5-1-5, Kashiwanoha, Kashiwa, Chiba 277-8581, Japan\n6National Synchrotron Radiation Research Center, 101 Hsin- Ann Road, Hsinchu 30077, Taiwan\n7Department of Quantum Matter, ADSM, Hiroshima University, Higashi-Hiroshima 739-8530, Japan\n(Dated: October 22, 2018)\nUsing x-ray absorption spectroscopy at the Rh- L2,3, Co-L2,3, and Fe- L2,3edges, we find a valence\nstate of Co2+/Rh4+in Ca 3CoRhO 6and of Fe3+/Rh3+in Ca 3FeRhO 6. X-ray magnetic circular\ndichroism spectroscopy at the Co- L2,3edge of Ca 3CoRhO 6reveals a giant orbital moment of about\n1.7µB, which can be attributed to the occupation of the minority-s pind0d2orbital state of the high-\nspin Co2+(3d7) ions in trigonal prismatic coordination. This active role of the spin-orbit coupling\nexplains the strong magnetocrystalline anisotropy and Isi ng-like magnetism of Ca 3CoRhO 6.\nPACS numbers: 78.70.Dm, 71.27.+a, 71.70.-d, 75.25.+z\nI. INTRODUCTION\nThe quasi one-dimensional transition-metal oxides\nCa3ABO6(A= Fe, Co, Ni, ...; B= Co, Rh,\nIr, ...) have attracted a lot of interest in recent\nyears because of their unique electronic and mag-\nnetic properties.1,2,3,4,5,6,7,8,9,10,11,12,13The structure of\nCa3ABO6contains one-dimensional (1D) chains consist-\ning of alternating face-sharing AO6trigonal prisms and\nBO6octahedra. Each chain is surrounded by six paral-\nlel neighboring chains forming a triangular lattice in the\nbasal plane. Peculiar magnetic and electronic behaviors\nare expected to be related to geometric frustration in\nsuch a triangle lattice with antiferromagnetic (AFM) in-\nterchaininteractionandIsing-likeferromagnetic(FM)in-\ntrachain coupling. Ca 3Co2O6, which realizes such a situ-\nation, showsstair-stepjumpsinthemagnetizationatreg-\nular intervals of the applied magnetic field of Ms/3, sug-\ngesting ferrimagnetic spin alignment. It has a saturation\nmagnetization of Ms= 4.8µBper formula unit at around\n4 T.14Studies on the temperature and magnetic-field de-\npendence of the characteristic spin-relaxation time sug-\ngest quantum tunneling of the magnetization similar to\nsingle-molecular magnets.15An applied magnetic field\ninduces a large negative magnetoresistance, apparently\nnotrelatedtothethree-dimensionalmagneticordering.11\nBand-structure calculations using the local-spin-density\napproximation plus Hubbard U (LSDA+U) predicted\nthat the Co3+ion at the trigonal site, being in the high-\nspin (HS) state ( S= 2), has a giant orbital moment of\n1.57µBdue to the occupation of minority-spin d2orbital,\nwhiletheCo3+ionattheoctahedralsiteisinthelow-spin\n(LS) state ( S= 0).16An x-ray absorption and magnetic\ncircular dichroism study at the Co- L2,3edge has con-\nfirmed this prediction.17Both studies explain well the\nIsing nature of the magnetism of Ca 3Co2O6.Ca3CoRhO 6and Ca 3FeRhO 6have the same crys-\ntal structure as Ca 3Co2O6, but different magnetic and\nelectronic properties: Neutron diffraction and mag-\nnetization measurements also indicated intrachain-FM\nand interchain-AFM interactions in Ca 3CoRhO 6like\nin Ca 3Co2O6.7In contrast, susceptibility data on\nCa3FeRhO 6reveal a single transition into a three-\ndimensional AFM.5,18Although Ca 3CoRhO 6has a sim-\nilar magnetic structure as Ca 3Co2O6, it exhibits consid-\nerable differences in the characteristic temperatures in\nthe magnetic susceptibility. The high-temperature limit\nof the magnetic susceptibility shows a Curie-Weiss be-\nhavior with a positive Weiss temperature of 150 K for\nCa3CoRhO 6,5while 30 K was found for Ca 3Co2O6.2,3\nThe measured magnetic susceptibility undergoes two\ntransitions at Tc1= 90 K and Tc2= 25 K for\nCa3CoRhO 6, and at Tc1= 24 K and Tc2= 12 K\nfor Ca 3Co2O6,3,5,7,8,12,18which were attributed to FM-\nintrachain and AFM-interchain coupling, respectively.\nIn contrast, Ca 3FeRhO 6has an AFM ordering below\nTN= 12 K.5,18,19Unlike Ca 3Co2O6, there is only one\nplateau at 4 T and no saturation even at 18 T in the\nmagnetization of Ca 3CoRhO 6at 70 K.7A partially dis-\nordered state in Ca 3CoRhO 6has been inferred by the\nprevious work of Niitaka et al.8\nIn order to understand the contrasting magnetic prop-\nerties of Ca 3CoRhO 6and Ca 3FeRhO 6, and, partic-\nularly, the type and origin of the intrachain mag-\nnetic coupling of these quasi 1D systems, the va-\nlence, spin, and orbital states have to be clarified.\nHowever, these issues have been contradictorily dis-\ncussed in previous theoretical and experimental stud-\nies. The general-gradient-approximated (GGA) density-\nfunctional band calculations20suggest a Co3+/Rh3+\nstate in Ca 3CoRhO 6, while LSDA+U calculations with\ninclusion of the spin-orbit coupling favor a Co2+/Rh4+2\nstate and, again, a giant orbital moment due to the\noccupation of minority-spin d0andd2orbitals.21Neu-\ntron diffraction experiments on Ca 3CoRhO 68,22suggest\nthe Co3+/Rh3+state. However, based on the magnetic\nsusceptibility5and x-ray photoemission spectroscopy23\ntheCo2+/Rh4+statewasproposed. ForCa 3FeRhO 6, the\nFe2+/Rh4+state was suggested in a magnetic suscepti-\nbility study,5whereas M¨ ossbauer spectroscopy indicates\na Fe3+state,19and thus Rh3+.\nIn order to settle the above issues, in this work we\nfirst clarify the valence state of the Rh, Co, and Fe ions\nin Ca3CoRhO 6and Ca 3FeRhO 6using x-ray absorption\nspectroscopy (XAS) at the L2,3edges of Rh, Co, and Fe.\nWe reveal a valence state of Co2+/Rh4+in Ca3CoRhO 6\nand of Fe3+/Rh3+in Ca3FeRhO 6. Then, we investigate\nthe orbital occupation and magnetic properties using x-\nray magnetic circular dichroism (XMCD) experiments at\nthe Co-L2,3edge of Ca 3ChRhO 6. We find a minority-\nspind0d2occupation for the HS Co2+ground state and,\nthus, a giant orbital moment of about 1 .7µB. As will\nbe seen below, our results account well for the previous\nexperiments.\nII. EXPERIMENTAL\nPolycrystalline samples were synthesized by a solid-\nstate reaction and characterized by x-ray diffraction to\nbe single phase.5The Rh- L2,3XAS spectra were mea-\nsured at the NSRRC 15B beamline in Taiwan, which is\nequipped with a double-Si(111) crystal monochromator\nfor photon energies above 2 keV. The photon-energy res-\nolution at the Rh- L2,3edge (hν≈3000–3150 eV) was\nset to 0.6 eV. The Fe- L2,3XAS spectrum of Ca 3FeRhO 6\nwas measured at the NSRRC Dragon beamline with a\nphoton-energy resolution of 0.25 eV. The main peak at\n709 eV of the Fe- L3edge of single crystalline Fe 2O3\nwas used for energy calibration. The Co- L2,3XAS and\nXMCD spectra of Ca 3CoRhO 6were recorded at the ID8\nbeamline of ESRF in Grenoble with a photon-energy res-\nolution of 0.2 eV. The sharp peak at 777.8 eV of the\nCo-L3edge of single crystalline CoO was used for energy\ncalibration. TheCo- L2,3XMCDspectrawererecordedin\na magnetic field of 5.5 T; the photons were close to fully\ncircularly polarized. The sample pellets were cleaved in\nsituin order to obtain a clean surface. The pressure was\nbelow 5×10−10mbar during the measurements. All data\nwere recorded in total-electron-yield mode.\nIII. XAS AND VALENCE STATE\nWe first concentrate on the valence of the rhodium\nions in both studied compounds. For 4 dtransition-metal\noxides, the XAS spectrum at the L2,3edge reflects ba-\nsically the unoccupied t2g- andeg-related peaks in the\nOhsymmetry. This is due to the larger band-like char-\nacter and the stronger crystal-field interaction of the 4 d3000 3010Rh-L2Ca3FeRhO6Ca3CoRhO6Intensity\nPhoton Energy (eV)Rh-L3 \nRh4+  4d 5\neg\nt2g\nRh3+ 4d 6\neg\nt2g\n3140 3150 3160\nFIG. 1: The Rh- L2,3XAS spectra of Ca 3CoRhO 6and\nCa3FeRhO 6and a schematic energy level diagram for Rh3+\n4d6and Rh4+4d5configurations in octahedral symmetry.\nstates as well as due to the weaker intra-atomic interac-\ntions as compared with 3 dtransition-metal oxides, where\nintra-atomic multiplet interactions are dominant. The\nintra-atomic multiplet and spin-orbit interactions in 4 d\nelements only modify the relative intensity of the t2g-\nandeg-related peaks. Fig. 1shows the XAS spectra\nat the Rh- L2,3edges of Ca 3FeRhO 6(dashed line) and\nCa3CoRhO 6(solid line). The Rh- L2,3spectrum shows a\nsimple, single-peaked structure at both Rh- L2and Rh-\nL3edges for Ca 3FeRhO 6, while an additional low-energy\nshoulder is observed for Ca 3CoRhO 6. Furthermore, the\npeak in the Ca 3CoRhO 6spectrum is shifted by 0.8 eV to\nhigher energies compared to that of the Ca 3FeRhO 6.\nThe single-peaked spectral structure for Ca 3FeRhO 6\nindicates Rh3+(4d6) with completely filled t2gorbitals,\ni.e. only transitions from the 2 pcore levels to the\negstates are possible. The results are in agreement\nwith M¨ ossbauer spectroscopy.19The shift to higher en-\nergies from Ca 3FeRhO 6to Ca3CoRhO 6reflects the in-\ncrease in the Rh valence from Rh3+to Rh4+as we\ncan learn from previous studies on 4 dtransition-metal\ncompounds.24,25,26,27Furthermore, for Ca 3CoRhO 6the\nspectrum shows a weak low-energy shoulder, which is\nweaker at the Rh- L2edge than at the Rh- L3edge. This\nshouldercan be attributedto transitionsfrom the2 pcore\nlevels to the t2gstate, reflecting a 4 d5configuration with\none hole at the t2gstate. Such spectral features were\nfound earlierforRu3+in Ru(NH 4)3Cl6.24,28Detailed cal-\nculations reveal that the multiplet and spin-orbit inter-\nactions suppress the t2g-related peak at the L2edge for\na 4d5configuration.24,25,26,27Thus, we find a Rh4+(4d5)\nstate for Ca 3CoRhO 6. Having determined a Rh3+state\nin Ca3FeRhO 6and a Rh4+state in Ca 3CoRhO 6, we turn\nto the Fe- L2,3and the Co- L2,3XAS spectra to further\nconfirm the Fe3+state and the Co2+state, as expected\nfor charge balance.3\n700 705 710 715 720 725 730Oh: ∆CF = 0.9 eV\nFeO (experimental)(i)  Fe2+\n(j)  Fe2+(h)  Fe2+(g)  Fe3+(f)  Fe3+(e)  Fe3+(d)  Fe3+(c)  Fe3+(b)  Fe3+\n∆10 = 0.9 eV; Vmix = 0.4 eV∆10 = 0.9 eV; Vmix = 0.4 eVD3h: ∆10 = 0.9 eVspherical ( ∆CF = 0)Oh: ∆CF = 1.0 eV\nCa3FeRhO6 (experimental)Oh: ∆CF = 1.6 eV  \nEnergy (eV)Fe2O3 (experimental)\n(a)  Fe3+\nFIG. 2: (color online) Experimental XAS spectra at the Fe-\nL2,3edge of (a) Fe 2O3(Fe3+), (g) Ca 3FeRhO 6, and (j) FeO\n(Fe2+), taken from Park,29together with simulated spectra\n(b, c) in Oh, (d) spherical, and (e, f) D3hsymmetry for Fe3+\nand simulated spectra in (h) D3hand (i)Ohsymmetry for\nFe2+.\nFigure2shows the experimental Fe- L2,3XAS spectra\nof (g) Ca 3FeRhO 6, along with those of (a) single crys-\ntalline Fe 2O3as a Fe3+reference and of (j) FeO, taken\nfrom Ref. 29, as a Fe2+reference. Additionally, calcu-\nlated spectra for different symmetries using purely ionic\ncrystal-field multiplet calculations24,30,31,32are shown.\nIt is well known that an increase of the valence state\nof the 3dtransition-metal ion by one causes a shift of\nthe XAS L2,3spectra by about one eV towards higher\nenergies.33,34,35The main peak of the Fe L3structure of\nthe Ca 3FeRhO 6lies 0.8 eV above the main peak of the\ndivalent reference FeO and only slightly lower in energy\nthan the one of Fe 2O3(Fe3+). This indicates trivalent\niron ions in Ca 3FeRhO 6. The slightly lower energy shift\nof Ca3FeRhO 6relative to Fe 2O3can be attributed to the\nweak trigonal crystal field in the former as compared an\noctahedral field in the later, as we will show below.\nThe experimental spectra of the reference compounds,\ncurve (a) for Fe 2O3and curve (j) for FeO, can be well un-\nderstood using the multiplet calculations. For Fe 2O3wefind a good simulation taking a Fe3+ion in an octahedral\nsymmetry with a t2g–egsplitting of 1.6 eV, which is de-\npicted in curve(b) in Fig. 2. ForFeO, agoodmatch with\nthe experiment can be found for the Fe2+in an octahe-\ndral environment with a splitting of 0.9 eV, see curve (i).\nThe weaker crystal field in FeO, compared with Fe 2O3,\nis consistent due to its larger Fe–O bond length.\nIn order to understand the experimental Fe L2,3spec-\ntrum of Ca 3FeRhO 6, we first return to the Fe 2O3spec-\ntrum. When we reduce the t2g–egsplitting from 1.6\neV (curve b) via 1.0 eV (curve c) to 0.0 eV (curve\nd), we observe that the the low-energy shoulder be-\ncomes washed out, while the high-energy shoulder be-\ncomes more pronounced.30Going further to a trigonal\ncrystal field, the high-energy shoulder looses its intensity\nasshownin curve(e)forasplitting of0.9eVbetween d±1\n(dyz/dzx) andd0/d±2(d3z2−r2/dxy/dx2−y2). The experi-\nmentalFe- L2,3XASspectrumofCa 3FeRhO 6inFig.2(g)\ncan be well reproduced with this trigonal crystal field of\n0.9 eV and in addition a mixing parameter Vmix= 0.4\neV, which mixes the d±2with the d∓1orbitals; the re-\nsult for this Fe with the 3 d5high-spin configuration is\npresented in curve (f).\nWe note that curve (f) has been generated with the\nFe in the trivalent state. As a check, we have also tried\nto fit the experimental spectrum of Ca 3FeRhO 6using a\ndivalent Fe ansatz. However, the simulation does not\nmatch, as is illustrated in curve (h), in which we have\nused the same trigonal crystal field splitting of 0.9 eV\nand mixing parameterof0.4 eV. To conclude, the Fe- L2,3\nand Rh-L2,3XAS spectra of Ca 3FeRhO 6firmly establish\nthe Fe3+/Rh3+scenario.\nFor the Ca 3CoRhO 6system, the Rh- L2,3XAS spec-\ntra suggest that the Rh ions are tetravalent, implying\nthat the Co ions should be divalent. To confirm this\nCo2+/Rh4+scenario we have to study explicitly the va-\nlence of the Co ion. Fig. 3shows the Co- L2,3XAS spec-\ntra of Ca 3CoRhO 6together with CoO as a Co2+and\nCa3Co2O6as a Co3+reference.17Again we see a shift to\nhigher energies from CoO to Ca 3Co2O6by about one eV.\nThe Ca 3CoRhO 6spectrum lies at the same energy po-\nsition as the CoO spectrum confirming the Co2+/Rh4+\nscenario21and rulingoutthe Co3+/Rh3+scenario.20The\nresult is fully consistent with the above finding from the\nRh-L2,3edge of Ca 3CoRhO 6and in agreement with pre-\nvious results from x-ray photoemission spectroscopy.23\nIV. XMCD AND ORBITAL\nOCCUPATION/MOMENT\nAfter determining the valence states of Rh, Fe, and\nCo ions we turn our attention to the orbital occupation\nand magnetic properties of the Co2+ion at the trigonal-\nprism site. This is motivated by the consideration that\nCo2+ions may have a large orbital moment,36whose size\ndepends ondetails ofthe crystalfield, whilethe high-spin\nFe3+(3d5) and low-spin Rh3+(4d6) ions in Ca 3FeRhO 64\n780 790 800Ca3CoRhO6\ncalculation\n     d↓\n2d↓\n0\n(e)(d)(c)(b)\nCa3CoRhO6\ncalculation\n    d↓\n2d↓\n−2Intensity (arb. units)Ca3Co2O6 \n Co-L2Co-L3\nCa3CoRhO6 (40 K)CoO\nPhoton Energy (eV)(a)\nFIG. 3: The Co- L2,3spectra of (a) Ca 3Co2O6(Co3+), (b)\nCoO (Co2+), and (c) Ca 3CoRhO 6. The simulated spectra\nof high-spin Co2+(3d7) in trigonal prismatic symmetry are\nshown in (d) for a d0d2and in (e) for a d2d−2minority-spin\norbital occupation.\n(a) (b)\nd0d±2d±1\nd3z2−r2dyz,dzx\ndxy,dx2−y2\nd±2d0d±1\n∆10\n−∆02∆10\n∆02d3z2−r2dyz,dzx\ndxy,dx2−y2\nFIG. 4: Scheme of the two possible 3 doccupations for a high-\nspin Co2+ion in trigonal prismatic symmetry, ignoring the\nfive up spins. (a) The d0d2minority-spin occupation allows\nfor a large orbital magnetic moment, whereas (b) for d2d−2\nthe orbital moment vanishes./s55/s55/s53 /s55/s56/s48 /s55/s56/s53 /s55/s57/s48 /s55/s57/s53 /s56/s48/s48/s40/s99/s41/s40/s98/s41/s40/s97/s41\n/s32 /s32/s40 /s100/s32\n/s50/s32/s100/s32\n/s48/s44/s32/s99/s97/s108/s99/s117/s108/s97/s116/s101/s100/s41\n/s32 /s32/s40 /s100/s32\n/s50/s32/s100/s32\n/s50/s44/s32/s99/s97/s108/s99/s117/s108/s97/s116/s101/s100/s41/s32 /s32/s40/s101/s120/s112/s101/s114/s105/s109/s101/s110/s116/s97/s108/s41\n/s32/s67/s111/s45 /s76\n/s50/s67/s111/s45 /s76\n/s51/s67/s97\n/s51/s67/s111/s82/s104/s79\n/s54\n/s53/s48/s32/s75/s44/s32/s53/s46/s53/s32/s84\n/s32\n/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121 \n/s80/s104/s111/s116/s111/s110/s32/s69/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\nFIG. 5: (color online) (a) Measured soft x-ray absorption\nspectra with parallel ( µ+, red dotted curve) and antiparal-\nlel (µ−, black solid curve) alignment between photon spin\nand magnetic field, together with their difference (XMCD)\nspectrum ( µ+−µ−, blue dashed curve); simulated XMCD\nspectra for (b) d0d2(olive curve) and (c) d2d−2(magenta\ncurve) minority-spin occupation of the high-spin Co2+.\nhave a closed subshell without orbital degrees of freedom\nand thus no orbital moment.\nIn trigonal-prism symmetry the 3 dorbitals are split\nintod±1,d0, andd±2states, see Fig. 4. In terms of\none-electron levels, the d±1orbitals lie highest in energy,\nwhile the lower lying d0, andd±2usually are nearly de-\ngenerate. For aCo3+d6system, it is a priorinot obvious\nfrom band structure calculations to say which of these\nlow lying orbitals gets occupied. Details, such as the\ninclusion of the spin-orbit interaction, can become cru-\ncial. Indeed, for Ca 3Co2O6, it was found from LDA+U\ncalculations16and confirmed by XMCD measurements17\nthat the spin-orbit interaction is crucial to stabilize the\noccupation of the d2orbital, thereby giving rise to gi-\nant orbital moments and Ising-type magnetism. For a\nCo2+d7ion, however, the situation is quite different. As\nwe will explain below, the double occupation of the d0d2\norbitals is energetically much more favored than that of\nthed2d−2: the energy difference could be of order 1 eV\nwhile the d0andd±2by themselves could be degenerate\non a one-electron level. The consequences are straight-\nforward: the double occupation of d0d2, see Fig. 4(a),\nshould lead to a large orbital moment of 2 µB(neglect-\ning covalent effects) and Ising type of magnetism with\nthe magnetization direction fixed along the chains.7,21In\ncontrast, the d2d−2, see Fig. 4(b), would have given a\nquenched orbital moment.\nIn order to experimentally establish that the Co2+ion\nhasthed0d2configuration, wehaveperformed anXMCD\nstudy at the Co- L2,3edges of Ca 3CoRhO 6. Fig.5shows\nthe Co-L2,3XMCD spectrum of Ca 3CoRhO 6taken at\n50 K under 5.5 T. The spectra were taken, respectively,\nwith the photon spin parallel ( µ+, red dotted curve) and5\nantiparallel ( µ−, black solid curve) to the magnetic field.\nOne can clearlyobserve largedifferences between the two\nspectra with the different alignments. Their difference,\nµ+−µ−, is the XMCD spectrum (blue dashed curve).\nAnimportantfeatureofXMCDexperimentsisthat there\nare sum rules, developed by Thole and Carra et al.,37,38\nto determine the ratio between the orbital ( morb=Lz)\nand spin ( mspin= 2Sz) contributions to the magnetic\nmoment, namely\nmorb\nmspin=2\n3∆L3+∆L2\n∆L3−2∆L2, (1)\nhere, ∆L3and ∆L2are the energy integrals of the L3\nandL2XMCD intensity. The advantage of these sum\nrules is that one needs not to do any simulations of the\nspectra to obtain the desired quantum numbers. In our\nparticular case, we can immediately recognize the pres-\nence of a large orbital moment in Fig. 5(a), since there is\na large net (negative) integrated XMCD spectral weight.\nUsing Eq. ( 1) we find morb/mspin= 0.63. Since the\nCo2+ion is quite ionic, mspinis very close to the ex-\npected ionic value of 3 µB. For example, our LDA+U\ncalculations yield 2 .72µBfor the Co2+ion (2.64µBfor\nLDA) and Whangbo et al.obtained 2 .71µBfrom GGA\ncalculations.20Using a value of 2 .7µBfor the spin mo-\nment, we estimate morb= 1.7µB, in nice agreement with\nour LDA+U result of 1 .69µB, for the d0d2minority-spin\norbital occupation.21\nTo critically check our experimental and previ-\nous LDA+U results21regarding the d0d2orbital oc-\ncupation and the giant orbital moment, we explic-\nitly simulate the experimental XMCD spectra us-\ning a charge-transfer configuration-interaction cluster\ncalculation,24,31,32whichincludesnotonlythefullatomic\nmultiplet theory and the local effects of the solid, but\nalso the oxygen 2 p–cobalt 3 dhybridization. The results\nof the calculated Co- L2,3XAS and XMCD spectra are\npresented in Figs. 3(d) and 5(b), respectively. We can\nclearly observe that the simulations reproduce the ex-\nperimental spectra very well. The parameters39used\nare those which indeed give the d0d2orbital occupation\nfor the ground state. The magnetic quantum numbers\nfound are morb= 1.65µBandmspin= 2.46µB, yielding\nmorb/mspin= 0.67 and a total Co magnetic moment of\n4.11µB. With the Rh in the S= 1/2 tetravalent state,\nthe total magnetic moment per formula unit should be\naround 5 µB. This is not inconsistent with the results\nof the high-field magnetization study by Niitaka et al.7:\nthey found a total moment of 4 .05µB, but there the satu-\nrationofthemagnetizationhasnotyetbeenreachedeven\nunder 18.7 Tesla. This can now be understood since the\nmagnetocrystallineanisotropy,associatedwith the active\nspin-orbit coupling, is extremely strong and makes it dif-\nficult to fully magnetize a powder sample as was used in\ntheir study.\nWe also have simulated the spectra for the d2d−2sce-\nnario. These are depicted in Figs. 3(e) for the XAS/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54/s49/s46/s56/s50/s46/s48/s83 /s40/s83 /s43/s49/s41/s32/s40 /s104/s50\n/s47/s52/s50\n/s41/s109\n/s115/s112/s105/s110/s32/s40/s181\n/s66/s41 /s109\n/s111/s114/s98/s32/s40/s181\n/s66/s41\n/s48/s50/s32/s40/s101/s86/s41/s48/s50/s32/s40/s101/s86/s41\n/s32/s79/s114/s98/s105/s116/s97/s108/s32/s111/s99/s99/s117/s112/s97/s116/s105/s111/s110\n/s48/s50/s32/s40/s101/s86/s41/s32/s100\n/s50\n/s32/s100\n/s48\n/s32/s100\n/s50/s76 /s40/s76 /s43/s49/s41/s32/s40 /s104/s50\n/s47/s52/s50\n/s41/s74 /s40/s74 /s43/s49/s41/s32/s40 /s104/s50\n/s47/s52/s50\n/s41/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48\n/s32/s32\n/s109\n/s111/s114/s98\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 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48/s48/s49/s50/s51\n/s109\n/s115/s112/s105/s110\n/s32/s32\n/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53\n/s74 /s40/s74 /s43/s49/s41\n/s32/s32\n/s48/s53/s49/s48\n/s83 /s40/s83 /s43/s49/s41/s76 /s40/s76 /s43/s49/s41\n/s32/s32\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 /s49/s46/s54 /s49/s46/s56 /s50/s46/s48/s48/s49/s50/s51\n/s32/s32\nFIG. 6: (color online) Top panel: Occupation number of the\nd0,d2, andd−2orbitals as function of the d0–d±2splitting\n∆02[Fig.4(b)]. Middle panel: Orbital and spin moments\n(morbandmspin) as function of ∆ 02. Bottom panel: J(J+1),\nL(L+1), and S(S+1) as function of ∆ 02.\nand5(c) for the XMCD. It is obvious that the experi-\nmental spectra are not reproduced. The simulated line\nshapes are very different from the experimental ones and\nthe integral of the simulated XMCD spectrum yields a\nvanishing orbital moment. We therefore can safely con-\nclude that the ground state of this material is not d2d−2.\nForcompletenesswementionthat the magneticquantum\nnumbers found for this d2d−2ansatz are morb= 0.03µB\nandmspin= 2.86µB, yielding morb/mspin= 0.01 and a6\ntotal Co magnetic moment of 2 .89µB.\nV. STABILITY OF THE d2d0STATE\nHaving established that the ground state of\nCa3CoRhO 6has the Co2+d7ion in the doubly oc-\ncupiedd0d2orbital configuration and not in the d2d−2,\nit is interesting to study its stability in more detail. As\nalready mentioned above, for a Co3+d6ion, thed0and\nd±2states can be energetically very close to each other.\nFor a Co2+d7ion, however, the d0d2andd2d−2states\nare very much different in energy. This is illustrated in\nthe top panel of Fig. 6, in which we have calculated the\noccupation numbers of the d0,d2, andd−2orbitals as a\nfunction of ∆ 02, the one-electron level splitting between\nthed0andd±2orbitals. The d0d2ground state which\ngives the best simulation to the experimental XAS and\nXMCD spectra was obtained with ∆ 02≈0.4 eV. We\ncan observe that the d0d2situation is quite stable for a\nwide range of ∆ 02values, certainly up to 0.8 eV. With\na transition region between ∆ 02= 0.8–1.2 eV, we find\na stable d2d−2situation only for ∆ 02values larger than\n1.2 eV. (For the d2d−2simulations above we have used\n∆02= 1.4 eV.) This is a very large number indeed, and\nit can be traced back to the multiplet character of the\non-site Coulomb interactions: an occupation of d2d−2\nmeans a much stronger overlap of the electron clouds\nas compared to the case for a d0d2. This results in a\nhigher repulsion energy, which is not a small quantity in\nview of the atomic-like values of the F2andF4Slater\nintegrals determining the multiplet splitting.31,40\nIn the middle panel of Fig. 6we also show the ex-\npectation values for morbandmspinwhen varying ∆ 02.\nAgain we clearly observe that the large orbital-moment\nsituation is quite stable. To quench the orbital momentone would need much higher ∆ 02values. Important is\nthat the spin state does not change here. Bottom panel\nof Fig.6demonstrates that the high-spin state of the\nCo2+ion is not affected by ∆ 02: the expectation value\n/angbracketleftS2/angbracketrightremains constant throughout at a value consistent\nwith an essentially S= 3/2 state. Obviously, the L2and\nJ2quantum numbers are strongly affected by ∆ 02.\nVI. CONCLUSION\nTo summarize, the Rh- L2,3, Co-L2,3and Fe-L2,3XAS\nmeasurements indicate Co2+/Rh4+in Ca3CoRhO 6and\nFe3+/Rh3+in Ca3FeRhO 6. The magnetic properties of\nCa3FeRhO 6are relatively simple as both the HS Fe3+\nand LS Rh3+ions have a closed subshell and thus no\norbital degrees of freedom and no orbital moment. The\nweak intrachain AFM coupling between the HS Fe ions\ncan be understood in terms of the normal superexchange\nvia the intermediate non-magnetic O–Rh–O complex.\nFor Ca 3CoRhO 6, the combined experimental and the-\noretical study of the Co- L2,3XAS and XMCD spectra\nreveals a giant orbital moment of about 1 .7µB. This\nlarge orbital moment is connected with the minority-spin\nd0d2occupation for HS Co2+(3d7) ions in the pecu-\nliar trigonal prismatic coordination. The high FM or-\ndering temperature in Ca 3CoRhO 6, compared with that\nof Ca3Co2O6, can be attributed to the distinct octahe-\ndral sites (which mediate the Co–Co magnetic coupling):\nthe magnetic Rh4+ion (S= 1/2) in the former and the\nnonmagnetic Co3+ion (S= 0) in the latter.\nWe would like to thank Lucie Hamdan for her skillful\ntechnical and organizational assistance in preparing the\nexperiment. The research in K¨ oln is supported by the\nDeutsche Forschungsgemeinschaft through SFB 608.\n1H. Fjellv˚ ag, E. Gulbrandsen, S. Aasland, A. Olsen, and\nB. C. Hauback, J. Solid State Chem. 124, 190 (1996).\n2S. Aasland, H. Fjellv˚ ag, and B. Hauback, Solid State Com-\nmun.101, 187 (1997).\n3H. Kageyama, K. Yoshimura, K. Kosuge, H. Mitamura,\nand T. Goto, J. Phys. Soc. Jpn. 66, 1607 (1997).\n4H. Kageyama, K. Yoshimura, K. Kosuge, M. Azuma,\nM. Takano, H. Mitamura, and T. Goto, J. Phys. Soc. Jpn.\n66, 3996 (1997).\n5S. Niitaka, H. Kageyama, M. Kato, K. Yoshimura, and\nK. Kosuge, J. 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/CB/CW/CX/D2/B8 /C1/BX/BX/BX /CC /D6/CP/D2/D7/BA /C5/CP/CV/D2/BA /BF/BJ /B8 /BD/BE/BI/BK /B4/BE/BC/BC/BD/B5/BA/BJ(a)\nmCurrent sourceNM\nM\nӪ\nMxMz\nxyz\nOӪMx'y'm\nӰ\nI\nO'(b)\nJ/BY/C1/BZ/BA /BD/BM /B4/CP/B5 /D7\r /CW/CT/D1/CP/D8/CX\r /D3/CU /CP /CC/C8/B9/CB/CC/C7/BA /C5 /CX/D7 /D8/CW/CT /AS/DC/CT/CS /D0/CP /DD /CT/D6 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CX/D0/D8/CT/CS /D3/D6/CX/CT/D2 /D8/CP/D8/CX/D3/D2/BA /CC/CW/CT/CU/D6/CT/CT /D0/CP /DD /CT/D6 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D1 /CX/D7 /D7/CT/D4/CP/D6/CP/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT /AS/DC/CT/CS /D0/CP /DD /CT/D6 /CQ /DD /CP /D2/D3/D2/D1/CP/CV/D2/CT/D8/CX\r /D0/CP /DD /CT/D6 /B4/C6/C5/B5/BN /B4/CQ/B5 /D8/CW/CT\r/D3 /D3/D6/CS/CX/D2/CP/D8/CT /D7/DD/D7/D8/CT/D1 /D9/D7/CT/CS /CX/D2 /D8/CW/CX/D7 /DB /D3/D6/CZ/BA /C5 /D0/CX/CT/D7 /CX/D2 /D8/CW/CT /DC /B9 /DE /D4/D0/CP/D2/CT /DB/CX/D8/CW /CP/D2/CV/D0/CTβ /DB/BA/D6/BA/D8/BA /D8/CW/CT /DC /B9/CP/DC/CX/D7/BA/BK102030\nF\nE\nDCB\nE=36q\nE=45qFrequency (GHz)A(a)W\r\nM\n-1.0 -0.5 0.0 0.5 1.00.00.20.40.60.81.0\nF=4E=45q\nF=0\nF=2\n\u0003F=4\nJ(108 A/cm2)Effective MR(b)\nABCDE\nFE=36qMr180 00.51\n90\nF=4F=2F=0\n0.51\n0 90 180 \nxz\nMA\nB\nC\nD\nE\nF(c)/BY/C1/BZ/BA /BE/BM /B4/CP/B5 /C8/D6/CT\r/CT/D7/D7/CX/D3/D2 /CU/D6/CT/D5/D9/CT/D2\r/DD /DA/D7/BA /CS/D6/CX/DA /CT \r/D9/D6/D6/CT/D2 /D8 /CU/D3/D6β /BP/BF/BI◦/B4/D7/D3/D0/CX/CS /D0/CX/D2/CT/B5 /CP/D2/CSβ /BP/BG/BH◦/B4/CS/CP/D7/CW /CS/D3/D8 /D0/CX/D2/CT/B5/BA/C1/D2/D7/CT/D8/BM /C6/D3/D6/D1/CP/D0/CX/DE/CT/CS /D7/D4/CX/D2 /D8/D3/D6/D5/D9/CTτ∗/BP/BGedτ//planckover2pi1J /DA/D7/BAϕ /BA /B4/CQ/B5 /BX/AR/CT\r/D8/CX/DA /CT /C5/CA /DA/D7/BA /C2 /CU/D3/D6β /BP/BF/BI◦/B4/D7/D3/D0/CX/CS /D0/CX/D2/CT/B5 /CP/D2/CS\nβ /BP/BG/BH◦/B4/CS/CP/D7/CW/CT/CS /D0/CX/D2/CT/B5/BA /C1/D2/D7/CT/D8/BM /CA/CT/CS/D9\r/CT/CS /C5/CA /DA/D7/BAϕ /BA /B4\r/B5 /C8/D6/CT\r/CT/D7/D7/CX/D3/D2 /D3/D6/CQ/CX/D8/D7 /D3/D2 /D8/CW/CT /D9/D2/CX/D8 /D7/D4/CW/CT/D6/CT /CU/D3/D6 /CS/CX/AR/CT/D6/CT/D2 /D8/C2 /CP/D2/CSβ /BP/BF/BI◦/BA/BL10-210-11001011021030.00.20.40.60.81.0\nxz\nM\n6\n5\n41\n2\n3\n25\nJ>0E=45°E=36°r\n|J| (108 A/cm2)J<0346\n1/BY/C1/BZ/BA /BF/BM /C5/CP/CV/D2/CT/D8/D3/D6/CT/D7/CX/D7/D8/CP/D2\r/CT /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU \r/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8 /DD /BA /C1/D2/D7/CT/D8/BM /D8/CW/CT /CT/D5/D9/CX/D0/CX/CQ/D6/CX/D9/D1 /D7/D8/CP/D8/CT/D7 /D3/CUˆm /CP/D8/CS/CX/AR/CT/D6/CT/D2 /D8 \r/D9/D6/D6/CT/D2 /D8 /CS/CT/D2/D7/CX/D8/CX/CT/D7 /DB/CW/CT/D2β /BP/BF/BI◦/BA /BD/BMJ /BP/B9/BC/BA/BH×108/BT/BB\r/D12/BN /BE/BMJ /BP/B9/BD×1011/BT/BB\r/D12/BN /BF/BMJ /BP/BC/BA/BJ/BH×108/BT/BB\r/D12/BN /BG/BMJ /BP/BJ×108/BT/BB\r/D12/BN /BH/BMJ /BP/BD×109/BT/BB\r/D12/BN /BI/BMJ /BP/BD×1011/BT/BB\r/D12/BA/BD/BC"    },    {        "title": "0808.0841v1.Surface_Magnetoelectric_Effect_in_Ferromagnetic_Metal_Films.pdf",        "content": "Surface Magnetoelectric Effect in Ferromagnetic Metal Films \n  \nChun-Gang Duan,1 Julian P. Velev,2,3 R. F. Sabirianov,3,4 Ziqiang Zhu,1 Junhao Chu,1\nS. S. Jaswal,2,3 and E. Y. Tsymbal2,3\n1Key Laboratory of Polarized Materials and Devices, Ministry of E ducation, East China Normal University, Shanghai 200062, China  \n2Department of Physics and Astronomy, University of Nebraska, Lincoln, Nebraska 68588, USA \n3Nebraska Center for Materials and Nanoscience, Uni versity of Nebraska, Lincoln, Nebraska 68588,USA  \n4Department of Physics, University of  Nebraska, Omaha, Nebraska 68182, USA \n  \nA surface magnetoelectric effect is revealed by density-functional calculations that are applied to \nferromagnetic Fe(001), Ni(001) and Co(0001) films in the presen ce of external electric fiel d. The effect originates \nfrom spin-dependent screening of the electric field which leads to notable changes in the surface magnetization \nand the surface magnetocrystalline anisotropy. These results are of considerable interest in the area of electrically-\ncontrolled magnetism and ma gnetoelectric phenomena.  \n \nPACS: 75.80.+q, 75.70.Ak , 77.84.-s, 75.75.+a \n \n \n \nThe coupling between ferroelectric and ferromagnetic \norder parameters in thin-film heterostructures is an exciting \nnew frontier in nanoscale science.1-5 The underlying \nphysical phenomenon controlling properties of such materials is the magnetoelectric (ME) effect\n6,7 that \ndetermines the induction of magnetization by an electric \nfield or electric pola rization by a magnetic field. The recent \ninterest in ME materials is stimulated by their significant \ntechnological potential. A prominent example is the control \nof the ferromagnetic state by an electric field.8,9 This \nphenomenon could yield entirely new device concepts, such \nas electric field-controlled magnetic data storage. In a \nbroader vision, ME effects include not only the coupling between the electric and magnetic polarizations, but also \nrelated phenomena such as an  electrically-controlled \nexchange bias,\n10,  11and magnetocrystalline anisotropy,12-14 \nand the effect of ferroel ectricity on spin transport.15-19  \nTwo mechanisms for the ME effect have been \nestablished: (1) in single-phase ME materials (including \nmultiferroics) an external elec tric field displaces ions from \nequilibrium positions which changes the magnetostatic and exchange interactions affecting the magnetization;\n20 (2) in \ncomposite multiferroic material s, piezoelectric strain in the \nferroelectric constituent of the multiferroic heterostructure induces changes in the  magnetic properties of the \nferromagnetic constituent due to magnetostriction.\n9, ,1314 \nRecently, two additional mechan isms for magnetoelectricity \nhave been proposed theoretically: (1) in a heterostructure comprising a ferroelectric insu lator and a magnetic material, \nferroelectric displacements of atoms at the interface may be \nreversed by an external electri c field resulting in the sizable \nchange of the interface magnetic moment\n21 and surface \n(interface) magnetic anisotropy;22 (2) in the insulator/ \nferromagnetic heterostructure, an external electric field \npolarizes the insulator resulting in the carrier-mediated \ninterface magnetoelectricity.23In this study we explore the ME effect due to the direct \ninfluence of an external elect ric field on magnetic properties \nof ferromagnetic Fe(001), Ni(001) and Co(0001) films. We \nshow that spin-dependent sc reening of the electric field \nleads to spin imbalance of the excess surface charge resulting in notable changes in the surface magnetization \nand the surface magnetocrystalline anisotropy. We argue that the effect may be used to switch the magnetization \nbetween in-plane and out-of-p lane orientations, thereby \nsignifying the potential of electrically-controlled magnetism. \nWhen a metal film is exposed to an electric field, the \nconduction electrons screen the electric field over the \nscreening length of the metal. In ferromagnetic metals, the screening charge is spin-d ependent due to exchange \ninteractions.\n24 The spin dependence of the screening \nelectrons leads to the induced  surface magnetization of the \nferromagnet, i.e. the ME effect . Since the electric field does \nnot penetrate into the bulk of metals and the induced electric \ncharge is confined to a depth of the order of atomic dimensions from the surface, this  ME effect is limited to the \nmetal surface. Therefore we name it a surface \nmagnetoelectric effect . \nIn order to elucidate the surface magnetoelectric effect \nquantitatively we carry out density-functional calculations on free-standing ferromagnetic films under the influence of \na uniform electric field applied perpendicular to the film \nsurface. The studied systems are bcc Fe(001) ( a = 2.87 Å), \nhcp Co(0001) (a  = 2.51 Å, c/a =1.622), and fcc Ni(001) ( a = \n3.52 Å) films with thicknesses ranging from 1 to 15 monolayers (MLs). The calculations are based on the projector augmented wave (PAW) method implemented in \nthe Vienna Ab-Initio  Simulation Package (VASP)\n25 and \ninclude spin-orbit interactions. The exchange-correlation \npotential is treated in the generalized gradient \napproximation (GGA). We use the energy cut-off of 500 eV \nfor the plane wave expansion of the PAWs and a 10 x 10 x 1 \n 1Monkhorst-Pack grid for k-point sampling in the self-\nconsistent calculations. All the structural relaxations are \nperformed until the Hellman-Feynman forces on the relaxed \natoms become less than 1 meV/Å. The external electric field is introduced by planar dipole layer method.\n26  \n-1.0-0.50 . 00 . 51 . 03.063.083.103.12M agnetic Mo me n t (μΒ)\nElectric Field (V/Å) M | |  [001]\n M | |  [100]\n  \n \n \nFIG. 1 (color online) Total magnetic moments on the (001) Fe surface as a \nfunction of applied electric field for the magnetic moment lying in the \nplane of the film (along the [100] dir ection) and perpendicular to the plane \n(along the [001] direction). The solid lin es are a linear fit to the calculated \ndata.  \n \nResults of our calculations demonstrate that magnetic \nmoments do change under applied electric field. As \nexpected, the magnetoelectric effect is almost solely a \nsurface effect and has little dependence on film thickness. \nFig. 1 shows  the calculated magnetic moment on the \nsurface Fe atom of a free-standi ng 15-ML Fe (001) film as a \nfunction of an applied electric field for magnetization lying \nin the plane of the film ([100] direction) and perpendicular \nto the plane ([001] direction). The spin and orbital \ncontributions to the total moment are of the order of 3.0 and \n0.1 μB respectively. It is seen that the magnetic moment \nchanges nearly linearly  with the electric field, so that the \ninduced surface magnetization ∆M depends on the applied \nelectric field E as follows: \n 0μ α Δ=S M E, (1) \nwhere αS denotes  the surface magnetoelectric coefficient.  \nHere a positive electric field is defined to be pointed away \nfrom the metal film surface. Therefore results for both positive and negative electric fields are obtained at two \nsurfaces within one simulation . From the linear fit to the \ncalculated data shown in Fig. 1 we find that for \nmagnetization in the plane \n100αS ≈ 2.4 ×  10-14 Gcm2/V and \nfor magnetization perpendicular to the plane 001αS ≈ 2.9 × \n10-14 Gcm2/V. The vertical separation between the two lines \nin Fig. 1 for a given applied electric field is the difference in \nthe orbital moments in the two magnetization directions (as \nshown later in Fig. 4) because the corresponding spin moments are essentially the same. 0 4 8 1 21 62 02 42 83 2-1.0-0.50.00.51.0\n z(Å)Δρ (arb. units)  majority spin\n  minority spin \n \nE\n \nFIG. 2 (color online) Induced xy-averaged electron charge densities, Δρ = \nρ(E) – ρ (0), along the z direction normal to the xy film plane for a 21 Å-\nthick Fe film (located between about 5.5 and 26.5 Å) for majority- (solid \nline) and minority- (dashed line) sp in electrons. The applied external \nelectric field is E = 1.0 V/Å, pointing from right to left. \n \nThe origin of this ME  effect stems from induced spin-\ndependent charge densities on th e surfaces of the film. Fig. \n2 displays the differences in the charge densities between \nthe disturbed ( E = 1.0 V/Å) and undisturbed (E = 0) Fe film. \nIt is seen that the surface dipoles are formed to screen the electric field inside the film and the induced charge density is strongly spin-polarized. We note the presence of the \nFriedel-like oscillation of the change density in Fig. 2, \nwhich is typical for the electron screening effect.\n27 However, \nunlike the case of normal metals,  the charge oscillations \nhave multiple periods, and the oscillations of the majority- \nand minority-spin electrons are di fferent. This is because the \nmajority- and minority-spin electrons have different Fermi \nwave vectors, and they are coupled in the dielectric \nresponse.24\nDue to the spin imbalance of the screening charge, there \nis an induced surface magnetization reflecting the presence of the surface ME effect. Fig. 3 visualizes the induced spin density across the Fe film. It is clearly seen that the ME \neffect is confined to the surfaces of the film. The net \ninduced spin densities at th e two opposite surfaces have \ndifferent signs, as the applied electric fields are oppositely \noriented with respect to the two surfaces of the film. \nIt is interesting to compare the calculated magnitude of \nthe surface ME coefficient due to the direct influence of an external electric field with those predicted earlier for BaTiO\n3/Fe  and SrTiO 3/SrRuO 3 structures.  In the later case, \nthe ME effect originates from the capacitive accumulation of spin-polarized carriers at the SrTiO\n3/SrRuO 3 interface \nunder an external electric fiel d. The calculations of ref. 23 \nsuggest that for a 7 unit cell thick SrTiO 3 (a = 3.904 Å) \nlayer the applied voltage of 27.8 mV induces a net magnetic \nmoment of 2.5 ×10-3 μB per surface unit cell. The \ncorresponding surface ME coefficient is αS ≈ 2×10-12 \nGcm2/V. This value is higher by two orders of magnitude \n 2than what we obtained for Fe f ilm. This difference is related \nto the dielectric constant of the dielectric at the \ndielectric/metal interface. For a given applied electric field, \nthe screening charge in the metal is proportional to the \ndielectric constant. According to ref. 17, SrTiO 3 has a very \nlarge dielectric constant, 490ε≈ , compared to 1ε= in the \npresent calculations. Therefore, the presence of a dielectric with large dielectric constant can significantly amplify the \nsurface ME effect. \n \n \nFIG. 3 (color online) Induced spin densities, ∆ σ＝σ (E) – σ (0), in arbitrary \nunits for a 15-ML-thick Fe film (about 21 Å) under the influence of an \nelectric field of 1 V/Å. \n \nIn the case of BaTiO 3/Fe structure,  the predicted ME \neffect occurs as a result of the reversal of the polarization \norientation due to an applied electric field and originates \nfrom the change in the interface bonding strength which \ndominates over the screening charge contribution. This ME \neffect is strongly non-linear and does not follow a simple \nrelation given by Eq. (1). Nevertheless, to have a crude estimate of the surface ME coef ficient we assume that the \npolarization of BaTiO\n3 can be switched at the coercive field \nof Ec = 100 kV/cm resulting in the change of the interface \nmagnetic moment of more than 0.3 μB per unit surface cell. \nWe find that the surface ME coefficient is then αs ≈ 2×10-10 \nGcm2/V. This value is much larger than the electric-field-\ninduced effects discussed above.  \nCalculations performed for magnetic Co (0001) and Ni \n(001) films confirm qualitatively  predictions obtained for Fe. \nWe find that the surface ME coefficients (in units of 10-14 \nGcm2/V) are100αS ≈ 1.6, 001αS ≈ 1.7 for 9-ML Co (0001) film and 100αS ≈ 3.0, 001αS ≈ 2.4 for 9-ML Ni (001) film. Note that \nthe predicted surface ME coefficients have the same sign for the three kinds of metal films. This is in disagreement with the prediction of ref. \n24 suggesting that Fe film has different \nsign of the induced magnetic moment from that in Co and Ni films. Obviously a simple free-electron-like model adopted in ref. \n24 is unable to take into account all the \nelectronic effects which occur in the 3 d metals due to the \npresence of the exchange-splitting d  bands, their \nhybridizations with s and p  bands, and the difference \nbetween bulk and surface electronic structures. \nThe origin of the predicted positive  ME coefficient for \nall the ferromagnetic films can be understood within a simple model. The model assumes a localization of the screening charge within the first atomic ML on the metal \nsurface and a rigid shift of the chemical potential on the \nsurface in response to the applied electric field E. According \nto this model the surface charge density \nσε=E is \nunequally distributed between the surface majority- and minority-spin states resulting in the surface ME coefficient  \n 2B\nsnn\necnnεμα↑↓\n↑↓−=−\n+. (2) \nHere ↑n and  are majority- and minority-spin surface \nDOS at the Fermi energy and ↓n\nε is the dielectric constant of \nthe dielectric adjacent to th e ferromagnetic surface (in our \ncase of vacuum 1ε=). As is seen from Eq. (2), αs is \nproportional to the spin polari zation of the surface DOS. It \nis known that at the Fe (001), Co(0001) and Ni(001) surfaces, the minority-spin stat es are dominant near the \nFermi level.\n28 Therefore the accumulation of electrons in \nresponse to the application of an inward (negative) electric \nfield results in a decrease of the surface spin moment, due to \nthe dominating minority-spin char acter at the Fermi energy. \nUsing Eq.(2) and the calculated surface DOS, we find αS ≈ \n5.2, 4.9 and 5.5 × 10-14 Gcm2/V  for Fe (001), Co (0001), \nand Ni (001) respectively. These values are in a qualitative agreement with the results of our density-functional \ncalculations.  \nAnother important consequence of applying electric \nfield to the ferromagnetic metal films is the change of their magnetocrystalline anisotropy energy (MAE), which may be \nconsidered as another manifestation of the ME effect. The MAE, whose physical origin is the spin-orbit coupling,\n29 \nplays an important role in high anisotropy  materials used in \nmodern magnetic storage technologies.30 Recently we have \ndemonstrated that the magnetic anisotropy of the \nferromagnetic film can be altered by switching the \npolarization of the adjacent fe rroelectric through applied \nbias voltage.  The effect occurs due to the change of the \nelectronic structure at the interface region, which is \nproduced by ferroe lectric displacements and mediated by \ninterface bonding. Here we dem onstrate the direct impact of \nan external electric field on the MAE. \n 3Following our previous study,  we decompose the MAE \nfor the whole film into individual contributions from each \nconstituent atom. By doing this, we find that the MAE \nchanges for the ferromagnetic film under various electric fields, again, mainly occur at the surface. To be specific, an \noutward (positive) electric field enhances and an inward \n(negative) electric field reduces the individual MAE contribution from surface atoms. As  is seen from Fig. 4, the \norbital moment  anisotropy ( M\nL [001] - M L[100]) of the \nsurface Fe atoms increases monot onically with the increase \nof applied electric field. This leads to a significant increase \nof the surface MAE when the electric field changes from inward (negative) to outward (positive) direction, evident \nfrom Fig. 4. In particular, wh en the electric field of 0.5 V/Å \nswitches from negative to positiv e, there is a change of \nabout 30% in the surface MAE.  \nThe predicted phenomenon can be used for switching \nthe magnetization by an applie d electric field. The total \nmagnetic anisotropy energy per unit area of the film involves the magnetostatic shape anisotropy energy K\nm = -\n2πM2t, where M is the magnetization and t is film thickness. \nThe shape anisotropy favors in-plane alignment of \nmagnetization, whereas positive MAE favors out-of-plane \nalignment. Thus, with the ME control of the surface magnetocrystalline anisotropy and thickness dependent shape anisotropy, it is possible to design ferromagnetic \nfilms with the anisotropy switchable between in-plane and \nout-of-plane orientations.  \n1012141618\n-1.0-0.50 . 00 . 51 . 00.50.60.70.80.9Orbital Mo me n t Anisotropy (10−3mB) \n Electric Field (V/Å)\n MAE (erg/cm2)\n \nFIG. 4 (color online) Electric field induced changes in calculated orbital \nmoment anisotropy (∆ ML = ML [001] -  M L [100], in units of 10-3μB) of the \nsurface Fe atom and surface magnetocrystalline anisotropy energy (MAE) \nfor 15 ML thick Fe (001) slab. \n \nThe predicted ME effects in ferromagnetic metal films \nare significant only when the applied field is very large, e.g., \nof a few hundred mV/Å. However, such strong electric \nfields could be achieved experimentally using a scanning tunneling microscope tip over the magnetic metal film \nsurface. Since the screening effect can be dramatically \nenhanced by high- κ dielectrics, the applied electric field \ncould be much smaller to have observable effects on the related devices. \nTo summarize, our first-prin ciple calculations show that \nthe surface magnetoelectric effect exists in ferromagnetic metal films due to spin-depende nt screening of electric field \nat the metal surfaces. The external electric field induces \nnotable changes in the surface  magnetization and the surface \nmagnetocrystalline anisotropy of a ferromagnetic metal. The \nmagnitude and sign of the surface magnetoelectric \ncoefficient depends on the dens ity and spin polarization of \nthe charge carriers near the Fermi level of the ferromagnetic \nmetal film. This purely electric field driven magnetoelectric \neffect may be interesting for application in advanced \nmagnetic and electronic devices. \nThe research at East China Normal University was \nsupported by the NSFC (50771072), 973 Program No. 2007CB924900, and Shanghai Basic Research Program No. \n07JC14018.  The research at University of Nebraska was \nsupported by NSF-MRSEC, the Nanoelectronics Research Initiative, the Office of Naval Research, and the Nebraska \nResearch Initiative. C.-G. D. thanks Nicola Spaldin for \nstimulating discussions. \n \n                                                 \n1. T. Kimura  et al. , Nature 426, 55 (2003). \n2. T. Lottermoser et al., Nature 430 , 541 (2004). \n3. W. Eerenstein, N. D. Mathur  and J. F. Scott, Nature 442, \n759 (2006). \n4. R. Ramesh and N. A. Spaldin, Nature Mater. 6, 21 \n(2007). \n5. C.-W. Nan  et al. , J. Appl. Phys. 103, 031101 (2008). \n6. M. Fiebig, J. Phys. D 38, R123 (2005). \n7. N. A. Spaldin and M. Fiebig, Science 309, 391 (2005). \n8. T. Zhao et al., Nature Mater. 5, 823 (2006) \n9. F. Zavaliche et al. , Nano Lett.  7, 1586 ( 2007) . \n10. P. Borisov  et al. , Phys. Rev. Lett. 94, 117203 (2005). \n11. V. Laukhin et al. , Phys. Rev. Lett. 97 , 227201 (2006). \n12. M. Weisheit  et al. , Science 315, 349 (2007). \n13. W. Eerenstein  et al. , Nature Mater. 6, 348 (2007). \n14. S. Sahoo et al. , Phys. Rev. B 76, 092108 (2007). \n15. E. Y. Tsymbal and H. Kohlstedt, Science 313, 181 \n(2006). \n16. M. Y. Zhuravlev et al. , Phys. Rev. Lett.  94, 246802 \n(2005). \n17. Ch. Binek and B. Doudin, J. Phys.: Cond. Matt. 17, L39 \n(2005). \n18. M. Gajek  et al. , Nature Mater . 6, 296 (2007). \n19. J. P. Velev  et al. , Phys. Rev. Lett. 98, 137201 (2007). \n20. I. Dzyaloshinskii, Sov. Phys. J. Expt. Theor. Phys. 10, \n628629 (1960). \n21. C.-G. Duan, S. S. Jaswal, and E. Y. Tsymbal, Phys. Rev. \nLett. 97 , 047201 (2006). \n22. C.-G. Duan et al., Appl. Phys. Lett.  92, 122905 (2008). \n23. J. M. Rondinelli, M. Stengel,  and N. A. Spaldin, Nature \nNanotech. 3, 46 (2008). \n24. S. Zhang, Phys. Rev. Lett. 83, 640 (1999). \n25. G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (1999). \n26. J. Neugebauer and M. Sc heffler, Phys. Rev. B 46, 16067 \n(1992). \n27. K. M. Indlekofer and H. Kohlstedt, Europhys. Lett. 72, 282 \n(2005).  \n28. O. Hjortstam et al. , Phys. Rev. B 53, 9204 (1996). \n29. J. H. Van Vlec k, Phys. Rev. 52 , 1178 (1937). \n30. R. C. O’Handley, Modern Magnetic Materials: \nPrinciples and Applications (Wiley-VCH, 1999). \n 4"    },    {        "title": "0809.4289v1.The_role_of_magnetic_anisotropy_in_the_Kondo_effect.pdf",        "content": "The role of magnetic anisotropy in the Kondo effect  \n \nAlexander F. Otte1,2, Markus Ternes1, Kirsten von Bergmann1,3, Sebastian Loth1, Harald \nBrune1,4, Christopher P. Lutz1, Cyrus F. Hirjibehedin1,5 and Andreas J. Heinrich1 \n \n1IBM Research Division, Almaden  Research Center, San Jose, CA 95120, USA  \n2Kamerlingh Onnes Laboratorium, Universiteit Leiden, 2300 RA Leiden, The Netherlands  \n3Institute of Applied Physics, University of Hamburg, Jungiusstr. 11, 20355 Hamburg, Germany  \n4Institut de Physique des Nanostruct ures, Ecole Polytechnique Fédérale de Lausanne, CH -1015 Lausanne, \nSwitzerland  \n5London Centre for Nanotechnology, Department of Physics & Astronomy, Department of Chemistry, \nUniversity College London, London WC1H OAH, UK  \n \nIn the Kondo effect, a localized ma gnetic moment is screened by forming a correlated \nelectron system with the surrounding conduction electrons of a non -magnetic host1. Spin \nS = 1/2 Kondo systems have been investigated extensively in theory and experiments , but \nmagnetic atoms often  have a larger spin2. Large r spins are subject to the influence of \nmagneto -crystalline anisotropy , which  describes the dependence of the energy on the \norientation of the spin  due to the surrounding atomic environment3,4. Here  we \ndemonstrate the decisive ro le of magnetic anisotropy in the physics of Kondo screening.  \nA scanning tunnelling microscop e is used  to simultaneously determine the magnitude of \nthe spin, the magnetic  anisotropy, and the Kondo properties of individual  magnetic \natom s on a surface . We fin d that a Kondo resonance emerges for large -spin atoms only \nwhen the magnetic anisotropy creates degenerate ground state levels that are connected \nby the spin -flip of a screening electron. The magnetic anisotropy also determines how \nthe Kondo resonance evol ves in a magnetic field: the resonance peak splits at rates that are strongly direction -dependent . These rates are well -described by the energies of the \nunderlying unscreened spin states.  \n \nA low density of magnetic impurities in a non -magnetic host metal c an have dramatic effect s \non the magnetic, thermodynamic, and  electrical properties of the material due to the Kondo \neffect, a many -body interaction between the metal’s conduction electrons and the electron \nspin of the localized magnetic impurity5. This interaction gives rise to a narrow, pronounced \npeak in the density of states close to the Fermi energy1. The last decade has seen a surge of  \ninterest in the Kondo screening of individual atomic spins , as a result of  experimental \nadvances in  probing  individ ual magnetic atoms by using  scanning tunnelling microscopes \n(STM)6,7 and single -molecule transistors8,9. When magnetic atoms are placed on metal \nsurfaces the Kondo interaction between the localized spin and the conduction electrons is very \nstrong, leading to high Kondo temperatures  TK, in the range from  40 to 200 K 10. In order to \nprobe the Kondo physics i t is desirable to reduce this interaction so that the Kondo screening \ncompetes on equal footing with external influences  such as magnetic fields. It was s hown \nrecently that this can be achieved by incorporating  a decoupling layer between the atomic spin \nand the screening conduction electrons11. \n \nIn the current study , individual Co and Ti atoms are separated from a Cu(100) crystal by a \nmonolayer of copper nitride (Cu 2N) 12 as sketched in the left inset of Fig. 1. We used a home -\nbuilt ultra -high vacuum scanning tunnelling microscope (STM) with 0.5  K base temperature \nand magnetic fields up to 7  T to explore this system (see Methods). We probed the local \nelect ronic excitations by measuring the differential conductance d I/dV, where I is the \ntunnelling current and V is the sample voltage.  \n A single Co  atom bound to the Cu 2N surface  exhibits  a sharp zero-voltage peak in its \nconductance spectrum (Fig.  1). This peak  is due to  an increase in the density of states near the \nFermi level that  result s from  Kondo screening1. Such a K ondo peak has been observed in \nquantum dots13,14 and in atoms  or molecules on surfaces6,7. The peak is well described by a \nthermally broadened Fano lineshape15 that results from interference between tunnelling into \nthe Kondo resonance and tunnelling directly into the substrate6,7. For Co on Cu 2N the Fano \nlineshape is nearly Lorentzian , indicating that tunnelling into the Kondo resonance dominates \nover tunnelling directly into the substrate, presumably because  the decoupling Cu 2N layer  \ninhibits the tunnelling to the substrate .    \n \nTo confirm that the conductance peak is due to a Kondo r esonance, we measured the spectra \nat higher temperatures (Fig.  1) and observe d a rapid reduction of the Kondo peak height.  \nThese spectra are broadened due to the Fermi -Dirac distribution of the tunnelling electrons in \nthe tip, but the intrinsic temperatur e behaviour of the Kondo system can be determined by \nperforming a deconvolution of the measured spectra15 to remove the effect of the finite tip \ntemperature  (see Methods). As expected for the Kondo effect14, the intrinsic width of the \nresonance grows linea rly with temperature at higher temperatures , but saturates at a finite \nvalue at low temperature  (Fig.  1 right inset) . The half -width at half -maximum  of the zero \ntemperature peak defines the Kondo temperature TK =  / kB, where kB is Boltzmann’s \nconstant.  We find a Kondo temperature for Co on Cu 2N of TK = 2.6 ± 0.2 K.  \n \nTo determine the effect of magnetic anisotropy  on the Kondo screening , we applied a \nmagnetic field B along each of the three high -symmetry directions of the sample (Fig.  2b and \nc). We find that the Kondo resonance splits into a double peak as in previous studies of other \nKondo systems8,9,11,13,14,16,17. In contrast to previous studies , we observed that the peak splits at a rate that depends strongly on the direction of the field : when  B is oriented along the \ndirection in which the binding site is neighboured by two hollow sites (the ‘hollow direction’) \nthe splitting is much less than for the perpendicular in -plane ‘N  direction’  (Fig.  2a). The \nstrong directional dependence of the Kondo peak splitting suggests the presence of large \nmagneto -crystalline anisotropy3,4. With B oriented out -of-plane the spectra are essentially \nidentical to the spectra for the N  direction. We note that this is true even though there is no \nsymmetry between these  two axes.  \n \nWe are able to determine the magnitude and the magnetic anisotropy of the local spin by \ndetecting  inelastic spin excitations performed by the tunnelling electrons. In addition to the \nKondo resonance peak , the spectra show  an ‘outer’ step in the conductance near ±  5 meV \n(Figs.  1, 2b and c). This step can be explained within the framework of Inelastic Electron \nTunnelling Spectroscopy (IETS)  as the onset of an inelastic spin excitation , which opens  a \nnew conductance channel11. The evolution of the step energy as a function of the magnitude \nand direction of an external magnetic field  (Figs.  2b, c) can be used to quantify the magnetic \nanisotropy and the spin of the Co atom18. Since results obtained for B oriented in the N and \nout-of-plane directions are indistinguishable , we model the system as having uni -axial \nanisotropy along the hollow direction, which will be designated as the z axis in the following . \nWe describe  the single particle anisotropic spin behaviour – ignoring the m any-body Kondo \nproperties – with a spin Hamiltonian  which is the sum of the Zeeman energy and the \nanisotropy energy4: \n2 ˆˆ ˆ\nBz H g DS   BS\n,         ( 1 ) \nwhere g is the Landé g-factor and B the Bohr magneton. The longitudinal anisotropy \nparameter D partly breaks the zero -field degeneracy of the eigenlevels ; different magnitudes of m, the eigenvalue of the z-projection Ŝz of the spin vector Ŝ, lead to different energies \n(Fig.  3). \n \nWe find good agreement between the measured outer steps  and the corresponding excitation \nof this spin Hamiltonian  (i.e. the second -lowest excitation)  by using S = 3/2. The fit then \nyields g = 2.19 ± 0.09, which lies close to the free -electron g -value of 2.00, and \nD = 2.75 ± 0.05 meV, similar to the large magnetic anisotropy seen in prior studies of \nindividual magnetic atoms on surfaces3,18. The uncertainties quoted are mainly  due to slight \nvariations in the atoms’ local environments, possibly  due to position -dependent strain in the \nCu2N islands19. We note that S = 5/2 and higher half -integer values of S also give an adequate \ndescription of the measured spectra. However , all the conclusions drawn in this work about \nthe Kondo and low-energy spin-excitation behaviour would remain unchanged. When an \natom is adsorbed on a surface  the spin is generally  unchanged or reduced from the free atom \nvalue .  Since the spin of a free Co atom  is 3/2, the spin of Co on this surface is likely 3/2 . \n \nThe sign of the anisotropy parameter  D determines whether states with large or small | m| form \nthe ground state. The positive D observed here yields hard-axis (easy-plane ) anisotropy , in \nwhich  the two m = ±1/2 states have the lowest energy and are degenerate in the absence of B. \nWe note that even in the case of finite transverse anisotropy th e twofold zero -field degeneracy \nof these states would not be broken , due to Kramers’ theorem. The observed outer step, used \nto obtain the fitting results discussed above, thus correspond s to m = ±1/2 → ±3/2 transitions \nwhen B is applied along the z directi on, see Fig.  3. \n \nWe calculate d the inelastic tunnelling spectra in the absence of Kondo screening  (orange \ncurves in Figs.  2b and c). The transition energies were  derived by diagonalizing the spin Hamiltonian , equation  (1), using the parameter values found above . For finite B a low -energy \nspin excitation becomes possible as m = +1/2 and m = –1/2 split in energy . In addition, w e \nused a scattering intensity operator that describes spin scattering by tunnelling electrons18 to \nmodel the relative step height s of the inner and outer transition s.  \n \nWe show the effect of Kondo screening on the spectra by subtracting the calculated inelastic \ntunnelling spectra from the measured spectra  to provide  a measure of the shapes and positions \nof the split Kondo peak . This procedure results in symmetrically shaped peaks on a nearly flat \nbackground  (Fig.  2d, e) . Here we have treated  the conductance as the sum of an IETS \nconductance channel and a separate Kondo channel in a manner similar to Ref. 20. We stress \nthat the underly ing many -body physics may be more complex  than the sum of the se two \nchannels21,22. We also note that t he single -particle model of the inelastic tunnelling channel \nneglect s any interference in the elastic channel due to the presence of the inelastic channel23. \n \nThe position of the split Kondo peak coincides with the calculated low -energy transition \nm = +1/2 → –1/2 of the non -Kondo Hamiltonian of equation  (1) for all measured fields and \nfield orientations (Fig.  3). Here it makes no substantial difference whet her we use the \npositions of the peaks in the original measurements  (open circles  in Fig.  3) or those that \nremain after subtracting the calculated IETS curves (crosses  in Fig.  3). It is worth highlighting \nthat the spin exhibits direction dependent splitting  of the Kondo peak only because the spin \n(S = 3/2) is greater than 1/2. In the case of an S = 1/2 impurity such anisotropic behaviour due \nto crystal field effects is expected to be absent. We note that the precise nature of the splitting \nof the Kondo peak , especially for magnetic fields that are small compared to the Kondo \ntemperature,  is still under theoretical debate21,22,24.  \n In contrast to the hard -axis anisotropy, which we find to be the case for Co on Cu 2N, easy -\naxis anisotropy ( D < 0) would favour the m = ±3/2 doublet as the ground state of an S = 3/2 \nimpurity. In this situation Kondo screening is inhibited, as it would require ∆ m = 3 transitions \nto be made through electron scattering. A Kondo resonance can only be formed if the \nanisotropy creates a  degenerate ground state with levels connected by ∆ m = 1 transitions (i.e. \nthe flipping of a conduction electron spin). This picture agrees well with previously studied \natomic spins Fe ( S = 2) and Mn ( S = 5/2) on Cu 2N that showed no Kondo effect. Each of t hese \nwas found to have easy axis anisotropy18 and therefore do not have a ground state doublet that \nis linked via ∆m = 1 spin excitations. Similar mechanisms, where crystalline anisotropy is \nresponsible for creating a Kondo system from a large spin (i.e. S > 1/2), have been suggested \nfor bulk impurities25,26. Recent theoretical investigations have shown that a Kondo effect can \nalso occur in systems with easy -axis magnetic anisotropy, exemplified by single -molecule \nmagnets, if an additional strong transverse  anisotropy sufficiently mixes states with different \nm values to create a degenerate ground state which is linked via ∆ m = 1 transitions27. \n \nUnlike the measurements on Co, spectra taken on individual Ti atoms (Fig.  4) show a clear \nKondo peak but no conductance steps due to additional spin-excitations. Consequently Ti on \nCu2N can be modelled  as an S = 1/2 system. We note that a free Ti atom has S = 1 in the 3 d2 \nconfiguration , so in this case the binding  to the surface presumably changes the atom’s spin . \nTi binds at the same location as Co on Cu 2N so comparable magnetic anisotropy may be \nexpected. However, as shown in Fig.  4, measurements at finite magnetic fields do not show \ndirection -dependent splitting . This observation confirms  that a crystal field can only affect the \nKondo resonance of an impurity that has a spin larger than 1/2.  \n In summary, access to a single large -spin atomic impurity provides a new opportunity for \nstudying the interplay between th e Kondo effect and crystalline anisotropy. We find that a \nthorough characterization of magnetic anisotropy is essential to understand ing the emergence \nof the Kondo effect. For various atom s on Cu 2N with spins larger than 1/2, the presence or \nabsence of Kon do screening can be explained solely based on their magneto -crystalline \nanisotropy: Kondo screening can occur only if the anisotropy results in degenerate ground \nstate levels connected by Δ m = 1 transitions. Our result will be applicable  to the Kondo effect \nin other systems with large spin  as well , such as magnetic atoms placed directly on a metal \nsubstrate  or single -molecule magnets with transverse magnetic anisotropy27. Further studies \nmay also reveal a more quantitative understanding of t he link between anisotropy and the \nstrength of the Kondo screening. In addition, the ability to tune the Kondo effect by varying \nthe magnitude and orientation of the magnetic anisotropy would create a new class of Kondo \nsystems in which the screening could  be manipulated directly through control of the local \nenvironment.  \n \nWe thank M.  F. Crommie, D.  M. Eigler, A. C. Hewson, B. A. Jones, J.  E. Moore and J.  M. \nvan Ruitenbeek for stimulating discussions and B.  J. Melior for his expert technical \nassistance. A.  F. O. acknowledges support from the Leiden University Fund;  M. T. from the \nSwiss National Science Foundation;  K. v. B. from the German Science Foundation ( DFG \nForschungsstipendium ); S. L. from the Alexander von Humboldt Foundation;  C. F. H. from \nthe Enginee ring and Physical Sciences Research Council (EPSRC) Science and Innovation \nAward; and M.  T., C.  P. L., and A.  J. H. from the Office of Naval Research. H.  B. \nacknowledges EPFL for supporting his Sabbatical stay with IBM.  \n \nMethods  Details of the experimental  setup can be found in Ref. 11. Co and Ti  atoms were deposited \nonto the Cu 2N surface at low temperature by thermal evaporation from a metal target. These \natoms were subsequently placed on specific binding sites by means of vertical atom \nmanipulation. The d ifferential conductance d I/dV was measured using lock -in detection with a \n50 μV RMS modulation at 745  Hz.  \n \nThe tip was verified to have a flat density of states in the energy ranges presented, by \nobserving essentially constant conductance spectra when the  tip is placed over the bare Cu 2N \nand over the bare Cu surfaces.  We determined that presence of the tip does not influence the \nKondo system, but merely probes its density of states, by observing that the spectrum does not \nchange when the junction resistanc e is increased.  \n \nThe temperature of the sample Tsample was regulated using a heater , while the tip was cooled \nvia strong  thermal contact directly to the 3He refrigerator.  Different refrigerator operating \nmodes , each corresponding to a particular tip temperature Ttip, were used for different sample \ntemperature ranges . For the deconvolution of the spectra measured when Tsample < 1.4 K we \nused Ttip = 0.5 K, and  for 1.4  K < Tsample  < 5.0 K we used Ttip = 1.8 K. With this assignment of \ntip temperatures the intrinsic peak width was found to increase without a discontinuity at \nTsample = 1.4 K. Uncertainties in Ttip up to 0.3  K were taken into account for determining the \nerror bars in the right inset of Fig.  1, and h ence the uncertainty in TK. \n \nThe precise parameter values found for the Co atom s of Fig.  2 are g = 2.16 and D = 2.71 meV \nfor the  atom with B along the hollow direction, and g = 2.22 and D = 2.79 meV for  the atom \nwith B along the N direction. These values w ere used to calculate the energy levels for the \ncorresponding spins in Fig.  3 as well as the inelastic tunnelling spectra in Fig s. 2b and c.   \nFigure captions  \nFigure 1 | Temperature dependence of the Kondo resonance of a Co atom. The left inset \nshows a schematic drawing of a single Co atom bound on top of a Cu atom of the Cu 2N \nsurface. Through the Cu 2N layer the Co atom is coupled to the electron sea of the bulk copper \nwhich can Kondo screen the localized spin  on the atom . The main panel shows differenti al \nconductance (dI/dV) spectra  measured with the tip positioned over a Co atom at 0.5  K and \nhigher temperatures. For each spectrum the tip height was adjusted to give a 10  M tunnel \njunction ( V = 10 mV, I = 1 nA). The red curves show fits to Fano functions  broadened by the \ntemperature of the probing tip (Fano lineshape  parameter  q = 17 ± 2). The right inset shows \nthe intrinsic full width at half maximum of the peak as a function of the sample temperature  T. \nThe errors in these values are dominated by the un certainties in the tip temperatures (see \nMethods) . The solid black line shows a best fit to the function [(kBT)2+(2kBTK)2]1/2 which \ndescribes the intrinsic width of the Kondo resonance in a Fermi liquid  model15. For T >> TK \nthe width approaches linear behaviour with a slope  = 5.4 ± 0.1 (red line) . \n \nFigure 2 | Anisotropic field dependence of the Kondo resonance. a , Topographic STM \nimage (10 nm  10 nm, 10  mV, 1  nA) of four Cu 2N islands with single Co atoms. The two \nfigures on the side s sketch the adsorption site of the two marked Co atoms (Cu and N atoms \nare depicted as yellow and green circles, respectively) . Labels indicate the  two non -equivalent \nin-plane directions , referring  to neighbouring N atoms or hollow sites  along the  direction of \nthe magnetic field . b, c, Black curves: differential conductance spectra taken with the tip \npositioned over the two Co atoms of panel a when a field up to 7  T was applied in the \ndesignated directions. Successive spectra are offset by 0.15  nA/mV for clarity. The blue \ncurves in panel c show similar measurements on a Co atom where the magnetic field was directed perpendicular to the surface. All spectra recorded at T = 0.5 K. Orange curves: \ncalculated inelastic tunnelling spectra based on the par ameters obtained by fitting the \npositions of the outer conductance steps to equation  1. d, e, Result of subtracting the orange \ncurves from the black curves in panels b and c respectively, showing the change in the spectra \ndue to Kondo interactions.  \n \nFigure  3 | Energy eigenlevels for different field directions.  a, Solid lines show the \ncalculated energy levels based on equation  1 with  magnetic field B parallel to the hollow \ndirection  (see Methods) . Full and open circles indicate the energies of the steps and the peaks, \nrespectively, of the spectra shown  in Fig.  2b. The positions of the peaks in Fig.  2d are \nrepresented by crosses. Values are plotted relative to the calculated ground state as illustr ated \nby the arrows in b. b, Same as a but with B parallel to the N direction and data taken from \nFigs. 2c and e.  \n \nFigure 4 | Kondo effect of a Ti atom: a S = 1/2 system.  Differential conductance spectra on \nindividual Ti atoms on Cu 2N in the absence of a ma gnetic field (black curve) and with an \nexternal field of 7  T applied along  the two in -plane directions (red and orange curves , offset \nby 0.28 and 0.30  nA/mV ) and oriented out -of-plane (blue curve , offset by 0.32  nA/mV ). The \n7 T spectra are essentially iden tical because an S = 1/2 system cannot show magnetic \nanisotropy. All curves measured at T = 0.5 K. \n \nReferences  \n1. Hewson, A. C. The Kondo Problem to Heavy Fermions  (Cambridge University Press, \nCambridge, 1997)  2. Owen, J., Browne, M. E., Arp, V.  & Kip, A. F. Electron -spin resonance and magnetic -\nsusceptibility measurements on dilute alloys of Mn in Cu, Ag and Mg. J. 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Zero -bias tunneling anomalies – temperature, voltage, and \nmagnetic field dependence. Phys. Rev.  165, 566 –577 (1968)  \n17. Kogan, A. et al.  Measurements of Kondo and spin splitting in single -electron transistors. \nPhys. Rev. Lett. 93, 166602 -1–166602 -4 (2004)  \n18. Hirjibehedin, C. F. et al.  Large magnetic anisotropy of a single atomic spin embed ded in a \nsurface molecular network. Science  317, 1199 –1203 (2007)  \n19. Komori, F., Ohno, S. -Y. & Nakatsuji, K.  Lattice deformation and strain -dependent atom \nprocesses at nitrogen -modified Cu(001) surfaces. Prog. Surf. Sci.  77, 1–36 (2004)  \n20. Appelbaum , J. A. Exchange model of zero -bias tunneling anomalies. Phys. Rev.  154, 633 –\n643 (1967)  \n21. Costi, T. A. Kondo effect in a magnetic field and the magnetoresistivity of Kondo alloys. \nPhys. Rev. Lett.  85, 1504 –1507 (2000)  \n22. Moore, J. E. & Wen, X. -G. Anomal ous magnetic splitting of the Kondo resonance. Phys. \nRev. Lett.  85, 1722 –1725 (2000)  \n23. Lorente, N. Mode excitation induced by the scanning tunnelling microscope.  Appl. Phys. \nA 78, 799 –806 (2004)  \n24. Grobis, M., Rau, I. G., Potok, R. M. & Goldhaber -Gordon , D. \"Kondo effect in \nmesoscopic quantum dots \" in: Handbook of Magnetism and Advanced Magnetic Materials , \nKronmüller , H. & Parkin , S., eds. (Wiley , 2007)  25. Schlottmann, P. Effects of crystal fields on the ground state of a Ce atom. Phys. Rev. B 30, \n1454 –1457 (1984)  \n26. Újsághy, O., Zawadowski, A. & Gyorffy, B. L. Spin -orbit -induced magnetic anisotropy \nfor impurities in metallic samples of reduced dimensions: finite size dependence in the Kondo \neffect. Phys. Rev. Lett. 76, 2378 –2381 (1996)  \n27. Romeike , C., Wegewijs, M. R., Hofstetter, W. & Schoeller , H. Quantum -tunneling -\ninduced Kondo effect in single molecular magnets. Phys. Rev. Lett.  96, 196601 -1–196601 -4 \n(2006)  \n   \n \n  \n \n \n  \n \n \n  \n \n"    },    {        "title": "0810.4679v1.The_magnetoresistance_tensor_of_La_0_8_Sr_0_2_MnO_3_.pdf",        "content": "arXiv:0810.4679v1  [cond-mat.mtrl-sci]  26 Oct 2008The magnetoresistance tensor of La0.8Sr0.2MnO3\nY. Bason1,∗J. Hoffman2, C. H. Ahn2, and L. Klein1\n1Department of Physics, Nano-magnetism Research Center,\nInstitute of Nanotechnology and Advanced Materials,\nBar-Ilan University, Ramat-Gan 52900, Israel and\n2Department of Applied Physics, Yale University, New Haven, Connecticut 06520-8284, USA\n(Dated: November 3, 2018)\nWe measure the temperature dependence of the anisotropic ma gnetoresistance (AMR) and the\nplanar Hall effect (PHE) in c-axis oriented epitaxial thin fil ms of La 0.8Sr0.2MnO3, for different\ncurrent directions relative to the crystal axes, and show th at both AMR and PHE depend strongly\non current orientation. We determine a magnetoresistance t ensor, extracted to 4thorder, which\nreflects the crystal symmetry and provides a comprehensive d escription of the data. We extend the\napplicability of the extracted tensor by determining the bi -axial magnetocrystalline anisotropy in\nour samples.\nPACS numbers: 75.47.-m, 75.47.Lx, 72.15.Gd, 75.70.Ak\nThe interplay between spin polarized currentand mag-\nnetic moments gives rise to intriguing phenomena which\nhave led to the emergence of the field of spintronics [1].\nIn most cases, the materials used for studying these phe-\nnomena have been amorphous alloys of 3d itinerant fer-\nromagnets (e.g., permalloy), while much less is known\nabout the behavior in materials which are crystalline and\nmore complicated. Manganites, which are magnetic per-\novskites, serve as a good example for such a system. As\nwe will show, elucidating these phenomena in this ma-\nterial system provides tools for better theoretical under-\nstanding of spintronics phenomena and reveals opportu-\nnities for novel device applications.\nThe magnetotransport properties of manganites\nknown for their colossal magnetoresistance have been\nstudied quite extensively; nevertheless, along numerous\nstudies devoted to elucidating the role of the magni-\ntude ofthe magnetization, relativelyfew reportshavead-\ndressed the role of the orientation of the magnetization,\nwhich is known to affect both the longitudinal resistivity\nρlong(anisotropic magnetoresistance effect - AMR) and\ntransverse resistivity ρtrans(planar Hall effect - PHE).\nForconductorsthat areamorphousmagnetic films, the\ndependence of ρlongandρtranson the magnetic orienta-\ntion is given by:\nρlong=ρ⊥+(ρ/bardbl−ρ⊥)cos2ϕ (1)\nand\nρtrans= (ρ/bardbl−ρ⊥)sinϕcosϕ (2)\nwhereϕis the angle between the current Jand the mag-\nnetization Mandρ/bardblandρ⊥are the resistivities parallel\nand perpendicular to M, respectively [3, 4]. Eqs. 1 and\n2 are not expected to apply to crystalline conductors,\nas they are independent of the crystal axes [5]. Never-\ntheless, they have been used to describe AMR and PHEin epitaxial films [6, 7, 8, 9]; qualitative and quantita-\ntive deviations were occasionally attributed to extrinsic\neffects.\nHere, we quantitatively identify the crystalline con-\ntributions to AMR and PHE in epitaxial films of\nLa0.8Sr0.2MnO3(LSMO) and replace Eqs. 1 and 2 with\nequations that provide a comprehensive description of\nthe magnetotransport properties of LSMO. The equa-\ntions are derived by expanding the resistivity tensor to\n4thorder and keeping terms consistent with the crystal\nsymmetry.\nAMR and PHE in manganites constitute an impor-\ntant aspect of their magnetotransport properties; hence,\nquantitative determination of these effects is essential for\ncomprehensive understanding of the interplay between\nmagnetism and transport in this class of materials. In\naddition, when the dependence of AMR and PHE on lo-\ncal magnetic configurations is known, the two effects can\nbe used as a powerful tool for probing and tracking static\nand dynamic magnetic configurations in patterned struc-\ntures. Moreover,as the magnitude of the AMR and PHE\nchanges dramatically with current direction, the elucida-\ntion of the appropriate equations is crucial for designing\nnovel devices with optimal properties that are based on\nthese phenomena.\nOur samples are epitaxial thin films ( ∼40 nm) of\nLSMO with a Curie temperature (T c) of∼290 K grown\non cubic single crystal [001] SrTiO 3substrates using off-\naxismagnetronsputtering. θ−2θx-raydiffractionreveals\nc-axis oriented growth (in the pseudocubic frame), with\nan out-of-plane lattice constant of ∼0.3876 nm, and an\nin-plane lattice constant of ∼0.3903 nm, consistent with\ncoherently strained films. No impurity phases are de-\ntected. Rocking curves taken around the 001 and 002\nreflections have a typical full width at half maximum\nof 0.05o. The film surfaces have been characterized us-\ning atomic force microscopy, which shows a typical root-\nmean-squaresurfaceroughnessof ∼0.2 nm. Thesamples2\nwere patterned using photolithography to create 7 pat-\nterns on the same substrate. Eachpattern has its current\npath at a different angle θrelative to the [100] direction\n(θ= 0◦,15◦,30◦,45◦,60◦,75◦,90◦), with electrical leads\nthat allow for AMR and PHE measurements.\nFig. 1 presents ρlongandρtransdata obtained by ap-\nplying a field of H=4 T in the film plane and rotating\nthe sample around the [001] axis. The figure shows the\ndata for all seven patterns at T=5, 125 and 300 K. At\nT = 300 K both ρlongandρtransseem to behave accord-\ning to Eqs. 1 and 2. However, contrary to these equa-\ntions, the amplitude of ρlongdiffers from the amplitude\nofρtrans; moreover, they both change with θ, the angle\nbetween Jand [100].\nThe discrepancies increase as the temperature de-\ncreases, and at T=125K the variationsin the amplitudes\nfor measurements taken for different θincrease. Further-\nmore, the location of the extremal points are dominated\nbyα, the angle between Mand [100]. At T = 5 K, the\ndeviations are even more evident as the AMR measure-\nmentsarenolongerdescribedwithasinusoidalcurve. All\nthese observations clearly indicate the need for a higher\norder tensor to adequately describe the magnetotrans-\nport behavior of LSMO.\nThe resistivity tensorin a magnetic conductordepends\non the direction cosines, αi, of the magnetization vector,\nand can be expressed as a series expansion of powers of\ntheαi[10], giving:\nρij(α) =3/summationdisplay\nk,l,m...=1(aij+akijαk+aklijαkαl+\n+aklmijαkαlαm+aklmnijαkαlαmαn+...) (3)\nwherei,j= 1,2,3 and the a’s are expansion coefficients.\nAs usual ρij(α) =ρs\nij(α) +ρa\nij(α) whereρs\nijandρa\nijare\nsymmetric and antisymmetric tensors, respectively. As\nboth AMR and PHE are symmetric, we use only ρs\nijfor\ntheir expression. As we are interested only in the in-\nplane properties, we use the tensor expansion for crystals\nwith m3m cubic-crystal structure [11]. The 4thorder\nsymmetric resistivity tensor ρsfor this class of materials\nin the xy plane (as M,Jand the measurements are all\nin the plane of the film) is given by:\nρs=/parenleftbiggC′+C′\n1α2\n1+C′\n2α4\n1C′\n4α1α2\nC′\n4α1α2C′+C′\n1α2\n2+C′\n2α4\n2/parenrightbigg\n.(4)\nWhenJis alongθwe obtain:\nρlong=Acos(2α−2θ)+Bcos(2α+2θ)+Ccos(4α)+D\n(5)\nand\nρtrans=Asin(2α−2θ)−Bsin(2α+2θ) (6)\nwith:A= (C′\n1+C′\n2+C′\n4)/4\nB= (C′\n1+C′\n2−C′\n4)/4\nC=C′\n2/8\nD=C′+C′\n1/2+3C′\n2/8\nEquations 5 and 6, which take into account the crys-\ntal symmetry, have 4 independent parameters ( A,B,C\nandD) with which we fit (as shown in Figure 1) at any\ngiven temperature and magnetic field a set of 14 different\ncurves (7 AMR curves and 7 PHE curves).\nThe parameter Ais a coefficient of a term describing\na non-crystalline contribution since ( α−θ) is the angle\nbetween MandJirrespectiveoftheir orientationrelative\nto the crystal axes. On the other hand, the parameters\nBandCare coefficients of terms that depend on the\norientation of Mand/orJrelative to the crystal axes.\nWe note that adding the terms with the coefficient B\n(in both Eq. 5 and 6) to the ” A” term changes only the\namplitude and the phase of the signal compared to Eqs.\n1 and 2: Eq. 5 can be written (for C=0) as:\nρlong=Ecos(2α−φlong)+D (7)\nwhereE2=A2+B2+ 2ABcos4θand sinφlong=\nA−B\nEsin(2θ); and Eq. 6 can be written as:\nρtrans=Fsin(2α−φtrans) (8)\nwhereF2=A2+B2−2ABcos4θand sinφtrans=\nA+B\nFsin2θ. The amplitude of ρtrans(α),F, varies with θ\nbetween a maximal value of |A+B|forθ=±45◦and a\nminimal value of |A−B|forθ= 0,±90◦. On the other\nhand, the amplitude of ρlong(α),E, obtains its maximal\nvalue|A+B|atθ= 0,±90◦and itsminimal value |A−B|\natθ=±45◦. When the Cterm isaddedit doesnotaffect\nρtrans; however, ρlongbehaves qualitatively differently.\nWethus observethat the currentdirectionaffectsquite\ndramatically the amplitude of the effect. At 125 K, for\ninstance, the PHE amplitude for current at 45◦relative\nto [100] is more than 20 times larger than the PHE for\ncurrent parallel to [100]. This means that appropriate\nselection of the current direction that takes into consid-\neration crystalline effects is important for designing de-\nvices that use the PHE for magnetic sensor or magnetic\nmemory applications [12].\nFigure 2 presents the temperature dependence of B/A\nandC/A. Close to T cbothBandCare negligible rel-\native toA; therefore, AMR and PHE measurements ap-\npear to fit Eqs. 1 and 2. At intermediate temperatures\nwhereCis still much smaller than A(whileBandA\nare of the same order), the signal remains sinusoidal, al-\nthough its deviation from Eqs. 1 and 2 becomes quite\nevident. At low temperatures, Cis on the order of B,\nand the AMR signal is no longer sinusoidal.\nWhen AMR and PHE measurements are performed\nwith low applied fields, Mis no longer parallel to H, due\nto intrinsic magnetocrystalline anisotropy. Our LSMO\nfilms exhibit bi-axial magnetocrystalline anisotropy with3\neasy axes along /angbracketleft110/angbracketrightdirections, a manifestation of in-\nplane cubic symmetry. When a field His applied,\nthe total free energy consists of the magnetocrystalline\nanisotropy energy and the Zeeman energy:\nE=K1\n4cos22α−MHcos(α−β) (9)\nwhereK1is the magnetocrystalline anisotropy energy\nandβis the angle between Hand [100]. The first term is\nresponsibleforthebi-axialmagnetocrystallineanisotropy\nwith easy axes along α=±π\n4andα=±3π\n4. We have\ndetermined the value of K1at various temperatures (see\nFig. 2) by switching the magnetization between the two\neasy axes (see Fig. 3). The extracted value of K1allows\nus by using Eqs. 5, 6 and 9 to fit the AMR and PHE\ndata obtained with relatively low applied fields (e.g., 500\nOe), where Mdoes not follow H(see Fig. 3).\nIn summary, we have expanded the magnetoresistance\ntensor to 4thorder keeping terms consistent with the\nsymmetry of epitaxial films of LSMO and derived equa-\ntions that provide a comprehensive description of AMR\nand PHE in LSMO films in a wide range of tempera-\ntures. The results shed new light on the interplay be-\ntween magnetism and electrical transport in this class of\nmaterials and may serve as a basis for further study of\nthe microscopic origin of magnetotransport properties of\nLSMO and other manganites. The results contribute to\nthe ability to monitor magnetic configurations via mag-\nnetotransport properties, a feature of particular impor-\ntance in studying nano-structures, and will facilitate the\ndesign of novel devices that use AMR and PHE.We acknowledge useful discussions with E. Kogan.\nL.K. acknowledgessupport by the Israel Science Founda-\ntion founded by the Israel Academy of Sciences and Hu-\nmanities. WorkatYalesupportedbyNSF MRSECDMR\n0520495, DMR 0705799, NRI, ONR, and the Packard\nFoundation.\n∗Electronic address: basony@mail.biu.ac.il\n[1] G. A. Prinz, Science 282, 1660 (1998).\n[2] M. McCormack, S. Jin, T H. Tiefel, R. M. Fleming, J.\nM. Phillips and R. Ramesh, Appl. Phys. Lett. 64, 3045\n(1994);\n[3] C. GoldbergandR.E. Davis, Phys.Rev. 94, 1121(1954);\nF. G. West, J. Appl. Phys. 34, 1171 (1963); W. M. Bullis,\nPhys. Rev. 109, 292 (1958).\n[4] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. 11,\n1018 (1975).\n[5] D¨ oring, W., Ann. Physik (32) (1938) 259.\n[6] X. Jin, R. Ramos, Y. Zhou, C. McEvoy, and I. V. Shvets,\nJ. Appl. Phys. 99, 08C509 (2006);\n[7] H. X. Tang, R. K. Kawakami, D. D. Awschalom, and M.\nL. Roukes, PRL 90,107201 (2003);\n[8] Y. Bason, L. Klein, J.-B. Yau, X. Hong, and C. H. Ahn,\nAppl. Phys. Lett. 84, 2593 (2004).\n[9] I.C. Infante, V.Laukhin,F.S´ anchez, J. Fontcuberta, M a-\nterials Science and Engineering B 126(2006) 283-286;\n[10] T. T. Chen, V. A. Marsocii, Physica 59(1972) 498-509.\n[11] Birss, R. R., Symmetry and Magnetizm, North-Holland\nPubl. Comp. (Amsterdam, 1964).\n[12] Y. Bason, L. Klein, J.-B. Yau, X. Hong, J. Hoffman, and\nC. H. Ahn, J. Appl. Phys. 99, 08R701 (2006).4\n18070 18090 18110 \n0 45 90 135 180 T=300 K \nα [Deg.] 354.2 354.6 355 T=5 K \n882 883 884 885 ρlong  [ µΩ  cm] T=125 K \n-0.2 00.2 T=5 K \n-20 020 \n0 45 90 135 180 θ=0 o\nθ=15 o\nθ=30 o\nθ=45 o\nθ=60 o\nθ=75 o\nθ=90 o\nα [Deg.]T=300 K -1 01ρtrans  [µΩ  cm]T=125 K \nFIG. 1: Longitudinal resistivity ρlong(left) and transverse resistivity ρtrans(right) vs. α, the angle between the magnetization\nand [100], for different angles θ(the angle between the current direction and [100]) at differ ent temperatures with an applied\nmagnetic field of 4 T. The solid lines are fits to Eqs. 5 and 6. Ins et: Sketch of the relative orientations of the current densi ty\nJ, magnetization M, and the crystallographic axes.\n-0.2 00.2 0.4 0.6 0.8 11.2 \n-50 050 100 150 200 250 \n0 50 100 150 200 250 300 B/A \nC/A K1K1 [kJ/m 3]\nT [K] Coefficients ratio \nFIG. 2: The ratios of the coefficients from Eqs. 5 and 6 (B/A and C /A) (left axis) and the coefficient K 1from Eq. 9 (right\naxis) as a function of temperature.5\n-0.15 -0.1 -0.05 00.05 0.1 0.15 \n0 90 180 270 360 ρtrans  [ µΩ  cm] \nβ [Deg.] -0.014 -0.012 -0.01 -0.008 -0.006 -0.004 -0.002 00.002 \n0 20 40 60 80 100 ρ\ntrans  [ µΩ  cm] \nMagnetic Field [Oe] \nFIG. 3: Left: PHE signal as a function of β, the magnetic field direction relative to the [100] (H=500 Oe and T=50 K). The\nline is a fit to Eq. 6 with αextracted using Eq. 9. Right: PHE as a function of magnetic fie ld. The sample is prepared with\nMalong [100], and the field is applied along [1 ¯10]. The line is a fit using Eq. 9."    },    {        "title": "0812.3797v1.Domain_structure_of_epitaxial_Co_films_with_perpendicular_anisotropy.pdf",        "content": " 1 Domain structure of epitaxial Co films with perpend icular anisotropy \n \nJ. Brandenburg a, R. Hühne, L. Schultz, V. Neu b \n \nIFW Dresden, Institute for Metallic Materials, P.O.  Box 270116, 01171 Dresden, Ger- \nmany, and \nInstitute for Solid State Physics, University of Te chnology Dresden, 01062 Dresden, \nGermany \n \nAbstract \nEpitaxial hcp Cobalt films with pronounced c-axis t exture have been prepared by \npulsed lased deposition (PLD) either directly onto Al 2O3 (0001) single crystal \nsubstrates or with an intermediate Ruthenium buffer  layer. The crystal structure \nand epitaxial growth relation was studied by XRD, p ole figure measurements and \nreciprocal space mapping. Detailed VSM analysis sho ws that the perpendicular \nanisotropy of these highly textured Co films reache s the magnetocrystalline ani- \nsotropy of hcp-Co single crystal material. Films we re prepared with thickness t of \n20 nm < t < 100 nm to study the crossover from in-p lane magnetization to out-of-\nplane magnetization in detail . The analysis of the periodic domain pattern ob- \nserved by magnetic force microscopy allows to deter mine the critical minimum \nthickness below which the domains adopt a pure in-p lane orientation. Above the \ncritical thickness the width of the stripe domains is evaluated as a function of the \nfilm thickness and compared with domain theory. Esp ecially the discrepancies at \nsmallest film thicknesses show that the system is i n an intermediate state between \nin-plane and out-of-plane domains, which is not des cribed by existing analytical \ndomain models.  \n \n                                                 \na Present address: Max Plank Institute for Chemical Physics of Solids, Nötnitzer Str. 40, 01187 Dresden , \nGermany \nb Corresponding author. Email: v.neu@ifw-dresden.de  2 I. Introduction \n \nThin magnetic films with perpendicular anisotropy r ecently received a renewed scien- \ntific interest due to the change from in-plane to p erpendicular recording media in the \nmagnetic data storage technology. For this applicat ion, the effective uniaxial anisotropy \nof the material and the orientation of its easy mag netization axis perpendicular to the \nsurface has to be large enough to overcome the natu ral demagnetizing effect arising \nfrom the shape anisotropy of a thin film. Polycryst alline Co films typically possess a \npure in-plane magnetization as the uniaxial magneto crystalline anisotropy averages out \nand the domain structure is dominated by the large demagnetizing energy of K d = \n(1/2µ 0)J s2 = 1.35 MJ/m 3. However, if prepared as a single crystal film and  with perpen- \ndicular orientation of the c-axis, the uniaxial ani sotropy of hcp-Co (K u = 0.45 MJ/m 3) \ncan compensate at least partially the in-plane anis otropy. From films with such compet- \ning anisotropies a non trivial domain structure is expected. Vice versa, the study of the \ndomain structure is a useful approach to examine th e participating energy terms present \nin the material. In films with large perpendicular anisotropy domain theory and experi- \nmental observation often lead to good qualitative a nd quantitative agreement, as dem- \nonstrated e.g. for L1 0-ordered FePt films1 or epitaxial Nd 2Fe 14 B films. 2 Reliable domain \ninterpretation in Co films with competing shape and  magnetocrystalline anisotropy is, \nhowever, complicated by the limited applicability o f existing domain theories. 3 The \nfocus of this work is on thin, epitaxially grown Co  films with precise control of the \ncrystal structure and magnetocrystalline anisotropy , where the epitaxial relation to the \nsubstrate guarantees a perpendicular orientation of  the c-axis with respect to the film \nplane. For such films a transition from pure in-pla ne domains into stripe domains with \nperpendicular magnetization component is expected. In extension to the few works \nwhich report on similar samples,3,4 the thickness range from 20 to 50 nm has been stud - \nied in detail to determine the crossover thickness and domain width for above men- \ntioned transition. Furthermore in all samples the e ffective uniaxial anisotropy of single \ncrystal Co was proven by an analysis of the global magnetization process. In these well  3 characterized films the evolution of the domain str ucture with film thickness can be \ncritically compared with prediction from domain the ory. \n \n \nII. Experimental \n \nThin Co films with varying thickness t (20 nm < t <  100 nm) were prepared on heated \n(100°C –500°C) single crystal Al 2O3(0001) substrates by pulsed laser deposition (PLD) \nin a UHV chamber with a background pressure of 10 -9 mbar. In few cases, an additional \n5 to 10 nm thick Ru film was deposited as a buffer layer. Elementary targets (Co and \nRu) were moved into the beam of an KrF excimer lase r (248 nm, 25 ns, 3 J/cm 2) and the \nablated material was deposited in direct on-axis ge ometry onto the substrate placed at a \ndistance of 72 mm opposite to the target. Details o n the imaging conditions of the laser \nbeam and the ablation of Co can be found in Branden burg et al .5 The film thickness was \nadjusted by an appropriate choice of pulses onto th e target and determined after the \ndeposition by means of energy dispersive x-ray anal ysis (EDX). For these measure- \nments, comparative EDX spectra of film and elementa l bulk standards were analysed \nwith a thin film software. Details are given in Neu  et al .6 The crystallinity of the films \nwas measured by standard θ-2θ x-ray diffraction (XRD) in Bragg-Brentano geometry  \nwith a Philips X’Pert diffractometer and Co-K α radiation. Pole figures were measured \nwith a Philips X'Pert texture goniometer and Cu-K α radiation to analyze the epitaxial \ngrowth of the individual layers. In case of the Ru buffer layer, the resolution of the \nstandard pole figure goniometer is not sufficient t o separate the reflections of the Al 2O3 \nsubstrate and the buffer layer. Therefore, reciproc al space mapping was applied in a \nhigh resolution Philips X'Pert MRD goniometer using  Cu-Kα radiation, a Ge mono- \nchromator in the incoming beam and 0.09° soller sli ts on the detector side. The effective \nmagnetic anisotropy of each film was determined by measuring the magnetic hysteresis \nwith a Quantum Design PPMS vibrating sample magneto meter. For this, the area en- \nclosed by the initial magnetization curves, recorde d in in-plane and out-of-plane field  4 orientation, was compared to the value obtained fro m an isotropic reference sample \n(prepared at room temperature on a Si substrate wit h amorphous SiN surface layer). The \nmagnetic domain structure was imaged with a Digital  Instruments Dimension 3100 \nscanning force microscope using high resolution MFM  tips with single side coating \n(SC-20 by SmartTip B.V). The tip magnetization in t he z-direction leads to a sensitivity \nto gradients in the z-component of magnetic stray f ields above the sample surface. \n \n \nIII. Results and Discussion \n \nA. Epitaxial growth of Co(0001) on Al 2O3 \n \nFig. 1 shows the XRD data for about 80 nm thick Co films deposited at different sub- \nstrate temperatures directly onto the sapphire subs trate. For temperatures of 100°C and         \n  \nFIG. 1. (Color online) X-ray diffraction patters of  Co films prepared at different \nsubstrate temperatures. \n  5 below only two sharp substrate reflections (Al 2O3 (0 0 0 6) and (0 0 0 12)) are visible. \nAt 200 °C two additional peaks evolve, which can be  identified with the  (0002) and \n(0004) reflection of the hcp-Co phase, demonstratin g a textured growth with the c-axis \nperpendicular to the surface. A further increase of  substrate temperature leads to an in- \ncreased intensity of the Co reflections, which leve ls off at about 400 °C. At 500 °C a \nslight shift of the second Co peak is visible, whic h can be attributed to the growth of fcc \nrather than hcp Co, in agreement with the transform ation temperature of 422 °C known \nfrom the equilibrium phase diagram of Co.  \n \nPole figure measurements on the Co (1 0 –1 1) refle ction (2 θCu  = 47.42°) in comparison \nwith the Al 2O3 (20-22) pole (2 θCu  = 46.17°) were performed on all films to determine  \nthe in-plane texture and a possible epitaxial relat ion to the substrate. Fig. 2a shows the \nresult obtained on an 80 nm thick film deposited at  400 °C, where both measurements \nare overlayed graphically. Besides the expected sha rp reflections of the sapphire single \ncrystal with threefold symmetry, the signal of the Co reflection is seen as six evenly \nspaced discrete poles arranged on a ring correspond ing to a 61° tilt away from the sub- \nstrate normal. This is in agreement with the c-axis  textured growth of the hexagonal Co \ngrain and a unique epitaxial orientation within the  basal plane. Furthermore a 30° rota- \ntion in the plane with respect to the substrate is observed. The full epitaxial relation is \ngiven as: \n  Co (0 0 0 1)[1 0 -1 0] // Al 2O3 (0 0 0 1)[1 1 –2 0] \n \n \n \n \n \n \n \n  6  \nThis growth mode holds for all films in which cryst alline reflections can be observed, \nindicating, that the epitaxial relation dictates th e grain orientation as soon as the sub- \nstrate temperature is sufficiently large to result in a film with hexagonal crystal struc- \nture. The texture spread as determined from the ful l width at half maximum (FWHM) \nof the (1 0 -1 1) pole manifests a rather small til t in out-of-plane direction of ∆Ψ = 3° \nand a rotational variation in-plane of ∆Φ  = 6°. Fig. 2b illustrates again the orientation \nrelation between sapphire substrate and Co film. No te that of course no lateral orienta- \ntion configuration can be deduced from the x-ray ex periments and the illustration is \nthus purely schematic. It shows, however, the size relation between the Al 2O3 and Co \nunit cells and allows to estimate the lattice misma tch to be about -9%. 7  \nThe epitaxial growth has to be understood from the matching symmetry between the 6 \nfold Al 2O3 surface and that of the hexagonal basal plane of h cp-Co, and from a tolerable \nlattice mismatch  and justifies the choice of the A l 2O3 (0001) substrate. Whether these \nconditions are decisive cannot be decided from this  study. Interestingly, in other works \nreporting hcp-Co films with perpendicular c-axis or ientation the films were grown on \n \nFIG 2. (Color online) (a) Pole figure measurements of Al 2O3 and Co, and (b) sketch \nillustrating the epitaxial relation between film an d substrate. \n  7 either Al 2O3 (11-20) [8] or onto rigid mica substrates,4 however a Ru (0001) buffer was \nutilized to support the desired c-axis texture. \n \n \nB. Epitaxial growth of Co(0001) on Ru(0001)/Al 2O3(0001) \n \nTo investigate the role of a Ru buffer layer on the  epitaxial growth of Co on sapphire, a \n5 to 10 nm thick Ru film was grown at 500 °C onto A l 2O3 (0001), before the Co layer \nwas deposited at 400°C. Judged from a work by Yamad a et al .,9where Ru was depos- \nited by sputtering at 500 °C on Al 2O3 (0001), an epitaxial growth with c-axis texture \nand 30° in-plane rotation is expected. Fig. 3a show s a pole figure measurement of a \nCo/Ru/Al 2O3 sample, where the Co (1 0 –1 1) pole, the Ru (1 0 –1 3) pole (2 θCu  = \n78.55) and the Al 2O3 (2 0 –2 2) pole are overlaid graphically. The iden tical 6 discrete \n+\nAl O2 3\na) b)\n \nFIG 3. (Color online) (a) Pole figure measurement, (b) XRD reciprocal space map on a \nlogarithmic intesity scale of the Al 2O3 (1 1 -2 9) reflection at Q x = 0.325 rlu, Q y = 0.5324  \nrlu with the neighboring Ru(1 0 -1 3) pole at Q x = 0.330 rlu, Q y = 0.5385 rlu. The recipr o- \ncal length unit rlu is defined as 2 λ/d. \n  8 poles measured at 2 θ/uniF020 = 78.55° were identified by Yamada et al . in their experiment as \nRu reflections, as no 6-folded symmetry was expecte d from the rhombohedral substrate. \nHowever, although a measurement at fixed 2 θ-value of 78.55° results in above pole \nfigure, a θ-2θ scan on these poles results in a peak at 2 θ/uniF020 = 77.2°, not explainable by \nhexagonal Ru, but by the (1 1 –2 9) Al 2O3 reflection, which, as a substrate reflection \nstill contributes to the x-ray intensity at 2 θ/uniF020= 78.55°. This explanation was easily proven \nby measuring a pole figure at 2 θ = 77.2° on an uncovered substrate, which lead to a n \nidentical pole figure as shown in Fig. 3a. It is cl ear, that the experiment of Yamada et \nal . cannot verify the claimed epitaxial relation, but  it is nevertheless a possible growth \nmode. Therefore, reciprocal space mapping was perfo rmed using a high resolution dif- \nfractometer in order to study the epitaxial relatio nship between Ru and the sapphire \nsubstrate in detail. The Al 2O3 (1 1 –2 9) peak having a 2 θ value of 77.22° (Cu-K α radia- \ntion) was chosen for these investigations. The resu lt of the measurement is shown in \nFig. 3b. The sharp peak with very high intensities corresponds to the Al 2O3 (1 1 –2 9) \nsubstrate reflection whereas the broader peak with low intensity was identified as Ru (1 \n0 –1 3). Additional measurements using different in -plane orientations of the substrate \nrevealed that the Ru peak has a 6-fold symmetry sim ilar to the Al 2O3 peak, and thus the \nepitaxial growth relation of Ru on Al 2O3 is determined correctly as: \n \n Ru(0 0 0 1)[1 0 -1 0]  //  Al 2O3(0 0 0 1)[1 1 –2 0] \n \nThe lattice mismatch between Ru and sapphire is –1. 5%. 10  The subsequent Co film now \ngrows in a 1:1 fashion onto the Ru buffer, as can b e seen in Fig. 3a from comparing the \n(hidden) Ru (1 0 –1 3) poles with Co (1 0 –1 1). Th e epitaxial orientation towards the \nsubstrate is thus identical to that observed on the  non-buffered sapphire substrate. The \nmismatch is slightly reduced to –8% and the overall  epitaxial relation reads: \n \nCo (0 0 0 1)[1 0 –1 0]  //  Ru(0 0 0 1)[1 0 -1 0]  //  Al 2O3(0 0 0 1)[1 1 –2 0]  9  \nThe c-axis texture and magnetocrystalline anisotrop y (see next section) of Co/Ru/Al 2O3 \nis not improved over that of Co films grown directl y on Al 2O3. Therefore, in the follow- \ning only simple Co/Al 2O3 films were investigated. A suitable hexagonal buff er, which \nmediates c-axis growth on sapphire, may however bec ome of interest for the deposition \nof more oxygen sensitive films, e.g. Rare-earth –Co  intermetallics, which need to be \nprotected with a diffusion layer. 11  \n \n \nC. Effective uniaxial anisotropy \n \n \nFig. 4 displays the magnetic hysteresis of two epit axial Co films prepared at room tem- \nperature and 400°C substrate temperature, respectiv ely. In both cases the magnetization \nshows only very little hysteresis but a large quali tative difference when measuring ei- \nther with the field parallel (in-plane: ip) or perp endicular (out-of-plane: oop) to the film \nplane. In both samples the in-plane direction leads  to a steeper hysteresis and a larger \nabsolute polarization over the whole field range, d espite the perpendicular uniaxial \n \nFIG 4. (Color online) Hysteresis loops of two 40 nm  thick epitaxial Co films pre- \npared at (a) room temperature and (b) 400°C showing  strong anisotropy when meas- \nured in two orthogonal directions (in plane and out  of plane). \n  10  magnetocrystalline anisotropy of the epitaxially gr owing Co grains. Thus the shape ani- \nsotropy dominates the overall anisotropy of the sam ple. However, this effective in-\nplane anisotropy reduces when films are grown at hi gher temperature, as can be qualita- \ntively deduced from the reduced saturation field in  oop direction (Fig. 4b). Due to the \ncompeting anisotropies none of the hysteresis measu rements are pure easy axis or hard \naxis procesess and the anisotropy can not be estima ted simply from that saturation field. \nTorque measurements performed in saturation would l ead to purely reversible rotational \nmagnetization processes and are therefore the recom mended choice for the analysis of \nanisotropies in such a case, but were not possible with the available experimental \nequipment. Therefore an alternative integral method  was used, which was tested to give \ncomparable values with a 3% precision.12  For that, initial curves in the first quadrant of \na magnetization loop are measured in both, ip and o op direction (Fig. 5a) and the en- \nclosed integral area is determined. A fully isotrop ic film (grown at room temperature on \nan amorphous buffer) serves as a reference sample f or a purely shape anisotropy driven \nbehavior.  \n \n \nFIG 5. (Color online) (a) First quadrant initial lo ops of a 40 nm thick textured Co \nfilm in comparison to an isotropic reference film; (b) summary of the effective \nmagnetocrystalline anisotropy as a function of depo sition temperature. \n  11  When defining the above mentioned area as ∫− dH )J J (oop ip , the shape anisotropy con- \ntributes positively to this value, whereas the perp endicular anisotropy arising from the \nmagnetocrystalline contribution of the textured Co- grains counts negatively. The area \nenclosed by the textured film is thus reduced over that of the isotropic sample. By sub- \ntracting the (in-plane) shape anisotropy (i.e. the demagnetizing energy K d) the remain- \ning perpendicular anisotropy can be evaluated. Fig.  5b shows the values for roughly 40 \nnm thick Co films prepared at different substrate t emperatures from RT to 500°C (full \ncircles), for films with thickness between 20 nm to  100 nm, all prepared at a substrate \ntemperature of 400°C (open squares), and for the is otropic reference sample (triangle). \nIn all cases, the such determined perpendicular ani sotropy depends on both, the uniaxial \nanisotropy constant K 1 of the individual grains and the c-axis distributi on function of \nthe whole grain ensemble in the film and is thus al so denoted as the texture averaged \nanisotropy constant <K 1>text . As expected, <K 1>text  of the isotropic reverence sample is \nclose to zero. For films prepared on sapphire subst rate <K 1>text  increases steadily with \nthe substrate temperature and reaches a maximum of about 0.45 – 0.5 MJ/m 3 at a depo- \nsition temperature of 400°C. As in the pole figure measurements no significant change \nof the c-axis spread with substrate temperature was  observed, the increasing <K 1>text  \nmust be largely due to an improved uniaxial anisotr opy K 1 of the individual Co grains. \nThe maximum at 400°C reaches the anisotropy value f or hexagonal single crystal Co, \nthus the anisotropy of the Co grains in the epitaxi al film is fully developed. Comparably \nhigh values are furthermore obtained for all films within the thickness series (open \nsquares), so that in the later interpretation of th e domain structure of these films a uni- \naxial anisotropy constant of single crystal hcp-Co can be assumed, irrespective of the \nfilm thickness. No systematic dependency of <K 1>text  on the film thickness is observed, \nindicating that surface anisotropy effects do not y et play a role. This is in agreement \nwith an estimated surface contribution of less than  20% (10%) for films thicker than 10 \nnm (20 nm). For surface anisotropy contributions se e C. Chappert et al .13  The 40 nm \nthick film prepared at a substrate temperature of 5 00 °C has a <K 1>text  value reduced   12   \n \nover that of single crystal Co. This is explained b y the occurrence of the fcc-Co phase \nobserved in the x-ray investigation. \n \n \nD. Domain structure  \n \n        \n \nFIG 6. (Color online) MFM images of four films with  30, 36, 44 and 80 nm thick- \nness. For films of 36 nm and above clear perpendicu lar MFM contrast on a length \nscale below 100 nm is observed. The film with 30 nm  thickness shows a cross tie \nwalls – a clear indication of in-plane magnetizatio n (see text). \n  13  The magnetic domain structure of four exemplary Co films with thickness 30 nm, 36 \nnm, 44 nm, and 80 nm is seen in Fig. 6. For films o f 36 nm and above a pattern of \nstripes with alternating dark and bright contrast i s observed. The stripe width varies \nfrom 60 to 100 nm. They run roughly parallel on a l ength scale of some micrometer, but \ntheir orientation is randomly distributed on the la rger scale of the 10 µm x 10 µm sized \nimage. The contrast can be interpreted as coming fr om micron sized domains with a \npredominant in-plane component due to the shape ani sotropy, which are oriented isot- \nropically in all directions. The perpendicular magn etocrystalline anisotropy of the epi- \ntaxially grown Co grains leads to an additional per pendicular magnetization component, \nwhich alternates on a smaller length scale and give s rise to the observed stripe pattern. \nSuch domains have been suggested by Saito et al. 14  for thin Ni-Fe films with (shape \ndominated) in-plane magnetization and a small addit ional strain induced perpendicular \nanisotropy. In films with smaller thickness no stri pe domains can be observed. As the \nlateral resolution of the MFM measurement is by far  not reached for the film with the \nsmallest imaged domain width, the vanishing domain contrast has to be interpreted as a \ndying out of the perpendicular magnetization compon ent. Instead, features with a length \nof some micrometers are seen in the MFM image which  are correlated with topography \nand are thought to arise from the domain boundaries  of the mentioned in-plane do- \nmains. In some areas cross-tie walls are observed ( Fig. 6, t = 30 nm), which are known \nin soft magnetic thin films with in-plane magnetiza tion and arise from a sequence of \n90° Néel walls to avoid energetically more costly 1 80° Néel walls. By examining a \nlarge number of films in the thickness range betwee n 20 nm and 50 nm the critical \nthickness for stripe nucleation is found as t cr = (36 ± 3) nm, with the error being deter- \nmined by the precision of the thickness measurement . The average stripe domain width \nfor all films with thickness 36 nm and larger is su mmarized in Fig. 7. Thinner films, \nwhich do not develop a stripe domain pattern, are l isted with an arbitrarily chosen do- \nmain width of zero to demonstrate the sharpness of the transition. Within the accuracy \nof the measurement the domain width increases monot onously from about 60 nm to 95 \nnm. (red squares). Included in Fig. 7 are the data by Hehn 3 (black triangles), which  14  cover the film thickness range from 50 nm to 150 nm . In the overlapping thickness \nrange the domain width data compare well. \n \n \n \nA good understanding of the domain structure is ach ieved if the experimentally ob- \nserved thickness dependency can be explained by dom ain theory, i.e. by a prediction of \nan average domain width as a function of the film t hickness, assuming a simplified, but \nrealistic domain structure, that considers the mate rials parameters of the investigated \nmagnetic thin film and carries the characteristic f eatures of the domain pattern. Fur- \nthermore, such a model domain structure has to be e nergetically favored over other pos-      \n  \nFIG 7. (Color online) Average stripe domain width a s a function of the film thick- \nness in comparison with predictions from analytical  domain models. Films which do \nnot develop stripe domains are displayed with an ar bitrarily set domain width of \nzero. \n \n  15  sible domain configurations. The stripe domain mode l by Saito et al.  considers a case of \ncompeting anisotropies very similar to that of the epitaxial Co films. It predicts a pure \nin-plane magnetization for very thin films and the occurrence (nucleation) of the al- \nready mentioned stripes above a critical film thick ness. Critical thickness and the do- \nmain width at the stripe nucleation depend on the m aterials parameters, the saturation \npolarization J s, the perpendicular anisotropy constant K u and the exchange constant A. \nFor J s = 1.8 T, K u = 0.45 MJ/m 3 and A = 28·10 -12  J/m critical thickness and domain \nwidth are calculated to be t cr  = 37 nm and w cr  = 52 nm, respectively. The predicted on- \nset (critical thickness) of stripe nucleation agree s very well with the experimental be- \nhavior. The observed domain width is about 20% too large. A further comparison is, \nhowever, not possible, as at present no theory of t he stripe domain width as a function \nof the film thickness is available. So far, only fe w cases have been treated numeri- \ncally 15,16  and found an increased domain width with increasin g film thickness. It is thus \ndesirable to test simpler models and evaluate, to w hich extend they can be applied to the \npresent situation. A domain model for parallel stri pes with alternating perpendicular \nmagnetization has been developed by Kittel,17  which is valid under the assumption of a \nnegligible domain wall width, a pure perpendicular magnetization, and a film thickness \nlarger than the domain width. 18  Although its application to a domain structure wit h \nstrong in-plane component is doubtful, the square r oot functional dependency of do- \nmain width of the film thickness was used in Hehn e t al. 3 to compare it with their ex- \nperimental data. Good agreement was found with a va lue of the exchange stiffness con- \nstant A = 85·10 -12  J/m, but neither is this value reasonable nor is a n agreement between \nmodel calculations and experiment satisfactory, if essential characteristics of the model \nare not met by the reality. If domain width calcula tions based on the Kittel model are \nperformed with a realistic exchange stiffness const ant A = (10 – 28)·10 -12  J/m (hatched \narea in Fig. 7), it becomes obvious, that the exper imental behavior can not be described. \nA more realistic model has to include the in-plane component present in the domain \nstructure. The calculations by Kittel for perpendic ular stripes have been modified by \nMalek and Kambersky 19  to cover also films with thickness t comparable to  the domain  16  width w, and by Kooy and Enz, 20  to describe films which possess an additional in-p lane \ncomponent in the magnetization. In the latter ansat z, a ratio Q = K u/K d <1 between per- \npendicular uniaxial anisotropy and shape anisotropy , stabilizes a stripe domain struc- \nture, where the magnetization in each domain is til ted away from the perfect perpen- \ndicular orientation expected in films with Q ≥ 1. Consequently, the equilibrium domain \nwidth of such a magnetization structure possesses a  modified dependency on the film \nthickness. The dashed curve in Fig. 7 is a calculat ion after Malek and Kambersky with \nJs = 1.8 T, K u = 0.45 MJ/m 3 and A = 28·10 -12  J/m. For large film thickness it approaches \nthe square root dependency of the Kittel model, as expected. For smaller film thick- \nnesses the refined model leads to increasingly smal ler domain values with decreasing \nthickness and but also predicts a strong upturn in the equilibrium domain width for film \nthicknesses below 25 nm. When the shape anisotropy is considered by performing the \ncalculations after Kooy and Enz with the experiment ally given quality factor Q = 0.35 \nthe domain width function modifies again (solid lin e). The minimum domain width is \nreduced and it occurs at a lower film thickness val ue. The increase in domain width \nwith increasing film thickness is, however, steeper  as in the pure perpendicular case. \nFor t = 150 nm a 20% larger domain width is expecte d. Including an in-plane compo- \nnent in the predominant perpendicular domain struct ure thus leads to a more realistic \ndescription of the experimentally observed stripe d omains. Nevertheless, the agreement \nis still far from being satisfactory. Qualitatively , in the stripe domain model by Saito et \nal.  and in the experimental observation, at low film t hickness the stray field minimiza- \ntion leads to a pure in-plane domain structure, whe reas by insisting on a perpendicular \nmagnetization component such as in [19] and [20] th e domains simply grow larger. \nQuantitatively, in the region where stripe domains are observed, their domain width is \nstill larger than predicted by Kooy and Enz. Possib le reasons for the discrepancy are \ndeviations of the experimentally observed domain wi dth from the equilibrium width of \nlong parallel stripes considered in the models. Suc h differences in the domain width of \nmaze-like stripe domains and parallel stripe domain s are e.g. reported in Co/Pt multi- \nlayers with strong perpendicular anisotropy,21  but are not found in the Co films studied  17  by Hehn et al. 3 A more serious reason for the discrepancy between experiment and \nmodel is thought to be the assumption of a homogene ous magnetization state within \neach domain and a vanishing domain wall width. Impr ovements over such simplifying \nassumptions are so far only attempted by micromagne tic methods. \nMicromagnetic simulations of stripe domains in thin  films have only been performed \nfor few cases, such as Co-Pt alloy films 15  and most recent for Co films.16  In that nu- \nmerical study, stripe domain nucleation and the evo lution of domain width with film \nthickness was calculated based on an energy minimiz ation scheme (OOMMF) for a \nfixed set of Co material parameters (J s = 1.78 T, K u = 0.46 MJ/m 3 and A = 13·10 -12  \nJ/m). They find a critical thickness for stripe nuc leation of t cr  = 22 nm, which is smaller \nthan the experimentally observed value of 36 nm and  the value calculated after Saito et \nal.  (t cr  = 37 nm).14  At t cr , the numerically obtained stripe domain width is w cr  = 35 nm \nand above this critical thickness the equilibrium d omain width is observed to closely \nfollow a square root dependency as a function of fi lm thickness, which is quantified as \n)nm ( d nm 5 . 7 w21⋅ ⋅ = . This dependency nicely matches the experimentally  observed \nincrease in domain width concerning the functional dependency, but still leaves a dis- \ncrepancy between the simulated and measured absolut e values in domain width. Given \nthe large spread in reported values for the exchang e constant A in Co, better quantita- \ntive agreement might be achieved with slightly modi fied materials constants in the nu- \nmerical simulations. The predictions from the diffe rent analytical models and the mi- \ncromagnetic simulation are compared once more with the experimental observations in \ntable I.  \n \n \n \n \n \n \n  18  Table I: Critical thickness for stripe nucleation ( if available) and equilibrium stripe do- \nmain width for film thicknesses of 36 nm and 80 nm as obtained from the experiment, \nthe four analytical models and the micromagnetic si mulation. In the evaluation of the \nanalytical models the materials parameters were cho sen as K u = 0.45 MJ/m 3, J s = 1.8 T \nand A = 28 x 10 -12  J/m. The numerical results were taken from referen ce 15 and were \nbased on K u=0.46 MJ/m 3, J s = 1.78 T and A= 13 x 10 -12  J/m. \n \n \n \n \nIV. Conclusions \n \nCo films have been grown epitaxially on pure and Ru -buffered Al 2O3 (0001) substrates \nwith hexagonal crystal structure and perpendicular alignment of the c-axis with respect \nto the film plane. The epitaxial growth of Ru on Al 2O3 has been confirmed by high \nresolution reciprocal space mapping, and regular po le figure measurements revealed a \nsingle epitaxial growth mode for hcp-Co on both, pu re and buffered substrate with a 30° \nrotation in the basal plane. The texture averaged p erpendicular anisotropy reaches the \nmagnetocrystalline anisotropy value of bulk hcp-Co,  when the films are prepared at a \nsubstrate temperature of 400°C. Stripe domain nucle ation is found for films with thick- \nness of 36 nm and above, in agreement with the theo ry of Saito et al . Existing analytical \ndomain models can only describe the functional depe ndency of the domain width on \nfilm thickness, but fail to reach satisfactory quan titative agreement. Still, including an  experiment Saito Kittel Malek Kooy numerical \nCritical thickness \n(nm)  36 37 - - - 22 \nDomain width (nm) \n(t = 36 nm) 63 52 39 47 49 45 \nDomain width (nm) \n(t = 80 nm) 95 - 58 61 71 67 \n  19  in-plane tilt originating from the large demagnetiz ing energy, improves the perpendicu- \nlar domain model after Kittel substantially. It is expected, that further micromagnetic \nsimulations are required to fully describe the doma in formation in this system with \ncompeting perpendicular magnetocrystalline anisotro py and in-plane shape anisotropy, \nwhich will eventually allow to deduce materials par ameters from the evaluation of \nnanoscaled domain images.   20  References \n                                                 \n1  J.U. Thiele, L. Folks, M.F. Toney, D. Weller, J. Ap pl. Phys. 84 , 5686 (1998). \n2  V. Neu, S. Melcher, U. Hannemann, S. Fähler, L. S chultz, Phys. Rev. B 70 , 144418 \n(2004).  \n3  M. Hehn, S. Padovani, K. Ounadjela, J.P. Pucher.,  Phys. Rev. B 54 , 3428 (1996). \n4  D M. Donnet, K.M. Krishnan, Y. Yajima., J. Phys. D: Appl. Phys. 28 , 1942 (1995). \n5  J. Brandenburg, V. Neu, H. Wendrock, B. Holzapfel , H.-U. Krebs, S. Fähler Appl. \nPhys. A: Mater. Sci & Proc 79 , 1005 (2004). \n6  V. Neu, S. Fähler, A. Singh, A.R. Kwon, A.K. Patr a, U. Wolff, K. Häfner, B. Hol- \nzapfel, and L. Schultz, J. Iron Steel Res. Int. 13 , 102 (2006). \n7  nm  76 . 4 asapphire = , nm  51 . 2 aCo = , with nm  35 . 4 a3Co = ⋅ the relative difference \ncalculates to %4 . 9 35 . 4 / ) 76 . 4 35 . 4 ( −= − =δ .  \n8  M. Hehn, K. Ounadjela, S. Padovani, J. P. Bucher,  J. Arabski, N. Bardou, B. Bar- \ntenlian, C. Chappert, F. Rousseaux, D. Decanini, F.  Carcenac, E. Cambril, and M. F. \nRavet, J. Appl. Phys. 79 , 5068 (1996). \n9  S. Yamada, Y. Nishibe, M. Saizaki, H. Kitajima, S . Ohtsubo, A. Morimoto, T. Shi- \nmizu, K. Ishida, and Y. Masaki, Jpn. J. Appl. Phys.  41 , L206 (2002). \n10   nm  76 . 4 asapphire = , nm  708 . 2 aRu = , nm  69 . 4 a3Ru = ⋅    \n11   M. Seifert, L. Schultz, V. Neu, Epitaxial SmCo 5 film with perpendicular anisotropy, \nto be published \n12    V.W. Guo, B. Li, X. Wu, G. Ju, B. Valcu, D. Welle r, J. Appl. Phys 99 , 08E918 \n(2006). \n13   C. Chappert, J. Magn. Magn. Mater 93 , 319 (1991). \n14   N. Saito, H. Fujiwara, Y. Sugita., J. Phys. Soc. Jpn 19 , 1116 (1964).  21                                                                                                                                                 \n15   M. Ghidini, G. Zangari, I. L. Prejbeanu, G. Patta naik, L. D. Buda-Prejbeanu, G. Asti, \nC. Pernechele, and M. Solzi, J. Appl. Phys.  100 , 103911 (2006). \n16   M. Kisielewski,  A. Maziewski, V. Zablotskii, J. Magn. Magn. Mater. 316 , 277 \n(2007). \n17   C. Kittel, Phys. Rev. 70 , 965 (1946). \n18   To distinguish such perpendicular domains from th e formerly discussed stripe do- \nmains with predominant in-plane magnetization they are also often named “band do- \nmains”, see e.g. A. Hubert and S. Schäfer, Magnetic Domains (Springer, Berlin, 1998). \n19   Z. Malek and V. Kambersky, Czech. J. Phys. 8, 416 (1958). \n20   C. Kooy and U. Enz, Philips Res. Rep. 15 , 7 (1960). \n21   O. Hellwig, A. Berger, J.B. Kortright, E.E. Fulle rton, J. Magn. Magn. Mater. 319 , 13 \n(2007). "    },    {        "title": "0902.3684v1.Residual_attractive_force_between_superparamagnetic_nanoparticles.pdf",        "content": "arXiv:0902.3684v1  [cond-mat.other]  20 Feb 2009Residual attractive force between\nsuperparamagnetic nanoparticles\nJohn F. Dobson and Evan MacA. Gray\nNanoscale Science and Technology Centre, Griffith Universit y,\nNathan, Queensland 4111, Australia\nNovember 1, 2018\nAbstract\nAsuperparamagneticnanoparticle(SPN)isananometre-siz edpiece\nof a material that would, in bulk, be a permanent magnet. In th e\nSPN the individual atomic spins are aligned via Pauli effects i nto a\nsingle giant moment that has easy orientations set by shape o r mag-\nnetocrystalline anisotropy. Above a size-dependent block ing tempera-\ntureTb(V,τobs) , thermal fluctuations destroy the average moment by\nflippingthe giant spin between easy orientations at a rate th at is rapid\non the scale of the observation time τobs. We show that, depite the\nvanising of the average moment, two SPNs experience a net att ractive\nforce of magnetic origin, analogous to the van der Waals forc e between\nmolecules that lack a permanent electric dipole. This could be rele-\nvant for ferrofluids, for the clumping of SPNs used for drug de livery,\nand for ultra-dense magnetic recording media.\n1 Introduction\nIn many areas of physics, forces are effectively suppressed in the interac-\ntion between separated fragments of matter, because of the ne utrality of\neach fragment with respect to the appropriate charge quantity. Nevertheless\n”residual” forces still occur between these fragments, typically w ith a decay\n(as a function of the spatial separation Dbetween the fragments) that is\ndifferent from that of the ”bare” interaction.\n1For example, ordinary matter consisting of atoms and molecules is ty pi-\ncally neutral with respect to electrical charge, but two well-separ ated charge-\nneutral fragments always experience at least the van der Waals or dispersion\ninteraction. This is a residual force that arises because the zero- point mo-\ntions of the electrons on the two fragments are correlated via the Coulomb\ninteraction, leading to a non-zero time-averaged force of Coulomb ic origin,\ndespite overall charge neutrality of each fragment. For neutral molecules\ndistantD, this leads to an interaction energy varying as −D−6. This is to\nbe compared with the bare Coulomb interaction proportional to Q1Q2D−1\nthat acts between between fragments with nonzero electric char gesQ1,Q2.\n(D−6is replaced by D−7whenDis large enough that retardation of the\nelectromagnetic interaction needs to be considered [1, 2]).\nSimilarly the nuclear force between two nucleons has sometines been re-\ngarded as a residual color interaction between color-neutral obj ects.\nHere we propose a similar residual force, of magnetic dipolar origin, a ct-\ning between two ”superparamagnetic nanoparticles (SPNs)”. By t his we\nmean that each nanometre-sized particle is composed of a material that is\nferromagnetic in its bulk state [3, 4]. Typically at the temperatures o f in-\nterest, the elementary electron spins inside an individual nanopart icle re-\nmain locked together by the microscopic exchange interaction, yield ing ef-\nfectively a single giant spin with a magnetic dipole moment d0. If the di-\nrections of the giant moments remain steady over time, two such na noparti-\ncles experience a conventional magnetic dipole-dipole energy propo rtional to\nd(1)\n0d(2)\n0f(θ1,φ1;θ2,φ2)/R3. HereRis the spatial separation of the nanopar-\nticles, and fis a dimensionless function of the angles between each fixed\nmoment and the vector /vectorRjoining the spatial locations of the nanoparticles.\nHowever each particle has one or more ”easy axes” in directions det ermined\nby magnetocrystalline or, more typically, shape anisotropy. The lat ter effect\narises in the strong angular dependence of the magnetostatic self -energy of\na non-spherical magnetised particle. We will consider the simplest ca se, in\nwhich the particle is sufficiently elongated that it has a single easy axis, i.e.\ndominant uniaxial shapeanisotropy. Then theenergyofasingle nan oparticle\nis lowest when its giant spin (dipole moment, /vectord) lies parallel or antiparallel\nto this easy axis. Because the energy barrier E0for rotation of /vectordbetween\neasy orientations (not mechanical rotation of the particle) derive s from the\nmagnetic self-energy of the nanoparticle, it decreases with decre asing volume\nof the nanoparticle. For very small particles, therefore, the pro jection of /vectordon\n2a measurement axis averages to zero over time, because of repea ted thermal\nflipping of the giant spin [3], caused by thermal agitation from the hea t bath\n(e.g. a fluid or solid matrix) that surrounds the nanoparticle. Thus o n time\naverage the nanoparticle is ”neutral” i.e. it has a zero magnetic mome nt.\nWhen the thermal agitation of the giant spin is insufficient to flip it be-\ntween easy orientations within the observation time, τobs, the nanoparticle\nis ”blocked”, i.e. apparently frozen as to its magnetism. This occurs below\nthe blocking temperature of this nanoparticle, Tb, which depends on E0and\ntherefore on the volume Vof the nanoparticle. If the relaxation time of /vectord\nover the barrier E0isτ, thenTbis defined by τ(Tb) =τobs. Blocking is thus\na purely dynamic phenomenon: extending the observation time, or lo wing\nthe frequency, lowers Tband vice-versa [3]. For the present case of SPNs\nsuspended in a fluid, the observation time τobswill be a relevant time for\nmechanical motion of the SPN through the fluid. - e.g. a rotational o r trans-\nlational diffusion time. Note that the direct dipolar magnetic interact ion\nbetween SPNs could in principle lead them to clump. However when T > T b\nthemotionof theSPNs throughthe fluid will not ”see” thebare dipola rmag-\nnetic interaction between the SPNs, as it has been averaged away b etween\nattractive and repulsive values during the thermal flipping of the sp ins. It\ncould lead to additional Brownian type of damping and difusion of cour se,\nbut we show here that there is also a net attractive force between SPNs even\nabove the blocking temperature.\nThe destruction of the permanent magnetic moments by thermal fl ucta-\ntions is highly undesirable in the case of a magnetic data recording med ium,\nwhere very fine magnetic particles in the nanometer size range will be needed\nin order to pack the magnetically stored data as densely as possible f or the\nnext generation of devices. The thermal destruction of the perm anent mo-\nments means that data cannot be stored over long times.\nOn the other hand, as will be discussed below, the same thermal flipp ing\noccuring for T > T bis beneficial in the case of nanoparticles deliberately\nsuspended in human blood as carriers for drug or thermal therapie s, since\nnow the clumping of the nanoparticles from magnetic dipole interactio n is\nsuppressed because each particle has effectively a zero magnetic m oment.\nThe strong clumping that would occur for fully ferromagnetic partic les from\ntheirR−3dipole-dipole interactions could be clinically dangerous, poten-\ntially causing blockage of blood vessels, difficulty of elimination etc. We w ill\nshow below, however, that despite the vanishing of the average ind ividual\nmoments, there is a residual attractive interaction between two s uperparam-\n3agnetic nanoparticles separated by distance R,that falls off as ( const)/R6. It\nis the magnetic analog of the van der Waals or dispersion force that a rises via\nthe Coulomb interaction between fluctuating electricdipoles on two electri-\ncally neutral molecules [5] lacking permanent dipole moments. This res idual\nforce could also lead to clumping of the nanoparticles, and so its analy sis\ncould be signifiant in modern magnetic-particle therapies [4].\n2 Simple preliminary model\nThe model described here is based on an argument frequently used to explain\ntheattractivevanderWaalsenergyproportionalto −R−6thatarisesbetween\ntemporary electric dipoles occurring on a pair of electronically neutr al atoms\nseparated by distance R(see e.g. [5]). It is not rigorous derivation, but may\nhelp to elucidate the more careful and general mathematical trea tment to be\nprovided in later Sections. Consider two superparamagnetic nanop articles\nSPN1 and SPN2 as defined above. While averaging to zero over time as\ndescribed above, the magneticmoment /vectord(1)onSPN1 canexhibit ashort-lived\nthermal (or quantal) fluctuation so that its value /vectord(1)(t) is nonzero at some\nparticular time t. For simplicity we will assume that only magnetizations of\nSPN1 and SPN2 along one axis (say ˆ z) are possible so that /vectord(1)(t) =d(1)(t)ˆz,\nand we will consider the case that the spatial separation /vectorRbetween SPN1\nand SPN2 is parallel to ˆ x. Then the spontaneous moment /vectord1(t) produces a\ndipolar magnetic induction (B-field)\n/vectorb(2)(t) =−µ0\n4πR−3d(1)(t)ˆz\nat the position of SPN2. Responding to this field, SPN2 produces its o wn\nmagnetic moment\n/vectord(2)(t) = ¯χ(2)/vectorb(2)(t) =−χ(2)µ0\n4πR−3d(1)(t)ˆz (1)\nwhere ¯χ(2)is the dynamic magnetic susceptibility of SPN2, assumed for now\nto represent an instantaneous response to the field. (Note that here ¯χrep-\nresents the response of the total magnetic moment of the SPN to a small\napplied magnetic induction /vectorb. By contrast, the symbol χis normally used\nfor the response of the magnetic moment per unit volume to a small applied\nmagnetic field /vectorh. Thus for a single SPN of volume V,\nd= ¯χb;d\nV=χh;χ=µ0¯χ\nV(2)\n4The dipole (1) in turn produces a dipolar magnetic induction back at th e\nposition of SPN1:\n/vectorb(1)(t) =−µ0\n4πR−3d(2)(t)ˆz=/parenleftbigg\n−µ0\n4πR−3/parenrightbigg/parenleftbigg\n−χ(2)µ0\n4πR−3d(1)(t)/parenrightbigg\nˆz\n=/parenleftbiggµ0\n4π/parenrightbigg2\nR−6χ(2)d(1)(t)ˆz.\nThe interaction energy of this back-field with the original moment /vectord(1)(t) is\nE=−/vectorb(1)(t)./vectord(1)(t)\nand this energy has a time or thermal ensemble average\n/angbracketleftE/angbracketright=−/parenleftbiggµ0\n4π/parenrightbigg2\nR−6χ2/angbracketleftig\n(d(1)(t))2/angbracketrightig\nwhich is non-zero because/angbracketleftig\n(d(1)(t))2/angbracketrightig\n/negationslash= 0 even though/angbracketleftig\nd(1)(t)/angbracketrightig\n= 0.\nThis negative energy produces, upon differentiation with respect t oR, a\nnet time-averaged attractive force between SPN1 and SPN2 that falls off as\nR−7.\nThe above simplified theory produces the basic physics andthe R−7force,\nbut it glosses over a number of issues, such as the role of entropic e ffects at fi-\nnitetemperature, thetensor natureofthemagneticdipole-dipole interaction,\nthe quantal aspects of the problem, and the retardation of the e lectromag-\nnetic field. Also, the response χ2has been assumed to be instantaneous,\nwhereas there can be a strong and important frequency depende nce (time-\ndelayed aspect) to the linear response of a SPN. All of these consid erations\nare treated in detail in the theory given the next Section.\n3 Detailed theory\nThe magnetic dipolar energy (hamiltonian) between two particles with mag-\nnetic dipoles /vectord(1)and/vectord(2), separated in space by a nonzero vector /vectorR=RˆR,\nis of form\nH(12)=−R−33/summationdisplay\nij=1d(1)\niTij(ˆR)d(2)\nj (3)\nwhere\nTij=µ0\n4π3RiRj−δijR2\nR2.\n5We assume that we are above the blocking temperature, T > T b, i.e. that\nthe temperature is high enough (compared with the anisotropy ene rgy bar-\nrier), that each isolated giant magnetic dipole has zero thermal exp ectation\ntaken over the time-scale of interest\n</vectord(1)>0=/vector0 =</vectord(2)>0.\nThe theory to be developed here is meaningful provided that the th ermal\nfluctuations of the moment occur on a time-scale τthat is short compared to\nthe time τobs≡Tmechfor the nanoparticle to change its spatial position (or\nphysical angular orientation) appreciably, within its fluid medium. Und er\nthese conditions we will derive a residual attractive force between the two\nsuperparamagnetic nanoparticles, that could for example be used to study\nresidual clumping effectsinfluidsuspension attemperatures above theblock-\ning temperature .\nThe quantum-thermal expectation, denoted < >, of the interaction en-\nergy between the giant spins is\nE(12)≡< H(12)>=−R−33/summationdisplay\nij=1Tij(ˆR)< d(1)\nid(2)\nj>\nHowever at finite temperature it is not this energy but the corresp onding\nthermal Helmholz free energy\nA=< H(12)>−TS\nthat must be considered, where Sis the entropy. We achieve an expression\nforAvia a Feynman-theorem argument in Appendix A for a classical treat -\nment of the fluctuations, and in Appendix B for the fully quantal cas e. In\neither case the result is\nA(λ= 1,T,R)−A(λ= 0,T,R) =/integraldisplay1\n0E(12)\nλdλ\nλ(4)\nHere the subscript λmeans that the quantity is evaluated in the thermal\nensemble with modified interaction\nH(12)\nλ=λH(12)=−λR−33/summationdisplay\nij=1d(1)\niTij(ˆR)d(2)\nj,0≤λ≤1 (5)\nE(12)\nλ=< H(12)\nλ>λ. (6)\n6Since the coupling will be zero (equivalent to λ= 0) at infinite separation\nR→ ∞, we can write Eq (4) as an expression for just the free energy of\ninteraction between the two nanoparticles:\nA(T,R)−A(T,R→ ∞) =/integraldisplay1\n0E(12)\nλdλ\nλ\n=−R−33/summationdisplay\nij=1Tij(ˆR)/integraldisplay\n< d(1)\nid(2)\nj>λdλ(7)\nTheproblemnowreducestothecalculationoftheequal-timecross- correlation\nfunction < d(1)\nid(2)\nj>λbetween the moments in a thermal ensemble with λ-\nreduced interaction.\nThe equal-time correlation function < d(1)\nid(2)\nj>λcan be recovered from\nthe time Fourier transform\ngλ(ω) =/integraldisplay∞\n−∞Gλ(t)exp(−iωt)dt=/integraldisplay∞\n−∞< d(1)\ni(0)d(2)\nj(t)>λexp(−iωt)dt\nof the time-displaced correlation function Gλ,\nGλ(t)≡< d(1)\ni(0)d(2)\nj(t)>λ.\nWe can use the finite-temperature fluctuation-dissipation theore m (see\ne.g. [6]) to relate the fluctuation quantity < d(1)\nid(2)\nj>to the dipole-dipole\nresponse function ¯ χ(12)\nijof the combined interacting system, defined in Eq\n(10) below:\n< d(1)\nid(2)\nj>λ,ω+< d(2)\njd(1)\ni>λ,ω=2¯h\n1−exp(β¯hω)Im/braceleftig\n¯χ(12)\nij,λ(ω+i0)+ ¯χ(21)\nji,λ(ω+i0)/bracerightig\nwhereβ=1\nkBT. Then\n< d(1)\nid(2)\nj+d(2)\njd(1)\ni>λ,equal time =1\n2π/integraldisplay∞\n−∞gλ(ω)exp(iωt)\n=/integraldisplay∞\n−∞¯h\nπ(1−exp(β¯hω))Im/braceleftig\n¯χ(12)\nij,λ(ω+i0)+ ¯χ(21)\nji,λ(ω+i0)/bracerightig\ndω\nThis can also be expressed as a Matsubara sum by closing upwards in t he\ncomplex ωplane, using Cauchy’s theorem to obtain a sum of residues at the\npolesωn=iun=i2nπ/(β¯h), but we will not make explicit use of this here.\n7The interaction energy Eand Helmholtz free energy Athen become\nE(12)\nλ=−λR−33/summationdisplay\nij=1Tij(ˆR)/integraldisplay∞\n−∞¯h\nπ(1−exp(β¯hω))Im/braceleftig\n¯χ(12)\nij:λ(ω+i0)/bracerightig\ndω(8)\nA(R)−A(∞) =/integraldisplay1\n0E(12)\nλdλ\nλ(9)\nWe assume we know the dipole responses ¯ χ(1)\nij,λ(ω),¯χ(2)\nij,λ(ω) of each iso-\nlated giant dipole SPN1, SPN2 to an external B field /vectorbsuch that\nd(1)\ni(ω)exp(−iωt) =/summationdisplay\nj¯χ(1)\nij(ω)b(1)exp(−iωt).\nThese individual responses must express the known superparama gnetic prop-\nerties of individual systems. In general it should also describe any B rownian\ntumbling aspects of the response, in the case that the time scale of these\ntumbling motions overlaps that of the magnetic reponse behaviour o f each\nSPN. Fornow we assume that thetumbling is slow so that onlythe magn etic\nresponse of a SPN oriented in a fixed spatial orientation is required t o appear\nin ¯χ. (The interaction energy may of course depend on the details of th is\norientation, which will be manifested in the particular values of ¯ χ(12)\nijin the\nchosen cartesian frame.) In Appendix C we discuss a simple model for the ¯χ\nof a single isolated SPN. However, to calculate the interaction of two SPNs,\nEq (8) requres knowledge of the cross-response function (cros s-susceptibility)\n¯χ(12)\nλfor the interacting pair of SPNs. This is defined as the linear respons e\nof SPN1’s moment to an alternating B field that acts upon SPN2 only:\nd1\ni(ω) = ¯χ(12)\nij,λ(ω)b(2)\nj, (10)\nwhere the subscripts i,jlabel caretesian components of the vectors. To\ncalculate ¯ χ(12)we now consider the slightly more general situation where\nindependently-specified small external B fields /vectorb(1)exp(−iωt),/vectorb(2)exp(−iωt)\nare applied to the individual dipoles, in the presence of the dipolar cou pling\nbetween the two systems.\nIn time-dependent mean-field theory (RPA), the equations of mot ion of\nthe coupled systems are (all at arbitrary frequency ωand with Einstein sum-\nmation convention for repeated indices):\nd(1)\nα= ¯χ(1)\nαµ/parenleftig\nb(1)\nµ+λR−3Tµβd(2)\nβ/parenrightig\n(11)\nd(2)\nβ= ¯χ(2)\nβε/parenleftig\nb(2)\nε+λR−3Tεγd(1)\nγ/parenrightig\n(12)\n8These equations describe the evolution of each giant spin in an effect ive B\nfield containing a time-dependent contribution due to the polarizato n of the\nother giant spin. Using (12) to eliminate d(2)\nβin(11), we get\nd(1)\nα= ¯χ(1)\nαµ/parenleftig\nb(1)\nµ+λR−3Tµβ¯χ(2)\nβε/parenleftig\nb(2)\nε+λR−3Tεγd(1)\nγ/parenrightig/parenrightig\n/parenleftig\nδαγ−λ2¯χ(1)\nαµR−3Tµβ¯χ(2)\nβεR−3Tεγ/parenrightig\nd(1)\nγ= ¯χ(1)\nαµb(1)\nµ+λ¯χ(1)\nαµR−3Tµβ¯χ(2)\nβεh(2)\nε\nThen for b(1)\nµ= 0 (i.e. an external oscillating B field applied only to moment\nSPN2) we have\nd(1)\nα=χ(12)\nαε,λb(2)\nε\nwhere\n¯χ(12)\nαε,λ=/parenleftig\nε−1\nλ/parenrightig(12)\naβ,λS(12)\nβε (13)\nS(12)\nβε= ¯χ(1)\nβµλR−3Tµν¯χ(2)\nνε\nελ=\n1−λ2R−6¯χ(1)\n1µTµβ¯χ(2)\nβεTε1−λ2R−6¯χ(1)\n1µTµβ¯χ(2)\nβεTε2−λ2R−6¯χ(1)\n1µTµβ¯χ(2)\nβεTε3\n−λ2R−6¯χ(1)\n2µTµβ¯χ(2)\nβεTε11−λ2R−6¯χ(1)\n2µTµβ¯χ(2)\nβεTε2−λ2R−6¯χ(1)\n2µTµβ¯χ(2)\nβεTε3\n−λ2R−6¯χ(1)\n3µTµβ¯χ(2)\nβεTε1−λ2R−6¯χ(1)\n3µTµβ¯χ(2)\nβεTε21−λ2R−6¯χ(1)\n3µTµβ¯χ(2)\nβεTε3\n\nThis becomes simpler if we have strictly uniaxial responses of the indiv idual\nspins along (say) the x axis, i.e.\n¯χ(1)\nβµ=δβ1δµ1¯χ(1)\nand similarly for ¯ χ(2). Then we can ignore the 2 and 3 components d(1)\n2, d(1)\n3\nand only need solve a scalar equation, giving\n¯χ(12)\nλ=λ¯χ(1)R−3T11¯χ(2)\n1−λ2R−6¯χ(1)T11¯χ(2)T11(14)\nand then from (8)\nE(12)\nλ=−λ2R−6T2\n11/integraldisplay∞\n−∞¯h\nπ(exp(β¯hω)−1)Im¯χ(1)¯χ(2)\n1−λ2R−6¯χ(1)T11¯χ(2)T11dω\nFrom (9), the corresponding free energy of the residual interac tion is\n9A(R)−A(∞) =/integraldisplay1\n0E(12)dλ\nλ\n=−1\n2/integraldisplay∞\n−∞¯h\nπ(exp(β¯hω)−1)Im/braceleftig\nln(1−R−6¯χ(1)¯χ(2)T2\n11)/bracerightig\ndω (15)\nThe corresponding force between SPN1 and SPN2 is\nFµ=−∂A(/vectorR)\n∂Rµ(16)\nEq(15)isvalidfortheuniaxialcasebutisreadilygeneralized: thereis ingen-\neral a sum of logarithms of the eigenvalues of the matrix 1 −R−6T¯χ(1)T¯χ(2).\nNote that both ¯ χ(1)and ¯χ(2)in (14, 15) are frequency-dependent. If the\ndenominator of (14) vanishes for some frequency ω0jthen we have a finite\noscillation of the magnetic moments for zero driving field - i.e. a free ma gnon\ncollective oscillation mode of the coupled giant spins. Indeed the free energy\n(15) can be related to a sum of the thermal free energies of these magnons.\nActually for the present model, namely ¯ χ(ω) = ¯χ0/(1−iωτ) (see Appendix\nC) these frequencies will have a large imaginary part (damping), so t here\nare really no magnons in the absence of an applied DC magnetic field. Th e\nexception is the case ω≈0, where the damping vanishes. If one of the\nmagnon frequencies vanishes, ω0J= 0, then we have an instability and the\nsystem will try to ”feeze in” the magnon. This means that the denom inator\nin (14) vanishes for zero frequency, which, as the coupling is increa sed, will\nhappen first for λ= 1, i.e.\n1−R−6¯χ(1)(0)¯χ(2)(0)T2\n11= 0. (17)\nIn time-dependent mean-field theories such as this, this behaviour is usually\ntaken to indicate a transition to a broken-symmetry state - in this c ase the\nmoments presumably freeze into a permanent ordering in the antipa rallel\nconfiguration.\n4 Energy in second order (weak coupling)\nNote that if we only want the energy to second order in the interact ion then\nfrom (8) we only need ¯ χ(12)to first order in Tij, so we can take ελ=Iin\n10(13), giving\n¯χ(12)\nαελ≈λ¯χ(1)\nβµR−3Tµν¯χ(2)\nνε. (18)\nso that (8) becomes, since/integraltext1\n0λ2dλ\nλ=1\n2,\nAresidual=A(R)−A(∞) =/integraldisplay1\n0E(12)\nλdλ\nλ(19)\n=−1\n2R−63/summationdisplay\ni,j,µ,ν=1Tij(ˆR)Iijµν, where (20)\nIijµν=/integraldisplay∞\n−∞¯h\nπ(exp(β¯hω)−1)Im/bracketleftig\n¯χ(1)\niµ(ω+i0)Tµν(ˆR)¯χ(2)\nνj(ω+i0)/bracketrightig\ndω.(21)\nTheR−6dependence is apparent. In the case of uniaxial response\nAresidual=−1\n2R−6T11(ˆR)T11(ˆR)I, where (22)\nI=/integraldisplay∞\n−∞¯h\nπ(exp(β¯hω)−1)Im/bracketleftig\n¯χ(1)(ω+i0)¯χ(2)(ω+i0)/bracketrightig\ndω(23)\nFrom Appendix C, a simple model for a superparamagnetic susceptib ility\nis ¯χ(ω) = ¯χ0/(1−iωτ), and the frequency integral Iin (22) can be estimated\nanalytically in two limits depending on the thermal flipping time τof the\ngiant spins (see Eqs (36) and (35) of Appendix C). This gives a residu al free\nenergy\nAresidual=−1\n2R−6T2\n11¯χ2\n0/braceleftigg\nkBT, k BT >>¯hτ−1\n1\nπ¯hτ−1, kBT <<¯hτ−1 (24)\nand a residual van-der-Waals-like force\nFresidual=−∂Aresidual\n∂R= 3R−7T2\n11¯χ2\n0/braceleftigg\nkBT, k BT >>¯hτ−1\n1\nπ¯hτ−1, kBT <<¯hτ−1(25)\n5 Orders of magnitude\n5.1 SPNs below the blocking temperature\nFirst consider the energy and force of interaction between two SP Nsbelow\ntheir blocking temperature so that each has a permanant magnetic moment\n11of magnitude d0=nµB. At separation Rthe direct dipole-dipole energy is\ndependent on orientation but is of order\n/vextendsingle/vextendsingle/vextendsingleEdirect/vextendsingle/vextendsingle/vextendsingle≈µ0\n4πd2\n0R−3= (10−7)(9×10−24n)2(109)3/bracketleftig\nR/10−9/bracketrightig−3\n= 8.1×10−27n2/parenleftigg10−9m\nR/parenrightigg3\nJoule (26)\nFor example if n= 1000 and R= 1nm,Edirect≈10−20J.AtT= 300Kthe\nthermal energy is kBTroom= 4×10−21J, soEdirect≈2kBT. Thus if the two\nSPNs are not thermally suppressed at T= 300Kand are able to approach\nto within a nanometer, they will not be prevented by thermal effect s from\nrotating to the antiparallel configuration and binding (clumping).\nThe corresponding force Fdirectbetween the SPNs is highly orientation-\ndependent but is of order\n/vextendsingle/vextendsingle/vextendsingleFdirect/vextendsingle/vextendsingle/vextendsingle≈µ0\n4π3d2\n0R−4≈3×8.1×10−27(10−9)−1n2/parenleftigg10−9m\nR/parenrightigg4\nN\n= 2×10−17n2/parenleftigg10−9m\nR/parenrightigg4\nN.(27)\nForn= 1000 and R= 1nmthis gives a force of order 20 pN, which is small\nbut should be directly detectable via Atomic Force Microscopy (AFM) with\nsingle SPNs attached to substrate and tip.\n5.1.1 SPNs above the blocking temperature\nNow consider a similar system but with a blocking temperature below ro om\ntemperature so that at 300 Kthere are no permanent moments. Then the\nvdW-like theory derived above gives the free energy of interaction . For\nnumerical estimates we assume uniaxial susceptibilites and work in th e weak-\ncoupling limit. We also assume that the giant spins have a zero-freque ncy\nsusceptibility\n¯χ0=(nµB)2\nkBT(28)\ncorresponding to a giant moment of nBohr magnetons. Then (24) gives\nAresidual≈\n\n−2×10−7n4\nT2/parenleftig\n10−9m\nR/parenrightig6¯hτ−1Joule,¯hτ−1>> kBT\n−2×10−7n4\nT2/parenleftig\n10−9m\nR/parenrightig6kBT Joule, ¯hτ−1<< kBT(29)\n12Forexample, let n= 1000,R= 1nm,T= 300K, and¯hτ−1<< kBT. Then\nAresidual≈ −2×10−7(1000)4\n(300)2kBT= 2kBT.This means that for the present case\nthe residual energy predicted by the perturbative theory is abou t the same as\nthe direct energy (26), which is unphysical and simply means that th e weak-\ncoupling condition is not met and we need (at least) the full RPA theor y here\n(Eq. (15)). If we are in the limit ¯ hτ−1>> kBTthe residual interaction will\nbe even larger. In this case the system of two SPNs, despite the th ermal\naveraging of an individual SPN, is most probably near to a trasition to a\nspin-locked configuration. In the RPA theory the onset of this con dition\nwould correspond to a zero denominator in (14). This would occur fo r\nR−6¯χ(1)\n0T11¯χ(2)\n0T11≈1, i.e.\nRlock≈(T11¯χ0)1/3≈/parenleftigg10−7(nµB)2\nkBT/parenrightigg1/3\n≈10−2/parenleftbigg300\nT/parenrightbigg1/3\nn2/3nm\nFor the present case with n= 1000 and T= 300K, the crossover occurs\nat about Rlock= 1nm, which is consistent with the above finding that the\nperturbative calculation of the attraction at this separation was u nphysical.\nTo give another example, suppose that n= 100,R= 10nm,T=\n300K, and ¯hτ−1<< kBT.ThenAresidual≈ −2×10−6(100)4\n(300)2(1\n10)6kBT= 2×\n10−10kBT,whereas the direct interaction between permanent moments unde r\nthe same conditions from (26) is Edirect= 8.1×10−271002/parenleftig\n1\n10/parenrightig3/(300(1.24×\n10−23))kBT= 2×10−5kBT.So for this example, neither the direct nor the\nresidual interaction would tend to lock the SPNs into an antiferroma gneti-\ncally aligned pair. The mechanical forces on the SPNs due to the spin- spin\ninteraction in either the direct or the thermally smeared residual ca se would\nbe negligible in the context of normal Brownian motion.\n6 Clumping considerations in fluid suspen-\nsion of SPNs\n(i) Consider SPNs with n= 1000 and with a bocking temperature satisfying\nTb>300K. From the numbers shown above, at T= 300K, if they are able\nto aproach one another within about a nanometer, these particles will form\npairsor larger clusters (”clumping”) that aredue to thedirect (no tthermally\nsuppressed) magnetic dipolar interaction, and that are not readily broken by\n13thermal processes. Furthermore under these same conditions t he demagneti-\nzation field inside a single SPN might be significant, so that the SPN would\nno longer contain a single domain as assumed so far. At larger separa tionsR\nthe binding energy falls off as R−3, and so the direct magnetic energy, as R\nis increased, will soon be less than the thermal energy kBT. The interaction\nat these larger separations will not immediately cause binding, but ma y well\ndetermine the kinetics of closer approach between nanoparticles, resulting\nultimately in clumping when shorter separations are attained. This pr ocess\nis complicated by the strong orientational dependence of the direc t interac-\ntion (3). SPNs will tend to rotate mechanically within in the fluid, in orde r\nto minimize the free energy in the ”antiferromagnetic” relative orien tation,\nafter which their mutual force is attractive. Thus the kinetics of c lumping\nwill be far from straightforward.\nIf clumping is undesirable, the n2R−3dependence of of the direct SPN-\nSPN magnetic binding energy suggests that smaller SPNs (e.g. n= 100)\nwill be desirable because they are less susceptible to clumping, i.e. the y\ncan approach to smaller distances (e.g. R=3√\n10−2nm= 0.2nm) before\nclumping occurs. In fact, at such small separations R, the point dipole\napproximation used here may break down, softening the interactio n and pos-\nsibly leading to the conclusion that the binding energy even at contac t is less\nthan the thermal energy. This would imply minimal clumping.\n(ii) Consider SPNs in suspension at T= 300K, withn= 1000, but now\nwithTb<300K.Here, despite the thermal suppression of the net individual\nmoments, there is a uniformly attractive residual magnetic SPN-SP N free\nenergyAresidual. This varies as n4R−6within the perturbative approxima-\ntion (see (29)), and so becomes much weaker than the direct inter action at\nlarge separations R. However at shorter separations, the stronger nand\nRdependence of the perturbative residual energy expression (29 ) suggests\nthatEresidcould exceed Edirect.This is of course unphysical: the correlations\nbetween the orientations of giant moments that give rise to Eresidcannot be\ngreater than perfect correlation, corresponding to the direct in teraction in\nthe antiferromagnetic configuration of the two giant moments. Th us in gen-\neral/vextendsingle/vextendsingle/vextendsingleEresid/vextendsingle/vextendsingle/vextendsingle≤/vextendsingle/vextendsingle/vextendsingleEdirect,max/vextendsingle/vextendsingle/vextendsingle.In fact the perturbative approximation breaks\ndown in small- Rregime, and the full RPA expression (15) will be needed\ninstead of (29). We do not yet have analytic energy and force expr essions\nin this regime. However it is clear that this approach can yield a residua l\ninteraction/vextendsingle/vextendsingle/vextendsingleEresid/vextendsingle/vextendsingle/vextendsingleof a strength approaching/vextendsingle/vextendsingle/vextendsingleEdirect,max/vextendsingle/vextendsingle/vextendsingle. It seems likely,\n14therefore, that because of the residual interaction, there will n ot always be\na discontiunous cessation of clumpimg as the temperature is raised a bove\nthe blocking temeparure Tb. However the direct interaction can be repulsive\nwhereas the residual interaction is always attractive, so there is s cope for\nsome quite rich behaviour.\n7 Prospects for experimental verification of\nthe theory\n7.1 Direct measurement of the force between two in-\ndividual SPNs\nIn Section 5.1 above, the direct interaction between permanently m agne-\ntized SPNs with n= 1000 at separation R= 1nmwas estimated to exceed\nkBTroom, and the force was estimated as 20 pN. A force of this magnitude is\nlikely to be observable, with some care, via atomic Force Microscopy. The\nsimplest configuration might involve one SPN attached to a non-magn etic\nsubstrate, and another SPN attached to the AFM tip. One could th en\nmeasure the force as a function of temperature. One might expec t a reduc-\ntioninthemeasured forceas Tisincreased above Tb. As discussed above, the\nforce could even change from repulsive to attractive, depending o n the initial\norientation of the giant moments prior to heating and subsequent d estruction\nof the net moments. The need for a measurably large force puts us out of\nthe perturbative regime for the residual interaction, so more str aightforward\nbut messy theoretical work will be required in order to predict the w ay in\nwhichFvaries with distance and temperature near ( R,T) = (1nm,Tb). It\nis not clear whether the force will be large enough for AFM detection in the\nregime of larger separations where the perturbative analysis (29) is valid.\n7.2 Indirect measurement via observation of structure\nfactors in fluid suspension\nHere we propose (e.g.) small-angle xray diffraction measurements on SPNs\nin suspension in a viscous fluid such as glycerine. The metallic SPNs shou ld\nprovide good Xray contrast. The measured structure factor of the array of\nSPNs should reveal evidence of positional corrrelations between t he SPNs,\n15which in turn is related to the forces between the SPNs as predicted here.\nAgain, one hopes to see some changes as the temperature is raised through\nthe blocking temperature Tb.\n7.3 Magnetic resonance experiments\nAlthough the present theory did not predict any lightly damped magn ons\n(combined oscillations of the magnetic moments) for a pair of adjace nt SPNs,\nthere might be the possibility of such modes if a strong DC magnetic fie ld\nis applied. Magnetic resonance experiments might then be able to det ect\nshifts in the single-SPN resonance frequency due to the proximity o f a pair\nof SPNs. Even without the external DC field, an analysis of the linewid th\nof the zero-frequency ”resonance” might reveal information ab out SPN-SPN\ncoupling.\n8 Summary and future directions\nWe have predicted a residual force between superparamagnetic n anoparticles\nthat persists above the blocking temperature. The force is the ma gnetic ana-\nlogue of the electrically-driven van der Waals interaction between ele ctrically\nneutral molecules. Our theory also deals with the dynamic spin respo nse\nof coupled SPNs to small ac external magnetic fields. Our results ma y\nbe experimentally testable, and may have implications for ferro-fluid s, for\nnanoparticle-based medical therapies, and for magnetic recordin g technol-\nogy. The new force is most likely to be significant for nanoparticles th at\napproach one another quite closely, at separations of O(nm). At t hese sep-\narations the point-magnetic-dipole approximation used here will nee d to be\nreplaced by a theory that attributes a finite spatial size and definit e physi-\ncal shape to the nanoparticles. A good starting model will be an ellips oidal\nshape, and fortunately the full electrodynamic theory of Casimir in teractions\nis quite well developed for this geometry. A theory along these lines w ill be\nthe next step.\n169 Appendix A: How to deal with the entropic\npart (classical angle-distribution approach)\nThe joint state of two interacting superparamagnets is specified b y a clas-\nsical distribution f(2)(Ω1,Ω2) in the two solid angles Ω 1,Ω2defining spatial\ndirections where the 2 giant spins point:\nThe reduced-strength interaction λEbetween the superparamagnets is\ngiven by (6). Then from general thermodynamic principles, at a give n tem-\nperature T,coupling strength λand separation R, the correct distribution\nf(2)\nλ(T,R) is that which minimizes the trial free energy:\nA(λ,T,R) =Minf(2):|f(2)|=1A(λ,T,R: [f(2)])\nso that the following functional derivative is zero\n0 =δA\nδf(2), where (30)\nA(λ,T,R: [f(2)]) =λE−TS\n=λR−33/summationdisplay\nij=1Tij(ˆR)/summationdisplay\nΩ1,Ω2d(1)\ni(Ω1)d(2)\nj(Ω2)f(2)(Ω1,Ω2)\n+kBT/summationdisplay\nΩ1,Ω2f(2)(Ω1,Ω2)lnf(2)(Ω1,Ω2)\nConsider an infinitesimal increase in the coupling strength from λtoλ+∆λ.\nAs a result, f(2)changes by an amount ∆ f(2)and noting that E=< H > λ=\nEλ/λwe have a resulting change in A:\n∆A= ∆λE+/summationdisplay\nΩ1,Ω2δA\nδf(2)∆f(2)= ∆λEλ\nλ+0\nwhere the zero comes from (30).\nNotice that we only have to know the interaction Eand not the entropic\npart, to find the change in A.\nThen the change in Ain switching on the interaction adiabatically is\nAλ=1−Aλ=0=/integraldisplay1\n0∆A=/integraldisplay1\n0Eλdλ\nλ\n17Wehavealreadyshownhowtocalculate Eλbyusingthefluctuation-dissipation\ntheorem and the mean-field (RPA) assumption. Also note that the λ= 0\nvalue of the free energy is independent of separation R:\nf(2)\nλ=0(Ω1,Ω2) =f(a)\n0(Ω1)f(b)\n0(Ω2),\nA(λ= 0,T,R) =kBT/summationdisplay\nΩ1,Ω2f(2)\nλ=0(Ω1,Ω2)lnf(2)\nλ=0(Ω1,Ω2)\n=kBT\n/summationdisplay\nΩ1f(a)\nλ=0(Ω1)lnf(a)\nλ=0(Ω1)+/summationdisplay\nΩ2f(b)\nλ=0(Ω2)lnf(b)\nλ=0(Ω2)\n.\nThus the entire Rdependence of A(λ= 1,R,T) is captured by the integral/integraltext1\n0Eλdλ\nλ.\n10 Appendix B: How to deal with the en-\ntropic part (fully quantal approach)\nOur quantum mechanical basis (NOT the eigenstates) for the comb ined mag-\nnetic state of the two systems together consists of the factoris ed states\n|ij/angbracketright=|i/angbracketright|j/angbracketright.\nwhere the first ket refers to quantum state of SPN1 and the seco nd ket to\nSPN2.\nThe thermal density matrix operator of a pair of magnetically intera cting\nnanoparticles has matrix elements in this basis denoted by\nρ(2)\nij:kl\nand traces can be taken over this or any other basis with the same r esult.\nWeconsiderstartingfromthethermalequilibriumnoftwoisolatedna nopar-\nticles, and consider the effect on the free energy of turning on the interac-\ntion by replacing the inter-nanoparticle interaction hamiltonian ˆH(12)(R) by\nλˆH(12)(R), and then increasing λfrom 0 to 1 while holding the inter-particle\nseparation Rfixed.\nFor coupling strength λthe Helmholtz free energy is a trace:\nA(λ,T,R,[ˆρ(2)]) =E−TS=Tr/parenleftig/parenleftigˆH0+λˆH(12)(R)/parenrightig\nˆρ(2)/parenrightig\n−kBTr/parenleftig\nˆρ(2)lnˆρ(2)/parenrightig\n18For fixed Hamiltonian, at thermal equilibrium Ais stationary with respect\nto arbitrary variations in density matrix that preserve Trˆρ(2)( see e.g. []):\nA(λ,T,R) =Minρ(2):trρ(2)=1A(λ,T,R: [ˆρ(2)])\nso that\n0 =δA\nδˆρ(2). (31)\nThen the first-order change in the equilibrium free energy, when th e coupling\nis increased from λto ∆λ, is\n∆A= ∆λTr/parenleftigˆH(12)(R)ˆρλ/parenrightig\n+λTr/parenleftiggδA\nδˆρ(2)∆ˆρ(2)/parenrightigg\n= ∆λE(12)\nλ\nλ+0\nThen the change in free energy in switching on the interaction betwe en the\ntwo systems is\nA(λ= 1,T,R)−A(λ= 0,T,R) =/integraldisplay1\n0E(12)\nλdλ\nλ(32)\nThe same formula can be derived for the classical case, by consider ing\na pair distribution f(Ω1,Ω2) of angular orientations Ω of the two giant mo-\nments. (See Appendix A).\n11 AppendixC:simplesuperparamagneticmodel\nfor¯χαµ(ω)\nAssume that the individual giant moment has its easy axis along ˆ e. Then\nthe response to a field /vectorhis only via /vectorh.ˆe, and the response is along ˆ e\ni.e.\n/vectord= ¯χ(ω)(/vectorb.ˆe)ˆe\n¯χαµ(ω) = ˆeαˆeµ¯χ(ω)\nHere we assume a widely-used model [3] for the frequency-depend ent mag-\nnetic susceptibility of an individual SPN in the absence of a d.c. extern al\nmagnetic field:\n¯χ(ω) =¯χ0\n1−iωτ, τ=τ0exp(E0/(kBT))\n19whereτis the thermal flipping time of the giant moment, assumed to arise\nfrom an intrinsic attempt time τ0and a thermally-activated Boltzmann suc-\ncess rate in surmounting the anosotropy energy barrier E0. Then\nIm/bracketleftig\n¯χ(1)(ω+i0)¯χ(2)(ω+i0)/bracketrightig\n=Im\n¯χ(1)\n0\n1−iωτ(1)¯χ(2)\n0\n1−iωτ(2)\n\n=Im\n¯χ(1)\n0(1+iωτ(1))\n1+ω2τ(1)2¯χ(2)\n0(1+iωτ(2))\n1+ω2τ(2)2\n\n=¯χ(1)\n0¯χ(2)\n0ω(τ(1)+τ(2))\n(1+ω2τ(1)2)(1+ω2τ(2)2)\nIf the two superparamagnetic nanoparticles have the same param eters, we\nget\nIm/bracketleftig\n¯χ(1)(ω+i0)¯χ(2)(ω+i0)/bracketrightig\n=2¯χ2\n0ωτ\n(1+ω2τ2)2(33)\nThen the second order energy (20) between two nanoparticles wit h easy\naxes ˆe(1), ˆe(2)becomes\n< E12\nλ>=λ2R−33/summationdisplay\nij=1Tij(ˆR)/integraldisplay∞\n−∞¯h\nπ(exp(β¯hω)−1)\n×Im¯χ(1)\niµ(ω+i0)R−3Tµν(ˆR)¯χ(2)\nνj(ω+i0)dω\n=λ2R−63/summationdisplay\nij=1Tij(ˆR)ˆe(1)\niˆe(1)\nµTµν(ˆR)ˆe(2)\nνˆe(2)\nj\n×/integraldisplay∞\n−∞¯h\nπ(exp(β¯hω)−1)¯χ(1)(ω+i0)¯χ(2)(ω+i0)dω\nFor two similar SPNs the frequency integral is\nI=/integraldisplay∞\n−∞¯h\nπ(exp(β¯hω)−1)2¯χ2\n0ωτ\n(1+ω2τ2)2dω (34)\nIfβ¯h/τ << 1then the Im¯χ2factor cuts the integral off for |ω|> τ−1, i.e.\nforβ¯hω >1 and we can Taylor-expand the denominator to 1st order giving\nI=/integraldisplay∞\n−∞¯h\nπβ¯hω2¯χ2\n0ωτ\n(1+ω2τ2)2dω=kBT2¯χ2\n0\nπτ/parenleftigg/integraldisplay∞\n−∞1\n(1+ω2|τ|2)2dω/parenrightigg\n=kBT2¯χ2\n0\nπτ/parenleftbigg1\n2πτ\nτ2/parenrightbigg\n= ¯χ2\n0kBT (35)\n20On the other hand if β¯h/τ >>1 then\nI≈¯h\nπ/integraldisplay0\n−∞−2¯χ2\n0ωτ\n(1+ω2τ2)2dω=−2¯h¯χ2\n0τ\nπ/integraldisplay0\n−∞ω\n/parenleftig\n1+ω2|τ|2/parenrightig2dω\n=−2¯h¯χ2\n0τ\nπ/parenleftbigg\n−1\n2τ2/parenrightbigg\n= ¯χ2\n0¯h\nπτ−1(36)\nReferences\n[1] J. Mahanty and B. W. Ninham, Dispersion forces (Academic Press, Lon-\ndon, 1976).\n[2] V. A. Parsegian, Van der Waals Forces (Cambridge University Press,\nCambridge, 2006).\n[3] W. F. Brown, Phys. Rev. 130, 1677 (1963).\n[4] S. C. McBain, H. Yiu, andJ. Dobson, Int. J. Nanomedicine 3, 169 (2008).\n[5] J. F. Dobson et al., Int. J. Quantum Chem. 101, 579 (2005).\n[6] L. D. Landau and E. Lifshitz, Statistical Physics (Addison-Wesley, Read-\ning, Massachusetts, 1969).\n21"    },    {        "title": "0903.2348v1.Connection_Between_Magnetism_and_Structure_in_Fe_Double_Chains_on_the_Ir_100__Surface.pdf",        "content": "arXiv:0903.2348v1  [cond-mat.mtrl-sci]  13 Mar 2009Connection Between Magnetism and Structure in Fe Double\nChains on the Ir (100)Surface\nRiccardo Mazzarello1,2∗and Erio Tosatti1,2,3\n1SISSA, Via Beirut 2/4,\n34014 Trieste, Italy\n2DEMOCRITOS-INFM,\nVia Beirut 2/4, 34014 Trieste, Italy\n3ICTP, Strada Costiera 11,\n34014 Trieste, Italy\n(Dated: October 31, 2018)\nAbstract\nThe magnetic ground state of nanosized systems such as Fe dou ble chains, recently shown to\nform in the early stages of Fe deposition on Ir(100), is gener ally nontrivial. Using ab initio density\nfunctional theory we find that the straight ferromagnetic (F M) state typical of bulk Fe as well\nas of isolated Fe chains and double chains is disfavored afte r deposition on Ir(100) for all the\nexperimentally relevant double chain structures consider ed. So long as spin-orbit coupling (SOC)\nis neglected, the double chain lowest energy state is genera lly antiferromagnetic (AFM), a state\nwhich appears to prevail over the FM state due to Fe-Ir hybrid ization. Successive inclusion of SOC\nadds two further elements, namely a magnetocrystalline ani sotropy, and a Dzyaloshinskii-Moriya\n(DM) spin-spin interaction, the former stabilizing the col linear AFM state, the second favoring a\nlong-period spin modulation. We find that anisotropy is most important when the double chain is\nadsorbed on the partially deconstructed Ir(100) – a state wh ich we find to be substantially lower in\nenergy than any reconstructed structure – so that in this cas e the Fe double chain should remain\ncollinear AFM. Alternatively, when the same Fe double chain is adsorbed in a metastable state\nonto the (5 ×1) fully reconstructed Ir(100) surface, the FM-AFM energy d ifference is very much\nreduced and the DM interaction is expected to prevail, proba bly yielding a helical spin structure.\nPACS numbers: 71.70.Ej, 73.20.At, 75.70.Rf, 79.60.Bm\n1I. INTRODUCTION\nControlling the magnetic order of materials is a long standing goal of a pplied solid state\nphysics, with a tremendous impact on the information technology ind ustry. The onset of\na magnetic moment in a transition metal atom arises primarily out of int ra-atomic Hund’s\nrules, which are poorly structure-dependent even in a solid. Inter atomic magnetic order,\nhowever, depends very critically on structure. As is well known, fo r example, bcc Fe is a\nprototypical ferromagnetic metal, but the magnetic properties d o change with the crystal\nstructure and the Fe-Fe interatomic distance, so that bulk Fe can support AFM configura-\ntions in the metastable fcc structure.1,2,3,4,5Low-dimensional and mesoscopic systems offer\nnew possibilities to control the magnetic order of Fe. In particular, the heteroepitaxial\ngrowth of Fe films and nanowires on non-magnetic transition metal s ubstrates is expected\nto yield novel magnetic structures due to the combined effects of a ) lattice mismatch, b) re-\nduced dimensionality, c) hybridization of Fe d-orbitals with the substrate, and d) spin-orbit\nrelated effects for heavy metal substrates.\nNovel experimental techniques have been developed, such as spin -polarized scanning tun-\nneling microscopy (SP-STM),6capable of resolving the magnetic structure of nanosized\nsystems at the atomic level. This technique has recently shown that the ground state of a\nsingle monolayer (ML) of Fe on W(001) is a collinear AFM state rather t han a FM one.7\nOne way to partly rationalize the demise of ferromagnetism in this sys tem could be the ob-\nservation that the density of states (DOS) at the Fermi energy n(EF) is strongly depressed\nupon adsorption.7Since the FM susceptibility is essentially proportional to n(EF), while\nthe AFM susceptibility is not, antiferromagnetism might happen to su ffer less from interac-\ntion with the substrate, and prevail over ferromagnetism becaus e of that. This hypothetical\npossibility fits the additional experimental observation that Fe mon olayers remain FM on\nW(110), where adsorption is weaker, this different tungsten face being better packed and\nless reactive than W(001).8Single MLs of Fe on Ir(111) have also been shown to be AFM\nand to form complex, collinear mosaic structures.9\nHere we are concerned with deposited Fe nanostructures rather than monolayers. The\ninitial steps of Fe deposition on the (1 ×5) reconstructed Ir(100) surface of Ref.10 showed\nthat Fe deposition initially forms metastable double chains, which appe ar to occupy the\ntrough-like double minima of the quasi-hexagonal Ir(100) top layer height profile. While\n2the presence of the (1 ×5) periodicity suggests the permanence of reconstruction or at le ast\nsome amount of reconstruction, it does not actually certify that t he pristine quasi hexagonal\nreconstruction of the Ir(100) substrate remains unaltered upo n Fe adsorption. The existing\ndata do not permit to resolve the detailed structure of the underly ing Ir substrate.10The Fe\ndouble chains might deposit without altering the initial reconstructio n, or they may alter it\nto someextent. Indeed, it isfoundthat the(1 ×5)Ir(100)reconstruction becomes eventually\nlifted at high Fe coverage and high temperature.10\nThe scope of the present calculations is to analyze and possibly pred ict the magnetic\nstate of Fe double chains adsorbed on Ir(100). As an added bonus , we wish to establish\nwhether something can be learnt from the relationship between mag netism and structure,\nalso in view of the ongoing SP-STM experiments on these systems at lo w-temperatures.11\nThis is not the first theory work on Fe double chains on Ir(100). In R ef.12 the structure\nand energetics of this system was already investigated by first-pr inciples density-functional\ntheory (DFT). Different adsorption sites were considered and str uctural relaxations were\nperformed for both FM and non-magnetic (NM) configurations. Th e FM configuration was\nshown to be always preferred over the NM one, which is consistent w ith Fe’s strong Hund’s\nrule intra-atomic parameters. However, these studies did not exa mine other interesting\npossibilities such as AFM or non-collinear orderings. Furthermore th e effects of SOC were\nnotconsidered, sothatnospecificeasymagnetizationaxisandmag netocrystallineanisotropy\nparameters were established.\nWe will present here two sets of calculations. The first set will invest igate collinear spin\nstructures only and, for that purpose, we will use a realistic model of the substrate consist-\ning of a seven Ir layer slab. The ground state energy and optimal st ate of magnetization of\nfree standing and Ir(100) deposited Fe double chains will be compar ed without SOC, i.e.,\nwithin the scalar relativistic approximation. Here only two possible mag netization states\nare considered, namely FM and AFM (same magnetization sign of two F e atoms across the\ndouble chain, alternating sign between first neighbors parallel to th e chains). Crucially, dif-\nferent structures will be considered for the underlying Ir(100) s ubstrate, and their relative\nenergetics compared. In a second set of calculations, the SOC will b e included by switching\nto the more time-consuming fully relativistic approximation, and here different AFM spin\ndirections will be considered, so as to extract magnetic anisotropy energies (MAEs). For\nMAE calculations the same realistic seven Ir layer slab will be used to mo del the surface.\n3Noncollinear spin structures with opposite chirality will also be conside red, so as to extract\nthe DM coupling energy. However, because of the larger supercells required along the chain\ndirection to model spin spirals, this set of calculations is limited by comp uter time econ-\nomy to smaller and simpler slabs. Eventually, a fairly complete scenario of the structures,\nenergies, and magnetization geometries will emerge, allowing a discus sion, and a tentative\nprediction subject to our rather limited accuracy, of the relations hip between them. Our\ntentative conclusion is that Fe double chains metastably deposited o n fully reconstructed\nIr(100) may develop long-pitched helical spin structures whereas the same double chains\non the partly reconstructed surface, a state of much lower ener gy, should exhibit a simple,\ncollinear AFM ground state.\nII. COMPUTATIONAL METHODS\nStandard DFT electronic structure calculations were carried out w ithin a GGA approx-\nimation with a PBE exchange-correlation functional,13as implemented in the plane-wave\nPWscf code included in the Quantum-Espresso package.14,15We employed ultrasoft pseudo-\npotentials generated according to the Rappe-Rabe-Kaxiras-Joa nnopoulos scheme.16The\nwavefunctions were expanded in plane waves with a kinetic energy cu toff of 30 Ry, whereas\nthe charge density cutoff was 300 Ry for slab calculations and 800 Ry for free-standing wires.\nIn all the structural optimization runs, Hellmann-Feynman forces were calculated with high\naccuracy (at each step, the allowed error in the total energy was set to 10−7Ry) and a strin-\ngent convergence criterion was used for structural energy minim ization (all components of\nall forces required to be smaller than 10−3atomic units and the change in the total energy\nbetween two consecutive steps required to be less than 10−4atomic units). Convergence\nwith respect to k-points, smearing parameters, wavefunction an d density cutoff was checked\nvery carefully. Furthermore, the total energies and forces of t he optimized structures were\nrecalculated within the projector-augmented wave (PAW) method (same method used in\nRef. 12), very recently implemented in the PWscf code, and found t o be in good agreement\nwith the ultrasoft pseudopotential calculations. Free-standing s ingle and double Fe wires in\nthe initial test calculations were modeled by chains parallel to the ˆ z-axis and periodically\nrepeated along the xandydirection. The minimum distance between periodic images was\n20 a.u. For single chains, the intra-chain spacing was allowed to vary s o as to determine\n4the equilibrium spacing. For double chains, the intra-chain spacing wa s set at 2.758 ˚A, cor-\nresponding to the substrate-imposed intra-chain spacing of depo sited chains which we will\nneed to adopt in later calculations.\nThe reconstructed Ir(100) surface has (1 ×5) periodicity and a ∼20% higher lateral\ndensity (in its quasi-hexagonal top layer) than a regular bulk (100) plane with its square\nlattice. A (1 ×5) supercell was used for the clean Ir(100) and for FM Fe double ch ains\non Ir(100), whereas a (2 ×5) supercell was required for the AFM case. In the following,\nthe ˆy-axis will be taken perpendicular to the surface, the ˆ z-axis parallel to the chains, and\nthe ˆx-axis in the plane and normal to the chains. In all scalar relativistic ca lculations, and\nin the SO calculations of magnetic anisotropy, the Ir substrate was modelled as a seven\nlayer slab periodically repeated across 13 ˚A wide vacuum regions. Fe double chains were\ndeposited on one face of the slab, while the other face was a perfec t (100) surface. Both the\nreconstruction of the clean surface and the relaxation of Fe/Ir( 100) systems were treated by\nallowing the four topmost Ir layers to relax, and starting from a six- atom/cell Ir topmost\nlayer. A few scalar-relativistic test calculations were repeated with a 9-layer, symmetric slab\nwith Fe double chains on both faces of the slab. The agreement betw een these calculations\nand those carried out with asymmetric 7-layer slabs was very good. A 2×10×1 Monkhorst-\nPack mesh17of special points was used for the integration over the Brillouin Zone for the\n(1×5)cell andan equivalent mesh was used for the (2 ×5) cell. The Fermi function smearing\napproach of Ref. 18 was used to deal with electron occupancy nea r the Fermi level, with a\nsmearing parameter of 0 .01 Ry. As a test of our pseudopotential, we calculated the lattice\nparameter, a0, andthe bulk modulus Bof bulk fcc Ir. Our results, a0= 3.90˚AandB= 3.42\nMbar, compare very well with the experimental values, a0= 3.84˚A andB= 3.55 Mbar.\nIII. RESULTS: STRUCTURE\nWe started off with ideal, free-standing Fe single chains. Similar to pre vious work,19,20,21\nwe found first of all that the lowest energy state of free standing Fe chains is FM. In Table I\nthecalculatedequilibrium Fe-FedistanceandthemagnetizationperF eatomofNM,FMand\nAFM chains are compared with those given in the literature. The calcu lated total energy\nof free single chains in the FM and AFM configuration as a function of F e-Fe distance is\nshown in Fig.1. The epitaxial Fe chains on Ir(100) are stretched len gthwise and the energy\n5difference between AFMandFMchains, ∆ E=EAFM−EFM, shrinks forstretched wires. At\nthe theoretical intrachain Fe-Fe distance of wires deposited on Ir (100), 2.758 ˚A, ∆E= 0.142\neV/Fe atom.\nFor double chains, we restricted ourselves to the 2.758 ˚A intrachain Fe-Fe distance only.\nIt was found that free standing Fe double chains are also FM, althou gh the energy difference\nper Fe atom between FM and AFM is smaller than for isolated chains. En ergies and magne-\ntizations of Fe atoms are shown in Table II. The magnetization of an a tom is conventionally\ncalculated by integrating the up-down spin density difference in a sph ere centered on the\natom, with radius equal to half the distance between the atom and it s nearest neighbor.\nSeparately we considered the clean, reconstructed (1 ×5) Ir(100) surface. The calculated\nsurface energy Esand work function Wof the surface are 1 .31 eV and 5 .51 eV respectively,\nin excellent agreement with experiments22,23and with previous theoretical work.12,24The\ncalculated surface energy difference between the perfect (1 ×1) and the reconstructed (1 ×\n5) surface was 0 .05 eV / ((1 ×1) area), which also compares well with previous GGA\ncalculations.24Note that, had we used the LDA approximation, the (1 ×5) reconstructed\nphase would instead have been unstable,24in contrast to experiments.\nThe structural parameters of the reconstructed Ir(100)sur face are shown in Table III(the\nnotation of Ref. 12 is used). Our results are in good agreement with experimental structure\nparameters as measured by LEED,25as well as with previous calculations.12,24\nAll basic ingredients ready, we proceeded to investigate the prope rties and energetics of\nFe double chains deposited on the Ir substrate. There are three d ifferent energy scales at\nplay in this system. The first is the structural scale, involving energ y differences of the\norder of 100 meV/Fe atom. The second is the magnetic intersite exc hange scale, involving\ndifferences of the order of 10 meV/Fe atom. The third is the spin orie ntational scale (spin\norbit related), involving differences of the order of 1 meV/Fe atom. We stress that the size\nof inter-site exchange interactions between magnetic Fe atoms is t wo orders of magnitude\nsmaller than the intra-atomic “magnetic” exchange energy scale, o f order of 1 eV/Fe atom,\ndue to the very strong Hund’s rule intra-atomic interactions.\nWe proceeded to examine structures first. Several configuratio ns of Fe double wires\non Ir(100) were considered, corresponding to different adsorpt ion sites for the Fe atoms\n(see Fig.2). Adopting the notation of Ref. 12 we considered C 1, C2and C 4configurations.\nConfigurations C 1and C 4correspond to Fe chains adsorbed onthe troughs of (1 ×5) Ir(100),\n6whereas C 2corresponds to Fechains sitting onthehills of(1 ×5)Ir(100). Thezig-zag shaped\nconfiguration denoted as C 3in Ref. 12 was not considered, for STM images fail to suggest\nzig-zag shaped chains.10\nWe found, interestingly, that C 1, C2and C 4configurations were all metastable. This is\nbecause the Fe chains should lift the reconstruction of Ir(100), r ather than adsorb on the\n(1×5)fullyreconstructedstructure. Infactthecalculatedadsorb tionenergyofdoublechains\non perfect, unreconstructed (1 ×1) Ir(100), where the top layer is a square lattice, is 0.57\neV/Featomlargerthanon(1 ×5)Ir(100), wherethetoplayerisadistortedtriangularlattice.\nThis energy difference was calculated in a grand-canonical definition , i.e. by subtracting the\nsumoftheenergy ofthefullydeconstructed structureandthee nergyofa bulkIr atomtothe\nenergy of the reconstructed structure. However, the Ir surf ace deconstruction from (1 ×5)\nto (1×1) implies the removal of 20 % of the first layer Ir atoms, which may no t readily\ntake place when the double chains are experimentally deposited at low temperature and low\ncoverage.10At high T and/or high Fe coverages, full experimental deconstruc tion of Ir(100)\ntakes place, with expulsion of the excess Ir atoms from the first lay er and formation of Ir\nchains on top of the surface. Similar superstructures consisting o f Ir rows on Ir(100) have\nrecently been seen also in case of adsorption of H atoms on this surf ace, at sufficiently high\ntemperatures.26However, at low T and coverage, the (1 ×5) structure may be kinetically\nfrozen, given the massive atomic migration and rearrangement req uired to produce the\n(1×1). A second possibility is a partial deconstruction of the surface, taking place without\nremoval of any Ir atoms. A simple displacement of Ir atoms from ben eath the Fe double\nchains (where the Ir layer structure may be locally altered) to besid es the chains should be\nmuch less kinetically hindered than a full deconstruction. To investig ate this possibility, we\nstarted fromC 1andC 4structures and looked for concerted displacements of Feand Ir a toms\nthat would spontaneously lower the energy. The structures were perturbed by moving the\nFe atoms midway between the C 1and C 4adsorption sites: then they were relaxed using\na standard Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-N ewton method. It became\napparent in this way that both C 1and C 4structures are unstable against a lateral shifting\nmotion of Ir atoms underneath the Fe chains. To lower the energy, the Ir atoms shifted\nsideways so as to restore a square ideal (100) geometry underne ath the double chains, and\naccumulating besides them. This Ir rearrangement yielded a partially deconstructed (1 ×5)\nstructurewhereintheFeatomssitonthehollowsitesofaquasi-squ are, locallydeconstructed\n7Ir(100) surface (see Fig.2, where the structure is denoted as DE C), slightly compressed along\nthe direction perpendicular to the Fe chains (the Ir-Ir distance alo ng this direction is 2.67\n˚A, to be compared with the equilibrium distance of 2.76 ˚A). The atomic coordinates of this\nstructure in the FM phase are listed in Table IV. This structure is lowe st in energy among\nthe(1×5)Fe/Irsystems explored, withalargeenergygainofabout0.46eV perFeatomwith\nrespect to C 4, which is the lowest energy reconstructed structure. Since STM im ages do not\nyield information on the position of Ir atoms beneath or besides the F e chains, this partially\ndeconstructed (1 ×5) Fe/Ir (DEC) structure seems as compatible as reconstructed (REC)\nC1and C 4structures with available data, and thus deserves to be investigat ed on similar\ngrounds. We remark finally that boththe REC and DEC surface geometries are strictly\nspeaking metastable. We calculate in fact the total energy of doub le chains (coverage 0.4\nML) on a completely deconstructed Ir(100) surface to be still 0.11 eV /Fe atom lower than\nthe energy of the DEC structure, and 0.57 eV lower than that of th e REC C 4structure.\nThe larger extent of the latter difference indicates however that m ost of the energy gain is\nobtained as soon as the Ir rearrangemnent is actuated locally bene ath the Fe double chain,\nsuggesting that structures like DEC should be taken in serious cons ideration as structural\ncandidates. The structural parameters for non-magnetic, FM a nd AFM wires on Ir(100) are\nshown in Table V. We note that, contrary to Ref. 12, and surprising ly given the similarity of\napproaches, the C 2structure is highest in energy amongst all REC structures, rathe r than\nthe lowest. We repeatedly checked all possible sources of error in o ur calculation but found\nnone.\nA. Ferromagnetism versus Antiferromagnetism\nIn agreement with Ref. 12, we found for all structures that nonm agnetic configurations\nare always disfavored over the magnetic ones, reflecting Fe’s stro ng Hund’s rule coupling.\nWe then considered in parallel the REC and DEC structures. In our c alculations the lowest\nenergy FM structure among the REC ones is C 1, whereas in Ref. 12 it was reported to be\nC2, which is least favored in our calculations. In the AFM case, on the ot her hand, C 4is\nlower in energy than either C 1and C 2, although the structural energy difference between\nC1and C 4is quite small, only 0.02 eV. C 2is always the highest energy structure, which\nagrees with the experimental evidence that double chains appear t o sit in the troughs of\n8the (1×5) Ir(100).10However, the structural interchain distances of the C 1, C4and DEC\nstructures are 2.42, 4.17 and 2.52 ˚A respectively, all different from the apparent distances in\nthe STM pictures, 3.3 ±0.2˚A. As pointed out in Ref. 12, the error of this “experimental”\nvalue may well be much larger than 0.2 ˚A because apparent maxima in STM images may\nstrongly deviate from actual centers of the Fe atoms. In conclus ion, the STM pictures do\nnot really discriminate between various structures. The calculated magnetic moments of Fe\natoms are of the order of 3.1 +/- 0.1 µBin both FM and AFM configurations. The Ir atoms\nmoments neighbouring the magnetic Fe chains generally become magn etically polarized,\nwith moments of order 0.1-0.3 µBin the FM case; in the AFM case, on the other hand,\nsome of the moments of nearby Ir atoms vanish by symmetry (in the DEC structure, they\nallvanish by symmetry).\nWe also considered AFM structures wherein Fe atoms transverse t o the double chain\nhave opposite magnetization sign (and the coupling between first ne ighbors parallel to the\nchain direction is still AFM). The energy of this tranverse AFM config uration is higher than\nthe longitudinal AFM one considered above by 0 .04−0.1 eV per atom depending on the\nstructure, with the exception of C 4, where the two configurations are practically degenerate.\nThis is not unexpected, for the distance between chains is large in C 4so they weakly interact\nwith each other. Since these transverse AFM structures are in ge neral energetically higher\nthan the longitudinal AFM ones, we will not investigate them further .\nThe demise of FM in favor of AFM in Ir-deposited chains is due to Fe-Ir hybridization,\nsince free-standing double chains are always FM. Following a reasonin g parallel to that of\nBl¨ ugel et al.7this could tentatively be rationalized in terms of changes in the respe ctive\nFM and AFM susceptibilities. The FM susceptibility should be approximat ely proportional\nto the electronic density of states (DOS) at the Fermi level evalua ted in the NM state\nand projected (PDOS) on the Fe atoms, nFe(EF). We calculated the NM DOS for all the\nrelevant structures and compared their projected value onto Fe atoms with the NM DOS of\nfree-standing, coupled chains (see Figs. 3- 4, where the NM PDOS o f the C 1and the DEC\nstructures are shown together with the FM and AFM ones; the PDO S of the C 2and C 4\nstructures show a qualitatively similar behavior). Confirming expect ations, we note a clear\ndecrease of PDOS upon deposition on the Ir(100) surface. One mig ht now be tempted to\nsurmise that since nFe(EF) is reduced upon deposition on Ir(100) due to hybridization with\nIr, ferromagnetism might be disfavored relative to AFM due to a sele ctive decrease of the\n9FM susceptibility relative to the AFM one. A PDOS decrease could redu ce the violation\nof Stoner’s FM criterion 1 −Un(EF)<0 (where U is an exchange energy parameter),\nwhile the AFM susceptibility need not do exactly the same, as there is n o straight a priori\nproportionality between PDOS and AFM susceptibility. To investigate that aspect, we\nconducted constrained magnetization calculations allowing a numeric al evaluation of the\nzero-field FM and AFM susceptibilities. To reduce computational time s, we did that for a\n“toy” DEC structure consisting of Fe atoms and nearest neighbor Ir atoms only (in total\n2+3 atoms in the FM cell and 4+6 atoms in the AFM cell). For this struct ure the AFM\nstructure is lower in energy than the FM one by about 60 meV per Fe a tom. The constraint\non the Fe local magnetic moments (calculated by integrating the mag netization density in a\nsphere centered on the Fe atoms, as explained at the beginning of S ection III) was imposed\nby adding a penalty functional to the total energy. As Fig. 5 shows , for small magnetizations\nthere is a quadratic energy decrease with magnetization, which mea sures separately the FM\nand AFM susceptibilities. The FM and AFM energies remain however ext remely close at\nall small magnetizations, and do not indicate appreciable differences between FM and AFM\nsusceptibilities. So while there is an Ir-induced FM susceptibility decre ase connected with\nthe Ir-induced decrease of nFe(EF), that does not seem to explain the switch from FM to\nAFM. The AFM energy gain is realized at large magnetization magnitude s, not revealed at\nthe perturbative level.\nThe main notable Ir-related difference between FM and AFM states is the finite magnetic\npolarization required for the nearby Ir atoms in the FM case, contr asted by the symmetry-\ninduced zero magnetic polarization for some (REC structures) or a ll (DEC structure) of the\nnearby Ir atoms in the AFM case. Due to an interplay between magne tism and structure,\nthe Fe magnetic orbitals delocalize over the Ir substrate atoms in th e FM case, but less, or\nnot atall (depending onthestructure), inthe AFMcase. As aresu lt theIr-relatedreduction\nof magnetic energy gain is less important in the AFM case than in the FM case. If this\nindeedisthemechanismthatcausestheswitchfromFMtoAFM,then itcouldholdforother\nmagnetic elements as well. To explore this hypothesis, we studied the magnetic properties of\nMn, Co, Ni double chains on (1 ×5) Ir(100) (restricting to C 1and DEC configurations). The\nstarting unsupported Mn chains were found to be AFM, whereas Co and Ni chains were FM.\nEnergy differences between FM and AFM Mn, Co and Ni double chains ( free-standing and\ndeposited on Ir(100)) are shown in Table VI, with Fe also shown by co mparison. Similar to\n10the Fe chains, the Ir surface was unstable against deconstructio n when Mn, Co or Ni chains\nare adsorbed on the surface. In the end, it turned out that Ir-d eposited Co and Ni double\nchains were still FM, unlike Fe. However, the energy difference betw een FM and AFM\nstructures was substantially reduced when Co and Ni chains were a dsorbed on Ir(100) for\nall geometries (except for Co chains in C 1geometry, where it increased by 0.03 eV). Double\nMn chains remained AFM when deposited on Ir(100), but the energy gap between the AFM\nand FM configurations again increased. On the whole, these results seem to confirm our\nstarting hypothesis. We may tentatively conclude therefore that the selective spillout of\nmagnetization to Ir atoms near the Fe chains present in the FM stat e but reduced or absent\nin the AFM state should play an important role in shifting the energetic balance from FM\ntowards AFM, although other, subtler and more specific effects sh ould be invoked in order\nto explain the dependence of the relative stability of FM and AFM confi gurations upon the\ntransition metal element and the adsorption structure. On this as pect there is room for\nfurther work addressing the physical mechanism in more detail, may be resorting to some\nsimplified and more transparent schemes such as tight-binding.\nB. Magnetic Anisotropy\nMagnetic anisotropy energies were calculated for both unsupport ed and deposited Fe\ndouble chains, where the REC (C 1and C 4) and the DEC configurations have been consid-\nered. In free AFM Fe double chains, the easy axis was found to lie alon g ˆy, perpendicular\nto the plane containing the chains for chain-chain distances corres ponding to the C 1and\nDEC structures (see Table VII). For large chain-chain distances, the easy axis switched to\nˆz, along the chains, in agreement with the single chain limit.27,28\nIn Ir-deposited AFM Fe double chains the easy magnetization axis of both C 1and C 4\nREC structures was ˆ x, parallel to the surface and perpendicular to the chains. In the\nDEC structure, ˆ xwas instead the hard axis, whereas the easy axis was ˆ z, parallel to the\nchains (see Table VII). These magnetic anisotropy results hold for FM configurations as\nwell, as could be expected from phenomenological on-site anisotrop y parameters. From\nthe predicted opposite magnetic anisotropies of REC and DEC struc tures, it follows that\nthe detection of the easy magnetization axis of the double chains on Ir(100) by SP-STM\ntechniques could yield indirect but important information on the unkn own local structure\n11of the Ir(100) surface.\nIn principle, we note, magnetostatic effects due to magnetic dipolar interactions could\nalso give rise to magnetic anisotropy effects. However, for our two -chain AFM system\nthese dipole-dipole energies can be estimated to be less than 0.1 meV, much smaller than\nmagnetocrystalline energies due to SOC, and can be neglected.\nC. Dzyaloshinskii-Moriya Interactions\nThe second important effect of spin orbit interaction on magnetism is the onset of a\nDzyaloshinskii-Moriya inter-site interaction term of the form29,30\nHDM=Dij·Si×Sj (1)\nwhereDijis the Dzyaloshinskii vector. The DM interaction is chiral and is due to the\nconcerted effect of spin-orbit coupling and a lack of structural inv ersion symmetry at the\nsurface. Thedirectionof Dijisdeterminedsolelybystructuralsymmetry.30Morespecifically,\nthe intrachain inter-site Dijmust be orthogonal to a mirror plane containing sites Riand\nRj, and parallel to a mirror plane bisecting Rij. For the Fe double chains on Ir(100), where\na pair of (magnetically parallel) Fe atoms is the effective magnetic site, the vector Dijlies\non the surface plane and normal to the double chain, i.e. parallel or a ntiparallel to the ˆ x\naxis. The sign of Dij, a vector which breaks the left-right structural symmetry, will s witch\nby switching the sign of magnetization, in accordance with time rever sal symmetry. It is\notherwise fully determined microscopically by the asymmetry of the s elfconsistent potential\ngradient in the surface region.\nWe calculated the magnitude and sign of D, assumed to be restricted to first neighbors,\nby direct energy difference between two noncollinear magnetic stru ctures of the deposited\ndouble chain, each composed of four Fe pairs, or eight Fe atom/cell. The magnetization was\nconstrained to be orthogonal between one Fe pair to the next dow n the double chain. In the\nfirst magnetic structure, the magnetization direction was taken t o rotate in the sense y, z,\n-y, -z; in the second, it counter-rotated in the sense y, -z, -y, z . These two magnetic spirals\nhave identical structural, exchange and anisotropy energies, so that their energy difference\nidentifies precisely the DM term alone.\nSince heavy computational cost restricted us to relatively small sy stems, we considered\n12two successive sizes, comprising respectively 12 and 36 Ir atoms, c orresponding to Fe nearest\nneighbor atoms and Fe nearest and next nearest neighbor atoms, respectively. This allowed\nan appreciation of the kind of finite size error involved, as well as som e level of extrapolation\ntowards ideally larger sizes. Atomic relaxations of the small systems were not taken into\naccount, i.e. atoms were frozen at the positions obtained by relaxin g the corresponding\n7-layer slabs. Moreover, only two experimentally relevant structu res were considered: the\nC1(REC) structure and DEC structure. (As discussed above, the d istance betweeen chains\nin C4is very large and somewhat less likely than C 1).\nIt turns out that DM favors right-handed cycloidal spin spirals for both structures. As\nfar as the REC structure is concerned, the magnitude Dof the Dzyaloshinskii vector slightly\nincreases for the larger size systems, from 2 to 3 meV, whereas an isotropy energies decrease\nsomewhat from 3-4 meV to 1-2 meV. We conclude that for the REC de posited Fe double\nchain, MAEs are of the order of 1 meV (Table VII), whereas Dis about 3 meV. In the DEC\nstructure, both Dand MAEs are large but do get significantly smaller in the larger size\nsystem:Ddrops from 12 to 7 meV and K=Kz−Kyfrom 8 to 2 meV. Extrapolating, we\nconclude that in the DEC deposited double chain the anisotropy ener gyKcould be about\n1 meV,Dof order 5 meV.\nIV. ROTATING MAGNETISM VERSUS COLLINEAR ANTIFERROMAG-\nNETISM\nIf anisotropy were ideally zero but at the same time the DM term were finite, no mat-\nter how small, the collinear AFM magnetic structure would spontaneo usly transform to a\nrotating magnetic structure, whose pitch would diverge as Dtends to zero.32On the other\nhand, once anisotropy is large enough, the collinear AFM state will pr evail over noncollinear\nmagnetism. The relatively large anisotropies and DM values reported in the previous sec-\ntions indicate that the competition between AFM and helical spin stru ctures needs to be\nconsidered in quantitative detail, as was recently done for other sy stems by Bl¨ ugel and\ncollaborators.33,34,35\nIn the following we will describe our system by a micromagnetic continu ous model:36for\nFM systems, this approximation is justified if the magnetic moment va riations are small on\na length scale where exchange and DM interactions are significant. F or our AFM double\n13chains, given two intrachain nearest neighbor sites iandi+1 with magnetization miand\nmi+1, a micromagnetic model is valid if the difference between miand−mi+1is small.\nWithin this approximation, taking into account the quasi one-dimens ional nature of our\nsystems, the energy functional is given by\nE=/integraldisplay+∞\n−∞\nA/parenleftBiggdm\ndz/parenrightBigg2\n+¯D·/parenleftBigg\nm×dm\ndz/parenrightBigg\n+m†·¯K·m\ndz, (2)\nwhereAis the spin stiffness, ¯Dis an effective Dzyaloshinskii vector and ¯Kis an effec-\ntive anisotropy energy tensor. Following the convention usually ado pted in micromagnetic\ncalculations, we assume that m2\nx+m2\ny+m2\nz= 1. These three quantities depend on the\ncrystal structure and can be expressed in terms of the exchang e constants Jij,Dijvectors\nand anisotropy energy tensor Kof the discrete model. We may assume that only nearest\nneighbor exchange and DM interactions are important: then A∼dintraJ/2,¯D∼Dand\n¯K∼K/dintra, wheredintrais the Fe-Fe intrachain distance and JandDare the nearest\nneighbor exchange and DM parameters. The nearest neighbor Jis straightforwardly eval-\nuated from the energy difference between FM and AFM phases.31In the following we will\nseparately address two cases, namely:\na)D=Dˆxparallel to the hard axis. This is the case in the DEC surface.\nb)D=Dˆxparallel to the easy axis. This is the case in the REC surface.\na) IfDis parallel to the hard axis, then the magnetic moments lie in the ( y,z) plane,\nperpendicular to D, which contains the double chain and is orthogonal to the surface. In\nthiscase acollinear ortwo-dimensional non-collinear structurewill a ppear, depending onthe\nrelative strength of Dand the in-plane anisotropy parameter K≡Kz−Ky, whereKyand\nKzare the ˆyand ˆzcomponents of the anisotropy energy tensor. This problem is cons idered\nin detail in the literature32,37and excellently summarized in the thesis of M. Heide.38For\nspin structures lying in the ( y,z) plane, Formula (2) simplifies to\nE=/integraldisplay+∞\n−∞\nA/parenleftBiggdφ\ndz/parenrightBigg2\n+¯Ddφ\ndz+¯Ksin2φ\ndz, (3)\nwhereφis the anglebetween the local magnetization and the easy axis, ˆ z, and¯K≡¯Ky−¯Kz.\nA non-collinear, helical state will appear if the DM-related energy ga in is higher than twice\nthe formation energy of an optimal domain wall in the ( y,z) plane.32,37This is equivalent\nto:\nD >4\nπ/radicalBigg\nJK\n2. (4)\n14Inserting the numerical values J= 29 meV and K= 1.7 meV, we obtain the inequality\nD >6.3 meV, estimated for the occurrence of a helical state in the DEC st ructure. From\nour calculations, we estimate Daround 5 meV, smaller than the critical value, although\ngenerally of the same order of magnitude. Therefore, we tentativ ely conclude that the\nAFM collinear state is most likely in the DEC structure. In view of our po or accuracy,\nhowever, we cannot totally exclude a helical state with a very long pit ch, consisting of wide\nantiferromagnetic domains separated by well separated domain wa lls.\nb)ifDisparalledtotheeasyaxis, thenthree-dimensionalnon-collinearst ructuresarealso\npossible, besides collinear and two-dimensional non-collinear ones. A thorough description\nof this case can be found in Ref. 38. Since the condition m2\nx+m2\ny+m2\nz= 1 holds, Formula\n(2) can be written as\nE=/integraldisplay+∞\n−∞\nA/parenleftBiggdm\ndz/parenrightBigg2\n−¯D/parenleftBigg\nmydmz\ndz−mzdmy\ndz/parenrightBigg\n+/parenleftBig¯Ky−¯Kx/parenrightBig\nm2\ny+/parenleftBig¯Kz−¯Kx/parenrightBig\nm2\nz\ndz,\n(5)\nwhere¯Kx,¯Kzand¯Kyare the easy, intermediate and hard components of the anisotrop y\nenergy tensor respectively (see anisotropy energies for the C 1REC structure in Table VII).\nExpanding the integrand of (5) around the AFM solution, my=mz= 0, one gets the\nfollowing Euler-Lagrange equations:\nAd2my\ndz2+¯Ddmz\ndz−/parenleftBig¯Ky−¯Kx/parenrightBig\nmy= 0 (6)\nAd2mz\ndz2−¯Ddmy\ndz−/parenleftBig¯Kz−¯Kx/parenrightBig\nmz= 0. (7)\nConsidering again only nearest neighbor JandD, these equations have a periodic solution,\nmy=αycos(ωz+βy), (8)\nmz=αzcos(ωz+βz), (9)\nif and only if38\nD >/radicalBigg\nJ\n2(Kz−Kx)+1. (10)\nMoreover, when the above inequality is fulfilled, the non-collinear sta te is always lower in\nenergy than the AFM collinear solution. Therefore, when D=/radicalBig\nJ\n2(Kz−Kx)+1, a second-\norder phase transition to a 3-dimensional state takes place. The s ystem undergoes a second-\norder transition to a 2-dimensional helical state in the ( y,z) plane at slightly higher values of\n15D, but this critical point cannot be determined analytically.38We should emphasize that, in\nourcase, therangeof Dvalueswhere the3-dimensional stateisstable(which depends onth e\nmagnitude of the components of the magnetic anisotropy tensor, see Ref. 38) is very narrow.\nInserting the numerical values J= 0.5 meV and Kz−Kx= 0.7 meV corresponding to C 1in\nFormula (10), we obtain that the AFM will be destabilized if D >1.4 meV. Since our REC\nsurface calculations suggest Dvalues around 3 meV, which is larger than this threshold,\nwe conclude that in a REC structure like C 1, where the double chains do not deconstruct\nthe underlying Ir(100) surface, the magnetic ground state shou ld be non-collinear, and in\nparticular a ( y,z) helical state is the most likely outcome. In conclusion, a sketch of t he\npredicted magnetic ground state for the DEC and REC (C 1) structures is shown in Fig. 6.\nV. DISCUSSION AND CONCLUSIONS\nWe studied by ab initio electronic structure and total energy calculations Fe double\nchains on (1 ×5) Ir(100). Several different structures with the experimentally observed\n(1×5) periodicity were considered, particularly one, C 1REC, where the underlying Ir\nsurface remains quasi-hexagonally reconstructed, and another , DEC, where it is partially\ndeconstructed, with a large decrease of total energy. By addre ssing magnetism first without\nspin orbit effects, we find that in all structures considered the dep osited Fe double chains\ndo not remain FM as in vacuum, but generally adopt an AFM ground sta te. The demise\nof ferromagnetism is attributed to Fe-Ir hybridization. The hybrid ization of Fe with the\nIr substrate brings about first of all a drop of the Fe-projected density of electronic states\nnearEFin the non-magnetic state, which reduces the FM susceptibility. How ever, we\nfind that the AFM susceptibility is also reduced by the same amount up on adsorption.\nAt large magnetization, AFM appears eventually to be favored by a m agnetization node\nintervening by symmetry in the bridging Ir atoms, a node which is abse nt in the FM case.\nBy including spin orbit in the calculations, the magnetic anisotropy ene rgies of relevant\nREC and DEC structures have been determined. The easy axis is fou nd to lie in the\nsurface plane and perpendicular to the Fe double chain in the REC str ucture, and parallel\nto the chains in the DEC structure. Finally, we calculated the Dzyalos hinskii-Moriya\nspin-spin interaction energy, and found it to be generally of a compe titive magnitude when\ncompared to anisotropy. The different possibilities arising for the re sulting ground state\n16magnetization pattern are examined. Within the substantial uncer tainties connected with\nour estimated computational and finite size errors, we conclude th at a collinear AFM state\nwith in-plane magnetization vector is likely to prevail in the DEC struct ure, whereas a long\nperiod rotating magnetization in an orthogonal plane could instead p revail in the REC\nstructure. These predictions and clear signatures should be of va lue for future experimental\nobservations by SP-STM techniques.\nVI. ACKNOWLEDGMENTS\nWe are grateful to R. Wiesendanger for providing the initial motivat ion to conduct this\nstudy. We warmly acknowledge many instructive discussions with, an d much help by, An-\ndrea Dal Corso. Discussions with S. Bl¨ ugel and R. Gebauer were a lso stimulating. The\nwork was directly sponsored by MIUR PRIN/Cofin Contract No. 200 6022847 as well as\nby INFM/CNR “Iniziativa transversale calcolo parallelo”. The resear ch environment pro-\nvided by the independent ESF project CNR-FANAS-AFRI was also ve ry useful. Part of the\ncalculations were carried out on the SP5 machine at CINECA, Casalec chio.\n17∗Corresponding author. Present Address: Computational Sci ence, Department of Chemistry\nand Applied Biosciences, ETH Zurich, USI Campus, via Giusep pe Buffi 13, CH-6900 Lugano,\nSwitzerland. Email address: riccardo.mazzarello@phys.c hem.ethz.ch\n1S. C. Abrahams, L. Guttman, and J. S. Kasper, Phys. Rev. 127, 2052 (1962).\n2U. Gonser, C. J. Meechan, A. H. Muir, and H. Wiedersich, J. App l. 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Sci. 163, 435 (1985); K. Johnson, Q. Ge, S. Titmuss, and D. A.\nKing, J. Chem. Phys. 112, 10460 (2000); A. Schmidt, W. Meier, L. Hammer, and K. Heinz, J.\nPhys.: Condens. Matter 14, 12353 (2002).\n26L. Hammer, W. Meier, A. Klein, P. Landfried, A. Schmidt, and K . Heinz, Phys. Rev. Lett. 91,\n156101 (2003).\n27G. Autes, C. Barreteau, D. Spanjaard, and M.-J. Desjonquere s, J. Phys.: Condens. Matter 18,\n6785 (2006).\n28Y. Mokrousov, G. Bihlmayer, S. Heinze, and S. Bl¨ ugel, Phys. Rev. Lett. 96, 147201 (2006).\n29I. E. Dzyaloshinskii, Sov. Phys. JETP 5, 1259 (1957).\n30T. Moriya, Phys. Rev. 120, 91 (1960).\n31It should be noted that the value of J is very small, especiall y in the C1 REC structure. The\nreason why AFM competes with and narrowly wins over with FM is discussed earlier in the\npaper. Our tentative conclusion is that it is related to the u nfavourable circumstance that a\nbridging Ir atom must acquire a spin polarization in the FM st ate, whereas spin polarization\nhas a node at that atom in the AFM state. Of course with J values in the meV range there can\nbe no order, even of the Kosterlitz-Thouless type, unless te mperature is hundreds of milliKelvin\n– which however is where spin-polarized STM experiments are usually performed.\n32Yu. A. Izyumov, Sov. Phys. Usp. 27, 845 (1984); Usp. Fiz. Nauk 144, 439 (1984).\n33M. Bode, M. Heide, K. von Bergmann, P. Ferriani, S. Heinze, G. Bihlmayer, A. Kubetzka, O.\nPietzsch, S. Bl¨ ugel, and R. Wiesendanger, Nature 447, 190 (2007)\n34P. Ferriani, K. von Bergmann, E. Y. Vedmedenko, S. Heinze, M. Bode, M. Heide, G. Bihlmayer,\nS. Bl¨ ugel, and R. Wiesendanger, Phys. Rev. Lett. 101, 027201 (2008).\n35M. Heide, G. Bihlmayer, and S. Bl¨ ugel, Phys. Rev. B 78, 140403(R) (2008).\n36W. F. Brown, “Micromagnetics”, Wiley, New York, 1963.\n37I. E. Dzyaloshinskii, Sov. Phys. JETP 20, 665 (1965).\n38M. Heide, ”Magnetic Domain Walls in Ultrathin Films: Contri bution of the Dzyaloshinskii-\nMoriya Interaction”, PhD thesis, RWTH-Aachen (2006).\n192.0 2.2 2.4 2.62.8 3.0 3.2\nFe-Fe distance (A)0.20.40.60.81.01.2Energy (eV)FM\nAFM\no\nFIG. 1: Total energy per atom of FM and AFM free standing Fe sin gle chains as a function of\nFe-Fe distance.The dashed vertical line corresponds to the theoretical interatomic distance of the\nFe chain deposited on Ir(100).\nour work Ref. 19 Ref. 20 Ref. 21\na0(˚A)m(µB)a0(˚A)m(µB)a0(˚A)m(µB)a0(˚A)m(µB)\nFM2.28 3.32 2.25 3.34 2.28 2.98 2.25 3.41\nAFM2.40 3.14 2.38 3.05 - - 2.15 1.82\nNM1.91 0.00 1.94 0.00 - - 1.94 0.0\nTABLE I: Equilibrium Fe-Fe distance a0and magnetization per atom mfor FM and AFM free-\nstanding single wires: comparison between our results and r ecent results. All results in this Table\nwere obtained by using GGA functionals.\n20C1 C4\nC2 DECBA A A\nA\nA B AB ABA\nFIG. 2: Top and side view of the reconstructed Ir(100) surfac e with the studied configurations\nfor the dimer chain. Configurations C 1and C4correspond to Fe chains adsorbed on the troughs\nof (1×5) Ir(100), whereas C 2corresponds to Fe chains sitting on the hills of (1 ×5) Ir(100).\nDEC is the partially deconstructed structure, wherein the F e atoms sit on the hollow sites of a\nquasi-square, locally deconstructed Ir(100). A and B indic ate the Ir atoms nearest neighbors of Fe.\nVertical displacements have been exaggerated for clarity p urposes.\n21-10 -5 0\nEnergy (eV)020PDOS (arb. units)C1 AFM Fe-10 -5 0\nEnergy (eV)-20020PDOS (arb. units)C1 FM Fe-10 -5 0\nEnergy (eV)02040PDOS (arb. units)deposited chains\nfree chainsC1 NM Fe\nFIG. 3: Ferromagnetic, antiferromagnetic and non-magneti c density of states of the C 1REC\nstructure projected onto Fe atoms. Dashed lines indicate th e DOS of free-standing, double Fe\nwires. In the AFM case, the PDOS were calculated by projectin g onto all of the Fe atoms, i.e.\nboth those with positive magnetic moments and those with neg ative ones: as a consequence, the\nPDOS of spin-up and spin-down electrons are the same.\n22-10 -5 0\nEnergy (eV)020PDOS (arb. units)DEC AFM Fe-10 -5 0\nEnergy (eV)-20020PDOS (arb. units)DEC FM Fe-10 -5 0\nEnergy (eV)02040PDOS (arb. units)deposited chains\nfree chainsDEC NM Fe\nFIG. 4: Ferromagnetic, antiferromagnetic and non-magneti c density of states of the partially de-\nconstructed structure projected onto Fe atoms. Dashed line s indicate the DOS of free-standing,\ndoubleFewires. IntheAFM case, thePDOSwerecalculated byp rojectingontoall of theFeatoms,\ni.e. both those with positive magnetic moments and those wit h negative ones: as a consequence,\nthe PDOS of spin-up and spin-down electrons are the same.\n230 1 2 3\nm (µB)-1.2-0.8-0.40   Energy (eV)FM\nAFM\nFIG. 5: Total energy perFeatom of”toy” DECFM andAFM structu resconsistingofFeatoms and\nnearestneighborIratoms asafunctionofthemagneticmomen tmonaFeatom. Thesecalculations\nwere performed by adding a penalty functional to the total en ergy in order to constrain the local\nmagnetic moment around a Fe atom. For small m, which corresponds to small magnetic fields\nand small staggered magnetic fields for the FM and AFM case res pectively, the dependence of the\nenergy on mis quadratic and the coefficient of the quadratic term is inver sely proportional to the\nFM or AFM susceptibility.\n24C1\nDEC\nFIG. 6: View of the spin-structure of the reconstructed C 1and the partially deconstructed DEC\nIr(100) configuration: Fe magnetic moments in C 1are expected to form a right-handed cycloidal\nspin spiral, whereas in DEC a collinear AFM state with moment s parallel to the chains should\nprevail.\n25dinter(˚A) magnetic struct. ∆ E(eV)Eint(eV)m(µB)\n- FM 0.000 - 3.45\n(SC) AFM 0.142 - 3.38\nNM 2.228 - 0.00\n2.33 FM 0.000 0.835 3.14\nAFM 0.070 0.907 3.16\nNM 1.544 1.519 0.00\n2.40 FM 0.000 0.803 3.18\nAFM 0.092 0.854 3.21\nNM 1.631 1.400 0.00\n2.52 FM 0.000 0.733 3.25\nAFM 0.116 0.759 3.29\nNM 1.748 1.212 0.00\n4.14 FM 0.000 0.083 3.43\nAFM 0.090 0.136 3.37\nNM 2.211 0.101 0.00\nTABLE II: Calculated energy differences per atom, ∆ E, and magnetizations per atom, m, for\nNM, FM and AFM free-standing single chains (SC) and NM, FM and AFM double chains at\ninterchain distances dinterof 2.33, 2.40, 2.52 and 4.14˚A (corresponding to the interchain distances\nof Fe double chains in the C 2, C1, DEC and C 4configurations respectively). The intrachain Fe-Fe\ndistance is 2.758 ˚A for all the structures. For each structure, ∆ Eis given with respect to the\npreferred FM solution. For double chains the interaction en ergy between chains, Eint, is shown as\nwell.\n26Present Work LEEDRef. 12Ref. 24\nd0 1.95 1.9201.9431.916\nd12 1.96 1.942.001.97\n< d12>2.26 2.252.25 -\nb13\n1 0.22 0.250.20 -\nb23\n1 0.54 0.550.470.47\nb34\n1 -0.21 -0.20-0.17-0.20\np2\n1 -0.04 -0.05-0.03-0.05\np3\n1 -0.07 -0.07-0.07-0.03\nd23 1.83 1.791.851.92\n< d23>1.91 1.881.89 -\nb13\n2 0.05 0.070.03 -\nb23\n2 0.10 0.100.05 -\np2\n2 0.03 0.010.00 -\np3\n2 0.01 0.020.01 -\nd34 1.88 1.831.91 -\n< d34>1.96 1.931.96 -\nb13\n3 0.08 0.100.05 -\nb23\n3 0.04 0.050.02 -\np1\n3 -0.01 -0.01 -\np2\n3 0.00 -0.00 -\nd45 1.94 1.891.92 -\n< d45>1.96 1.911.93 -\nb13\n4 0.05 0.060.03 -\nb23\n4 0.02 0.030.01 -\np2\n4 0.00 --0.01 -\np3\n4 -0.01 --0.01 -\nTABLE III: Calculated and experimental structural paramet ers (in˚A) of the reconstructed Ir(100)\nsurface. The notations of Ref. 12 are used: d0is the bulk interlayer distance, dijand< dij>\nare the shortest and average distance between layer iandj,bkl\niandpk\niare vertical and lateral\ndisplacement amplitudes of atoms kandlin layeri.\n27Element x ( ˚A) y (˚A) z (˚A)Element x ( ˚A) y (˚A) z (˚A)\nFe 9.537 13.424 1.365 Ir 1.379 9.686 1.374\nFe 7.015 13.424 1.367 Ir 11.055 7.823 -0.004\nIr 13.043 12.477 1.370 Ir 8.273 7.781 -0.003\nIr 8.274 11.664 2.747 Ir 5.492 7.823 -0.004\nIr 3.505 12.476 1.370 Ir 2.773 7.819 -0.004\nIr 10.948 11.700 -0.011 Ir -0.015 7.819 -0.004\nIr 5.600 11.701 -0.010 Ir 12.413 5.852 1.376\nIr 1.379 11.905 -0.009 Ir 9.665 5.866 1.377\nIr 12.424 9.780 1.371 Ir 6.881 5.866 1.377\nIr 9.656 9.703 1.373 Ir 4.133 5.852 1.376\nIr 6.891 9.704 1.373 Ir 1.379 5.872 1.375\nIr 4.124 9.780 1.372\nTABLE IV: Coordinates of the DEC structure in the FM phase: on ly Fe atoms and Ir atoms\nbelonging to the four uppermost layers are listed. The y-axis is perpendicular to the surface,\nwhereas the z-axis is along the chains. In the NM and AFM DEC phases, positi ons of the Ir atoms\ndo not differ appreciably from those in the FM phase.\n28∆E(eV) chain-chain d ( ˚A) Fe- Ir Ad (˚A) Fe- Ir Bd (˚A)m(µB)Eint(eV)\nC1NM 1.543 1.97 2.53 2.60 0.00 4.277\nFM 0.481 2.40 2.51 2.62 3.07 3.112\nAFM 0.480 2.42 2.49 2.59 3.06 3.254\nC2NM 1.731 1.97 2.32 2.71 0.00 4.089\nFM 0.903 2.33 2.40 2.71 3.02 2.690\nAFM 0.895 2.31 2.38 2.70 2.99 2.840\nC4NM 1.642 3.77 2.30 2.45 0.00 4.179\nFM 0.521 4.14 2.52 2.59 3.14 3.071\nAFM 0.459 4.17 2.49 2.55 3.15 3.275\nDEC NM 0.949 2.40 2.53 2.47 0.00 4.872\nFM 0.059 2.52 2.62 2.57 3.06 3.533\nAFM 0.000 2.57 2.58 2.55 3.02 3.734\nTABLE V: Calculated structural parameters and energetics f or NM, FM and AFM double chains\non the (1 ×5) Ir(100) surface. C 1, C2and C4configurations, where the Ir surface is reconstructed\n(REC), are considered, as well as the structure where Fe chai ns sit on a partially deconstructed\nsurface (DEC). The energy differences ∆ Eare given with respect to AFM DEC, which is the\nlowest energy configuration. In columns 3 and 4 the distances between Fe atoms and their nearest\nneighbor Ir Aand IrBatoms (as indicated in Fig.2) are provided. The interaction energy (Eint) for\na given structure and magnetic configuration is defined as the difference between the total energy\nof the structure and the sum of the energies of the clean recon structed (1 ×5) Ir(100) and twice\nthe energy of the isolated Fe chain (with the same magnetic co nfiguration).\nMn Fe Co Ni\nfree dep. free dep. free dep. free dep.\nC10.085 0.169 -0.092 0.001 -0.057 -0.088 -0.041 -0.012\nDEC0.090 0.143 -0.116 0.059 -0.098 -0.068 -0.045 -0.011\nTABLE VI: Energy difference (per adsorbed metal atom) between FM and AFM configurations\nof free-standing and deposited Mn, Fe, Co and Ni double wires . C1and DEC configurations have\nbeen considered. Energies are in units of eV.\n29Ez- Ex(10−3eV)Ez- Ey(10−3eV)\nfree FM single chain 1.9 1.9\nfree AFM double chain ( dinter= 2.4˚A) 0.5 -0.6\ndep. AFM double chain (C 1) -0.7 0.6\ndep. AFM double chain (C 4) -0.8 0.7\ndep. AFM double chain (DEC) 1.9 1.7\nTABLE VII: Magnetic anisotropy energies (per atom) of unsup ported FM single chains, unsup-\nported double AFM Fe chains at dinter= 2.4˚A and double AFM Fe chains deposited onto the\nIr(100) surface for REC (C 1and C4) and DEC configurations. The intrachain Fe-Fe distance is\n2.758˚A for all the structures. The z-axis is along the chains, whereas the x-axis is perpendicular\nto the chains and parallel to the plane containing the chains .\n30"    },    {        "title": "0904.0993v1.Magneto_crystalline_anisotropies_in__Ga_Mn_As__A_systematic_theoretical_study_and_comparison_with_experiment.pdf",        "content": "arXiv:0904.0993v1  [cond-mat.mtrl-sci]  6 Apr 2009Magneto crystalline anisotropies in (Ga,Mn)As: A systemat ic theoretical study and\ncomparison with experiment\nJ. Zemen1, J. Kuˇ cera1, K. Olejn´ ık1, T. Jungwirth1,2\n1Institute of Physics ASCR, v. v. i., Cukrovarnick´ a 10, 162 0 0 Praha 6, Czech Republic and\n2School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK\n(Dated: November 3, 2018)\nWe present a theoretical survey of magnetocrystalline anis otropies in (Ga,Mn)As epilayers and\ncompare the calculations toavailable experimental data. O ur model is based on an envelope function\ndescription of the valence band holes and a spin representat ion for their kinetic-exchange interaction\nwith localised electrons on Mn2+ions, treated in the mean-field approximation. For epilayer s with\ngrowth induced lattice-matching strains we study in-plane to out-of-plane easy-axis reorientations as\na function of Mn local-moment concentration, hole concentr ation, and temperature. Next we focus\non the competition of in-plane cubic and uniaxial anisotrop ies. We add an in-plane shear strain\nto the effective Hamiltonian in order to capture measured dat a in bare, unpatterned epilayers, and\nwe provide microscopic justification for this approach. The model is then extended by an in-plane\nuniaxial strain and used to directly describe experiments w ith strains controlled by postgrowth\nlithography or attaching a piezo stressor. The calculated e asy-axis directions and anisotropy fields\nare in semiquantitative agreement with experiment in a wide parameter range.\nI. INTRODUCTION\nDilute moment ferromagnetic semiconductors, such as\n(Ga,Mn)As, are particularly favourable systems for the\nresearch in basic spintronics phenomena and towards\npotential applications in memory and information pro-\ncessing technologies. For typical doping levels 1-10% of\nMn the magnetic dipole interactions and corresponding\nshapeanisotropiesare10-100timesweakerin(Ga,Mn)As\nthan in conventional dense-moment ferromagnets. Con-\nsequently, magnetocrystalline anisotropy plays a decisive\nrole in the process of magnetisation reversal. Despite the\nlow saturation magnetisation the magnetic anisotropy\nfields reach ∼10-100mT due to the large spin-orbit cou-\npling.\nThe dependence of magnetic properties of (Ga,Mn)As\nepilayers on doping, external electric fields, temperature,\nand on strain has been explained by means of an effective\nmodel of Mn local moments anti-ferromagnetically cou-\npled to valence band hole spins. The virtual crystal k·p\napproximation for hole states and mean-field treatment\nof their exchange interaction with Mn d-shell moments\nallow for efficient numerical simulations.1,2,3,4The ap-\nproach has proved useful in researching many thermody-\nnamic and magneto-transport properties of (Ga,Mn)As\nsamples with metallic conductivities,3such as the mea-\nsured transition temperatures,5,6,7,8the anomalous Hall\neffect,9,10,11,12anisotropic magneto resistance,9,11,12,13,14\nspin-stiffness,15ferromagnetic domain wall widths,16,17\nGilbert damping coefficient,18,19and magneto-optical\ncoefficients.1,12,18,20,21In thisstudy wesystematicallyex-\nplore the reliability of the effective model in predicting\nthe magnetocrystalline anisotropies of (Ga,Mn)As epi-\nlayer and micro-devices. In our comparisons to experi-\nment we include an extensive collection of available pub-\nlished and unpublished measured data.\nSec. II reviews key elements of the physical model of\n(Ga,Mn)As and of the corresponding effective Hamilto-nian used in our study. Special attention is given to\nmechanisms breaking the cubic symmetry of an ideal\nzinc-blende(Ga,Mn)Ascrystal. Thelatticemismatchbe-\ntweentheepilayerandthesubstrate,producingagrowth-\ndirection strain, is responsible for the broken symmetry\nbetween in-plane and out-of-plane cubic axes. Micro-\nscopic mechanism which breaks the remaining in-plane\nsquare symmetry in unpatterned epilayers is not fully\nunderstood. However, it can be modelled by introducing\nanadditionaluniaxial in-planestrainin the Hamiltonian.\nIn Sec. IIA we discuss the correspondence of this effec-\ntive approach and a generic k·pHamiltonian with the\nlowered symmetry of the p-orbital states which form the\ntop of the spin-orbit coupled valence band. Sec. IIB pro-\nvides brief estimates of the shape anisotropy in thin-film\n(Ga,Mn)As epilayers and micro(nano)-bar devices.\nSections III and IV give the survey and analysis of\ntheoretical and experimental data over a wide range of\nstrains, Mn moment concentrations, hole densities, and\ntemperatures. Sec. IIIA focuses on the easy-axis switch-\ning between the in-plane and out-of-plane directions.\nSec. IIIB studies the competition of cubic and uniax-\nial in-plane anisotropies. Sec. IIIC provides compari-\nson based on anisotropy fields extracted by fitting the\ncalculated and experimental data to the phenomenologi-\ncal formula for the magnetic anisotropy energy. Sec. IV\nstudies in-plane easy axis reorientations in systems with\nadditionalin-planeuniaxialstrainintroducedexperimen-\ntally by post-growth treatment of epilayers. Finally, in\nSec. V we draw conclusions and discuss the limitations\nofour theoretical understanding of magnetic anisotropies\nin (Ga,Mn)As.\nII. MAGNETIC ANISOTROPY MODELLING\nWe use the effective Hamiltonian approachto calculate\nthe magneto-crystalline anisotropy energy of a system of2\nitinerantcarriersexchangecoupledto Mn localmoments.\nThek·papproximation is well suited for the descrip-\ntion of hole states near the top of the valence band in a\n(III,Mn)V semiconductor. The strong spin-orbit interac-\ntion makes the band structure sensitive to the direction\nof the magnetisation. The Hamiltonian reads:\nH=HKL+Jpd/summationdisplay\nISI·ˆs(r)δ(r−RI)+Hstr.(1)\nHKLis the six-band Kohn-Luttinger Hamiltonian22in-\ncluding the spin-orbit coupling (see Appendix A). We\nuse GaAs values for the Luttinger parameters.23.Hstr\nis the strain Hamiltonian discussed in the following sec-\ntion. The second term in Eq. (1) is the short-range\nantiferromagnetic kinetic-exchange interaction between\nlocalised spin SI(S= 5/2) on the Mn2+ions and\nthe itinerant hole spin ˆs, parametrised by a constant23\nJpd= 55 meVm−3. In the mean-field approximation it\nbecomesJpdNMn/angbracketleftS/angbracketrightˆM·ˆs. The explicit form of the 6 ×6\nspin matrices ˆsis given in Ref. [2]. ˆMis the magneti-\nsation unit vector and NMn= 4x/a3\n0is the concentra-\ntion of Mn atoms in Ga 1−xMnxAs (a0is the lattice con-\nstant). Note that the Fermi temperature in the studied\nsystemsismuchhigherthantheCurietemperaturesothe\nsmearing of Fermi-Dirac distribution function is negligi-\nble. Therefore, finite temperature enters our model only\nin the form of decreasing the magnitude of magnetisa-\ntion|M|=SBS(Jpd/angbracketleftˆs/angbracketright/kBT), whereBSis the Brillouin\nfunction, /angbracketleftˆs/angbracketrightis the hole spin-density calculated from the\nmean-field form of Eq. (1).\nWe emphasise that the above model description is\nbased on the canonical Schrieffer-Wolf transformation of\nthe many-body Anderson Hamiltonian. For (Ga,Mn)As\nthe transformation replaces the microscopic hybridisa-\ntion of Mn d-orbitals with As and Ga sp-orbitals by the\neffective spin-spin kinetic-exchange interaction of L=\n0,S= 5/2 local Mn-moments with host valence band\nstates.3Therefore, the local moments in the effective\nmodel carry zero spin-orbit interaction and the magneto-\ncrystalline anisotropy is entirely due to the spin-orbit\ncoupled valence-band holes. The ˆM-dependent total en-\nergy density, which determines the magneto-crystalline\nanisotropy, is calculated by summing one-particle ener-\ngies for all occupied hole states in the valence band,\nEtot(M) =m/summationdisplay\nn=1/integraldisplay\nEn(k,M)f(En(k,M))d3k,(2)\nwhere 1 ≤m≤6 is the number of occupied bands\nf(En(k)) is the Fermi distribution function at zero tem-\nperature.\nA. Beyond the cubic symmetry of the GaAs host\nThek·pmethod provides straightforward means of\nincorporating elastic strains,1,24,25which we now discussin more detail. Small deformation of the crystal lattice\ncan be described by a transformation of coordinates:\nr′\nα=rα+/summationdisplay\nβeαβrβ, (3)\nwhereeαβis the strain tensor. Expressing HKLin\nr′coordinates leads to extra terms dependent on the\nstrain that can be treated perturbatively. The resulting\nstrain Hamiltonian has the same structure as the Kohn-\nLuttingerHamiltonianwith kikjreplacedby eij. (Forde-\ntailed description of Hstrsee Eq. (B2) in the Appendix.)\nLattice matching strain induced by the epitaxial\ngrowth breaks the symmetry between in-plane and out-\nof-planecubic axes. Correspondingnon-zerocomponents\nof the strain tensor read exx=eyy≡e0=−c11\n2c11ezz=\n(as−a0)/a0whereasanda0are the lattice constant of\nthe substrate and the relaxed epilayer, respectively, and\nc12,c11are the elastic moduli.23Typical magnitudes are\ne0∼10−4−10−2.\nAs we discuss in Sec. IV, relaxing the growth strain\nin microbars in transverse direction produces a uniaxial\nsymmetrybreakingin the plane, described by acombina-\ntion ofexx/negationslash=eyyandexystrains, depending on the crys-\ntal orientation of the microbar.4,26,27,28The magnitudes\nrangebetween zero and the growth strain. Additional in-\nplane uniaxial anisotropy effects can be also induced by\npiezo stressors.29,30,31,32The typical magnitude achieved\nby commercial stressors33at low temperature is of the\norder of 10−4.\nAn unpatterned bulk (Ga,Mn)As epilayer can also\nshowbrokenin-planesymmetry, mostfrequentlybetween\nthe [110] and [1 10] directions (see e.g. Refs. [34,35,36,37,\n38,39,40,41,42,43]). For convenience and for direct com-\nparison with effects mentioned in the previous paragraph\nwe model this “intrinsic” in-plane uniaxial anisotropy by\neint\nxy. We fix its sign and magnitude for a given wafer by\nfitting to the corresponding measured anisotropy coeffi-\ncients. To narrow down the number of fitted values for\neint\nxyin the extensive set of experimental data which we\nanalyse, we assume that eint\nxydescribes effectively a sym-\nmetry breaking mechanism induced during growth and\nits value does not change upon the post-growth treat-\nments, including annealing, hydrogenation, lithography\nor piezo-stressing.\nWe point out that an in-plane strain has not\nbeen detected experimentally in the bare unpatterned\n(Ga,Mn)As epilayers. It is indeed unlikely to occur as\nthe substrate imposes the cubic symmetry. The possi-\nbility of transfer of the shear strain from the substrate\nto the epilayer was ruled out by the following test ex-\nperiment. A 50 nm (Ga,Mn)As film was grown on GaAs\nsubstrate. An identical film was grown on the opposite\nsideofthe neighbouringpartofthe samesubstrate. Both\nsamples developed uniaxial magnetic anisotropy along a\ndiagonalbut the easyaxeswereorthogonalto each other.\nIf there were a uniaxial strain in the substrate responsi-\nble for the uniaxial anisotropy in the epilayer, the easy\naxes in the two samples would be collinear. Nevertheless,3\n/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s45/s48/s46/s49\n/s48/s46/s48\n/s48/s46/s49\n/s48/s46/s50/s32/s101\n/s120/s121 /s32/s61/s32/s48/s46/s48/s49/s37\n/s32\n/s52/s32/s61/s32/s48/s46/s48/s50/s32/s126/s32\n/s50/s47/s49/s48/s48/s40/s69\n/s70/s45/s69\n/s91/s49/s48/s48/s93/s41/s32/s91/s107/s74/s32/s109/s45/s51\n/s93/s91/s48/s49/s48/s93\n/s91/s49/s48/s48/s93/s91/s49/s49/s48/s93\n/s91/s49/s49/s48/s93/s95\nFIG. 1: (Color online) Modification of the originally cubic\nin-plane magnetic anisotropy by adding a uniaxial anisotro py\ndue to the shear strain exyor due to the local potential V=\nxyξ.e0=−0.3%,p= 3×1020cm−3,x= 3%,γ4is the\nadditional Luttinger parameter resulting from the in-plan e\nsymmetry lowering and γ2is one of the Luttinger parameters\nfor GaAs (see text and Eq. (A12) in the Appendix).\nwe argue below that the effective modelling via eint\nxypro-\nvides a meaningful description of the “intrinsic” uniaxial\nanisotropy.\nWe compare the effective Hamiltonian corresponding\nto theeint\nxystrain with a k·pHamiltonian in which,\nwithout introducing the macroscopic lattice distortion,\nthe [110]/[110] symmetry is broken. In the derivation of\nthe6-bandKohn-LuttingerHamiltonianoriginatingfrom\nthe Asp-orbitals(denoted by |X/angbracketright,|Y/angbracketright, and|Z/angbracketright), thek·p\nterm is treated perturbatively to second order:\n/angbracketlefti|Hkp|j/angbracketright=¯h2\nm2\n0/summationdisplay\nl/∈{X,Y,Z}/angbracketlefti|k·p|l/angbracketright/angbracketleftl|k·p|j/angbracketright\nEi−El,(4)where the diagonal terms of the unperturbed 6-band\nHamiltonian corresponding to atomic orbital levels are\nset to zero. The symmetries of the tetrahedron (zinc-\nblend) point group Tdnarrow down the number of non-\nvanishing independent matrix elements, represented by\nKohn-Luttinger parameters. The summation over neigh-\nbouring energy levels runs only through the Γ 1and Γ 4\nstatesofthe conduction band asother levelsareexcluded\ndue to the parity of the wave functions or by the large\nseparation in energy. After including the spin-orbit in-\nteraction and transforming to a basis of total momen-\ntum eigen-states we obtain the Hamiltonian HKL(see\nEqs. (A9) and (A10) in the Appendix) with three in-\ndependent Luttinger parameters γ1,γ2, andγ3, plus a\nspin-orbit splitting parameters ∆ so.25,44\nIf the tetrahedral symmetry of the GaAs lattice is bro-\nkenthenumberofindependentparametersincreases. Let\nus consider a perturbation to the crystal potential that\nremoves two of the C2elements of group Td(rotations by\n180◦about the [100] and [010] axes). The corresponding\npotential takes a form V=xyξ, which mixes the Γ 1\nand Γ 4(z) states of the conduction band considered in\nthe summation in Eq. (4) and leaves Γ 4(x) and Γ 4(y)\nstates unchanged. ( ξis a fast decreasing radial func-\ntion.) Such inter-mixing of surrounding states represents\nthe local symmetry lowering of the environment of the\nvalence band p-orbitals. The summation over the per-\nturbed states, αΓ1+βΓ4(z),−βΓ1+αΓ4(z), Γ4(x), Γ4(y)\nin Eq. 4 gives rise to extra terms in the Hamiltonian ˜Hkp.\n(The original form Hkpis given in Eq. (A2) in the spin\ndegenerate basis listed by Eq. (A1) in the Appendix.)\nAssuming a weak local potential V,α>>β, we can ne-\nglect terms of quadratic and higher order dependence on\nVand obtain:\n˜Hkp=\nAk2\nx+B(k2\nz+k2\ny)+2DkxkyCkxky+D(k2\nx+k2\ny) Ckxkz\nCkykx+D(k2\nx+k2\ny)Ak2\ny+B(k2\nz+k2\nx)+2DkxkyCkykz\nCkzkx Ckzky Ak2\nz+B(k2\nx+k2\ny)\n, (5)\nwhere\nD∼ /angbracketleftX|py|Γ4(z)/angbracketright/angbracketleftΓ1|px|X/angbracketright. (6)\nSee Eq. (A8) in the Appendix giving the full expression\nfor the paramater D. Elements containing the parameter\nDchange the dependence of the original Kohn-Luttinger\nHamiltonian on the k-vector. After considering the spin-\norbit coupling we find that the original Kohn-Luttinger\nHamiltonian with Hstrcorresponding to eint\nxyhas the\nsame form as the corrected Kohn-Luttinger Hamiltonian\n˜HKLwith the microscopic symmetry breaking potentialVincluded if weneglect the contributionofthis potential\nto the diagonalelements and replace the term D(k2\nx+k2\ny)\nby a constant term proportional to exy.\nFig. 1 illustrates that the in-plane anisotropy energy\nprofile due to the local potential Vcan indeed be ac-\ncurately obtained by the mapping on the effective shear\nstrain Hamiltonian. For the particular set of material\nparameters and eint\nxy= 0.01% considered in Fig. 1, the\nnew Luttinger parameter γ4≈γ2/100, where γ4=\n−2Dm0/3¯h2(see Eq. (A12) in the Appendix for the def-\ninition ofγ2and the other Luttinger parameters). As we4\ndiscuss in the following section, effective modelling us-\ning the strain Hamiltonian with the constant eint\nxyterm is\nsufficient to capture semiquantitatively many of the ob-\nserved experimental trends. Here we have demonstarted,\nthat the model effectively describes a microscopic sym-\nmetry breaking mechanism yielding quantitatively the\nsame in-plane anisotropy energy profiles without the pre-\nsumption of a macroscopic lattice distortion.\nB. Shape anisotropy evaluation\nWe conclude this theoretical modelling section by\nbriefly discussing the role of shape anisotropy in\n(Ga,Mn)As thin films and microstructures. Magnetic\nshape anisotropy is due to the long range dipolar interac-\ntion. Surface divergence of magnetisation Mgives rise to\ndemagnetising field HD(M,r). In homogeneously mag-\nnetised bodies of general shape the demagnetising field\nis a function of magnetisation magnitude and direction\nwith respecttothe sample. Inellipsoidalbodiesthe func-\ntion becomes linear in MandHD(M) is uniform in the\nbody:\nHD\ni(M) =−/summationdisplay\njNijMj. (7)\nTensorNijis the so called demagnetising factor. In rect-\nangular prisms the linear formula (7) is a good approxi-\nmation and the non-uniform demagnetising factor can be\nreplacedbyitsspatialaverage. Forthemagnetostaticen-\nergy density of a homogeneously magnetised rectangular\nprism we get:\nED(M) =−1\n2µ/summationdisplay\nijNij(a,b,c)MiMj,(8)\nwhere we assume a prism extending over the volume\n−a < x < a ,−b < y < b and−c < z < c in a Carte-\nsian coordinate system. Ref. [45] shows the expression\nforNij(a,b,c) in such prism.\nFig. 2 shows the calculated shape anisotropy energy\nEA=ED(M1)−ED(M2) for a (i) thin film with a=b>\ncand with magnetisation out-of-plane or in-plane ( M1=\n(0,0,M),M2= (M,0,0)), and (ii) for a bar with a >\nb∼cand with magnetisation in-plane ( M1= (0,M,0),\nM2= (M,0,0)). In the former case the shape anisotropy\nfavours in-plane easy-axis direction while in the latter\ncase the easy-axis tends to align along the bar.\nAs a result of the relatively low saturation magnetisa-\ntion of the dilute magnetic semiconductor, the in-plane\nvs. out-of-plane shape anisotropy EAis only about 1 .4\nkJ/m3(0.06 T) for Mn doping x= 5% andc < a/100.\nThis is in agreement with the limit of infinite 2D sheet,\nwhere the formula for shape anisotropy energy per unit\nvolume simplifies to EA=µ0\n2M2cos2θ.θis the an-\ngle that the saturation magnetisation Msubtends to the\nplane normal. The in-plane anisotropy of a bar is even\nweaker and decreases with relative widening of the bar.\nFIG. 2: Shape anisotropy EA=ED(M1)−ED(M2) of a film\nof a thickness cand a long bar of length aand widthbas a\nfunctionofthedimension-less ratio rasdefinedinthecaption.\nThe curves were obtained using the demagnetising factor ap-\nproximation of Ref. [45] for |M|= 0.06T which corresponds\nto Mn doping of x= 5% atT= 0K.\nIn general, the shape anisotropies in the (Ga,Mn)As\ndilute-moment ferromagnet are weak compared to\nthe spin-orbit coupling induced magneto-crystalline\nanisotropies and can be often neglected.\nIII. MAGNETIC EASY AXES IN\nUNPATTERNED SAMPLES\nA large amount of experimental data on magnetic\nanisotropy in (Ga,Mn)As has accumulated over the past\nyears. Comparison of these results with predictions of\nthe effective Hamiltonian model is not straightforward\ndue to the presence of unintentional compensating de-\nfects in (Ga,Mn)As epilayers. Most importantly, a frac-\ntion of Mn is incorporated in interstitial positions. These\nimpurities tend to form pairs with Mn Gaacceptors in as-\ngrown systems with approximately zero net moment of\nthe pair, resulting in an effective local-moment doping\nxeff=xs−xi.8Herexsandxiare partial concentra-\ntions of substitutional and interstitial Mn, respectively.\nIn as-grown materials, the partial concentration xiin-\ncreases with the total Mn concentration, xtot=xs+xi.\nForxtot>1.5%,dxi/dx≈0.2.8We emphasise that in\ntheorytheMnlocalmomentdopinglabelledas” x”corre-\nsponds to the density of uncompensated local moments,\ni.e., toxeffin the notation used above. Mn doping ” x”\nquoted in experimental works refers typically to the total\nnominalMndoping, i.e., to xtot. Whencomparingtheory\nand experiment this distinction has to be considered.\nAlthough interstitial Mn can be removed by low-\ntemperature annealing, xeffwill remain smaller than\nthe total nominal Mn doping. The interstitial Mn im-\npurities are double donors. Assuming no other sources\nof charge compensation the hole density is given by5\np= (xs−2xi)4/a3\n0.8\nThe concentration of ferromagnetically ordered Mn lo-\ncal moments and holes is not accurately controlled dur-\ning growth or determined post growth.7We acknowledge\nthis uncertainty when comparing available magnetome-\ntry results with theory. Throughout the paper we test\nthe relevance of our model over a wide parameter range,\nfocusing on general trends rather than on matching re-\nsults directly based on the material parameters assumed\nin the experimental papers.\nA. In-plane vs. out-of-plane magnetic easy axis\nIn this sectionwestudy the switchingbetween in-plane\nand perpendicular-to-plane directions of the magnetic\neasy axis. (Anisotropies within the growth plain of a\nsamplearestudiedinSec. IIIB.) Earlyexperimentswere\nsuggesting that the in-plane vs. perpendicular-to-plane\neasy axis direction is determined exclusively by the sign\nof the growth induced strain in the sample. The in-plane\neasy axis (IEA) develops for compressive growth strain\ne0= (as−a0)/a0<0. Tensile growth strain, e0>0, re-\nsults in the perpendicular-to-planeeasyaxis(PEA). This\nsimple picture was subsequently corrected by experimen-\ntal results reported for example in Refs. [34,46,47,48,49].\nSign changes in the magnetic anisotropy for the same\nsign of the growth strain were observed with varying Mn\nconcentration, hole density, and temperature.\nAn overview of theoretical easy axis reorientations\ndriven by changes of the material parameters is given\nin Figs. 3 - 6. In the plots we show the difference ∆ Ebe-\ntween total hole energy density for the magnetisation ly-\ning in-plane ( Etot(M||)) and out of plane ( Etot(M⊥)) as a\nfunction of the hole density and temperature. ( Etot(M||)\nis always the smaller of Etotfor magnetisation along the\n[100] and the [110] axis.) We include calculations for four\nMnlocalmomentconcentrationstofacilitatethecompar-\nison with experimental data ofdifferent nominal Mn con-\ncentrations and different degree of annealing, which also\nincreasesthenumberofuncompensatedlocalmomentsas\ndiscussed above. We note that the calculated magneto-\ncrystalline anisotropies are almost precisely linear in the\ngrowth strain and therefore the boundaries between IEA\nandPEAintheFigs.3-6dependonlyveryweaklyonthe\nmagnitude of the growth strain, certainly up to the typ-\nical experimental values |e0|<1%. Magneto-crystalline\nanisotropy diagrams presented in this section for a com-\npressive strain e0=−0.2% are therefore generic for all\ntypical strains, with the IEA and PEA switching places\nfor tensile strain.\nSolid arrows in Figs. 3 - 6 mark easy-axis behaviour as\na function of temperature and doping that has been ob-\nserved experimentally. The dashed arrows correspond to\ntheoretical anisotropy variations that have not been ob-\nserved experimentally. At low hole densities, increasing\ntemperature (marked by arrow (1)) induces a reorienta-\ntion of the easy axis from a perpendicular-to-plane to\nFIG. 3: (Color online) Anisotropy energy ∆ E=E(M||)−\nE(M⊥) [kJm−3] calculated for x= 8%,e0=−0.2%,exy=\n0. Positive(negative) ∆ Ecorresponds to IEA(PEA). Arrows\nmark anisotropy transitions driven by change of temperatur e\nor hole density.\nFIG. 4: (Color online) Anisotropy energy ∆ E=E(M||)−\nE(M⊥) [kJm−3] calculated for x= 6%,e0=−0.2%,exy=\n0. Positive(negative) ∆ Ecorresponds to IEA(PEA). Arrows\nmark anisotropy transitions driven by change of temperatur e\nor hole density.\nan in-plane direction. With decreasing xthis transition\nshifts to lower hole densities; at x= 2% the theoretical\ndensities allowing for such a transition reach unrealisti-\ncally low values for a ferromagnetic (Ga,Mn)As mate-\nrial with metallic conduction. Warming up the partially\ncompensated samples (marked by arrow (2)) has no re-\norientation effect and the easy axis stays in-plane. There6\nFIG. 5: (Color online) Anisotropy energy ∆ E=E(M||)−\nE(M⊥) [kJm−3] calculated for x= 4%,e0=−0.2%,exy=\n0. Positive(negative) ∆ Ecorresponds to IEA(PEA). Arrows\nmark anisotropy transitions driven by change of temperatur e\nor hole density.\nare no exceptions to this behaviour at different Mn con-\ncentrations. Finally, increasing temperature of a very\nweakly compensated (fully annealed) sample can cause\nswitching of the theoretical easy direction from in-plane\ntoperpendicular-to-plane(markedbyarrow(3)),withthe\nexception of the low Mn concentrations.\nThe techniques used to increase the hole density in\nthe experimental works discussed in this section are the\npostgrowth sample annealing and annealing followed by\nhydrogenpassivation/depassivation.48Thelattermethod\nyields solely a change of hole density, whereas the former\nis associated also with an increase of the effective Mn\nconcentration and a decrease of the growth strain. The\ngrowth strain is caused to a large extent by Mn atoms in\ninterstitial positions,50which are removed by the anneal-\ning. The simultaneous increase of hole density and effec-\ntive Mn concentrationdue to annealingimplies atransfer\nbetween the phase diagrams of Figs. 3 - 6 accompanying\nthe transitionsmarkedbyarrows(4) -(6). We arguethat\nthe remarkable similarity of the four diagrams assures a\nmeaningfull qualitative comparison with the effect of an-\nnealing even within a given diagram.\nWe now discuss individual measurements and compare\nwith theoretical diagrams in Figs. 3 - 6. Ref. [48] reports\nexperiments in a 50 nm thick (Ga,Mn)As epilayer nomi-\nnallydopedto x= 6−7%andgrownonaGaAssubstrate\nunder compressivestrain. The sample is first annealed to\nlower the number of interstitial Mn, then hydrogenated\nto passivate virtually all itinerant holes and finally de-\npassivated in subsequent steps by annealing. The hole\ndensity was not measured but for the given Mn doping\nweexpectthe densityinthe rangeof p∼1020−1021cm−3\nFIG. 6: (Color online) Anisotropy energy ∆ E=E(M||)−\nE(M⊥) [kJm−3] calculated for x= 2%,e0=−0.2%,exy=\n0. Positive(negative) ∆ Ecorresponds to IEA(PEA). Arrows\nmark anisotropy transitions driven by change of temperatur e\nor hole density.\nafter depassivation. The low temperature ( T= 4 K) re-\norientation from PEA to IEA induced by successive de-\npassivations and detected indirectly by anomalous Hall\neffect measurement in Ref. [48] matches the transition\nmarked by arrow (4) in Figs. 3 - 5.\nMagnetic hysteresis loops measured by the Hall resis-\ntivity in Ref. [49] reveal easy axis reorientations induced\nby annealing or increasing temperature in material with\nnominal Mn doping x= 7%. This (Ga,Mn)As epilayer\nwas grown on a (In,Ga)As buffer which leads to a ten-\nsile strain. (Recall that the anisotropy energy ∆ Eis an\nodd function of the growth strain so the IEA and PEA\nregions have to be interchanged in Figs. 3-6 when con-\nsidering tensile strain.) Again, the hole density is not\nknown and can be estimated to p∼1020−1021cm−3.\nAfter annealing, the material exhibits perpendicular-to-\nplane easy axis at 4 K and no reorientationoccurs during\nheating up to 115 K ( TC≈120−130K in this material).\nSuch behaviourcorrespondsto arrow(2) of Fig. 4 or Fig.\n3. The as-grown sample has IEA at 4 K and PEA at\n22 K. This easy axis reorientation corresponds to arrow\n(1), again considering a tensile strain. The as-grown and\nannealed samples both share PEA at elevated temper-\nature. Such a stability of the easy axis while changing\nthe hole density corresponds to arrow (5). Theoretical\nanisotropy variations described by arrows (3) and (6) are\nnot observed in Ref. [49]\nRef. [34] presents measurements in compressively\nstrained (Ga,Mn)As epilayers grown on a GaAs sub-\nstrate. The reported nominal Mn concentrations are\nx= 5.3% andx= 3% with compressive growth strain\ne0=−0.27% ande0=−0.16%, respectively, as inferred7\nfrom x-ray diffraction measurement of the lattice param-\neter. The higher doped material was partially annealed\nfor several different annealing times. The hole density\nwas not measured but likely increases substantially with\nannealing. The as-grown x= 5.3% sample at 5 K ex-\nhibits PEA, which changes to IEA upon warming up to\n22 K. This anisotropy variation is not observed for sam-\nples subject to long annealing times. Such a result is\nconsistent with Ref. [49] and corresponds to the theoret-\nical predictions marked by arrows (1) and (2) of Fig. 5\nfor increasing temperature of the as-grown and annealed\nsample, respectively. Again, the effect of annealing is in\ngood agreement with anisotropy behaviour predicted for\nlow (high) temperature represented by arrow (4) (arrow\n(5)), however, there is no experimental counterpart of\ntransitions marked by arrows (3) and (6). The sample\ndoped tox= 3% was not annealed and no transition\nfrom PEA to IEA is observed upon warming. The be-\nhaviour corresponds to arrow (2) in Fig. 6 or 5.\nRef. [51] already reports a successful compari-\nson of measured magnetic anisotropy and theoretical\npredictions.1Among other samples, it presents a com-\npressively strained (Ga,Mn)As epilayer with nominal Mn\nconcentration x= 2.3% (inferred from x-ray diffraction\nmeasurement). A superconducting quantum interference\ndevice (SQUID) measurement of this as-grown sample\nshows PEA at 5 K and IEA at 25 K, corresponding to\nanisotropy variation marked by arrow (1) in Fig. 6 (oc-\ncurring only for a very narrow hole density interval).\nRef. [52] presents (Ga,Mn)As epilayers with compres-\nsive and tensile strain grown on GaAs and (In,Ga)As\nbuffers, respectively, with nominal Mn concentration\nx= 3% inferred from reflection high energy electron\ndiffraction (RHEED) oscillations measured during the\nmolecular-beamepitaxy(MBE)growth. Twoofthe sam-\nples areannealedand magneticanisotropyis investigated\nat 5 K. The tensile strained sample has its easy axis\naligned perpendicular to the growth plane and the com-\npressivelystrainedsamplehasanin-planeeasyaxis. This\nobservation is in good agreement with our theoretical\nmodelling.\nFinally, Ref. [46] shows a transition from PEA to IMA\nupon increasing temperature or change of hole concen-\ntration (induced by gating in this case). The sample is a\n(In,Mn)As epilayer grown on an InAs, and its magnetic\nanisotropy is described consistently by our model when\nthe appropriate band parameters are used.\nB. In-plane anisotropy: Competition of cubic and\nuniaxial components\nAs we discussed in the previous section, the magnetic\neasy axis(axes) is in the plane of (Ga,Mn)As/GaAs films\noverawiderangeofdopings. Experimentalworksinbare\n(Ga,Mn)As epilayers discussed in this section show that\nthe in-plane magnetic anisotropy has cubic and uniax-\nial components. Typically, the strongest uniaxial term is/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53\n/s45/s49/s46/s53\n/s45/s49/s46/s48\n/s45/s48/s46/s53\n/s48/s46/s48\n/s48/s46/s53/s40/s69\n/s70/s45/s69\n/s91/s49/s48/s48/s93/s41/s32 /s91/s107/s74/s32/s109/s45/s51\n/s93\n/s32/s112/s32/s61/s32/s49/s117\n/s32/s112/s32/s61/s32/s51/s117\n/s32/s112/s32/s61/s32/s53/s117/s91/s49/s48/s48/s93/s91/s49/s49/s48/s93/s91/s48/s49/s48/s93\n/s91/s49/s49/s48/s93/s95\nFIG. 7: (Color online) Magnetic anisotropy energy ∆ E=\nEφ−E[100]as a function of the in-plane magnetisation orien-\ntationM=|M|[cosφ,sinφ,0] and its dependenceon material\nparameters. Magnetic easy axes (marked by arrows) change\ntheir direction upon change of hole density pgiven in units\nu≡1020cm−3at Mn local moment concentration x= 5%,\nshear strain exy= 0, and zero temperature.\nalong the in-plane diagonal ([110]/[1 10]) direction. (A\nweak uniaxial component along the main crystal axes\n([100]/[010])has alsobeen detected.42,43) The theoretical\nmodel used so far to describe the easy axis reorientation\nbetween the in-plane and out-of-plane alignment, assum-\ning the growth strain, can account only for the cubic\nin-plane anisotropy component. In this case we find two\neasy axes perpendicular to each other either along the\nmain crystal axes or along the diagonals depending on\nthe Mn concentration and hole density, as shown in Fig.\n7. In order to account for the uniaxial component of the\nin-plane[110]/[1 10]anisotropyinbare(Ga,Mn)Asepilay-\ners the elastic shear strain exyis incorporated into our\nmodel as discussed in Sec. II. (For brevity we omit the\nindex ”int” in the following text and reintroduce the in-\ndex only when additionalreal in-planestrains arepresent\nduetomicro-patterningorattachedpiezo-stressors.) The\nsuperpositionofthetwocomponentsresultsinarichphe-\nnomenologyofmagneticeasyaxisalignmentsasreviewed\nin Fig. 8 - 10.\nFig.8showsanexamplewith easyaxesalignedcloseto\nthe main crystal axes [100]and [010] at Mn local moment\nconcentration x= 5%, hole density p= 3×1020cm−3,\nand a weak shear strain exy= 0.01%. For a stronger\nshearstrain exy= 0.03%thecubicanisotropyisnolonger\ndominant and the easy axes “rotate” symmetrically to-\nwards the diagonal [1 10] direction until they merge for\nexy>∼0.05%. As explained in detail in Sec. II, the mag-\nnitude and sign of the intrinsic shear strain exyenter as\nfree parameters when modelling in-plane anisotropies of\nbare epilayers.\nThe relative strength of uniaxial and cubic anisotropy\nterms depends also on the hole density and Mn concen-8\n/s45/s50/s45/s49/s48/s49/s50\n/s45/s50\n/s45/s49\n/s48\n/s49\n/s50/s40/s69\n/s70/s45/s69\n/s91/s49 /s48/s48 /s93/s41/s32/s91/s107/s74/s32/s109/s45/s51\n/s93\n/s32/s101\n/s120/s121/s32/s61/s32/s48/s46/s48/s53/s37\n/s32/s101\n/s120/s121/s32/s61/s32/s48/s46/s48/s51/s37\n/s32/s101\n/s120/s121/s32/s61/s32/s48/s46/s48/s49/s37/s91/s49/s48/s48/s93/s91/s49/s49/s48/s93/s91/s48/s49/s48/s93\n/s91/s49/s49/s48/s93/s95\nFIG. 8: (Color online) Magnetic anisotropy energy ∆ E=\nEφ−E[100]as a function of the in-plane magnetisation ori-\nentation M=|M|[cosφ,sinφ,0] and its dependence on ma-\nterial parameters. Magnetic easy axes (marked by arrows)\nchange their direction upon change of magnitude of shear\nstrainexy>0 at Mn local moment concentration x= 5%,\nhole density p= 3×1020cm−3, and zero temperature.\n/s45/s48/s46/s51/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s45/s48/s46/s51\n/s45/s48/s46/s50\n/s45/s48/s46/s49\n/s48/s46/s48\n/s48/s46/s49/s40/s69\n/s70/s45/s69\n/s91/s49/s48/s48/s93/s41/s32/s91/s107/s74/s32/s109/s45/s51\n/s93\n/s32/s112/s32/s61/s32/s49/s117\n/s32/s112/s32/s61/s32/s51/s117\n/s32/s112/s32/s61/s32/s53/s117/s91/s49/s48/s48/s93/s91/s49/s49/s48/s93/s91/s48/s49/s48/s93\n/s91/s49/s49/s48/s93/s95\nFIG. 9: (Color online) Magnetic anisotropy energy ∆ E=\nEφ−E[100]as a function of the in-plane magnetisation orien-\ntationM=|M|[cosφ,sinφ,0] and its dependenceon material\nparameters. Magnetic easy axes (marked by arrows) change\ntheir direction upon change of hole density pgiven in units\nu≡1020cm−3, at Mn local moment concentration x= 3%,\nshear strain exy= 0.01%, and zero temperature.\ntration as shown by Fig. 9 and 10, respectively. Both\nanisotropies are non-monotonous functions of xandp,\ncomparedto the linear dependence ofuniaxialanisotropy\non the shear strain. We do not show explicitly the effect\nof increasing temperature which in the mean-field theory\nis equivalent to decreasingthe effective Mn concentration\nwhile keeping the hole density constant (as explained in\nSec. II)./s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50\n/s45/s48/s46/s49\n/s48/s46/s48\n/s48/s46/s49\n/s48/s46/s50/s40/s69\n/s70/s45/s69\n/s91/s49/s48/s48/s93/s41/s32 /s91/s107/s74/s32/s109 /s45/s51\n/s93\n/s32/s120/s32/s61/s32/s49/s37\n/s32/s120/s32/s61/s32/s51/s37\n/s32/s120/s32/s61/s32/s53/s37/s91/s49/s48/s48/s93/s91/s49/s49/s48/s93/s91/s48/s49/s48/s93\n/s91/s49/s49/s48/s93/s95\nFIG. 10: (Color online) Magnetic anisotropy energy ∆ E=\nEφ−E[100]as a function of the in-plane magnetisation orien-\ntationM=|M|[cosφ,sinφ,0] and its dependenceon material\nparameters. Magnetic easy axes (marked by arrows) change\ntheirdirectionuponchangeofMnlocal momentconcentratio n\nxat hole density p= 3×1020cm−3, shear strain exy= 0.01%,\nand zero temperature.\nWe begin the comparison of theory and experiment\nby analysing experimental studies of in-plane magnetic\nanisotropy in bare samples without lithographically or\npiezo-electrically induced in-plane uniaxial strain. Ex-\nperimental results aresummarised in Tab. I. Samples are\nidentified by nominal Mn concentration and hole density\nor annealing as given by the authors. Typically, the hole\ndensity is in the range 1020- 1021cm−3. All samples are\nthin (Ga,Mn)As epilayers deposited by MBE on a GaAs\nsubstrate. According to our calculations, the compres-\nsive growth strain has a negligible effect on the interplay\nof cubic and uniaxial in-plane anisotropies.\nTab. I shows the largest measured projection of the\neasy axis (axes) on the main crystal directions ([100],\n[010], [110], [1 10]) in the corresponding sample. (Note\nthatunlikeourtheoreticalcalculationsofthefullin-plane\nanisotropy profile, most experiments listed in Tab. I re-\nport only projections of the magnetisation to the main\ncrystal directions. Studies using anisotropic magneto-\nresistance (AMR) to map the easy axis direction pre-\ncisely are discussed in Sec. IIIC and IV.) Tab. I includes\na column labelled as EA 0giving the largest easy axis\nprojection at low temperatures (typically 4 K) and a col-\numn labelled as EA TCcorrespondingto measurementsat\ntemperatures close to TC. This simplified overviewof the\ntemperature-dependence of the in-plane anisotropies re-\nflects the nature of available experimental data. The fer-\nromagnetic resonance (FMR) spectra are typically pro-\nvided only at one high and one low temperature. More-\nover, available SQUID data reveal at most one transi-\ntionbetweenmaincrystaldirectionscorrespondingtothe\nlargest projection of the magnetisation in the whole tem-\nperature interval. Sample No. 25 in Tab. I which shows9\ntwo transitions is the only exception to this trend.\nFrom Tab. I we infer the following general trend in\nthe experimentally observed in-plane anisotropies: At\nlow temperatures the in-plane anisotropy is dominated\nby its cubic component. In most cases, this leads to two\nequivalenteasyaxesalignedcloseto [100]and[010]direc-\ntions. Only in a few samples the cubic anisotropy yields\neasy-axis directions along the [110]/[1-10] diagonals at\nlow temperature. The two diagonals are not equiva-\nlent, however, due to the additional uniaxial anisotropy\ncomponent.40,53,54,55At high temperatures the uniaxial\nanisotropy dominates giving rise to only one diagonal\neasy axis. Finally we note that Refs. [37,41] do not iden-\ntify the correspondence between the in-plane diagonal\neasy-axis and one of the two non-equivalent crystallo-\ngraphic axes [110] and [1-10] (these measurements are\nmarked as ⊗in Tab. I). This ambiguity does not affect\nthe comparison with our modelling of unpatterned bare\nfilms since the shear strain exydetermining which of the\ntwo diagonals is magnetically easier is a free effective pa-\nrameterofthe theory. Possibilityoferrorin assigningthe\ntwo non-equivalent diagonal crystallographic axes is ac-\nknowledged by the authors of Ref. [34], where switching\nroles of the diagonals makes the results consistent with\nlater works of the group.\nFollowing the strategy for presenting experimental\ndata in Tab. I, we plot in Figs. 11 - 16 theoretical di-\nagrams indicating crystallographic axes ([100],[110] or\n[110]) with the largest projection of magnetisation as a\nfunction of the hole density and temperature. The com-\nparison with experimental results in Tab. I is facilitated\nby numbered arrowsadded to the diagrams, which corre-\nspond to switchings between crystallographic directions\nwith the largest projection of the easy-axis, driven by in-\ncreasing temperature (horizontal arrows) and hole den-\nsity (vertical arrows).\nFigs. 11 - 14 present diagrams for different Mn con-\ncentrations and for exy= 0.01%. Anisotropy transitions\nseen in the figures are consistent with majority of the re-\nviewedexperimentalworks, i.e., thearrowscorrespondto\nthe experimentally observed transitions and their place-\nment in the diagrams is reasonably close to the relevant\nexperimental parameters. Figs. 11 - 14 also demonstrate\nhow the transition from the [100] to the [1 10] direction\nmoves to higher temperatures with increasing Mn local\nmoment concentration.\nFigs. 15 and 16 address samples where the observed\ntransition cannot be modelled by exy= 0.01%. Four of\nthe low doped samplesin Refs. [35,36,37] aremodelled by\na weaker strain, whereas one of the highly doped samples\nin Ref. [54] is modelled by a stronger strain.\nNow we discuss in detail the theoretical diagrams in\nFigs. 11 - 14 and compare to individual samples from\nTable I, referred to as TI-No. Fig. 11 maps in-plane\nmagnetic anisotropy at Mn local moment concentration\nx= 3% and shear strain exy= 0.01%. The easy axis\nreorientation of the as-grownsample TI-7 corresponds to\narrow (1) in Fig. 11. Arrow (2) in Fig. 11 highlights theNo.Ref.x[%]p[∗]EAlTEAhTFig.ApAlTAhT\n1.[35]2ag+տ15(1)(2)(3)\n2.[35]2an+տ15(1)\n3.[36]23.5+տ15(1)\n4.[37]2ag+⊗15(1)\n5.[56]21.1+ր15(1)n\n6.[39]24+ր15(1)n\n7.[34]3ag+տ11(1)\n8.[38]3ag+ր11(1)n\n9.[53]43.5+ 11 (2)\n10.[53]45+ 11\n11.[55]5ag+տ12(2)(5)(6)\n12.[55]5anրր12(3)\n13.[35]5ag+տ12(2)(4)(6)\n14.[35]5an+ր12(2)n\n15.[40]6ag+տ12(2)(5)(6)\n16.[40]6anրր12(3)\n17.[41]70.75+⊗13(3)\n18.[41]72+⊗13(3)\n19.[41]78.8+⊗13(4)\n20.[41]712+⊗13(4)\n21.[53]73.6+ 13 (6)\n22.[53]711ր 13\n23.[54]8ag+տ16(1)(3)(4)\n24.[54]8anտտ16(2)\n25.[35]8an+ր14(4)\nTABLE I: Experimental in-plane magneto-crystalline\nanisotropies at low temperature EAlT, and high temperature\nEAhTextracted from SQUID or FMR measurements: largest\neasy axis projection along [100] and [010] axes (+), along\n[110] axis ( տ), along [110] axis ( ր), and along one of the\n[110]/[110] diagonals not distinguished in the experiment\n(⊗). Nominal Mn concentrations xreported in experimental\nstudies are rounded down to percents. Hole density p[∗] is\ngiven in units of 1020cm−3. If the hole density is unknown\nthe as-grown and annealed samples are indicated by “ag”\nand “an”, respectively. Samples are ordered according\nto Mn concentration and hole density (annealed sample\nfollows the as-grown counterpart when it exists). The last\nfour columns label the experimental data in a way which\nfacilitates direct comparison with transitions highlight ed by\narrows in the theory Figs. 11 - 16. Numbers in columns\nAp, AlT, and AhTpoint to corresponding theory transitions\nmarked by horizontal arrows, vertical arrows at low T, and\nvertical arrows at high T, respectively. The indexnindicates\ncorrespondence of the given arrow to modelling with negativ e\nvalue ofexy.\nfinite range of hole densities for which the largest pro-\njection of the easy-axes stays along the [100] and [010]\ndirections at low temperature, consistent with the be-\nhaviour of the as-grown and annealed sample TI-9 and10\nFIG. 11: (Color online) Theoretical hole density - temper-\nature diagrams of crystal directions with the largest proje c-\ntion of the magnetic easy axis at x= 3%,exy= 0.01%,\ne0=−0.2%. Arrows mark anisotropy behaviour driven by\nchange of temperature or hole density explaining experimen -\ntally observed behaviour surveyed in Tab. I.\nFIG. 12: (Color online) Theoretical hole density - temper-\nature diagrams of crystal directions with the largest proje c-\ntion of the magnetic easy axis at x= 5%,exy= 0.01%,\ne0=−0.2%. Arrows mark anisotropy behaviour driven by\nchange of temperature or hole density explaining experimen -\ntally observed behaviour surveyed in Tab. I.\nTI-10. (Note that hole densities in samples TI-9 and\nTI-10 weremeasured by the electrochemical capacitance-\nvoltage profiling.) The transition from the largest easy\naxis projection along the cube edges to the [110] diag-\nonal observed in as-grown sample TI-8 with increasing\ntemperature has no analogy in Fig. 11 or Fig. 12. The\nFMR measurement does not indicate switching of the\neasy axis alignment between the diagonals at any inter-\nmediate temperature. This behaviour can be explained\nonly if the opposite sign of the shear strain is used to\nmodeltheintrinsicsymmetrybreakingmechanism. Then\nthe easy axis transition of TI-8 would correspond to ar-\nFIG. 13: (Color online) Theoretical hole density - temper-\nature diagrams of crystal directions with the largest proje c-\ntion of the magnetic easy axis at x= 7%,exy= 0.01%,\ne0=−0.2%. Arrows mark anisotropy behaviour driven by\nchange of temperature or hole density explaining experimen -\ntally observed behaviour surveyed in Tab. I.\nFIG. 14: (Color online) Theoretical hole density - temper-\nature diagrams of crystal directions with the largest proje c-\ntion of the magnetic easy axis at x= 9%,exy= 0.01%,\ne0=−0.2%. Arrows mark anisotropy behaviour driven by\nchange of temperature or hole density explaining experimen -\ntally observed behaviour surveyed in Tab. I.\nrow (1) in Fig. 11.\nThe behaviour of as-grown samples TI-11,13,15 cor-\nresponds to arrow (2) in Fig. 12. The annealed sam-\nples TI-12,16 exhibit the rarely experimentally observed\ndomination ofuniaxial anisotropyfor the whole tempera-\nture range. This behaviour is also consistently captured\nby the theory as highlighted by arrow (3) in Fig. 12.\nSample TI-14 has a dominant cubic anisotropypreferring\n[100]/[010] magnetisation directions at low temperature\nand the easy axis aligns closer to the [110] direction at\nhigh temperatures. Similarly to sample TI-8, this transi-\ntion has no analogy in Fig. 11 or Fig. 12, however, it can11\nbe explained assuming that the [110]/[1-10] symmetry\nbreaking mechanism has opposite sign in this material\nand therefore should be modelled by a negative value of\nthe effective strain exy. Then the easy axis transition of\nTI-14 would correspond to arrow (2) in Fig. 12. Another\npossibility is to assume the same sign of exyas for the\nabove samples and associate the transition in sample TI-\n14 with arrow(4) in Figs. 13 and 14. Note, however, that\nthe intermediate-temperature anisotropy state with the\nlargest magnetisation projection along the [1 10] diagonal\nseen when following the theory trend along arrow (4) has\nnot been reported in the experimental study of sample\nTI-14. Arrows (4)-(6) in Fig. 12 correspond to measured\nanisotropy behaviour driven by increasing hole density\nin pairs of as-grown and annealed samples TI-11,12, TI-\n13,14, and TI-15,16.\nAt the upper end of the investigated effective Mn con-\ncentration interval the theoretical alignment of magnetic\neasyaxesismapped byFigs.13and14. SamplesTI-17to\nTI-20 nominally doped to x= 7% were all annealed after\ngrowth, passivated by hydrogen plasma, and then gradu-\nally depassivated to achieve different hole densities (mea-\nsured by high-field Hall effect). Magnetic anisotropies\nweredeterminedbyFMR.Theassignmentofthein-plane\ndiagonal directions to the non-equivalent [110] and [1 10]\ncrystallographic axes is not specified in this experimen-\ntal work; recall that this ambiguity is not crucial for the\npresent discussion. The transition observedin these sam-\nples from a cubic ([100]/[010] easy directions) dominated\nanisotropy at low temperatures to a uniaxial behaviour\nat high temperatures is captured by arrows (3) and (4)\nin Figs. 13 and 14. Importantly, the depassivated higher\nhole density samples TI-19 and TI-20 show an additional\nswitching of the easy-axis from one to the other diagonal\ndirection at intermediate temperatures, consistent with\nthe theoretical temperature dependence along the arrow\n(4). This double transition behaviour was also detected\nin the annealed sample TI-25, where the temperature\ndependent magnetisation projections were measured by\nSQUID. In this experiment it is identified that the easy-\naxis first rotates towards the [1 10] direction at interme-\ndiate temperatures and then switches to the [110] direc-\ntion at high temperatures, consistent with the behaviour\nmarked by arrow (4) in Figs. 13 and 14.\nSamples TI-21,22 are measured only at low tempera-\nture. Easy axis reorientationfrom [100]to [110] direction\nisdrivenbyincreaseofholedensity, whichcorrespondsto\narrow(6)inFig.13or14. Theholedensitywasmeasured\nby the electrochemical capacitance-voltage method.\nIn-plane anisotropiesof samples with x≈2% are mod-\nelled in Fig. 15. To obtain the cubic anisotropy domi-\nnated region at low temperatures and a transition to the\nuniaxial behaviour at high temperatures, as observed in\nsamples TI-1 to TI-6, we take for this low Mn doping\nexy= 0.005%. (The effective strain exy= 0.01% would\nlead to easy axis along [1 10] over the entire temperature\nrange and for exy= 0.001% the cubic anisotropy region\nwould extend up to very high temperatures.) Arrow (1)\nFIG. 15: (Color online) Theoretical hole density - temper-\nature diagrams of crystal directions with the largest proje c-\ntion of the magnetic easy axis at x= 2%,exy= 0.005%,\ne0=−0.2%. Arrows mark anisotropy behaviour driven by\nchange of temperature or hole density explaining experimen -\ntally observed behaviour surveyed in Tab. I.\nFIG. 16: (Color online) Theoretical hole density - temper-\nature diagrams of crystal directions with the largest proje c-\ntion of the magnetic easy axis at x= 7%,exy= 0.03%,\ne0=−0.2%. Arrows mark anisotropy behaviour driven by\nchange of temperature or hole density explaining experimen -\ntally observed behaviour surveyed in Tab. I.\nin Fig. 15 corresponds to easy axis switching from the\n[100] to the [1 10] direction in samples TI-1,2,3. Arrows\n(2) and (3) in Fig. 15 mark the behaviour of the easy\naxisdriven by increasinghole density when annealingthe\nsample TI-1 to obtain the sample TI-2 at low and high\ntemperature,respectively. SampleTI-4assumesthe[1 10]\ndiagonal always harder than the [110] diagonal. A tran-\nsition from cubic to uniaxial dominated anisotropy is ob-\nserved upon increasing the temperature. This behaviour\ncorresponds to arrow (1) in Figs. 15. (The hole density\nof sample TI-3, p= 3.5×1020cm−3, was determined\nby low-temperature high-field Hall effect measurements,12\nhowever, it was not measured for samples TI-1,2,4.)\nSamples TI-5 and TI-6 have their easy axis aligned\ncloser to the [100]/[010] directions at low temperatures\nand to the [110] direction at higher temperatures, simi-\nlarly to sample TI-8. The SQUID measurement of mag-\nnetisation projectionsfor the whole range of temperature\ndoes not indicate the easy axis alignment close to the\n[110] direction at any intermediate temperature. The\nhole density of the sample TI-5, p= 1.1×1020cm−3,\nis measured by Hall effect (at room temperature) and\nits Mn concentration is inferred from X-ray diffraction\nmeasurement of the lattice constant. The hole density\nof the sample TI-6 is p= 4×1020cm−3(measured by\nthe electrochemical capacitance-voltage method at room\ntemperature)andweestimatetheMnconcentrationfrom\nthe reported critical temperature, TC= 62 K, after an-\nnealing. The described experimental behaviour does not\ncorrespond to predicted anisotropy transitions for rele-\nvantholedensities, Mnlocalmomentconcentrations,and\npositive shear strain. The behaviour can be explained,\nhowever, if the opposite sign of the shear strain is used\nto model the intrinsic symmetry breaking mechanism at\nlow Mn concentration. Then the easy axis transition of\nTI-5,6 would correspond to arrow (1) in Fig. 15.\nFinally we comment on the less frequent behaviour\nobserved in the annealed sample TI-24. While its as-\ngrown counterpart TI-23 shows the commonly seen tran-\nsition from the cubic dominated anisotropy to the uni-\naxial anisotropy with increasing temperature, marked by\narrow (1) in Fig. 16, the annealed material has its easy\naxis aligned close to the [1 10] direction over the entire\nstudied temperature range. Arrow (2) in Fig. 16 pro-\nvidesaninterpretationofthisbehaviourifweincreasethe\nmagnitude of the effective shear strain. At exy= 0.03%\nthecubicanisotropydominatedregionisalreadystrongly\ndiminished and for exy= 0.05% it vanishes completely.\nArrows (3) and (4) then highlight within the same dia-\ngram the consistent description of the evolution of the\nexperimental anisotropies, both at low and high temper-\natures, from the as-grown low hole density sample TI-23\nto the annealed high hole density sample TI-24.\nTo summarise this section, our theoretical modelling\nprovides a consistent overall picture of the rich phe-\nnomenology of magneto-crystalline anisotropies in un-\npatterned (Ga,Mn)As epilayers. Our understanding is\nlimited, however, to only a semiquantitative level, ow-\ning to the approximate nature of the mean-field kinetic-\nexchange model, ambiguities in experimental material\nparameters of the studied films, and unknown micro-\nscopic origin of the in-plane uniaxial symmetry breaking\nmechanism. We remark that the effective shear strain\nwe include to phenomenologically account for the exper-\nimental [110]/[1 10] uniaxial anisotropy scales with Mn\ndoping (exy≃0.005x). It brings additional confidence\nin this modelling approach as it is most likely the incor-\nporation of Mn which breaks the cubic symmetry of the\nlattice. The magnitude of the effective strain parame-\nter falls into the range 0 .005%< exy<0.05% and the0 2 4 6 8 10 12\nHole density [1020 cm-3]-15-10-505E[110] - E[110] [kJm-3]\nBG = 10 meV\nBG = 20 meV\nBG = 40 meV\nBG = 60 meV0 2 4 6 8 10 12-6-4-20246E[110] - E[110] [kJm-3] BG = 10 meV\nBG = 20 meV\nBG = 40 meV\nBG = 60 meV[110] easy\n[110] easy(a)\n(b)__\nFIG. 17: (Color online) In-plane uniaxial anisotropy as a\nfunction hole density at zero temperature, exy= 0.05%,\nande0= 0 calculated in this work (a) and in Ref. [35] (b).\nCurves are labelled by the valence-band spin-splitting par am-\neter BG≡JpdNMnS/6 to allow for simple comparison with\nRef. [35]. (B G= 4.98xin meV and in percent, respectively.)\nDashed intervals of the horizontal axis mark regions where\na change of temperature (inversely proportional to B G) can\nlead to the [ 110]↔[110] easy axis reorientation.\nanisotropy behaviour consistent with most experimental\nworks is modelled with positive sign of exy.\nWe conclude this section by a remark on numerical\nsimulations of the [110] to [1 10] easy axis transition per-\nformed in Ref. [35]. The physical model employed by\nthe authors of Ref. [35] is identical to ours, neverthe-\nless, the results of the calculations do not quantitatively\nmatch ours, as illustrated in Fig. 17. We have clarified\nwith the authors of Ref. [35] the numerical origin of the\ndiscrepancy. This helpful exercise has provided an inde-\npendent confirmation of the accuracy, within the applied\nphysical model, of the theoretical results presented in\nthe current paper. (To compare Fig. 17 to the original\nplot in Ref. [35] use the conversionto units of normalised\nanisotropy field Hun/M= 2(E[110]−E[110])/(µ0M2).)13\nC. Anisotropy fields\nHaving analysed the in-plane and out-of-plane\nanisotropies based on the direction of easy axes, we turn\nour attention to the relative strength of the anisotropy\ncomponents, i.e., to the anisotropy energies. The compo-\nnents of magnetocrystalline anisotropy can be described\nin terms of a simple phenomenological model separat-\ning the free energy density F(ˆM) into components of\ndistinct symmetry. Each component is described by a\nperiodic function with a corresponding coefficient. We\nfind that angular dependencies of the energies obtained\nfrom our microscopic modelling can be approximated ac-\ncurately even in the first and second order of expansion\ninto periodic functions of uniaxial and cubic symmetry,\nrespectively.\nThe coefficients can be determined experimentally,\ne.g., by analysing the FMR spectra,41,52,56,57from\nAMR58,59or by fitting SQUID magnetometry data to\nan appropriate phenomenological formula for anisotropy\nenergy.36,60In this subsection we extract the relevant co-\nefficients from the calculated anisotropies, track their de-\npendence on material parameters and compare theory to\nexperiment on this level.\nWe startwith identifying the types ofanisotropyterms\nconsidered in our expansion of the anisotropy energy.\nThe cubic anisotropy due to the crystal symmetry of the\nzinc-blende structure is described using terms invariant\nunder permutation of the coordinate indices x,y, and\nz. The independent first, second and third order cubic\nterms read: Kc1/parenleftbig\nn2\nxn2\ny+n2\nxn2\nz+n2\nzn2\ny/parenrightbig\n,Kc2/parenleftbig\nn2\nxn2\nyn2\nz/parenrightbig\n,\nandKc3/parenleftbig\nn4\nxn4\ny+n4\nxn4\nz+n4\nyn4\nz/parenrightbig\n, respectively, where nx=\ncosφsinθ,ny= sinφsinθ, andnz= cosθare compo-\nnents of the magnetisation unit vector ˆM(the angles θ\nandφare measured from the [001] and [100] axis, respec-\ntively). See the Appendix C for details on the mutual\nindependence of all cubic terms.\nAs mentioned in previous sections, the cubic\nanisotropy of the host crystal lattice is accompanied by\ndifferent types of uniaxial anisotropy. A generic term\ncorresponding to uniaxial anisotropy along a given unit\nvectorˆUdepends on the even powers of the dot prod-\nuct (ˆM·ˆU). The first and second order terms read:\nKu1(ˆM·ˆU)2andKu2(ˆM·ˆU)4. The particular cases\nof uniaxial anisotropy terms and their correspondence to\nlattice strains will be described later in this section.\nBefore we present the calculated values of the cu-\nbic anisotropy coefficient, we introduce the so called\nanisotropy fields which are often used in literature in-\nstead of the energy coefficients. In this section we plot\nthe anisotropy fields in 0ersteds (Oe) to make the com-\nparison with experiment more convenient. The relation\nof the anisotropy fields Hato the energy coefficients Ka\nreads:Ha= 2Ka/M.\nFig. 18 shows Hc1andHc2as functions of hole density\npand Mn local moment concentration xat zero tem-\nperature. Both coefficients oscillate as function of thehole density p. As discussed in detail in Ref. [2] the\nanisotropiestendtoweakenwithincreasingpopulationof\nhigher bands which give competing contributions. Con-\nsistent with this trend the amplitude of the oscillations\nincreases with increasing xand decreasing p. The up-\nper limit of the hole density p=NMncorresponds to no\ncharge compensation (Recall, NMn≈2.21xin 1020cm−3\nforxin percent).\nFIG. 18: (color online) Lowest order cubic anisotropy field\nHc1and second order cubic anisotropy field Hc2calculated\nas functions of hole density p(up to zero compensation\np=NMn) and Mn local moment concentration xat zero\ntemperature;\nOur modelling predicts the extremal magnitude of the\nsecond order cubic term Hc2a factor of two smaller than\ntheextremalmagnitudeofthefirstorderterm Hc1. Upon\nincreasing the hole density the amplitude of oscillations\nofHc2decreases faster than in case of Hc1. The third\norder cubic anisotropy field Hc3is negligible compared\ntoHc1andHc2for all studied combinations of the ma-\nterial parameters. To our knowledge, Hc2andHc3have\nnot been resolved experimentally. We emphasise that\nthe second order cubic term does not contribute to the14\nanisotropy energy for magnetisation vectors not belong-\ning to the main crystal plains. The dependence of all\nthree calculated cubic terms on the lattice strains of typ-\nical magnitudes (up to 1%) is negligible.\nNow, we focus on classification of distinct uniax-\nial anisotropy components and their relation to lattice\nstrains lowering the underlying cubic symmetry of the\nzinc-blende structure. We have already mentioned that\ntypically the strongest symmetry breaking mechanism is\nthe growth strain (introduced in Sec. II). It is relevant\nforthein-planeversusout-of-planealignmentofthemag-\nnetic easyaxis. We have alsomentioned the in-plane uni-\naxial anisotropy between the [110] and the [1 10] axes. Its\norigin is not known, however, we have modelled it using\nthe shear strain which is about a factor of ten weaker\nthan the typical growth strain.\nSome (Ga,Mn)As epilayers42,43also show a very\nweakly broken symmetry between the main crystal axes\n[100] and [010]. We will introduce here a uniaxial strain\nthat can account for this type ofanisotropy, however, our\nmain motivation for introducing this third strain tensor\nis to complete an in-plane strain basis. This basis is used\nin Sec. IV to describe all types of lattice in-plane strains\ninduced experimentally by growth and post-growth pro-\ncessing of the (Ga,Mn)As epilayers. Once the strain ten-\nsors and corresponding anisotropy contributions to the\nfree energy are introduced, it will be shown that the cho-\nsen basis has the advantage of collinearity of the strain\nand of the resulting anisotropy component. Finally, in\nthis subsection the numerical data and comparison with\nexperiment will be presented for the bare unpatterned\nepilayers. The patterned structures will be discussed in\nSec. IV.\nFirstly, we recall the growth strain introduced in\nEq. (3). It is usually referred to as the biaxial pseu-\ndomorphic strain as it is due to the lattice missmatch\nbetween the substrate and the epilayer. The doped crys-\ntal is forced to certain dimensions by the substrate in\nthe two in-plane directions whereas it can relax in the\nperpendicular-to-planedirectionkeepingtherequirement\nof zero net force acting on the crystal: 0 = c12exx+\nc12eyy+c11ezz. The corresponding strain tensor:\neg=\ne00 0\n0e00\n0 0−2c12\nc11e0\n (9)\ndescribes an expansion (contraction) along the [100] and\n[010] axes for positive (negative) e0accompanied by a\ncontraction (expansion) along the [001] axis. Parameters\nc11andc12are the elastic moduli. The growth strain\nenters our model via the strain Hamiltonian Hstr(see\nEq. (1)) and induces a uniaxial anisotropy component\nwhich can be described in the lowest order by an energy\nterm−K[001]n2\nz=−K[001]cos2θ.\nThe shear strain, first introduced in Sec. IIA, is rep-resented by a tensor:\nes=\n0κ0\nκ0 0\n0 0 0\n. (10)\nPositive(negative) κcorrespondstoturningasquareinto\na diamond with the longer (shorter) diagonal along the\n[110] axis. We have used this type of strain as the “in-\ntrinsic“shearstrain eint\nxytomodelthe difference inenergy\nfor magnetisation aligned with the two in-plane diago-\nnals. It results in uniaxial anisotropy along the diago-\nnals, described in analogy to the growth strain by a term\n−K[110](ny−nx)2/2 =−K[110]sin2(φ−π/4)sin2θ.\nFinally,wewritedownthethirdelementofthein-plane\nstrain basis:\neu=\nλ0 0\n0−λ0\n0 0 0\n. (11)\nPositive(negative) λcorrespondstoturningasquareinto\na rectanglewhere expansion(contraction) alongthe [100]\naxis is accompanied by a contraction (expansion) along\nthe[010]axisofthesamemagnitude. Muchlikeincaseof\nthe growth strain and the shear strain, the requirement\nof zero net force acting on the crystal is kept but this\ntime it results in ezz= 0. The strain euinduces uniaxial\nanisotropy along the main crystal axes, described by a\nterm−K[100]n2\ny=−K[100]sin2φsin2θ.\nLet us remark that strain tensors in Eqs. (9-11) are ex-\npressed in Cartesian coordinates fixed to the main crys-\ntallographicaxes. Strains esandeuforκ=λare related\nby a rotation about the [001] axis by π/4, however, the\ncubic crystal is not invariant under such rotation so the\ntwo strains induce anisotropies with magnitudes K[100]\nandK[110]which are different in general. The growth\nstraineg, the shear strain es, and the uniaxial strain eu\ncan be characterised by a single direction of deformation\nand induce uniaxial anisotropy components aligned with\nthat particular direction. We found that higher order\nuniaxial terms are small unless we approach experimen-\ntally unrealistic large values of exchange splitting (large\nx) and hole compensation (low p).\nIn total, we can write our phenomenological formula\napproximating accurately the calculated free energy den-\nsity of an originally cubic system subject to three types\nof strain as a sum of distinct anisotropy components:\nF(ˆM) =Kc1/parenleftbig\nn2\nxn2\ny+n2\nxn2\nz+n2\nzn2\ny/parenrightbig\n+Kc2/parenleftbig\nn2\nxn2\nyn2\nz/parenrightbig\n−\n−K[001]n2\nz−K[110]\n2(ny−nx)2−K[100]n2\ny.(12)\nBy definition of the terms, a positive coefficient K[001]\nprefers perpendicular-to-plane easy axis (PEA); positive\nK[110]andK[100]prefer easy axis lying in-plane (IEA)\naligned closer to [1 10] and [010] axis, respectively. Note15\nthat the anisotropy terms entering the phenomenologi-\ncal formula follow a sign convention consistent with with\nexisting literature.41,52,56,57\nWe now provide the microscopic justification for the\nchoiceofthe elements esandeuofthe in-plane strainba-\nsis and corresponding phenomenological uniaxial terms.\nThis will be based on symmetries of the Kohn-Luttinger\nHamiltonian HKLand the strain Hamiltonian Hstras\nshown in Eqs. (A11) and (B2), respectively, which re-\nlates the band structure to a general in-plane strain with\nthe components exx,eyy, andexy.\nFirst let us point out that the basis element eg(the\ngrowth strain) is invariant under rotation about the [001]\naxis and according to our calculation does not influ-\nence the in-plane direction of the easy axis (in the linear\nregime of small deformations). We continue by show-\ning that for esandeu, the strains and the correspond-\ningmagnetocrystallineanisotropycomponentsareindeed\ncollinear and that this collinearity applies only for the\nspecial cases of uniaxial symmetries along the in-plane\ndiagonals or main axes. Let us assume a rotation of the\ntensoreuby an arbitrary angle ωabout the [001] axis:\neu(ω) =RT\nω\nλ0 0\n0−λ0\n0 0 0\nRω (13)\n=\nλcos2ω λsin2ω0\nλsin2ω−λcos2ω0\n0 0 0\n,\nwhereRωis the rotation matrix. (The same analysis ap-\nplies to a rotation of es). The parameters exx=−eyy=\nλcos2ωandexy=λsin2ωenter the strain Hamiltonian\n(see Eq. (B2) in the Appendix) only via the matrix ele-\nment:\ncs=a2\n2√\n3(eyy−exx)+ia3exy\n=−λ/bracketleftig\na2√\n3cos(2ω)−ia3sin(2ω)/bracketrightig\n,(14)\nwherea2√\n3/negationslash=a3are strain Luttinger constants. More-\nover, the strain component exyquantifying the shear\nstrain enters only Im( cs), whereas the components exx=\n−eyyenter only Re( cs). According to our calculation\nthe imaginary and real part of csgenerate independent\nuniaxial anisotropy components along the [110] and [100]\naxis, respectively. Their combined effect can be under-\nstood based on an analogy of the in-plane rotation of the\nstrain tensor euand an in-plane rotation of a k-vector.\nAs mentioned in Sec. II the Kohn-Luttinger Hamilto-\nnianHKLandthestrainHamiltonian Hstrhavethesame\nstructure. Wewritehereexplicitlythematrixcomponent\ncoftheHamiltonian HKLanalogousto csasafunctionof\nthe in-plane angle of the k-vectork=|k|[cosφ,sinφ,0].\nThe element reads:\nc=√\n3¯h2\n2m/bracketleftbig\nγ2(k2\nx−k2\ny)−2i(γ3kxky)/bracketrightbig=√\n3¯h2\n2mk2/bracketleftbig\nγ2cos2φ−iγ3sin2φ/bracketrightbig\n,(15)\nwhere again γ2/negationslash=γ3are Luttinger constants describing\na cubic crystal. For γ2=γ3the Hamiltonian HKLhas\nspherical symmetry. Similarly, if a2√\n3 =a3, the strain\nHamiltonian Hstris spherically symmetric and the con-\ntributions of Im( cs) and Re(cs) to the anisotropy of the\nsystem combine in such a way that the resulting uniax-\nial term is collinear with the strain eu(ω) rotated with\nrespect to the crystallographic axes by an arbitrary in-\nplane angle ω.\nClearly, the underlying cubic symmetry of the host\ncrystal causes a non-collinearity of the uniaxial strain\nalong a general in-plane direction and the corresponding\nanisotropy component. Moreover, the misalignment is a\nfunction of Mn local moment concentration, hole density\nand temperature. We discuss further this misalignment\nin more detail in Sec. IV. Here we point out the distinct\nexception when ωis an integer multiple of π/4 and ei-\nther the real or the imaginary part of csvanish rendering\nthe strain Hamiltonian effectively spherically symmetric.\nWe choosequite naturallythe simple forms of eu(ω) with\nω= 0 andω=π/4 as elements of the in-plane strain ba-\nsis. For a different choice of the basis elements than in\nEqs. (10) and (11), setting up the phenomenological for-\nmula would be more complicated.\nWe can now resume our discussion of the interplay of\nthe cubic and uniaxial anisotropy components. Adding\nthe uniaxial terms leads to rotation or imbalance of the\noriginal (cubic) easy axes as shown in Sec. IIIB in Fig. 7.\nFig. 19 shows H[110]= 2K[110]/MandH[100]=\n2K[100]/Mas functions of hole density pand Mn lo-\ncal moment concentration xat zero temperature. Both\nanisotropy fields denpend on material parameters in a\nqualitatively very similar manner. Moreover, we observe\nsimilar dependence on the doping parametersalsoin case\nof the field H[001](not plotted). All the three fields os-\ncillate as functions of hole density. The period of the\noscillation is longer than in case of Hc1. In general, the\namplitude of the oscillations decreases with decreasing\nMn local moment concentration.\nThe uniaxial fields are linearly dependent on the strain\nfrom which they originate, unless the strains are very\nlarge (>1%). For the shear strain of the value exy=κ≈\n0.01%, which is the typical magnitude in our modelling,\nand zero temperature, the extremal values of H[110]are\nan order of magnitude smaller than the extremal values\nofHc1∼103Oe. For typical compressive growth strain\ne0≈ −0.2% of an as-grown 5% Mn doped epilayer and\nzero temperature the extremal values of H[001]are of the\nsame order as Hc1. When the magnitude of the uniaxial\nstrain along [100] axis is set to ( exx−eyy)/2 =exy, or\nequivalently κ=λ,H[100]is approximately a factor of\ntwo smaller than H[110].\nTo quantify the observed similarity in the calculated\ndependencies of the uniaxial anisotropy coefficients on x,16\nFIG. 19: (color online) Calculated anisotropy fields H[110]and\nH[100]as functions of hole density p(up to zero compensation\np=NMn) and Mn local moment concentration xat zero\ntemperature and e0=−0.2%. ForH[110]the in-plane strains\nareκ= 0.01% andλ= 0 (exy= 0.01%,exx=eyy=e0),\nwhileH[100]is found for κ= 0 andλ= 0.01% (exy= 0,\nexx=e0+0.01%,eyy=e0−0.01%).\np, and strains, we can write approximate relationships:\nK[001](x,p,e0)≃q[001](x,p)e0,\nK[100](x,p,λ)≃q[100](x,p)λ,\nK[110](x,p,κ)≃q[110](x,p)κ. (16)\nNote, that each anisotropy component depends only on\none type of strain, which is due to the choice of the basis\nin the strain space (see Eqs. (9), (10), and (11)). (Such\nexclusive dependence of a particular uniaxial anisotropy\ncomponent on the corresponding strain is, indeed, ob-\ntained also from simulations of systems subject to com-\nbinations of all three types of strain.) The linearity of\nanisotropy coefficients as functions of lattice strains is\nlimited to small elastic deformations of the lattice. The\napproximationcannotbeusedforstrainsgreaterthan1%as revealed also by calculations in Ref. [1]. Experiment\nconfirms the linear behaviour in case of the growth strain\nup toe0≈ ±0.3%.58Linear dependence on in-plane uni-\naxial strains is corroborated by experiments discussed in\nSec. IV.\nIn addition to the linearity with respect to strain,\nwe observe universal dependence of the three uniax-\nial anisotropy coefficients on hole density and Mn local\nmoment concentration. It can be expressed using the\nanisotropy functions:\nq[001](x,p)≃q[100](x,p)≃0.43q[110](x,p).(17)\nThe anisotropy function q[110](x,p) due to shear strain is\napproximately twice as large as the anisotropy functions\nq[100](x,p) andq[001](x,p). A general property of these\nfunctions is that at medium hole densities a relative com-\npression yields a tendency of the easy axis to align with\nthat direction. On the other hand, for very low and high\nhole densities, the magnetisation prefers alignment par-\nallel to the direction of lattice expansion.\nWe caution that Eqs. (16) and (17) are included to\npromote the general understanding of the anisotropic be-\nhaviour of the strained crystal but are not precise. The\nrelative error of the approximation given by Eq. (17) av-\neraged over the x−pspace shown in Fig. 19 is less than\n20%, however, the relative error can be much larger at a\ngiven combination of xandpwhere the anisotropy coef-\nficients fall to zero.\nTo finish the analysis of the theoretical results we in-\nclude Fig. 20 to improve the legibility of the data. The\nindividual curves correspond to cuts through the 3D\nplots in Figs. 18 and 19 at fixed Mn local moment con-\ncentrations. As already mentioned, the dependence of\nanisotropy fields on hole density is oscillatory. Note that\nthe critical hole densities, where the sign inversion oc-\ncurs, shift away from the extremal values, i.e., zero hole\ndensityand zerocompensation p=NMn, with increasing\nx.\nNeglecting the complexity of the dependence of the\nband structure on M (whether changed by doping or\ntemperature), one would expect the cubic anisotropy\ncoefficientKc1to be proportional to M4and uniaxial\nanisotropy coefficients K[001],K[100], andK[110]toM2.\nIn Fig. 20 we can identify intervals of hole density where\nanychangeinMnconcentration,andthereforeinM,does\nnot induce a sign change of the anisotropy fields and the\nfunctional forms of Ka(M) are roughly consistent with\nthe above expectations. For other hole density intervals,\nhowever,the behaviourishighlynon-trivialandthe func-\ntionKa(M) can even change sign.\nWe now proceed to the discussion of how the theoreti-\ncally expected phenomenology detailed above is reflected\nin experiments in bare unpatterned (Ga,Mn)As epilay-\ners. The experimental results41,52,56,57are often anal-\nysed using the following version of the phenomenological\nformula:\nF(ˆM) =−2πM2sin2θ−K2⊥cos2θ−1\n2K4⊥cos4θ17\n0.5 NMn\nHole density [1020cm-3]-1500-1000-500050010001500Hc1 [Oe]\nx=2%\nx=4%\nx=6%\nx=8%Magnetization prefers [100] and [010]\nM prefers [110] and 110](a)\n-\n   0.5 NMn\nHole density [1020cm-3]-200-1000100200H[110] [Oe]x=2%\nx=4%\nx=6%\nx=8%\n[110] easier(b)\n[110] easier-\n   0.5 NMn\nHole density [1020cm-3]-100-50050100H[100] [Oe]\nx=2%\nx=4%\nx=6%\nx=8%[100] easier(c)\n[010] easier\nFIG. 20: Anisotropy fields Hc1,H[110], andH[100]as func-\ntion of hole density (up to zero compensation p=NMn) at\nfour Mnlocal momentconcentrations x, zerotemperature and\ngrowth strain e0=−0.2%. ForH[110]the in-plane strains are\nκ= 0.01% andλ= 0, while H[100]is found for κ= 0 and\nλ= 0.01%. (The field Hc1is not a function of lattice strains\nbut the same values as for calculation of H[110]were used.)\n”Critical“ hole densities, where the anisotropy fields chan ge\nsign, are dependent on Mn local moment concentration.−1\n2K4/bardbl3+cos4φ\n4sin4θ−K2/bardblsin2(φ−π/4)sin2θ,(18)\nwhere angle θandφare measured, as above, from the\n[001] and [100] axis, respectively. The first term in\nEq. (18) corresponds to the shape anisotropy described\nin Sec. IIB and not included in Eq. (12). The uniax-\nial anisotropy coefficients K2⊥andK2/bardblcorrespond to\nthe coefficients K[001]andK[110]in the phenomenologi-\ncal formula Eq. (12), respectively. To identify the third\nand fourth term in Eq. (18) we rewrite those terms as\n(see also Eq. C1):\n−1\n2K4/bardbl/parenleftbigg3+cos4φ\n4sin4θ+cos4θ/parenrightbigg\n−(19)\n−1\n2/parenleftbig\nK4⊥−K4/bardbl/parenrightbig\ncos4θ=\n=−1\n2K4/bardbl/parenleftbig\nn4\nx+n4\ny+n4\nz/parenrightbig\n−\n−1\n2/parenleftbig\nK4⊥−K4/bardbl/parenrightbig\nn4\nz=\n≡Kc1/parenleftbig\nn2\nxn2\ny+n2\nxn2\nz+n2\nzn2\ny/parenrightbig\n−\n−1\n2K[001]2n4\nz+c,\nwherecis an angle independent constant. From here\nwe see that the coefficient K4/bardblcorresponds to the low-\nest order cubic coefficient Kc1in Eq. (12) and K4⊥−\nK4/bardbl≡K[001]2corresponds to the second order uniax-\nial anisotropy coefficient Ku2forˆU/bardbl[001]. We point\nout that omission of the second order cubic term (and\nother higher order terms) can make the determination of\nK[001]2from fitting the data to the phenomenologicalfor-\nmula in Eq. (18) unreliable. Moreover, the accurate ex-\ntraction of the coefficient K[001]2can be difficult in sam-\nples with large value of the first order coefficient K[001].52\nWe therefore only note that K[001]2extracted from the\nexperiment41,56,57never dominates the anisotropy, con-\nsistent with our calculations, and do not discuss the co-\nefficient further in more detail.\nThe predicted strong dependence of K[001],K[110], and\nKc1on hole density, Mn local moment concentration and\ntemperature is consistently observedin many experimen-\ntal papers. We start with experiments where the out-\nof-plane anisotropy is studied. Measurements focusing\nmainly on the in-plane anisotropies are discussed at the\nend of this section and in Sec. IV for patterned or piezo-\nstrained samples.\nThe coefficient K[001]is extracted in Ref. [58] using\ndetailed angle-resolved magnetotransport measurements\nat 4 K for different growth strains in as-grown and an-\nnealed, 180 nm thick samples with identical nominal\nMn concentration x≈5%. The growth strain rang-\ning frome0=−0.22% (compressive) to e0= 0.34%\n(tensile) is achieved by MBE growth of (Ga,Mn)As on\n(In,Ga)As/GaAs templates. The observed linear depen-\ndence ofK[001]one0agrees on the large range of e0with\nthe prediction given in Eq. (16). The calculated and18\nmeasured gradients are of the same order of magnitude\nand sign, and depend on the hole density. The off-set at\nzero strain in the measured dependence of K[001]one0\nin Ref. [58] is due to the shape anisotropy.\nRef. [41] presents 50 nm thick, annealed samples with\nnominal Mn doping x= 7%. All the samples are first\npassivated by hydrogen and then depassivated for differ-\nent times to achieve different hole densities while keeping\nthe growth strain the same. The FMR spectroscopy is\ncarried out for in-plane and out-of-plane configurations.\nThere is qualitative agreement of calculation and mea-\nsurement on the level of the directions of the easy axes as\ndiscussed in the previous subsection. The sign change of\nthe uniaxial anisotropy fields driven by increase of tem-\nperature is observed. The measured coefficients K[001]\nandKc1are of the same order of magnitude as the calcu-\nlated ones and K[001]≈Kc1is consistent with the weaker\ngrowth strain in annealed samples.\nRef. [57] presents an as-grown, 6 nm thick film nomi-\nnallydopedwithMnto x= 6%,grownonGa 0.76Al0.24As\nbarrier doped with Be. Increasing the Be doping in-\ncreases the hole density without changing the Mn local\nmoment concentration. The fitting of the FMR spectra\nis done using the coefficients K[001]andKc1and the g-\nfactor of the Mn. The anisotropy field corresponding to\nthe coefficient K[001]reaches value as high as ≈6000 Oe\nat 4 K. Large values of K[001]is consistent with expected\nlarge growth strain in a thin as-grown sample.50,61How-\never, for the measured K[001]our calculations would im-\nply straine0∼1% which is an order of magnitude larger\nthan typical strains in as-grown x= 6% (Ga,Mn)As ma-\nterials. Other effects are therefore likely to contribute\ntoK[001]in this sample. (Confinement effect or inhomo-\ngeneities are among the likely candidates.) The experi-\nmentalK[001](Kc1) increases (decreases) with increasing\nhole density which is in agreement with our modelling of\nhighly compensated samples.\nObservation of qualitatively consistent behaviour of\nthe anisotropies with the theory but unexpectedly large\nmagnitudes of the anisotropy fields applies also to thick\nsamples studied by FMR in Refs. [52,56]. Temperature\ndependence of the anisotropy fields is studied by FMR\nin Ref. [56] for a low doped ( x≈2%), as-grown, 200 nm\nthick (Ga,Mn)As film. Only the combined contribution\nof shape anisotropy and K[001]was resolved. The easy\naxis stays in-plane for all studied temperatures which is\nconsistentwith predicted crystallineanisotropyaswellas\nthe shape anisotropy dominating at weak growth strains.\nThe uniaxial in-plane anisotropy is of the predicted mag-\nnitude but its sign corresponds to modelling by the less\nfrequent negative intrinsic shear strain.\nRef. [52] discussed in Sec. IIIA on the level of easy\naxis orientation shows, among other samples, 300 nm\nthick annealed epilayers with nominal Mn concentration\nx= 3% deposited on GaAs and (Ga,In)As substrate un-\nder compressive and tensile growth strain, respectively.\nThe strain is measured by x-ray diffraction, however,\nthe predicted linear dependence of K[001]on the growthstrain (Eq. (16)) cannot be tested due to different sat-\nuration magnetisation and TCin both samples. Both\nRefs. [52,56] report the coefficient Kc1in the 300 nm and\n200 nm thick samples an order of magnitude larger than\nthe calculated one which can62be attributed to sample\ninhomogeneities in these thick epilayers. Ref. [52] stud-\nies also 120 nm thick, annealed and as-grown epilayers\nwithx= 8% deposited on GaAs. The coefficient K[001]\ndoubles its value at low temperature on annealing. Both\nK[001]andKc1in the thinner samples have values of the\norder predicted by theory for material with Mn doping\nx= 8%.\nFIG. 21: (Color online) Angle ψof the easy axis with respect\nto the [1 10] axis as function of hole density p(up to zero\ncompensation p=NMn) and Mn local moment concentration\nxat zero temperature, e0=−0.2%, andκ= 0.01%;\nNow we analyse experiments focusing on the in-plane\nanisotropy where the relevant anisotropy coefficients are\nKc1andK[110]. Note that the experimental papers dis-\ncussedbelowmostly17,31,36,59,60usethenotationwiththe\nin-plane magnetisation angle ψmeasured from the [1 10]\naxis. To avoid any confusion we write the in-plane form\nof Eq. (12) using the original anisotropy coefficients and\nthe angleψ=φ+π/4:\nF(ˆM) =−Kc1\n4sin22ψ+K[110]sin2ψ.(20)\nTo facilitate the comparison with experiment we use the\nnotation of Eq. (20) consistently in the remaining parts\nof this paper.\nThe magnetic easy axes lie closer to the [100] or\n[010] direction than to any diagonal when Kc1>0 and√\n2K[110]<Kc1. Negative Kc1always leads to diagonal\neasy axes. We include Fig. 21 to elucidate the combined\neffect ofKc1andK[110]on the in-plane direction of the\neasy axes. The angle ψEA(x,p), plotted as a function\nof Mn concentration and hole density at zero tempera-\nture minimises the free energy F(ˆM). The local minima\natψ= 0◦(black) and ψ= 90◦(white) are formed for\nnegativeKc1. WhenK[110]is positive (negative), the19\nglobal minimum is at ψ= 0◦(ψ= 90◦). The higher\nenergy local minimum disappears for |Kc1|=|K[110]|.\nOnly one energy minimum forms for |Kc1|<|K[110]|and\nfor positive (negative) K[110]the easy axis is at ψ= 0◦\n(ψ= 90◦). The interface of black and white regions is an\nevidence of a discontinuity of the function ψEA(x,p) due\nto switching ofthe sign of K[110]whenKc1<0. The grey\n(coloured online) regions in Fig. 21 correspond to com-\npetition of cubic and uniaxial anisotropy when Kc1>0\nand|Kc1|>|K[110]|. Then there are two easy axes at\nψEAand 180◦−ψEAforming “scissors” closing at the\n[110] axis. (The darker the colour, the more closed the\nscissors.)\n0 0.9\nT/Tc020406080100K[110], Kc1 [J/m3]Kc1 (p=1.5u)\nKc1 (p=2.5u)\nK[110] (p=1.5u)\nK[110] (p=2.5u)\n0.3 0.4 0.50.6 0.7 0.8 0.9 1\nM/M(T=0)020406080100K[110], Kc1 [J/m3]\n0.2 0.4 0.60.8 10204060\nKc1 ((M/M0)4)\nK[110] ((M/M0)2)\nFIG. 22: Calculated anisotropy fields Kc1andK[110]as func-\ntion of temperature and magnetisation at two hole densities\n(given in units u ≡1020cm−3), Mn concentration x= 2%,\nstrainse0=−0.2%,κ= 0.005%. Irregular behaviour is ob-\nserved for the lower hole density.\nTo demonstrate the typical scaling of in-plane\nanisotropy components with temperature, we discuss the\n50 nm thick as-grown (Ga,Mn)As epilayer with Mn con-\ncentrationx= 2.2% determined by x-ray diffraction and\nsecondary ion mass spectrometry, presented in Ref. [36].\nThe anisotropy coefficients K[110]andKc1are obtained\nby fitting to the M(H) loop with magnetic field along\nthe hard direction. They can be compared to Fig. 22which shows the calculated anisotropy fields as func-\ntions of temperature for two values from the interval of\nhole densities corresponding to the as-grown sample. For\np= 2.5×1020cm−3both the calculated and measured\nKc1is greater than K[110]at low temperatures but be-\ncomes smaller than K[110]atT≈TC. The calculated\nKc1is an order of magnitude smaller than the experi-\nmental one, however, there is agreement on the level of\nthe temperature dependent ratio of Kc1andK[110]. On\nthe contrary, Fig. 22 shows a non-monotonous depen-\ndence ofKc1on temperature for p= 1.5×1020cm−3.\nThis singular behaviour is not measured in Ref. [36] but\nit is reported in a more systematic study in Ref. [41].\nThe temperature dependence of anisotropy coefficients\nK[110]andKc1isstudiedbyplanarHalleffectinRef.[63].\nThe mutual behaviour of the two coefficients observed\nin the as-grown (Ga,Mn)As epilayer with nominal Mn\nconcentration x≈4% andTC= 62 K is qualitatively the\nsame as in Ref. [36]. Kc1becomes smaller than K[110]\natT= 26 K which is in agreement with our modelling.\nNo sign change of Kc1is reported in this experimental\nwork. Again, the calculated Kc1is an orderofmagnitude\nsmaller than the experimental one.\nRef. [41] resolves the in-plane coefficients Kc1and\nK[110]in four samples with nominal Mn doping x= 7%\nand different hole densities. In samples with lower hole\ndensities the dependence of Kc1andK[110]is qualita-\ntivelyconsistentwith Ref. [36], however,both coefficients\nchange sign when temperature is increased in samples\nwith higher hole densities ( p∼1021cm−3,TC= 130 K).\nOur model predicts such sign change for a short interval\nof high hole compensations and a larger interval of low\nhole compensations as shown in Fig. 20(a).\nAnother type of temperature scaling of Kc1andK[110]\nis observed in a 50 nm thick, annealed sample with nom-\ninal Mn doping x= 7% and TC= 165 K.60K[110]\nis larger than Kc1on the whole temperature interval\n(T= 4−165 K). Both coefficients are positive, de-\ncrease on increasing temperature, and their magnitudes\nare of the same order of magnitude as the calculated\nanisotropies. The stability of sign of K[110]is observed\ntheoretically for higher “intrinsic” shear strain as dis-\ncussed in Fig. 16 in Sec. IIIB.\nThe temperature dependence of domain wall proper-\nties of a 500 nm, as-grown (Ga,Mn)As film with Mn dop-\ningx= 4% is studied by means of the electron hologra-\nphyin Ref. [17]. The width andangleofthe domainwalls\nwere determined directly from the high-resolution im-\nages. The ratio of the anisotropy coefficients K[110]/Kc1\nwas extracted from these observations combined with\nLandau-Lifshitz-Gilbert simulations. The N` eel type do-\nmain walls evolve from near-90◦-walls at low tempera-\ntures (T= 10 K) to large angle [1 10]-oriented walls and\nsmall angle [110]-oriented walls at higher temperatures\n(T= 30 K). The angles of domain walls aligned with\nparticular crystallographic directions reveal positions of\nthe magnetic easy axes. The “scissors” of the easy axes\n(described in discussion ofFig. 21) are closingaroundthe20\n[110] axis on increasing temperature consistent with our\nmodelling.\nThe domain-wall width is inversely proportional to the\neffective anisotropy energy barrier between the bistable\nstates on respective sides of the domain wall: Keff\n[110]≡\nKc1/4−K[110]/2 ([110]-oriented walls) and Keff\n[110]≡\nKc1/4 +K[110]/2 ([110]-oriented walls). The width of\nthe [110]-oriented wall in Ref. [17] initially increases\nwith temperature and then saturates at high tempera-\nture while the [110]-oriented wall width keeps increasing\nwith temperature until it becomes unresolvable. This\nobservation corresponds well to the theoretical predic-\ntion and can be qualitatively understood by consider-\ning the approximate magnetisation scaling of Kc1∼M4,\nK[110]∼M2, and magnetic stiffness ∼M2.\nFinally, Refs. [59,64] present (Ga,Mn)As field-effect\ntransistors (FETs), where hole depletion/accumulation\nis achieved by gating induced changes of the in-plane\neasy axis alignment. In Ref. [59] the Mn doped layer\nis 5 nm thick with Mn doping x= 2.5% and hole density\np∼1×1019−1020cm−3. The direction of magnetic easy\naxeswasdetectedbyAMRat T= 4K.The20%variation\nof the hole density achieved by applying the gate voltage\nfrom−1Vto3Visdeterminedfromvariationofthechan-\nnel resistance near TC. This value was a starting point\nfor simulations of the depletion at T= 4 K giving hole\ndensity changes ∆ p≈5×1019cm−3. The measured Kc1\nis negative and its magnitude decreases with depletion.\nThe theoretical magnitude ( ∼10 mT) and sign of Kc1\nfor the relevant hole density range, as well as the varia-\ntion ofKc1with varying hole density, are consistent with\nthe experiment. Recall that negative Kc1corresponds to\ndiagonal easy axes captured by two black/white regions\nin Fig. 21. Samples reported in Refs. [40,53,54,55] (see\nalso Sec. IIIB) and in Ref. [60] with diagonal easy axes\nat low temperatures fall into the right region with lower\nhole compensations, whereas the sample in Ref. [59] is\na rarely observed example of diagonal easy axes at high\ncompensation and low temperature corresponding to the\nleft black/white region in Fig. 21.\nIV. SAMPLES WITH POST-GROWTH\nCONTROLLED STRAINS\nIn the previous section, we discussed three types of\nlattice strain and calculated corresponding types of uni-\naxial anisotropy components. In the bare, unpatterned\nepilayers we could analyse and compare to experiment\nonly anisotropies induced by the growth strain and by\nthe unknown symmetry breaking mechanism modelled\nby the “intrinsic” shear strain. The calculations includ-\ning the model shear strain allow us also to estimate the\nmagnitudeofrealin-planelatticestrains, controlledpost-\ngrowth by patterning or piezo stressing, that can induce\nsizable changes of anisotropy. In this section we inves-\ntigate samples where these post-growth techniques areused to apply additional stress along any in-plane direc-\ntion. We will focus primarily on stresses along the main\ncrystal axes and in-plane diagonals. We will also com-\nment on the procedure for determining the lattice strain\nfrom specific geometrical parameters of the experimental\nsetup. Where necessary, we distinguish the externally in-\nduced strain and the “intrinsic” shear strain, which mod-\nels the in-plane symmetry breaking mechanism already\npresent in the bare epilayers. Returning to the nota-\ntion of Sec. II we denote the latter strain by the symbol\neint\nxy. Forbetterphysicalinsightandtorelatewith discus-\nsion in previous section we will map the anisotropies on\nthe phenomenological formulae by decomposing the total\nstrain matrix into the three basis strains (Eqs. (9-11)).\nWe will then write the corresponding anisotropy energy\nterms as in Sec. IIIC, assuming linearity between the re-\nspective basis strains and anisotropy energy components\n(see Eq. (16)). Experiments will be discussed based on\nmicroscopic anisotropy calculations with the total strain\ntensor directly included into the Hamiltonian.\nWe begin this section by discussion of the in-plane uni-\naxial strain induced by post-growth lithography treat-\nment of Mn-doped epilayers grown under compressive\nlattice strain. Narrow bars with their width comparable\nto the epilayer thickness allow for anisotropic relaxation\nof the lattice matching strain present in the unpatterned\nfilm. An expansion of the crystal lattice along the di-\nrection perpendicular to the bar occurs while the epi-\nlayer lattice constant along the bar remains unchanged.\nParameters sufficient for determination of the induced\nstrain are the initial growth strain e0and the thickness\nto width ratio t/wof the bar. In the regime of small\ndeformations the components of the induced strain are\nlinearly proportional to the growth strain. The strain\ntensor for a bar oriented along the [100] axis reads:\ner\n[100]=e0\n−ρ+1 0 0\n0 1 0\n0 0c12\nc11(ρ−2)\n,(21)\nwhere the lattice relaxation is quantified by ρwhich is a\nfunction of t/wand can vary over the bar cross-section.\nWe calculate the distribution of ρover the cross-section\nof the bar using Structural Mechanics Module of Com-\nsol (standard finite element partial differential equation\nsolver, www.comsol.com). Since the macroscopic simu-\nlations ignore the microscopic crystal structure, they ap-\nply to bars oriented along any crystallographic direction.\nWe therefore introduce a coordinate system fixed to the\nbar:x′-axis lies along the relaxation direction transverse\nto the bar, y′-axis along the bar, and z′-axis along the\ngrowth direction. We approximate the bar by an infinite\nrectangular prism with translational symmetry along the\ny′-axis, attached to a thick substrate.\nFig. 23 shows the spatial dependence of the function\nρ(x′,z′) for a given thickness to width ratio and com-\npressive growth strain e0<0. Only the area of the bar\nis plotted, whereas the strain induced in the patterned21\nFIG. 23: Spatial dependence of the strain coefficient ρdue\nto lattice relaxation in a narrow bar with t/w= 0.4 and\ncompressive growth strain e0<0, simulated values of ρare\nplotted for the cross-section of the bar.\npart of the substrate is not shown. (The substrate re-\nlaxation is not directly related to the microscopic simu-\nlation of the anisotropy energy). In wide bars ( t/w≪1)\nthe relaxation is very non-uniform, whereas narrow bars\n(t/w≫1) are fully relaxed. Fig. 24 shows still a fairly\nnon-uniform relaxation for t/w= 0.4 with large relax-\nation at the edges. We point out in this case that the\nresulting anisotropy can be very sensitive to the details\nof the etching (vertical under-cut/over-cut profile).\nThe non-uniform strain distribution in wider bars can\nin principle force the system to break into magneti-\ncally distinct regions. However, experiments show rather\nthat the whole bars behave as one effective magnetic\nmedium. Because of the linearity between the strain and\nthe anisotropy (see Eq. (16)) we can model the mean\nmagnetic anisotropy by considering the spatial average\nofer\n[100]over the bar cross-section. The inset of Fig. 24\nshows the averaged value ρas a function of the width to\nthickness ratio. It confirms that the effect of relaxation\ncan reach magnitudes necessary to generate significant\nchanges in the magnetic anisotropy. In very narrow bars\nthe induced uniaxial anisotropycan overridethe intrinsic\nanisotropies of the unpatterned epilayer and determine\nthe direction of the easy axis.\nIf the bar is aligned with the [100] or [010] crystal\naxis, the strain er\n[100]in Eq. (21) with the average re-\nlaxation magnitude ρcan be used directly as input pa-\nrameter of the microscopic calculation (see Eq. (B2) in\nthe Appendix). Alternatively, the total strain tensor can\nbe decomposed into the growth basis strain from Eq. (9)\nand the uniaxial basis strain introduced in Eq. (11):\ner\n[100](e0,ρ) =eg(˜e0)+eu(˜λ), (22)\n˜e0=e0/parenleftbigg\n1−ρ\n2/parenrightbigg\n, (23)\n˜λ=−e0ρ\n2. (24)\nTheir effects on the magnetic anisotropy can be consid-\nered separately utilising the results shown in Sec. IIIC.\nNow we discuss the introduction of uniaxial in-plane\nanisotropies by a piezo actuator attached to the sam-\nple. In this case, the (Ga,Mn)As film is assumed to0 0.2 0.4 0.6 0.8 1\nx'0.40.60.811.2\n0.0250.050.10.20.40.8\nt/w0.10.46ρ\n0.1\n0.010.20.30.4ρ_\nFIG. 24: (Color online) Sections of ρ(x′,z′) in Fig. 23 at fixed\nvalues ofz′(given next to the curves in relative units) of a\nthin bar. Inset shows the average strain ρ(t/w) as a function\nof the thickness to width ratio.\nfollow the deformation of the stressor. (The substrate\nis usually thinned to achieve better transmission of the\npiezo-strain to the studied epilayer. Macroscopic Comsol\nsimulationspredicttransmissionofapproximately70%of\nthe piezo-strain in a substrate with thickness to lateral\nsize ratiot/l≈0.1 and transmission of approximately\n90% of the piezo-strain for t/l≈0.02.) The net effect\nof the piezo-stressing on normal GaAs epilayers has been\ninvestigated experimentally for example in Ref. [33] for a\nstandard PbZrTiO 3(PZT) piezo actuator. The induced\nstraincanreachmagnitudes ∼10−4at lowtemperatures,\nwhich are sufficient to induce observable anisotropies in\n(Ga,Mn)As, as shown in Sec. IIIC. The deformation is\nlinearlyproportionaltoappliedvoltageonthetransducer\nand increases with increasing temperature.\nThe dependence of uniaxial anisotropies due to addi-\ntional piezo-strains is analogous to the behaviour of re-\nlaxed microbars, however, the form of the strain tensor\ninduced by the stressor is typically more complex. Let\nus first assume a strain tensor with components in the\nCartesian coordinate system fixed to the orientation of\nthe piezo stressor: x′-axis lies along the principal elon-\ngation direction, z′-axis is perpendicular to plane of the\nthinfilm. Wedenotethedeformationalongthe x′-axisby\nσand the simultaneous deformation along the y′-axis by\nσ′. Note that shear strains are typically not considered\nwhen describing the action of a piezo-stressor. The third\nparameter describing the strained (Ga,Mn)As epilayer is\nthe growth strain e0. Our analysis takes into account\nonly structures that can be parametrised by these three\nvalues. The strain tensor in the dashed coordinate sys-\ntem reads:\nep\n[100]=\nσ+e00 0\n0σ′+e0 0\n0 0 −c12\nc11(2e0+σ+σ′)\n(25)\nComponents of this tensor are considered uniform in the22\nstudied epilayer. If the principal elongation direction\nof the piezo stressor is aligned with the [100] crystallo-\ngraphic axis the strain tensor ep\n[100]can be used directly\nas an input of the microscopic simulation. Similarly to\nthe strain induced by lattice relaxation, ep\n[100]can be de-\ncomposed into the growth basis strain and the uniaxial\nbasis strain:\nep\n[100](e0,σ,σ′) =eg(˜e0)+eu(˜λ),(26)\n˜e0=e0+1\n2(σ+σ′),(27)\n˜λ=1\n2(σ−σ′). (28)\nAgain, the results shown in Sec. IIIC can then be\nused when analysing the resulting magnetocrystalline\nanisotropies. Recall that eghas a minor effect on the\nin-plane anisotropy and can therefore be omitted when\ndiscussing in-plane magnetisation transitions.\nSo far we have described induced strains aligned with\nthe [100] crystal axis. In case of a lattice relaxation or\npiezo stressor aligned at an arbitrary angle ω, the fol-\nlowing transformation of the total strain tensor er\n[100]or\nep\n[100]to the crystallographic coordinate system applies:\ner(p)\nω=RT\nωer(p)\n[100]Rω (29)\nwhere the rotation matrix reads:\nRω=\ncos(ω−π/4) sin(ω−π/4) 0\n−sin(ω−π/4) cos(ω−π/4) 0\n0 0 1\n.(30)\nThe angular shift by −π/4 is because we measure the\nangleωfrom the [1 10] axis. This convention was intro-\nduced in Sec. IIIC before Eq. (20) and is used consis-\ntently in this section for all in-plane angles. The rotated\ntotal induced strain can be used directly as the input\nstrain matrix for the microscopic calculation or it can be\ndecomposed into all three elements of the in-plane strain\nbasis. In caseofthe relaxation-inducedstrain, we obtain:\ner\nω(e0,ρ) =eg(˜e0)+eu(˜λ)+es(˜κ),(31)\n˜e0=e0/parenleftbigg\n1−ρ\n2/parenrightbigg\n, (32)\n˜λ=−e0ρ\n2sin2ω, (33)\n˜κ=e0ρ\n2cos2ω. (34)\nIn case of the rotated piezo stressor, the same decompo-\nsition follows, however, the effective strain magnitudes ˜λ\nand ˜κdepend on different real experimental parameters:\nep\nω(e0,σ,σ′) =eg(˜e0)+eu(˜λ)+es(˜κ),(35)\n˜e0=e0+(σ+σ′)\n2, (36)˜λ=(σ−σ′)\n2sin2ω, (37)\n˜κ=−(σ−σ′)\n2cos2ω. (38)\nConsidering the linear dependence of the anisotropy\ncoefficients on the corresponding strain elements (see\nEq. (16)), we can write the part due to post-growth in-\nduced strains of the phenomenological formula for the\nfree energy as a function of angles ψandω:\nFu(ˆM) =K[110](ω)sin2ψ+ (39)\n+K[100](ω)sin2(ψ+π/4)\n≃q[110]˜κ(ω)sin2ψ+\n+q[100]˜λ(ω)sin2(ψ+π/4),\nwhere we use the notation analogous to Eq. (16) in\nSec. IIIC. The relation of the effective parameters ˜λ\nand ˜κto the experimental parameters of microbars or\nstressors oriented along arbitrary crystallographic direc-\ntion is given by Eqs. (33-34) or (37-38), respectively. The\nlinearity of the anisotropy constants K[100],K[110], and\nK[001]on corresponding strain coefficients and the form\nof the strain tensors in Eqs. (31) and (35) allow us to fac-\ntor out the ω-dependence of Ku’s. Figs. 18, 19, and 20\ntogether with Eqs. (31) and (35) can therefore be used\nforanalysingmagneticanisotropiesinduced bymicropat-\nterning or piezo stressors oriented along any crystallo-\ngraphic direction.\nThe full angular dependencies of the anisotropy en-\nergy calculated directly from the total strain tensor in-\ncluded into the Kohn-Luttinger kinetic-exchange Hamil-\ntonian for several combinations of ˜ κand˜λare plotted in\nFig. 25. Recall that analogous in-plane angular depen-\ndencies of the anisotropy energy were presented in Fig. 8\n- 10, where only the competition of the growth strain eg\nand shear strain eswith the cubic anisotropy of the host\nlattice was considered.\nFig. 25(a) shows four angular dependencies of the\nanisotropy energy for x= 3% and p= 3×1020cm−3.\nThe curves are marked by the values of the effective\nstrain components. The solid curve for weak shear strain\n˜κ= 0.01% and no uniaxial strain ˜λ= 0 has two local\nminima close to the main crystal axes indicating domi-\nnant cubic anisotropy with Kc1>0 for the considered\nxandp. The easy axes are shifted due to the positive\nshear strain towards the [1 10] axis which is the direc-\ntion of relative lattice compression, consistently with the\ndiscussion in Sec. IIIC for samples with medium hole\ndensities. Additional uniaxial strain ˜λ=−0.05% results\nin only one global minimum easy axis rotating towards\nthe [100] direction which is again the direction of relative\nlattice compression.\nThe dashedcurvein Fig.25(a) correspondingtostrong\nshear strain ˜ κ= 0.09% and no uniaxial strain ˜λ= 0 has\nonlyoneglobalminimumatthe[1 10]diagonal,indicating\ndominationoftheuniaxialanisotropyovertheunderlying\ncubic anisotropy. Addition of the uniaxial strain ˜λ=23\n(a)x= 3%,p= 3×1020cm−3, ˜e0=−0.3% when\n˜λ/2 = 0, ˜e0=−0.25% when ˜λ/2 = 0.05%\n/s45/s48/s46/s50/s53/s48/s46/s48/s48/s48/s46/s50/s53\n/s45/s48/s46/s50/s53\n/s48/s46/s48/s48\n/s48/s46/s50/s53/s32/s120/s61/s51/s37/s44/s32/s112/s61/s52/s117\n/s32/s120/s61/s53/s37/s44/s32/s112/s61/s56/s117\n/s32/s120/s61/s53/s37/s44/s32/s112/s61/s49/s50/s117/s40/s69\n/s121/s45/s69\n/s91/s49/s48/s48/s93/s41/s32/s91/s107/s74/s32/s109/s45/s51\n/s93/s91/s48/s49/s48/s93\n/s91/s49/s48/s48/s93/s91/s49/s49/s48/s93/s91/s49/s49/s48/s93/s95\n(b)˜λ=−0.05%, ˜κ= 0.01%, ˜e0=−0.25%\nFIG. 25: (Color online) Magnetic anisotropy energy ∆ E=\nEψ−E[100]as afunctionofthein-planemagnetisation orienta-\ntionM=|M|[cosψ,sinψ,0] and its dependence on material\nparameters. Effects of the shear strain and the uniaxial stra in\ncombine linearly (a). Magnetic easy axes (marked by arrows)\nchange their direction upon change of Mn local moment con-\ncentrationx, and hole density p(in units u ≡1020cm−3) for\na fixed uniaxial and shear strain (b). Both plots assume zero\ntemperature.\n−0.05% leads to rotation of the easy axis towards the\ndirection of relative compression ([100] for ˜λ<0).\nCurves plotted in Fig. 25(b) differ in the material pa-\nrametersbut sharethesameweakshearstrain ˜ κ= 0.01%\nand the same uniaxial strain ˜λ=−0.05%. The solid\ncurve forx= 3% and p= 4×1020cm−3falls into the\nrange of hole densities where the cubic anisotropy coef-\nficientKc1is positive so the easy axes in the absence of\nin-plane strains align parallel to the main crystal axes.\nAdding the uniaxial strain ˜λyields only one global mini-\nmum along the [100] direction and the shear strain shifts\nthe easy axis towards the [1 10] diagonal. Again, for both\nstrains the easy axes tend to align along the direction oflattice compressionfor these medium doping parameters.\nThe dashed curve in Fig. 25(b) for x= 5% and\np= 8×1020cm−3can be described by a negative Kc1cor-\nresponding to diagonal easy axes in the unstrained bulk\nepilayer. The additional shear strain ˜ κmakes the [110]\ndirection the global minimum easy axis. Note that for\nthese values of xandpthe easy axis prefers to align with\nthe direction of lattice expansion. Consistently, the uni-\naxial strain ˜λrotates the easy axis towards the direction\nof relative lattice expansion, i.e., towards the [010] axis.\nFinally, the dash-dotted curve for x= 5% and high hole\ndensityp= 12×1020cm−3corresponds to positive Kc1\nandagain,whenthein-planestrainsareincludedtheeasy\naxes prefer the direction of relative lattice expansion. To\nsummarise the discussion of Figs. 25(a) and (b), the pre-\nferred alignment of the in-plane easy axis with either the\nlattice contraction or expansion direction depends on x\nandp. For a given doping it has always the same sense\nfor both the shear strain ˜ κand the uniaxial strain ˜λand\nis uncorrelatedwith the signofthe cubic anisotropycom-\nponent. These conclusionsareindependent ofthe growth\nstrain, at least for its typical values e0<1%.\nNow we analyse experimental studies that control the\nin-plane strain by means of post-growth lithography.\nRefs. [28] and [27] present structures with the shear and\nuniaxial strain induced locally by anisotropic relaxation\nof the compressive growth strain. Ref. [27] studies an L-\nshaped channel with arms aligned along the [1 10] and\n[110] directions patterned by lithography in a 25 nm\nthick (Ga,Mn)As epilayer with nominal Mn concentra-\ntionx= 5%. Hole density p= 5×1020cm−3was es-\ntimated from high-field Hall measurements. This pat-\nterning allows relaxation of the growth lattice matching\nstrain in direction perpendicular to the channel. There-\nfore, the generated uniaxial strains in each arm of the L-\nshaped channel have opposite signs. The induced shear\nstrain is added to (subtracted from) the “intrinsic” shear\nstrain in the arm fabricated along the [1 10] ([110]) axis.\nThe magnitude of the induced strain increases with de-\ncreasing width of the channel. A large effect on magnetic\neasy axes orientation has been observed in a 1 µm wide\nchannel while only moderate changes have been found in\na 4µm bar. In both cases the easy axes of the unpat-\nterned epilayer rotated in the direction perpendicular to\nlattice expansion. The sense and magnitude of the easy-\naxisreorientationsintherelaxedmicrobarsareconsistent\nwith theory prediction for the relevant values of x,p, and\nmicrobar geometry.\nRefs. [4] and [26] show lithographically induced uniax-\nialanisotropyalongthe[100]or[010]axisinarraysofnar-\nrow bars. Ref. [4] presents 200 nm wide bars fabricated\nin an as-grown 70 nm thick film with Mn concentration\nx= 2.5% determined by x-ray diffraction. Ref. [26] re-\nports lattice relaxation in 200 nm wide, 20 nm thick bars\nin an as-grown material with nominal Mn concentration\nx= 4%. In both studies the unpatterned epilayers have\ntwo equivalent easy axes close to main crystal axes. Af-\nter the anisotropic relaxation of the growth strain in the24\nnanobarstheeasyaxiscorrespondingtotherelaxationdi-\nrection is lost, whereas the other easy axis is maintained.\nThis behaviour is in agreement with our simulations on\nthe relevant interval of dopings and patterning induced\nstrains.\nThe anisotropies induced in the relaxed structures in\nRefs. [4,26,27,28] can be predicted using the results of\nSec. IIIC directly. Bearing in mind the negligible effect\nof the growth strain, the relevant part of the strain ten-\nsor describing the relaxation along the main crystal axes\nhas the form of the uniaxial basis strain eu, as shown in\nEq. (22), and corresponds to the anisotropy component\nwith the previously calculated coefficient K[100]. The re-\nlaxation along the diagonals is described by the strain\ntensor:er\n[110](e0,ρ) =es(˜κ) with ˜κ=−1\n2e0ρ, where we\nagain neglected the contribution from the growth strain\neg. It induces uniaxial anisotropy component quantified\nby the coefficient K[110]. Note that the “intrinsic” shear\nstraineint\nxyin the modelling is independent of the exter-\nnally introduced lattice distortion and needs to be added\nto the total strain tensor if the corresponding anisotropy\nis present in the unpatterned epilayer. As mentioned be-\nfore, the simulated rotation of easy axis directions in the\nrelaxed microbars is in good agreement with the mea-\nsured behaviour.\nThe piezo-strain is also applied in most cases along the\nmain crystal axes or diagonals. In Ref. [29] a PZT piezo-\nelectric actuator is attached to a 30 nm thick (Ga,Mn)As\nepilayer grown on a GaAs substrate thinned to 100 µm.\nThe principal elongation direction of the actuator is\naligned with the [110] crystallographic direction. The\nnominal Mn concentration of the as-grown epilayer is\n4.5%. The relative actuator length change is approxi-\nmately 4 ×10−4atT= 50 K (measured by a strain\ngauge) for the full voltage sweep (from -200 V to 200 V).\nSuch piezo-strain induces a rotation of the easy axis by\n∆ψEA≈65◦. Our modelling predicts ∆ ψEAof the same\norder for relevant material and strain parameters. The\neasy axis rotates towards the [110] ([1 10]) direction upon\ncontraction(elongation)alongthe[110]axisinagreement\nwiththebehaviourobservedintherelaxedmicrobarsand\nwith our modelling.\nRef. [30] extends the piezo-stressed (Ga,Mn)As study\nin Ref. [29] to low temperatures. Again, PZT piezo ac-\ntuator is attached to a Hall bar along the [110] crystallo-\ngraphicdirection. The30nmthick, as-grown(Ga,Mn)As\nepilayer grown on GaAs substrate has nominal Mn con-\ncentration 4 .5% andTC= 85 K. A strain gauge measure-\nment shows almost linear dependence of the piezo-strain\nintheHallbarontemperature(intherange5Kto50K).\nThe anisotropy coefficients K[110]andKc1are extracted\nfrom the angle-dependent magnetoresistance measure-\nment as a function of temperature for three voltages (-\n200 V, 0 V, and 200 V). At high temperatures the rel-\native elongation of the structure is again approximately\n4×10−4and the corresponding uniaxial anisotropy dom-\ninates over the intrinsic uniaxial anisotropy along the\n[110] axis. Close to 5 K the action of the piezo is negli--0.3-0.2-0.100.1E   - E[110]  [kJ/m-3]\n    = 1.77e-4 (-150V)\n    = 2.14e-4 (-120V)\n    = 2.39e-4 (-100V)\n    = 2.63e-4 (-80V)\n    = 3.62e-4  (0V)\n    = 5.47e-4 (150V)\n[110] [110] [010] [100] [110]−>ψ\nσ\n- -σ\nσ\nσ\nσ\nσ\nψ-\nFIG. 26: (Color online) Calculated magnetic anisotropy en-\nergy ∆E=Eψ−E[110]as a function of the in-plane magneti-\nsation angle ψmeasured from the [1 10] axis at T= 5 /8TC,\ne0= 0,eint\nxy= 0.017%,x= 5%, and p= 5×1020cm−3. The\ncurves are labelled by σ, the induced strain along the princi-\npal elongation direction of the piezo tilted by angle ω=−10◦,\nand by the corresponding voltage. (The relationship of σand\nvoltage is inferred from Ref. [31] to allow for direct compar i-\nson with experiment.) The easy axis rotates smoothly upon\nsweeping the voltage. For -100 V a shallow local energy min-\nimum forms due to the underlying cubic anisotropy (marked\nby arrow).\ngible so the intrinsic uniaxial anisotropy is stronger than\nthe induced one, however, the total in-plane anisotropy\nis dominated by the cubic anisotropy. The measured and\ncalculated induced anisotropy along the [110] direction\nare of the same sign and order of magnitude for the con-\nsidered temperatures.\nRef. [32] presents a 15 nm thick, annealed sample\ndoped tox= 8%, subject to piezo stressing along the\n[010] axis. The anisotropy coefficients are extracted\nfrom transverse AMR. The PZT actuator induces rela-\ntive elongation ranging from 1 .1×10−3for voltage 200 V\nto 0.7×10−3for -200V, measuredby astrain gauge. The\ndifference of the limits is again approximately 4 ×10−4\nbutallvaluesareshiftedtowardstensilestrainmostlikely\ndue to different thermal dilatation in the sample and the\nactuator. The lattice expansion along the [010] direc-\ntion leads to alignment of the easy axis along the [100],\nin agreement with our modelling and with the experi-\nmental studies discussed in this section. The extracted\ncubic anisotropy field is roughly a factor of two lower\ncompared to studies of samples with high hole compen-\nsation sharing the value ≈1000 Oe at different nominal\nMn concentrations.36,57,63The low critical temperature\nTC= 80 K suggests lower effective Mn concentration\nin Ref. [32]. Our calculations for lower Mn local mo-\nment concentration and high hole compensation predict\nthe anisotropy coefficients Kc1andK[100]induced by the\npiezo strain in correspondence with the measured coeffi-\ncients.\nFinally, we discuss a piezo-strain induced along a gen-25\neral in-plane direction. In Ref. [31] the principal elon-\ngation direction of the PZT piezo actuator is tilted by\nangleω=−10◦(with respect to the [1 10] axis). The\n25 nm thick, as-grown (Ga,Mn)As epilayer with nomi-\nnal Mn concentration x= 6% is grown on a GaAs sub-\nstrate, which was thinned before attaching of the stres-\nsor to≈150µm. The anisotropies are determined from\nSQUID and AMR measurements at 50 K. The uniaxial\nstrain caused by differential thermal contraction of the\nsample and the piezo on cooling (at zero applied voltage)\nis of the order 10−4. The uniaxial strains generated at\nthe voltage ±150 V areσ≈ ±2×10−4andσ′≈ −σ/2 at\n50 K. The tilt of the piezo with respect to the crystal di-\nagonal results in a complicated interplay of the intrinsic\nand induced anisotropy. The easy axisofthe baresample\naligns with the [1 10] axis due to strong intrinsic uniaxial\nanisotropy with K[110]>Kc1>0. This easy axis rotates\nto an angle ψEA= 65◦upon attaching of the piezo and\ncooling to 50 K. Application of +150 V to the stressor\ncauses the easy axis to rotate further to ψEA= 80◦while\nfor -150 V the axis rotates in the opposite direction to\nψEA= 30◦. Note that the negative voltage weakens the\ntotal piezo-strain and allows domination of the intrinsic\nanisotropy with easy axis closer to the [1 10] axis.\nThe hole compensations expected in Ref. [31] are in\nthe rangep/NMn= 0.6−0.4 and the relevant range of ef-\nfective Mn concentrationsis x= 3−5%.K[110]measured\nin the bare epilayer is modelled by eint\nxy= 3−2×10−4\n(slightly weaker than the strain induced in the structure\nat zero piezo-voltage). The in-plane anisotropy ener-\ngies calculated on this parameter interval using the total\nstraintensor(induced and“intrinsic”components)arein\ngood quantitative agreement with the easy axis orienta-\ntions measured at the three piezo voltages. Fig. 26 shows\ncalculatedcurvesforonerepresentativecombinationof x,\np, andeint\nxyfrom the relevant interval, for the fixed tilt of\nthe stressor ω=−10◦, and for a range of induced strains\nσ. The curves are marked also by the voltagesas we infer\na simple linear relationship between σand the voltage to\nfacilitate comparison with the experimental paper.\nThe anisotropy behaviour shown in Fig. 26 can be de-\nscribed as a smooth rotation of the global energy mini-\nmum upon increase of σrather than the ”scissors” effect\nshown in Fig. 8 in Sec. IIIB. The total induced strain\nnow contains both components esandeuas written in\nEq. (26). The uniaxial basis strain eupresent due to the\ntilt of the stressor diminishes significantly one of the lo-\ncal minima typically occurring because of interplay of a\npositivecubic and asmalluniaxialanisotropycomponent\nalong a crystal diagonal. The remainder of the weaker\nlocalminimumisobservedtheoreticallyfor σcorrespond-\ning to voltages ≈ −100 V when the escomponent of the\ninduced strain and the “intrinsic” shear strain compen-\nsate each other. One would expect domination of cu-\nbic anisotropy with two equivalent local minima close to\nthe main crystal axes if the stressor had purely diago-\nnal alignment. The eucomponent of the total strain of\nthe tilted stressor makes the local minimum closer to the[010] axis less pronounced (marked by arrow in Fig. 26).\nFor completeness, we discuss the free energy phe-\nnomenological formula used in Ref. [31] to describe the\nin-plane angular dependence of the induced anisotropy.\nThe decomposition of the total induced strain in Eq. (26)\ninto the strain basis introduced in Eqs. (9-11) is not con-\nsidered in that work. Instead, the induced anisotropy is\ndescribed by a single uniaxial term KΩsin2(ψ−Ω) added\nto the phenomenological formula rather than terms with\ncoefficients K[110]andK[110]from Eq. (39). Effectively,\nthis corresponds to a change of variables from K[110]\nandK[110]toKΩand Ω. The angle Ω is measured\nfrom the [1 10] axis and it rotates the additional uniax-\nial anisotropy term so that it describes the effect due to\nthe tilted stressor. One may assume collinearity of the\nresulting anisotropy component with the principal elon-\ngation direction of the piezo. However, this simple situ-\nation is observed both theoretically and experimentally\nonly when the stressor is aligned with the main crys-\ntal axes or diagonals. The missalignment for arbitrary\norientation of the induced strain is due to the under-\nlying cubic symmetry of the system incorporated into\nour microscopic band structure calculation in the form\nof the band parameters γ2,γ3,a2, anda3. It has been\nexplained in Sec. IIIC that the collinearity of the in-\nplane strain and corresponding anisotropy occurs only\nfor the strains esoreu(see Eqs. (10) and (11)). For\nany other stressor orientation, Ω /negationslash=ω, which is reflected\non the level of the anisotropy functions by the inequality,\nq[100](x,p)/negationslash=q[110](x,p). It expressesthedifferenceinthe\neffect on magnetic anisotropy between straining the lat-\ntice along the main crystal axis and along the diagonals\n(see Eq. (17) in Sec. IIIC).\nThe transformation from variables K[110](x,p,ω) and\nK[110](x,p,ω) toKΩ(x,p,ω) and Ω(x,p,ω) in the phe-\nnomenological formula in Eq. (39) for −π/2< ω < π/ 2\nreads:\nFu(ˆM) =K[110](ω)sin2ψ+K[100](ω)sin2(ψ+π/4)\n=KΩsin2(ψ−Ω), (40)\nwhere:\nΩ(x,p,ω) =1\n2arctan/parenleftbigg\n−K[100]\nK[110]/parenrightbigg\n, (41)\nKΩ(x,p,ω) =−K[110]cos2Ω+K[100]sin2Ω.\nConsidering the approximate relation q[100]= 0.43q[110]\nthe formulae in Eq. (41) simplify to:\nΩ(x,p,ω) =1\n2arctan/parenleftbiggq[100](x,p)sin2ω\nq[110](x,p)cos2ω/parenrightbigg\n(42)\n=1\n2arctan(0.43tan2ω),\nqΩ(x,p,ω)≡q[110](x,p)cos2ωcos2Ω+\n+0.43q[110](x,p)sin2ωsin2Ω,\nwhereKΩ=qΩ(σ−σ′)/2. (The same transformation\nof variables can be used in case of strains induced along26\narbitraryin-plane direction by relaxationin a narrowbar\n(see Eqs. (33-34)). Then we obtain KΩ=−qΩe0ρ/2.)\nNote that in the representation of Fu(ˆM) viaK[110]\nandK[110]the dependence on ωcan be simply factored\nout and the dependence on xandpis contained only in\nthe functions functions q[110]andq[100]. For our general\ndiscussion presented in this paper it is therefore the more\nconvenient form than Fu(ˆM) expressed via KΩand Ω.\nWe conclude that the in-plane alignment of the easy\naxis in patterned or piezo-stressed samples can be de-\nscribedonasemi-quantitativelevelbyourmodellingsim-\nilarly to the bare (Ga,Mn)As epilayers.\nV. SUMMARY\nThe objective of this work was to critically and thor-\noughly inspect the efficiency of a widely used effective\nHamiltonian model in predicting the magneto-crystalline\nanisotropies in (Ga,Mn)As. We have provided overview\nof the calculated anisotropies which show a rich phe-\nnomenology as a function of Mn concentration, hole den-\nsity,temperatureandlatticestrains,andcomparedittoa\nwide rangeofexperimentalworksonthe levelofthe mag-\nnetic easy axis direction and on the level of anisotropy\nfields. The large amount of analysed results compensates\nfor the common uncertainty in sample parameters as-\nsumed in experiment and allowed us to make systematic\ncomparisons between theory and experiment on the level\nof trends as a function of various tunable parameters.\nGenerically, we find this type of comparisonbetween the-\nory and experiment in diluted magnetic semiconductors\nmuch more meaningful than addressing isolated samples,\ngiven the complexity of these systems and inability of\nany theoretical approach applied to date to fully quan-\ntitatively describe magnetism in these random-moment\nsemiconducting ferromagnets.\nIn Sec. II we introduced the mean-field model used\nthroughout the study, estimated the relative strength of\nthe shape anisotropy, and discussed the correspondence\nof the shear strain, modelling the broken in-plane sym-\nmetry measured in most (Ga,Mn)As epilayers, with a\nmicroscopic symmetry breaking mechanism.\nIn Sec. III we focused on modelling and experiments\nin bare unpatterned epilayers. The in-plane and out-of-\nplane magnetisation alignment was studied. For com-\npressively strained samples the generally assumed in-\nplane anisotropy is found to be complemented by regions\nof out-of-plane anisotropy at low hole densities and low\ntemperatures. This observation is corroborated by avail-\nable experimental data showing in-plane anisotropy in\nmost of the studied epilayers but also the occurrence of\nthe out-of-plane easy axis in materials with high hole\ncompensation. At the same time, the model predicts\nout-of-plane easy axis for high hole densities at all Mn\nconcentrations which has yet not been observed experi-\nmentally.\nNext, the competition of cubic and uniaxial in-planeanisotropy components was investigated. Wealth of ex-\nperimentally observed easy axis transitions driven by\nchange of temperature or hole density finds correspond-\ning simulated behaviour. The following general trend\nis observed in most samples: at low temperatures the\neasy axes are aligned close to the main crystal axes,\nwhileathightemperaturesthereisalwaysdiagonalalign-\nment. This trend is in good agreement with our calcu-\nlation, however, at low hole densities the calculated and\nmeasured easy axis transitions are more consistent than\nat higher hole densities where the measured phenomena\nmatch the predictions assuming hole densities typically\na factor of two lower than in the experiment.\nWe next introduced anisotropy fields corresponding\nto the crystal symmetry and to three distinct uniaxial\nstrains. We extracted these anisotropy fields from the\ncalculated data and found their dependence on mate-\nrial parameters. We observed linear dependence of the\nuniaxial anisotropy fields on the corresponding strains.\nAnalysing experiments which determine the anisotropy\nfields from FMR, AMR or SQUID measurements allowed\nfor detailed comparison of the cubic anisotropy compo-\nnent and two uniaxial anisotropy components (due to\ngrowth and the [110 /[110] symmetry breaking). The\nmeasuredand calculatedanisotropyfields areofthe same\norder of magnitude ( ∼102−103Oe) in most samples.\nFinally, in Sec. IV we investigated structures where\nthe post-growth patterning or piezo stressing was used\nto induce additional strains along any in-plane direction.\nThe interplay of the intrinsic and induced anisotropies\nwas studied. We discussed the procedure for obtaining\nthe strain Hamiltonian from the parameters describing\nthe experimental setup and a finite element solver was\nemployed to find the inhomogeneous lattice relaxation in\nthe patterned epilayers. Induced anisotropies were cal-\nculated directly using the total strain tensor. Alterna-\ntively, we also introduced a decomposition of the total\nstrain matrix for any of the studied materials and device\nconfigurations into three basis strains and their additive\neffect on the total anisotropy. We found an overall semi-\nquantitative agreement of theory and experiment on the\nlevel of easy axis reorientations due to induced strains.\nThe limitations of the theory approach employed in\nthis paper have been thoroughly discussed in Ref. [2].\nThe model, which treats disorder in the virtual crystal\napproximation and magnetic interactions on the mean-\nfield level is expected to be most reliable at lower tem-\nperatures and in the (Ga,Mn)As materials with metallic\nconductivity. We have shown that despite the limita-\ntions, the model captures on a semi-quantitative level\nmost of the rich phenomenology of the magnetocrys-\ntalline anisotropies observed in (Ga,Mn)As epilayers and\nmicrodevices over a wide parameter range. We hope\nthat our work will provide a useful guidance for future\nstudies of magnetic and magnetotransport phenomena in\n(Ga,Mn)As based systems in which magnetocrystalline\nanisotropies play an important role.27\nAcknowledgments\nWe thank K. Y. Wang for providing us with previously\nunpublished experimental data. We acknowledge fruitful\ndiscussions with Richard Campion, Tomasz Dietl, Kevin\nEdmonds, Tom Foxon, Bryan Gallagher, V´ ıt Nov´ ak,\nElisa de Ranieri, Andrew Rushforth, Mike Sawicki,\nJairo Sinova, Laura Thevenard, Jorg Wunderlich. The\nwork was funded through Præmium Academiæ and con-\ntracts number AV0Z10100521, LC510, KAN400100652,\nFON/06/E002 of GA ˇCR, of the Czech republic, and by\nthe NAMASTE (FP7 grant No. 214499) and SemiSpin-\nNet projects (FP7 grant No. 215368).\nAPPENDIX A: SYMMETRIES OF THE\nKOHN-LUTTINGER HAMILTONIAN\nDifferent representations of the six-band Kohn-\nLuttinger Hamiltonians are used in literature. Here, thenotation of Ref. [2] is used and extended.\nThe states at the top of the valence band have p-like\ncharacter and can be represented by the l=1 orbital mo-\nmentum eigenstates |l,ml/angbracketright. In the basis of combinations\nof orbital angular momentum eigenstates:\n|X/angbracketright=1√\n2/parenleftbig\n|1−1/angbracketright+|11/angbracketright/parenrightbig\n,\n|Y/angbracketright=i√\n2/parenleftbig\n|1−1/angbracketright−|11/angbracketright/parenrightbig\n,\n|Z/angbracketright=|10/angbracketright (A1)\nthe Kohn-Luttinger Hamiltonian for systems with no\nspin-orbit coupling can be written as:\nHkp=\nǫv+Ak2\nx+B(k2\ny+k2\nz)Ckxky Ckxkz\nCkykxǫv+Ak2\ny+B(k2\nx+k2\nz)Ckykz\nCkzkx Ckzkyǫv+Ak2\nz+B(k2\nx+k2\ny)\n, (A2)\nwhere\nA=¯h2\n2m0+¯h2\nm2\n0/summationdisplay\ni/∈{X,Y,Z}|/angbracketleftX|px|i/angbracketright|2\nǫ1−ǫi, (A3)\nB=¯h2\n2m0+¯h2\nm2\n0/summationdisplay\ni/∈{X,Y,Z}|/angbracketleftX|py|i/angbracketright|2\nǫ1−ǫi, (A4)\nC=¯h2\nm2\n0/summationdisplay\ni/∈{X,Y,Z}/angbracketleftX|px|i/angbracketright/angbracketlefti|py|Y/angbracketright+/angbracketleftX|py|i/angbracketright/angbracketlefti|px|Y/angbracketright\nǫ1−ǫi, (A5)\nandǫvis the energy of the valence band p-orbitals.\nThe simple form is due to the symmetry of the zinc-\nblende crystal structure. The summation in elements\nA,B,Cruns only through the Γ 1and Γ 4states of\nthe conduction band as other levels are excluded by the\nmatrix element theorem combined with the tetrahedron\nsymmetry.65The only nonzero momentum operator ex-\npectation values with neighbouring states are:\n/angbracketleftX|py|Γ4(z)/angbracketright=/angbracketleftY|pz|Γ4(x)/angbracketright=/angbracketleftZ|px|Γ4(y)/angbracketright\n/angbracketleftX|px|Γ1/angbracketright=/angbracketleftY|py|Γ1/angbracketright=/angbracketleftZ|pz|Γ1/angbracketright.(A6)\nDue to the reflection symmetry with respect to the (110)\nplanes it holds also:65\n/angbracketleftX|py|Γ4(z)/angbracketright=/angbracketleftY|px|Γ4(z)/angbracketright/angbracketright (A7)If the tetrahedral symmetry of the GaAs lattice is bro-\nken by potential V=xyξas described in Sec. IIA the\nstates Γ 1and Γ 4(z) of the conduction band considered\nin the summation in Eq. (4) are mixed, whereas states\nΓ4(x) and Γ 4(y) are left unchanged. In the perturbed ba-\nsisαΓ1+βΓ4(z),−βΓ1+αΓ4(z), Γ4(x), Γ4(y) we obtain\nterms containing the parameter Din the Hamiltonian\n˜Hkp(See Eq. (5) in Sec. IIA.) A weak local potential V,\nα >> β was assumed so terms of quadratic and higher\norderdependence on Vcould beneglected. Thereforethe\nexpression for parameters A,B, andCdoes not change.\nUsing Eqs. (A6) and (A7) allows also for a compact ex-\npression of the parameter D:\nD=ζ/angbracketleftX|py|Γ4(z)/angbracketright/angbracketleftΓ1|px|X/angbracketright, (A8)28\nζ=¯h2\nm2\n0αβ/bracketleftbigg1\nǫv−(ǫc1+∆)−1\nǫv−(ǫc4−∆)/bracketrightbigg\n,\nwhereǫc1andǫc4are the energies of the conduction band\nΓ1and Γ 4states, respectively. The small energy ∆ is\nquadratically dependent on the size of the potential V\nbut we include it to express the shift of the perturbed\nenergy levels.\nTo include spin-orbit coupling we use the basis formed\nby combinations of orbital angular momentum:\n|1/angbracketright ≡ |j= 3/2,mj= 3/2/angbracketright\n|2/angbracketright ≡ |j= 3/2,mj=−1/2/angbracketright\n|3/angbracketright ≡ |j= 3/2,mj= 1/2/angbracketright\n|4/angbracketright ≡ |j= 3/2,mj=−3/2/angbracketright\n|5/angbracketright ≡ |j= 1/2,mj= 1/2/angbracketright\n|6/angbracketright ≡ |j= 1/2,mj=−1/2/angbracketright (A9)\nThe spin-orbit correction to the 6-band Hamiltonian is\ndiagonal in this basis and can be parametrised only by\na single parameter ∆ so.2The 6-band Kohn-Luttinger\nHamiltonian in the representation of vectors (A9) reads:\nHKL=\nHhh−c−b0b√\n2c√\n2\n−c∗Hlh0b−b∗√\n3√\n2−d\n−b∗0Hlh−cd−b√\n3√\n2\n0b∗−c∗Hhh−c∗√\n2b∗\n√\n2\nb∗\n√\n2−b√\n3√\n2d∗−c√\n2Hso0\nc∗√\n2−d∗−b∗√\n3√\n2b√\n20Hso\n\n(A10)\nThe 4-band Hamiltonian is highlighted. The Kohn-\nLuttinger eigen-energies are hole energies (measured\ndown from the top of the valence band). The matrix\nelements of HKLare listed in Ref. [2]. Here we focus on\nthe modification of these elements due to incorporating\nthe microscopic potential V=xyξ:\n˜Hhh=¯h2\n2m/bracketleftbig\n(γ1+γ2)(k2\nx+k2\ny)+(γ1−2γ2)k2\nz+6γ4kxky/bracketrightbig\n˜Hlh=¯h2\n2m/bracketleftbig\n(γ1−γ2)(k2\nx+k2\ny)+(γ1+2γ2)k2\nz+2γ4kxky/bracketrightbig\n˜Hso=¯h2\n2m/bracketleftbig\nγ1(k2\nx+k2\ny+k2\nz)+4γ4kxky/bracketrightbig\n+∆so\n˜b=√\n3¯h2\nmγ3kz(kx−iky)\n˜c=√\n3¯h2\n2m/bracketleftbig\nγ2(k2\nx−k2\ny)−2i(γ3kxky+γ4\n2(k2\nx+k2\ny))/bracketrightbig\n˜d=−√\n2¯h2\n2m/bracketleftbig\nγ2(2k2\nz−k2\nx−k2\ny)−2γ4kxky/bracketrightbig\n(A11)\nwhere we neglect the higher order effect of broken sym-\nmetry on standard Luttinger parameters:\nγ1=−2m0\n3¯h2(A+2B)\nγ2=−m0\n3¯h2(A−B)\nγ3=−m0\n3¯h2Cand add a new parameter:\nγ4=−2m0\n3¯h2D. (A12)\nAPPENDIX B: LATTICE STRAINS AND\nMICROSCOPIC POTENTIAL\nWe incorporate the lattice strain into the k·pthe-\nory following Ref. [25], which shows that HKLand the\n6-band strain Hamiltonian have the same structure given\nin Eq. (A10). Components of the strain tensor eαβin-\ntroduced in Eq. (3) play role of the k-vector components.\nThe replacements in of matrix elements of Eqs. (A11) (or\nrather of Eq. (A9) in Ref. [2]) read:\nkαkβ→eαβ (B1)\n−¯h2\n2m0γ1→a1,−¯h2\n2m0γ2→a2\n2,−¯h2\n2m0γ3→a3\n2√\n3,\nwherea1,a2, anda3are the elastic constants. Their val-\nuesaredifferenttoLuttingerparameters γ1,γ2, andγ3as\nthey originate from the first order momentum operator\nperturbation due to strain and second order perturba-\ntion treatment of the k·pterm, respectively. The strain\nHamiltonian has the following elements (in the hole pic-\nture):\nHs\nhh=−/parenleftig\na1+a2\n2/parenrightig\n(exx+eyy)−(a1−a2)ezz\nHs\nlh=−/parenleftig\na1−a2\n2/parenrightig\n(exx+eyy)−(a1+a2)ezz\nHs\nso=−a1(exx+eyy+ezz)\nbs=−a3(ezx−iezy)\ncs=a2\n2√\n3(eyy−exx)+ia3exy\nds=√\n2\n2a2(2ezz−(exx+eyy)). (B2)\nNow we compare the effect of microscopic symmetry\nbreaking described by including the γ4dependent terms\ninto the Hamiltonian HKLto the effect of a uniform lat-\ntice strain incorporated as Hstrwith matrix elements\ngiven in Eqs. (B2). First, we write the strain Hamil-\ntonianHstras a sum of a contribution corresponding\nto the in-plane shear strain along the [110] axis and the\ngrowth strain introduced in Sec. IIIC by Eq. (9) and\nEq. (10), respectively. Their magnitudes are denoted by\nexyandexx=eyy≡e0. Then we write the correc-\ntionHV=˜HKL− HKLto the 6-band Kohn-Luttinger\nHamiltonian due to the microscopic potential V=xyξ\nbreaking the tetrahedral symmetry of the crystal as a\nsum of terms with different dependence on the in-plane\ndirection of the k-vector:29\nHstr=a3exy\n0−i0 0 0 i√\n2\ni0 0 0 0 0\n0 0 0 −i0 0\n0 0i0i√\n2 0\n0 0 0 −i√\n2 0 0\n−i√\n2 0 0 0 0 0\n+a2e0c11+2c12\nc11\n0 0 0 0 0 0\n0 2 0 0 0√\n2\n0 0 2 0 −√\n2 0\n0 0 0 0 0 0\n0 0−√\n2 0 1 0\n0√\n2 0 0 0 1\n, (B3)\nHV=√\n3¯h2\n2m0γ4(k2\nx+k2\ny)\n0−i0 0 0 i√\n2\ni0 0 0 0 0\n0 0 0 −i0 0\n0 0i0i√\n2 0\n0 0 0 −i√\n2 0 0\n−i√\n2 0 0 0 0 0\n+¯h2\nm0γ4kxky\n3 0 0 0 0 0\n0 1 0 0 0 −√\n2\n0 0 1 0√\n2 0\n0 0 0 3 0 0\n0 0√\n2 0 2 0\n0−√\n2 0 0 0 2\n,(B4)\nwherec11,c12are the elastic moduli.23,25\nNote that by resetting the reference energy in Eq. (B4)\nby subtracting3¯h2\nm0γ4kxkyfrom the Hamiltonian HVand\nthe following substitutions:\na3exy→√\n3¯h2\n2m0γ4(k2\nx+k2\ny)\n−a2e0c11+2c12\nc11→¯h2\nm0γ4kxky (B5)\nwe can identify the two components of the strain Hamil-\ntonianinEq.(B3)withthetwocomponentsoftheHamil-\ntonianHVin Eq. (B4). The important difference, how-\never, is the dependence on k-vector in case of HV. The\nfirst term of HVdepends on the magnitude of the k-\nvector, not on its in-plane orientation. The second term\nofHVhas the same structure as the second term of Hstr\n(which incorporatesthe effect of the growth strain), how-\never, it does depend on the in-plane direction of the k-\nvector so it generates a uniaxial in-plane anisotropycom-\nponent that contributes to the energy profile (shown in\nFig. 1) similarly to the first term of Eq. (B4) (contrary to\nthe negligible uniaxial in-plane anisotropies correspond-\ning to the growth strain).\nAPPENDIX C: CUBIC ANISOTROPY TERMS\nThe angular dependence of the magnetocrystalline\nanisotropy energy can be approximated by a series of\nterms of distinct symmetry. In Sec. IIIC we introduced\na simple phenomenological formula consisting of the low\norder terms of the cubic and uniaxial symmetry. Here we\nexplain the choice of the independent cubic terms.\nWe write the terms using the components of the\nmagnetisation unit vector ˆM:nx= cosφsinθ,ny=\nsinφsinθ,nz= cosθ, where our angles θandφare mea-\nsured from the [001] and [100] axis, respectively. The cu-\nbic symmetry requires invariance under permutation ofthecoordinateindices x,y, andz. Thesimplest termsat-\nisfying the condition is equal to unity: n2\nx+n2\ny+n2\nz= 1.\nThe first order cubic term can be derived from its second\npower:\n/parenleftbig\nn2\nx+n2\ny+n2\nz/parenrightbig2= (C1)\n= 2/parenleftbig\nn2\nxn2\ny+n2\nxn2\nz+n2\nyn2\nz/parenrightbig\n+n4\nx+n4\ny+n4\nz.\nWe obtained two lowest order cubic terms which are mu-\ntually dependent. Therefore it is enough to choose only\none of them. In case of Eq. (12) the lowest order cu-\nbic anisotropy term reads: Kc1/parenleftbig\nn2\nxn2\ny+n2\nxn2\nz+n2\nzn2\ny/parenrightbig\n,\nwhereKc1is an energy coefficient.\nThe second order term can be derived from the first\norder term:\n/parenleftbig\nn2\nxn2\ny+n2\nxn2\nz+n2\nyn2\nz/parenrightbig/parenleftbig\nn2\nx+n2\ny+n2\nz/parenrightbig\n= (C2)\n=n4\nxn2\ny+n4\nxn2\nz+n2\nxn4\ny+n4\nyn2\nz+n2\nxn4\nz+n2\nyn4\nz+\n+n2\nxn2\nyn2\nz.\nThe two second order terms and the first order term are\nmutuallydependent. Again,onlyonetermdescribesfully\nthe second order component of the cubic anisotropy. We\nchooseKc2/parenleftbig\nn2\nxn2\nyn2\nz/parenrightbig\ntobeincludedintoourapproximate\nformula in Eq. (12).\nThe independent third orderterm is derivedas follows:\n/parenleftbig\nn2\nxn2\ny+n2\nxn2\nz+n2\nyn2\nz/parenrightbig/parenleftbig\nn2\nx+n2\ny+n2\nz/parenrightbig2= (C3)\n= 3(n4\nxn2\nyn2\nz+n2\nxn4\nyn2\nz+n2\nxn2\nyn4\nz)\n+ 2(n4\nxn4\ny+n4\nxn4\nz+n4\nyn4\nz)\n+n6\nxn2\ny+n6\nxn2\nz+n2\nxn6\ny+n6\nyn2\nz+n2\nxn6\nz+n2\nyn6\nz.\nNote that the first part of the product is proportional to\nthe second order cubic term. Again, we can choose one\nof the two dependent third order terms. 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Schmidt email: georg.schmidt@physik .uni-halle.de\nWe report the observation of tunneling anisotropic magneto resistance (TAMR) in an organic\nspin-valve-like structure with only one ferromagnetic ele ctrode. The device is based on a new high\nmobility perylene diimide-based n-type organic semicondu ctor. The effect originates from the tun-\nneling injection from the LSMO contact and can thus occur eve n for organic layers which are too\nthick to support the assumption of tunneling through the lay er. Magnetoresistance measurements\nshow a clear spin-valve signal, with the typical two step swi tching pattern caused by the magne-\ntocrystalline anisotropy of the epitaxial magnetic electr ode.\nPACS numbers:\nOver the past years a number of spin-valves based on various orga nic semiconductors and contact materials have\nbeen demonstrated (e.g. [1–8]). Although some experiments indicat e injection of spin polarized carriers[7] and\nsome clearly show tunneling [6, 8], it is still unclear for a number of othe r results whether their data show tunneling\nmagnetoresistance(TMR) oractualspininjection andconsequen tlygiantmagnetoresistance(GMR). Moreover,recent\nresults[9] suggest that spin-valve effects are only observed in dev ices whose resistance only slightly changes during\ncooldown. This behavioris typicalfor devices operatedin the tunne ling regime, however,it is less commonfor electron\ntransport in polycristalline or amorphous organic semiconductors.\nNonetheless, both GMR and TMR are related to the switching of a pair of magnetic electrodes between parallel\nand antiparallel magnetization. For both effects at least some spin c onservation throughout the whole thickness of\nthe organic interlayer is necessary. It is, however, known that sim ilar magnetoresistance traces can also be caused\nby charge carrier injection from a single magnetic electrode into ano ther material by a tunneling process (like charge\ncarrier injection into an OSC), if the magnetic electrode exhibits a su itable magnetocrystalline anisotropy. In that\ncase tunneling anisotropic magnetoresistance (TAMR) can appear . [10]. TAMR can cause the magnetic switching\ncharacteristics of the electrodes to modulate the tunneling resist ance of the injection contact. This effect is much\nlarger than the well known anisotropic magnetoresistance observ ed in the electrode itself and can erroneously be\nidentified as a true spin-valve signal, unless suitable control measur ements are carried out.\nWe have fabricated a number of different spin-valve like structures (Fig. 1a) using the organic semiconductor\n(OSC)N,N’-bis(n-heptafluorobutyl)-3,4:9,10-perylene tetracarboxylic diimid e (PTCDI-C4F7, Fig. 1b) and three dif-\nferent combinations of magnetic and non-magnetic contact mater ials. While most spin-valves reported until now are\neither based on AlQ 3[2], which is an amorphous low-mobility n-type semiconductor, or on p- type semiconductors such\nas P3HT[3] or TPP[4], PTCDI-C4F7 is a new air-stable, high-mobility n-t ype semiconductor[11] from the perylene\ndiimide/dianhydride family whose most common member is PTCDA. Owing t o its favorable electron transport prop-\nerties, PTCDI-C4F7 should be well-suited for the demonstration of spin-polarized electron transport. Three different\ndevice types were investigated, distinguished by the combinations o f contact materials as listed in table I.\ndevice bottom contact PTCDI-C4F7 layer top contact\n1 15nmLSMO 100nm 30nmCoFe\n2 10nmAl 250nm 30nmAl\n3 10nmLSMO 150nm 15nmAl\nTABLE I: Contact materials and layer thicknesses of the thre e investigated multilayers.\nType-1 devices which have an LSMO bottom- and a ferromagnetic me tal top-contact are similar to those reported\nfor most organic spin-valves (OSVs) in the literature and serve to d emonstrate the suitability of PTCDI-C4F7 for\nspin-valve operation. Type-2 devices are control samples with only non-magnetic electrodes and help to exclude\npossible artifacts caused by the OSC and the well known Organic mag netoresistance (OMAR)[12] which is also2\nobserved in completely non-magnetic layer stacks. Type-3 devices have one magnetic LSMO electrode and a non-\nmagneticAlcounterelectrode. Thesesamplesareintendedforinv estigationsontheoccurrenceoftunnelinganisotropic\nmagnetoresistance (TAMR). All layer stacks are deposited in a UHV chamber designed to allow for the deposition of\nOSC and metal layers in direct sequence without breaking the UHV.\nThe LSMO bottom contacts are fabricated from 10 or 15 nm thick LS MO layers, grown by pulsed plasma\ndeposition[13] on Strontium Titanate substrates. For device fabr ication, first Ti/Au metal stripes are deposited\non the LSMO, using optical lithography and lift-off. These stripes se rve as alignment marks and later as bondpads.\nA rectangular bottom contact is then patterned into the LSMO laye r by optical lithography and dry etching, leaving\nthe metal contact at one side of the rectangle. Subsequently, th e sample is inserted into the UHV-deposition chamber\nwhere a bake-out procedure is performed at 450◦C for 1 hour at an oxygen pressure of 10−5mbar, in order to\ncompensate under-oxygenation which may occur during the proce ssing. Subsequently, the PTCDI-C4F7 layer and\nthe metal top electrode are deposited under different angles of inc idence through a shadow mask with a rectangular\nopening. After removing the sample from the UHV chamber, Ti/Au st ripes are deposited through a second shadow\nmask with striped windows. These metal stripes are later used as bo nd pads for the top contacts and also serve\nas an etch mask for the removal of the top electrode material bet ween the stripes by dry etching. Samples with\naluminum bottom electrode are fabricated on a Si substrate with a 2 00 nm thick thermal SiO 2cover layer. A Ti/Au\ncontact pad is deposited before inserting the sample into the UHV ch amber where the bottom aluminum layer, the\nPTCDI-C4F7 layer and the top aluminum layer are evaporated throu gh the shadow mask at three different angles\nof incidence. The different angles are necessary in order to allow for insulation between top and bottom electrode.\nAfter removing the sample from the UHV chamber the processing co ntinues in the same way as for the LSMO based\nsamples. This approach provides clean, oxygen-free, and reprod ucible interfaces. The samples are characterized at\nvarious temperatures between 4.2 K and room temperature, eithe r in a flow cryostate with an external room tempera-\nture electromagnet (600 mT) or in a4He bath cryostate with a vector field magnet in which magnetic fields u p to 400\nmT can be applied in any direction. In the measurements all three sam ples show less than one order of magnitude\nincrease of resistance during cooldown from room temperature to 4.2 K.\nA typical magnetoresistance trace of a type I sample 4.2 K is shown in Fig. 2a. The B field is applied along the long\naxis of the stripe-like device. The magnetoresistancehas at least t wo distinct components. The first comprises the two\nswitching events for each of the scan directions, which are usually a ttributed to spin-valve operation. For this device\nthis effect is negative (as often described in literature e.g. [2, 4, 5]) a nd its size is approx. 8%. The other component is\na continuous increase of the resistance with increasing magnetic fie ld also observed in various experiments [4, 14–16]\nwhich may be attributed to the magnetic saturation of the electrod es. When the B-field is again applied in the plane\nbut perpendicular to the stripe (Fig. 2b) the shape of the curve ch anges. This change in shape can be explained\nby shape anisotropy of the electrodes leading to a rotation rather than switching. Although the zero field resistance\nof both measurements is identical, it should be noted that the resist ance at 350 mT is different, indicating traces of\nTAMR.\nThe type-2 layer stack, which has no magnetic electrodes, is used t o identify any magnetoresistancein the pure OSC\nwhich might be misinterpreted as a spin-valve signal. The type-2 devic es exhibit no detectable magnetoresistance,\nneither spin-valve nor OMAR (between -400 mT and +400 mT, with an e rror of less than ±0.05%). We can thus\nexclude OSC-related magnetoresistance effects as explanation fo r the effects found in samples with magnetic contact\nlayers.\nThe 3rd type of layer stack, with a ferromagnetic LSMO bottom con tact and a non-magnetic aluminum counter\nelectrode, has an OSC layer thickness of 150 nm. Given the thicknes s of the OSC layer we can exclude any transport\nby direct tunneling through the OSC. Even multi step tunneling which has recently been discussed [8] can not explain\nthe observed transport phenomena. Nevertheless, the high res istance at room temperature and the weak temperature\ndependence (increase in resistance by one order of magnitude bet ween room temperature and 4.2 K) indicates that\nthe I/V characteristics are dominated by the charge injection pro cesses at the metal/OSC Schottky contacts[17],\nexplaining why the I/V characteristiscs we observe (insert Fig. 4) a re strongly reminiscent of tunneling processes.\nBecause there is only one magnetic layer present in type-3 stacks, no genuine spin-valve signal can be expected.\nNevertheless, the magnetoresistance scans show the two-stat e behaviour characteristics of a spin-valve with two\nferromagnetic layers. The magnetoresistance curves taken at a bias voltage of 255 mV (Fig. 3) clearly exhibit two\nspin-valve like switching events superimposed to the background. B oth, background and switching are well known\nfrom organic spinvalves. In this structure, however, they can ne ither originate from GMR nor from TMR, both TMR\nand GMR requiring two ferromagnetic electrodes.\nAs TMR and GMR as well as any intrinsic magnetoresistanceof the LSM O layer[18] can be excluded here, we inter-\npret these data as tunneling anisotropic magnetoresistance (TAM R), an effect which was first observed in (Ga,Mn)As\n[10] and has since been reported to occur in various tunnel contac ts on ferromagnetic semiconductors and metals[19–\n22]. This effect allows for spin-valve functionality in layer stacks with o nly one ferromagnetic contact.\nTAMR originates from spin orbit coupling in a ferromagnetic electrode with crystalline anisotropy. In these elec-3\ntrodes spin orbit coupling can translate any change of the magnetiz ation vector into a change in the density of states\n(DOS). The k-dependence of the DOS in the electrode has a (weak) dependence on the relative orientation of the\nmagnetization vector with respect to the crystal axes. If the ele ctrode is part of a tunneling contact, the DOS com-\nponent of the tunneling matrix element of any state strongly depen ds on the k-vector of the respective state and\nthus the change in magnetization is reflected in the tunneling resista nce. The magnitude of the effect increases with\nincreasing anisotropy and increasing spin orbit coupling.\nThe typical spin-valve like signature with two pronounced switching e vents can appear for example in systems with\na biaxial magnetocrystalline anisotropy. In such a system a magnet ic field sweep at an angle between the two easy\naxes usually leads to a magnetization reversal via a two step switchin g process combined with small rotations [23].\nHere, symmetry breaking for example from the growth strain can c ause the tunneling matrix element to be different\nfor magnetization alignment along the two different easy axes. In th is case the system has four stable magnetization\nstates with two different tunneling resistance values (one for each easy axis) which we may call R1 and R2. The\ntwo step switching process then leads either from R1 to R2 and back or from R2 to R1 and back. Depending on\nthe starting point of the magnetoresistance scan the magnetore sistance trace corresponds to a positive or a negative\nspin-valve effect. The respective switching fields which are observe d depend on the field angle with respect to the\neasy axes. Only for a magnetic field applied exactly along one of the ea sy axes single step switching occurs and the\nspin-valve effect vanishes. It is noteworthy that in many (however , not all) organic spin-valves with LSMO electrodes\nin literature a negative spin-valve effect is observed.\nObviously this phenomenology can be used to identify TAMR. We have p erformed magnetoresistance scans with\ndifferent directions of the magnetic field in the plane. In these scans , the coercive fields of the two switching events\n(HC1andHC2) vary depending on the relative orientation of the magnetic field with respect to the easy axes as\nexpected. Experimental data obtained on our type-3 sample are s hown in Fig. 3 (0◦and 90◦are the two sample\ndiagonals, see also Fig.1).\nThe effect is even clearer in a magnetoresistance plot where the two scans for field orientations of 0◦and 90◦are\nshown together on the same scale (Fig. 4). For 90◦the resistance is in the high-state in magnetic saturation[24], and\nit is low between HC1andHC2, while for 0◦orientation the resistance in saturation is in its low-state and it is high in\nbetween H C1and H C2. These two directions are the main axes which determine the minimum a nd maximum value\nof the tunneling resistance. The observation that the maximum res istance value between H C1and H C2(0◦curve) is\nhigher than the maximum resistance in saturation (90◦curve) can be explained by the occurrence of multi-domain\nbehavior in which parts of the sample are magnetized out of plane due to a weak out-of-plane easy axis. SQUID\nmeasurements with the B-field perpendicular to the surface indeed show a remanent magnetization component.\nStill further information comes from a magnetoresistance scan on the type-3 sample in which the direction of the\nB-field is rotated by 360◦in the plane while its magnitude is kept constantclose to saturation[2 4] (full saturationis not\npossible in our magnet). For ordinary TMR or GMR with two ferromagn etic electrodes this scan must always show\nthe same resistance value because the two layers are always aligned in parallel. For our sample we see an anisotropic\nresistance distribution (Fig. 5) with a biaxial symmetry, clearly indica ting the presence of TAMR.\nIt should be noted that in order for TAMR to be observed, electron tunneling through the organic layer is not\nnecessary. It is sufficient to have carrier injection into the OSC thr ough a barrier (as in most OSV) and thus to\nhave a tunneling contribution in the injection process. For a sufficien tly high contact resistance, the TAMR will\nalways be visible above the normal device resistance, especially at low bias voltages. If however, the resistance of the\nsemiconductor massively increases at low temperatures the effect will no longer be detectable.\nWe have thus demonstrated that TAMR can exist in OSV structures with at least one LSMO electrode. This effect\nopens up new perspective for organic spintronics. However, as th e effect can be either positive or negative in sign,\ndepending on the direction of the applied magnetic field, our observa tions imply that careful measurements on any\nOSV are necessary in order to distinguish between TAMR and real sp in-valve operation.\nWe thank the EU for funding the research in the project OFSPIN.\n[1] V. Dediu et al., Solid State Commun. 122, 181-184 (2002).\n[2] Z. H. Xiong et al., Nature 427, 821-824 (2004).\n[3] S. Majumdar et al., J. Alloys Compd. 423, 169-171 (2006).\n[4] W. Xu et al., Appl. Phys. Lett. 90, 072506 (2007).\n[5] F.J. Wang et al., Synth. Met. 155, 172-175 (2005).\n[6] T.S. Santos et al., Phys. Rev. Lett. 98, 016601 (2007).\n[7] T.D. Nguyen et al., Nature Materials 9,345 (2010)\n[8] J.J.H.M. Schoonus et al., Phys. Rev. Lett. 103, 146601 (2009).\n[9] R. Lin et al., Phys. Rev. B 81, 195214 (2010)4\n[10] C. Gould et al., Phys. Rev. Lett. 93, 117203 (2004).\n[11] J.H. Oh et al., Appl. Phys. Lett. 91, 212107 (2007).\n[12] T.L. Francis et al., New J. Phys. 6, 185 (2004).\n[13] I. Bergenti et al., Org. Electron. 5, 309-314 (2004).\n[14] F.J. Wang et al., Phys. Rev. B 75, 245324 (2007).\n[15] S. Majumdar et al., Appl. Phys. Lett. 89, 122114 (2006).\n[16] H. Vinzelberg et al., J. Appl. Phys. 103, 093720 (2008).\n[17] M.A. Baldo and S.R. Forrest, Phys. Rev. B 64, 085201 (2001).\n[18] The in-plane magnetoresistance of the LSMO layer deter mined to be less than 1 Ω which is three orders of magnitude\nsmaller than the effect observed.\n[19] C. R¨ uster et al., Phys. Rev. Lett. 94, 027203 (2005).\n[20] K. I. Bolotin, F. Kuemmeth, and D.C. Ralph, Phys. Rev. Le tt.97, 127202 (2006).\n[21] J. Moser et al., Phys. Rev. Lett. 99, 056601 (2007).\n[22] B.G. Park et al., Phys. Rev. Lett. 100, 087204 (2008).\n[23] R.P. Cowburn et al., J. Appl. Phys. 78, 7210 (1995).\n[24] By saturation, we mean a field large enough to perfectly a lign the magnetization to the direction of the external field .5\nFIG. 1: Schematic drawing of the vertical transport structu re (a). A is the bottom contact material and B is the top contac t\nmaterial as listed in table 1 for the respective samples. (b) Structure of the PTCDI-C4F7 molecule (b).\nFIG. 2: Magnetoresistance trace of a PTCDI-C4F7-based vert ical OSV structure with an LSMO bottom electrode and a CoFe\ntop electrode (type-1sample). ApplyingtheB-field along th e stripe (a) yields pronounced switchingevents while ameas urement\nwith B-field perpendicular to the stripe (b) indicates a rota tion of the stripes magnetization.\nFIG. 3: Magnetoresistance sweeps for the TAMR test sample wi th the magnetic field applied in different directions in the pl ane.\nAfter each scan the field direction is rotated by 30◦. 0◦and 90◦are the sample diagonals. For 90◦and 270◦the switching\nevent is towards negative values. For clarity, scans for diff erent directions are offset by 850 kΩ.\nFIG. 4: Magnetoresistance traces (type-3 sample) with the m agnetic field aligned along 0◦and 90◦. At large positive or\nnegative fields two different resistance states can be distin guished, a clear signature of TAMR. For the scan in 0◦direction, the\nspin-valve signal is positive while it is negative for the 90◦sweep. The insert shows the I/V-characteristics of the type -3 device\nat room temperature and at 4.2 K.\nFIG. 5: Magnetoresistance scan (type-3 sample) taken at con stant field while the angle of the applied field is slowly rotat ed\nby 360◦. The scan clearly shows the minimum and maximum resistance s tate at Φ=0◦/180◦and Φ=90◦/270◦, respectively,\ncorresponding to the saturation states in Fig. 4.6\n45°135°\nABOO O\nOFC73CF37 N N\nab\nFig.1 Gruenewald et al.7\nFig.2 Gruenewald et al.8\nFig. 3 Gruen ewa ld et al.9\nFig.4 Gruenewald et  al.10\nFig.5 Gruenewald et al."    },    {        "title": "0906.5365v1.Adaptive_modulations_of_martensites.pdf",        "content": "Adaptive modulations of martensites \n \nS. Kaufmann1,2, U.K. Rößler1, O. Heczko3,1, M. Wuttig4, J. Buschbeck1, L. Schultz1,2 and S. Fähler1,2 \n1 IFW Dresden, P.O. Box: 270116, 01171 Dresden, Germany \n2 Institute for Solid State Physics, Department of Physics, Dresden University of Technology, \n01062 Dresden, Germany \n3 Institute of Physics, Academy of Science of Czech Republic, Na Slovance 2, 182 02 Prague, \nCzech Republic \n4 Department of Material Science, University of Maryland, College Park, MD, 20742, USA \n \nModulated phases occur in numerous functional materials like giant ferroelectrics and magnetic shape memory alloys. \nTo understand the origin of these phases, we review and generalize the concept of adaptive martensite. As a starting point, we \ninvestigate the coexistence of austenite, adaptive 14M phase and tetragonal martensite in Ni-Mn-Ga magnetic shape memory \nalloy epitaxial films. The modulated martensite can be constr ucted from nanotwinned variants of a tetragonal martensite \nphase. By combining the concept of adaptive martensite with branching of twin variants, we can explain key features of modulated phases from a microscopic view. This includes phase stability, the sequence of 6M-10M-NM intermartensitic \ntransitions, and magneto crystalline anisotropy. \nDOI:     PACS: 81.30.Kf, 61.50.Ks, 68.55.-a, 64.60.My, 77.84.-s \nIn numerous materials, el ectric and magnetic fields \ncan distort lattice unit cells, resulting in an associated strain which is commonly below 0.3 %. Recently, \nferroelectric [\n1] and magnetic shape memory materials [ 2] \nhave been found where applied fields control the \norientation low symmetry unit cells. Thus giant strains of \nseveral percent are achieved by a rearrangement in \ntwinned microstructures. Why relatively weak electrical or \nmagnetic fields can change the microstructure of a solid \nmaterial is not fully understood. However, in both material \nclasses, these effects take place only in phases with a \nmodulated structure [ 3,4]. \nCommonly, such modulated structures are considered as thermodynamically stable phases. The displacive transition from a high-symmetry austenite to a low-\nsymmetry phase is of the martensitic type. The \ntransformation requires that this martensite is accommodated on a habit plane as lattice invariant \ninterface  which fixes the geometrical relationship between \nthe two crystal structures [\n5]. The lattice mismatch is \ncompensated by twinning of th e martensite. Hence a large \nnumber of twin boundaries, connecting differently aligned \nmartensitic variants are introduced. Extrapolating this \ngeometrical continuum approach to the atomic scale, \nKhachaturyan et al. [ 6] argue that the modulated \nstructures observed in materials with lattice instabilities \nshould be understood as ultrafinely twinned metastable \nstructures and not as thermodynamically stable phases. In this view, the large and complex apparent unit cell of the \nmodulated phase is composed of nano-twin lamellae of a \nsimpler, thermodynamically stable, martensitic phase. The twinning periodicity and, hence, the modulation is \ndetermined by geometrical constraints and the \ntransformation path. The key requirement for the validity \nof this explanation is very low nano-twin boundary \nenergy. The power of this con cept is demonstrated for the \nNi-Mn-Ga system, where the sequence of phase \ntransitions, magneto-crystalline anisotropy and \nmicroscopic models for intermartensitic transitions as well \nas stress induced martensite are explained and generalized. \nThis concept of adaptive martensite competes with \nalternative theoretical ideas, e.g., emphasizing the relevance of Fermi surface nesting for modulated phases \nin metallic martensites [\n7,8]. Direct experimental proofs for the adaptive nature of modulated martensite structures are difficult because thermodyn amic measurements do not \neasily identify metastable phases. Furthermore, diffraction experiments cannot directly distinguish a regular nano-\ntwinned microstructure from a long-period modulated phase [\n9]. For lead-based ferroe lectric perovskites, the \nconcept of adaptive phases has been employed to explain the transitional region at the morphotropic phase boundary [\n3,10,11]. Still, the adaptive concept is strongly debated \n[12]. One approach explains anomalous phenomena at the \nmorphotropic phase boundary by the existence of low symmetry equilibrium phases as bridging structures [\n13]. \nThis assumption, however, cannot explain the transformation paths between, and the co-existence of these different modulated and non-modulated structures.  \nIn this letter, starting from experimental observations \non epitaxial NiMnGa films, we demonstrate that the 14M modulated phase observed in bulk [\n14] is a metastable \nadaptive phase. Since this modulated phase exhibits an anomalously large strain due to twin re-arrangement under magnetic fields, adaptivity seems to be crucial for the \ngiant strain effects not only in the ferroelectrics but also in \nmagnetic shape memory materials. We identify epitaxial \nfilms as a suitable experiment al setting to decide on the \norigin of modulated phases. We exploit two key \nadvantages of epitaxial films as compared to bulk: First, \nthe geometrical constraint at  the interface to the rigid \nsubstrate stabilizes otherwise thermodynamically unstable \nphases rendering frozen intermediate states accessible to \nexperimention. Second, the single crystalline substrate \nacts as a reference system which allows probing crystallographic orientations of all phases in absolute \ncoordinates. \nAs a model system to test Khachaturyan’s concept \nwe selected the Ni-Mn-Ga magnetic shape memory alloy. For the chosen alloy composition, the modulated 14M \nlattice cell is built from unit cells of the thermodynamically stable non-modulated (NM) phase. \nThe geometrical martensite theory [\n5] predicts a periodic \ntwinning of the tetragonal ma rtensite lattice, expressed \nthrough the fraction of the twin lamella widths d 1 and d2: \nd1/d2 = (aNM-aA)/(aA-cNM). Here, aNM and cNM represent the \nlattice constants of the tetragonal martensite  and aA the \nlattice constant  of the cubic austenite. This ratio directly  \nFIG. 1: (color online) Comparison of lattice constants for the three \ndifferent phases. The blue dashed  lines mark the calculated lattice \nconstants of the adaptive martensite phase. For each phase, film and bulk \n[14] lattice constants are shown. \n \ndetermines the minimal number of stacked atomic layers \nforming the adaptive phase. In  diffraction experiments, \none expects an orthorhombic lattice with: a14M = c NM + \naNM – a A , b14M = a A and c14M= a NM. These relationships \ncan be understood by describing the modulated martensite \nas a periodic superlattice [ 9]. Diffraction on such a \nperiodic structure results in satellite reflections appearing \nlike a new phase. The connection between the \nnanotwinned modulated structure and a macroscopic amount of NM phase can be described by branching [\n15] \nwhich preserves the orientation relationship between the austenite and the NM twins. The adaptive nature of 14M explains why this structure do es not correspond to a global \nenergy minimum in first-pr inciples calculations, which \nconsistently find the tetragonal  NM phase as ground state \n[\n16].  \nWe have investigated epitaxial Ni-Mn-Ga films with \na thickness of about 500 nm, deposited by DC magnetron \nsputtering on MgO(100) substrates at 250°C. Structural \ncharacterization was performed by X-ray diffraction \n(XRD) in a Philips X’Pert 4-circle setup with Cu-K α \nradiation. θ−2θ-scans of the {400}-planes, performed at \nroom temperature [ 17], reveal the coexistence of three \nphases: the cubic austenite ( aA = 0.578 nm), the pseudo-\northorhombic 14M martensite ( a14M = 0.618 nm,  \nb14M = 0.578 nm,  c14M = 0.562 nm) and the tetragonal NM \nmartensite ( aNM = 0.542 nm,  c NM = 0.665 nm). Lattice \nconstants of all phases are described with reference to the \ncubic L2 1 Heusler unit cell. In equilibrium, the coexistence \nof three phases at one temperature is contradicting the \nGibbs phase rule, confirming that the interface to the rigid \nsubstrate is influencing these diffusionless martensitic transformations. \nFor both, bulk [ 14] and the present thin film, the measured \nlattice parameters of the 14M phase agr ee with the \npredicted lattice constants from the concept of adaptive \nmartensite ( FIG. 1 ). Although in thin films the tetragonal \ndistortion of the adaptive phase is less than expected from theory, for bulk and thin films b\n14M is almost identical to \naA. In a simplified geometrical model [ 6], this equality is \nthe key precondition for a coherent austenite-martensite interface. Thus the most important relation between the \nlattice constants of the adaptive martensite and austenite is fulfilled. Using the measured lattice constants the concept \nalso predicts a twinning periodicity of d\n1/d2 = 0.428 for bulk and d 1/d2 = 0.417 for the thin film. This is close to \nthe ideal value of d1/d2 = 2/5 = 0.4  expected for the 14M \nphase consisting of twin variants with widths of 2 and 5 \natomic layers, respectively. The difference in d1/d2 \nbetween 0.4 for a perfect ( 2 5)2 stacking and the value \ncalculated from the measured  lattice constants suggests \nthat the structure exhibits stacking faults [ 6]. This could \nexplain the difference between measured and expected \nlattice constants. \nApplying these results on the lattice geometry we \ncan construct the 14M unit cell by using NM unit cells as building blocks. In addition to a projection shown in \nFIG. \n2, a foldable 3D model is available as EPAPS document \n[17]. We start from NM martensite unit cells, originating \nfrom tetragonal deformed and slightly rotated L2 1 \naustenite. One NM unit cell is exemplarily marked with grey background; the different variants are connected by \n(101)-type twin boundaries. In \nFIG. 2  one also finds the \ncommonly used 14M unit cell described in the “bct” \nsystem [ 16] (rotated by about 45° to the NM cells and \nmarked with yellow background).  \nThe only difference of our expanded picture \ncompared to the common pictur e is, that one can directly \nidentify the nanotwinned NM variants. Using the NM \nlattice constants and the ( 2 5)2 twinning periodicity one \ncan calculate the lattice cons tants of the adaptive 14M \nphase by basic geometry. Additionally the angles between \ncrystal axes of the tetragonal NM twin variants and the axes of the 14M unit cell can be  calculated (as sketched in \nFIG. 1 ). Lattice constants ( aadbct = 0.428 nm , badbct = 2.955 \nnm, cadbct = 0.542 nm) and monoclinic angle (β = 95.3 °) \nin the commonly used bct reference system are close to bulk measurement data ( a\n14Mbct = 0.426 nm , b 14Mbct = \n2.954 nm , c14Mbct = 0.543 nm , β = 94.3°) [ 14].  \nThe identification of a 14M martensite with a \nnanotwinned NM martensite suggests that macroscopic \nNM variants are connected to the nanotwinned NM variants by a branching mechanism [\n15], which does not \nchange the orientation of the NM variants. Thus, the angles between axes of 14M and its NM building blocks should be the same as those between the 14M and the \nmacroscopic NM martensite. \nThis can be proved using an advantage of epitaxial films \nwhere the substrate provides a fixed reference frame. \nThereby, it is possible to study the different \ncrystallographic orientations in absolute coordinates by \ninvestigating {004} pole figures of all different phases and \norientations.  These pole figures give the real-space \norientation of the different unit cells selected by their \nlattice spacing. Following our previous report [ 18] we \nknow that the orientation of the austenite is determined by \nthe epitaxial growth. Moreover some austenite phase \nremains at the interface to the rigid substrate below the \nmartensitic transformation temperature [ 19].For the \npresent film the austenite contributes to the (400) A + \n(040) 14M pole figure which exhibits one intense peak at \nzero tilt ( FIG. 3  (a)). Since aA and b14M are equal, as \npredicted from the adaptive martensite concept, both \nlattices contribute to the same pole figures. \nIn contrast to the austenite, the 14M pole figures also \nexhibit peaks at tilted positions. The orientations of these  \nFIG. 2: (color) 14M structure constructed by periodic ( 2 5)2 twinning of \ntetragonal NM building blocks. One of the NM cells is exemplarily \nmarked with grey background. The directions of the three different 14M \nlattice parameters are sketched with brown colour. The angles of the NM \nunit cells subtended with the 14M supercell (thick lines) are given. The \nconventional unit cell used to descri be 14M within the bct reference \nsystem is marked with a yellow background at the right. \n \npoles are almost identical to the ones reported previously \nfor an epitaxial 14M film [ 18]. These orientations agree \nwith the predictions of Wech sler, Liebermann and Read \n(WLR) theory of an almost exact habit plane [ 5]. Hence \nthe orientation of the 14M martensite variants is \ncompletely determined by the requirement of a coherent interface to the austenite. \nCompared to 14M, the pole figures of NM martensite (\nFIG. 3 (d),(e)) exhibit significantly more peaks. This is \nexpected since single NM vari ants cannot form an exact \ninterface to the austenite. Peak positions are summarized in \nTable I  Using the orientation of 14M and the angles \ngiven in FIG. 2  we can directly calculate the orientation of \nNM variants and thus the peak positions in the pole figures (\nFIG. 3  (d),(e)). The calculated and measured \nangles agree within the accur acy of the texture device for \nψ of ~1°. This confirms that the orientation of the NM \nvariants does not change during coarsening from nanotwinned to macroscopically twinned NM variants. Each reflection in the pole figures of the NM martensite \ncan be assigned to a well defined variant of the \nnanotwinned structure, which forms the 14M. Starting at macroscopic NM variants, branching of twin boundaries \noccurs within the NM phase when the habit plane is \napproached. Since bo th phases coexist, branching must \ncontinue down to the atomic scale.  \nThese experiments confirm that  14M is an adaptive phase. \nIn the following we will use the concept of building \nmodulated phases from a nanotwinned martensite to analyse several peculiarities of systems exhibiting giant strain. To allow a direct comparison with the present \nexperiments, the focus will be on the Ni-Mn-Ga system.\n  \nDensity functional calculations for the energy curve of Ni\n2MnGa as a function of the tetragonal distortion [ 16] \nshow a global energy minimum at a c/a = 1.25. Compared \nto this NM martensite, the 14M structure exhibits a higher energy. Using the concept of adaptive martensite, we can \ninterpret this energy difference as twin boundary energy γ \nof the NM phase. On the basis of these calculation [\n16] we \nderive a twin boundary energy of about γ = 2 meV/Å2, \nclose to γ = 0.87 meV/Å2 recently observed in NiTi \nnanocrystals [ 20]. This low value of the twin boundary \nenergy fulfils the key requirement for the formation of \nadaptive martensite in Ni-Mn-Ga.   \nThese above energy considerations suggest that the 14M \nphase is not thermodynamically stable. Since, however, \nbulk single crystals exist and can actuate for several 106 \ncycles [ 21], 14M can be considered a metastable phase. \nMetastability requires the existence of an energy barrier hindering the transition from the nanotwinned to the \nmacroscopically twinned NM martensite. This energy barrier may be related to the repulsive forces between twin \nboundaries and lattice defects like dislocations required \nfor the annihilation of twin boundaries. Since the nanotwinned 14M can easily ad apt internal and external \nstress by variations of the stacking sequence, appropriate \nprocessing (e.g. cooling under load) may be a precondition \nfor the formation of a meta stable 14M phase. For the \npresent thin films the constraint by the rigid substrate \nadditionally hinders detwinning and thus explains the \ncoexistence of austenite, 14M and NM martensites. \nTemperature dependent XRD measurements [\n17] indeed \nreveal that all phases  \n \nTable I: Comparison between calculated and measured crystal \norientations for the tetragonal NM martensite variants. The angles ψ and \nϕ are sketched in FIG. 3 (a). The grey columns mark the three different \nunderlying 14M martensite variants, from which the two differently \noriented macroscopic NM varian ts originate by coarsening. \n \n \n \nFIG. 3: (color) Pole figure measurements of the {400}-planes of  the pseudo-orthorhombic 14M ((a)-(c)) and the tetragonal NM mar tensite ((d)-(e)) from ψ = \n0…10°. The four-fold symmetry verifies epitaxial growth on the Mg O(100) substrate. The shift of reflections with respect to the  centre is due to a slight \nmisalignment of the sample during measurement.  coexist over a temperature range of more than 100 K. \nThe adaptive phase concept enables to estimate the \nmagnetocrystalline anisotropy of the Ni-Mn-Ga 14M \nmartensite. Since the thickness of the nanotwin variants is \nsignificantly below the magnetic exchange length, the \nmagnetocrystalline anisotropy equals the weighted mean of the differently aligned NM variants. Referring to the \nL2\n1 system and using data at 300 K [ 22], the building \nblock concept predicts Ka14M = -5/7·K 1NM = 1.63·105 J/m3 \nand Kb14M = -2/7 ·K1NM = 0.65·105 J/m3, using the \nanisotropy constant K1NM = -2.28·105 J/m3 of the NM \nphase. These values agree with the measured constants K\na14M = 1.72·105 J/m3 and Kb14M = 0.83·105 J/m3. This \nfavourable comparison also holds for their temperature \ndependence [ 22]. The alignment of the hard axis in 14M is \ncorrectly predicted as inclined by 45° with respect to the twinning planes of the NM nanotwins and not by 0° or 90°, as expected by interface anisotropy. \nIn Ni-Mn-Ga, the phase sequence austenite, 6M \n(premartensitic), 10M (5-layer), 14M (7-layer), NM martensite is commonly observed with increasing electron \ndensity [\n23]. Detailed diffraction experiments on the \nmodulated structures reveal that they own orthorhombic or \nmonoclinic unit cells [ 24, 25]. In analogy to 14M all these \nstructures can be assembled from tetragonal building blocks. Their tetragonality increases with the electron density (c/a\nNM = 1.0152 (for 6M), 1.16 (for 10M), 1.26 (for \n14M) [ 17]. Since typically Ni-Mn-Ga alloys exhibit the \nsame phase sequence during cooling [ 23], we suggest that \nthe electron density is the key parameter which controls \nthe tetragonal distortion. This parameter is systematically \nvaried by stoichiometry and/or thermal expansion. All the \nmodulated phases can form an almost exact habit plane to \nthe austenite [ 17]. This indicates that the austenite-\nmartensite interface energy σ and the nano-twin boundary \nenergy γ are small and similar in magnitude for all phases. \nHence, the interface energies do not appreciably influence \nthe relative stability of the modulated phases. Therefore, \nall modulations observed in the Ni-Mn-Ga system are \nadaptive. The sequence of the modulated structures 6M – 10M – 14M is determined by the electron density e/a via \nthe variation of the tetragonal distortion in the equilibrium \nNM phase. In the metastable modulated phases, the nano-\ntwin widths are minimized for (c/a)\nNM values which \nresults in a fraction of small integer numbers for d1/d2. \nMetastability of the adaptive phase can explain \nirreversibilities of intermartensitic transitions under \nexternal stress. As observed in Ni-Mn-Ga alloys, the \napplication of a sufficient external compression selects NM variants with their short axes in compression \ndirection. Under sufficient load nanotwin boundaries \nvanish, resulting in a transformation to macroscopic NM variants, while a clear reverse transition is not observed \n[\n26]. The nanotwinned nature of a modulated martensite \ncan also affect the thermal hysteresis of the martensitic \ntransition. As reported by Cui et al. [ 27], the mismatch \nbetween austenite and martensite lattice constants determines the width of the hysteresis. For an adaptive phase, the precondition of an exact habit plane is fulfilled, \nthus, a small hysteresis is expected and measured [\n4].  \nTo conclude, our investigations show that modulated \nphases in Ni-Mn-Ga originat e from the adaption of a \nthermodynamically stable martensite to the austenite. This \nestablishes magnetic shape-memory alloys as an important metallic counterpart to ferroelectrics near the \nmorphotropic phase boundary. The similarity between \nthese systems suggests that adaptivity is crucial for field-induced giant strains in martensitic functional materials. \nThe modulated phases facilitate adaption to external forces \nand fields by a redistribution of nano-twin boundaries, in \ncontrast to a thermodynamically stable, stiff martensite. \n \nThe authors would like to thank A.N. Bogdanov, P. \nEntel, M. E. Gruner and A. G. Khachaturyan for \ndiscussions. We gratefully acknowledge funding by the DFG via the priority program SPP1239 \nwww.MagneticShape.de.  \n                                                 \n[1] S. E. Park, T. R. Shrout, J. Appl. Phys.  82, 1804 (1997) \n[2] Ullakko K., Huang J.K., Kantner C ., O’Handley R.C., Kokorin V.V., \nAppl. Phys. Lett. 69, 13 (1996) \n[3] Y. M. Jin, Y. U. Wang, A. G. Khachaturyan, J. F. Li, and D.  \nViehland, Phys. Rev. Lett.  91, 197601 (2003) \n[4] Sozinov A., Likhachev A., Lanska L., Ullakko K., Appl. Phys. Lett. \n80, 10 (2002) \n[5] Wechsler M.S., Lieberman D.S., Read T.A., Transactions of the \nAIME 197, 1503 (1953) \n[6] Khachaturyan A.G., Sh apiro S.M., Semenovskaya S., Phys. Rev. B \n43, 10832 (1991) \n[7] Bungaro C., Rabe K.M., Dal Corso A., Phys. Rev. B 68, 134104 \n(2003) \n[8] Opeil C.P. et al., Phys. Rev. Lett. 100, 165703 (2008) \n[9] Wang Y.U., Phys. Rev. B  74, 104109 (2006)   \n[10] Glazer A.M., Thomas P.A., Baba-Kishi K.Z., Pang G.K.H., Tai \nC.W., Phys. Rev. B 70, 184123 (2004) \n[11] Schönau K.A. et al., Phys. Rev. B 75, 184117 (2007) \n[12] Ahart M. et al., Nature 4 51, 545 (2008)  \n[13] B. Noheda, D. E. Cox, Phase Trans. 79, 1029 (2006).  \n[14] Pons J., Chernenko V.A, Santamarta R., Cesari E., Acta Mat. 48, \n3027 (2000) \n[15] Kohn V., Müller S., Philos. Mag. A  66, 5 (1992) \n[16] Entel P. et al., J. Phys. D: Appl. Phys.  39, 865 (2006) \n[17] See EPAPS Document No. [ ] for 3D model and structure details. \nFor more information on EPAPS, see \nhttp://www.aip.org/pubservs/epaps.html . \n[18] Thomas M. et al., New J. Phys . 10, 023040 (2008) \n[19] Buschbeck J., Niemann R., Heczko O., Thomas M., Schultz L. \nFähler S., Acta Mater . 57, 2516 (2009) \n[20] Waitz T., Spisak D., Hafner J., Karnthaler H.P., Europhys. Lett. 71, \n98 (2005) \n[21] Müllner P., Chernenko V.A., Mukherji D., Kostorz G., Materials \nResearch Society Symposium Proceedings 785, 415 (2004) \n[22] Straka L., Heczko O., J. Appl. Phys . 93, 8636 (2003) \n[23] Heczko O. et al. in Magnetic Shape Memory Phenomena  (Ch. 14), \nin J.P. Liu et al. (eds) Nanoscale Magnetic Materials and \nApplications , Springer Science (2009) in print, DOI 10.1007/978-0-\n387-85600-1 p.14 \n[24] Ohba T., Miyamoto N., Fukuda K. , Fukuda T., Kakeshita T., Kato \nK., Smart Mater. Struct. 14, 197 (2005) \n[25] Righi L.,Albertini F., Pareti L., Paoluzi A., Calestani G., Acta Mat. \n55, 5237 (2007)  \n[26] Martynov V.V., J. de Phys. IV 5, 91 (1995) \n[27] Cui J. et al., Nat. Mater. 5, 286 (2006) EPAS document for: Adaptive  modulations of martensites \nS. Kaufmann, U.K. Rößler, O. Heczko, M. Wu ttig, J. Buschbeck, L. Schultz and S. Fähler \n \n1) 3-D model of 14M martensite \n \nFIG S1: 3-D model of one 14M supercell of Ni 2MnGa, consisting of two different NM variants. For a better understanding of \nthe different tilts determining the orientations of the NM marten site variants, the two counterparts of the figure can be cut o ut \nand should be glued following the numbers. Folding edges are marked by dashed lines. One unit cell of the NM martensite is \nexemplarily marked in grey. Additionally, the directions of the 14M unit cell axis are sk etched in brown. Th e twin boundaries \nconnecting two different NM variants are marked in green. Mn and Ga atoms are not in plane but shifted by ¼ times the NM \nlattice constant into the interior of the cell .\n The assembled 3D model for the 14M supercell can be used for direct visualization of severa l key features of the adaptive \nphase. First however it is helpful to recognize that the comple te 14M unit cell is built from NM building blocks. In the five \natomic layer thick variant, one NM unit cell has a grey backgro und and its lattice axes are mark ed. The unit cell is selected i n a \nway, that Ni atoms occupy the edges. Mn and Ga atoms are not in plane but shifted by ¼ of the NM lattice constant into the interior of the cell. The NM unit cell is tetragonal ( c/a\nNM = 1.22). The (101) NM-type twin boundaries between the neighbouring \nNM nanovariants are marked with green lines. The inherent twinning angle results in the characteristic modulated structure. \nThe neighbouring variant is only two layers thick, hence no comp lete NM unit cell fits into this  nanotwin lamella. Therefore it  \nis more convenient to consider half a NM un it cell (framed in black) as building block. Since Ni 2MnGa is an ordered L2 1 \nHeusler alloy, composition of the two twin lamella into ( 25) stacking does not preserve the translation symmetry of the \nordered lattice. Therefore, a comp lete unit cell of the nanotwinned superstructure comprises two  (25) stacking sequences and \nhas to be described as a 14M modulation. The parameters that fully determine the 14M supercell are chemical order, lattice \nconstants of NM and the ( 25) stacking sequence. \nThe relevance of this building block principle becomes evident when estimating the magnetocrystalline anisotropy of the 14M \nmartensite. The NM martensite has easy plane an isotropy. The easy plane is spanned by the two aNM axes, while cNM is the hard \naxis. The magnetocrystalline anisotropy of 14M can now be derived from anisotropy of NM by counting NM building blocks \nwith the hard cNM-axis parallel to each crystallographic direction (along a14M, b14M and c14M) and dividing by the overall number \nof building blocks. The favoured magnetisation axis of 14M is in c14M direction since no hard cNM axis is aligned in parallel. \nParallel to the b14M and a14M directions, fractions of 2/7 and 5/7 of the hard cNM axis are aligned, respectively. Consequently \na14M is the hard axis and b14M is semi-hard. With these weighted mean values, one can not only derive the right order of hard, \nsemi-hard and easy axis of 14M, but also the magnitude of th e magnetocrystalline anisotropy energies, which agree well with \nexperiments (values are given in the main paper). \n \n2) Film composition and Structure Analysis \n \nComposition was determined to be Ni 54.8Mn 22.0Ga23.1 by EDX using a stoichiometric standard. Commonly, bulk samples with \nsimilar composition are in NM martensite phase at room temperat ure [1]. Hence, in this film on MgO(100),  austenite and 14M \nmartensite are stabilized by the interface to the rigid substrate.  \nThe coexistence of austenite, 14M and NM martensite was confirmed by XRD θ-2θ measurements. Despite epitaxial growth of \naustenite at elevated temperature, the martensitic transition results in certain tilts of the martensitic unit cells (See fig. 3). For \nmeasuring the lattice constants, in analogy to a previous report [2], sample alignm ent was optimized for each variant to obtain  \nmaximum intensity. In Fig. S1 all five independent measurements  are shown in one graph. From  these, the lattice parameters \ngiven in the main paper were determined . Due to equality of the lattice constants aA and b14M (expected from the theory of \nadaptive martensite), the respective peaks coincide. \n \n \nFIG S2: Summary of X-Ray diffraction patterns (Cu-K α) measured for the different {400}-planes of martensite and austenite, \nrespectively. \n \n \n \n \n3) Temperature dependence \n \nThe differences of an adaptive phase compared to a common thermodynamic phase are apparent in temperature dependent X-\nray analysis. A qualitative idea about the phase content can be obtained from integrated intensities of reflections of the thre e \ndifferent phases. The temperature dependence of the integrated  intensities of the {220} reflections for all three phases are \ndisplayed in FIG S3  for the temperature range from 20 to 140°C. In this broad temperature ra nge, all three phases coexist. This \nobservation is in contrast to the properties in bulk system s, where sometimes a well defined sequence of first order \n(inter)martensitic transitions from austenite to 14M to NM is observed with decreasing temperature [3]. For the present film, \nthe intensity of the reflection from austenite increases con tinuously with rising temperature. This is expected when \napproaching the austenite to martensite  transformation temperature, however in  the accessible temperature range, this \ntransformation is not complete. For the NM martensite the intens ity is highest at low temperatures, which is expected for NM \nbeing the ground state. The intensity of the 14M martensite shows an unexpected behaviour as it first decreases and then re-increases with rising temperat ure. For a usual intermediate thermodynamic phase, one would expect phase content and, \ntherefore,  maximum intensity near the midpoint of the existence range. The anomalous observation of a minimum intensity at \nan intermediate temperature indicates that, for the present thin film sample, 14M is not a thermodynamically stable phase, but a \nunstable, adaptive phase which is sandwiched  between austenite and NM martensite.  \n20 40 60 80 100 120 1402025303540455055sqrt Intensity\nTemperature (°C)NM martensite14M martensiteaustenite\n \nFIG S3: Transformation of the different phases in the temperat ure range between 20°C and 140°C. The temperature dependent \nintensity of the integrated X-ray reflections of the {220}-planes of austenite, 14M and NM martensite, is shown for each phase.  \nThe measurements were performed during heating.   \n 4) Model of film architecture an d calculation of orientation relat ionship between different phases \n \n \nFIG S4: Sketch of the orientation relationship between Austenite, adaptive 14M and NM phase in constrained epitaxial films. \nThe orientation of Austenite is fixed by epitaxial growth on the MgO(100) substrate (the edges of the  MgO unit cell are  \nparallel to the axis system to the left). The geometrical Wech sler-Liebermann-Read (WLR) theory determines the habit plane \nbetween Austenite and 14M and hence the orientation of the 14M unit cell. As the nanotwinned 14M structure and the \nmacroscopically twinned NM plate on the same habit plane are distinguished only by the number density of twin boundaries, \nthe orientations of the nano- and macroscopic twin variants are identical. The orientations of the NM crystal axes with respect  \nto the 14M supercell are obtainable by basic geometry, as sketched in Fig. 2 of the main paper.   \n \nIn the following we present an approach to de scribe the film architecture step by step ( FIG S4 ). Based on this, the \ncrystallographic orientations between the th ree observed phases are calculated. In th e pole figures, these orientations are \nquantitatively represented by the peak positions (expressed through the angles ψ and ϕ). Due to epitaxial growth, the 14M pole \nfigures exhibit 4-fold symmetry, thus it is sufficient to discuss one of the four equivalent reflections. \nFrom our previous experiments [5] we know that films grow within the austenite state on heated substrates and their \norientation is fixed by the epitaxial re lationship (MgO(100)[001] || Ni -Mn-Ga (100)[011]). The subs trate-film interface hinders  \nthe martensite transformation si nce the constraint at a rigid interface does not allow a varia tion of the lattice constants. He nce, \nwhen cooling below the martensite transformation temperature, so me austenite remains close to the substrate [5]. This agrees \nwith the measured pole figure for (400) A plane, which exhibits one intense peak at zero tilt (Fig 3 (c) in main article). (The \nadditional, weak peripheral reflections in this pole figure are due to second order twinning of a14M and b14M variants).  \nIt is expected that only in direct proximity to the substrat e interface, the martensitic transformation is suppressed. Since th e \nfilm is about 0.5 μm thick, most of the film volume transforms to the martensite state (Fig. S2). For a coherent martensite-\naustenite interface, the theory of Wechsl er, Liebermann and Read (WLR), which is based on the assumption of an invariant \nplane (the habit plane), can be used to calculate the relative orientation of austenite and mart ensite [4]. Since the austenite  is \nfixed by the substrate, our pole figure measurements give the orientation relationships in an absolute manner. In a previous work on an epitaxial 14M film we used the measured lattice constants in orthorhombic approximation to calculate the \norientation of 14M [5]. These calculated orientations were confirmed by pole figure measurements. The 14M phase of the \npresent film exhibits almost identical lattice constants and orie ntations (Fig. 4 (a),(b),(c) in the main article). We can conc lude \nthat the orientation of 14M is determined by the invariant plane to the austenite at the interface, described by WLR theory. \nSince an almost exact habit plane is formed, the 14M unit cell is tilted only by small \nψ and ϕ angles.  \nTo obtain the orientation of the macroscopi c NM variants, we will show in the followi ng that it is sufficient to consider the \nnanotwinned NM variants forming the 14M supercell. Since the twin boundary angle α = 90° - 2 arctan( aNM/cNM) = 11.8° \nbetween NM variants is fixed by the lattice constants and not by  the variant length, annihilation of twin boundaries does not \nchange their orientation. Therefore, the orientation of the NM nanotwin variants forming 14M and those of the macroscopic NM variants are identical, independent of the actual lengths scal e of the twin lamellae. As sketched in Fig. 4S, nanotwinned \n14M and macroscopically twinned NM can be connected by a branching mechanism [6]. The density of twin boundaries can \nbe reduced successively. This branching mechanism preserves the invariant habit plane on the macroscopic scale. Branching \ndoes not leave any degree of freedom for the orientation of the macroscopic NM variants – the orientations of all NM variants \nare already determined by WLR theory. \nThe crystallographic orientations of macr oscopic NM variants  are characterized by two characteristic tilt operations. The firs t \ntilt is induced by the orientation of the na notwinned adaptive lattice with  respect to the austenite and described by WLR theor y \nas outlined above. The second tilt is determined by the tilt a ngle between the crystallographic axes of the nanotwinned NM \nvariants and the 14M supercell axes (as sketched in Fig. 2 of  the main paper). Since each 14M variant is built from two NM \nnanotwin variants, two different sets of peaks are observed in th e NM pole figures, whereas only  one is observed for 14M. All \ndifferent combinations of both tilt operations result in the expected reflection positions in the (400) NM and (004) NM pole \nfigures, respectively (Fig. 4 (d),(e) in the main article).  \nTo substantiate this general idea, one can use the 3D model in order to visualize the crystallographic orientations leading to the \npeaks in the different pole figures of 14M and NM. For this, it is helpful to consider  the desk as substrate and place the mode l \nwith the c14M axis perpendicular to this “substrate plane” and the a14M and equilvalently the b14M axes directions rotated by 45°  \nwith respect to the desk edges (e.g. as sketched for 14M in FIG S4 ). In order to link the crystallographic orientations with the \npeak position in the pole figures, please note that in Fig. 4, ψ is only varied by 10°, hence th ese measurements always reflect \nthe orientation of the axis aligned approximately perpendicu lar to the substrate. One can illustrate the two different tilt \nmechanisms as follows: The tilt angles of the 14M variants are always realized by a tilt of the complete 14M supercell around \nspecific axes of the 14M martensite. This tilt mechanism is sufficient to describe the 14M martensite pole figures (FIG. 3 (a-c ) \nin main article). To understand the peak positions in the NM pol e figures, additionally, one has to consider the angles between  \nthe b14M axis and the inherent NM unit cells. These angles are diffe rent for the 5 layer thick (+3.91°) and the 2 layer thick (-\n7.95°) variant and result from the building block model by basic geometry. \nAs an example, we will describe the orient ation of one particular variant of the 14M martensite and the expected peak positions  \nin the pole figures for the two different NM variants it is connected with. We select the 14M martensite variant with a14M \npointing out-of-plane, the orientation of this variants is imaged in the (400) 14M pole figure (FIG. 3 (b) in main article). The \nmodel has to be rotated by 45°  and placed on the desk with a14M pointing out-of-plane. To obtain the tilt towards the substrate \nnormal observed in the (400) 14M pole figure, the sample should be slightly tilted around b14M (more precisely by a tilt angle \n~2.7° around [010] 14M). The origin of this tilt is the interface to the austenite as discussed above. By this simple tilt operation, \nthe pole figure of the (400) 14M plane is completely described. \nThe situation becomes more complex when now looking at the macroscopic NM vari ants, which originated from this 14M \nvariant by branching. Since, as described above, branching does not change the variant orientations, it is sufficient to use th e \npresent 3D model. The only difference is that we now consider  the orientations of both NM unit cells within the tilted 14M \nsupercell, since both inherent NM variants contribute to differ ent pole figures. We start with the 5 layer thick variant, for w hich \nthe cNM axis is pointing out-of-plane. Hence this variant contributes to the (004) NM pole figure only. The operations \ndetermining the orientation of this NM variant are:  \n(1) Tilt of the 14M supercell (~ 2.7° around [010] 14M) \n(2) Tilt of the 5-layered variant within the 14M supercell (= 3.91° tilt around [001] 14M) \nSince both tilt operations can also be described in the frame of  the 14M unit cell, both tilt axes are perpendicular to each ot her. \nApplying both tilt operations su ccessively, one obtains an expected peak position at ψ  = 4.8° and ϕ = ±10.4° in the (004) NM \npole figure (see also Table 1 in main article). Considering the 4-fold symmetry, the four measured peak positions at ψ  = 5.3° \nand ϕ ~0° in the (004) NM pole figure are explained. The argumentation for the 2-layer thick variant is very similar. As in this \ncase aNM is pointing out-of-plane, this variant contributes to the (400) NM pole figure. The tilt of the 14M supercell is the same \nas described before, only the tilt of the 2 layer NM variants within the 14M supercell is different (= 7.95° around [001] 14M). If \none applies again both tilt operations, one expects a peak at ψ  = 8.4° and ϕ = ±26.3° in the (400) NM pole figure. This \ncorresponds to the peaks measured at ψ  = 7.4° and ϕ = ±18°. \nThe absolute values of the tilt angles can be calcula ted using the rotation matrix around the unit vector ()Tvvvv3 2 1=G: \n⎟⎟⎟\n⎠⎞\n⎜⎜⎜\n⎝⎛\n− + + − − −− − − + + −+ − − − − +\n=\n) cos1( cos sin ) cos1( sin ) cos1(sin ) cos1( ) cos1( cos sin ) cos1(sin ) cos1( sin ) cos1( ) cos1( cos\n)(~\n2\n3 1 32 2 311 322\n2 3 212 31 3 212\n1\nα α α α α αα α α α α αα α α α α α\nα\nv v vv v vvv vv v v vvv vv v vv v\nDvG  \nFor a direct comparison with the angles ψ and ϕ measured in the pole figure, the substrate edges are used as reference system \n(x ≙ MgO[100], y ≙ MgO[010], z ≙ MgO[001]). As rotation axis for the first tilt of the complete 14M supercell, one has to \nuse ()Tv 01 1\n21\n1 − =G. This vector is identical to the [010] 14M-axis described in the paragraph above using the Ni-Mn-Ga \naustenite unit cell as reference system. With 1vG, one obtains )(~\n1αvDG  with α = 2.7° from WLR theory. This consideration for \nthe first tilt is valid for two of the three 14M variants. The second tilt operation is different for the 2- and 5-layer thick NM \nvariants. The following detailed calculation is carried out for the 5-layer thick NM variant contributing to the (004) NM pole \nfigure (same example as in the paragraph above). First, we ha ve to determine the rotation axis for the second tilt operation. \nThis  [001] 14M-axis depends on the first tilt and can thus be described by the vector ()T\nvD v 011)(~\n21\n1 2 αGG=  in the MgO \nreference system. With this rotation axis, one obtains the second matrix ),(~\n2βαvDG , with β =+ 3.9° being the angle between \n14M crystal axes and the axes of the inherent 5-layer thick NM variant. The final orientation of the NM martensite variant \n(described by the vector wG) can be calculated by applying both tilt operations on a vector ()Tg 100=G pointing out-of-\nplane (MgO[001] direction). Thus, one obtains: \n=wGg D Dv vG\nG G )(~),(~\n1 2α βα . \nAs last step, wG is transformed from the Cartesian substrate coordinate system to spherical coordinates. The two angle \ncoordinates of wG directly give the expected peak position ( ψ and ϕ) in the pole figure. For the 5-layer thick NM variant connected with the (400)14M variant, one obtains ψ = 4.7° and ϕ = 280.4°. Considering the 4-fold symmetry given by the \nsubstrate, each calculated peak position results in 4 peaks in th e respective pole figure. In order to reduce the experimental \nerrors originating from slight sample misalignment, values for the measured angles are averaged over all 4 quadrants. In an analogous manner, we can determine the orientations of the other two 14M variants and the orientations of the connected \nmacroscopic NM variants. The results are su mmarized in Table 1 in the main article. \n \n5) Constructing modulated phases by tetragonal building blocks \n \nAt the first glance, this approach alread y seems to fail when considering 10M martensite since there is no way to construct \n10M by using the experimentally observed lattice parameters for the NM phase. As sketched in the following, this however is \npossible when assuming that NM phases with different c/a ratio may exist. In FIG S5 , measured and calculated lattice \nparameters are summarized. As solid horizontal  lines the orthorhombic lattice constants of  10M as measured by Righi et al are \ngiven [7]. The c-axis lattice constant of a virtual NM unit cell is varied whereas its a-axis lattice constant is kept constant at \na\nNM = c 10M, using the direct adaption of the geometrical model for 14M. Assuming a ( 23)-stacking periodicity, the diagonal \nlines for a10M and b10M are the expected lattice constants for the 10M structure built from the virtual NM. The best agreement of \ncalculation and measurement is obtained at about cNM = 0.642 nm. The additionally calculated volume difference of austenite \nand martensite vanishes for this lattice constant (right y-axis). Hence it seems to be plausible to build 10M from NM building \nblocks with a c/a NM ratio of 1.161, a significantly lower value than for reported NM martensites, even compared to the large \nscatter experimentally observed for different NM samples [8]. \nAdditionally, the same idea can be applied on the reported pr emartensitic phase, which exhibits a 3-layered structure and \nshould thus be called 6M. Investigations by high energy synchrotron radiation revealed an orthorhombic structure with an only very slightly distorted unit cell [9]. Following the approach sk etched above, we can calculate the lattice constants of the \ntetragonal NM unit cells, which can be used as building blocks for the 6M. Comparing the calculated lattice constants with the \nmeasured orthorhombic structure, we can determine the lattice constants of the NM building blocks to be a\nNM = 0.579 nm  and \ncNM = 0.586  nm leading to a c/a NM ratio of 1.0152 . \n0.40 0.45 0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.900.400.450.500.550.600.650.70\nmeas. c10M= 0.553 nm\nΔ a10M : 0.06 nm\nvirtual NM : c = 0.642 nm\n      a = 0.553 nm      c/a = 1.16110M: c/a = 0.926\nmeas. b10M= 0.592 nmcalc. a10M\ncalc. b10M\nmeas. a10M= 0.597 nm\nvirtual cNM (nm)10M lattice constants (nm)\nΔVAust-NM(nm3)\n0.000.020.040.060.080.100.120.140.160.180.20\n \nFIG S5: Approach to construct 10M from NM building blocks with different c/a NM ratio. \n \nWith increasing electron density (e/a ratio) typically the followi ng sequence of martensite phases is observed: A – 6M – 10M – \n14M – NM [10]. This indicates that the underlying mechanism for this specific phase sequence is  associated with an increase \nof c/a NM with electron density.  \n \n \n \n 6) Martensite-austenite interfaces of the modulated structures in Ni  –Mn -Ga \n \nIn this section, the interface conditions between the different, modulated phases in the Ni-Mn-Ga system with respect to the \ncubic austenite are considered. It is shown that all  phases allow for the formation of a co mpatible exact interface, thus, \nfulfilling the geometrical key requirement to identif y these phases as adaptive metastable structures. \n \nThis compatibility condition is ma thematically expressed by an e quation to be solved [11,12], \n \n ˆii−=⊗ QU I b m , (0.1) \nwhere iU describes the deformation from the austenite to one varian t of the martensite crystal structure. In Eq. (1.1) the \nunknowns are the rotation matrix iQ and the vectors band ˆm. These vectors are the so-ca lled shape-strain and the unit \nnormal to the habit  plane, respectively . A necessary and sufficient condition that (1.1) has a solution is that the symmetric  \nmatrix 2\niU has an eigenvalue exactly equal one, 21λ=, one eigenvalue smaller than one, an d one eigenvalue larger than one, \n11λ< and31λ>.  \nIn the following we analyse crystallographic structure data for th e 6M (3layer premartensite) [9], 10M (5 layer) [7,8], and 14M  \n(7 layer) [8,13] within this formalism. For convenience, we here use the setting fo r the transformation from  a cubic B2-like \nunit cell into a monoclinically distorted bct unit cell,  as in Refs. [12,14]. In Tabl e ST1, we list the lattice parameters of the \nmonoclinic unit cell of the martensite aXMbct, bXMbct, cXMbct, and the monoclinic angle βXM  for the modulated phases with X=6, \n10, and 14 and  the corresponding lattice parameter of the L2 1 cubic lattice cell aC. Then, the resulting eigenvalues,  \n,1 , 2 , 3ii λ = , the vector components of the corresponding habit plane normal 123ˆ(, , )mmm=m and the shape strain \n123(, , )bbb=b are given that correspond to the variant 1Uin the notation of Ref. [12]. The other habit planes can be obtained \nby permutation of  vector components as detailed in Ref. [12]. From the data in Table ST1  it is obvious that the parameters of \nthe fundamental bct-like lattice cells are very similar in all modulated phases, a fact already noticed by Pons et al. [8]. Thu s, all \nmodulated structures can form a compatible, i.e., almost exact habit plane with the austenite with rather similar orientation \nˆmand shape-strain b. \n \nTable  ST1: Compatibility of the austenite-modulated martensite  interface for different modulated phases in the Ni-Mn-Ga \nsystem according to Eq. (1.1) . \n \nXM: 6M 10M 14M bulk 14M epitaxial film \nReference [9] [7] [13] this investigation \naC [nm] 0.5828 0.5825 0.5825 0.578 \naXMbct [nm] 0.4121 0.4228 0.422 0.428 \nbXMbct[nm] 0.2913 0.2788 0.270 0.271 \ncXMbct [nm] 0.4093 0.4200 0.420 0.422 \nβXM  [1°] 90.0 90.3 92.7 95.3 \n1λ 0.707  0.677 0.654 0.663 \n2λ 0.993 1.020 1.017 1.024 \n3λ 1.41 1.452 1.450 1.487 \nb1 0.408 0.430 0.422 0.431 \nb2 -0.409 -0.458 -0.497 -0.512 \nb3 0.409 0.454 0.458 0.437 \nm1 0.816 0.819 0.810 0.813 \nm2 0.408 0.401 0.376 0.335 \nm3 -0.408 -0.410 -0.450 -0.477                      \n \n     \n \n \n References  \n  \n[1]  Lanska N., Söderberg O., Sozinov A., Ge Y., Ullakko K., Lindroos V.K, J Appl. Phys . 95, 8074 (2004) \n[2] Buschbeck J., Niemann R., Heczko O., Thom as M., Schultz L., Fähler S., Acta Mater . 57, 2516 (2009) \n[3] Sozinov A., Likhachev A., Lanska L., Ullakko K., Appl. Phys. Lett. 80, 1746 (2002)  \n[4] Wechsler M., Lieberman D., Read  T., Transactions of the AIME 197, 1503 (1953) \n[5]  Thomas M. et al., New J. Phys.  10, 023040 (2008) \n[6]  Kohn V., Müller S., Philos. Mag. A  66, 697 (1992) \n[7]  Righi L.,  Albertini F., Pareti L., Paoluzi A., Calestani G., Acta Mat. 55, 5237 (2007) \n[8]  Pons J., Chernenko V.A, Sant amarta R., Cesari E., Acta Mat. 48, 3027 (2000) \n[9]  Ohba T.,  Miyamoto N., Fukuda K.,  Fukuda T., Kakeshita T., Kato K., Smart Mater. Struct. 14, 197 (2005) \n[10]   Heczko O. et al. in Magnetic Shape Memory Phenomena  (Ch. 14), in J.P. Liu et al. (eds) Nanoscale Magnetic \nMaterials and Applications , Springer Science (2009) in print, DOI 10.1007/978-0-387-85600-1 p.14 \n[11]  Ball, J.M., James R.D., Arch. Ration. Mech. Analysis 100, 13 (1987) [12]  Bhattacharya K., Microstructure of Martensite  (Oxford University Press, Oxford 2003) chap 7.1, p.106ff. \n[13] Righi L et al., Acta Mat. 56, 4529 (2008). \n[14] Hane, K.F., Shield T.W., Acta Mat. 47, 2603 (1999) \n "    },    {        "title": "0908.4017v1.Magnetocrystalline_anisotropy_and_antiferromagnetic_phase_transition_in_PrRh___2__Si___2__.pdf",        "content": "arXiv:0908.4017v1  [cond-mat.str-el]  27 Aug 2009Magnetocrystalline anisotropy and antiferromagnetic pha se\ntransition in PrRh 2Si2\nV K Anand∗and Z Hossain\nDepartment of Physics, Indian Institute of Technology, Kan pur 208016, India\nG Behr\nInstitute for Solid State and Materials Research, Dresden, Germany\nG Chen, M Nicklas, and C Geibel\nMax Planck Institute for Chemical Physics of Solids, 01187 D resden, Germany\n(Dated: February 6, 2020)\nAbstract\nWe present magnetic and transport properties of PrRh 2Si2single crystals which exhibit anti-\nferromagnetic order below T N= 68 K. Well defined anomalies due to magnetic phase transitio n\nare observed in magnetic susceptibility, resistivity, and specific heat data. The T Nof 68 K for\nPrRh2Si2is much higher than 5.4 K expected on the basis of de-Gennes sc aling. The magnetic\nsusceptibility data reveal strong uniaxial anisotropy in t his compound similar to that of PrCo 2Si2.\nWith increasing pressure T Nincreases monotonically up to T N= 71.5 K at 22.5 kbar.\n∗Electronic address: vivekkranand@gmail.com\n1Introduction\nYbRh 2Si2hasbeenwidelyinvestigated duetoitsproximity toaquantumphaset ransition\n[1, 2, 3, 4]. We show in this paper that its Pr-homolog PrRh 2Si2also presents unique\nmagnetic properties. All the investigated RRh 2Si2(R = rare earth) compounds have been\nfound to order antiferromagnetically [1, 5, 6, 7, 8, 9, 10, 11, 12]. A mong them GdRh 2Si2has\nthe highest ordering temperature, T N∼106 K [6]. EuRh 2Si2exhibits complex magnetic\norder with an antiferromagnetic ordering below 25 K [11]. CeRh 2Si2and YbRh 2Si2have\nunusual and interesting magnetic properties which are discussed b elow.\nThe antiferromagnetic ordering temperature T N∼36 K in CeRh 2Si2is very high com-\npared to the de-Gennes expected ordering temperature of 1.2 K [ 12, 13]. One more transi-\ntion is observed at 24 K. The exact nature (localized versus itineran t) of the magnetism of\nCeRh2Si2is not yet settled. The pressure dependence of T Nand of the magnetic moment\nindicates an itinerant nature of the magnetism [13]. The itinerant cha racter of magnetism\nin CeRh 2Si2has also been suggested from a systematic study of doping at Rh sit es in\nCe(Rh 1−xPdx)2Si2[14]. However, the dHvA study suggests local moment magnetism in\nCeRh2Si2at ambient pressure. Under the application of pressure the Fermi surface topology\nchanges discontinuously leading to an itinerant moment magnetism ab ove the critical pres-\nsure of around 1 GPa [15]. Pressure induced superconductivity has been observed around 1\nGPa below 0.5 K [16, 17].\nHeavy-fermion YbRh 2Si2has an antiferromagnetic ordering temperature T Nof∼70 mK\n[1]. The antiferromagnetic order can be suppressed very easily by a pplication of magnetic\nfield or by substitution of Si by Ge, leading to a quantum critical point [2, 3, 4]. Electrical\ntransport, thermodynamic and thermal expansion data reveal t hat quantum critical point\nin YbRh 2Si2is of local nature in contrast to the spin density wave type quantum critical\npoint in CeCu 2Si2[18, 19].\nCrystal field effects can have strong influence on the properties o f Pr-compounds. For\nexample, thelowlying crystalfieldexcitations areresponsible forth eheavyfermionbehavior\nin unconventional superconductor PrOs 4Sb12[20, 21, 22]. Despite numerous investigations\non RRh 2Si2, we did not find any discussion in literature on the properties of PrRh 2Si2. In\nthispaperwereportmagnetization, specificheat, electrical resis tivity andmagnetoresistance\nof PrRh 2Si2single crystals. In addition, we also carried out pressure dependen t electrical\n2resistivity measurements.\nSample preparation and measurements\nSingle crystals of PrRh 2Si2were grown from indium flux as well as using floating zone\nmethod in a mirror furnace (CSI Japan). Appropriate amounts of h igh purity elements\n(Pr: 99.99%, La: 99.9%, Rh: 99.999% and Si: 99.9999%) were arc melte d several times\non a water cooled copper hearth under argon atmosphere. The ar c melted polycrystalline\nPrRh2Si2and indium were taken in a molar ratio of 1:20 in an alumina crucible, which\nwas then sealed inside a tantalum crucible with a partial pressure of a rgon gas. The sealed\ntantalum crucible was heated to 1450oC under argon atmosphere for two hours and then\ncooled down to 900oC at a rate of 5oC/hour. Below 900oC rate of cooling was increased\nto 300oC/hour. Indium flux was removed by etching with dilute hydrochloric a cid. We\nobtained single crystals of about 2.5 mm x 1.5 mm x 0.4 mm by this method. We also\nsucceeded in growing PrRh 2Si2single crystal using float zone mirror furnace using 10 mm/h\ngrowth rate and counter-rotation of seed and feed rods. The dia meter of the float zone\ngrown crystal was about 6 mm.\nSamples were characterized by copper K αX-ray diffraction and scanning electron micro-\nscope (SEM) equipped with energy dispersive X-ray analysis (EDAX) . Laue method was\nused to orient the single crystals. A commercial SQUID magnetomet er was used to measure\nmagnetization. Specific heat was measured using relaxation method in a physical property\nmeasurement system (PPMS–Quantum design). Electrical resistiv ity was measured by stan-\ndard ac four probe technique using AC-transport option of PPMS. Pressure studies of the\nelectrical resistivity up to 2.3 GPa and in the temperature range 3 K <T<300 K were\ncarried out utilizing a clamp-type double layer pressure cell consistin g of an inner cylinder\nmade of NiCrAl and an outer body of Cu:Be. Silicone oil served as pres sure transmitting\nmedium. The pressure inside the cell was determined at low temperat ure by the inductively\nmeasured shift of the superconducting transition temperature o f lead.\n3/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s48/s51/s48/s48/s54/s48/s48/s57/s48/s48/s49/s50/s48/s48/s49/s53/s48/s48\n/s40/s51/s50/s49/s41/s40/s51/s49/s52/s41/s40/s51/s49/s50/s41/s40/s48/s48/s56/s41/s40/s49/s49/s50/s41\n/s40/s50/s48/s48/s41\n/s40/s50/s49/s51/s41\n/s40/s50/s48/s52/s41\n/s40/s49/s49/s54/s41\n/s40/s50/s49/s53/s41/s40/s50/s50/s50/s41/s40/s50/s50/s48/s41/s40/s50/s49/s49/s41/s40/s49/s48/s53/s41/s40/s49/s49/s52/s41\n/s40/s50/s48/s50/s41/s40/s49/s48/s51/s41/s40/s49/s48/s49/s41/s40/s48/s48/s50/s41/s99/s111/s117/s110/s116/s115\n/s50 /s32/s40/s100/s101/s103/s41/s80/s114/s82/s104\n/s50/s83/s105\n/s50\nFIG. 1: Indexed powder x-ray diffraction pattern of ThCr 2Si2-type body-centered-tetragonal pul-\nverized PrRh 2Si2single crystal (flux grown).\nResults and Discussion\nFrom the analysis of powder X-ray diffraction data of the crushed s ingle crystals (figure\n1), we find that PrRh 2Si2crystallizes in ThCr 2Si2-type tetragonal structure (space group\nI4/mmm) with the lattice parameters a= 0.4079 nm, c= 1.0138 nm, and the unit cell\nvolume = 0.16876 nm3for the flux grown sample, and a= 0.4078 nm, c= 1.0138 nm, and\nthe unit cell volume = 0.16858 nm3for the float zone grown sample. The X-ray diffraction\nand SEM image confirmed the samples to be single phase. The EDAX com position analysis\nconfirmed the desired stoichiometric composition of 1:2:2.\nThe temperature dependence of the magnetic susceptibility of PrR h2Si2single crystal is\nshown in figure 2 for magnetic field applied along the a-bplane and the c-axis. A large\nanisotropy in the magnetic susceptibility χ(T) is observed. The susceptibility data have\nmuch larger values for B// ccompared to that for B// a-bimplying the easy axis to be\nthec-axis. This anisotropic behavior is similar to the strong uniaxial anisot ropy along c-\n4/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/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s48/s46/s48/s53/s48/s46/s48/s54\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s46/s48/s48/s51/s48/s46/s48/s48/s52/s48/s46/s48/s48/s53/s48/s46/s48/s48/s54/s48/s46/s48/s48/s55/s48/s46/s48/s48/s56/s48/s46/s48/s48/s57/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s32/s84/s32/s40/s75/s41/s66/s47/s47 /s97/s45/s98/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41/s66/s47/s47 /s99\n/s66/s47/s47 /s97/s45/s98\nFIG. 2: Temperature dependence of magnetic susceptibility of PrRh 2Si2single crystal (flux grown)\nmeasured in a field of 3.0 T. Inset shows the enlarged view of B/ /a-bdata. Thesolid lines represent\nfit to Curie-Weiss behaviour.\naxis in CeRh 2Si2[10] but different from the easy plane behavior observed in YbRh 2Si2[1].\nWithin a series of R-T-X compound, the change of the magnetic aniso tropy with changing\nthe R-element is governed by the change in αJsecond order Stevens factor within the CEF\nHamiltonian [23]. A change from a uniaxial behavior in CeRh 2Si2and PrRh 2Si2to an easy\nplanebehavior inYbRh 2Si2isinfull accordancewith αJ<0forCe- andPr-while αJ>0for\nYb-compound. Since in all three cases the anisotropy is very prono unced, it indicates a very\nlarge and positive A0\n2CEF-parameter in the whole RRh 2Si2series. An antiferromagnetic\ntransition is observed in the susceptibility data at 68 K for both B// a-band B//c. As\nexpected for an antiferromagnet T Ndecreases with increasing magnetic field (T N= 66.5 K\nat B = 5 T). The susceptibility data exhibit slight deviation from the Cur ie-Weiss behaviour\nχ(T)=C/(T- θp)forbothB// a-bandB//cduetotheeffect ofcrystal fields. Fromthelinear\nfit of inverse susceptibility data (100 K – 300 K) at 3 T we obtain the eff ective magnetic\nmoment µeff= 3.48µB(very close to the theoretical value of 3.58 µBfor Pr3+ions) and\n5/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s66/s47/s47 /s97/s45/s98\n/s56/s48/s32/s75/s54/s48/s32/s75/s77/s32/s40\n/s66/s47/s80/s114/s41\n/s66/s32/s40/s84/s41/s66/s47/s47 /s99/s54/s48/s32/s75\n/s56/s48/s32/s75\nFIG. 3: Field dependence of isothermal magnetization of PrR h2Si2single crystal (flux grown) at\n60 and 80 K along B// cand B//a-b.\nthe Curie-Weiss temperature θa\np= -103.2 K for B// a-b, andµeff= 3.63µBandθc\np= +57.9\nK for B// c. Further, we note a very pronounced peak and a rapid decrease o f magnetic\nsusceptibility to essentially zero value below 20 K for B// cand a much weaker temperature\ndependence for B// a-b, which suggests strongly anisotropic Ising-type antiferromagne tism\nin PrRh 2Si2similar to that of PrCo 2Si2[24].\nThe isothermal magnetization data exhibit a linear dependence on fie ld at 60 K (mag-\nnetically ordered state) and 80 K (paramagnetic state) for both B //a-band B//c(figure\n3). The magnetic moments at 5 T are very small in both the directions (0.07µB/Pr for\nB//a-band 0.39 µB/Pr for B// c) and the maximum value attained is only 12% of the sat-\nuration magnetization for Pr3+ion (3.2µB/Pr). Measurements at higher fields are required\nto observe the metamagnetic transitions which are expected in ant iferromagnets with strong\nmagneto-crystalline anisotropy.\nThe specific heat data of single crystal PrRh 2Si2(indium flux grown) together with that\nof the nonmagnetic reference compound LaRh 2Si2are shown in figure 4. The specific heat\n6/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48 /s57/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48\n/s50/s48 /s52/s48 /s54/s48 /s56/s48/s48/s50/s52/s54/s56/s83\n/s109/s97/s103/s32/s40/s74/s47/s109/s111/s108/s101/s45/s75/s41\n/s84/s32/s40/s75/s41\n/s32/s80/s114/s82/s104\n/s50/s83/s105\n/s50\n/s32/s76/s97/s82/s104\n/s50/s83/s105\n/s50/s67/s32/s40/s74/s47/s109/s111/s108/s101/s45/s75/s41\n/s84/s32/s40/s75/s41\nFIG. 4: Temperature dependence of the specific heat of single crystal PrRh 2Si2(flux grown)\nand polycrystalline LaRh 2Si2in the temperature range 2 to 90 K. The inset shows the magneti c\ncontribution to the entropy of PrRh 2Si2.\nof PrRh 2Si2exhibits a pronounced λ-type anomaly at 68 K, which confirms the intrinsic\nnature of antiferromagnetic ordering in this compound. The float z one grown single crystal\nof PrRh 2Si2also exhibits a similar well defined anomaly at 68 K due to antiferromagn etic\norder. The specific heat data of PrRh 2Si2and LaRh 2Si2hardly differ from each other\nbelow 20 K showing that the magnetic excitations have vanished expo nentially below 20 K.\nThis indicates a large gap in the magnetic excitation spectra in the ord ered state, which\ncan obviously be attributed to the strong Ising-type anisotropy o bserved in the magnetic\nsusceptibility data. The linear coefficient to the specific heat is γ∼18 mJ/mole-K2. The\ntemperature dependence of the magnetic entropy is shown as inse t in figure 4. At 70 K the\nmagnetic entropy attains a value of 7.85 J/mole-K, which is 36% more t hanRln2 and 14\n% lower than Rln3. Thus, either three singlets or one singlet and one doublet CEF lev els\nare in the energy-range below 80 K and involved in the magnetic order ing. Because of the\nhuge uniaxial anisotropy and the general trend of the CEF parame ters within the RRh 2Si2\n7/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s40 /s45/s99/s109/s41\n/s84/s32/s40/s75/s41\nFIG. 5: Temperaturedependenceofelectrical resistivity ( I//a-b)of fluxgrownPrRh 2Si2singlecrys-\ntal in the temperature range 1.8 – 210 K. Solid line shows the fi t to gapped magnon characteristics\nin the ordered state, i.e. ρ(T) =ρ0+AT2+C/braceleftbigg\n1\n5T5+△T4+5\n3△2T3/bracerightbigg\nexp(−△/T).\nseries, one can suspect that these lowest CEF levels are the two Γ 1singlets and either the\nΓ2singlet or the Γ 5doublet [25].\nThe electrical resistivity measured with ac current flowing in the a-bplane is shown in\nfigure 5. The resistivity shows typical metallic behavior with room tem perature resistivity\nρ300Kof 85µΩ-cm, residual resistivity ρ0∼9.6µΩ-cm and residual resistivity ratio (RRR)\n∼9. A linear decrease of resistivity is observed with decreasing tempe rature until it meets\nthe antiferromagnetic transition at 68 K, below which the resistivity shows a large decrease.\nIn the ordered state the resistivity data present gapped magnon characteristics and fit well\nto the relation [26]\nρ(T) =ρ0+AT2+C/braceleftbigg1\n5T5+∆T4+5\n3∆2T3/bracerightbigg\nexp(−∆/T)\nbelow 65 K (inset of figure 5) where ρ0= 9.8µΩ-cm is the residual resistivity, A =\n8/s48 /s52 /s56 /s49/s50 /s49/s54 /s50/s48 /s50/s52/s54/s56/s46/s53/s54/s57/s46/s48/s54/s57/s46/s53/s55/s48/s46/s48/s55/s48/s46/s53/s55/s49/s46/s48/s55/s49/s46/s53\n/s52/s53 /s54/s48 /s55/s53 /s57/s48/s51/s48/s52/s53/s54/s48/s55/s53/s57/s48/s32/s40 /s45/s99/s109/s41\n/s84/s32/s40/s75/s41/s48/s46/s54/s32/s107/s98/s97/s114/s50/s50/s46/s52/s32/s107/s98/s97/s114/s84\n/s78/s32/s40/s75/s41\n/s80/s32/s40/s107/s98/s97/s114/s41\nFIG. 6: T Nof PrRh 2Si2as a function of externally applied pressure. The inset show s temperature\ndependence of resistivity under pressure.\n0.00241µΩ-cm/K2is the coefficient to the Fermi liquid term, C = 8.96 x 10−9µΩ-cm/K5\nis the prefactor to the magnon contribution, and ∆ = 37.8 K is the mag non energy gap.\nAs both CeRh 2Si2and YbRh 2Si2exhibit strong pressure dependence in the electrical\nresistivity we have performed resistivity measurement on PrRh 2Si2under externally applied\npressure. Up to 22.5 kbar there is no pronounced effect of extern ally applied pressure on\nthe resistivity except an increase of T Nfrom 68.5 K at p = 0 to 71.5 K at p = 22.5 kbar\n(figure 6). Similar weak effect of pressure on the magnetically order ed state was also found\nin PrCo 2Si2[27].\nFrom the de-Gennes scaling in the family of RRh 2Si2(R = rare earths) one would expect\nan ordering temperature of 5.4 K in PrRh 2Si2. While in CeRh 2Si2the anomalously high\nTNmight be a result of the mixture of localized and itinerant character o f the magnetic\norder we can not offer any clear reason for the high T Nof PrRh 2Si2. Enhanced density of\nstates as in the case of GdRh 2Si2and large value of exchange constant (as evidenced by large\nθp) definitely contribute to higher value of T N. It is also found that RRh 2Si2compounds\nwhich have higher values of T Nthan expected on the basis of de-Gennes scaling have their\n9moments aligned along c-axis below T N. PrRh 2Si2also has higher T Nthan expected and\nthe magnetic susceptibility data suggest that Pr moments lie along c-axis in this case also.\nWe suspect the uniaxial anisotropy which forces the moment to lie alo ng thec-axis is also\nresponsible for the high T Nin PrRh 2Si2. System with uniaxial anisotropy has much larger\nvalue of magnetic susceptibility for B//(easy-axis) which helps in the process of magnetic\nordering. Thus T Nfor a system with uniaxial anisotropy will be higher than that of an\nisotropic system or a weakly anisotropic system.\nConclusion\nWe succeeded in growing single crystals of PrRh 2Si2which forms in ThCr 2Si2-type body-\ncentered tetragonal structure. Temperature dependent mag netic susceptibility, electrical\nresistivity, specific heat data reveal strongly anisotropic Ising ty pe antiferromagnetic order\nbelow 68 K in this compound. Application of pressure up to 22.5 kbar do es not stabilize\nany new ordered phase but T Nincreases from 68 K to 71.5 K.\nAcknowledgement\nTechnical assistance from Mr. Jochen Werner is gratefully acknow ledged.\n[1] Trovarelli O, Geibel C, Mederle S, Langhammer C, Grosche F M, Gegenwart P, Lang M,\nSparn G and Steglich F 2000 Phys. Rev. Lett. 85626\n[2] Gegenwart P, Custers J, Geibel C, Neumaier K, Tayama T, Te nya K, Trovarelli O and Steglich\nF 2002Phys. Rev. Lett. 89056402\n[3] Plessel J, Abd-Elmeguid M M, Sanchez J P, Knebel G, Geibel C, Trovarelli O and Steglich F\n2003Phys. Rev. B 67180403(R)\n[4] Custers J, Gegenwart P, Wilhelm H, Neumaier K, Tokiwa Y, T rovarelli O, Geibel C, Steglich\nF, P´ epln C and Coleman P 2003 Nature424524\n[5] Felner I and Nowik I 1983 Solid State Commun. 47831\n[6] Tung L D, Franse J J M, Buschow K H J, Brommer P E and Thuy N P 19 97J. Alloys Comp\n26035\n10[7] Szytula A, `Slaski M, Ptasiewicz-Bak H, Leciejewicz J and Zygmunt A 1984 Solid State Com-\nmun.52395\n[8]`Slaski M, Leciejewicz J and Szytula A 1983 J. Magn. Magn. Mater. 39268\n[9] Melamud M, Pinto H, Felner I and Shaked H 1984 J. Appl. Phys. 552034.\n[10] Quezel S, Rossat-Mignod J, Chevalier B, Lejay P and Etou rneau J 1984 Solid State Commun.\n49685\n[11] Hossain Z, Trovarelli O, Geibel C and Steglich F 2001 J. Alloys Comp. 323-324 396\n[12] Graf T, Hundley M F, Modler R, Movshovich R, Thompson J D, Mandrus D, Fisher R A and\nPhillips N E 1998 Phys. Rev. B 577442\n[13] Kawarazaki S, Sato M, Miyako Y, Chigusa N, Watanabe K, Me toki N, Koike Y and Nishi M\n2000Phys. Rev. B 614167\n[14] G´ omez Berisso M, Pedrazzini P, Sereni J G, Trovarelli O , Geibel C and Steglich F 2002 Eur.\nPhys. J. B 30343\n[15] Araki S, Settai R, Kobayashi T C, Harima H and ¯Onuki Y 2001 Phys. Rev. B 64224417\n[16] Movshovich R, Graf T, Mandrus D, Thompson J D, Smith J L an d Fisk Z 1996 Phys. Rev. B\n538241\n[17] Araki S,Nakashima M, Settai R, Kobayashi TC and ¯Onuki Y 2002 J. Phys.: Condens. Matter\n14L377\n[18] Gegenwart P, Langhammer C, Geibel C, Helfrich R, Lang M, Sparn G, Steglich F, Horn R,\nDonnevert L, Link A and Assmus W 1998 Phys. Rev. Lett. 811501\n[19] Stockert O, Faulhaber E, Zwicknagl G, St¨ ußer N, Jeevan H S, Deppe M, Borth R, K¨ uchler R,\nLoewenhaupt M, Geibel C and Steglich F 2004 Phys. Rev. Lett. 92136401\n[20] Bauer E D, Frederick N A, Ho P -C, ZapfV Sand Maple MB2002 Phys. Rev. B 65100506(R)\n[21] Goremychkin EA, OsbornR, BauerED, Maple MB, Frederick NA, YuhaszWM,Woodward\nF M, and Lynn J W 2004 Phys. Rev. Lett. 93157003\n[22] Thalmeier P 2006 Physica B 378-380 261\n[23] see e.g. Buschow K H J and de Boer F R 2003 Physics of Magnetism and Magnetic Materials\n(Kluwer academic/Plenum publishers) p 56\n[24] Shigeoka T, Fujii H, Yonenobu K, Sugiyama K and Date M 198 9J. Phys. Soc. Japan 58394\n[25] Santini P, Amoretti G, Blaise A and Caciuffo R 1993 J. Appl. Phys. 736560\n[26] Jobiliong E, Brooks J S, Choi E S, Lee H and Fisk Z 2005 Phys. Rev. B 72104428\n11[27] Kawano S, Onodera A, Achiwa N, Nakai Y, Shigeoka T and Iwa ta N 1995 Physica B 213-214\n321\n12"    },    {        "title": "0909.1728v3.Tuning_of_crystal_structure_and_magnetic_properties_by_exceptionally_large_epitaxial_strains.pdf",        "content": "1 \nTuning of crystal structure and magnetic properties  by \nexceptionally large epitaxial strains \nJ. Buschbeck 1,2 , I. Opahle 1,3 , M. Richter 1, U.K. Rößler 1, P. Klaer 4, M. Kallmayer 4, H. J. \nElmers 4, G. Jakob 4, L. Schultz 1,2 , S. Fähler 1 \n1 IFW Dresden, P.O. Box 270116, 01171 Dresden, German y \n2 Department of Mechanical Engineering, Institute for  Materials Science, Dresden \nUniversity of Technology, 01062 Dresden, Germany \n3 Institut für Theoretische Physik, Universität Frank furt, 60438 Frankfurt/Main, \nGermany \n4 Institut für Physik, Johannes Gutenberg-Universitä t Mainz, 55099 Mainz, Germany \n \nAbstract \nHuge deformations of the crystal lattice can be ach ieved in materials with inherent \nstructural instability by epitaxial straining. By c oherent growth on seven different \nsubstrates the in-plane lattice constants of 50 nm thick Fe 70 Pd 30  films are continuously \nvaried. The maximum epitaxial strain reaches 8.3 % relative to the fcc lattice. The in-\nplane lattice strain results in a remarkable tetrag onal distortion ranging from c/a bct  = \n1.09 to 1.39, covering most of the Bain transformat ion path from fcc  to bcc  crystal \nstructure. This has dramatic consequences for the m agnetic key properties. \nMagnetometry and X-ray circular dichroism (XMCD) me asurements show that Curie \ntemperature, orbital magnetic moment, and magnetocr ystalline anisotropy are tuned \nover broad ranges.  2 \nStrain effects on functional materials are of great  current interest for improving \nmaterials properties. By strained epitaxial film gr owth, physical properties can be \ncontrolled and improved e.g. in semiconductors 1, multiferroic materials 2 and \nferromagnets 3 - 7. Moreover, there are even materials exhibiting the ir functional \nproperties like ferroelectricity only in strained f ilms 8.  \nDuring epitaxial growth, the film orientation is co ntrolled by the substrate onto which \nthe material is deposited. In addition, a thin film  may also adapt its in-plane lattice \nparameters to the substrate, even if their equilibr ium lattice parameters differ \nconsiderably. The particular case when the lattice parameters of the film material are \nstrained such that they are identical to those of t he substrate is called strained coherent \nfilm growth. In common rigid metals, straining of t he crystal lattice by coherent film \ngrowth requires substantial elastic energy. As a co nsequence, already at strains of a few \npercent, coherent growth is limited to ultrathin fi lms with thicknesses of up to several \natomic layers. For the growth of coherent epitaxial  films with high strains and large \nthickness soft materials should be used, as suggest ed by van der Merwe  9. Exceptional \nsoftening is observed in crystals with lattice inst abilities, e.g. in materials with a \nmartensitic transformation. As an example how to ex ploit such a martensitic instability, \nGodlevsky and Rabe 10  predicted the possibility to induce a cubic to tet ragonal \ndistortion with c/a bct  ratios from 0.95 to 1.18 in the magnetic shape mem ory material \nNi 2MnGa. In fact, in experiments Dong et al. 11  demonstrated a considerable epitaxial \nstrain of 3% in a Ni-Mn-Ga film.  \nThis shows that for improved tunability of crystal lattice and functional properties in \nthicker films it is advantageous to exploit lattice  instabilities. Here, we report the \npreparation of seven evenly distributed stages alon g the Bain path, between face \ncentered cubic fcc  and body centered cubic bcc  structure in 50 nm thick Fe 70 Pd 30  films \n(Fig. 1). The martensitic and ferromagnetic Fe 70 Pd 30  alloy is well known for anomalous 3 \nmagnetomechanical effects such as the magnetic shap e memory effect 12 . We \ndemonstrate the inverse magnetomechanical behavior of this compound: By varying \nc/a bct , Curie temperature, orbital magnetic moment, and m agnetocrystalline anisotropy \nare tuned in wide ranges. The approach can be exten ded to a multitude of materials with \nferroelastic or martensitic lattice instabilities e nabling exceptionally large strains. In \nparticular, the possibility to stabilize intermedia te lattice geometries and vary smoothly \nbetween stable phases along Bain transformation pat hs offers new routes to adjust and \nunderstand structure-property relations in function al materials.  \nFe-Pd has been suggested as a model system for mart ensitic transformations, already in \n1938 13 . For the present experiments it is in particular f avorable that the Fe 70 Pd30  alloy \nexhibits a structural instability resulting in soft ening of the crystal lattice near room \ntemperature 14 . In addition, growth at ambient temperature freque ntly enables \npreparation of coherent films with larger thickness  due to the reduced mobility of \ndislocations 15 . In the composition range from Fe 70 Pd 30  to Fe 75 Pd 25  and in the vicinity of \nroom temperature four phases with different tetrago nal distortions have been observed \nin quenched, chemically disordered bulk: fcc , “ fct ”, bct  and bcc  16 . These structures \nfollow the Bain transformation path 17 . The Bain path is a geometrical description for th e \ntransformation from fcc  to bcc  lattice. Considering a body centered tetragonal ( bct ) unit \ncell with the lattice parameters abct  and cbct  (marked by colored atoms and thicker lines \nin Fig. 1B) one can describe the transformation fro m fcc  (top) to bcc  lattice (bottom) by \na continuous variation of the tetragonal distortion  c/a bct  from 2 to 1. In accordance \nwith Bain we use the bct  description of the unit cells here. The four phase s are described \nby c/a bct  ratios of 1.41  ( fcc ), 1.33  (“ fct ”), 1.02  ( bct ) and 1 ( bcc ). \nFe 70 Pd 30 films of 50 nm nominal thickness were deposited ont o different substrates at \nroom temperature by magnetron sputtering from a 2 i nch Fe 70 Pd 30 alloy target. The \nsubstrates consist of a MgO(100) single crystal wit h different epitaxial metallic buffer 4 \nlayers (Rh, Ir, Pt, Pd, Au-Cu, Fe and Cr). Details on the growth of the epitaxial films are \ndescribed in the auxiliary material available onlin e 18 . The epitaxial growth on the \ndifferent substrates induces considerable distortio n of the crystal lattice. X-ray \ndiffraction (XRD) was measured at room temperature.  As an overview over the crystal \nstructure, θ-2θ diffraction patterns were recorded in Bragg-Brenta no geometry (Co K α, \n0.1789 nm). The diffraction patterns show a (002) bct  reflection of the epitaxial Fe 70 Pd 30  \nfilm (Fig. 1A, reflection marked by an arrow). When  varying the substrate this \nreflection shifts by 12.3 degrees. However, it alwa ys lies within the boundaries of the \nBain-transformation, between the reflection positio ns expected for fcc  and bcc  structure. \nIntensity measured at 2θ ≤ 56° is due to the (002) reflection of the fcc  Au-Cu, Pt, Pd, Ir \nor Rh layer. \nDetermination of the lattice constants was performe d in a 4-circle set-up with Euler \nstage (Cu K α, 0.1540 nm) 18 . The in-plane lattice parameters of the films are identical to \nthe substrate lattice spacings (Fig. 1C). This prov es a coherent epitaxial growth in spite \nof a large variation of the substrate lattice spaci ng. The epitaxial strain reaches a \nmaximum value of 8.3 % relative to the Fe 70 Pd 30 fcc lattice 19 . Due to the coherent \ngrowth, our films are in a single variant state wit h both in-plane lattice parameters abct  \nof the crystal structure’s basal plane fixed by the  cubic substrate lattice, while no \nconstraint exists for the lattice parameter cbct  in the out-of-plane direction. The \nmeasured c/a bct  ratio ranges from c/a bct  = 1.09 to 1.39, covering most of the Bain path \nfrom bcc  to fcc . This corresponds to a variation of tetragonal dis tortion by 27% which \nwe could stabilize in films with a remarkable thick ness of 50 nm. The dotted line in Fig. \n1C represents the ideal behavior at constant unit c ell volume ( c/a bct  = Vbct /2 a3 , Vbct  = \n0.0265 nm 3 [19]). The experimental data points show that the a ssumption of constant \nvolume is justified even for such large deformation s.  5 \nThe achieved strained film growth indicates very lo w energy differences between the \ncrystal structures along the Bain path. For the pre sent material, this is confirmed by \ndensity functional calculations (Fig. 2A). Electron ic structure calculations for \ndisordered Fe-Pd alloys were performed with the ful l potential local orbital (FPLO) \ncode 20  in the framework of density functional theory. Dis order was treated in the \ncoherent potential approximation (CPA) 21  and the exchange and correlation potential \nwas treated in the local spin density approximation  (LSDA). In the calculations, the \nc/a bct  ratio is varied, while the volume of the unit cell  is held constant, as justified by \nour experiments. Numerical details of the calculati ons are identical to those given in \nRef. 22. In comparison to the rigid pure Fe 23 and Pd 24 , the energy landscape is very flat \nalong the entire Bain path, i.e., energy difference s between the different c/a bct  ratios are \nindeed small. Hence, the film-substrate interaction  is sufficient to stabilize large \ntetragonal distortions in films of bulk-like thickn ess. \nThe key property of magnetic shape memory alloys is  their magnetocrystalline \nanisotropy. Strained epitaxial growth holds the pro mise of adjusting this property, \nbeyond what is possible by conventional inverse mag netostriction. On the atomic scale, \nthe origin of the magnetocrystalline anisotropy is the spin-orbit coupling of the valence \nelectrons in combination with exchange-splitting an d with the particular electronic \nstructure. We studied the effect of lattice distort ion on the spin-orbit related properties \nby measuring the ratio of orbital- to spin magnetic  momentum (µ orb /µ spin  ) of the iron 3d \nelectrons by means of X-ray magnetic dichroism alon g the [001] crystal direction of \nfour films with considerably different tetragonal d istortion. In addition we determined \nthe anisotropy constants K1 and K3 of these films by magnetometer measurements along \ndifferent crystallographic directions [001]; [100];  [110] (Fig. 2 B). Magnetic \nmeasurements with both methods were performed at 30 0K 18 . While K1 defines the \nwork required to magnetize the material along the h ard magnetization axis [001], K3 is a \nmeasure of the anisotropy in the basal plane of the  tetragonal unit cell. All three 6 \nquantities ( µorb /µ spin  and K1, K3) show considerable changes along the Bain path but  \nsimilar trends. At c/a bct = 1.33 maximum anisotropy is observed. This value o f \ntetragonal distortion coincides with the “fct”  phase that exhibits high \nmagnetocrystalline anisotropy and shows the magneti c shape memory effect in bulk. \nFor the magnetic shape memory effect, the anisotrop y constant K1 is a crucial quantity. \nWe find that due to the strain and tetragonal disto rtion K1 is strongly varying in our \nseries of epitaxial films. When approaching cubic s tructures the magnetic anisotropy is \nreduced as expected due to their high symmetry. The  maximum value of K1 = \n1.5*10 5 J/m 3 at c/a bct =1.33 is identical with the literature value that h as been reported at \nT=100 K for a “fct”  single crystal with c/a fct  = 0.94 (corresponding to c/a bct = 1.34) 25 .  \nMoreover, the tetragonal distortion of the lattice also significantly changes the Curie \ntemperature. Temperature dependent spontaneous magn etization was measured in the \nfilm plane (Fig. 3). The applied field of 1 T is su fficient to saturate the sample along this \ndirection. Increase in curvature is observed with i ncreasing c/a bct  ratio. To explain this \nbehavior, the Curie temperature TC was evaluated. According to Kuz’min’s model 26 , TC \ncan be determined by a fit of the relative magnetiz ation curves (inset in Fig. 3). The \nadditional shape parameter s that is included in this model, depends on the spi n wave \nstiffness 27 . While the shape parameter does not vary significa ntly with tetragonal \ndistortion, the ferromagnetic transition temperatur e increases remarkably from 652 K at \nc/a bct = 1.39 to a value of 829 K at c/a bct = 1.09. The extrapolated value of TC = 650 K at \nc/a bct  = 1.41 ( fcc ) is similar to the literature value of 600 K repor ted for the fcc-phase in \nFe 70 Pd 30  bulk 28, 29 .  \nIn conclusion, by strained coherent growth of 50 nm  thick Fe 70 Pd 30 films with inherent \nstructural instability, we achieve a quasi continuo us variation of the lattice distortion \nalong the Bain path. The magnetic properties of the  Fe 70 Pd 30 films display large \nchanges: The Curie temperature is increased more th an 25% with respect to the value 7 \nfor Fe 70 Pd 30  with fcc structure. The ratio of orbital vs. spin magnetic momentum \nchanges by a factor of two. This is accompanied by a large modification of the magnetic \nanisotropy from near zero to values close to those of “fct” bulk Fe 70 Pd 30 . Softening of \nthe crystal lattice and a flat energy landscape alo ng the Bain path are not a unique \nfeature of this alloy. Similar lattice instabilitie s may be exploited in various functional \nmaterials including (magnetic) shape memory, ferroe lectric, multiferroic, or \nmagnetocaloric materials for extended adjustability  of their crystal structure in strained \nepitaxial films. This must cause severe changes in the electronic structure of the \nmaterials and, thus, enables to wider tune their ma gnetic, transport, optical, or even \ncatalytic properties.  \n \n \n 8 \n \n \nFIG. 1 : A: A considerable shift of the Fe 70 Pd 30  (002) bct  peak (arrow) observed in X-ray \ndiffraction patterns is measured on the films grown  on different substrate materials \n(CoK α). Peak positions expected for fcc  and bcc  structure are marked by dotted lines. \nThey represent the boundaries of the Bain transform ation. B:  Sketch of the different \nstages during the Bain transformation between fcc  (top, dark blue) and bcc  structure \n(bottom, red). The black lines mark the bct  unit cell used to describe this transformation. \nC:  Variation of the Fe 70 Pd 30  crystal lattice (substrate materials are marked on  top). The \nin-plane lattice parameter abct  (open symbols) follows the straight line represent ing \nidentity with the substrate’s lattice spacing d. The c/a bct  ratio (solid symbols) is varied \nalmost from fcc  to bcc  structure by the coherent epitaxial growth. The do tted curve \nillustrates the expected change of c/a bct  at constant volume of the unit cell. 9 \n \nFIG. 2: A: Calculated energy required to induce tetragonal d istortions ( c/a bct ) along the \nBain path. Compared to literature data of the commo n metals Fe [23] and Pd [24] only \nlittle elastic energy is required for a tetragonal distortion of Fe-Pd. B:  Change of the \nmagnetocrystalline anisotropy constants K1 and K3 and the ratio of orbital to spin \nmomentum ( µorb /µ spin ) of Fe 3d electrons along the Bain path. For each quantity the \nexperimental error is represented by an error bar. 10 \n \nFIG. 3: The curvature of the relative spontaneous m agnetization versus temperature \ndecreases with increasing tetragonal distortion. Inset:  Curie temperature TC and shape \nparameter s that have been extracted using Kuz’min’ s model 26 . TC increases \nconsiderably when approaching the bcc  structure. The shape parameter only shows \nminor changes. \n 11 \nAcknowledgements  The authors thank O. Heczko, M. E. Gruner, J. McCo rd and S. \nKaufmann for helpful discussions and U. Besold and T. Eichhorn for experimental \nsupport. This work is funded by DFG via the priorit y program on magnetic shape \nmemory alloys (www.magneticshape.de). \n \n                                                 \nReferences \n1 J. C. Bean, Science 230 , 127 (1985). \n2 J. Wang, J. B. Neaton, H. Zheng, V. Nagarajan, S. B. Ogale, B. Liu, D. Viehland, V. \nVaithyanathan, D. G. Schlom, U. V. Waghmare, N. A. Spaldin, K. M. 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Mat. 15 , \n1201 (1980). 1/13 Auxiliary Material \nHere we provide experimental details of the film de position as well as the \nmethods for structural and magnetic characterizatio n and density functional calculations. \nThis is followed by figures supporting the results of the structural characterization \npresented in the main paper. Film texture is charac terized by means of pole figure \nmeasurements. By these, epitaxial film growth is pr oven and the orientation relationships \nof film, buffer and substrate are determined. In ad dition, the positions of the poles in the \npole figure verify the tetragonal distortion in an independent measurement. Finally results \nof the reciprocal space mapping are presented which  are used to exemplarily verify the \ncoherent epitaxial growth of a Fe 70 Pd 30  film deposited on Pt buffer.  \nIn the main paper, for better readability, the term  “substrate” describes the metals \nwhere the Fe 70 Pd 30  films were deposited on. These metals themselves a re prepared as \nepitaxial thin layers on MgO(100) single crystals. Thus, for precise designation of the \nparts of these composite substrates, we will name t his metal layer “buffer” in the \nfollowing. \n \nMethods \n \nFilm deposition \nUsing the sputtering method for Fe 70 Pd 30 film deposition we could overcome the \nlimitations of our previous experiments where pulse d laser deposition (PLD) was applied \n[26]. For PLD films we observed that in addition to  the intended stabilization of fct  \nmartensite severe stress of up to 4 GPa caused (111 ) deformation twinning in the film, 2/13 resulting in the loss of epitaxy at film thickness exceeding 20 nm. We could attribute the \norigin of the stress to high kinetic energy of the deposited ions characteristic for PLD \n[27]. To reduce film stress, we changed to the sput tering method for deposition of \nFe 70 Pd 30 films where the kinetic energy of the ions is at le ast one order of magnitude \nlower compared to PLD [28]. \nThe base pressure in the sputtering chamber was les s than 10 -8 mbar and Argon \nsputtering gas of 6N  purity was used. Epitaxial metallic buffer layers (Rh, Ir, Pt, Pd, Au-\nCu, Fe and Cr) serving as substrates for the strain ed growth of Fe 70 Pd 30 films were \ndeposited at 300°C onto MgO(100) single crystals of  10 mm x 10 mm x 0.15 mm in size. \nBuffer layers (Cr, Fe, Au-Cu, Pd, Pt) were prepared  using magnetron sputtering from 2 \ninch targets. In order to limit materials cost, Rh and Ir buffers were prepared using on-\naxis pulsed laser deposition from smaller targets a nd subsequently transferred ex-situ to \nthe sputtering chamber. The nominal film architectu re is: MgO / 50 nm buffer / 50 nm \nFe 70 Pd 30 . When metals with fcc crystal structure are used as buffer (Rh, Ir, Pt, P d or Au-\nCu), prior deposition, a thin layer of 5 nm Cr was grown on MgO to promote epitaxial \ngrowth. Buffer layer of metals with bcc-structure ( Cr and Fe) grow epitaxial without a \nfurther interlayer.  \nTo promote subsequent Fe 70 Pd 30 film growth on a relaxed buffer surface, the \nthickness of the buffer layer was chosen such that it exceeds the common critical \nthickness of metal layers by more than one order of  magnitude. Deposition at 300°C \nenables epitaxial growth of the buffer layers with bulk-like lattice parameters. By \ndepositing Fe 70 Pd 30  at room temperature undesirable decomposition into  ordered L1 0 \nphase and iron rich Fe(Pd) phase is avoided [29]. 3/13 Film composition was checked for films on Rh, Ir, P t and Cr substrate by energy \ndispersive x-ray spectroscopy using a bulk Fe 70 Pd 30  standard. The measured film \ncomposition of Fe 70.4 Pd 29.6  ( ±0.7) is constant and identical to the target compos ition. \n \nStructural Characterization \nFor diffraction pattern measurements in Bragg-Brent ano geometry a 2-circle \nPhillips set-up is used (CoK α). Since the bcc  lattice normally does not exist at a \ncomposition of Fe 70 Pd 30  under normal conditions, the lattice spacing of th e bcc  structure \nwas calculated assuming constant volume by Vbcc =Vfcc /2 = 0.0265 nm 3 [30]. For \ndetermination of the crystal structure of the Fe 70 Pd 30  films, as shown in Fig. 1C of the \nmain paper, 2θ of (002) bct  and  2θ of (101) bct -type lattice planes were measured in 4-circle \ngeometry. By this two independent, non-collinear la ttice parameters are probed. Since \nboth in-plane lattice parameters are identical this  is sufficient to describe the tetragonal \nunit cell. Using elementary geometry we calculated the out-of-plane lattice constant cbct  \nand the in-plane lattice constants abct  by  bct bct d(002) 2  c ⋅= and 2\nbct 2\nbct 2\nbct 2\nbct \nbct d(101) - cc d(101)  a⋅= . \nFrom these values the c/a bct  ratio was calculated. For Fe 70 Pd 30  grown on Rh buffer, the \n(101) bct  and (111) fcc  lattice planes of epitaxial Fe 70 Pd 30 film and Rh buffer, respectively, \nexhibit very similar lattice spacing and spatial al ignment. Since film and buffer could not \nbe distinguished in the Phillips 4-circle device in  this case, the lattice parameters of this \nfilm were measured in the reciprocal space mapping setup having a higher resolution. \nReciprocal space measurements were performed in a n on-commercial 4- circle set-up \nusing a rotating anode system ( Cu K α), equipped with a focussing multilayer X-ray 4/13 mirror. Besides conventional angular scans, the sof tware allows direct mapping of \nscattered intensity in reciprocal space coordinates . \n \nMagnetic Characterization \nX-ray magnetic circular dichroism (XMCD) was measur ed in transmission at the \nGerman synchrotron light source BESSY II (beam line  UE56/1-SGM) at normal \nincidence. XMCD results are obtained from transmiss ion measurements detecting the X-\nray intensity transmitted through the Fe 70 Pd 30  films via the luminescence light the X-ray \nlight induces in the MgO(100) substrate [31],[32]. This transmission data measures film \nproperties averaged along the film normal. The line ar background subtraction was \nadjusted in order to achieve the same relative post -edge absorption intensity at 750 eV as \nfor the transmission data. The magnetization of the  sample was switched at each energy \nstep by applying a magnetic field of 1.22 T at oppo site directions perpendicular to the \nsample. Measurements were performed at 300 K.  \nMagnetic anisotropy was characterized by hysteresis  curve measurements at \n300 K up to fields of 4.5 T using a Quantum Design Physical Property Measurement \nSystem (PPMS) with vibrating sample magnetometer (V SM) add-on.  \nThe magnetocrystalline anisotropy energy of a tetra gonal unit cell can be \ndescribed by the following equation [33]: \nα φ φ φ 44\n34\n22\n1 cos sin K sin K sin K EA ⋅ + + =  [1] \nwhere K1…K 3 are the anisotropy constants;  φ is the angle between magnetization \ndirection and c-axis and α is the angle to the a-axes in the basal plane of t he tetragonal \nunit cell. It is known from literature that in Fe 70 Pd 30  “fct” bulk single crystals the hard 5/13 magnetization direction is aligned along the c-axis  while the a-axes are the easy \nmagnetization directions [30]. \nFinite values of K1 cause a linear dependence of the magnetization on the applied \nmagnetic field along the hard magnetization axis. F inite values of the second anisotropy \nconstant K2 cause a deviation from this linear dependence. Sin ce we observe almost \nlinear behavior we assumed K2≈0. The 4-fold anisotropy in the basal plane of the \ntetragonal unit cell is described by K3.  \nK1 was determined from the anisotropy field obtained by linear extrapolation of \nhysteresis loop measurements along the [001] crysta l direction of the film by K1=-H A/2J S. \nShape anisotropy was corrected by assuming a demagn etization factor of N=1 of an \ninfinite film. K3 was determined from the area A enclosed by the hysteresis loops \nmeasured along the [100] and [110] directions accor ding to 2K3=W[100] bct -W[110] bct =A \nwith W[hkl] bct  being the work to magnetize the sample along the r espective direction of \nthe bct unit cell. K3 of a “fct”  single crystal at different temperatures and appli ed stresses \nhas been reported by Cui et al. [30]. Accordingly, in “fct”, a  and b axes of the unit cell \nare the easy magnetization directions. In agreement  with the Bain transformation we \nobserve the [110] direction being the easy axis in bct  description. However, due to the \ndifferent descriptions of the anisotropy energy and  unit cell used in the work of Cui et al \nand here, the absolute values of K3 have to be converted. A factor of 1/8 has to be ap plied \nto the values published by Cui et al. [30] in order  to make them comparable [34]. Cui et \nal. report K3(Cui) /4 ≈1.2*10 4 J/m 3 at c/a fct = 0.935 (-20°C) under applied stresses of 2 and 8 \nMPa. Due to the applied stress, the single crystal was in the single variant state. In bct  \ndescription c/a fct  = 0.93 corresponds to c/a bct =1.32. When considering the different 6/13 measurement temperatures of 253 K and 300 K, respec tively, the values of 1/8 K3(Cui) =K 3 \n=6*10 3 J/m 3 and K3 =1.3*10 3 J/m 3 measured here for c/a bct =1.33, are in reasonable \nagreement. \nSpontaneous magnetization J S was determined at a field of 1 T from the \ndemagnetizing branch of the in-plane hysteresis mea surements, after magnetizing the \nsample in a magnetic field of 2 T. Film thickness w as determined by X-ray reflectometry. \nIn-plane, the samples saturate at fields below 0.5 T. \n \nEnergy calculation along the Bain path \nExperimentally, the martensitic transformation from  fcc (c/a bct =1.41) to “fct”  \n(c/a bct ≈1.34) in disordered Fe-Pd alloys is found for compo sitions where the ground state \nchanges from fcc  to bcc with increasing Fe content. In the vicinity of this  transition, the \nenergy curve along the Bain path becomes flat. In t he LSDA calculations of the internal \nenergy this transition is found at somewhat higher Fe concentrations between 78 and 82 \nat.% in the vicinity of the experimental volume, co mpared to the experimental \nconcentration of 70 at.% Fe. The deviation between the experimental phase diagram and \nthe calculations for the transition between the fcc  and bcc  structures can be mainly \nattributed to the entropy contributions in the ther modynamic potential at finite \ntemperatures that are not included in the electron- theoretical calculation at zero \ntemperature. Fig.1 shows the calculated total energ y along the Bain path for Fe 78 Pd 22  with \nthe atomic volume fixed to 0.013256 nm 3. The specific shape of the energy curve as \nfunction of the distortion along the Bain path depe nds on the chosen composition, but the \nvery flat energy landscape is characteristic for al loys with an “fct”  martensite phase.  7/13  \nAuxiliary Structural Characterization \n \nTexture measurements \nEpitaxial film growth on the buffer is proven by th e existence of a single pole in \neach quadrant of the Fe 70 Pd 30  (101) bct  pole figure measurements (Fig. S2). The four-fold \nsymmetry of the used (001) oriented MgO single crys tal surface makes it sufficient to \ndepict just a quarter of the pole figure.  \nOrientation relationships of film||buffer||MgO(001)  have been determined as follows: \nFe 70 Pd 30 bct (001)[110]|| fcc -buffer(001)[100]||Cr(001)[110]||MgO(001)[100] or \nFe 70 Pd 30 bct (001)[110]|| bcc -buffer(001)[110|| MgO(001)[100]  \nfor the films on fcc -buffers (Rh, Ir, Pt, Pd, Au-Cu) and bcc -buffer (Fe,Cr), respectively. \nThe orientation relationships are sketched in Fig. S3. The bct  unit cell of the Fe 70 Pd 30  film \nis 45° rotated with respect to the MgO fcc unit cel l independent of the buffer layer. In \ncontrast, the relative orientation of the bct  unit cell of the Fe 70 Pd 30  film with respect to the \nbuffer, changes from [100] bct (film)|| [110] fcc  on fcc -buffer to [100] bct (film)||[100] bcc  on \nbcc -buffer. However, the epitaxial orientation relatio nship of film and buffer remains \nidentical when the buffers crystal structure is des cribed according to the Bain concept and \na bct  unit cell with c/a =1.41 is used for fcc -buffers (not shown in the sketch). When doing \nso, the orientation relationship simplifies to [100 ] bct (film) || [100] bct (buffer). Thus, as done \nin the paper, we can use the “substrate lattice spa cing d”, equal to d[100] bct (buffer), as the \nkey parameter for coherent epitaxial growth, indepe ndent of the actual crystal structure of \nthe used buffers. In the common fcc  and bcc  description the lattice spacing [100] bct  is 8/13 equal to [220] fcc  and [100] bcc . At c/a bct =1.41 ( fcc ) the (101) bct  pole is identical to the \n(111) fcc  pole. \nThe four-fold symmetry of the buffer forces both in -plane lattice constants to be \nidentical, resulting in the observed tetragonal dis tortion. In addition to texture and the \nepitaxial relationship, pole figures also give info rmation about the crystal structure of the \nfilm itself. The position of the (101) bct  pole along the tilt angle Ψ is a measure of the \ntetragonal distortion of the bct  unit cell. As sketched in Fig. S2, Ψcorresponds to the \nangle between the (101) bct  lattice plane and the substrate plane. When the str ucture \nchanges from c/a bct =1.41 ( fcc ) to c/a bct =1 ( bcc ), a position change of the (101) bct  poles \nfrom Ψ = 54.7° to Ψ= 45° is expected. Indeed, this shift of position i s already visible in \nexemplarily shown pole figures in Fig. S2. The meas ured angle Ψ for Fe 70 Pd 30 on \ndifferent buffers is summarized in Fig. S4. When va rying the buffer lattice spacing, a \ncontinuous shift of the (101) bct  poles is observed, between the Ψ angles expected for fcc  \nand bcc  structure. The variation of the (101) bct  pole positions shows that, following the \nBain path, bct  crystal structures have been stabilized with inter mediate tetragonal \ndistortions between c/a bct =1.41 and 1. As an independent measurement, this be haviour \nconfirms the previously depicted dependence of the c/a bct -ratio versus substrate lattice \nspacing depicted in Fig. 1B.  \n \n 9/13 \n \nFigure S1 : (101) bct  pole figures measured on Fe 70 Pd 30  films deposited on Ir, Pt and Cr buffers verify the  \nepitaxial growth. A variation of the tetragonal dis tortion causes a variation of the tilt angle Ψ of the (101) bct  \nlattice planes (sketch). In result, a shift of the pole position along Ψ is measured in the pole figure. Blue and \nred dotted lines within the pole figures mark the t ilt angles Ψ expected for cubic fcc  or bcc  structure, \nrespectively. 10/13 \n \nFigure S2: Orientation relationship of the layer sy stems and the MgO single crystal as determined from  \npole figure measurements. Two types have to be dist inguished: A) Fe 70 Pd 30 on fcc -buffer using a bcc  Cr-\nadhesion layer and B) Fe 70 Pd 30 directly deposited on bcc -buffer. In the Fe 70 Pd 30 unit cell one (101) bct  plane \nis marked by white dotted lines.     \n \n  \nFigure S3: Tilt angle  Ψ of the (101) bct  reflections extracted from pole figure measurement s of Fe 70 Pd 30 on \ndifferent buffers. The behaviour verifies the exist ence of a tetragonal distortion following the Bain path \nbetween fcc  and bcc  structure. For the film on Rh, we could not measur e Ψ due to the overlapping of film \nand buffer reflections.   11/13 Reciprocal Space Mapping \nBesides the determination of the crystal structure of the film grown on Rh buffer, \nwe used reciprocal space mapping in order to probe the structure of the Fe 70 Pd 30  film on \nPt buffer in more detail. This film was chosen sinc e its c/a ratio is approximately in the \nmiddle of the Bain path. Figure S5 shows the plane scan of the substrate, buffer and film \nreflections measured with varying h and l along a (101) bct plane. The coordinates of the \nreciprocal space are based on the bct  unit cell of the film. In this particular measurem ent \nplane, all three reflections of Fe 70 Pd 30  film (101) bct , Pt buffer (111) fcc  and MgO(111) fcc  are \nvisible. Due to the different lattice spacings of f ilm, buffer and MgO they appear at \ndifferent positions in reciprocal space. Buffer and  film reflections are much broader \ncompared to the MgO(111) fcc . In agreement with the less perfect crystal struct ure \nexpected for the epitaxial films compared to bulk s ingle crystals this can be interpreted as \na wider spread of the lattice spacings in the films . Together with further measurements \nalong different measurement planes, the single crys talline state, the orientation as well as \nthe previously calculated tetragonal structure of t he Fe 70 Pd 30  unit cell could be verified. \nNo evidence for stress relaxation by a modulated cr ystal structure or lateral splitting into \nvariants (regions) with different c/a  ratios was observed.  12/13 \n \nFigure S4: Reciprocal Space Mapping of Fe 70 Pd 30  deposited on Pt, showing the (111) fcc  reflections of MgO \nand Pt and a (101) bct  reflection of bct  Fe 70 Pd 30  measured by varying h and l in the reciprocal space \ncoordinate system of the bct  Fe 70 Pd 30 . The h and l coordinate give the in-plane and out-of-plane latt ice \nconstants, respectively. 13/13 References \n                                                 \n26  J. Buschbeck, I. Lindemann, L. Schultz, S. Fähler,  Phys. Rev. B 76,  205421 (2007). \n27  T. Edler, J. Buschbeck, C. Mickel, S. Fähler, S. G . Mayr, New J. Phys. 10 , 063007 (2008). \n28  S. Fähler, H. U. Krebs, Appl. Surf. Sci. 96-98 , 61 (1996).  \n29  J. Buschbeck, O. Heczko, A. Ludwig, S. Fähler, L. Schultz, J. Appl. Phys. 103 , 07B334 (2008). \n30  J. Cui, T. W. Shield, R. D. James, Acta Mater. 52 , 35 (2004).  \n31  M. Kallmayer, H. Schneider, G. Jakob, H. J. Elmers , K. Kroth, H. C. Kandpal, U. Stumm, S. Cramm,  \nAppl. Phys. Lett.  88 , 072506 (2006). \n32  M. Kallmayer, A. Conca, M. Jourdan, H. Schneider, G. Jakob, B. Balke, A. Gloskovskii, H. J. Elmers, J . \nPhys. D - Appl. Phys. 40 , 1539 (2007). \n33  K. H. J. Buschow, F. R. de Boer Physics of Magnetism and Magnetic Materials  (Kluwer, New York \n2004) pp. 97-100 \n34  Explanation for the origin of the factor of 1/8: T he formulation of the anisotropy energy used by Cui  et \nal. is E A=K 1sin 2φ+K 2sin 4φ+ K3sin 2α*sin 2β. Compared to our formulation, only the description o f the \nanisotropy in the basal plane differs ( φ=90°). In an orthogonal system, instead of using bo th angles α and \nβ to describe the magnetization direction relative t o the crystal axes a and b it is sufficient to consider \none of the angles. Hence sin 2α*sin 2β can be equally expressed by sin 2α*cos 2α = -1/8 cos4 α. Thus, if the \nunit cell remains the same, the conversion factor i s -1/8. However, instead of “fct” we used the bct  unit \ncell here. In the bct  description of the unit cell, [100] fct =[110] bct  while [110] fct =[100] bct . Thus the change \nfrom “ fct” description to bct  description, causes a further change in sign, resu lting in a conversion factor \nof 1/8. "    },    {        "title": "0910.4285v1.Interplay_between_the_magnetic_anisotropy_contributions_of_Cobalt_nanowires.pdf",        "content": "arXiv:0910.4285v1  [cond-mat.mtrl-sci]  22 Oct 2009Interplay between the magnetic anisotropy contributions o f Cobalt nanowires\nJ. S´ anchez-Barriga1,2,∗, M. Lucas3, F. Radu2, E. Martin1, M. Multigner4,5, P. Marin1, A. Hernando1and G. Rivero1\n1Instituto de Magnetismo Aplicado (UCM-ADIF-CSIC), P.O.Bo x 155, 28230, Las Rozas, Madrid, Spain\n2Helmhotz-Zentrum Berlin f¨ ur Materialien und Energie,\nAlbert-Einstein-Strasse 15, D-12489 Berlin, Germany\n3Technische Universit¨ at Berlin, Institut f¨ ur Theoretisc he Physik, Hardenbergstr. 36, D-10623 Berlin, Germany\n4Centro Nacional de Investigaciones Metal´ urgicas (CENIM- CSIC),\nAvd. Gregorio del Amo 8, 28040, Madrid, Spain\n5Centro de Investigaci´ on Biom´ edica en Red en Bioingenier´ ıa,\nBiomateriales y Nanomedicina (CIBER-BBN), Madrid, Spain\n(Dated: October 20, 2021)\nWe report on the magnetic properties and the crystallograph ic structure of the cobalt nanowire\narrays as a function of their nanoscale dimensions. X-ray di ffraction measurements show the ap-\npearance of an in-plane HCP-Co phase for nanowires with 50 nm diameter, suggesting a partial\nreorientation of the magnetocrystalline anisotropy axis a long the membrane plane with increasing\npore diameter. No significant changes in the magnetic behavi or of the nanowire system are observed\nwith decreasing temperature, indicating that the effective magnetoelastic anisotropy does not play a\ndominantrole intheremagnetization processes ofindividu alnanowires. Anenhancementofthetotal\nmagnetic anisotropy is found at room temperature with a decr easing nanowire diameter-to-length\nratio (d/L), a result that is quantitatively analyzed on the basis of a simplified shape anisotropy\nmodel.\nPACS numbers: 75.75.+a, 75.50.Tt, 75.60.Jk, 75.60.-d, 75. 60.Ej\nThe discovery of interlayer exchange coupling [1] and\ngiant magnetoresistance [2, 3] have promoted a tremen-\ndousadvanceofstoragemedia, readoutsensors,andmag-\nnetic random access memory (MRAM). By further re-\nducing the lateral size of the magnetic structures an in-\ncreased performance of the devices is achieved together\nwith a sustained interest for fundamental understand-\ning oflow dimensionalmagnetism. Forantiferromagnetic\nmaterials the finite-size effects lead to a scaling of the in-\ntrinsic magnetic properties like the ordering temperature\nand anisotropy as-well as to an enhanced contribution of\nthe magnetically disordered surface to the macroscopic\nmagnetization [4, 5]. For ferromagnetic materials with\nreduced dimensions, the presence of a finite magnetiza-\ntion leads to an increased shape anisotropy which de-\npends strongly on the geometry of the objects [6]. This\nmanifestsstrongerwhenthelateraldimensionsaretouch-\ning to the nanoscale regime. Besides the enhanced shape\nanisotropies, the remagnetization processes like coherent\nrotation and curling modes are more favorable against\nthe domain wall movements. Ultimately, the manipula-\ntion of the dynamical magnetic properties of such struc-\ntures in ultrashort time scales [7] needs to be comple-\nmentary to an appropriate characterization and control\nof their magnetic and geometrical properties.\nIn recent years, with the aim of exhaustively tai-\nloring and controlling properties such as perpendicular\nmagnetic anisotropy (PMA) [8], a large variety of ge-\nometries for small-sized elements is being produced by\ndifferent techniques. Magnetic nanoparticles [9], nan-\notubes [10], micron-sized rectangular patelets and dots\n[11], nanowires [12] or ultrathin films are typically fab-\nricated by using various combinations of state-of-the-art\nmodern experimental tools. Techniques such as electronbeam lithography or imprint lithography [13], molecu-\nlar beam epitaxy (MBE) or magnetron sputtering are\ncombined to produce high-quality magnetic structures of\nultrasmall sizes [6]. Among them, lithography represents\na top-down fabrication technique where a bulk material\nis reduced in size to a nanoscale pattern. Alternatively,\nelectrodeposition [14] is a very simple and inexpensive\nbottom-up technique which allows fabrication of large\narrays of magnetic nanowires. Combined with the use\nof self-assembling methods for the deposition of mem-\nbranes it has the particular advantage of producing sys-\ntemswithhighPMAandultra-highdensities. Inorderto\ncontrolthe anisotropyofsucha system, it isimportantto\nunderstand how the magnetic properties depend on the\nfabrication parameters, and thus on the geometrical and\nstructural properties of the nanowires. Changes in the\nelectrodeposition parameters like chemical composition,\ntemperature and pH of the electrolyte [15], deposition\ntime or electrodeposition voltage [16] will result in arrays\nof nanowires with different intrinsic magnetic properties.\nWe focus our study on the influences of the reduced\ndiameter-to-length ratio (d/L) on the magnetic effective\nanisotropies. Ourmainfindingisthat the otherwisecom-\nplex demagnetization treatment of the shape anisotropy\ncan be reduced to a much simpler expression for the co-\nercive fields of nanoscale Co arrays, which is probed ex-\nperimentally. The Co nanowires with lengths between\n1 and 6 µm have been electrodeposided into the pores\nof track-etched polycarbonate membranes with nominal\nporediametersof30and50nm. Previoustoelectrodepo-\nsition, the membranes were covered by a 100 nm Cu film\nby sputtering technique. This acts as an electrode during\nthe fabrication process. An agitation system was used in\norder to avoid hydrogen evolution over the sample and2\nto obtain a homogeneous growth inside the nanopores.\nCobalt nanowires were grown in a mixed solution [18]\n(pH=4.5) of CoSO 4·7H2O (252 g/L), H 3BO3(50g/L)\nand NaCl (7 g/L) at 25oC with a constant applied volt-\nage of -0.95 V. A saturated calomel electrode was used\nas reference in potentiostatic mode, and a Co film as\ncounter electrode. By measuring the experimental devel-\nopment of the deposition current in the electrochemical\ncell, we interrupted the growth process at different times\nbefore the complete filling of the nanopores. This allows\na reliable control of the average nanowires length. The\ncrystallographicstructureandmorphologyofthesamples\nwere investigated by x-ray diffraction and scanning elec-\ntron microscope (SEM). The field dependent magnetiza-\ntion hysteresis of the nanowires was measured at room\ntemperature in a LDJ Vibrating Sample Magnetometer\n(VSM) with a external magnetic field of 1 T applied par-\nallel (H||) and perpendicular (H⊥) to the nanowire sym-\nmetry axis. Temperature dependent measurements were\nperformed in a Quantum Design XL7 superconducting\nquantuminterferencedevicemagnetometerinthe(5-300)\nK range.\nFigures1aand1bshowthehighresolutionSEMmicro-\ngraphs of the membrane surface and nanowires morphol-\nogy, respectively. Both measurements were performed\nwith a JEOL Scanning Microscope JSM-6400 working at\n24 kV. Figure 1a corresponds to a polycarbonate mem-\nbranewith6 µmthicknessand50nmnominalporediam-\neter. Some defects are observed, probably due to stress-\ning effects, as well as a low ordering degree of nanopores.\nThe pore density and pore diameter distributions can\nbe estimated from this type of images, as it has been\ndescribed in details elsewhere [19]. The measured mean\nporedensity of(6.1 ±2.4)·108nanopores ·cm−2isin good\nagreement with the value of 6 ·108nanopores ·cm−2given\nbythe manufacturer. Figure1b showsConanowireswith\n50 nm nominal diameter and ∼3µm length after disso-\nlution of the polycarbonate membrane. While inside the\nmembrane all nanowires are parallel to each other, af-\nter its dissolution they reorganize in different positions\nall over the sample area. In this way, the length of the\nnanowires can be measured more accurately and com-\npared to the chronoamperometric measurements as ex-\nplained in Ref. [19]. Note that in the Fig. 1a, the ap-\nparent deviations from a circular shape of the nanowires\noccurs only near the surface. This is due to the fabrica-\ntion process of the polycarbonate membranes which typ-\nically leads to a conical opening of the nanopores which\nextends a few nanometers deep from the membrane sur-\nface. Therefore, we have taken particular care and ended\nthe deposition process before the nanopores were com-\npletely filled with the magnetic material.\nForthe x-raydiffraction(XRD) measurementswehave\nutilized a SIEMENS D-5000 diffractometer providing a\nCu-Kαradiation ( λ= 1.5418˚A) under working condi-\ntions of 40 kV and 30 mA. The measurements were done\nprior to dissolution of the polycarbonate membrane (as\nseen in Fig. 1a), when all nanowires are parallel to each\nFIG. 1: SEM images of (a) a small area of a track-etched\npolycarbonate membrane and (b) Co nanowires after dissolu-\ntion of the substrate. The images were recorded at 24KV and\n43000X and 6000X respectively.\nother and the sputtered Cu layer on one side of the mem-\nbrane was still present. Figure 2 shows the θ−2θdiffrac-\ntion patterns of Co nanowires with ∼6µm length and\nnominal pore diameters of 30 nm and 50 nm. Contribu-\ntions from the Cu substrate appear as intense peaks cor-\nrespondingto Cu-FCC (111) and (200)orientations. The\ntwo diffractogramsexhibit different crystalline structures\nas a function of the nanowire diameter. The diffraction\npattern of nanowires with 30 nm diameter is dominated\nby a highly textured Co-FCC phase oriented along the\n[111] and [200] directions. The nanowires with 50 nm\ndiameter exhibit, however, a more complicated diffrac-\ntion pattern which indicates that a mixture of Co-HCP\nand Co-FCC phases is present. This suggests that Co\nnanowires are composed of hcp and fcc crystalline Co\nsegments [20]. We basically observe two new promi-\nnent structures corresponding to a Co-HCP phase ori-\nented along the [100] and [101] directions. Their coex-\nistence gives rise to an asymmetric peak appearing near3\nthe expected angular positions of the Co-HCP (002) and\nCo-FCC (111) phases. The double structure behind this\npeak cannot be resolved in this case, indicating that the\ndegree of coexistence between the two crystalline phases\nis dominated by the Co-HCP contribution.\nIntensity (arb.units)\n525048464442 /s32/s67/s111\n/s32/s70/s67/s67/s32\n/s40/s49/s49/s49/s41 /s32/s67/s111\n/s32/s72/s67/s80/s32\n/s40/s49/s48/s48/s41/s32/s67/s111\n/s72/s67/s80/s32\n/s40/s49/s48/s49/s41\n Cu\nFCC \n(111) Cu\nFCC \n(200)\n2θ (deg) 50nm\n 30nm\n /s32/s67/s111\n/s32/s70/s67/s67/s32\n/s40/s50/s48/s48/s41 /s32/s67/s111\n/s32/s72/s67/s80\n/s40/s48/s48/s50/s41/s32/s32/s32/s32\n  Co\n FCC \n (200)    Co\n+ FCC \n   (111)\nFIG. 2: Comparison of the x-ray diffraction patterns of elec-\ntrodeposited Co nanowires with 30 nm and 50 nm nominal\ndiameter.\nIn addition, other features, as the one corresponding\nto Co-FCC (200) phase, are barely distinguishable from\nthe background signal. Since the final texture of the\nnanowires also depends on the plating procedure, it is\nstill unclear why the amount of FCC phase increases\nwith decreasing nanowire diameter. In recent studies,\na coexistence of two different Co phases with different\norientations for nanowires with 200 nm in diameter was\nalso found [20]. A transition to a Co-HCP crystalline\nstructure was shown for nanowires with 65 nm in diame-\nter and a pH and diameter-dependent phase diagram for\nelectroplated Co nanowires was proposed. Our results\ngive complementary information to the proposed phase\ndiagram in Ref. [20], indicating that by further reduction\nof the nanowire diameter a second transition to a Co-\nFCC phase occurs for nanowire diameters of ∼30 nm.\nOther investigations of Co nanoparticles [21] show that\nthe presence of a HCP phase or a FCC phase can be\nrelated to the particle diameter. In that case, the transi-\ntion from HCP to FCC was ascribed to a size effect due\nto the lower surface energy of the FCC phase.\nFigures 3a-f show the resulting hysteresis loops mea-\nsured for nanowire arrays with different diameters and\nlengths. Figures 3g-h show the extracted coercive field\nvalues and the corresponding theoretical fits (see below)\nas a function of the nanowire diameter-to-length ratio\nfor both (H||) and (H⊥) externally applied fields, respec-\ntively. A better understanding of the observed depen-\ndence of the coercive field with the nanoscale dimensionsobserved in figures 3g-h can only be achieved by analyz-\ning the effects of the different anisotropy contributions\nto the total anisotropy of the system, as it will be shown\nin the following. The first observation is a pronounced\nreduction of the coercivity and remanent magnetization\nvalues for nanowires with 50 nm in diameter (Figs. 3d-f).\nBesides, by comparingthe axialand transversehysteresis\nloopsforeachcase,weobservehowthe effectivemagnetic\nanisotropyalongthenanowireaxisincreaseswithincreas-\ning length for both diameters. The longer the nanowires,\nthe larger the remanent magnetization and coercive field\nvalues for an applied field parallel to the nanowire axis\n(H||). Thiseffectappearsmorepronouncedfornanowires\nwith 30 nm in diameter (Figs. 3a-c).\nWe argue below that the effective magnetic anisotropy\nis mainly oriented along the nanowire axis in all cases,\nand that the shape anisotropy of the nanowire arrays is\nplaying a more dominant role than the dipolar interac-\ntion for our particular system. The shape anisotropy of\naninfinitelylongwirewouldresultinasquaredhysteresis\nloop exhibiting a finite coercive field for an applied field\nalong the wire H||and zero coercive field for a perpen-\ndicular applied field to the wire H⊥. Within the Stoner-\nWohlfarthmodel, aparticularcaseisthesocalledprolate\nspheroid geometry of magnetic objects. If the ellipsoid\nis uniformly magnetized, the magnetic field inside the el-\nlipsoid depends directly on the demagnetization tensor\n(/vectorHin∼−N/vectorM). Since the trace of the tensor N must be\n1, for a limiting case of a infinitely long cylinder in which\nthe shape anisotropy plays the dominant role one would\nobtainNx=Ny= 1/2 andNz= 0. In a general case,\nthe demagnetizing factor along the long axis of the el-\nlipsoid (or, by analogy, along the cylinder ) depends on\nsecond order terms of the type ( d/L)2and it approaches\n0 when L → ∞. Since/vectorHc∼/vectorHa−N/vectorM, this leads to the\nfollowing expectation for the coercive field: /vectorHc→/vectorHa\nwhen N→0, where Hais the intrinsic anisotropy field.\nTherefore, a system with finite dimensions which is dom-\ninated by shape anisotropy ( d≪L) exhibits a finite Nz\n(Nz/negationslash= 0). Our observations in Fig. 3, where coercive\nfield values for H||are larger than the for H⊥, are fully\nconsistent with shape anisotropy origin of the enhanced\ncoercivity as discussed qualitatively within the Stoner-\nWohlfart model. Nevertheless, other contributions to the\neffective anisotropy of the whole nanowire array, other\nthan the shape induced anisotropy, may influence the\nobserved magnetic behavior: dipolar interactions among\nnanowires, magnetocrystallineanisotropyand magnetoe-\nlasticanisotropy[22]. Asit hasbeen shownfornanowires\ngrown in alumina membranes, the former certainly de-\npends strongly on the inter-wire distances and typically\nmanifests through a decreasing coercive field along H||\nwith an increasing ratio of nanowire diameter to inter-\nwire spacing [23]. Although the use of randomly dis-\ntributed array of nanopores which is characteristic of the\npolycarbonate membranes does not allow an exhaustive\ncontrol of the inter-wire distances, an average value of\n∼500 nm can be deduced, which is one order of magni-4\nFIG. 3: Room temperature field dependent magnetization hyst eresis of Co nanowires with different lengths and diameters.\nFigs. 3a, 3b and 3c: Nanowires with 30 nm in diameter and lengt hs of∼4.4µm,∼3.3µm,∼2.3µm, respectively; figs. 3d,\n3e and 3f: Nanowires with 50 nm in diameter and ∼3.3µm,∼1.5µm,∼1µm in length. Figs 3g and 3h: General dependence\nof the coercive field as a function of the ratio (d/L) of the nan owires when the magnetic field is applied parallel (H||) and\nperpendicular (H⊥) to the nanowire axis.\ntude larger than the typical nanowire diameter. There-\nfore, for our case the dipolar interactions have a minor\neffect for sufficiently long nanowires. A reminiscent in-\nfluence of the dipolar interaction on the magnetic behav-\nior would manifest through a small contribution to the\nreduced remanent magnetization for shorter wires. Fur-\nther discussion about the role of the magnetocrystalline\nanisotropy is provided below by interpreting the mea-\nsured hysteresis loops in combination with the different\norientations of the crystalline structure observed in the\nx-ray diffraction patterns of Fig. 2.\nFor nanowires which are 30 nm in diameter, the [111]\ndirection of the Co-FCC phase is mainly oriented along\nthe axis of the nanowires, with the basal (111) planes\nbeing parallel to the membrane plane. For nanowires\nwhich are 50 nm in diameter, the Co-HCP [100] and\n[002] directions are oriented perpendicular and paral-\nlel to the nanowire axis, respectively. Moreover, the\nc-axis of the HCP (101) phase makes an angle of 32o\nwith respect to the nanowire axis. As a result, the ef-\nfective crystalline alignment resulting from the differ-\nent Co-HCP orientations observed in the diffraction pat-\ntern would lead to a reinforcement of the total mag-\nnetocrystalline anisotropy of the system along the H⊥\ndirection. For a Co-FCC(111) phase oriented alongthe nanowire axis, the magnetocrystalline anisotropy en-\nergy density K 1=6.3·105erg/cm3is almost one order\nof magnitude smaller than the shape anisotropy energy\ndensity K S=π·M2\nS=6.0·106erg/cm3, whereas for a Co-\nHCP structure, they are of the same order of magnitude\n(K1=5·106erg/cm3∼KS=π·M2\nS=6.0·106erg/cm3) [24].\nHence, fornanowireswith 50nmdiameter, wecanexpect\na more pronounced interplay of both magnetocrystalline\nand shape anisotropies that results in a slightly weaker\neffective anisotropy along the nanowire axis. This leads\nto a further reduction of the coercive field and remanent\nmagnetizationfor H||as compared to the 30nm wires. In\nthis respect, the reduced coercivity for larger diameters\ndepends not only on the geometrical properties, i.e, the\nnanowire diameter dependence of the coercive field [25],\nbut also on the partial influence of the crystalline orien-\ntation of the Co-HCP phase, i. e., on magnetocrystalline\nanisotropy.\nAs to further investigate the validity of the aforemen-\ntioned effects, i.e, the interplay between magnetocrys-\ntalline and shape anisotropies, we present in Fig. 4 tem-\nperature dependent measurements for nanowires with 30\nand 50 nm diameter and ∼4 and∼3µm in length, re-\nspectively. Such type of measurements allow us to deter-\nmine if other contributions, particularly the ones arising5\nFIG. 4: Field dependent magnetization hysteresis of Co nano wires at different temperatures. Figs. 3a, 3b and 3c: Nanowir es\nwith 30 nm in diameter and ∼4µm in length; figs. 3d, 3e and 3f: Nanowires with 50 nm in diamete r and∼3µm in length.\nFigs 3g and 3h: General dependence of the coercive and irreve rsible fields as a function of temperature for both parallel ( H||)\nand perpendicular (H⊥) appied magnetic fields. Dotted and dashed lines result from linear fits of the experimental data for\nnanowires with nominal diameters of 30 and 50 nm respectivel y.\nfrom magnetoelastic effects may influence the magnetic\nbehavior of the nanowires system at room temperature.\nFigures 4a-fshow the resulting hysteresis loops measured\nfor both systems of nanowire arrays at different temper-\natures. Figures 4g-h show the extracted coercive field\n(Hc) and irreversible switching field ( Hirr) values as a\nfunction of temperature for both parallel (H||) and per-\npendicular (H⊥) applied magnetic fields. Hirris the field\nwhere the magnetization changes irreversibly (i.e, where\nthe hysteresis opens). In Figures 4a-f we do not observe\nsignificant changes in the hysteresis loops when the tem-\nperature decreases down to 5 K. The main observation is\na quasi-monotonic decrease of HcandHirrwith increas-\ning temperature, as summarized in figures 4g-h. This\neffect which is observed for both applied field directions,\nappears to be less pronounced in the case of the Hcval-\nues extracted for H⊥. As shown in figures 4g-h, we per-\nformed linear fits to the experimental data from which\nwe estimate the relative changes in the angle of orienta-\ntion of the anisotropy axis (Θ), HcandHirr. We obtain\nΘ∼35oand a partial reorientation of ∼6otowards the\nnanowire axis when the temperature is decreased. For\nH||, the changes of Hcwith respect to its low temper-\nature value are ∼23% and ∼38% for nanowires with 30\nand 50 nm in diameter, respectively. For an applied fieldH⊥, these changes are ∼15% and ∼24%, respectively.\nThe relative increase in Hirris∼47% in all cases. Note\nthat in general, (i) this effect is more pronounced in the\nHirrvalues and that (ii) Hcis larger for H||than for\nH⊥, whereas Hirris smaller for H||than for H⊥. This\ndifferences are expected from the Sthoner-Wohlfarth the-\nory, and they can be qualitatively explained by consid-\nering the calculated azimuthal dependence of both Hirr\nandHcfor a magnetic system with uniaxial anisotropy\n[26]. Our results indicate that only a moderate increase\nof the effective anisotropy of the nanowire system along\nthenanowireaxisoccursatlowtemperatures,resultingin\na rather small reorientation of the anisotropy axis along\nthe nanowires. Previous studies of Ni nanowires showed\nthat magnetoelastic effects can have profound influences\non the magnetic properties of the arrays [27, 28], typi-\ncally leading to a reorientation of the magnetic easy axis\nof the nanowire system from the parallel to perpendic-\nular configuration with decreasing temperature. At low\ntemperatures, the nanowires are under stress due to the\nlarge mismatch between the thermal expansion coeffi-\ncients of Co ( αCo∼13·10−6K−1) and the polycarbonate\nmembrane ( αPolyc∼67·10−6K−1). Since αPolyc>αCo,\nthe polycarbonate template tends to contract more than\nthe Co during cooling, resulting in a transverse com-6\npressive force which acts perpendicular to the nanowire\naxis and leads to an expansion of the nanowires along\ntheir axis. The axial strain produced in this process\ncan be calculated as ǫ||=-ν∆T(αCo-αPolyc), where νis\nthe Poisson’s ratio of Co ( νCo=0.31) and ∆T=295 K\nwhen the temperature varies from 300 to 5 K. This is\nequivalent to an axial tensile stress of σ||=ECoǫ||∼1.03\nGPa, where E Co∼209 GPa is the Young’s modulus of\nCo. Considering that the saturation magnetostriction\nconstant of Co is negative ( λCo\ns∼-55·10−6) [29, 30], such\nan axial effective tensile stress will decrease the effec-\ntive anisotropy along the nanowire axis and, therefore\ntheHcandHirrvalues, in contrast to our observations.\nIn some instances, when the magnetoelastic effects are\nvery strong [30], the interplay between magnetoelastic\nand shape anisotropies may lead to a reorientation of\nthe anisotropy axis at certain crossover temperature be-\nlow which H⊥\nc>H||\nc. The maximum contribution of\nthe transverse magnetoelastic anisotropy energy density\ncan be estimated as K me=3λCo\nsσ||/2∼-8.4·105erg/cm3\nwhich is one order of magnitude smaller than the shape\nanisotropy energy density K S=6.0·106erg/cm3. There-\nfore, the effective magnetoelastic anisotropy is not play-\ning a dominant role in the remagnetization processes of\nindividual Co nanowires. Note that the calculated value\nof Kmerepresents an upper limit since we assume per-\nfect adhesion of the nanowires to the pore walls. In this\nsimplified calculation, we have neglected (i) the weak\ndependence of the E Cowith the nanowire length, (ii)\nthe losses in the compression energy due to the large\nmismatch between the Young’s modulus of Co and the\npolycarbonate membrane (E Polyc∼2GPa<<ECo),(iii)\nnonlinear effects in the temperature dependent expan-\nsion coefficients and (iv) other second order effects in the\ntemperature dependence of the elastic properties. We\nattribute the observed effects to a preferential increase\nof the effective magnetocrystalline anisotropy with de-\ncreasing temperature. While cooling down to 5 K, the\nHcandHirrare increasing due to the thermal energy\nwhich acts against the anisotropy energy. At low tem-\nperatures, the relative changes in HcandHirrwill de-\npend on the size and temperature dependent behavior\nof the magnetocrystalline anisotropy, and therefore on\nthe orientation of the different crystalline structures ap-\npearing in the x-ray diffraction patterns of Fig. 2. For\nnanowires with 30 nm in diameter, the relative increase\nof the magnetocrystalline anisotropy along the nanowire\naxis is∼27%, whereas ∼40% for nanowires with 50 nm\nin diameter. This indicates that the dominant contri-\nbutions in the diffraction patterns are the Co-HCP(002)\nand Co-FCC(111) phases for nanowires with 50 and 30\nnm in diameter, respectively.\nIn order to gain more insight on the interplay be-\ntween both magnetocrystalline and shape anisotropies,\nwe would like to come back to the results shown in Figs.\n3g and 3h and provide further theoretical analysis of the\nobserved room temperature dependence of the coercive\nfield as a function of the diameter-to-length ratio (d/L)of the nanowires. Such type of analysis gives more infor-\nmation about the origin of the dominant anisotropy con-\ntributions to the total magnetic anisotropy of the system\nand, particularly, how strongly the remagnetization pro-\ncess of the nanowires depend on the shape anisotropy,\ni.e, on the nanowire dimensions. Based on the afore-\nmentioned argumentations, our theoretical analysis does\nnot consider contributions to the effective anisotropy of\nthe nanowire system arising from magnetoelastic effects\nand dipolar interactions amongnanowires. As previously\nmentioned, the results of Figs. 3g and 3h are presented\ntogether with a fit function H||,⊥\nc=f||,⊥(d/L) which\nexplicitly shows a size-dependent behavior as it will be\ndeduced in the following. The physical origin of this de-\npendence are demagnetizing fields [31, 32], which give\nrise to the shape anisotropy of the nanowires and, there-\nfore, it allow us to check the validity of a simple model\nin which the shape anisotropy is the dominant contribu-\ntion to the observed size-dependent coercive field behav-\nior. The relation between the demagnetizing factors N||\nor N⊥and the structure dimensions can be expressed in\nterms of the hypergeometric Gauss function2F1[33]. In\nthe limit in which d≪Land taking into account that any\ngiven finite circular cylinder with diameter d and length\nL possesses rotational symmetry about its geometrical\naxis, for κ=d/L, N||can be written as [34]:\nN||=4\n3πκ−8κ2+O(κ4) (1)\nwhich represents, a second order dependence of the de-\nmagnetizing factor on the diameter-to-length ratio of the\nnanowires. We can considerthat the expressionsfor both\ncoercive fields H c||and H c⊥are given by [26]:\nH||\nc=2Keff\nµ0Ms|cosΘ|;H⊥\nc=2Keff\nµ0Ms|sinΘ|(2)\nwithKeffbeing the effective anisotropy constant of the\nferromagnet, Msthesaturationmagnetization,andΘthe\nanglebetween the applied field and the anisotropyaxisof\nthe system. Considering that N||−N⊥= (3N||−1)/2,\na widely used phenomenological expression for Keffis\ngiven by:\nǫkKM−(1−3N||)µ0Ms2\n4ifd < dc (3)\nǫkKM+c(N||)A\n(d/2)2−N||µ0Ms2\n2ifd > dc(4)\nwhereǫkis the real-structure-dependent Kronm¨ uller pa-\nrameter [35], KMis the effective magnetocrystalline\nanisotropy constant, d cis the critical coherence diameter\nandAistheexchangestiffnessconstant(A=30 ·10−12J/m\nfor Co). If d <dc, then the dominant exchange interac-\ntion yields coherent (uniform) rotation (equation (3)),\nwhereas if d >dcmagnetostatic interactions give rise to\ncurling reversal (equation (4)). The critical diameter,\ndefined as d c=3.68/radicalbig\nA/πM s2yields to d c∼32 nm for7\nCo. The second term on the right hand of equation (3)\nrepresents the shape anisotropy contribution to the ef-\nfective anisotropy of the system, while in equation (4)\nthe exchange term partly compensates for the absence\nof shape anisotropy in a proper sense. Considering that\nc(N||) = (Msdc)2N⊥and by substituting (3) or (4) into\n(2) and then introducing (1), after some manipulation\nthe coercive field becomes:\nH||,⊥\nc=α||,⊥−β||,⊥/parenleftbiggd\nL/parenrightbigg\n+γ||,⊥/parenleftbiggd\nL/parenrightbigg2\n(5)\nThe relation between the physical constants and the fit-\nted parameters α||,⊥,β||,⊥andγ||,⊥can be written as\nfollows:\n(α||,β||,γ||)\n|cosΘ|=(α⊥,β⊥,γ⊥)\n|sinΘ|= (A,B,C)\nwith A, B and C given by:\nA=α/parenleftbigg2ǫkKM\nµ0Ms+gMs\n2/parenrightbigg\n;B=4βMs\n3π/parenleftBig\n1+g\n2/parenrightBig\n;\nC= 8γMs/parenleftBig\n1+g\n2/parenrightBig\nwithg=/parenleftbiggdc\nd/parenrightbigg2\nwhereg=1forcoherentrotation. We haveintroducedthe\nparameters α,βandγin order to account for quantita-\ntive deviations from the situation in which the magneti-\nzation reversal processes are driven by coherent rotation\nor curling mechanisms. Certainly, since the diameters of\nthefabricatednanowiresareneartothelimitforcoherent\nrotation it is not straightforward to deduce which mag-\nnetization reversal process is the preferred mechanism.\nFor curling processes and within the present approxima-\ntion, we assume that dcis not decreasing strongly with\nincreasing d/L. That means that 0 .20≤g/2≤0.56 for\nnanowires with 50 and 30 nm respectively. The results\nof the fit using the function (5) show qualitative agree-\nment to the experiment and are represented in Figs. 3g\nand 3h. The fitted parameters are given in Table I. From\nthe experimental data obtained at maximum and min-\nimum d/L values we estimate a nearly constant angle\nof rotation respect to the nanowire axis of Θ ∼32o, and\nits variation is below 8ofor a decreasing structure size.\nIn agreement to it, the theoretical fit gives Θ=(28 ±3)o.\nTherefore, we suggest that the observed changes in the\ncoercive field are mainly due to an increase of the ef-\nfective magnetic anisotropy Keffof the system, rather\nthan due to a rotation of the anisotropy axis towards\nthe nanowire axis. Considering ǫk=1 and introducing\nMs(Co)∼1424Oe, for α=1 and g=1 we estimated K M\n∼(1.8±0.4)·106erg/cm3, in agreement with the aver-\nage value of ∼2.8·106erg/cm3obtained from the mag-\nnetocrystalline anisotropy energy densities of Co-FCC\nand HCP crystalline structures. The estimated value\nof KMindicates that the changes in the magnetocrys-\ntalline anisotropy density of the system are mostly due\nto a mixture of both FCC and HCP crystal orientations,α||,⊥(Oe)β||,⊥(Oe) γ||,⊥(Oe)\nH||805±108(2.5±0.6)·104(2.7±1.3)·105\nH⊥472±60(1.2±0.3)·104(1.4±0.6)·105\nTABLEI:Valuesofthefittedparameters α||,⊥,β||,⊥andγ||,⊥\nfor both applied field directions H||and H⊥respectively.\nin agreement with the x-ray diffraction patterns of Fig.2.\nFor g=1, the deviations from the first and second or-\nder terms yield to β∼(30±8) andγ∼(18±9), whereas\nfor 0.20≤g/2≤0.56,β∼(34±10) and γ∼(20±11).\nAlthough a qualitative agreement is reached in the gen-\neral dependence of the coercive field with d/L, this de-\nviations indicate that the Stoner-Wohlfarth approxima-\ntion is overestimating the coercive field values by a fac-\ntor of∼3 for both applied directions. In this respect,\nit was shown that at the wire ends a butterfly-type ar-\nrangement of the magnetization exists which reduces the\nswitching field considerably [36]. Micromagnetic simula-\ntions for Ni nanowires of 40 nm diameter, also showed\nthat the nucleation field can be typically reduced by a\nfactor of 2 [37]. In addition, in the case of an array of\nnanowires, collective demagnetization modes have to be\ntaken into account which lead to a further decrease of\nthe coercive field. The reduced squareness observed in\nthe hysteresis loops of Fig. 3 might result from the ex-\nistence of polycrystalline and surface-related structural\nimperfections along the nanowires, leading to a magnetic\nlocalization of the reversal modes [38]. The coherent-\nrotation and curling modes are delocalized, which means\nthat the spatial variations of the magnetization along the\nnanowirearesmall. However,localvariationsofthe mag-\nnetization cost some exchange energy but they may be\nenergetically favorable from the point of view of a local\nanisotropy, Keff(r). This competition leads not only to\na reduction of the nucleation field but also to a local-\nization of the magnetization reversal modes and to the\nformation of magnetic domains along the wires. Further\nunderstanding of the real-space character of the mag-\nnetization mechanisms can be achieved by considering\ncooperative effects between coherent-rotation and curl-\ning modes [39]. Actually this type of processes are of\ngreat importance in advanced technology, because they\nlead to the formation of interactive magnetic domains\nwhich may improve the thermal stability of the samples\nor reduce the storage densities in the case of extended\nmagnets or ultrathin films.\nIn conclusion, we have shown that the magnetic prop-\nerties of the fabricated Co nanowire arrays are domi-\nnated by the shape anisotropy contributions more than\nthe magnetocrystallineanisotropy, magnetoelasticeffects\nand dipolar interactions among nanowires. A partial in-\nfluence of the different crystallographic orientations on\nthe magnetic hysteresis loops at room temperature is de-\nduced by analyzing the x-ray diffraction patterns. Low\ntemperature measurements indicate that the magnetoe-\nlastic anisotropy is not playing a dominant role in the8\nremagnetization process of the nanowires. The changes\nare attributed to a temperature dependent behavior of\nthe magnetocrystalline anisotropy. 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/D8/D0/DD /D7/CT/CP/D0/CT/CS /CX/D2 /CP/D2 /CT/DA /CP\r/D9/CP/D8/CT/CS /D5/D9/CP/D6/D8/DE/D8/D9/CQ /CT/BA /CC/CW/CT /D1/CX/DC/D8/D9/D6/CT /DB /CP/D7 /CW/CT/CP/D8/CT/CS /D9/D4 /D8/D3 /BD/BD/BC/BCo/BV /CP/D2/CS /D1/CP/CX/D2/B9/D8/CP/CX/D2/CT/CS /CP/D8 /D8/CW/CP/D8 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CU/D3/D6 /BF/BC /CW/D6/D7 /B8 /CU/D3/D0/D0/D3 /DB /CT/CS /CQ /DD /D7/D0/D3 /DB\r/D3 /D3/D0/CX/D2/CV /B4/BDo/BV /BB /CW/B5 /D8/D3 /BH/BH/BCo/BV /CP/D2/CS /D8/CW/CT/D2 /D6/CP/D4/CX/CS/D0/DD \r/D3 /D3/D0/CT/CS /CS/D3 /DB/D2/D8/D3 /D6/D3 /D3/D1 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /CC/CW/CT \r/D6/DD/D7/D8/CP/D0/D7 /DB /CT/D6/CT /D7/CT/D4/CP/D6/CP/D8/CT/CS /CU/D6/D3/D1/D8/CW/CT /AT/D9/DC /CQ /DD /D1/CT/CP/D2/D7 /D3/CU \r/CT/D2 /D8/D6/CX/CU/D9/CV/CX/D2/CV/BA /CC/CW/CT /D8 /DD/D4/CX\r/CP/D0 /D7/CX/DE/CT /D3/CU /D8/CW/CT/D3/CQ/D8/CP/CX/D2/CT/CS \r/D6/DD/D7/D8/CP/D0/D7 /DB /CP/D7 /BJ× /BG× /BD /D1/D13/BA /CC/CW/CT \r/D6/DD/D7/D8/CP/D0/D7 /D2 /D9\r/D0/CT/B9/CP/D8/CT/CS /D3/D2 /D8/CW/CT /DB /CP/D0/D0/D7 /D3/CU \r/D6/D9\r/CX/CQ/D0/CT /CP/D2/CS /CU/D3/D6/D1/CT/CS /CP/D7 /D8/CW/CX/D2 /D4/D0/CP/D8/CT/D0/CT/D8/D7/D7/D8/CX\r /CZ/CX/D2/CV /D8/D3/CV/CT/D8/CW/CT/D6 /D8/D3 /CU/D3/D6/D1 /CP /CQ/CX/CV /D7/CX/D2/CV/D0/CT \r/D6/DD/D7/D8/CP/D0/BA /BT/D2 /CT/D2/CT/D6/CV/DD/CS/CX/D7/D4 /CT/D6/D7/CX/DA /CT /DC/B9/D6/CP /DD /CP/D2/CP/D0/DD/D7/CX/D7 /BX/BW /BT/CG /DB /CP/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /D3/D2 /CP/D0/D0 /D3/CU/D8/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /D7/CX/D2/CV/D0/CT \r/D6/DD/D7/D8/CP/D0/D7 /D8/D3 /CX/CS/CT/D2 /D8/CX/CU/DD /D8/CW/CT/CX/D6 /D4/CW/CP/D7/CT/BA /CC/CW/CT/BX/BW /BT/CG /D6/CT/D7/D9/D0/D8/D7 \r/D3/D2/AS/D6/D1/CT/CS /D8/CW/CT \r/D6/DD/D7/D8/CP/D0/D7 /D8/D3 /CQ /CT /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/B9/D7/CX/D8/CX/D3/D2 /BD/BM/BE/BM/BE/BA /C1/D2 /D3/D9/D6 /D4/D6/CT/D7/CT/D2 /D8 /D7/D8/D9/CS/DD /DB /CT /CW/CP /DA /CT /D9/D7/CT/CS /D8/CW/CT /D7/CX/D2/CV/D0/CT\r/D6/DD/D7/D8/CP/D0/D7 /CV/D6/D3 /DB/D2 /CU/D6/D3/D1 /D8/CW/CT /BU/CX /AT/D9/DC/BA /CC /D3 \r /CW/CT\r /CZ /CU/D3/D6 /D8/CW/CT /D4/CW/CP/D7/CT/D4/D9/D6/CX/D8 /DD /B8 /D4 /D3 /DB/CS/CT/D6 /DC/B9/D6/CP /DD /CS/CX/AR/D6/CP\r/D8/CX/D3/D2 /D4/CP/D8/D8/CT/D6/D2 /D3/CU /D8/CW/CT/D7/CT \r/D3/D1/B9/D4 /D3/D9/D2/CS/D7 /DB /CT/D6/CT /D6/CT\r/D3/D6/CS/CT/CS /CQ /DD /D4 /D3 /DB/CS/CT/D6/CX/D2/CV /CP /CU/CT/DB /D7/D1/CP/D0/D0 /D4/CX/CT\r/CT/D7/D3/CU /D7/CX/D2/CV/D0/CT \r/D6/DD/D7/D8/CP/D0/D7/BA /CC/CW/CT /CV/D6/D3 /DB/D2 \r/D6/DD/D7/D8/CP/D0/D7 /DB /CT/D6/CT /D8/CW/CT/D2 /D3/D6/CX/CT/D2 /D8/CT/CS/CP/D0/D3/D2/CV /D8/CW/CT /D4/D6/CX/D2\r/CX/D4/CP/D0 \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /CQ /DD /D1/CT/CP/D2/D7/D3/CU /C4/CP/D9/CT /CS/CX/AR/D6/CP\r/D8/CX/D3/D2/BA /CF /CT/D0/D0 /CS/CT/AS/D2/CT/CS /C4/CP/D9/CT /CS/CX/AR/D6/CP\r/D8/CX/D3/D2 /D7/D4 /D3/D8/D7/B8/D8/D3/CV/CT/D8/CW/CT/D6 /DB/CX/D8/CW /D8/CW/CT /D8/CT/D8/D6/CP/CV/D3/D2/CP/D0 /D7/DD/D1/D1/CT/D8/D6/DD /D4/CP/D8/D8/CT/D6/D2/B8 /CX/D2/CS/CX\r/CP/D8/CT/CS/D8/CW/CT /CV/D3 /D3 /CS /D5/D9/CP/D0/CX/D8 /DD /D3/CU /D7/CX/D2/CV/D0/CT \r/D6/DD/D7/D8/CP/D0/D7/BA /CC/CW/CT \r/D6/DD/D7/D8/CP/D0/D7 /DB /CT/D6/CT /D8/CW/CT/D2\r/D9/D8 /CP/D0/D3/D2/CV /D8/CW/CT \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /D9/D7/CX/D2/CV /CP /D7/D4/CP/D6/CZ/CT/D6/D3/D7/CX/D3/D2 \r/D9/D8/D8/CX/D2/CV /D1/CP\r /CW/CX/D2/CT /D8/D3 /D7/D8/D9/CS/DD /D8/CW/CT/CX/D6 /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX\r /D4/D6/D3/D4/B9/CT/D6/D8/CX/CT/D7/BA /CC/CW/CT /BW/BV /D1/CP/CV/D2/CT/D8/CX\r /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /DB /CT/D6/CT /D4 /CT/D6/CU/D3/D6/D1/CT/CS/CX/D2 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D6/CP/D2/CV/CT /BD/BA/BK /B9 /BF/BC/BC /C3 /CP/D2/CS /CX/D2 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/AS/CT/D0/CS/D7 /D9/D4 /D8/D3 /BD/BE/BC /C3/C7/CT /CP/D0/D3/D2/CV /D8/CW/CT /D8 /DB /D3 /D4/D6/CX/D2\r/CX/D4/CP/D0 /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/D9/D7/CX/D2/CV /CP /CB/C9/CD/C1/BW /D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/CT/D6 /B4/C9/D9/CP/D2 /D8/D9/D1 /BW/CT/D7/CX/CV/D2/B5 /CP/D2/CS/CP /DA/CX/CQ/D6/CP/D8/CX/D2/CV /D7/CP/D1/D4/D0/CT /D1/CP/CV/D2/CT/D8/D3/D1/CT/D8/CT/D6 /B4/CE/CB/C5/B8 /C7/DC/CU/D3/D6/CS /C1/D2/D7/D8/D6/D9/B9/D1/CT/D2 /D8/D7/B5/BA /CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /CP/D2/CS /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /DB /CT/D6/CT /D1/CT/CP/D7/D9/D6/CT/CS/D9/D7/CX/D2/CV /CP /D4/CW /DD/D7/CX\r/CP/D0 /D4/D6/D3/D4 /CT/D6/D8 /DD /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8 /D7/DD/D7/D8/CT/D1 /B4/C8/C8/C5/CB/B8/C9/D9/CP/D2 /D8/D9/D1 /BW/CT/D7/CX/CV/D2/B5/BA/C1 /C1 /C1/BA /CA/BX/CB/CD/C4 /CC/CB /BT/C6/BW /BW/C1/CB/BV/CD/CB/CB/C1/C7/C6/CC/CW/CT /CA/BT/D92\n/BZ/CT2\n/B4/CA /BP /C4/CP/B8 /BV/CT /CP/D2/CS /C8/D6 /B5 \r/D3/D1/D4 /D3/D9/D2/CS/D7 /CU/D3/D6/D1/CX/D2 /CP /D8/CT/D8/D6/CP/CV/D3/D2/CP/D0 /D7/D8/D6/D9\r/D8/D9/D6/CT /DB/CX/D8/CW /CP /D7/D4/CP\r/CT /CV/D6/D3/D9/D4 /C1/BG/BB/D1/D1/D1 /BA/CC /D3 \r/D3/D2/AS/D6/D1 /D8/CW/CT /D4/CW/CP/D7/CT /CW/D3/D1/D3/CV/CT/D2/CT/CX/D8 /DD /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS/D7/DB/CX/D8/CW /D4/D6/D3/D4 /CT/D6 /D0/CP/D8/D8/CX\r/CT /CP/D2/CS \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/B8 /CP/CA/CX/CT/D8/DA /CT/D0/CS /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS /DC/B9/D6/CP /DD /D4/CP/D8/D8/CT/D6/D2 /D3/CU /D8/CW/CT/D8/CW/D6/CT/CT \r/D3/D1/D4 /D3/D9/D2/CS/D7 /DB /CP/D7 \r/CP/D6/D6/CX/CT/CS /D3/D9/D8 /D9/D7/CX/D2/CV /D8/CW/CT /BY/CD/C4/C4/C8/CA /C7/BY/D4/D6/D3/CV/D6/CP/D1 /CJ/BD/BJ ℄/BA /CC/CW/CT /D0/CP/D8/D8/CX\r/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D2/CS /D8/CW/CT /D9/D2/CX/D8 \r/CT/D0/D0/DA /D3/D0/D9/D1/CT /D8/CW /D9/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CP/D6/CT /D0/CX/D7/D8/CT/CS /CX/D2 /CC /CP/CQ/D0/CT /C1 /CP/D2/CS /CP /D6/CT/D4/D6/CT/B9/D7/CT/D2 /D8/CP/D8/CX/DA /CT /CA/CX/CT/D8/DA /CT/D0/CS /D6/CT/AS/D2/CT/CS /D4/D0/D3/D8 /D3/CU /BV/CT/BT/D92\n/BZ/CT2\n/CX/D7 /D7/CW/D3 /DB/D2 /CX/D2/BY/CX/CV/BA /BD /BA /CF /CT /D3/CQ/D8/CP/CX/D2/CT/CS /CPχ2/DA /CP/D0/D9/CT /D3/CU /BE/BA/BI/B8 /CV/D3 /D3 /CS/D2/CT/D7/D7 /D3/CU /AS/D8 /D3/CU8\n6\n4\n2\n0\n-2  Intensity (arb. units)\n8070605040302010\n 2θ (degree) Observed\n Calculated\n Difference\n Bragg Positions\n***CeAu2Ge2/BY/CX/CV/D9/D6/CT /BD/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /C8 /D3 /DB/CS/CT/D6 /DC/B9/D6/CP /DD /CS/CX/AR/D6/CP\r/D8/CX/D3/D2 /D4/CP/D8/D8/CT/D6/D2/D6/CT\r/D3/D6/CS/CT/CS /CU/D3/D6 \r/D6/D9/D7/CW/CT/CS /D7/CX/D2/CV/D0/CT \r/D6/DD/D7/D8/CP/D0/D7 /D3/CU /BV/CT/BT/D9 2\n/BZ/CT2\n/CP/D8 /D6/D3 /D3/D1/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /CC/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /CS/CP/D8/CP/D4 /D3/CX/D2 /D8/D7 /CX/D7 /D8/CW/CT /CA/CX/CT/D8/DA /CT/D0/CS /D6/CT/AS/D2/CT/D1/CT/D2 /D8 /D4/D6/D3/AS/D0/CT \r/CP/D0\r/D9/D0/CP/D8/CT/CS /CU/D3/D6 /D8/CW/CT/D8/CT/D8/D6/CP/CV/D3/D2/CP/D0 /BV/CT/BT/D9 2\n/BZ/CT2\n/BA /CC/CW/CT /D7/D8/CP/D6/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /D8/CW/CT /DC/B9/D6/CP /DD /D4 /CT/CP/CZ/D7\r/D3/D6/D6/CT/D7/D4 /D3/D2/CS/CX/D2/CV /D8/D3 /BU/CX/BA/BV/D3/D1/D4 /D3/D9/D2/CS /CP /B4/FH/B5 \r /B4/FH/B5 /CE /B4/FH3/B5 /CCN\n/B4/C3/B5/C4/CP/BT/D9 2\n/BZ/CT2\n/BG/BA/BG/BE/BE /BD/BC/BA/BG/BH /BE/BC/BG/BA/BF /C8/B9/C8/BV/CT/BT/D9 2\n/BZ/CT2\n/BG/BA/BF/BK/BH /BD/BC/BA/BG/BG/BG /BE/BC/BC/BA/BK /BD/BF/BA/BH/C8/D6/BT/D92\n/BZ/CT2\n/BG/BA/BF/BI/BI /BD/BC/BA/BG/BG/BF /BD/BL/BL /BL/BA/BC/CC /CP/CQ/D0/CT /C1/BM /C4/CP/D8/D8/CX\r/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CU /CA/BT/D92\n/BZ/CT2\n\r/D3/D1/D4 /D3/D9/D2/CS/D7 /DB/CX/D8/CW /D9/D2/CX/D8\r/CT/D0/D0 /DA /D3/D0/D9/D1/CT /CP/D2/CS /C6`e /CT/D0 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /C8/B9/C8/BM /C8 /CP/D9/D0/CX /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r/BA/BD/BA/BJ/B8 /CP/D2/CS /BU/D6/CP/CV/CV /CA /CU/CP\r/D8/D3/D6 /D3/CU /BC/BA/BC/BK/BD/BA /CC/CW/CT /D0/CP/D8/D8/CX\r/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/CP/D6/CT \r/D3/D1/D4/CP/D6/CP/CQ/D0/CT /D8/D3 /D8/CW/D3/D7/CT /D6/CT/D4 /D3/D6/D8/CT/CS /CT/CP/D6/D0/CX/CT/D6 /CU/D3/D6 /D8/CW/CT /D4 /D3/D0/DD\r/D6/DD/D7/B9/D8/CP/D0/D0/CX/D2/CT /D7/CP/D1/D4/D0/CT/D7 /CJ/BD/BH /B8 /BD/BI ℄/BA /CC/CW/CT /D0/CP/D8/D8/CX\r/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CS/CT\r/D6/CT/CP/D7/CT/D7/CP/D7 /DB /CT /D1/D3 /DA /CT /CU/D6/D3/D1 /C4/CP /D8/D3 /C8/D6/B8 /CP/D8/D8/D6/CX/CQ/D9/D8/CT/CS /D8/D3 /DB /CT/D0/D0 /C3/D2/D3 /DB/D2 /D0/CP/D2/B9/D8/CW/CP/D2/CX/CS/CT \r/D3/D2 /D8/D6/CP\r/D8/CX/D3/D2/BA /CC/CW/CT /D9/D2/CX/D8 \r/CT/D0/D0 /DA /D3/D0/D9/D1/CT /D3/CU /CA/BT/D92\n/BZ/CT2\r/D3/D1/D4 /D3/D9/D2/CS/D7 /CX/D7 /CW/CX/CV/CW/CT/D6 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CP/D8 /D3/CU /CA /BV/D92\n/BZ/CT2\n\r/D3/D1/B9/D4 /D3/D9/D2/CS/D7 /CP/D2/CS /D0/CT/D7/D7 /D8/CW/CP/D2 /D8/CW/CP/D8 /D3/CU /CA/BT/CV2\n/BZ/CT2\n\r/D3/D1/D4 /D3/D9/D2/CS/D7/BA /CC/CW/CX/D7/D1/CP /DD /CQ /CT /CS/D9/CT /D8/D3 /D8/CW/CT /CX/D2 /D8/CT/D6/D1/CT/CS/CX/CP/D8/CT /D7/CX/DE/CT /D3/CU /D8/CW/CT /BT/D9 /CP/D8/D3/D1 \r/D3/D1/B9/D4/CP/D6/CT/CS /D8/D3 /BV/D9 /CP/D2/CS /BT/CV/BA/BT/BA /C4/CP/BT/D9 2\n/BZ/CT2/CF /CT /AS/D6/D7/D8 /CS/CT/D7\r/D6/CX/CQ /CT /D8/CW/CT /D4/CW /DD/D7/CX\r/CP/D0 /D4/D6/D3/D4 /CT/D6/D8/CX/CT/D7 /D3/CU /C4/CP/BT/D92\n/BZ/CT2/DB/CW/CX\r /CW \r/CP/D2 /CQ /CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /CP/D7 /D8/CW/CT /D6/CT/CU/CT/D6/CT/D2\r/CT/B8 /D2/D3/D2/B9/D1/CP/CV/D2/CT/D8/CX\r/CP/D2/CP/D0/D3/CV /CU/D3/D6 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CA/BT/D92\n/BZ/CT2\n\r/D3/D1/D4 /D3/D9/D2/CS/D7/BA /CC/CW/CT/D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /C4/CP/BT/D92\n/BZ/CT2\n/B4/BY/CX/CV/BA /BE /CP/B5 /D7/CW/D3 /DB/D7 /CP /C8 /CP/D9/D0/CX/B9/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6 /CP/D8 /D6/D3 /D3/D1 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 /DB/CX/D8/CW /CP/D2/CP/CQ/D7/D3/D0/D9/D8/CT /DA /CP/D0/D9/CT /D3/CU /D2/CT/CP/D6/D0/DD /BE/BA/BF× /BD/BC−4/CT/D1 /D9/BB/D1/D3/D0/BA /C1/D8 /D6/CT/D1/CP/CX/D2/D7/D2/CT/CP/D6/D0/DD /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CS/D3 /DB/D2 /D8/D3 /BH/BC /C3 /CP/D2/CS /D7/CW/D3 /DB/D7/CP/D2 /D9/D4/D8/D9/D6/D2 /CP/D8 /D0/D3 /DB /CT/D6 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /B4/BY/CX/CV/BA /BE /CP/B5/BA /CC/CW/CT /D0/D3 /DB /D8/CT/D1/B9/D4 /CT/D6/CP/D8/D9/D6/CT /D9/D4/D8/D9/D6/D2 /CX/D7 /D1/D3/D7/D8 /D0/CX/CZ /CT/D0/DD /CS/D9/CT /D8/D3 /D8/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU/D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /CX/D3/D2/D7 /CX/D2 /D8/CW/CT \r/D3/D2/D7/D8/CX/D8/D9/CT/D2 /D8/D7 /D9/D7/CT/CS /D8/D3 /D4/D6/CT/D4/CP/D6/CT/D8/CW/CT /CP/D0/D0/D3 /DD/D7/BA /BT /AS/D8 /B4/D7/CW/D3 /DB/D2 /CQ /DD /D8/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT /CX/D2 /BY/CX/CV/BA /BE/CP/B5 /D3/CU/BF\n5\n4\n3\n2\n1\n0  χ (10-4emu/mole)\n300250200150100500LaAu2Ge2\n(a) Data\n Fit\n120\n90\n60\n30\n0  C (J/mole K)\n200 150 100 50 0\n Temperature (K) Data\n Fit(b)LaAu2Ge2\n3.0\n2.0\n1.0\n0 C/T3 (J/mole K3)\n100806040200 T (K)LaAu2Ge2\nTmax/BY/CX/CV/D9/D6/CT /BE/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /C5/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU/C4/CP/BT/D9 2\n/BZ/CT2\n/DB/CX/D8/CW /CP /AS/D8 /CS/CT/D7\r/D6/CX/CQ /CT/CS /CX/D2 /D8/CT/DC/D8/BA /CC/CW/CT /CX/D2/D7/CT/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT/BV/BB/CC /DA/D7 /CC2/D4/D0/D3/D8 /DB/CX/D8/CW /CP /D0/CX/D2/CT/CP/D6 /AS/D8/BA /CQ/B5 /C0/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD \r/D9/D6/DA /CT /D3/CU/C4/CP/BT/D9 2\n/BZ/CT2\n/DB/CX/D8/CW /CP /AS/D8 /CS/CT/D7\r/D6/CX/CQ /CT/CS /CX/D2 /D8/CW/CT /D8/CT/DC/D8/BA /CC/CW/CT /CX/D2/D7/CT/D8 /D7/CW/D3 /DB/D7/D8/CW/CT /BV/BB/CC3/DA/D7 /CC /D4/D0/D3/D8/BA/D8/CW/CT /D1/D3 /CS/CX/AS/CT/CS /BV/D9/D6/CX/CT/B9/CF /CT/CX/D7/D7 /D0/CP /DB\nχ=χ0+Nµ2\neff\n3kB(T−θp)\n/B4/BD/B5/D8/D3 /D8/CW/CT /CS/CP/D8/CP/B8 /DB/CW/CT/D6/CT /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CW/CP /DA /CT /D8/CW/CT/CX/D6 /D9/D7/D9/CP/D0/D1/CT/CP/D2/CX/D2/CV/B8 /DD/CX/CT/D0/CS/D7χ0\n/BP /BE/BA/BF/BD/BE /DC /BD/BC−4/CT/D1 /D9/BB/D1/D3/D0 /CP/D2/CSµeff\n/BP/BC/BA/BDµB\n/BA /CC/CW/CT /D1/CP/CV/D2/CX/D8/D9/CS/CT /D3/CUχ0\n/CX/D7 /D8 /DD/D4/CX\r/CP/D0 /CU/D3/D6 /D8/CW/CT /C4/CP \r/D3/D1/B9/D4 /D3/D9/D2/CS/D7 /CP/D2/CS /CP /D0/D3 /DB /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT /CT/AR/CT\r/D8/CX/DA /CT /D1/D3/D1/CT/D2 /D8 /CX/D2/CS/CX\r/CP/D8/CT/D7/D8/CW/CP/D8 /D8/CW/CT /D9/D4/D8/D9/D6/D2 /CP/D8 /D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /CX/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /D4/D6/CT/D7/B9/CT/D2\r/CT /D3/CU /D7/D3/D1/CT /B4/BO /BD /B1/B5 /D1/CP/CV/D2/CT/D8/CX\r /CX/D1/D4/D9/D6/CX/D8 /DD /CX/D3/D2/D7 /CX/D2 /D8/CW/CT \r/D3/D2/B9/D7/D8/CX/D8/D9/CT/D2 /D8/D7/BA /CC/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS /B4/BY/CX/CV/BA /BE/CQ/B5/CX/D2\r/D6/CT/CP/D7/CT/D7 /D1/D3/D2/D3/D8/D3/D2/CX\r/CP/D0/D0/DD /DB/CX/D8/CW /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /CC/CW/CT /D1/CP/CV/D2/CX/B9/D8/D9/CS/CT /D3/CU /D8/CW/CT /CT/D0/CT\r/D8/D6/D3/D2/CX\r \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 γ /B8 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT/D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CS/CP/D8/CP /CX/D7≈ /BJ /D1/C2/BB/D1/D3/D0 /C32/BA/C1/D2/D7/CT/D8 /D3/CU /BY/CX/CV/BA /BE/CQ /D7/CW/D3 /DB/D7 /CP /D4/D0/D3/D8 /D3/CU /BV/BB/CC3/DA/D7 /CC /B8 /CP /D6/CT/D4/D6/CT/D7/CT/D2/B9/D8/CP/D8/CX/D3/D2 /D8/CW/CP/D8 /CX/D7 /D3/CU/D8/CT/D2 /D9/D7/CT/CS /D8/D3 /CP/D7/D7/CT/D7/D7 /D8/CW/CT /D4 /D3/D7/D7/CX/CQ/D0/CT /D4/D6/CT/D7/CT/D2\r/CT/D3/CU /D0/D3 /DB/B9/CU/D6/CT/D5/D9/CT/D2\r/DD /BX/CX/D2/D7/D8/CT/CX/D2 /D1/D3 /CS/CT/D7 /CX/D2 /D8/CW/CT /D7/D4 /CT\r/CX/AS\r /CW/CT/CP/D8 /CJ/BD/BL ℄/BA/CC/CW/CT /D1/CP/DC/CX/D1 /D9/D1/B8 /CX/D2 /D8/CW/CX/D7 \r/CP/D7/CT /CP/D8 /CCmax\n/BP /BD/BF /C3/B8 \r/D3/D9/D0/CS /CQ /CT/CX/D2 /D8/CT/D6/D4/D6/CT/D8/CT/CS /CP/D7 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CQ /CT/D0/D3 /DB /DB/CW/CX\r /CW /D8/CW/CT /BX/CX/D2/D7/D8/CT/CX/D2/D1/D3 /CS/CT/D7 /CP/D6/CT /CU/D6/D3/DE/CT/D2 /D3/D9/D8/BA /CC/CW/CX/D7 /CX/D7 /CP/D0/D7/D3 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /DB/CW/CT/D6/CT/D8/CW/CT /CS/CT/DA/CX/CP/D8/CX/D3/D2 /CU/D6/D3/D1 /CP /D4/D9/D6/CT /BW/CT/CQ /DD /CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D7/D4 /CT/B9\r/CX/AS\r /CW/CT/CP/D8 /CQ /CT\r/D3/D1/CT/D7 \r/D3/D2/D7/D4/CX\r/D9/D3/D9/D7/BA /BV/D3/D2/D7/CX/CS/CT/D6/CX/D2/CV /D8/CW/CX/D7/B8 /D8/CW/CT\n/D8/CW/CT/D6/D1/CP/D0 /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS/DB /CP/D7 /AS/D8/D8/CT/CS /D8/D3 /D8/CW/CT \r/D3/D1 /CQ/CX/D2/CT/CS /BX/CX/D2/D7/D8/CT/CX/D2 /CP/D2/CS /BW/CT/CQ /DD /CT \r/D3/D2 /D8/D6/CX/CQ/D9/B9/D8/CX/D3/D2/D7 /CP/D7 /D7/CW/D3 /DB/D2 /CQ /DD /D8/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT /CX/D2 /BY/CX/CV/BA /BE /CQ/BA /CC/CW/CT /D8/D3/D8/CP/D0 /CW/CT/CP/D8\r/CP/D4/CP\r/CX/D8 /DD /CX/D2 /D7/D9\r /CW /CP \r/CP/D7/CT /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nCTot=γT+(CE+CD) /B4/BE/B5/DB/CW/CT/D6/CT /D8/CW/CT /AS/D6/D7/D8 /D8/CT/D6/D1 /D6/CT/D4/D6/CT/D7/CT/D2 /D8/D7 /D8/CW/CT /CT/D0/CT\r/D8/D6/D3/D2/CX\r \r/D3/D2 /D8/D6/CX/CQ/D9/B9/D8/CX/D3/D2/B8 /D8/CW/CT /D7/CT\r/D3/D2/CS /D8/CT/D6/D1 /CX/D2\r/D0/D9/CS/CT/D7 /BX/CX/D2/D7/D8/CT/CX/D2 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /BVE/CP/D2/CS /BW/CT/CQ /DD /CT \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /BVD\n/BA /CC/CW/CT /BX/CX/D2/D7/D8/CT/CX/D2 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nCE=/summationdisplay\nn′3nEn′Ry2ey\n(ey−1)2\n/B4/BF/B5/DB/CW/CT/D6/CTy /BPΘEn′/T /B8ΘE\n/CX/D7 /D8/CW/CT /BX/CX/D2/D7/D8/CT/CX/D2 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 n′/CX/D7/D8/CW/CT /D7/D9/D1/D1/CP/D8/CX/D3/D2 /D3 /DA /CT/D6 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8 /BX/CX/D2/D7/D8/CT/CX/D2 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7/B8\nR /CX/D7 /D8/CW/CT /CV/CP/D7 \r/D3/D2/D7/D8/CP/D2 /D8 /CP/D2/CSnE\n/CX/D7 /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /BX/CX/D2/D7/D8/CT/CX/D2/D3/D7\r/CX/D0/D0/CP/D8/D3/D6/D7/BA /CC/CW/CT /BW/CT/CQ /DD /CT \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nCD= 9nDR/parenleftbiggT\nΘD/parenrightbigg3ΘD/T/A2\n0x4exdx\n(ex−1)2\n/B4/BG/B5/DB/CW/CT/D6/CTx /BPΘD/T /BAΘD\n/CX/D7 /D8/CW/CT /BW/CT/CQ /DD /CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CSnD/CX/D7 /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6 /D3/CU /BW/CT/CQ /DD /CT /D3/D7\r/CX/D0/D0/CP/D8/D3/D6/D7/BA /C1/D8/CT/D6/CP/D8/CX/DA /CT /AS/D8 /D8/D3 /D8/CW/CT/BX/D5/BA /BE /DB /CP/D7 /D4 /CT/D6/CU/D3/D6/D1/CT/CS /CQ /DD /D9/D7/CX/D2/CV /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU /CT/D0/CT\r/D8/D6/D3/D2/CX\r\r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 γ /CP/D7 /CT/D7/D8/CX/D1/CP/D8/CT/CS /CP/CQ /D3 /DA /CT /CP/D2/CS /AS/DC/CX/D2/CV /D8/CW/CT /D2 /D9/D1 /CQ /CT/D6/D3/CU /CP/D8/D3/D1/D7nD\n/CP/D2/CSnE\n/CU/D3/D6 /CP /D4/CP/D6/D8/CX\r/D9/D0/CP/D6 /AS/D8/B8 /CP/D0/D0/D3 /DB/CX/D2/CV /CQ /D3/D8/CW\nΘEn′/CP/D2/CSΘD\n/D8/D3 /DA /CP/D6/DD /CP/D7 /AS/D8/D8/CX/D2/CV /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA /BT /CV/D3 /D3 /CS /AS/D8/D8/D3 /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /C4/CP/BT/D92\n/BZ/CT2\n/D3 /DA /CT/D6 /D8/CW/CT /CT/D2 /D8/CX/D6/CT /D6/CP/D2/CV/CT/D3/CU /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /DB /CP/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CQ /DD /CP/D7/D7/CX/CV/D2/CX/D2/CV /D8/CW/D6/CT/CT /BW/CT/CQ /DD /CT\r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CP/D8/D3/D1/D7 /B4nD\n/BP /BF/B5 /DB/CX/D8/CWΘD= 305 /C3 /D4/D0/D9/D7/D8 /DB /D3 /BX/CX/D2/D7/D8/CT/CX/D2 \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /CP/D8/D3/D1/D7 /B4nE1=nE2= 1 /B5 /DB/CX/D8/CW\nΘE1= 74 /C3 /BA /CC/CW/CT /CS/CT/D7\r/D6/CX/D4/D8/CX/D3/D2 /D3/CU /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CX/D2 /D8/CT/D6/D1/D7/D3/CU /CP \r/D3/D1 /CQ/CX/D2/CP/D8/CX/D3/D2 /D3/CU /CP\r/D3/D9/D7/D8/CX\r /CP/D2/CS /D3/D4/D8/CX\r/CP/D0 /D1/D3 /CS/CT/D7 \r/CP/D2 /CQ /CT/D6/CT/CP/CS/CX/D0/DD /D9/D2/CS/CT/D6/D7/D8/D3 /D3 /CS /CQ /DD /CP/D7/D7/CX/CV/D2/CX/D2/CV /D8/CW/CT /C4/CP /CP/D2/CS /BZ/CT /CP/D8/D3/D1/D7 /CX/D2/D8/CW/CT /D9/D2/CX/D8 \r/CT/D0/D0 /D8/D3 /D8/CW/D6/CT/CT /BW/CT/CQ /DD /CT \r /CW/CP/D6/CP\r/D8/CT/D6/CX/D7/D8/CX\r /D1/D3 /CS/CT/D7 /CP/D2/CS /BT/D9/D8/D3 /D8/CW/CT /D6/CT/D1/CP/CX/D2/CX/D2/CV /BX/CX/D2/D7/D8/CT/CX/D2 /D1/D3 /CS/CT/D7/BA /CB/CX/D2\r/CT /BT/D9 /CX/D7 /D1 /D9\r /CW /D0/CP/D6/CV/CT/D6/CX/D2 /D7/CX/DE/CT \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 /D3/D8/CW/CT/D6 /CP/D8/D3/D1/D7/B8 /CX/D8 \r/CP/D2 /CQ /CT /CT/DC/D4 /CT\r/D8/CT/CS /D8/D3/DA/CX/CQ/D6/CP/D8/CT /DB/CX/D8/CW /CP /D0/D3 /DB /CT/D6 /D2/CP/D8/D9/D6/CP/D0 /CU/D6/CT/D5/D9/CT/D2\r/DD /BA /C1/D8 \r/CP/D2 /CQ /CT /D7/CW/D3 /DB/D2/D8/CW/CP/D8 /D8/CW/CT /BX/CX/D2/D7/D8/CT/CX/D2 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT ΘE\n/CX/D7 /D6/CT/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /D4 /CT/CP/CZ/CX/D2 /BV/BB/CC3/CP/CV/CP/CX/D2/D7/D8 /CC /D4/D0/D3/D8 /B4/CCmax\n/B5 /CQ /DD /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CCmax\n/BP\nΘE/5 /CJ/BE/BC ℄/B8 /DB/CW/CX\r /CW /CX/D7 /CX/D2 /CU/CP/CX/D6 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS/DA /CP/D0/D9/CT /D3/CU /CCmax\n/BP /BD/BF /C3/BA/CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D3/CU /C4/CP/BT/D92\n/BZ/CT2\n/DB/CX/D8/CW \r/D9/D6/D6/CT/D2 /D8 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3/CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BF/BA /CC/CW/CT /D6/CT/D7/CX/D7/B9/D8/CX/DA/CX/D8 /DD /CP/D0/D3/D2/CV /CQ /D3/D8/CW /D8/CW/CT \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /CT/DC/CW/CX/CQ/CX/D8/CP /D1/CT/D8/CP/D0/D0/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6 /CS/D3 /DB/D2 /D8/D3 /BD/BH /C3 /CP/D2/CS /D0/CT/DA /CT/D0/D7 /D3/AR /CP/D8 /D0/D3 /DB/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /DB/CX/D8/CW /CP /D6/CT/D7/CX/CS/D9/CP/D0 /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D3/CU /BD/BKµΩcm /CP/D2/CS/BD/BA/BGµΩcm /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /CU/D3/D6 /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /CC/CW/CT/D3/CQ/D7/CT/D6/DA /CT/CS /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D7 /CX/D2 /D8/D9/D2/CT /DB/CX/D8/CW /D8/CW/CT /D4/CW/D3/D2/D3/D2 /CX/D2/CS/D9\r/CT/CS/D7\r/CP/D8/D8/CT/D6/CX/D2/CV /D3/CU /D8/CW/CT \r /CW/CP/D6/CV/CT \r/CP/D6/D6/CX/CT/D6/D7 /CP/D7 /CT/DC/D4 /CT\r/D8/CT/CS /CU/D3/D6 /CP /D2/D3/D2/D1/CP/CV/D2/CT/D8/CX\r \r/D3/D1/D4 /D3/D9/D2/CS/BA /CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /CP/D0/D3/D2/CV /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/B9/D8/CX/D3/D2 /CX/D7 /CU/D3/D9/D2/CS /D8/D3 /CQ /CT /D0/D3 /DB /CT/D6 \r/D3/D1/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CT /CX/D2/B9/D4/D0/CP/D2/CT /CJ/BD/BC/BC℄/D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /CQ /DD /CP /CU/CP\r/D8/D3/D6 /D3/CU /D2/CT/CP/D6/D0/DD /BE/BC /CP/D8 /BF/BC/BC /C3 /D4 /D3/CX/D2 /D8/D7 /D3/D9/D8/D7/CX/CV/D2/CX/AS\r/CP/D2 /D8 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /BA /CC/CW/CT /D7/CX/D1/CX/D0/CP/D6 /CP/D2/CX/D7/D3/D8/D6/D3/D4/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6/CX/D2 /D8/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /DB /CP/D7 /CP/D0/D7/D3 /CU/D3/D9/D2/CS /CU/D3/D6 /BV/CT /CP/D2/CS /C8/D6 \r/D3/D1/D4 /D3/D9/D2/CS/D7/BG\n100\n80\n60\n40\n20\n0  ρ (µΩ cm)\n300250200150100500\n Temperature (K)5\n4\n3\n2\n1LaAu2Ge2\n J // [100]\n J // [001]\n Fit/BY/CX/CV/D9/D6/CT /BF/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CA/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D3/CU /C4/CP/BT/D9 2\n/BZ/CT2\n/DB/CX/D8/CW \r/D9/D6/D6/CT/D2 /D8/D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /CP/D0/D3/D2/CV /D8/CW/CT/D8 /DB /D3 /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /CX/D7 /AS/D8/D8/CT/CS /D8/D3 /BU/D0/D3 \r /CW/B9/BZ/D6 ¨ u /D2/D2/CT/CX/D7/CT/D2 /D6/CT/D0/CP/D8/CX/D3/D2/BA/CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /CT/D0/D3 /DB /CP/D2/CS /D1/CP /DD /CP/D6/CX/D7/CT /CS/D9/CT /D8/D3 /D8/CW/CT /CX/D2/CW/CT/D6/CT/D2 /D8 /D7/D8/D6/D9\r/B9/D8/D9/D6/CP/D0 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS/BA /CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /DB/CX/D8/CW\r/D9/D6/D6/CT/D2 /D8 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /D8/CW/CT /D8 /DB /D3 \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /DB /CP/D7/AS/D8/D8/CT/CS /D8/D3 /D8/CW/CT /D1/D3 /CS/CX/AS/CT/CS /BU/D0/D3 \r /CW/B9/BZ/D6 ¨ u /D2/CT/CX/D7/CT/D2 /D6/CT/D0/CP/D8/CX/D3/D2 /CV/CX/DA /CT/D2 /CQ /DD\nρ(T) =ρ0+4ΘDR/parenleftbiggT\nΘD/parenrightbigg5ΘD/T/A2\n0x5dx\n(ex−1)(1−e−x)−KT3/B4/BH/B5/DB/CW/CT/D6/CTx /BPΘD/T /B8ΘD\n/CX/D7 /D8/CW/CT /BW/CT/CQ /DD /CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 ρ0/CX/D7 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /D6/CT/D7/CX/CS/D9/CP/D0 /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /CP/D2/CS\nRandK /CP/D6/CT /D8/CW/CT \r/D3 /CTꜶ\r/CX/CT/D2 /D8/D7 /D3/CU /D8/CW/CT /D4/CW/D3/D2/D3/D2 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D8/D3 /D8/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /B4/D7/CT\r/D3/D2/CS /D8/CT/D6/D1/B5 /CP/D2/CS /D8/CW/CT /C5/D3/D8/D8 /D7 /B9 /CS /CX/D2 /D8/CT/D6/CQ/CP/D2/CS /D7\r/CP/D8/D8/CT/D6/CX/D2/CV /B4/D8/CW/CX/D6/CS /D8/CT/D6/D1/B5 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA /CC/CW/CT /AS/D8 /D8/D3 /D8/CW/CT/D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD \r/D9/D6/DA /CT /D7 /DD/CX/CT/D0/CS/D7ΘD= 126 /C3/B8ρ0= 18µΩcm /B8/CP/D2/CSR= 0.273µΩcm /CU/D3/D6 /C2 /BB/BB /CJ/BD/BC/BC℄ /CP/D2/CSΘD= 125 /C3/B8\nρ0= 1.4µΩcm /CP/D2/CSR= 0.011µΩcm /CU/D3/D6 /C2 /BB/BB /CJ/BC/BC/BD℄/B8/D8/CW/CT /DA /CP/D0/D9/CT /D3/CUK /DB /CP/D7 /CU/D3/D9/D2/CS /D8/D3 /CQ /CT /DE/CT/D6/D3 /CU/D3/D6 /CQ /D3/D8/CW /D8/CW/CT /CS/CX/B9/D6/CT\r/D8/CX/D3/D2/D7/BA /CC/CW/CT /BW/CT/CQ /DD /CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D6/CT/D1/CP/CX/D2/D7 /D2/CT/CP/D6/D0/DD /D7/CP/D1/CT/CQ/D9/D8 /D8/CW/CT/D6/CT /CX/D7 /CP/D2 /D3/D6/CS/CT/D6 /D3/CU /CS/CT\r/D6/CT/CP/D7/CT /CX/D2 /D8/CW/CT /D1/CP/CV/D2/CX/D8/D9/CS/CT /D3/CU/D4/CW/D3/D2/D3/D2 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /CU/D3/D6 /C2 /BB/BB /CJ/BC/BC/BD℄ /CX/D2/B9/CU/CT/D6/D6/CT/CS /CU/D6/D3/D1 /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU /CA/BA /CC/CW/CT /BW/CT/CQ /DD /CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D2/D8/CW/CT /BU/D0/D3 \r /CW/B9/BZ/D6 ¨ u /D2/CT/CX/D7/CT/D2 /D6/CT/D0/CP/D8/CX/D3/D2 /D3/CU/D8/CT/D2 /CS/CX/AR/CT/D6/D7 /CU/D6/D3/D1 /D8/CW/CT /DA /CP/D0/D9/CT/D3/CUΘD\n/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CS/CP/D8/CP/BA /C1/D2 /D4/D6/CX/D2\r/CX/D4/D0/CT/B8/D8/CW/CT /DA /CP/D0/D9/CT /D3/CUΘD\n/D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /BU/D0/D3 \r /CW/B9/BZ/D6 ¨ u /D2/CT/CX/D7/CT/D2 /D6/CT/D0/CP/D8/CX/D3/D2/D7/CW/D3/D9/D0/CS \r/D3/D2/D7/CX/CS/CT/D6/CP/CQ/D0/DD /CS/CX/AR/CT/D6 /CU/D6/D3/D1 /D8/CW/CP/D8 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /CW/CT/CP/D8\r/CP/D4/CP\r/CX/D8 /DD /CQ /CT\r/CP/D9/D7/CT /D8/CW/CT /CU/D3/D6/D1/CT/D6 /D8/CP/CZ /CT/D7 /CX/D2 /D8/D3 /CP\r\r/D3/D9/D2 /D8 /D3/D2/D0/DD /D8/CW/CT/D0/D3/D2/CV/CX/D8/D9/CS/CX/D2/CP/D0 /D4/CW/D3/D2/D3/D2/D7 /CJ/BE/BD ℄/BA/BU/BA /BV/CT/BT/D9 2\n/BZ/CT2/BY/CX/CV/D9/D6/CT/BG /CP /D7/CW/D3 /DB/D7 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU/BV/CT/BT/D92\n/BZ/CT2\n/CU/D6/D3/D1 /BD/BA/BK /C3 /D8/D3 /BF/BC/BC /C3 /CX/D2 /CP /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D3/CU/BF /CZ/C7/CT /CP/D0/D3/D2/CV /D8/CW/CT /D8 /DB /D3 \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /B4/CJ/BD/BC/BC℄/CP/D2/CS /CJ/BC/BC/BD℄/B5/BA /CC/CW/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /DB/CX/D8/CW /AS/CT/D0/CS /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3/CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2 /D7/CW/D3 /DB/D7 /CP/D2 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2/CP/D8 /CCN\n/BP /BD/BF/BA/BH /C3/B8 /D0/CT/D7/D7 /D8/CW/CP/D2 /D8/CW/CP/D8 /D3/CQ/D7/CT/D6/DA /CT/CS /CU/D6/D3/D1 /D2/CT/D9/D8/D6/D3/D20.10\n0.08\n0.06\n0.04\n0.02\n0  χ (emu/mole)\n200 150 100 50 0\n Temperature (K) H // [001]\n H // [100]CeAu2Ge2\n(a)\n2.0\n1.5\n1.0\n0.5\n0  M (µΒ/f.u.)\n120100806040200\n Magnetic Field (kOe) H // [001]\n H // [100]CeAu2Ge2\n \n T = 2 K\n(b)/BY/CX/CV/D9/D6/CT /BG/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /C5/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU/BV/CT/BT/D9 2\n/BZ/CT2\n/DB/CX/D8/CW /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /B4/BF/CZ/C7/CT/B5 /CP/D4/D4/D0/CX/CT/CS /CP/D0/D3/D2/CV /D8/CW/CT /D8 /DB /D3\r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /CQ/B5 /C5/CP/CV/D2/CT/D8/CX\r /CX/D7/D3/D8/CW/CT/D6/D1 /CP/D8 /BE /C3 /CU/D3/D6/D8/CW/CT /D7/CP/D1/CT /DB/CX/D8/CW /AS/CT/D0/CS /CP/D0/D3/D2/CV /CQ /D3/D8/CW /D8/CW/CT \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA/CS/CX/AR/D6/CP\r/D8/CX/D3/D2 /D3/D2 /CP /D4 /D3/D0/DD\r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /D7/CP/D1/D4/D0/CT /B4/BD/BI /C3/B5 /CJ/BD/BH ℄/BA /CC/CW/CT/D7/CW/CP/D6/D4 /CS/D6/D3/D4 /D3/CU /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /CQ /CT/D0/D3 /DB /D8/CW/CT /C6`e /CT/D0 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CP/D8 /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D7 /CP/D6/CT /CP/D0/CX/CV/D2/CT/CS /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/B9/D2/CT/D8/CX\r/CP/D0/D0/DD /CP/D0/D3/D2/CV /D8/CW/CT /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2 /CX/D2 /D4 /D3/D7/D7/CX/CQ/D0/DD /CP \r/D3/D0/D0/CX/D2/CT/CP/D6/CP/D6/D6/CP/D2/CV/CT/D1/CT/D2 /D8/BA /CC/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CP/D8 /D8/CW/CT /CJ/BC/BC/BD℄/CP/DC/CX/D7 /CX/D7 /D8/CW/CT /CT/CP/D7/DD /CP/DC/CX/D7 /D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CU/D3/D6 /BV/CT/BT/D92\n/BZ/CT2/CP/D7 /D6/CT/D4 /D3/D6/D8/CT/CS /CQ /DD /C4/D3/CX/CS/D0 /CT/D8 /CP/D0 /CJ/BD/BH ℄/BA /CF/CX/D8/CW /AS/CT/D0/CS /CP/D0/D3/D2/CV /CJ/BD/BC/BC℄/CS/CX/D6/CT\r/D8/CX/D3/D2/B8 /D8/CW/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D6/CT/D1/CP/CX/D2/D7 /CQ /CT/D0/D3 /DB /D8/CW/CP/D8 /D3/CU /CJ/BC/BC/BD℄/CS/CX/D6/CT\r/D8/CX/D3/D2 /CX/D2 /D8/CW/CT /CT/D2 /D8/CX/D6/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D6/CP/D2/CV/CT /CU/D3/D0/D0/D3 /DB /CT/CS /CQ /DD/CP /C3/CX/D2/CZ /CP/D8 /D8/CW/CT /D3/D6/CS/CT/D6/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 /CX/D2/CS/CX\r/CP/D8/CX/D2/CV /D8/CW/CT /CW/CP/D6/CS/CP/DC/CX/D7 /D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/BA /BV/D9/D6/CX/CT/B9/CF /CT/CX/D7/D7 /AS/D8/D7 /D3/CU /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT/D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /CX/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT /CV/CX/DA /CT/D7 /CT/AR/CT\r/D8/CX/DA /CT/D1/D3/D1/CT/D2 /D8 /B4µeff\n/B5 /CP/D2/CS /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /BV/D9/D6/CX/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B4θp\n/B5 /CP/D7 /BE/BA/BH/BGµB\n/BB/BV/CT /CP/D2/CS /BE/BA/BH/BGµB\n/BB/BV/CT /CP/D2/CS /B9/BI/BF /C3 /CP/D2/CS/BE/BJ /C3 /CU/D3/D6 /AS/CT/D0/CS /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/B8/D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA /CC/CW/CT /D3/CQ/D8/CP/CX/D2/CT/CS /CT/AR/CT\r/D8/CX/DA /CT /D1/D3/D1/CT/D2 /D8 /CU/D3/D6 /CQ /D3/D8/CW/D8/CW/CT /CP/DC/CT/D7 /CX/D7 /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT /CT/DC/D4 /CT\r/D8/CT/CS /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0 /DA /CP/D0/D9/CT/B4/BE/BA/BH/BGµB\n/BB/BV/CT/B5/BA /CC/CW/CT /D4 /D3/D0/DD\r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /CP /DA /CT/D6/CP/CV/CT /D3/CUθp\n/CX/D7 /B9/BF/BG /C3/B8/DB/CW/CX\r /CW /CX/D7 /CX/D2 /D8/D9/D2/CT /DB/CX/D8/CW /D8/CW/CT /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CX/D7\r/D3/D1/D4 /D3/D9/D2/CS/BA /CC/CW/CT /D0/CP/D6/CV/CT /D2/CT/CV/CP/D8/CX/DA /CT /DA /CP/D0/D9/CT /CP/D0/D7/D3 /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT/D4/D6/CT/D7/CT/D2\r/CT /D3/CU /D4 /D3/D7/D7/CX/CQ/D0/CT /C3 /D3/D2/CS/D3 /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/BA /CC/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/CX/D7/D3/D8/CW/CT/D6/D1/D7 /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS /DB/CX/D8/CW /AS/CT/D0/CS /CP/D0/D3/D2/CV /CQ /D3/D8/CW /D8/CW/CT\r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /B4/CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄/B5 /CP/D6/CT /D7/CW/D3 /DB/D2/CX/D2 /BY/CX/CV/BA /BG /CQ/BA /CC/CW/CT /D0/CX/D2/CT/CP/D6 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /CP/D8/BH\n50 \n40 \n30 \n20 \n10 \n0  Magnetic Field (kOe) \n16 14 12 10 8 6 4 2 0\n Temperature (K) CeAu2Ge 2\nH // [001] Ferromagnet \nAntiferromagnet /BY/CX/CV/D9/D6/CT /BH/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /C5/CP/CV/D2/CT/D8/CX\r /C8/CW/CP/D7/CT /BW/CX/CP/CV/D6/CP/D1 /D3/CU/BV/CT/BT/D9 2\n/BZ/CT2\n/BA/BE /C3 /CP/D2/CS /D9/D4 /D8/D3≈ /BG/BF /CZ/C7/CT /DB/CX/D8/CW /D8/CW/CT /AS/CT/D0/CS /CP/D0/D3/D2/CV /CJ/BC/BC/BD℄ /CP/DC/CX/D7\r/D3/D2/AS/D6/D1/D7 /D8/CW/CT /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS/CX/D2 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /D3/D6/CS/CT/D6/CT/CS /D7/D8/CP/D8/CT/BA /CC/CW/CT /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/D9/D2/CS/CT/D6/CV/D3 /CT/D7 /CP /D7/D4/CX/D2 /AT/CX/D4 /D8 /DD/D4 /CT /D1/CT/D8/CP/D1/CP/CV/D2/CT/D8/CX\r /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CP/D8 /D8/CW/CT\r/D6/CX/D8/CX\r/CP/D0 /AS/CT/D0/CS /C0C≈ /BG/BF /CZ/C7/CT /CU/D3/D0/D0/D3 /DB /CT/CS /CQ /DD /D2/CT/CP/D6 /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2/CP/D8 /CW/CX/CV/CW /AS/CT/D0/CS/D7/BA /CC/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D1/D3/D1/CT/D2 /D8 /D3/CQ/D8/CP/CX/D2/CT/CS /CP/D8 /BE /C3/CP/D2/CS /BD/BE/BC /C3/C7/CT /CX/D7 /BD/BA/BK/BIµB\n/B8 /D0/CT/D7/D7 /D8/CW/CP/D2 /D8/CW/CT /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0/D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D1/D3/D1/CT/D2 /D8 /D3/CU /BE/BA/BD/BGµB\n/CP/D2/CS /CX/D7 /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW/D8/CW/CT /D2/CT/D9/D8/D6/D3/D2 /CS/CX/AR/D6/CP\r/D8/CX/D3/D2 /D6/CT/D7/D9/D0/D8/D7 /CJ/BD/BH ℄/BA /BY /D6/D3/D1 /D8/CW/CT /CS/CX/AR/CT/D6/CT/D2 /D8/CX/CP/D0/D4/D0/D3/D8/D7 /D3/CU /D8/CW/CT /CX/D7/D3/D8/CW/CT/D6/D1/CP/D0 /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 \r/D9/D6/DA /CT/D7 /B4/D2/D3/D8 /D7/CW/D3 /DB/D2/CW/CT/D6/CT/B5/B8 /DB /CT /CW/CP /DA /CT \r/D3/D2/D7/D8/D6/D9\r/D8/CT/CS /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1/CP/D7 /CS/CT/D4/CX\r/D8/CT/CS /CX/D2 /BY/CX/CV/BA /BH /BA /CC/CW/CT \r/D6/CX/D8/CX\r/CP/D0 /AS/CT/D0/CS /C0C\n/CS/CT\r/D6/CT/CP/D7/CT/D7/DB/CX/D8/CW /CX/D2\r/D6/CT/CP/D7/CT /CX/D2 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CS /AS/D2/CP/D0/D0/DD /DA /CP/D2/CX/D7/CW/CT/D7 /CP/D8 /CCN\n/BA/BT /D8 /D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /CP/D2/CS /CU/D3/D6 /AS/CT/D0/CS/D7 /D0/CT/D7/D7 /D8/CW/CP/D2≈ /BG/BH /C3/C7/CT/B8/D8/CW/CT /D7/DD/D7/D8/CT/D1 /CX/D7 /CX/D2 /CP /D4/D9/D6/CT/D0/DD /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT /CP/D7/CX/D2/CS/CX\r/CP/D8/CT/CS /CP/D2/CS /CT/D2 /D8/CT/D6/D7 /CX/D2 /D8/D3 /D8/CW/CT /AS/CT/D0/CS /CX/D2/CS/D9\r/CT/CS /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r/D7/D8/CP/D8/CT /CP/D8 /CW/CX/CV/CW/CT/D6 /AS/CT/D0/CS/D7/BA/CC/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS /CX/D2 /DE/CT/D6/D3/CP/D2/CS /CP/D4/D4/D0/CX/CT/CS /AS/CT/D0/CS/D7 /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BI /BA /BT/D2 /CP/D2/D3/D1/CP/D0/DD /CX/D2/D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CP/D8 /BD/BF/BA/BH /C3 /DB/CX/D8/CW /CP /D4 /CT/CP/CZ /CW/CT/CX/CV/CW /D8 /D3/CU≈ /BD/BD/C2/BB/D1/D3/D0 /C3 \r/D3/D2/AS/D6/D1/D7 /D8/CW/CT /CQ/D9/D0/CZ /D1/CP/CV/D2/CT/D8/CX\r /D3/D6/CS/CT/D6/CX/D2/CV /D3/CU /BV/CT3+/CX/D3/D2/D7/BA /CC/CW/CT /D4 /CT/CP/CZ /CW/CT/CX/CV/CW /D8 /CX/D7 \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /D1/CT/CP/D2 /AS/CT/D0/CS /DA /CP/D0/D9/CT/D3/CU /BD/BE/BA/BH /C2/BB/D1/D3/D0 /C3 /CU/D3/D6 /D7/D4/CX/D2S=1/2 /BA /CC/CW/CT /D1/CP/CV/D2/CX/D8/D9/CS/CT /D3/CU/D8/CW/CT /CB/D3/D1/D1/CT/D6/AS/CT/D0/CS \r/D3 /CTꜶ\r/CX/CT/D2 /D8 γ /DB /CP/D7 /CT/D7/D8/CX/D1/CP/D8/CT/CS /D8/D3 /CQ /CT≈/BD/BH /D1/C2/BB/D1/D3/D0 /C32/CU/D6/D3/D1 /D8/CW/CT /DD /CX/D2 /D8/CT/D6\r/CT/D4/D8 /D3/CU /D8/CW/CT /BV/BB/CC /DA/D7 /CC2\r/D9/D6/DA /CT /B4/D2/D3/D8 /D7/CW/D3 /DB/D2/B5/BA /BT/D4/D4/D0/CX\r/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /D3/CU/BE/BC /C3/C7/CT /D7/CW/CX/CU/D8/D7 /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D4 /CT/CP/CZ /D8/D3 /DB /CP/D6/CS/D7 /D0/D3 /DB /D8/CT/D1/B9/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CS /D6/CT/CS/D9\r/CT/D7 /CX/D8/D7 /CW/CT/CX/CV/CW /D8/BA /BT /D8 /CW/CX/CV/CW/CT/D6 /AS/CT/D0/CS/D7 /B4/BI/BC /CP/D2/CS/BK/BC /C3/C7/CT/B5 /D8/CW/CT /D4 /CT/CP/CZ /DA /CP/D2/CX/D7/CW/CT/D7 /CP/D0/D8/D3/CV/CT/D8/CW/CT/D6 \r/D3/D2/D7/CX/D7/D8/CT/D2 /D8 /DB/CX/D8/CW/D8/CW/CT /D4/D6/CT/D7/CT/D2\r/CT /D3/CU /D1/CT/D8/CP/D1/CP/CV/D2/CT/D8/CX\r /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /B4/BG/BF /C3/C7/CT/B5/BA /CC/CW/CT/BG /CU \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /BV/CT/BT/D92\n/BZ/CT2\n/BV4f\n/B8/BY/CX/CV/BA /BI /CQ /DB /CP/D7 /CT/DC/D8/D6/CP\r/D8/CT/CS /CQ /DD /D7/D9/CQ/D8/D6/CP\r/D8/CX/D2/CV /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU/C4/CP/BT/D92\n/BZ/CT2\n/BA /BU/CT/D7/CX/CS/CT/D7 /D8/CW/CT /D4 /CT/CP/CZ /CP/D8 /CCN\n/B8 /BV4f\n/CT/DC/CW/CX/CQ/CX/D8/D7 /CP /CQ/D6/D3/CP/CS/D4 /CT/CP/CZ \r/CT/D2 /D8/CT/D6/CT/CS /CP/D6/D3/D9/D2/CS /BI/BC /C3 /CP/D6/CX/D7/CX/D2/CV /CS/D9/CT /D8/D3 /D8/CW/CT /CB\r /CW/D3/D8/D8/CZ/DD\r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/CT /D8/CW/CT/D6/D1/CP/D0 /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /D4 /D3/D4/D9/D0/CP/B9/D8/CX/D3/D2 /D3/CU /CT/DC\r/CX/D8/CT/CS /BV/BX/BY /D0/CT/DA /CT/D0/D7/BA /CC/CW/CT /CT/D2 /D8/D6/D3/D4 /DD \r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D7/CX/D2/CV/D8/CW/CT /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 S4f=\n/A1T\n0C4f\nTdT /B8 /D4/D0/D3/D8/D8/CT/CS /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU\n/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /D8/CW/CT /CX/D2/D7/CT/D8 /D3/CU /BY/CX/CV/BA /BI/CP/BA/CC/CW/CT /CT/D2 /D8/D6/D3/D4 /DD\n40\n30\n20\n10\n0  C (J/mole K)\n302520151050CeAu2Ge2\n 0   kOe\n 20 kOe\n 60  kOe\n 80  kOe(a)12\n8\n4\n0 S (J/mole K)\n150100500\n T (K)CeAu2Ge2\nR ln 2\n12\n10\n8\n6\n4\n2\n0  C4f (J/mole k)\n1801501209060300\n Temperature (K) C4f\n Schottky FitCeAu2Ge2 (b)/BY/CX/CV/D9/D6/CT /BI/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /C0/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /BV/CT/BT/D9 2\n/BZ/CT2\n/CX/D2/D4/D6/CT/D7/CT/D2\r/CT /CP/D2/CS /CP/CQ/D7/CT/D2\r/CT /D3/CU /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS/D7/BA /CC/CW/CT /CX/D2/D7/CT/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT/BG /CU /CT/D2 /D8/D6/D3/D4 /DD /CP/CV/CP/CX/D2/D7/D8 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CU/D3/D6 /D8/CW/CT /D7/CP/D1/CT/BA /CQ/B5 /C5/CP/CV/D2/CT/D8/CX\r\r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /B4/BV4f\n/B5 /D8/D3 /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /BV/CT/BT/D9 2\n/BZ/CT2\n/DB/CX/D8/CW /CP/CB\r /CW/D3/D8/D8/CZ/DD /AS/D8/BA/CX/D7 /BH/BA/BH /C2/BB/D1/D3/D0 /C3 /CP/D8 /CCN\n/B8 \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /DA /CP/D0/D9/CT /CU/D3/D6 /CP /DB /CT/D0/D0 /CX/D7/D3/D0/CP/D8/CT/CS/CS/D3/D9/CQ/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CP/D2/CS /CP/D8/D8/CP/CX/D2/D7 /CP /DA /CP/D0/D9/CT /D3/CU /BD/BF/BA/BG /C2/BB/D1/D3/D0 /C3/CP/D8 /BD/BL/BC /C3/B8 \r/D3/D1/D4/CP/D6/CP/CQ/D0/CT /D8/D3 /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0/D0/DD /CT/DC/D4 /CT\r/D8/CT/CS /DA /CP/D0/D9/CT/D3/CU /CA /D0/D2 /BI /B4/BD/BG/BA/BL /C2/BB/D1/D3/D0 /C3/B5/BA/C1/D2 /D3/D6/CS/CT/D6 /D8/D3 /D9/D2/CS/CT/D6/D7/D8/CP/D2/CS /D8/CW/CT /D1/CP/CV/D2/CT/D8/D3 \r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT/CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CP/D2/CS /D8/D3 /C3/D2/D3 /DB /CP/CQ /D3/D9/D8 /D8/CW/CT \r/D6/DD/D7/D8/CP/D0 /AS/CT/D0/CS /CT/D2/CT/D6/CV/DD/D0/CT/DA /CT/D0 /D7/D4/D0/CX/D8/D8/CX/D2/CV/D7 /D3/CU /D8/CW/CT /CA3+/CX/D3/D2 /CX/D2 /CA/BT/D92\n/BZ/CT2\n/B8 /DB /CT /CW/CP /DA /CT/D4 /CT/D6/CU/D3/D6/D1/CT/CS /D8/CW/CT /BV/BX/BY \r/CP/D0\r/D9/D0/CP/D8/CX/D3/D2/D7 /D9/D7/CX/D2/CV /D8/CW/CT /D4 /D3/CX/D2 /D8 \r /CW/CP/D6/CV/CT/D1/D3 /CS/CT/D0/BA /CC/CW/CT /D6/CP/D6/CT/B9/CT/CP/D6/D8/CW /CP/D8/D3/D1 /CX/D2 /D8/CW/CX/D7 /D7/CT/D6/CX/CT/D7 /D3/CU \r/D3/D1/D4 /D3/D9/D2/CS/D7/D3 \r\r/D9/D4 /DD /D8/CW/CT2a /CF /DD\r /CZ /D3/AR /D4 /D3/D7/CX/D8/CX/D3/D2 /DB/CW/CX\r /CW /CW/CP/D7 /D8/CW/CT /D8/CT/D8/D6/CP/CV/D3/D2/CP/D0/D4 /D3/CX/D2 /D8 /D7/DD/D1/D1/CT/D8/D6/DD /BA /CC/CW/CT /BV/BX/BY /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2 /CU/D3/D6 /CP /D8/CT/D8/D6/CP/CV/D3/D2/CP/D0/D7/DD/D1/D1/CT/D8/D6/DD /CX/D7 /CV/CX/DA /CT/D2 /CQ /DD /B8\nHCEF=B0\n2O0\n2+B0\n4O0\n4+B4\n4O4\n4+B0\n6O0\n6+B4\n6O4\n6, /B4/BI/B5/DB/CW/CT/D6/CTBm\nℓ\n/CP/D2/CSOm\nℓ\n/CP/D6/CT /D8/CW/CT /BV/BX/BY /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D2/CS /D8/CW/CT/CB/D8/CT/DA /CT/D2/D7 /D3/D4 /CT/D6/CP/D8/D3/D6/D7/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /CJ/BE/BE /B8 /BE/BF ℄/BA /CC/CW/CT /BV/BX/BY /D7/D9/D7/B9\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /CX/D7 /CS/CT/AS/D2/CT/CS /CP/D7/BI\n400\n300\n200\n100\n0  χ−1(mole/emu)\n300250200150100500\n Temperature (K) H // [001]\n H // [100]\n CEF Fit\n CeAu2Ge2/BY/CX/CV/D9/D6/CT /BJ/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /C1/D2 /DA /CT/D6/D7/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /BV/CT/BT/D9 2\n/BZ/CT2/CU/D3/D6 /C0 /BB/BB /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /CC/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT /D8/CW/D6/D3/D9/CV/CW /D8/CW/CT/CS/CP/D8/CP /D4 /D3/CX/D2 /D8/D7 /D6/CT/D4/D6/CT/D7/CT/D2 /D8 /D8/CW/CT \r/D6/DD/D7/D8/CP/D0 /CT/D0/CT\r/D8/D6/CX\r /AS/CT/D0/CS /AS/D8/BA\nχCEFi=N(gJµB)21\nZ\n/summationdisplay\nm/negationslash=n| /angbracketleftm|Ji|n/angbracketright |21−e−β∆m,n\n∆m,ne−βEn+/summationdisplay\nn| /angbracketleftn|Ji|n/angbracketright |2βe−βEn\n, /B4/BJ/B5/DB/CW/CT/D6/CTgJ\n/CX/D7 /D8/CW/CT /C4/CP/D2/CS/GHg /B9 /CU/CP\r/D8/D3/D6/B8En\n/CP/D2/CS|n/angbracketright /CP/D6/CT /D8/CW/CTn /D8/CW/CT/CX/CV/CT/D2 /DA /CP/D0/D9/CT /CP/D2/CS /CT/CX/CV/CT/D2/CU/D9/D2\r/D8/CX/D3/D2/B8 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /BA Ji\n/B4i /BPx /B8y/CP/D2/CSz /B5 /CX/D7 /D8/CW/CT \r/D3/D1/D4 /D3/D2/CT/D2 /D8 /D3/CU /D8/CW/CT /CP/D2/CV/D9/D0/CP/D6 /D1/D3/D1/CT/D2 /D8/D9/D1/B8 /CP/D2/CS\n∆m,n=En−Em\n/B8Z=/summationtext\nne−βEn/CP/D2/CSβ= 1/kBT /BA/CC/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /CX/D2\r/D0/D9/CS/CX/D2/CV /D8/CW/CT /D1/D3/D0/CT\r/D9/D0/CP/D6 /AS/CT/D0/CS\r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 λi\n/CX/D7 /CV/CX/DA /CT/D2 /CQ /DD\nχ−1\ni=χ−1\nCEFi−λi. /B4/BK/B5/BY /D3/D6 /BV/CT3+/CX/D3/D2/D7 /D8/CW/CTO6\n/D8/CT/D6/D1/D7 /CX/D2 /D8/CW/CT /CP/CQ /D3 /DA /CT /C0/CP/D1/CX/D0/D8/D3/D2/CX/CP/D2/DA /CP/D2/CX/D7/CW/CT/D7 /D6/CT/D7/D9/D0/D8/CX/D2/CV /CX/D2 /D3/D2/D0/DD /D8/CW/D6/CT/CT \r/D6/DD/D7/D8/CP/D0 /AS/D0/CT/CS /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/BA/CC/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /BV/CT/BT/D92\n/BZ/CT2\n/DB/CX/D8/CW /AS/CT/D0/CS /CP/D0/D3/D2/CV/CQ /D3/D8/CW /D8/CW/CT \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /DB /CP/D7 /AS/D8/D8/CT/CS /D8/D3 /D8/CW/CT/D1/CT/D2 /D8/CX/D3/D2/CT/CS /BV/BX/BY /D1/D3 /CS/CT/D0 /CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BJ/BA /CC/CW/CT /BV/BX/BY/D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D3/D6 /D8/CW/CT /CQ /CT/D7/D8 /AS/D8 /CP/D6/CTB0\n2\n/BP /B9/BI/BA/BG /C3/B8B0\n4/BP /B9/BC/BA/BE/BJ /C3 /CP/D2/CSB4\n4\n/BP /BE/BA/BI /C3 /DB/CX/D8/CW /CP /D1/D3/D0/CT\r/D9/D0/CP/D6 /AS/CT/D0/CS \r/D3/D2 /D8/D6/CX/B9/CQ/D9/D8/CX/D3/D2 /D3/CUλ(100) /BP /B9/BG/BD /C3 /CP/D2/CSλ(001) /BP /B9/BK /C3 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD/CU/D3/D6 /AS/CT/D0/CS /CP/D0/D3/D2/CV /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /CC/CW/CT /D2/CT/CV/CP/D8/CX/DA /CT/DA /CP/D0/D9/CT /D3/CUλ /D7/D9/D4/D4 /D3/D6/D8/D7 /D8/CW/CT /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /CT/DC\r /CW/CP/D2/CV/CT /CX/D2/B9/D8/CT/D6/CP\r/D8/CX/D3/D2 /CP/D1/D3/D2/CV /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D7/BA /CC/CW/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD \r/D3/D9/D0/CS/CQ /CT /AS/D8/D8/CT/CS /D9/D7/CX/D2/CV /CP /D7/CT/D8 /D3/CU /DA /CP/D0/D9/CT/D7 /D3/CU /BV/BX/BY /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CQ/D9/D8 /D3/D2/D0/DD/D8/CW/D3/D7/CT /DA /CP/D0/D9/CT/D7 /DB /CT/D6/CT \r/D3/D2/D7/CX/CS/CT/D6/CT/CS /DB/CW/CX\r /CW \r/D3/D9/D0/CS /CP/D0/D7/D3 /AS/D8 /D8/CW/CT /CT/DC/B9/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0/D0/DD /D3/CQ/D8/CP/CX/D2/CT/CS /CB\r /CW/D3/D8/D8/CZ/DD /CP/D2/D3/D1/CP/D0/DD /BA /CC/CW/CT /BV/BX/BY /D7/D4/D0/CX/D8/CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /D3/CQ/D8/CP/CX/D2/CT/CS /CU/D6/D3/D1 /D8/CW/CT /CP/CQ /D3 /DA /CT /BV/BX/BY /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7/CP/D6/CT /D8/CW/D6/CT/CT /CS/D3/D9/CQ/D0/CT/D8/D7 △0\n/BP /BC /C3/B8△1\n/BP /BD/BE/BK /C3 /CP/D2/CS△2\n/BP /BD/BL/BL /C3/DB/CW/CX\r /CW /CP/D6/CT /CX/D2 /CT/DC\r/CT/D0/D0/CT/D2 /D8 /CP/CV/D6/CT/CT/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /DA /CP/D0/D9/CT/D7 /CS/CT/D6/CX/DA /CT/CS/CU/D6/D3/D1 /D2/CT/D9/D8/D6/D3/D2 /D7\r/CP/D8/D8/CT/D6/CX/D2/CV /D6/CT/D7/D9/D0/D8/D7 /D3/D2 /D4 /D3/D0/DD\r/D6/DD/D7/D8/CP/D0/D0/CX/D2/CT /D7/CP/D1/D4/D0/CT/BA/CC/CW/CT/D7/CT /CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /DB /CT/D6/CT /D9/D7/CT/CS /D8/D3 \r/CP/D0\r/D9/D0/CP/D8/CT /D8/CW/CT /CB\r /CW/D3/D8/D8/CZ/DD\n\r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D9/D7/CX/D2/CV /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2\nCSch(T) =R/bracketleftBigg/summationtext\nigie−Ei/T/summationtext\nigiE2\nie−Ei/T−/bracketleftbig/summationtext\nigiEie−Ei/T/bracketrightbig2\nT2/bracketleftbig/summationtext\nigie−Ei/T/bracketrightbig2/bracketrightBigg/B4/BL/B5/DB/CW/CT/D6/CTR /CX/D7 /CP /CV/CP/D7 \r/D3/D2/D7/D8/CP/D2 /D8/B8 Ei\n/CX/D7 /D8/CW/CT /CT/D2/CT/D6/CV/DD /CX/D2 /D9/D2/CX/D8/D7 /D3/CU/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D2/CSgi\n/CX/D7 /D8/CW/CT /CS/CT/CV/CT/D2/CT/D6/CP\r/DD /D3/CU /D8/CW/CT /CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/BA/CC/CW/CT \r/CP/D0\r/D9/D0/CP/D8/CT/CS /CB\r /CW/D3/D8/D8/CZ/DD /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CX/D7 /CX/D2 /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/B9/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /D3/CQ/D7/CT/D6/DA /CT/CS /D3/D2/CT /CP/D7 /D7/CT/CT/D2 /CX/D2 /BY/CX/CV/BA /BI /CQ/BA /BT \r\r/D3/D6/CS/CX/D2/CV/D8/D3 /D8/CW/CT /D1/CT/CP/D2 /AS/CT/D0/CS /D8/CW/CT/D3/D6/DD /B8 /D8/CW/CT /BV/BX/BY /D4/CP/D6/CP/D1/CT/D8/CT/D6 B0\n2\n/CX/D7 /D6/CT/B9/D0/CP/D8/CT/CS /D8/D3 /D8/CW/CT /CT/DC\r /CW/CP/D2/CV/CT \r/D3/D2/D7/D8/CP/D2 /D8 /CP/D2/CS /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /BV/D9/D6/CX/CT/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CQ /DD /D8/CW/CT /D6/CT/D0/CP/D8/CX/D3/D2 /CJ/BE/BG ℄\nθ[001]\np=J(J+1)\n3kBJex[001]−(2J−1)(2J+3)\n5kBB0\n2, /B4/BD/BC/B5\nθ[100]\np=J(J+1)\n3kBJex[100]+(2J−1)(2J+3)\n10kBB0\n2. /B4/BD/BD/B5/CB/D9/CQ/D7/D8/CX/D8/D9/D8/CX/D2/CV /D8/CW/CT /DA /CP/D0/D9/CT/D7 /D3/CU /D8/CW/CT /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /DB /CT /D3/CQ/D8/CP/CX/D2\nJ[100]\nex\n/BP /B9/BD/BG/BA/BH /C3 /CP/D2/CSJ[001]\nex\n/BP /B9/BG/BA/BJ /C3/BA /CC/CW/CT /D2/CT/CV/CP/D8/CX/DA /CT /DA /CP/D0/D9/CT/D3/CU /D8/CW/CT /CT/DC\r /CW/CP/D2/CV/CT \r/D3/D2/D7/D8/CP/D2 /D8 /CP/D0/D3/D2/CV /CQ /D3/D8/CW /D8/CW/CT /D4/D6/CX/D2\r/CX/D4/CP/D0 /CS/CX/B9/D6/CT\r/D8/CX/D3/D2/D7 /CX/D1/D4/D0/DD /CP/D2 /D3 /DA /CT/D6/CP/D0/D0 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2/CP/D1/D3/D2/CV /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D7/BA/BY/CX/CV/BA /BK/CP /D7/CW/D3 /DB/D7 /D8/CW/CT /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CS/CT/D4 /CT/D2/CS/CT/D2\r/CT /D3/CU /D6/CT/D7/CX/D7/B9/D8/CX/DA/CX/D8 /DD /CU/D3/D6 /BV/CT/BT/D92\n/BZ/CT2\n/DB/CX/D8/CW \r/D9/D6/D6/CT/D2 /D8 /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /CJ/BD/BC/BC℄ /CP/D2/CS/CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D7/CW/D3 /DB/D7 /CP /D1/CT/D8/CP/D0/D0/CX\r /CQ /CT/CW/CP /DA/B9/CX/D3/D6 /CS/D3 /DB/D2 /D8/D3≈ /BD/BF/BC /C3 /CU/D3/D0/D0/D3 /DB /CT/CS /CQ /DD /CP /CQ/D6/D3/CP/CS /CW /D9/D1/D4 /CP/D2/CS /D8/CW/CT/D2/CS/D6/D3/D4/D7 /CU/CP/D7/D8/CT/D6 /CP/D8 /D8/CW/CT /D3/D6/CS/CT/D6/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /B4 /CCN\n/BP /BD/BF/BA/BH /C3/B5/BJ\n160\n120\n80\n40\n0  ρ (µΩ cm)\n300250200150100500\n Temperature (K) J // [100]\n J // [001]\n \n  CeAu2Ge2\n(a)40\n35\n30\n25  ρ (µΩ cm)\n302520151050\n T (K) 0   kOe\n 90 kOe\n \n CeAu2Ge2\n \n H // [001]: J // [100](I)\n-20-1001020  MR (%)\n80 60 40 20 0\n Magnetic Field (kOe) 2    K\n 3.5 K\n 5.5 K\n 8    K\n 12  K\n 16  K\n 20  K\n CeAu2Ge2\n 2   K\n 8   K\n 16 K\n J // [100] : H // [001] J // [001] : H // [100]\n(b)46\n44\n42  ρ (µΩ cm) \n302520151050\nT (K)CeAu2Ge2\nAu:Ge Flux\nJ // [100]\n(II) /BY/CX/CV/D9/D6/CT /BK/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /CA/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D3/CU /BV/CT/BT/D9 2\n/BZ/CT2\n/DB/CX/D8/CW /C2/BB/BB /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ 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\r/D3/D1/B9/D4/CP/D6/CT/CS /D8/D3 /D8/CW/CP/D8 /CP/D8 /BE /C3/BA /CC/CW/CT /CQ /CT/CW/CP /DA/CX/D3/D6 /CX/D7 /CP/D0/D7/D3 /CX/D2 /CP/CV/D6/CT/CT/D1/CT/D2 /D8/DB/CX/D8/CW /D8/CW/CT /C3 /D3/D2/CS/D3 /CQ /CT/CW/CP /DA/CX/D3/D6 /DB/CW/CX\r /CW /CT/DC/CW/CX/CQ/CX/D8 /CP /D1/CX/D2/CX/D1 /D9/D1 /CX/D2/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /DA /CP/D6/CX/CP/D8/CX/D3/D2 /D3/CU /C5/CA /CJ/BE/BH ℄/BA /BT /D8 /CW/CX/CV/CW /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B4/CP/CQ /D3 /DA /CT /BH/BA/BH /C3/B5 /D8/CW/CT /C5/CA /CX/D2\r/D6/CT/CP/D7/CT/D7 /DB/CX/D8/CW /AS/CT/D0/CS /CS/D9/CT /D8/D3 /D8/CW/CT/CX/D2\r/D6/CT/CP/D7/CX/D2/CV /D4 /D3/D7/CX/D8/CX/DA /CT \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CU/D6/D3/D1 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r\r/D3/D9/D4/D0/CX/D2/CV /CP/D2/CS /CS/CT\r/D6/CT/CP/D7/CX/D2/CV \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CU/D6/D3/D1 /C3 /D3/D2/CS/D3 /CT/AR/CT\r/D8/BA/CC/CW/CT /D1/CP/DC/CX/D1 /D9/D1 /D3 \r\r/D9/D6/D7 /CP/D8 /BK /C3 /CP/D2/CS /D8/CW/CT/D2 /CS/CT\r/D6/CT/CP/D7/CT/D7 /DB/CX/D8/CW/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D7 /CT/DC/D4 /CT\r/D8/CT/CS/BA /CC/CW/CT /D2/CT/CV/CP/D8/CX/DA /CT \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CX/D2/D8/CW/CT /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT /CP/D8 /CW/CX/CV/CW /AS/CT/D0/CS/D7 /CX/D7 /CS/D9/CT /D8/D3 /D8/CW/CT /CS/CT/B9\r/D6/CT/CP/D7/CT /CX/D2 /D8/CW/CT /D7/D4/CX/D2 /CS/CX/D7/D3/D6/CS/CT/D6 /D7\r/CP/D8/D8/CT/D6/CX/D2/CV/BA /CC/CW/CT /C5/CA /DB/CX/D8/CW /C2/BB/BB /CJ/BC/BC/BD℄ /CP/D2/CS /C0 /BB/BB /CJ/BD/BC/BC℄ /CX/D7 /CX/D2 /D7/CW/CP/D6/D4 \r/D3/D2 /D8/D6/CP/D7/D8 /D8/D3 /D8/CW/CT /CU/D3/D6/B9/D1/CT/D6 /D3/D2/CT /D7/CW/D3 /DB/CX/D2/CV /CP /D4 /D3/D7/CX/D8/CX/DA /CT \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CP/D8 /CP/D0/D0 /AS/CT/D0/CS/D7 /CP/D2/CS/D2/CT/CP/D6/D0/DD /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CQ /CT/CW/CP /DA/CX/D3/D6/BA /CC/CW/CT /D8/CT/D1/D4 /CT/D6/B9/CP/D8/D9/D6/CT /CX/D2/CS/CT/D4 /CT/D2/CS/CT/D2 /D8 /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT /C5/CA /CX/D2/CS/CX\r/CP/D8/CT/D7 /D8/CW/CT /CP/CQ/B9/D7/CT/D2\r/CT /D3/CU /D1/CP/CV/D2/CT/D8/CX\r \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/BA /BV/D3/D1/D4/CP/D6/CX/D2/CV /D8/CW/CT /C5/CA /CQ /CT/B9/CW/CP /DA/CX/D3/D6 /DB/CX/D8/CW /D8/CW/CP/D8 /D3/CU /D8/CW/CT /C4/CP \r/D3/D9/D2 /D8/CT/D6/D4/CP/D6/D8 /CP/D2/CS /CP/D7/D7/D9/D1/CX/D2/CV /D8/CW/CP/D8/CP /D7/CX/D1/CX/D0/CP/D6 /CP/D4/D4/D0/CX/CT/D7 /CW/CT/D6/CT/B8 /CX/D8 /CX/D7 /D4 /D3/D7/D7/CX/CQ/D0/CT /D8/CW/CP/D8 /CP/D2 /CP/D2/CX/D7/D3/D8/D6/D3/D4 /DD /CX/D2/D8/CW/CT /BY /CT/D6/D1/CX /D7/D9/D6/CU/CP\r/CT \r/CP/D9/D7/CT /CP \r/D3/D1/D4/CP/D6/CP/D8/CX/DA /CT/D0/DD /D0/CP/D6/CV/CT /C5/CA /DB/CX/D8/CW /C2/BB/BB /CJ/BC/BC/BD℄ /D7/D9/D4/D4/D6/CT/D7/D7/CX/D2/CV /D8/CW/CT /D3/D8/CW/CT/D6 /D7/D1/CP/D0/D0/CT/D6 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/BA /CF/CX/D8/CW/C2 /BB/BB /CJ/BD/BC/BC℄/B8 /CQ /CT\r/CP/D9/D7/CT /D3/CU /D8/CW/CT /D7/D1/CP/D0/D0/CT/D6 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /CU/D6/D3/D1 /D8/CW/CT/BY /CT/D6/D1/CX /D7/D9/D6/CU/CP\r/CT/B8 /D8/CW/CT /D3/D8/CW/CT/D6 /D1/CP/CV/D2/CT/D8/CX\r \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2/D7 /CP/D6/CT /CS/D3/D1/B9/CX/D2/CP/D2 /D8/BA0.20\n0.15\n0.10\n0.05\n0  χ (emu/mole)\n100 80 60 40 20 0\n Temperature (K) H // [100]\n H // [001]\n PrAu2Ge2\n  3 kOe\n(a)\n2.0\n1.5\n1.0\n0.5\n0  M (µΒ/f.u.)\n120100806040200\n Magnetic Field (kOe)PrAu2Ge2\n H // [100]\n H // [001]\n(b) 2 K/BY/CX/CV/D9/D6/CT /BD/BC/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /C5/CP/CV/D2/CT/D8/CX\r /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU/C8/D6/BT/D92\n/BZ/CT2\n/DB/CX/D8/CW /D1/CP/CV/D2/CT/D8/CX\r /AS/CT/D0/CS /B4/BF/CZ/C7/CT/B5 /CP/D4/D4/D0/CX/CT/CS /CP/D0/D3/D2/CV /D8/CW/CT /D8 /DB /D3\r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA /CQ/B5 /C5/CP/CV/D2/CT/D8/CX\r /CX/D7/D3/D8/CW/CT/D6/D1 /CP/D8 /BE /C3 /CU/D3/D6/D8/CW/CT /D7/CP/D1/CT /DB/CX/D8/CW /AS/CT/D0/CS /CP/D0/D3/D2/CV /CQ /D3/D8/CW /D8/CW/CT \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA/BV/BA /C8/D6/BT/D9 2\n/BZ/CT2/C8/D6/BT/D92\n/BZ/CT2\n/D3/D6/CS/CT/D6/D7 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /CP/D8 /BL /C3 /DB/CX/D8/CW/CJ/BC/BC/BD℄ /CP/D7 /D8/CW/CT /CT/CP/D7/DD /CP/DC/CX/D7 /D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /B4/BY/CX/CV/BA /BD/BC /CP/B5 /D7/CX/D1/CX/D0/CP/D6/D8/D3 /BV/CT/BT/D92\n/BZ/CT2\n/BA /C1/D2 /D8/CW/CT /D4/CP/D6/CP/D1/CP/CV/D2/CT/D8/CX\r /D7/D8/CP/D8/CT/B8 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r/D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /DB /CP/D7 /AS/D8/D8/CT/CS /D8/D3 /D8/CW/CT /BV/D9/D6/CX/CT/B9/CF /CT/CX/D7/D7 /D0/CP /DB/BA /CC/CW/CT /AS/D8/CV/CX/DA /CT/D7µeff\n/CP/D2/CSθp\n/CP/D7 /BF/BA/BH/BJµB\n/BB/C8/D6 /CP/D2/CS /B9/BD/BC /C3 /CP/D2/CS /BD/BC /C3/CU/D3/D6 /AS/CT/D0/CS /D4/CP/D6/CP/D0/D0/CT/D0 /D8/D3 /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/B8 /D6/CT/D7/D4 /CT\r/B9/D8/CX/DA /CT/D0/DD /BA /CC/CW/CT /DA /CP/D0/D9/CT /D3/CUµeff\n/CX/D7 /CT/D5/D9/CP/D0 /D8/D3 /D8/CW/CT /D8/CW/CT/D3/D6/CT/D8/CX\r/CP/D0/D0/DD/CT/DC/D4 /CT\r/D8/CT/CS /DA /CP/D0/D9/CT /D3/CU /C8/D63+/CX/D3/D2/BA /CC/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CX/D7/D3/D8/CW/CT/D6/D1 /CP/D8/BE /C3 /B4/BY/CX/CV/BA /BD/BC /CQ/B5 /DB/CX/D8/CW /C0 /BB/BB /CJ/BC/BC/BD℄ /D7/CW/D3 /DB/D7 /CP /D0/CX/D2/CT/CP/D6 /CQ /CT/CW/CP /DA/B9/CX/D3/D6 /D9/D4 /D8/D3 /BE/BC /C3/C7/CT /CX/D2 \r/D3/D2/AS/D6/D1/CP/D8/CX/D3/D2 /DB/CX/D8/CW /D8/CW/CT /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/B9/D2/CT/D8/CX\r /D2/CP/D8/D9/D6/CT /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS/BA /BT /D8≈ /BE/BE /C3/C7/CT /B4/C0 /BB/BB /CJ/BC/BC/BD℄/B5/D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS /D9/D2/CS/CT/D6/CV/D3 /CT/D7 /CP /D7/D4/CX/D2 /AT/CX/D4 /D8 /DD/D4 /CT /D1/CT/D8/CP/D1/CP/CV/D2/CT/D8/CX\r/D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /CU/D3/D0/D0/D3 /DB /CT/CS /CQ /DD /D7/D0/D3 /DB /CX/D2\r/D6/CT/CP/D7/CT /DB/CX/D8/CW /AS/CT/D0/CS /CP/D8/D8/CP/CX/D2/CX/D2/CV/CP /D1/CP/CV/D2/CT/D8/CX\r /D1/D3/D1/CT/D2 /D8 /D3/CU≈ /BE/BA/BEµB\n/BB/CU/BA/D9/BA /CP/D8 /BD/BE/BC /C3/C7/CT/BA /CC/CW/CT/D1/D3/D1/CT/D2 /D8 /CX/D7 /D0/CT/D7/D7 /D8/CW/CP/D2 /D8/CW/CT /D7/CP/D8/D9/D6/CP/D8/CX/D3/D2 /D1/D3/D1/CT/D2 /D8 /D3/CU /C8/D63+/CX/D3/D2/BA/CC/CW/CT /D0/CT/D7/D7 /D1/D3/D1/CT/D2 /D8 /D1/CP /DD /CQ /CT /CS/D9/CT /D8/D3 /D8/CW/CT \r/D6/DD/D7/D8/CP/D0 /AS/CT/D0/CS /CT/AR/CT\r/D8/BA/CC/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CX/D7/D3/D8/CW/CT/D6/D1 /DB/CX/D8/CW /AS/CT/D0/CS /CP/D0/D3/D2/CV /CJ/BD/BC/BC℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/CX/D2\r/D6/CT/CP/D7/CT/D7 /D0/CX/D2/CT/CP/D6/D0/DD /DB/CX/D8/CW /CP /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2 /D3/CU≈ /BD/BA/BGµB\n/BB/CU/BA/D9/BA/CP/D8 /BD/BE/BC /C3/C7/CT/B8 /CX/D2/CS/CX\r/CP/D8/CX/D2/CV /D8/CW/CT /CW/CP/D6/CS /CP/DC/CX/D7 /D3/CU /D1/CP/CV/D2/CT/D8/CX/DE/CP/D8/CX/D3/D2/BA/CC/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /D4/CW/CP/D7/CT /CS/CX/CP/CV/D6/CP/D1 \r/D3/D2/D7/D8/D6/D9\r/D8/CT/CS /CP/D7 /CS/CX/D7\r/D9/D7/D7/CT/CS/CQ /CT/CU/D3/D6/CT /B4/BY/CX/CV/BA /BD/BD /CP/B5 /D7/CW/D3 /DB/D7 /D8/CW/CT /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /CP/D2/CS /D8/CW/CT/BL\n25 \n20 \n15 \n10 \n5\n0  Magnetic Field (kOe)  Ferromagnet \nAntiferromagnet PrAu2Ge 2\n(a)\n10 8 6 4 2 0\n200 \n150 \n100 \n50 \n0  χ−1 (mole/emu) \n300 250 200 150 100 50 0\nTemperature (K)  H // [001] \n H // [100] \n CEF Fit \nPrAu2Ge 2\n(b)H // [001] /BY/CX/CV/D9/D6/CT /BD/BD/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /C5/CP/CV/D2/CT/D8/CX\r /C8/CW/CP/D7/CT /BW/CX/CP/CV/D6/CP/D1 /D3/CU/C8/D6/BT/D92\n/BZ/CT2\n\r/D3/D2/D7/D8/D6/D9\r/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT /D1/CP/CV/D2/CT/D8/CX\r /CX/D7/D3/D8/CW/CT/D6/D1/D7 /CP/D8 /DA /CP/D6/CX/B9/D3/D9/D7 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7/BA /CQ/B5 /C1/D2 /DA /CT/D6/D7/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /D3/CU /C8/D6/BT/D92\n/BZ/CT2\n/DB/CX/D8/CW\r/D6/DD/D7/D8/CP/D0 /CT/D0/CT\r/D8/D6/CX\r /AS/CT/D0/CS /AS/D8/BA/AS/CT/D0/CS /CX/D2/CS/D9\r/CT/CS /CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS/BA/BY /D3/D6 /CS/CT/D8/CP/CX/D0/CT/CS /CX/D2 /DA /CT/D7/D8/CX/CV/CP/D8/CX/D3/D2 /CP \r/D6/DD/D7/D8/CP/D0 /AS/CT/D0/CS /CP/D2/CP/D0/DD/D7/CX/D7 /D3/CU /D8/CW/CT\r/D3/D1/D4 /D3/D9/D2/CS /DB /CP/D7 /CS/D3/D2/CT /CQ /DD /AS/D8/D8/CX/D2/CV /D8/CW/CT /CX/D2 /DA /CT/D6/D7/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD/CP/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BD/BD /CQ/BA /CC/CW/CT /C7/CQ/D8/CP/CX/D2/CT/CS /DA /CP/D0/D9/CT /D3/CU /D8/CW/CT \r/D6/DD/D7/D8/CP/D0/AS/CT/D0/CS /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CTB0\n2\n/BP /B9/BD/BA/BE /C3/B8B0\n4\n/BP /BC/BA/BC/BK /C3/B8B4\n4\n/BP/BC/BA/BE/BH /C3/B8B0\n6\n/BP /B9/BC/BA/BC/BC/BC/BD/C3 /CP/D2/CSB4\n6\n/BP /BC/BA/BC/BC/BI /C3 /DB/CX/D8/CW /CP /D1/D3/D0/CT\r/B9/D9/D0/CP/D6 /AS/CT/D0/CS \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D3/CUλ(100) /BP /B9/BD/BC /C3 /CP/D2/CSλ(001) /BP/BC /C3 /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/D0/DD /CU/D3/D6 /AS/CT/D0/CS /CP/D0/D3/D2/CV /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/D7/BA/C7/D2/D0/DD /D8/CW/D3/D7/CT /DA /CP/D0/D9/CT /D3/CU \r/D6/DD/D7/D8/CP/D0 /AS/CT/D0/CS /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT \r/D3/D2/D7/CX/CS/B9/CT/D6/CT/CS /DB/CW/CX\r /CW /AS/D8/D7 /D8/CW/CT /D7/D9/D7\r/CT/D4/D8/CX/CQ/CX/D0/CX/D8 /DD /CP/D7 /DB /CT/D0/D0 /CP/D7 /D4/D6/D3 /DA/CX/CS/CT /CP /D7/CP/D8/B9/CX/D7/CU/CP\r/D8/D3/D6/DD /D6/CT/D4/D6/CT/D7/CT/D2 /D8/CP/D8/CX/D3/D2 /D3/CU /D8/CW/CT /CB\r /CW/D3/D8/D8/CZ/DD /CP/D2/D3/D1/CP/D0/DD /CX/D2/CU/CT/D6/D6/CT/CS/CU/D6/D3/D1 /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D1/CT/CP/D7/D9/D6/CT/D1/CT/D2 /D8/D7 /B4/CS/CT/D7\r/D6/CX/CQ /CT/CS /CQ /CT/D0/D3 /DB/B5/BA/CC/CW/CT \r/D6/DD/D7/D8/CP/D0 /AS/CT/D0/CS /D7/D4/D0/CX/D8 /CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 \r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D7/CX/D2/CV /D8/CW/CT/CP/CQ /D3 /DA /CT \r/D6/DD/D7/D8/CP/D0 /AS/CT/D0/CS /D4/CP/D6/CP/D1/CT/D8/CT/D6/D7 /CP/D6/CT△0\n/BP /BC /C3 /B4/CS/D3/D9/CQ/D0/CT/D8/B5/B8\n△1\n/BP /BF/BL /C3/B8△2\n/BP /BL/BK /C3/B8△3\n/BP /BD/BD/BH /C3/B8△4\n/BP /BD/BG/BK /C3/B8△5\n/BP/BD/BK/BI /C3/B4/CS/D3/D9/CQ/D0/CT/D8/B5 /CP/D2/CS△6\n/BP /BE/BI/BD /C3/BA /CC/CW/CT /CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0 /D7\r /CW/CT/D1/CT/D7/CW/D3 /DB/D7 /CP /CS/D3/D9/CQ/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /CU/D3/D6 /D8/CW/CT /C8/D63+/CX/D3/D2/BA /CC/CW/CT /CT/DC/B9\r /CW/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 \r/D3/D2/D7/D8/CP/D2 /D8 /D3/CQ/D8/CP/CX/D2/CT/CS /D9/D7/CX/D2/CV /BX/D5/BA /BD/BD /CP/D2/CS /BD/BC/CP/D6/CTJ[100]\nex\n/BP /B9/BC/BA/BD/BD/BG /C3 /CP/D2/CSJ[001]\nex\n/BP /B9/BD/BA/BE/BJ /C3/BA /CC/CW/CT /D2/CT/CV/CP/D8/CX/DA /CT/DA /CP/D0/D9/CT /D3/CU /D8/CW/CT /CT/DC\r /CW/CP/D2/CV/CT /CX/D2 /D8/CT/D6/CP\r/D8/CX/D3/D2 \r/D3/D2/D7/D8/CP/D2 /D8 /CP/D0/D3/D2/CV /CQ /D3/D8/CW /D8/CW/CT/CS/CX/D6/CT\r/D8/CX/D3/D2 /CX/D2/CS/CX\r/CP/D8/CT/D7 /CP/D2 /D3 /DA /CT/D6/CP/D0/D0 /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /CX/D2 /D8/CT/D6/CP\r/B9/D8/CX/D3/D2 /CP/D1/D3/D2/CV /D8/CW/CT /D1/D3/D1/CT/D2 /D8/D7/CC/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CQ /CT/CW/CP /DA/CX/D3/D6 /D3/CU /C8/D6/BT/D92\n/BZ/CT2\n/CX/D2 /BC/B8 /BE/BC /CP/D2/CS12\n10\n8\n6\n4\n2\n0  C4f (J/mole K)\n150 100 50 0\n Temperature (K)PrAu2Ge2(b)\n C4f\n Schottky Fit25\n20\n15\n10\n5\n0  C (J/mole K)\n20 15 10 5 0PrAu2Ge2\n  0 kOe\n 20 kOe\n 40 kOe(a)15\n10\n5\n0  S4f  (J/mole K)\n12080400 T (K)PrAu2Ge2/BY/CX/CV/D9/D6/CT /BD/BE/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /C0/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /C8/D6/BT/D92\n/BZ/CT2\n/CX/D2/BC/B8 /BE/BC /CP/D2/CS /BG/BC /C3/C7/CT/BA /CC/CW/CT /CX/D2/D7/CT/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT /BG /CU /CT/D2 /D8/D6/D3/D4 /DD /CP/CV/CP/CX/D2/D7/D8/D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /CQ/B5 /C5/CP/CV/D2/CT/D8/CX\r \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /B4/BV4f\n/B5 /D8/D3 /D8/CW/CT /CW/CT/CP/D8\r/CP/D4/CP\r/CX/D8 /DD /D3/CU /C8/D6/BT/D92\n/BZ/CT2\n/BN /D8/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT /D7/CW/D3 /DB/D7 /D8/CW/CT /CB\r /CW/D3/D8/D8/CZ/DD /CW/CT/CP/D8\r/CP/D4/CP\r/CX/D8 /DD \r/CP/D0\r/D9/D0/CP/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT /BV/BX/BY /D0/CT/DA /CT/D0/D7/BA/BG/BC /C3/C7/CT /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BD/BE /CP/BA /CC/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D7/CW/D3 /DB/D7 /CP/D2/CP/D2/D3/D1/CP/D0/DD /CP/D8 /D8/CW/CT /CP/D2 /D8/CX/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r /D3/D6/CS/CT/D6/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D3/CU /D8/CW/CT \r/D3/D1/D4 /D3/D9/D2/CS/BA /BT/D2 /CT/DC/D8/CT/D6/D2/CP/D0 /AS/CT/D0/CS /D3/CU /BE/BC /C3/C7/CT /D7/CW/CX/CU/D8/D7 /D8/CW/CT/D4 /CT/CP/CZ /D8/D3 /DB /CP/D6/CS/D7 /D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CP/D7 /CT/DC/D4 /CT\r/D8/CT/CS /CU/D3/D6 /CP/D2 /CP/D2 /D8/CX/B9/CU/CT/D6/D6/D3/D1/CP/CV/D2/CT/D8/CX\r/CP/D0/D0/DD /D3/D6/CS/CT/D6/CT/CS \r/D3/D1/D4 /D3/D9/D2/CS/BA /BT /D8 /CP /CW/CX/CV/CW/CT/D6 /AS/CT/D0/CS/D3/CU /BG/BC /C3/C7/CT/B8 /CP/CQ /D3 /DA /CT /D8/CW/CT /D1/CT/D8/CP/D1/CP/CV/D2/CT/D8/CX\r /D8/D6/CP/D2/D7/CX/D8/CX/D3/D2 /AS/CT/D0/CS/B8 /D8/CW/CT/D4 /CT/CP/CZ /DA /CP/D2/CX/D7/CW/CT/D7 /CP/D2/CS /D8/CW/CT/D6/CT /CX/D7 /CP /CQ/D6/D3/CP/CS /CW /D9/D1/D4/BA /CC/CW/CT /D1/CP/CV/B9/D2/CT/D8/CX\r \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D8/D3 /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /B4/BY/CX/CV/BA /BD/BE /CQ/B5 /DB /CP/D7/CX/D7/D3/D0/CP/D8/CT/CS /CQ /DD /D7/D9/CQ/D8/D6/CP\r/D8/CX/D2/CV /D8/CW/CT /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /D3/CU /C4/CP/BT/D92\n/BZ/CT2\n/BA/C1/D8 /D7/CW/D3 /DB/D7 /CP /D7/CW/CP/D6/D4 /D4 /CT/CP/CZ /CP/D8 /D8/CW/CT /D3/D6/CS/CT/D6/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CU/D3/D0/B9/D0/D3 /DB /CT/CS /CQ /DD /CP /CB\r /CW/D3/D8/D8/CZ/DD /CP/D2/D3/D1/CP/D0/DD /CP/D8 /CW/CX/CV/CW /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/BA /CC/CW/CT/CT/D2 /D8/D6/D3/D4 /DD \r/CP/D0\r/D9/D0/CP/D8/CT/CS /D9/D7/CX/D2/CV /D8/CW/CT /CT/D5/D9/CP/D8/CX/D3/D2 /CP/D7 /D1/CT/D2 /D8/CX/D3/D2/CT/CS /CQ /CT/B9/CU/D3/D6/CT /CX/D7 /D4/D0/D3/D8/D8/CT/CS /CP/D7 /CP /CU/D9/D2\r/D8/CX/D3/D2 /D3/CU /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /CX/D2 /D8/CW/CT /CX/D2/D7/CT/D8/D3/CU /BY/CX/CV/BA /BD/BE /CP/BA /C1/D8 /CP/D8/D8/CP/CX/D2/D7 /CP /DA /CP/D0/D9/CT /D3/CU≈ /BJ /C2/BB/D1/D3/D0 /C3 /CP/D8 /D8/CW/CT /D3/D6/B9/CS/CT/D6/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/B8 /DB/CW/CX\r /CW /CT/DC\r/CT/CT/CS/D7 /D7/D9/CQ/D7/D8/CP/D2 /D8/CX/CP/D0/D0/DD /D8/CW/CT /CT/D2/B9/D8/D6/D3/D4 /DD /CU/D3/D6 /CP /CS/D3/D9/CQ/D0/CT/D8 /CV/D6/D3/D9/D2/CS /D7/D8/CP/D8/CT /DB/CX/D8/CW /CT/AR/CT\r/D8/CX/DA /CT /C2 /BP /BD/BB/BE/B4/BH/BA/BJ/BI /C2/BB/D1/D3/D0 /C3/B5/BA /CC/CW/CT /CT/DC\r/CT/D7/D7 /CT/D2 /D8/D6/D3/D4 /DD /CP/D4/D4 /CT/CP/D6/D7 /CQ /CT\r/CP/D9/D7/CT /D3/CU/CP /D7/D9/CQ/D7/D8/CP/D2 /D8/CX/CP/D0 \r/D3/D2 /D8/D6/CX/CQ/D9/D8/CX/D3/D2 /D8/D3 /BV4f\n/CS/D9/CT /D8/D3 /D8/CW/CT /CB\r /CW/D3/D8/D8/CZ/DD/CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD /CP/D8 /D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /CP/D6/CX/D7/CX/D2/CV /CU/D6/D3/D1 /D8/CW/CT /AS/D6/D7/D8/CT/DC\r/CX/D8/CT/CS /BV/BX/BY /D0/CT/DA /CT/D0 /D0/DD/CX/D2/CV /CP/D8 /BF/BL /C3/BA /CC/CW/CT /D8/D3/D8/CP/D0 /CT/D2 /D8/D6/D3/D4 /DD /D3/CQ/B9/D8/CP/CX/D2/CT/CS /CP/D8 /BD/BH/BC /C3 /CX/D7 /BD/BJ/BA/BI /C2/BB/D1/D3/D0 /C3/B8 \r/D0/D3/D7/CT /D8/D3 /D8/CW/CT /CT/DC/D4 /CT\r/D8/CT/CS/BD/BC\n12\n9\n6\n3\n0  MR (%)\n80 60 40 20 0\n Magnetic Field (kOe) 2    K\n 3.5 K\n 5.5 K\n 8    K\n 12  KPrAu2Ge2\n J // [100] : H // [001] 2 K J // [001] : H // [100]\n(b)150\n100\n50\n0  ρ (µΩ cm)\n300 200 100 0\n Temperature (K)15\n10\n5 J // [100]\n J // [001]PrAu2Ge2(a)40.0\n38.0\n36.0\n34.0 ρ (µΩ cm)\n20 15 10 5 0\n T (K)4.6\n4.4\n4.2\n4.0\n3.8 J // [100]\n J // [100]_50 kOe\n J // [001]\n PrAu2Ge2/BY/CX/CV/D9/D6/CT /BD/BF/BM /B4/BV/D3/D0/D3/D6 /D3/D2/D0/CX/D2/CT/B5 /CP/B5 /CA/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D3/CU /C8/D6/BT/D92\n/BZ/CT2\n/DB/CX/D8/CW /C2/BB/BB /CJ/BD/BC/BC℄ /CP/D2/CS /CJ/BC/BC/BD℄ /CS/CX/D6/CT\r/D8/CX/D3/D2/BA /CC/CW/CT /CX/D2/D7/CT/D8 /D7/CW/D3 /DB/D7 /D8/CW/CT /CT/DC/D4/CP/D2/CS/CT/CS/D0/D3 /DB /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT /D4 /D3/D6/D8/CX/D3/D2 /DB/CX/D8/CW /CP/D6/D6/D3 /DB/D7 /CX/D2/CS/CX\r/CP/D8/CX/D2/CV /D8/CW/CT /D6/CT/D7/D4 /CT\r/D8/CX/DA /CT/CP/DC/CX/D7/BA /CQ/B5 /CC/CW/CT /D8/D6/CP/D2/D7/DA /CT/D6/D7/CT /D1/CP/CV/D2/CT/D8/D3/D6/CT/D7/CX/D7/D8/CP/D2\r/CT /D3/CU /D8/CW/CT /D7/CP/D1/CT /DB/CX/D8/CW/C2 /BB/BB /CJ/BD/BC/BC℄/BM /C0 /BB/BB /CJ/BC/BC/BD℄ /CP/D8 /DA /CP/D6/CX/D3/D9/D7 /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/D7 /CP/D2/CS /C2 /BB/BB /CJ/BC/BC/BD℄/BM/C0 /BB/BB /CJ/BD/BC/BC℄ /CP/D6/CT /D7/CW/D3 /DB/D2/BA/DA /CP/D0/D9/CT /D3/CU /BD/BK/BA/BE /C2 /BB/D1/D3/D0 /C3/BA /CC/CW/CT /D7/D3/D0/CX/CS /D0/CX/D2/CT /CX/D2 /BY/CX/CV/BA /BD/BE /CQ /D7/CW/D3 /DB/D7/D8/CW/CT /CB\r /CW/D3/D8/D8/CZ/DD /CW/CT/CP/D8 \r/CP/D4/CP\r/CX/D8 /DD \r/CP/D0\r/D9/D0/CP/D8/CT/CS /CU/D6/D3/D1 /D8/CW/CT /BV/BX/BY /D7/D4/D0/CX/D8/CT/D2/CT/D6/CV/DD /D0/CT/DA /CT/D0/D7 /D9/D7/CX/D2/CV /CT/DC/D4/D6/CT/D7/D7/CX/D3/D2 /BL/BA /BT/D2 /D3 /DA /CT/D6/CP/D0/D0 /CV/D3 /D3 /CS /CP/CV/D6/CT/CT/B9/D1/CT/D2 /D8 /DB/CX/D8/CW /D8/CW/CT /CT/DC/D4 /CT/D6/CX/D1/CT/D2 /D8/CP/D0 /BV4f\n/CX/D2\r/D6/CT/CP/D7/CT/D7 /D8/CW/CT \r/D3/D2/AS/CS/CT/D2\r/CT/CX/D2 /D8/CW/CT /D4/D6/D3/D4 /D3/D7/CT/CS /BV/BX/BY /D0/CT/DA /CT/D0 /D7\r /CW/CT/D1/CT/BA/CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /D3/CU /C8/D6/BT/D92\n/BZ/CT2\n/DB/CX/D8/CW \r/D9/D6/D6/CT/D2 /D8 /CP/D0/D3/D2/CV /D8/CW/CT /D8 /DB /D3/D4/D6/CX/D2\r/CX/D4/D0/CT \r/D6/DD/D7/D8/CP/D0/D0/D3/CV/D6/CP/D4/CW/CX\r /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /CX/D7 /D7/CW/D3 /DB/D2 /CX/D2 /BY/CX/CV/BA /BD/BF /CP/BA/CC/CW/CT /D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /DB/CX/D8/CW /C2 /BB/BB /CJ/BD/BC/BC℄ /CX/D7 /CW/CX/CV/CW/CT/D6 /D8/CW/CP/D2 /D8/CW/CP/D8 /DB/CX/D8/CW/C2 /BB/BB /CJ/BC/BC/BD℄ /CP/D7 /D7/CT/CT/D2 /CU/D3/D6 /CQ /D3/D8/CW /C4/CP /CP/D2/CS /BV/CT \r/D3/D1/D4 /D3/D9/D2/CS/D7/BA /CC/CW/CT/D6/CT/D7/CX/D7/D8/CX/DA/CX/D8 /DD /DB/CX/D8/CW \r/D9/D6/D6/CT/D2 /D8 /CP/D0/D3/D2/CV /CQ /D3/D8/CW /D8/CW/CT /CS/CX/D6/CT\r/D8/CX/D3/D2/D7 /D7/CW/D3 /DB/CP /D1/CT/D8/CP/D0/D0/CX\r /CQ /CT/CW/CP /DA/CX/D3/D6 /CS/D3 /DB/D2 /D8/D3 /D8/CW/CT /D3/D6/CS/CT/D6/CX/D2/CV /D8/CT/D1/D4 /CT/D6/CP/D8/D9/D6/CT/CU/D3/D0/D0/D3 /DB /CT/CS /CQ /DD /CP 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/DD/D7/BA /BD/BD /B8 /BE/BD/BG/BJ /B4/BD/BL/BK/BD/B5/CJ/BE/BI℄ /CA/BA /C2/BA /BX/D0/D0/CX/D3/D8 /CP/D2/CS /BY/BA /BT/BA /CF /CT/CS/CV/DB /D3 /D3 /CS/BM /C8/D6/D3 \r/BA /C8/CW /DD/D7/BA /CB/D3 \r/BA /BK/BD/B8/BK/BG/BI /B4/BD/BL/BI/BF/B5/BA"    },    {        "title": "0911.2879v1.Field_induced_resistivity_anisotropy_in_SrRuO3_films.pdf",        "content": "arXiv:0911.2879v1  [cond-mat.str-el]  15 Nov 2009Field induced resistivity anisotropy in SrRuO 3films\nYishai Shperber1, Isaschar Genish1, James W. Reiner2, and Lior Klein1\n1Department of Physics, Nano-magnetism Research Center,\nInstitute of Nanotechnology and Advanced Materials,\nBar-Ilan University, Ramat-Gan 52900, Israel and\n2Department of Applied Physics, Yale University,\nNew Haven, Connecticut 06520-8284\nAbstract\nSrRuO 3is anitinerant ferromagnet withorthorhombicstructurean duniaxial magnetocrystalline\nanisotropy - features expected to yield resistivity anisot ropy. Here we explore changes in the\nresistivity anisotropy of epitaxial SrRuO 3films due to induced magnetization in the paramagnetic\nstate by using the planar Hall effect. We find that the effect of the induced magnetization on the\nin-plane anisotropy is strongly angular dependent and we pr ovide a full description of this behavior\nat 160 K for induced magnetization in the (001) plane.\nPACS numbers: 73.50.Jt, 75.30.Gw\n1The itinerant ferromagnet SrRuO 3has attracted considerable experimental and theoret-\nical effort for its intriguing properties [1, 2, 3, 4]. Being almost cubic , thin films of SrRuO 3\nexhibit rather small anisotropy which is difficult to characterize by co mparing resistivity\nmeasurements taken on patterns with current flowing along differe nt directions relative the\ncrystallographic axes. Consequently, we have determined the zer o field anisotropy of epi-\ntaxial films of this compound, by measuring the planar Hall effect (PH E) [5], a method\nwhich provides a good local measurement of the anisotropy with high accuracy. To explore\nthe contribution of the magnetization Mto the anisotropy we examine how changing the\nmagnitude and orientation of the magnetization affects the resistiv ity anisotropy. We find\nthat the anisotropy is proportional to M2for any angle in the (001) plane; however, the\nproportionality coefficient is strongly angular dependent.\nOur samples are epitaxial films of SrRuO 3grown on slightly miscut (2◦) SrTiO 3. The\nfilms are orthorhombic ( a= 5.53˚A,b= 5.57˚A,c= 7.85˚A) [6] and their Curie temperature\nis∼150 K. The films grow with the in-plane caxis perpendicular to the miscut, and a\nandbaxes at 45◦relative to the film plane. The films exhibit uniaxial magnetocrystalline\nanisotropy with the easy axis along the baxis at T ≥Tc[6].\nThe data presented here are from a 27 nm thick film with resistivity ra tio of∼12. The\nmeasurements were done at 160 K on a pattern with current direct ion along [1 ¯11] for which\nthe PHE attains its maximal value [5].\nFigure 1(a) and Figure 1(b) show the transverse signal (symmetr ic (a) and antisymmetric\n(b)) vsθ, the angle between the applied magnetic field and the normal to the fi lm. Positive\nθcorresponds to an anticlockwise rotation. The PHE is the symmetric part of the transverse\nsignal [7, 8, 9] and in our configuration ρPHE=ρ[1¯10]−ρ[001]whereρ[1¯10]is the longitudinal\nresistivity along the [1 ¯10] direction and ρ[001]is the longitudinal resistivity along [001]. The\nantisymmetric part ( ρAS) consists of the ordinary Hall effect (OHE) and the extraordinary\nHall effect (EHE) [10].\nSubtracting the OHE contribution from ρASyields the EHE which we present in Fig-\nure 2(a). As is well known, the EHE is proportional to M⊥, the component of Mwhich\nis perpendicular to the plane of the film. However, the magnetocrys talline anisotropy of\nSrRuO 3allows us to determine the full vector M. As the easy axis is at 45◦to the film\nnormal, a field H, applied in the (001) plane at angle θrelative to the normal of the sample’s\nplane, creates magnetization Mat angleα(Figure 2(b)-inset), where the same field at angle\n21.5 2.5 3.5 4.5 \n0.2T \n0.6T \n1T \n2T \n3T \n4T \n5T \n7T \n9T ρPHE (µΩ  cm) (a) \n00.2 \n-45 0 45 90 135 ρAS  ( µΩ  cm) \nθ (deg.) (b) \nFIG. 1:ρPHE(a) andρAS(b) vsθ, the angle between the applied field and the film normal.\n90−θcreatesMwith the same magnitude but at angle 90 −α. Consequently, by measuring\ntheρEHEat angle θ(ρEHE(θ)) and 90- θ(ρEHE(90−θ)) we obtain ρEHE(θ)∝M⊥(θ) and\nρEHE(90−θ)∝M/bardbl(θ), where M/bardblandM⊥are the in-plane and perpendicular components\nofM, respectively. Thus we obtain a full description of the magnetizatio n vector [11].\nFigure 2(b) shows the vector of magnetization in arbitrary units ob tained for some of the\nfields we used in our measurements. In this polar graph we see that t he magnetization has\nits maximal value when it is aligned along the easy axis and its minimal value when it lies\nalong the hard axis. The data indicate that with H= 1 T the induced magnetization along\nthe easy axis is larger than the induced magnetization along the hard axis by a factor of\n∼5.5, in good agreement with a previous report [12].\nFigure 3 shows the dependence of ρPHEon m2wheremis defined as M/|/tildewiderM|and/tildewiderMis\nthe magnetization obtained with a field of 1 T applied along the easy axis . Based on the\nconstruction of the vector of magnetization presented in Figure 2 , we can extract the change\ninρPHEas a function of mfor a given orientation. We clearly see that:\nρPHE=ρo\nPHE+f(α)m2(1)\nwhereρ0\nPHEis the PHE measured at zero field and f(α) is the coefficient of m2whenm\n3-0.34 00.34 \n-45 0 45 90 135 ρEHE  ( µΩ  cm) \nθ (deg.) (a) \n015 30 45 60 75 90 105 \n120 \n195 \n210 \n225 \n240 \n255 270 285 300 315 330 345 0.2T \n0.6T \n1T \n2T \n3T \n4T \n5T \n7T \n9T \n00.5 11.5 2(b) \nα (deg.) \nFIG. 2: a) The EHE part of the antisymmetric signal vs θthe angle between the applied field and\nthe film normal. b) The magnitude (arbitrary units) and direc tion of the magnetization obtained\nfor rotating different fields in the (001) plane. Inset: Crysta llographic directions of the film and\ndefinitions of the angles θandα.\nis alongα. Figure 4 shows f( α) for some of the α’s. We see that there is an excellent fit of\nf(α) with:\nf(α) =a+bsin2α (2)\nThis means that one can write ρPHEas:\n43.6 3.7 3.8 3.9 44.1 4.2 \n0 0.2 0.4 0.6 0.8 1-45 \n-15 \n0\n30 \n45 \n60 \n90 \n105 \n135 \nm2ρPHE  (µΩ cm )\nFIG. 3:ρPHEvsm2. The lines are linear fits .\n-0.4 -0.3 -0.2 -0.1 00.1 0.2 \n-45 0 45 90 135 \nα (deg.) f(α) \nFIG. 4:f(α) whereαis the angle between mand the film normal. The solid line is a fit to Eq. 2.\nρPHE=ρ0\nPHE+am2+bm2\n/bardbl (3)\nwhere a=0.16 and b=-0.54.\nThe dependence we have found indicate that there are two effects of the induced magneti-\nzationontheresistivityanisotropy. First, thereisacontributionin dependent ofthedirection\nofMwith a positive coefficient awhich means that the anisotropy increases with increasing\nmagnetization. Second, there is another contribution sensitive to the in-plane projection of\nthe magnetization with a negative coefficient bwhich means that the anisotropy decreases\nwith increasing magnetization. The extremal points of f(α) are at α= 0◦, 90◦. When\nα= 0◦the induced magnetization is normal to J along both [1 ¯10] and [001]. At this angle\nthe anisotropy contribution is minimal and f(α) attains its maximal positive value. When\n5α= 90◦, the magnetization is parallel to [1 ¯10] and perpendicular to [001] and therefore its\nanisotropic contribution is maximal and f(α) attains its maximum negative value.\nL. K. acknowledges support by the Israel Science Foundation fou nded by the Israel\nAcademy of Sciences and Humanities. J. W. R. grew the samples at St anford University in\nthe laboratory of M. R. Beasley.\n[1] L. Klein, J. S. Dodge, C. H. Ahn, G. J. Snyder, T. H. Geballe , M. R. Beasley, and A.\nKapitulnik, Phys. Rev. Lett. 77, 2774, (1996).\n[2] M. S. Laad, and E. M¨ uller-Hartmann Phys. Rev. Lett., 87, 246402 (2001); Carsten Timm, M.\nE. Raikh, and Felix von Oppen, ibid.94, 036602, (2005).\n[3] P. Kostic, Y. Okada, N. C. Collins, Z. Schlesinger, J. W. R einer, L. Klein, A. Kapitulnik, T.\nH. Geballe, and M. R. Beasley, Phys. Rev. Lett., 81, 2498, (1998).; J. S. Dodge, C. P. Weber,\nJ. Corson, J. Orenstein, Z. Schlesinger, J. W. Reiner, and M. R. Beasley, ibid.85, 4932 (2000).\n[4] L. Klein, Y. Kats, N. Wiser, M. Konczykowski, J. W. Reiner , T. H. Geballe, M. R. Beasley,\nand A. Kapitulnik Europhys. Lett. 55, 532 (2001).\n[5] Isaschar Genish, Lior Klein, James W. Reiner and M. R. Bea sley, Phys. Rev. B 75, 125108\n(2007).\n[6] A. F. Marshall, L. Klein, J. S. Dodge, C. H. Ahn, J, W. Reine r, L. Mieville, L. Antognazza,\nA. Kapitulnik, T. H. Geballe, and M. R. Beasley, J. Appl. Phys .85, 4131 (1999).\n[7] Colman Goldberg, and R. E. Davis, Phys. Rev. 94, 1121 (1954); F. G. West, J. Appl. Phys.\n34, 1171 (1963).\n[8] W. M. Bullis, Phys. Rev. 109, 292 (1958).\n[9] T. R. McGuire and R. I. Potter, IEEE Trans. Magn. MAG, 11, 1018 (1975).\n[10] J. Smit, Physica 21, 877 (1955). J. M. Luttinger, Phys. Rev. 112, 739 (1958). J. Smit, Physica\n24, 39 (1958). L. Berger, Phys. Rev. B. 2, 4559 (1970).\n[11] Isaschar Genish, Lior Klein, James W. Reiner and M. R. Be asley, J. App Phys. 95, 6681\n(2004).\n[12] Yevgeny Kazs, Isaschar Genish, Lior Klein, James W. Rei ner and M. R. Beasley, Phys. Rev.\nB.71, 100403-1 (2005).\n6"    },    {        "title": "0911.3526v1.Structurally_driven_magnetic_state_transition_of_biatomic_Fe_chains_on_Ir_001_.pdf",        "content": "Structurally-driven magnetic state transition of biatomic Fe chains on Ir(001)\nYuriy Mokrousov1;2,\u0003Alexander Thiess1;2, and Stefan Heinze1;3\n1Institute of Applied Physics, University of Hamburg, D-20355 Hamburg, Germany\n2Institut f ur Festk orperforschung and Institute for Advanced Simulation,\nForschungszentrum J ulich, D-52425 J ulich, Germany and\n3Institute of Theoretical Physics and Astrophysics,\nChristian-Albrechts-University of Kiel, D-24098 Kiel, Germany\n(Dated: November 29, 2021)\nUsing \frst-principles calculations, we demonstrate that the magnetic exchange interaction and\nthe magnetocrystalline anisotropy of biatomic Fe chains grown in the trenches of the (5 \u00021) re-\nconstructed Ir(001) surface depend sensitively on the atomic arrangement of the Fe atoms. Two\nstructural con\fgurations have been considered which are suggested from recent experiments. They\ndi\u000ber by the local symmetry and the spacing between the two strands of the biatomic Fe chain.\nSince both con\fgurations are very close in total energy they may coexist in experiment. We have\ninvestigated collinear ferro- and antiferromagnetic solutions as well as a collinear state with two mo-\nments in one direction and one in the opposite direction ( \"#\"-state). For the structure with a small\ninterchain spacing, there is a strong exchange interaction between the strands and the ferromagnetic\nstate is energetically favorable. In the structure with larger spacing, the two strands are magneti-\ncally nearly decoupled and exhibit antiferromagnetic order along the chain. In both cases, due to\nhybridization with the Ir substrate the exchange interaction along the chain axis is relatively small\ncompared to freestanding biatomic iron chains. The easy magnetization axis of the Fe chains also\nswitches with the structural con\fguration and is out-of-plane for the ferromagnetic chains with small\nspacing and along the chain axis for the antiferromagnetic chains with large spacing between the\ntwo strands. Calculated scanning tunneling microscopy images and spectra suggest the possibility\nto experimentally distinguish between the two structural and magnetic con\fgurations.\nI. INTRODUCTION\nDriven by the wish to realize the proposed concepts of\nfuture spintronic devices1,2,3the development of novel\nnanostructures and nanomaterials with tailored elec-\ntronic and magnetic properties has become a key chal-\nlenge of today's research. A promising path to control the\nmagnetic properties of matter is to use low-dimensional\nsystems and to reduce their size down to the nanome-\nter or even atomic scale. For nanoscale systems, how-\never, an essential requirement is to enhance the magnetic\nanisotropy in order to stabilize magnetic order against\nthermal \ructuations or quantum tunneling. The manip-\nulation of exchange interactions opens another path to\ncreate new materials with a magnetic state that may be\ntunable by external magnetic or electric \felds.4E.g. the\noccurrence of chiral spin spiral states at surfaces has\nbeen demonstrated,5,6their manipulation by electrical\ncurrents was suggested5,7and a way to grow \flms with\nmultiple metastable magnetic states has been proposed.8\nThe ability to create one-dimensional monoatomic\nmagnetic chains of transition-metals on surfaces by self-\norganization9,10or by manipulation with a scanning tun-\nneling microscope11has recently opened new vistas to\nexplore and manipulate arti\fcial magnetic nanostruc-\ntures even atom-by-atom. E.g. the pioneering work of\nGambardella et al.9,10demonstrated that the magnetic\nanisotropy of atomic transition-metal chains, consist-\ning of Co atoms on a stepped Pt(111) surface, is dra-\nmatically enhanced with decreasing dimensionality from\ntwo-dimensional \flms to quasi-one-dimensional chains\nand depends sensitively on the number of Co strandsin a chain. These experimental observations were ex-\nplained based on electronic structure calculations which\nemphasized the crucial role played by the substrate,\nreduced symmetry, and structural relaxations for the\nmagneto-crystalline anisotropy.12,13,14,15The large mag-\nnetic anisotropies led to slow relaxation dynamics of the\nmagnetization and the observation of magnetic hystere-\nsis loops at low temperatures indicative of ferromagnetic\ncoupling. In another experiment, Mn chains of up to 10\natoms were created by manipulation with a scanning tun-\nneling microscopy tip on an insulating CuN layer grown\non Cu(001). Experimentally, the exchange interaction\nbetween individual spins was obtained by measuring the\nexcitation spectrum via inelastic tunneling spectroscopy\nwhich showed the quantum behavior of the entire chain.11\nEven the sign and size of the exchange interaction be-\ntween the Mn atoms could be extracted from the exper-\nimental data. Calculations based on density-functional\ntheory (DFT) clari\fed that a superexchange mechanism\nalong the Mn-N-Mn bond is responsible for the weak an-\ntiferromagnetic coupling.16,17\nRecently, Hammer et al. have used a combination\nof IV-LEED and scanning tunneling microscopy (STM)\nmeasurements to demonstrate that the Ir(001) surface\ncan serve as an ideal template to grow defect-free,\nnanometer long transition-metal nanowires of di\u000berent\nstructure, chemical composition, and length depend-\ning on the preparation conditions.18,19,20E.g. biatomic\nFe chains can be created on the (5 \u00021) reconstructed\nIr(001) surface and lifting the surface reconstruction by\nhydrogen opens the possibility to produce Fe-Ir-Fe tri-\natomic chains. While the biatomic Fe chains are a veryarXiv:0911.3526v1  [cond-mat.mes-hall]  18 Nov 20092\npromising system to study magnetism of (quasi-) one-\ndimensional transition-metal chains, there is little un-\nderstanding so far. Experimentally, it is extremely chal-\nlenging as for laterally averaging measurements samples\nwith a homogeneous distribution of chains are needed\nor a technique must be applied which allows to locally\nprobe the magnetic properties of individual Fe chains.\nAnother key di\u000eculty is the detailed characterization of\nthe chains structure. From combined STM and LEED ex-\nperiments it is only known that the biatomic chains grow\nin the trenches of the (5 \u00021) reconstructed Ir(001) sur-\nface, however, the adsorption sites in the trenches could\nnot be deduced. As structure and magnetism are closely\ncorrelated in such systems, their magnetic properties are\nan open issue.\nA theoretical study using \frst-principles calculations21\nreported an excellent agreement with the structural pa-\nrameters of the (5 \u00021) reconstructed Ir(001) surface and\nconcluded that the Fe chains are strongly ferromagnetic\nat low temperatures but were probably non-magnetic\nin the room temperature measurements of Hammer et\nal.However, this theoretical study considered only fer-\nromagnetic solutions and did not determine the magne-\ntocrystalline anisotropy energy which is crucial for an ex-\nperimental veri\fcation of the proposed ferromagnetism in\nthese chains. On the other hand, a 5 dtransition-metal\nsubstrate such as Ir can have dramatic consequences on\nthe exchange coupling in a deposited Fe nanostructure as\nis apparent from the observation of a complex nanoscale\nmagnetic structure for an Fe monolayer on Ir(111),22\nthe antiferromagnetic ground state of an Fe monolayer\non W(001),23to name just a few. Recent studies on\nFe stripes on Pt(997)24and FePt surface alloys25report\non strong correlation between complex magnetic ground\nstates and the details of structural arrangement.\nHere, we use \frst-principles calculations based on\ndensity-functional theory to study the structural, elec-\ntronic, and magnetic properties of biatomic Fe chains\ndeposited in the trenches of the (5 \u00021) reconstructed\nIr(001) surface.26We focus on two structural arrange-\nments of the biatomic chains which di\u000ber by the ad-\nsorption sites of the Fe atoms. In one con\fguration the\ndistance between the two strands of the biatomic chain\nis smaller than the atom spacing along the chain direc-\ntion (denoted as C1 in accordance with Ref. 21), while\nin the other their separation is clearly larger than the\ninterchain spacing (denoted as C4), see Fig. 1. We con-\nsider collinear ferro- and antiferromagnetic arrangements\nalong the chains and the \"#\"-state with two moments in\none and one moment in the opposite direction. We \fnd\nthat the energetically favorable magnetic state as well as\nthe easy magnetization axis of the Fe chains depend sen-\nsitively on their atomic arrangement and local symmetry.\nFor the Fe chains in the C1-structure, the exchange cou-\npling along the chain is ferromagnetic, however, due to\nhybridization with the Ir substrate it is much weaker than\nfor freestanding biatomic Fe chains with the same atom\nspacing. Due to their small separation, the two strandsof the C1-chains are also ferromagnetically coupled. Sur-\nprisingly, for the C4-con\fguration, we \fnd a transition\nfrom ferromagnetic state for free-standing chains to anti-\nferromagnetic order along the biatomic chains after depo-\nsition on the Ir substrate. In this case, the hybridization\nwith the Ir surface is strong enough to invert the sign of\nexchange coupling, while the two Fe strands are nearly\nexchange decoupled. The interplay of the Fe interstrand\ndistance and the hybridization with the Ir substrate re-\nsults also in a di\u000berent easy axis of the magnetization\nfor the two structures: while in the C1-FM state the\neasy axis is out-of-plane, it switches into the chain axis\nfor the C4-AFM con\fguration. The total energy of the\nC1- and C4-structure are quite close and therefore both\nchain types could occur in an experiment depending on\nthe growth conditions. We simulate measurements by\nSTM and observe that the two strands of the biatomic\nFe chain in the C1-ferromagnetic state are too close to be\nindividually resolved, while they can be distinguished in\nthe C4-structure with an antiferromagnetic ground state.\nIn the latter case, spin-polarized STM (SP-STM) should\nfurther allow to directly resolve the two-fold magnetic\nperiodicity along the chain.\nThe structure of this paper is as follows. In the next\nsection we present details of our method and the calcu-\nlations. Then we discuss the structural relaxations of\nthe pure (5\u00021) reconstructed Ir(001) substrate and of\nthe biatomic Fe chains in the two structural con\fgura-\ntions on the (5\u00021) reconstructed Ir(001) surface. In sec-\ntion IV, we analyze the magnetic ground state con\fgura-\ntion and the e\u000bects of hybridization with the Ir substrate,\nbefore we turn to the magnetocrystalline anisotropy in\nsection V. We explore the feasibility to experimentally\nresolve the di\u000berent structural and magnetic properties\nby SP-STM and to verify our predictions of a structure-\ndependent magnetic ground state. Finally, a conclusion\nand summary is given.\nII. COMPUTATIONAL DETAILS\nWe employed the \flm-version of the full-potential lin-\nearized augmented plane-wave (FLAPW) method, as im-\nplemented in the J ulich density-functional theory (DFT)\ncode FLEUR . We used inversion-symmetric \flms with 7\nand 9 layers of the (5 \u00021) reconstructed Ir(001) sur-\nface and Fe biatomic chains on both sides of the slab.\nThe whole system possesses spatial inversion symme-\ntry. We calculate the biatomic Fe chains in a (5 \u00021)\nsupercell along the y-direction ([0 \u001611]-axis), perpendicu-\nlar to the chain x-axis ([011]-axis), which results in a\ndistance of 13.75 \u0017A between the axes of two adjacent\nbiatomic chains, c.f. Fig. 1 for structural arrangements\nand de\fnition of the axes. The theoretical Ir lattice\nconstant we used for calculations constituted a value of\n3.89 \u0017A. We used the generalized-gradient approximation\n(GGA, revPBE functional27) of the exchange-correlation\npotential for the structural relaxations and tested total3\nFIG. 1: (color online) Geometrical structure of the biatomic Fe chains on the (5 \u00021) reconstructed Ir(001) substrate in C1\n((a)-top view, (c)-side view) and C4 ((b)-top view, (d)-side view) con\fgurations. Fe atoms are marked in blue, while Ir atoms\nare marked in gold. In (e) and (f) a schematic top view of respectively C1 and C4 biatomic Fe chains is shown (positions of\nthe atoms do not correspond to realistic calculated values). In the latter graphs the numbering of the Ir atoms corresponds to\nthat in Fig. 2 and table II. The x- andy-axis de\fned in (a) correspond to [011] and [0 \u001611] directions, respectively.\nenergy di\u000berences between the two magnetic con\fgura-\ntions also within the local-density approximation (LDA,\nVWN functional28). We used 18 k-points in a quarter\nof the full two-dimensional Brillouin zone (2D-BZ) for\nself-consistent calculations. The calculated total energy\ndi\u000berences between di\u000berent magnetic ground states were\ncarefully tested with respect to the number of k-points.\nFor the basis functions, we used a cut-o\u000b parameter of\nkmax= 3:6 a.u.\u00001for relaxations and 3.7 a.u.\u00001for com-\nparing the total energies of di\u000berent magnetic con\fgura-\ntions.\nWe considered two possible structural arrangements of\nthe Fe atoms denoted as C1 and C4 according to the no-\ntation of Ref. 21, which are shown in Fig. 1. Experimen-\ntally, it has been observed by STM18that the biatomic\nchains grow in the trenches of the (5 \u00021) reconstructed\nIr(001) surface and we therefore focus on these two con-\n\fgurations. Relaxations were performed until the forceschanged by less than 3 \u000110\u00004htr/a.u. The convergence\nof the relaxed atomic positions was carefully tested with\nrespect to the computational parameters. Relaxations\nwere performed only for the ferromagnetic state of both\nC1 and C4 con\fgurations and the antiferromagnetic and\nthe\"#\"-states were calculated on these atomic positions.\nIII. STRUCTURE AND RELAXATIONS\nAs a \frst step, we performed a structural relaxation of\nthe pure (5\u00021) reconstructed Ir(001) surface. As can be\nseen in table I, the results we obtain with a \flm of 9 layers\nagree very well with the experimental data measured by\nIV-LEED.29In particular, the experimentally observed\ntrench-like structure is reproduced. Our values are also\nin close agreement with those obtained with the VASP4\nFIG. 2: (color online) Schematical representation of the\nstructural parameters given in table I. Note that in this \fgure\nthe buckling of the Ir substrate is greatly exaggerated and the\npositions of the Fe atoms (blue) and Ir atoms in the \frst and\nsecond layer of the substrate (orange) do not correspond to\nrealistic calculated values.\ncode by Spi\u0014 s\u0013 ak et al.21For deposited biatomic Fe chains\nwe have also found a very good agreement between the\nvalues of the relative atomic positions obtained by relax-\ning a slab with 9 and 7 layers of the Ir substrate (see\ntable I).\nWe performed relaxations of the Fe biatomic chains\nonly for the ferromagnetic solutions to reduce the com-\nputational e\u000bort, presenting the results in table I. While\nthe distance \u0001 between the atomic strands in C1-\ncon\fguration is 2.35 \u0017A, and thus smaller than along the\nchain (see Figs. 1 and 2), it is almost twice larger in the\nC4-con\fguration, i.e. \u0001 = 4 :23\u0017A, and the two strands\nare well separated. In the latter structure, the Fe atoms\nare also more embedded into the Ir surface which is il-\nlustrated by the smaller vertical distance \u000efrom the Ir\nsurface atoms between the two Fe chain atoms (denoted\nas Ir4 in Fig. 2). The in\ruence of the C4-chains on the\nbuckling of the Ir substrate is also more pronounced as\ncan be seen from the increased vertical separations of the\nsurface Ir atoms. The di\u000berent structural relaxations of\nthe two chain con\fgurations already hint at a larger hy-\nbridization of the Fe 3 dand Ir 5dstates and a stronger\nin\ruence in the C4 arrangement. For both con\fgurations\nthe distance between the Fe dimers along the chain's axis\nwas imposed by the Ir substrate and constituted 2.75 \u0017A.\nIV. MAGNETIC ORDER AND EXCHANGE\nINTERACTIONS\nNow we study the magnetic order and exchange inter-\naction of the chains in the two structural arrangements.LEED (5\u00021) Ir (5\u00021) Ir C1-FM C4-FM\n9 layers Ref. 29 Ref. 21\n\u0001\u0000 2.35 4.23\n(2.65) (4.35)\n\u000e\u0000 1.90 1.53\nd12 1.94 2.00 1.97 1.93 1.82\nb13 0.25 0.20 0.20 0.22 0.42\nb23 0.55 0.47 0.54 0.62 0.77\nb34 0.20 0.17 0.18 0.30 0.55\np2 0.05 0.03 0.04 0.07 0.04\np3 0.07 0.07 0.07 0.12 0.11\n7 layers\n\u0001\u0000 2.40 4.19\n\u000e\u0000 1.90 1.54\nd12 1.94\u0000 \u0000 1.93 1.81\nb13 0.25\u0000 \u0000 0.22 0.42\nb23 0.55\u0000 \u0000 0.63 0.79\nb34 0.20\u0000 \u0000 0.30 0.53\np2 0.05\u0000 \u0000 0.07 0.05\np3 0.07\u0000 \u0000 0.13 0.12\nTABLE I: Relaxations of the Fe biatomic chains and the\nuppermost layer of the (5 \u00021) reconstructed Ir(001) substrate\nfor 9- and 7-layer slabs. All values are given in \u0017Angstrom. The\ndistancesd;b;andpcorrespond to those in Ref. 29 and are\ndepicted in Fig. 2. For comparison the relaxations of the bare\nIr(001) substrate from Ref. 29 (IV-LEED data) and Ref. 21\nare given. For the interchain distance \u0001 values is brackets\ncorrespond to those calculated in Ref. 21.\nWe considered the ferro- (FM) and antiferromagnetic\n(AFM) solution (with antiparallel magnetic moments be-\ntween adjacent Fe atoms along the chain) for both types\nof chains. For comparison, we have also calculated free-\nstanding biatomic Fe chains with the same interatomic\nspacings as in the C4 and C1 con\fgurations. As can be\nseen in table II, the magnetic moments of the Fe atoms\nare slightly larger in the C4 con\fguration where the Fe\natoms are further apart. The di\u000berence in the magnetic\nmoments between FM and AFM state is very small. The\ninduced moments in the Ir surface layer depend much\nmore sensitively on the magnetic state of the Fe chains.\nFor the FM solutions, these moments are signi\fcant in\nboth structures and decay slightly faster for the C1- than\nfor the C4-con\fguration. In the AFM state, due to sym-\nmetry some of the Ir atoms do not carry an induced mo-\nment. In the C4 structure, the Ir surface atom between\nthe Fe atoms has a rather large moment of 0.18 \u0016Bindi-\ncating a strong hybridization.\nThe total energy di\u000berences between the two mag-\nnetic con\fgurations given in table II reveal a surprising\nresult. In particular, we \fnd that the preferred mag-\nnetic state is FM for the Fe chains in the C1 struc-\nture (dFe\u0000Fe=2.35 \u0017A), while the total energy di\u000berence\nis in favor of the AFM solution for the C4 structure\nwith Fe atoms further apart ( dFe\u0000Fe=4.23 \u0017A). For free-\nstanding biatomic Fe wires with the same spacing be-\ntween Fe atoms, the energy di\u000berence between FM and\nAFM states is 74 meV/Fe and 164 meV/Fe in favor of5\nthe FM state for the C1 and C4 con\fguration, respec-\ntively. For supported chains the corresponding energy\ndi\u000berences are of the order of rather small 30 meV/Fe,\nindicating a strong in\ruence of the Ir substrate on the\nmagnetic coupling and weakening of the FM interaction\nin the free-standing chains due to hybridization with the\nIr atoms.30Because of the large separation of the two\nstrands of the biatomic chain in the C4 con\fguration,\nthis e\u000bect is dramatic and leads to the AFM ground\nstate. This notion is further supported by the magni-\ntude of the exchange interaction between the two strands.\nWe have calculated the total energy di\u000berence between\na FM and AFM alignment of the two strands of the de-\nposited Fe chains which is 166 meV/Fe and 4 meV/Fe\nin favor of ferromagnetic coupling in the C1- and C4-\ncon\fguration, respectively. The extremely small value in\nthe C4-con\fguration, indicating that the two strands are\nmagnetically nearly decoupled, is probably due to a small\nindirect exchange interaction via the Ir substrate.\nWe have also calculated the FM and AFM total en-\nergy di\u000berences for both C1 and C4 con\fgurations using\n7 layers of Ir substrate, and the results are compared\nin table II to those obtained with 9 layers. In the C1-\nstructure the FM-AFM energy di\u000berence of 20.6 meV/Fe\nis in good agreement with the value of 21.1 meV/Fe for 9-\nlayers of substrate. In the C4-con\fguration, on the other\nhand, the energy di\u000berence between the AFM ground\nstate and the FM state increases by only 7 meV/Fe for\nthe thinner substrate. These results show that the favor-\nable magnetic state within each structural arrangement\nis independent of the substrate thickness. However, in\ncase of 9 layers the C4-AFM state is lower in energy\nthan the C1-FM solution by 9.8 meV/Fe, while in case\nof 7 layers this energy di\u000berence reverses sign and consti-\ntutes 8.5 meV/Fe in favor of C1-FM state (see table II).\nThis discrepancy probably arises due to quantum well\nstates in the Ir substrate which do not in\ruence the to-\ntal energy di\u000berence between di\u000berent magnetic states\nwithin the same structural arrangement. Overall, our\ncalculations reveal that, judging only from this energy\ndi\u000berence, both con\fgurations might appear in experi-\nment and can be observed via, e.g., scanning tunneling\nmicroscopy measurements.\nWe have further checked the in\ruence of the exchange-\ncorrelation potential on the FM-AFM energy di\u000berences\nand found very similar results within the local-density\napproximation LDA. Using the LDA, a 7 layer (5 \u00021)\nIr(001) substrate and the relaxed atomic positions found\nwith GGA, in the C1 con\fguration the FM state is by\n25.1 meV/Fe lower in energy than the AFM state (c.f. a\nvalue of 20.6 meV/Fe in GGA). In the C4 structure the\nAFM state is by 23.7 meV/Fe lower than the FM state\n(c.f. a value of 40.3 meV/Fe in GGA).\nWithin GGA, we also performed calculations of a\ncollinear magnetic state with a spin arrangement of \"#\"\n(periodically repeated along the chain axis) for the Fe\nbiatomic chains on 7 layers of Ir substrate in the C1 and\nC4 structure. Our calculations reveal that the C4- \"#\"state is by 16.0 meV/Fe higher in energy than the C4-\nAFM state and by 24.3 meV/Fe lower in energy than\nthe C4-FM state. In the C1 con\fguration, the \"#\"-state\nis 9.4 meV/Fe higher in energy than the FM state, and\nby 11.2 meV/Fe lower in energy than the AFM state.\nFrom these three collinear magnetic solutions we can es-\ntimate the nearest-neighbor and next-nearest neighbor\nexchange constants of an e\u000bective 1D Heisenberg model\nto beJ1=\u000010 meV and J2= +1 meV for the C4 struc-\nture andJ1= +5 meV and J2=\u00001:3 meV for the C1\nstructure. These values illustrate the strong tendency to-\nwards antiferromagnetic coupling due to the Ir substrate.\nEven in the C1-con\fguration, the FM nearest-neighbor\ncoupling has become very weak. Note, that a similar in-\n\ruence of the Ir surface has recently been reported for Fe\nmonolayers on Ir(111) and for other 4 d/5dsubstrates.31\nIn order to understand the sensitive dependence of the\nmagnetic coupling in the Fe chains upon the structural\narrangement we take a look at the density of states (DOS)\nfor the two con\fgurations, shown in Fig. 3 in compari-\nson with the free-standing chains. In the non-magnetic\nstate, the DOS of the supported Fe chains displays a large\npeak at the Fermi energy for both structures, however,\nthed-band width is smaller in the C4 structure due to\nthe larger separation and weaker hybridization between\nthe Fe atoms\u0000an e\u000bect even more pronounced in free-\nstanding wires.\nIn the C1 structure, the direct interaction between the\nFe atoms perpendicular to the chain is much stronger\nthan in C4-wires and the free-standing chains are a better\napproximation. Correspondingly, the FM DOS of the\nfree-standing chains in C1-geometry is very similar to the\nsupported chains, while larger changes are visible in the\nC4-FM con\fguration. This becomes even more evident\nfrom the comparison of the bandstructures of the C1-FM\nfree-standing and C1-FM supported biatomic Fe chains\npresented in Fig. 4. In this plot the electronic states of\nthe free-standing C1 chain (small red and small green\ncircles) display a close correspondence to the states of\nthe C1 supported chain (large black circles) which are\nlocalized mainly inside the mu\u000en-tin spheres of Fe atoms.\nRemarkably, for many of the bands of the two systems a\ndirect correspondence in terms of symmetry can be made.\nFor the AFM solution in both structural con\fgura-\ntions the modi\fcations in the DOS due to interaction\nwith the substrate are quite signi\fcant. The electronic\nbands in free-standing AFM chains are normally very \rat\nand corresponding peaks in the DOS are very sharp30\u0000\nin this case the e\u000bect of the hybridization of the local-\nized 3d-orbitals with extended states of Ir atoms on the\nDOS can be very strong. For both FM and AFM mag-\nnetic states, a slightly larger exchange splitting can be\nobserved in C4 deposited chains, as compared to the C1\ncon\fguration, which leads also to larger spin moments of\nFe atoms in C4 arrangement (c.f. table II). Overall, an in-\nterplay of decreasing Fe-Fe hybridization with increasing\nFe-Ir hybridization when going from C1 to C4 structural\ncon\fguration leads to somewhat larger localization of Fe6\nC1-FM C1-AFM C4-FM C4-AFM\nUBC 3.12 3.14 3.28 3.22\nEnergy 0 +74.1 +751.1 +914.6\n9 layers\nEnergy +9.8 +30.9 +33.7 0\nTotal 3.21 0.00 3.41 0.00\nFe 2.97 2.98 3.01 3.04\nIr1 0.00 0.01 0.00 0.00\nIr2\u00000.02 0.00 0.07 0.06\nIr3 0.08 0.06 0.14 0.00\nIr4 0.29 0.00 0.16 0.18\n7 layers\nEnergy 0 +20.6 +48.8 +8.5\nTotal 3.23 0.00 3.34 0.00\nFe 2.98 2.98 3.01 3.04\nIr1 0.00 0.00 0.00 0.00\nIr2\u00000.03 0.00 0.08 0.05\nIr3 0.08 0.06 0.14 0.00\nIr4 0.29 0.00 0.15 0.18\nTABLE II: Relative total energies obtained in GGA (in\nmeV/Fe-atom), spin moments in the mu\u000en-tin spheres of the\nFe atoms in unsupported (UBC) and supported bichains, as\nwell as for Ir surface atoms, and total moments in the unit\ncell (in\u0016B) for calculations with 9 and 7 layers of the (5 \u00021)\nreconstructed Ir(001) substrate. The Ir surface atoms are de-\nnoted as in Fig. 2.\nelectronic states in C4 biatomic chains.\nV. MAGNETO-CRYSTALLINE ANISOTROPY\nAs mentioned in the introduction, the magneto-\ncrystalline anisotropy energy (MAE) is a key quantity\nfor nanoscale magnets as it determines the preferred di-\nrection of the magnetic moments and is crucial to sta-\nbilize the magnetic order against thermal \ructuations.\nWe have calculated the MAE for the Fe chains within\nthe GGA employing the force theorem and found that\nits value is stable against the chosen ground state upon\nwhich we perform the perturbation. For the C1 structure,\nwe considered the FM ground state and started from a\nconverged ground state with an out-of-plane magneti-\nzation from which the MAE was obtained by applying\nthe force theorem for three possible high-symmetry di-\nrections: perpendicular to the surface and the chain axis\n(along the z-axis), parallel to the axis of the chain ( k-\ndirection) and in the surface plane perpendicular to the\nchain's axis (?-direction). We de\fne two principal ener-\ngies: MAEk= E tot(k)\u0000Etot(z), and MAE?= E tot(?)\n\u0000Etot(z). The number of k-points used for calculations\nconstituted 144 in the full 2D-BZ. We carefully tested\nthe MAE values with respect to the number of k-points.\nUsing 144 k-points results in an accuracy of not less\nthan 0.3 meV/Fe. We \fnd that in the C1-FM con\fg-\nuration the calculated values are 2.1 meV/Fe for MAE k\nand 1.8 meV/Fe for MAE ?which corresponds to a mag-netization along the z-axis in the ground state, i.e. per-\npendicular to the surface and chain axis, see Fig. 5(a).\nFor the calculations of the MAE in the C4-AFM con-\n\fguration we used a slab of 7 layers of the (5 \u00021) Ir(001)\nsubstrate with 8 Fe chain atoms in the unit cell and 144\nk-points in the full 2D-BZ (again the stability of the MAE\nwith respect to the number of k-points was carefully\nchecked). We obtain values of \u00000.2 meV/Fe for MAE k\nand 1.2 meV/Fe for MAE ?corresponding to a ground\nstate with the magnetic moments along the chain axis,\nsee Fig. 5(b). Considering the accuracy of our calcula-\ntions we conclude that the easy axis of the magnetization\nfor the Fe biatomic chains in the C4-AFM con\fguration\nconstitutes a plane which cuts through the chain axis and\nis perpendicular to the substrate.\nThe MAE and its dependence on the con\fguration can\nbe qualitatively related to the anisotropy of the orbital\nmoments in the system.32For this purpose for di\u000ber-\nent magnetization directions we compare the orbital mo-\nments inside the atomic spheres of Fe chain atoms, \u0016Fe\nL,\nof the Ir atoms in the surface layer, \u0016Ir\nL, and the total\norbital moment per unit cell, \u0016tot\nL, de\fned as the sum\nover the moments of the two Fe chain atoms and the \fve\nIr surface atoms on one side of the slab. For the evalua-\ntion of\u0016tot\nLwe do not take into account the much smaller\ncontributions from Ir atoms deeper inside the slab. For\nthe antiferromagnetic ground state in the C4 geometry,\none should note that the sign of the orbital moments\nswitches for atoms of antiparallel magnetization. There-\nfore, we de\fne \u0016tot\nLin this case as the orbital moment\nsummed over atoms in one half of the (2 \u00025) unit cell on\none side of the slab.\nIn the C1-FM state the anisotropy of \u0016tot\nLis in quali-\ntative agreement with the anisotropy of the total energy:\nwhile for an out-of-plane magnetization \u0016tot\nLreaches\na value of 0.126 \u0016B, it constitutes only 0.083 \u0016Band\n0.035\u0016Bfor the?- andk-directions of the magnetiza-\ntion, respectively. Therefore, the easy axis coincides with\nthat along which the orbital moment is largest.32This\nanisotropy of \u0016tot\nLcan be explained based on the depen-\ndence of the Ir contributions on the magnetization di-\nrection. For an out-of-plane magnetization, the values\nof\u0016Ir\nLare small and the total orbital moment is domi-\nnated by the Fe atoms. In contrast, for a magnetization\nalong the?- and thek-direction the Ir orbital moments,\n\u0016Ir\nL, reach signi\fcant values, however, of opposite sign\nwith respect to \u0016Fe\nL. For the?-magnetization direction\nwith\u0016Fe\nL= 0:077\u0016Bthe value of the orbital moment\nof the Ir4 atom (see Fig. 2) is \u00000.013\u0016B, while for the\nk-magnetization it even reaches \u00000.029\u0016Band the cor-\nresponding value of \u0016Fe\nLis only 0.058 \u0016B.\nIn the C4-AFM con\fguration of the Fe chains the\nagreement between the anisotropy of the orbital moment\nand of the total energy is even better. While for the\n?-magnetization direction the value of \u0016tot\nLis 0.160\u0016B,\nit reaches much larger values of 0.239 \u0016Band 0.253\u0016B\nfor thez- andk-direction, respectively. The direction of\nthe smallest total orbital moment, \u0016tot\nL, coincides with7\nFIG. 3: (color online) Density of states (DOS) for the nonmagnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM)\nstate of Fe biatomic chains on 9 layers of the (5 \u00021) reconstructed Ir(001) surface in C4, (a)-(c), and C1, (d)-(f), con\fguration.\nThe DOS is given inside the mu\u000en-tin spheres of Fe atoms. The dashed lines show the DOS for unsupported Fe biatomic\nchains with corresponding spacings between the Fe atoms in the chain. In (b), (c), (e) and (f) left (green) and right (red)\ncurves stand for spin-up and spin-down channels, respectively.\nthe hard axis and, moreover, a very small di\u000berence be-\ntween the values of \u0016tot\nLfor the two other directions cor-\nresponds to a very small energy di\u000berence MAE k. In this\nstructural arrangement the anisotropy of the Fe orbital\nmoments,\u0016Fe\nL, dominates the anisotropy of \u0016tot\nL. The\ncontribution of the Ir atoms to the total orbital moment\nis nearly independent of the magnetization direction. For\nthez- andk-magnetization directions \u0016Fe\nLis 0.100\u0016Band\n0.106\u0016B, respectively, while it is only 0 :060\u0016Band thus\nmuch smaller for the hard ?-magnetization axis.\nVI. SIMULATION OF STM EXPERIMENTS\nIn the previous sections, we have demonstrated that\nthe magnetic properties of biatomic Fe chains on the\n(5\u00021) Ir(001) surface depend crucially on the atomic\narrangement of the atoms. Both easy magnetization di-\nrection and magnetic order change upon displacements\nof the Fe atoms which changes the interaction between\nthe two strands of the chain and their hybridization with\nthe Ir substrate. In order to verify our predictions ex-\nperimentally, a technique with a high lateral resolution\nseems indispensable. Therefore, we study theoreticallythe possibility to use spin-polarized scanning tunneling\nmicroscopy (SP-STM) to resolve the atomic and mag-\nnetic structure of these chains.\nFig. 6 displays the calculated local density of states\n(LDOS) in the vacuum at a distance of about 6.8 \u0017A from\nthe Fe chain atoms. Within the Terso\u000b-Hamann model\nof STM the vacuum LDOS is directly comparable with\nmeasureddI=dU tunnel spectra. The comparison of dif-\nferent magnetic states shows distinct features in the FM\nsolution for both chain structures. Strong peaks appear\nin the minority spin channel at +0.7 eV and +0.4 eV in\nthe C4 and C1 con\fguration, respectively, see Figs. 6(c)\nand (d). In the AFM state, we observe a structure of\ntwo broad peaks at \u00000:5 eV and +1.0 eV for the C1\nchains, Fig. 6(f), while the C4 chains display a relatively\nfeatureless structure with a small peak at the Fermi en-\nergy. Considering the two favorable C1-FM and C4-AFM\nstates of the chains, it seems that the two types can be\ndistinguished by the strong peak of the C1-FM con\fgu-\nration.\nA direct way of verifying the magnetic structure of the\nFe chains might be feasible by imaging them in the to-\npography mode of SP-STM. In addition, the structural\narrangement of the atoms might be detectable. Our sim-8\nFIG. 4: (color online) Bandstructure of the Fe biatomic\nchains in the C1-FM con\fguration on 9 layers of the (5 \u00021) re-\nconstructed Ir(001) surface. Left panel shows majority bands\nand right panel the minority bands. k-vector has been cho-\nsen along the chain direction. Large black \flled circles denote\nstates which are localized on the Fe chain. For comparison the\nbands for an unsupported biatomic Fe chain with the same\ninteratomic spacing are given by green (majority) and red\n(minority) \flled small circles. Grey circles on the background\nmark the states of the whole Fe+Ir system.\nulations of STM and SP-STM images are displayed in\nFigs. 7 and 8 for the FM ground state of the C1 chains\nand the AFM ground state of the C4 chains, respectively.\nWe have chosen an energy window corresponding to the\nunoccupied states close to the Fermi energy, but the oc-\ncupied states lead to very similar results.\nThe two strands of the biatomic Fe chains in the C1\nstructure are only 2.35 \u0017A apart which makes it impos-\nsible to resolve them in cross-sectional scans as seen in\nFig. 7(a) and (b) for both spin channels and also in their\nsummation. The corrugation amplitudes which we calcu-\nlate from these plots are 1.9 and 2.5 \u0017A and the apparent\nwidth (full width at half maximum using the topmost\nlines of constant charge density) amounts to 7.7 and 6.9\n\u0017A for the majority and minority spin channel, respec-\nFIG. 5: (color online) Sketches of the energy landscape as\na function of magnetization direction for the two types of\nchains. (a) C1-FM con\fguration with an easy axis pointing\nperpendicular to the surface, hard axis along the chain (+2.1\nmeV/Fe atom), and middle axis perpendicular to the chain\nand in the surface plane (+1.8 meV/Fe atom). (b) C4-AFM\ncon\fguration with an easy axis along the chain axis but only\na small energy di\u000berence with respect to the middle axis per-\npendicular to the surface (+0.2 meV/Fe atom), and a hard\naxis perpendicular to the chain and in the surface plane (+1.2\nmeV/Fe atom). In this plot, we construct the energy land-\nscape based on the lowest power of the directional cosines of\nthe magnetization with respect to the crystallographic axes\nwhich are allowed by symmetry. The coe\u000ecients are obtained\nfrom the values of MAE kand MAE?.\ntively. Interestingly, the chains appear wider if major-\nity states of the chain are imaged than for the minority\nstates which can be explained based on the orbital char-\nacter of the dominating states. From the DOS of the\nFe atoms, Fig. 3(e), we see that the minority channel\nis dominated by d-electrons while the majority d-band\nis far below the Fermi energy and therefore, s- andp-\nstates provide a large contribution. The d-character of\nthe states for spin down electrons is clearly visible in the\ncross-sectional plot, Fig. 7(b), and leads to a sharper im-\nage of the chains. The more delocalized s- andp-states\ndominate the majority spin channel, Fig. 7(a), and lead\nto a larger apparent width and smaller corrugation am-9\n5\n4\n3\n2\n1\n0LDOS (arb. units)\n5\n4\n3\n2\n1\n0LDOS (arb. units)\n5\n4\n3\n2\n1\n0LDOS (arb. units)\n-1.0 0.0 1.0\nEnergy relative to EF (eV)-1.0 0.0 1.0\nEnergy relative to EF (eV)(a) (b)\n(c) (d)\n(e) (f)C4, NM C1, NM\nC4, FM C1, FM\nC4, AFM C1, AFM\nFIG. 6: (color online) Local density of states (LDOS) in\nthe vacuum for the NM, FM and AFM states of Fe biatomic\nchains on 9 layers of the (5 \u00021) reconstructed Ir(001) surface in\nC4 (left column) and C1 (right column) con\fgurations. The\nLDOS is given at a distance of \u00196.8\u0017A above the Fe atoms.\nplitude of the chains.\nIn the C4 con\fguration, the Fe chain atoms are much\nfurther apart in the perpendicular direction, d= 4:23\u0017A,\nwhich is large enough to allow the resolution of the two\nstrands as can be seen in our simulations of STM images,\nshown in Fig. 8. The corrugation amplitude of the C4-\nAFM chains in both spin channels is 1.5 and 2.2 \u0017A in the\nspin up and down channel, respectively, the corrugation\nbetween the two Fe chain atoms is 0.15 and 0.3 \u0017A, and\nthe apparent widths are about 10.5 \u0017A, i.e. signi\fcantly\nwider than the C1 FM chains. With a spin-polarized\nSTM it should further be possible to resolve the antiferro-\nmagnetic spin alignment along the chains as can be seen\nfrom the STM images for the two separate spin channels\nas seen from Figs. 8(c) and (d). For a spin-polarization\nof the tip of Pt= (n\"\u0000n#)=(n\"+n#) = 0:4, wheren\"and\nn#are the majority and minority spin LDOS of the tip\nat the Fermi energy, we obtain a corrugation amplitude\nof \u0001z= 0:15\u0017A along the chain, i.e. the maximum height\nchange as the tip scans along the chain.\nVII. CONCLUSIONS\nWe performed extensive \frst-principles calculations to\nelucidate the interplay of structure and magnetism in bi-atomic Fe chains on the (5 \u00021) reconstructed Ir(001)\nsurface. We \fnd a crucial in\ruence of the hybridiza-\ntion of Fe chains with the Ir substrate on the magnetic\nground state of the wires. Depending on the particu-\nlar structural arrangement, the magnetic ground state\nswitches from along-the-chain ferromagnetic for the C1\ncon\fguration with a smaller ( \u00192.4\u0017A) distance between\nthe two strands, to antiferromagnetic for the C4 state\nfor which this distance constitutes an almost twice larger\nvalue (\u00194.2\u0017A). In the C4 con\fguration, the two strands\nof the chain are nearly decoupled in terms of exchange\ninteraction, while we \fnd strong ferromagnetic coupling\nin the C1 con\fguration. We also \fnd that the direc-\ntion of the magnetization in these two con\fgurations is\ndi\u000berent: while in C1-FM chains the Fe spin moments\npoint out of plane with a value of the magnetic anisotropy\nof\u00192 meV/Fe with respect to in-plane directions, the\nmagnetization in C4-AFM chains can freely rotate in the\nplane of along-the-chain and out-of-plane directions at\nsu\u000eciently small temperatures, protected from switch-\ning to the in-plane perpendicular to the chain direction\nby a value of\u00191.2 meV/Fe.\nThe two di\u000berent magnetic types of chains are very\nclose in total energy, which facilitates their experimen-\ntal observation and provides a considerable challenge for\nexperimentalists to verify their magnetic ground state.\nWith our calculations we provide theoretical evidence\nfor the feasibility to use spin-polarized STM to resolve\nthe atomic arrangement and magnetic order. Consider-\ning the rather small calculated total energy di\u000berences\nbetween the di\u000berent magnetic collinear solutions of the\norder of 20\u000030 meV/Fe in both types of chains, we also\ncannot exclude the occurrence of noncollinear magnetic\nstates either due to exchange interactions or due to the\nDzyaloshinskii-Moriya interaction driven by spin-orbit\ncoupling.5,6Such non-collinear calculations were beyond\nthe scope of the present work due to the large supercell\nrequired for realistic modeling of chains on the (5 \u00021)\nreconstructed Ir(001) surface. However, future investi-\ngations of this system need to address this open ques-\ntion. Qualitatively and quantitatively di\u000berent energy\nlandscapes of the magnetization direction in real space\ncould also result in a di\u000berent response of the magneti-\nzation with respect to external magnetic \feld or temper-\nature, providing an additional channel for tackling the\nmagnetism in these two types of chains experimentally.\nFinancial support of the Stifterverband f ur die\nDeutsche Wissenschaft and the Interdisciplinary\nNanoscience Center Hamburg are gratefully acknowl-\nedged. We would like to thank Matthias Menzel, Kirsten\nvon Bergmann, Andr\u0013 e Kubetzka, Paolo Ferriani, Gustav\nBihlmayer, Stefan Bl ugel and Roland Wiesendanger for\nmany fruitful discussions.10\nFIG. 7: (color online) Partial charge density plots for the FM ground state of the C1 structure (with 7 layers of Ir substrate)\nwith an out-of-plane easy magnetization axis in an energy regime of ( EF;EF+ 0:5) eV. 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Enders,11\nFIG. 8: (color online) Partial charge density plots for the AFM ground state of the C4 structure (with 7 layers of Ir substrate)\nwith an easy magnetization axis along the chains in an energy regime of ( EF;EF+ 0:3) eV. (a) shows a cross-section plot of\nthe majority states and (b) the minority states up to a distance of 5 \u0017A from the Fe chains (c.f. Fig. 1). (a) and (b) cut through\nthe middle green Fe atoms in (f) and majority and minority states are de\fned with respect to these Fe atoms. STM images at\na distance of 5 \u0017A above the Fe chains are shown in (c) for majority electrons, (d) for minority electrons, and in (e) for the sum\nof both contributions. The part of the surface displayed in the STM images is given in (f) where arrows indicate the direction\nof the Fe spin moments. Alternating color of Fe chain atoms and arrows are introduced to emphasize opposite direction of Fe\nspin moments. 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Bl ugel, G.\nBihlmayer and S. Heinze, Phys. Rev. B 79, 094411 (2009).\n32P. Bruno, Phys. Rev. B 39, 865 (1989)"    },    {        "title": "0911.4137v2.Magnetisation_dynamics_in_exchange_coupled_spring_systems_with_perpendicular_anisotropy.pdf",        "content": "arXiv:0911.4137\nMagnetisation dynamics in exchange coupled spring systems with perpendicular\nanisotropy.\nPedro M. S. Monteiro\u0003, D. S. Schmooly\nDepartamento de F\u0013 \u0010sica and IFIMUP, Universidade do Porto,\nRua do Campo Alegre 687, 4169-007 Porto, Portugal\nMagnetisation dynamics in exchange spring magnets have been studied using simulations of the\nFePt/Fe bilayer system. The FePt hard layer exhibits strong perpendicular magnetocrystalline\nanisotropy, while the soft (Fe) layer has negligible magnetocrystalline anisotropy. The variation of\nthe local spin orientation in the Fe layer is determined by the competition of the exchange coupling\ninteraction with the hard layer and the magnetostatic energy which favours in-plane magnetisation.\nDynamics were studied by monitoring the response of the Fe layer magnetisation after the abrupt\napplication of a magnetic \feld which causes the systems to realign via precessional motion. This\nprecessional motion allows us to obtain the frequency spectrum and hence examine the dynamical\nmagnetisation motion. Since the rotation of the spins in the soft layer does not have a well de\fned\nmagnetic anisotropy, the system does not present the usual frequency \feld characteristics for a thin\n\flm. Additionally we obtain multi-peaked resonance spectra for the application of magnetic \felds\nperpendicular to the \flm plane, though we discount the existence of spin wave modes and propose\nthat this arises due to variations in the local e\u000bective \feld across the Fe layer. The dynamic response\nis only considered in the Fe layer, with the FePt layer held \fxed in the perpendicular orientation.\nI. INTRODUCTION\nIncreasing attention has been paid to the so-called\nexchange-spring systems, which couple the hard and soft\nmagnetic properties of the two magnetic components via\nthe exchange interaction between them. Recent stud-\nies have indicated that the magnetic energy product can\nbe theoretically as high as 120 MGOe [1]. The exchange\nspring system is in general characterised by the reversible\ndynamics in the soft component arising from the compe-\ntition between the magnetic anisotropies in the two mag-\nnetic materials. Goto et al. [2] were the \frst to consider\nsuch systems based on the assumption that the exchange\nand anisotropy energies are large enough to overcome\nthe magnetostatic energy of the system. The speci\fc\nspin con\fguration for such a system will depend on the\nrelative strengths and directions of the anisotropies, as\nwell as the exchange interaction between the magnetic\nlayers (phases) and the orientation of any applied mag-\nnetic \feld. The spin con\fguration is then determined\nby the energy minimisation, taking into account the var-\nious magnetic contributions in the system as a whole.\nKneller and Hawig [3] applied these principles to the con-\nstruction of a new class of permanent magnetic materials\nwhich are referred to as the exchange springs. These au-\nthors analysed the reversible component of the hysteresis\nloop. Fullerton et al [4] considered layered systems of\nSm-Co/Fe and Sm-Co/Co, performing simulations based\non a 1D model. Bowden et al. [5] have made exten-\nsions to Goto's 1D model for discrete systems and have\nobtained a relation between the bending and exchange\n\felds. Bowden et al. [1] have also performed numerical\n\u0003Electronic address: pedrmonteiro@gmail.com\nyElectronic address: dschmool@fc.up.ptsimulations of 1, 2 and 3D systems, using point defects in\nthe chain to determine the behaviour of the system and\npredict a collapse of the exchange spring system. The\nmodel of Asti et al. [6] considers a FePt (hard) mag-\nnetic layer with perpendicular anisotropy coupled to a\nFe (soft) magnetic layer. Among the various consider-\nations, the authors have studied the equilibrium phase\ndiagram as a function of the thicknesses of the two mag-\nnetic layers, predicting a transition from a rigid mag-\nnet (RM) to an exchange spring (ES). The rigid magnet\nregime concerns the state where the soft magnetic layer\nmagnetisation strictly follows that of the hard layer. The\nexchange spring e\u000bect is manifest in the local variation\nof the e\u000bective \feld. In the bilayered system that we\nare considering, the FePt layer has dominant perpendicu-\nlar magnetocrystalline anisotropy, while the Fe layer has\ndominant in-plane anisotropy due to shape e\u000bects. As\nthe Fe layer thickness increases, the local \feld re\rects\nthe varying in\ruences of the two contributions and the\nlocal spin orientation is seen to rotate from the perpen-\ndicular direction at the FePt/Fe.\nFerromagnetic resonance can provide explicit infor-\nmation regarding interlayer interactions in multilayered\nstructures and has been successfully applied to many sys-\ntems [7] and [8]. There are very few papers that deal\nexplicitly with FMR in exchange spring systems, so it\nis therefore a new challenge that needs to be studied.\nGrimsditch et al [9] studied experimentally the magnon\nfrequency dependence of a Sm-Co/Fe system suing Bril-\nlouin scattering and found a good agreement with the\ntheory developed for the one-dimensional model. Also,\nPechan et al [10] investigated the frequency dependence\nof the ferromagnetic modes as a function of the in-plane\nangle.\nCrew and Stamps [11] have simulated a bilayered sys-\ntem of CoPt/Co, to study the resonance frequency spec-\ntrum and anisotropy for a particular applied \feld direc-arXiv:0911.4137v2  [cond-mat.mes-hall]  19 Apr 20102\ntion. However, this paper o\u000bers a more detailed descrip-\ntion of the dynamical behaviour of these kind of systems,\ncontributing to a better understanding.\nExperimentally, FePt/Fe system has been studied by\nCasoli et al. [12], who have performed measurements of\nthe hysteretic behaviour, con\frming the transition from\nrigid magnet to exchange spring behaviour. Schmool et\nal.[13] have performed ferromagnetic resonance mea-\nsurements on the same FePt/Fe interface system and\nhave observed signi\fcant changes in the FMR spectra\nfor RM and ES samples.\nExchange spring systems o\u000ber various potential tech-\nnological applications, for example, in the form of per-\nmanent magnets [3], mag-MEMS [6] and MRAM devices\n[14]. In the current paper we consider the exchange\nspring system which corresponds to the FePt/Fe bilayer\nsamples discussed above, in which the magnetocrystalline\nanisotropy of the FePt layer aligns its magnetisation in\nthe direction perpendicular to the \flm plane.\nThe paper is organised as follows: in the following sec-\ntion we discuss the basic theory and micromagnetic simu-\nlations used to study the system, in III we present results\nand discussions of these simulations and in the \fnal sec-\ntion we will give the conclusions of our study.\nII. THEORETICAL MODEL\nA. Basic theoretical considerations\nThe free energy of a magnetic system can be expressed\nin the most general form as the sum of various contri-\nbutions: exchange energy, Eex, demagnetising energy,\nEdem, magnetocrystalline anisotropy energy, EKand the\nZeeman energy, E0. This can then be expressed as:\nEtotal=Eex+Edem+E0+EK: (1)\nThe equilibrium condition of the magnetic system can\nfrequently then be evaluated by minimising this energy\nwith respect to the orientation of the magnetisation for\nexample. The dynamics of coherent magnetisation be-\nhaviour is typically described using the Landau - Lifshitz\n- Gilbert (LLG) equation and takes the form:\nd\u0000 !M\ndt=\u0000\r\u0000 !M\u0002\u0000 !Heff+\u000b\nMs\u001a\u0000 !M\u0002@M\n@t\u001b\n;(2)\nThis equation describes the temporal evolution of the\ndirection of the magnetisation vector due to the preces-\nsional motion induced by the magnetic torque associated\nwith the e\u000bective applied \feld, Heffand its subsequent\nrelaxation (viscous damping) which is controlled by the\nmagnitude of the Gilbert damping parameter, \u000b. The\ne\u000bective magnetic \feld will have various contributions:\n\u0000 !Heff=\u0000 !Hk+\u0000 !H0+\u0000 !Hex\u0000\u0000 !Hdem; (3)and is related to the free energy of the system as de-\nscribed by the relation:\n~Heff=\u0000@Etotal\n@~M: (4)\nThe boundary conditions of a magnetic \flm can have\na signi\fcant in\ruence on the dynamic response of the\nmagnetisation to an excitation \feld or torque [15]. This\nis because these boundary conditions are related to the\nsurface and interface anisotropies and can e\u000bectively pin\nthe magnetic spins at such magnetic discontinuities. For\nexample, in a thin magnetic \flm with a magnetic \feld\napplied perpendicular to the plane standing spin waves\ncan be excited, giving a resonance condition which is de-\nscribed by:\n!n=\r\u0002\nH0\u0000Hk\u0000M0+Dk2\nn\u0003\n: (5)\nwhereknis the spin wave wavevector of mode n. Restric-\ntions imposed by the boundary conditions on the allowed\nvalues of the wavevectors can be expressed as [16]:\nk(p1+p2)\np1p2\u0000k2= tan (kL); (6)\np1andp2are the pinning parameters which depend on\nthe surface or interface energies and hence will be related\nto surface and interface anisotropies.\nB. Simulations\nIn this work we concentrated on the study of the static\nand dynamic properties of a bilayer system where, for\nthe most part, the interface has a rigid interface condi-\ntion. This means that the interface spins at the Fe/FePt\nboundary are \fxed in the perpendicular direction (which\nwe take as the y-axis of the coordinate system). The\ndynamic response comes entirely from the Fe layer for\nour simulations (which we introduce in light of experi-\nmental results, which show that this is active in the fre-\nquency range studied [13]). The simulations were per-\nformed by constructing a bilayer of FePt, with perpen-\ndicular anisotropy (in the L1 0phase [12, 17]), coupled to\nan Fe layer using the OOMMF (Object Oriented Micro-\nmagnetic Framework) [18] software. The Fe \flm thick-\nness was variable, in general, but for the dynamics we\nhave maintained this constant, once we have prede\fned\nthe thickness necessary to obtain an exchange spring sys-\ntems, which corresponds to 25 \u000210\u000010m. We note that\nthe lateral dimensions were arbitrarily chosen and are\nmuch larger than the \flm thickness. The Fe layer is con-\nsidered to have shape anisotropy and negligible magne-\ntocrystalline anisotropy (as compared to the FePt layer).\nA Zeeman contribution will arise from the application of\na magnetic \feld.\nThe precision of the numerical results are governed by\nthe micromagnetic cell parameters; (10000 ;1;10000)\u00023\n10\u000010m. This essentially means that we are taking the\ncell to be one spin in the direction perpendicular to the\n\flm plane, where we measure the direction with respect\nto the xz plane orientation, as illustrated in \fg 1. The\nexchange constant between Fe/Fe and FePt/Fe spins is\n1\u000210\u000012J/m, the uniaxial anisotropy in Fe/Pt layer\nis 4\u0002106J/m3(in y direction) [19], the damping coef-\n\fcient is 0:01 and the time step is 1 \u000210\u000014s. Finally\nthe magnetisation saturation for Fe and Fe/Pt layers are\n1700\u0002103and 690\u0002103A/m [19], respectively.\nFIG. 1: Schematic representation of the position of the spins\nin the sample.\nIn order to perform the dynamical calculations we need\nto obtain the equilibrium spin con\fguration, which is ob-\ntained by an energy minimisation routine without any\nstatic applied \feld. This is then taken as the initial con-\ndition for any subsequent dynamical calculations. The\ndynamics of interest in the Fe layer then arise due to the\nperturbation or excitation of the system upon the abrupt\napplication of a magnetic \feld along some speci\fc direc-\ntion with respect to the \flm plane. The evolution of the\nmagnetisation is then evaluated as a function of time,\nfrom which we perform a Fast Fourier Transform (FFT)\nto obtain the relevant frequency spectrum. We have used\nthe z-component of the magnetisation for all calculations\nof the spin dynamics since it is transversal to the direc-\ntion of the applied \feld, which is maintained in the x-y\nplane.\nIII. RESULTS AND DISCUSSION\nA. Static con\fguration\nThe simulations were performed using the conjugate\ngradient method in the OOMMF package to evaluate the\nspin orientation as a function of its position in the Fe\nlayer. We have varied the Fe layer thickness (number of\nspins) in order to \fnd an optimum thickness for which to\nobtain an exchange spring (ES) system (i.e. a rotational\nvariation across the \flm thickness). The results for the\nrigid interface (\frst Fe spin \fxed in the perpendicular\ndirection) are shown in \fgure 2(a) . Here we note thatfor thicknesses corresponding to 7 Fe spins, the system\nforms a rigid magnet (RM), whereby the Fe spins align\nalong the FePt anisotropy (easy) axis as de\fned along\nthe \flm normal. Adding one more spin we see that there\nis a signi\fcant relaxation to the spins in the Fe layer\ntowards the in-plane direction. This relaxation further\nincreases as further Fe spins are added. Almost full in-\nplane rotation (of the top Fe spin) occurs for around 25\nFe spins. The transition from RM to ES is expected from\nenergy considerations since at a certain thickness there\nare su\u000ecient spins to allow the competition between the\nexchange coupling energy between the layers (causing an\ne\u000bective perpendicular anisotropy in the Fe layer) and\nthe magnetostatic energy to be spread among the spins\ncreating a domain wall. Full in-plane rotation can be seen\nas the construction of a 90\u000edomain wall (DW). Such a\ntransition has been predicted, for example by Asti et al.\n[6] and observed experimentally by Casoli et al. [12]. In\naddition to considering the rigid interface conditions, we\nhave performed the corresponding calculations for a re-\nlaxed interface. In this case we de\fne the FePt layer to\nconsist of 5 spins. This means that the domain wall can\nnow penetrate the hard magnetic layer, helping to fur-\nther reduce the total energy of the system; the results\nare shown in \fgure 2(b). Once again a transition from\nRM to ES is observed, though the number of Fe spins is\nreduced compared to the rigid interface condition. This\ncan be expected since the extension of the DW into the\nFePt layer helps shift some of the energy from the Fe layer\nto that of the FePt. The static con\fguration is a sensi-\ntive function of the number of spins in both the Fe and\nFePt layers. We have performed extensive studies of such\nconditions and will be reported elsewhere [20]. For our\npurposes, we have chosen to study the Fe layer consisting\nof 25 spins using rigid interface conditions. This o\u000bers a\nsystem in which an almost complete 90\u000eDW wall exists.\nWe now proceed to study the magnetisation dynamics in\nthis system.\nB. Magnetisation dynamics\nTo study the magnetisation response to an abruptly\napplied magnetic \feld along some speci\fc direction, we\nuse the LLG, which is solved using the Runge-Kutta\nmethod, also in the OOMMF software. To understand\nthe importance of material parameters and response of\nthe system, we have made a series of simulations with\napplied magnetic \felds of di\u000berent strengths applied in\nvarious directions. The perpendicular anisotropy of the\nFePt layer provides the system with a unidirectional char-\nacter which will be transmitted to the Fe layer via the\nexchange coupling between the layers. Such properties\nwill be expected to in\ruence the dynamic properties of\nthe Fe layer. This can be illustrated, for example, in the\nspectra obtained for the +20 and \u000020\u000edirections, which\nare shown in \fgure 3(a) and 3(b).\nWe note that the principle position (frequency) of the4\n(a)Angle with respect to the bydirection as a function of the number of\nspins for a rigid interface.\n(b)Angle with respect to the bydirection as a function of the number\nof spins for a 5 spins damped interface.\nFIG. 2: (a), (b) Show the spin angle as function of position in\nFe layer. The \frst spin on each plot corresponds to the \frst\nun\fxed spin in the chain.\npeaks in the spectra are the same; however, the relative\nintensities are very di\u000berent - the principal peak in \fgure\n3(b) shows a higher intensity than in \fgure 3(a). There-\nfore this camou\rages the importance of the other relevant\npeaks (frequency modes). Some of the principle charac-\nteristics of the dynamical properties can be elucidated\nfrom the study of the frequency \feld behaviour, which\nwe have done as a function of the direction of the applied\nmagnetic \feld. In 4(a) and 4(b) we show the dispersion\nrelations of the dominant resonance frequency for various\nangles of the applied \feld. (We note that the 0\u000edirection\ncorresponds to the \feld applied in the plane of the \flm.)\nWhile the positive and negative directions appear to give\n(a)FFT for magnetic \feld of 1 T and 20\u000e.\n(b)FFT for magnetic \feld of 1 T and \u000020\u000e.\nFIG. 3: (a), (b) show the FFT spectrum of the Fe layer.\nvery similar results, there are some important di\u000berences\nwhich are related to the unidirectional character of the\nsystem. The variation in each case appears as for a sys-\ntem with almost vanishing magnetic anisotropy, which is\nclearly not the case.\nIn a system with a well de\fned anisotropy axis, the\nfrequency \feld behaviour has a characteristic shape and\ndepends on the type of magnetic anisotropy; uniaxial, cu-\nbic, etc. For example, in the case of uniaxial anisotropy,\nwith the \feld applied along the hard axis, the disper-\nsion relation shows a marked cusp with a minimum cor-\nresponding to the anisotropy \feld, HK, of the system.\nWhile with the \feld applied along the easy axis the in-\ntercept with the frequency axis occurs to a resonance\ncondition in which a \feld corresponding to the anisotropy\n\feld is applied; i.e. !=\rHK, see for example Vonsovskii\n[21]. In the present case we see no well de\fned axis of\nanisotropy. This is because in our case this does not\nexist, since the directions of the spins in the Fe layer\nvary as a function of position across the thickness of the5\n(a)Plot of the resonance frequency as a function of the applied\nmagnetic \feld for positive angles (measured in the \frst quadrant).\n(b)Plot of the resonance frequency as a function of the applied\nmagnetic \feld for negative angles (measured in the fourth quadrant).\nFIG. 4: (a), (b) depict the spectrum of frequencies as a func-\ntion of the applied magnetic \feld for several angles.\n\flm. This will mean that we cannot, strictly speaking,\ntreat the system in the usual manner, despite the usual\nasymptotic behaviour still being apparent [22]. In \fgure\n5(a) we illustrate the resonance frequency as a function\nof angle (of the applied magnetic \feld) for 0 :1, 1 and 2\nT. This shows a reasonably symmetric variation about\nthe x-axis. However, a closer inspection reveals that for\nnegative angles the frequencies are higher.\nA more sensitive display of this asymmetry is shown\nin the variation of the asymptotic slope of the frequency\n\feld data as a function of angle, see \fgure 5(b). The dis-\nparity between positive and negative directions becomes\nmore marked for larger angles. In fact it would appear\nthat there is a discontinuity in the variation in the region\nof 5\u000e. This asymmetry and discontinuity, as well as the\nasymmetry in the angular dependence of the frequency\nare related to the unidirectional behaviour transmitted\nto the Fe layer from the FePt layer via the exchange in-\n(a)Resonance frequency as a function of the angle for\n\fxed applied magnetic \feld.\n(b)Resonance frequency per applied magnetic \feld as a function of the\nangle.\nFIG. 5: (a), (b) show the symmetry properties of the system.\nteraction. Added to this the rigid boundary conditions\nwill be expected to accentuate this behaviour.\nTo gain a fuller understanding of the dynamical re-\nsponse of the system to the abruptly applied \felds in the\nsimulation, it is instructive to consider the initial and \f-\nnal con\fgurations; i.e. that of the initial state without an\napplied \feld (which will be the same for all simulations)\nand the \fnal equilibrium state to which the system will\nrelax. In \fgure 6 we show the spin con\fgurations for spe-\nci\fc cases: the 0 T case corresponds to the initial spin\ncon\fguration used for each simulation. We also show\nthe \fnal state con\fgurations for 1 and 2 T applied along\nthe \flm plane ( x-axis) and along the normal direction to\nthe \flm plane (y-axis). In all cases the lower (Fe) spin\nis always in the perpendicular direction and arises due\nto the \fxed (or rigid) boundary condition that we have\nimposed. In the case of a 2 T \feld applied along the\n\flm normal, all spins align along the \feld giving a sat-\nurated state. It will be noted that 1 T is not su\u000ecient\nto push the DW out of the \flm and a small rotation is6\nFIG. 6: Initial spins con\fguration (0 T); \fnal spin con\fgu-\nration in by(90\u000e) and bx(0\u000e) direction. The \frst spin, in the\nbotton of each chain, is the \fxed spin from the FePt layer.\nstill evident. Applying the \feld in the plane of the \flm\ne\u000bectively squeezes the 90\u000eDW, reducing its thickness.\nThe transition from initial to \fnal state will de\fne the\ndynamics of the magnetisation process via the preces-\nsional motion of each spin. There will be two main factors\nwhich will govern this motion; the e\u000bective \feld; Heff,\nfor each spin, which will depend on its position within the\nlayer, and the overall angle di\u000berence between the initial\nand \fnal states. The fact that the e\u000bective \feld varies\nas a function of position can be seen by the equilibrium\nspin distribution across the Fe layer. The di\u000berence be-\ntween initial and \fnal state will govern in large part the\namplitude of the motion of each spin in its movement;\n\fgure 7 shows these variations for the parallel and per-\npendicular \felds illustrated in \fgure 6. It will be noted\nthat for in-plane \felds the variation is negative, while\nfor out of plane \felds this will be positive (due to the\nconvention we have chosen). This will therefore deter-\n(a)Variation of the angle between initial and \fnal con\fgurations\nas a function of the position of the spin in the sample in the bx\ndirection for 1 T and 2 T.\n(b)Variation of the angle between initial and \fnal con\fgurations\nas a function of the position of the spin in the sample in the by\ndirection for 1 T and 2 T.\nFIG. 7: (a), (b) show the relative deviations from the initial\nangle. The \ructuations observed in \fgure 7(a) are due to\nthe error in determining the angle. Only the line of tendency\nshould be considered and not the particular value of each\npoint in the curve.\nmine the character of each spectrum and can be related\nto the resonance frequency and the relative intensities of\nthe various peaks in the spectra. In addition to the am-\nplitude or precessional angle of motion, the intensity of a\nparticular peak will also depend on the number of spins\nwhich are involved in that motion. To illustrate this in-\nterpretation we can consider some speci\fc cases. Let us\n\frst consider the \feld applied in the plane of the sample\n(along the x-axis). The initial and \fnal spin con\fgura-\ntions are illustrated in \fgure 6. The \frst thing to note\nis that there is only a small change in the con\fguration\nitself, with the largest changes occurring at around the\n10th spin from the interface between Fe and FePt. For Fe\nlayers of di\u000berent thicknesses this will be di\u000berent. We\nalso note that the di\u000berences should increase with the\nstrength of the applied \feld; see \fgure 7(a). When the\nmagnetic \feld is applied along the \flm normal we notice\nthat the character of the spectrum becomes much more\nregular; the spectrum for a 2 T \feld applied in the per-\npendicular direction is shown in \fgure 8. Similar spectra\nare obtained for other \feld values. In this case the dif-\nferences from initial to \fnal state are much greater, and\nincrease as we move from the interface to the outer sur-\nface of the Fe layer, as shown in \fgure 7(b). In traditional7\nFIG. 8: Spectrum of frequencies obtained by FFT for an ap-\nplied magnetic \feld of 2 T and 90 (with respect to the by\ndirection).\nmagnetodynamic measurements performed by ferromag-\nnetic resonance (FMR), for example, multi-peaked spec-\ntra are usually associated with the existence of standing\nspin wave resonance modes which arise from the mag-\nnetic con\fnement in the perpendicular direction and the\nspeci\fc boundary conditions which apply. Such consid-\nerations lead to a resonance equation for the resonance\nfrequency in function the applied magnetic \feld and the\nspin wave wave-vector of the form of the Kittel equation\n[23] - see equation 5.\nIf we consider that the spins at the lower interface are\n\fxed in the perpendicular direction, while those on the\nouter surface are free to move (as in the bulk), we can\neasily arrive at the allowed values for the spin wave modes\nas given by:\nk=(2n\u00001)\n2L\u0019 ; n = 1;2;3::: ; (7)\nwherendenotes the mode number and Lthe thickness\nof the Fe layer. In this case both odd and even modes\nshould be allowed since the boundary conditions are non-\nsymmetric. A plot of frequency against the square of the\nwave-vector should yield a linear variation - according to\nequation 5. Using our data we have made such plots for\nvarious applied \felds along the perpendicular direction;\nthese are shown in \fgure 9. We note that while much of\nthe data \ft on linear portions, the extremal values are\nclearly not in agreement. Therefore we believe that the\nvarious peaks in the spectra do not arise from the excita-\ntion of standing spin wave modes. Rather, we think that\nthe existence of the multi-peaked spectra is due to the\ndi\u000berent local conditions (local e\u000bective \felds) of each\nspin, there will be a distribution of resonance conditions\nwhich are su\u000eciently spread out as to be distinguishable\nresonances. These local e\u000bective \felds will re\rect the\ndi\u000bering conditions and the local environment, where alocal energy minimum is encountered, giving rise to the\ndirection of the spin at that position. The intensity dis-\ntribution also re\rects the initial to \fnal state directions,\nwhich in much of the layer are quite large (see \fgures\n6 and 5). This will mean that the initial precessional\nangles will be large and hence give increased intensities,\nwhich are proportional to the transversal component of\nthe magnetisation.\nFIG. 9: Frequency as a function of the squared wavenumber\nfor di\u000berent magnetic \felds using the Kittel theory for spin\nwaves.\nIV. CONCLUSIONS\nWe have made a detailed study of the model system\ncomprised of a hard magnetic layer (FePt) with perpen-\ndicular anisotropy exchange coupled to a soft magnetic\nlayer (Fe). The \frst step to the study of the magneti-\nsation dynamics in this system concerns the determina-\ntion of the equilibrium condition. This is achieved by\nminimising the free energy density of the system and ob-\ntaining the orientation of the individual local spin ori-\nentations. We have performed such calculations for the\nsystem with varying Fe thicknesses. Since we are inter-\nested in studying the exchange spring system, we use\nthese calculations to choose the Fe thickness we require;\ni.e. where there is signi\fcant rotation of the magnetic\nmoment through the \flm. We see the transition from\nthe RM to ES occurs for a system with a thickness cor-\nresponding to 6 and 7 atomic spins for rigid and free\nboundary conditions, respectively. This corresponds to a\nthickness of around 0 :86 and 1:0 nm and compares rea-\nsonably well with the calculations of Asti et al. [6], who\nobtain a value of between 0 :6 and 0:8 nm, depending on\nthe thickness of the FePt layer.\nThe dynamics of the system - in response to an\nabruptly applied magnetic \feld - were studied as a func-\ntion of the strength and direction of the applied \feld.8\nThis allowed us to obtain the frequency spectrum and\ntherefore construct the dispersion relation. We have con-\nsidered only the dynamics of the Fe layer, since this\nshould be separate from the FePt dynamic response due\nto its elevated anisotropy, and is supported by FMR mea-\nsurements which only detect Fe resonance features at low\nfrequencies [13]. We note that the lack of a well de\fned\nanisotropy axis in the Fe layer means that the angular\nvariation of the frequency \feld characteristic di\u000ber from\nthat of a normal \flm, and always pass through the origin.\nThe high \feld asymptotic gradient varies with direction,\nwhose angular dependence is related to the exchange cou-\npling with the hard layer. A small asymmetry is noted\naround the x-axis which is due to the unidirectional char-\nacter of this exchange coupling. For magnetic \feld ap-plied in the perpendicular direction we obtain a rich fre-\nquency spectrum which can be understood in terms of\nthe variation of the local e\u000bective magnetic \feld across\nthe Fe layer.\nAs an aside, it is worth noting that the OOMMF soft-\nware provides su\u000ecient resolution to allow us to treat the\nsystem as atomic layers of spins in the direction perpen-\ndicular to the \flm plane. This is supported, for example,\nby the fact that we were able to reproduce the RM-ES\ntransition, and is supported by experimental measure-\nments [12] and other calculations [20].\nAcknowledgements . We thank OOMMF mailing\nlist for insightful discussions.\n[1] G. J. Bowden, K. N. Martin, B. D. Rainford and P. A. J.\nde Groot, J. Phys.: Condens. Matter 20, 015209 (2008)\n[2] E. Goto, N. Hayashi, T. Miyashita and K. Nakagawa, J.\nAppl. Phys. 36, 2951 (1965)\n[3] E. F. Kneller and R. Hawig, IEEE Trans. Magn. 27, 3588\n(1991)\n[4] E. E. Fullerton, J. S. Jiang, M. Grimsditch, C. H. Sowers\nand S. D. Bader, Phys. Rev. B 58, 12193 (1998)\n[5] G. J. Bowden, J. M. L. Beaujour, S. Gordeev, P. A. J.\nde Groot, B. D. Rainford and M. Sawicki, J. Phys.: Con-\ndens. Matter 12, 9335 (2000)\n[6] G. Asti, M. Ghidini, R. Pellicelli, C. Pernechele, M. Solzi,\nF. Albertini, F. Casoli, S. Fabbrici, and L. Pareti, Phys.\nRev. B 73, 094406 (2006)\n[7] B. Heinrich, Ultrathin Magnetic Structures II, edited by\nB. Heinrich and J. Bland (Springer-Verlag, Berlin, 1994),\np. 195.\n[8] B. Heinrich, S. T. Purcell, J. R. Dutcher, K. B. Urquhart,\nJ. F. Cochran and A. S. Arrott, Phys. Rev. B 38 12 879\n(1988)\n[9] M. Grimsditch, R. Camley, E. E. Fullerton, S. Jiang, S.\nD. Bader, and C. H. Sowers, J. Appl. Phys. 85, 5908\n(1999)\n[10] M. J. Pechan, N. Teng, J-D Stewart, J. Z. Hilt, E. E.\nFullerton, J. S. Jiang, C. H. Sowers, and S. D. Bader, J.\nAppl. Phys. 87, 6686 (2000)\n[11] D.C. Crew and R. L. Stamps, J. Appl. Phys. 96, 10 (2003)\n[12] F. Casoli, F. Albertini, S. Fabbrici, C. Bocchi, L. Nasi,R. Ciprian, and L. Pareti, IEEE Trans. Magn. 41, 3877\n(2005)\n[13] D. S. Schmool, A. Apolinario, F. Casoli and F. Albertini,\nIEEE Trans. Magn. 44, 3087 (2008)\n[14] K. Li, Y. Wu, G. Han, P. Luo, L. An, J. Qiu, Z. Guo and\nY. Zheng, J. Appl. Phys. 94, 5905 (2003)\n[15] A. Maksymowicz, Phys. Rev. B 33, 6045, (1986)\n[16] D. S. Schmool and J. M. Barandiaran, J. Phys.: Condens.\nMatter 10, 10679 (1998)\n[17] X.-H. Xu, H.-S. Wu, F. Wang and X.-L. Li, Appl. Surf.\nSci. 233, 1 (2004)\n[18] NIST technical report, M. J. Donahue and D. G. Porter,\n\\OOMMF Users Guide, Version 1.0,\" NISTIR 6376, Na-\ntional Institute of Standards and Technology, Gaithers-\nburg, MD (Sept 1999)\n[19] F. Casoli, L. Nasi, F. Albertini, S. Fabbrici, C. Bocchi,\nF. Germini, P. Luches, A. Rota, S. Valeri, J. Appl. Phys.\n103, 043912 (2008)\n[20] N. Sousa, A. Apolinario, P. M. S. Monteiro, F. Casoli, F.\nAlbertini, H. Kachkachi, F. Vernay and D. S. Schmool,\nto be published\n[21] S. V. Vonsovskii, Ferromagnetic Resonance, Pergamon\nPress, Oxford (1966)\n[22] H. Kachkachi and D. S. Schmool, Eur. Phys. J. B 56, 27\n(2007)\n[23] C. Kittel, Phys. Rev. 110, 1295 (1958)"    },    {        "title": "1001.1405v1.Correlation_induced_half_metallicity_in_a_ferromagnetic_single_layered_compound__Sr__2_CoO__4_.pdf",        "content": "arXiv:1001.1405v1  [cond-mat.str-el]  9 Jan 2010Correlation induced half-metallicity in a ferromagnetic s ingle-layered compound:\nSr2CoO4\nSudhir K. Pandey∗\nUGC-DAE Consortium for Scientific Research, University Cam pus, Khandwa Road, Indore - 452001, India\n(Dated: November 21, 2018)\nThe electronic and magnetic properties of Sr 2CoO4compound have been studied using ab initio\nelectronic structurecalculations. Asopposed toGGAcalcu lation, whichgives ferromagnetic metallic\nsolution, GGA+ Ucalculations provide two kind of ferromagnetic solutions: (i) half-metallic and (ii)\nmetallic. The half-metallic solution is a ground state of th e system and the metallic one is a\nmetastable state. The strong hybridization between Co 3 dand O 2porbitals decides the electronic\nand magnetic properties of the compound. The total magnetic moment per formula unit is found to\nbe∼3µB(S= 3/2). Our calculations give the magnetocrystalline aniso tropy energy of ∼2.7 meV,\nwhich provides a good description of experimentally observ ed large magnetocrystalline anisotropy.\nThe Heisenberg exchange parameters up to fourth nearest nei ghbours are also calculated. The\nmean-field theory gives the TC= 887 K. The possible physical implications of the ferromagn etic\nhalf-metallic ground state are also discussed.\nPACS numbers: 71.27.+a, 71.20.-b, 75.20.Hr\nI. INTRODUCTION\nThe single-layered compounds with a general formula\nofA2BO4(AandBstandforrare-earth/alkalimetalsand\ntransition metals, respectively) have attracted a great\ndeal of attention after the discovery of high temperature\nsuperconductor in La 2−xSrxCuO4.1These compounds\nshow many exotic physical phenomena like spin/charge\nstripes formation in nickelates and manganites1; and\nspin-triplet superconductivity in ruthenates.2\nAmong the single-layered compounds recently synthe-\nsized Sr 2CoO4has been reported to show ferromagnetic\n(FM) and metallic behaviours, which have never been\nfound in any other such materials.3,4The magnetiza-\ntion data show TCaround 250 K. The large magnetic\nanisotropy where magnetic easy axis is the c-axis is also\nobserved. Moreover,the resistivity data reveal the quasi-\ntwo-dimensional electronic nature for the system. In-\nterestingly, there are contradicting reports on the mag-\nnetic moments. The Matsuno et al.3and Wang et al.4\nhave reported the saturation magnetization of about 1.8\nand 1.0µB/Co, respectively. The effective magnetic mo-\nment of 3.72 µBhas also been found by fitting Curie-\nWiess behaviour in the paramagnetic phase, which sug-\ngests the magnetic moment corresponds to S≈3/2 spin\nconfiguration.4\nLeeet al.5tried to understand the above contradic-\ntory reports on the saturation magnetization by using\nLDA+Ucalculations. Their studies suggest that such\nvalues of magnetization can be intrinsic to the system as\nthey got similar two values for the total magnetic mo-\nment for different ranges of U. However, if we read the\nworks of Matsuno et al. and Wang et al. carefully we\ncan easily make out that the different values of satura-\ntion magnetization they got may not be intrinsic to the\nsystem. The M(H) data of both the groups look similar.\nFor example, the values of total magnetization at 5 T\nfound by Wang et al. and Matsuno et al. are about 1.4and1.6µB/Co, respectively, which arenotverydifferent.\nThe different values of saturation magnetization quoted\nin the last paragraph appear to arise due to two different\napproaches the authors used in extracting them.\nThe partial replacement of Sr by rare-earth elements\nleads to the formation of interesting magnetic phases\nkeeping the crystal structure intact.6–8For example,\nSr1.5La0.5CoO4compound retains the FM ground state\nof Sr2CoO4, whereas SrLaCoO 4manifests spin glass\nstate.6Contrary to FM state for Sr 1.5La0.5CoO4, the\nground state of Sr 1.5Pr0.5CoO4compound is found to be\nspin glass.7Similarly, the formation of spin glass state is\nalso reported in Sr 1.25Nd0.75CoO4compound.8The ex-\nistence of spin glass state in the doped compounds ap-\npears to be a generic phenomenon. It is well known that\nthe spin glass state arises due to the presence of com-\npeting ferromagnetic and antiferromagnetic interactions.\nTherefore, it would be interesting to calculate the na-\nture of magnetic interactions between the neighbouring\nCo atoms in the Sr 2CoO4compound.\nHere, we report the detailed electronic and magnetic\nstates ofSr 2CoO4usingab initio electronic structure cal-\nculations. The ground state solution is found to be a fer-\nromagnetic half-metallic state having S= 3/2 spin con-\nfiguration, which is different from that reported by Lee\net al. The strong hybridization between Co 3 dand O 2p\norbitals decides the electronic and magnetic properties of\nthe compound. The calculation gives magnetocrystalline\nanisotropy energy of ∼2.7 meV. The nature of mag-\nnetic interactions between the Co atoms is calculated up\nto fourth nearest neighbours, which shows mixed ferro-\nmagnetic and antiferromagneticcoupling. The estimated\nferromagnetic transition temperature within mean-field\ntheory is ∼887 K.2\nII. COMPUTATIONAL DETAILS\nThe nonmagnetic and ferromagnetic electronic struc-\nture calculations of Sr 2CoO4compound were carried\nout using LmtArt 6.61.9For calculating charge density,\nfull-potential linearized Muffin-Tin orbital (FP-LMTO)\nmethod working in plane wave representation was em-\nployed. In the calculations, we have used the Muffin-Tin\nradii of 2.915, 1.965, 1.608, and 1.778 a.u. for Sr, Co,\nO1, and O2, respectively. The charge density and effec-\ntive potential were expanded in spherical harmonics up\ntol= 6 inside the sphere and in a Fourier series in the\ninterstitial region. The initial basis set included 5 s, 4p,\nand4dvalence, and4 ssemicoreorbitalsofSr; 4 s, 4p, and\n3dvalence, and 3 psemicore orbitals of Co, and 2 sand 2p\norbitals of O. The exchange correlation functional of the\ndensity functional theory was taken after Vosko et al.10\nand GGA calculations were performed following Perdew\net al.11\nThe effect of on-site Coulomb interaction ( U) under\nGGA+Uformulation of the density functional theory is\nalso considered in the calculations. The detailed descrip-\ntion of the GGA+ Umethod implemented in the code\ncan be found in Ref. [12]. The double counting scheme\nused in the code is normally called as fully localized limit\nin the literature. To study the role of orbital degrees of\nfreedom, wehavealsoincludedspin-orbitcoupling(SOC)\nin the calculations. The Self-consistency was achieved\nby demanding the convergence of the total energy to be\nsmaller than 10−5Ry/cell. (10, 10, 10) divisions of the\nBrillouin zone along three directions for the tetrahedron\nintegration were used to calculate the density of states\n(DOS).\nIII. RESULTS AND DISCUSSIONS\nThe atomic arrangement in the unit cell is shown in\nFig. 1, which displaysbodycentertetragonallattice. The\nCo atoms occupy the corners and bodycenter positions\nand each Co atom is surrounded by six O atoms forming\na distorted octahedron. In this structure there are two\nkinds of O represented by O1 and O2. O1 lies in the ab\nplane and O2 along the c-axis. The bond distance be-\ntween the corner and bodycenter Co atoms is almost 1.8\ntimes larger than that between nearest Co atoms sitting\nat the corners. This suggests the quasi-two-dimensional\nnature of the system where its electronic properties are\nexpected to be decided by transport of electrons in the\nabplane.\nIn order to know the crystal-field effect and nature\nof Co-O bonding in Sr 2CoO4we have plotted the par-\ntial density states (PDOS) of Co 3 dand O 2pobtained\nfrom nonmagnetic solution in Fig. 2. In the octahe-\ndral symmetry Co 3 dstates split into t2gandegstates\nand the separation between these states is found to be\n∼1.5 eV. The degeneracies of these states are further\nlifted in tetragonal symmetry as evident from Figs. 2(a)\nO1 O2 \nSr \nCo \nFIG. 1: (Color online) Atomic arrangements in the unit cell.\nSr, Co and O atoms are represented by spheres with decreas-\ning radii.\nand (b). The triply degenerate t2gstates split into dou-\nbly degenerate ( dxz,dyz) states and nondegenerate dxy\nstate. Similarly, doubly degenerate egstates split into\nnondegenerate dx2−y2anddz2−r2states. The O 2 por-\nbitalsalsosplit indoublydegenerate( px,py)orbitalsand\nnondegenerate pzorbital as evident from Figs. 2(c) and\n(d).\nThet2gandegsectorsarespreadoverthe energyrange\nof about 5 and 10 eV, respectively. Roughly two times\nlarger extent of the egstates is due to larger overlap be-\ntweenegandporbitalsin comparisonto that between t2g\nandporbitals. In the present situation one can get max-\nimum overlaps of dx2−y2and O1 ( px,py) orbitals; and\ndz2−r2and O2pzorbitals which can lead to larger band-\nwidth of egsymmetric states. The occupied dz2−r2states\nare mostly found in the narrow region of 1 eV whereas\noccupied dx2−y2states are spread over an energy window\nof about 4 eV. Total number of delectrons is found to be\n∼6.5, which is about 1.5 more than the expected nom-\ninal value for Co4+ion. The above observations can be\nconsidered as a signature of the covalent nature of the\nCo-O bonds. There are large Co 3 dPDOS at the EF(∼\n8.4 states/eV/atom) in the nonmagnetic solution which\nmaybeconsideredasasignatureofferromagneticground\nstate based on the Stoner theory.3\n-6 -4 -2 0 21(d) O2  px/py\n pzPDOS (states eV -1  atom -1 )\nEnergy (eV) -6 -4 -2 0 21(c) O1 Co eg\n px/py\n pz-6 -4 -2 0 212(b) PDOS (states eV -1  atom -1 )\n dx2-y 2\n dz2-r 2-6 -4 -2 0 212(a) Co t2g  dxz /dyz \n dxy \nFIG. 2: (Color online) The partial density of different\nsymmetric states obtained from nonmagnetic solution. (a)\ndxz/dyz(thin lines) and dxy(thick lines) states, (b) dx2−y2\n(thin lines) and dz2−r2(thick lines) states. The px/py(thin\nlines) and pz(thick lines) symmetric partial density of states\nof planar oxygen (O1) and apical oxygen (O2) are shown in\n(c) and (d), respectively.\nIn order to know the electronic and magnetic struc-\ntures of the compound we have carried out FM calcu-\nlation. The energy of the FM solution is found to be\n∼336 meV less than that of nonmagnetic solution in-\ndicating the ferromagnetic ground state for the system.\nThis result is in accordance with the experimental obser-\nvation and earlier theoretical calculations.3,5The total\nand partial density of states obtained for the FM config-\nuration are plotted in Fig. 3. There is an asymmetric\ndistribution of states in both the spin channels. The dif-\nference between number of electrons in up and down spin\nchannels is a measure of net magnetization and found to\nbe∼2. Under the rigid band picture one can estimate\nthe effective value of the exchange parameter ( J) by con-\nsidering the energy difference between the up and down\nband edges, which is estimated to be ∼0.4 eV. The t2g\nup-spin bands are fully occupied whereas t2gdown-spin\nandegbands are partially occupied.\nSimilar to the nonmagnetic case, egbands are highly\nextended in comparisonto t2gbands. The strength ofex-\nchange parameters for t2gandegelectrons are estimated-6 -4 -2 0 2-1 01PDOS (states eV -1  atom -1 )\n(d)  O1 2 p\n O2 2 pPDOS (states eV -1  atom -1 )\nEnergy (eV) -1 01(c) Co eg dx2-y 2\n dz2-r 2Co t2g \n -1 01(b)  dxz /dyz \n dxy DOS (states eV -1  fu -1 )\n-10 -5 0510 (a)  \nTDOS \nFIG. 3: (Color online) The density of states obtained from\nferromagnetic GGA calculation. (a) total density of states\n(TDOS), (b) dxz/dyz(thin lines) and dxy(thick lines) states,\nand (c) dx2−y2(thin lines) and dz2−r2(thick lines) states.\nIn (d) partial density of psymmetric states for planar oxygen\n(O1)andapical oxygen(O2)arerepresentedbythinandthick\nlines, respectively.\nto be∼0.6 and 0.4 eV, respectively. The larger value\nofJfort2gelectrons can be attributed to more localized\nt2gstates leading to higher Coulomb interaction. Inter-\nestingly, O 2 pPDOS also show spin polarization and the\nestimated value of Jforpelectrons is also found to be ∼\n0.4 eV. The values of magnetic moment for Co, O1 and\nO2 atoms are found to be about 1.52, 0.17 and 0.07 µB,\nrespectively. The total magnetization per formula unit is\n∼2µB, which is almost same to the earlier reports.3,5\nThe appearance of magnetic moments at the O sites may\nbe attributed to the strong hybridization between dx2−y2\nand O1 ( px,py) orbitals; and dz2−r2and O2pzorbitals.\nThe values of magnetic moment for O1 and O2 atoms\nalso appear to support this conjecture as the magnetic\nmoment of O1 is roughly 2 times larger than that of O2.\nIn order to know the role of orbital degrees of free-\ndom on the magnetic properties of the system we have\nincluded SOCin the calculationsby consideringmagneti-\nzation directions along (001) and (100) axes. Our calcu-\nlations indicate that the c-axis is an easy axis as the total\nenergy of the system for (001) direction is found to be ∼\n2.7meV/Colessthanthatfor(100)direction. Thisresult4\nis in line with the experimentally observedmagnetic easy\naxis.3This value ofmagnetocrystallineanisotropyenergy\n(MAE) is ∼2.1 meV/Co more than that obtained by Lee\net al.5BasedonthemagnetizationdataofRef. [3], Lee et\nal. have given a rough estimate of ∼1.5 meV/Co for the\nMAE. This value is about three times larger than that of\nthe Leeet al. and about half of our value. The inclusion\nof SOC keeps the spin magnetic moments almost intact\nand induces 0.1 µBorbital magnetic moment at the Co\nsite.\nThe above GGA calculations provide the good repre-\nsentation of experimentally observed electronic and mag-\nnetic properties of the system including magnetocrys-\ntalline anisotropy. However, the on-site Coulomb inter-\naction parameter Uhas been found to be important in\nunderstandingthedetailed electronicand magneticprop-\nerties of the 3 dtransition metals oxies.12Therefore, it\nwould be interesting to see the effect of Uon electronic\nand magnetic states ofthe compound under GGA+ U. In\nthis approximation UandJare used as parameters.\nThe values of UandJrequired to reproduce the elec-\ntronic structure of a compound are sensitive to the ap-\nproximationsused in the calculations.13,14Moreover,it is\nwell known that the GGA+ Ucalculations often converge\nto the local minima depending on the starting electronic\nconfigurations used.15–17In the light of these facts we\nhave varied Ufrom 3 to 5 eV and fixed the value of J\nto 0.4 eV as estimated for the egelectrons. To know the\nexact ground state we have also used different starting\nconfigurations for the Co 3 delectrons. This range of U\nis found to provide the good representation of the exper-\nimentally observed electronic and magnetic properties of\ndifferent Co based systems.13,18–21\nBased on symmetry consideration we used 10 differ-\nent starting electronic configurations for the Co atom\nin the calculations. For these configurations only two\nkind of FM solutions S1 and S2 are found to exist. The\nS1 and S2 give half-metallic and metallic states, respec-\ntively. The difference in the electronic structure of Co\nin both the solutions can be found from Fig. 4, where\nwe have plotted the Co 3 dPDOS calculated for U= 4\neV. The half-metallicity with a gap of about 1 eV in the\ndown-spin channel is evident for the S1. The S2 shows\nmetallic state with a large t2gsymmetric states around\ntheEFin the down-spin channel. In the up-spin channel\nboth the solutions provide similar PDOS of egcharacter\nin the vicinity of EF. The comparison of Co 3 dPDOS\nobtained from the S2 and FM GGA solutions indicates\nthatS2isoriginatingfromtheFMGGAsolution. TheS1\nsolution is quite different and it has origin to the orbital\npolarization.\nThe PDOS closer to EFalong with occupancies of the\nCo 3dorbitals listed in Table 1 can be used to charac-\nterize both the solutions. Numbers written in normal\nand italics are corresponding to the S1 and S2, respec-\ntively. The t2g↑states are fully occupied in both the\nsolutions. The main difference between the two solutions\narises from the t2g↓sector. In the S1 and S2, dxy↓state-8 -6 -4 -2 0 2 4-2 -1 012S1 \n Energy (eV)  t2g \n eg \nDown Up PDOS (states eV -1  atom -1 )\n-8 -6 -4 -2 0 2 4S2 \nDown Up  t2g \n eg\n Energy (eV) \nFIG. 4: (Color online) The partial density of Co 3 dsymmet-\nric states obtained from ferromagnetic GGA+ U(U= 4 eV\nandJ= 0.4 eV) calculations corresponding to two conversed\nsolutions represented by S1 and S2. t2gandegorbitals are\ndenoted by solid and dashed lines, respectively.\ncan be considered as fully occupied and unoccupied, re-\nspectively. The contribution of dxz↓/dyz↓state below\ntheEFis small ( ∼0.17 electron) and can be considered\nas fully unoccupied in the S1, which gives rise to the\nhalf-metallic state. However, dxz↓/dyz↓state is roughly\nhalf-filled ( ∼0.6) in the S2 and mainly responsible for\nthe metallic state. Thus, we can represent the S1 and S2\nbydxy↓1anddxz↓1/2/dyz↓1/2electronic configurations,\nrespectively.\nTable 1: Electronic occupancies of different Co 3 dor-\nbitals along with the total number of delectrons for both\nthe spin channels obtained from ferromagnetic GGA+ U\n(U= 4 eV and J= 0.4 eV) calculations correspondingto\ntwo self consistent solutions represented by S1 (normal)\nand S2 (italics).\ndxz/dyzdxydx2−y2dz2−r2Totald\nUp0.95,0.940.94,0.950.84,0.900.88,0.824.56,4.55\nDown 0.17,0.610.92,0.140.22,0.250.36,0.251.84,1.86\nUp - Down 0.78,0.330.02,0.810.62,0.650.52,0.572.72,2.69\nBoth the solutions give roughly same magnetic mo-\nment (∼2.7µB) for the Co. The S1 induces magnetic\nmoment of ∼0.08µBat the O2 site whereas the S25\nprovides ∼0.18µBat the O1 site. The total magnetic\nmoment per formula unit for both the solutions is found\nto be∼3µB/fu representative of S= 3/2 spin state. In-\nterestingly, the energy of the S1 is ∼234 meV less than\nthat of the S2 indicating that the S1 corresponds to the\ntrue ground state of the system. Present work clearly\nshows the importance of Uin stabilizing a half-metallic\nstate which turns out to be the lowest energy state of the\nsystem. Thus, the ground state of Sr 2CoO4is found to\nbe half-metallic with S= 3/2 spin configuration.\nOur half-metallic ground state is quite different from\nthat obtained in earlier work.5The half-metallic solution\nof Leeet al. givesS= 1/2 spin configuration. This so-\nlution corresponds to almost occupied dxz↓/dyz↓states\nand half-filled dxy↓state whereas our S1 corresponds to\nthe opposite situation where dxz↓/dyz↓states are almost\nvacant and dxy↓state is filled. Here it is worth mention-\ning that we have also included the generalized gradient\napproximationin thecalculationswhereasLee et al. have\nonly considered the local density approximation (LDA).\nThis may give an impression that the different formula-\ntions of the exchange correlation functional can be the\nreason for the difference between our results and that of\nthe Leeet al. To address this issue we have also carried\nout detailed calculations using spin polarized LDA+ U\nmethod. The results obtained from these calculations\nare found to be almost the same as those obtained by\nGGA+U. This indicates that the existence of two solu-\ntions has nothing to do with the approximationsinvolved\nin treating different exchange correlation functional and\nsolely related to the orbital polarization.\nThe total and partial densities of states correspond\nto the half-metallic solution are shown in Fig. 5. The\nlarge spin polarization is evident from the total den-\nsity of states plotted in Fig. 5(a). The lowest occupied\nband shows the signature of van Hove singularity aris-\ning from the egsymmetric states as evident from Fig. 5\n(c). The dxyorbital is fully occupied and a gap of ∼1\neV is created between it and the unoccupied dxz/dyzor-\nbitals in the down-spin channel. In the up-spin channel\nonlyegsector provides the conducting electrons with a\ndominating contribution from the dz2−r2electrons. The\ndx2−y2symmetric PDOS in the vicinity of the EFis al-\nmost flat suggesting that the temperature dependent be-\nhaviour of the compound should be decided by dz2−r2\nelectrons. Fig. 5(b) shows a huge exchange splitting ( ∼\n7.5 eV) for the degenerate dxz/dyzorbitals. This result\nalso suggests that the Coulomb correlation in the pres-\nence of strong hybridization between Co 3 dand O 2 p\norbitals along with a tetragonal distortion makes full oc-\ncupation of nondegenerate dxyorbital energetically more\nfavorable. This may be considered as a signature of a\nferrodistortive Jahn-Teller ordering.\nIn orderto study the detailed nature ofmagnetic inter-\nactions we have calculated the Heisenberg exchange pa-\nrameters J0ibetweentheCoatomsusingFLEURcode,22\nwhere 0 and istand for the central atom and coordina-\ntion shell index, respectively. The Co atom sitting at the-8 -6 -4 -2 0 2 4-1 01PDOS (states eV -1  atom -1 )\n(d)  O1 2 p\n O2 2 pPDOS (states eV -1  atom -1 )\nEnergy (eV) -1 01\n(c) Co eg dx2-y 2\n dz2-r 2Co t2g \n -2 02\n(b)  dxz /dyz \n dxy DOS (states eV -1  fu -1 )\n-5 05\n(a)  \nTDOS \nFIG. 5: (Color online) The density of states corresponds\nto the ferromagnetic half-metallic solution obtained from\nGGA+U(U= 4 eV and J= 0.4 eV) calculation. (a) to-\ntal density of states (TDOS), (b) dxz/dyz(thin lines) and dxy\n(thick lines) states, and (c) dx2−y2(thin lines) and dz2−r2\n(thick lines) states. In (d) partial density of psymmetric\nstates for planar oxygen (O1) and apical oxygen (O2) are\nrepresented by thin and thick lines, respectively.\norigin is considered as the central atom, Fig. 1. The\nvalues of J0iup to 4thnearest neighbours (coordination\nshells) are shown in Table 2. It is evident from the table\nthat thefirst andthird nearestneighbourinteractionsare\nferromagnetic whereas second and fourth nearest neigh-\nbour interactions are antiferromagnetic in nature. The\nvalue ofJ01is found to be 19.06 meV which is about 15\ntimes larger than the values of neighbouring interactions.\nThis suggests that the effective magnetic interaction in\nSr2CoO4is ferromagnetic in nature, which is in accor-\ndance with the experimentally observed ferromagnetic\nground state.3Under the mean-field approximation we\ncalculated the Curie temperature ( TC) of∼887K, which\nis roughly 3.5 times larger than the experimentally mea-\nsured value. Such discrepancy between the experimental\nand calculated TCis not surprising as the overestimation\nofTCunder mean-field approximation is well known in\nthe literature.23–25\nTable 2: The Heisenberg exchange parameters ( J0i)\nup to fourth nearest neighbours (coordination shells) ob-\ntained by ferromagnetic GGA calculations using FLEUR6\ncode. 0 and istand for central atom and coordination\nshell index, respectively.\nNearest neighbour Coordination number J0i(meV)\n1st4 19.06\n2nd4 -1.11\n3rd8 1.25\n4th4 -0.33\nFinally, we discuss the physical implications of our re-\nsults. The half-metallic ferromagnet has potential appli-\ncation in the sprintonics industry.26This compound can\nbe considered as a good candidate as it shows TCcloser\nto the room temperature. The dPDOS at the EFare\nquite low (0.19 states/eV/atom) as evident from the S1\nof Fig. 4. This suggests that even a moderate hole dop-\ning at the Co sites can lead to insulating ground state.\nThe present work predicts the saturation magnetization\nof∼3µB/fu. The 5K M(H) data of Wang et al. do not\nshow any sign of saturation at maximum measured field\nof 5 T where the value of magnetization is found to be\nabout 1.4 µB/Co.4However, their M(T) data suggest to\nsupport our calculated S= 3/2 spin state as they have\nreported the effective magnetic moment of ∼3.72µB.\nHere it is important to note that M(H) data of Matsuno\net al. show almost saturation of magnetization at 7 T\nwhere the value of magnetization is found to be about\n1.8µB/Co.3However, this work has been carried out on\nthe thin films, which may suffer from the effects of sub-\nstrate induced strain and/or defects present in the films.Moreover, substrate contribution to the magnetization\nand the weight of the material in the thin film samples\nare difficult to measure accurately. All these things may\nlead to different magnetization value than bulk sample.\nInterestingly, Matsuno et al. haveexplainedtheir satura-\ntion magnetizationvalue by assumingCo4+in intermedi-\nate spin state and that would lead to the same magnetic\nmoment we obtained.\nIV. CONCLUSIONS\nThe detailed electronic and magnetic properties of a\nsingle-layered compound - Sr 2CoO4- have been studied\nusingab initio calculations with different formulations of\nthedensityfunctionaltheory. TheGGAcalculationgives\na ferromagnetic metallic state with total magnetic mo-\nment of ∼2µB/fu. The GGA+ Ucalculations provide\ntwo ferromagnetic solutions with S= 3/2 spin config-\nuration: (i) half-metallic and (ii) metallic. The ferro-\nmagnetic half-metallic state is the lowest energy state\nand representative of the ground state. Our results also\nsuggest that the compound may show metal to insulator\ntransition when a small amount hole is doped at the Co\nsites. In the present work we have also estimated the\ncrystal-field energy, magnetic moments, Heisenberg ex-\nchange parameters, transition temperature, etc., which\nprovidegooddescriptionofexperimentallyobservedelec-\ntronic and magnetic properties of the system.\n∗Electronic mail: sk−iuc@rediffmail.com\n1M. Imada, A. Fujimori, and Y. Tokura, Rev. Mod. Phys.\n70, 1039 (1998).\n2Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fu-\njita, J. G. Bednorz, and F. Lichtenberg, Nature 372, 532\n(1994).\n3J. Matsuno, Y. Okimoto, Z. Fang, X. Z. Yu, Y. Matsui, N.\nNagaosa, M. Kawasaki, and Y. Tokura, Phys. Rev. Lett.\n93, 167202 (2004).\n4X. L. Wang and E. Takayama-Muromachi, Phys. Rev. B\n72, 064401 (2005).\n5K. -W. Lee and W. E. Pickett, Phys. Rev. B 73, 174428\n(2006).\n6Y. Shimada, S. Miyasaka, R. Kumai, and Y. Tokura, Phys.\nRev. B73, 134424 (2006).\n7R. Ang, Y. P. Sun, X. Luo, C. Y. Hao, X. B. Zhu, and W.\nH. Song, Appl. Phys. Lett. 92, 162508 (2008).\n8S. Huang, K. Ruan, Z. Lv, L. 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Treger, Science 294, 1488 (2001)."    },    {        "title": "1001.4403v1.Solution_growth_of_Ce_Pd_In_single_crystals__characterization_of_the_heavy_fermion_superconductor_Ce2PdIn8.pdf",        "content": " 1Solution growth of Ce-Pd-In single cryst als: characterization of the heavy-fermion \nsuperconductor Ce 2PdIn 8 \n Klára Uhlí řová, Jan Prokleška, Vladimír Sechovský, Stanislav Daniš \n \nDepartment of Condensed Matter Physics, F aculty of Mathematics and Physics, Charles \nUniversity in Prague, Ke Karl ovu 5, 12116 Prague 2, Czech Republic  \n \nAbstract \nSolution growth of single crystals of the recently reported new compound Ce 2PdIn 8 was \ninvestigated. When growing from a stoich iometry in a range 2:1:20 - 2:1:35, single \ncrystals of CeIn 3 covered by a thin (~50 μm) single-crystalline layer of Ce 2PdIn 8 were \nmostly obtained. Using palladium richer  compositions the thickness of the Ce 2PdIn 8 \nlayers were increased, which allowed mechani cal extraction of single-phase slabs of the \ndesired compound suitable for a thorough st udy of magnetism and superconductivity. In \nsome solution growth products also CePd 3In6 (LaNi 3In6 –type of structure) and traces of \nphases with the stoichiometry CePd 2In7, Ce 1.5Pd1.5In7 (determined only by EDX) have \nbeen identified. Magnetic measurements of the Ce\n2PdIn 8 single crystals reveal paramagnetic behaviour of \nthe Ce3+ ions with significant magnetocrystalline anisotropy. Above 70 K the magnetic \nsusceptibility follows the Curie-Weiss law with  considerably different values of the \nparamagnetic Curie temperature, for the magnetic field applied along the a- (-90 K) and \nc-(-50 K) axis. Below the reported critical temperature for superconductivity Tc (0.69 K) \nthe electrical resistivity drops to zero. Co mparative measurements of the electrical \nresistivity, heat capacity and AC susceptibility  of several crystals reveal that the \nsuperconducting transition is strongly sample-dependent.  \n \n1. Introduction \nIn some Ce-based compounds, the interac tion of the Ce ions with the conduction \nelectrons often leads to a large enhancemen t of the effective electron mass. These so-\ncalled heavy-fermion (HF) compounds exhibit attractive electronic properties, such as \nstrongly enhanced paramagnetism, non-Fermi liquid behaviour, interplay between magnetism and superconductivity (SC), etc. Duri ng the last decade, the attention of many \nresearchers has been focused on the series of HF materials of the general chemical \nformula Ce\nmTnIn3m+2n where m, n = 1 or 2 and T = Co, Rh, Ir. These compounds \ncrystallize in the Ho mConGa3m+2n-type tetragonal structures with the space group \nP4/mmm . These structures are built by n layers of distorted cuboctahedra [CeIn 3] and one \nmonolayer of rectangular parallelepipeds [ TIn2] stacked sequentially in the [001] \ndirection [1] making the structure to be qua si-2D. The main building block of these \ncompounds, the cubic CeIn 3, is a heavy-fermion antiferromagnet (AF) with a Néel \ntemperature TN = 10.2 K [2] and becomes superconducti ng (SC) under pressure [3]. The \nwhole Ce mTnIn3m+2n series is characterized by interplay between magnetism and \nunconventional superconductivity, which makes these compounds suitable for thorough studies of varieties of the two cooperativ e phenomena in strongly correlated electron \nsystems. The quasi-2D R\nmTnIn3m+2n crystal structure provides i nvestigation of the effect \nof varying dimensionality on magnetism and unconventional superconductivity because  2the structures become less 2D-like with increasing n. The respective compounds are often \ncalled 115 ( m, n = 1) and 218 ( m = 2, n =1) compounds. \nCeCoIn 5, CeIrIn 5 and Ce 2CoIn 8 [4-7] are ambient-pressure superconductors, CeRhIn 5 \nand Ce 2RhIn 8 are antiferromagnetic (AF), tuned to the SC phase by applying pressure or \nby doping [8-10]. In a certain pressure range , coexistence of AF and SC has been \nobserved. While the superconducting transition temperature Tc is increasing with \nincreasing applied pressure the Néel temperature TN is decreasing and \nantiferromagnetism vanishes when TN = Tc. In the range of pressures where TN < Tc, the \nmagnetism became hidden and can be recovered by magnetic field [11]. On the other hand the superconductivity can influence devel opment of field-induced magnetic order as \nobserved for CeCoIn\n5 [12]. \nMost recently the group of Ce nTIn3n+2 was extended to Ce 2PdIn 8. Ce 2PdIn 8 was first \nreported by Shepa et al. [13], while an independent st udy has shown a non-Fermi liquid \nbehaviour [14] and HF SC where the SC em erges out of the long-range AF state at Tc = \n0.68 K and coexist with the AF at ambient- pressure [15]. Based on our results and \nbecause the reported Néel temperature e quals to the Néel temperature of CeIn 3, we have \nascribed the AF origin to the intergrowth of CeIn 3 phase into Ce 2PdIn 8 [16]. \nLast year, Kurenbaeva et al . [17] reported a new structure type, CePt 2In7, with a \ntetragonal structure (space group I4/ mmm ) formed by two layers of [PtIn 2] and one layer \nof [CeIn 3] connecting this compound with the Ce nTIn3n+2 family. According to recent \nresults presented by E. D. Bauer [18], this compound should be another Ce-based \nsuperconductor.  \nBoth the 115 and 218 compounds were mostly  prepared in single crystalline form \nusing the solution growth technique [1, 4, 19]. Unfortunately, in many cases the \ndescription of sample preparation is limited by citing a source where only general information about the method is presente d, or even by unpublished work. Although this \nmight be a strategy of the authors, we would like to show that the sample quality and its \nhistory might be a crucial parameter for the subsequent physical behaviour. In this paper \nthe SC of single-crystalline Ce\n2PdIn 8 will be presented together with detailed description \nof sample preparation.  \n \n2. Experimental \n2.1. Crystal growth of Ce 2PdIn 8 \nThe Ce-Pd-In phase diagram as reported by Shepa et al.  [13] is rather rich and according \nto our results probably still not completed. Unfortunately there are many compounds existing in rather broad Ce-Pd and Pd-In sol ubility ranges, which complicates the sample \nsynthesis namely when high quality of the samples is required.   The solution growth technique [20, 21] was used for the synthesis of single crystal \nsamples. The high purity alumina (99.8 %) crucibles or, for the best-found growth conditions, ultrahigh purity alumina (99.99 %) crucibles by Almath crucibles ltd . have \nbeen used to reach the maximum sample purity.     \nIn the first experiments, the single crystals of Ce\n2PdIn 8 were grown from In flux using \nthe starting stoichiometry ratio of Ce, Pd a nd In 2:1:20-30. The starting elements of high \npurity (In – 99.999 %, Pd – 99.95 %, Ce – 99.9 %) were put into alumina crucibles and \nsealed in quartz glass under high vacuum. Af ter that the samples were put into a \nprogrammable furnace and a thermal process wa s started; the system was heated up to  3950 °C hold there for 120 minutes and than slowly (2-4 °C/min) cooled down to \n350-400 °C. The crucibles were than replaced into new quartz glass ampoule with a quartz-wool stopper and evacuated again to h eat it up to 350 °C at which the remaining \nindium was centrifuged.  \nWhile crystals of the other Ce\n2TIn8 (T = Co, Rh, Ir) compounds can be prepared rather \neasily, the case of growing Ce 2PdIn 8 is rather complicated but also even more interesting. \nNumerous discrepancies were observed duri ng the characterization process; while the \nmicroprobe analysis from the sample surface confirmed the Ce 2PdIn 8 composition, the X-\nray powder diffraction (XRPD) resulted mostly in CeIn 3. It was found that after growing \nout of the above mentioned stoichiometry, multiphase products were formed. Mainly \ncuboid-shaped single crystals of CeIn 3 covered by a very thin layer (50-100 μm) of \nCe2PdIn 8 were obtained. This lead us to the idea that CeIn 3 grew until part of cerium was \nconsumed, than, in a narrow concentration region, Ce 2PdIn 8 was grown and finally, from \nthe reaming palladium in the melt, Pd 3In7 was formed.  \nBased on these results, a Pd-richer compositi on was used to suppress the initial growth \nof CeIn 3. Many experiments in a broad concentrati on range of Pd and In have been done \nfollowing the estimation of products ratio in previous experiments. The best composition \nhas been found to be CePd 2-3In35. From the higher Pd content, CePd 3In6 (LaNi 3In6 [22]) \nwas formed covering the surface of Ce 2PdIn 8 (Supplementary Fig. 5). The details of \ncharacterization of CePd 3In6 will be published elsewhere. The pure Ce 2PdIn 8 has not \nbeen reached yet but the Ce 2PdIn 8 layers in the CeIn 3-Ce 2PdIn 8 sandwiches were thick \nenough (~ 200 μm) to be mechanically separated. The border between the two phases is \nvery well defined and sharp as demonstrated  in Fig. 1 and Supplementary Figs. 1-3. The \nelement mapping Fig.1 (left) shows a cut polished surface of the so called CeIn 3 - \nCe2PdIn 8 sandwich. The line scan in Fig. 1 (right) pr esents the relative intensity of the Ce \nand Pd spectra along the marked arrow. A mi nority of a third phase with the composition \ndetermined from the EDX analysis as Ce 1.5Pd1.5In7 was detected. In some cases it \nappeared that a \"multilayered\" system with rather thick (50-200 nm) well defined layers \nof CeIn 3 and Ce 2PdIn 8 was formed as presented in supplementary Fig. 5. In some \nbatches, traces of CePd 2In7 were detected by EDX analysis. One should be also careful of \ninclusions of Pd 3In7 in Ce 2PdIn 8 single crystals. \nSince CeIn 3 oxidizes much faster than Ce 2PdIn 8 the boundaries between these phases \nare observable by naked eye or optical micros cope (Supplementary Fig. 1). The Ce-oxide \non the CeIn 3 phase makes also the back scattered electron (BSE) contrast more clear. The \nBSE contrast between CeIn 3 and Ce 2PdIn 8 at just polished (no oxide) samples is hardly \nvisible because palladium and indium have  similar atomic masses and difference in \nvolume densities of both compounds (7.820 g.cm-3 and 8.062 g.cm-3 for CeIn 3 and \nCe2PdIn 8, respectively) is only 3%. The separation process is described in Supplementary \ninformation. \n  4\n \n \nFig. 1 . Element mapping (left) of the cut and polished CeIn 3-Ce 2PdIn 8 sandwich. CeIn 3 \n(central, red) is covered by layer of Ce 2PdIn 8, a small region of new phase with nominal \ncomposition Ce 1.5Pd1.5In7. The line scan (right) along the bl ue arrow show a sharpness of \nthe boundary between phases \n \n2.2 Characterization and measurement techniques \nThe samples were checked by X-ray pow der diffraction (XRPD) on a Bruker D8 \nAdvance diffractometer, and by a scanning el ectron microscope (SEM) Tescan Mira I \nLMH equipped by an energy dispersive X-ray detector (EDX) Bruker AXS.  \n Magnetic properties, resistivity and heat capacity were measured on MPMS and \nPPMS (using 3He option) devices (Quantum Design). The AC magnetic susceptibility \n(χAC) at low temperatures (0.35<T<2.5K) was meas ured using custom made extension to \nthe PPMS apparatus allowing to measure χAC using the ACMS option with the 3He insert \n(including the ACMS preamplifier). The se lf-made coil set-up with two counter-wounded \nsecondary coils (Cu wire) and primary coil (N bTi wire) was used for this measurement. \nDetails about the set-up, necessary modifica tion and test measurements are described \nelsewhere [23].  \nThe low temperature part of resistivity wa s measured in a magnetic field of 0.03 T, \nbecause in zero magnetic field a SC transiti on of indium was detected (a kink around 3.2 \nK). \nAll measured samples were plate like single-crystals with a thickness ~100 μm and a \nmass of ~1.5 mg; the c-axis was perpendicular to the surface. The measurements were \ndone on several crystals of the composition Ce 2PdIn 8 (confirmed by EDX). For this work, \nthree representative crystals, marked as the sample “A”, “B” and “C”, respectively, were \nchosen.   \n3. Results \nBy the XRPD the Ho\n2CoGa 8-type of structure was confirmed, the lattice parameters \nhave been determined as: a = 0.4695 nm and c = 1.220 nm. The parameter a is almost \nequal to the lattice parameter of the cubic CeIn 3 (a = 0.4689 nm) which supports the \nstability of this system. On the other hand, the lattice parameter a = 0.4691 nm of \nCe2IrIn 8 matches the one of CeIn 3 even better and no evidence of forming such \nsandwiches has been observed.   5Magnetic measurements on Ce 2PdIn 8 single crystals show paramagnetic behaviour with \nsignificant magnetocrystalline anisotropy. The te mperature dependence of the reciprocal \nDC susceptibility (1/ χ), presented in Fig. 2, is lin ear above ~70 K, i.e. the χ (= M/H ) vs. \nT dependence of the susceptibility fo llows the Curie-Weiss law with μeff = 2.6 μB/Ce3+\n, \nΘp = -90 K, and μeff  = 2.6  μB/Ce3+, Θp = -50 K for the magnetic field B||a and B||c, \nrespectively. At lower temperatures the crys tal field effects become important and the \nmagnetic susceptibility departs from the Curie-Weiss law; in the c-direction, the magnetic \nsusceptibility departs from the C-W behaviour at about 70 K and become constant below \n23 K, while for the a-direction it departs form the C-W behaviour at 22 K and became \nalmost constant below 15 K, see Fig. 2 (inset ). Below 8 K, there is an increase of the \nsusceptibility in both directions. Such beha viour is usual for many Ce-compounds and is \nfrequently ascribed to a paramagnetic contribu tion of other rare earth elements presented \nin the Ce metal in a very small amount (< ~100 ppm). Of course, it can also signify a \nphase transition in the temperature region be low 2 K, but no anomaly has been observed \nin resistivity and heat capacity data. \n \n \nFig. 2. Temperature dependence of reci procal magnetic susceptibility of Ce 2PdIn 8  \nmeasured along both principal crystallographi c directions, black lines are the C-W law \nfits. The inset shows a low temperature part of the magnetic susceptibility.    6\n \nFig. 3. Temperature dependence of  electrical resistivity of Ce 2PdIn 8. The low temperature \npart (inset) was measured in magnetic fiel d of 0.03 T to suppress the SC transition of \nindium.     In Fig. 3 the temperature dependence of the resistivity measured with an AC current \napplied in the basal plane is presented. A maximum is evident at the temperature T\nmax ~ 40 K, this is the typical behaviour of other members of this family of compounds \n(more pronounced in 218’s [9, 10, 24]). The ma ximum is caused by the magnetic part of \nthe resistivity and corresponds to onset of low-temperature coherent behaviour of a \nKondo lattice, and the characteristic temperature is generally marked as T* [25]. A rough \nestimation of T* can be also obtained from the departure of the susceptibility data from \nthe Curie-Weiss behaviour [26] yielding the value T*χ = 40 K (polycrystalline average). \nAlthough both values are in agreement, we w ould be careful to draw conclusion, since \nthey were found to be sample dependent. The presented values were determined from sample “A” whereas e.g. sample “B” shows to the values of T\n* ~ 30 K.  \nBelow ~ 10 K a linear temperature dependence of the resistivity was observed (Fig. 3 \ninset) until the resistivity drops to zero va lue manifesting the SC transition (Fig. 5). \nLinear dependence of resistivity confirms  the non-Fermi liquid behaviour reported in \n[14], which may indicate that the compound appears on the verge of magnetism and the \nsuperconductivity is magnetically driven. Below critical temperature Tc < 0.69 K a SC \ntransition was observed. The critical temperat ure was determined by resistivity, heat \ncapacity and AC magnetic susceptibility measur ements see Figs. 4-6. As presented in \nFig. 6 and Fig. 7, the heat capacity and AC su sceptibly measurements detected more SC \ntransitions. The highest value of TC = 0.69 K (determined from the temperature derivative \nof the AC susceptibility) was observed in sa mple “C”; however heat capacity and AC \nmagnetic susceptibility data  evidenced anothe r transition at 0.46 K in this sample. The  7latter transition was detected in all measur ed samples with various intensity. A weak \npronounced heat capacity anomaly at ~ 0.46 K wa s responsible for the broad resistivity \ndecrease to zero in sample “A”; however the main transition in sample “A” seems to be \nbelow 0.36 K as hinted in Fig. 6 (inset). The last case is sample “B”, for which three \nanomalies were observed, the resistivity showed obviously a drop to zero value around the highest critical temperature. \nThe discrepancies in the low temperature behaviour of Ce\n2PdIn 8 were also observed \nby Kaczorowski et al . [14,15], where for the polycrystalline sample they presented \nnormal paramagnetism down to the lowest reached temperature T ~ 0.35 K, while the \nsingle crystal was superconducting below Tc = 0.68 K. Passing over the other presented \n(magnetic) transitions above Tc, which are of controversial or igin, they detected one clear \nSC phase transition both by resistivity and heat capacity measurements, the highest Tc \nobserved for Ce 2PdIn 8. \n \n \nFig. 4. Low temperature region of electrical re sistivity of three different samples of \nCe2PdIn 8 (A, B and C) showing on sensitivity on sample quality (measured at magnetic \nfield of 0.03 T).    8\n \nFig. 5. Low temperature region of heat capacity of the samples A, B and C. \n \nFig. 6. Heat capacity, resistivity and real part of AC magnetic susceptibility for sample B. \n \n \n4. Conclusions \nAlthough we have not found optimal c onditions for the growth of the Ce 2PdIn 8 single \ncrystal, we were able to separate single-phased samples for basic physical measurements.  9The study of Ce 2PdIn 8 showed that the compound is paramagnetic down to the SC \ntransition. The SC transition was found to  be sample dependent; the highest Tc value has \nbeen determined to be 0.69 K.  \nOur samples were carefully (faces and edge s of the plate-like crystals) checked by \nEDX analysis. Nevertheless, also this method has its limits and therefore we cannot 100 % exclude a presence of small amount of impurity phase or small variation of \ncomposition in our samples below the sensitivity of the EDX detector. With respect to the magnitude of the effects we exclude the in fluence of (completely different) impurity \nphase and ascribe the differences in the low- temperature behaviour to structural defects \nsuch as stacking faults typical for layered systems or small variation of composition or \nboth. This is motivated by results on Ce\n2RhIn 8 [19] where high resolution neutron \ndiffraction revealed Ce 2RhIn 8 to be “a complex mixture of two layered mosaic, polytypic \nphases affected by non-periodic partially correla ted planar defects”. Also high-resolution \nelectron microscopy may provide important data for resolving these questions. Although the EDX analysis is not able to detect a po ssible small variation in the composition, it has \nproven to be a crucial tool for exploring this attractive system, especially when \nidentifying and separating CeIn\n3 and other spurious phases. \nConsidering that samples with the highest Tc ~ 0.69 K have the sharpest SC transition \n(resistivity measurement) and the same value of Tc has been confirmed by AC \nsusceptibility and heat capacity measurements (and also by [15]) one might guess that it is a property of the “best” Ce\n2PdIn 8 crystal, while structural  defects or deviation of \nstoichiometry shift the Tc to lower values. Further studies of the sample behaviour \ndepending on sample fabrication such as e ffects of cooling rate or annealing are \nnecessarily to be done. \nFrom our point of view, CeIn 3 and Ce 2PdIn 8 form a beautiful sandwich-like system; \nhowever, for physical property studies isolat ed well defined single crystals would be \npreferred. On the other hand, the evident te ndency for forming such layered systems \nmight be interesting for epitaxial grow studies as reported e.g. for CeCoIn 5 [27]. \nWe have also synthesized new compounds, CePd 3In6 and traces of Ce 1.5Pd1.5In7 and \nCePd 2In7. Detailed characterization of these materials will be an object of our further \nstudies; the results will be published elsewhere.  \n \nAcknowledgments  \n \nThis work is a part of the research plan MSM 0021620834 that is financed by the Ministry of Education of the Czech Republic  and has been also partly supported by the \nCzech Science Foundation grants no. 202/09/1027, 202/09/H041 and by the Charles University Grant Agency grant no. 109907. We would like to thank to our colleague Ji ří \nPospíšil for his help with building our solution-growth laboratory.   Reference List  [1]  Moshopoulou EG, Fisk Z, Sarrao JL, Thompson JD. J Solid State Chem \n2001;158:25-33. \n [2]  Vandiepe AM, Craig RS, Wallace WE. J Phys Chem Solids 1971;32:1867-1872.  [3]  Walker IR, Grosche FM, Freye DM, Lonzarich GG. Physica C 1997;282:303-306.  10 [4]  Chen GF, Ohara S, Hedo M, Uwatoko Y, Saito K, Sorai M, Sakamoto I. J Phys Soc \nJpn 2002;71:2836-2838. \n [5]  Movshovich R, Jaime M, Thompson JD, Petrovic C, Fisk Z, Pagliuso PG, Sarrao \nJL. Phys Rev Lett 2001;86:5152-5155. \n [6]  Kim JS, Moreno NO, Sarrao JL, Thompson JD, Stewart GR. Phys Rev B \n2004;69:024402. \n [7]  Hedo M, Kurita N, Uwatoko Y, Chen  G, Ohara S, Sakamoto I. J  Magn Magn \nMater 2004;272:146-147. \n [8]  Nicklas M, Sidorov VA, Borges HA, Pa gliuso PG, Petrovic C, Fisk Z, Sarrao JL, \nThompson JD. Phys Rev B 2003;67:020506(R)-020510(R). \n [9]  Hegger H, Petrovic C, Moshopoulou EG, Hundley MF, Sarrao JL, Fisk Z, \nThompson JD. Phys Rev Lett 2000;84:4986-4990. \n[10]  Ferreira LM, Park T, Sidorov V, Ni cklas M, Bittar EM, Lora-Serrano R, Hering \nEN, Ramos SM, Fontes MB, Baggio-Sait ovich E, Lee HO, Sarrao JL, Thompson \nJD, Pagliuso PG. Phys Rev Lett 2008;101:017005-017009. \n[11]  Park T, Ronning F, Yuan HQ, Salamon MB, Movshovich R, Sarrao JL, Thompson \nJD. Nature 2006;440:65-68. \n[12]  Kenzelmann M, Strassle T, Niedermayer C, Sigrist M, Padmanabhan B, Zolliker M, \nBianchi AD, Movshovich R, Bauer ED, Sarrao JL, Thompson JD. Science 2008;321:1652-1654. \n[13]  Shtepa DV, Nesterenko SN, Tursina AI, Murashova EV, Seropegin YD. Moscow \nUniv Chem Bull 2008;63:162-166. \n[14]  Kaczorowski D, Pikul A, Belan B,  Sojka L, Kalychak Y. Physica B 2009;404:2975-\n2977. \n[15]  Kaczorowski D, Pikul AP, Gnida D, Tran VH. Phys Rev Lett 2009;103:017005. [16]  Uhlí řová K, Prokleška J, Sechovský V, To be published. \n[17]  Kurenbaeva Z, Murashova EV, Seropegi n YD, Nodl H, Tursina AI Intermetallics \n2008;16:979-981. \n[18]  Bauer ED, Kurita N, Sidorov VA,  Ronning F, Movshovich R, Thompson JD, \npresented at ICM 2009. \n[19]  Moshopoulou EG, Ibberson RM, Sarrao JL , Thompson JD, Fisk Z Acta Crystallogr \nSect B 2006;62:173-189. \n[20]  Canfield PC, Fisk Z. Philos Mag B 1992;65:1117-1123. [21]  Kanatzidis MG, Pöttgen R, Jeitschko W. Angew Chem Int Ed 2005;44:6996-7023. [22]  Kalychak YM, Zaremba VI, Pöttgen R, Lukachuk M, Hoffmann RD. In: \nGschneider KA Jr, Bunzli J-CG, Pech arsky VK (Eds), Handbook on the Physics \nand Chemistry of Rare Earths,  Elsevier B.V., 2005;34:1-133 \n[23]  Prokleška J, Šebek J, To be published. [24]  Chen GF, Ohara S, Hedo M, Uwatoko Y, Sakamoto I J. Phys. Condens. Matter \n2003;15:S2175-S2178. \n[25]  Nakatsuji S, Yeo S, Balicas L, Fisk Z, Schlottmann P, Pagliuso PG, Moreno NO, \nSarrao JL, Thompson JD. Phys Rev Lett 2002;89:106402. \n[26]  Yang YF, Fisk Z, Lee HO, Thompson JD, Pines D. Nature 2008;454:611-613. [27]  Zaitsev AG, Beck A, Schneider R, Fr omknecht R, Fuchs D, Geerk J, Löhneysen H, \nPhysica C 2009;469:52-54. \n  11 \n  "    },    {        "title": "1001.4618v1.Competing_magnetic_anisotropies_in_atomic_scale_junctions.pdf",        "content": "arXiv:1001.4618v1  [cond-mat.mes-hall]  26 Jan 2010Competing magnetic anisotropies in atomic-scale junction s\nAlexander Thiess1, Yuriy Mokrousov1,∗and Stefan Heinze2\n1Institut f¨ ur Festk¨ orperforschung, Institute for Advanc ed Simulation,\nForschungszentrum J¨ ulich, D-52425 J¨ ulich, Germany and\n2Institute of Theoretical Physics and Astrophysics,\nChristian-Albrechts-University of Kiel, D-24098 Kiel, Ge rmany\n(Dated: November 21, 2021)\nUsingfirst-principles calculations, we studythemagnetis m of5dtransition-metal atomic junctions\nincluding structural relaxations and spin-orbit coupling . Upon stretching monatomic chains of W,\nIr, and Pt suspended between two leads, we find the developmen t of strong magnetism and large\nvalues of the magnetocrystalline anisotropy energy (MAE) o f up to 30 meV per chain atom. We\npredict that switches of the easy magnetization axis of the n anocontacts upon elongation should\nbe observable by ballistic anisotropic magnetoresistance measurements. Due to the different local\nsymmetry, the contributions to the MAE of the central chain a toms and chain atoms in the vicinity\nof the leads can have opposite signs which reduces the total M AE. We demonstrate that this effect\noccurs independent of the chain length or geometry of the ele ctrodes.\nPACS numbers: 73.63.Nm, 72.25.-b, 75.30.Gw, 75.47.Jn, 75. 75.+a\nI. INTRODUCTION\nFascinating insights into the formation of atomic\nchains consisting only of a few atoms suspended be-\ntween two electrodes have been obtained by transmission\nelectron microscopy and mechanically controllable break\njunction techniques.1–3These experiments triggered the\nimagination of scientists to use such atomic-scale junc-\ntions as future electronic devices by exploiting their\nunique properties, e.g. ballistic electronic transport.1,4,5\nDue to their reduced dimensions, atomic chains have\nalso been predicted to develop magnetic moments even\nfor elements nonmagnetic in bulk such as Ir, Pt, or\nPd.6,7Recently, an indirect proof has been given that\ntransition-metal chains in break junctions are in general\nmagnetic.8Mastering the intriguing magnetic properties\nof suspended chains would enable spintronic applications\nbased on the unique possibility to probe, control and\nswitch the magnetic states by spin-polarizedcurrents.9,10\nThe orientation of the magnetic moments is stabi-\nlized against thermal fluctuations by the magnetocrys-\ntalline anisotropy energy (MAE) arising from spin-orbit\ncoupling. Theoretical studies of this key quantity per-\nformedforidealizedsystemssuchasfree-standinginfinite\nmonowires (MWs) and small clusters suggest giant val-\nues for 4d- and 5d-transition-metals and switching of the\neasy axis upon stretching the wires.11–13Even the size\nof the magnetic moment itself can crucially depend on\nthe magnetization direction, an effect coined as colossal\nmagnetic anisotropy.4The ballistic conductance in such\nwires varies with the orientation of the magnetization di-\nrection with respect to the chain axis givingrise to ballis-\ntic anisotropic magnetoresistance (BAMR).1,5While the\npredictedeffectsprobablyoccurinrealatomic-scalejunc-\ntions, the theoretical studies have either focused on ide-\nalized systems or neglected varying interatomic distances\nin the chains.14–16However, for small suspended chains\nvariations in the interatomic distance due to stretchingthe electrodes are unavoidable and can provide a unique\nway to control the properties of the system, not available\nin higher dimensions.4,6,11,17\nHere, we go beyond such theoretical studies by ex-\nplicitly taking the contacts into account and by inves-\ntigating the development of magnetism in atomic-scale\njunctions upon increasing the distance between the elec-\ntrodes. We perform first-principles calculations for W, Ir\nand Pt junctions including spin-orbit coupling and struc-\ntural relaxations and focus on atomic chains of three\natoms suspended between two bulk-like bcc-(001) elec-\ntrodes as shown in Fig. 1. These transition-metals are\ncommon tip materials in scanning tunneling microscopy,\nandIrandPthavebeenshowntoformlongatomicchains\nin break junction experiments.2In particular we concen-\ntrate on the influence of varying tip-to-tip separation on\nthemagneticpropertiesofthejunctions. Weobservethat\nrelaxations of the chain atoms upon stretching are in ac-\ncordance to observed expermentally trends of chain for-\nmation for these elements.17We also find that large spin\nmoments develop in the suspended trimers upon stretch-\ning and on the central atom they reach similar values as\nin infinite monowires.6–8Interestingly, the W trimer be-\nhaves as a single spin impurity as only the central atom\ndevelops a sizable magnetic moment.\nBy including spin-orbit coupling (SOC), we obtain the\nMAE of the suspended chains and observe large values\nof 10−30 meV per atom. Surprisingly, we find that the\ncontribution to the MAE from central chain atoms and\natoms in contact with the electrodes can have opposite\nsigns and are of similar giant magnitude of a few tens\nmeV per atom. The magnitude of this effect, unknown\nin bulk or on surfaces may lead to a strong tendency to-\nwards non-collinear magnetic order in atomic-scale junc-\ntions even when the exchange interactions are small or\nprofoundly collinear.18We prove the generality of this\nphenomenonbyshowingthatitoccursforchainsofdiffer-\nent length with different geometry of the contacts. Fur-2\nFIG. 1: (color online) Upper panel: three-atoms chain sus-\npended between two bcc(001)-contacts. Ldenotes the length\nof the unit cell consisting of 17 atoms, while daanddcgive\nthe distance between the first plane of the contact to the\napex atom and from the apex atom to the central atom of the\ntrimer, respectively. Lower panel: relaxation of the dcandda\nbonds in suspended trimers of W, Ir, and Pt upon increasing\nL. For comparison, the bond length for ideal stretching with\nfixed apex atom positions is given (dashed line).\nther, the MAE may exhibit changes of sign upon stretch-\ning, i.e. the easy magnetization direction switches from\nalong the chain to perpendicular to it. We demonstrate\nthatswitchingmaybeobservableexperimentallybymea-\nsuring the ballistic conductance of the junctions.\nII. METHOD AND STRUCTURE\nWe have performed first-principles calculations within\nthe generalized gradient approximation (GGA) to the\ndensity functional theory (DFT). We employed the full-\npotential linearized augmented plane-wave method for\none-dimensional (1D) systems (1D-FLAPW),19as imple-\nmented in the FLEURcode.20Basis functions were ex-\npanded up to kmax= 4.0 a.u.−1and we have used 8\nk-points in one half of the 1D Brillouin zone. Consid-\nering more k-points does not change the values of the\nspin moments and magnetic anisotropy energies signifi-\ncantly. The contact structure has first been relaxed sep-\narately in all three dimensions and then kept fixed for\nall contact separations L(Fig. 1). The apex atoms of\nthe trimer have been relaxed along the chain direction\n(z) upon varying L, while the central atom is fixed by\nsymmetry (Fig. 1). Calculations have been performed\nfor junctions stretched from the equilibrium up to ∆ L=\n3 a.u. For Pt and Ir trimers we considered the ferromag-netic ground state, while W trimer always converges to\nthe antiferromagnetic solution.\nSpin-orbit coupling SOC was included into the calcu-\nlations in a non-perturbative way.21We define the mag-\nnetic anisotropy energy (MAE) as the total energy differ-\nence between configurations with a magnetization along\nthe chain axis ( z-direction), and perpendicular to it ( r-\ndirection). The MAE was calculated employing the force\ntheorem, and for several electrode stretchings, ∆ L, we\nchecked the force theorem results for the trimer’s MAE\nagainst self-consistent calculations, finding good qualita-\ntive agreement. Taking into account the magnitude of\nthe obtained MAE we can safely neglect the magnetic\ndipole-dipole contribution.\nIn order to check the applicability of our 1D geometry\nfor the electrodes, which allows us to efficiently perform\nthe demanding calculations with structural relaxations\nand including SOC, we have also considered a geometry\nwith electrodes represented by surfaces, c.f. Fig. 2(a).\nDue to the immense computational cost, we have re-\nstricted our test to the most critical case of a Pt trimer\nand chose an elongation of ∆ L= 2.4 a.u. For these\ncalculations we used the film version of the FLEURcode.\nWe performed a single super-cell calculation in which the\ntrimer was suspended between two electrodes consisting\nof a four Pt atom square base deposited onto three lay-\ners of fcc Pt(001). This configuration will be referred to\nas ”2D” in the following, Fig. 2. The ideal bulk values\nof Pt with an in-plane lattice constant of 5.23 a.u. were\ntaken for the four contact atoms and the bulk-like leads.\nThe positions of the trimer’s atoms were set to their re-\nlaxed values at ∆ L= 2.4 a.u. obtained within the wire\ngeometry of Fig. 1. The in-plane separation between the\nneighboring trimers was set to 15.7 a.u. We used a basis\nfunctions cut-off of kmax= 3.7 a.u.−1and 15k-points in\nthe irreducible part of the 2D Brillouin zone, while 144\nk-points in the whole Brillouin zone were used in order\nto determine the MAE with the force theorem. Along\nthe trimer’s axis only the Γ-point was used.\nIII. RELAXATIONS AND MAGNETISM\nUnder the condition of reduced coordination of transi-\ntion metal atoms in a monoatomic chain suspended be-\ntween thicker leads, a situation realized e.g. in a break\njunction experiment, the distance dbetween the chain\natoms becomes of utter importance. Upon pulling apart\nthe leads during the chain elongation process3,17dis con-\nstantlychangingwhichhasdramaticconsequencesforthe\nmagneticpropertiesofsuspendedchains, inparticularfor\nheavy 4dand 5dtransition-metals.4,6,7,11By increasing\nthe separation ∆ Lbetween the leads and accounting at\nthe same time for the changes in the chain’s interatomic\ndistances −which was neglected in most of the previous\nab initio studies of magnetism in suspended chains −we\naim at mimicking a real break junction experiment.\nThe relaxations of the trimers along the chain’s sym-3\nFIG. 2: (color online) 2D configuration, and B1 and B2 cases\nof the Pt trimer considered for the separation of Pt electrod es\nof ∆L= 2.4 and 1.8 a.u., respectively. In 2D configuration\nthe trimer is suspended between two electrodes consisting o f\na four Pt atom square base deposited onto three layers of fcc\nPt(001). Green color in B1 and B2 marks the Pt atoms of the\ncontacts in which the spin moment was artificially quenched.\nIn accordance to Fig. 1, red color marks the central atom and\nthe blue color marks the apex atoms of suspended trimer. For\nmore details see text.\nmetryz-axis are shown in Fig. 1, where the bond lengths\ndcanddaare presented as a function of contact separa-\ntionL. ForWwithanalmosthalf-filled5 d-shellandlarge\nbulk cohesiveenergyalmost the entireincreasein contact\nseparation enters the dcbond length, while the dabond\nbetween the apex atom and the contact is nearly unaf-\nfected. In contrast, for Ir and Pt, some of the stretching\ntransfers into an increased bond length at the contact.\nThe apex atom of the Ir trimer has a stronger bond to\nthe contact than to the centralatom, which is reflected in\nthe linear regime of dc(L) for ∆L >1.5 a.u. For Pt, the\ndcbond is significantlystrongerresulting in a remarkable\nstretching of the dabond even at large electrode separa-\ntions, which in a realistic break junction will eventually\nresult in chain elongation.17Overall, our predicted trend\nis in accordance with experimental observations of in-\ncreased probability for chain formation when going from\nW to Pt.2,22\nThe magnetic properties of suspended 5 dTM atoms\nare very sensitive to the local environment. In most pre-\nvious studies it was common to model the contacts by\nsemi-infinite surface electrodes.14–16This restriction on\nthe geometry of the contacts is quite strong in particular\nfor 5dTM break junctions, as the surface geometry leads\nto suppressed spin moments of the atoms in suspended\nwire and contact atoms closest to it.14However, it is\nFIG. 3: (color online) Local spin moments of the central ( µc\nS)\nand apex ( µa\nS) atoms (denoted as ”c-atom” and ”a-atom”,\nrespectively) of thesuspended (a) W, (b)Ir and (c)Pt trimer s\nas a function of the stretching of the contacts (without SOC) .\nFor comparison the values for an infinite free monowire (MW)\nare given by dashed lines. The interatomic distance in the\ninfinite MW corresponds to the distance dcfor a given ∆ L.\nFor Pt triangles down and triangles up stand for the values of\nthe apex (filled) and central (open) spin moments in B2 and\n2D cases, respectively. For details see text.\nwell-established that the shape of the contacts in a break\njunction is rather a thinning wire-like geometry, with the\nreduced coordination number of the contact atoms adja-\ncent to the chain which greatly enhances their tendency\ntowards magnetism.23,24In this respect our contact ge-\nometry is probably closer to the real situation, although\nit might overestimate the tendency of the contacts to\nmagnetism. In order to investigate this point quantita-\ntively, we haveperformed calculationsin the setup shown\nin Fig. 2, which are discussed further below.\nThe calculated spin moments µSinside the atomic\nspheres of the suspended trimers upon increasing the dis-\ntance between the contacts ∆ Lare shown in Fig. 3. For\nW, the central atom reveals a sizable magnetic moment\nof 1.7µBalready at ∆ L= 0 which further increases upon\nstretching, while the moments of all other atoms are neg-4\nligible. In this respect the W trimer presents a unique\nsystem of a single spin impurity between nonmagnetic\nleads. At small stretching, the spin moments of the Ir\ntrimer are rather small with significantly larger moments\nof the central atom. Upon further stretching the apex\nand central moments rise rapidly, reachinglargevalues of\n1.4µBand 2µB, respectively. Both, the µc\nSspin moments\nof the central W and Ir chain atom follow the behavior\nin an infinite monowire very closely as seen in Fig. 3(a)\nand (b).\nThe Pt trimer showsa trend similar to Ir upon stretch-\ning, onlywith smallermoments −thisis in contrastto an\ninfinite Pt MW, wherethe spinmoment iszeroforalarge\nintervalofinteratomicdistanceswithoutSOC.Itisresur-\nrected upon including SOC, but only when the magneti-\nzation points along the z-axis, an effect coined as colossal\nmagnetic anisotropy.4In contrast, for the suspended Pt\ntrimer, the moments of the trimer atoms are non-zero al-\nready without SOC for a wide range of dc. Upon includ-\ning SOC the spin moments do not change dramatically\nand the moments are non-zero for both magnetization\ndirections in the whole range of ∆ L, which manifests the\nsubtle magnetism of Pt chains and its sensitivity to local\nenvironment.\nIn the critical case of Pt, we analyze in more detail the\ninfluence of the contact geometry on the spin moments\nof the trimer. At the stretching of ∆ L= 1.8 a.u. we per-\nformedacalculationin whichweartificiallyquenchedthe\nspin moments in the contacts by applying a small mag-\nnetic field inside the muffin-tin spheres of the Pt atoms,\nsituation referred to as ”B2” in the following (see Fig. 2).\nWe observe that µa\nSdrops significantly from 0.55 µBto\n0.18µB, whileµc\nSdropsfrom0.89 µBto0.54µB(Fig.3(c))\n−a value, very close to that reported in Ref. [14], where\ncontacts were modelled by infinite bulk electrodes. How-\never, quenching the spin moment in the deeper parts of\nthe contacts only (”B1”-case, Fig. 2) −the situation\nwhich is probably closer to that in real experiments −\nalmost does not affect the values of µc\nSandµa\nS.\nFinally, we comparethe resultsobtained within our1D\ngeometry of the electrodes with the situation of surface-\nlike electrodes, as shown in Fig. 2(a). For a separation\nof ∆L= 2.4 a.u.µc\nSis reduced in the 2D geometry by\nonly 25% from 1.03 µBin the wire geometry to 0.79 µB,\nwhile the moment of the apex atom is affected stronger,\ndropping from 0.64 µBto 0.26µB(Fig. 3(c)). In a more\nrealistic situation the coordination of the contact atoms\nnext to the apex atoms is more reduced and the moments\nofthe trimer’satomswill increaseapproachingthe values\nobtained in our wire geometry. However, we have to con-\nclude that magnetism in suspended short Pt chains with\ntheir small moments is extremely sensitive to slightest\nchanges in the contact geometry. Therefore, observing\nthis magnetism experimentally will be a hard task, even\nconsidering the cluster’s large magnetic anisotropy ener-\ngies.\nFIG. 4: (color online) Magnetic anisotropy energies of sus-\npended trimers of (a) W, (b) Ir and (c) Pt as a function of the\nstretching of the contacts ∆ L. The total trimer’s anisotropy\n∆Etri(black half-filled circles) is decomposed into the con-\ntribution from the central ∆ Ec(red open circles) and apex\natoms ∆ Ea(blue filled circles) (for definitions of the quanti-\nties see text). Black dashed line stands for the MAE of the\ninfinite MWs (per atom). For Pt squares stand for the ∆ Ea\n(filled) and ∆ Ec(open) tetramer values, while triangles up\nand triangles down stand for the corresponding values in B2\nand 2D cases. For details see text.\nIV. MAGNETIC ANISOTROPY ENERGIES\nRecent theoretical studies,17,25which are in accor-\ndance with experiments,3,26clearly state that the in-\nteratomic distance dcin suspended chains of 5 dele-\nments consisting of several atoms can reach large val-\nues of 5.0 −5.5 a.u. upon stretching, which corresponds\nto ∆L≈2−3 a.u., c.f. Fig. 1. Therefore, in a real Pt, Ir,\nor W break junction experiment the central atom will re-\nvealstrongfingerprintsofspin-polarizationatsufficiently\nsmall temperatures. In order to study the thermal sta-\nbility of magnetism in the suspended trimers, next we\nconcentrate on their magnetic anisotropy energies.\nIn a transport break junction experiment, the data is5\nobtained by averaging over thousands of measurements,\nwhich differ from each other by the thinning history of\nthe wire and the contact geometry. Therefore, in order\nto analyze this data with respect to the magnetism of the\nsuspended chain, it is necessaryto disentangle the contri-\nbutions of intrinsic trimer’s magnetic anisotropy energy\n∆Etrifromthatoriginatingfromthecontacts. Inourcal-\nculations,wedothisbyswitchingSOCoffinthecontact’s\natomic spheres. Moreover, we individually switch SOC\non and off in the apex and central atoms, to determine\ntheir separatecontributions, ∆ Eaand ∆Ec, respectively,\ntothe totalMAE ofthe trimer, ∆ Etri. ForWwith essen-\ntially one magnetic atom in the trimer we define ∆ Etrias\n∆Ec+2∆Ea, whileforIrandPtwedefine thetotalMAE\nof the trimer per atom as ∆ Etri= (∆Ec+2∆Ea)/3.27\nThe absolute value of trimer’s MAE, shown in Fig. 4\nwith a black solid line for W, Ir and Pt, reach large ab-\nsolute values of 10 to 30 meV per atom at most electrode\nseparations. These values, which are one to two orders of\nmagnitude larger than those in most of transition-metal\nnanostructures, give us confidence that the magnetism of\nsuspended trimers can be tackled experimentally. For all\nthree elements the calculated values of anisotropy ener-\ngies ∆Etriare of the order of magnitude of those pre-\ndicted for infinite monowires (dashed line in Fig. 4 and\nRefs. [4,11]). Interestingly, the trimer’s MAE displays a\nnon-trivial dependence on ∆ Lquite different from that\nin ideal chains for Ir and W, while for Pt both anisotropy\nenergies are close in their trend.\nFor W, Fig. 4(a), the only contribution to ∆ Etristems\nfrom the strongly spin-polarized central atom. Start-\ning at about 35 meV and a magnetization along the\nchain, there is a switch to a perpendicular direction upon\nstretching by ∆ L= 0.6 a.u. Upon further increasing the\ncontact separation, the perpendicular magnetization is\nstabilized by a sizable MAE of 10 −20 meV. A switch of\nthe magnetization direction can be alsoobserved for Ir at\nsignificant stretching of around ∆ L= 3.0 a.u., Fig. 4(b).\nIn general, for Ir, the behavior of ∆ Etriupon stretching\nis quite smooth, and the z-direction of the magnetiza-\ntion is favored by considerable 15 −30 meV over a wide\ninterval of contact separations.\nDetecting traces of magnetism in Pt breakjunctions at\nsmallstretchingwill present asignificantchallengeascan\nbe seenfromFig.4(c). In theregimeof∆ L <1.3a.u.the\nvalue of ∆ Etriis on the order of a few meVs, posing the\nquestion of whether the sensitive magnetization will sur-\nvive temperature fluctuations. Moreover, in this regime,\nas pointed out above, the spin moments of the apex and\ncentral atoms depend strongly on contact geometry and\ndetails of the thinning process. This will influence the\ncontribution of these atoms to ∆ Etri, causing frequent\nmeasurement-to-measurement oscillations in its sign and\nmagnitude. Beyond a distance of ∆ L= 1.5 a.u. the well-\ndeveloped magnetization of the trimer is pointing rigidly\nalong the chain axis with a MAE of about 15 meV −a\nvalue somewhat smaller than that of an infinite MW of\ncorresponding interatomic distance, c.f. Fig. 4(c).Asageneraltrend, weobserveinFig.4competingcon-\ntributionstothe MAEfromcentralandapexatomsforIr\nand Pt atomic junctions. At most separations a positive\nvalue from the apex atom, ∆ Ea>0, favors a perpendic-\nular magnetization direction, while negative contribution\nfrom the central atom, ∆ Ec<0, forces the magnetiza-\ntion along the trimer’s axis. Due to this competition the\ntrimer’s MAE behaves qualitatively differently from that\nin the infinite MW as a function of the interatomic dis-\ntance for Ir. It significantly quenches the central atom’s\nMAE, so that the resulting total anisotropy is on average\nsmaller than anticipated from the infinite MW both for\nIr and Pt.\nThe origin of this competition is the different local\nsymmetry of apex and central atoms and it exists also\nin longer chains or chains made of other elements. To\ndemonstrate this point we additionally calculated the\nMAE of a suspended four-atom chain of Pt atoms with\nthe contacts as in Fig. 1. The ∆ Eaand ∆Ecanisotropy\nenergies for the two apex and two central atoms, respec-\ntively, are displayed by squares in Fig. 4(c) (per atom)\nand reveal the same effect: ∆ Eaand ∆Ecare close in\ntheir values but opposite in sign. Due to its large mag-\nnitude, this effect may even compete with exchange in-\nteractions in suspended small chains and lead to a non-\ncollinear magnetic order. With increasing chain length\nthe anisotropy energies of the atoms in the center of the\nchainwilleventuallyapproachthat ofanatominthe infi-\nnitemonowire(this canbealreadyseeninFig.4(c)forPt\nwhen going from a trimer to a tetramer), and the whole\ncluster will behave as a superparamagnet. For smaller\nsuspended clusters, however, which are much more prob-\nable to occur in an experiment, the effect of competition\nbetween central and apex atoms will be dramatic.\nWe demonstrate that the effect of the competition be-\ntween the apex and central atoms for the MAE value is\nalso stable with respect to the geometry of the contacts.\nFor this purpose we calculate the MAE of the Pt trimer\nin 2D-configuration, as well as in B1 and B2 cases, and\npresent the calculated values in Fig. 4(c). As far as the\n2D-case is concerned, we observe that despite significant\nchanges in the spin moments, c.f. Fig. 3(c), the values of\n∆Eaand ∆Ecare very close to those calculated within\nthe wire geometry, moreover, they are also opposite in\nsign. Their competition results in rather close values for\nthe total ∆ Etriof 7 and 18 meV for the 2D and wire-\nlike geometry of the contacts, respectively, both favoring\nthez-direction of the trimer’s magnetization. Notably,\nboth values are very close to that of 12 meV reported by\nSmogunov et al.in Ref. [14] for the suspended between\nsemi-infinite electrodes Pt trimer, although one has to\nkeep in mind the difference between the geometries of all\nthree cases.\nIn the B1-situation, in which we do not quench the\nspin moments of the contact atoms of the electrode, the\nMAE is almost not affected compared to the wire setup\nof Fig. 1. On the other hand, in the B2-case, in which we\nquench the spin moments inside the entire electrode, the6\nFIG. 5: (color online) Spin-decomposed local density of sta tes\n(LDOS) for a Pt trimer including SOC for a separation be-\ntween the electrodes of ∆ L= 1.8 a.u. (a) and (b): ∆ 3\n(dxz,dyz) and ∆ 4(dx2−y2,dxy) 3d-LDOS of the central (”c-\natom”) and apex (”a-atom”) atoms for the magnetization\nalong the z-axis. (c) LDOS of the central atom for two differ-\nent magnetization directions ( zandr).\nspin moments of the trimer atoms are strongly reduced,\nc.f. Fig. 3(c), and so is the MAE: ∆ Ecand ∆Eadrop to\nalmost half their value. However, the latter two contri-\nbutions to the total MAE ∆ Etriare still of opposite sign,\nproving the generality of the phenomenon of competing\nanisotropies for this type of atomic junctions. The latter\neffect is responsible for the value of the total MAE ∆ Etri\nof 9 meV, smaller than the value of 20 meV obtained\nwithin the wire-geometry.\nTo understand the effect, we analyze the local density\nofstates(LDOS)andchoosethePttrimerasanexample.\nIn an infinite MW, the position of the localized d-states\nwith respect to the Fermi energy ( EF) is responsible for\nthe formation of the orbital moment and the direction\nof the magnetization.11,28,29According to the symmetrythesed-states can be subdivided into ∆ 3(dxz,dyz) and\n∆4(dx2−y2,dxy) groups, not taking into account ∆ 1sd-\nhybrid.11In a Pt (Ir) infinite MW, the position of the X-\nedge (Γ-edge) ofthe ∆ 4degenerate dxyanddx2−y2bands\natEFis responsible for the easy magnetization direc-\ntion along the chain.28,29In a trimer at larger stretching,\nthe 5d-LDOS of a central atom qualitatively follows that\nof an atom in an infinite monowire (not shown), which\nexplains the same sign of the MAE (Fig. 4). A small\nsplitting between the dx2−y2anddxystates due to the\npresence of the leads can be already seen for the cen-\ntral atom (Fig. 5(a)), however, for the apex atom the\neffect is dramatic and has crucial consequences for the\napex MAE. Interaction with the contacts locally breaks\ntheC1∞symmetry of an infinite chain which results in\nlarge splitting between the dxy- anddx2−y2-orbitals in\nthe ∆ 4-band and their shift towards the lower energies\n(Fig. 5(b)). On the other hand, the degeneracy between\nthedxzanddyz-states (∆ 3-band) is locked by symmetry.\nThis leads to the rearrangement of the states around EF\nand as a result the in-plane direction of the apex spin\nmoment becomes favorable (Fig. 4).\nIn general, it should be possible to deduce the pre-\ndicted switches in the magnetization direction of the sus-\npended chains fromtransportmeasurementsin these sys-\ntems. As an example, in Fig. 5(c) we present the LDOS\nof the central atom in a Pt trimer at ∆ L= 1.8 a.u. Upon\nchanging the magnetization direction significant changes\nin the LDOS around the Fermi energy EFcan be ob-\nserved (shaded area in Fig. 5(c)), which will inevitably\nresult in large changes of the experimentally measured\nconductance. As far as the 5 dTMs are concerned, gi-\nant values of the ballistic anisotropic magnetoresistance\nshould be achievable due to strong spin-orbit coupling\nin these metals, which is able to modify the electronic\nstructure significantly in response to the changes of the\nmagnetization direction in real space.\nV. CONCLUSIONS\nBy performing ab initio calculations of suspended\ntrimers of W, Ir and Pt including structural relaxations\nas a function of the separation between the leads, we\ndemonstratethatthechainatomsdevelopsignificantspin\nmoments upon stretching. We investigate the stabil-\nity of these spin moments via evaluating the magnetic\nanisotropy energy of the trimers. Our calculations re-\nveal large MAE of the whole trimers of the order of 10\nto 30 meV per atom. Interestingly, we observe that the\ntotal MAE presents a competition between large contri-\nbutions from the apex and central atoms. We argue that\nthis effect is general and can occur in suspended chains\nof different elements and different length, leading to non-\ntrivial real space textures of magnetic anisotropy energy\nwhich might even lead to complex magnetic ordering in\natomic-scale junctions.\nUnder the condition of large predicted values of MAE7\nhuge magnetic fields would be required to change the\nmagnetization of the chains, therefore, new ways to\nachieve this goal have to be established. One of the pos-\nsibilities lies in using the ability to control the magneti-\nzation direction in the junction by changing the distance\nbetween the leads. We show that such switches of the\nmagnetization can happen and will result in strong fea-tures in the measured conductance. Distinguishing these\nfeatures from the changes in conductance due to chain\nelongation or atomic rearrangements presents a consid-\nerable challenge.\nFinancial support of the the Stifterverband f¨ ur die\nDeutsche Wissenschaft is gratefully acknowledged. We\nalso thank Phivos Mavropoulos for discussions.\n∗Electronic address: y.mokrousov@fz-juelich.de\n1A. Sokolov, C. Zhang, E.Y. Tsymbal and J. Redepenning,\nNat. Nanotech. 2, 171 (2007)\n2R.H.M. Smit, C. Untied, A.I. Yanson and J.M. van Ruiten-\nbeek, Phys. Rev. Lett. 87, 266102 (2001)\n3T. Kizuka, Phys. Rev. B 77, 155401 (2008).\n4A. Smogunov, A. DalCorso andE. Tosatti, Nat.Nanotech.\n3, 22 (2008)\n5J. Velev, R.F. Sabirianov, S.S. Jaswal and E.Y. Tsymbal,\nPhys. Rev. Lett. 94, 127203 (2005).\n6A. Delin and E. Tosatti, Phys. Rev. B 68, 144434 (2003)\n7A. Delin, E. Tosatti and R. Weht, Phys. Rev. Lett. 92,\n057201 (2004)\n8A. Thiess, Y. Mokrousov, S. Heinze and S. Bl¨ ugel, Phys.\nRev. Lett. 103, 217201 (2009).\n9P. Lucignano, R. Mazzarello, A. Smogunov, M. Fabrizio\nand E. Tosatti, Nat. Materials 8, 563 (2009).\n10M.R. Calvo, J. Fern´ andez-Rossier, J.J. Palacios, D. Jacob ,\nD. Natelson and C. Untied, Nature 458, 1150 (2009).\n11Y. Mokrousov, G. Bihlmayer, S. Heinze and S. Bl¨ ugel,\nPhys. Rev. Lett. 96, 147201 (2006)\n12T.O. Strandberg, C.M. Canali andA.H.MacDonald, Phys.\nRev. B77, 174416 (2008).\n13L. Fern´ andes-Seivane and J. Ferrer, Phys. Rev. Lett. 99,\n183401 (2007).\n14A. Smogunov, A. Dal Corso and E. Tosatti, Phys. Rev. B\n78, 014423 (2008).\n15K.M. Smelova, D.I. Bazhanov, V.S. Stepanyuk, W. Hegert,\nA.M. Saletsky and P. Bruno, Phys. Rev. B 77, 033408\n(2008).16M. Czerner, B.Yu. Yavorskiy and I. Mertig, Phys. Rev. B\n77, 104411 (2008).\n17A. Thiess, Y. Mokrousov, S. Bl¨ ugel and S. Heinze, Nano\nLetters8, 2144 (2008).\n18S. Mankovsky, S. Bornemann, J. Min´ ar, S. Polesya,\nH. Ebert, J.B. Staunton and A.I. Lichtenstein,\narXiv:0902.3336v1 (2009).\n19Y. Mokrousov, G. Bihlmayer and S. Bl¨ ugel, Phys. Rev. B\n72, 045402 (2005)\n20http://www.flapw.de .\n21C. Li, A.J. Freeman, H.J.F. Jansen and C.L. Fu, Phys.\nRev. B42, 5433 (1990)\n22C. Untiedt, M.J. Caturla, M.R. Calvo, J.J. Palacios, R.C.\nSegers and J.M. van Ruitenbeek, Phys. Rev. Lett. 98,\n206801 (2007).\n23L.G.Rego, A.R.Rocha, V.RodriguesandD.Ugarte, Phys.\nRev. B67, 045412 (2003).\n24D.Cheng, W.Y.Kim, S.K.Min, T.NautiyalandK.S.Kim,\nPhys. Rev. Lett. 96, 096104 (2006).\n25E. Z. da Silva, A. J. R. da Silva, A. Fazzio, Phys. Rev.\nLett.87, 256102 (2001)\n26S.B.Legolas, D.S.Galv˜ ao, V.Rodrigues, D.Ugarte, Phys.\nRev. Lett. 88, 076105 (2002)\n27In reality the value of the apex MAE is obtained by sub-\nstracting the MAE of the central atom from the MAE of\nthe whole trimer.\n28G. vanderLaan, J. Phys.Condens. Matter 10, 3239 (1998)\n29A. Thiess, Y. Mokrousov and S. Heinze (in preparation)."    },    {        "title": "1003.2449v2.High_temperature_multiferroicity_and_strong_magnetocrystalline_anisotropy_in_3d_5d_double_perovskites.pdf",        "content": "arXiv:1003.2449v2  [cond-mat.mtrl-sci]  25 Jan 2011High temperature multiferroicity and strong magnetocryst alline anisotropy in 3d−5d\ndouble perovskites\nMarjana Leˇ zai´ c1,2and Nicola A. Spaldin2\n1Peter Gr¨ unberg Institut, Forschungszentrum J¨ ulich, D-5 2425 J¨ ulich and JARA-FIT, Germany∗\n2Department of Materials, ETH Zurich, Wolfgang-Pauli Strass e 27, CH-8093 Zurich, Switzerland\nUsing density functional calculations we explore the prope rties of as-yet-unsynthesized 3 d−5d\nordered double perovskites ( A2BB′O6) with highly polarizable Bi3+ions on the Asite. We find that\nthe Bi 2NiReO 6and Bi 2MnReO 6compounds are insulating and exhibit a robust net magnetiza tion\nthat persists above room temperature. When the in-plane lat tice vectors of the pseudocubic unit\ncell are constrained to be orthogonal (for example, by coher ent heteroepitaxy), the ground states\nare ferroelectric with large polarization and a very large u niaxial magnetocrystalline anisotropy with\neasy axis along the ferroelectric polarization direction. Our results suggest a route to multiferroism\nand electrically controlled magnetization orientation at room temperature.\nPACS numbers: 75.85.+t,77.55.Nv\nINTRODUCTION\nThere is increasing current interest in developing mul-\ntiferroic materials with ferromagnetic and ferroelectric\norder in the same phase for future spintronic or mag-\nnetoelectronic devices [1–4]. Although new multiferroics\nare being predicted and synthesized at an accelerating\npace, a major obstacle for their adoption in applications\nremainstheir low magneticorderingtemperatures, which\nare generally far below room temperature. The problem\nmore generally lies in the scarcity of insulators with any\nnet magnetization – either ferro- or ferrimagnetic – at\nroom temperature, even before the additional require-\nment of polarization is included. Most work on novel\nmultiferroics has been restricted to 3 dtransition metal\noxides (see, for example, Refs. [5–12]), motivated by the\nexpectation that 4 dor 5dcompounds would likely be\nmetallic. However, recent work on new 3 d−5ddouble\nperovskites [13, 14] showed this to be a misconception,\nandidentifiedaferrimagneticinsulator,Sr 2CrOsO 6, with\nmagnetic ordering temperature well above room tem-\nperature. First-principles calculations have been shown\nto accurately reproduce the measured magnetic ordering\ntemperatures, [15, 16] in the Sr 2CrMO6series (M= W,\nRe, Os), [17] and have been invaluable in explaining the\norigin of the robust ferrimagnetic ordering. [18–21]\nWe build here on these recent developments to pro-\npose a route to achieving multiferroics with higher mag-\nnetic ordering temperatures. We use 3 d−5ddou-\nble perovskites to achieve high temperature magnetism\ncombined with insulating behavior, and introduce lone-\npair active cations on the A sites to induce polarizabil-\nity [22, 23]. The heavy Bi and 5 delements provide an\nadditional desirable feature: The spin-orbit interaction\nstrongly couples the magnetic easy axis to the ferroelec-\ntric polarization direction. Specifically, we explore two\ncompounds, Bi 2MnReO 6and Bi 2NiReO 6. We use den-\nsityfunctionaltheorytocalculatetheirzeroKelvinstruc-tureandmagneticordering,andMonteCarlosimulations\nwith parameters extracted from the first-principles cal-\nculations to calculate their magnetic ordering tempera-\ntures. We find that both materials are magnetic insula-\ntors, with ordering temperatures well above room tem-\nperature. While the global ground state in both materi-\nals is anti-polar, we show that when the in-plane lat-\ntice vectors form an angle close to 90◦a ferroelectric\nphase results. In practice this could be achieved using\ncoherent heteroepitaxy and the resulting ferroelectric-\nity could be manipulated using strain. We show that\nthe magnetic behavior results from an antiferromagnetic\nsuperexchange [17, 24] and the ferroelectricity from the\nBi3+lone pairs, reminiscent of other Bi-based multifer-\nroics. Finally, we investigate the effect of strong spin-\norbit coupling (SOC) due to the presence of heavy Bi3+\nand Re4+cations on the structural, magnetic, and ferro-\nelectric properties of these compounds.\nMETHOD OF CALCULATION\nStructural optimizations were performed using the Vi-\nennaAb-initio Simulation Package (VASP) with projec-\ntor augmented wave (PAW) potentials; [25] the semicore\npstates of Mn, Ni, and Re and the dstates of Bi were\nincluded in the valence. We used an energy cutoff of 500\neV and 6 ×6×6, 4×4×6, and 4×4×4Γ-point centered\nk-point meshes for unit cells containing 10, 20, and 40\natoms respectively. The exchange-correlation functional\nwas treated within the generalized gradient approxima-\ntion (GGA); [26] while extension to the GGA+ Umethod\nhadonlyasmallinfluenceontheelectronicproperties,we\ndiscuss later its effect on the structural behavior. Mag-\nnetic ordering temperatures were obtained using a finite-\ntemperature Monte Carlo scheme within a Heisenberg\nHamiltonian H=−1\n2/summationtext\ni,jJi,jMi·Mj, whereMiand\nMjare the magnetic moments on sites iandjof the\ncrystal lattice. The exchange constants Ji,jwere cal-2\nculated in the frozen-magnon scheme [27] using the all-\nelectron full-potential linearized augmented plane-wave\ncodeFLEUR[28], with a plane-wave cutoff of 4.2 hartrees\nand 15×15×15k-points and a 6 ×6×6 spin-spiral\ngrid in a 10-atom unit cell. Ferroelectric polarizations\nwere extracted from the shifts of the centers of Wannier\nfunctions [29]. Themagnetocrystallineanisotropyenergy\nwas extracted from the self-consistent total-energycalcu-\nlations within the FLEURcode, on a 13 ×13×13k-point\ngrid.\nFirst, we calculate the lowest energy structure for both\nBi2MnReO 6and Bi 2NiReO 6within the GGA approxi-\nmation. We proceed by successively freezing in the 12\ncentrosymmetric combinations of tilts and rotations of\nthe oxygen octahedra that can occur in ordered double\nperovskites[30]. We then further reducethe symmetryof\nthese lowest energy tilt systems by displacing the anions\nand cations relative to each other in the manner of a po-\nlar displacement. Next we optimize the atomic positions\n(by minimizing the Hellmann-Feynmanforces), aswell as\nthe unit cell shape and volume within each symmetry. In\nthis first series of optimizations we assume ferrimagnetic\nordering and do not include SOC; the influence of SOC\non the lowest-energy structures is examined later.\nGROUND-STATE STRUCTURE AND\nINFLUENCE OF EPITAXIAL CONSTRAINTS\nOur calculations yield the same monoclinic, cen-\ntrosymmetric P21/nsymmetry ground state for both\ncompounds. It is characterized by an a−a−c+tilt pat-\ntern of the oxygen octahedra. Two lattice vectors are\nidentical with angles of 93.6◦and 93.5◦between them\nin the Mn and Ni compounds, respectively (in the fol-\nlowing we refer to these as the “in-plane” lattice vec-\ntors); the third lattice vector is perpendicular to the first\ntwo and of different length. Importantly, in both com-\npounds there isa slightlyhigherenergyferroelectric(FE)\nstructure with rhombohedral R3 symmetry (see Table I).\nTheR3 structure corresponds to an a−a−a−tilt pattern\nyielding lattice vectorsof equal length and rhombohedral\nangles of 60◦and 61◦for the Mn and Ni compounds re-\nspectively, combined with polar relative displacements of\nanions and cations along the [111] rhombohedral axis in-\nSymmetry Tilt system Total energy\n(in Glazer notation [31]) Bi2MnReO 6Bi2NiReO 6\nFm¯3m a0a0a01.86 eV 2.02 eV\nR¯3 a−a−a−248 meV 333 meV\nR3a−a−a−+ FE shift 32 meV 18 meV\nTABLE I: Total energies (per formula unit) of the lowest-\nenergy phases of Bi 2MnReO 6and Bi 2NiReO 6and the cubic\nFm¯3mphase with respect to their ground state P21/nphase.(a)\n(b)\nFIG. 1: (Color online) (a) Ground-state P21/nstructure and\n(b) the ferroelectric R3 structure of the double perovskites\nBi2MnReO 6and Bi 2NiReO 6. Bi3+cations are depicted as\nthe large (brown) spheres and O2−as the small (red) spheres.\nRe4+and Mn2+(Ni2+) are alternating in the shaded octa-\nhedra, in a three-dimensional checkerboard manner (this is\nindicated by the color of the octahedra). While in the P21/n\nsymmetry the Bi3+cations form an “antipolar” pattern, their\ncoherent shift along the [111] direction can be clearly seen in\ntheR3 structure.\nduced by the well-established stereochemically active Bi\nlone pair [22] (see Fig. 1). The energy of the FE phase\nis significantly lower than that of the corresponding cen-\ntrosymmetric R¯3 phase, suggesting a high ferroelectric\nordering temperature if this symmetry could be stabi-\nlized. For comparison, in the prototypical multiferroic\nBiFeO 3withTCof 1103 K, we obtain a difference of 515\nmeV per two formula units between the corresponding\nstates. The calculated polarization in both cases is along\nthe [111] direction, and comparable in size to that of\nBiFeO 3(see Table II).\nWhile the ground state P21/nphase is of interest in\nits own right as a possible high temperature magnetic\ninsulator, we next explore whether it is possible to iden-\ntify conditions that stabilize the R3 ferroelectric phase.\nWe use the fact that in the FE R3 phase all three lattice\nvectors have equal length, and the rhombohedral angle is\nequal orcloseto the ideal value of 60◦. In contrast, in the\nP21/nphase, only the lengths of the two in-plane lattice\nvectors are equal, and the inter-in-plane angles deviate3\n(a)7.95 8 8.05 8.1 8.15 8.2\nIn-plane lattice constant [Å]020406080100120Energy [meV] /  formula unit\n7.95 8 8.05 8.1 8.15 8.2\nIn-plane lattice constant [Å]020406080100120\nFEAPAP\nFEBi2MnReO6Bi2NiReO6\nP21/nR3\nP21/nR3\n(b)7.95 8 8.05 8.1 8.15 8.2 8.25\nIn-plane lattice constant [Å]020406080100120140Energy [meV] /  formula unit\n7.95 8 8.05 8.1 8.15 8.2 8.25\nIn-plane lattice constant [Å]020406080100120140\nFEAP\nAPFEBi2MnReO6Bi2NiReO6\nU(Ni)=6eV\nP21/nR3\nP21/nR3U(Mn)=3eV\nFIG. 2: (Color online) (a) Distorted Bi 2MnReO 6and Bi 2NiReO 6obtained from the R3 (red circles) and P21/n(blue triangles)\nstructure by constraining the two in-plane lattice vectors to be equal and the angle between them to 90◦. The out-of-plane\nparameter and the ionic positions were relaxed. The energie s are positioned with respect to the bulk unconstrained P21/n\nground state (blue squares). The red diamonds show the energ ies for the unconstrained R3 structure. A crossover between the\nantipolar (AP) and the ferroelectric (FE) state occurs at 1. 8% (with respect to the minimum of the FE curve) of compressiv e\nstrain in Bi 2MnReO 6. Note that the energy of the R3 state lies on the strain curve for Bi 2MnReO 6, but is somewhat lower for\nBi2NiReO 6, due to the choice of a square in-plane lattice: the rhombohe dral angle in R3 phase of Bi 2MnReO 6is 60◦, while in\nBi2NiReO 6it is 61◦. (b) Similar to (a), with an addition of a Hubbard Uon Mn (3 eV) and Ni (6 eV) 3 dstates. The shaded\narea shows the AP-FE crossover region.\nstrongly from the ideal 90◦. Motivated by these observa-\ntions, we first constrain all three lattice parameters to be\nequal in length and indeed find that this constraint sta-\nbilizes the FE phase. Next we enforce that only two of\nthe three lattice parameters are equal in length, but con-\nstrain them to be perpendicular to each other and again\nfind that the FE phase is the lowest energy [37]. Such a\nconstraintcouldbe achievedinpracticethroughcoherent\nheteroepitaxy between a thin film and a substrate, and\nis often described as a biaxial strain state in the litera-\nture. In Fig. 2(a), we show the calculated energies of the\npreviously R3 andP21/nstructure types subject to this\nadditional constraint, with the in-plane lattice parame-\ntersvariedoverarangeofrealisticsubstratestrainscorre-\nspondingto(001)epitaxialgrowth. Foreachin-planelat-\ntice parameter we allow the length and angle of the out-\nof-plane lattice parameter and atomic positions to relax\nto their lowest-energy configurations. Our main finding\nis that, under the investigated constraint, the FE phase\nis lower in energy than the antipolar (AP) phase. At the\noptimized value of the in-plane lattice parameter (8.16 ˚A\nfor Bi2MnReO 6and 8.07 ˚A for Bi 2NiReO 6) the energy\ndifferences between the FE and AP phases are 36 and\n30 meV per formula unit, respectively. The FE phases\nare robustly insulating with calculated band gaps of 0.7\nand 0.3 eV, respectively, and have large polarizations of\n84µC/cm2(Bi2MnReO 6) and 81 µC/cm2(Bi2NiReO 6)\nalong the pseudocubic [111] direction. Notice, however,\nthat as in-plane compressive strain is applied, the dif-\nferences in energies reduce, and in fact in Bi 2MnReO 6a\ncrossoverisreachedatacompressivestrainvalueof1.8%.We therefore expect that both compounds, although AP\ninbulk, will infact be ferroelectricandhencemultiferroic\nin thin-film form, over a range of experimentally accessi-\nble strains. We see later that SOC tips the scale toward\nthe FE phase, while electron correlations favor the AP\nphase.\nELECTRONIC AND MAGNETIC PROPERTIES\nOF THE FERROELECTRIC PHASE\nNext, we analyze the magnetic behavior of the two\nferroelectric compounds. We find that in both cases\nthe lowest-energy ordering is ferrimagnetic, with the mo-\nments on the 3 dand 5dtransition metal ions antialigned\n(see Fig. 3). In both compounds, Re is in formal oxida-\ntion state 4+corresponding to a filled d−t2gmanifold in\nthe minority spin-channel. The oxidation states of Mn in\nBi2MnReO 6and of Ni in Bi 2NiReO 6are 2+:t3\n2ge2\ng(high\nspin) for Mn and t6\n2ge2\ngfor Ni. Due to these specific con-\nfigurations of the outer electronic shells, and the nearly\n150◦Mn-O-Re (Ni-O-Re) angle, the magnetic moments\non Re and Mn/Ni are coupled via an antiferromagnetic\nsuperexchange mechanism. [33–35] The result is a ferri-\nmagnetic configuration with a total spin moment of 2 µB\nin Bi2MnReO 6and 1µBin Bi2NiReO 6.\nOur calculated magnetic ordering temperatures in the\nferroelectric phase are 330 K for Bi 2MnReO 6and 360 K\nfor Bi 2NiReO 6, both significantly above room temper-\nature. As a comparison, using the same method, we\ncalculate a magnetic ordering temperature of 670 K for4\n9630369DOS [states/eV]Total\nMn/Ni\nRe\n-8 -6 -4 -2 0 2\nE-EF [eV]9630369DOS [states/eV]Bi2MnReO6\nBi2NiReO6\nmajority spin minority spin majority spin minority spin(a)\n(b)\nFIG. 3: (Color online) Density of states (DOS) of the ferro-\nelectric phase of Bi 2MnReO 6(a) and Bi 2NiReO 6(b). The\nlocal magnetic moments on Mn/Ni and Re are antialigned.\nThin black line, total DOS; green dashed line, Re; orange\nsolid line, Mn/Ni.\nSr2CrOsO 6(the experimental value is 725 K). Note that\nif growth can only be achieved in the form of ultra-thin\nfilms, the ordering temperatures will likely be reduced.\nSPIN-ORBIT COUPLING AND CORRELATIONS\nEFFECTS\nSince the heavy Re and Bi atoms in our double per-\novskites are also likely to exhibit strong SOC, and con-\nsequently significant magnetostructural coupling, [18] we\nrepeat ourcalculationswith SOC explicitly included. We\nfindthatthegroundstateremains P21/nAP,butnowits\nenergy difference from the FE state with R3 symmetry\nis reduced to 10 meV in Bi 2MnReO 6and only 2 meV in\nBi2NiReO 6. Both compounds still have semiconducting\ngaps which are now somewhat reduced while the ferro-\nelectric polarization of the R3 phase is slightly increased\n(see Table II). The total magnetic moments (spin + or-\nbital) amount to 2.34 and 0.58 µB, respectively. Note\nthat, in contrast to half-metallic ferromagnets where the\nSOC introduces states in the minority-spin gap yielding\na noninteger total magnetic moment, here the noninte-\nger moment is the consequence of the mixing of spin-up\nand spin-down states (spin is no longer a good quantum\nnumber with SOC included), while the gap is preserved.\nIn both compounds the magnetic easy axis lies along\nthe ferroelectric polarization direction. The associated\nanisotropy energy is large and comparable to that in cur-SOCCompound Band gap [eV] P(µC/cm2)M(µB/f.u.)\nOffBi2MnReO 60.7 84 2\nBi2NiReO 60.3 78 1\nOnBi2MnReO 60.4 86 2.34\nBi2NiReO 60.2 80 0.58\nTABLE II: Properties of the bulk ferroelectric R3 phase in\nBi2MnReO 6and Bi 2NiReO 6with and without the SOC in-\ncluded: band gap, ferroelectric polarization Pand the total\nmagnetic moment M(spin+orbital in case of SOC) per for-\nmula unit (f.u.).\nrent magnetic recording media: it amounts to 7 and 5.5\nmeV per formula unit in Bi 2MnReO 6and Bi 2NiReO 6,\nrespectively. This suggests an exciting possibility of elec-\ntrical control of the magnetization direction.\nFinally, we investigate the influence of electronic cor-\nrelations by adding a Hubbard Uon thedstates of Mn\nand Ni. In order to reduce the computational effort, we\ndo not include the SOC. We keep in mind, however, that\na consequence of its inclusion, as we have seen, is a re-\nduction in the energy of the FE state with respect to\nthe AP one. Introduction of Hubbard Ucorrections (on\nMnU=3 eV and J=0.87 eV, and on Ni U=6 eV and\nJ=0.90 eV) does not change the ground state structure\n(P21/nstill has the lowest energy). Interesting quanti-\ntative changes occur in the strain dependence however\n[Fig. 2(b)]. While within the GGA the ferroelectricity\nwas robust to the choice of substrate lattice parameters\nover a likely range of accessible strains provided that the\nin-plane lattice parameters were constrained to be per-\npendicular to eachother, in GGA+ Uwe find a cross-over\nbetween the FE and AP states at moderate strain values.\nThis suggests the intriguing possibility that an external\nstrain, for example from a piezoelectric substrate [36]\ncould induce a “dipole-flop” transition from a non-polar\nstructure in which the dipoles are antialigned to a ferro-\nelectric one. For the assumed value of U, the crossover\nregion [shaded area in Fig 2(b)] in Bi 2MnReO 6is within\nless than 1% from either of the two minima: a small\ncompressive strain will push the system toward the AP\nstate, while a small tensile strain will favor a FE state.\nIn Bi2NiReO 6the applied Upushes the AP state lower\nin energy, but the crossoverregion to the FE state is still\nwithin an experimentally accessible range of 1.5% tensile\nstrain. While we can not make a quantitative prediction,\nsince the relative energies of the AP and the FE state de-\npend strongly on the value of the applied Ucorrection,\nwithin a reasonable range of Uthe crossover region lies\nwithin 2% strain from the ground state structure.5\nCONCLUSION\nIn summary, we predict from first-principles calcu-\nlations that the 3 d−5ddouble-perovskite compounds\nBi2MnReO 6and Bi 2NiReO 6are high-T Cferrimagnetic\ninsulators. Moreover,itislikelypossibletostabilizethem\nin thin-film form in a ferroelectric phase with high polar-\nization along the [111] direction. The magnetic easy axis\nlies along the ferroelectric polarization direction with a\nvery large associated anisotropy energy. The different\nsizes and charges of the 3 dand 5dtransition metal ions\nsuggest that the required B-site ordering should be ex-\nperimentally feasible; we hope that our findings will ini-\ntiate experimental efforts to synthesize these novel mul-\ntiferroic compounds.\nWe thank Drs. Kris Delaney, Phivos Mavropoulos,\nStefan Bl¨ ugel, Frank Freimuth and Sergey Ivanov for\nmany valuable discussions. M.L. gratefully acknowledges\nthe support of Deutsche Forschungsgemeinschaft, grant\nLE 2504/1-1, and the Young Investigators Group Pro-\ngramme of Helmholtz Association, contract VH-NG-409,\nas well as the J¨ ulich Supercomputing Centre. NS ac-\nknowedges support from the NSF NIRT program, grant\nnumber 0609377.\n∗Electronic address: m.lezaic@fz-juelich.de\n[1] Xi Chen, A. Hochstrat, P. Borisov, and W. Kleemann,\nAppl. Phys. Lett. 89, 202508 (2006).\n[2] M. Gajek, M. Bibes, S.Fusil, K.Bouzehouane, J. Fontcu-\nberta, A. Barth´ el´ emy, and A. Fert, Nature Mat. 6, 296\n(2007).\n[3] Ying-HaoChu, LaneW. Martin, MikelB. Holcomb, Mar-\ntin Gajek, Shu-Jen Han, Qing He, Nina Balke, Chan-Ho\nYang, Donkoun Lee, Wei Hu, Qian Zhan, Pei-Ling Yang,\nArantxa Fraile-rodrguez, Andreas Scholl, Shan X. Wang\nand R. Ramesh, Nature Mat. 7, 478 (2008).\n[4] D. Lebeugle, A. Mougin, M. Viret, D. Colson, and L.\nRanno, Phys. Rev. Lett. 103, 257601 (2009)\n[5] Kunihiko Yamauchi, Tetsuya Fukushima, and Silvia Pi-\ncozzi, Phys. Rev. 79, 212404 (2009).\n[6] S. Picozzi, K. Yamauchi, B. Sanyal, I. A. Sergienko, and\nE. Dagotto, Phys. Rev. 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Rev. 79, 350 (1950).\n[25]The Vienna ab-initio simulation package , G. Kresse and\nJ. Hafner, Phys.Rev. B 47, R558(1993); G. Kresse, The-\nsis, Technische Universit¨ at Wien (1993); G. Kresse andJ.\nFurthm¨ uller, Comput. Mat. Sci. 6, 15 (1996); G. Kresse\nand J. Furthm¨ uller, Phys. Rev. B 54, 11169 (1996); G.\nKresse, and D. Joubert, Phys. Rev. B 59, 1758 (1999);\nP.E. Bl¨ ochl, Phys. Rev. B 50, 17953 (1994).\n[26] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett.77, 3865 (1996).\n[27] M. Leˇ zai´ c, PhD Thesis, RWTH Aachen (2005)\n[http://darwin.bth.rwth-aachen.de/opus/volltexte/20 06/1359/].\n[28] www.flapw.de\n[29] F. Freimuth, Y. Mokrousov, D. Wortmann, S. Heinze,\nand S. Bl¨ ugel, Phys. Rev. B 78, 035120 (2008).\n[30] C. J. Howard, B. J. Kennedy, and P. M. Woodward, Acta\nCryst. B 59, 463 (2003).\n[31] A. M. Glazer, Acta Cryst. B 28, 3384 (1972)\n[32] P. Baettig and N. A. Spaldin, Appl. Phys. Lett. 86,\n012505 (2004).\n[33] J. B. Goodenough, Phys. Rev. 100, 564 (1955).\n[34] J. Kanamori, J. Phys. Chem. Solids 1087 (1959).\n[35] E. O. Wollan, Phys. Rev. 117, 387 (1960).\n[36] A. D. Rata, A. Herklotz, K. Nenkov, L. Schultz, and K.\nD¨ orr, Phys. Rev. Lett. 100, 076401 (2008).\n[37] Note that the lowest energy P21/nstructure would not\nbe readily identified as the ground state through a search\nfor unstable phonons starting from the prototype cubic\nperovskite structure because it only becomes lowest in\nenergy when the lattice parameters are allowed to be\nunequal. Indeed, when we perform full structural opti-\nmizations within the LDA+U method for the previously\nstudied double perovskite, Bi 2FeCrO 6[32], we find the\nP21/nstructure to be lower in energy than the previ-\nously reported R3 when we allow for lattice parame-\nters of different length. Interestingly, in this case, the\nGGA+U method yields a ferroelectric ground state, con-6\nsistent with some experimental reports [10]."    },    {        "title": "1003.5870v1.Electric_Field_Effect_on_Magnetization_and_Magnetocrystalline_Anisotropy_at_the_Fe_MgO_001__Interface.pdf",        "content": "1Electric Field Effect on Magnetization and Magnetocrystalline Anisotropy \nat the Fe/MgO(001) Interface\nManish K. Niranjan1*, Chun-Gang Duan2, Sitaram S. Jaswal1and Evgeny Y. Tsymbal1†\n1Department of Physics and Astronomy & Nebraska Center for Materials and Nanoscience, \nUniversity of Nebraska, Lincoln, Nebraska 68588, USA\n2Key Laboratory of Polarized Materials and Devices, Ministry of Education, East China Normal University, Shanghai 200062, China\nDensity-functional calculations are performed to explore magnetoelectric effects originating from the influence of \nan external electric field on magnetic properties of the Fe/MgO(001) interface. It is shown that the effect on the \ninterface magnetization and magnetocrystalline anisotropy can be substantially enhanced if the electric field is applied \nacross a dielectric material with a large dielectric constant. In particular, we predict an enhancement of the interface \nmagnetoelectric susceptibility by a factor of the dielectric constant of MgO over that of the free standing Fe (001) \nsurface. We also predict a significant effect of electric field on the interface magnetocrystalline anisotropy due to the \nchange in the relative occupancy of the 3 d-orbitals of Fe atoms at the Fe/MgO interface. These results may be \ninteresting for technological applications such as electrically controlled magnetic data storage. \nMaterials that have high magnetocrystalline anisotropy \n(MCA), especially nanometer-thick ferromagnetic films, are \nwidely utilized in modern perpendicular magnetic recording \ntechnology (see, e.g., ref. 1). The large coercivity of such \nmaterials requires, however, high magnetic field to “write” bit \ninformation on them. In the past decade, various methods have \nbeen proposed to solve this problem, e.g., heat-assisted \nrecording or the application of anisotropy-graded and \nexchange-coupled media. An alternative way to tailor the \nMCA (that has not yet been realized in practical devices) is to \nexploit the magnetoelectric (ME) effect.2 , 3The ME effect \nallows changing the bulk magnetization of a material via \napplying an electric field.4In a broader vision, ME effects also \ninvolve the electrically-controlled surface (interface) \nmagnetization,5-11magnetic order,12,13exchange bias,14-16spin \ntransport,17 - 22and magnetocrystalline anisotropy.8, 23 - 31Since \nthe MCA determines stable orientations of magnetization, the \nlatter approach is especially promising – tailoring the \nmagnetic anisotropy by electric fields may yield entirely new \nparadigms for magnetic data storage. \nFor metallic ferromagnets, electronically-driven ME\neffects are confined to the interface and originate from spin-\ndependent screening.32Consequently the electric field affects \nonly the surface (interface) MCA. Experimentally, a change in \nthe surface MCA of a few percent was observed in \nFePt(Pd)/electrolyte films.23Magnetic easy axis manipulation \nby electric field was also demonstrated in the dilute magnetic \nsemiconductor (Ga,Mn)As.25Recently, a strong effect of \nelectric field on the interface MCA was demonstrated for the \nFe/MgO (001) 26,27and FeCo/MgO (001) 31interfaces. It was \nfound that the application of a relatively small field 0.1 V/nm \nleads to a 40% change of MCA for Fe films.26, 27First-\nprinciples calculations have been performed and shown that \nrather large fields are required to observe a sizable change in \nthe surface magnetization and MCA.8, 28-30\nIn this article we demonstrate that the electric field effect \non the surface magnetization and MCA can be substantially \nenhanced if the electric field is applied across a dielectric \nmaterial with a sufficiently large dielectric constant . Since the induced screening charge scales with the dielectric \nconstant 0E  , for high permittivity dielectrics the ME \neffect may be enhanced by orders of magnitude. To illustrate \nthe significance of this prediction we explore the ME effect at \nthe Fe/MgO (001) interface using density-functional \ncalculations. We demonstrate the enhancement of the surface \nME susceptibility by a factor of the dielectric constant of MgO \nover that of a free standing Fe (001) surface. We also find a \nsignificant increase in the electric field effect on the surface \nMCA due to the change in the relative occupancy of the 3 d-\norbitals of Fe atoms at the Fe/MgO interface. \nTo study the effect of electric field on the interface \nmagnetization and MCA energy of the Fe/MgO (001) \ninterface we perform density-functional calculations using a \nMgO/Fe/Cu(001)/Vacuum supercell. We employ the projected \naugmented wave (PAW) method33and the generalized \ngradient approximation (GGA) for exchange and correlation, \nas implemented within Vienna Ab-Initio Simulation Package \n(VASP).34We use standard plane wave basis set with a kinetic \nenergy cutoff of 500 eV and the k-mesh sampled using \n10×10×1 k-points in the full Brillouin zone. Along the [001] \ndirection the supercell consists of nine monolayers of bcc Fe \non top of nine monolayers of MgO followed by four \nmonolayers of bcc copper and a vacuum layer. At the Fe/MgO \ninterface, the O atoms are placed atop Fe atoms, consistent \nwith the experimental data.35The electric field is introduced \nby the dipole layer method 36with the dipole placed in the \nvacuum region of the supercell. The electric field points away\nfrom the Fe layer at Fe/MgO interface. The Cu layer is used to \neliminate the screening charge at the otherwise Fe/Vacuum \ninterface. The in-plane lattice constant of the supercell is kept \nfixed at the experimental lattice constant of Fe ( a= 2.87 Å). \nThe structures are relaxed in the absence of the electric field \nuntil the largest force becomes less than 5.0 meV/Å. The \nMCA energy is obtained by taking the difference between the \ntotal energy calculated within the force theorem corresponding \nto magnetization pointing along the [100] and [001] directions \nin the presence of spin-orbit interaction.    2First, we consider the effect of electric field on the \nmagnetization at the Fe/MgO interface. Here and below the \nME effects are discussed as a function of the electric field \nwithin the MgO layer relevant to experiments where the field \nis applied between two metal electrodes across the dielectric. \nOur calculations find that the electric field in MgO is reduced \nby a factor of 3.1 as compared to that in vacuum due to \ndielectric screening. In the absence of ionic relaxations this \nreduction is entirely caused by the electronic contribution to \ndielectric susceptibility of MgO which is associated with the \nhigh frequency dielectric constant ε∞. The calculated value of \nε∞ ≈ 3.1 is in agreement with the experimental value ε ∞ ≈ 3.0.37\nThe electric field is screened by free charges in Fe at the \nFe/MgO interface. Due to ferromagnetism of Fe this screening \nis spin-dependent and leads to the induced interface \nmagnetization, resulting in the surface (interface) ME effect.32\nThe induced interface magnetization is evident from Fig. 1a\nwhich shows the change in the spin density Δ= (E) –(0) \ndue to electric field E= 1 V/nm projected on the x-zor (010) \nplane of the supercell. It is seen that only the spin density at\nthe interfacial Fe atoms is changed significantly. This result is \nqualitatively similar to that found for the Fe(001) surface.8\n0.0 0.4 0.8 1.2 1.6 2.02.7602.7642.7682.772z\nx-0.016-0.00800.0080.016\nMgOFe\nE(b)m (B)\nE (V/nm)(a)\nFig. 1 : (a) Induced spin density Δ= (E) –(0), in units of e/Å3, projected \nto the x-zor (010) plane around the Fe/MgO interface under the influence of \nelectric field E= 1.0 V/nm in MgO. The dashed line indicates the interfacial \nFe monolayer at the Fe/MgO interface. (b) Magnetic mom ent (in units of μ B) \nof Fe at Fe/MgO interface as a function of the electric field in the MgO.\nFig. 1b shows the magnetic moment mof the Fe atom at \nthe Fe/MgO interface versus the electric field Ein MgO. It is \nseen that within the computational error mchanges linearly\nwith E. The induced interface magnetization ∆Mis given by \n0ΔM = α SE, where αSis the surface (interface) ME \ncoefficient.8We find that αS  ≈ 1.1×10-13Gcm2/V. This value is \nlarger than that for the Fe(001) surface8by a factor of 3.8, \nwhich is approximately equal to the calculated ε ∞. The larger \nαSis due to the enhancement of the screening charge at the \nFe/MgO interface. Within a rigid band model the surface ME \ncoefficient is given by 2B\ns Pec ,8where P is the spin \npolarization of the interface density of states at the Fermi \nenergy. This result indicates that αSis scaled linearly with \ndue to the enhanced surface charge density 0E \nunequally distributed between the interface majority- and \nminority-spin states. 0.0 0.5 1.0 1.5 2.0 2.52.02.22.42.62.83.0\n1.31.41.51.61.71.81.9MCA(erg/cm2)ml (B)\nE (V/nm)\nz\n-0.00500.005\nEx(b)\nExy(a)\nFig. 2 : (a) Magnetic anisotropy energy (MCA, squares) and orbital moment \nanisotropy ( Δml, triangles) of the Fe/MgO(001) interface as a function of \nelectric field in MgO. (b) Induced charge density Δρ =  ρ(E) –ρ(0), in units of \ne/Å3, at the interfacial Fe atom for E= 1.0 V/nm in the  x-z(010) plane (top \npanel) and the x-y(001) plane (bottom panel). \nNext we discuss the electrically induced MCA at the \nFe/MgO (001) interface. For a given Ewe calculate MCA of \nthe MgO/Fe/Cu/Vacuum structure and subtract the MCA \nvalue of the Fe/Cu interface obtained in a separate calculation \nfor a Fe/Cu(001) supercell. Since the contribution of the Fe/Cu \ninterface to the total MCA energy does not depend on E, due \nto the screening charge being confined at the Cu surface, this \nallows us to obtain the Fe/MgO(001) interface MCA energy as \na function of E. The results are shown in Fig. 2a. It is seen that \nwithin the accuracy of our calculation the MCA of the \nFe/MgO interface changes linearly with electric field. We can \ndefine the surface (interface) MCA coefficient βSaccording to \nΔK= β SE, where ΔKis the change in the MCA energy. From \nthe slope in Fig. 2a, we find that the surface MCA is changed \nby 0.10 erg/cm2for E= 1 V/nm resulting in βS ≈ 10-8erg/Vcm. \nThis value is larger by a factor of 5 than that obtained for the \nFe(001) surface, i. e. βS ≈ 2×10-9erg/Vcm.8\nIt is instructive to compare the variation in the MCA and \nthe orbital moment anisotropy Δ ml= ml[001] – ml[100], since \nthey both arise from the spin-orbit interaction. Fig. 2a \nindicates that the two anisotropies are linearly related which is \npossible only if the majority dband is fully occupied so that\nspin-flip matrix elements of the spin-orbit interaction can be \nneglected.38This fact allows us to discuss the behavior of \nMCA energy in terms of the orbital magnetic moment. \nThe decrease in MCA at the Fe/MgO (001) interface with \nelectric field pointing away from the Fe layer originates from \nthe redistribution of electron charge between different d-\norbitals. Fig. 2b shows the change in the charge density, Δρ= \nρ(E) –ρ(0), at the interface Fe atom induced by electric field E\n= 1 V/nm. From the x-z(010) plane and the x-y(001) plane \nprojections we see a reduced occupation of the dxz(and by \nsymmetry dyz) orbitals (top panel in Fig. 2b) and the enhanced \noccupation of the dxyorbitals (bottom panel in Fig. 2b). The Lz\noperator has positive non-zero matrix elements xz z yz dL d , \nwhile 0xz z xy yz z xy dL d dL d  . Thus, the reduced \noccupation of the dxzand dyzorbitals leads to the reduction of\nzL, and hence a decrease in the associated orbital magnetic \nmoment ml[001]. The latter is evident from the calculated3ml[001] shown in Fig. 3a (squares), indicating a reduction of \nml[001] with electric field. On the other hand, as seen from \nFig. 3a (triangles), ml[100] is weakly dependent on E. \nApparently the reduced occupation of the dxz(dyz) orbitals and \nenhanced occupation of the dxyorbitals (Fig. 2b) are largely \ncanceled out in the contribution to xL (and hence to\nml[100]) through non-vanishing matrix elements xz x xy dL d\nand yz x xy dL d . As pointed out earlier and shown in Fig. 2a, \nthe difference of the two orbital moments and hence MCA\ndecrease with increasing electric field. This behavior is \nopposite to that found previously for the Fe (001) surface.8\nFor the latter, as seen from Fig. 3b, the two orbital magnetic \nmoments diverge with increasing electric field (pointing away \nfrom Fe) resulting in the increase of the MCA energy.  \n0.0 0.5 1.0 1.5 2.0 2.50.080.090.100.110.12\n0 2 4 60.080.090.100.110.12\n(a) (b) ml[001]\nml[100]ml ()\nE (V/nm) E (V/nm)\nFig. 3 : Orbital magnetic moment of Fe atom at (a) the Fe/MgO (001)\ninterface and (b) the Fe(001) surface, as a function of electric field, for the \nmagnetic moment pointing along the [001] direction (squares) and along the \n[100] direction (triangles). \nWe have explored the effect of ionic relaxations driven by \nelectric field. The calculation of the static dielectric constant \nof MgO in fields ranging from 0 to 3 V/nm predicts the \naverage value of ~ 8.5 in agreement with the experimental \nvalue of ≈ 9.5. For a given field in vacuum , changes in the\ninterface magnetization and MCA are qualitatively similar to \nthose for the unrelaxed structure. This is because the \nmacroscopic screening charge primarily responsible for the \nME effects depends only on the vacuum field due to the \ncancellation of the dielectric effects of the insulator. Since the \nelectric field in MgO is scaled with its dielectric constant, this \nresult implies that the ME effects for the ideal Fe/MgO(001) \ninterface seen in static measurements should be proportionally \nenhanced. We note however that the interface MCA is very \nsensitive to the interface electronic and atomic structure. In \nparticular, we find that MCA of the Fe/MgO(001) interface \nwith additional O adsorbed in the Fe interfacial monolayer \n[35] becomes negative (about –0.7 erg/cm2), hence suggesting \nan in-plane anisotropy. Thus the amount of oxygen at the \nFe/MgO interface may significantly influence the interface \nMCA, which may explain much smaller values of the Fe/MgO \ninterface anisotropy observed experimentally. [31] On the \nother hand, there are indications that the interface MCA of \nFe/MgO (001) interfaces with controlled oxygen \nstoichiometry may exceed 1 erg/cm2consistent with our calculations. [39] These experimental results in conjunction \nwith our predictions are very exciting in view of using \nFe/MgO interfaces to control the interface magnetic \nanisotropy by electric fields. We therefore hope that our \nresults will stimulate further experiments in this field. \nThis work was supported by the Nebraska MRSEC (NSF \nGrant No. 0820521), the Nanoelectronics Research Initiative, \nNSFC (Grant No. 50771072 and 50832003), and the Nebraska \nResearch Initiative. 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A Molegraaf et al., Adv. Mater. 21, 3470 (2009).\n12. J. D. Burton and E. Y. Tsymbal, Phys. Rev. B 80, \n174406 (2009). \n13. C. A. F. Vaz et al. , Phys. Rev. Lett. 104, 127202 (2010). \n14. P. Borisov et al. , Phys. Rev. Lett. 94, 117203 (2005).\n15. V. Laukhin et al. , Phys. Rev. Lett. 97, 227201 (2006).\n16. Y.-H. Chu et al. , Nature Mat. 7, 478 (2008).\n17. M. Y. Zhuravlev et al. , Appl.  Phys.  Lett. 87, 222114 (2005).\n18. E. Y. Tsymbal and H. Kohlstedt, Science 313, 181 (2006). \n19. M. Gajek et al. , Nature Mater. 6, 296 (2007).\n20. J. P. Velev et al. , Nano Lett. 9, 427 (2009).\n21. V. Garcia et al. , Science 327, 1106 (2010).\n22. M. Ye. Zhuravlev, S. Maekawa, and E. Y. Tsymbal, Phys. Rev. \nB 81, 104419 (2010).\n23. M. Weisheit et al. , Science 315, 349 (2007).\n24. C.–G. Duan et al. , Appl. Phys. Lett. 92, 122905 (2008).\n25. D. Chiba et al. , Nature 455, 515 (2008).\n26.  T. Maruyama et al. , Nature Nanotech. 4, 158 (2009).\n27.  Y. Shiota et al. , Appl. Phys. Exp. 2, 063001 (2009).\n28. K. Nakamura et al. , Phys. Rev. Lett. 102, 187201 (2009); Phys. \nRev. B 80, 172402 (2009).\n29. M. Tsujikawa and T. Oda, Phys. Rev. Lett. 102, 247203 (2009).\n30.  H. Zhang et al. , New J. Phys. 11, 043007 (2009).\n31.  T. Nozaki et al. , Appl. Phys. Lett.  96, 022506 (2010).\n32. S. Zhang, Phys. Rev. Lett. 83, 640 (1999).\n33. P. E. Blöchl, Phys. Rev. B 50, 17953 (1994).\n34. G. Kresse and D. Jourbert, Phys. Rev. B 59, 1758 (1999).\n35. H. L. Meyerheim et al. , Phys. Rev. Lett. 87, 076102 (2001).\n36. J. Neugebauer and M. Scheffler, Phys. Rev. B 46, 16067 (1992).\n37.  J. R. Jasperse et al. , Phys. Rev. 146, 526 (1966).  \n38.  G. van der Laan, J. Phys.: Condens. Matt. 10, 3239 (1998).\n39. B. Dieny, private communication."    },    {        "title": "1007.2766v1.Strain_control_of_magnetic_anisotropy_in__Ga_Mn_As_microbars.pdf",        "content": "Strain controll of magnetic anisotropy in (Ga,Mn)As microbars\nC. King1, J. Zemen2, K. Olejn\u0013 \u0010k2;3, L. Hor\u0013 ak4, J. Haigh1, V. Nov\u0013 ak2,\nJ. Ku\u0014 cera2, V. Hol\u0013 y4, R. P. Campion1, B. L. Gallagher1, and T. Jungwirth2;1\n1School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, UK\n2Institute of Physics ASCR, v. v. i., Cukrovarnick\u0013 a 10, 162 00 Praha 6, Czech Republic\n3Hitachi Cambridge Laboratory, Cambridge CB3 0HE, United Kingdom and\n4Charles University in Prague, Ke Karlovu 3, 121 16 Prague 2, Czech Republic\n(Dated: July 12, 2021)\nWe present an experimental and theoretical study of magnetocrystalline anisotropies in arrays of\nbars patterned lithographically into (Ga,Mn)As epilayers grown under compressive lattice strain.\nStructural properties of the (Ga,Mn)As microbars are investigated by high-resolution X-ray di\u000brac-\ntion measurements. The experimental data, showing strong strain relaxation e\u000bects, are in good\nagreement with \fnite element simulations. SQUID magnetization measurements are performed to\nstudy the control of magnetic anisotropy in (Ga,Mn)As by the lithographically induced strain re-\nlaxation of the microbars. Microscopic theoretical modeling of the anisotropy is performed based\non the mean-\feld kinetic-exchange model of the ferromagnetic spin-orbit coupled band structure of\n(Ga,Mn)As. Based on the overall agreement between experimental data and theoretical modelling\nwe conclude that the micropatterning induced anisotropies are of the magnetocrystalline, spin-orbit\ncoupling origin.\nI. INTRODUCTION\nDilute moment ferromagnetic semiconductors, such as\n(Ga,Mn)As, are favorable systems for studying and utiliz-\ning controllable magnetic anisotropy since micromagnetic\nparameters of this ferromagnet are very sensitive to Mn\ndoping, hole concentration, lattice strains, and temper-\nature. The magnetic moment density is small in these\nferromagnets and therefore the spin-orbit coupling in-\nduced magnetocrystalline anisotropy typically dominates\nthe dipolar-\feld shape anisotropy.\nThe control of the magnetocrystalline anisotropy in\n(Ga,Mn)As epilayers has been achieved by choosing dif-\nferent substrates and therefore di\u000berent growth induced\nstrain in the magnetic layer, by varying the growth pa-\nrameters of the (Ga,Mn)As \flm, and by postgrowth\nannealing.1,2Reversible electrical control of the mag-\nnetocrystalline anisotropy has been demonstrated by\nutilizing piezo-electric stressors3{5or by electrostatic\ngating in thin-\flm (Ga,Mn)As \feld e\u000bect transistor\nstructures.6,7Recently, a local control of the magne-\ntocrystalline anisotropy has been reported, which pro-\nvides the possibility for realizing non-uniform magneti-\nzation pro\fles and which can be utilized, e.g., in studies\nof current induced magnetization dynamics phenomena\nor non-volatile memory devices.8,9In these studies an ef-\n\fcient method of local strain control has been used which\nis based on lithographic patterning that allows for the re-\nlaxation of the lattice mismatch between the (Ga,Mn)As\nepilayer and the GaAs substrate.8{12The modi\fcation\nto the strain distribution can cause strong changes of the\nmagnetic anisotropy for strains as small as 10\u00004. The\nhigh e\u000eciency and practical utility of the lithographic\npattering control of magnetic anisotropy in (Ga,Mn)As,\ndemonstrated in the previous works, have motivated our\nthorough investigation of the phenomenon which is pre-\nsented in this paper. Our study is based on combinedhigh-resolution X-ray di\u000braction and magnetization mea-\nsurements and on macroscopic modeling of the strain re-\nlaxation and microscopic calculations of the correspond-\ning magnetic anisotropies.\nWe investigate two sets of lithographically patterned\n(Ga,Mn)As microbars which di\u000ber in the thickness to\nwidth ratio, Mn doping, and hole concentration. First,\nwe study the structural properties by high resolution\nX-ray di\u000braction of microbars patterned in the thicker,\nhigher Mn doped as-grown (Ga,Mn)As material which\nhas a large growth induced strain. The spatial distribu-\ntion of the lattice relaxation in the stripe cross-section is\ndetermined by comparing the measured intensity maps to\nmaps simulated using the theory of elastic deformations\nand the kinematic scattering theory. The good agree-\nment of the measurement and simulation shows that the\napplied model is quantitatively reliable in predicting the\nlocal lattice relaxation in patterned epilayers subject to\nsmall lattice mismatch. This allows us to infer the much\nweaker lattice relaxation in stripes fabricated in the thin-\nner and lower Mn concentration (Ga,Mn)As by perform-\ning only the elastic theory simulations.\nIn the next step, we measure the magnetic properties of\nour samples by Superconducting Quantum Interference\nDevice (SQUID) and extract the anisotropy coe\u000ecients.\nStronger focus is on stripes fabricated in the thinner,\nannealed (Ga,Mn)As epilayer where the SQUID magne-\ntometry data allow for a reliable extraction of the tem-\nperature dependence of the anisotropy coe\u000ecients and\nfor direct comparison with the microscopic model. We\nassumed a linear superposition of the in-plane uniaxial\nanisotropies and the presence of a single magnetic do-\nmain when analyzing the SQUID magnetometry data.\nWe show that the easy axis can be rotated by 90\u000eby the\nmicropatterning, completely over-writing the underlying\nmaterial anisotropy at all studied temperatures.\nFinally, we calculate the anisotropy coe\u000ecients forarXiv:1007.2766v1  [cond-mat.mtrl-sci]  16 Jul 20102\na range of material parameters and temperatures be-\nlowTC. The lattice relaxations determined form the\nX-ray di\u000braction measurement and from \fnite element\nsimulations are the inputs of the microscopic calcula-\ntions of the magnetocrystalline anisotropy. The micro-\nscopic model we use is based on an envelope function\ndescription of the valence-band holes and a spin repre-\nsentation for their kinetic-exchange interaction with lo-\ncalized moments on Mn2+ions, treated in the mean-\feld\napproximation.10,13{15\nII. SAMPLES\nWe study two sets of patterned (Ga,Mn)As epilayers\ngrown on GaAs substrate. The samples in set A are\ndoped nominally to 5% of Mn, annealed for approxi-\nmately 75 minutes at 180\u000eC, and the epilayer is 25 nm\nthick. The Curie temperature TC\u0019120 K corresponds to\noptimal annealing of the wafer.16The control sample A 0\nwas not patterned. Samples A[110]and A [110]were pat-\nterned into 25 mm2arrays of stripes at an angle \u000b\u0019140\u000e\nand\u000b\u001950\u000e, respectively. Here the angle \u000bis measured\nfrom the [100] crystallographic direction. The uninten-\ntional 5\u000emisalignment from the crystal diagonals caused\nby the microfabrication is accounted for when analyzing\nthe data. The stripes are 750 nm wide, 100 \u0016m long,\nand separated by 450 nm gaps, as measured by Atomic\nForce Microscope (AFM). The fabrication was done by\nelectron beam lithography and wet chemical etching us-\ning a solution of phosphoric acid and hydrogen perox-\nide. The AFM measurements revealed an etch depth\nof\u001960 nm, and cross-sectional Scanning Electron Mi-\ncroscope (SEM) imaging con\frmed that the wet etching\nleads to anisotropic stripe cross-sections, with the A [110]\nstripes being undercut and the A[110]stripes overcut, as\nshown in Fig. 1.\nThe samples in set B are doped nominally to 7%, not\nannealed, the epilayer is 200 nm thick, and the Curie\ntemperature TC\u001985 K. The control sample B 0was not\npatterned. Samples B[110]and B [010]were patterned into\narrays of 1\u0016m wide stripes with 1 \u0016m wide gaps along the\n[110] and [010] crystallographic directions, respectively.\nThe fabrication was done by electron beam lithography\nand dry chemical etching with an etch depth \u0019700 nm\n(B[110]) and \u0019900 nm (B [010]). The sides of the stripes\nare slightly overcut in both cases owing to the symmetric\ndry etching.\nWith respect to our theoretical modelling of the mag-\nnetic anisotropies of our samples, we recall that relating\nthe prediction to the measurement based on the material\nparameters is not straight forward due to the presence\nof unintentional compensating defects in the epilayers.\nMost importantly, a fraction of Mn is incorporated in in-\nterstitial positions. These impurities tend to form pairs\nwith Mn Gaacceptors in as-grown systems with approxi-\nmately zero net moment of the pair, resulting in an e\u000bec-\nFIG. 1: (color online) Cross-sectional scanning electron mi-\ncroscope (SEM) images of the stripes in set A. (a) Image of\nsample A [110]showing both the cleaved face and the top sur-\nface. Although di\u000ecult to discern, the pro\fle is undercut.\nThe curvature is due to the sample stage drifting during the\nexposure of the image. Introduction of coordinates \fxed to\nthe crystallographic axes and dashed coordinates \fxed to the\nstripe geometry: the relaxation direction perpendicular to the\nstripes, the x0axis, is rotated by angle !\u000045\u000ewith respect\nto the [100] crystallographic direction, the xaxis. The angle\n!\u0011\u000b\u000045\u000edescribes the rotation of x0with respect to the\n[110] axis. (b) Image of sample A[110]showing a cut through\nthe stripes and substrate in the x0\u0000z0plane revealing the\novercut sides of the stripes.\ntive local-moment concentration xeff=xs\u0000xi.16Here\nxsandxiare partial concentrations of substitutional and\ninterstitial Mn, respectively. We emphasize that in \fg-\nures presenting calculated data the Mn concentration la-\nbelled asxcorresponds to the density of uncompensated\nlocal moments, i.e., to xeff.\nAnother input parameter of the theoretical modeling is\nthe lattice mismatch which is di\u000berent in set A and B as\nit depends on the partial concentrations of Mn atoms in\nsubstitutional and interstitial positions in the lattice and\nof other unintentional impurities.17The lattice mismatch\nis determined by direct X-ray measurement as detailed\nin the following section.\nFig. 1 introduces the coordinate system \fxed to the\ncrystallographic axes: x-axis along the [100] direction, y-\naxis along the [010] direction, and z-axis along the [001]\ndirection which is the frame of reference for the micro-\nscopic magneto-crystalline anisotropies. The dashed co-\nordinate system is \fxed to the stripe geometry: x0-axis\nlies along the relaxation direction transverse to the stripe,\ny0-axis along the stripe, and z0-axis along the growth di-\nrection coinciding with the z-axis. The dashed coordi-\nnates are the natural reference for the macroscopic lattice\nrelaxation simulations.3\nIII. LATTICE RELAXATION\nThe lattice of thin (Ga,Mn)As \flms grown epitaxially\non GaAs substrates is strained compressively due to a\nlattice mismatch e0= (as\u0000a0)=a0<0 whereasand\na0are the lattice constant of the substrate and of the\nrelaxed free-standing (Ga,Mn)As epilayer, respectively.\nThe narrow stripes allow for anisotropic relaxation of the\ncompressive strain present in the unpatterned epilayer.\nAn expansion of the crystal lattice along the direction\nperpendicular to the bar occurs while the epilayer lattice\nconstant along the bar remains unchanged. Parameters\nsu\u000ecient for determination of the induced strain are the\nlattice mismatch e0and the shape of the stripe, mainly\nthe thickness to width ratio of the stripe. In the regime of\nsmall deformations the components of the induced strain\nare linearly proportional to the lattice mismatch. The\nstrain tensor in the coordinate system \fxed to the stripe\nreads:\ner=e00\n@\u0000\u001a+ 1 0 0\n0 1 0\n0 0c12\nc11(\u001a\u00002)1\nA; (1)\nwhere the lattice relaxation is quanti\fed by \u001a(x0;z0)\nwhich varies over the stripe cross-section, c12andc11\nare the elastic moduli. The strain components in this\nwork are expressed with respect to a relaxed free-standing\n(Ga,Mn)As epilayer. Note that for \u001a= 0 the strain ten-\nsorerdescribes the growth strain of the unpatterned epi-\nlayer. In this section we investigate experimentally and\ntheoretically the geometry of the stripes, the size of the\nlattice mismatch and the spatial dependence of the lat-\ntice relaxation \u001a(x0;z0). The results are used as an input\nof the microscopic modeling of the magnetic anisotropies\nin Sec. V.\nMicrobars in set B have larger thickness to width ra-\ntio than microbars in set A. Therefore the relaxation is\nexpected to be larger in set B. At the same time, the\n(Ga,Mn)As epilayer has larger volume in set B, primarily\ndue to a larger number of interstitial Mn in this higher\ndoped unannealed material. The larger \flm thickness\nand larger growth strain in set B make these materials\nsigni\fcantly more favorable for an accurate X-ray di\u000brac-\ntion analysis of the strain pro\fle in the patterned micro-\nbars.\nThe lattice relaxation in samples B[110]and B [110]was\nmeasured by high-resolution X-ray di\u000braction using the\nsynchrotron source at ESRF Grenoble (beamline ID10B,\nphoton energy 7.95 keV). For a reliable determination\nof both in-plane ( u0\nx) and vertical ( u0\nz) components of\nthe elastic displacement \feld we measured the reciprocal-\nspace distribution of the di\u000bracted intensity around the\nsymmetric 004 and asymmetric 404 reciprocal lattice\npoints. The asymmetric di\u000braction was chosen so that\nthe in-plane component of the corresponding reciprocal\nlattice vector hwas perpendicular to the stripes. The\ndi\u000bracted radiation was measured by a linear X-ray de-\ntector lying in the scattering plane.\nFIG. 2: (Color online) The measured (upper left panel) and\nsimulated (upper right panel) reciprocal-space maps in the\nsymmetric 004 di\u000braction of sample B [010]. In the bottom\nrow, the measured (points) and simulated (lines) intensities\nintegrated along the horizontal (left) and vertical (right) di-\nrections are plotted. In the intensity maps, the color scale is\nlogarithmic.\nFIG. 3: (Color online) The measured (upper left panel) and\nsimulated (upper right panel) reciprocal-space maps in the\nasymmetric 404 di\u000braction of sample B [010]. In the bottom\nrow, the measured (points) and simulated (lines) intensities\nintegrated along the horizontal (left) and vertical (right) di-\nrections are plotted. In the intensity maps, the color scale is\nlogarithmic.\nFigs. 2 and 3 present examples of the measured (upper\nleft panels) and simulated (upper right panels) recipro-\ncal space maps, showing two maxima corresponding to\nthe reciprocal lattice points of the GaAs substrate and\nthe (Ga,Mn)As layer. The bottom panels show the mea-\nsured and simulated integrated intensities for two direc-\ntions in the reciprocal space. Since the lateral stripe pe-\nriod was larger than the coherence width of the primary\nradiation, di\u000berent stripes were irradiated incoherently,4\nso that the lateral intensity satellites stemming from the\nlateral stripe periodicity could not be resolved. The mea-\nsured intensity distribution is therefore proportional to\nthe intensity scattered from a single microbar.\nWe \ftted the measured intensity maps to numerical\nsimulations based on the kinematic scattering theory and\nthe theory of anisotropic elastic medium. We used a\n\fnite-element simulation (implemented in Structural Me-\nchanics Module of Comsol Multiphysics, standard partial\ndi\u000berential equation solver) to obtain the local relaxation\ndistribution \u001a(x0;z0) in the stripes and derived the corre-\nsponding reciprocal space map. The angle of the sides of\nthe stripes and the lattice mismatch e0of the (Ga,Mn)As\nand GaAs lattices were the two \ftting parameters. The\nleft column of Figs. 2 and 3 shows the measured di\u000brac-\ntion maps and projections. The right column shows the\nsimulated results. The lateral and vertical projections\nof the measured and simulated intensity maps as well as\nthe whole maps are used in the \ftting. The coordinates\nq0\nxandq0\nzspan the reciprocal space conjugate to the real\nspace with coordinates x0andz0\fxed to the stripe. They\nare measured with respect to the reciprocal lattice point\n004 and 404.\nThe remarkable agreement of the measured and simu-\nlated di\u000braction maps shows that our model of the lattice\ndeformations is quantitatively relevant in determining\nthe local lattice relaxation \u001a(x0;z0) in the stripes shown\nin Fig. 4, the lattice mismatch between the epilayer and\nthe substrate, e0=\u00000:38\u00060:03% for set B, and the\nstripe geometry, a trapezoidal cross-section of the stripe\nalso shown in Fig. 4. The largest relaxation is observed\nin the corners of the stripes.\nThe slopes of the sides in set B are few degrees larger\nthan angles typically occurring when dry etching is used\nduring the patterning process. Note that the X-ray\ndi\u000braction reveals only the regions with regular lattice\nstructure whereas the dry etching can leave a thin non-\nuniform amorphous coating on the stripes which leads to\nthe unexpected non-rectilinear shape of the stripe cross-\nsection resulting from the \ftting.\nIn the next step, we use our modelling of the lattice re-\nlaxation also for stripes of set A where the X-ray di\u000brac-\ntion would be less accurate due to the small volume of\nthe epilayer, however, the relaxation mechanism should\nbe of the same nature as in set B. Fig. 5 shows the spa-\ntial dependence of the function \u001a(x0;z0) for two di\u000berent\ngeometries relevant to samples in set A. The shape of\nthe stripe cross-section cannot be determined from the\nSEM image of Fig. 1 with nanometer accuracy. This\nuncertainty cannot be neglected in the undercut stripes\nA[110]. Therefore, more geometries (slopes of the sides)\nwere simulated and one representative example is given\nin the upper panel of Fig. 5. On the other hand, the\nprecise shape of the sides does not play such an impor-\ntant role in case of the overcut stripes A[110]shown in the\nlower panel of Fig. 5. In all geometries, the local induced\nstrain is stronger closer to the edges of the stripes.\nThe comparison of the macroscopic simulations and X-\nFIG. 4: (Color online) Finite element calculation of the lat-\ntice relaxation, \u001a(x0;z0), on the cross-section perpendicular\nto the slightly overcut stripes B[110](upper panel) and B [110]\n(lower panel). The cross-section of one stripe and the under-\nlying substrate is plotted. The relaxation \u001a= 1 and\u001a= 0\ncorresponds to a full relaxation of the lattice and to a lattice\nunder a compressive strain of the unpatterned layer, respec-\ntively. Both stripes are close to full relaxation.\nFIG. 5: (Color online) Finite element calculation of the lat-\ntice relaxation, \u001a(x0;z0), on the cross-section perpendicular to\nthe undercut stripes A [110](upper panel) and overcut stripes\nA[110](lower panel). The cross-section of one stripe and the\nunderlying substrate is plotted. The relaxation \u001a= 1 and\n\u001a= 0 corresponds to full relaxation of the lattice and to a\nlattice under a compressive strain of the unpatterned layer,\nrespectively. All stripes show weaker net relaxation than the\nstripes in set B.\nray di\u000braction measurements are done on the level of the\nfull spatial distribution of the relaxation \u001a(x0;z0). The\nmagnetic characteristics, considered in this work in the\nsingle domain approximation, are analyzed based on the\nnet lattice relaxation. Here we take advantage of the di-\nrect proportionality of the magnetocrystalline anisotropy\nto the corresponding strain18,19and calculate the mean\nanisotropy from the spatial average of \u001a(x0;z0) over the\nstripe cross-section. We will denote this average quantity\nby ^\u001ain the rest of the paper.\nThe last step in obtaining the input parameters for the\nmicroscopic modelling is writing the net in-plane compo-5\nnents of the total strain tensor introduced in Eq. (1) in\nthe coordinate system \fxed to the main crystallographic\naxes introduced in Fig. 1:\nexx=e0\u0012\n1\u0000^\u001a\n2\u0000^\u001a\n2sin 2!\u0013\n; (2)\neyy=e0\u0012\n1\u0000^\u001a\n2+^\u001a\n2sin 2!\u0013\n;\nexy=e0^\u001a\n2cos 2!;\nwhere the angle !is measured from the [1 10] axis and\nthe angle!\u000045\u000edescribes the rotation of the relax-\nation direction (the dashed coordinates) with respect to\nthe crystalline coordinate system. Note that the above\nstrain components coincide with those in Eq. (1) when\n!= 45\u000e, i.e., the relaxation direction is aligned with the\n[100] axis. We emphasize that the average relaxation ^ \u001a\ndepends on !. We rotate the elasticity matrix describing\nthe cubic crystal when simulating the lattice relaxation\nalong di\u000berent directions.\nThe strain components exx,eyy, andexyfor the stripes\nin set A are obtained from the macroscopic simulations\nand considering e0\u0019 \u00000:22%.16,20Table I summarizes\ne0[%] ^\u001a\nA[110]\u00000:22\u00060:030:184\u00060:005\nA[110]\u00000:22\u00060:030:24\u00060:05\nB[110]\u00000:38\u00060:030:79\u00060:01\nB[010]\u00000:38\u00060:030:99\u00060:01\nTABLE I: The lattice mismatch e0and the lattice relaxation\n^\u001afor the patterned samples as entering the microscopic cal-\nculations in Sec. V. The value of e0in set B is determined\nfrom the X-ray di\u000braction experiment, whereas e0in set A is\ninferred from the partial Mn concentrations using the analysis\nof Refs. [20,16].\nthe parameters determined in this section.\nIV. EXPERIMENTAL MAGNETIC\nANISOTROPIES\nIn-plane magnetic anisotropies in thin (Ga,Mn)As\n\flms are often analyzed using the lowest order decom-\nposition of the free energy pro\fle into separate terms of\ndistinct symmetry.21{23In this study, we follow this track\nby adopting the following phenomenological formula:\nF( ) =\u0000Kc\n4sin22 +Kusin2 \u0000K\nsin2( \u0000\n):(3)\nThe cubic symmetry of the underlying zinc-blende struc-\nture is described by the \frst term with minima along the\n[100] and [010] directions in case of Kc>0. The sec-\nond term quanti\fed by the coe\u000ecient Kudescribes the\nso called \\intrinsic\" uniaxial anisotropy along the crystal\nFIG. 6: (Color online) Remanent magnetization along the\nmain crystallographic directions for sample A 0(25 nm thick\nunpatterned epilayer).\ndiagonals present in the unpatterned (Ga,Mn)As epilay-\ners. The last term quanti\fed by K\ndescribes the uniax-\nial anisotropy with an extremum at an angle \n induced\nby the relaxation of the lattice mismatch of the doped\nepilayer and the substrate. The angle \n is in general not\nequal to the angle of the corresponding lattice relaxation\n!.19Both angles are measured from the [1 10] axis.\nA. Remanent magnetization\nRemanent magnetization along the main crystallo-\ngraphic directions was measured by SQUID for both sets\nof samples. The obtained values include the magneto-\ncrystalline anisotropies described in the previous para-\ngraph as well as the shape anisotropy which always\nprefers the magnetization alignment with the longest side\nof a rectangular prism such as the stripes.24\nFig. 6 shows that in the control sample A 0the intrinsic\nuniaxial anisotropy dominates over the cubic anisotropy\non a large temperature range and the easy axis along\nthe [1 10] diagonal. The ratio of the remanent magneti-\nzation projections to the [1 10] and [100] directions below\n60 K reveals that the system is almost purely uniaxial.\nThe behavior of the anisotropy components at T >60 K\ncannot be described within the single domain approxima-\ntion. However, the anisotropies of unpatterned samples\nare relevant to our microscopic analysis of measurements\nin the microbars only at the lowest temperatures where\nwe extract intrinsic anisotropy coe\u000ecients and deduce\nthe material parameters as detailed in Sec. V.\nFig. 7 shows that the patterning of the sample A[110]\nstrengthens the uniaxial anisotropy present in the par-\nent wafer. The [1 10] diagonal becomes the easiest of the\ninvestigated directions at all temperatures and the [110]\ndiagonal becomes the hardest axis at all temperatures\nbelowTC.\nFig. 8 shows that in the sample A [110], the two diago-6\nFIG. 7: (Color online) Remanent magnetization along the\nmain crystallographic directions for sample A[110](750 nm\nwide stripes along the [1 10] direction).\nFIG. 8: (Color online) Remanent magnetization along the\nmain crystallographic directions for sample A [110] (750nm\nwide stripes along the [110] direction).\nnals switch roles and in analogy with the previous case\nthe easy axis prefers alignment close to the stripe direc-\ntion, which is the hard axis over most of the temperature\nrange in the parent wafer. This means that a rotation\nof the easy axis by as much as 90\u000eis achieved by the\npost-growth patterning. Note that the di\u000berence of the\nprojection of the remanent magnetization to the [100]\nand [010] directions in the two patterned samples is due\nto a 5\u000emisalignment between the stripes and the crystal\ndiagonals introduced during the fabrication.\nThe samples in set B posses stronger cubic anisotropy.\nFig. 9 shows that in the control sample B 0the intrinsic\nuniaxial anisotropy dominates over the cubic anisotropy\nonly at temperatures above 20 K and the [1 10] diagonal is\neasier than the [110] diagonal at all temperatures below\nTC.\nFig. 10 shows a strengthening of the uniaxial\nanisotropy along the stripe direction in the sample\nB[110], although not large enough to overcome the cubic\nFIG. 9: (Color online) Remanent magnetization along the\nmain crystallographic directions for sample B 0(200 nm thick\nunpatterned epilayer).\nFIG. 10: (Color online) Remanent magnetization along the\nmain crystallographic directions for sample B[110](1\u0016m wide\nstripes along the [1 10] direction).\nanisotropy at the lowest temperatures. The transition\nfrom cubic to uniaxial anisotropy occurs at a lower tem-\nperature than in the control sample. The [110] direction\nis hardened. The main crystal axes [100] and [010] re-\nmain equal due to the more accurate alignment of the\nstripes with the crystal diagonal.\nFig. 11 shows a di\u000berentiation of the [100] and [010]\nprojections in the sample B [010]. The uniaxial anisotropy\nalong the stripe direction now dominates at all tempera-\ntures. The intrinsic anisotropy di\u000berentiating the diago-\nnal directions is less pronounced than in case of B 0as it\nhas to compete also with the induced uniaxial anisotropy.\nWe can conclude that the universal e\u000bect seen in all\npatterned (Ga,Mn)As/GaAs samples is the preference\nof the easy axis to align parallel to the stripe which7\nFIG. 11: (Color online) Remanent magnetization along the\nmain crystallographic directions for sample B [010](1\u0016m wide\nstripes along the [010] direction).\nis the direction in which the growth induced compres-\nsive strain cannot relax, i.e., the direction of the rel-\native lattice contraction in (Ga,Mn)As. This is remi-\nniscent of the magnetocrystalline anisotropy of unpat-\nterned (Ga,Mn)As epilayers which typically yields easy-\naxis oriented also along the direction of contraction, i.e.,\nin-plane for compressively strained (Ga,Mn)As epilayers\nand out-of-plane for (Ga,Mn)As \flms grown under ten-\nsile strain.19We point out that the measured magnitudes\nof magnetic anisotropies in the microbars are an order\nof magnitude larger than the shape anisotropy contribu-\ntion for given concentration of magnetic moments and\nthickness to width ratio. The microfabrication e\u000bects\nin the (Ga,Mn)As stripes are therefore primarily due\nto the spin-orbit coupling induced magnetocrystalline\nanisotropy.\nB. Anisotropy coe\u000ecients\nAfter investigating the reorientations of the easy axis\nwe focus on the magnitude of the individual anisotropy\ncomponents. We measure the hysteresis loops using the\nSQUID magnetometry and \ft the results to the following\nequation:\nF( )=\u00160=\u00001\n4MSHcsin22 +MSHusin2 \u0000 (4)\n\u0000MSH\nsin2( \u0000\n)\u0000MSHcos ( \u0000\u001eH);\nwhereKi=\u00160MSHiwere introduced in Eq. (3), MSis\nthe saturation magnetization, His the external magnetic\n\feld applied at the angle \u001eH, and the last term is the Zee-\nman energy. All angles in Eq. (4) are measured from the\n[110] axis. In case of a general alignment of the induced\nuniaxial strain, the angle \n of the corresponding uniaxial\nanisotropy is an independent \ftting parameter. However,in case of the main crystallographic axes and their small\nsurrounding we can set \n = !, i.e., the anisotropy term\nis aligned with the corresponding uniaxial strain.19An\noverview of the resulting angles \n for the di\u000berent align-\nments of stripes in sets A and B is given in Table II.\nKc[kJ/m3]Ku[kJ/m3]K\n[kJ/m3]\n [deg]\nA0 0.412 0.404 0.0\nA[110]0.412 0.404 0.83 95\nA[110] 0.412 0.404 1.037 5\nB0 2.213 0.381 0.0\nB[110]2.213 0.381 0.935 90\nB[010] 2.213 0.381 0.696 45\nTABLE II: The anisotropy coe\u000ecients obtained by \ftting the\nhysteresis loops at T= 2 K to Eq. (4) and the angular shift\nof the anisotropy term induced by the lattice relaxation as\nintroduced in Eq. (3). Note that the lattice relaxes perpen-\ndicular to the stripe direction. The error of the anisotropy\ncoe\u000ecients is approximately 10 \u000020%, approaching the up-\nper limit in case of the thick inhomogeneous samples in set B.\nWhen determining the anisotropy coe\u000ecients in the\nstripes we use the assumption of linear superposition of\nthe anisotropies present in the unpatterned samples with\nthe anisotropies induced by the patterning and lattice\nrelaxation: the coe\u000ecients KcandKuare obtained \frst\nin the control samples and kept \fxed when \ftting the\nstripes fabricated from the same epilayer. The assump-\ntion is justi\fed on the qualitative level by the rema-\nnent magnetization measurement discussed in the pre-\nvious subsection which revealed the persistence of the\nbulk anisotropies in all patterned samples. Its valid-\nity has been corroborated also by studies of epilayers\nsubject to post-growth piezo straining3and lithographic\npatterning.8We emphasize that our approach is appro-\npriate only when the lattice relaxation direction is very\nclose to the main crystallographic axes or when the angle\n\n is also treated as a \ftting paramater.19\nAnother assumption concerns the magnetization reori-\nentation mechanism determining the shape of the hys-\nteresis loops. In case of a dominant uniaxial anisotropy\nwe \ft the hysteresis loops obtained for external \felds ap-\nplied along the hard axis. In case of a dominant cubic\nanisotropy there is no completely hard direction. We\nnevertheless still consider a single domain model in the\n\ftting.\nAnisotropy coe\u000ecients for all six samples at the low-\nest temperature are summarized in Table II. Recall that\nthese energies include also the contribution of the shape\nanisotropy which amounts to \u00180:1 kJ/m3in samples\nA[110]and A [110]and\u00180:3 kJ/m3in the samples B[110]\nand B [010]with the higher thickness to width ratio. Note\nthat the smaller coe\u000ecient K45leads to the formation of\na strongly uniaxial system as shown in Figs. 11, whereas\nthe larger coe\u000ecient K90cannot overcome the cubic\nanisotropy component, at least at low temperatures as8\nFIG. 12: (Color online) Anisotropy coe\u000ecients as functions of\ntemperature obtained by \ftting the hysteresis loops to Eq.(3)\nfor the three samples of set A. The uniaxial coe\u000ecients K\n(denoted by KAfor set A) due to the growth strain relaxation\nin the patterned samples dominate the total anisotropy.\nshown in Fig. 10. It is because in case of sample B [010],\nthe induced anisotropy is added along the [010] axis\nwhich was already the easy (together with [100]) direc-\ntion in the unpatterned epilayer.\nFor the thinner and more homogeneous epilayers in\nset A we were able to extract the temperature depen-\ndence of the anisotropy coe\u000ecients from the hysteresis\nloops up to T= 60 K as shown in Fig. 12. The uni-\naxial coe\u000ecients due to lattice relaxation dominate the\nanisotropy at all temperatures. At low temperatures the\nrelative size of the induced anisotropies corresponds well\nto the simulated relaxations ^ \u001a: Sample A[110]with over-\ncut sides (weaker relaxation) shows smaller anisotropy\nthan sample A [110] with undercut sides (stronger relax-\nation). The cubic anisotropy remains positive for all\nstudied temperatures T < 60 K which is in good agree-\nment with the remanent magnetization data shown in\nFig. 6. We do not discuss measurements above 60 K for\nwhich, as mentioned above, the single domain model is\nnot applicable.\nV. COMPARISON WITH THEORY\nIn this section we build on macroscopic calculations\nof the lattice relaxation presented in Sec. III, perform\nthe microscopic calculations of the magnetic anisotropy\nenergy, and analyze its correspondence with the experi-\nmental results on the level of anisotropy coe\u000ecients. We\nextract the coe\u000ecients by \ftting the calculated total en-\nergies to Eq. (3) for di\u000berent magnetization directions.\nThe comparison of the experimental and theoretical\nresults involves a number of material parameters. The\nmost important inputs of the microscopic calculations\nare the concentration of the ferromagnetically ordered\nMn local moments ( x) and the hole density ( p). Unfor-\ntunately, these two parameters cannot be accurately con-\nFIG. 13: (Color online) Correspondence of the hole density,\np, and the intrinsic shear strain, ein, to the e\u000bective Mn con-\ncentration, x, based on the agreement of the calculated Kc\nandKuwith the measured values. Samples A and B at zero\ntemperature.\ntrolled during the growth or determined post growth.25\nThe measured saturation magnetization, the conductiv-\nity, and the Curie temperature of the control samples\nprovide only estimates of these input parameters with\nlimited accuracy.\nAnother independent input parameter of the micro-\nscopic simulations is the \\intrinsic\" shear strain which\nhas been used successfully to model19,26the in-plane uni-\naxial anisotropy in the unpatterned samples. We re-\ncall that such modelling for small strains (the typical\nvalues19areein\u001810\u00004) complies well with the assump-\ntion that the \\intrinsic\" uniaxial anisotropy superposes\nlinearly with anisotropy components induced by the lat-\ntice relaxation, as mentioned in the previous section. The\nintrinsic shear strain is added to the o\u000b-diagonal ele-\nment of the total strain tensor written in Eq. (2) giving:\nexy=ein+e0^\u001a\n2cos 2!.\nFig. 13 shows the combinations of x,p, andeinfor\nwhich the calculated intrinsic KuandKcof the con-\ntrol samples A 0and B 0agree with the measured values\nat zero temperature. By this we limit the intervals of\nx,p, andeinvalues considered in the modeling of the\ntemperature dependent anisotropy coe\u000ecients in all mea-\nsured samples. Note, that we have also imposed an upper\nbound toxgiven by the nominal Mn doping in the par-\nticular material and a bound to pensuring a maximum of\none hole per Mn ion and in-plane easy axis (axes). This\nmethod allows for predicting the induced anisotropy co-\ne\u000ecients in the microbars without any adjustable param-\neters in the microscopic model.\nA. Low temperatures\nUsing parameter combinations shown in Fig. 13 we cal-\nculate the induced uniaxial anisotropies in the microbars\nat zero temperature. The left and right vertical axis of9\n3 4 5 6 7\nx [%]00.10.20.30.40.50.60.70.80.91KA [kJ/m3]A[110], e0 = -0.22%, ^ = 0.239\nA[-110], e0 = -0.22%, ^ = 0.184\n012345678910\nKB [kJ/m3]\nB[-110], e0 = -0.38%, ^ = 0.79\nB[010], e0 = -0.38%, ^ = 0.999ρ\nρρ\nρ\nFIG. 14: (Color online) Calculated anisotropy coe\u000ecients due\nto the lattice relaxation in the patterned samples A and B at\nzero temperature for \fxed combinations of xandpshown in\nFig. 13.\nFig. 14 shows the extracted anisotropy coe\u000ecients for\nstripes in sets A and B, respectively. The combinations\nofx,p, andeinare indexed only by xfor simplicity. The\nplotted values can be compared to the measured coe\u000e-\ncients summarized in Table II. The relations K95< K 5\nandK90> K 45hold both in theory and in experiment.\nWe observe a semi-quantitative agreement in samples\nA[110]and A [110]where the measured values are roughly\na factor of 2 larger than the calculated ones. The ra-\ntio of the calculated coe\u000ecients K\nfor samples A[110]\nand A [110],K95=K5, is in excellent agreement with ex-\nperiment (the di\u000berence is only 4%). These agreements\njustify the interpretation of the measured e\u000bects in the\nmicrobars based on the strain-relaxation controlled mag-\nnetocrystalline anisotropy. Note that they also support\nthe assumption of the linear superposition of individual\nuniaxial anisotropies terms used in our analysis.\nFig. 14 shows also extracted anisotropy coe\u000ecients for\nsamples B[110]and B [010]. In this case, the calculated\nratio of coe\u000ecients extracted for the two stripe align-\nments,K90=K45, is approximately 20% larger then the\ncorresponding experimental ratio, i.e., still in a very good\nagreement. We note, however, that the absolute values\nof the measured coe\u000ecients are about a factor of 10 lower\nthan the calculated ones. A possible source of this dis-\ncrepancy is the large value of the experimentally inferred\nKcdue to inaccurate subtraction of the paramagnetic\nand diamagnetic backgrounds from the measured hys-\nteresis curves. In general, we also expect that the theo-\nretical modelling is less reliable in the thicker, as-grown\nsamples B due to stronger disorder and inhomogeneities\nin the material.\nAs a consequence of the almost complete relaxation\nof the lattice mismatch in the thicker samples the cal-\nculated anisotropy coe\u000ecients are larger than the cubic\ncoe\u000ecient at all studied temperatures which is not in\nagreement with the measured coe\u000ecients in set B at low\ntemperature (see Table II).\n0 20 40 60 80 100 120\nT [K]00.10.20.30.4Kc, Ku [kJ/m3]Kc, x = 3%, p = 3.01u\nKu, x = 3%, ein= 0.0204%\nKc, x = 4%, p = 4.45u\nKu, x = 4%, ein= 0.0201%\nKc, x = 5%, p = 5.12u\nKu, x = 5%, ein= 0.0163%FIG. 15: (Color online) Calculated cubic a uniaxial intrinsic\nanisotropy coe\u000ecients present in all samples A as functions\nof temperature for \fxed combinations of x,p, andeinshown\nin Fig. 13.\nB. Temperature dependence\nWe now select six representative combinations of x,\np, andeinfrom the relevant interval shown in Fig. 13,\ncalculate the temperature dependence of all anisotropy\ncoe\u000ecients for each set of parameters, and discuss the\ncomparison with the measured anisotropies. We recall\nthat in our mean-\feld modeling at \fnite temperatures\nthe calculated TCis uniquely determined by xandp.\nNote that for the entire interval of relevant xandpdeter-\nmined from the low-temperature analysis in the previous\nsection, we obtain Curie temperatures which are in agree-\nment with the experimental values in materials A and B\nwithin a factor of 2. This provides an additional support\nfor the overall consistency of our microscopic theoretical\nanalysis of the measured data.\nFig. 15 shows the calculated intrinsic anisotropy coef-\n\fcientsKcandKuof samples in set A for three \fxed\nparameter combinations. At zero temperature the values\ncoincide with data in Fig. 13. The cubic anisotropy com-\nponent is stronger than the intrinsic uniaxial component\nat lowest temperatures but it quickly becomes weaker\nas temperature is increased for all parameter combina-\ntions. This temperature dependence is in agreement with\nthe experimental anisotropies measured below 60 K, as\nshown in Fig. 12. The comparison cannot be extended\nto higher temperatures because, as explained above, the\nexperimental behavior at these temperatures is not cap-\ntured by the single domain model.\nFig. 16 shows the calculated anisotropy coe\u000ecients K\nof samples A[110]and A [110]again for the three \fxed pa-\nrameter combinations. The calculated anisotropy com-\nponents decrease monotonously with increasing temper-\nature in agreement with the measured dependencies pre-\nsented in Fig. 12. The comparison provides additional\nsupport for the interpretation of the experimental data,\nsuggested already by the analysis at low-temperature,\nwhich is based on the strain relaxation induced magne-10\n0 20 40 60 80100 120 140\nT [K]00.10.20.30.40.50.60.7KA [kJ/m3]A[110], x = 3%\nA[-110], x = 3%\nA[110], x = 4%\nA[-110], x = 4%\nA[110], x = 5%\nA[-110], x = 5%\nFIG. 16: (Color online) Calculated anisotropy coe\u000ecients due\nto the lattice relaxation in the patterned samples A as func-\ntions of temperature for \fxed combinations of xandpshown\nin Fig. 13 and for the induced strain given in Fig. 14.\n0 20 40 6080100 120 140 160 180\nT [K]-0.500.511.522.5Kc, Ku [kJ/m3]Kc, x = 5%, p = 3.83u\nKu, x = 5%, ein= 0.0083%\nKc, x = 6%, p = 4.98u\nKu, x = 6%, ein= 0.0082%\nKc, x = 7%, p = 5.83u\nKu, x = 7%, ein = 0.0076%\nFIG. 17: (Color online) Calculated cubic a uniaxial intrinsic\nanisotropy coe\u000ecients present in all samples B as functions\nof temperature for \fxed combinations of x,p, andeinshown\nin Fig. 13.\ntocrystalline ansisotropy e\u000bects.\nFig. 17 shows the calculated intrinsic anisotropy coef-\n\fcientsKcandKuof samples in set B again for three\n\fxed parameter combinations. At zero temperature the\nvalues coincide with data in Fig. 13. The calculated cubic\nanisotropy dominates over the uniaxial anisotropy at low\ntemperatures in agreement with the experiment. The\ncross-over in the theory curves to the dominant uniax-\nial anisotropy occurs at higher temperatures than TC=3\nobserved in experiment (see Fig. 9); at the upper part\nof the relevant interval of Mn concentrations the theo-\nretical crossover occurs at TC=2. We again attribute this\nquantitative discrepancy to inhomogeneities and stronger\ndisorder in the thick as-grown material B.\nFig. 18 shows the anisotropy coe\u000ecients K\nof sam-\nples B[110]and B [010]for the same \fxed parameter com-\nbinations as in Fig. 17. Again, we observe very simi-\nlar dependence of the uniaxial anisotropy coe\u000ecients on\ntemperature as in experiment. The monotonous decrease\nof the coe\u000ecients with growing temperature is in agree-\n0 20 40 6080100 120 140 160 180\nT [K]012345678KB [kJ/m3]B[-110], x = 5%\nB[010], x = 5%\nB[-110], x = 6%\nB[010], x = 6%\nB[-110], x = 7%\nB[010], x = 7%FIG. 18: (Color online) Calculated anisotropy coe\u000ecients due\nto the lattice relaxation in the patterned samples B as func-\ntions of temperature for \fxed combinations of xandpshown\nin Fig. 13 and for the induced strain given in Fig. 14.\nment with the measured remanent magnetization data\nin Figs. 11 and 10. Both induced anisotropy coe\u000ecients\nare predicted to be larger than the cubic coe\u000ecient at all\nstudied temperatures. This complies with the measured\nremanence data of sample B [010]. Sample B[110]shows\nagreement above 20 K. Its behavior at temperatures be-\nlow 20 K, is not captured by the theory data as we have\nalready discussed in the previous subsection.\nVI. SUMMARY\nWe have performed a detailed experimental and theo-\nretical analysis of magnetic anisotropies induced in litho-\ngraphically patterned (Ga,Mn)As/GaAs microbar ar-\nrays. Structural properties of the microbars have been\nstudied by X-ray spectroscopy showing strong strain re-\nlaxation transverse to the bar axis. The relaxation in-\nduced lattice distortion in stripes with thickness to width\nratio as small as \u00180:1 induces additional uniaxial mag-\nnetic anisotropy components which dominate the mag-\nnetic anisotropy of the unpatterned (Ga,Mn)As epilayer,\nas revealed by SQUID magnetization measurements. The\neasy axis can be rotated by the micropatterning by 90\u000e\nat all temperatures below the Curie temperature.\nWe have carried out systematic macroscopic and mi-\ncroscopic modeling of the structural and magnetic char-\nacteristics of the microbars and analyzed in detail the ex-\nperimental results. The agreement of the measured and\nsimulated X-ray di\u000braction maps shows that the applied\nelastic theory model is quantitatively reliable in predict-\ning the local lattice relaxation in patterned epilayers with\nthe growth induced strain. The overall good agreement\nof the microscopically calculated and measured magnetic\nanisotropies conclusively demonstrate that the pattern-\ning induced anisotropies are of the magnetocrystalline,\nspin-orbit coupling origin.11\nAcknowledgments\nWe acknowledge fruitful discussions with A. W.\nRushforth and K. V\u0013 yborn\u0013 y. The work was funded\nthrough Pr\u001amium Academi\u001a and contracts numberAV0Z10100521, LC510, KAN400100652, FON/06/E002\nof GA \u0014CR, of the Czech republic, and by the NAMASTE\n(FP7 grant No. 214499) and SemiSpinNet projects (FP7\ngrant No. 215368).\n1K. W. Edmonds, P. Boguslawski, K. Y. Wang, R. P. Cam-\npion, N. R. S. Farley, B. L. Gallagher, C. T. Foxon, M. Saw-\nicki, T. Dietl, M. B. 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B 71 ,\n121302 (2005), arXiv:cond-mat/0410544."    },    {        "title": "1008.2142v1.Rotational_Doppler_Effect_in_Magnetic_Resonance.pdf",        "content": "arXiv:1008.2142v1  [cond-mat.other]  12 Aug 2010Rotational Doppler Effect in Magnetic Resonance\nS. Lend´ ınez1,2, E. M. Chudnovsky1,3, and J. Tejada1,2\n1Departament de F´ ısica Fonamental, Facultat de F´ ısica,\nUniversitat de Barcelona, Avinguda Diagonal 645, 08028 Bar celona, Spain\n2Institut de Nanoci` encia i Nanotecnologia IN2UB,\nUniversitat de Barcelona, c. Mart´ ı i Franqu` es 1, 08028 Bar celona, Spain\n3Physics Department, Lehman College, The City University of New York,\n250 Bedford Park Boulevard West, Bronx, NY 10468-1589, U.S. A.\n(Dated: November 6, 2018)\nWe compute the shift in the frequency of the spin resonance in a solid that rotates in the field of\na circularly polarized electromagnetic wave. Electron spi n resonance, nuclear magnetic resonance,\nand ferromagnetic resonance are considered. We show that co ntrary to the case of the rotating LC\ncircuit, the shift in the frequency of the spin resonance has strong dependence on the symmetry\nof the receiver. The shift due to rotation occurs only when ro tational symmetry is broken by the\nanisotropyofthegyromagnetic tensor, bytheshapeofthebo dy,orbymagnetocrystalline anisotropy.\nGeneral expressions for the resonance frequency and power a bsorption are derived and implications\nfor experiment are discussed.\nPACS numbers: 76.30.-v, 76.50.+g, 76.60.-k, 32.70.Jz\nI. INTRODUCTION\nThe term Rotational Doppler Effect (RDE) is used to\ndescribe a frequency shift encountered by a receiver of\nelectromagnetic radiation when either the receiver or the\nsource of the radiation are rotating. The effect is illus-\ntrated in Fig. 1. The frequency of the wave, ω= 2πf,\nmeasured at a given point in space, corresponds to the\nangular velocity of the rotation of the electric (magnetic)\nfield due to the wave. If the receiver is rotating mechan-\nically at an angular velocity Ω about the axis parallel\nto the wave vector k, than the frequency of the wave\nperceived by the receiver equals\nω′=ω±Ω. (1)\nThe sign, plus or minus, depends on the helicity of the\nwave and the direction of the rotation of the receiver.\nThe RDE is less commonly known than the conven-\ntional Doppler effect. One reason is that it is more\ndifficult to observe. M¨ ossbauer technique provides the\nmostsensitivemethod forthe studyofthefrequencyshift\ndue to the conventional Doppler effect, δω= (v/c)ωfor\nv≪c. The limiting velocity has been a fraction of a\nmillimeter per second and is due to the finite very small\nlinewidth of gamma radiation, δω/ω∼10−13−10−12.\nSuch a small linewidth has even permitted observation of\nthetransverseDopplereffect1,2byperformingM¨ ossbauer\nexperiment on a rotating platform. This effect, not to be\nconfused with the RDE, consists of the frequency shift\nδω/ω=−v2/(2c2) due to the relativistic time dilation\nfor a receiver moving tangentially with respect to the\nsource of the radiation. It is easy to see, however, that\nthe frequency shift as little as Ω /ω∼10−13−10−12due\nthe RDE would require angular velocity of the emitter\nor the receiver in the M¨ ossbauer experiment on the order\nof a few MHz or even a few tens of MHz. The latter is\nstill one-two orders of magnitude greater than the angu-FIG. 1: Color online: Rotational Doppler effect. The fre-\nquencyωof the circularly polarized electromagnetic wave\n(ω,k) is the angular velocity of the rotation of the electric\n(magnetic) field due to the wave at a given point in space.\nThe rotation of the receiver at an angular velocity Ω, de-\npending on the direction of the rotation and the helicity of\nthe wave, adds or subtracts Ω to the frequency of the wave ω,\nrendering ω′=ω±Ω in the coordinate frame of the receiver.\nlar velocities of high-speed rotors used for magic angle\nspinning in NMR applications.\nThe RDE frequency shift caused by a rotating plate\ninserted into a beam of circularly polarized light was re-\nported in Refs. 3–7. The RDE was predicted for rotat-\ning light beams8and subsequently observed using mil-\nlimeter waves9as well as in the optical range10(see Ref.\n11 for review). In solid state experiments the RDE has\nproved surprisingly elusive. Frequencies of the ferromag-\nnetic resonance (FMR) are typically in the GHz range or\nhigher, which is far above achievable angular velocities of\nmechanical rotation of macroscopic magnets. However,\nsmall magnetic particles in beams12or in nanopores132\nmay rotate very fast. Eq. (1) was recently applied to the\nanalysis of the observed anomalies in the FMR data on\nrotatingnanoparticles13. TheRDEmaybeespeciallyim-\nportant for the NMR technology that uses rapidly spin-\nning samples. Frequency shifts of the quadrupole line in\nthe nuclear magnetic resonance (NMR) experiment with\na rotating sample were reported in Ref. 14 and analyzed\nin terms of Berry phase15. It was never fully explained,\nhowever, why such shifts do not persist in the NMR ex-\nperiments in which the angular velocity of the magic-\nangle-spinning rotor with the sample often exceeds the\nlinewidth by an order of magnitude. Some hint to an-\nswering this question can be found in Ref. 16 that stud-\nied the effect of the rotation on radiation at the atomic\nlevel. The authors of this work correctly argued that\nthe RDE can only be seen in the radiation of atoms and\nmolecules placed in the environment that destroys rota-\ntional symmetry.\nSituation depicted in Fig. 1 rather obviously leads to\nthe frequency shift by Ω when the emitter and the re-\nceiver are based upon LC circuits. This has been tested\nby the GPS for the case of a receiving antenna mak-\ning as little as 8 revolutions per second as compared to\nthe carrier frequency of the electromagnetic waves in the\nGHz range17. Eq. (1) has been also applied to the ex-\nplanation of the frequency shift encountered by NASA\nin the communications with Pioneer spacecrafts18. One\nessential difference between conventional and rotational\nDoppler effects is that the first refers to the inertial sys-\ntems while the second occurs in the non-inertial systems.\nThis prompted works that considered RDE in the con-\ntext of nonlocal quantum mechanics in the accelerated\nframe of reference19. Relativity (or Galilean invariance\nforv≪c) makes the conventional Doppler effect quite\nuniversal. As we shall see below, such a universality\nshould not be expected for the RDE. Indeed, the argu-\nment behind the RDE is based upon perception of a cir-\ncularly polarized wave by a rotating observer. Through\nthe Larmortheorem20the mechanicalrotationofthe sys-\ntem of charges is equivalent to the magnetic field. Con-\nsequently, when making the argument, one has to check\nwhether the resonant frequency of the receiver is affected\nby the magnetic field. Resonant frequencies of LC cir-\ncuits are known to be insensitive to the magnetic fields,\nthus making the argument rather solid. On the contrary,\nthe frequency of the receiver based upon magnetic reso-\nnance would be sensitive to the fictitious magnetic field\ndue to rotation, thus making the argument incomplete.\nIn this paper we develop a rigorous theory of the RDE\nfor magnetic resonance. We show that the frequency\nshift due to rotation is always different from Ω. Bro-\nken rotational symmetry is required for the shift to have\na non-zero value, in which case the magnetic resonance\nsplits into two lines separated by 2Ω. For the electron\nspin resonance (ESR) violation of the rotational sym-\nmetry would naturally arise from the anisotropy of the\ngyromagnetic tensor. In a solid state NMR experiment\nwith a rotating sample, violation of symmetry would bemore common in the presence of the magnetic order that\nprovides anisotropy of the hyperfine interaction. For a\nferromagnetic resonance (FMR) the asymmetry comes\nfrom the shape of the sample and from magnetocrys-\ntalline anisotropy. The paper is organized as follows.\nThe physics of spin-rotation coupling is reviewed in Sec-\ntion II. Frequency shift of the ESR in a rotating crys-\ntal with anisotropic gyromagnetic tensor is computed in\nSection III. The effect of rotation on the NMR spectra\nis discussed in Section IV. FMR in a rotating sample is\nstudied in Section V. Power absorption by the rotating\nmagnet is considered in Section VI. Section VII contains\nsome suggestions for experiment and discussion of possi-\nble application of the RDE in solid state physics.\nII. SPIN-ROTATION COUPLING\nIn classical mechanics the Hamiltonian of the system\nin a rotating coordinate frame is given by21\nH′=H−L·Ω. (2)\nHereHis the Hamiltonian at Ω = 0 and Lis the me-\nchanical angular momentum of the system. For a system\nof charges one can write\nL=M\nγ, (3)\nwhereMis the magnetic moment and γis the gyro-\nmagnetic ratio. Eq. (2) then becomes equivalent to the\nHamiltonian,\nH′=H−M·B, (4)\nin the fictitious magnetic field,\nB=Ω\nγ, (5)\nwhich is the statement of the Larmor theorem20.\nNeither classical mechanics nor classical field theory\ndealswith the conceptofaspin. The questionthen arises\nwhether Eq. (2) should contain spin Salongside with the\norbital angular momentum L. Eq. (4) hints that since\nthe magnetic moment can be of spin origin this should\nbe the case. Also it is known from relativistic physics\nthat the generator of rotations is\nJ=L+S. (6)\nIt should be, therefore, naturally expected that in the\npresence of a spin Eq. (2) should be generalized as\nH′=H−(L+S)·Ω. (7)\nIn quantum theorythis relationcanbe rigorouslyderived\nin the following way. Rotation by an angle φtransforms\nthe Hamiltonian of an isolated system into22\nˆH′= exp/bracketleftbiggi\n¯h(L+S)·φ/bracketrightbigg\nˆHexp/bracketleftbigg\n−i\n¯h(L+S)·φ/bracketrightbigg\n.(8)3\nTo the first order on a small rotation φone obtains\nˆH′=ˆH−i\n¯h(L+S)·[ˆH,φ], (9)\nwhere we have taken into account that for an isolated\nsystemJis conserved, that is L+Scommutes with ˆH.\nThis equation becomes Eq. (7) if one takes into account\nthe quantum-mechanical relation\nΩ=dφ\ndt=i\n¯h[ˆH,φ] (10)\nandreplacesoperator Ωbyitsclassicalexpectationvalue.\nFor an electron Eq. (7) can be also formally derived as a\nnon-relativistic limit of the Dirac equation written in the\nmetric of the rotating coordinate frame23. The answer\nfor the corresponding Schr¨ odinger equation reads\ni¯h∂Ψ\n∂t=ˆH′Ψ,ˆH′=ˆp2\n2m−/parenleftbigg\nr׈p+1\n2¯hˆσ/parenrightbigg\n·Ω,(11)\nwhererandp=−i¯h∇are the radius-vector and the\nlinear momentum of the electron, respectively, and σx,y,z\nare Pauli matrices.\nThere has been some confusion in literature regarding\nthe term −S·Ωin the Hamiltonian of the body stud-\nied in the coordinate frame that rotates together with\nthe body24–26. To elucidate the physical meaning of this\nterm, let us consider the resulting equation of motion for\na classical spin-vector27\ndS\ndt=−S×δH′\nδS. (12)\nIfHdoes not depend on spin, then the spin cannot be\naffected in any way by the rotation of the body. In this\ncaseδH′/δS=−Ωand Eq. (12) simply describes the\nprecession of SaboutΩ:\ndS\ndt=S×Ω. (13)\nIt shows how a constant vector S(or any other vector\nto this matter) is viewed by an observer rotating at an\nangular velocity Ω. This has nothing to do with the\nspin-orbit or any other interaction. Such interactions\nshould be accounted for in the ˆHpart of the Hamilto-\nnianˆH′. The effect of rotations on various magnetic\nresonances is considered in the next sections.\nIII. FREQUENCY SHIFT OF THE ELECTRON\nSPIN RESONANCE DUE TO ROTATION\nIn this Section we consider an electron in a rotating\ncrystal or in a rotating quantum dot characterizedby the\nanisotropic gyromagnetic tensor, gij. The effect of local\nrotations due to transverse phonons on the width of the\nESR has been studied in Ref. 28. Here we are interestedin the effect of the global rotation on the ESR frequency.\nTo deal with the stationary states we shall assume that\nthe axis of rotation Ωis parallel to the applied magnetic\nfieldBand will compute the energy levels of the electron\nas measured by the observer rotating together with the\nsystem. In the rotating frame the spin Hamiltonian of\nthe electron is\nˆH′=1\n2µBgijσiBj−1\n2¯hσ·Ω. (14)\nPositive sign of the first (Zeeman) term is due to the\nnegativegyromagneticratio γforthe electron( µB= ¯h|γ|\nbeing the Bohr magneton).\nThe geometryofthe problem is illustrated in Fig. 2. In\nthe rotating frame the solid matrix containing the elec-\ntron is stationary. It is convenient to choose the coordi-\nnate axes of that matrix along the principal axes of the\ntensorgij. Thengijis diagonal,\ngij=giδij, (15)\nrepresented by three numbers, gx,gy, andgzthat can be\ndirectly measured when the system is at rest. Eq. (14)\nthen becomes\nˆH′=1\n2[(µBgxBx−¯hΩx)σx+(µBgyBy−¯hΩy)σy\n+ (µBgzBz−¯hΩz)σz]. (16)\nDiagonalization of this Hamiltonian with the account of\nthe fact that Ωwas chosen parallel to Bgives the follow-\ning energy levels of ˆH′:\nE±=±1\n2µBB\n/summationdisplay\ni=x,y,z/parenleftbigg\ngi−¯hΩ\nµBB/parenrightbigg2\nn2\ni\n1/2\n(17)\nHerenis the unit vector in the direction of the axis of\nrotation,\nn=Ω\nΩ=B\nB. (18)\nIn practice, the angular velocity of the mechanical ro-\ntationwill alwaysbe sufficiently smalltoprovidethe con-\ndition ¯hΩ≪µBB. Contribution of the rotation to the\nESR frequency in the rotating frame,\n¯hω′\nESR=E+−E−, (19)\nwill, therefore, be small compared to the ESR frequency\n¯hωESR=µBB(g2\nxn2\nx+g2\nyn2\ny+g2\nzn2\nz)1/2(20)\nunperturbed by rotation. Expanding Eq. (17) to the first\norder in Ω one obtains\nω′\nESR=ωESR−κΩ, (21)\nκ=gxn2\nx+gyn2\ny+gzn2\nz/radicalBig\ng2xn2x+g2yn2y+g2zn2z. (22)4\nFIG. 2: Color online: Spin in the magnetic field parallel to\nthe rotation axis of the crystal. The rotating coordinate ax es\nx,y,zare chosen along the principal axes of the gyromagnetic\ntensor.\nHere Ω can be positive or negative depending on the di-\nrection of rotation.\nFew observations are in order. Firstly, according to\nEq. (22), the frequency shift for the observer rotating to-\ngether with the sample containing the electron is never\nzero. Secondly, when the rotation is about one of the\nprincipal axes of the gyromagnetic tensor, Eq. (22) gives\nκ= 1, so that the frequency shift for the rotating ob-\nserver is exactly Ω. The ESR occurs when the frequency\nω′of the circularly polarized electromagnetic wave per-\nceived by the rotating observer and given by Eq. (1) co-\nincides with ω′\nESR. If the rotation is about one of the\nprincipal axes of gij, thenκ= 1 and the angular velocity\nΩ cancels exactly from the equation ω′=ω′\nESRfor the\npolarization of the wave that corresponds to ω′=ω−Ω,\nthus, resulting in no RDE frequency shift for an experi-\nmentalist workingin the laboratoryframe. For the oppo-\nsitepolarizationofthewave,correspondingto ω′=ω+Ω,\nthe shift in the rotationallyinvariantcaseformallyequals\n2Ω. However, such photons would have their spin pro-\njection in the direction opposite to the one necessary to\nproduce the spin transition. They can be absorbed only\nwhen the rotational symmetry is broken so that the elec-\ntron spin in the direction of the wave vector is no longer\na good quantum number (see Section VI).\nIV. FREQUENCY SHIFT OF THE NUCLEAR\nMAGNETIC RESONANCE DUE TO ROTATION\nLet us consider a nuclear spin Iin the magnetic field\nparallel to the axis of rotation of the sample. It is clear\nfrom the previous section that the mechanical rotation\ncombined with the rotationally invariant Zeeman inter-\naction of the nuclear magnetic moment with the field,\nˆH′=−γngnI·B−I·Ω, (23)(withγn>0 andgnbeing nuclear gyromagnetic ratio\nand gyromagnetic factor, respectively) are not sufficient\nto produce the RDE. Isotropic hyperfine interaction with\nanatomicspin Softheform −AI·Swouldnotchangethis\neither. However, an anisotropic hyperfine interaction,\nˆHhf=−AijIiSj, (24)\nin principle, can do the job. If there is a ferromagnetic\norder in the solid, then Sdevelops a non-zero average,\n/angbracketleftS/angbracketright. Replacing Sjin Eq. (24) with /angbracketleftSj/angbracketrightand adding the\nhyperfine interaction to Eq. (23), one obtains\nˆH′=−γngnI·B−AijIi/angbracketleftSj/angbracketright−I·Ω.(25)\nTo work with the stationary energy states in the ro-\ntating frame, we shall assume that all three vectors B,\n/angbracketleftS/angbracketright, andΩare parallel to each other. Let us study the\ncase ofI= 1/2. Choosing the coordinate axes along the\nprincipal axes of tensor Aij=Aiδij, it is easy to see that\nEq. (25) is equivalent to the Zeeman Hamiltonian,\nˆH′=−1\n2µn/bracketleftbig\ngeff\nxσxBx+geff\nyσyBy+geff\nzσzBz/bracketrightbig\n(26)\nwith an effective gyromagnetic tensor whose principal\nvalues are given by ( i=x,y,z)\ngeff\ni=gn+Bhf\ni\nB+¯hΩ\nµnB, (27)\nwhere we have introduced the nuclear magneton, µn=\n¯hγn, and the hyperfine field, Bhf, with components\nBhf\ni=¯hAi|/angbracketleftS/angbracketright|\nµn. (28)\nThe energy levels of the Hamiltonian (26) are\nE±=±1\n2µnB\n/summationdisplay\ni=x,y,z/parenleftBig\ngeff\ni/parenrightBig2\nn2\ni\n1/2\n,(29)\nwheren=B/B.\nLet us consider the case of small Ω. Making the series\nexpansion of Eq. (29) one obtains to the first order on Ω\nω′\nNMR=E+−E−\n¯h=ωNMR+κΩ (30)\nwithκgiven by\nκ=/summationtext\ni=x,y,z/parenleftBig\ngn+Bhf\ni/B/parenrightBig\nn2\ni/radicalbigg\n/summationtext\ni=x,y,z/parenleftBig\ngn+Bhf\ni/B/parenrightBig2\nn2\ni.(31)\nIn the case of the isotropic hyperfine interaction, Bhf\nx=\nBhf\ny=Bhf\nz(that is, Ax=Ay=Az), Eq. (31) gives\nκ= 1. Same situation occurs when the direction of the\nfield and the axis of rotation coincide with one of the5\nprincipal axes of the tensor of hyperfine interactions. For\narbitrary rotations Eq. (31) gives κ→1 whenB≫Bhf,\nmaking the frequency shift defined by ω′=ω′\nNMRneg-\nligible for the polarization ( ω′=ω+Ω) that is predom-\ninantly absorbed due to the selection rule. Is is likely,\ntherefore, that a significant RDE in the NMR can be\nobserved only in magnetically ordered materials, in the\nfield comparable or less than the hyperfine field, for rota-\ntions about axes that do not coincide with the symmetry\naxes of the crystal. If these conditions are satisfied, and\nthe width of the resonance is not very large compared to\nΩ, the NMR produced by linearly polarized waves would\nsplit into two lines of uneven intensity separated by 2Ω.\nIn fact, the existing experimental techniques permit ob-\nservation of this effect (see Section VII).\nV. FREQUENCY SHIFT OF THE\nFERROMAGNETIC RESONANCE DUE TO\nROTATION\nWe now turn to the rotating ferromagnets. We begin\nwith a simplest model of ferromagnetic resonance stud-\nied by Kittel29. In this model one neglects the effects of\nmagnetocrystalline anisotropy and considers a uniformly\nmagnetized ferromagnetic ellipsoid in the external mag-\nnetic field B=µ0H(withµ0being the magnetic perme-\nability of vacuum). The energy density of such a ferro-\nmagnet is determined by its Zeeman interaction with the\nexternal field and by magnetic dipole-dipole interactions\ninside the ferromagnet:\nH=µ0/bracketleftbigg\n−M·H+1\n2NijMiMj/bracketrightbigg\n.(32)\nHereMis the magnetization and Nijis tensor of demag-\nnetizing coefficients. The principal axes of Nijcoincide\nwith the axes of the ellipsoid. Choosing the coordinate\naxes along the principal axes and taking into account\nthat for a ferromagnet\nM2=M2\nx+M2\ny+M2\nz=M2\n0 (33)\nis a constant, one can rewrite Eq. (32) as\nH=−µ0/bracketleftbigg\nM·H+1\n2(Nx−Nz)M2\nx+1\n2(Ny−Nz)M2\ny)/bracketrightbigg\n,\n(34)\nwhere we have omitted unessential constant. For, e.g.,\nan infinite circular cylinder Nx=Ny= 1/2,Nz= 0. In\ngeneral, for an ellipsoid elongated along the Z-axis one\nhasNx−Nz>0,Ny−Nz>0, so that in the absence\nof the field the minimum of Eq. (34) corresponds to M\nin theZ-direction. This will still be true in the external\nfield if the latter is applied in the Z-direction, which is\nthe case we consider here. Note that a finite field is al-\nways needed to prevent the magnet from breaking into\nmagnetic domains.The FMR frequency, ωFMR, can be obtained from ei-\nther classical or quantum mechanical treatment27. Clas-\nsically, itisthe frequencyoftheprecessionof Maboutits\nequilibrium direction. To find ωFMRone should linearize\nthe equation,\ndM\ndt=γM×B(eff),B(eff)=−δH\nδM,(35)\naroundM=M0ez(γ <0 being the gyromagneticratio).\nThe answer reads29\nωFMR=√ωxωy, (36)\nwhere\nωx=|γ|[B+(Nx−Nz)µ0M0]\nωy=|γ|[B+(Ny−Nz)µ0M0].(37)\nTo study the RDE we should now solve the same prob-\nlem in the coordinate frame rotating about the Z-axis at\nan angular velocity Ω. In the presence of rotation the\nHamiltonian becomes\nH′=H−M\nγ·Ω. (38)\nIt is easy to see that for Ω= Ωezthis effectively adds\nΩ/γto the external field. Consequently, the FMR fre-\nquency in the rotating frame becomes\nω′\nFMR=/radicalBig\nω′xω′y (39)\nwith\nω′\nx=|γ|/bracketleftbigg\nB+Ω\nγ+(Nx−Nz)µ0M0/bracketrightbigg\nω′\ny=|γ|/bracketleftbigg\nB+Ω\nγ+(Ny−Nz)µ0M0/bracketrightbigg\n.(40)\nOur immediate observation is that for a symmetric el-\nlipsoid (Nx=Ny)\nω′\nFMR=ωFMR−Ω, (41)\nso that the RDE frequency shift determined by the equa-\ntionω′=ω−Ω =ω′\nFMRis exactly zero. For an asym-\nmetric ellipsoid ( Nx/negationslash=Ny), expanding Eq. (39) into a\nseries on Ω one obtains to the first order\nω′\nFMR=ωFMR−κΩ, (42)\nwith\nκ=1\n2/parenleftbigg/radicalbiggωx\nωy+/radicalbiggωy\nωx/parenrightbigg\n. (43)\nIt is easy to see that κ≥1. At large fields, B≫µ0M0,\nequations (37) and (43) give κ→1, that is, no frequency\nshift due to the RDE. Sizable frequency shift of the FMR\nobserved in the laboratory frame due to the rotation of6\nFIG.3: Color online: Geometry oftheFMRstudiedinthepa-\nper. Ferromagnet uniformly magnetized by a static magnetic\nfield,B, is rotating at an angular velocity Ωin the radia-\ntion field of circularly polarized photons of wave vector kand\nspins. (Due to the negative gyromagnetic ratio, the equilib-\nrium spin of the magnet, S0, is antiparallel to its equilibrium\nmagnetic moment M0.)\nthe sample should occur only at Bnot significantly ex-\nceedingµ0M0and only in a sample lacking the rotational\nsymmetry.\nOne can easily generalize the above approach to\ntake into account any type of the magnetocrystalline\nanisotropy. The formulas look especially simple in the\ncase of the second-order anisotropy. Such anisotropy\nadds the term\n−1\n2µ0βijMiMj (44)\nto the Hamiltonian of the magnet, with βijbeing some\ndimensionless symmetric tensor. Consider, e.g., an or-\nthorhombic crystal whose axes ( a,b,c) coincide with the\naxes of the ellipsoid and whose easy magnetization axis,\nc, is parallel to the Z-direction. In this case all the above\nformulas remain valid if one replaces the demagnetizing\nfactors with\nN′\ni=Ni−βi, i=x,y,z, (45)\nwhereβx,βy, andβzare the principal values of βij. Due\nto the orthorhombic anisotropy ( a/negationslash=b→βx/negationslash=βy) the\nRDE may now occur even in a sample of the rotationally\ninvariant shape ( Nx=Ny).\nVI. POWER ABSORPTION BY A ROTATING\nMAGNET\nFor non-relativistic rotations the radiation power ab-\nsorbed by the magnet should be the same in the labora-\ntory frame and in the rotating frame. Calculation in the\nrotatingframe is easier. We shall assume that the dimen-\nsions ofthe sample aresmall comparedto the wavelength\nof the radiation, so that the field of the wave at the posi-\ntion of the ferromagnet is nearly uniform. The geometry\nstudied below is illustrated in Fig. 3. Within the model\nof Eq. (38), the rotating magnet placed in the field of\na circularly polarized wave feels the oscillating magnetic\nfield that can be represented by a complex function\nh(t) =h0e±iω′t, ω′=ω∓Ω (46)giving the components of the field as\nhx= Re(h), hy= Im(h). (47)\nHereh0is the complex amplitude of the wave, ±sign\nin Eq. (46) determines the helicity of the wave, while\nthe sign of Ω determines the direction of rotation of the\nmagnet. Due to the wave the magnetization acquires\na small ac-component m(t) (whose real and imaginary\nparts represent mxandmy, respectively),\nm(t) = ˆχ(ω)h(t), (48)\nwhere ˆχis the susceptibility tensor. The absorbed power\nis given by27\nP=±iµ0ω′h∗\n0(ˆχ−ˆχ†)h0. (49)\nThe problem has, therefore, reduced to the computation\nof the susceptibility in the rotating frame. The latter can\nbe done by solving the Landau-Lifshitz equation,\ndM\ndt=γM×B(eff)−η\nM0|γ|M×/bracketleftBig\nM×B(eff)/bracketrightBig\n,(50)\nin the rotating frame, that is, with B(eff)=−δH′/δM\nand\nH′=H−M\nγ·Ω−M·h. (51)\nThe parameter ηin Eq. (50) is a dimensionless damping\ncoefficient that is responsible for the width of the FMR\nin the absence of inhomogeneous broadening.\nSubstituting M=M0ez+minto Eq. (50) and solving\nfor ˆχone obtains for the power\nP±=1\n2η|γ|M0µ2\n0|h0|2f±(ω′), (52)\nwhere\nf±=ω′2[2(ω′2−ω′2\nFMR)±2ω′(ω′\nx+ω′\ny)+(ω′\nx+ω′\ny)2]\n(ω′2−ω′2\nFMR)2+η2ω′2(ω′x+ω′y)2.\n(53)\nNotice that when there is a full rotational symmetry,\nω′\nx=ω′\ny=ω′\nFMR, the absorbed power at the resonance\nis non-zero only for one polarizationof the wavethat cor-\nresponds to the upper sign in Eqs. (46) and (53). This\nis a consequence of the selection rule due to conserva-\ntion of the Z-component of the total angular momentum\n(absorbed photon + excited magnet).\nLet us now consider a rotating ferromagnet in the ra-\ndiation field of a linearly polarized electromagnetic wave.\nIn the rotating frame the complex magnetic field of such\na wave is\nh(t) =h0\n2/bracketleftBig\nei(ω−Ω)t+e−i(ω+Ω)t/bracketrightBig\n=h0e−iΩtcos(ωt)\n(54)7\n/SolidCircle\n– –Κ/Equal1—Κ/Equal1.11\n/CapOmega/OverTilde/Equal0.01\nB/OverTilde/Equal0\nΩ/OverTilde\nFMR/Equal0.3162\nΗ/Equal0.003\n0.2900.2950.3000.3050.3100.3150.3200100200300400500600\nΩ/OverTildePower/LParen1a.u/RParen1\nFIG. 4: Color online: Absorption of power of linearly po-\nlarized electromagnetic radiation by a rotating magnet. Fr e-\nquencies are given in the units of γµ0M0. As the rotational\nsymmetry is violated the FMR becomes shifted and the sec-\nond FMR line emerges separated by 2Ω from the first line.\nRepeating the above calculation, one obtains for the\npower averaged over the period of rotation\nP=1\n8η|γ|M0µ2\n0|h0|2[f+(ω−Ω)+f−(ω+Ω)].(55)\nWhen the rotational symmetry of the magnet is broken,\nωx/negationslash=ωy,κ >1, the absorption has two maxima of un-\neven height at\nω=ωFMR−(κ∓1)Ω. (56)\nAs the rotational symmetry is gradually restored, ωx→\nωy,κ→1, the rotational shift in the position of the\nmain maximum disappears. In that limit the shift in the\nposition of a smaller maximum approaches 2Ω while the\nheight of that maximum goes to zero, see Fig. 4.\nVII. DISCUSSION\nWe have computed the frequency shift of the magnetic\nresonance due to rotation of the sample. The effect of\nrotation on the ESR, NMR, and FMR has been studied.\nWe found that it is, generally, quite different from the ro-\ntational Doppler effect reported in other systems11. The\ndifferences stem from the observation that the spin of an\nelectron or an atom would be insensitive to the rotation\nof the body as whole if not for the relativistic spin-orbit\ncoupling. Even with account of spin-orbit interactions\nthe spin would not simply follow the rotation of the body\nbut would exhibit more complex behavior described by\nthe dynamics of the angular momentum. Everyone who\nwatched the behavior of a gyroscope in a rotating frame\nwould easily appreciate this fact.\nWe found the following common features of the mag-\nnetic resonance in a rotating sample.\n•If the spin Hamiltonian is invariant with respect to\nthe rotation, then the rotation of the body has noeffect on the frequency of the resonant absorption\nof a circularly polarized electromagnetic wave.\n•As the rotationalinvariance is violated, the absorp-\ntion line shifts. The shift is different from the an-\ngular velocity of rotation, Ω. It depends on the de-\ngree of violation of the rotational symmetry. The\nfrequency shift goes to zero when the symmetry is\nrestored.\n•In the case of a linearly polarized radiation a sec-\nond resonance line emerges, separated by 2Ω from\nthe first line. The intensity of that line depends\non the degree of violation of rotational symmetry.\nIt disappears when the rotational symmetry is re-\nstored.\nESR and FMR measurements are usually performed\nin the GHz range, with the width of the resonance\nbeing sometimes as low as a few MHz. Currently\navailable small mechanical rotors can rotate as fast as\n100kHz, which, nevertheless, is still low compared to the\nlinewidths of ESR and FMR. Note, however, that the po-\nsition of the ESR or FMR maximum can be determined\nwith an accuracy of a few hundred kHz. It is then not\nout of question that under appropriate conditions the\nRDE frequency shift and the splitting of the resonance\ncan be observed in high precision ESR and FMR exper-\niments even when the rotation frequency is significantly\nlower than the linewidth. Since anisotropy of the sample\nis needed to provide rotational asymmetry, the measure-\nments should be performed on single crystals. Crystals\nwith significant anisotropy of the gyromagnetic tensor\nshould be selected for ESR experiments. When the mag-\nnetocrystalline anisotropy is weak, the RDE in FMR can\nbe induced by the asymmetric shape of the sample alone\ndue to the anisotropy of dipole-dipole interactions. Even\nin this case, however, a single crystal would be preferred\nto provide a narrow linewidth. Same applies to experi-\nments on RDE in solid state NMR. The NMR frequency\nrange is much lower than that used in ESR and FMR\nexperiments. The width of the NMR line can be as low\nas a few kHz, that is, well below the available rotational\nangular velocities. The key to the observation of RDE in\na solid state NMR must be the use of a crystal having\nmagnetic order and strong anisotropy of the hyperfine\ninteraction.\nA separate interesting question is magnetic resonance\nin small magnetic particles that are free to rotate. Parti-\ncles of size in the nanometer range can easily be excited\ninto rotational states with Ω of hundreds of MHz. Con-\ntrary to the rotational quantum states of molecules that\nhave been studied for decades, analytical solution of the\nproblem of a quantum-mechanical rotator does not exist\nevenwithoutaspin. Presenceofthespininteractingwith\na mechanical rotation complicates this problem even fur-\nther. Rigorous solution has been recently found for the\nlow energy states of a rotator that can be treated as a\ntwo-state spin system30. General solution is very diffi-\ncult to obtain. In the case when a particle consists of8\na large number of atoms, one can develop a semiclassi-\ncal approximation in which Ωis replaced with L/I(with\nIbeing the moment of inertia). This suggests that the\nmagnetic resonance in nanoparticles that are free to ro-\ntate would split into many lines related to the quanti-\nzation of L. Some evidence of this effect has been re-\ncently found in the FMR studies of magnetic particles in\nnanopores13. Rapid progress in measurements of single\nmagnetic nanoparticles31may shed further light on their\nquantized rotational states and related spin resonances.VIII. ACKNOWLEDGEMENTS\nS.L. acknowledges financial support from Grupo de\nInvestigaci´ on de Magnetismo de la Universitat de\nBarcelona. The work of E.M.C. has been supported by\nthe grant No. DMR-0703639 from the U.S. National Sci-\nence Foundation and by Catalan ICREA Academia. J.T.\nacknowledges financial support from ICREA Academia.\n1H. J. Hay, J. P. Schiffer, T. E. Cranshaw, and P. A. Egel-\nstaff , Phys. Rev. Lett. 4, 165 (1960); D. C. Champeney\nand P. B. Moon, Proc. Phys. Soc. 77, 350 (1961); W.\nK¨ undig, Phys. Rev. 129, 2371 (1963).\n2A.L.Kholmetskii, T.Yarman, andO.V.Missevitch, Phys.\nScr.77, 035302 (2008); A. L. Kholmetskii, T. Yarman, O.\nV. Missevitch, and B. I. Rogozev, Phys. 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Lett. 72, 3433 (1994); E.\nM. Chudnovsky and X. Mart´ ınez Hidalgo, Phys. Rev. B\n66, 054412 (2002).\n25F. Hartman-Boutron, P. Politi, and J. Villain, Int. J. Mod.\nPhys. B 10, 2577 (1996).\n26E. M. Chudnovsky, D. A. Garanin, and R. Schilling, Phys.\nRev. B72, 094426 (2005).\n27E. M. Chudnovsky and J. Tejada, Lectures on Magnetism\n(Rinton Press, Princeton, NJ, 2008).\n28C. Calero, E. M. Chudnovsky, an D. A. Garanin, Phys.\nRev. Lett. 95, 166603 (2005).\n29C. Kittel. Phys. Rev. 73, 155 (1948).\n30E. M. Chudnovsky and D. A. Garanin, Phys. Rev. B 81,\n214423 (2010).\n31L. Bogani and W. Wernsdorfer, Nature Materials 7, 179\n(2008)."    },    {        "title": "1009.0815v1.Uniaxial_contribution_to_the_magnetic_anisotropy_of_La0_67Sr0_33MnO3_thin_films_induced_by_orthorhombic_crystal_structure.pdf",        "content": "arXiv:1009.0815v1  [cond-mat.mtrl-sci]  4 Sep 2010Uniaxial contribution to the magnetic anisotropy of La 0.67Sr0.33MnO 3thin films\ninduced by orthorhombic crystal structure\nHans Boschker, Mercy Mathews, Peter Brinks, Evert Houwman,∗\nGertjan Koster, Dave H. A. Blank, and Guus Rijnders\nFaculty of Science and Technology and MESA+Institute for Nanotechnology,\nUniversity of Twente, 7500 AE, Enschede, The Netherlands\nArturas Vailionis\nGeballe Laboratory for Advanced Materials, Stanford Unive rsity, Stanford, California 94305, USA\n(Dated: February 28, 2013)\nLa0.67Sr33MnO3(LSMO) thin films under compressive strain have an orthorhom bic symmetry\nwith (110)oand (001) oin-plane orientations. (The subscript o denotes the orthor hombic symme-\ntry.) Here, we grew LSMO on cubic (LaAlO 3)0.3-(Sr2AlTaO 6)0.7(LSAT) substrates and observed\na uniaxial contribution to the magnetic anisotropy which is related to the orthorhombic crystal\nstructure. Since the lattice mismatch is equal in the two dir ections, the general understanding of\nanisotropy in LSMO, which relates the uniaxial anisotropy t o differences in strain, cannot explain\nthe results. These findings suggest that the oxygen octahedr a rotations associated with the or-\nthorhombic structure, possibly resulting in different Mn-O -Mn bond angles and therefore a change\nin magnetic coupling between the [1 10]oand [001] odirections, determine the anisotropy. We expect\nthese findings to lead to a better understanding of the micros copic origin of the magnetocrystalline\nanisotropy in LSMO.\nI. INTRODUCTION\nThe perovskite oxide La 1−xAxMnO3(A=Ca, Ba,\nSr) has initiated a substantial body of research due\nto its colossal magnetoresistance1,2. Extensive theo-\nretical studies and experimental investigations utiliz-\ning La 1−xAxMnO3perovskites in bulk form revealed\na strong coupling between lattice distortions and mag-\nnetism, which substantially modify magnetic proper-\nties such as magnetoresistanceand Curie temperature3,4.\nLa0.67Sr33MnO3(LSMO) has the highest Curie tempera-\nture (370K) and a 100% spin polarization5,6. LSMO can\nbe coherently grownon a range of commercially available\nperovskite substrates, such as e.g. NdGaO 3(NGO) and\nSrTiO 3(STO). The epitaxy stabilizes a different crystal\nstructure which modifies the magnetic properties. Espe-\ncially magnetic anisotropy is shown to be very sensitive\nto the LSMO crystal structure7–13. When anisotropic\nstrain is applied to the LSMO the magnetocrystalline\nanisotropybecomes stronglyuniaxial14,15, which is a use-\nful tool to tailor the magnetic properties for device ap-\nplications.\nIn the case of isotropic tensile strain, e.g. tetrago-\nnal LSMO thin films on cubic STO (001) csubstrates,\nthe magnetocrystalline anisotropy is biaxial with easy\naxes aligned with the <110>pclattice directions9,10.\n(We use subscript c, pc, o and t for cubic, pseu-\ndocubic, orthorhombic and tetragonal crystal struc-\ntures, respectively.) Next to the magnetocrystalline\nanisotropy a uniaxial anisotropy is present as well,\nwhich is stepedge induced13,16. Here we investigate the\ncase of isotropic compressive strain, which can be real-\nized with LSMO thin films on the cubic (LaAlO 3)0.3-\n(Sr2AlTaO 6)0.7(LSAT) (001) csubstrate. LSMO thin\nfilms under compressive strain adopt an orthorhombiccrystal structure17,18, which is characterized by the pres-\nence ofoxygenoctahedrarotationsaround all three pseu-\ndocubic crystal axes. As the magnetic coupling depends\non the Mn-O-Mn bond angle19,20, it is an interesting\nquestion whether the magnetic properties are anisotropic\nin the different orthorhombic directions. Note that for\nanothercase, orthorhombicLSMOgrownonNGO(110) o\nthe difference in lattice mismatch between the two in-\nplane directions determines the anisotropy14, so this sys-\ntem is not suitable to study the effect of the orthorhom-\nbicityonthemagneticproperties. ForLSMOfilmsgrown\non NGO (110) othe [110]olattice direction is subjected\nto less compressivestrain than the [001] olattice direction\nandisthereforetheeasyaxisduetothestrainanisotropy.\nFor LSMO films grown on LSAT the lattice mismatch is\nequalandtheanisotropyisduetotheintrinsicanisotropy\nof the orthorhombic crystal structure between the [1 10]o\nand [001] olattice directions.\nHere, we show that LSMO thin films can be grown\ncoherently and untwinned on LSAT substrates and that\nthe orthorhombicity induces anisotropic magnetic prop-\nerties. Next to a biaxial component of the magnetic\nanisotropy, we observed a uniaxial component to the\nanisotropy which is aligned with the principal crystal di-\nrections and became more pronounced for increasing film\nthickness. We found no correlation between the uniax-\nial anisotropy and the stepedge direction. We obtained\ntwinned samples, by growth on surfaces with reduced\ncrystalline quality, for which the uniaxial anisotropy\nwas reduced. Therefore we conclude that the uniaxial\nanisotropy is caused by the orthorhombic crystal struc-\nture.2\na) b)\nFIG. 1. (Color online) Surface analysis of the LSAT substrat e\nby atomic force microscopy. a) after annealing at 1050◦C for\n12 hours. b) after annealing at 950◦C for 1 hour, The images\nare 5 by 5 µm and the color scale is 2 nm. The insets show a\nclose-up of the roughness of the terraces.\nII. SAMPLES AND SUBSTRATE\nPREPARATION\nThe as-received LSAT substrates were cleaned with\nacetone and ethanol before they were subjected to an an-\nneal treatment. Two anneal treatments were used to ob-\ntain respectively surfaces with smooth terraces and sur-\nfaces with sub unit cell roughness on the terraces. The\nfirst treatment consisted of an annealing step at 1050◦C\nfor 12 hour in 1 bar of O 2gas pressure. For the second\ntreatment both the anneal time and temperature were\ndecreased to 1 hours and 950◦C respectively. The sur-\nfaces were characterized with atomic force microscopy\n(AFM). Typical results are shown in figure 1. For the\nsubstrates subjected to the first anneal treatment a step\nand terrace structure with 4 ˚A (a single unit cell) step\nheight was observed. The stepedges were not straight\nbut meandering and 4 ˚A deep holes are observed near the\nstepedges. Note that the miscut of these substrates is\nvery small, approximately 0.02◦, giving a terrace width\nof more than 1 µm. Between the stepedges areas with\natomically smooth morphology were observed. The sub-\nstrates subjected to the second treatment show terraces\nwith reduced crystalline quality, but still single unit cell\nstep heights.\nLSMO thin films were grown on the LSAT (001) sub-\nstrates by pulsed laser deposition (PLD) from a stoichio-\nmetric target in an oxygen background pressure of 0.35\nmbar with a laser fluence of 3 J/cm2and at a substrate\ntemperature of 750◦C. After LSMO deposition, the films\nwere cooled to room temperature at a rate of 10◦C/min\nin a 1 bar pure oxygen atmosphere. The growth settings\nwere previously optimized and were identical to the ones\nused for LSMO films on other substrates14,21.\nIn this paper four samples are described, see table I.\nSample U12 and U40 were grown on substrates with a\nsmooth surface and have a thickness of 12 and 40 nm\nrespectively. Samples T29 and T50 were grown on sub-\nstrates with terraces with reduced crystalline quality and\nare respectively 29 and 50 nm thick. (The sample labels\nconsist of either the letter T or U for twinned/untwinnedandanumberwhichindicatesthesamplethickness.) The\nsample thicknesses were measured with x-ray reflectiv-\nity measurements. AFM measurements (not shown) re-\nvealed surfaces of the thin film where the morphology of\nthe substrate was still visible. The Curie temperature of\nthe films was larger than 350 K (350 K was the measure-\nment limit of the vibrating sample magnetometer) and\ndid not depend on film thickness and the twinning, as\ndiscussed in the next section, of the films.\nIII. STRUCTURAL CHARACTERIZATION\nThe top panel of figure 2 shows reciprocal space maps\nof LSMO and LSAT around the (204) c, (024) c, (204)c\nand (024)cLSAT reflections. These results were ob-\ntained from sample U40. The LSMO has a slightly dis-\ntorted orthorhombic (monoclinic) unit cell with (110) o\nout-of-plane orientation and (1 10)oand (001) oin-plane\norientations. The orthorhombicity can be deduced from\nfigures 2a and 2c which show a difference in lattice\nspacing for the (260) oand (620) oLSMO reflections,\nwhich represent a dissimilarity between the LSMO a\nand b lattice parameters. The lattice parameters are as\nfollows: a=5.47 ±0.01˚A, b=5.51 ±0.01˚A, c=7.74 ±0.01˚A,\nα=90±0.1◦,β=90±0.1◦andγ=89.6±0.1◦.\nNext to the LSMO (444) oand (444)oreflections satel-\nlites are observed. In a previous paper18we have dis-\ncussed the orthorhombic crystal structure and the pres-\nence of satellite peaks in the [001] odirection for LSMO\ngrown on NGO (110) osubstrates. The satellites result\nfrom periodic lattice modulations which partially relieve\nthe applied strain22–24. As the LSMO crystal structure\ncaneasilyrelievestraininthe [1 10]olatticedirectionwith\na change in the γangle, the lattice modulations are only\npresent in the [001] odirection. We conclude that LSMO\nfilms on LSAT behave similarly as LSMO on NGO.\nThe panels e-h) of figure 2 show the same reciprocal\nspace maps, but obtained for sample T50. Dissimilar\nlattice spacings are observed in both in-plane directions\nand the satellite peaks are visible in all reciprocal space\nmaps. Zhouet al. observedsimilarsatellitepeaksaround\nreflections of LSMO on LSAT and attributed the satel-\nlites to an in-plane superlattice with alternating [1 10]o\nand [110]oorientations25. This cannot be the case for\nour samples as the [ 110]oin-plane orientation would re-\nsult in reciprocal space maps with both the (620) oand\nthe (260) opeak visible in the same plot. In e.g. figure 2f\nthis is clearly not the case. Together with the presence of\nthe satellites in figure 2b and 2d only in the [001] odirec-\ntion, we conclude that sample D is twinned in domains\nwith different [1 10]oand [001] oorientations.\nFor the two thinner samples we observed the same be-\nhavior as the thicker samples with equal substrate treat-\nment (not shown). Sample U12 was untwinned as con-\ncluded from the positions of the Bragg reflections, but\nhere no satellites could be observed. Sample T29 was\ntwinned with satellites in both directions in reciprocal3\nSampleThickness Substrate Crystal Satellites φeasy(◦)ku/k1ku\n(nm) surface (J/m3)\nU12 12smooth untwinned not observed 40±10.18±0.05110±30\nU40 40smooth untwinned along [001] o12±10.91±0.03540±5\nT29 29 rough twinned both directions 42±10.1±0.0560±30\nT50 50 rough twinned both directions 31±10.48±0.03290±25\nTABLE I. The ratio between the anisotropy constants for the v arious samples. The angle of the easy axis is obtained from th e\nfitting procedure and the ratio between the anisotropy const ants is calculated with equation 2.\nspace.\nTo explain the absence of twinning in the samples with\nsmooth surfaces we compare our results to the more\nwidely studied SrRuO 3(SRO) thin films on STO sub-\nstrates. SROisorthorhombicandsingledomainfilmscan\nbe realized by growth on smooth vicinal substrates with\nstepedges approximately aligned with the main crystal\naxis. In that case the [001] olattice directions aligns\nwith the stepedge direction26. This has been explained\nin three different ways. Gan et al. suggested that single\ndomain growthis relatedto the observedstepflow growth\nmode26. Maria et al. suggested that stepedge strain is\nthe dominant mechanism as the films are not orthorhom-\nbic at deposition temperature27. Finally Vailionis et al.\nsuggestedthat the films aretetragonalat depositiontem-\nperature and that the [001] tlattice direction aligns pref-\nerentially with the stepedges28. In contrast to SRO the\nLSMO films have a preferred orientation with the [001] o\nlattice direction aligned perpendicular to the stepedges\nand does not grow in stepflow mode. Therefore the ex-\nplanation by Maria et al. is the most likely candidate\nto explain the single domain growth. During cooldown\northorhombic domains nucleate at the stepedges and the\nstepedgestrainfavorsoctahedrabucklinginonedirection\nand the domains continue to grow across the terraces re-\nsulting in a single domain film. For the samples with\na rough surface the orthorhombic domains can nucleate\nat defects at the substrate surface and no preferential\norientation exists, resulting in twinned films.\nIn summary, growth on a relatively smooth surface re-\nsults in untwinned LSMO films, while growth on terraces\nwith reducedcrystallinequalityresultsin twinned LSMO\nfilms. We used the difference in magnetic properties be-\ntweentwinnedanduntwinned filmstoidentify the contri-\nbution from the orthorhombicity as it should be reduced\nfor the twinned samples.\nIV. MAGNETIC CHARACTERIZATION\nThe samples were characterized with vibrating sam-\nple magnetometer (VSM, Model 10 VSM by Microsense)\nmeasurements at room temperature. The in-plane angle\nof the applied field was varied to determine the mag-\nnetic anisotropy. For all field angles a full magnetization\nloop was measured and the remanent magnetization wase) \n 3.2 3.25 3.3 6.4 6.42 6.44 6.46 6.48 6.5 6.52Qout-of-plane  (A-1)f)\n 3.2 3.25 3.3\nQin-plane (A-1)g)\n 3.2 3.25 3.3h)\n 3.2 3.25 3.3a)\n 3.2 3.25 3.3 6.4 6.42 6.44 6.46 6.48 6.5 6.52Qout-of-plane  (A-1)b)\n 3.2 3.25 3.3\nQin-plane (A-1)c)\n 3.2 3.25 3.3d)\n 3.2 3.25 3.3\nb a/c103\n[110]o [001]o[110]o\n[110]o~30 nmi) j)T50U40\nFIG. 2. [Color online] Top panel: Reciprocal space maps\naround the a) (204) c, b) (024) c, c) (204)cand d) (0 24)cLSAT\nreflections of the 40 nm thick sample grown on a smooth sub-\nstrate (sample U40). In a) and c) the dissimilar spacing of\nthe (260) oand (620) oLSMO reflections is clearly observed\nwhile in b) and d) satellites are present next to the (444) o\nand (444)oLSMO reflections. Middle panel: Reciprocal space\nmaps around the e) (204) c, f) (024) c, g) (204)cand h) (0 24)c\nLSAT reflections of the 50 nm thick sample grown on a rough\nsubstrate (sample T50). The satellites are present in all ma ps\nand both e) and h) show intensity at the position for the\n(260)oLSMO reflection. Therefore this sample is twinned.\nBottom panel: Schematic of the real space crystal structure\nof an untwinned sample viewed along i) [001] oand j) [1 10]o.4\nobtained from the loop. Figure 3a shows the magneti-\nzation loops of sample U12 with the field aligned with\nthree high symmetry directions. In figure 3b we plotted\nthe dependence of the remanent magnetization on the in-\nplanefieldangle. Thelargestremanencewasfoundforan\napplied field at approximately 40 degrees with respect to\nthe [001] oin-planelattice direction. Apredominantbiax-\nial behavior is observed with easy axes aligned with the\n[110]pcand symmetry related crystal directions. Next\nto this biaxial anisotropy a small uniaxial anisotropy is\npresent as well which can be seen in the difference in re-\nmanent magnetization at 0 and 90 degrees and the shift\nof the easy axes to ±40◦. This uniaxial contribution be-\ncomes more pronounced in the thicker film (sample U40)\nshown in figure 3c and 3d. The remanent magnetization\nof sample B at 0 degrees approaches the easy axes value\nand the remanent magnetization at 90 degrees is much\nsmaller, only 20 percent of the easy axes value. Due to\nthe combination of uniaxial and biaxial anisotropy the\neasy axes are shifted to ±15◦. The hard axis magnetiza-\ntion loop does not show a switch of the magnetization,\nbut an almost linear dependence of the magnetization on\nthe applied field which is characteristic for a hard axis\nloop in a sample with a uniaxial anisotropy.\nA similar thickness dependence of the anisotropy was\nalso found for the samples T29 and T50, which are\ntwinned. The results are plotted in figure 3e-3h. A biax-\nial and a uniaxial contribution are present and the uni-\naxial contribution is more pronounced in the thicker film.\nSample T50 shows easy axes which are shifted to ±30◦\nand the hard axis remanence is 50 percent of the easy\naxis value. Comparing samples U12 and U40 with sam-\nples T29 and T50 we find that the uniaxial contribution\nis more pronounced in the samples U12 and U40.\nV. DISCUSSION\nIn order to quantify the biaxial and uniaxial contribu-\ntion to the anisotropy we start with a general anisotropy\nenergy equation which contains both a biaxial and a uni-\naxial contribution29.\nEa/V=kusin2(φ−φ1)+k1\n4sin2(2(φ−φ2)),(1)\nin which Eais the anisotropy energy, Vthe volume of\nthe sample, ku(k1) the uniaxial (biaxial) anisotropy con-\nstant,φthe angle of the magnetization, φ1=0◦(φ2=45◦)\nthe angle of the easy axis of the uniaxial (biaxial)\nanisotropy. The easy axes are found by minimizing the\nenergy with respect to φ. This results in:\n/braceleftbigg\ncos(2φeasy) =ku/k1forku< k1,\nφeasy= 0 for ku≥k1,(2)\nfrom which the easy axes, φeasy, can be obtained. The\nmeasured remanence versus field angle dependencies are\nthe projectionsofthemagnetization, whichatremanence/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s53/s48/s48/s45/s52/s48/s48/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s45/s51/s48 /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s53/s48/s48/s45/s52/s48/s48/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s45/s51/s48 /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s53/s48/s48/s45/s52/s48/s48/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s45/s51/s48 /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48\n/s45/s49/s46/s53 /s45/s49/s46/s48 /s45/s48/s46/s53 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53/s45/s53/s48/s48/s45/s52/s48/s48/s45/s51/s48/s48/s45/s50/s48/s48/s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48/s52/s48/s48/s53/s48/s48\n/s45/s51/s48 /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48/s48/s53/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48/s50/s53/s48/s51/s48/s48/s100/s41 /s99/s41/s98/s41\n/s32/s48/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s52/s53/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s57/s48/s32/s100/s101/s103/s114/s101/s101/s115/s32/s48/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s52/s53/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s57/s48/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s32\n/s97/s41\n/s84/s53/s48/s84/s50/s57/s85/s52/s48/s82/s101/s109/s97/s110/s101/s110/s99/s101/s32/s40/s107/s65/s47/s109/s41\n/s82/s101/s109/s97/s110/s101/s110/s99/s101/s32/s40/s107/s65/s47/s109/s41/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s107/s65/s47/s109/s41\n/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s107/s65/s47/s109/s41/s32/s68/s97/s116/s97\n/s32/s70/s105/s116/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s107/s65/s47/s109/s41\n/s82/s101/s109/s97/s110/s101/s110/s99/s101/s32/s40/s107/s65/s47/s109/s41/s82/s101/s109/s97/s110/s101/s110/s99/s101/s32/s40/s107/s65/s47/s109/s41/s77/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110/s32/s40/s107/s65/s47/s109/s41/s32\n/s32\n/s32/s32\n/s85/s49/s50/s32/s32/s32\n/s32\n/s65/s112/s112/s108/s105/s101/s100/s32/s102/s105/s101/s108/s100/s32/s40/s107/s65/s47/s109/s41/s32/s68/s97/s116/s97\n/s32/s70/s105/s116\n/s70/s105/s101/s108/s100/s32/s97/s110/s103/s108/s101\n/s32/s32/s32\n/s32\n/s101/s41\n/s32/s48/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s52/s53/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s57/s48/s32/s100/s101/s103/s114/s101/s101/s115/s32/s48/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s52/s53/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s57/s48/s32/s100/s101/s103/s114/s101/s101/s115\n/s32/s32/s32\n/s32/s102/s41\n/s32/s68/s97/s116/s97\n/s32/s70/s105/s116\n/s32\n/s32/s32/s32\n/s32/s68/s97/s116/s97\n/s32/s70/s105/s116\n/s103/s41\n/s32/s32/s32\n/s32/s104/s41\n/s32/s32\n/s32/s32\nFIG. 3. [Color online] a) Magnetization loops with the field\naligned with 3 high symmetry directions and b) remanence\nversus field angle dependence obtained from sample U12. The\nlatter graph shows the data from the measurements (black\ndots) and the result from the fit procedure described in the\ntext (blue line). c) and d) The corresponding graphs for sam-\nple U40. e) and f) Sample T29. g) and h) Sample T50. The\nfield angle is defined with respect to the crystal structure, 0◦\n(90◦) corresponds to the [001] o([110]o) lattice direction.\nis aligned with the easy axis, onto the measurement di-\nrection:\nMrem(θ) =M0cos(θ−φeasy), (3)\nwhereMremis the remanent magnetization, θis the field\nangle,M0istheremanenceintheeasydirectionand φeasy\nis the closest easy axis. Equation 3 has been used to ob-\ntainthefitsinfigure3. Themeasureddata, andtherefore\nthe anisotropy, is well described by equation 1-3. This\nallows us to extract the ratio between the uniaxial and\nbiaxial anisotropy energies ku/k1. The results are pre-\nsented in table I. For all samples the uniaxial anisotropy\nenergy is found to be smaller than the biaxial anisotropy\nenergy. Between the different samples the ratio ku/k1\nchanges by a factor of 10.\nThe anisotropy field Hancan be obtained from the\nslope dM/dHatH=0 in the hard axis magnetization5\nloop of a material with uniaxial anisotropy by the rela-\ntion\nHan=Msat\ndM/dH. (4)\nHere,Msatis the saturation magnetization, Mis the\nmagnetization and His the applied field. The anisotropy\nenergy is given by\n2ku=Hanµ0Msat, (5)\ninwhich µ0isthepermeabilityoffreespace. Weobtained\navalue for kuof540±5J/m3forthe almostuniaxialsam-\nple U40. This would imply a biaxial anisotropy constant\nof 600±10 J/m3which corresponds well with earlier ob-\ntained results9,21. Therefore one can assume that the k1\nvalue is the same for all films and independent of thick-\nness. The uniaxial anisotropy constants are calculated\nfrom the ku/k1ratios and presented in table I as well.\nThe uniaxial anisotropy is very weak for these samples,\nonly in the range 50-600 J/m3.\nNext we turn to the origin of both contributions to\nthe anisotropy. The biaxial contribution with easy axes\naligned with the <110>pclattice directions corresponds\nwell with earlier results of magnetic anisotropy of LSMO\non STO (001) csubstrates9,10. This represents the intrin-\nsic magnetocrystalline anisotropy of LSMO strained to\nin-plane cubic symmetry. For the uniaxial contribution\ndifferent explanations exist. It is well known that a weak\nuniaxial anisotropy in LSMO can be the result of ste-\npedge induced anisotropy13,16. However, the thickness\ndependence of the uniaxial anisotropy is at odds with an\ninterpretation in terms of stepedge induced anisotropy.\nThe stepedge induced anisotropy compared to the biax-\nial anisotropy should scale with the ratio of the volume\nof the surface layers containing the stepedges (in practice\nthe monolayers at the interfaces) to the volume of the\nfilm. Therefore the stepedge induced anisotropy should\nbe most pronounced for the thinnest films as the miscut\nof the samples was comparable. Here the opposite is ob-\nserved. Also the uniaxial easy axis was not aligned with\nthestepedgedirectionsinmostofthesamples, sometimes\neven 90 degrees perpendicular to the stepedges. This\nrulesoutthecontributionofstepedgeinducedanisotropy.\nThe uniaxial easy axis is aligned with the [001] olattice\ndirectionwhilethehardaxisisalignedwiththe[1 10]olat-\ntice direction. This, together with the reduced uniaxial\nanisotropy for the twinned samples, relates the observed\nanisotropy to the orthorhombicity of the LSMO films. It\nis unclear what the origin of the magnetic anisotropy in\nthe orthorhombic crystal structure is. In general mag-\nnetic anisotropy in manganites is explained by a global\nstrain field which relates the easy axis to the maximum\nstrain direction30. This cannot be applied for these sam-\nples, as the LSMO has equal strain in the [001] oand the\n[110]olattice directions due to the cubic symmetry of the\nsubstrate. Note that for the case of orthorhombic LSMO\non NGO (110) othe uniaxial easy axis is aligned with the[110]olattice direction and is due to strain anisotropy14,\nincontrasttotheobservedanisotropyofLSMOonLSAT.\nThe difference between the twolattice directions in the\northorhombic symmetry is due to the different oxygen\noctahedra rotations. We therefore suspect that the mi-\ncroscopic origin of the magnetocrystalline anisotropy in\nLSMO is effected by the oxygen octahedra rotations. A\nplausible scenario would be that the difference in octahe-\ndrarotationsalongthe[1 10]oand[001] odirectionsresults\nin a different Mn-O-Mn bond angle ( <Mn-O-Mn >)in\nthese two directions. For manganites it is well known\nthat the bandwidth, due to the double exchange mech-\nanism, depends on the Mn-O-Mn bond angle according\nto20\nW=cosω\nd3.5\nMnO(6)\nin which Wis the bandwidth, ω= 0.5(π−<\nMn-O-Mn >is proportional to the amount of octahedra\nrotation and dMnOis the Mn-O distance. This suggests\nthat for LSMO on LSAT the Mn-O-Mn bond angle is\nlarger in the [001] odirection than in the [1 10]odirection,\nresulting in an increase in the bandwidth and therefore\nthe easy axis is aligned with this direction. This explana-\ntion is alsoconsistent with the increaseof ku/k1with film\nthickness. Due to structural reconstructions at the inter-\nface and the surface31, the oxygen octahedra rotations in\nthe surface and interfacial regions deviate from those of\nthe bulk of the film. The biaxial anisotropy is intrinsic\nto MnO 6octahedra and not sensitive to the structural\nreconstructions. In a follow up work we will investigate\nthe crystal structure, and especially the octahedra rota-\ntions, in more detail, in order to resolve the microscopic\norigin of the magnetocrystalline anisotropy in LSMO.\nAn alternative interpretation of the magnetic data is\nthat the anisotropy is somehow related to the lattice\nmodulations observed with the satellites in the XRD\nmeasurements. As the lattice modulations only occur\nin the orthorhombic [001] odirection, it is not possible\nto discriminate between anisotropy induced by the or-\nthorhombic crystal structure and anisotropy induced by\nthe lattice modulations in our experiment. Neverthe-\nless, we expect that the lattice modulations result in mi-\ncrotwins which have reduced magnetic coupling at the\nmicrotwin boundaries. The shape anisotropy of each mi-\ncrotwin would then be aligned with the [1 10]odirection,\ncontrarytotheobservationofamagnetic[001] oeasyaxis.\nAlthough one would suspect that the uniaxial\nanisotropywould disappear for the twinned samples, this\nis not the case. We assume that this is due to the domi-\nnant presence of grains with one orientation.\nVI. CONCLUSION\nLSMO films with an orthorhombic crystal structure\ncan be grown coherently and untwinned on cubic LSAT\nsubstrates. The magnetic anisotropy of the films is6\ndescribed by a combination of biaxial anisotropy with\neasy axes along the <110>pcdirections and a uniaxial\nanisotropy with easy axis along the [001] odirection. For\nthicker films the uniaxial anisotropy becomes more pro-\nnounced. The uniaxial part of the anisotropy is induced\nby the orthorhombic symmetry of the LSMO. We expect\nthese findings to lead to a better understanding of themicroscopic origin of the magnetocrystalline anisotropy\nin LSMO.\nThis research was financially supported by the Dutch\nScience Foundation, by NanoNed, a nanotechnology pro-\ngram of the Dutch Ministry of Economic Affairs and by\nthe NanOxide program of the European Science Founda-\ntion.\n∗e.p.houwman@utwente.nl\n1R. Vonhelmolt, J. Wecker, B. Holzapfel, L. Schultz, and\nK. Samwer, PHYSICAL REVIEW LETTERS 71, 2331\n(1993), ISSN 0031-9007.\n2S. Jin, T. Tiefel, M. Mccormack, R. Fastnacht, R. Ramesh,\nand L. Chen, SCIENCE 264, 413 (1994), ISSN 0036-8075.\n3Y. Moritomo, A. Asamitsu, and Y. Tokura, PHYSICAL\nREVIEW B 51, 16491 (1995), ISSN 0163-1829.\n4H. Hwang, T. Palstra, S. Cheong, and B. Batlogg, PHYS-\nICAL REVIEW B 52, 15046 (1995), ISSN 0163-1829.\n5J. Park, E. Vescovo, H. Kim, C. Kwon, R. Ramesh, and\nT. Venkatesan, NATURE 392, 794 (1998), ISSN 0028-\n0836.\n6M. Bowen, M. Bibes, A. Barthelemy, J. Contour,\nA. Anane, Y. Lemaitre, and A. Fert, APPLIED PHYSICS\nLETTERS 82, 233 (2003), ISSN 0003-6951.\n7C. Kwon, M. Robson, K. Kim, J. Gu, S. Lofland, S. Bha-\ngat, Z. Trajanovic, M. Rajeswari, T. Venkatesan, A. Kratz,\net al., JOURNAL OF MAGNETISM AND MAGNETIC\nMATERIALS 172, 229 (1997), ISSN 0304-8853.\n8Y. Suzuki, H. Hwang, S. Cheong, T. Siegrist, R. vanDover,\nA. Asamitsu, and Y. Tokura, JOURNAL OF APPLIED\nPHYSICS 83, 7064 (1998), ISSN 0021-8979.\n9K. Steenbeck and R. Hiergeist, APPLIED PHYSICS LET-\nTERS75, 1778 (1999), ISSN 0003-6951.\n10F. Tsui, M. Smoak, T. Nath, and C. Eom, APPLIED\nPHYSICS LETTERS 76, 2421 (2000), ISSN 0003-6951.\n11R. Desfeux, S. Bailleul, A. Da Costa, W. Prellier, and\nA. Haghiri-Gosnet, APPLIED PHYSICS LETTERS 78,\n3681 (2001), ISSN 0003-6951.\n12J. Dho, Y.Kim, Y. Hwang, J. Kim, and N. Hur, APPLIED\nPHYSICS LETTERS 82, 1434 (2003), ISSN 0003-6951.\n13M. Mathews, F. Postma, J. Lodder, R. Jansen, G. Rijn-\nders, and D. Blank, APPLIED PHYSICS LETTERS 87\n(2005), ISSN 0003-6951.\n14H. Boschker, M. Mathews, E. P. Houwman, H. Nishikawa,\nA. Vailionis, G. Koster, G. Rijnders, and D. H. A. Blank,\nPHYSICAL REVIEW B 79(2009), ISSN 1098-0121.\n15M. Mathews, E. P. Houwman, H. Boschker, G. Rijnders,\nand D. H. A. Blank, JOURNAL OF APPLIED PHYSICS\n107(2010).\n16Z. Wang, G. Cristiani, and H. Habermeier, APPLIED\nPHYSICS LETTERS 82, 3731 (2003), ISSN 0003-6951.17A. Vailionis, W. Siemons, and G. Koster, APPLIED\nPHYSICS LETTERS 93(2008), ISSN 0003-6951.\n18A. Vailionis, H. Boschker, E. Houwman, G. Koster, G. Ri-\njnders, and D. H. A. Blank, APPLIED PHYSICS LET-\nTERS95(2009), ISSN 0003-6951.\n19H.Hwang, S.Cheong, P.Radaelli, M. Marezio, andB. Bat-\nlogg, PHYSICAL REVIEW LETTERS 75, 914 (1995),\nISSN 0031-9007.\n20P.Radaelli, G.Iannone, M.Marezio, H.Hwang, S.Cheong,\nJ. Jorgensen, and D. Argyriou, PHYSICAL REVIEW B\n56, 8265 (1997), ISSN 0163-1829.\n21M. Mathews, Ph.D. thesis, University of Twente (2007).\n22A. Vigliante, U. Gebhardt, A. Ruhm, P. Wochner,\nF. Razavi, and H. Habermeier, EUROPHYSICS LET-\nTERS54, 619 (2001), ISSN 0295-5075.\n23N. Farag, M. Bobeth, W. Pompe, A. Romanov, and\nJ. Speck, JOURNAL OF APPLIED PHYSICS 97(2005),\nISSN 0021-8979.\n24U. Gebhardt, N. V. Kasper, A. Vigliante, P. Wochner,\nH. Dosch, F. S. Razavi, and H. U. Habermeier, PHYSI-\nCAL REVIEW LETTERS 98(2007), ISSN 0031-9007.\n25T. F. Zhou, G. Li, X. G. Li, S. W. Jin, and W. B. Wu,\nAPPLIED PHYSICS LETTERS 90(2007), ISSN 0003-\n6951.\n26Q. Gan, R. Rao, and C. Eom, APPLIED PHYSICS LET-\nTERS70, 1962 (1997), ISSN 0003-6951.\n27J. Maria, H. McKinstry, and S. Trolier-McKinstry, AP-\nPLIED PHYSICS LETTERS 76, 3382 (2000), ISSN 0003-\n6951.\n28A. Vailionis, W. Siemons, and G. Koster, APPLIED\nPHYSICS LETTERS 91(2007), ISSN 0003-6951.\n29S. Chikazumi, Physics of ferromagnetism (John Wiley and\nSons, Inc, New York, 1964).\n30K. Terakura, I. Solovyev, and H. Sawada, in Colossal mag-\nnetoresistive oxides , edited by Y. Tokura (Gordon and\nBreach Science Publishers, the Netherlands, 2000), pp.\n119–148.\n31R. Herger, P. R. Willmott, C. M. Schlepuetz, M. Bjoerck,\nS. A. Pauli, D. Martoccia, B. D. Patterson, D. Kumah,\nR. Clarke, Y. Yacoby, et al., PHYSICAL REVIEW B 77\n(2008), ISSN 1098-0121."    },    {        "title": "1009.1520v1.Controllable_modification_of_the_anisotropy_energy_in_Laves_phase_YFe2_by_Ar__ion_implantation.pdf",        "content": "Controllable modification of the anisotropy energy in Laves phase \nYFe 2 by Ar+ ion implantation\nA. R. Buckingham1, G. J. Bowden1, D. Wang1,2, G. B. G. Stenning1, I. Nandhakumar3, R. C. C. Ward4 and \nP. A. J . de Groot1 \n1School of Physics and Astronomy, University of Southampton, Southampton, SO17 1BJ, UK.  \n2Department of Physics, National University of Defense Technology, Changsha, Hunan 410073, China.  \n3School of Chemistry, University of Southampton, Southampton, SO17 1BJ, UK .  \n4Cla rendon Laboratory, University of Oxford, Oxford, OX1 3PU, UK.  \n8th September  2010  \n Implanted 3.25 keV Ar+ ions have been  used to modify the in-plane bulk anisotropy in thin \nfilms of epitaxially grown Laves phase YFe 2. The magneto optical Kerr effect , vibrating \nsample magnetometry  and computational modeling have been used to show that the dominant \nsource of anisotropy changes from magneto elastic in as-grown sa mples  to magnetocrystalline \nin ion implanted samples. This change occurs a t a critical fluence of order  1017 Ar+ ions cm-2. \nThe change in source of the anisotropy  is attributed to a relaxation of the strain inherent in the \nepitaxially grown thin -films. Atomic force microscopy shows that the samples’ topography \nremain s unchanged after ion implantation. The ability to control  the dominant source of \nmagnetic anisotropy without affecting the sample surface could have important consequences \nin the fabrication of patterned media for high use in density magnetic data storage devices.\n Due to the ever increasing demands on the density of magnetic data storage media  there \nhas been a recent influx of work on patterned media. Patterned media is expected to allow \ndata storage densities to exceed the rapidly approaching limits of traditional g ranular media , \nthereby avoiding the superparamagnetic limit1. Patterned media behave as single magnetic \nbits, i.e. as single domain particles, or as a collection of strongly coupled grains, rather than \nas many weakly coupled grains as in existing granular media. This reduces the volume of \nmagnetic material required per bit of information stored whilst maintaining a high signal -to-\nnoise ratio, thereby circumventing superparamagnetism. Top-down lithographic, and bottom -\nup self -assembly techniques towards topographically patterned media have received  much \nattention  over the past decade2,3. Furthermore, a non-topographical, purely magnetic \npatterning process has recently become intensely investigated  following its first successful \ndemonstration in 19984. Magnetic patterning , using energetic ions which either irradiate or \nare implanted in the media , is especially attractive since it alleviates many issues surrounding \nthe need for planarization of the topographically patterned recording media5,6. In this wor k \nwe report on the affects of implanting Ar+ ions into epitaxi ally grown Laves phase YFe 2 samples.  \n The YFe 2 samples were grown via molecular beam epitaxy (MBE) in a Balzers UMS 630  \nultrahigh vacuum facility on epi -prepared (112ത0) sapphire substrates at 600 °C. A 10 nm \n(110 ) Nb chemical buffer layer and 2 nm iron seed layer were deposited on the substrates \nprior to the growth of the Laves phase film7,8. The YFe 2 was grown by the co -deposition of \nelemental fluxes  described elsewhere9,10, following the original pr ocedures developed by \nKwo et al.11. Finally, t he samples were capped with a 10 nm protective Y layer12. \n Ar+ ions from an Oxford Applied Research IG5  ion source  were implanted into the YFe 2 \nsamples. Th is was conducted under  vacuum with a base pressure of 1 × 10-6 mBar, which \nrose to 1 × 10-4 mBar during the implantation process. The samples were continuously  rotated  \nwith the incident Ar+ ions making an angle of 68 ° to the sample surface normal. The fluence \nof Ar+ ions was calculated from measurements of the c urrent density induced at the sample \nsurface (1 µAcm-2) and the acceleration voltage (3.25 keV). The incident fluence ranged from \nzero through  to 1.7 × 1017 Ar+ ions cm-2. Sample characterization was performed by \nlongitudinal magneto optical Kerr effect (M OKE) measurements, vibrating sample \nmagnetometry (VSM) ( Oxford Instruments Aersonic 3001 ) and atomic force microscopy \n(AFM) ( Digital Instruments Multimode SPM ) both before and after implantation with various \nAr+ ion fluences . Modelling of the implantation of ions was conducted using SRIM13 whilst \nmodelling of the sample magnetization was performed with OOMMF14. \n MOKE magnetization data for a sample in an as-grown state is presented in graphs a) and \nb) in Figure 1. These data clearly show that the sample exhibits an anisotropic response to an \napplied magnetic field. From the hysteresis loop obtained with the magnetic field applied \nalong the [1ത10] directi on (graph a) ), the sample has a coercivity ܪ=17.4 mT and a \nsquareness ெೃ\nெೄ=1.0 (with MR and MS the magnetization at remanence and saturation, \nrespectively). In graph b) the hysteresis loop obtained with the magnetic field applied along \nthe [001 ] direc tion is presented; h ere both the coercivity and squareness are greatly reduced , \nto ܪ=5.2 mT and ெೃ\nெೄ=0.2 respectively. This corresponds to a hard axis of magnetization \nwhere  the cubic anisotropy of the Laves phase structure is clear. There is good q ualitative and \nquantitative agreement between these data and the VSM data for the same sample ( Figure 2, \ngraphs a) and b)), confirming  that the observe d anisotropy is a bulk , rather than surface \neffect. Note that the skin depth of YFe 2 (ߜଢ଼ୣమ) at 633 nm (HeNe laser used for MOKE \nmeasurements) is approximately  20 nm using values for electrical resistivity at room temperature from Ikeda and Nakamichi15). \n \nFigure 1: (Colour online) MOKE hysteresis loops of a 100 nm thick YFe 2 sample  in the as -grown state (graphs \na) and b)) and for sample s implanted with increasing fluences of Ar+ ions (graphs c) – f)). The magnetic field \n(Bapp) has been applied along two orthogonal di rections as indicated by the upper labels.  \n \n \nFigure 2: (Colour online) VSM hysteresis loops of a 100 nm thick YFe 2 sample  in the as -grown state (graphs a) \nand b)) and for a sample  implanted with  a fluence of 1.7 × 1017 Ar+ ions cm-2 (graphs c) and d)). The magnetic \nfield ( Bapp) has been applied along two orthogonal directions as indicated by the upper labels.  \n It has been reported that epitaxially grown Laves phase YFe 2 exhibits little or no \nmagnetoelastic anisotropy29,16 since it is usually described within the context of multilayer \nexchange -spring YFe 2/REFe 2 magnetic materials ( e.g. RE = Dy, Er)17,18,19,20, which exhibit a \nconsiderably larger anisotropy than the soft YFe 2 alone . In these REFe 2 materials  the \nanisotropy is ascribed to second order crystal field terms originating from the RE atoms in the \nLaves phase str ucture. Previously Wang et al.21 presented anisotropic magnetization data for \nstrained Laves phase YFe 2 sample s, noting its evolution due to the increased dominance of -505\n-505\n-40 -20 0 20 40-505\n-40 -20 0 20 40 \n  a)\n  \n b)\nIon fluence  (x 1016 Ar+ ions cm-2)17 6.8 0 (As-grown)  \n c)MOKE Signal  (V)\n   \n d)B // [001] B // [110]\n   \ne)\nBapp (mT)\n   \n f)\n-0.50.00.5\n-30 -15 0 15 30-0.30.00.3\n-30 -15 0 15 30B // [110]\n \n a)M (memu)\n17 0 (As-grown)B // [001] \n \n  \nb)\nc)\n  \nBapp (mT)\nIon fluence  (x 1016 Ar+ ions cm-2)\nd)\n  magnetostatic effects induced by micron -scale patterning. Combining these data with the \nhysteresis loops in Figure 1 and Figure 2 provides conclusive evidence that there is a \nsignificant anisotropy energ y associated with the Laves phase YFe 2. \n It is well established that during the first stages of deposition in epitaxial REFe 2 systems  \nthe growth mechanism is island -like10,22, becoming  continuous once the REFe 2 thickness \nreaches 25 nm. Island growth is a classical signature of a growth mechanism which is \nevolving in an attempt to reduce strain within the system23,24. Indeed, the lattice parameters of \nbulk Laves phase REFe 2 materials differ significantly  from those of the MBE grown sample s. \nX-Ray diffraction data has confirmed that there is an in -plane expansion and perpendicular \ncontraction in the Laves phase unit cell16. Accordingly the MBE grown samples are said to be \nstrained  and thus are subject to a magnetoelastic energy term , the origin of which  is thermal \nin nature , induced during the post -growth cooling of the sample from 600 °C to room \ntemperature16. In the bulk, un -strained Laves phase REFe 2 compoun ds there is a dominant \nmagnetocrystalline anisotropy energy ( EMC)25,26,27, defining the easy and hard axes of \nmagnetization. However, in MBE grown thin -films of Laves phase REFe 2 compounds, the \nmagnetoelastic energy ( EME) induced during the sample growth is large r than EMC, serving to \nalter the easy and hard axes of magnetization22. It has been shown that the MBE grown Laves \nphase samples are subject t o a shear strain of ߝ௫௬=−0.5%16. This is determined \nexperimentally to be the dominant strain term, thus the standard equation describing EME in a \nsystem with cubic anisotropy ( e.g. see refs. 29 and 28) may be given by a much simpler \nrelationship;  \n   ܧொ=ܾଶߝ௫௬ߙ௫ߙ௬ (1)  \nwhere b2 is a magnetoelastic coefficient given in ref. 29 and αx and αy are direction cosines. In \nthin-film samples of YFe 2, EME is expected to dominate . The easy axis of magnetization is \nshown to be along equivalent 〈221〉 directions29, close to the  [1ത10] direction  shown in graphs \na) of Figure 1 and Figure 2. \n After implantation by a fluence of > 1017 Ar+ ions cm-2 (graphs e) & f ) in Figure 1 and \ngraphs c) and d) in Figure 2), it is clear that the easy and hard axes of magnetization have \nexchanged directions from those in the as -grown samples. From the MOKE data for the \nsample implanted with a fluence of 1.7 × 1017 Ar+ ions cm-2, the coercivity is reduced to \nܪ=4.0 mT and the squareness to ெೃ\nெೄ=0.3 when the magnetic field is applied along the  \n[1ത10] direction . When the field is applied along the [001 ] direction, the coercivity is increased to ܪ=12.5 mT and squareness to ெೃ\nெೄ=1.0. The decreases along the [1ത10] \ndirection and the increases along the [001 ] direction, and the accompanying change in shape \nof the hysteresis loops, corresponds  to a rotation of 90 ° in the easy and hard axis of \nmagnetization within the ion implanted sample. After ion implantation above a critical \nfluence of approximately 1017 Ar+ ions cm-2, there is a close to  easy axis of magnetization \nalong the [001 ] direction, whilst the hard axis lies close  the [1ത10] direction . It is worthwhi le \nnoting that at a fluence of 6.8 × 1016 Ar+ ions cm-2 (graphs c) and d), Figure 1), the hysteresis \nloops obtained for the magnetic field applied along orthogonal directions are equivalent.  This \ngradual progression to the rota tion of the easy and hard axes of magnetization introduces a \nuseful degree of controllability into engineering the magnetic properties via ion implantation.  \n Furthermore, we  note that the VSM data for the ion implanted sample agrees with the \nMOKE data . Fro m this we may conclude that the apparent re-orientation of the magnetic axes \nis throughout the sample, rather than being confined to a small region at the top of the sample \ncorresponding to the MOKE skin depth, although some material is removed via ion mil ling as \nillustrated by the reduction in MS in the VSM data.  These  data show a reduction in MS of \n~60% (0.65 to 0.25 memu for the easy axes of magnetization (graphs a) and d)) and 0.80 to \n0.30 memu for the hard axes of mag netization (graphs b) and c))), corresponding  to a \nreduction in sample thickness to ~40 nm. This is consistent with the MOKE data, i.e. \nߜଢ଼ୣమ<40 nm. Ion milling  also accounts for t he reduced  amplitude of the MOKE signal in \nFigure 1, graphs  a) and b) . The protective Y cap  is removed via ion milling, thus the  ion \nimplanted samples exhibit a larger amplitude of MOKE signal due to the experiment probing \na larger volume of the magnetic ma terial.  Note that the MOKE signal amplitude does not \nbegin to decrease, showing that the sample  thickness remains greater than ߜଢ଼ୣమ. \n OOMMF  modelling of 100 nm thick  layers of  strained YFe 2 has been conducted. A one \ndimensional spin -chain model was utilised with a mesh size of 1 nm3. The model was tailored \nto include a strain pre -factor ( SPF)20 in order for the theoretical work of Bowden et al.30 –\nwhich appears to under estimate the manetoelastic strain – to agree with the experimental \nwork of Zhukov et al.31. Hysteresis loops  from OOMMF  calculations to model an  as-grown \nsample  are presented in Figure 3, graphs a) and b) , with the [1ത10] and [001 ] directions \ncorresponding to the easy and hard axes respectively . OOMMF  simulations were also \nperformed with reduced SPF values  in order to model the effects of a reduced EME. These  \nhysteresis loops , generated with SPF = 1.0 and 0.0 , are shown in Figure 3, graphs c) – f). The \ndata in graphs e) and f) are the direct opposi te of those in graphs a) and b); t he easy and hard axes of magnetization are now along the [001 ] and [1ത10] direction s respectively . This is in \ngood agreement with the experimental MOKE and VSM data . Additionally , an intermediate \nstep is also apparent in Figure 3 (graphs c) and d)), where the OOMMF  calculate d hysteresis \nloops show that the response of the sample’s magnetization to Bapp to be almost equivalent \nwhether it is applied along the [1ത10] or [001 ] directions, just as evidenced in the MOKE data \nfor a sample subject to a fluence of 6.8 × 1016 Ar+ ions cm-2. As the strain within the sample \nis reduced, EME becomes less dominant until eventually EMC dominates, with the two being \napproximately equal at some intermediate crossing point.  \n \nFigure 3: (Colour online) OOMMF  calculated data fo r 100 nm thick YFe 2 layer  subject to a variation in the \ndominance of the magnetoelastic anisotropy energy. Graphs a) and b) represent a YFe 2 sample in its as -grown, \nstrained stat e.  Graphs c) – f) show the corresponding hysteresis loops as the magnetoelastic anisotropy energy \nis reduced. The magnetic field ( Bapp) has been applied along two orthogonal directions as indicated by the upper  \nlabels.  \n AFM was used to characterize the sample topography both in the as -grown state and after \nimplantation by a fluence of 6.8 × 1016 Ar+ ions cm-2. The AFM micrographs are \npresented in Figure 4, images a) and b) . Although a certain amount of material is ine vitably \nsputtered away during the ion implantation process, from these data it is clear that the \nimplanted Ar+ ions have had no other effects on the sample topography . The surface \nroughness remains as 3 ± 0.3 nm.  From this we may conclude that the reorient ation of easy \nand hard axes of magnetization are not due to an ion implantation induced shape anisotropy.  -4000400\n-4000400a)B // [110] B // [001]\nb)\n2.5 0.0\nStrain pre-factor1.0c)M (kAm-1)d)\n-40 -20 0 20 40-4000400e)\n   \n -40 -20 0 20 40\nBapp (mT)f)\n   \n  \nFigure 4: (Colour online) AFM micrographs of a YFe 2 sample in the as -grown state (image a)) and after \nimplantation by a fluence of 6.8 × 1016 Ar+ ions cm-2 (image b)).  \n Implantation profiles for 3.25 keV Ar+ ions have been calculated using  SRIM. The \nresults  confirm that all of the Ar+ ions are stopped within 10 nm of the sample surface. Thus \ndespite the effects of ion milling , a volume significantly larger than that penetrated by the Ar+ \nions remains after the implantation process, yet the effects of these Ar+ ions are felt \nthroughout the sample.  The Ar+ ions energy is deposited in the  YFe 2 sample  via \npredom inantly elastic collisions;  SRIM  calculations show  that the largest proportion of which \nis to phonons created within the crystalline lattice. The incident Ar+ ions suffer many recoil  \ncollisions with the YFe 2 lattice, but recoil cascades are rare due to the large mass difference \nbetween the incident ion and the tightly bound target atom32. Therefore the overall crystalline \nstructure of the Laves phase is maintained, with the recoiled YF e2 atoms relaxing only to \ntheir un -strained lattice positions. From the anisotropic magnetization data i t is clear that a \nwell ordered crystalline structure remains throughout the material after Ar+ ion implantation. \nWe suggest that it is due to the regula r epitaxial structure of the Laves phase sample, that the \neffects of ion implantation are much longer -range than predicted by SRIM  modelling. The \nrecoils induced within the YFe 2 crystalline lattice , which incidentally is a relatively brittle \nmaterial,  propagate readily throughout the entire epitaxial structure via the rapid propagation \nof ion implantation -induced dislocations, in much the same way a crack readily propagates \nthrough glass . \n In summary we present the first demonstration of ion implantati on induced changes to the \nmagnetic properties of epitaxial Laves phase YFe 2. Previous work has demonstrated the \nability to induce a uniaxial anisotropy in polycrystalline films via ion implantation33,34, and \nalso substantial work exists on modification of the plane in which the  easy axis of \nmagnetization  lies by ion irradiation induced modifications to interfacial anisotropy ( e.g. see \nrefs, 4, 35 and 36). However, this work is the first to demonstrate ion implantation as a tool to \nengineer  the dominant anisotropy energy of an epitaxial Laves phase system. 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Res. B  148, 872 ( 1999 ). \n36 T. Devolder , Phys. Rev. B  62, 5794 (2000).  \n37 T. Mewes, R. Lopusnik, J. Fassbender, B. Hillebrands, M. Jung, D. Engel, A. Ehresmann and H. Schmora nzer, Appl. Phys . Lett.  76, 1057 \n(2000).  "    },    {        "title": "1010.5264v2.High_temperature_magnetic_properties_of_noninteracting_single_domain_Fe3O4_nanoparticles.pdf",        "content": "arXiv:1010.5264v2  [cond-mat.mes-hall]  17 Dec 2010High-temperature magnetic properties of noninteracting\nsingle-domain Fe 3O4nanoparticles\nJun Wang1,a, Pieder Beeli2, L. H. Meng1, and Guo-meng Zhao1,2,b\n1Department of Physics, Faculty of Science,\nNingbo University, Ningbo, P. R. China\n2Department of Physics and Astronomy,\nCalifornia State University, Los Angeles, CA 90032, USA\nAbstract\nMagnetic measurements have been performed on 40-nm sphere- like Fe 3O4nanoparticles using\na Quantum Design vibrating sample magnetometer. Coating Fe 3O4nanoparticles with SiO 2ef-\nfectively eliminates magnetic interparticle interaction s so that the coercive field HCin the high-\ntemperature range between 300 K and the Curie temperature (8 55 K) can be well fitted by an\nexpression for noninteracting randomly oriented single-d omain particles. From the fitting parame-\nters, the effective anisotropy constant Kis found to be (1.68 ±0.17)×105erg/cm3, which is slightly\nlarger than the bulk magnetocrystalline anisotropy consta nt of 1.35 ×105erg/cm3. Moreover, the\ninferred mean particle diameter from the fitting parameters is in quantitative agreement with that\ndetermined from transmission electron microscope. Such a q uantitative agreement between data\nand theory suggests that the assemble of our SiO 2-coated sphere-like Fe 3O4nanopartles represents\na good system of noninteracting randomly-oriented single- domain particles.\n1Ensembles of magnetic nanoparticles in various forms have been at t he focus of scientific\ninterest [1] since the days of N´ eel [2] and Brown [3], who developed a theory for noninter-\nacting single-domain ferromagnetic particles. A complete understa nding of the magnetic\nproperties of ferromagnetic nanoparticles is not simple, in particula r because of the com-\nplexity of real nanoparticle assemblies, involving magnetic interpart icle interactions and\nmagnetic anisotropy. An important contribution to the understan ding of the magnetic be-\nhavior ofnanoparticles was given by Beanand Livingston (BL) [4] who assumed anassembly\nof noninteracting single-domain particles with uniaxial anisotropy. T his study was based on\nthe N´ eel relaxation time τ=τ0exp(KV/k BT), where τ0is the characteristic time constant,\nkBis the Boltzmann constant, Kis the uniaxial anisotropy constant, and Vis the particle\nvolume. KVrepresents the energy barrier between two easy directions. Acc ording to Bean\nand Livingston, at a given observation time τobs, there is a critical temperature, called the\nblocking temperature TB, given by [4]\nTB=KV\nkBln(τobs/τ0), (1)\nabove which the magnetization reversal of an assembly of identical single-domain particles\ngoes from blocked (having hysteresis) to superparamagnetic-ty pe behavior. Within this\nframework the coercive field HCis expected to decrease with the square root of temperature:\nHC=α2K\nMs[1−(T/TB)1/2], (2)\nwhereMsis the zero-temperature saturation magnetization and α= 1 if the particle easy-\naxes are aligned [4] or α= 0.48 if randomly oriented [5].\n200300400500600700\n20 30 40 50 60 70Intensity (arb. unit)\n2θ (degree)(220)\n(311)\n(222)\n(400)\n(422)\n(511)\n(440)\nFIG. 1: X-ray diffraction (XRD) spectrum of hydrothermally sy nthesized Fe 3O4nanoparticles.\n2The above equations have not been well tested by experiments due to the experimental\ndifficulties in producing assemblies of noninteracting sphere-like nano particles. When mag-\nnetic nanoparticles are closely packed and/or aggregate, the inte rparticle interactions are\nexpected to modify the magnetic behavior of the assembly. These in teractions can have a\ndipolar, Ruderman-Kittel-Kasuya-Yosida (RKKY), or a superexch ange character, depend-\ning on the character of an assemble. For magnetic nanoparticles em bedded in an insulating\nmatrix such as amorphous alumina [6] and amorphous SiO 2[7–9]), the dipolar interactions\nare the dominant ones [5]. Major theoretical and experimental effo rts have been focused\non the understanding of the role of the dipolar interactions [10]. In addition to granular\nmetal solids [6–9, 11], frozen ferrofluids have been used to investig ate the role of dipolar\ninteractions [12–15]. In these systems, the magnetic particles are held fixed in a frozen\ninsulating liquid. The degree of dilution in the liquid solvent controls the a verage particle\ndistance and therefore the strength of the interactions. Howev er, these studies have been\nlimited to ultra-fine particles with a low TBand to a temperature region well below the\nCurie temperature ( TC) of the magnetic nanoparticles. Since the particles are so fine, the\ncontribution of the surface anisotropy becomes significant and ev en dominant if they are not\nperfectly spherical particles [16].\nHere we report magnetic measurements on 40-nm sphere-like Fe 3O4nanoparticles using\na Quantum Design vibrating sample magnetometer. Coating Fe 3O4nanoparticles with SiO 2\neffectively eliminates magnetic interparticle interactions so that the coercive field HCin the\nhigh-temperature range between 300 K and TCfollows Eq. (2) for noninteracting randomly\noriented particles. FittingthedatawithEq. (2) yields K= (1.68±0.17)×105erg/cm3, which\nis slightly larger than the bulk magnetocrystalline anisotropy consta nt of 1.35 ×105erg/cm3\n(Ref. [17]). Moreover, the inferred mean particle diameter from th e fitting parameters is\nin good agreement with that determined from transmission electron microscope. Such a\ngood agreement between data and theory suggests that the ass emble of our SiO 2-coated\nsphere-like Fe 3O4nanopartles represents a good system of noninteracting random ly-oriented\nsingle-domain particles.\nSamples were synthesized by an improved hydrothermal route [18] using the following\nreagents and solvents: iron(III) chloride hexahydrate, diethyle ne glycol, sodium hydroxide,\nand iron(II) chloride tetrahydrate. The as-grown Fe 3O4nanoparticles were then coated\nwith SiO 2following the method of Ref. [19]. Magnetization was measured using a Quantum\nDesign vibrating sample magnetometer (VSM). The moment measure ment was carried out\n3a\n0510152025303540\n29 31 33 36 38 40 43 45 48 50 52 55 57 60Number of particles\nParticle diameter (nm)b\nFIG. 2: a) Transmission electron microscopy images of as gro wn Fe3O4nanoparticles. b) Histo-\ngraph of the particle size distribution. The mean diameter o f the particles is found to be about 43\nnm.\nafter the sample chamber reached a high vacuum of better than 9 ×10−6torr. The absolute\nmeasurement uncertainty in moment is less than 1 ×10−6emu.\nFigure 1 shows x-ray diffraction (XRD) spectrum of hydrothermally synthesized Fe 3O4\nnanoparticles. The positions and relative intensities of all diffraction peaks match well with\nthose of JCPDS card (19-0629) of magnetite with a lattice constan t of 8.367 ˚A.\nFigure2a shows Transmission electron microscopy (TEM) images of a s synthesized Fe 3O4\nnanoparticles. The TEM images were taken with a Hitachi model H-80 0 using an acceler-\nating voltage of 80 kV. The pictures demonstrate high quality and mo nodispersity of the\nas synthesized nanoparticles. Fig. 2b displays a histograph of the p article size distribution.\nFrom the histograph, we determine the mean diameter of the partic les to be about 43 nm.\n40.000.050.100.150.200.250.30\n300 400 500 600 700 800 900 1000χ (emu/g-Fe3O4)\nT (K)H = 50 Oea\n01020304050\n300 400 500 600 700 800 900 10001/χ (mole-Fe3O4/emu)\n T (K)b\nFIG. 3: a) Temperature dependencies of the ZFC and FC suscept ibilities for Fe 3O4nanoparticles\ncoated with SiO 2, which were measured in a field of 50 Oe. The downward arrow ind icates a Curie\ntemperature TCof 855 K. b) The reciprocal of the susceptibility 1 /χversus temperature. The\nCurie-Weiss fit (solid line) yields TC= 855 K.\nFigure 3a shows temperature dependencies of the ZFC and FC susc eptibilities for Fe 3O4\nnanoparticles coated with SiO 2, which were measured in a field of 50 Oe. The susceptibility\nwas calculated using the fact that the SiO 2-coated sample contains 75% Fe 3O4and 25%\nSiO2(in weight), which were determined from the measured room-tempe rature saturation\nmagnetizations of both as-grown and SiO 2-coated Fe 3O4samples. One can clearly see that\nthere are significant differences between the ZFC and FC susceptib ilities in the whole tem-\nperature range between 300 K and TC. This indicates that the blocking temperature of the\nnanoparticle assembly is higher than TC. A Curie temperature of 855 K is inferred from the\ndata (indicated by the downward arrow), which takes into account a small thermal lag. In\nFig. 3b, we plot the reciprocal of the susceptibility 1 /χversus temperature. It is apparent\nthat the susceptibility data above the Curie temperature can be we ll fitted by the Curie-\nWeiss law (solid line): χ=C/(T−TC) withTC= 855 K and the Curie-Weiss constant C\n5= 1.12 emu/K mole-Fe 3O4. It is remarkable that the TC’s determined from the data above\nand below TCare almost identical. The value of the Curie-Weiss constant corresp onds to\nan effecctive moment peff= 3.0µBper Fe 3O4(whereµBis the Bohr magneton).\n-100-500.050100\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5M (emu/g-Fe3O4)\nH (10 kOe)300 K\n456 K\n572 K\n687 Ka\n-20-15-10-505101520\n-300 -200 -100 0 100 200 300687 K\n572 K\n456 K\n300 KM (emu/g)\nH (Oe)b\nFIG. 4: a) Magnetic hysteresis loops at four different tempera tures. b) An expanded view of\nmagnetic hysteresis loops at four different temperatures.\nIn Fig. 4a we plot magnetic hysteresis loops at four different temper atures. The magne-\ntization at H= 10 kOe progressively decreases as the temperature increases a nd is almost\nsaturated at 10 kOe. In order to see the low field data more clearly, we show an expanded\nview oftheloopsinFig.4b. Itisclearthatthecocercive field HCalsodecreases progressively\nas the temperature increases.\nFigure 5a shows the saturation magnetization Msas a function of temperature. The Ms\nvalue at TC= 855 K is set to zero. The solid line is a fitted curve by Ms=A(T−TC)β\nwithβ= 0.354. The Msat room temperature is 86.6 emu/g-Fe 3O4, which is slightly lower\nthan the bulk value of 92 emu/g-Fe 3O4. Then the zero-temperature Msshould be about 90\n6emu/g-Fe 3O4.\n020406080100\n300 400 500 600 700 800 900 1000Ms (emu/g-Fe3O4)\nT (K)a\n050100150200250\n300 400 500 600 700 800 900 1000Hc (Oe)\nT (K)b\nFIG. 5: a) Temperature dependence of the saturation magneti zationMs. The solid line is a fitted\ncurve by Ms=A(T−TC)βwithβ= 0.354 and TC= 855 K. b) Temperature dependence of the\ncocercive field HC. The solid line is the fitted curve by Eq. (2) with the fitting pa rameters: TB=\n1243±153 K and K/Ms= 361±37 Oe.\nIn Fig. 5b, we plot HCversusTfor the sample. Since the HCvalues are found to\nbe slightly different from the positive and negative field data due to a r emanent magnetic\nfield of about 10 Oe in the superconducting magnet of the equipment , we take HCto be the\naverage of thetwo HCvalues obtainedfromthepositive andnegative field data, respectiv ely.\nThe solid line is a fitted curve by Eq. (2) with the fitting parameters TB= 1243±153 K\nandK/Ms= 361±37 Oe. Using Ms= 90 emu/g and K/Ms= 361±37 Oe, we find K=\n(1.68±0.17)×105erg/cm3, which is slightly larger than the bulk value of 1.35 ×105erg/cm3\n(Ref. [17]). If we use Ms= 77.8 emu/g (Ref. [20]), we obtain K= (1.45±0.14)×105erg/cm3,\nwhich is very close to the bulk value.\nIn order to further check whether our data are in quantitative ag reement with the the-\n7oretical predictions [Eqs. (1) and (2)], we use Eq. (1), the bulk K, and the inferred TBto\nestimate the average particlediameter. Since the average measur ing timefor each datapoint\nis 0.5 s, we can set τobs= 0.5 s. The value of τ0for Fe3O4nanoparticles was found [20] to be\n9×10−13s. Substituting τobs= 0.5 s,τ0= 9×10−13s, andTB= 1243±153 K into Eq. (1),\nwe findd= 40.4±2.1 nm, which is close to that (43 ±2 nm) deduced from TEM. Using\nthe diameter of 40 nm for as-grown Fe 3O4nanoparticles and the fact that the SiO 2-coated\nsample contains 75% Fe 3O4and 25% SiO 2(in weight), we estimate the average thickness of\nthe coated SiO 2layers to be about 6 nm.\nThe quantitative agreement between our data and the BL theory s uggest that the as-\nsembly of our 40-nm Fe 3O4nanoparticles coated with 6-nm SiO 2layers respresents a nearly\nideal system of noninteracting single-domain particles. The diamete r of 40 nm is well below\nthe maximum diameter of single-domain particles of about 128 nm (Ref s [21, 22]). The\nfact that the inferred Kvalue is close to the bulk one suggests that a contribution of the\nsurface anisotropy is small, in agreement with the sphere-like shape of particles (see the\nTEM pictures in Fig. 2a).\nIn summary, we have made high-temperature magnetic measureme nts on hydrothermally\nsynthesized Fe 3O4nanoparticles using a Quantum Design vibrating sample magnetomete r.\nCoating40-nmFe 3O4nanoparticleswithabout6-nmSiO 2effectively reducesmagneticinter-\nparticleinteractionssothatthecoercivefield HCfollowstheBLexpressionfornoninteracting\nsingle-domain magnetic particles. The quantitative agreement betw een our data and the BL\ntheory [4] suggests that the assemble of our SiO 2-coated sphere-like Fe 3O4nanopartles rep-\nresents a nearly ideal system of noninteracting randomly-oriente d single-domain particles.\nAcknowledgment: This work was supported by the National Natural Science Foun-\ndation of China (10874095), the Science Foundation of China, Zhej iang (Y407267,\n2009C31149), the Natural Science Foundation of Ningbo (2008B1 0051, 2009B21003), K.\nC. Wong Magna Foundation, and Y. G. Bao’s Foundation.\nawangjun2@nbu.edu.cn\nbgzhao2@calstatela.edu\n8[1]Magnetic Properties of Fine Particles , edited by J. L. Dormann and D. Fiorani (North-\nHolland, Amsterdam, 1992).\n[2] L. N´ eel, Ann. Geofis. 5, (1949) 99.\n[3] W. F. Brown, Jr., Phys. Rev. 130, (1963) 1677.\n[4] C. Bean and J. D. Livingston, J. Appl. Phys. 30, 120S (1959).\n[5] D. Kechrakos and K. N. Trohidou, Phys. Rev. B 58, (1998) 12169 and references therein.\n[6] J. L. Dormann, L. Bessais, and D. Fiorani, J. Phys. C 21, (1988) 2015.\n[7] F. G. Aliev, M. A. Correa-Duarte, A. Mamedov, J. W. Ostran der, M. Griersig, L. M. Liz-\nMarzan, and N. A. Kotov, Advanced Materials 11, (1999) 1006.\n[8] S. Mitra and K. Mandal, Materials and Manufacturing Proc esses22, (2007) 444.\n[9] H. T. Yang, D. Hasegawa, M. Takahashi, and T. Ogawa, Appl. Phys. Lett. 94, (2009) 013103.\n[10] J. L. Dormann, D. Fiorani, and E. Tronc, Adv. Chem. Phys. 98, (1997) 283.\n[11] C. L. Chien, in Science and Technology of Nanostructured Magnetic Materia ls, edited by G. C.\nHadjipanayis and G. A. Prinz, Vol. 259 of NATO Advanced Study Institute, Series B: Physics\n(Plenum, New York, 1991) p. 477.\n[12] R. W. Chantrell, M. El-Hilo, and K. OGrady, IEEE Trans. M agn.27, (1991) 3570.\n[13] M. El-Hilo, K. OGrady, and R. W. Chantrell, J. Magn. Magn . Mater. 114, (1992) 295.\n[14] W. Luo, S. R. Nagel, T. F. Rosenbaum, and R. E. Rosensweig , Phys. Rev. Lett. 67, (1991)\n2721.\n[15] S. Morup and E. Tronc, Phys. Rev. Lett. 72, (1994) 3278.\n[16] F. Bodker, S. Mrup, and S. Linderoth, Phys. Rev. Lett. 72, (1994) 282.\n[17] B. D. Cullity, in Introduction to Magnetic Materials (Addison-Wesley, Reading, MA, 1972) p.\n234.\n[18] J. Wang, M. Yao, G. J. Xu, P. Cui, and J. T. Zhao, Mater. Che m. Phys. 113, (2009) 6.\n[19] J. Wang, K. Zhang, and Q. W. Chen, J. Nanosci. Nanotechno .5, (2005) 772.\n[20] G. F. Goya, T. S. Berquo, F. C. Fonseca, and M. P. Morales, J. Appl. Phys. 94, (2003) 3520.\n[21] C. M. Sorensen, in Nanoscale Materials in Chemistry , edited by K. J. Klabunde (Wiley, New\nYork, 2001) p. 169.\n[22] C. Kittel, Phys. Rev. 70, (1946) 965.\n9"    },    {        "title": "1011.0939v1.Issues_with_J_dependence_in_the_LSDA_U_method_for_non_collinear_magnets.pdf",        "content": "Issues with J-dependence in the LSDA +Umethod for non-collinear magnets\nEric Bousquet1;2and Nicola Spaldin3\n1Materials Department, University of California, Santa Barbara, CA 93106, USA\n2Physique Théorique des Matériaux, Université de Liège, B-4000 Sart Tilman, Belgium and\n3Department of Materials, ETH Zurich, Wolfgang-Pauli-Strasse 10 CH-8093 Zurich, Switzerland\nWe re-examine the commonly used density functional theory plus Hubbard U(DFT +U) method\nfor the case of non-collinear magnets. While many studies neglect to explicitly include the exchange\ncorrection parameter J, or consider its exact value to be unimportant, here we show that in the case\nof non-collinear magnetism calculations the Jparameter can strongly affect the magnetic ground\nstate. We illustrate the strong J-dependence of magnetic canting and magnetocrystalline anisotropy\nby calculating trends in the magnetic lithium orthophosphate family LiMPO 4(M = Fe and Ni) and\ndifluoritefamilyMF 2(M=Mn, Fe, CoandNi). Ourresultscanbereadilyunderstoodbyexpanding\nthe usual DFT +Uequations within the spinor scheme, in which the Jparameter acts directly on\nthe off-diagonal components which determine the spin canting.\nKeywords: first-principles, LDA+U, non-collinear magnetism, magnetocrystalline anisotropy\nDensity functional theory (DFT) within the local den-\nsity (LDA) and generalized gradient (GGA) approxima-\ntions is widely used to describe a large variety of materi-\nals with good accuracy. The LDA and GGA functionals\noften fail, however, to correctly reproduce the properties\nof strongly correlated materials containing dandfelec-\ntrons. TheLDA +Uapproach–inwhichaHubbard Ure-\npulsion term is added to the LDA functional for selected\norbitals–wasintroducedinresponsetothisproblem,and\noften improves drastically over the LDA or GGA. Indeed,\nit provides a good description of the electronic properties\nof a range of exotic magnetic materials, such as the Mott\ninsulator KCuF 31and the metallic oxide LaNiO 22.\nTwo main LDA +Uschemes are in widespread use\ntoday: The Dudarev3approach in which an isotropic\nscreened on-site Coulomb interaction Ueff =U\u0000Jis\nadded, and the Liechtenstein1approach in which the U\nand exchange ( J) parameters are treated separately. The\nDudarev approach is equivalent to the Liechtenstein ap-\nproach with J= 04. Both the effect of the choice of\nLDA+Uscheme on the orbital occupation and subse-\nquent properties5, as well as the dependence of the mag-\nnetic properties on the value of U6, have recently been\nanalyzed. There has been no previous systematic study,\nhowever, of the effect of the Jparameter of the Liechten-\nstein approach in non-collinear magnetic materials. Here\nwe show that neither the approach of not explicitly con-\nsidering the Jparameter (as in the Dudarev implemen-\ntation), nor the assumption that its importance is bor-\nderline – a common approximation is to use J'10%U\nwithout careful testing – within the Liechtenstein imple-\nmentation are justified in the case of non-collinear mag-\nnets. We demonstrate that in the case of non-collinear\nantiferromagnets, the choice of Jcan strongly change the\namplitude of the spin canting angle (LiNiPO 4) or even\nmodify the easy axis of the system (LiFePO 4and FeF 2),\nwith consequent drastic effects on the magnetic suscep-\ntibilities and magnetoelectric responses.\nFirst we remind the reader how the UandJparame-\nters appear in the usual collinear spin LSDA +Uformal-ism. The LSDA +Ureformulation of the LSDA Hamilto-\nnian is usually written as:\nHLSDA +U=HLSDA +HU; (1)\nwhith\nH\u001b\nU=X\nm1;m2Pm1;m2V\u001b\nm2;m1; (2)\nwherePis the projection operator, \u001bis the spin index,\nand (on a given atomic site):\nV\"(#)\nm2;m1=\nX\n3;4\u0000\nVee\n1;3;2;4\u0000U\u000e1;2\u0000Vee\n1;3;4;2+J\u000e1;2\u0001\nn\"(#)\n3;4\n+\u0000\nVee\n1;3;2;4\u0000U\u000e1;2\u0001\nn#(\")\n3;4+1\n2(U\u0000J)\u000e1;2 (3)\nHereVee\n1;3;2;4=\nm1;m3\f\fVee\nm1;m3;m2;m4\f\fm2;m4\u000b\nare the\nelements of the screened Coulomb interaction (which can\nbe viewed as the sum of Hartree (direct) contributions\nVee\n1;3;2;4and Fock (exchange) contributions Vee\n1;3;4;2and\nn\u001b\ni;jare thed-orbital occupancies.\nIn the case of non-collinear magnetism, the formal-\nism is extended and the density is expressed in a two-\ncomponent spinor formulation:\n\u001a=\u0012\n\u001a\"\"\u001a\"#\n\u001a#\"\u001a##\u0013\n=\u0012\nn+mzmx\u0000imy\nmx+imyn\u0000mz\u0013\n(4)\nwherenis the charge density and m\u000bthe magnetiza-\ntion density along the \u000bdirection (\u000b=x;y;z). Using\nthe double-counting proposed by Bultmark et al.7, the\nLSDA +Upotential is then also expressed in the two-\ncomponent spin space as:\nVi;j= \nV\"\"\ni;jV\"#\ni;j\nV#\"\ni;jV##\ni;j!\n(5)\nwhereV\"\"andV##are equal to Eqs.3 and\nV\"#(#\")\nm2;m1=X\n3;4\u0000\n\u0000Vee\n1;3;4;2+J\u000e1;2\u0001\nn\"#(#\")\n3;4 (6)arXiv:1011.0939v1  [cond-mat.str-el]  3 Nov 20102\nFor collinear magnets, only V\"\"andV##(Eqs. 3) are\nrelevant since n\"#andn#\"are equal to zero, and Jaffects\nthepotentialmainlythroughaneffective U\u0000J. However,\nin the case of non-collinear magnetism, the n\"#andn#\"\nand hence the V\"#andV#\"(Eqs. 6) are non-zero. Then\nit is clear from Eqs. 6 that Jacts explicitly on the off-\ndiagonal potential components.\nNext, we show the effect of the choice of Jparameter\nin the family of lithium orthophosphates, LiMPO 4(M\n= Ni and Fe) and in the family of difluorites MF 2(M\n= Mn, Co, Fe and Ni). The orthophosphates crystallize\nin the orthorhombic Pnmaspace group with C-type an-\ntiferromagnetic (AFM) order. The difluorites crystalize\nin the tetragonal P42/mnmrutile structure with AFM\norder. We performed calculations within the Liechten-\nstein approach of the DFT +Uas implemented in the\nVASP code8,910withUandJcorrections applied to the\n3dorbitals of the M cations. In all cases we relaxed the\natomic positions until the residual forces on each atom\nwere lower than 10 \u0016eV/Å at the experimental volume\nand cell shape reported in Tab. I, taking into account\nthe spin-orbit interaction. We found good convergence\nof the non-collinear spin ground state with a cutoff en-\nergy of 500 eV on the plane wave expansion and a k-point\ngrid of 2\u00024\u00024for the orthophophates and 4\u00024\u00026for\nthe difluorites.\na b c Ref.\nLiFePO 410.332 6.010 4.692 11\nLiNiPO 410.032 5.854 4.677 12\nNiF 24.650 4.650 3.084 13\nFeF 24.700 4.700 3.310 14\nMnF 24.650 4.650 3.084 15\nCoF 24.695 4.695 3.179 16\nTABLE I. Experimental cell parameters (Å) used in the sim-\nulations of LiMPO 4phosphates and MF 2difluorites.\nFirst, we focus on LiNiPO 4, which is known experi-\nmentally to be C-type AFM, with an easy-axis along the\ncdirection and a small A-type AFM canting of the spins\nalong theadirection (CzAxground state with mm’m\nmagnetic point group)17. Performing calculations within\nthe LSDA +Umethod with J= 0, we find that we cor-\nrectly reproduce the CzAxground state with a rather\nsmallUsensitivity of the magnetocrystalline anisotropy\nenergy (MCAE) and the spin canting; this finding is\nconsistent with a previous report using the GGA func-\ntional18. However, our calculated canting angle of 1.6\u000e\nforU= 5eV andJ= 0eV severely underestimates\nthe experimental value of 7.8\u000e17. In Fig.1 (a) we show\nthe evolution of the canting angle with JatU= 5eV.\nWe find that the canting angle is extremely sensitive to\nthe value of J– in fact it is/J3– changing from 1.6\u000e\natJ= 0eV to 7.8\u000eatJ= 1:7eV. To reproduce the\nexperimental value of the canting angle we need to use\nthe rather large Jvalue of 1.7 eV. The dependence of\nthe canting angle on Jis consistent with Eqs. 6, as the\nFIG. 1. (a) Calculated LSDA +Ucanting angle of LiNiPO 4\nversus JforU= 5eV. The experimental value of the canting\nangle is equal to 7.8\u000e17. (b) Energy versus canting angle in\nLiNiPO 4forU= 5eV and J= 0eV (red circles), U= 5eV\nandJ= 1eV(bluetriangles), Ueff= 4eV(greencrosses)and\nU= 5eV and J= 1eV but by fixing J=0 eV in Eqs.6 (pink\nsquares). The zero energy reference is chosen at zero canting\nangle. (c) Magnetocrystaline anisotropy energy (MCAE) be-\ntween the aandborientations of the magnetic moments of\nLiFePO 4. The experimental borientations is taken as energy\nreference.\noff-diagonal elements n\"#andn#\"are non-zero when the\nspins cant away from the easy axis.\nIn Fig. 1 (b) we report the energy versus the canting\nangle in LiNiPO 4forU= 5eV and different values of J.\nWe see that as Jis increased from J= 0eV toJ= 1eV\n(red circles and blue triangles) the minimum of the en-\nergy shifts to larger canting angle, with a stronger gain\nof energy with respect to the uncanted reference. When\nperforming the same calculation with Ueff=4 eV (green\ncrosses in Fig. 1) we obtain results that are very similar\nto the case U= 5eV andJ= 0eV, which is formally\nequivalent to the Dudarev approach with Ueff= 5eV.\nThese comparisons confirm that varying Uhas a minimal\neffect on the canting angle in LiNiPO 4and also that the\nuseoftheLiechtensteintreatmentof Jisextremlyimpor-\ntant. To further confirm the direct relationship between\nthe spin canting and the Jparameter, we performed the\nsame calculations with U= 5eV andJ= 1eV but we\nartificially fixed J= 0eV only in Eqs. 6 (pink squares in3\nFig.1 (b)). We clearly see that the energy versus canting\nangle is strongly affected by this modification and in fact\nthe canting is almost removed.\nSimilarJdependence of the canting angle was also re-\nported previously for Ni2+in BaNiF 419; in Ref. 19 it was\nfound that at U= 5eV, the canting varies from 2\u000eto 3\u000e\nwhenJis varied from 0 eV to 1 eV. In both LiNiPO 4and\nBaNiF 4the Ni ion is divalent, with a d8configuration,\nand octahedrally coordinated. To investigate the gener-\nality of this behavior, we next consider the case of the\ncanted-spin antiferromagnet NiF 2, in which the Ni ion is\nin the same coordination environnement as in BaNiF 4.\nExperimentally, NiF 2has the spins aligned preferentially\nintheplaneperpendiculartothe caxiswithaslightcant-\ning from antiparallel alignment by an estimated \u00180.5\u000eat\nlow temperatures13. Performing LSDA +Ucalculations\nattheexperimentalvolumeandwith U= 5eVandJ= 0\neV we indeed obtain the easy axis perpendicular to the caxis and a small canting of 0.3\u000e, in excellent agreement\nwiththeexperiments. IncontrasttothecaseofLiNiPO 4,\nhowever, we find that the amplitude of the canting angle\nis almost insensitive to the value of Jwith just a small\ntendency to be reduced when Jincreased. This insen-\nsitivity of the canting angle to the value of Jin NiF 2\ncan be understood from the fact that in this compound\nthe magnetism is almost collinear, and therefore the off-\ndiagonal elements of the occupation matrix, n\"#andn#\",\nare close to zero. Inspection of Eqs. 3 then shows that\nthe effect of Jis reduced largely to the diagonal part of\nthe potential where the Uparameter is dominant.\nTo summarize our findings for the Ni-based com-\npounds, in cases where the experimental canting is large\n(2-3\u000e) we find a strong J-dependence of the canting an-\ngle, which increases with increasing J; when the canting\nisweakexperimentallythe J-dependenceismuchweaker.\nFIG. 2. Magnetocrystaline anisotropy energy versus the Jparameter of (a) FeF 2(Experimental value from Ref.20), (b) NiF 2,\n(c) MnF 2(Experimental value from Ref.21) and (d) CoF 2(“sc“ are calculations with Co semi-cores while ”no sc” are calculations\nwithout Co semi-cores). The MCAE reported here is the energy between the aandcorientation of the spins, the energy of the\ncorientation is taken as reference.\nNext we analyse the effect of Jon the behavior on\nthe corresponding divalent iron compounds. We begin\nwith LiFePO 4, which is known experimentally to be a C-\ntype AFM with an easy axis along the bdirection and no\nobserved canting of the spins22,23(Cyground state with\nmmm’magnetic point group). Our calculations within\nthe LSDA +Ufunctional at the commonly used values of\nU= 4eV andJ= 0eV for Fe2+yield the correct C-\ntype AFM order but find the easy axis incorrectly along\ntheadirection. Now we switch to J6= 0eV and report\nin Fig. 1.c the MCAE between the bandadirections,\ncalculated by turning all the spins homogenously from\ntheCyto theCxdirection. We find that the MCAE is\napproximately linear with J, but with rather dramatic\nqualitative dependence: while at J= 0eV the easy axis\nis along the adirection (negative MCAE) the MCAE is\nalmost reduced to zero around J= 0:5eV and the easy\naxis changes to the bdirection for J>\u00180:5eV (positive\nMAE). To reproduce the experimental easy axis ( Cy) a\nvalue ofJgreater than 0.58 eV is required. In the caseswhere the correct easy axis is reproduced ( Cy) we do not\nobserve any canting of the spins, in agreement with the\nexperimental magnetic point group mmm’.\nAsasecondexamplewithFe2+, weanalysetheeffectof\nJon the MCAE of FeF 2. Experimentally FeF 2is known\nto have its spin magnetization parallel to the tetragonal c\naxis with a rather large MCAE of about +4800 \u0016eV24,25.\nIn Fig. 2.a we report the LSDA +UMCAE energies with\nrespect toJat four different values of U(3, 4, 5 and 6\neV). All the calculations with J= 0eV give the wrong\neasy axis (spins are perpendicular to c) with a huge error\nin the MCA energy (MCAE from -16000 to -26000 \u0016eV\nforUgoing from 3 to 6 eV). Increasing the value of Jin\nthe range of 0–0.5 eV has the tendency to strongly reduce\nthis error with a linear increase of the MCAE with Jas\nwe found above for LiFePO 4. However beyond J'0:5\nthe increase of the MCAE is reduced and the evolution\nbecomes more complex with the appearance of two max-\nima before a drastic decrease beyond J'1:3eV. The\ncorrect easy axis (MCAE >0) is only obtained for a very4\nsmall range of UandJvalues, and the amplitude of the\nMCAE is correct over an even smaller range. This Jde-\npendence of the MCAE is again consistent with Eqs.3-6.\nFrom Eq.4 it is clear that when changing the orienta-\ntion of the spins from the zaxis to the xoryaxis the\noff-diagonal parts of Eq.4 become non-zero resulting in a\ndirect effect of Jon the MCAE from Eqs.6.\nWe also performed the same analysis of the MCAE\nfor NiF 2(Fig.2.b), MnF 2(Fig.2.c) and CoF 2(Fig.2.d).\nMnF 2and CoF 2have the same easy axis as FeF 2while\nNiF 2has its easy axis perpendicular to the cdirection.\nThe easy axis is well reproduced for all three compounds\natJ= 0eV. As for FeF 2, the amplitudes of the MCAE\ndepend strongly on Jbut with a completely different\ntrend in each compound. For MnF 2and FeF 2the experi-\nmentalvaluecanbereproducedbyadjustingthevaluesof\nUandJ. In the case of CoF 2and NiF 2no experimental\nvalues are available. For CoF 2we also performed calcula-\ntionswithandwithoutCosemi-coresstates(Fig.2.d)and\nfindastrongdifferenceinthemagnitudeoftheMCAEfor\nthe two cases. For FeF 2we also performed calculations\nwithin the GGA functional (black pentagons in Fig.2.a)\nand obtained a completely different Jdependence than\nthose calculated with the LDA functional. These com-\nparisons illustrate the difficulty of extracting a general\nrule about the Jdependence of the MCAE.\nOur results reveal a problem with the predictability of\nthe LSDA +Umethod for non-collinear magnetic materi-als: A strong dependence of the MCAE and spin canting\nangles on the values of Uand particularly Jthat are\nused in the calculation. Since properties such as magne-\ntostriction, piezomagnetic response, magnetoelectric re-\nsponse and exchange bias coupling are directly related to\nMCAEs and spin canting, it is of primary importance\nto reproduce these quantities accurately. At the mo-\nment, the most reliable, although not entirely satisfac-\ntory, option appears to be a fine tuning of the UandJ\nparameters by adjustment to reproduce experimentally\nmeasured anisotropies and canting angles; there is some\nevidence to suggest that properties such as magnetoelec-\ntric responses are then in turn well reproduced26. Future\nstudies might explore methodologies for self-consistent\ncalculation of the Jparameter, or the predictions of new\ndescriptions of the exchange and correlation such as the\nhybridfunctionals27. Ontheflipside, itisclearthatnon-\ncollinear magnetic systems provide a challenging case for\ntesting the correctness of new exchange correlation func-\ntionals within the density functional formalism.\nAcknowledgments: This work was supported by the\nDepartment of Energy SciDAC DE-FC02-06ER25794.\nWe made use of computing facilities of TeraGrid at the\nNational Cen-ter for Supercomputer Applications and of\nthe California Nanosystems Institute with facilities pro-\nvided by NSF grant No. CHE-0321368 and Hewlett-\nPackard. EB also acknowledges FRS-FNRS Belgium and\nthe ULg SEGI supercomputer facilities.\n1A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, Phys.\nRev. B 52, R5467 (Aug 1995).\n2K.-W. Lee and W. E. Pickett, Phys. Rev. B 70, 165109\n(Oct 2004).\n3S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J.\nHumphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (Jan\n1998).\n4P. Baettig, C. Ederer, and N. A. Spaldin, Phys. Rev. B 72,\n214105 (2005).\n5E. R. Ylvisaker, W. E. Pickett, and K. Koepernik, Phys.\nRev. B 79, 035103 (Jan 2009).\n6S. Y. Savrasov, A. Toropova, Kat, K. M. I., L. A. I., A. V.,\nand G. Kotliar, Z. Kristallogr. 220, 473 (2005).\n7F. Bultmark, F. Cricchio, O. Grånäs, and L. Nordström,\nPhys. Rev. B 80, 035121 (2009).\n8G.KresseandJ.Furthmüller,Phys.Rev.B 54,11169(Oct\n1996).\n9G. Kresse and D. Joubert, Phys. Rev. B 59, 1758 (Jan\n1999).\n10We note that LSDA +Udouble-couting term taking into\naccound the magnetization density as proposed by Bult-\nmark et al.7is mandatory within non-collinear magnetism\ncalculations. This is not necessarily done in the present\nimplementation of other codes.\n11V. A. Streltsov, E. L. Belokoneva, V. G. Tsirelson, and\nN. K. Hansen, Acta Cryst. B 49, 147 (Apr 1993).\n12I. Abrahams and K. S. Easson, Acta Cryst. C 49, 925\n(1993).\n13M. T. Hutchings, M. F. Thorpe, R. J. Birgeneau, P. A.Fleury, and H. J. Guggenheim, Phys. Rev. B 2, 1362 (Sep\n1970).\n14M. J. M. de Almeida, M. M. R. Costa, and J. A. Paixão,\nActa Cryst. B 45, 549 (1989).\n15T. Oguchi, Phys. Rev. 111, 1063 (Aug 1958).\n16N. J. O’Toole and V. A. Streltsov, Acta Cryst. B 57, 128\n(2001).\n17T. B. S. Jensen, N. B. Christensen, M. Kenzelmann, H. M.\nRønnow, C. Niedermayer, N. H. Andersen, K. Lefmann,\nJ. Schefer, M. v. Zimmermann, J. Li, J. L. Zarestky, and\nD. Vaknin, Phys. Rev. B 79, 092412 (2009).\n18K. Yamauchi and S. Picozzi, Phys. Rev. B 81, 024110\n(2010).\n19C. Ederer and N. A. Spaldin, Phys. Rev. B 74, 020401 (Jul\n2006).\n20M. E. Lines, Phys. Rev. 156, 543 (Apr 1967).\n21U. Gäfvert, L. Lundgren, P. Nordblad, B. Westerstrandh,\nand . Beckman, Sol. State Comm. 23, 9 (1977).\n22Zimmermann, A. S., Van Aken, B. B., Schmid, H., Rivera,\nJ.-P., Li, J., Vaknin, D., and Fiebig, M., Eur. Phys. J. B\n71, 355 (2009).\n23G. Liang, K. Park, J. Li, R. E. Benson, D. Vaknin, J. T.\nMarkert, and M. C. Croft, Phys. Rev. B 77, 064414 (2008).\n24C. Rudowicz, J. Phys. Chem. Solids 38, 1243 (1977).\n25R. C. Ohlmann and M. Tinkham, Phys. Rev. 123, 425\n(1961).\n26K. Delaney, E. Bousquet, and N. A. Spaldin,\narXiv:0912.1335v2(2010).\n27J. Heyd and G. E. Scuseria, The Journal of Chemical5\nPhysics 120, 7274 (2004)."    },    {        "title": "1012.4690v1.Detection_of_stacking_faults_breaking_the__110___1_10__symmetry_in_ferromagnetic_semiconductors__Ga_Mn_As_and__Ga_Mn__As_P_.pdf",        "content": "Detection of stacking faults breaking the [110]/[1 \u001610] symmetry in ferromagnetic\nsemiconductors (Ga,Mn)As and (Ga,Mn)(As,P)\nM. Kopeck\u0013 y,1J. Kub,1F. M\u0013 aca,1J. Ma\u0014 sek,1O. Pacherov\u0013 a,1\nB. L. Gallagher,2R. P. Campion,2V. Nov\u0013 ak,3and T. Jungwirth3, 2\n1Institute of Physics ASCR, v.v.i., Na Slovance 2, 182 21 Praha 8, Czech Republic\n2School of Physics and Astronomy, University of Nottingham, Nottingham NG7 2RD, United Kingdom\n3Institute of Physics ASCR, v.v.i., Cukrovarnick\u0013 a 10, 162 53 Praha 6, Czech Republic\nWe report high resolution x-ray di\u000braction measurements of (Ga,Mn)As and (Ga,Mn)(As,P) epi-\nlayers. We observe a structural anisotropy in the form of stacking faults which are present in the\n(111) and (11 \u00161) planes and absent in the ( \u0016111) and (1 \u001611) planes. The stacking faults produce no\nmacroscopic strain. They occupy 10\u00002\u000010\u00001per cent of the epilayer volume. Full-potential den-\nsity functional calculations evidence an attraction of Mn Gaimpurities to the stacking faults. We\nargue that the enhanced Mn density along the common [1 \u001610] direction of the stacking fault planes\nproduces su\u000eciently strong [110]/[1 \u001610] symmetry breaking mechanism to account for the in-plane\nuniaxial magnetocrystalline anisotropy of these ferromagnetic semiconductors.\nPACS numbers: 61.72.Dd, 75.50.Pp, 78.55.Cr\nThe rich phenomenology of magnetocrystalline\nanisotropies in ferromagnetic (III,Mn)V semiconduc-\ntors has been the prerequisite of numerous studies\nof magnetic, magneto-transport, and magneto-optical\nphenomena and of prototype semiconductor spintronic\ndevices [1]. The in-plane biaxial anisotropy as well\nas the out-of-plane uniaxial anisotropy terms are well\nunderstood based on the periodic crystal structure char-\nacteristics of the epilayers [2]. The former term is due\nto the cubic symmetry of the host III-V semiconductor\nand the latter term due to the lattice-matching strain\ninduced by the di\u000berence between lattice parameters of\nthe free-standing (III,Mn)V crystal and the substrate.\nThe most extensively studied material is (Ga,Mn)As\ngrown on GaAs, in which the compressive strain in\nthe ferromagnetic (Ga,Mn)As epilayer makes the out-of-\nplane orientation of magnetization energetically unfavor-\nable. It has been recognized from the early studies of\nthese in-plane ferromagnets that the biaxial anisotropy\nterm is complemented by an additional uniaxial term\nbreaking the symmetry between [110] and [1 \u001610] crystal\ndirections. While the presence of this term and its com-\npetition with the biaxial anisotropy term have played a\nkey role in the research of ferromagnetic semiconductors,\nincluding studies of electrical or optical manipulation of\nthe magnetic state [1], the microscopic origin of the in-\nplane uniaxial anisotropy has remained elusive.\nWe have considered the following guidelines when\nsearching for the uniaxial in-plane symmetry breaking\nmechanism: (i) The corresponding uniaxial magnetocrys-\ntalline anisotropy is not a surface or interface e\u000bect but\na bulk phenomenon. (ii) No strain component has been\ndetected in the epilayers that would break the symme-\ntry between the two in-plane diagonals. (iii) No system-\natic dependence has been identi\fed in the in-plane uni-\naxial magnetic anisotropy on the growth induced lattice-\nmatching strain, as seen e.g. from the comparison of com-pressively strained (Ga,Mn)As/GaAs epilayers and ten-\nsile strained (Ga,Mn)(As,P)/GaAs epilayers. (iv) From\nthe e\u000bective modeling of the in-plane uniaxial anisotropy\nit has been concluded that the symmetry breaking mech-\nanism is related to the high Mn Gadoping [2].\nIn this paper we report experimental observation and\ntheoretical investigation of stacking faults in (Ga,Mn)As\nand (Ga,Mn)(As,P) which break the in-plane [110]/[1 \u001610]\nsymmetry and whose characteristics are consistent with\nthe above guidelines. The experimentally estimated\ndensity of the stacking faults and the theoretically in-\nferred attraction of Mn Gato these lattice defects yields\na strength and sense of the symmetry breaking mecha-\nnism which we compare to the broken crystal symmetry\ndue to an in-plane uniaxial strain. The latter symme-\ntry breaking mechanism has been commonly used as an\ne\u000bective parameter to theoretically model the uniaxial\nmagnetocrystalline energy in unpatterned epilayers or as\na real tool to control the anisotropy in microstructured\n\flms or in epilayers attached to piezostressors [2].\nMeasured (Ga,Mn)As and (Ga,Mn)(As,P) samples\nwere grown by low-temperature (200-230\u000eC) molecu-\nlar beam epitaxy (LT-MBE) on a GaAs substrate and\nbu\u000ber layer. For more details on the sample growth see\nRefs. [3, 4]. X-ray experiments were carried out on the\ndi\u000braction beamline at the ELETTRA synchrotron fa-\ncility in Trieste. Samples were mounted on a two-axis\ntilt platform and aligned in the way that the normal to\nthe sample surface coincides with the rotation axis of the\ndi\u000bractometer; the rotation angle is \u001e. In this geometry,\nthe glancing angle \u0012between the beam and the sample\nsurface remained constant during the \u001e-scan. The energy\nof the incident beam was set to 10.3 keV, i.e. just below\ntheKabsorption edge of gallium, in order to minimize\nthe absorption in the sample and to avoid \ruorescence.\nThe beam size was set to 500 \u0016m (horizontal) and 20 \u0016m\n(vertical) using two pairs of slits in front of the sample.arXiv:1012.4690v1  [cond-mat.mtrl-sci]  21 Dec 20102\nA small vertical aperture guarantees the elimination of\nthe undesired scattering from sample borders. The area\ndetector Pilatus 2M (1475 \u00021679 pixels of the size of\n172\u0002172\u0016m2in 3\u00028 modules, dynamic range of 20\nbits) made it possible to collect the weak di\u000buse scatter-\ning patterns using the exposure rate of 5 s per frame. One\nscan was composed of 720 frames collected with the an-\ngular step \u0001 \u001e= 0:5\u000e. The frames were put together and\nthe intensity maps in the large volume of the reciprocal\nspace were constructed.\nIn Figs. 1(a),(b) we show measured data for 500 nm\nthick 5% Mn-doped (Ga,Mn)As epilayer. The large sam-\nple thickness allows us to set the glancing angle \u0012to 0.3\u000e,\nwhich is slightly above the critical angle \u0012c= 0:25\u000e. Un-\nder these conditions a large portion of the (Ga,Mn)As\nepilayer volume is illuminated while only a small por-\ntion of the beam penetrates into the substrate and the\nscattering from the substrate is negligible. Two cross-\nsections through the reciprocal space perpendicular to\neach other are shown in Figs. 1(a),(b). The cross-section\nin panel 1(a) contains only di\u000braction spots along with\ntruncation rods perpendicular to the sample surface. On\nthe other hand, the cross-section shown in panel 1(b)\ncontains additional di\u000buse streaks in directions [111] and\n[11\u00161], indicating the presence of stacking faults in planes\n(111) and (11 \u00161).\nIn Fig. 2(a) we present a detailed image of the area\nnear the -111 di\u000braction and compare in Fig. 2(b) with\nthe same area measured at \u0012= 0:2\u000e, i.e., slightly be-\nlow the critical angle. In the latter image the thick-\nness of the illuminated (Ga,Mn)As surface layer is only\n\u001810 nm. Consistently, the truncation rod perpendicular\nto the sample surface is more extended in Fig. 2(b) than\n2(a). The comparison of the two images demonstrates\nthat the stacking faults occupy a signi\fcant part of the\nepilayer volume since the intensity of the di\u000buse [111] and\n[11\u00161] streaks is signi\fcantly larger in Fig. 2(a) than 2(b).\nWe have performed additional measurements at \u0012=\n0:2\u000eon (Ga,Mn)As and (Ga,Mn)(As,P) epilayers of\nwidth ranging from 35 to 100 nm, as-grown and annealed,\nand with Mn-doping up to 10% and P-doping of 9%. In\nall cases we obtained results similar to Fig. 2(b) with the\ndi\u000buse streaks only in directions [111] and [11 \u00161]. No dif-\nfuse streaks in any direction were observed in a reference,\nLT-MBE grown undoped 500 nm thick GaAs epilayers.\nFrom these data we can conclude that the stacking faults\nspan across the epilayer volume, are related to the Mn-\ndoping, are present in the same planes in samples with\nsmaller (annealed GaMnAs) and larger (as-grown GaM-\nnAs) magnitude of the growth lattice-matching strain\nand in epilayers with both compressive (GaMnAs) and\ntensile (GaMnAsP) strain. We also point out that the\nstacking faults are present in thick epilayers as well as in\nthin, annealed high magnetic quality \flms.\nFrom the modeling of the measured x-ray scattering\npatters (Figs. 1(c) and 2(c)), we identi\fed the micro-\na)\nb)\nc)FIG. 1: (a,b)Cuts through the reciprocal space of the mea-\nsured di\u000braction intensity maps. The vertical truncation rods\nperpendicular to the sample surface are due to \fnite thickness\nof the illuminated \flm. The streaks in directions [111] and\n[11-1] represent the di\u000braction on stacking faults in planes\n(111) and (11-1), resp. (c)The calculated x-ray di\u000braction\nconsidering the distribution of the stacking faults described\nin the text for the cut through the reciprocal space as in (b).\nscopic nature of the stacking faults. In the unperturbed\nGaAs zinc-blende crystal structure, the plane stacking\nalong the [111] direction (or any other body diagonal)\nis characterized by a repeating a-b-c sequence of Ga-As\nplanes, as shown in Fig. 3(a). A stacking fault along the\n[111] diagonal shown in Fig. 3(b) corresponds to one of\nthe planes missing in the stacking. The fault can be also\nviewed as replacing the zinc-blende stacking by two in-\nterpenetrating wurtzite stackings highlighted by the two\nred (vertical) bars in Fig. 3(b). A stacking fault due\nto an extra plane, or more separated two wurtzite se-\nquences, is depicted in Fig. 3(c). A removed and im-\nmediately inserted plane creates a fault comprising two\nneighboring wurtzite stackings, as shown in Fig. 3(d).3\nc)\nqx/a*-‐2 -‐1 0 1 2log Intensity (arb. U.)\na)\nb)-111\n-111T>Tc\nT<Tcexp. data num. simulation\nFIG. 2: Detail of the di\u000buse scattering in the vicinity of the\nre\rection -111 measured at (a) \u0012= 0:3\u000eand (b)\u0012= 0:2\u000e. (c)\nMeasured intensity pro\fle along the [111] direction passing\nthrough the re\rection -111 and corresponding numerical sim-\nulation. Here qxis thexco-ordinate of the di\u000braction vector\nanda\u0003is the reciprocal lattice unit cell vector.\nOur experimental data could not be \ftted by the stack-\ning faults in 3(b) and 3(c). As seen from the compari-\nson with Fig. 3(a), these two faults yield a zinc-blende\nstacking after the fault which is shifted by one plane with\nrespect to the unperturbed lattice. It disrupts the match-\ning of the epilayer with these stacking faults to the sub-\nstrate on a macroscopic scale and, therefore, the faults\nare unlikely to form. Their absence is also consistent\nwith the absence of an in-plane uniaxial strain which we\ncon\frmed experimentally in our epilayers with accuracy\n\u001810\u00005. (We compared average lattice parameters in the\n[110] and [1 \u001610] directions by using relative position of 17\nBragg re\rections of the Mn-doped epilayer with respect\nto these of the GaAs substrate.) The stacking fault in\nFig. 3(d), on the other hand, produces no macroscopic\nstrain and indeed our \fttings show that this fault repre-\nsents\u001890% of the stacking faults present in the mea-\nsured epilayers. The remaining faults, producing also no\nmacroscopic strain outside the fault, have one or more\nzinc-blende a-b-c sequences between the removed and in-\nserted plane and their probability with increasing num-\nber of inner a-b-c sequences was \ftted by a \u0000-function\ndistribution.\nAs pointed out in the previous paragraph, there is no\nexperimentally observable strain in the (Ga,Mn)As epi-\nlayers, within the \u001810\u00005error bar which would break\nthe [110]/[1 \u001610] crystal symmetry. The uniaxial in-plane\nstrain has been used as a hypothetical symmetry break-\ning mechanism to account in microscopic theory calcula-\ntions for the experimentally observed uniaxial in-plane\nmagneto-crystalline anisotropy [2, 6]. Typical magni-\ntudes of the strain required by the modeling to reproduce\nthe measured magnitude of the magnetic anisotropy are\n\u001810\u00004, i.e., an order of magnitude larger than the ex-\nperimental error of our measurements. Based on this\nwe can exclude strain as the real microstructural origin\nof the [110]/[1 \u001610] magnetic anisotropy. Results of our\nab initio calculations, described in the following para-\na)\nb)c)d)\n123456789\n0\ne)FIG. 3: (a) Zinc-blende stacking in an ideal GaAs lattice.\n(b)-(d) Stacking faults with one removed, added, or removed\nand immediately inserted planes, resp. (e) Calculations of\nthe total energy of the supercell with the respective stacking\nfault as a function of the position of the substitutional Mn Ga.\nTotal energies are plotted with respect to the energy of the\nsupercell with Mn Gaat the 0-th layer in the supercell.\ngraphs, suggest that the observed stacking faults pro-\nvide the searched for microstructural symmetry breaking\nmechanism. Our supercell calculations were performed\nusing the full-potential linearized-augmented-plane-wave\nmethod with the improved local-density approximation\nby the generalized-gradient approximation (LAPW-GGA\nWIEN package [5]). The calculated formation energies\nof the stacking faults shown in Figs. 3(b)-(d) for an un-\ndoped GaAs are in the range of 13-15 mRy. In Fig. 3(e)\nwe plot the dependence of the formation energy of the\nstacking faults on the position of the substitutional Mn Ga\nimpurity along the [111] (or [11 \u00161]) crystal direction in the\nsupercell. By comparing the results for the three consid-\nered stacking faults (Figs. 3(b)-(d)) we can conclude that\nMnGais attracted to the wurtzite stacking perturbation\nin the lattice and the energy pro\fle can be understood\nas a convolution of attractive potentials due to the indi-\nvidual wurtzite perturbations. Remarkably, in our super-\ncells corresponding to 25% of Mn Gaimpurities, the depth\nof the attractive potential energy well is as large as 20%\nof the formation energy of the stacking fault in GaAs.\nWe conclude that Mn impurities have a strong tendency\nto decorate the stacking faults and that by this process\nthe Mn impurities signi\fcantly enhance the probability4\n[1-‐10][1-‐10]\n[110]\n[110]Anisotropic Mn distribution\nb)d)a) c)\nStrain\n[1-‐10][110]\nFIG. 4: (a) Schematic image of uniformly randomly dis-\ntributed Mn (red (dark) dots) in the epilayer plane in an\nunstrained lattice with cubic symmetry. (b) Schematics of\nthe planes in which the stacking faults are observed by x-ray\ndi\u000braction. (c) Schematic image of anisotropic Mn distribu-\ntion in the epilayer plane in an unstrained lattice which results\nfrom the tendency of Mn to decorate the stacking faults. (d)\nSchematic image of uniformly distributed Mn on the lattice\nsites in the presence of a uniaxial in-plane symmetry breaking\nstrain. (c) and (d) are analogous from the perspective of the\nanisotropic mean separation of Mn in the epilayer plane.\nof the formation of the stacking faults in the epilayer.\nIn schematic diagrams in Fig. 4 we illustrate the in-\nferred connection between the Mn Gadecorated stacking\nfaults and the uniaxial magnetocrystalline anisotropy.\nFig. 4(a) shows schematically the biaxial symmetry in\ncase of unperturbed lattice and uniform distribution of\nMnGaimpurities. Since the stacking faults are observed\nonly in the (111) and (11 \u00161) planes (Fig. 4(b)) and Mn Ga\nis attracted to the stacking faults, we can expect higher\nmean density of the Mn Gaimpurities along the com-\nmon [1 \u001610] direction of the two planes. This leads to an\nanisotropic distribution of Mn Gaimpurities in the epi-\nlayer plane with a smaller mean distance between impu-\nrities in the [1 \u001610] direction than [110] direction.\nIn Fig. 4(d) we illustrate that the non-uniform Mn Ga\ndistribution in an unstrained square lattice is from the\nperspective of the mean Mn Gaseparation analogous to\nthe case of a uniform Mn distribution and a strained\nlattice. The latter case has been considered previously\nin the e\u000bective modelings of the corresponding uniaxial\nmagnetic anisotropy. Since the density of the stacking\nfaults we observe is \u00180:01\u00000:1%, the corresponding\nsymmetry breaking mechanism is strong enough to ex-\nplain the uniaxial magnetic anisotropy. (Compare the\ndensity of the stacking faults with the \u001810\u00004strain used\nin the e\u000bective modeling of the magnetic anisotropy.) We\nnote that detection of the stacking faults and of the in-\ncreased Mn Gadensity on these defects is likely unachiev-\nable by direct atomic resolution scanning techniques.\nThis underlines the merit of our approach, combining\nthe macro-scale x-ray measurements with ab initio totalenergy calculations, for identifying the subtle symmetry\nbreaking mechanism.\nThe sign of the magnetic anisotropy in the Mn-doped\nmagnetic epilayers is not uniquely determined by the sign\nof the underlying broken structural symmetry; it can vary\nfor the given sense of the broken symmetry as a function\nof, e.g., the hole density in the magnetic \flm. Typi-\ncally, however, the [1 \u001610] direction is magnetically eas-\nier than [110] direction. The common [1 \u001610] direction of\nthe stacking fault planes then coincides with the typical\neasy magnetic direction. If again considering the anal-\nogy with the case of uniform Mn Gadistribution and in-\nplane strained lattice, this observation provides further\nsupport for the stacking-fault origin of the uniaxial in-\nplane anisotropy. From the modelings considering the\nin-plane strain, as well as from experiments on intention-\nally in-plane strained (Ga,Mn)As epilayers by lithograph-\nical pattering or piezostressors [2], it has been concluded\nthat the magnetic easy direction is along the smaller\nlattice parameter, i.e., along the direction with smaller\nmean separation between Mn Gaimpurities. Our results\nshow that in unstrained epilayers the stacking faults yield\nsmaller mean separation between Mn Gaimpurities along\nthe in-plane axis which also coincides with the typical\nuniaxial magnetic easy direction. We conclude that all\ncharacteristics of the experimentally observed stacking\nfaults in the studied (III,Mn)V epilayers are consistent\nwith the guidelines (i)-(iv) summarized in the introduc-\ntion for identifying the microstructural [110]/[1 \u001610] sym-\nmetry breaking mechanism in these magnetic semicon-\nductors.\nWe acknowledge support from EU Grants FP7-\n215368 SemiSpinNet, FP7-214499 NAMASTE,\nand Czech Republic Grants GAAV IAA100100915,\nGACR 202/07/0456, AV0Z10100520, AV0Z10100521,\nKAN400100652, LC510, Preamium Academiae.\n[1] eds. T. Dietl, D. D. Awschalom, M. Kaminska, and\nH. Ohmo, in Spintronics (Elsevier, 2008), vol. 82 of Semi-\nconductors and Semimetals .\n[2] J. Zemen, J. Kucera, K. Olejnik, and T. Jungwirth, Phys.\nRev. B 80 , 155203 (2009).\n[3] T. Jungwirth, P. Horodysk\u0013 a, N. Tesa\u0014 rov\u0013 a, P. N\u0014 emec,\nJ.\u0014Subrt, P. Mal\u0013 y, P. Ku\u0014 zel, C. Kadlec, J. Ma\u0014 sek, I. N\u0014 emec,\net al., Phys. Rev. Lett. 105, 227201 (2010).\n[4] A. W. Rushforth, M. Wang, N. R. S. Farley, R. C. Cam-\npion, K. W. Edmonds, C. R. Staddon, C. T. Foxon, and\nB. L. Gallagher, J. Appl. Phys. 104, 073908 (2008).\n[5] P. Blaha, K. Schwarz, G. K.H. Madsen, D. Kvasnicka,\nand J. Luitz, WEIN2k, FPLAPW package for calculating\ncrystal properties, TU Vienna.\n[6] M. Sawicki, K.-Y. Wang, K. W. Edmonds, R. P. Campion,\nC. R. Staddon, N. R. S. Farley, C. T. Foxon, E. Papis,\nE. Kaminska, A. Piotrowska, et al., Phys. Rev. B 71 ,\n121302 (2005)."    },    {        "title": "1012.4717v1.Magnetocrystalline_anisotropy_and_uniaxiality_of_MnAs_GaAs_100__films.pdf",        "content": "arXiv:1012.4717v1  [cond-mat.mtrl-sci]  21 Dec 2010Magnetocrystalline anisotropy and uniaxiality of MnAs/Ga As(100) films\nJ. Magnus Wikberg1, Ronny Knut2, Sumanta Bhandary2, Igor di Marco2, Mikael Ottosson3, Janusz Sadowski4,\nBiplab Sanyal2, P˚ al Palmgren2, Cheuk W. Tai5, Olle Eriksson2, Olof Karis2and Peter Svedlindh1\n1Department of Engineering Sciences, Uppsala University, P .O. Box 534, SE-751 21 Uppsala, Sweden\n2Department of Physics and Astronomy, Uppsala University, B ox 516, 751 20 Uppsala, Sweden\n3Department of Materials Chemistry, Uppsala University, P. O. Box 538, SE-751 21 Uppsala, Sweden\n4MAX-lab, Lund University, P.O. Box 118, SE-221 00 Lund, Swed en and\n5Department of Materials and Environmental Chemistry,\nArrhenius Laboratory, Stockholm University,SE-106 91 Sto ckholm, Sweden\nWepresentaninvestigation ofthemagnetic behavior ofepit axial MnAsfilms grownon GaAs(100).\nWe address the dependence of the magnetic moment, ferromagn etic transition temperature ( Tc)\nand magnetocrystalline anisotropy constants on epitaxial conditions. From thorough structural\nand magnetic investigations, our findings indicate a more co mplex relationship between strain and\nmagnetic properties in MnAs films than a simple stretch/comp ression of the unit cell axes. While\na small increase is seen in the anisotropy constants the enha ncement of the magnetic moment at\nsaturation is significant. X-ray magnetic circular dichroi sm results show a behavior of the spin-\nand orbital-moment which is consistent with a structural tr ansition at Tc. In particular, we find\nthat the ratio of the orbital to spin moment shows a marked inc rease in the coexistence region of\nthe ferromagnetic α- and paramagnetic β-phases, a result that is well in accord with the observed\nincreaseofthe c/a-ratiointhesame temperatureregion. The ab initio densityfunctional calculations\nreveal that the magnetic properties are more sensitive towa rds change in in-plane axis as compared\nto a change of the out-of-plane axis, which is explained by th e analysis of band structures. The\neffects ofelectron correlation in MnAsusing ab initio dynamical mean fieldtheory are also presented.\nPACS numbers: 75.30.Gw, 75.70.-i, 71.15.Mb\nI. INTRODUCTION\nThe search for suitable materials for spintronic appli-\ncation has spurred a large interest in materials that can\nbe deposited on ordinary semiconductors such as Si and\nGaAs. Bulk MnAs exhibits a ferromagnetic (FM) tran-\nsition temperature ( Tc) of 313 K,1where it is accom-\npanied by a structural phase transition going from the\nFMα-phase to the orthorhombic β-phase and as the\ntemperature is raised to 398 K MnAs exhibits an ad-\nditional phase transition back to a hexagonal γ-phase.\nThe long term interest in MnAs lies in the first order\nmagnetostructural phase transition with a Tcthat can\nbe altered by for instance applying a stress to the mate-\nrial resulting in tensile/compressive strain of the MnAs\nlattice.2When MnAs is depositedon aGaAssubstrateas\na thin film, MnAs undergoes a strain induced increase in\nTcand moreover develops a two phase region where the\nα- andβ-phases coexist between approximately 280 and\n340K.3Themagnetic transitiontemperature depends on\nchoice of substrate, e.g. MnAs/GaAs(001) has a Tc=320\nK while MnAs/GaAs(111)B has a Tc=346 K,4as well\nas on strain, e.g. stretching of the c-axis and compres-\nsion of thea-axis induce a reduction of Tc.5Even though\nthe films are fully relaxed after just a few monolayers6\nthe strain induced Tcincrease seems to prevail in thicker\nfilms and through application of high magnetic fields the\nβ-phase can be transformed back to the α-phase even\naboveTcdue to the structural strain.7The magnetic na-\nture of the β-phase has been under debate where some\nconsider it to be paramagnetic8while later suggestionspoint out that it has an antiferromagnetic (AFM) short\nranged structure,9albeit without long range order.10In-\nvestigations of the magnetic anisotropy have determined,\ncounter intuitively, that for MnAs the easy axis of mag-\nnetization lies along the a-axis ([11 20]-direction) and the\nhardaxisalongthe c-axis([0001]-direction).11Since both\ncandalie inthe film plane ofMnAs/GaAs(001)films the\nmagnetocrystalline anisotropy can be measured by sim-\nply rotating the film 90◦. An additional contribution to\nthe in-plane anisotropy is shape-anisotropy arising from\nthe striped structure with alternating α- andβ-stripes,\nstretching out along [0001], that decrease and grow in\nwidth, respectively, as the temperature increases.12Re-\nported values of the anisotropy constant8are close to\nvalues obtained for bulk MnAs (7 .4×105J/m3at room\ntemperature).13\nWe present an investigation of MnAs films grown on\nGaAs(100) substrates and describe how the magnetic\nproperties depend on strain and uniaxial structure of the\nthin film. The temperature dependent 1st and 2nd order\nterms (K1andK2) of the magnetocrystalline anisotropy\n(MCA) are derived from the free energy density equa-\ntion revealing an easy plane anisotropy in MnAs films\nat all temperatures. X-ray magnetic circular dichroism\n(XMCD) measurements reveal that the orbital moment\ndoes not strictly follow the spin magnetic moment in the\ntemperature range where both the α- andβ-phases co-\nexist.2\nII. EXPERIMENTAL\nThe MnAs films were grown in a KRYOVAK MBE\nsystem, using epi-ready GaAs wafers with (100) orien-\ntation as substrates. The substrates were attached to\nmolybdenum holders by liquid In gluing, which ensures\nuniformity of the substrate temperature during the MBE\ngrowth. After introduction into the MBE growth cham-\nber, the substrates were subjected to the typical proce-\ndures of thermal evaporation of native oxides and high\ntemperature growth of a GaAs buffer layer. An As\ncracking cell (by DCA Instruments) operated at 900oC\nwas used as the arsenic source. After the buffer layer\ngrowth the temperature of the As cracker was lowered\nto 600oC, which changes the As flux from As dimer rich\nto As tetramer (As 4) rich and the substrate tempera-\nture was lowered to about 250oC. This temperature was\nmeasured by an IR pyrometer operating in the 100 -\n700oC range. During the cooling time the substrates\nwere exposed to the As 4flux. The desired surface re-\nconstruction (c(4 ×4) or d(4 ×4)) of the GaAs substrate\nwas obtained by following established procedures.14,15\nThe MnAs growth was started after reaching and sta-\nbilizing the substrate temperature at 250oC; the MnAs\ngrowth rate was chosen to be about 200 ˚A/h. The growth\nwas monitored by a reflection energy electron diffraction\n(RHEED) system. The RHEED images evolved rather\nquickly from two-dimensional (2D) 4 ×4 streaky patterns\noriginating from the GaAs surface to diffraction spots\nindicating formation of MnAs islands during the initial\ngrowth stage. When the growth proceeded the RHEED\nimagesfromMnAs:GaAs(100)evolvedtoanadmixtureof\nthree-dimensional spotty and 2D streaky patterns, which\nwere visible up to a MnAs film thickness of about 500 ˚A.\nThe spotty patterns eventually disappeared for thicker\nMnAs films. To prevent MnAs surface oxidation after\nthe MBE growth the samples were capped with a 1000 ˚A\nthick amorphousAs layerdeposited in the MBEchamber\nafter switching off the substrate heater and cooling down\nthe sample to the lowest temperature possible to attain\nin the MBE system.\nOne of the investigated samples had a c(4 ×4) surface\nreconstruction of the GaAs(100) surface prior to MnAs\ngrowth, henceforth referred to as the c(4 ×4) sample,\nwhile the As- and de-cap samples had a d(4 ×4) surface\nreconstruction of the GaAs(100) surface prior to MnAs\ngrowth. The de-cap sample was subjected to an addi-\ntional post heat treatment at 310◦C for 60 minutes (tak-\ning place during removal of the As capping layer) in vac-\nuum.\nX-ray diffraction (XRD) θ−2θscans were carried out\non a Bruker D8 Bragg-Brentano system, equipped with\na CuKα1monochromator and a V˚ antec detector. Pole\nfigures were measured using a Phillips MRD system with\npoint focus and an X-ray poly capillary lens and a sec-\nondary 0.18◦parallel plate collimator and a graphite\nmonochromator. This set-up was also used to measure\nthec-axis ((0004) peaks) of the MnAs (10 10) orientationusing grazing incidence XRD, with an incoming angle\nof about 0.7-0.8◦. For the MnAs (10 11) orientation an\nasymmetric scan was used to measure the c-axis. The\nMRD system was also used for rocking curve measure-\nments, using a 2-bounce hybrid monochromator/mirror\nand secondary X-ray mirror set-up. Due to the differ-\nence in scattering power of the (10 10), (1011) and (10 12)\nplanes the peak intensities in the θ-2θdiffractogram\nshown in Fig. 1 are not proportional to the amount\nof each orientation. The (10 11) orientation is overrepre-\nsented due the high scattering factor of this plane (about\n40×higher than for the (10 10) plane). The proportion\nofeach orientationwas therefore estimated by comparing\nthe intensities in (11 20) pole figures, cf. Fig. 2.\nA transmission electron microscopy (TEM) study was\nperformed as a function of temperature between 89 and\n325K on a d(4 ×4) surface reconstructed sample using\na JEOL-2100 microscope operated at 200keV with a\ndouble-tilt cooling holder (Gatan CHDT3504). The\ncross-sectionTEM specimen wasobtained, aftermechan-\nical lapping and polishing, by ion-milling (JEOL Ion-\nslicer) thinning. The selected area electron diffraction\n(SAED) patterns along [101] GaAs (or [10 10] MnAs),\nwhichwererecordedonaCCDcamera(GatanES500W),\nwereused to determine the a- andc-axis at different tem-\nperatures. As a reference, the (0-20) GaAs reflection was\nalso measured. Two selected areas of the sample, each\nwith an effective size of 800 nm in diameter, were inves-\ntigated but no significant difference in the results were\nfound between the studied areas.\nX-ray magnetic circular dichroism (XMCD) measure-\nments were performed at beamline D1011 at the MAX-\nlab synchrotron facility.16All measurements were per-\nformed by magnetizing the sample with an in-plane mag-\nnetic field in the easymagnetizationdirectionafterwhich\nthe absorption spectrum was measured in remanence us-\ning total electron yield. The measurements were per-\nformed using 75 ±5% circularly polarized light and with\nthe sample 13◦from grazing incidence. The same pro-\ncedure was repeated after magnetizing the sample in the\nopposite direction. Saturation effects which are promi-\nnent at low angles are compensated for according to\nNakajima et al.17The spectra are normalized at low pho-\nton energies after which a step function is subtracted.\nWe have used a correction factor for the spin magnetic\nmoment of 1.47, which has been suggested to remedy\nproblems arising when attempting to separate overlap-\nping contributions (due to the relatively small spin-orbit\nsplitting of the Mn Lshell levels) from the L3andL2\nedges.18\nMagnetization measurements were performed in a Quan-\ntum DesignMPMS-XLSuperconductingQuantumInter-\nference Device (SQUID) magnetometer. Magnetization\n(M)versustemperature( T)wasstudiedbetween5Kand\n400 K following a field cooled protocol. A weak magnetic\nfield (H= 50 Oe) was applied along the crystallographic\n[1120], the easy axis of magnetization, at 400 K and the\nmagnetization was measured as the sample cooled down3\nto 5 K. Magnetization versus field measurements were\nperformed at different temperatures in the range 10 - 340\nK between 50 kOe and -50 kOe. The field was applied\neither along [11 20] or along [0001], the in-plane easy and\nhard axis of magnetization, respectively.\nTo corroborate the experimental studies we performed\ntheoreticalinvestigationsbasedondensityfunctionalthe-\nory. In these studies, we have used linearized muffin tin\norbital (LMTO) calculations with no shape approxima-\ntion for charge density or potential.19The lattice is de-\nscribed by a non-overlapping spherical region surround-\ning atomic sites called muffin-tin sphere and in between\nthem an interstitial part. The basis set is formed with\naugmented linear muffin-tin orbitals. In-side the muffin-\ntinspherebasisfunction, chargedensityandpotentialare\nexpanded in spherical harmonic functions up to lmax= 8\nalong with radial functions and in a Fourier series in\nthe interstitial region. In the interstitial region the basis\nfunctions are Bloch sums of spherical Hankel and Neu-\nmann functions. The Dirac equation is solved for a core\ncharge density which means that no frozen core approxi-\nmation is considered. A multi basis formalism is used to\nensure that all wave functions are well converged, i.e. we\nhave considered three 4s, three 4p and two 3d orbitals\nfor Mn and three 4s, three 4p and two 4d orbitals for As\nin the wave function. The modified tetrahedron method\nis used for integration in reciprocal space over 1152 k-\npoints in the whole Brillouin zone (BZ) for self-consistent\nground state calculations and to ensure convergence of\nthe magnetic anisotropy energy (MAE), we carried out\nup to 9216 k-points in the full BZ. The exchange correla-\ntion potential is approximated by the PBE96 functional\nwithin the generalized gradient approximation. We also\nperformed studies ofthe influence ofelectron correlations\nusing dynamical mean field theory.\nIII. RESULTS\nA. Structural properties\nSeveral out-of-plane and in-plane orientations have\nbeen reported for MnAs films deposited on GaAs (100)\nsubstrates. These orientations have been classified as A\norBdepending on the in-plane relationship with respect\nto the substrate.20–23TheAandBorientations have a\ntwin relationship of 90owhere in the A-type thec-axis of\nMnAs is parallel with the GaAs [1 10] direction and for\ntheB-type the MnAs c-axis is parallel to the GaAs [110]\ndirection. Moreover, an index in the notation for the ori-\nentation was introduced to distinguish between in-plane\nandout-of-planeorientationsoftheMnAs c-axis.25,26For\nA0andB0there is an in-plane orientation of the MnAs\nc-axis, while for A1,B1andA2,B2thec-axis is tilted\nout-of-plane with respect to the GaAs(001) surface (see\nTable I for details). The orientation of the MnAs film\ncan be controlled by the template layer. More in par-\nticular, a very As-rich template with a d(4 ×4) surfacereconstruction yields an A0orientation while reduction\nof the As-concentration on the template yields a B0or\neven aB1orientation.20log (Rel. Int.)\n\u0015T\u0003(deg)20 30 40 50 60 70 80c(4x4)de-capAs-cap1 1 10 0 1 10\n2 1 10\n0 2 20\n3 2 20\nFIG. 1: (Color online) X-ray diffraction θ-2θscans for the As-\ncap (red), de-cap (blue) and c(4 ×4) (black) samples obtained\nat 297 K. All samples show good epitaxy but with several\norientations present in the films, see Table I.\nThed(4×4)filmsshowadominating(10 10)orientation\n(A0). Smallamountsofadditional(10 11)and(10 12)out-\nof-plane orientations were also observed. The fraction of\n(1011) orientation was estimated to about 5-13 %, while\nthe (1012) orientation was less abundant, 0.4-2%. The\nde-cap film showsless (10 11) and (10 12) out-of-plane ori-\nentations, probably due to the heat treatment during the\nAs-layer removal. The A0orientation exhibits a rock-\ning curve Full Width Half Maximum (FWHM) value of\nabout 0.5◦for the de-cap sample and 0.65◦for the As-\ncap sample, while the B1orientations have rocking curve\nFWHM values of 0.4-0.5◦. TheB2orientation shows an\neven higher FWHM value, 1.2◦and 1.9◦for the As-cap\nand de-cap samples, respectively. These relatively high\nrocking curve FWHM values indicate a high degree of\nstrain relaxation in the a- andb-axis directions.\nThe c(4×4) film exhibits a dominating (10 11) orienta-\ntion with smaller amounts of the (10 12) and (20 23) ori-\nentations, 2% and 25%, respectively. The rocking curve\nFWHM value for the dominating orientation is about\n0.5◦, while the (10 12) orientation exhibits a higher value,\nabout 1.0◦. The (20 23) orientation exhibits two rocking\ncurve peaks, ±1.7◦from theθangle. The FWHM value\nof the (20 23) rocking curve peak is about 1.5-1.6◦. This\nindicates that the fraction of (20 23) orientation could be\nunderestimated due to the high FWHM value. The films\nare almost completely relaxed with aexhibiting values\nclose to the bulk value, while a strain up to 1% is ob-\nserved forc(see Table II). In order to investigate the\nin-plane orientation and the fraction of each orientation\npresent in the films, pole figures were measured. From\nthe (1120) pole figure, shown in Fig. 2 for the de-cap\nand c(4×4) samples, the relative fraction (see Table II)4\nTABLE I: Epitaxial orientations of MnAs films deposited on\nGaAs (100), using the notation defined in Refs. 21–24, as\nwell as the corresponding out-of-plane angles ( α) of thec-axis\nwith respect to the GaAs (001) surface.\nOrientation (hkl) Miller-Bravais αNotation\n(100) 10 10 0◦A0,B0\n(101) 10 11 29◦A1,B1\n(102) 10 12 41◦A2,B2\n(203) 20 23 50◦A3,B3\nofeachorientationcould be estimated throughthe height\nof each peak. It should be noted that A/B-orientations\nwith the same αwill appear on the same ψ-angle in the\n(1120) pole figure.\nFIG. 2: 11 20 pole figures in linear scale with indications of\norientations ( A0,B1,B2andB3) fora)the de-cap sample\nwith mostly A0orientation and a very small B1contribution\nandforb)thec(4×4)sample withclear B1andB3orientation\npeaks.\nInthec(4 ×4)sample,theout-of-planeorientations( B1\nandB3) dominate and almost no in-plane orientations\nofc(A0/B0) with respect to the substrate surface are\nfound. For the d(4 ×4) surface reconstructed samples the\nA0orientation dominates and through heat treatment a\nrelaxation of the c-axis occurs (0.3%) together with an\nincrease of the fraction of A0orientation. Both the de-\ncap and As-cap samples have a fraction of B-orientation,\nwhich is responsible for the small but finite value of the\nremanent magnetization when measuring the magnetiza-\ntion with the field applied along the hard magnetization\ndirection, cf. Fig. 6. Theout-of-plane A-orientations(A 1\nand A 2) will also decrease the remanent magnetization\nwhen applying the field along the easy axis of magneti-\nzation (to be discussed below).\nFigure3showsthe temperaturedependenceofthe c/a-\nratio as derived from the SAED patterns. Even though\nthe SAED patterns are dominated by the α-phase reflec-\ntions, someweakreflectionsdue to the β-phase canbe re-\nsolved at the highest temperatures. The c/a-ratio starts\nto increase at about 250 K, which is where we expect the\nlow temperature limit of the α−βcoexistence region to\nbe (see below), and reaches a value close to 1.55 at the\nhighesttemperature. Thisincreaseofthe c/a-ratioisdue\nto a 1% increase of cand a somewhat smaller decrease/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s49/s46/s53/s50/s49/s46/s53/s51/s49/s46/s53/s52/s49/s46/s53/s53\n/s32/s32/s99/s32/s47/s32/s97\n/s84/s32/s40/s75/s41\nFIG. 3: Temperature dependence of the c/a-ratio. The c- and\na-axis data were extracted from SAED patterns.\n(0.7%) ofa. The results also indicate that the increase\nofcstarts at slightly lower temperature compared to the\ndecrease of a.\nB. Magnetic properties\nFromthetemperaturedependentmagnetizationcurves\nshown in Fig. 4 it is seen that the highest Tcis obtained\nfor the c(4 ×4) sample followed by the de-cap and As-\ncap samples (see Table II for values). For comparison,\nthe temperature dependence of the remanent magnetiza-\ntion for the de-cap sample ( Mr) along with the magnetic\nmoment obtained from XMCD measurements ( mXMCD)\nnormalized with respect to Mrare plotted in Fig. 4. The\narchetypical behavior of the field dependent magnetiza-\ntion in MnAs/GaAs(100) thin films at different temper-\natures and fields applied along [11 20] and [0001], here\nrepresented by the results obtained for the de-cap sam-\nple, is seen in Fig. 5. When measured along the easy\naxis of magnetization (closed symbols) the film shows\na very square like and narrow hysteresis curve, with a\ncoercivity around 160 Oe at 10 K. The hard axis mag-\nnetization exhibits a nearly linear relationship with the\napplied field up to saturation with a saturating field of\n≈35 kOe at 10 K. That the hard axis magnetization ex-\nhibits a small but finite remanent magnetization as well\nas magnetic hysteresis is related to the small fraction\nof out-of-plane Borientation in the film, as seen in the\nXRD-measurements, i.e. a small part of the film will\nhave easy axis or close to easy axis alignment.\nMnAs has its easy axis of magnetization along the\na-axis ([11 20]-direction) and the hard axis along the c-\naxis ([0001]-direction) in the hexagonal unit cell. When\nMnAs is deposited on GaAs(100) both the a- andc-axis\nwill be oriented in the film plane and thus the magnetic\nanisotropy can be studied by performing MversusH5\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48 /s52/s48/s48/s48/s49/s50/s51/s52\n/s32/s32/s77/s32/s40\n/s66/s32/s47/s32/s77/s110/s41\n/s84/s32/s40/s75/s41/s32/s100/s101/s45/s99/s97/s112\n/s32/s65/s115/s45/s99/s97/s112\n/s32/s77\n/s114\n/s32/s99/s40/s52/s120/s52/s41\n/s32/s109\n/s88 /s77/s67/s68\nFIG. 4: (Color online) Magnetization ( M) versus tempera-\nture (T) for the de-cap (blue squares), As-cap (red circles),\nand c(4×4) (black triangles) samples. The remanent magne-\ntization ( Mr, purple diamonds) for the de-cap sample and\nthe magnetic moment ( mXMCD, green triangles) obtained\nfrom XMCD measurements are included for comparison. The\nmXMCDdata have been normalized with respect to Mrat 100\nK.\nmeasurements along the in-plane directions defined by\nthese two axes. In case of uniaxial crystal structure, the\nexpression for the magnetic free energy density ( F) is\ngiven by:\nF=K1[a0cos2α0+a1cos2α1+...]sin2θ+\n+K2[a0cos2α0+a1cos2α1+...]sin4θ−\n−µoHMscosθ, (1)\nwhereK1andK2are the magnetic anisotropy constants,\na0,a1...are the fractions of A0,A1...orientations, re-\nspectively, in the sample and α0,α1...are the corre-\nsponding out-of-plane angles of the c-axis with respect to\nthe film plane. The last term is the Zeeman contribution\ntothefreeenergyand θistheanglebetweenthemagneti-\nzation and applied field directions. Through the relation-\nshipsM=Mscosθand∂\n∂θF= 0 an expression for field\ndependence of the hard axis magnetization, Mhard/Ms\nvs.Hcan be derived, which when fitted to the corre-\nsponding experimental data yields the anisotropy con-\nstants. The values for K1andK2, given in Table II,\nhave been obtained by subtracting the easy-axis magne-\ntization contribution in the Mhardvs.Hdata before fit-\nting theMhard/Msexpression to the experimental data.\nAs seen in Table II the As-cap and de-cap samples show\na somewhat higher anisotropy and Tcthan obtained for\nbulk MnAs,13while thec(4×4) sample displays lower\nK-values but a higher Tcvalue.\nA difference in magnetic behavior can also be seen\nin the field dependence of the magnetization, see Fig. 6/s45/s51/s48 /s45/s50/s48 /s45/s49/s48 /s48 /s49/s48 /s50/s48 /s51/s48/s45/s52/s45/s51/s45/s50/s45/s49/s48/s49/s50/s51/s52\n/s32/s77/s32/s40\n/s66/s32/s47/s32/s77/s110/s41\n/s72/s32/s40/s107/s79/s101/s41/s32/s49/s48/s32/s75\n/s32/s49/s48/s48/s32/s75\n/s32/s50/s48/s48/s32/s75\n/s32/s50/s53/s48/s32/s75\n/s32/s50/s56/s48/s32/s75\n/s32/s50/s57/s48/s32/s75\n/s32/s51/s48/s48/s32/s75\n/s32/s51/s48/s53/s32/s75/s100/s101/s45/s99/s97/s112\n/s99/s108/s111/s115/s101/s100/s32/s115/s121/s109/s98/s111/s108/s115/s58/s32/s101/s97/s115/s121/s32/s97/s120/s105/s115\n/s111/s112/s101/s110/s32/s115/s121/s109/s98/s111/s108/s115/s58/s32/s104/s97/s114/s100/s32/s97/s120/s105/s115\nFIG. 5: (Color online) Magnetization ( M) versus field ( H) at\ndifferent temperatures in the range 10 −305 K, for the de-\ncap sample with filled symbols corresponding to field applied\nalong the easy axis ([11 20]-direction) of magnetization and\nopen symbols to field applied along the hard axis ([0001]-\ndirection) of magnetization.\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s48/s49/s50/s51/s52\n/s32/s77/s32/s40\n/s66/s32/s47/s32/s77/s110/s41\n/s72/s32/s40/s107/s79/s101/s41/s32/s65/s115/s45/s99/s97/s112\n/s32/s100/s101/s45/s99/s97/s112\n/s32/s99/s40/s52/s120/s52/s41\n/s84/s32/s61/s32/s49/s48/s32/s75\nFIG. 6: Magnetization ( M) versus field ( H) measured at 10\nK along the easy axis (filled symbols) and hard axis (open\nsymbols) of magnetization, with the As-cap sample in red\ncircles, de-cap sample in blue squares and c(4 ×4) sample in\nblack triangles.\nwhere the closed and open symbols correspond to mea-\nsurements along [11 20] and [0001], respectively. A clear\ndifference in the values for the saturation magnetization\n(Ms) is seen between the samples. We believe that the\ndifferences in Tc,Msand MCA between samples are\nstrain related and connected to the existence of multi-\nepitaxial orientations in the films. The c(4 ×4) sample\nexhibits a dominant out-of-plane orientation that is re-\nsponsible for the more rounded shape of the Mhardvs.\nHcurve as well as for the higher Tc.20Even though the6\nTABLE II: Anisotropy constants K1andK2measured at different temperatures ( Tm),Tc, room temperature values for c-\naxis length and c/a-ratio, and fraction of each in- and out-of-plane orientati on for the different samples of 1000 ˚A thick\nMnAs/GaAs(100), compared with bulk values from Ref. 13.\nsample Tm(K)K1(×105J/m3)K2(×105J/m3)Tc(K)chex(˚A)c/a A0/ B1/ A2/ B2/ B3\nAs-cap 10 -14.0 1.4 317 5.772 1.555 0.86 / 0.13 / 0.017 / - / -\n100 -13.9 1.3\n200 -12.4 1.2\n250 -10.3 0.9\n280 -8.4 0.9\n290 -7.5 0.9\n300 -6.2 0.6\n305 -5.9 0.4\nde-cap 10 -14.4 1.7 320 5.755 1.551 0.95 / 0.050 / - / 0.004 / -\n100 -14.4 1.7\n200 -12.8 1.5\n250 -10.6 1.2\n280 -8.8 1.1\n290 -7.9 1.1\n300 -6.5 0.8\n305 -5.8 0.6\nc(4×4) 10 -7.3 1.5 335 5.746 1.544 0.008 / 0.72 / - / 0.024 / 0.25\n100 -5.6 1.1\n200 -4.8 0.9\n300 -3.2 0.8\n320 -2.9 0.9\nBulk 77 -12.0 - 313 5.712 1.540\n198 -11.0 -\n273 -9.0 -\n299 -7.6 -\nout-of-plane angles are compensated for when deriving\nK1andK2, the obtained K1values are a factor of 2\nsmaller than the values obtained for the As-cap and de-\ncap samples as well as values derived for bulk samples.13\nIt should be noted that the easy-axismagnetization mea-\nsurements on the c(4 ×4) sample are made with the field\napplied along GaAs[1 10] instead of GaAs[110]. Although\nthe As-cap and de-cap samples have a more uniaxial ori-\nentation with 86% and 92% of A0-orientation, respec-\ntively, they still show some dissimilarities in Tc,Msas\nwellasinK1andK2. Thisclearlycorrespondswelltothe\nearly investigations on bulk MnAs describing changes in\nmagnetic properties with strain,2whereas in the present\nstudy the unit cell distortion comes from lattice mis-\nmatchandmulti-orientationinducedstrain. Theincrease\ninTcwith the relaxing cis well in line with reports from\nIikawaet al.,5who showed a relationship between com-\npression/stretch of the lattice and changes in Tc. We\nwould like to point out that the variations in Tcare due\nto two effects; the fraction and angle ( α) ofc-axis out-of-\nplane orientations20and the stretching/relaxation of the\nc-axis. The interdependence between αandTcis evident\nfor the c(4 ×4) sample where no in-plane alignment is de-\ntected. Although the c(4 ×4) sample has a close to bulk\nc/a-ratio (see Table II) the anisotropy seems to be more\ndependent on c-axis alignment along GaAs[1 10] than on\nc-axis strain. A conclusion that becomes even more ev-\nident when examining the c/a-ratio for the As-cap andde-cap samples.\n1. X-ray Magnetic Circular Dichroism\nMnL2,3x-ray magnetic circular dichroism spectra\nrecorded at 100 K for the de-cap sample are shown in\nFig. 7 together with the corresponding dichroic differ-\nence. The spectra were obtained by reversing the mag-\nnetization of the sample keeping the polarization fixed.\nSimilar spectra were recorded for severaltemperatures in\nthe interval 100-330 K. The spectra have been normal-\nized toacommon edgejump correspondingto aper atom\nnormalization of the absorption cross section.27The use\nof the XMCD sum rules28for Mn has been controversial\nsince there is a mixing between the L2andL3edges and\ntherefore a constant of 1.47 has been proposed to com-\npensateforthiseffect.18Usingad-holecountof4.9asthe\nnumber of empty dstates in the sum rules, a magnetic\nmoment of ∼5µB/Mn atom is obtained at low temper-\natures. This number is clearly too large as compared to\nthe numbers obtained from magnetization measurements\nand theory. This discrepancy might have several expla-\nnations. Wefirstnotethatthespin-quadrupoleterm /angbracketleftTz/angbracketright\nwas neglected in the sum rule analysis. A d5configura-\ntionwouldinalocalizedpicturegivethesameoccupation\nof all orbital levels resulting in a zero /angbracketleftTz/angbracketrightterm.29How-\never, one could of course argue that this approach might7\nnot be strictly valid for the MnAs system due hybridiza-\ntion effects andadeviation fromastrict d5configuration.\nFrom our calculations we find that the doccupation is\n5.13 electrons. We have therefore also made explicit cal-\nculations of the /angbracketleftTz/angbracketrightcorrection, following the recipe by\nWu and Freeman.29In no case we find that the correc-\ntion is greater than 0 .18µB, i.e., a correction less than\n5 percent of the total moment. We also note that with-\nout the corrective factor suggested in Ref. 18, we obtain\na spin moment that is very close to the value obtained\nfrom calculations. Errors in the estimate of the number\nof holes, obtained from theory, can safely be ruled out,\nas this would require a charge transfer of almost 1.5 elec-\ntronsnot accountedforby the theoreticalcalculations. A\nmore likely explanation for the discrepancy in magnetic\nmoments connects to the value used for the correction\nfactor and a more thorough investigation on the appli-\ncability of the corrective factor for the XMCD spin sum\nrule for MnAs is under preparation.30To avoid distract-\ning the focus from the more interesting point regarding\nthe behavior of the spin and orbital moments as a func-\ntion of temperature, we have chosen to present moments\nobtained from XMCD sum rule analysis as magnetic mo-\nments per hole. The results aresummarized in Fig. 8. As\nshown in Fig. 8b the orbital moment ( ml) behaves differ-\nentfromthespinmagneticmoment( ms)giveninFig.8a.\nAs most evident from the ml/ms-ratio, seen in Fig. 8c,\nthe orbital moment decreases more slowly than the spin\nmagnetic moment in the temperature range where the α-\nandβ-phases coexist. This behavior is not strictly ex-\ncluded by any physical principle as we are dealing with\na structurally driven first order magnetic transition. We\nnote howeverthat the result is an indication of structural\nchanges in the α-phase in the coexistence region. This\nconjecture is confirmed by the temperature dependence\nofc/ashown in Fig. 3; the very similar temperature de-\npendence of c/aandml/msis striking. Moreover, the\nobserved variation of the orbital magnetic moment is in\naccordance with the theoretical calculations, discussed\nlater in the text and seen in Fig. 9, where it is shown\nthat an increased c/a-ratio can yield a larger orbital mo-\nment anisotropy.\nC. Theoretical Calculations\nIn this section, we will present ab initio calculations\nto address some behavior observed in the experiments\ndiscussed above. Let us start with the discussions on\nthe magnetic anisotropy energy. This is a very delicate\nproperty to calculate, since a very dense sampling of the\nBrillouin-zone is needed in order to get accurate results.\nWe have done careful convergence tests, with respect to\nthe number of k-points, and our final calculation was\ndone for 9216 k-points in the full Brillouin-zone. The\nfinal self-consistent result of the MAE is shown in Fig. 9,\nas a function of c/a-ratio. In Fig. 9 we also show the\nenergy difference between a FM phase and an AFM one,\n/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s54/s56/s48 /s54/s55/s48 /s54/s54/s48 /s54/s53/s48 /s54/s52/s48 /s54/s51/s48 /s54/s50/s48\n/s80/s104/s111/s116/s111/s110/s32/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s68/s105/s99/s104/s114/s111/s105/s99/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41/s88/s77/s67/s68\n/s49/s48/s48/s32/s75\n/s32/s78/s101/s103/s46/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s32/s80/s111/s115/s46/s32/s109/s97/s103/s110/s101/s116/s105/s122/s97/s116/s105/s111/s110\n/s32/s68/s105/s99/s104/s114/s111/s105/s99/s32/s100/s105/s102/s102/s101/s114/s101/s110/s99/s101\nFIG. 7: Mn L 2,3XMCD spectra recorded at 100 K and their\ncorresponding difference spectrum. Spectra were obtained b y\nreversing the magnetization direction in the sample.\nms(µ B/n h) ml(10 -3 µB/n h) ml/m s(×10 -2 )\nT (K) a\nb\nc5\n23410 15 20 25 \nFIG. 8:a)Spin magnetic moment ( ms),b)orbital magnetic\nmoment ( ml) andc)the ratio ml/ms.8\n-0.2-0.100.1EFM - EAFM (eV)\nfixed c\nfixed a\n1.4 1.45 1.5 1.55 1.6 1.65\nc/a00.050.10.150.20.25MAE / f.u (meV)(a)\n(b)\n1.45 1.5 1.55 1.6 1.65\nc/a0.0050.010.015Orbital moment anisotropy ( µΒ)fixed c\nfixed a\n1.45 1.5 1.55 1.6 1.6522.533.54Mn spin moment ( µΒ)\n(c)\nFIG. 9: (Color online) Calculations for different c/a-ratio,\nwhere black symbols (red symbols) correspond to keeping the\nc-axis (a-axis) fixed while varying the a-axis (c-axis).a)dif-\nference in total energies between ferromagnetic and antife r-\nromagnetic alignments of Mn spins in the [0001]-direction.\nb)magnetic anisotropy energy (MAE) as a function of c/a-\nratio,c)orbital moment anisotropy (OMA) as a function of\nc/a-ratio on the left hand axis with filled symbols and solid\nline and spin magnetic moment as a function of c/a-ratio on\nthe right hand axis with open symbols and dashed line.\nas a function of c/a-ratio. Note from the figure that\nwe have made c/achanges while keeping either the c-\naxis ora-axis fixed at bulk experimental values ( c=5.712\n˚A anda=3.71˚A). The bulk experimental c/a-ratio is\n1.54, and for this geometry the FM phase is stable (see\nFig. 9) over the AFM phase with ∼0.1 eV per formula\nunit. We also note from the theoretical results that the\neasyaxislies in the ab-plane, and that the c-axisis ahard\nmagnetizationdirection. Thisresultisingoodagreement\nwith experimental observations, and the size of the MAE\n(0.25 meV/f.u. ≈12×105J/m3) is rather substantial and\nwell comparable to our experimental results, as can be\nseen by a comparison to data in Table II, as well as to\nbulk results.13From Fig. 9a it is seen that the AFM-8-6-4-20246E - EF (eV)\nΓK M ΓA L H A(a)\n-8-6-4-20246E - EF (eV)\nΓK M ΓA L H A(b)\n-8-6-4-20246E - EF (eV)\nΓK M ΓA L H A(c)\nFIG. 10: (Color online) Spin-polarized band structures whe re\nblack and red symbols indicate spin-up and spin-down bands,\nrespectively. a)c/a=1.45 for fixed cand varying a,b)\nc/a=1.55 and c)c/a=1.45 for fixed aand varying c.\nphase is stable when the c/a-ratio is 1.65 for a fixed c-\naxis. A rather interesting feature is displayed in Fig. 9b\nin that the dependence on the calculated MAE is rather\nweak when keeping the a-axis fixed and varying the c-\naxis, whereas the dependence is very strong when the\nc-axis is fixed and the a-axis is varied. The microscopic\nreason for the MAE lies in the electronic structure31and\nin order to analyze the behavior in Fig. 9, we show in\nFig. 10 the energy bands for different c/a-ratio. We also9\ndisplay energy bands when the c-axis is varied and the\na-axis kept constant, and vice versa, for c/a-ratios equal\nto 1.45 and 1.55 (experimental value). It is well known\nthat the contribution to the MAE from the spin-orbit\ncoupling in perturbation theory is from occupied bands\njustbelowtheFermileveltounoccupiedbandsjustabove\nthe Fermi level. Hence, it is the energy bands around the\nFermi level one should inspect if one wants to investigate\nthe MAE changes shown in Fig. 9. The energy bands\nin Fig. 10 demonstrate an overall larger sensitivity when\nmodifying the a-axiswhile keepingthe c-axisfixed, which\nis consistent with the data in Fig. 9. This is particularly\npronouncedalongthe symmetrydirectionΓ →A→L→\nM. Here entire segments of the spin-down Fermi surface\nare removedby the strain, since the spin-down band that\nfor the unstrained case crosses the Fermi level, becomes\ncompletely unoccupied. This only happens for a strain\nwhere thec-axis is fixed.\n0 0.005 0.01 0.015\nOMA (µΒ)0.050.10.150.20.250.3MAE / f.u (meV)c-fixed\na-fixed\nFIG. 11: (Color online) Magnetic anisotropy energy (MAE)\nvs. orbital moment anisotropy (OMA) for different c/a-ratio.\nThe calculations were done by keeping either the a-axis (red\ncircles) or the c-axis (black circles) fixed.\nThe final theoretical result we would like to discuss is\nthe relationship between MAE and the anisotropy of the\norbital moment when measureed along an easy and hard\nmagnetisation direction, the so called orbital moment\nanisotropy (OMA) shown in Fig. 9c. From perturbation\ntheory it has been shown32that there is a direct cou-\npling between the MAE and the OMA. The derivation is\nmade under the assumption that cross-terms in the spin-\norbit interaction (i.e. ξ(l+s−+l−s+)) can be neglected\ncompared to diagonal terms (i.e. ξlzsz). For a material\ncomposed ofone element this relationshipholds well, and\nthere is a direct proportionality between the MAE and\nOMA. However, a generalization of this theory to mate-\nrials composed of more than one element, shows a more\ncomplex behavior,33since both the MAE and OMA may\nhave contributions from spin-orbit effects of all atomic\nspecies of the material. In short, this means that for a-4-2024 U = 0 eV\n-4-202DOS (1/eV)U = 2 eV\n-10 -8 -6 -4 -2 0 2 4 6 8 10\nE - EF (eV)-202U = 4 eV\nFIG. 12: Spin-polarized DOS for (top) GGA with U=0 eV\n(top), (middle) DMFT with U=2 eV and (bottom) DMFT\nwith U=4 eV calculations.\nmaterial like MnAs, the MAE of the magnetic moment,\nwhich is primarily located on the Mn atom, is also influ-\nenced by the spin-orbit interaction of the As atom. Since\nthere are cross terms in atomic site index to both the\nMAE and OMA, which are large when there is strong hy-\nbridization of electron states centered on different atomic\nsites, there is no reason to expect a linear relationship\nbetween the MAE and OMA. Figure 11 shows a compar-\nisonbetweencalculatedvalues ofthe MAE andOMA, for\ndifferent values of the c/a-ratio. Again, the calculations\nwere done by keeping either the a-axisor the c-axis fixed.\nNote from the figure that the linear relationship between\nOMA and MAE is lost. Hence, for this material it is\nmisleading to measure the OMA and from this value try\nto draw conclusions about the MAE. This has previously\nbeen shown experimentally for Au/Co thin films.33\nThe electronic structure of MnAs has been analyzed\nin the past34–37using LSDA and/or GGA. A relevant\nquestion for this material concerns the influence of elec-\ntron correlations, which might be important for the elec-\ntronic structure and hence also for the magnetic proper-\nties. Mn compounds often display electronic structures\nwhich have a correlated electronic d-shell, that may even\nbecome localized due to strong electron-electron interac-\ntion. Examples of this are LSMO (i.e La 1−xSrxMnO3)38\nand MnO.39Electron correlation has recently been dis-\ncussed even for one of the elemental forms of Mn, the\nγ-phase.40Hence, it is important to investigate this as-\npect in MnAs, and for this reason we have compared the\nelectronic structure and the spin and orbital moments\nof MnAs using GGA and dynamical mean field theory\n(DMFT - a recent method to take into account electron\ncorrelations).41In this calculation we have used the im-\nplementationbyGrechnev et. al.42InFig.12wecompare\nfirst the electronic structure from GGA calculations with\nDMFTcalculations,usingaCoulombU(i.e. thestrength\nof the electron-electron interaction) of 0 for GGA and 210\nand 4 eV for DMFT. As seen in Fig. 12 one can notice\nsome changes in the electronic structure with the varia-\ntion of the value of U. The variation in the spin-up DOS\nis more prominent than the spin-down one. However, the\nresulting exchange splitting does not differ much as dis-\ncussed below. Next we turn our attention to the spin-\n(ms) and orbital ( ml) moments. From the GGA calcula-\ntion we obtain for the Mn d-orbital ams=3.12µB/atom\nand aml=0.026µB/atom. From DMFT calculations\nfor U=2 eV we obtain ms=3.12µB/atom and ml=0.028\nµB/atom, while for U=4 eV we get ms=3.11µB/atom\nandml=0.017µB/atom. Our calculations thus show a\nrather weak influence on the spin and orbital moments\nas a function of increasing values of U, implying that\nelectron correlations outside of what is included in GGA\nare not of major importance for MnAs. For the GGA\ncalculation we obtain a total moment of 3.13 µB/atom,\nwhich should be compared to earlier theoretical results\nof 3.06µB/atom35,36and to the bulk experimental value\nof 3.45µB/atom.43\nIV. DISCUSSION\nThe experimental Msvalues for the As-cap and de-cap\nsamples are at least 10-15% larger than what has been\nreported for thin film11or bulk MnAs.43Values similar\nto our experimental values were obtained by Sanvito\nand Hill44from calculations of the effect of cell volume\nexpansion where a Ms∼4µBwas obtained due to the\nreduction of p−dhybridization when then unit cell was\nstretched. The necessary increase of the cell volume\nrequired almost unphysical values of aandc, but the\nimportant observation is the large change in magnetic\nmoment with lattice distortion. Our theoretical results\nalso show that the magnetic properties depend strongly\non lattice strain, especially on variations of a, see Fig. 9c.\nSince XRD is a global probe, the measured aandc\ncorrespond to averages over the sample volume and thus\nlocally the tensile strain could be much larger. In fact,\nour SAED results, where the size of the selected area\nis 800 nm, indicate are more strained a-axis. However,\nwe refrain from depending solely on the TEM results\nsince thinning during sample preparation could induce\nadditional lattice strain. Still, changes in MsandA0\nwith annealing show that even small relaxations of thestructure play a significant role.\nOur calculated magnetic anisotropy energy and easy axis\ndirection are in good agreement with the experiments.\nMoreover, the theoretical results suggest that the change\nin magnetic properties is more sensitive by straining the\nin-plane axis than the out-of-plane one, an effect which\nhas been attributed to the change in band structures\ndue to strain. The calculated magnetic moments are\nobserved to vary significantly upon straining the system.\nFollowing this observation, one may argue that the\nexperimentally measured large moments may arise from\nthe locally strained regions of the samples. In short,\nour theoretical and experimental results are in line with\neach other.\nV. CONCLUSIONS\nIn conclusion, our findings indicate a more complex\nrelationship between strain and magnetic properties in\nMnAsfilmsthanasimplestretch/compressionoftheunit\ncell axes. A strong dependence of K1,K2andTcon the\nangle and fraction of out-of-plane orientation has been\nestablished. The ab initio calculations demonstrate a\nstrongdependence ofthemagneticpropertiesonthevari-\nationofthein-planeaxis. Webelievethatlocalstructural\nstrains that distort the lattice give rise to the enhanced\nMsand MCA measured in our samples. The possibility\nto alter the magnetic properties through controlled film\ngrowth with maintained epitaxy gives MnAs strong po-\ntential as a material for various spintronic applications.\nVI. ACKNOWLEDGMENTS\nThe authors like to thank the Swedish Research Coun-\ncil, Knut and Alice Wallenberg Foundation (KAW),\nSTINT, Swedish Foundation for Strategic Research\n(SSF), G¨ oran Gustafssson Foundation, Carl Tryggers\nFoundation for financial support. O.E. is also grateful\nto the ERC for support. We also acknowledge Swedish\nNational Infrastructure for Computing (SNIC) for the\nallocation of time in high performance supercomputers.\n1S. Hilpert and T. Dieckelman, Ber. Dtch. Chem. Ges. A 44,\n2378 (1911).\n2N. Menyuk, J. A. Kafalas, K. Dwight and J. B. Goode-\nnough, Phys. Rev. 177, 142 (1969).\n3A. K. Das, C. Pampuch, A. Ney, T. Hesjedal, L. D¨ aweritz,\nR. Koch and K. H. Ploog, Phys. Rev. Lett. 91, 087203\n(2003).\n4unpublished results, J.M. Wikberg et al.\n5F. Iikawa, et al., Phys. Rev. Lett. 95, 077203 (2005).6B. Jenichen , D. Satapathy, W. Braun, V.M. Kaganer, X.\nGuo, L. D¨ aweritz and K.H. Ploog, J. Appl. Phys. 96, 6103\n(2004).\n7A.Ney,T.Hesjedal, L.D¨ aweritz, R.KochandK.H.Ploog,\nJ. Magn. Magn. Mater. 288, 173 (2005)\n8J. Lindner, et al., J. Magn. Magn. Mater. 277, 159 (2004).\n9H. Yamaguchi, A. K. Das, A. Ney, T. Hesjedal, C. Pam-\npuch, D. M. Schaadt, and R. Koch, Europhys. Lett 72,\n479 (2005)11\n10M. K. Niranjan, B. 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B 62, 15553 (2000)."    },    {        "title": "1101.0980v1.Four_state_nanomagnetic_logic_using_multiferroics.pdf",        "content": "FOUR-STATE NANOMAGNETIC L OGIC USING MULTIFERROICS \n \n1,aNoel D’Souza , Jayasimha Atulasimhaa and Supriyo Bandyopadhyayb\n(a) Department of Mechanical and Nuclear Engineering,  \n(b) Department of Electrical a nd Computer Engineering,  \nVirginia Commonwealth Univer sity, Richmond, VA 23284, USA. \n \nAbstract \nThe authors show how to implement a 4-state uni versal logic gate (NOR) using three strain-\ncoupled magnetostrictive-piezoe lectric multiferroic nanomagnets  (e.g. Ni/PZT) with biaxial \nmagnetocrystalline anisotropy. Two of the nanomagne ts encode the 2-state input bits in their \nmagnetization orientations and the third na nomagnet produces the output bit via dipole \ninteraction with the input nanom agnets. A voltage pulse alternat ing between -0.2 V and +0.2 V is \napplied to the PZT layer of the third nanomagnet and genera tes alternating tensile and \ncompressive stress in its Ni layer to produ ce the output bit, while  dissipating ~ 57,000 kT (0.24 \nfJ) of energy per gate operation. \n__________________________ \n1Corresponding author. E-mail dsouzanm@vcu.edu \n \n \n \n \n1 \n Nanomagnetic logic (NML) is an emergi ng paradigm for low-power computing1-4 where \nlogic bits 0 and 1 are encoded in two stable magne tization directions parallel to the easy axis of a \nsingle domain nanomagnet with uniaxial shape an isotropy. Bits are switched by flipping the \nmagnetization of a nanomagnet with an ex ternal agent that acts like a clock. \nThe ultimate energy-efficiency of NML is de termined by the clocking methodology, i.e. how \nbits are flipped. If a magnetic field generated with a current5, or spin transfer torque6, is used to \nflip a nanomagnet’s magnetization, then the ener gy consumed can be exorbitant, which defeats \nthe very purpose of NML. However, stre ss-induced rotation of  magnetization in multiferroic  \nnanomagnets seems to be able to switch magnets with much higher energy efficiency7-9. This is \nonly now beginning to attract attention7-10.   \nThe magnetization vector of a multiferroic na nomagnet, consisting of a piezoelectric and a \nmagnetostrictive layer, can be flipped with an el ectrostatic potential app lied to the piezoelectric \nlayer. The voltage generates a strain that is el astically transferred to the magnetostrictive layer \ncausing its magnetization to flip. We can embellis h the functionality of the multiferroic element \nby introducing biaxial magnetocrystalline  anisotropy in the magnetostri ctive layer, giving it four \npossible stable magnetization directions (‘up’, ‘right’ ,‘down’, ‘left’) that are chosen to encode \nfour possible 2-bit combinations (00, 01, 11, 10), illustrated in Fig. 1(a). The choice is motivated \nby allowing only 1 bit to  change for every 90 º rotation of magne tization.  For single crystal Ni, \nwith magnetocrystallin e anisotropy constant K1< 0, the “easy” directions that encode these states \nin the (001) plane are the [110], [11−\n0], [1−\n1−\n0] and [1−\n10] directions, as shown in the energy \ncurves in Fig. 1(a) (saddle-shaped curve, with the four easy directions of magnetization having \nlower energy levels than the othe r principal crystallographic axes). \n2 \n When two nanomagnets are placed next to each ot her, two types of dipo le interaction arise. \nThe first type arises when the magnetizations are perpendicular to the array axis. In this case, \ndipole interaction favors anti-parallel arrangeme nt (or anti-ferromagnetic coupling) of adjacent \nmagnetizations. The second type aris es when the magnetizations are parallel to the array axis. In \nthis case, dipole-dipole interact ion favors a parallel alignment (or ferromagnetic coupling) of \nadjacent magnetizations. Using combinations of th is dipole-dipole interaction and a small global \nstatic magnetic field to resolve tie situations, we realize a 2-bit NOR uni versal logic gate as \nexplained below.  \nConsider the arrangement shown in Fig. 1 (b) through (e), where the two input nanomagnets, \nencoding bits AB and CD, are located on either side of the output nanomagnet encoding bits EF. \nFour different scenarios are inve stigated. The first scenario is  shown in Fig. 1(b), where the \nmagnetization direction of the input magnets is perpendicular to the axis of the nanomagnet \narray. When both input magnetizations are oriented up (or down) (first two rows of Fig. 1 b), the \ndipole coupling favors a down (or up) orientatio n of the output nanomagnet. When one input \nmagnetization points up and the other points down (t hird and fourth rows of  Fig. 1 b), the output \nnanomagnet is in a tied state, which we resolve us ing a global static magnetic  field that forces the \noutput magnetization to point “up”. The second s cenario is shown in Fig. 1(c), where input \nmagnetizations are parallel to the nanomagnet a rray axis. When both input magnetizations are \noriented left or right (first tw o rows of Fig. 1 c), the dipole coupling respectively favors left or \nright orientation of the output nanomagnet. When one input ma gnetization points left and the \nother points right (t hird and fourth rows of Fi g. 1 c) the output nanomagnet is in a tied state, and \nends up pointing neither left nor  right, but orie nting upward because of the global bias field. The \nthird (Fig 1 d) and fourth (Fig 1 e) scenario s are mixed inputs, where one input magnetization \n3 \n points perpendicular (“up/down”) a nd the other parallel (“left/right ”) to the axis of the array, \nrespectively favoring a “down/up” or “left/right ” orientation of the output magnetization. These \nresult in tied states. If one of the inputs is “up”, the static bias  field ensures th at the output is \n“up”, while if one of the inputs is  “down”, the bias field counters this, causing the output to point \neither “left” or “right” depe nding on the second input. The i nput (AB, CD) and output (EF) \nconfigurations, based on their bi nary bit representations, are tr ansferred to a Karnaugh map (K-\nmap) to simplify the logical expression of the output. The output table of the K-map is shown in \nFig. 2. On simplification of the K-map, the result obtained (in Boolean notation) is DBE=  \nand C A F= , which is NOR logic. \nWe present detailed simulation results to show that the output magnetization always \nrepresents the NOR function of th e inputs, independent of its init ial orientation, when we clock \nthe piezoelectric layer of the output multiferroic  nanomagnet with a voltage to generate the \nproper stress cycle. To show this, we consider the total energy of the output nanomagnet when its \nmagnetization vector subtends an angle2θ with the positive x-axis11: \n() () ( )\n() () []22 0\n22 1 3 2 1 3\n22 0 1\n2 100 2 22cos cos cos sin sin sin4\n3sin 2 4 cos 4 sin42 4total s\ns appliedEMR\nKMH3μθθ θθθθπ\nμθ\nθπλ σθ ππ⎡⎤=Ω − + + + ⎡⎤⎣⎦⎣⎦\nΩ+− − Ω − − Ω  (1)  \nθ\nwhere the first term is the dipole interaction en ergy with the neighbor s subtending angles and 1θ\n3θwith the positive x-axis, the second term is the magnetocrystalline anisotropy energy with  \nbeing the first-order magnetocrystalline anisot ropy constant, the third term is the stress \nanisotropy energy due to stress σ applied along the [100] direction (451K\n0 with the x-axis) with \nbeing the magnetostrictive consta nt in the direction of stress, and the last term is the 100λ\n4 \n interaction energy due to the static bias field  pointing in the “up” direction, or the [110] \ndirection. Here,appliedH\nΩ is the permeability of free space,  is the saturation magnetization, sM0μ  is \nthe nanomagnet volume, and R is the distance between the cent ers of two adjacent nanomagnets. \nThe stress σ is taken as positive for tension and negative for compression. \nThe energetic of such a system is briefly e xplained here.  For single crystal nickel with K1 < \n0, the magnetic easy axes are <111> and hard ax es are <100>. We tacitly assume that the two-\ndimensional geometry of the nanomagnet preclu des out-of-plane magnetization orientation due \nto a large magnetostatic energy penalty. Thus,  th e magnetization is confined to the (001) plane \nand [110], [11 0],    [1− −\n1−\n0] and [1 10] are the ground states which respectively correspond to the \n90°, 0°, -90° and 180° orientations in Fig. 1. The dipole and static magne tic field interaction \nenergies are not high enough to move the magne tization away from these minima. However, \nupon applying a stress, the magnetizations are pushed out of these minima. Since the \nmagnetostrictive coefficient −\n100λof Ni is negative, a tensile stress along [100] rotates the \nmagnetization to either the -45º or the +135º st ate (depending on which is closest to its initial \nstate), while a compressive stress along [100] dir ection rotates the magnetiz ation to either the -\n135º or +45º states. When this stress is released , the dipole interactions and the static bias field \ndetermine which of the two adjacent easy direct ions the output magne tization settles into. Thus, \nto rotate the magnetization through 180º one ne eds both a tensile and compressive stress cycle; \nwith each half-cycle producing a +90° rotation . \nFor numerical simulation, the multiferroic nano magnets were assumed to be made of two \nlayers: single crystal nickel a nd lead-zirconate-titanate (PZT) with the following properties for \nNi: -5 5 \n100λ= -2 × 10 , K1 = -4.5 x 103 J/m3 12, and Young’s modulus Y = 2 × , M= 4.84 × 10 A/m \ns \n5 \n 1011\n Pa. The PZT layer can transfer up to 500 ×10-6 \nstrain to the Ni layer13. The nanomagnets are \nassumed to be circular disks with a diameter d = 100 nm and thickne ss = 10 nm, while the \ncenter-to-center separation (or pitch) is 160 nm. The above parameters were chosen to ensure \nthat: (i) The magnetocrystalline energy barrier of  the nanomagnets is su fficiently high (~0.55 eV \nor ~22 kT at room temperature) so that the static bit error pr obability due to spontaneous \nmagnetization flipping is very low; (ii) The st ress anisotropy energy (~1.5 eV) generated in the \nmagnetostrictive Ni due to a strain of 500 ×10-6 \ntransferred from the PZT layer can clearly rotate \nthe magnetization against the magnetocrystalline an isotropy; (iii) The dipole interaction energy \nis limited to 0.2 eV, which is lower than the magn etocrystalline anisotropy  energy. This prevents \nthe magnetization from switching spontaneously w ithout the application of the electric-field \ninduced stress for clocking.     \nTo show that the “output” nanomagnet behaves as desired (for the va rious configurations \nshown in Fig. 1), its  total energy ( Etotal) is computed as a function of θ2 upon application of the \nstress clock cycle , in increments/decrements \nof 0.1 MPa up to a maximum amplitude of 100 MPa. For each value of the stress, the local \nenergy minimum closest to the previous state determines the final magnetization orientation θtension relaxation compression relaxation→→ →\n2. \nWe study a particular case: input  nanomagnets having magnetizati on directions as ‘right’ and \n‘up’, i.e. AB = ‘01’ and CD = ‘00’ with the output magnet ‘EF’ initially in the ‘down’ (‘11’) \nstate as shown in Fig 3. All other cases are ex haustively studied in the supplementary material \nthat accompanies this letter (see Figs S1 to S10 in the supplement, Ref 14). When tensile stress is \napplied to the nanomagnet along the [100] direction (i.e. at ) the output magnetization \nrotates right and settles at -45° as shown in Fig 3 (a). This is because Ni has negative \nmagnetostriction and a tensile stress tends to  rotate the magnetization away from the 45 º stress 45θ°=\n6 \n axis. Of the two perpendicular directions (+1 35° and -45°), -45° is closer and hence the \nmagnetization rotates from -90° to -45°. In the ne xt stage, the stress on ‘EF’ is stepped down to \nzero. The result, shown in Fig. 3(b), indicates rela xation of the magnet’s magnetization from -45° \nto 0° as expected. This can be  explained by understanding the eff ect of the bias field and dipole \ninteraction on the output magnet.  The left input (AB) favors th e output magneti zation orienting \nparallel to it, i.e. pointing right , while the right input (CD) favors the output aligni ng anti-parallel \nto it, i.e. pointing down. However, the global bias  field, pointing up, resolves the tie between the \n“down” (or -90º) state and “right” (or 0 º) state in favor of the “r ight” (or 0 º) state, causing the \noutput magnetization vector to se ttle to 0° when stress is relaxed to zero. Following this, a \ncompressive stress is applied to ‘EF’ at +45° th at rotates the magnetization from 0° to +45° (Fig. \n3(c)).  Subsequently, when stress  on ‘EF’ is relaxed to zero, th e final state of the output magnet \nsettles back to 0° (‘right’) under the influence of dipole interaction and the bias field as expected. \nThus at the end of the cycle, th e output EF = ‘01’ is realized, showing successful NOR operation.  \nFor universality, the initial st ate of the output nanomagnet should not affect the desired \nresult. To verify this, we performed simulations for the input combination AB = ‘01’, CD = ‘00’ \nwith the other three possible initial orientations of the output magnet (see Fig. S1, S2 and S3 in \nthe supplement, Ref 14) and have shown that the final output state EF = ‘01’ is achieved \nregardless of its initial orientat ion. Simulation of the output for a ll other input gate combinations \nare exhaustively covered in the supplementary material (see Fig. S4 - S10, Ref 14).  \nIn conclusion, we have shown the feasibility  of four state nanomagnetic logic using \nmultiferroic nanomagnets. It is obvious that if the initial state of any nanomagnet is not in one of \nthe four stable states, it will relax to the nearest stable state, thus behaving like an associative memory element\n15. They have applications in pattern recognition and other signal processing. \n7 \n Figure Captions \nFig. 1 (a): Ni/PZT multiferroic nanomagnet with single crystal Ni magnetostrictive layer having \na biaxial anisotropy that creates four possible magnetization direc tions (easy axes) – ‘up’ (00), \n‘right’ (01), ‘down’ (1 1) and ‘left’ (10). The bit assignments ( AB, A B−\n, B,A−\nA−\nB−\n) are also \nshown. The saddle-shaped curve represents th e energy profile of the nanomagnet in the \nground/unstressed state, in which the energy minima  are located along the eas y axes. As a result, \neach of these four axes is a possible direction for the magnetization. Nanomagnet array with dc \nbias magnetic field showing NOR logic realiza tion for all input combinations. The two input \nnanomagnets (AB, CD) are placed on either side  of the output magnet (EF). The dotted arrows \nindicate the occurrence of a tie-condition (output  has two equally possible choices) that is \nresolved by the influence of the bias field’ s direction. (b) The i nput combinations have \nmagnetization directions perpendicu lar to the magnet array axis, resu lting in the ou tput direction \nhaving two possible orientations – ‘up’ or ‘dow n’ (c) The magnetization direction of the inputs \nare parallel to the magnet axis. Cons equently, the output orientations ar e either ‘left’ or ‘right’ or \n‘up (tie condition)’. (d) The left input magnet, AB, is  either ‘left’ or ‘right’ while the right input, \nCD, is either ‘up’ or ‘down’. Th e output is, therefore, either ‘lef t’ or ‘right’ for non-tie cases and \n‘up’ (determined by bias field) when a tie condition arises. (e) AB is  either ‘up’ or ‘down’ while \nCD is ‘left’ or ‘right’. Similar to (d), the outputs  are either ‘left’, ‘right’ or ‘up (tie condition)’. \n \nFigure 2:  A Karnaugh-map representation of the input (AB, CD) combinations is illustrated, \nwith the output EF indicated in the dotted rectangle, which is then separated into individual ‘E’ \nand ‘F’ sub-K-maps in order to determine their logical expressions C A F=  and DBE= .  \n8 \n Figure 3:  Energy plots of the output nanomagnet (EF) as a function of the magnetization angle. \nThe initial conditions used are: AB = ‘01’, CD = ‘00’ and EF = ‘11’. (a) With no stress applied \nto magnet EF initially, the magnetization direction begins at -90°. The first stage of the stress \ncycle (Tension at +45°) is then initiated, which causes the magneti zation to rotate away from the \nstress axis to the closest energy  minima (-45°). (b) Upon relaxation of stress on EF (stage 2), the \nclosest energy minimum is at 0° and therefore, the magnetization rotates into that position. (c) \nStage 3 of the stress cy cle involves the application of a compressive stre ss (negative, +45°) on \nEF. The energy minima are located along the stress axis; therefore, the ma gnetization rotates and \nsettles at +45°. (d) Finally, the stress on EF is relaxed which cause s the magnetization to rotate to \nthe closest energy minimum (0°). The arrows indicate the direc tion of applied stress and the \nresulting magnetization rotation.  \n \n \n \n \n \n \n \n \n \n \n \n \n9 \n List of References \n1. R. P. Cowburn and M E Welland, Science, 287, 1466-1468 (2000). \n2. B. Behin-Aein, S. Salahuddin and S. Da tta, IEEE Trans. Nanotech., IEEE Trans. \nNanotech., 8 , 505 (2009). \n3. G. Csaba, A. Imre, G. H. Bernstein, W. Porod and V. Metlushko, IEEE Trans. Nanotech., 1, \n209 (2002). \nB. Behin-Aein, D. Datta, S. Sala huddin and S. Datta, Nature Nanotech., 5, 266 (2010). 4. \n5. M. T. Alam,  M. J. Siddiq, G. H. Bernstein, M. Niemier, W. Porod and X.S Hu , IEEE \nTrans. Nanotech., 9, 348 (2010). \n6. D.C. Ralph, M.D. Stiles, J. Magn. Mag. Mat., 320, 1190 (2008). \n7. J. Atulasimha and S. Bandyopadhyay, Appl. Phys. Lett. , 97, 173105, (2010). \n8. M. S. Fashami , K. Roy, J. Atulasimha  and S. Bandyopadhyay , arXiv:1011.2914v1 . \n9. K. Roy, S.Bandyopadhyay and J. Atulasimha, arXiv:1012:0819v1. \n10. S. A. Wolf, J. Lu, M. R. Stan, E. Chen and D. M. Treger, Proc. IEEE , 98, 2155 (2010). \n11. S. Chikazumi, Physics of Magnetism , Wiley New York, 1964. \n12. E. W. Lee, Rep. Prog. Phys., 18, 184 (1955). \n13. M. Lisca, L. Pintilie, M. Alexe a nd C.M. Teodorescu, Appl. Surf. Sci., 252, 30 (2006). \n14. Supplementary material located at….. \n15. V.P. Roychowdhury, D. B. Janes, S. Bandyopadhyay and X. Wang, IEEE Trans. Elec. \nDev., 43, 1688 , (1996); N. Ganguly, P. Maji, B. Sikdar and P. Chaudhuri, IEEE Trans. \nSyst. Man Cyber., Part B, 34, 672, (2004). \n \n \n10 \n 11 \n  \n \n \n \n \n \n \n \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n  AB00 01 11 10\nCD\n00 11 01 00 10\n01 01 01 00 00\n11 00 00 00 00\n10 10 00 00 10\n AB\nCD00 01 11 10\n00 1 0 0 1\n01 0 0 0 0\n11 0 0 0 0\n10 1 0 0 1 AB\nCD00 01 11 10\n00 1 1 0 0\n01 1 1 0 0\n11 0 0 0 0\nEF\nE F\n10 0 0 0 0\nFig.2\n12 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n  \n \n \n \n \n a) b)\n \n \n \n \n \n \n \n \n \nc) d)  \n  \nFig.3   \n13 \n SUPPLEMENTARY INFORMATION  \nFOUR-STATE NANOMAGNETIC L OGIC USING MULTIFERROICS \naNoel D’Souza , Jayasimha Atulasimhaa a , Supriyo Bandyopadhyay\nEmail: {dsouzanm, jatulasimha, sbandy}@vcu.edu  \n(a) Department of Mechanical and Nuclear Engineering,  \n(b) Department of Electrical a nd Computer Engineering,  \nVirginia Commonwealth Univer sity, Richmond, VA 23284, USA. \n \nIn the letter, we showed that dipole-coupled  Ni/PZT multiferroic nanomagnets with binary \nbits encoded in the four stable magnetization di rections (‘up’, ‘down’, ‘left’ and ‘right’) can \nimplement 4-state NOR logic with the proper clock sequence. We showed this for one particular \ninput bit combination and for one particular in itial state of the output. For the sake of \ncompleteness, we have to show it for all possibl e input combinations and for all possible initial \nstate of the output magnet.  This is shown here. \nOf the sixteen possible input conf igurations (input bits) and four  possible initial states of the \noutput magnet (output bit), the results and ener gy plots for one particul ar input configuration \n(AB = ‘right’, CD = ‘up’) and one initial state of the output (magnetization direction pointing \n‘down’) was shown in the letter. In this suppleme nt, we consider the other cases in order to be \nexhaustive. We first pick the i nput configuration (AB = ‘right’, CD = ‘up’)  discussed in the \nletter and show that for this input combination, the final state of the output is independent of the \ninitial state of the output. There are three other possible initial stat es of the output (‘left’, ‘right’ \nand ‘up’) and each of them is examined for the a bove input combination (Figs. S1 – S3). In each \ncase, the final output settles in the correct dire ction (‘right’) conforming to NOR logic. Thus, the \nfinal output is independent of the initial state of the output for this input combination. This \n14 \n means that the output is determined solely by the inputs and hence is a unique, single-valued \nfunction of the inputs. \n We also show the results obtained from the seven other unique input combinations when the \ninitial state of the output is EF = 11 (Figs. S4 – S10). It is  obvious that the fi nal state of the \noutput will be independent of the initial state fo r these input combinations  as well. The remaining \neight input combinations are not examined since they are equivalent to ones examined here due \nto symmetry. For all input combinations, the NOR function is always realized. \n \nSupplement Figure Captions \n \nFigure S1:  Energy plots of the output ma gnet representing the bits EF with input bits AB = ‘01’, \nCD = ‘00’ and EF = ‘00’ initial ly. (a) Rotation from +90° to +135°  as a consequence of tension \napplied along the +45° axis. (b ) Upon relaxation of the stress, the magnetization vector rotates \nback to +90°. (c) With compression applied on the output magnet along the +45° axis, its \nmagnetization rotates to +45°. (d) Finally, when the stress is relaxed to zero, the output magnet \nrotates its magnetization to 0°, completing the NOR logic operation. \n \nFigure S2: The input combinations AB=’01’, CD=’00’ are the same as in figures S1. The initial \nstate of the output magnet EF is set at ‘01’. The stress cycle (tension, relaxation, compression, \nrelaxation) is applied to the output magnet with  the magnetization rotatin g sequentially through -\n45°, 0°, +45° and finally settling at 0°, th ereby once again completing the NOR operation. \n \n15 \n Figure S3: AB = ‘01’, CD = ‘00’ as in figure S1. The in itial output state is set to EF = ‘10’. The \nstress cycle applied to the output magnet causes its  magnetization to rotate sequentially through \n+135°, +90°, +45° and 0°, thereby completing th e NOR operation. Figs. S1-S3 show that the \nfinal state of the output is ind eed independent of the initial state and hence determined uniquely \nby the inputs. \n \nFigure S4: AB = ‘00’, CD = ‘00’ and EF = ‘11’. Si nce the inputs are pointing ‘up’, the dipole \ninteraction pushes the output magnet’s magnetiza tion vector ‘down’. The stress cycle applied to \noutput magnet causes its magnetization vector to  rotate sequentially through -45°, -90°, -135° \nand back to -90°, completing the NOR operation. \n \nFigure S5: AB = ‘00’, CD = ‘11’ and EF = ‘11’. The input magnetization directions, on either \nside of the output magnet encodi ng EF, point in opposite directi ons (‘up’ and ‘down’). As a \nresult, the dipole interaction of the inputs on th e output cancels out. This would result in a tie-\ncondition when the stress cycle is applied (speci fically, during the relaxation phases, when the \nmagnetization would have two equally  likely directions to settle into ).  However, when a dc bias \nmagnetic field is applied [H appl = 1000 A/m (~12 Oe)], the energy profile is no longer symmetric \nand is slightly biased towards +90°. Now, when the stress cycle is app lied to the output magnet, \nits magnetization rotates through +135°, +90°, +45° and 90°, thereby once again completing the NOR operation. \n \nFigure S6: AB = ‘01’, CD = ‘01’ and EF = ‘11’. In this case, the inputs are both pointing \ntowards the ‘right’. Hence, the dipole intera ction shows a strong preference for ferromagnetic \n16 \n coupling (parallel arrangement). This can be seen in the energy profile of  the output magnet, EF, \nwhich has an absolute energy minimum located at  0° (‘right’). The magne tization rotation arising \ndue to the stress cycle applied to the output magne t is from the initial -90° direction to -45° \n(since the ferromagnetic coupling due to the di pole interaction is strong, the magnetization \neasily rotates to 0° at low values of applied stress. However, at higher stress values, the stress \nanisotropy energy is greater than the dipole en ergy and, consequently, the magnetization settles \nat -45°, 0°, +45° before settling to 0°. Ultimatel y, the NOR operation is once again realized. \n \nFigure S7:  AB = ‘01’, CD = ‘10’ and EF = ‘11’. Since the input magne tizations point in \nopposite directions, there is no net dipole interac tion on the output magnet encoding the bits EF \n(similar to the configuration of Fig. S5). Once  again, the applied bias magnetic field tips the \nenergy profile of the output magnet towards +90°. The stress cycle applied to the output magnet \ncauses its magnetization to rotate through -45°, 0°, +45° and + 90°, thus implementing the NOR \noperation. \n \nFigure S8: AB = ‘01’, CD = ‘11’ and EF = ‘11’. In this configuration, AB points ‘right’ while CD points ‘up’. In the first stage of the stress cycle (tension along +45°) on the output magnet, the magnetization rotates from -90° to -45° (s imilar to Fig. S6, at low tensile stresses, the \npreferred alignment is 0°, parallel to AB. Further increases in stress cause the magnetization to settle at -45°). Relaxation of the stress then ro tates it to 0°, compression takes it to +45° and \nultimately, relaxation causes it to settle at +90°. The NOR operation is realized. \n \n17 \n Figure S9: AB = ‘10’, CD = ‘00’ and EF = ‘11’. With  AB pointing ‘left’ and CD pointing ‘up’, \nthe dipole interaction is similar to that of the case in Fig. S7, with the output EF preferring a \nparallel alignment with AB. The stress cycl e applied to the output magnet causes its \nmagnetization to rotate from the initial direction of -90° to -45°, -90°, -135° and finally, -180°, \nthus implementing the NOR function. \n \nFigure S10:  AB = ‘10’, CD = ‘11’ and EF = ‘11’. This configuration (AB points ‘left’, CD \npoints ‘down’) is similar to that of Fig. S8. Th e applied bias field tries to align the output \nmagnet's magnetization vector along the +90° direction, without which a tie-condition would \narise (two equally possible directions). The stress cycle applied to the output magnet causes its \nmagnetization to rotate sequentially through -45°, 0°, +45° and +90°. This completes the NOR operation. \n \n18 \n  \n \nFig.S1c) d)a) b)\n19 \n Fig.S2c) d)a) b)\n20 \n  \n \n \n \n \n \n \n \n \n \n \nFig.S3c) d)a) b)  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n21 \n  \nFig.S4c) d)a) b)\n22 \n 22 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n  \nFig.S5c) d)a) b)\n \n \n \n \n \n \n \n \n \n \n23 \n  \n \n  \n \n \n \n \n \n \n \nFig.S6c) d)a) b)\n \n \n \n \n \n \n \n \n \n \n \n \n \n24 \n  \n \n  \n \n \n \n \n \n \n \n \n \n \n \nFig.S7c) d)a) b)\n \n \n \n \n \n \n \n \n \n25 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig.S8c) d)a) b)\n \n \n \n \n \n \n \n \n \n26 \n  \n \n  \n \n \n \n \n \n \n \n \n \n  \nFig.S9c) d)a) b)\n \n \n \n \n \n \n \n \n \n \n27 \n  \n \n \n \n \n \n \n \n \na) b)  \n \n \n \n \n \nc) d)\nFig.S10\n28 \n "    },    {        "title": "1103.3783v1.Magnetic_dynamics_of_single_domain_Ni_nanoparticles.pdf",        "content": "J. Appl. Phys.  93, 6531 (2003); doi:10.1063/1.1540032 ( 3 pages ) \n \nMagnetic dynamics of single domain Ni nanoparticles  \n \n \n \nG. F. Goya, F.C. Fonseca, and R. F. Jardim  \nInstituto de Física, Universidade de São Paulo, CP 66318, 05315 -970, São Paulo, \nBrazil  \n \nR. Muccillo  \nCentro Multidisciplinar de Desenvolvimento de Materiais Cerâmicos CMDMC, \nCCTM -Instituto de Pesquisas Energéticas e Nucleares CP 11049, 05422 -970, São \nPaulo, SP, Brazil  \n \nN. L. V. Carreño, E. Longo, and E.R. Leite  \nCentro Multidisciplinar de Desenvolvimento d e Materiais Cerâmicos CMDMC, \nDepartamento de Química, Universidade Federal de São Carlos CP 676, 13560 -905, \nSão Carlos, SP, Brazil  \n \n \nThe dynamic magnetic properties of Ni nanoparticles diluted in an amorphous \nSiO2 matrix  prepared from a modified sol -g e l  me thod have been studied by the \nfrequency f dependence of the ac magnetic susceptibility χ(T). For samples with \nsimilar average radii ~ 3 -4 nm, an increase of the blocking temperature from T B ~ 20 \nto ~ 40 K was observed for Ni concentrations of ~ 1.5 and 5 wt.%, respectively, \nassigned to the effects of dipolar interactions. Both the in -phase χ’(T) and the out -of-\nphase χ’’(T) maxima follow the predictions of the thermally activated Néel -Arrheni us \nmodel. The effective magnetic anisotropy constant K eff inferred from χ’’(T) versus f \ndata for the 1.5 wt.% Ni sample is close to the value of the magnetocrystalline \nanisotropy of bulk Ni, suggesting that surface effects are negligible in the present \nsamples. In addition, the contribution from dipolar interactions to the total anisotropy \nenergy E a in specimens with 5 wt.% Ni was found to be comparable to the intrinsic \nmagnetocrystalline anisotropy barrier.  \n AF-13 \n  2  \nThe dynamics of ferromagnetic nanoparticles with different interaction \nstrengths has been widely studied in recent years.1,2 The model describing the \nmagnetic behavior of a system of monodispersed and noninteracting single -domain \nparticles proposed by Néel3 has been successfully tested by numerous e xperiments \nwith increasing sophistication, as the delicate series of works that have recently \nconfirmed its applicability at the single -particle level.4  \nFor a single -domain particle, the energy barrier between magnetic states may \nbe considered to be propo rtional to the particle volume V. In the case of uniaxial \nanisotropy, the anisotropy energy E a in the absence of external magnetic field is \ndescribed by θ2\neffVsin K=aE , where K eff is an effective magnetic anisotropy \nconstant and θ is the angle betw een the magnetic moment of the particle and its easy \nmagnetization axis. On the other hand, the dynamic response of such particles to an \nalternating external magnetic field is determined by the measuring time τm of each \nexperimental technique. Since revers ion of the magnetic moments over the anisotropy \nenergy barrier E a is assisted by thermal phonons, the relaxation time τ of each \nmagnetic particle exhibits an exponential dependence on temperature characterized \nby an Néel -Arrhenius law  \n√√\n↵\u0000\n\u0000\u0000\n\u0000=\nTexp0\nBa\nkEττ        (1) \nwhere f0 = τ0−1 is an attempt frequency. Typical values for τ0 are in the 10-9 - 10-11 s \nrange for superparamagnetic (SPM) systems.  \nWhen an ensemble  of single -domain magnetic particles is considered, the \nabove description is still valid provid ed that the particles are non interacting.  3 However, as the concentration of the magnetic phase increases, interparticle \ninteractions alter the single -particle energy barrier, and concurrent effects involving \ndipolar interactions, particle size distribution , and aggregation make the application of \nEq. (1) not obvious. To better understand how dipolar interactions affect the SPM \nrelaxation rates it is therefore desirable to prepare samples near the infinite -dilution \nlimit of the magnetic phase, settling the s ingle -particle properties of a specific \nmagnetic system, and then gradually increase the particle density. In this work we \nhave used the above approach to study the dynamics of magnetic properties in high -\nquality Ni nanoparticles. The samples were prepared  by a modified sol -gel technique \nand characterized by ac magnetic susceptibility χ(T) measurements as a function of \ntemperature, applied field, and excitation frequency.  \nNanocomposites of Ni:SiO 2 were synthesized by using tetraethylorthosilicate \n(TEOS), citric acid, and nickel (II)  nitrate. The citric acid was dissolved in ethanol \nand the TEOS and the nickel nitrate were added together and mixed for \nhomogenization at room temperature. After the polymerizing reaction adding \nethylene glycol, the solid resin  was heated at 300 °C, ground in a ball mill, and then \npyrolyzed at 500 °C. Further details of the method employed can be found \nelsewhere.5 In the present work we will concentrate our discussion in two samples \nhaving ~1.5 - and 5 -wt.% Ni which will be refer red as S1 and S2, respectively. The \nstructure and morphology of the magnetic powders were examined by transmission \nelectron microscopy with a 200 -kV, high -resolution transmission microscope. \nMagnetization and ac magnetic susceptibility measurements were pe rformed in a \ncommercial SQUID magnetometer both in zero -field-cooling (ZFC)  and field -cooling \n(FC) modes, between 1.8 K < T < 300 K and under applied fields up to 7 T. The  4 frequency dependence of both in -phase χ'(T) and out -of-phase χ''(T) components of \nthe ac magnetic susceptibility was measured by using an excitation field of 2 Oe and \ndriving frequencies between 20 mHz < f < 1.5  kHz.  \nWe have previously characterized these two samples of Ni nanoparticles \nembedded in SiO 2 by several techniques and have observed some features which are \nsummarized as follows: (1) a log -normal distribution of particle sizes with average \nradius close to ~ 3 -4 nm for both samples; (2) the occurrence of a SPM behavior \nabove T B > 20 K and 40 K, f or samples S1 and S2, respectively; (3) a nearly spherical \nmorphology for both samples; and (4) the absence of a shell -core NiO -Ni \nmorphology, where an antiferromagnetic layer of NiO (shell) surrounds the \nferromagnetic Ni (core) particles.6 \nTurning now to the dynamics of the magnetic particle systems, Fig. 1 displays \nthe temperature dependence of χ'(T) and χ''(T) of the more diluted sample S1 and for \ndifferent frequencies f . The data for both components χ'(T) and χ''(T) exhibit the \nexpected behavior of a S PM system, i. e., the occurrence of a maximum in \ntemperature for both χ'(T) and χ''(T) components, and a shift of this maximum \ntowards higher temperatures with increasing frequency. The freezing of the magnetic \nmoments from the SPM to a blocked state occur s at the blocking temperature, T B, at \nwhich the relaxation time τ of the Ni nanoparticles is equal to the experimental time \nwindow τe = 1/f of the ac measurement, T B = β Ea / kB ln(1/ f τ0).7 In this expression β \nrepresents the effect of the particle size d istribution g(D), being β = 1 for a \nmonodispersed sample (i.e., a delta g(D) = δ(D-D0) size distribution). However, spin -\nglass systems also display features similar to the ones described above and it seems \nconvenient to classify first our Ni nanoparticles.  An empirical and model -independent  5 criterion used for classifying a transition to a frozen state is the relative shift of the \ntemperature of the maximum in χ”(T), T m, with the measuring frequency f as \n)( log TT\n10 mm\nf ∆∆=Φ        (2) \nwhere ∆Tm is the difference between T m measured in the ∆log 10(f) frequency interval.  \nExperimentally, the Φ values found for SPM systems are in the range ~0.10 -\n0.13, whereas a much smaller dependence of T m with f is observed in spin glasses ( Φ \n~ 5x10-3 – 5x10-2). 2,8 Theref ore, Eq. (2) provides a model -independent classification \nof the kind of freezing transition. However, it is well known that intermediate \nsituations (0.001< Φ < 0.05) are often reported, usually related to non -diluted \nparticulate systems. 2,9 Our calculated  values of Φ = 0.12 and 0.13 for samples S1 and \nS2, respectively, show unambiguously that the shift in T m with increasing f \ncorresponds to a thermally activated Néel -Arrhenius model for superparamagnets.  \nThis behavior was confirmed by the linear dependence  of ln[ τ] versus 1/T B \nshown in Fig. 2 for both samples. It can be further seen that both curves are fitted \nvery well by using Eq. 1 and show the same extrapolated value of τ0 = 8x10-10 s, \nconsistent with a SPM system. The frequency dependence of T B in Eq. 1 is \ndetermined by the effective activation energy barrier E a. Contributions to E a can \noriginate from intrinsic anisotropies of the particles (shape, magnetocrystalline, or \nstress ansotropies) or interparticle interactions (dipolar or exchange). Inasmuch a s \nthese two mechanisms contribute to modify the energy barrier, it is usually quite \ndifficult to separate both kind of effects.  \nThe values of K eff of our samples were extracted from the activation energies \nby using the average particle radii from TEM data (rm = 4.2 and 3.3 nm for S1 and S2,  6 respectively) and then compared to the first -order magnetocrystalline anisotropy \nconstant at low temperature K 1bulk\n = -8x105 erg/cm3 of bulk Ni.10 For the present case, \nwith cubic anisotropy and K 1 < 0, the effective (un iaxial) anisotropy is related to K 1 \nthrough the relation K eff = K 1/12. 7 Therefore, from the K eff =1.3x105 erg/cm3 value \nobtained for S1 a magnetocrystalline anisotropy of K 1 = 15x105 erg/cm3 is extracted, \nwhich is only twice the value of K 1bulk. If shape anisotropy is assumed as the only \nsource of anisotropy, a small deviation from spherical shape (e.g., to prolate \nspheroidal) to an axis ratio c/a ~ 1.2 would be enough to explain the calculated value \nof K eff. On the other hand, it is useful to relate K eff = 1.3x105 erg/cm3 with the \nexpected coercive field for purely magnetocrystalline anisotropy of spherical particles \nHC = 2K eff/MS ≈ 500 Oe, a value in excellent agreement with H C ~ 520 Oe obtained \nfrom hysteresis curves at low temperatures. 6 Therefore, the above data suggest that \nthese Ni particles have indeed nearly spherical shape, with intrinsic magnetic \nanisotropy close to the Ni (fcc) bulk value.  \nReturning to the curves shown in Fig. 2, it is also clear that the energy barriers \nincrease with increasing Ni content, as inferred from the larger slope of ln τ vs. T B-1 \ncurves.  Such an increase in E a can not be related to a larger average volume of the Ni \nparticles in sample S2 since both radius distributions have similar mean values.  \nActually , the average radius r m extracted from the log -normal distribution of sample \nS2 is slightly smaller  than for sample S1. 6  Similarly, from our previous discussion \nregarding the value of K eff obtained for sample S1, a significant contribution from \nsurface e ffects to E a in sample S2 seems to be unlikely.  Therefore, the increase of the \neffective energy barrier for the more concentrated sample should be related to dipolar \ninteractions.   7 Following this discussion, we have estimated this dipolar contribution to t he \ntotal energy by comparing the values of E a for both samples S1 and S2.  Based on the \nsimilar volume distributions from TEM images, we assume that Ni nanoparticles in \nboth samples have similar intrinsic anisotropies.  Within this context, the only effect  \nof increasing concentration is thus to add a dipolar term E dip to the effective \nactivation energy E a.  Following Luis et al.11 we have used a modified Arrhenius -Néel \nexpression for the relaxation time including the contribution of the dipolar energy as τ \n= τ0 exp{(U 0 + E dip)/kBT}, where U 0 is the single -particle energy barrier.  From this \nrelationship, we can write  \nTkE\nBdip=√√\n↵\u0000\n\u0000\u0000\n\u0000\n12lnττ\n       (3) \nwhere τ1 and τ2 are the relaxation times of samples S1 and S2, respectively.  From the \nEa values fitted for samples S1 and S2, we obtained E dip = 247 K.  This value is \ncomparable to U 0 = 282 K for single -domain and isolated Ni nanoparticles as \nestimated from S1 sample.  \n In conclusion, we have studied the dynamics of ferromagnetic Ni \nnanopartic les with similar radius distributions and different concentrations via ac \nmagnetic susceptibility measurements.  The general behavior of these nanoparticles is \nwell described by the Néel -Arrhenius model for single -domain, noninteracting \nparticles.  For the  more diluted sample with 1.5 wt.% Ni, the estimated magnetic \nanisotropy of the particles was similar to the value of the magnetocrystalline \nanisotropy for bulk (fcc) Ni, suggesting that both shape and surface anisotropies are \nnegligible.  For the more con centrated sample with 5 wt.% Ni, the increase of the \nenergy barrier E a could be described by an additional contribution E dip coming from  8 dipolar interactions.  We estimated E dip ≈ 247 K, a value comparable to the intrinsic \nmagnetic anisotropy U 0 ≈ 282 K fo r single -domain nanoparticles.  \n \n This work was supported in part by the Brazilian agency Fundação de Amparo \nà Pesquisa do Estado de São Paulo (FAPESP) under Grant Nos. 99/10798 -0; \n01/02598 -3; 01/04231 -0; 98/14324 -0; and the Conselho Nacional de \nDesenvolvim ento Científico e Tecnológico (CNPq) under Grant Nos. 300569/00 -9 \nand 304647/90 -0.  \n \n   9 REFERENCES  \n \n \n1. X. Batlle and A. Labarta, J. Phys. D 35, R15 (2002).  \n \n2. J. L. Dormann, D. Fiorani, and E. Tronc, in Advances in Chemical Physics , Ed. By \nI. Prigogine and S. A. Rice. Vol. XCVIII (1997) p.326.  \n3. L. Néel, Ann. Geophys. 5, 99 (1949).  \n4. E. Bonet, W. Wernsdorfer, B. Barbara, A. Benôit, D. Mailly, and A. Thiaville, \nPhys. Rev. Lett. 83, 4188 (1999).  \n5. E. R. Leite, N. L. V. Carreño, E. Longo, A. Valentini, a nd L. F. D. Probst, J. \nNanosci. Nanotechnol. 2, 89 (2002).  \n6. F. C. Fonseca, G. F. Goya,  R. F. Jardim, R. Muccillo, N. L. V. Carreño, E. Longo, \nand E. R. Leite, Phys. Rev. B 66, 104406 (2002).  \n7. J. I. Gittleman, B. Abeles and S. Bozowski, Phys. Rev. B 9, 3891 (1974).  \n8. J. A. Mydosh, Spin Glasses: an experimental introduction,  (Taylor & Francis, \nLondon, 1993)Chap.3.  \n9. J. A. De Toro, M. A. López de la Torre, M. A. Arranz, J. M. Riveiro, J. L. Marínez, \nP. Palade, and G. Filoti, Phys. Rev. B 64, 094438 (200 1). \n10. H. J. Williams and R. M. Bozorth, Phys. Rev. 56, 837 (1939).  \n11. F. Luis, F. Petroff, L. M. García, J. Bartolomé, J. Carrey, and A. Vaurès, Phys. \nRev. Lett. 88, 217205 (2002).  \n  10 Figure Captions  \n \nFigure 1.  Temperature dependence of the real component χ´(T) of the magnetic \nsusceptibility for 1.5 wt.% Ni (sample S1) at different excitation \nfrequencies. Inset: Imaginary part χ´´(T) for the same sample shown in  \nan expanded T -scale. The data were taken with an external magnetic \nfield H of 50 Oe.  \n \nFigure 2.  Arrhenius plots of the relaxation time τ vs. blocking temperature T B \nobtained from the imaginary component χ´´(T) of the ac magnetic \nsusceptibility. Dashed lin es are the best fit using Eq. (1) with a single \nτ0 value and E a(S1) = 282 K, E a(S2) = 529 K.  \n   11  \n \n \n  \n0 20 40 60 80 100 120 14001234\n 1488 Hz\n 947 Hz\n 400 Hz\n 150 Hz\n 80 Hz\n 11 Hz\n 2 Hz\n 0.7 Hz\n  χ' (10-3 emu/g)\nT (K)0204060801000123\nFigure 1\nGoya et al.  \nT (K)χ'' (10-4 emu/g) 12  \n \n \n0 10 20 30 40 50 60 70 80-20-15-10-50\nFigure 2\nGoya et al.\nτ0 = 8 10-10 s\n  \n S1\n S2ln τ\nT -1 (x10 -3 K -1 )"    },    {        "title": "1103.3786v1.Magnetic_properties_and_energy_absorption_of_CoFe2O4_nanoparticles_for_magnetic_hyperthermia.pdf",        "content": "Magnetic properties and energy absorption of  C o F e 2O4 \nnanoparticles for magnetic hyperthermia  \n \nT.E. Torres1,2,3, A.G.  Roca1,4, M.P. Morales5, A. Ibarra1, C. Marquina2,3, M.R. \nIbarra1,2,3 and G.F. Goya1,2  \n \n1 Aragon Institute of Nanoscience (INA), University of Z aragoza,  Zaragoza  Spain.  \n2 Condensed Matter Department, Sciences Faculty, University of Zaragoza, Spain.  \n3 Instituto de Ciencia de Materiales de Aragón (ICMA), CSIC  - Universidad de \nZaragoza,   Zaragoza,  Spain . \n4 Networking Biomedical Research Center (CIBER -BBN), Zaragoza, Spain . \n5 Instituto de Ciencia de Materiales de Madrid, CSIC, Madrid, Spain.  \n \nAbstract.  We have studied the magnetic and power absorption properties of three samples of \nCoFe 2O4 nanoparticles with sizes from 5 to 12 nm prepared by thermal decomposition of Fe  \n(acac) 3 and Co(acac) 2 at high temperatures. The  blocking temperatures T B estimated from \nmagnetization M(T) curves spanned the range 180 ≤ TB ≤ 320 K, reflecting the large \nmagnetocrystalline anisotropy of these nanoparticles . Accordingly, high coercive fi elds H C ≈ 1.4 \n- 1.7 T were obser ved at low temperatures . Specific Power Absorption (SPA)  experime nts \ncarried out in ac magnetic fi elds indicat ed that, besides particle volume, the eff ective  magnetic  \nanisotropy is a key parameter determining the absorption effi ciency. SPA values as high as 98  \nW/g were obtained for nanoparticles with average size of  ≈ 12 n m . \n \n \n1. Introduction.  \nThe capability of magnetic nanoparticles (MNPs) to act as effective heating agents for Magnetic \nHyperthermia (MHT) was demonstrated many years ago [1]. Considerable efforts have been \nmade  in the synthesis and characterisation of MNPs  to assess their capaci ties as heat  generating \nagents and to establish the  mechanisms governing  heat generation  at the nanoscale . Several \nstudies have shown a link between the MNPs energy absorption in ac magnetic fields and the ir \nsize [2]. We present a study of CoFe 2O4 single -domain  MNP s, having sizes  from 5 to 12 nm, \nand relate their structural and magnetic properties with the power absorption efficiency .  \n \n2. Experimental Procedure  \nCoFe 2O4 nanoparticles of different sizes were prepared by high temperature decomposition of \niron and cobalt organic precursors  as described elsewhere [3,4]. The samples were synthesized \nusing iron and cobalt acetylacetonate  and different solvents ( phenyl ether and 1-octadecene ) \nwhich led to  different synthesis te mperatures. To control the final particle size, different \nprecursor/surfactant mola r ratios  were used.  Table  summarizes the different synthesis \ncond itions. The resulting nanoparticles were washed several times  with ethanol after \nmagnetically -assisted precipitation , and the final product w as re-dispersed in hexane . \n   \nTable 1: Molar precursor/surfactant ratio  Rmol, reflux  temperature Tref and growth time t G at T ref, used for \nthe synthesis of the CoFe 2O4 nanoparticles.  \nSample  solvent  Rmol Tref dTEM  σ \n[precursor]/[surfactant]  (ºC) ( n m )   \nM01  Diphenyl -ether  1:3 260 5.7(3)  0.26(3)  \nM03  Diphenyl  -ether 1:10 260 7.0(4) 0.25( 4) \nS01 1-Octadecene  1:3 360 12.7(4)  0.16(2)  \n Particle size and shape were studied by Transmission Electronic Microscopy (TEM) using a \nthermoionic 200 kV Tecnai T20  microscope. The mean particle size ( dTEM) and statistical size \ndistribution were evaluated  by measuring the largest internal dimension of at least  100 particles. \nX-ray diffraction of powders were made between 10º < 2 θ < 70º using Cu -Kα radiation.   Iron \nand cobalt concentrations were determined by Atomic Emission Spectroscopy -Inductively \nCoupled  Plasma (AES -ICP).  \n \n10 20 30 40 50 60 70050100150\n(111)(220)(331)\n(222)(400)\n(422)(551)\n  Relative Intensity/cps\n2θ (theta)(440)\n \nFigure 1. (a) TEM image s of CoFe2O 4 nanoparticles obtained from de composition of Fe(III)  and Co(II) \nacetylacetonate in Fenhyl ether (sample M03, left) and  in 1-Octadecene   (sample S01, right ); Inset s: log-normal  size-\ndistribution s.  (b) X -ray pattern of sample S01.  \n \nMagnetization and ac susceptibility measurements were performed from 5 to 340 K , in \nmagnetic  fields up to 5 Tesla, in a commercial SQUID magnetometer. The temperature \ndependence of the magnetization was measured following zero -field-cooling (ZFC) and field -\ncooling (FC) protocols (applied field H= 100 Oe).  AC susceptibility measurements  were \nperformed at frequencies from 0.1 to 103 Hz and a field amplitude of  3 Oe. SPA was measured \nusing a commercial applicator (nB nanoscale Biomagnetics) working at 260 kHz and field \namplitudes from 0 to 160 mT . \n  \n3. Experimental Results  and Discussion.  \nSize histograms obtained from TEM images  fit to  a log -normal distribution . As an example, \nTEM images and size distributions of samples M03 and S01 are shown in figure 1a. The \nresulting  dT E M and polidispersity , σ, of the three synthesized samples  are shown in Table . As \nseen in  figure  1, thermal decomposition of iron –organic precursors resulted in  very uniform \nnanoparticles . Particles obtained by reflux in phenyl ether  (samples M01 and M03)  have a mean \nparticle size of 5 -7 nm and are less spherical than particles of sample S 01, synthesi zed in 1-\noctadecene , which have an average particle size dTEM = 12.7 nm . The XRD pattern of the as \nprepared powder of sample S01  was indexed w ith the cubic spinel structure (figure 1 b). In \nagreement with TEM data, t he average crystallite size extracted  from the most intense (3 1 1) \nreflection by applying the Scherrer formula was dXRD = 12 nm .  \n \nTable 2: Average particle diameter dTEM, saturation magnetization MS, coercive field HC, Effective \nmagnetic anisotropy Keff and Specific Power Absorption (SPA) of CoFe 2O4 nanoparticles.  \nS a m p l e  dTEM  ΤΒ HC (Oe) MS (emu/g)  Keff SPA  \n ( n m )  (K) 5 K 280 K  5 K 280 K  (erg/cm3) ( W / g )  \nM01  5.7(3)  174 14000  9.3 77.05  62.6 19×106 10.08  \nM03  7.0(4) 212 17100  8.3 73.19  60.0 14×106 18.59  \nS01 12.7(4)  315 17000  264 79.26  70.11  2.5×106 98.51  \n \nZFC and FC curves are shown in f igure 2 a. In spite of the small size of the MNPs, large \nblocking temperatures T B were derived from the  ZFC results , reflecting the large \nmagnetocrystalline anisotropy of th e particles . T B shifts to higher  values as the particle size  \nincreases ( see table 2 ). In the  case of the S03 sample, an abrupt increase of the magnetization \nbelow TB was observed, which has not a clear explanation so far.  In agree ment with the  high TB \nvalues, hyste resis loops measured at 5 K (see figure 2b) display high coercive fields, HC, listed \nin table II. A bove T B, HC drops to zero  as expected for the superparam agnetic regime. The \nvalues of saturation magneti zation M S (see table II) are close  to those of bulk CoFe 2O4 [5]. \n \n036912036912\n0 50 100 150 200 250 300061218M01\n \n M03 M(emu/g CoFe2O4)(a)H=100 Oe \n \n \n  \n \nT(K)S01-60-3003060\n-60 -40 -20 0 20 40 60-80-4004080\nT = 5 K\n  \n M01\n M03\n S01T = 280 K(b)\n  M(emu/gCoFe2O4 )\nH(kOe) \nFigure 2. (a) M(T) data taken in Zero -field-cooled (ZFC) and field -cooled (FC) modes for CoFe 2O4 samples . \n(b) M (H) curves measured at 280 K ( top) and  5K (bottom ). \n \n \nIn a single -domain particle  with volume V and effective magnetic anisotropy Kef f, the \nreversion of the magnetic moment over the anisotropy energy barrier Ea = Keff V is assisted by \nthermal phonons, and the relaxation time τ is given by a Neél –Arrhenius law\n)/ exp(0 TkEB a ττ= . AC susceptibility results ( figure 3a ) showed that χ´(T) and χ´ ´(T) exhibit \nthe expected behaviour  from  SPM systems, i.e., the occurrence of a maximum at Tm, related to \nthe unblocking of the magnetic moments. For  the imaginary component χ´´(T), the peak at  Tm \ncorresponds to the temperature for which fexp = τ, where fexp is the ac excitation frequency . \nTherefore  as the excitation frequency increases, Tm shifts to higher values [6]. \nThe linear dependence of ln( τ) vs. TB-1 observed in figure  3b indicates that the Neé l–\nArrhenius model correctly suits  the behavio ur of the three  s a mples . The fitting of the \nexperimental  data using the average  particle diameter  dTEM yielded  the values of  Keff listed in \nTable 2 . The values for the smaller particles  M01 and M03 are one order of magnitude larger  \nthan the first-order magnetocrystalline anisotropy constant  of bulk  magnetite  (K1= 2.0x106 \nerg/cm3) [5], as found in other CoFe 2O4 nanoparticles  [7]. This anisotropy enhancement  \nobserved in  MNPs  is customarily associated to surface  effects  [8]. Accordingly, the larger \nparticles S01 have Keff values close to the bulk value.  \n \n0 50 100 150 200 250 30001234567\n60 120 180 240 300010203040\n  χ´ (x 10-2 emu/gOe)\nT(K) 0.1 Hz\n 1Hz\n 10Hz\n 80 Hz\n 300 Hz\n 1000 Hz(a)  χ´´(x10-4 emu/gOe) \nT(K)\n3,0 3,5 4,0 4,5 5,0 5,5 6,0-8-6-4-2024\nEa= 18430 KEa= 14604 K Ea= 11929 KLn(τ)\nTB-1(x10-3 K-1) \n  \nM01 \nM03 \nS01  (b)\n Figure 3. (a) Temperature dependence of the in -phase (real) component χ´(T) of the magnetic  ac susceptibility for \nthe M01  sample, at different excitation frequencies. Inset: Out of phase  (imaginary ) component χ´´(T). (b) Arrhenius \nplot of the relaxation time τ logarithm vs. the inverse blocking temperature  TB-1. \n \nTo assess the influence of the size and magnetic  parameters on heat generation, the SPA was \nmeasured  from the temperature increase  (∆T )  o f  a  g i v e n  m a s s  o f  MNPs  diluted in the liquid \ncarrier  during the time interval ( ∆t) of the experiment . The expression for power absorption Π \nper unit mass of the magnet ic material is given by  \n∆∆=ΠtT cLIQ LIQ\nφδ, where cLIQ and δLIQ \nare the specific heat capacit y and density  of the liquid carrier, respectively , and φ is the weight \nconcentration of the MNPs in the colloid. [2] The SPA values listed in t able 2 (in W/g of  \nCoFe 2O4) show  that particles with d TEM = 12 n m (S01) are much more efficient for power \nabsorption , whereas  particles with 5 -6 nm  show poor heating aptitude . This results  support the  \nNéel relaxation -based mechanism  of magnetic relaxation, in which  the opti mu m size  of \nnanoparticles for hyperthermia lays within a narrow size range, whose mean value strongly \ndepends on the magnetic anisotropy of the system. [9] The large anisotropy of our NPs implies \nthat good absorption can be obtained using smaller particles  than for Fe 3O4 particles, thus \nhelping to the colloidal stability and size limitations for biomedical applications.  \n \n4. Conclusion s \nWe have succeeded in producing highly stable , nearly monodisperse  magnetic NPs with large \neffective magnetic anisotropy . Due to this anisotropy, high blocking temperatures were \nobserved even in small (5 -6 nm) particles . Moreover , excellent heating efficiency was observed \nfor NPs with d =12-13 nm, much smaller  than the optimal values needed to obtain similar SPA \nvalues using magnetite NPs (25 -35 nm). Our results suggest that CoFe 2O4 NPs could overcome \nsome of the problems derived from colloidal instability  of large Fe 3O4 nanoparticles for \nbiomedical applications . \n \n5. Acknowledgements  \nThe authors acknowledge financial support f rom the Spanish Ministry of Science and \nInnovation (MICINN) (projects CONSOLIDER CSD2006 -00012, MAT2008 -0274/NAN, \nMAT2008 -06567 -C02-02/NAN) and Diputación General de Aragón (project PI118/08). GFG \nthanks MICINN through the Ramon y Cajal program. AGR thanks  CIBER -BBN center of the \nSpanish Ministry of Health  through projects IMAFEN, NANOMAG, MICROPLEX and \nMONIT .  \nReferences  \n[1] Jordan A, Wust P, Fahling H, John W, Hinz A and Felix R 1993 Int. J. Hyperthermia  9 \n51  \n[2] G o y a  G  F ,  L i m a  J r .  G F, ArelaroA D, Torres T, Rechenberg H R, Rossi L, Marquina C, \n  and Ibarra M R, 2008  IEEE Trans. Magn. Magn. , 44 4444  \n[3] Sun S, Zeng H, Robinson D B, Raoux S, Rice M, Wang S X and Li G 2004 J. Am. Chem. \nSoc. 126 273  \n[4] Roca A G, Morales M P, O’Grady K  and Se rna C J 2006 Nanotechnology  17 2783  \n[5] Cullity B D 1972 Introduction to Magnetic Materials (Reading, MA: Addison -Wesley)  \n[6] Dormann J L, Fiorani D and  Tronc E 1997 Adv. Chem. Phys.  XCVIII  326 \n[7] López J L, Pfannes H D, Paniago R, Sinnecker J P, Novak M A 2008 J. Magn. Magn. \nMater. 320 e327  \n[8] Batlle X and Labarta A 2002,  J. Phys. D  35, R15  \n[9] Rosensweig R E 2002 J.Magn.Magn.Mater.  252 370 "    },    {        "title": "1103.5024v1.Temperature_dependence_of_spin_resonance_in_cobalt_substituted_NiZnCu_ferrites.pdf",        "content": "PreprintfromAppl.Phys.Lett. 97,182502(2010)\nTemperature Dependence Of Spin Resonance In Cobalt Substituted  \nNiZnCu Ferrites\nA. Lucas 1,2, R. Lebourgeois 1, F. Mazaleyrat 2,E. Labouré2\n1. THALES R&T, Campus Polytechnique, 1 avenue Augustin Fresnel, 91767 Palaiseau, France\n2. SATIE, ENS de Cachan, 61 av. du Président Wilson, 94235 Cachan, France\nAbstract : Cobalt substitutions were investigated in Ni 0.4Zn0.4Cu0.2Fe2O4 ferrites, initial complex  \npermeability was then measured from 1 MHz to 1 GHz. It appears that cobalt substitution led to a  \ndecrease of the permeability and an increase of the µ s×fr factor. As well, it gave to the permeability  \nspectrum  a  sharp  resonance  character.  We  also  observed  a  spin  reorientation  occuring  at  a  \ntemperature depending on the cobalt content. Study of the complex permeability versus temperature  \nhighlighted that the most resonant character was obtained at this temperature. This shows that cobalt  \ncontribution to second order magneto-crystalline anisotropy plays a leading role at this temperature. \nKeywords : NiZnCu ferrites, cobalt substitutions, complex permeability, induced anisotropy\nNickel-zinc-copper ferrites are interesting materials because of their high permeability in MHz range. Moreover,  \ntheir low sintering temperatures make them suitable for the realization of integrated components in power  \nelectronic. As for nickel-zinc ferrites, cobalt substitution is an efficient technique to decrease permeability [1]  \nand magnetic losses of nickel-zinc-copper ferrites [2]. It has been proposed that the effect of cobalt is to produce  \npinning of the domain walls because of anisotropy enhancement due Co2+ ions ordering [3]. The aim of this  \npaper is to study the effect of cobalt substitution and particularly the role of the cobalt contribution to anisotropy.  \nFerrites of formula (Ni0.4Zn0.4Cu0.2)1-xCoxFe1.98O4 were studied for cobalt substitutions up to 0.035 mol.\nFerrites were synthesized using the conventional ceramic route. The raw materials (Fe 2O3, NiO, ZnO, CuO) were  \nball milled for 24h hours in water. Co 3O4 was then added before the calcination at 760°C in air for 2 hours. The  \ncalcined ferrite powder was then milled by attrition for 30 min. The resulting powder was compacted using axial  \npressing.  The sintering was performed at 935°C for 2 hours in air. Magnetic characterizations were done on ring  \nshaped samples with the following dimensions: external diameter = 6.8 mm; internal diameter = 3.15 mm; height  \n= 4 mm. Initial complex permeability (µ’ and µ’’) was measured versus frequency between 1 MHz and 1 GHz  \nusing an impedance-meter HP 4291. Static initial permeability (µ s) was defined as µ’ at 1 MHz because for these  \nferrites µ’(1 MHz) = µ’(100 Hz). For permeability versus temperature measurements, the rings were wound with  \na copper wire and placed in an oven going from –70°C to 200°C. µ s was deduced from the inductance measured  \nat 100 kHz by an impedance-meter Agilent 4194A.\nThe samples sintered at 935°C have all the pure spinel crystalline structure and a density higher than 96% of the  \ntheoretical density. Table I shows evolution of the permeability of the  (Ni0.4Zn0.4Cu0.2)1-xCoxFe1.98O4 ferrites. \nCobalt substitutions lead to a decrease of the initial complex permeability and an increase of the µ s×fr factor \nwhich is maximum for Co = 0.021 mol (f r is the frequency resonance defined as the maximum of µ’’). The raise  \nof this factor shows that f r increases faster than µ s decreases.\nFigure 1 shows the initial complex permeability versus frequency for (Ni 0.4Zn0.4Cu0.2)1-xCoxFe1.98O4 ferrites with \ndifferent cobalt content. One can see that the spectra become sharper when the cobalt rate increases. It is  \naccepted  that  the  permeability  has  two  contributions  :  at  low  frequency  wall  domain  displacements  are  \npreponderant and at higher frequency,  permeability is mainly due to the spin rotation [4].  In general, the  \nrelaxation behavior of domain walls essentially hides the spin resonance, but cobalt is known to inhibit the  \ndomain wall displacements [5], which produces two effect: (i) the initial permeability is decreasing with cobalt  \ncontent; (ii)  at higher frequency,  the  cobalt seems to promote  the  spin rotation by shifting the frequency  \nresonance (maximum of µ”) toward higher frequencies. Consequently, the magnetic losses due to domain wall  \ndisplacements are lowered, leading to a stronger dissymmetry in the shape of µ” peak. The magnetic losses rise  \nat higher frequency but with a steeper slope.\nThe cobalt has also an effect on the temperature variation of the permeability. In order to understand this  \nphenomenon, the permeability dependence on temperature has been studied (figure 2). \n\n1PreprintfromAppl.Phys.Lett. 97,182502(2010)\nThe cobalt free  ferrite  (curve A) shows a monotonous increase in this temperature range.  In contrast, the \nbehavior is different for the cobalt-substituted ferrites, for which a local maximum in the initial permeability  \nappears. This is the consequence of the magneto-crystalline anisotropy compensation due to the cobalt ions  \ncontribution. Indeed, the first order anisotropy constant of the Ni(ZnCu) ferrite host crystal is negative, whereas  \nCo ferrite has a positive one. As previously described by Van Den Burgt [6] for a certain amount of Co within  \nthe order of 0.1/u.f., it results that a spin reorientation transition occurs ( SRT, i.e. a change in easy axis)  at a  \ntemperature  T0, increasing with cobalt content. The permeability is described by the following relation :\neffs\nKMµ2\n'α [6]\nMs is the saturation magnetization and K eff the effective anisotropy. K eff consists of three components due to :  \nmagneto-crystalline anisotropy (K 1) of the host crystal, the cobalt ions contribution to anisotropy, higher order  \ncontributions and magneto-elastic energy [8]. This  spin reorientation  leads to an increase of the permeability  \ncharacterised by a local maximum around T 0. Below the SRT, K1>0 and K2>0 as the Co contribution dominates,  \nand  above  SRT  K1<0,  corresponding  to  a  change  from  [100]  to  [110]  easy  axis.  Figure  2,  shows  that  \n(Ni0.40Zn0.40Cu0.20)0.979Co0.035Fe1.98O4 ferrite has a SRT close to room temperature. The strong resonant character of  \nthe permeability at this point could be explained by the strong pinning of domain wall due to high second order  \nanisotropy contribution. \nIn order to go deeper insight the effect of cobalt on the magnetic behavior, the initial complex permeability  \nspectra were recorded near the SRT on two ferrites : \n-a (Ni0.4Zn0.4Cu0.2)0.965Co0.035Fe1.98O4 ferrite, which has a SRT around 10°C. \n-a Ni0.4Zn0.4Cu0.2Fe1.98O4 ferrite which doesn’t exhibits SRT. \nFigure 3 shows µ’(f) spectra between –50°C and 180°C of these two materials .  To quantify the resonant  \ncharacter of the µ’ spectrum versus frequency, we defined the following resonance factor F res :\n100')''(max×−=\nstaticstaticresµµµF\nwhich variation for the two compositions is shown on figure 4.\nThe cobalt free ferrite has a low F res (around 15%) slightly decreasing with temperature. This behavior can be  \nexplained by the decrease of the magnetization saturation. In contrast, t he permeability spectra of the cobalt-\nsubstituted ferrite strongly depend on the temperature. Figures 3 and 4 show that the sharpest resonant character  \nis obtained around 10°C, corresponding to the SRT. It can also be noted that, in this range of temperature, the  \nresonance factor of the cobalt-substituted ferrites is always higher than one of the cobalt free ferrite. As the  \nmagnetization is constantly decreasing in this temperature range, such a change in the permeability behaviour is  \nnecessarily due to a change in the effective anisotropy.\nIn the case of Co substituted ferrite, the resonance should be explained if one considers that domain wall are  \npinned (case of strong anisotropy) and the spins rotates freely (case of vanishing anisotropy), so there is an  \napparent  contradiction.  However,  near  the  SRT,  the  situation  is  much  different  compared  to  usual:  K 1 is \nvanishing but K2 is not necessarily dropping to zero accordingly. In the limit of  K1 = 0, there arte two main  \nconsequences. Firstly, the domain wall energy is not vanishing but γ = 2(AK2)1/2 [9], so the pinning energy may  \nbe still important. Secondly, development of anisotropy energy reduces to 2322212αααKEA=Δ. If one of the \ndirection cosines is null (case of {100} and equivalent planes) there is no anisotropy variation. So, a possible  \nexplanation of the strong resonance observed only in the cobalt-substituted sample would be nearly free spin  \nrotation in the  {100}, {010} and {001} planes [10]. If the magnetization turns in the {110} from 110 to \n111, the anisotropy change is at most K2/8. As the static permeability is relatively small, this would mean that  \nK2 is relatively strong in this material near the SRT, explaining also why the µ S(T) maximum is relatively smooth  \nIn  conclusion,  the  study  of  permeability  spectra  of  cobalt  substituted  NiZnCu  ferrites  as  a  function  of  \ntemperature shows that cobalt substitution can shift the spin reorientation transition close to room temperature  \ndue to cancellation of first order anisotropy constants of Co ion and the host crystal. As a consequence of the  \n2PreprintfromAppl.Phys.Lett. 97,182502(2010)\nnon-zero second order anisotropy constant, spin resonance damping is very small resulting in a strong resonance  \nrevealed by a very large overshoot (60%) of µ’ spectra and a sharp absorption peak in µ” spectra in the vicinity  \nof SRT.\nReferences\n[1] T. Y. Byun, S. C. Byeon, K. S. Hong, Factors affecting initial permeability of Co-substituted Ni-Zn-Cu  \nferrites, IEEE, vol 35, Issue 5, Part 2, pages : 3445-3447, sept 99\n[2] R. Lebourgeois, J. Ageron, H. Vincent and J-P. Ganne, low losses NiZnCu ferrites (ICF8), Kyoto and Tokyo,  \nJapan 2000\n[3] J. G. M. De Lau, A. Broese van Groenoul, Journal de  Physique 38, 1977, page Cl-17\nL. Néel, J. Phys. Radium 13, 249, 1952\n[4] T. Tsutaoka, M. Ueshima and T. Tokunaga, J. Appl. Phys. 78 (6), p 3983-3991, 1995\n[5] A.P. Greifer, V. Nakada, H. Lessoff, J. Appl. Phys., vol 32, 382-383, 1961\n[6] C. M. van der Burgt, Philips Research Report 12, 97-122, 1957\n[7] H. Pascard, SMM13, J. Phys. IV France, 8, 1998\n[8] A. Globus and P. Duplex, J. Appl. Phys., vol 39, no.2 (part I) 727-729, 1968\n[10] R. Skomski, J.M.D. Coey,Permanent Magnetism, Taylor & Francis, 1999, p. 156\n[11] S. Chikazumi, Physics of Ferromagnetism, 2nd ed., Oxford Science Pub., 1997, p. 252\nTable I : Static permeability and resonance frequency of (Ni 0.40Zn0.40Cu0.20)1-єCoєFe1.98O4 ferrites\nmol CobaltStatic µ’fr (MHz)µs×fr (GHz)\nCo = 027022.16\nCo = 0.00718536.36.7\nCo = 0.01416045.67.3\nCo = 0.02114550.77.35\nCo = 0.02813056.57.35\nCo = 0.03511563\nList of figu  res :   \nFigure 1 : Complex permeability versus frequency of (Ni 0.4Zn0.4Cu0.2)1-xCoxFe1.98O4 ferrites\n3\nPreprintfromAppl.Phys.Lett. 97,182502(2010)\nFigure 2 : Initial permeability versus temperature of  (Ni 0.4Zn0.4Cu0.2)1-xCoxFe1.98O4 ferrites.\nFigure 3 : (A) : Ni0.4Zn0.4Cu0.2Fe1.98O4  µ’(f) spectrum from 1 MHz to 110 MHz for temperature between –40°C to  \n130°C.\n4PreprintfromAppl.Phys.Lett. 97,182502(2010)\n        (B) : (Ni0.4Zn0.4Cu0.2)0.965Co0.035Fe1.98O4  µ’(f) spectrum from 1 MHz to 110 MHz for temperature  \nbetween –50°C to 180°C.\nFigure 4 : Fres versus temperature of Ni 0.4Zn0.4Cu0.2Fe1.98O4 and (Ni0.4Zn0.4Cu0.2)0.965Co0.035Fe1.98O4 ferrites.\n5\n"    },    {        "title": "1103.5303v1.The_effect_of_chemical_disorder_on_the_magnetic_anisotropy_of_strained_Fe_Co_films.pdf",        "content": "arXiv:1103.5303v1  [cond-mat.mtrl-sci]  28 Mar 2011physica statussolidi, 9 March 2021\nThe effect of chemical disorder on\nthe magnetic anisotropy of strained\nFe-Cofilms\nC.Neise1, S.Sch¨onecker1,M. Richter1, K.Koepernik1, H. Eschrig*,1\n1IFWDresden, P.O.Box270116, D-01171 Dresden, Germany\nReceivedXXXX, revisedXXXX,accepted XXXX\nPublishedonline XXXX\nKeywords: magnetic anisotropy, Fe-Coalloys, strainedfilms,chemica l disorder, perpendicular recording.\n∗Corresponding author: e-mail h.eschrig@ifw-dresden.de , Phone: +49-351-4659-569, Fax: +49-351-4659-750\nStrained Fe-Co films have recently been demonstrated\nto exhibit a large magnetocrystalline anisotropy (MCA)\nand thus to be of potential interest as magnetic storage\nmaterial. Here, we show by means of density-functional\n(DF) calculations, that chemical order can remarkably\nenhancetheMCA.We also investigate the effect of relaxation perpendicu-\nlar to the applied strain and evaluate the strain energy\nas a function of Co concentration and substrate lattice\nparameter. On this basis, favourable preparation routes\nfor films with a large perpendicular anisotropy are sug-\ngested.\nCopyrightlinewillbe provided by the publisher\n1 Introduction Materials for high-density magnetic\nrecordingmedia (hard disc drives) have to obey two com-\npetingrequirements[1]. On the onehand,stability against\nthermally activated switching is guaranteed only if the\nvolume-integrated MCA of a single storage bit is larger\nthan about 50kBT, wherekBis the Boltzmann constant\nandTis the operatingtemperature[2]. On the otherhand,\nthe magnetic field needed to write a bit is, apart from de-\nmagnetisation effects, proportional to Ku/Ms[3], where\nKudenotes the (uniaxial)MCA energyand Msis the sat-\nuration magnetisation. Since the write field is constrained\nby the construction of the write head [1], a large Msof\nthestoragematerialisdesirablealongwithalarge Ku,the\nlatterdemandresultingfromthestability requirement.\nFe-Co alloys are well-knownto have a high saturation\nmagnetisation at ambient conditions, but their bulk MCA\nis minute due to their cubic symmetry. It is however pos-\nsible to achieve high anisotropy values without sacrificing\nthe advantageously large magnetisation by designing arti-\nficial structures that break the cubic symmetry. One way\nto achieve this goal would be to manufacture Fe |Co su-\nperlattices. It has been demonstrated experimentally and\nconfirmed by DF calculations, that [(110)-Fe |Co]nsuper-\nlatticesshowalargein-planemagneticanisotropy[4].Un-\nfortunately,advancedrecordingtechniquesrequire a mag-netisation orientation perpendicular to the surface, i.e. , an\nout-of-planeanisotropy.Asecondpossibilitywouldbethe\npreparation of ultrathin films. Monolayers of Fe-Co on\nPt(111) show a maximum out-of-plane anisotropy energy\nof about500µeV per atom both in DF theory and in low\ntemperature experiments. However, they have Curie tem-\nperatures close to room temperature [5] and are, thus, not\nsuitedforthediscussedapplication.\nA third, recently suggested route is to fabricate\nstrained, bulk-like Fe 1−xCoxfilms by epitaxial growth\non a suitable substrate [6]. This idea relies on the fact\nthat alloys which are cubic in their bulk phase can be\ngrownas metastable tetragonalfilms, if they are deposited\non substrates with a fourfold symmetric surface, e.g., the\n(001) surface of cubic crystals. It was shown in a recent\nstudy that the Fe 0.7Pd0.3alloy can be prepared to form\n50nm thick, i.e., bulk-like, epitaxial films on a number of\nsubstrates. Thus, the centred-tetragonal structure of the se\nfilms with lattice-parameter ratios c/abetween1.09and\n1.39spansalmostthewholerangefromBCC( c/a= 1)to\nFCC (c/a=√\n2) [7].\nTurningbacktoFe 1−xCoxalloys,theout-of-planeval-\nues of∆EMCA=Ku(T= 0)predicted by DF calcu-\nlations reach up to 800µeV per atom for x= 0.6and\nc/abetween1.20and1.25[6].Wewilldemonstratebelow,\nCopyrightlinewillbe provided by the publisher2 C. Neise et al.:MCA of Fe-CoFilms\nthat the elastic strain energyof Fe 1−xCoxis small enough\nin the interesting region of xto allow the stabilisation of\nfilmswith a decentthickness.(Note,that quantum-sizeef-\nfects strongly influence the MCA of thin films at least up\nto10monolayers(ML)orabout 1.5nmthickness[8].)\nMeanwhile,the mentionedpredictionby Burkert et al.\n(abbreviatedbelowBNEH[6])hasbeenconfirmedbysev-\neralexperiments[9,10,11,12].Theseconfirmationsshould\nbe viewed with caution, since in all quantified cases the\nmeasured values of Kuseem to be lower than the pre-\ndictions for comparable parameter values of c/aand of\nxby factors of two to four. In detail, Ku= 108µeV\nper atom was found for Fe 0.5Co0.5|Pd(001) films with 3-\n10ML thickness, about half the predicted value [10]; for\n[Fe0.36Co0.64|Pt(001)] nsuperlattices, only one half [9] or\nonequarter[11]ofthetheoreticalvaluewasmeasured,de-\npending on the way of comparison (dedicated calculation\nforacertainsuperlattice[9]orevaluationofbulkdatafro m\nvarioussuperlatticedata [11]). One shouldnote that in the\nlatter case the anisotropy was measured at room tempera-\nture,whichexplainsa partofthediscrepancy.\nLooking for a reason of the possibly systematic dis-\nagreement between theory and experiment, a first thought\nmight blame the mentioned quantum oscillations. How-\never, data evaluation in two of the three mentioned cases\nconsidered fits to a number of systems with varying film\nthickness, and the third comparisoninvolved a calculation\nfor the specific geometry. Another possible source of de-\nviation might be the local density approximation (LSDA)\napplied by BNEH. We argue, that this is unlikely since in\nmost known cases LSDA results for ∆EMCAunderesti-\nmatetheexperimental Ku-values.\nHere, we are going to advocate a third idea: While\nthe experiments in all cases known to us were performed\non presumably chemically disordered (though structurally\nwell-ordered)alloys, the calculationsby BNEH employed\ntheso-calledvirtualcrystalapproximation(VCA)tosimu-\nlatethealloy.Inthisapproximation,theelectronnumberi s\nadjusted to its correct valueaccordingto the alloy compo-\nsition by allowingthe atomic nuclei to carry a non-integer\ncharge.Inthisway,aperfectchemicalorderisintroduced.\nInourparticularcase,all Fe andCo atomsarereplacedby\nonly one kind of atoms with atomic charge 26 +x. This\nmeans, the VCA describes a chemically ordered structure\nwith the correct electron number of the chemically disor-\nderedalloy.\nAs the main result of the present work, we will show\nthat strained Fe-Co films with chemical order can have\na much larger magnetic anisotropy than chemically dis-\nordered films. Beyond this main point, we will contrast\nthe strain-energy landscape with the related magnetocrys-\ntalline anisotropyenergyin orderto find promisingprepa-\nration parameters for films with a large perpendicular\nanisotropy.\nThe following sections contain computational details,\nnumericalresultsandtheirdiscussion,andtheconclusion s.2.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85\na [Å]33.54c [Å]x = 0.0\nx = 0.2\nx = 0.4\nx = 0.6\nx = 0.8\nx = 1.0 CVBP - experimental\n             Fe volume\nEBP\nmixed - CVBPx = 0.0\nx = 1.0FCC\nBCC\nFigure 1 Geometries used in the calculations: Constant\nvolume Bain path using the experimental volume of BCC\niron (thick dashed lines, Model I) and epitaxial Bain path\ncalculated in VCA for different cobalt concentrations x\n(symbols, Model II). The shaded area with dashed mar-\ngins forx= 0andx= 1denotes an x-weighted aver-\nagebetweenthecalculatedvolumesofBCCironandFCC\ncobalt.Thickfulllinesindicate c/aratiosof√\n2(FCC)and\n1 (BCC).\n2 Computational details The DF calculations were\ncarriedoutwiththeall-electron,full-potentiallocal-o rbital\ncodeFPLO, version8.00-31[13,14]. Thegeneralisedgra-\ndient approximation (GGA) in the parameterisation by\nPerdew,Burke,andErnzerhof[15]wasused.Wepreferred\nthisapproachagainsttheLSDApreviouslyusedbyBNEH,\nsince we are interested in the evaluation of strain energies\ninadditiontothemagneticquantities.Thevalencebasisse t\ncomprised 3s,3p,3d,4s,4p,4d, and5sstates. Brillouin\nzone integrationswere performedusing the linear tetrahe-\ndronmethodwithBl¨ ochlcorrections.Structuralproperti es\nand spin magnetism were evaluated in a scalar relativistic\nmode, while a four-componentfully relativistic mode was\nusedforthecalculationoftheMCA.\nFive different structural models were employed to de-\nscribe the atomistic geometry and the chemical order of\nthe strained alloy films. All these models are based on the\nassumptionsthat(i)thefilmsarethickenoughtodisregard\ntheinfluenceofthesurfaceandofthesubstrateontheelec-\ntronicstructureandthat(ii)thesubstrate-filminteracti onis\nnonethelessstrongenoughtoletthein-planelatticeparam -\neterbedictatedbythesubstrate.Theseassumptionsarein-\nherentto the conceptofthe epitaxialBain path(EBP) [16,\n17] thatwe appliedina partofthe calculations.\nModel I: A centred-tetragonal (BCT) structure with\none kind of atoms of atomic number 26 +x(VCA). The\nlattice parameter cwas determined by the assumption of\nconstantvolume,wherethelatter waschosento be theex-\nperimental volume of BCC Fe at room temperature, i.e.,\nc= 2×11.78˚A3/a2(Figure 1, thick dashed line, the so-\ncalledconstantvolumeBainpath,CVBP).\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 3\nModelII: Samestructure(BCT)andchemistry(VCA)\nas in Model I, but the lattice parameter cwas calculated\nby minimisation of the total energy E(a,c,x)foraandx\nfixed,usingspinpolarisedGGA. Related data aregivenin\nFigure1withsymbolsfordifferent x.Thesedataconstitute\nthe EBP. For comparison, CVBPs are given (dashed lines\nincludingshadedarea)usingtheGGAvolumesofBCCFe\n(x= 0)andofFCCCo( x= 1).TheEBPvaluesof c/aare\nclose to the related CVBP values exceptfor small xin the\nregion close to the FCC structure. This can be understood\nby the strong x-dependenceof the spin moment µsin this\nregion,see Figure4 below.\nContour plots presented below are based on data cal-\nculated with Model I or II on grids with ∆x= 0.1and\n∆a= 0.05˚A.\nModelIII: L10structurewithstacksofsinglequadratic\nFe- and Co-layers, x= 0.5. This structure is the strained\nvariantoftheknown α′-Fe-Co(B2)phase,where c/a= 1.\nThe lattice parameter cwas calculated by minimisation of\nthetotalenergy E(a,c)forafixed(EBP).\nModel IV: Stacks of single FCC-like (Fe, Co)- and\n(Co, Co)-layers, x= 0.75, and EBP condition for c. This\nstructure is derived from the L1 2structure which is ob-\ntainedfor c/a=√\n2.\nModel V: A2×2×2BCT supercell with 16 atoms,\nusedto describechemicaldisorderbymeansof an ensem-\nble average.Fe 8Co8(x= 0.5),Fe6Co10(x= 0.625),and\nFe4Co12(x= 0.75) were considered. The atoms were ar-\nranged such that the nearest neighbour patterns match the\ncompletelydisorderedalloyascloselyaspossible.Starti ng\nfrom a single atomic configuration, a symmetry-adapted\nensemble average was constructed in such a way that its\nMCA correctly vanishes for a=c(BCC structure). De-\ntails of the methodare describedin Reference[18], where\n32-atomsupercellsmodelledthecaseofaslightlydistorte d\nFCCstructure.\nThe space groups 139,139,123,123,1were used in\nthe calculations for structure Models I ...V, respectively.\nStructureoptimisationwascarriedoutforModelsI ...IV,\nwherek-mesheswith 24×24×24pointsin thefull Bril-\nlouin zone (BZ) for all models were used. For the more\ncomplex Model V, internal relaxation is expected to be of\nminorimportance,since Fe and Co atoms have almost the\nsame atomic volumes. Thus, we abstained from a relax-\nation of the inner degrees of freedom. Further, the atomic\nvolume obtained with Model II at the appropriate x-value\nanda= 2.65˚A (a value of particular interest, see next\nsection)wastakentoconstructa CVBP forModelV.\nFortheModelsI ...IV,theMCAenergywasevaluated\nfrom independent self-consistent total energy calculatio ns\nfor magnetic moment orientationsalong the x-axis (E100)\nandalongthe z-axis(E001),\n∆EMCA=E100−E001. (1)\nThein-planeanisotropy( E100−E110)wasnotconsidered.\nIn these calculations, finer k-meshes than in the structureoptimisations were used: 48×48×48points in the full\nBZ forallmodels.\nTo cope with Model V, we relied on the so-called\nmagnetic force-theorem [19]: the total energy difference\n∆EMCAwas approximated by the difference of band en-\nergy sums, evaluated with 8×8×8k-points in the full\nBZ.InEqn.(1) E100istobereplacedby( E100+E010)/2,\nwhereE010is the band energy calculated for a magnetic\nmomentorientationalongthe y-axis.\n3 Results anddiscussion\nE − E0 [meV/atom]\n+\n55\n10\n1020\n204040\n80\n120\n160180190\n 0  0.2  0.4  0.6  0.8  1\nx2.452.502.552.602.652.702.752.802.85a [Å]\nFigure 2 Calculated strain energy (Model II) as a func-\ntion of the cobalt concentration xand of the in-plane lat-\ntice parameter a. For each value of x, the total energy is\nreferred to the lowest energy E0(x)along the respective\nEBP, i.e., there is a line a0(x)for which the presented en-\nergy is zero (not shown). The (unstrained) ground state is\nBCCforx <0.76andFCCfor x >0.76.(Duetointerpo-\nlationfroma gridthe jumpfromBCC to FCCat x= 0.76\nslightly deviates from being vertical in the picture.) The\nsymbol+ labelsthepointofthe largestMCA accordingto\nFig. 5b.\n3.1 Strain energy Figure 2 shows a contour plot of\nthestrain energy,\nEstrain(a,x) =E(a,x)−min\naE(a,x) :=E(a,x)−E0(x),\n(2)\nwithx-dependent ground-state energy E0, evaluated with\nModel II (VCA, EBP). There are two valleys correspond-\ning to the BCC ( x <0.76) and FCC ( x >0.76) ground\nstate, separatedbya saddlepointat x=0.78;a= 2.66˚A.\nThe calculated critical Co concentration of about 76% for\nthe FCC-BCC transition is consistent with the known ex-\nistence range of the BCC-like Fe-Co B2 phase up to 72%\nCo and two-phase behaviour (FCC + BCC) at higher Co\nconcentration[20].\nThe strain energy is an important quantity that deter-\nmines the feasibility of epitaxial growth. In a simplified\nCopyrightlinewillbe provided by the publisher4 C. Neise et al.:MCA of Fe-CoFilms\npicture,theachievablethicknessofametastablefilmisin-\nverselyproportionalto Estrain.RecentworkonFe 0.7Pd0.3\nfilms demonstrated the possibility to grow 50nm films\n(about300ML) with a strain energy of 6meV per atom,\nestimatedbyDFcalculations[7].ConsideringFe-Cofilms\non a Rh(001) substrate ( a= 2.69˚A), we predict a strain\nenergy between 40and80meV per atom for x <0.65.\nIndeed,experimentsindicatethatepitaxialgrowthisstab le\nup to about 15ML, while thicker films seem to become\ncrystallographicallydisordered[10].\nFrom this perspective, it seems worthwhile to try the\ngrowthofthickerfilmsinthesaddlepointregionofFigure\n2. The related in-plane lattice parameters define a region\nofveryhighstrain(eitherBCCorFCCserveaszero-strain\nreference),but the strain energy is comparably low. As an\nadditional advantage of this area the in-plane stress is low\n(zero at the saddle point) which might additionally facili-\ntate the epitaxialgrowth.Wenote,that in reality thesaddl e\npoint area might appear at about 0.04˚A higher a-values,\nsincetheGGAcalculationsfindsomewhatsmallerground-\nstate volumesthanknownfromexperiment,see Figure1.\n2.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85\na [Å]0102030E - E0 [meV/atom]x = 0.75 (VCA)\nx = 0.75 (EAVG)\nx = 0.75 (L12)\nc/a≈ 21/2c/a≈ 1.0\nFigure 3 Calculated strain energy for cobalt concentra-\ntionx= 0.75as a function of the in-plane lattice pa-\nrametera. Results for two different chemically ordered\nstructures(ModelII,denotedVCA,andModelIV,denoted\nL12) are comparedwith resultsfor ensemble-averagedsu-\npercells simulating chemical disorder (Model V, denoted\nEAVG). The total energies are referred to the respective\nlowestvalue E0(x=0.75).TheVCA dataarethesameas\nthose used in Figure 2, but were evaluated here on a finer\ngrid.\nTocheck,whetherdifferenttypesofstructuralorchem-\nical order have an influence on the film stability, we com-\npare strain energies obtained with three different struc-\nturemodels,Figure3.Thecalculationswereperformedfor\nx= 0.75, a line cutting the a−x-plane close to the sad-\ndle point. All three models show very similar dependence\nofEstrainona, with minima at the FCC- and BCC-(like)-\nstructuresandamaximum(closetothesaddlepointinFig-ure 2) of 20...25meV per atom. We conclude, that the\na-dependenceofthestrainenergyismainlydeterminedby\nthe electron number featured in the VCA. Structural de-\ntails and chemical disorder only slightly modulate the be-\nhaviour.\nµs [µB/atom]\n+\n1.01.22.2 2.2 2.0 1.8\n2.4\n 0  0.2  0.4  0.6  0.8  1\nx2.452.502.552.602.652.702.752.802.85a [Å]\nFigure 4 Calculated spin magnetic moment (in VCA,\nModel II) as a function of cobalt concentration xand in-\nplanelatticeparameter a. Thesymbol+labelsthepointof\nthelargestMCA accordingtoFig. 5b.\n3.2 Spinmoment Formagneticstorageapplications,\na largesaturationmagnetisationisneededtoallowswitch-\ning with a magnetic field of limited strength. Figure 4\nshows the VCA spin moment, which accounts for about\n90% [21] of Ms(T= 0), as a function of xanda. We\nfind high values of µs, depending only weakly on xand\nbeing almost insensitive to a variation of a, except in the\nregionx <0.2;a <2.60˚A. The weak x-dependence\ncan be understood by assuming constant atomic moments\nforx >0.2and transition from weak to strong ferro-\nmagnetism of iron in the region 0< x < 0.2(Slater-\nPaulingbehaviour).ForthecaseofFe-CofilmsonCu(001)\n(a= 2.55˚A), thisbehaviourhasbeenconfirmedin exper-\niment[22].\nThe pronouncedmoment reduction in the Fe-rich area\nbelowa= 2.60˚A is related to the instability of fer-\nromagnetism in FCC iron. In the present GGA calcula-\ntions, restricted to collinear ferromagnetic states, we fin d\na low-moment solution for very small Co concentrations\nand a high-moment solution for x≈0.2. This strong x-\ndependence of the spin moment has an effect on the EBP\n(Figure 1) by magneto-volume coupling: for x= 0, the\nvolume is reduced in comparison to the constant-volume\nassumption,whileitisconsiderablyenhancedfor x= 0.2.\nTheearlierresultsfor µs(x,c/a)byBNEHagreequal-\nitatively with our data but do not show any low-spin be-\nhaviour, since they do not include the region very close to\nFCC. Furthermore, those data were obtained for slightly\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 5\nlarger(experimental)volume[6]whichstabilisesthehigh -\nmomentsolution.\n−300−200−100 0100200300400500600700∆EMCA [µeV/atom]\n 0  0.2  0.4  0.6  0.8  12.452.502.552.602.652.702.752.802.85a [Å]\n−300−200−100 0100200300400500600700\n 0  0.2  0.4  0.6  0.8  1\nx2.452.502.552.602.652.702.752.802.85a [Å](a)\n(b)\nFigure 5 Calculated MCA (in VCA) as a function of\ncobaltconcentration xandin-planelatticeparameter a:(a)\nCVBP-geometry(ModelI)and(b)EBP-geometry(Model\nII). Dark (brown and red) areas denote large MCA values\nwith an easy axis along [001], light (yellow) areas denote\nsmallMCA. BlacklinesindicatezeroMCA.\n3.3 Magnetocrystalline anisotropy: chemically\nordered films We now turn to the key quantity for mag-\nnetic materials applications, the MCA. Recall, that for\nchemically ordered Fe-Co layers BNEH predicted a peak\nin∆EMCAwith a maximum height of about 800µeV in\nthe vicinity of x= 0.6;1.20< c/a < 1.25. Figure 5a\nshows our data for ∆EMCA(x,a), evaluated with Model\nI (VCA, CVBP). They agree well with the data published\nby BNEH which were confirmed qualitatively by experi-\nments[9,10,11,12].\nTocheckwhetherconsiderationof a-andx-specificre-\nlaxation of the film perpendicular to the film plane might\naffect the MCA, we repeated the calculation for the EBP\n(Model II, Figure 5b). By comparison of the two panels\nof Figure 5 we find that both geometries yield qualita-\ntively the same results. A cut at x= 0.6(approximately\nthrough the peak maximum) shows, that both structure\nmodels yield almost the same maximum MCA energy of\n800µeV, see Figure 6. The main difference consists of a\nshift of the MCA-maximum from a= 2.68˚A (CVBP) to\na= 2.64˚A (EBP).Further,theEBP-peakhasa somewhat\nsmallerwidth.Inbothcases,the c/aratioliesbetween 1.222.45 2.5 2.55 2.6 2.65 2.7 2.75 2.8 2.85\na [Å]0200400600800∆EMCA [µeV/atom]CVBP\nEBP\nFigure 6 CalculatedMCA(inVCA)forcobaltconcentra-\ntionx= 0.6asafunctionofthein-planelatticeparameter\na: comparisonbetweenCVBP (ModelI) andEBP (Model\nII)(samedataasinFigure5, butonafinergrid).\nand1.24(asfoundbyBNEH).Thus,thedifferentoptimum\na-valueoriginatesfromthevolumedifference(experimen-\ntal vs.calculatedvolume).\n2.5 2.6 2.72.80100200300400500600700∆EMCA [µeV/atom]VCA\nEAVG\nL10,2\n2.5 2.6 2.72.8\na [Å]2.5 2.6 2.72.8VCA\nEAVG\nL10\nL12x = 0.500 x = 0.625 x = 0.750\nPd(001)\nFigure 7 Calculated MCA for cobalt concentrations x=\n0.5,0.625,and0.75asafunctionofthein-planelatticepa-\nrametera: comparison between chemically ordered struc-\ntures, Models II, III, and IV (VCA, L1 0, and L1 2-derived\nstructure, respectively) and ensemble-averaged supercel ls\nsimulating disorder(EAVG, Model V). The VCA data are\nthesameasinFigure5b,butwereevaluatedhereonafiner\ngrid. The red cross denotes recent experimental data ob-\ntainedforFe 0.5Co0.5|Pd(001)[10].\n3.4 Magnetocrystalline anisotropy: the effect of\nchemical disorder Finally, we investigate how chemi-\ncal disorder of the films might influence their magnetic\nanisotropy. To this end, VCA results are compared with\nCopyrightlinewillbe provided by the publisher6 C. Neise et al.:MCA of Fe-CoFilms\nresults for two other structure types with perfect chemical\norder on the one hand and with an ensemble average sim-\nulating disorder on the other hand. We recall, that VCA\nis a model imposing perfect chemical order. The calcula-\ntions were performed for the most interesting concentra-\ntionrangebetween x= 0.5andx= 0.75.\nFigure 7 compiles calculated data for all four men-\ntioned structure models. Obviously, the specific type of\ncrystallographic structure is, in the present case, of mino r\nimportance.Whatmattersischemicalorder.Thisfindingis\nnotatall trivial,sincethemagneticanisotropyisknownto\nbe sensitiveto the detailed electronicstructureand,henc e,\nto the specific crystal structure. It was already mentioned\nby BNEH that L1 0results for ∆EMCAclosely resemble\nthe VCA results for x= 0.5. We here confirm this point\n(leftpanelofFigure7) andadd,thatalso byusinganL1 2-\nderivedstructure the VCA data ( x= 0.75) are rather well\nreproduced(rightpanel).\nHowever, a striking difference is immediately visible\nbetween the MCA of the chemically ordered structure\nmodels and the MCA of Model V, the chemically disor-\ndered case. The latter yields ∆EMCApeaks smaller than\nthose of the ordered structures by factors of 1.5to3,\ndependingon x. We suggest,that the relativelylow exper-\nimental values (compared with the prediction by BNEH)\nmightbecausedbythelackofchemicalorder.\nAsaconcludingremark,theexperimental Kuvaluefor\nFe0.5Co0.5|Pd(001)[10],redcrossin theleft panelofFig-\nure 7, seems to compare quite well with our VCA or L1 0\ndata. One has to bear in mind, however, that renormalisa-\ntion of the calculated curves to the experimental volume\nwould shift all calculated data by 0.04˚A to the right, thus\nproviding coincidence of the experimental value with the\nsimulateddisorder.\n4 Conclusions We have shown, that neither detailed\nstructuralrelaxationnorthespecificstructuretypehavea n\nimportantinfluence on the magnetic and elastic properties\nofstrainedFe-Cofilms.Ontheotherhand, chemicaldisor-\ndercan reduce the magnetic anisotropy energy by factors\nof1.5to3. We suggest that this might be a reason for the\nrelativelysmall Ku-valuesmeasured,hitherto.\nEpitaxial growth of thick, highly strained Fe 1−xCox\nfilms on substrates with lattice parameter ais predicted to\nbe most stable in the parameter range 0.67< x <0.86;\n2.64˚A< a <2.76˚A, independentofchemicalorder.The\nmaximum possible values of Kuare expected for almost\nthe same range of in-plane lattice parameters, 2.60˚A< a\n<2.75˚A but somewhatlowerCo concentration, 0.45< x\n<0.70.\nBased on these findings, we suggest two promising\nroutesto fabricate strained Fe-Co films with large perpen-\ndicularanisotropy:\n(1) Preparation of relatively thin orderedFe-Co L1 0-\nfilms (x= 0.5) on a substrate with a≈2.64˚A. The\nmaximum thickness of these films is probably limited toabout10ML by a relatively large strain energy. On the\nother hand, very large Kuvalues up to 500µeV per atom\nare expected (Figure 7). Thus, a large volume-integrated\nanisotropycanbeachievedwith relativelythinfilms.\n(2)Preparationofrelativelythickfilmsclosetothesad-\ndle pointofthe strainenergy,e.g., x= 0.75and2.60˚A<\na<2.65˚A.Fortheseparameters, Kuofchemicallydisor-\nderedfilmsislimitedtoabout 100µeVperatom,toosmall\na value to overcome the in-plane shape anisotropy. There\nis, however, a chance to enhance the magnetic anisotropy\nofthesefilmsbycarefulannealing:calculationsperformed\nby a group including Manfred F¨ ahnle point to the pos-\nsibility to stabilise an ordered Co 3Fe phase by epitaxial\nstrain[23].\nWe hope that our results will add another important\nroute to the experimental search for Fe-Co based mag-\nnetic recording media: the preparation of chemically or-\nderedstrainedfilms.\nAcknowledgements We thank Hongbin Zhang, Sebastian\nF¨ ahler,andIngoOpahlefordiscussionandwethankIngoOpa hle\nforprovidingacodegeneratingstochasticensemblesinMod elV.\nReferences\n[1] D. Weller, A. Moser, L. Folks, M.E. Best, W. Lee, M.F.\nToney, M. Schwickert, J.U. Thiele, and M.F. Doerner,\nIEEETransactions onMagnetics 36, 10–15 (2000).\n[2] S.H.Charap,P.L.Lu,andY.J.He,IEEETrans.Magn. 33,\n978 –983 (1997).\n[3] H. Kronm¨ uller, in: Science and Technology of Nanostruc -\ntured Magnetic Materials, edited by G.C. Hadjipanayis\nand G.A.Prinz,(Plenum, New York,1991), p.657.\n[4] V.A.Vas’ko,M.Kim,O.Mryasov, V.Sapozhnikov, M.K.\nMinor,A.J.Freeman,andM.T.Kief,Appl.Phys.Lett. 89,\n092502 (2006).\n[5] G. Moulas, A. Lehnert, S. Rusponi, J. Zabloudil, C. Etz,\nS.Ouazi,M. Etzkorn,P.Bencok, P.Gambardella,P.Wein-\nberger, and H.Brune, Phys.Rev. B 78, 214424 (2008).\n[6] T. Burkert, L. Nordstr¨ om, O. Eriksson, and O. Heinonen,\nPhys. Rev. Lett. 93, 027203 –027206 (2004).\n[7] J.Buschbeck,I.Opahle,M.Richter,U.K.R¨ oßler,P.Kla er,\nM. Kallmayer, H.J. Elmers, G. Jakob, L. Schultz, and\nS.F¨ ahler,Phys. Rev. Lett. 103, 216101 (2009).\n[8] H.Zhang,M.Richter,K.Koepernik, I.Opahle,F.Tasn´ ad i,\nandH.Eschrig,NewJournalofPhysics 11,043007(2009).\n[9] G. Andersson, T. Burkert, P. Warnicke, M. Bj¨ orck,\nB. Sanyal, C. Chacon, C. Zlotea, L. Nordstr¨ om, P. Nord-\nblad, and O. Eriksson, Phys. Rev. Lett. 96, 037205 –\n037208 (2006).\n[10] F.Luo,X.L.Fu,A.Winkelmann,andM.Przybylski,Appl.\nPhys. Lett. 91, 262512 (2007).\n[11] P. Warnicke, G. Andersson, M. Bj¨ orck, J. Ferr´ e, and\nP. Nordblad, J. Phys.: Condens. Matter 19, 226218 –\n226227 (2007).\n[12] F.Yildiz,M.Przybylski,X.D.Ma,andJ.Kirschner,Phy s.\nRev. B80, 064415 (2009).\n[13] K.KoepernikandH.Eschrig,Phys.Rev.B 59,1743–1757\n(1999), see alsohttp://www.fplo.de.\nCopyrightlinewillbe provided by the publisherpss header willbeprovided by thepublisher 7\n[14] H. Eschrig, M. Richter, and I. Opahle, Relativitic Soli d\nState Calculations, in: Relativistic Electronic Structur e\nTheory, Part 2: Applications, edited by P. Schwerdtfeger,\n(Elsevier,2004), pp.723 –776.\n[15] J.P.Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett .\n77, 3865 –3868 (1996).\n[16] E.C. Bain,Trans. AIME 70, 25(1924).\n[17] P.Alippi, P.M. Marcus, and M. Scheffler, Phys. Rev. Lett .\n78, 3892 –3895 (1997).\n[18] J. Buschbeck, I. Opahle, S. F¨ ahler, L. Schultz, and\nM.Richter, Phys.Rev. B 77, 174421 (2008).\n[19] G.H.O. Daalderop, P.J. Kelly, and M.F.H. Schuurmans,\nPhys.Rev. B 41, 11919 –11937 (1990).\n[20] T.NishizawaandK.Ishida,Bull.AlloyPhaseDiagrams 5,\n250–259 (1984).\n[21] F. Yildiz, F. Luo, C. Tieg, R.M. Abrudan, X.L. Fu,\nA. Winkelmann, M. Przybylski, and J. Kirschner, Phys.\nRev. Lett. 100, 037205 (2008).\n[22] M. Zharnikov, A. Dittschar, W. Kuch, K. Meinel, C.M.\nSchneider, and J. Kirschner, Thin Solid Films 275, 262 –\n265(1996).\n[23] A. D´ ıaz-Ortiz, R. Drautz, M. F¨ ahnle, H. Dosch, and J.M .\nSanchez, Phys.Rev. B 73, 224208 (2006).\nCopyrightlinewillbe provided by the publisher"    },    {        "title": "1104.4766v2.Spin_canted_magnetism__decoupling_of_charge_and_spin_ordering_in_NdNiO__3_.pdf",        "content": "arXiv:1104.4766v2  [cond-mat.str-el]  19 Mar 2013Spin canted magnetism, decoupling of charge and spin orderi ng in NdNiO 3\nDevendra Kumar∗and K. P. Rajeev†\nDepartment of Physics, Indian Institute of Technology Kanp ur 208016, India\nJ. A. Alonso and M. J. Martínez-Lope\nInstituto de Ciencia de Materiales de Madrid, CSIC, Cantobl anco, E-28049 Madrid, Spain\nWe report detailed magnetization measurements on the perov skite oxide NdNiO 3. This system\nhas a first order metal-insulator (M-I) transition at about 2 00 K which is associated with charge\nordering. There is also a concurrent paramagnetic to antife rromagnetic spin ordering transition in\nthe system. We show that the antiferromagnetic state of the n ickel sublattice is spin canted. We also\nshow that the concurrency of the charge ordering and spin ord ering transitions is seen only while\nwarming up the system from low temperature. The transitions are not concurrent while cooling the\nsystem through the M-I transition temperature. This is expl ained based on the fact that the charge\nordering transition is first order while the spin ordering tr ansition is continuous. In the magnetically\nordered state the system exhibits ZFC-FC irreversibility, as well as history-dependent magnetization\nand aging. Our analysis rules out the possibility of spin-gl ass or superparamagnetism and suggests\nthat the irreversibility arises from magnetocrystalline a nisotropy and domain wall pinning.\nPACS numbers: 75.60.-d , 75.60.Ej , 71.30.+h\nKeywords: Nickelates, magnetic ordering, magnetization, hysteresis, magnetocrystalline anisotropy\nI. INTRODUCTION\nThe rare earth nickelates (RNiO 3, R/negationslash=La) have been\nunder active investigation for the past two decades be-\ncause of the interesting electronic and magnetic proper-\nties exhibited by these systems.1,2These oxides undergo\na bandwidth controlled metal-insulator (MI) transition\non changing the temperature, chemical or hydrostatic\npressure.3–7In the metallic state the structure of these\nnickelates is that of an orthorhombic distorted perovskite\nwith space group Pbnm .8The metal to insulator transi-\ntion occurs with a structural transition which consists of\nan increase in the unit cell volume, a decrease in Ni-O-Ni\nbond angle and a symmetry lowering from orthorhom-\nbicPbnm to monoclinic P21/n. The symmetry low-\nering is understood in terms of charge ordering with a\ncharge disproportionation 2Ni3+→Ni3+δ+Ni3−δwith\nδ≈0.2−0.3.9–14In the early reports, the M-I transition\nof these compounds was attributed to the opening of an\nNi-O charge transfer gap created by band narrowing.3\nBut the occurrence of charge ordering at the M-I tran-\nsition and some recent theoretical calculations suggest\nthat the M-I transition owes its origin to the opening\nof a gap between the spin up egband of Ni3−δand the\nhardly spin polarized egband of Ni3+δ.15In these com-\npounds the higher temperature phase is metallic and the\nlower temperature phase is insulating. The M-I transi-\ntion is of first order and is associated with a large thermal\nhysteresis and time dependent effects in transport prop-\nerties such as resistivity and thermopower.16–19During\nthe cooling process, in the temperature window where\nhysteresis is seen, these compounds phase separate into\ninsulating and supercooled metallic regions. The super-\ncooled regions are metastable and they switch over to the\ninsulating state stochastically giving rise to time depen-\ndence and hysteresis in transport properties.17–19The nickelates also undergo a temperature driven mag-\nnetic transition, which is relatively less studied, be-\ncause the higher magnetic moment of rare earth ion\n(e.g. Nd3+moment ≈3.6µB) makes it difficult to get\nany information about the magnetic ordering of the Ni\nsublattice (Ni3+moment ≈1µB) through magnetization\nmeasurements.1,20,21Muon spin rotation experiments of\nTorrance et al. show that these compounds undergo a\nmagnetic ordering from paramagnetic to an antiferro-\nmagnetic state on lowering the temperature.3The mag-\nnetic ordering temperature ( TN) coincides with the M-I\ntransition temperature ( TMI) for PrNiO 3and NdNiO 3,\nwhile it is lower than TMIfor all the other nickelates.\nThe magnetic transition is of second order for all nicke-\nlates having TMI> TN,22,23but for NdNiO 3and PrNiO 3\nwhereTMI=TN, the nature of the magnetic transition\nis difficult to probe independently. While one would ex-\npect the magnetic transition to be continuous as seen in\nother members of the series we note that there is at least\none report which goes against this expectation and claim\nthat the said transition is of first order.23\nNeutron diffraction experiments show that, below TN,\nthe magnetic arrangement of Ni moments is characterized\nby the propagation vector (1\n2,0,1\n2) which suggests three\npossible magnetic structures, of which, two are collinear\nand one is non-collinear.9,24,25,27,28The collinear mag-\nnetic structure consists of up-up down-down stacking of\nNi magnetic moments, where each Ni moment is anti-\nferromagnetically coupled to three of its nearest neigh-\nbors and ferromagnetically to the remaining three near-\nest neighbours. This magnetic structure implies that the\norbital degeneracy of Ni3+e1\ngelectrons should be lifted by\nan orbital ordering, a prediction which has not gathered\nany experimental support so far.25Soft X-ray resonant\nscattering experiments at the Ni L2,3edges show that\nthe (1\n2,0,1\n2) reflections are purely of magnetic origin with2\nno orbital contribution whatsoever thus more or less rul-\ning out collinear magnetic order in the system.29In fact,\nthe orbital degeneracy of the Ni3+e1\ngelectron is found to\nbe lifted by charge-ordering15and this supports the ex-\nistence of a non-collinear magnetic structure which does\nnot require orbital ordering. The low temperature spe-\ncific heat data and the resonant soft X-ray diffraction\ndata of induced Nd magnetic moment in NdNiO 3indi-\ncate that, in all likelihood, the ordering of Ni moments\nin NdNiO 3is non-collinear.30,31\nIn this work, for the first time, we report the mag-\nnetization of the Ni sublattice , which we extracted after\ncarefully subtracting the contribution of the Nd moments\nfrom the total magnetization. The magnetization of the\nNi sublattice shows weak ferromagnetism which indicates\nthat the magnetic arrangement of the Ni moments is per-\nhaps canted. The existence of weak ferromagnetism can-\nnot be understood in terms of the magnetic structures re-\nferred to in the previous paragraph, even the noncollinear\nones. This suggests that those magnetic structures do\nnot represent the true picture and the actual magnetic\narrangement of Ni moments could be quite different from\nwhat has been thought of so far. Further, we found that\nthe supercooled metallic phase is magnetically ordered\nwhich indicates that the transition, on cooling, from the\nparamagnetic to the antiferromagnetic state happens at\nthe nominal transition temperature ( ≈200 K) unlike the\nmetal-insulator transition which is broadened and hap-\npens at lower temperatures as the supercooled metallic\nregions switch to the insulating phase stochastically. Thi s\nshows that the connection between the magnetic transi-\ntion and the metal-insulator transition is rather weak and\nthey do decouple if the system is supercooled. Also, the\nmagnetization of the Ni sublattice shows features such as\nFC-ZFC irreversibility which is indicative of the presence\nof frustration in the weak ferromagnetic state.\nII. EXPERIMENTAL DETAILS\nHigh quality polycrystalline NdNiO 3pellets were\nprepared by a liquid mixture technique described\nelsewhere.32\nAll the magnetic measurements were performed in a\nSQUID magnetometer (Quantum Design, MPMS XL).\nSince, in this work, we are trying to extract the small sig-\nnal from the Ni moments buried under the much larger\nsignal from the Nd moments it is a sine qua non that\nwe are absolutely sure about the quality of the data.\nThe magnetic signal from the samples of NdNiO 3and\nNdGaO 3, each of mass about 120mg, is 0.00262 emu and\n0.00159 emu respectively at 150 K and 500 G. These num-\nbers are more than three orders of magnitude higher than\nthe level where artifacts start distorting the measured\ndata.33Further, the sample holders used in SQUID mea-\nsurements can give rise to misleading results when the\nbackground signal from the sample holder becomes large\nenough so that it can no longer be ignored compared/s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s52/s54/s56/s49/s48\n/s48 /s49/s53/s48 /s51/s48/s48/s50/s52/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41/s32/s78/s100/s78/s105/s79\n/s51/s32/s70/s67/s67\n/s32/s78/s100/s78/s105/s79\n/s51/s32/s70/s67\n/s32/s78/s100/s78/s105/s79\n/s51/s32/s90/s70/s67\n/s32/s78/s100/s71/s97/s79\n/s51/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41\nFigure 1: (Color Online) The temperature variation of the\nmagnetization of NdNiO 3in FCC (circles), FC (triangles),\nand in ZFC (squares) protocols at 500 Oe. The stars show the\nmagnetization of NdGaO 3at the same field. For NdGaO 3the\nFCC, FC and ZFC magnetizations coincide. The inset shows\nthe difference in magnetization of NdNiO 3and NdGaO 3down\nto 10 K at 1000 Oe in FC (upper curve, filled squares) and\nZFC (lower curve, open squares) protocol. We used 119 mg\nof NdNiO 3and 118 mg of NdGaO 3for these measurements.\nto the signal from the sample.34In our case, the sam-\nple holder is a piece of straw which gives a temperature\nindependent signal of about −4×10−6emu at 500 G\nwhich is about 600 times smaller than the signal from\nthe NdNiO 3sample at 150 K. From the aforementioned\nwe see that artifacts or extraneous contributions are neg-\nligible compared to the magnetic signal of NdNiO 3, and\nthus, our SQUID data can be confidently used for the\ncritical analysis we are setting out to do.\nThe field dependent resistivity measurements were per-\nformed in a home made cryostat placed between the pole\npieces of a large electromagnet. More details on the re-\nsistivity measurements are available in one of our earlier\npublications.17\nIII. RESULTS AND DISCUSSION\n1. Magnetization measurements\nFigure 1 shows the magnetization of NdNiO 3and\nNdGaO 3in FC, ZFC and FCC protocols at 500 Oe. In\nthe FC protocol we cool the sample in the presence of a\nspecified field and then record the magnetization while\nslowly warming up the sample keeping the field fixed. In\nthe ZFC protocol we cool the sample in zero field to the\nlowest temperature and then apply the specified field and\nrecord the magnetization while warming up. In FCC pro-\ntocol the magnetization is recorded while cooling in the\nspecified field. The magnetization plots of NdNiO 3show\na shoulder around 200 K attributable to the ordering of\nNi moments. We see that below 200 K the magnetization3\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53\n/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s49/s69/s45/s51/s48/s46/s48/s49/s48/s46/s49/s49/s49/s48/s49/s48/s48\n/s32/s40 /s32/s99/s109/s41\n/s84/s32/s40/s75/s41/s32/s72/s32/s61/s32/s48/s32/s79/s101\n/s32/s72/s32/s61/s32/s49/s48/s48/s48/s32/s79/s101/s48 /s49/s48/s48/s48 /s50/s48/s48/s48/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s49/s48/s48/s49/s53/s48/s50/s48/s48\n/s72/s32/s40/s79/s101/s41\n/s84\n/s80/s32/s40/s75/s41/s84\n/s73/s82/s82/s32/s40/s75/s41\n/s32\n/s40/s105/s105/s41/s40/s105/s41\n/s90/s70/s67/s32/s50/s48/s48/s48/s32/s79/s101/s70/s67/s32/s50/s48/s48/s48/s32/s79/s101/s90/s70/s67/s32/s49/s48/s48/s48/s32/s79/s101/s70/s67/s32/s49/s48/s48/s48/s32/s79/s101/s77/s47/s72/s32/s40/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s40/s109/s111/s108/s101/s45/s79/s101/s41/s41\n/s84/s32/s40/s75/s41/s40/s98/s41/s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48/s50/s46/s53/s53/s46/s48/s55/s46/s53/s49/s48/s46/s48\n/s50/s52/s48 /s50/s56/s48/s50/s51/s52/s32/s70/s67/s32/s32/s32/s49/s48/s48/s48/s32/s79/s101\n/s32/s90/s70/s67/s32/s49/s48/s48/s48/s32/s79/s101\n/s32/s70/s67/s32/s32/s32/s50/s48/s48/s48/s32/s79/s101\n/s32/s90/s70/s67/s32/s50/s48/s48/s48/s32/s79/s101/s40 /s45\n/s48/s41/s45/s49\n/s40/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s109/s111/s108/s101/s45/s79/s101/s41/s45/s49\n/s84/s32/s40/s75/s41/s40/s97/s41\n/s90/s70/s67/s32/s49/s48/s48/s32/s79/s101/s90/s70/s67/s32/s50/s48/s48/s32/s79/s101/s90/s70/s67/s32/s53/s48/s48/s32/s79/s101/s70/s67/s32/s53/s48/s48/s32/s79/s101/s70/s67/s49/s48/s48/s32/s79/s101/s77/s47/s72/s32/s40/s49/s48/s45/s51\n/s32/s101/s109/s117/s47/s40/s109/s111/s108/s101/s45/s79/s101/s41/s41\n/s84/s32/s40/s75/s41/s70/s67/s32/s50/s48/s48/s32/s79/s101\nFigure 2: (Color Online) The dc magnetic susceptibility of t he\nNi sublattice versus temperature for FC and ZFC protocols\nat various fields. The inset of (a) shows that the susceptibil ity\nabove 220 K follows the modified Curie-Weiss law shown in\nequation (1) quite closely. The top-right inset of (b) shows\nthe temperature dependence of resistivity at zero field and\n1000 Oe during cooling as well as warming. The bottom-left\ninset of (b) shows how TIRRandTPdepend on the applied\nfield.TPis determined by Gaussian fitting of the ZFC curves\nclose to their maxima.\nof NdNiO 3depends on the experimental protocol. The\nFCC magnetization is slightly higher than the FC mag-\nnetization while ZFC magnetization is lower than both\nFCC and FC magnetizations. Above 200 K, the FCC,\nFC, and ZFC curves overlap and are indistinguishable.\nThe existence of thermal and magnetic history depen-\ndence in magnetization suggests that the system is not\nin thermodynamic equilibrium below 200 K. In contrast,\nfor the reference sample NdGaO 3, the magnetization val-\nues in FCC, FC, and ZFC protocol coincide and follow\nthe Curie law.\nTo extract the magnetization of Ni sublattice from\nthe experimental data we subtract the contribution of\nNd moments from that of NdNiO 3. The Nd magnetic/s45/s52/s48/s48/s48/s48 /s45/s50/s48/s48/s48/s48 /s48 /s50/s48/s48/s48/s48 /s52/s48/s48/s48/s48/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s40/s98/s41\n/s45/s49/s48/s48/s48 /s48 /s49/s48/s48/s48/s45/s52/s45/s50/s48/s50/s52/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s72/s32/s40/s79/s101/s41/s32/s54/s48/s32/s75\n/s32/s49/s53/s48/s32/s75/s45/s52/s48/s48/s48/s48 /s48 /s52/s48/s48/s48/s48/s45/s52/s48/s48/s48/s52/s48/s48\n/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s72/s32/s40/s79/s101/s41/s32/s78/s100/s78/s105/s79\n/s51\n/s32/s78/s100/s71/s97/s79\n/s51\n/s32/s78/s105/s32/s115/s117/s98/s108/s97/s116/s116/s105/s99/s101/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s72/s32/s40/s79/s101/s41/s32/s54/s48/s32/s75\n/s32/s49/s53/s48/s32/s75\n/s40/s97/s41\nFigure 3: (Color Online) M-H curves for the Ni sublattice at\n150 K and 60 K. The inset (a) displays the magnetization of\nNdNiO 3, NdGaO 3, and their difference at 150 K. The inset (b)\nshows an expanded view of the low field data of Ni sublattice.\nField (Oe) C θ χ0χ2/DOF R2\n1000(FC) 0.043(4) 125(6) 0.00095(2) 1.314 0.99916\n1000(ZFC) 0.043(4) 126(6) 0.00095(2) 1.271 0.99920\nTable I: Fit parameters obtained from the fitting of equation\n(1) to the 1000 Oe magnetic susceptibility data of figure 2\nabove 220 K. The quality of the fit is clear from the fitted\nline to the red squares in the inset of fig 2(a) as well as from\nthe lowχ2/DOF values and the R2values very close to unity\npresented in this table. For other field values the number of\ndata points above 220 K and their span are not good enough\nto warrant comparable quality of fitting.\nmoment is estimated from the magnetization data of\nNdGaO 3which has the same crystal structure and al-\nmost the same lattice parameters as NdNiO 3.19Since\ngallium and oxygen ions have no magnetic moment, the\nmagnetization of NdGaO 3arises only from the contri-\nbutions of the Nd moments sitting at the A sites of the\nperovskite structure. By subtracting the NdGaO 3mag-\nnetization (per mole) from that of NdNiO 3we should be\nable to calculate the magnetization of Ni sublattice, pro-\nvided Nd moments behave in the same fashion in both\nNdGaO 3and NdNiO 3. Unfortunately this method runs\ninto rough weather because the Nd moments in NdNiO 3\ntend to order at low temperature aided by the ordering\nof the Ni sublattice.\nNeutron diffraction measurements on bulk NdNiO 3\nshow that the magnetic ordering of Nd moments starts\nbelow 40 K,25,26while the synchrotron radiation data on\nthin films of NdNiO 3suggest that magnetic ordering of\nNd moments starts at T MIbut becomes significant only\nat low temperatures below 70 K.31The higher Nd order-\ning temperature seen in the thin films may have a pos-\nsible connection with the epitaxial strain in the films.35\nThe ordering of Nd moments is thought to be induced\nby the direct exchange interaction with the neighboring\nNi moments and is antiferromagnetic in nature while the4\nNd moments in NdGaO 3remain paramagnetic through-\nout the temperature range (See Ref. 31 and Fig. 1). So\non cooling below the magnetic ordering temperature of\nNd, the difference in the magnetization of NdNiO 3and\nNdGaO 3would drop drastically because the contribution\nof Nd moments to the magetization of NdNiO 3would\nfall due to their antiferromagnetic ordering. In our case,\nsuch a drastic drop in the difference in magnetization of\nNdNiO 3and NdGaO 3is seen to occur below about 50 K\nas is clear from the inset of Fig. 1. This suggests that\nthe effect of Nd ordering becomes quite significant be-\nlow 50 K, and sufficiently above this temperature, the\nmagnetization of Ni sublattice could be obtained, to a\nreasonable degree of confidence, by the subtraction of\nNdGaO 3magnetization from that of NdNiO 3.\n2. Magnetic ordering of the Ni sublattice\nIn figure 2 we show the temperature dependence of\nZFC and FC dc magnetic susceptibility of Ni sublattice\nbetween 100 Oe to 2000 Oe. Above 220 K, as is clear from\nthe inset of figure 2(a), the data fit well to the modified\nCurie Weiss equation\nχ=C/(T−θ)+χ0 (1)\nwhereCandθare Curie and Weiss constants respec-\ntively, and χ0is a constant arising from Van Vleck\nand Pauli paramagnetism and Landau and core diamag-\nnetism. The parameters obtained from the fitting of\nequation (1) to the 1000 Oe susceptibility data of fig-\nure 2, in the temperature range of 220-300 K, is shown\nin table I. The R2values very close to unity and the low\nχ2/DOF values indicate that fit quality is very good.\nThe presence of possible defects in the crystalline lat-\ntice structure may also give a contribution to magnetic\nsusceptibility, but that contribution is generally around\n103times smaller than our measured signal,36,37and this\nfact allows us to ignore them.\nThe subtraction of NdGaO 3magnetic susceptibility\nfrom that of NdNiO 3cancels the temperature indepen-\ndent Van-Vleck and core contribution of Nd ions, and\nsoχ0is free of these two. The core diamagnetic sus-\nceptibility of Ni ions is around −68×10−6emu/mole38\nand the Landau diamagnetic susceptibility is connected\nto Pauli paramagnetic susceptibility by the equation\nχLandau=−(1/3)[m/m∗]2χPauli, where mis the free\nelectron mass and m∗is the effective mass of an elec-\ntron in the conduction band. Since m∗is found to be\nsignificantly larger than min this family of oxides,39the\nχLandaucan be neglected in comparison to χPauli.38Thus\ntheχ0values shown in table I arise predominantly from\nthe Pauli paramagentism of itinerant electrons, and they\nare in good agreement with the values reported in Refs.\n21 and 38. The Pauli paramagnetic susceptibility of\nNdNiO 3is around two orders of magnitude larger than\nthat calculated using the free-electron value which sug-\ngests that the electron correlation in these systems is verystrong.21It is to be noted that we get a positive Weiss\nconstant θwhich is indicative of a ferromagnetic interac-\ntion in the magnetically ordered state. This is surprising\nconsidering the fact that neutron and resonant soft X-ray\ndiffraction measurements show that the system has anti-\nferromagnetic order below TMI.9,24,25,27,28In consonance\nwith the above observation of a positive Weiss constant\nwe point out that below 195 K, in FC measurements, the\nmagnetic susceptibility increases on decreasing the tem-\nperature as would be expected in the case of ferrimagnets\nor canted antiferromagnets which behave as weak ferro-\nmagnets. See figure 2.\nIn figure 3 we have shown the field dependence of\nthe magnetization of the Ni sublattice. The inset (a)\nof the figure shows the magnetization versus field for\nNdNiO 3, NdGaO 3and the Ni sublattice at 150 K. The Ni\nsublattice magnetization is obtained by subtracting the\ncontribution of Nd moments (obtained from NdGaO 3)\nfrom that of NdNiO 3. In the main panel of figure 3\nand its inset (b) we show the magnetization versus field\nfor the Ni sublattice at 150 K and 60 K. The M-H\ncurves show a small hysteresis at small fields, while at\nhigher fields, the M-H curves behave as that of a typ-\nical antiferromagnet, with Mvarying linearly with H,\nwhich leads to the conclusion that this system is a spin-\ncanted antiferromagnet.40The presence of spin canted\nmagnetism (weak ferromagnetism) cannot be explained\non the basis of the magnetic structures proposed in the\nliterature (Refs. 23, 24, 26). This is because the sum\nof the Ni magnetic moments in the proposed collinear as\nwell as the non-collinear magnetic structure is zero (See\nfigure 5 of Ref. 28). Thus our experimental data clearly\nshow that the magnetic structures proposed in the lit-\nerature are not the true magnetic picture of NdNiO 3.\nFurther investigations are required to confirm this new\nexperimental finding.\nReferring to the inset (b) of figure 3, we see that, the\ncoercivity ( HC) is temperature dependent below TN, and\nit increases on lowering the temperature. Since coerciv-\nity is related to magnetic anisotropy, this suggests that\nthe magnetic anisotropy increases on decreasing the tem-\nperature.\n3. Magnetic state of the supercooled phase\nThe transport properties of NdNiO 3show thermal hys-\nteresis which is attributed to the presence of supercooled\nmetallic regions below the transition temperature.17–19\nNow the question we would like to ask is this: What is\nthe magnetic state of the supercooled metallic regions?\nAre they paramagnetic or antiferromagnetic? In other\nwords we are asking whether the paramagnetic to antifer-\nromagnetic transition, when we cool the system through\nits magnetic transition temperature (200 K ), takes place\nat that temperature or does it take place along with the\nM-I transition of the metastable phase at a lower temper-\nature? In order to throw some light on this issue we mea-5\n/s52/s48 /s56/s48 /s49/s50/s48 /s49/s54/s48 /s50/s48/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51/s48/s46/s52\n/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48\n/s40/s86\n/s67/s45/s86\n/s72/s41/s77\n/s70/s67/s67/s45/s77\n/s70/s67/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41/s32/s49/s48/s48/s32/s79/s101\n/s32/s50/s48/s48/s32/s79/s101\n/s32/s53/s48/s48/s32/s79/s101\n/s32/s50/s48/s48/s48/s32/s79/s101\nFigure 4: (Color Online) The temperature variation of the\ndifference in magnetization, MFCC−MFC, of NdNiO 3between\ncooling and heating runs at 100 Oe, 200 Oe, 500 Oe and\n2000 Oe (solid symbols). The open circles show the difference\nin the metallic volume fraction, VC−VH, between cooling and\nheating runs.\nsured the thermal hysteresis of magnetization. In figure 4\nwe show the difference in cooling and heating cycle mag-\nnetization, MFCC−MFC, of NdNiO 3at a few field values\nin the range 100 Oe to 2000 Oe. The data show that be-\ntween 200 K and 120 K, the magnetization of the cooling\ncycle is higher than that of the heating cycle. The dif-\nference in the magnetization is maximum around 170 K.\nFigure 4 also shows the difference in the metallic volume\nfractions between the cooling and heating runs VC−VH,\ntaken from reference 17. The difference in the magnetiza-\ntions and the difference in the metallic volume fractions\nhave remarkably similar temperature dependence which\nsuggests that they originate from a common underlying\nphysical mechanism. In a cooling run, below TMI, the\nsystem contains supercooled metallic and insulating re-\ngions, while in a heating run, it is mostly insulating.17–19\nTherefore VC−VHrepresents the volume fraction of su-\npercooled metallic regions. So the correlation between\nthe thermal hysteresis in magnetization and the super-\ncooled metallic volume fraction indicates that the super-\ncooled metallic regions have a higher magnetic moment\ncompared to the insulating regions.\nThe Ni moments are paramagnetic in the normal\nmetallic state ( T > T MI) while they show a spin-canted\nantiferromagnetic ordering in the insulating state. Also,\nthe spin-canted insulating state has a higher suscepti-\nbility than the paramagnetic metallic state (see figure\n2). This suggests that if the supercooled metallic re-\ngions were paramagnetic, as above TMI, then the mag-\nnetization of NdNiO 3in a cooling run, where below TMI\nthe system consists of supercooled metallic and insulat-\ning regions, should be lower than that in a heating run\nwhere the system is expected to be almost fully insulat-\ning. But the experimental results discussed in the pre-vious paragraph contradict this which indicates that the\nsupercooled metallic regions are not paramagnetic. To\nmake things more concrete, we compare the observed dif-\nference in the magnetization of cooling and heating runs\nto the expected value of the difference if the supercooled\nregions were paramagnetic. In the cooling run, at 170 K,\nthe volume fraction of the supercooled metallic regions\nis around 0.9 from figure 4. The dc magnetic suscep-\ntibility of the paramagnetic metallic phase at 2000 Oe\n(Figure 2), extrapolated down to 170 K, is about 20%\nsmaller than that of the insulating phase which suggests\nthat if the thermal hysteresis in the magnetization is be-\ncause of paramagnetic ordering of supercooled metallic\nregions, then, according to our estimate, the difference in\nthe magnetization of the cooling and heating runs should\nbe around −0.9emu/mole. But the observed difference\nin the magnetization is +0.4emu/mole which has the\nwrong sign and is smaller in magnitude than the expected\nvalue. This observation strongly suggests that the super-\ncooled metallic regions are antiferromagnetic with canted\nspins just like the insulating state. The small positive dif -\nference in magnetization between cooling and heating is\nproportional to the volume fraction of supercooled metal-\nlic regions and hence we conclude that this difference in\nsusceptibility is temperature independent. This suggests\nthat the observed difference in cooling and heating cy-\ncle magnetization is coming from itinerant electrons in\nthe supercooled metallic state through Pauli paramag-\nnetic and Landau diamagnetic contributions.41Thus we\nsee that the metallic state is paramagnetic above TMIand\non cooling below TMI, while a fraction of the high tem-\nperature metallic phase exists in its supercooled state,\nthe magnetic ordering of the whole sample switches to\nan antiferromagnetic state at TN.\nFrom the above discussion, we conclude that in\nNdNiO 3, even though the charge ordering and magnetic\nordering occur at the same temperature (in equilibrium)\nthey are not strongly coupled and occur independently of\neach other. Incidentally, we note that except in PrNiO 3\nand NdNiO 3of the nickelate series, the two transitions\noccur at different temperatures which supports the con-\nclusion that the two transitions are only weakly coupled.\nThe antiferromagnetic order of the supercooled metallic\nregions rules out the presence of any metastable mag-\nnetic phase associated with the magnetic transition and\nsuggests that the magnetic transition is continuous in na-\nture. This result removes the ambiguity associated with\nthe nature of the magnetic transition in nickelates where\nTMI=TN; the magnetic transition is continuous which\nis consistent with the other members of the series where\nTMI>TN.\n4. The FC-ZFC irreversibility\nThe FC and ZFC magnetic susceptibilities show a\nhistory dependence with a bifurcation between the two\ncurves at a temperature known as the temperature of6\n/s49/s55/s50 /s49/s55/s52 /s49/s55/s54 /s49/s55/s56/s49/s50/s48/s49/s50/s51/s49/s50/s54/s32/s32/s82/s101/s102/s101/s114/s101/s110/s99/s101\n/s32/s32/s67/s111/s111/s108/s105/s110/s103\n/s32/s32/s72/s101/s97/s116/s105/s110/s103/s77/s32/s40/s49/s48/s45/s50\n/s32/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s84/s32/s40/s75/s41\nFigure 5: (Color Online) Memory experiment in the FC pro-\ntocol with intermediate stops of one hour at 175, 150, 125 and\n110 K. The field is switched off during each stop. The data\nclose to 175 K is shown here. The black squares show the FC\nreference which is the magnetization in FCC protocol(after\nremoving the contribution of thermal hysteresis).\n/s48 /s49/s48/s48/s48 /s50/s48/s48/s48 /s51/s48/s48/s48/s48/s46/s55/s53/s51/s48/s46/s55/s53/s54/s48/s46/s55/s53/s57/s77/s32/s40/s101/s109/s117/s47/s109/s111/s108/s101/s41\n/s116/s105/s109/s101/s32/s40/s115/s41/s116\n/s87/s61/s48\n/s116\n/s87/s61/s49/s104\nFigure 6: (Color Online) The time decay of thermorema-\nnent magnetization of NdNiO 3at 80 K (red circles). The\nblue squares show the decay of thermoremanent magnetiza-\ntion with a one hour wait time.\nirreversibility ( TIRR). See inset (i) figure 2(b). The\ntemperature of irreversibility depends on the magnetic\nfield and it decreases on increasing the magnetic field.\nFor fields greater than 2 kOe the FC and ZFC curves\nsuperpose. Behavior such as this where the mag-\nnetic susceptibility depends on measurement history has\nbeen observed in non-equilibrium systems such as spin-\nglasses,42–44superparamagnets,45cluster-glasses,46,47su-\npercooled systems,48,49and also in anisotropic ferromag-\nnets and ferrimagnets.50–54The ZFC data show a peak,\nand the peak broadens and shifts to low temperatures on\nincreasing the magnetic field. We analysed the nature of\nthis peak and found that the peak temperature ( TP) as\na function of field ( H) does not behave as in the case of\nspin-glasses, cluster-glasses,55or superparamagnets56,57\nwhich indicates that the system is neither a spin-glassnor a superparamagnet. We also rule out supercooling\nas a possible reason for the FC-ZFC irreversibility by the\nfollowing argument. The resistivity measurements show\na thermal history dependence which is attributed to the\npresence of supercooled metallic regions below TMI. We\ndid not observe any significant magnetic field or mag-\nnetic history dependence in transport properties which\nsuggests that the volume fraction of supercooled metal-\nlic regions is not altered by the application of a magnetic\nfield. See inset (ii) of figure 2(b). The lack of dependence\nof resistivity on applied magnetic field has also been re-\nported earlier by Mallik et al.58From these results, we\ninfer that the magnetic history dependence of the dc\nmagnetic susceptibility (see figure 2(a) and (b)) cannot\nbe originating from the supercooled metallic phases. So\nfar our analysis has shown that the magnetic hysteresis\ndoes not arise from spin-glass or cluster-glass nature, su-\nperparamagnetism or supercooling. This leaves us with\nthe only possibility that the magnetic hysteresis in this\nsystem is arising from magnetic anisotropy of the spin\ncanted magnetic domains.\nTo be doubly sure that the history dependent FC and\nZFC susceptibility of the Ni sublattice has nothing to do\nwith superparamagnetism or spinglass nature, we per-\nformed FC, ZFC memory and aging experiments. Since\nthe Nd moments are paramagnetic, they would not have\nany role in the memory and aging of NdNiO 3. Thus if\nany such effect is seen in this system it would have to\nbe attributed to the Ni sublattice. The FC memory ex-\nperiments were performed with intermediate stops of one\nhour at 175, 150, 125, and 100 K. In these experiments\nthe system is cooled in a 100 Oe field from 220 K to 80 K\nand then heated back to 180 K to remove the influence\nof supercooled metastable regions on dynamic behavior.\nSubsequently the system is cooled from 180 K to 80 K\nwith intermediate stops of 1 hour at 175, 150, 125, and\n100 K. The field was switched off during the intermediate\nstops. The magnetization is recorded while cooling and\nthen during the subsequent heating. The FC memory\ndata at 175 K is shown in figure 5. We can see that im-\nmediately after an intermediate stop the magnetization\ndoes not go back to its pre-stop value after switching on\nthe field. In the subsequent heating run, we did not find\nany memory of the intermediate stops and this rules out\nthe possibility of superparamagnetism or spin-glass be-\nhavior in the system.44We also carried out ZFC memory\nexperiments on the system at 170 K and the result was\nnegative. This confirms the conclusions we arrived at\nfrom the FC memory experiments and once again rules\nout a spin-glass state.44\nIn figure 6 we show the results of the FC ageing exper-\niment. In this experiment one essentially measures the\ntime decay of thermoremanent magnetization along with\nwait time dependence. To begin with we cool the system\nfrom 250 K to 80 K in the presence of 100 Oe field, wait\nfor the duration twat 80 K with the field on, and then\nswitch off the field and record the magnetization as a\nfunction of time. It is clear from the figure that the sys-7\nCµH\nSµ) H+H(Sµ) H+H(S) µ(      +       CH H+Hµ(_)_S CHH H\nCµH\n)µ_(SHH ) µ(_\nSCHH +Hµ ) (C+S_HH H\n)µ_\nSH(H\nCH >  H  + H  ( )S H H_  \nSCµH\nSµHµC(_\nS)H H\nCµ ) (       +       SH H\nSµH\n+µ −µ\n  = 0H H HSH H C S S H  +H    > H > H( )_  \nS)µ_(SHH) µ(_\nSCHH +HCµH\n−µ +µ+µ −µ(a) (b) (c) (d)\n+µ −µ\nFigure 7: (Color Online) The free energy profile of a bistable subsystem at various applied fields\ntem does not show any noticeable wait time dependence\nin FC ageing and this yet again rules out the possibility\nof the system being a spin glass or a superparamagnet.44\nThe irreversibility of the FC and ZFC magnetic sus-\nceptibility in a system which is neither a spin-glass nor\nsuperparamagnetic can be understood in terms of a com-\npetition between the magnetocrystalline anisotropy and\ndomain wall pinning on the one hand and applied field\nand thermal energy on the other.50–54Below the temper-\nature of magnetic ordering, a magnetically ordered ma-\nterial consists of uniformly magnetized regions which are\nknown as magnetic domains. At any temperature Tand\napplied field H, the free energy of the magnetic systems\nhave a number of local minima which are determined by\nthe arrangement of the domains inside the magnetic ma-\nterial. These local minima states are separated by energy\nbarriers which arise due to magnetocrystalline anisotropy\nand domain wall pinning. When the thermal energy is\ngreater than the energy barrier of the metastable state\nin which the system is trapped, the system can explore\nthe neighboring states in search of the global minimum\nor the equilibrium state. The free energy configuration is\na function of applied magnetic field Hand temperature\nTand on changing HorT(which changes the mag-\nnetocrystalline anisotropy) the system evolves from one\nconfiguration to another.59We shall make an attempt to\nunderstand our system on the basis of the Preisach model\nin which the free energy configuration is decomposed into\nan ensemble of bistable subsystems.59A bistable subsys-\ntem consists of two metastable states separated by an\nenergy barrier. The two states have moments oriented in\nopposite directions and are termed as ±µstates. The free\nenergy of these states in the absence of applied magnetic\nfield is determined by the local interaction field ( HS) and\nthe the coercive field ( HC).HSis the net magnetic field\nproduced at the location of the moment µby the mag-\nnetic moments of all the neighboring domains. If HS= 0\nthenµHCrepresents the anisotropy energy barrier that\nhas to be crossed to go from +µto−µstate or vice versa.\nThe barrier height seen from the +µside isµ(HC+HS)while from the −µside it is µ(HC−HS). See figure 7\n(a). The application of a magnetic field ( H) changes the\nfree energy of the metastable states which in turn affects\nthe effective height of the energy barrier. We also note\nthat a change in the temperature can also affect the free\nenergy barrier through its effect on magnetocrystalline\nanisotropy.53,54,59\nIn the following paragraphs we discuss qualitatively\nthe FC-ZFC irreversibility and the remanent magnetiza-\ntion using the standard Preisach model. Thereafter we\napply it to understand the observed results of aging ex-\nperiments.\nIn ZFC protocol when the system is cooled below TN\neach subsystem will be in its lower energy state which is\ndetermined by HS(Figure 7 (a)). On applying a mag-\nnetic field, depending on the direction and strength of the\napplied field, the low energy state of the subsystem may\nremain as the low energy state (Figure 7 (b)), or may\nbecome metastable or unstable (Figure 7 (c) and (d)).\nIf/vectorH/bardbl −/vectorHS, the subsystems for which His larger than\nHC+HS, will flip to their new low energy state (Fig-\nure 7 (d)). It is this flipping that gives rise to the initial\nvalue of the ZFC magnetization of the system. The sub-\nsystems for which His less than (HC+HS), are now\nin a metastable state (Figure 7 (c)). These subsystems\nwill undergo a thermally activated transformation, which\ngives rise to a slowly rising time dependent ZFC mag-\nnetization even if the magnetic field is held fixed. On\nincreasing the temperature, HCdecreases and because\nof this more number of subsystems will flip to their new\nlow energy state and this increases the ZFC magnetiza-\ntion further. As one increases the temperature the ZFC\nmagnetization curve will attain a peak when the most\nprobable HCvalue of the Barkhausen moment ( µ) be-\ncomes equal to the applied field H.\nIn the FC protocol the subsystems get trapped in their\nlow energy states, as the sample is cooled through the\nmagnetic ordering temperature in the presence of an ap-\nplied field. At a constant field, a decrease in temperature\nincreases the energy barrier (because of increase in HC),8\nbut this does not affect the relative positions of the +µ\nand−µstates. Thus in the FC protocol there is hardly\nany change of state of the bistable subsystems when cool-\ning through TN. The temperature dependence observed\nin the FC magnetization is because of temperature de-\npendence of the Barkhausen moment µ(T). That is why\nthe shape of an FC magnetization curve is nearly the\nsame for all fields.\nIf we switch off the applied field in the FC protocol,\nthe subsystems for which applied field /vectorHis opposite and\ngreater in magnitude than /vectorHSwill result in their low en-\nergy state becoming a high energy state and vice versa.\nThis can be understood looking at figure 7 where the ini-\ntial states shown in figures 7 (c) or (d) switch to the final\nstate shown in figure 7 (a) on removal of the applied field.\nOf these subsystems, those which have HS≥HC, will\nbecome unstable on removing the field, and their change\nof state constitutes the initial loss of FC magnetization.\nThe other subsystems (which have HS< H C) will be-\ncome metastable and their thermally activated transfor-\nmation from a metastable to a new lower energy state\ngives a further slow decay in FC magnetization.\nAt this point let us examine the effect of aging (wait\ntime dependence) on the system. All the subsystems\noccupy their lower energy state on cooling through TN.\nThus, after cooling, if we wait for a few hours before\nswitching off (or on) the field, it will not affect the pop-\nulation of the ±µstates and hence we would not get any\neffect of aging on magnetic relaxation.\nIV. CONCLUSION\nWe performed detailed magnetization measurements\non NdNiO 3and extracted the magnetization of Ni sub-lattice after removing the contribution of the rare earth\nNd ion. Our results indicate the presence of weak fer-\nromagnetism coexisting with antiferromagnetic order in\nthe Ni sublattice. We argued that the weak ferromag-\nnetism is due to canting of antiferromagnetic spins. Fur-\nther we found that in contrast to the normal metallic\nstate, the supercooled metallic regions are magnetically\nordered. This shows that while cooling the metal insula-\ntor transition occurs over a temperature range of 200 K\nto 110 K, the magnetic ordering is sharp and occurs at\n200 K. 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The nanomagnets are assu med to be made of 2-phase strain-coupled \nmagnetostrictive/piezoelectric multiferroic elements , such as nickel and lead zirconate titanate \n(PZT). A small voltage of 200 mV applied acro ss the piezoelectric layer can generate enough \nmechanical stress in the magnetostrictive layer to  rotate its magnetization away from one of the \nfour stable orientations and implement Bennett cl ocking. We show that a particular sequence of \npositive and negative voltages will propagate 4-st ate logic bits unidirectionally down a chain of \nsuch multiferroic nanomagnets for logic flow.  \n \nIndex terms – Nanomagnetic logic, multiferroics, straintronics, Bennett clocking  \n \n \n 1.  INTRODUCTION  \nAn emerging technology in the field of digita l computing is nanomagne tic logic (NML) that \npromises significantly lower power  dissipation than conventional transistor-based electronics [1, \n2, 3, 4, 5, 6]. NML is non-volatil e, which permits implementation in both logic and memory, and \nit has no standby power dissipa tion unlike transistors. \nIn conventional  binary NML, bits 0 and 1 are encoded in two stable magnetization directions of \nsingle-domain nanomagnets with uniaxial shape anisotropy [1, 2]. Data  transmission between \nthem requires: (i) dipole  interaction between neighbors, and (i i) a Bennett clock that temporarily \nreorients the magnetization of every nanomagnet away  from one of the stable directions to allow \na bit to propagate through it [ 3, 7, 8]. The re-orienta tion can be carried out with an effective \nmagnetic field that is generated either with an  external current that does not pass through the \nnanomagnet but produces a local magnetic field in it s vicinity [6, 9], or with a spin-polarized \ncurrent that passes through the na nomagnets and generates a spin tr ansfer torque [10] (or perhaps \ndomain wall motion [11, 12]), or with a volta ge that produces mechanical strain in a \nmagnetostrictive-piezoelectric multiferroic  nanomagnet [13, 14, 15, 16] and rotates its \nmagnetization vector. The latter switching m odality, termed “straintronics”, promises \nunprecedented energy efficiency and is the subject of  this paper. We show how straintronics can \nbe employed to “Bennett clock” unconventional  multi-state  (specifically 4-state) logic circuits in \nNML with extremely low energy dissipation. \nOne way to implement Bennett clocking in traditional binary NML is to arrange shape \nanisotropic nanomagnets in a line along their hard  axis as shown in Fig. 1(a). The ground state of \nthe array will be “anti-ferromagnetic” whereby each nanomagnet's magnetization will align along the easy axis, but nearest neighbors will ha ve anti-parallel magnetizations, representing a sequence of binary bits 0 1 0 1 K. This anti-ferromagnetic ordering happens because of dipole \ninteraction between nei ghbors. If we now flip the first na nomagnet's magnetization (first bit) \nwith some external agent and expect all suc ceeding nanomagnets to sequentially flip in a \ndomino-fashion to re-assume the anti-ferromagnetic order because of dipole interaction, that will \nnot happen. What prevents its occurrence is that immediately after switching the first \nnanomagnet, the second nanomagnet finds itself in  a frustrated state wh ere its left neighbor's \ndipole interaction and right neighbor's dipole interaction exactly cancel. Therefore, this \nnanomagnet does not flip and the input  bit does not propagate further. \nIn order to break this logjam and make the i nput bit propagate, one needs a clock [17] to \nmanipulate the dipole interactions between ne ighboring pairs of nanomagnets. For example, \nprior to flipping the first bit, a global magnetic field could break the anti -ferromagnetic ordering \nand align every nanomagnet's magnetization along the common hard axis. This field is then \nwithdrawn and the magnetization of  the first nanomagnet is oriented by an external agent to \nconform to the input bit [18]. Dipole interaction will then flip the magnetization of all the \nsucceeding nanomagnets sequentially in a domi no-like fashion since every nanomagnet now \nexperiences non-zero dipole interac tion that restores the anti-fe rromagnetic order. This is an \nexample of propagating bits us ing Bennett clocking. Here the glob al magnetic field acts as the \nclock. The same type of clock can propagate an  input bit down a chain if the nanomagnets are \narranged in a line parallel to the easy axis as sh own in Fig. 1(b). In this case, dipole coupling will \ncause ferromagnetic ordering. \nUnfortunately, the use of a global magnetic fiel d makes the architecture non-pipelined and hence \nunacceptably slow and error-prone [2]. A superior strategy is to employ local clocking where the \norientation of every nanomagnet is  turned along the hard axis with  a local agent one  at a time to implement Bennett clocking [3]. This increases  the lithography overhead significantly since \nevery nanomagnet needs to be contacted, but it allows pipelining of data and makes the \narchitecture much faster [2]. Since non-pipelined and error prone architectures are unacceptable, \nwe will consider only the local clocking scheme. The issue then is what constitutes a suitable \nagent for local clocking, i.e. what is the most energy efficient way to rotate the magnetization of \na nanomagnet from the easy to the hard axis? Obviously, this can be achieved with a local \nmagnetic field directed along the hard axis that is generated with a current passing through a \nnearby wire [6, 9], or with sp in transfer torque (STT) curr ent passing through the nanomagnet \n[10, 12], or with a small voltage generating a stress in a multiferroic nanomagnet [13, 14, 15, 16, 19, 20, 21, 22, 23]. \nThe three possible methods for local clocking differ vastly in their energy-efficiencies. The first \nconsumes an exorbitant amount of energy and is most energy inefficient. Ref. [9] employed this \nscheme and dissipated at least 1012 kT (4.2×10  Joules) per bit flip while operating at a clock \nrate of ~ 1 MHz at room temperature. This is an  experimental result, but even theoretically, the \nenergy dissipation is not likely to be any less than 108 kT per bit flip for this methodology. For \nthe second method (STT), ref. [12] reported a dissipation of at least 10  kT at a clock rate of 500 \nMHz. Once again, this is an experimental result, but the theoretical estimate is not significantly better. Finally, the third method (“straintronics”) may turn out to be most energy-efficient. \nAlthough no experimental result is available (except for demons trating the basic mechanism of \nswitching the magnetization of a multiferroic with  voltage-generated strain [19]), there are \ntheoretical estimates that claim energy dissipa tion of only ~ 200 kT to switch a multiferroic \nnanomagnet at clock rates exceeding 1 GHz [1 3, 14, 16]. Here, we study straintronic clocking \nschemes to propagate composite logic bits (2-bit states) in 4-state logic circuits. 9−\n4 \n2. THEORY  \nA typical multiferroic nanomagnet that can encode 4-state logic bits is illustrated in Fig. 2, with \nthe magnetostrictive layer (nickel) on top and the piezoelectric layer (lead zirconate titanate, or \nPZT) at the bottom. The shape is circular, so  that there is no shap e anisotropy. The biaxial \nmagnetocrystalline anisotropy in the magnetostrict ive layer then creates four possible stable \nmagnetization directions (“up”, “right”, “down”, “left”) in which 2-bit states (00, 01, 11, 10) are \nencoded, as shown in Fig. 2. This encoding scheme  results in a change of only a single bit for \nevery 90° magnetization rotation. \nBecause of magnetocrystalline anisotropy, unstre ssed single-crystal Ni has its “easy” axis \nof magnetization along the  direction, a “medium” axis along the  direction and a \n“hard” axis along the  direction. In our study, we a ssume that the two-dimensional \ngeometry of the Ni layer suppresses out-of-plane excursion of the magne tization vector because \nof the large magnetostatic energy penalty, so that the magnetization vector always lies in the (001) plane. In that case, the “easy” axes of single -crystal Ni in the unstre ssed state are [110],   \n[> 111 < > 110 <\n> 100 <\n1 1 0], [ 1 10] and [1 1 0] in Miller notation, as  illustrated in the saddl e-shaped curve in Fig. 2 \n[15]. The coordinate axis system was rotated by 45° in order to have the stable states/bits point \n“up”, “right”, “down”, “left” along the x- and y- axes. Thus, the [100] axis lies along the +45° \ndirection. \nThis paper studies a synchronous Bennett clocking scheme where each 4-state \nmultiferroic nanomagnet is subjected to a particular  stress cycle that will allow 4-state logic bits \nto be propagated unidirectionally along a data pa th. We develop a novel scheme for such logic \npropagation and demonstrate its feasibility by modeling the rotation of  magnetization of each nanomagnet due to a cycle of tensile and compressive stresses generated by positive and negative \nelectrostatic potentials applied across the piezoe lectric layer of each multiferroic nanomagnet. \n \n3.  RESULTS AND DISCUSSION  \nWhen magnetizations of two adjacent nanomagnets are parallel to the line joining their centers, \nthe ordering will be ferromagnetic, but when the magnetizations are perpendicular to this line, \nthe ordering will be anti-ferroma gnetic because of dipole interacti on. Thus, if the first bit in a \nlinear array of circular 4-state multiferroics is sw itched from its initial state to one of the three \nother stable states, three possible arrangements result. Since each nanomagnet has four possible \nmagnetization orientations, there are twelve distinct  configurations that may arise when the first \nbit is switched, as illustrated in Fig. 3. \nConsider the anti-ferromagnetic arrangement of Fig. 3(a), with the first nanomagnet's \nmagnetization orientation acting as the input bit to the line. In this configuration, the input \nmagnetization can be switched from its initial “up” state to the “down”, “ri ght” or “left” state. \nThe corresponding nanomagnet states are shown in the  Final State  column (Fig. 3(a)) based on \nthe fact that coupling will be ferromagnetic (F) along the nanomagnet-array axis and anti-\nferromagnetic (AF) perpendicular to this axis. Ther efore, when the input b it is flipped from “up” \nto “down”, the change is propagate d along the array if it is appropr iately clocked, with the input \nmagnetization direction replicated in every odd- numbered nanomagnet from the left. This is a \nconsequence of anti-ferromagnetic ordering. If the i nput is switched to either  “left” or “right”, \nferromagnetic coupling will ensure that all the nanomagnets assume the “left” or “right” orientation, respectively. Similar considerations a pply to the other three configurations in Figs. \n3(b) – 3(d). Here, we only present the numer ical results corresponding to row I in the arrangements of Figs. 3(a) which pertains to the anti-ferromagnetic arrangement with the input \nmagnetization oriented “up”. All other cases have been exhaustively examined to confirm \nsuccessful operation, but are not presen ted here due to space limitations. \nFig. 3(a) shows that in an anti-ferromagnetically coupled line, the first bit will be \nreplicated in every odd-numbered nanomagnet (and has therefore propagated through the line) if \nthe array  can reach ground state  after the first bit is flipped. This can happen only if the array \ndoes not get stuck in a metastab le state and fail to reach the ground state [17, 24]. It can be \nshown that dipole interaction al one cannot guarantee that the ground state will be reached, which \nis why multi-phase clocking is needed to nudge the system out of any metastable state should the \nsystem get stuck in one [17]. A dditionally, the dipole interaction energy is usually not sufficient \nto overcome the magnetocrystalline anisotropy ener gy and rotate a nanomagne t out of its current \norientation to a different orient ation in order to propagate the bit. Thus, once again, a clock is \nneeded to supply the energy needed to overcome the magnetocrystalline anisotropy energy. In \nBennett clocking schemes, the clocking agent (local  magnetic field, spin transfer torque, strain, \netc.) will rotate the magnetization into an unsta ble state, perching it at the top of the energy \nbarrier, and then the dipole inte raction of its neighbors will push it into the desired stable state, \nthus ensuring unidirectional propagation of a logic bit. All this  can happen reliab ly if we neglect \nthermal fluctuations that can i nduce errors in switching. The eff ect of thermal fluctuations is \nbeyond the scope of this paper, but  preliminary considerations s how that they will undoubtedly \ninduce errors at room temperature, but not to  the point where the scheme is invalidated. \nConsider the nanomagnet array of Fig. 4 consisting of four nanomagnets in the collective \nground state of the array (row I). The magnetization of nanomagnet 1 on the fa r left is the input \nbit. If it is flipped from its in itial “up” to “down” state at time  = 0, then at time t = 0+, we treach the situation shown in row II where nanomagnet 2 experiences equal and opposite dipole \ninteractions from its two near est neighbors (magnets 1 and 3) which are magnetized in opposite \ndirections. As a result, the net dipole interaction experienced by nanomagnet 2 is zero. Thus, this \nnanomagnet does not flip its magne tization in response to  the first nanomagne t's flip, preventing \npropagation of the input logic bit down the chai n. In other words, the array is stuck in a  \nmetastable state  and cannot reach the ground state. \nIn order to break this logjam and allow the logic bit to flow past nanomagnet 2, we have to apply the following clock cycle. We will assume th at nanomagnets 1 and 4 remain stiff while \nnanomagnets 2 and 3 rotate when stressed. This is a good approximation if the magnetocrystalline anisotropy energy  is significantly larger than  the dipole interaction energy. \nStage 1: Tension (T)/Compression (C) (row III) : After the input nanoma gnet has been switched, \nnanomagnet 2 is subject to a tensile stress (gra dually increased to a maximum value of +100 \nMPa), applied along the [100] direction (+45° to the +x-axis) (row III). Since Ni has negative \nmagnetostriction, a tensile stress tends to raise the energy along the axis of applied stress while \nlowering the energy along the axis perpendicular to this direction. A compressive stress does the \nexact opposite [27]. As a result, tension applied on nanomagnet 2 along the [100] direction will \nprefer to align the magnetizati on along either -45° or +135° (- 225°) directions while raising the \nenergy barrier in the +45° and - 135° (+225°) directions. Since the in itial state of nanomagnet 2 is \nalong the -90° direction and the energy barrier is raised along the -135° (+225°) direction, the \nonly possible magnetization rotation that can take  place is from -90° to -45°. Energy profiles \nshowing the raising and lowering of energy levels  of nanomagnets 2 and 3 are presented later. \nAt the same time, a compressive stress (gradually increased to a maximum value of -100 MPa) is \napplied on nanomagnet 3 along the [100] axis, whic h causes its magnetization to rotate from the initial +90° state to the +45° state (row III). In  all cases studied in th is paper, stresses are \nsimultaneously applied on nanomagnets 2 and 3. \nStage 2: Relaxation(R)/C ompression(C) (row IV) : Next, the tensile stre ss on nanomagnet 2 is \ngradually reduced to zero while keeping the compressive stress on nanomagnet 3 fixed. The \nmagnetization of nanomagnet 3 remains oriented in  the +45° direction, but the magnetization of \nnanomagnet 2 rotates from -45° to 0°. This can  be understood from the energy profiles of the \nnanomagnets under stress, which we discuss later.  Rotations take place to lower the energy of a \nnanomagnet to the minimum energy state. \nStage 3: Compression(C)/Tension (T) (row V) : A compressive stress (up to -100 MPa) is now \napplied on nanomagnet 2 and simultaneously the compressive stress on nanomagnet 3 is relaxed \nto zero. This is immediately followed by the app lication of a tensile stre ss (up to +100 MPa) on \nnanomagnet 3 while keeping nanomagnet 2 unstre ssed. Nanomagnet 2 rotates to its preferred \nlowest-energy state along +45°. The relaxa tion of stress on nanomagnet 3 pushes its \nmagnetization towards 0° (ferromagnetic coupling is preferred over anti-ferromagnetic coupling \nsince the former has a stronger di pole interaction) while the subsequent tensile stress results in \nrotation of the magnetization to -45°. \nStage 4: Relaxation(R) /Tension(T) (row VI) : Finally, the compressive stress on nanomagnet 2 is \nrelaxed while keeping the tensile stress on nanoma gnet 3 fixed. This results in the magnetization \nof nanomagnet 2 rotating to the fi nal desired state of +90° (“up”). \nThe above clocking sequence successfully flips the magnetization of nanomagnet 2 in response \nto the flipping of the input nanom agnet 1 and allows the logic bi t to propagate past nanomagnet \n2. The same sequence of stresses is then applied to the next set of nanom agnets (3 and 4, with 2 \nand 5 now assumed to be stiff), which results in  nanomagnet 3 eventually settling in the “down” orientation (-90°), mirroring the state of the input bit. By continuing this cycle, the input bit can \nbe propagated down the entire chain, re sulting in successful logic propagation. \nIn order to prove rigorously that the magnetiza tions of the stressed multiferroic nanomagnets \norient as described, a theoretical analysis is  performed to determine the energy profiles of \nnanomagnets 2 and 3 under stress. The total ener gy of any nanomagnet is given by the equation  \n , =anisotropy stress talline magntocrys dipole total E E E E−+ + , (1) \nwhere  is the dipole-interaction ener gy due to neighboring nanomagnets,  is \nthe intrinsic magnetocrystalline anisotropy energy and  is the stress anisotropy \nenergy introduced by stress applie d along the [100] dir ection. Since the shape of the nanomagnet \nis isotropic, there is no shape anisotropy energy. dipoleEstalline magnetocryE\nanisotropy stressE−\nAfter nanomagnet 1 is switched, and nanomagnets  2 and 3 are stressed, their magnetizations \nrotate in order to reach the minimum energy state.  Let us assume that their magnetization vectors \nsubtend angles 2θ and 3θ with the x-axis. In order to find these angles for the minimum energy \nstate under a given stress, we make two simplifying assumptions: First, we assume that the \nmagnetocrystalline anisotropy energy is so much larger than the dipole interaction energy that \nnanomagnets 1 and 4 are immune to dipole influe nces of their neighbors and do not rotate when \nnanomagnets 2 and 3 rotate under stress. Sec ond, we will assume that  the stresses on the \nnanomagnets are changed slowly enough that th eir magnetization vectors can follow quasi-\nstatically. In that case, it is  sufficient to compute the energy minima of nanomagnets 2 and 3 \n(  and ) under any arbitrary stress to find the angles 2− totalE3− totalE2θ and 3θ. There is also a third \nassumption here; namely, that we neglect effect s of thermal fluctuations that may drive the \nsystem out of its minimum energy state randomly. \nThe total energies of nanomagnets 2 and 3 are given by  ()  2 [\n4=1 3 232 2\n0\n2 θ θ θ\nπμcos cos cos\nRMEs\ntotal + −Ω\n− ( )]  1 3 2θθθ sin sin sin+ + \n () ⎟\n⎠⎞⎜\n⎝⎛− Ω −Ω+4 232422\n100 22 1 πθ σ λ θ cos cosK\n \n()  2 [\n4=2 4 332 2\n0\n3 θ θ θ\nπμcos cos cos\nRMEs\ntotal + −Ω\n− ( )]  2 4 3θθθ sin sin sin+ +  \n () ,4 232432\n100 32 1⎟\n⎠⎞⎜\n⎝⎛− Ω −Ω+πθ σ λ θ cos cosK (2) \nwhere the first term is the dipole-interaction energy of a nanomagnet with its neighbors, the \nsecond term is the magnetocrystalline anisotropic energy, and the third term is stress anisotropy \nenergy resulting from a stress σ applied along the [100] direction (45° with the x-axis). Here, \n is the saturation magnetization, sM Ω is the nanomagnet's volume, 0μ is the permeability of \nfree space, R is the center-to-center separati on between neighboring nanomagnets, nθ is the \nangle subtended by the n-th nanomagnet's magnetization vector with the x-axis,  is the first \norder magnetocrystalline anisotropy constant, and 1K\n100λ is the magnetostriction constant. \nThe material parameters for nickel are given in  Table I. We adopt the convention that tensile \nstress is positive and compressive stress is negative . The PZT layer can transfer up to a strain of \n500 ppm to the Ni layer [27], so that the maximum stress that can be genera ted in that layer is \n100 MPa. The shape of the nanomagnets is that of  a circular disk of diameter of 100 nm and \nthickness 10 nm, while the center-to-cente r separation between the nanomagnets is R = 160 nm. \nThese dimensions ensure that the nanomagnet is  single-domain [28]. The parameters are chosen \nsuch that: (i) the magnetocrystalline an isotropy energy barrier is 0.55 eV (or 22 ) at room \ntemperature. This makes the static error probab ility associated  with spontaneous flipping of \nmagnetization very small, (ii) th e dipole interaction energy is 0.2 eV, which is nearly 3 times kTsmaller than the magnetocrystall ine anisotropy energy, and (iii) th e stress anisotropy energy at \nthe maximum stress of 100 MPa is 1.5 eV which is enough to overcome the magnetocrystalline \nenergy barrier and make the nanomagnet swit ch from one orientation to another. \nIn order to show that the magne tizations of nanomagnets 2 and 3 indeed rotate when the input \nnanomagnet is switched and the stress cycle on nanomagnets 2 and 3 is executed, and to find the \nnew orientations of these nanoma gnets, we follow the procedure be low. For each value of stress, \nfind  for every 2− totalE2θ while holding 1θ and 3θ constant at their in itial values. Next, find \n versus 3− totalE3θ while holding 4θ constant at the intial value and 2θ constant at the value \ncorresponding to the minimum of . Next, we re-evaluate  versus 2− totalE2− totalE2θ while changing \n3θ to the value corresponding to the minimum of . This process is iterated until \nconvergence is reached. 3− totalE\nWe now consider the arrangement in row I of Fi g. 4, where no stress is applied initially. This is \nan anti-ferromagnetic arrangement with the inpu t nanomagnet 1 in the “up” state. Accordingly, \nthe initial conditions are 1θ = +90°, 2θ = -90°, 3θ = +90° and 4θ = -90°. When the input is \nflipped, from “up” to “down” (1θ = -90°), nanomagnet 2 finds itself in a tie-state (frustrated) \nsince it experiences equal and opposite dipole magnetic fields from  magnet 1 and nanomagnet 3. \nThis can be seen in the energy profile of nano magnet 2 in Fig. 2(a) (the bottom curve) before \nstress is applied. The profile is symmetric about 2θ = 0°; hence 2θ = ±90° are degenerate in \nenergy. In other words, magnet 2 cannot lower its energy by responding to th e input, so that it \ndoes not respond. At this point, th e clocking cycle is initiated to break the tie. The energy \nprofiles of nanomagnets 2 and 3 as a function of  their orientation are shown in Fig. 2 with \nincreasing or decreasing compressi on or tension. The stress cycle consists of Tension (Fig. 2(a)) → Relaxation (Fig. 2(c)) → Compression (Fig. 2(e))  Relaxation (Fig. 2(g)) on nanomagnet \n2, and simultaneously Compression (Fig. 2(b))  Compression (Fig. 2(d))  Tension (Fig. \n2(f))  Tension (Fig. 2(h)) on nanomagnet 3. As noted  earlier, the stress is applied along the \n[100] direction (+45°). This can be accomplishe d by applying a voltage across the piezoelectric \nlayer, which generates the st rain in this layer through  coupling. Most of this strain is \ntransferred to the nickel layer which is much thin ner than the piezoelectric layer. Furthermore, to \nensure uniaxial stress along the +45° axis, the multiferroic nanoma gnet is mechanically \nrestrained to prevent expansi on and contraction along the direct ion perpendicular to the +45° \naxis. The two stress sequences (TRCR, CCTT; where T=tension, C=compression, and \nR=relaxation) are applied on nanomagnets 2 and 3 simultaneously. Stre ss is increased or \ndecreased in steps of 0.1 MPa. The ‘*’ markers indicate the magnetization orientations of nanomagnets 2 and 3 in their energy minima for any given stress. The squa res identify the final \norientation into which the nanomag net settles at the end of the stressing or relaxation cycle, \nwhile circles identify initial orientations. The thin (thick) solid curve represents the energy \nlandscape of a nanomagnet at the ons et (end) of a stage of the clock cycle while the dotted lines \nrepresent the intermediate energy profiles. →\n→ →\n→\n31d\nIn the first stage of the clock, a tensile stress is applied on nanomagnet 2 (Fig. 2(a)) while a \ncompressive stress is applied on nanomagnet 3 (F ig. 2(b)). The magnetization of nanomagnet 2 \nrotates from its initial -90° orie ntation as the tensile stress on it is increased and finally settles to \n~ -40° at +100 MPa stress; nanoma gnet 3 rotates from +90° to ~ + 45° as it is compressed to -100 \nMPa. It can be seen that at a certain stress (~ 50 MPa), both nanomagnets are drawn towards 0°. \nThis is due to the dipole coupling between th e magnets which prefers ferromagnetic coupling \nover anti-ferromagnetic coupling. Further increase in the stress (tension in nanomagnet 2, compression in nanomagnet 3) results in the nanoma gnets settling in their final states at the end \nof the stage (100 MPa) because the stress an isotropy energy dominates both the dipole and \nmagnetocrystalline energies. \nThe next stage of the clock cy cle involves relaxing the tensile stress on nanomagnet 2 to zero \n(Fig. 2(c)) while holding the compressive stress on  nanomagnet 3 at -100 MPa (Fig. 2(d)). As the \nstress anisotropy energy in nanomagnet 2 subsides  to zero, the relative influence of the dipole \nenergy (due to interaction w ith neighboring nanomagnets 1 a nd 3) increases and causes a \nmagnetization rotation from 2θ = ~ -40° to 0°. This rotation  towards 0° is preferred over a \nrotation back to -90° since the orientation at 0° is at a lower energy state.  Another way to explain \nthis rotation is by resolving the magnetic field components of the neighboring nanomagnets \nalong the x- and y- axes and recalling the pr eference for ferromagnetic coupling over anti-\nferromagnetic coupling. Since nanomagnet 3 is still compressed at -100 MPa, its magnetization remains at ~ +45°. Therefore, the x-component (~ +1050 A/m) of the ma gnetic field due to its \ninteraction with nanomagnet 2 is twice that of the y-component (~ -525 A/m). Magnet 1 is at -\n90° and, so, its interac tion with nanomagnet 2 produces a magne tic dipole field along the +y-axis \nwith magnitude ~ +750 A/m. The net dipole field on nanomagnet 2 is  +1050 A/m along the +x \ndirection (\n2θ = 0°) and +225 A/m along the +y direction (2θ = ~ +90°). This results in the \nmagnetization strongly favoring a rotation to 0°. \nIn the third stage of the clock, a compressive  stress, up to a maximum of -100 MPa, is \nincrementally applied on nanomagnet 2 (Fig. 2(e)).  At the same time, the compressive stress on \nnanomagnet 3 is relaxed (Fig. 2(f)), following whic h a tensile stress (up to  +100 MPa) is applied. \nThe magnetization of nanomagnet 2 ro tates from ~ 0° to ~ +45° since this is the closest energy \nminimum created by the compressive stress along th e +45° direction (the raising of the energy barrier at -45° prevents  a rotation to the other energy minimu m at -135°). Upon relaxation of the \ncompressive stress on nanomagnet 3, the x-compone nt of the magnetic field it experiences owing \nto its dipole interacti on with nanomagnet 2 exc eeds the y-component owi ng to interaction with \nnanomagnets 2 and 4. This can be seen in the sli ght tilt towards 0° in the energy profiles of Fig. \n2(f) which results in a magnetization rotation towa rds 0°. The tensile stress applied subsequently \ninduces a rotation from 0° to ~ -45° as the raising of the energy barrier along +45° prevents the \nmagnetization from rotating to th e other energy minimum at +135°. \nThe final stage consists of rela xing the compressive stress on nanomagnet 2 to zero (Fig. 2(g)), \nwhile holding the tensile stress on magnet 3 consta nt (Fig. 2(h)). Upon examination of the dipole \nfield experienced by nanomagnet 2 owing to  its interaction with nanomagnet 1 (1θ = -90°) and \nnanomagnet 3 (3θ = ~ -45°), it can be determined that  the +y-component of the dipole magnetic \nfield (compelling it to rotate “up” to satisfy anti-ferromagnetic ordering) is greater than the +x-\ncomponent (forcing it to rotate  “right” to assume ferromagnetic ordering). Therefore, the \nmagnetization rotates to the desired “up” or 2θ = +90°. Note that the energy profiles of \nnanomagnet 2, when undergoing relaxa tion in this final stage, appear to show an equal tendency \nfor the magnetization to rotate to either 0° or  +90°. This occurs due to the preference for \nferromagnetic coupling over anti-f erromagnetic coupling. The +90°  orientation is ultimately \npreferred in this case si nce the dipole magnetic field that would induce a rotation to the “up” \nstate is stronger, albeit slightly, than that  which forces a rotation to the “right”. \nThe clocking scheme describe d above is then repeated on the next set of nanomagnets \n(nanomagnets 3 and 4) starting with nanomagne t 3 being held under tensile stress. Successive \nrepetition of the clocking cycle on successive sets propagates th e input bit unid irectionally down \nthe chain. In this example, we have shown that the clocking cycle can indeed propagate bits \nunidirectionally in one case, whic h corresponds to the first case in Fig. 3. There are eleven more \ncases to consider. We have considered each one of them and found that the same stress cycle \nworks for all of them. \n \n4.  CONCLUSION  \nIn conclusion, we have demonstrated an eff ective clocking scheme that propagates the \nmagnetization state (two logic bits) of a four-state multiferroic nanomagnet unidirectionally \nalong a linear chain by applying a sequence of  stresses pairwise on succeeding nanomagnets. \nThis makes it possible to implement multistate l ogic circuits with wiring connections, fan-out \nand fan-in. This type of logic circuit is attractive not just because  of the higher logic density (4-\nstate versus the usual 2-state), but also because the 4-state elements can be used for associative \nmemory and neuromorphic computing. \nIn this work, we have neglecte d the effect of thermal fluctu ations that can induce switching \nerrors. Those studies will be reported elsewhere. \nIn our past work, we had shown [13, 14, 16] that a tiny voltage of V = 200 mV is sufficient to \ngenerate the maximum stress of  100 MPa in the nick el layer, if we choose the PZT layer \nthickness as 40 nm and the nickel layer thickness as 10 nm. The capacitance C o f  s u c h  a  \nstructure with circular cross-sec tion of 100 nm diameter is ~ 2 fF if we assume that the relative \ndielectric constant of PZT is 1000. Hence, the energy dissipated in a clock cycle to alternate \nbetween no stress to compressive to tensile to no stress is (1/2) CV  + 2  + (1/2)  = 3 \n = 0.24 fJ of energy. Based on previous results  [13, 14, 16], we estimat e that the switching \ndelay will be less than 1 ns. Hence the clock rate can exceed 1 GHz, even when the energy 2 2CV2CV\n2CVdissipation is so small. The energy dissipati on can be reduced even more – down to ~ 10  kT – \nby appropriate choice of material s [13]. This makes this scheme a fast and high density logic \nscheme with extremely low energy  dissipation. That, coupled w ith the fact that nanomagnets \nhave no standby power dissipation unlike transistors, makes it an attractive scheme for \ncomputing and signal processing. 2Fig.  1:  (a) Planar nanomagnets with uniaxial shap e anisotropy are arranged in a line along the \nin-plane hard axis: (i) The array has anti-f erromagnetic ordering in  the ground state where \nnearest neighbors have anti-parallel magnetizatio ns; (ii) all magnetizations are forcefully \nreoriented along the in-plane hard  axis by a global magnetic fiel d whose direction is indicated \nwith the thick arrow; (iii) the global field is  withdrawn and the orientation of the first \nnanomagnet is aligned along a chos en direction along the easy ax is by an extern al agent to \nprovide an input bit to the arra y; (iv) dipole inter action flips the second nanomagnet to assume \nanti-ferromagnetic ordering, an d this effect propagates in  a domino-fashion until all \nmagnetizations orient along the easy axes w ith nearest neighbors having anti-parallel \nmagnetizations. (b) When nanomagnets are arrange d in a line along their easy axes, they couple \nferromagnetically with nearest neigh bors having parallel magnetizations. \n \nFig. 2:   A multiferroic nanomagnet consisting of strain-coupled Ni/PZT layers viewed from the \ntop. The shape is circular so that the nanomagnet has no shape anisotropy, but it has biaxial \nmagneto-crystalline anisotropy in th e nickel layer which is assumed to be a single crystal. This \nanisotropy produces four stable (minimum ener gy) magnetization directions (or “easy axes”): \n“up” (00), “down” (11), “left” (10) and “right” (01), with thei r respective bit assignments shown \nin parentheses. The saddle-shape d curve within the circle represents the energy landscape of the \nnanomagnet in the unstressed ground state. St ress is applied along the +45° [100] axis. \n \nFig.  3:  The twelve distinct scenarios encountered during logic propagation. The “Initial State” \ncolumn shows the the ground state magnetizations  of a four-magnet array or “wire” with \nnanomagnet 1 acting as the input bi t to the array. The “Final St ate” column shows the expected state of the wire when the input nanomagnet is switched from its initial state to any of the three \nother possible states. (a) Anti-ferromagnetic arra ngement with input = “up”, (b) ferromagnetic \narrangement with input = “right ”, (c) anti-ferromagnetic arrange ment with input = “down”, and \n(d) ferromagnetic arrangement with input = “left”. \n \nFig. 4: The clock cycle and stress sequences involved in propagating a logic bit unidirectionally \nare illustrated for the anti-ferroma gnetic case when the input bit is switched from its initial “up” \n(row I) to the “down” state, which results in a tie-condition (row II). To counteract this, a 4-stage \n“clock” cycle is applied to nanomagnets 2 and 3 (rows III – VI) consis ting of tension (T), \ncompression (C) and relaxation (R). The stre ss sequence applied to nanomagnet 2 is T → R  \nC  R, while nanomagnet 3 undergoes a C → C  T → T stress cycle. At the end of a single \nclock cycle (row VI), the magnetizat ion of magnet 2 is rotated from  its initial “down” state (rows \nI and II) to the “up” state. The same clock cycle is then repeated on the next set of nanomagnets \nin the array (3 and 4) to propa gate the logic bit further down. →\n→ →\n \nFig. 5: Energy plots of nanomagnets 2 and 3, as a function of the magnetization angles 2θ and \n3θ. The array is initially in th e anti-ferromagnetic configurati on and the input nanomagnet 1 is \nflipped from “up” to “down”. Nanomagnets 2 a nd 3 are then clocked with the stress cycles \n(TRCR on nanomagnet 2 and CCTT on nanomagnet 3, simultaneously). The ‘*’ markers indicate \nthe magnetization orientation at a particular stre ss while the square indicates the orientation at \nthe end of a clocking stage. The starting magnetization angles are 1θ = -90°, 2θ = -90°, 3θ = \n+90° and 4θ = -90°. (a) Initially, with no stress applied, nanomagnet 2 is at an angle 2θ = -90°. A \ngradually increasing tensile stress is applied on this nanomagnet, which makes its magnetization rotate away from the stress axis toward s the closest energy minimum which is at 2θ = ~ -40°. (b) \nA gradually increasing compressive stress is applied on nanomagnet 3 causing its magnetization \nto rotate to ~ 45°. (c) The tensile  stress on nanomagnet 2 is gradually relaxed to zero, resulting in \nits magnetization settling at ~ 0°  because of the dipole interac tion with nanomagnet 1 (pointing \n“down”) and nanomagnet 3 which is held unde r compressive stress and whose energy landscape \nis shown in (d). (e) Nanomagnet 2 is now subjected to a compressive stress which causes it to \nrotate to ~ +45°, while (f) stress  on nanomagnet 3 is relaxed, maki ng it rotate towards 0°. 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SUPPLEMENTARY INFORMATION  \nAN ENERGY-EFFICIENT BENNETT CLOCKING SCHEME FOR \n4-STATE MULTIFERROIC LOGIC \nNoel D’Souza1, Jayasimha Atulasimha1, Supriyo Bandyopadhyay2 \n1) Department of Mechanical and Nuclear Engineering,  \n2) Department of Electrical and Computer Engineering,  \nVirginia Commonwealth University, Richmond, VA 23284, USA. \nEmail: {dsouzanm, jatulasimha, sbandy}@vcu.edu  \n \nIn the main paper, we proposed a clocking scheme to propagate four-state nanomagnet \nlogic in Ni/PZT multiferroic nanomagnets havi ng biaxial magnetocrystalline anisotropy. This \nclocking scheme involves the application of a unique sequence of Compression (C), Tension (T) \nand Relaxation (R) stresses to the magnetostrictive Ni laye r by applying an appropriate \nelectrostatic potential to the PZT layer. \n Since each nanomagnet has four stable ma gnetization orientations, there are twelve \npossible nanomagnet-array configur ations that can arise when th e input magnet’s orientation is \nchanged from its initial state to any of the three other possible stab le states. The possible states of \nthe magnet-array, based on the ex pectation of ferromagnetic dipol e coupling along the array axis \nand anti-ferromagnetic coupling perpendicular to th e array axis, are shown in the ‘Final state’ \ncolumn of Fig. 3 in the main pa per. Additionally, ther e are four cases corre sponding to the input \nmagnet’s magnetization not being changed. In th ese circumstances, the final magnet states \nshould remain in their original ‘ground’ states upon completion of  the clock cycle. The results \nincluded in the main paper described one particul ar case (Figs. 3(a) – row I), demonstrating the propagation of the magnetization orientati on along a nanomagnet array (anti-ferromagnetic \narrangement) when the input was flipped from “up” to “down”. \nWe now consider the configuration in which the input magnet is changed from “up” to \n“right” (Fig. 3(a), row II – main  paper). The orientations are θ1 = 0°, θ2 = -90°, θ 3 = +90° and θ 4 \n= -90°. At this point, the clock cycle we described in the main paper is applied on magnets 2 \n(TRCR) and 3 (CCTT), whose energy profiles and magnetization rotations are shown in Fig. S1. \nIn the first stage, the tension applied on magnet 2 rotates its magnetization from -90° to ~ -40° \n(Fig. S1(a)). The compression on magnet 3 rotates its magnetization to +45° (Fig. S1(b)). The \nstrong preference for a rotation towards 0° (hig her preference for ferromagnetic coupling than \nanti-ferromagnetic coupling) can be seen at intermediate stress values (~ 50 MPa). Next, the tensile stress on magnet 2 is relaxed while holding magnet 3 under co mpression. The magnetic \ndipole field on magnet 2 due to its interactio ns with magnets 1 and 3 has a larger + x-component \n(contributed by magnets 1 and 3) than a – y-component (due to magnet 3 alone) which results in \nits rotating to 0° (Fig. S1(c)). The next stage of  the clock cycle exerts a compressive stress on \nmagnet 2 which rotates its magnetization to from 0° to ~ +40° (Fig. S1(e)). Simultaneously, the \ncompressive stress on magnet 3 is gradually redu ced to zero, which rotates its magnetization \nfrom +45° to 0°. This is followed by the immediat e application of a tensile stress to rotate the \nmagnetization to -45° (Fig. S1(f)). In the last stage of the clock cycle, the stress on magnet 2 is \nrelaxed to zero (Fig. S1(g)) while holding the te nsile stress on magnet 3 constant (Fig. S1(h)). \nAgain, the +x -direction is strongly favored  and the magnetization of magnet 2 settles into the \ndesired “right” state, reproducing the state of the input bit (magnet 1). By repeating this sequence \nof stresses on the next set of magnets (3 and 4), the logic (magnetization orientation) is \npropagated to magnet 3 and beyond when the clock cy cle is applied to subsequent magnet pairs. The next two sets of results correspond to the initial/ground states of the nanomagnet array \npointing “right” (θ 1 = θ 2 = θ 3 = θ 4 = 0°) in a ferromagnetic-coupled arrangement. \n \nFig. S2 shows the energy profiles of magnets 2 and 3 when subjected to the clock cycle \nfollowing a change in the input magnetization (m agnet 1) from “right” to “down”. The same \nstress cycle (TRCR on magnet 2, CCTT on magnet 3) achieves the desired magnetization rotation. In the first stage, magnet 2 rotates from 0° to ~ -40° as  a tensile stress of up to +100 \nMPa is applied to it (Fig. S2(a)). Simultaneousl y, magnet 3 is gradually compressed to -100 MPa \nand its magnetization rotates from 0° to ~ +40° (Fig. S2(b)). It can be observed that although the \nstresses are applied along the +4 5° direction, the magnetizations of  magnets 2 and 3 do not settle \nat -45° and +45°, respectively. This is because, even at the maximum stress magnitude of 100 \nMPa, the dipole interaction between nearest neighbors (which prefers a ferromagnetic \narrangement) has small but adequate contributi on to the total energy of the nanomagnet that \nbiases the orientation slightly away from the -45°  and +45° states towards the 0° states. A higher \nstress will align the ma gnetization closer to the ±45° axis. When the tensile stress on magnet 2 is \nrelaxed to zero, while keeping magnet 3 compressed, its magnetization settles to ~ 0° (Fig. S2(c)). As explained earlier, this occurs since the x-component of the magnetic dipole field of \nmagnet 2 (favoring a parallel alignment with the + x-axis) has a stronger contribution to the \ndipole energy term than the y-component (favoring an anti-parallel alignment along the + y-axis). \nNext, magnet 2 is compressed whic h rotates its magnetization from ~ 0° to ~ +45° (Fig. S2(e)). \nAt the same time, the stress on magnet 3 is re laxed, which causes its magnetization to rotate \ntowards 0°, to be immediately followed by a tensile stress that rotates it to ~ -40° (Fig. S2(f)). \nFinally, while tensile stress is held on magnet 3 and magnet 2 is relaxed, its magnetization settles to the desired orientation of ~ +9 0° (Fig. S2(g)). It is driven to this “up” state due to the + y-\ncomponent of the dipole magnetic  field contributed by magnets 1 and 3. Magnet 3 also adds a \n+x-component that makes the magnetization of ma gnet 2 want to rotate towards 0°, but its \nmagnitude is smaller than that of the + y-component, which ultimately  results in the rotation \ntowards +90°. \n \nThe next case considered is when the input ma gnetization is switched fr om “right” to “left”. \nPropagation of the input bit th rough the array due to appropriate magnetization rotations is \nillustrated in the energy profiles of magnets 2 a nd 3 in Fig. S3. Prior to initiation of the clock \ncycle, magnet 2 experiences no net dipole interac tion and is in a frustrated state since magnet 1 \nwants it to flip to the “left” while magnet 3 wa nts it to stay pointing to the “right”. Both \ninfluences are equally strong and exactly cancel. This can be seen in the symmetric energy \ncurves of Fig. S3(a) (solid dark blue curve). In contrast, magnet 3 expe riences a strong dipole \nfield towards the “right” (towards 0°). When the cl ock cycle is initiated, a tensile stress is applied \nto magnet 2 that rotates its magnetization from 0° to ~ -45°, while a compressive stress on magnet 3 causes a rotation from 0° to ~ +45°. In  the second stage, relaxing magnet 2 while \nholding the compressive stress on magnet 3 consta nt results in a rotation towards -90° (Fig. \nS3(c)). Next, magnet 2 is compressed to take it s magnetization from ~ -90° to ~ -135° (Fig. \nS3(e)). At the same time, the compressive stre ss on magnet 3 is relaxe d (causing a rotation \ntowards 0°), followed by a tensile stress that ro tates its magnetization to ~ -45° (Fig. S3(f)). \nFinally, magnet 2 is relaxed and its magnetizati on rotates from ~ -135° to ~ 180°, while the \ntensile stress on magnet 3 is held at +100 MPa. Th is is expected since magnet 2 expereinces both \na –x-component and a + y-component of the dipole magnetic fi eld that causes a rotation that settles at ~ 180°. The tension held on magnet 3 in the final stage also serves as a transition to the \nfirst stage of the next clock cycle (tensi on on magnet 3, compression on magnet 4). Repeated \napplication of these stress seque nces in the clock cycle to s ubsequent magnet pairs propagates \nthe magnetization state of th e input magnet along the array. \n \nThe magnetization rotations occurring in the a dditional configurations shown in Fig. 3 (main \npaper) are also studied (Figs. S4 – S11) to confirm proper propagation of logic following \nswitching of the input magnetization. It is also essential that the clock cycle does not cause \nspurious rotations when the input magnetization is not changed. These cases have also been \ninvestigated and are illu strated in this supplement (Figs. S12 – S15), through which we confirm \nthat the magnetizations remain unchanged and in th e initial ‘ground’ states at the end of the clock \ncycle, if the input magnet is unchanged. \n \nFor any arbitrary initial arrangement of the magnetizations, whenever the first magnet is \nswitched,  the clock cycle is init iated on the next two magnets (T ÆRÆCÆR on magnet 2, \nCÆCÆTÆT on magnet 3). This is then repeated on succeeding pairs. By continuing this \nsequence, the input logic bit can be propagated unidirectiona lly down the entire array. Figure Captions \n \nFig. S1 : Energy profiles of magnets 2 and 3  when subjected to the stress sequences for the case \nin which for the magnets are in an anti-ferromag netic configuration with input magnet 1 initially \n“up” and flipped from “up” to “right”. The initial magnetization orientations are: θ1 = 0°, θ2 = -\n90°, θ3 = +90° and θ4 = -90°. (a) Applying a gradually increasing tensile stress on magnet 2 \nrotates its magnetization from -90° to ~ -40°. (b) Simultaneously, a compressive stress on magnet \n3 causes its magnetization to rotate from +90° to ~ +45°. (c) Relaxing the stress on magnet 2 \nresults in its magnetization settling to ~ 0° owing to dipole interactions with magnet 1 ( θ1 = 0°) \nand (d) magnet 3 which is held under compression. (e) Next, an increasing compressive stress is \napplied to magnet 2 making it rotate to ~ 40°, while (f) magnet 3 is re laxed, inducing a rotation \ntowards 0°. This is immediately followed by a tens ile stress that rotates its magnetization to ~ -\n40°. (g) The final stage involves relaxing the stress on magnet 2 that results in its settling at the \ndesired orientation of ~ 0° while (h) the tensile stress is held on magnet 3, setting it up for the \nnext clock cycle which would be  applied to magnets 3 and 4. \n \nFig. S2 : Energy profiles of magnets 2 and 3 when subjected to the stress sequences for the case \nin which the magnets have a ferromagnetic initi al state (pointing “right”) and the input \nmagnetization is switched from “right” to “down”. (a ) A tensile stress is applied to magnet 2 that \ncauses its magnetization to rotate from 0° to ~ -40°. (b) Magnet 3 is co mpressed, resulting in a \nmagnetization rotation to ~ +40°. (c) Relaxing the stress on magnet 2 makes its magnetization \nsettle back to ~ 0° while (d) the compressive stress on magnet 3 is held at  -100 MPa. (e) In the \nthird stage of the clock cycle, a compressive stre ss on magnet 2 results in a rotation to ~ +45°. (f) \nAt the same time, the stress on magnet 3 is relaxed, causing its magnetization to rotate towards 0°. This is immediately followed by a tensile stre ss that swings the magnetization to ~ -40°. (g) \nThe final stage involves gradua lly relaxing the stress on magnet 2 and the desired outcome is \nachieved when the magnetization rotates to ~ +90°, while (h) the tensile stress on magnet 3 is \nheld at +100 MPa. \n \nFig. S3 : Energy profiles of magnets 2 and 3 as a function of magnetization angle for the \nferromagnetic arrangement (initially pointing “ri ght”) when the input magnet is flipped from \n“right” to “left” and the clock cycle is applied. (a) In the first stage, a tensile stress gradually applied on magnet 2 sees its magnetization rotate from 0° to ~ -45°, while (b) a compressive \nstress on magnet 3 causes a rotation to ~ +45°. (c ) Relaxing the stress on magnet 2 results in a \nrotation towards -90°, while (d) the compressive stress on magnet 3 is held at -100 MPa. (e) A \ncompressive stress on magnet 2 rotates its magnetization to ~ -135°. (f) Simultaneously, magnet 3 is relaxed, causing its magnetizat ion to settle near 0°. This is immediately followed by a tensile \nstress that rotates its magnetizat ion to ~ -45°. (g) The final st age of the clock cycle involves \nrelaxing the stress on magnet 2 which results in th e desired final state of ~ -180°, while (h)  \nmagnet 3 is kept under tension that holds its magnetization along -45°.\n \n \nFig. S4:  Energy plots of magnets 2 and 3 when the initial ordering was anti-ferromagnetic and \nthe input nanomagnet was switched from “up”  to “left”. At the end of the clock cycle, magnet 2 \nsettles to ~ +180° or the “left” direction, en suring successful unidirectional propagation of the \ninput bit.  \n \nFig. S5:  Energy plots of magnets 2 and 3 when the initial ordering was ferromagnetic and the \ninput nanomagnet was switched from “right”  to “up”. At the end of the clock cycle, magnet 2 settles to ~ -90° or the “dow n” direction, ensuring successful unidirectional propagation of the \ninput bit.  \nFig. S6:  Energy plots of magnets 2 and 3 when the initial ordering was anti-ferromagnetic and \nthe input nanomagnet was switched from “down”  to “up”. At the end of the clock cycle, magnet \n2 settles to ~ -90° or the “down” direction, en suring successful unidirec tional propagation of the \ninput bit.  \n \nFig. S7:  Energy plots of magnets 2 and 3 when the initial ordering was anti-ferromagnetic and \nthe input nanomagnet was switched from “down”  to “right”. At the e nd of the clock cycle, \nmagnet 2 settles to ~ 0° or the “right” direc tion, ensuring successful unidirectional propagation \nof the input bit. \n \nFig. S8:  Energy plots of magnets 2 and 3 when the initial ordering was anti-ferromagnetic and \nthe input nanomagnet was switched from “down”  to “left”. At the end of the clock cycle, \nmagnet 2 settles to ~ +180° or the “left” directi on, ensuring successful unidirectional propagation \nof the input bit. \n \nFig. S9:  Energy plots of magnets 2 and 3 when the initial ordering was ferromagnetic and the \ninput nanomagnet was switched from “left”  to “r ight”. At the end of the clock cycle, magnet 2 \nsettles to ~0° or the \"r ight” direction, ensuring successful un idirectional propagation of the input \nbit. \n Fig. S10:  Energy plots of magnets 2 and 3 when the initial ordering was ferromagnetic and the \ninput nanomagnet was switched from “left”  to “up”. At the end of the clock cycle, magnet 2 \nsettles to ~ -90° or the “dow n” direction, ensuring successful unidirectional propagation of the \ninput bit. \n \nFig. S11:  Energy plots of magnets 2 and 3 when the initial ordering was ferromagnetic and the \ninput nanomagnet was switched from “left”  to “ down”. At the end of the clock cycle, magnet 2 \nsettles to ~ +90° or the “up” direction, ensu ring successful unidirect ional propagation of the \ninput bit. \n \nFig. S12:  Energy plots of magnets 2 and 3 when the initial ordering was anti-ferromagnetic and \nthe input nanomagnet was not switched (remains pointing “up”). At the end of the clock cycle, \nmagnet 2 settles to ~ -90° or the “down” direction, ensuring su ccessful unidirectional \npropagation of the input bit. \n \nFig. S13:  Energy plots of magnets 2 and 3 when the initial ordering was ferromagnetic and the \ninput nanomagnet was not switched and remains poin ting “right\". At the end of the clock cycle, \nmagnet 2 settles to ~ 0° or the “right” direc tion, ensuring successful unidirectional propagation \nof the input bit. \n \nFig. S14:  Energy plots of magnets 2 and 3 when the initial ordering was anti-ferromagnetic and \nthe input nanomagnet was not sw itched and remains pointing “dow n\". At the end of the clock cycle, magnet 2 settles to ~ +90° or the “up”  direction, ensuring su ccessful unidirectional \npropagation of the input bit. \n \nFig. S15:  Energy plots of magnets 2 and 3 when the initial ordering was ferromagnetic and the \ninput nanomagnet was not switched and remains poin ting “left\". At the end of the clock cycle, \nmagnet 2 settles to ~ +180° or the “left” directi on, ensuring successful unidirectional propagation \nof the input bit.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \na) b) \nc) d) \ne) f) \nh) g) \nFig. S1  \n \n \n \n \n \n   \n \n \n \n \n \n \n \n \n \n \n \n   \n \na) b)\nc) d) \ne) f) \nh) g) \nFig. S2  \na) b) \n \n \n \n \n \n \nc) d) \n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h) \nFig. S3  \na) b) \n \n \n \n \n c) d)\n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h) \n \n \n \n \nFig. S4  \na) b) \n \n \n \n \n \n \nc) d) \n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)  \n \n \n \n Fig. S5  \na) b) \n \n \n \n \n c) d) \n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\n \n \n \n \n \nFig. S6  \na) b) \n \n \n \n \n \n c) d) \n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h) \n \n \n \n \n Fig. S7  \na) b) \n \n \n \n \n c) d) \n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\n \n \n \n \n Fig. S8  \na) b) \n \n \n \n \n c) d)\n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\n \n \n \n \n Fig.S9 \na) b) \n \n \n \n \n c) d)\n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\n \n \n \n \n Fig.S10 \na) b) \n \n \n \n \n c) d)\n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\n \n \n \n \n  Fig.S11 \na) b) \n \n \n \n \n c) d)\n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\n \n \n \n \n Fig.S12 \na) b)\n \n \n \n \n \n c) d)\n \n \n  \n \n \ne) f) \n \n \n \n \n \ng) h)\nFig.S13 \na) b) \n \n \n \n \n c) d)\n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\nFig.S14 \na) b) \n \n \n \n \n \nc) d) \n \n \n \n \n \ne) f)  \n \n \n \n \n \ng) h)\nFig. S15 "    },    {        "title": "1106.1917v1.Quantitative_model_for_anisotropy_and_reorientation_thickness_of_the_magnetic_moment_in_thin_epitaxially_strained_metal_films.pdf",        "content": "arXiv:1106.1917v1  [cond-mat.str-el]  9 Jun 2011arxiv\nQuantitative model for anisotropy and reorientation thick ness of the magnetic\nmoment in thin epitaxial strained metal films\nArtur Braun∗\nEnvironmental Energy Technologies\nErnest Orlando Lawrence Berkeley National Laboratory\nOne Cyclotron Road, MS 70-108 B\nBerkeley, CA 94720, USA†\n(Dated: November 23, 2018)\nA quantitative mathematical model for the critical thickne ss of strained epitaxial metal films is\npresented, at which the magnetic moment experiences a reori entation from in-plane toperpendicular\nmagnetic anisotropy. The model is based on the minimum of the magnetic anisotropy energy with\nrespect to the orientation of the magnetic moment of the film. Magnetic anisotropy energies are\ntaken as the sum of shape anisotropy, magnetocrystalline an isotropy and magnetoelastic anisotropy,\nthe two latter ones being present as constant surface and var iable volume contributions. Apart from\nanisotropy materials constants, readily available from li terature, only information about the strain\nin the films for the determination of the magnetoelastic anis otropy energy is required. Application\nof continuum elasticity theory allows to express the strain in the film in terms of substrate lattice\nconstant and film lattice parameter, and thus to obtain an app roximative closed expression for the\nreorientation thickness in terms of lattice mismatch. The m odel is successfully applied to predict\nthe critical thickness with surprising accuracy.\nPACS numbers: 75.70.-i, 75.30.Gw, 75.40.Mg\nI. INTRODUCTION\nThe efficiency of magnetic data storage media depends\ncritically on the presence of a perpendicular magnetiza-\ntion, this is, the easy magnetic axis is perpendicular to\nthe surface of the magnet in the absence of an external\nmagnetic field. Generally, the magnetization lies in some\npreferred direction with respect to the crystalline axes or\nto the external shape of the magnet - this is known as\nmagnetic anisotropy. The energy involvedin rotating the\nmagnetization from a direction of low energy (easy axis)\ntowards one of high energy (hard axis) is typically of the\norder10−6to10−3eV/atomandthusaverysmallcorrec-\ntion to the total magnetic energy [1]. The physical origin\nfor the anisotropy is the symmetry breaking of the rota-\ntional invariance of the dipole-dipole interaction and the\nspin-orbit coupling with respect to the spin quantization\naxis. Atoms at interfaces in ultrathin films or multilay-\ners show a much stronger anisotropy than atoms in the\nbulk. Also, atoms in strained lattices show a stronger\nanisotropy.\nWhile it has been an established fact for many years\nthat epitaxial strain in ultrathin metal films can have\nsignificant impact on the stabilization of perpendicular\n∗present address:Empa\nSwiss Federal Laboratories for Materials Science and Techn ology\nLaboratory for High Performance Ceramics\nUeberlandstrasse 129\nCH-8600 Duebendorf\nSwitzerland; Electronic address: artur.braun@alumni.ethz.ch\n†URL:http://http://www.empa.ch/plugin/template/empa/613/\n*/---/l=1magnetization [2–5], many relevant publications still ig-\nnore magnetoelasticity for the interpretation of magnetic\nanisotropy data.\nThe relationship between the magnetic anisotropy and\nelastic strain is also important in structural geology.\nElastic strain is a marker that permits to draw conclu-\nsions about deformation history of rocks. Since such\nstrain data are not always directly available, quantita-\ntive modeling helps to retrieve strain data from magnetic\nanisotropydata[6]. Reviews aboutthe interplaybetween\nmechanical stress and magnetic anisotropy in thin films\nand in tectonics are given by Sander [7], and by Bor-\nradaile and Henry [8], respectively.\nIn the next section, the specific contributions to mag-\nnetic anisotropy energy will be given and summarized.\nThe magnetoelastic energy will be expressed in terms of\nlattice spacings in film and substrate in order to account\nfor the strain in films. The derivative of the total mag-\nnetic anisotropy with respect to the film crystal axes,\ni.e. the minimum of this energy will yield a thickness at\nwhich the magnetization switches from in-plane to out-\nof-plane. The terms out-of-plane andperpendicular are\nused equivocally in this manuscript. The Bain path is a\nfunctionalrelationshipbetweenin-planeandout-of-plane\nlattice distances of a bulk metal phase and its corre-\nsponding tetragonally distorted equilibrium phases un-\nder epitaxial stress. It can be regarded as a sequence\nof tetragonal states produced by epitaxial stress on an\nequilibrium tetragonal phase [9–11] and serves as a crite-\nrion for the identification of equilibria phases of strained\nstructures and for the interpretation of LEED data [12].\nThe concept is being improved still to date and allows,\nfor instance, deeper insight in phase transitions during\nepitaxial growth [11], or just the conversion of film thick-2\nness data from Monolayers to Angstrom [13]. For the\npresent model, the Bain path plays a key role. To the\nbest of the authors knowledge, the present model is the\nfirst of its kind which gives a quantitative prediction of\nthe critical reorientation thickness.\nII. FORMULATION OF THE MODEL\nA. Magnetocrystalline anisotropy energy\nThe magnetocrystalline anisotropy energy GV\ncryst(ΩM)\nis written as expansion of the cosine of directions α1,α2,\nandα3of ΩM:\nGV\ncryst(ΩM) =b0(HM)+/summationdisplay\ni,jbi,j(HM)αiαj\n+/summationdisplay\ni,j,k.lbi,j,k,l(HM)αiαjαkαl+.... (1)\nΩMis the unit vector of magnetization direction with\ncomponents α1,α2,α3or the polar angles θandφ. For\ncrystals with cubic symmetry, the expression in Eq. (1)\nsimplifies to\nGV\ncryst(ΩM) =K0+K1(α2\n1α2\n2+α2\n2α2\n3+α2\n3α2\n1)\n+K2α2\n1α2\n2α2\n3+K3(α2\n1α2\n2+α2\n2α2\n3+α2\n3α2\n1)+....(2)\nwith the coordinate system being in line with the crystal\naxes, and: α1=sinθcosφ,α2=sinθsinφ, andα3=cosθ.\nThe vector of magnetization, M, includes with the sur-\nface normal of the (001) plane the angle θand with the\n[010]-direction the angle φ. The expression in Eq. (2)\nconverges well and thus permits to interrupt the expan-\nsion after the third order. K 0, K1etc. are the magne-\ntocrystalline anisotropy constants. Note, that these con-\nstantsare temperature dependent materials parameters.\nBut here we will treat the case for ambient temperature\nonly.\nEq. (2) describes the magnetocrystalline anisotropy\nenergy or a single atom of one monolayer in the volume\nof the film. This contribution grows proportional with\nthe film thickness. Atoms at surfaces and interfaces are\nnot described properly by Eq. (2) anymore. Because\nof the reduction of symmetry at surfaces and interfaces,\nsecond order terms come into play. For a surface with\ntetragonal symmetry, such as the cubic (001) plane, the\nsurface magnetocrystalline of a single atom is\nGS\ncryst(ΩM) =KS\n1sin2θ\n+ (KS\n2+KS′\n2cos(4φ))sin4θ+.... (3)\nB. Shape anisotropy energy\nThe shape anisotropy energy\nGshape(ΩM) =−1\n2/integraldisplay\nVdVM(r)·Hd(r),(4)with demagnetization field Hd(r) and magnetization\nM(r), depends on the external shape and geometry of\nthe magnet. Ultrathin films can be well approximated\nby a plate with infinite lateral extension, such as:\nGV\nshape(θ) =KV\nshapesin2θ . (5)\nEq. (5) describes the shape anisotropy energy of a sin-\ngle atom in a monolayer of such a film. Considering in-\nfinitesimally thin slices yield generally the same angular\ndependence:\nGS\nshape(θ) =KS\nshapesin2θ . (6)\nKV\nshapeand KS\nshapeare the corresponding anisotropy en-\nergy constants.\nC. Magnetoelastic anisotropy energy\nFor the strained lattice of a magnetized body, the en-\nergytermsmaydepend onthe straintensor ǫandonΩ M;\nthis is the magneto-elastic energy. For small lattice mis-\nmatches, the energy terms can be expanded in spherical\nharmonics and in powers of ǫ:\nGmagn.el(ΩM,ǫ) =/summationdisplay\ni,j,k,lBi,j,k,lǫi,jαiαj+....(7)\nThe crystalline symmetry manifests itself in a coupling\nof the expansion coefficients. For a cubic system, the\nexpression reads\nGmagn.el(ΩM,ǫ) =\nB1(ǫ11α2\n1+ǫ22α2\n2+ǫ33α2\n3)\n+ 2B2(ǫ12α1α2+ǫ23α2α3+ǫ31α3α1)+....(8)\nTheǫiidescribe the strain and the ǫijdescribe the shear\ndeformation.\nWe now want to derive an expression for the magneto-\nelastic anisotropy energy in terms of lattice deformation.\nLet us consider here a face centered cubic lattice. The\nnearest neighbor (NN) distance of the atoms in a cubic\nlattice with no strain be a0. For the lattice under strain,\nthe NN distance in the tetragonal plane be a. For pseu-\ndomorphic growth, the latter value is the NN distance of\nthe atoms in the substrate. The expansion is equal to\nthe lattice mismatch:\nǫ11=ǫ22=f=a−a0\na0. (9)\nFor the contraction, we find\nǫ33=c−a0√\n2\na0√\n2=c\na0√\n2−1. (10)\nThe variation of cwithais given by the Bain path,\nthis is [9]3\nc\na0=/parenleftBiga0\na/parenrightBigγ\n,c0=a0√\n2 (11)\nso that we may write\nǫ33=/parenleftBiga0\na/parenrightBigγ\n−1,γ= 2c12\nc11. (12)\nHere,c11andc12are the elastic constants for the corre-\nsponding film material. Some elastic constants and lat-\ntice parameters for Fe, Co, and Ni are listed in Table\nI.\nElement T(K) c 11[GPa] c 12[GPa] a [ ˚A]\nbcc Fe 520 230 135 2.87\nfcc Fe 1428 154 122 3.65\nfcc Co 300 242 160 3.54\nhcp Co 300 307 165 2.82\nfcc Ni 300 247 153 3.524\nbcc Ni 300 - - 2.78\nTableI:Elasticconstantsandlatticeparametersforsome\nFe, Co, and Ni phases [14, 15], hcp Co after [16].We neglect shear effects in the distorted films and in-\nsert the expressions from Eqs. (9)-(11) in Eq. (8) and\nobtain in first order approximation\nGmagn.el(a,a0,θ) =\nB1/braceleftbigg/parenleftbigga\na0−1/parenrightbigg\nsin2θ+/parenleftBig/parenleftBiga0\na/parenrightBigγ\n−1/parenrightBig\ncos2θ/bracerightbigg\n.(13)\nD. Summary of contributions\nAll contributions considered so far contain bulk- and\nsurface/interfacecontributions, except for the magnetoe-\nlasticanisotropyenergy,forwhichnosurface-orinterface\ncontributions were available from literature. For a film\nwithdmonolayer thickness, the anisotropy energy per\nunit surface is thus\nG=d·/parenleftbig\nGV\ncryst+GV\nshape+GV\nmagn.el/parenrightbig\n+GS\ncryst+GS\nshape\n(14)\nWith the exception of GV\ncryst, we insert the expressions\nfor the particular contributions in Eq. (14) and obtain\nG(a,a0,θ) =/parenleftbig\nKS\n1+KS\nshape/parenrightbig\nsin2θ+d·/parenleftbigg\nKV\nshapesin2θ+B1/braceleftbigg/parenleftbigga\na0−1/parenrightbigg\nsin2θ+/parenleftBig/parenleftBiga0\na/parenrightBigγ\n−1/parenrightBig\ncos2θ/bracerightbigg/parenrightbigg\n(15)\nOmission of GV\ncrystis justified for ultrathin films, because\nthe error inferred by this contribution is negligible. With\nincreasingthickness, however,thiscontributionwilldom-\ninate together with the shape anisotropy, also, because\nmagnetoelastic contributions will become smaller due to\nrelief of strain in thikcer films.\nWe can replace cos2θby (1−sin2θ) in Eq. (15), so\nthatG(a,a0,θ) becomes a function linear in sin2θ. It is\nofinterestnowtoknowtheconditionsforwhichtheangle\nθbecomes 0. Then, the magnetic anisotropy energy has\na minimum, and this state corresponds to perpendicular\nmagnetization. To determine, for which particularangles\nθthe magnetic anisotropy energy becomes a minimum,\nwe calculate the first derivative of G(a,a0,θ) in Eq. (15)\nwith respect to θ. The derivative vanishes for θ=0◦and\n90◦. For these cases we obtain an explicit expression for\nthe critical thickness, at which we have a reorientation of\nthe magnetization:\ndcrit=−/parenleftBig\nKS\nshape+KS\ncryst/parenrightBig\nKV\nshape+B1/parenleftBig\na\na0−a\na0−2c12\nc11/parenrightBig(16)\nThe expression in Eq. (16) thus allows the prediction\nof the critical magnetic reorientation thickness, withinthe limits and approximations made in the model. More\ngenerally, Eq. (15) allows to determine the direction of\nmagnetization in a thin film for any given set of param-\neters.\nIII. RESULTS AND DISCUSSION\nThe model is applied to thin films of Fe, Co, and Ni,\nand compared with experimental data taken from liter-\nature. Structure and magnetism of these three metals\nhave been subject of intense research for decades, c.f.\n[7, 17] and a whole body of experimental data is readily\navailable in literature.\nWe begin with Figure 1, which shows a schematic dia-\ngram of the critical reorientation thickness dcrit(Eq. 16)\nas a function of a/aofor four different sets and signs of\nKSandB. Phaserangesforperpendicularmagnetization\nare marked with ⊥; ranges of in-plane magnetization are\nmarked with /bardbl. The modulus for the anisotropy energies\nwas chosen arbitrarily to KS\ncryst= 4.0 x 10−4,B= 0.20\nx 10−4, andKS\nshape= 8.0 x 10−4eV/atom. The ratio\na/aois a measure for the strain in the film. For a/ao<\n1, the film in-plane lattice parameter is compressed with\nrespect to the substrate. For a/ao>1, it is expanded.4\nThe regions of in-plane and out-of-plane magnetization\nare displayed depending on the lattice mismatch and the\nsign of the anisotropy constants. GV\ncrystwas not taken\ninto account here.\nFor these systems, the magnetocrystalline anisotropy\nenergy is dominant in terms of modulus. The posi-\ntion a/a o=1 denotes the relaxed state of the film and\nis marked by a dotted vertical line. In addition, a solid\nvertical line marks the specific ratio a/a o, beyond which\na polar magnetizationis not possible anymore, regardless\nthe film thickness. This is because in Eq. (16), the shape\nanisotropy and magnetoelastic anisotropy energy cancel\nout in the denominator.\nP. Bruno [1] has compiled a number of anisotropy con-\nstants, displayed in Table II and III, which are used in\nthe present work.\nbcc Fe hcp Co fcc Ni\nK14.02 10−6(a) 5.33 10−5(b) -8.63 10−6(a)\nK21.44 10−8(a) 7.31 10−6(b) 3.95 10−6(a)\nK36.60 10−9(a) - 2.38 10−7(a)\nK3’ - 8.40 10−7(c) 6.90 10−7(c)\nKV\nshape-8.86 10−4(c) -5.85 10−4(c) -0.74 10−4(c)\nB1-2.53 10−4-5.63 10−46.05 10−4\nB25.56 10−4-2.02 10−36.97 10−4\nB3 - -1.96 10−3-\nB4 - -2.05 10−3-\nTable II: Volume contributions of anisotropy energy\nconstants in [eV/atom]. Data for magnetocrystalline\nanisotropy energy are valid for T=4.2 K. (a) [18], (b)\n[19], (c) [20], (e) [1]. Magnetoelastic constants were ob-\ntained from [1] and [21–24] and are valid for 297 K.\nThe magnetocrystalline anisotropy energy is of the or-\nder 10−6eV/atom and is always dominated by the shape\nanisotropy energy by one (Ni) or two (Fe) order of mag-\nnitudes. They have alwaysa negative sign and thus favor\nmagnetization in the film plane. If we consider a Ni film\nwith a lattice mismatch of f=1%, the negative magne-\ntoelastic anisotropyenergy of Ni will overcompensatethe\nshape anisotropy energy by a factor of ten. The surface\nand interface contributions to the anisotropy energy are\nconstants that add up in the overall anisotropy energy\nof the thin film. Table III gives an overview of the sur-\nface/interface anisotropy constants of a number of sys-\ntems studied in the past, mostly one single crystalline\nsubstrates.System K S[mJ/m2] Reference\nFe(001)/UHV 0.96 [25]\nFe(001)/Au 0.47, 0.40, 0.54 [25]\nFe(001)/Cu 0.62 [25]\nCo/Au(111) 0.42 [26]\nCo/Cu(111) 0.53 [4, 27]\nCo/Ni(111) 0.31, 0.20, 0.22 [28–30]\nNi(111)/UHV -0.48 [31]\nNi/Au(111) -0.15 [32]\nNi(111)/Cu -0.22, -0.3, -0.12 [32–34]\nNi/Cu(001) -0.23 [34]\nSystem KS\nShape[erg/cm2] Reference\nFe(001)/UHV -0.27 [35]\nFe/Ag(001) -0.12 [35]\nNi(001)/UHV -0.017 [36]\nNi/Cu(001) -0.025 [36]\nTableIII: Surface contributionsofanisotropyenergycon-\nstants [1]. Proper conversion yields the order of approx-\nimately 10−4eV/atom.\nUltrathin films often exhibit perpendicular magnetiza-\ntion up to several monolayers, which is stabilized by the\npositive surface and interface anisotropy energy. For Fe\nandCo, elasticstrainworksagainstthis effect, and there-\nforeperpendicularmagnetizationisnot favoredanymore.\nThis is not the case for Ni. Here, we always register a\nnegativemagnetocrystallinesurface/interfaceanisotropy,\nand a perpendicular magnetization is favored. Elastic\nstrain, such as in the case of epitaxial growth, causes\na positive contribution of the magnetoelastic anisotropy\nenergy and thus enhances this effect. Since the magne-\ntoelastic anisotropy energy is a volume contribution, this\nis, it increases to the same extent as the volume or the\nthickness of the film, thick films of arbitrary thickness\nwith perpendicular magnetization can be grown- at least\ntheoretically. The practical limitation is that thick films\nwill not maintain the necessary lattice mismatch, i.e. the\nelastic strain, and relief most of the elastic energy by\ndislocations [37, 38].\nInFig. 2,weshowthevariationofthecriticalthickness\n(curved lines) for Fe, Co, and Ni as a function of a/a o.\nSolid lines indicate dcritafter Eq. (16), while dashed\nlines take also the volume contribution of the magne-\ntocrystalline anisotropy into account. The vertical solid\nand dashed lines indicate the ranges where the denomi-\nnator in Eq. (16) vanishes and beyond which, according\nto the model, no perpendicularmagnetization ispossible.\nFirst, we note that thin Fe films have a perpendicular\nmagnetization ( ⊥). Thick Fe films have the magnetiza-\ntion vector in the film plane ( /bardbl). This is the situation as\nrepresented by Figure 1, plot c). The critical thickness\nfor the reversal of the magnetization vector is increas-\ning with decreasing strain in the film. According to our\nmodel, particularly an ultrathin Fe film has a perpendic-\nular magnetization. The dotted line was obtained after\nEq. (16). The volume contribution of the magnetocrys-\ntallineanisotropyenergywasnottakenintoaccount. But\nwe can estimate the influence of this contribution on the5\n05101520 dcrit. [ML]KS\ncryst < 0\nB1 < 0\nKS\nshape < 0a)KS\ncryst < 0\nB1 > 0\nKS\nshape < 0b)\n05101520\n0.8 0.9 1 1.1 1.2dcrit. [ML]\na/a0KS\ncryst > 0\nB1 < 0\nKS\nshape < 0c)\n0.8 0.9 1 1.1 1.2\na/a0KS\ncryst > 0\nB1 > 0\nKS\nshape < 0d)\nFIG. 1: Schematic diagram of the critical reorientation thi ckness. Each sub-figure represents a specific case with combi nations\nof sign of magnetoelastic anisotropy constant B1and surface/interface anisotropy constant KS\ncryst. Shape anisotropy is taken\nnegative for all cases.\ncritical film thickness. For a 10 ML thick film, the con-\ntribution does not exceed 4.02 10−5eV/atom. The solid\nline in Fig. 2 takes this contribution into account and is\nthus an upper approximation for the critical film thick-\nness.\nFigure2, plot a), contains four experimentally ob-\ntained values for the critical reorientation thickness of\nbcc-Fe films, i.e. Fe/Cu(001) [39], Fe/Cu 3Au(001) [40],\nFe/Ag(001) [41], and Fe/Pd(001) [12]. All of them are\nlocated below the curves for the calculated critical thick-\nness. For the system Fe/Cu(001) and Fe/Cu 3Au(001) itis believed that the magnetic properties depend on the\npreparation conditions of the films, which could not be\ntaken into account in the model. In addition, for the\nlatter case, interdiffusion of Au atoms in the Ni film is\nbelieved to have changed the magnetic properties of the\nNi film [12]. It also should be mentioned that the tem-\nperature dependence of the anisotropy constants was not\ntaken into account in the present model. But it is strik-\ning that the experimentally obtained critical thickness\nincreases with decreasing film strain, as predicted by our\nmodel, i.e. experimental data and calculations show the6\n05101520\n2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3dcrit. [ML]\nafilm [Å]aNN\nCu(001)aNN\nCu3Au(001)\naNN\nbcc Fe aNN Pd\naNN Aga)\n1234567\n3.5 3.55 3.6 3.65 3.7 3.75dcrit. [ML]\nafilm [Å]Co/Cu(001)\nhcp Co/Au(111)b)\n010203040\n3.5 3.55 3.6 3.65 3.7 3.75dcrit. [ML]\nafilm [Å]fcc aNi\nNi/Cu(001)\nNi/Cu3Au(001)fcc aCu3Auc)\nFIG. 2: Critical thickness for Fe (plot a), Co (plot b), and Ni (plot c).7\nsametrend. Onethoroughconfirmationoftheclearsepa-\nration between surface and volume contributions is given\nby Thomas et al. [42]. They found that the uniaxial\nmagnetic anisotropy of very thin Fe/GaAs(001) is not a\nresult of magnetoelastic coupling nor a result of shape\nanisotropy, but a result of interface anisotropy. On the\nother hand, they find that the further evolution of mag-\nnetic anisotropy with increasing Fe film thickness is a re-\nsult of the competition between magnetoelastic coupling\nand interface anisotropy. It is worthwhile to mention\nthat their work was based on a thorough X-ray structure\nanalysis of the films.\nIn the middle plot in Fig. 2, plot b), the critical thick-\nnesses for Co films with and without correction for the\nvolumecontributionofthemagnetocrystallineanisotropy\nare given. Ultrathin Co films have an easy magnetic axis\nperpendiculartothefilmplane,andthickplanesaremag-\nnetized in the film plane. This is a situation similar as\nwith Fe films, as represented by Figure 1, plot c). The\nexperimental data are taken from Co/Au(111) [4] and\nCo/Cu(001) [43]. In the former case, elastic strain in\nthe film is anticipated, but not quantified. The lattice\nmismatch for this system would be theoretically about\n14%. But it is unlikely that the film would maintain\nsuch a large mismatch; pseudomorphic growth is thus\nruled out for the face centered phase of Co. Therefore,\nthe data point (3.1 ML) was set to the natural lattice pa-\nrameter of Co (3.544 ˚A). The lattice mismatch of fcc-Co\non Cu(001) is approximately 2%, and the critical thick-\nness was found to be 3.4 ML. Thus, both data points\nare in the permissible region between the two extreme\ncases of the model. Chen et. al. [44] observe in-plane\nanisotropy for Co/Pt(111) films larger than 3.5 ML, and\nperpendicular anisotropy for thinner Co films. Capping\nthese Co/Pt(111) films with thick enough Ag layershow-\never increased the reorientation thickness to as far as 7\nML, depending on the thickness of the Ag capping layer.\nLee et al. [45] found for the systems Co/Pt(111) and\nCo/Pd(111) reorientation thickness from perpendicular\nto in-plane of 5 ML and 12 ML, respectively. Sandwiches\nof Pd(111)/Co/Pd were shown to have perpendicular\nmagnetization up to 13 - 15 ML [46], which translates\nto approximately 5 ML. For Co/Cu(111), a critical reori-\nentation thickness of 5.5 ML is reported [45]. Kohlhepp\net. al. explain the strong perpendicular anisotropy of\nCo films on Pd(111) substrates, capped with either Fe or\nCu, with the alloying of Co and Pd at the Co/Pd(111)\ninterface [47].\nHowever, in this thickness range, a gradual phase\ntransformation from the fcc phase to a hcp phase takes\nplace[17, 45]. It isalsopossible thata bcccobaltphase is\nsynthesized [48] in a particular thickness range, which is\nnot necessarily stable, however [49]. Cobalt was also one\nof the first systems where magnetic domain formation\nwas found and related with magnetic anisotropy [50–53].\nThecriticalthicknessofNi isshownin thebottom part\nofFig. 2. UltrathinNifilmsarein-planemagnetized, and\nthick films are out-of-plane magnetized. This situationis represented in Fig. 1 by plot b). So far we have not\nyet addressed the differences in the interface anisotropy\nenergy, which ariseswhen different chemical elements are\nused as substrates, or even when the surface of a film is\ncapped with a different element,\nTable III lists a number of surface and interface\nanisotropy constants for various systems and chemical\nspecies. Since these anisotropy constants are specifically\ngiven for the coordinationnumbers and packing densities\nof the constituents of the interface, including their chem-\nical species, they cannot simply be adopted for systems\nwith different coordination numbers etc.\nThe dashed and solid lines in the bottom plot in\nFig. 2 represent actually four lines for the critical thick-\nness, each with surface anisotropy constants for either\nNi/Cu 3Au(001) and Ni/Cu(001). The differences of\nthese curves are so minute, however, that the curves can-\nnot be distinguished anymore. This holds even for the\ncurves which take the magnetocrystalline anisotropyinto\naccount. The reported critical thickness for Ni/Cu(001)\nis 12 ML [54], in good agreement with our model.\nThere are no anisotropy constants data available for\nthe system Ni/Cu 3Au(001), but we can try to find a rep-\nresentative average for this system, built from the linear\ncombination of data for Ni(111)/UHV, Ni/Au(111) and\nNi/Cu(001), which are readily available (Table III). We\nassume that the Cu 3Au substrate surface is occupied to\n3/4 of Cu atoms and 1/4 of Au atoms. For simplicity, we\nignoreherethat the Cu 3Au substratemayhaveaAu-rich\nsegregation profile on its surface [55], and that alloying\nmay occur between Ni and Cu or Au.\nThe Cu(001) surface has two atoms per unit mesh and\na lattice parameter of 3.61 ˚A. The number density on\nthe surface is thus 1.535 1019atoms/cm2. The inter-\nface anisotropy energy constant for Ni/Cu(001) is thus\n-0.935 10−4eV/atom. Taking the 3/4 occupation prob-\nability of Cu on Cu 3Au(001) into account, we obtain -\n0.701 10−4eV/atom.\nNickel does not grow pseudomorphic on Au(111) due\nto the large lattice mismatch of 15%. Instead, we as-\nsume that Ni grows on Au(111) in (001) orientation\nwith its natural lattice parameter of 3.5238 ˚A. The\nsame procedure yields -0.581 10−4eV/atom for the\ninterface anisotropy energy of Ni/Au(001), or -0.145\n10−4eV/atom with 1/4 occupation of Au.\nFor Ni/Cu 3Au(001) we thus obtain an occupation\nweighted value of -0.846 10−4eV/atom for the interface\nanisotropy energy constant.\nBased on the same considerations, we yield -1.6107\n10−4eV/atom for the surface anisotropy of the Ni film\nvs. the UHV.\nThus, the surface anisotropy and interface anisotropy\nof Ni/Cu 3Au(001) amount together to -2.4572\n10−4eV/atom.\nEq. (16) neglects the magnetocrystalline anisotropy\nenergy, which has a dependence of the azimuthal angle φ\nand thus makes mathematical expressions less transpar-\nent then intended here.8\nThe model does not include contributions to the mag-\nnetic anisotropy energy that arise from inhomogeneities\nsuch as magnetic domain walls or steps and kinks on sur-\nfaces, or surface roughness. For thicker films, Eq. (16)\ntherefore becomes less accurate. However, Eqs. (13)-\n(16) are quite simple and yet elegant expressions which\nmake our anisotropy model transparent enough for the-\noretical and also practical considerations. It is antici-\npated that these expressions can be implemented with\nreasonable effort in other formulae, or that other formu-\nlae can be implemented in our anisotropy model. For\ninstance, the magnetoelastic anisotropy contribution is\nbased on the assumption, that the lattice parameters of\nthe film material remain unchanged during film growth.\nThis assumption does not really hold, because of disloca-\ntion formation in the film. Implementation of the force\nbalance between film stress and the tension of disloca-\ntions [37, 38] should improve the present model. Crys-\ntal parameters and anisotropy constants are parameters\nthat depend on the temperature. Crystallographic phase\ntransitions may take place during temperature changes,\nin particular when ultrathin films are concerned. Dif-\nfusion and interdiffusion processes may be activated at\nslightly elevated temperatures and thus facilitate alloy-\ning of film atoms with substrate atoms, which has in-\nfluence on the bandstructure of the film. Finally, the\nmagnetic anisotropy constants are throughout tempera-\nture dependent and may even undergo changes of their\nsign, which then manifests in the occurrence of so called\nHopkinson maxima in magnetic susceptibility measure-\nments [57]. While it is difficult to account for all these in-\nstances and circumstances, it should be appreciated that\nthe Bain path at least partially admits to quantitatively\nmodel the magnetic anisotropy behaviour of ultrathin or\nvery thin films on single crystalline substrates.\nIV. CONCLUSIONS\nA quantitative model for the dependence of the mag-\nnetic anisotropy of thin films was established. It is basedon anisotropy constants, lattice strain, and film thick-\nness and takes volume and surface contributions into ac-\ncount. Information about the lattice strain is directly\nimplemented in the model as a function of the lattice\nmismatch via the epitaxial Bain path. At the present\nstage, the model does not take into account contribu-\ntions that arise from magnetic domain walls, steps, or\nsurface and interface roughness. Also, effects of tem-\nperature variation (phase transformations, temperature\ndependent change of sign of anisotropy constants, etc.)\nare not accounted for yet. But the open design of the\npresent model generally allows implementation of these\ncontributions and further refinement, on the cost of con-\nceptual transparency, however. Minimization ofthe total\nmagneticanisotropyenergyofthefilmyieldsaclosedand\nsimple expression for the critical reorientation thickness\nof the magnetic moment of the film. Experimental data\nof Fe, Co and Ni films are in fair agreement with our\nmodel. The significance of magnetoelastic effects on the\nmagnetic anisotropy of thin films is quantitatively veri-\nfied. To account for the influence of magnetoelasticity,\nassessment of elastic strain in the films is indispensable.\nA complete structural characterization of the film is thus\ndesirable, which permits to assign the correct crystallo-\ngraphic phase as well as information about elastic strain.\nAcknowledgments\nThis work was in part carried out in 1995/96 at Insti-\ntutf¨ ur Grenzfl¨ achenforschungandVakuumphysik(IGV),\nForschungszentrum J¨ ulich GmbH, Germany, in the re-\nsearch group of Prof. Matthias Wuttig. I am grateful\nto Dr. Benedikt Feldmann (IGV) who encouraged me to\ndevelop the model.\n[1] P. Bruno, 24. IFF Ferienkurs 1993, Forschungszentrum\nJ¨ ulich GmbH, J¨ ulich, Germany, ISBN 3-89336-110-3.\n[2] A. Braun, B. Feldmann, and M. Wuttig, J. Magn. Magn.\nMater.17116 (1997).\n[3] G. Bochi, C. A. Ballentine, H. E. Inglefield, S. S. Bogo-\nmolov, C. V. Thompson, and R. C. O’Handley, Mat. Res.\nSoc. Symp. Proc. Vol. 313309 (1993).\n[4] C. Chappert and P. Bruno, J. Appl. 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Lotni kow 32/46, 02-668 Warsaw, Poland\nYu. Kharchenko and M. F. Kharchenko\nB. Verkin Institute for Low Temperature Physics and Enginee ring,\nNational Academy of Sciences of Ukraine, pr. Lenina 47, 6110 3 Kharkiv, Ukraine\nH. Schmid\nDepartment of Inorganic, Analytical and Applied Chemistry ,\nUniversity of Geneva, 30 quai Ernest-Ansermet, 1211 Geneva 4, Switzerland\nSpecific heat, magnetic torque, and magnetization studies o f LiCoPO 4olivine are presented.\nThey show that an unique set of physical properties of LiCoPO 4leads to the appearance of features\ncharacteristic of 2D Ising systems near the N´ eel temperatu re,TN=21.6 K, and to the appearance\nof an uncommon effect of influence of magnetic field on the magne tocrystalline anisotropy. The\nlatter effect manifests itself as a first-order transition, d iscovered at ∼9 K, induced by magnetic\nfield of 8 T. Physical nature of this transition was explained and a model describing experimental\ndependences satisfactorily was proposed.\nPACS numbers: 75.40.Cx, 75.30.Kz, 75.80.+q, 82.47.Aa\nINTRODUCTION\nLiCoPO 4olivine, crystallizing in the Pnmastructure,1\nFig. 1, exhibits a unique set of physical properties, which\nmakes it attractive for both basic and applied studies.\nThat means:\n(i) It shows an exceptionally large linear magneto-\nelectric effect2–4and a large Li-ionic conductivity\n(making it promising for application as cathodes in\nLi-ions batteries).5–7\n(ii) In its structure, (100) oriented, “corrugated” Co-\nO layers can be distinguished, within which the\nCo2+magnetic moments are strongly coupled by\nsuperexchange Co–O–Co interactions. The neigh-\nboring (100) layers are coupled weakly by higher\norder interactions,8,9e.g., Co–O–P–O–Co. Be-\nlowTN=21.6 K, an antiferromagnetic orderingap-\npearsinthe system. Duetolargeanisotropy,8,10Co\nmagneticmomentsareconfinedtothedirectionsly-\ning within the b–cplane, ca. 4.6◦away from the\nbaxis. The magnetoelectric effect studies2(reveal-\ning “butterfly” hysteresis loops and a possibility to\nproduce the single domain state by application of\na magnetic field alone), as well as the direct mag-\nnetization measurements11,12showed that the Co\nmagnetic moments do not compensate each other\ncompletely and a small net magnetic moment, par-\nallel to the baxis, is present. Thus, LiCoPO 4is\nan intriguing quasi - two-dimensional weakly ferro-\nmagnetic Ising system.\n(iii) The presenceofa spontaneousmagnetizationisnot\nconsistent with the Pnma’symmetry, for years as-\nsumed to be the magnetic symmetry of LiCoPO 4,\nFIG. 1. (Color online). Orthorhombic ( Pnma) olivine struc-\nture of LiCoPO 4. Three unit cells ( a=10.20˚A,b=5.92˚A,\nc=4.70˚A) stacked along the baxis are presented. Starting\nfrom the lowest one, they show, respectively, oxygen coordi -\nnations of Co (octahedral), Li (octahedral) and P (tetrahe-\ndral) ions, the examples of strong Co–O–Co and weak Co–O–\nP–O–Co superexchange couplings, and the antiferromagneti c\nordering of magnetic moments of the Co ions.\nbut is consistent with the P12′\n11 monoclinic sym-\nmetry, in which, additionally, a nonzero dielectric\npolarization and a nonzero toroidal moment are\nallowed (the monoclinic baxis coincides with the\npseudo-orthorhombic aaxis). Attempts at measur-\ning the dielectric polarization were unsuccessful.13\nA nonzero toroidal moment was derived13,14on\nthe microscopic level, based on the magnetic struc-2\n/s48 /s50 /s52 /s54 /s56 /s49/s48/s49/s56/s49/s57/s50/s48/s50/s49/s50/s50\n/s32/s32\n/s84\n/s78/s61/s32/s50/s49/s46/s54/s32/s43/s32\n/s32/s32/s32 /s45/s32/s48/s46/s48/s51/s55/s53/s32 /s66/s50\n/s32/s66/s32 /s40/s84/s41/s84\n/s78/s32/s40/s75/s41\n/s99/s98/s97\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48/s49/s48/s50/s48/s84 /s32/s60/s32 /s84\n/s78\n/s84 /s32/s62/s32 /s84\n/s78\n/s32/s40/s67\n/s109/s45/s32 /s70 /s41/s32/s40/s74/s47/s40/s109/s111/s108/s101/s32/s75/s41/s41\n/s108/s110/s40/s124 /s84 /s45/s84\n/s78/s124/s47 /s84\n/s78/s41/s32/s48/s84\n/s32/s49/s84\n/s32/s53/s84\n/s32/s55/s84\n/s32/s56/s84\n/s32/s57/s84\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48/s56/s48\n/s32\n/s66 /s32/s61/s32/s57/s32/s84/s66 /s32/s61/s32/s56/s32/s84/s66 /s32/s61/s32/s55/s32/s84/s66/s32 /s61/s32/s53/s32/s84/s66 /s32/s61/s32/s49/s32/s84\n/s32\n/s84 /s32/s40/s75/s41/s66/s32 /s61/s32/s48/s32/s84\n/s32/s32\n/s40/s43/s52/s48/s41\n/s40/s43/s51/s48/s41\n/s40/s43/s50/s48/s41\n/s40/s43/s49/s48/s41\n/s40/s43/s48/s41/s67\n/s109/s32/s40/s74/s47/s40/s109/s111/s108/s101/s32/s75/s41/s41/s40/s43/s53/s48/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s53/s48/s49/s48/s48\n/s48 /s50/s48 /s52/s48 /s54/s48/s48/s49/s48/s50/s48/s32\n/s32/s67\n/s112/s32/s40/s74/s47/s40/s109/s111/s108/s101/s32/s75/s41/s41\n/s84 /s32/s40/s75/s41/s76/s105/s67/s111/s80/s79\n/s52\n/s32/s32/s97/s84/s32/s51\n/s43 /s98/s84/s32/s53\n/s32\n/s32\nFIG. 2. (Color online). Specific heat of LiCoPO 4. (a) Temperature dependence of the total specific heat in zer o magnetic\nfield,B. Inset shows the λ-anomaly near the N´ eel temperature, TN. The parameters a = 1 .31×10−4J/(mole K4) and b\n=−1.06×10−8J/ (mole K6) determine lattice contribution. (b) Magnetic contributi on to the specific heat, Cm, as a function\nof temperature, measured on heating, in Bparallel to the baxis. Curves for different Bvalues are shifted along the Cmaxis\nby the values given in parentheses. Inset shows the dependen ce ofTNonB(experimental points and fitted parabola). The\nsolid lines present logarithmic dependences fitted to the ex perimental data near TN. (c)Cm−F±vs. ln(τ),τ=|T−TN|/TN.\nSolid lines are linear approximations valid for ∼e−5< τ <∼e−0.6above and below TN.\nture data.8On the macroscopic level, four domain\nstates were observed,13two of which were inter-\npreted as ”antiferromagnetic” and two other ones\nas ”ferrotoroidic”. However, detailed symmetry\nconsiderations15showed that all four domain states\nare equivalent and differ in orientation of the net\nmagneticmoment. Eachofthe domains bearsanet\nmagnetic and a toroidal moment, whose signs and\ndirections are mutually rigidly coupled.\n(iv) The studies of birefringence induced by magnetic\nfield16suggest that the magnetic structure can be\neven more complex. In addition to the large, uni-\nform in space, and parallel to the baxis compo-\nnent of the main antiferomagnetic vector, L2=\nm1−m2−m3+m4(miare magnetic moments\nof Co ions), small, modulated in space, perpendic-\nular to the baxis components of L2and of other\nantiferromagnetic vectors, defined in Ref. 12, can\nexist.\nDespite intensive studies of structural,1\nmagnetic,1,8,11,12magnetoelectric,2,3,17transport,18\nand optical13,16,19–21properties, actual magnetic and\nelectric structures of LiCoPO 4and their transforma-\ntions in magnetic field have not yet been elucidated\nsatisfactorily.\nSince specific heat is very sensitive to all phase transi-\ntions, thisworkwasaimedatstudyingthermalproperties\nof LiCoPO 4, at determining the order of observed phasetransitions (spontaneous and induced by magnetic field,\nB, applied along the baxis) and at investigating how the\nintermediate dimensionality of the magnetic structure of\nLiCoPO 4influences the critical behavior near TN.\nEXPERIMENT\nFor the present studies, a LiCoPO 4single crystal ob-\ntainedbyhightemperaturesolutiongrowthusinglithium\nchloride (LiCl) as flux was chosen. In Ref. 22, this\nmethod of crystal growth was shown to be applicable\nfor the entire crystal family LiMPO 4(M= Ni, Co, Fe,\nMn). In the present case, the synthesis was realized in\nfull analogy to that described in detail for LiNiPO 4in\nRef. 23, i.e., using a molar ratio 1:3 between LiCoPO 4\nand LiCl in the starting mixture and using sealed plat-\ninum crucibles with 30 ml volume, with a 50 µm hole in\nthe lid for equilibrating the pressure and minimizing loss\nof the highly volatile LiCl solvent. The growth parame-\ntersandthespecialtechniqueforseparatingthefluxfrom\nthecrystalswereidenticalwiththoseusedforLiNiPO 4.23\nNo impurity phases occur in the described synthesis pro-\ncess. The growth morphology of the LiCoPO 4crystals\nhas been described in detail in Ref. 2 and is character-\nized by the development of orthorhombic (100), (210),\n(011), and (101) facets, which may be used as reference\nfor the preparation of samples (even without X-ray ori-\nentation).3\n/s32/s32/s32/s32/s32/s32 /s98/s32/s32/s32/s32/s32/s32 /s97\n/s49/s48 /s49/s53 /s50/s48 /s50/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48\n/s76/s105/s67/s111/s80/s79\n/s52/s32\n/s66/s32/s61/s32/s57/s32/s84\n/s32\n/s84 /s32/s40/s75/s41/s66/s32/s61/s32/s48/s32/s84\n/s32/s32\n/s32/s104/s101/s97/s116/s105/s110/s103\n/s32/s99/s111/s111/s108/s105/s110/s103\n/s32/s104/s101/s97/s116/s105/s110/s103\n/s32/s99/s111/s111/s108/s105/s110/s103\n/s40/s43/s56/s41\n/s40/s43/s48/s41/s67\n/s109/s32/s40/s74/s47/s40/s109/s111/s108/s101/s32/s75/s41/s41\n/s48 /s53 /s49/s48 /s49/s53/s48/s50/s52/s54\n/s32/s104/s101/s97/s116/s105/s110/s103\n/s32/s99/s111/s111/s108/s105/s110/s103\n/s32/s104/s101/s97/s116/s105/s110/s103\n/s32/s99/s111/s111/s108/s105/s110/s103/s76/s105/s67/s111/s80/s79\n/s52\n/s66/s32/s61/s32/s57/s32/s84/s66/s32/s61/s32/s56/s32/s84/s66/s32/s61/s32/s55/s32/s84\n/s84 /s32/s40/s75/s41/s40/s43/s52/s41\n/s40/s43/s50/s41\n/s40/s43/s48/s41/s67\n/s109/s32/s40/s74/s47/s40/s109/s111/s108/s101/s32/s75/s41/s41\nFIG. 3. (Color online). Magnetic contribution to the specifi c\nheat of LiCoPO 4,Cm, measured on heating and on cooling.\nCurves for different magnetic field values are shifted along\ntheCmaxis by the values given in parentheses. (a) Lack of\nhysteresis around TN. (b) Thermal hysteresis near T= 9 K\nforB= 9 T.\nThe specific heat of the LiCoPO 4single crystal was\nmeasured by means of the relaxation method, using the\nPhysical Property Measurement System, PPMS, made\nby Quantum Design. Estimated uncertainty of the deter-\nmined specific heat values was ∼2% . InB=0, studies\nweredone from 2to 300K.Since nophase transitionsap-\npeared above TN, temperature dependences for nonzero\nBvalues, ranging from 1 up to 9 T, were measured up\nto 40 K only (the magnetic field, B, was applied along\nthebaxis). The experimental points were measured ev-\nery 0.3 K (for B/negationslash=0) or 0.2 K (for B=0) below 15 K\nand every 0.1 K within the critical region around TN. In\nall figures, not all experimental points are marked with\nsymbols to keep legibility. Supplementary magnetization\nand magnetic torque measurements have been performed\nby using respective measurement options of PPMS.\nRESULTS\nThe zero-field temperature dependence of specific heat\nis plotted in Fig. 2a. The inset shows the λ-shaped\nanomaly accompanying the paramagnetic - weakly fer-TABLE I. Parameters fittingthe logarithmic dependence, Eq.\n(2), totheexperimentaldatathebest. A±andF±aregivenin\nJ/(mole K). For each parameter, an estimated uncertainty of\nthe last digit (or of the two last digits) is given in parenthe ses,\ne.g., 21.63(3) means 21 .63±0.03.\nB(T)TN(K) A−F−A+F+\n0 21.63(3) 5.3(2) −2.6(3) 1.6(1) −0.9(5)\n1 21.63(1) 5.25(5) −2.6(2) 1.45(5) −0.8(2)\n5 20.74(3) 4.5(2) −1.75(10) 1.5(1) −0.8(1)\n7 19.82(2) 3.9(1) −1.2(1) 1.3(1) −0.35(10)\n8 19.23(5) 3.6(1) −1.0(1) 1.25(10) −0.25(10)\n9 18.50(1) 3.2(1) −0.67(10) 1.25(5) −0.05(10)\nromagnetic phase transition at TN. Due to low electric\nconductivity,18∼10−9Scm−1, the electronic contribu-\ntion to the specific heat is negligible and the total specific\nheat,Cp, consists of the lattice, Cph, and magnetic, Cm,\ncontributions only. Below 60 K, Cphcan be described by\nthe formula:\nCph(T) = aT3+bT5,a = 7NAkB12π4\n5θ3\nD,(1)\nwhere the term ∼T3represents the low-temperature de-\npendence in the Debye model and the term ∼T5is the\ncorrection of that model24for the nonlinear phonon dis-\npersion relation: ω= c1|k|+ c2|k|2. Based on Eq. (1),\nthe Debye temperature was estimated to be θD= 470 K.\nThe magnetic contributions to the specific heat, de-\ntermined by subtracting the Cph(T) calculated accord-\ning to Eq. (1) from the measured specific heat, as well\nas evolution of TNwithBare presented in Fig. 2b.\nTo determine the order of the phase transition occur-\nring atTN,Cp(T) forB=0 and 9 T was measured on\nheating and on cooling. For both field values, no ther-\nmal hysteresis was detected, Fig. 3a, which strongly sug-\ngests that this is a second order transition and that its\norder does not change in magnetic field. It was veri-\nfied that the experimental data can not be described\nappropriately by assuming the classical form25of criti-\ncal behavior: Cm∼(|T−TN|/TN)−α, where the criti-\ncal exponent αtakes a value between 0 (corresponding\nto the logarithmic divergence for the two-dimensional,\n2D, Ising system26) and∼0.119 (found for the three-\ndimensional Ising model25). Thus, the critical behaviour\nin the form:27\nCm(T) =\n\n−A+ln/parenleftBig\nT−TN\nTN/parenrightBig\n+F+forT > T N\n−A−ln/parenleftBig/vextendsingle/vextendsingle/vextendsingleT−TN\nTN/vextendsingle/vextendsingle/vextendsingle/parenrightBig\n+F−forT < T N(2)\nwasassumedand the satisfactorydescriptionofthe ex-\nperimental data has been achieved, Figs. 2b and 2c. On\nthe contrary to the ideal 2D Ising system,26the anomaly\natTNis evidently asymmetric with respect to TN, Fig.\n3a, which suggests that both A and F parameters have\ndifferent values for both sides of TN. This qualitative\nexpectation was confirmed by calculations and the A±,4\n/s32/s32/s32/s32/s32/s32 /s99 /s32/s32/s32/s32/s32/s32 /s100/s32/s32/s32/s32/s32/s32 /s98\n/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/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\n/s32/s40/s100/s101/s103/s41\n/s32/s32\n/s84/s32/s61/s32/s57/s32/s75\n/s66/s61/s32\n/s54/s32/s84/s32\n/s32\n/s32/s55/s32/s84\n/s32/s56/s32/s84\n/s32/s57/s32/s84/s84/s32/s61/s32/s49/s50/s32/s75/s32/s66/s61/s32\n/s54/s32/s84\n/s32/s32 /s32\n/s32/s32 /s32/s84/s111/s114/s113/s117/s101/s32/s32/s40/s49/s48/s45/s53\n/s32/s78/s32/s109/s41\n/s32/s57/s32/s84/s32/s32/s32/s32/s32/s32 /s97\n/s45/s57/s48 /s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s48/s50/s52/s54\n/s32/s32/s32/s84/s111/s114/s113/s117/s101/s32/s32/s40/s49/s48/s45/s53\n/s32/s78/s32/s109/s41\n/s54/s32/s84\n/s32\n/s32\n/s32\n/s32/s32/s40/s100/s101/s103/s41/s84/s61/s49/s50/s32/s75\n/s84/s32/s61/s32/s57/s32/s75/s40/s43/s49/s41\n/s32/s55/s32/s84/s40/s43/s50/s41\n/s32/s56/s32/s84/s40/s43/s51/s41\n/s32/s57/s32/s84/s40/s43/s53/s46/s53/s41\n/s54/s32/s84\n/s32\n/s32/s40/s43/s54/s46/s53/s41/s32/s66/s61\n/s32/s57/s32/s84\n/s48 /s53 /s49/s48 /s49/s53/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s48 /s52 /s56/s49/s53/s48/s51/s48/s48\n/s48 /s52 /s56/s50/s48/s52/s48\n/s84/s32 /s40/s75/s41/s32/s104/s101/s97/s116/s105/s110/s103\n/s32/s99/s111/s111/s108/s105/s110/s103/s40/s43/s56/s41\n/s57/s32/s84/s56/s32/s84/s55/s32/s84/s53/s32/s84\n/s32/s66 /s32/s61/s32/s48/s32/s84/s67\n/s109/s32/s40/s74/s47/s40/s109/s111/s108/s101/s32/s75/s41/s41/s40/s43/s54/s41\n/s40/s43/s52/s41\n/s40/s43/s50/s41/s32\n/s40/s43/s48/s41/s84/s51\n/s34/s101/s120/s112/s34/s66 /s32/s40/s84/s41/s32\n/s97\n/s49/s32\n/s32/s97\n/s50/s32\n/s32\n/s32/s32/s32\n/s66 /s32/s40/s84/s41/s32/s32/s32/s32/s32/s32 /s97\nrigid,□| |=const. M1M1\nM2 Bsusceptible\nFIG. 4. (Color online). Field-induced first-order phase tra nsition. Curves in panels (a) and (b) are shifted along the y-axes\nby the values given in parentheses. (a) Magnetic specific hea t vs. temperature. For B/greaterorequalslant8 T, the anomaly at ∼9 K is visible.\nVertical blue and red arrows indicate the temperatures, at w hich torque was measured. Solid lines present the calculate d\nmagnon contributions. The insets show experimental (circl es) and fitted theoretical (solid lines) dependences of two p arameters\nappearing in Eq. (3), a1(in J/(mole K1/2)) anda2(in K), on B. (b) Magnetic torque for Brotating within the b-cplane (θ\nis counted from the baxis). Curves plotted with full and open symbols were measur ed forBrotating in opposite directions.\nDependences calculated within the proposed model (solid li nes) are superimposed on the experimental curves. (c) Outli ne of\nthe proposed two-sublattice model of the magnetic structur e. (d) Superimposed magnetic torque dependences measured a t\nT= 9 and 12 K. The oblique arrows indicate directions along whi ch the torque maximum moves with increasing BatT= 9\nK and at T= 12 K.\nF±, andTNvalues fitting the experimental data the best\nare given in Table I. (It should be stressed that the un-\ncertainty of TNgiven in Table I is the uncertainty of\nthe theoretical, fitted parameter. For temperatures ∼30\nK, the uncertainty of the absolute temperature values\ndetermined in PPMS is ±1%, whereas relative temper-\nature changes ∼0.03% can be detected an stabilized).\nWe attribute the affect of asymmetry of the λ-anomaly\nto the quasi-2D character of the magnetic structure, i.e.,\nto the fact that the buckled (100) layers of strongly\ncoupled Co2+magnetic moments are not isolated butweakly coupled mutually. With increasing B,TNde-\ncreases parabolically, as illustrates the inset to Fig. 2b,\nand theλ-anomaly decreases, but the transition remains\nsharp. Such a behaviour is characteristic of a 2D anti-\nferromagnetic Ising system.28\nAn additional anomaly appears in B= 8 T at 8.8\nK, Figs. 2b, 3b, and 4a. For B= 9 T, it becomes more\npronounced and shifts to 9.2 K. Near the anomaly, a hys-\nteresis between the curves measured on heating and on\ncooling appears and the anomaly measured on cooling\nis smaller, Fig. 3b. Since these two effects are the basic5\ncharacteristics of first-order transitions29,30(the first one\nis related to overheating and overcoolingphenomena and\nthe second one is inherent in the relaxation method of\nmeasurement), we interpret the anomaly as the indica-\ntion of occurrence of a first order phase transition. It can\nbesupposedthatthetendencyobservedonincreasingthe\nmagnetic field from 8 to 9 T will be preserved and the\nanomaly will increase and shift to higher temperatures\nwith further increase of the magnetic field.\nTo estimate a change of magnetic entropy related to\nthis transition, we assumed that the magnon contri-\nbution,Cma, being the only constituent of Cmapart\nfrom the transition, can be described in frames of the\nmodel developed for anisotropic antiferromagnets.31For\nthe cases of low and high temperatures, that model pre-\ndicts, respectively:\nCma=a11√\nTexp/parenleftBig\n−a2\nT/parenrightBig\nforµB\nkBBa>µB\nkBB > T,(3)\nCma=a3T3forTN≫T≫µB\nkBBa>µB\nkBB,(4)\nwherea1=a0(Ba−B)2,a2=b0(Ba−B),a0,b0,\nanda3are constants, Bais a parameter of the order\nof anisotropy and exchange fields, µBis Bohr magne-\nton, and kBis Boltzmann’s constant. A good descrip-\ntion was achieved up to 14 K, Fig. 4a. For B <5 T,\nthe experimental dependences can be fitted with Eq. (3),\nwhereas for larger fields a crossover between the behav-\niors given by Eqs. (3) and (4) occurs at ∼9.5 K. To\nget a satisfactory description above the crossover tem-\nperature for B=7, 8, and 9 T, it is necessary to add\nto Eq. (4) a constant term, respectively, of 0.31, 0.41,\nand 0.53 J/(mole K). Tentatively, this can be ascribed\nto the fact that the model of noninteracting magnons,31\nwithin whichEqs. (3) and(4)werederived, isinadequate\nclose toTN(which falls down from ∼22 K for B=0 to\n∼18 K for B=9 T). A steep changes of the calculated\nmagnon contributions appearing at the transition point\nforB/greaterorequalslant8 T, Fig. 4a, suggest that the transition is re-\nlated to a change in the stiffness of the magnon system,\ni.e. to a change of anisotropy or exchange interactions.\nAfter subtracting the calculated magnon contributions\nfromCm, Fig. 4a, the entropy changeassociated with the\nfirst-order transition, ∆ S, was calculated using the for-\nmula: ∆ S=/integraltextT2\nT1[(Cm−Cma)/T]dT(the temperatures\nT1andT2must be chosen sufficiently far from the tran-\nsition temperature, i.e., at points at which Cm=Cma).\nIt was found ∆ S/(kBNA) = 0.010, which is a very small\nvalue (e.g., the entropy change related to disappearance\nof a long-range order in a system of 1/2 spins is equal to\nln(2)≈0.693).\nBased on Ref. 17, where the magnetoelectric effect was\nshown to vanish in B∼12 T, and on Ref. 32, where de-\nstruction of the antiferromagnetic ordering in the mag-\nnetic field parallel to the baxis was shown to occur via/s32/s32/s32/s32/s32/s32 /s97/s32/s32/s32/s32/s32/s32\n/s32/s32/s32/s32/s32/s32 /s98\n/s48/s53/s49/s48\n/s32/s32\n/s100 /s77 /s47/s100 /s66 /s32/s32/s40/s65/s109/s50\n/s47/s40/s107/s103/s32/s84/s41/s41\n/s32/s77 /s32/s40/s65/s109/s50\n/s47/s107/s103/s41/s32/s84 /s32/s61/s32/s49/s48/s32/s75\n/s32/s84 /s32/s61/s32/s57/s32/s75/s32\n/s32\n/s48 /s50 /s52 /s54 /s56/s49/s50/s32/s32/s32 /s84 /s32/s61/s32/s49/s48/s32/s75\n/s32/s32 /s84 /s32/s61/s32/s57/s32/s75/s32\n/s32\n/s66 /s32/s40/s84/s41/s32/s32/s109/s111/s100/s101/s108/s32\n/s49/s48/s50/s48\n/s32/s32/s66 /s32/s61/s32/s57/s32/s84/s32/s32\n/s32/s32/s77 /s32/s40/s65/s109/s50\n/s47/s107/s103/s41\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48 /s50/s53 /s51/s48/s48/s49/s50/s51/s32\n/s32/s100 /s77 /s47/s100 /s84 /s32/s32/s40/s65/s109/s50\n/s47/s40/s107/s103/s32/s75/s41/s41\n/s84 /s32/s40/s75/s41\nFIG. 5. (Color online). Magnetization of LiCoPO 4along the\nbaxis, measuredinthemagnetic fieldapplied alongthe baxis.\n(a) Magnetization in B= 9 T (and its derivative with respect\ntoT) as a function of temperature. (b) Magnetizations at\nT= 9 and 10 K (and their derivatives with respect to B) as\na function of magnetic field. Solid line superimposed on the\nexperimental dependence for 9 K was calculated within the\ntwo-sublattice model proposed.\na series of spin-flip transitions, starting from B∼12 T,\nwe claim that the magnetic field of 9 T: (i) can influence\nthe distribution of electric charges and, hence, the mag-\nnetocrystalline anisotropy of LiCoPO 4, (ii) is too weak\nto induce a spin-flip transition. In order to verify these\nclaims and to elucidate the physical nature of the discov-\nered transition, supplementary magnetization and mag-\nnetic torque measurements have been performed.\nThe magnetization studies, Fig. 5, confirmed that this\nis not a spin-flip transition, because neither in the tem-\nperature and field dependences of magnetization nor in\ntheir first derivatives any anomaly occurs at the transi-\ntion point.\nThe magnetic torque has been measured at T=9 and\n12 K, for the magnetic field of different value, rotating\nwithin the b-cplane, Figs. 4b and 4d. The angle θdeter-\nmining the orientation of the field Bwithin the b-cplane\nwas counted from the baxis. The torque measurements\nshowed unequivocally, Fig. 4d, that the transition is re-\nlatedtoachangeinthemagneticanisotropy. Thisfollows\nfrom the fact that at T= 9 K, i.e., below the transition\npoint, the positions of the torque maxima evolve mono-\ntonically with increasing B(up to the highest Bvalue of\n9 T) along the direction indicated in Fig. 4d by a blue\noblique arrow, while at T= 12 K, i.e., above the transi-\ntion point, we observe a steep, qualitative change of the\ntorque behavior. That means at T= 12 K, for B= 6 T,\nthe maxima fit into the tendency observed at T= 9 K,\nwhereas for B= 9 T, the maxima shift to the opposite\ndirection than at T= 9 K. This is illustrated in Fig. 4d\nby a red oblique arrow.6\nTABLE II. Parameters, which fit the best the measured depende nces of torque on θand of magnetization along the baxis\nonB. Numbers in parentheses give uncertainty of the last digit ( or of the two last digits) of the parameters, e.g., 21.72(4)\nmeans 21 .72±0.04. Parameters that changed as the result of the phase transi tion are given in bold. T0is a small constant\nvalue (approximately, two orders of magnitude smaller than the maximum torque values measured) that must be added to the\ntheoretical dependence of torque on θ, (A.13), to offset a background, inherent in the measurement technique applied.\nT B K 1Bm χ1b χ2b χ3b χ1c χ2c χ3c T0\n(K) (T) (106J/m3) (T) (103A/(mT)) (103A/(mT2)) (A/(mT3)) (103A/(mT)) (103A/(mT2)) (A/(mT3)) (10−7Nm )\n9 6 4.3(3) 20.0(2) 28.01(3) -1.304(2) 21.72(4) 8.7(3) -1.2( 1) 50(30) -2(1)\n9 7 4.3(3) 20.0(2) 28.01(3) -1.304(2) 21.72(4) 9.4(2) -1.25 (5) 50(20) -2(1)\n9 8 4.3(3) 20.0(2) 28.01(3) -1.304(2) 21.72(4) 9.9(1) -1.26 (2) 50(10) -2(1)\n9 9 4.3(3) 20.0(2) 28.01(3) -1.304(2) 21.72(4) 10.0(1) -1.1 6(2) 50(5) -2(1)\n12 6 4.3(3) 17.55(5) 29.90(2) -1.520(1) 28.87(8) 5.8(1) -0. 33(3) 140(20) -2(1)\n12 9 11.5(1.1) 17.55(5) 29.90(2) -1.520(1) 28.87(8) 6.36(2) -0.113(3) 150(5) -3(1)\nTo describe this effect theoretically, we propose a sim-\nplified model, Fig. 4c, details of which are presented in\nthe appendix, based on the following assumptions:\n(i) As it was suggested in Ref. 12, modulated in space,\nperpendicular to the baxis, nonzero components of\nmagnetic moments exist in the magnetic structure\nof LiCoPO 4.\n(ii) The two-sublattice model, in which Co2 and Co3\nions form the one sublattice, denoted as 1, and Co1\nand Co4 ions form the other one, denoted as 2, can\nbe applied.\n(iii) The modulated, perpendicular to the baxis com-\nponents average out to zero and only the net mag-\nnetizations of both sublattices are essential, Fig.\n4c.\n(iv) The deflection of the net magnetizations of both\nsublattices by 4.6◦from the baxis and the pos-\nsibility of existence of domains, in which the sign\nof this deflection is different, can be neglected and\nit can be assumed that the net magnetizations are\ndirected along the baxis.\n(iv) The sublattice magnetized “along” the field (de-\nnoted as M1in Fig. 4c) is rigid, has a well de-\nfined magnetization modulus, and its magnetic\nanisotropycan be analyzed by using the anisotropy\nconstant K1.\n(v) The sublattice magnetized “against” the external\nfield (denoted as M2) is “weak” and behaves as an\nanisotropic paramagnet located within an effective\nfield,Beff, composed of the external field, B, and\nthe exchange (i.e. molecular) field, Bm, produced\nby the “rigid” sublattice.\n(vi) The magnetization of the “weak” sublattice is\nequal to: M2σ=χ1σBeffσ+χ2σsign(Beffσ)B2\neffσ+\nχ3σB3\neffσ, whereσidentifies the components along\nthebandcaxes.\n(vii) For −90◦< θ <90◦, the sublattice 1 is the rigid\none and the sublattice 2 is the susceptible one. For90◦< θ <270◦, both sublattices exchange their\nbehavior, that means the sublattice 1 becomes the\nsusceptible one, while the sublattice 2 becomes the\nrigid one.\n(viii) For θ=±90◦both sublattices are indistinguish-\nable, therefore the torque goes through zero at\nthese angles.\nBy taking the Bm,χ1b,χ2b, andχ3bvalues, which fit\nthe best the dependences of net magnetization on Bap-\nplied along the baxis (measured in Ref. 12 and in the\npresent work, Fig. 5b), and by treating K1,χ1c,χ2c,\nandχ3cas fitted parameters, a good agreement between\nthe measured and the calculated angle dependences of\ntorque was achieved (Fig. 4b). The values of the pa-\nrameters fitting the experimental data the best are given\nin Table II. These values imply that at the transition\npointthesublatticemagnetized“along”thefieldbecomes\n“harder” ( K1grows), whereas the other sublattice be-\ncomes “weaker” (its total susceptibility along the baxis\nremains unchanged, whereas the susceptibility along the\ncaxis grows). Nevertheless, the baxis remains the easy\nmagnetization direction (the sign of K1, characterizing\nthe anisotropy of the sublattice 1, does not change and\nthe total susceptibility of the sublattice 2 remains larger\nalongthe baxisthan alongthe caxis). This fact explains\nwhy no anomalies are observed at the transition point\non temperature and field dependences of magnetization\nmeasured in the magnetic field applied along the baxis,\nFig. 5. The uncertainty of each parameter given in Table\nII was estimated by keeping all other parameters fixed\nand checking that no noticeable change of the theoretical\ncurve appears for the values of the examined parameter\nlyingwithinthe uncertaintyrange, whereasforthe values\nbeyond that range, fit quality deteriorates evidently. It\nshould be mentioned that the proposed method of anal-\nysis, in which 9 fitted parameters are involved, should be\ntreated rather as a qualitative method of elucidating the\nphysical processes occurring in the sample, not as an ac-\ncurate method for determining physical parameters, e.g.\nK1.7\nCONCLUSIONS\nIn conclusion, it was shown that in LiCoPO 4, the sec-\nond order phase transition from the paramagnetic to\nthe weakly ferromagnetic phase is accompanied by a λ-\nshaped anomaly of specific heat, which can be described\nas the logarithmic divergence (2), characteristic of a 2D\nIsing system. The deviation from the purely 2D be-\nhaviour was ascribed to the quasi-2D character of the\nmagnetic structure. The first-order phase transition in-\nduced by an external magnetic field B/greaterorequalslant8 T parallel\nto thebaxis, appearing at ∼9 K, was discovered and\nshown to be related to the change of magnetocrystalline\nanisotropy.\nACKNOWLEDGMENTS\nSupportofSwissNSFforcrystalgrowth(priorto1996)\nis gratefully acknowledged. This work was partly sup-\nported by the Polish Ministry of Science and Higher Ed-\nucation from funds for science for 2008-2011 years, as\na research project (2047/B/H03/2008/34), and by the\nEuropean Union, within the European Regional Devel-\nopment Fund, through the Innovative Economy grant\n(POIG.01.01.02-00-108/09).\nAppendix\nWe assumed that the simplest description of the\nLiCoPO 4antiferromagnet can be based on the two-\nsublattice approximation, in which Co2 and Co3 ions\nform the one sublattice, denoted as 1, and Co1 and Co4\nions form the other one, denoted as 2 (Figs. 1 and 4c).\nThen, we can apply a molecular field approximation,\nwhich must be modified in such a way that the presence\nof an extremely weak, net spontaneous magnetization,\nMex\nsp, found experimentally,11,12will be mimicked. An\napproach to the latter effect can be based on considera-\ntions concerninga muchsimpler caseofaferromagnet. It\nis known that within the molecular field approximation,\nthe magnetization of a ferromagnet can be determined\ngraphicallyasthe intersectionpoint ofthe Brillouinfunc-\ntion:\nBS(y) =2S+1\n2Scoth/parenleftbigg2S+1\n2Sy/parenrightbigg\n−1\n2Scoth/parenleftbigg1\n2Sy/parenrightbigg\n,\n(A.1)\nand of the linear relation between the yparameter and\nthe magnetization:\nM\nMS=kBT\ngµBSBmy−1\nBmB, (A.2)\nwhereBm,MS,g, andSdenote, respectively, molecu-\nlar field, saturation magnetization, g-factor and spin of\nthe magnetic ion. As shown in Fig. 6, for T= 0.5TC,\nin zero magnetic field the intersection point is located in/s48 /s50 /s52/s48/s48/s46/s50/s48/s46/s54/s49/s46/s48\n/s66/s61/s45/s48/s46/s50/s66\n/s109/s66/s61/s48/s46/s50/s66\n/s109/s66/s61/s48/s77/s47/s77\n/s83\n/s121/s66/s61/s45/s48/s46/s50/s66\n/s109\nFIG. 6. (Color online) Molecular field approximation for a\nstandard ferromagnet. Red curve presents the Brillouin fun c-\ntion. Straightsolid line presentsthecase ofzero external field.\nThe dash-dot and dashed lines show, respectively, the cases\nof the external field applied along and against the molecular\nfield.\nthe plateau ofthe Brillouinfunction and the spontaneous\nmagnetization reaches nearly the saturation value MS.\nThen, if the external field B= 0.2Bmis applied along\nBm, the magnetization changes only slightly, whereas if\nthe same field is applied against Bm, the intersection\npoint shifts to the region of noticeably smaller magneti-\nzation values, where also the curvature of the Brillouin\nfunction and the susceptibility are larger.\nSince for LiCoPO 4, the temperature of the first-order\ntransition is ∼0.5TN∼10 K,Bm(estimated based on\ntheTNvalue) is ∼18 T, and the applied fields ranging\nfrom 6 to 9 T are of the order from 0.3 to 0.5 Bm, we\ncan expect a similar behavior for this more complex case\nof weakly ferromagnetic antiferromagnet.\nThus, we assumed that the sublattice 1, magnetized\nalong the field, Figs. 4c and 7, is rigid, with well defined\nmodulus of its magnetization, M0= 2mCo/Vuc, where\nmCo=gµBS= 3.26µBis the magnetic moment of one\nCo+2ion (S= 3/2) andVucis the volume of the or-\nthorhombic unit cell (containing 4 formula units). As\nthe result, magnetic anisotropy of this sublattice can be\ndescribed by using the anisotropy constant K1. On the\ncontrary, the sublattice 2, magnetized against the field,\nis susceptible and the modulus of its magnetization is a\nfunction of value and direction of the field B. Thus, the\nformalism involving anisotropy constants is inapplicable\nand we can describe a magnetic anisotropy of the sublat-\ntice 2 by introducing different magnetic susceptibilities\nalong different crystallographic directions.\nAdditionally, the following experimental facts11,12\nmust be taken into account:\n(i) The dependence of spontaneous magnetization, Mex\nsp,\non temperature has a form found in Ref.12:\nMex\nsp(T) = N(0.122−6.5×10−4(T−10.4)2),(A.3)\nwhere N is a coefficient needed to convert the value ex-8\ná1\ná2cb\nB\nM1\nM2è\nFIG. 7. (Color online) Coordinate system and the symbols\nused.\npressed in G to desired units.\n(ii) The dependence of magnetization on the field applied\nalong the baxis contains terms linear and cubic in B, as\nit was found in Ref.11:\nM(B) =Mex\nsp(T)+χex\n1B+χex\n3B3.(A.4)\n(Coming beyond the molecular field approximation we\ncan say that the real LiCoPO 4magnetic structure can\nhave a form of a very weakly spread out ”fan”,16Fig. 4c.\nThen, we can imagine that the molecular field picture de-\nscribed aboveis a simplification of the fact that the ”fan”\nof the sublattice magnetized along the filed folds slightly,\nwhereas the fan of the sublattice magnetized against the\nfield spreadsoutconsiderablyunderinfluence ofthemag-\nnetic field.)\nBesides we assumed that the deflection of the magne-\ntization by 4.6◦away from the baxis and the possibility\nof existence of domains differing in sign of this deflection,\ni.e., +4.6◦or−4.6◦, has no noticeable effect on the an-\ngle dependences of torque and on the net magnetization\nvalue along the baxis and can be neglected. Thus, in our\nconsiderations was assumed that the baxis is the easy\nmagnetization direction for both sublattices. Validity of\nthis assumption has been verified by direct calculations.\nUnder the assumptions given above, for −90◦/lessorequalslantθ/lessorequalslant\n90◦the magnetizations of both sublattices are given by\nthe expressions:\nM1b=M0cosα1, M 1c=M0sinα1,(A.5)\nM2σ=χ1σBeffσ+χ2σsign(Beffσ)B2\neffσ+χ3σB3\neffσ(A.6)\nwhereσdenotesbandc. The effective field, Beff, acting\non the sublattice 2 consists of the applied, B, and the\nmolecular, Bm, field and is equal to:\nBeffb=Bcosθ−Bmcosα1\nBeffc=Bsinθ−Bmsinα1. (A.7)\nThen, the free energy of the system is given by the ex-pression:\nF(T,B,α 1,θ) =K1sin2α1−M0Bcos(θ−α1)\n−/summationdisplay\nσ=b,c/parenleftbigg1\n2χ1σB2\neffσ+1\n3χ2σ|Beffσ|3+1\n4χ3σB4\neffσ/parenrightbigg\n.\n(A.8)\nTo assure consistency with the experimental results of\nmagnetization measurements, we assume that for θ= 0\nandBsmaller than the spin-flip field (as it is in the con-\nsidered case), also α1= 0 and dα1/dB= 0. Then, the\ntheoretical resultant magnetization of the sample, calcu-\nlated according to the formula:\nM(θ= 0,B) =−∂F\n∂B=M0+M2b,(A.9)\nshould be equal to the experimental one, given by Eq.\n(A.4). By comparing coefficients at different powers of B\n(in particular, the coefficient at B2should be equal to 0)\nwe receive the following relations:\nχ1b=χex\n1+3χex\n3B2\nm, χ2b=−3χex\n3Bm,\nχ3b=χex\n3.(A.10)\nBy substituting B= 0 into (A.9), we receive the equa-\ntion:\nM0−χex\n1Bm−χex\n3B3\nm=Mex\nsp(T). (A.11)\nNext, for T= 9 and 12 K, the Mex\nsp(T) parameter was\ncalculatedaccordingto Eq. (A.3) and the coefficients χex\n1\nandχex\n3were determined by fitting the function (A.4) to\ntheM(B) dependences (since wehad accessto the M(B)\ndependences measured for T= 9 and 10 K only, the\nχex\n1andχex\n3coefficients for T= 12 K were determined\nby extrapolating the ones found for T= 9 and 10 K).\nThen, knowing these parameters, Bmwas determined by\nsolving Eq. (A.11) and the susceptibilities χib, fori= 1,\n2, and 3, were calculated using Eq. (A.10). (We found\nBm= 20 T for T= 9 K and Bm= 17.55 T for T= 12\nK. At the first glance it seems strange that Bmvaries so\nconsiderably, but this can be attributed to the unusual\nparabolic dependence of the spontaneous magnetization\nonT(A.3)).\nIn order to determine angle dependences of the mag-\nnetic torque for Brotating within the b−cplane, for the\ncase−90◦/lessorequalslantθ/lessorequalslant90◦, theα1parameter was determined\nfor each θvalue by solving (numerically) the entangled\nequation:\n∂F\n∂α1=/parenleftbigg2K1\nM0/parenrightbigg\nM0sinα1cosα1−M0Bsin(θ−α1)\n−/parenleftbig\nχ1b+sign(Beffb)χ2bBeffb+χ3bB2\neffb/parenrightbig\nBeffbBmsinα1\n+/parenleftbig\nχ1c+sign(Beffc)χ2cBeffc+χ3cB2\neffc/parenrightbig\nBeffcBmcosα1\n= 0.\n(A.12)9\nNext,M1σandM2σvalues were calculated by using the\nformulae (A.5) - (A.7) and the magnetic torque acting\non the whole sample was calculated according to the for-\nmula:\nTa=V(M×B)a=m\nmmNA\n4Vuc/bracketleftBig\n(M1b+M2b)Bc\n−(M1c+M2c)Bb/bracketrightBig\n=m\nmmNA\n4VucB/bracketleftBig\nM0sin(θ−α1)\n+/parenleftbig\nχ1b+sign(Beffb)χ2bBeffb+χ3bB2\neffb/parenrightbig\nBeffbsinθ\n−/parenleftbig\nχ1c+sign(Beffc)χ2cBeffc+χ3cB2\neffc/parenrightbig\nBeffccosθ/bracketrightBig\n,\n(A.13)\nwhereV,m,mm, and N Adenote, respectively, the vol-\nume of the sample, the mass of the sample, the LiCoPO 4\nmolar mass, and the Avogadro number. The α2andM2\nvalues were determined by using the formulae:\nα2= arctan/parenleftbiggM2c\nM2b/parenrightbigg\n, M2=/radicalBig\nM2\n2b+M2\n2c.(A.14)Forthe case90◦< θ <270◦, both sublattices exchange\ntheirbehavior,thatmeansthesublattice2becomesrigid,\nwhereas the sublattice 1 becomes susceptible. Thus, the\nappropriatefreeenergyandtorquevalueswerecalculated\nby using Eqs. (A.7), (A.8), (A.12), and (A.13), in which\nθwas replaced with θ′=θ−180◦and theα1parameter\ndetermined by solving Eq. (A.12) was treated as α2.\nIt should be mentioned that for the special values\nθ=±90◦both sublattices are indistinguishable and the\ntorque is equal to zero.\nAs the result of applying the procedure described\nabove, in which K1,χ1c,χ2c, andχ3cwere treated as\nfitted parameters, the theoretical dependences of Taon\nθ, plotted in Fig. 4b with (black) solid lines superim-\nposed on the experimental data, have been obtained.\n∗szewc@ifpan.edu.pl\n1R. P. Santoro, D. J. Segal, and R. E. Newnham,\nJ. Phys. 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H1111 Budapest, Hungary and\n3Department of Materials Science and Engineering,\nNational University of Singapore, 117576, Singapore\n(Dated: March 1, 2022)\nWe perform \frst principles calculations of the magnetocrystalline anisotropy energy in the ve\nL10FePt samples studied experimentally by Ding et al. [J. App. Phys. 97, 10H303 (2005)]. The\ne\u000bect of temperature-induced spin \ructuations is estimated by scaling the MAE down according\nto previous Langevin dynamics simulations. Including chemical disorder as given in experiment,\nthe experimental correlation between MAE and lattice mismatch is qualitatively well reproduced.\nMoreover we determine the chemical order parameters that reproduce exactly the experimental\nMAE of each sample. We conclude that the MAE is determined by the chemical disorder rather\nthan by lattice distortion.\nPACS numbers: 75.30.Gw 75.50.Ss 71.15.Mb 71.15.Rf\nDue to its extraordinarily high magnetocrystalline\nanisotropy energy (MAE), L1 0FePt is of considerable in-\nterest to the development of ultrahigh density magnetic\nrecording applications and spintronics devices. From a\ntheoretical point of view, there is an obvious need for\na complete \frst principles model of FePt to be used in\ngenerating e\u000bective spin Hamiltonians for the purpose\nof atomistic and multiscale modelling. Amongst many\nother issues, this requires an understanding of the role of\ninterfacial e\u000bects and chemical disorder. The large e\u000bect\nof chemical disorder on the MAE of FePt has already\nbeen outlined both experimentally1and theoretically2.\nRecently, the experiments were extended to thin \flms\nof FePt deposited on di\u000berent substrates3. A strong cor-\nrelation was revealed between the MAE of the FePt sam-\nple and the lattice mismatch of the FePt \flms with re-\nspect to the substrate3. The experimental data are sum-\nmarised in Table I. The chemical order parameter, s, is\nde\fned as the probability of \fnding an Fe atom on a nom-\ninal Fe site or, equivalently, as the probability of \fnding\na Pt atom on a nominal Pt site. In the experiment, the\nchemical order parameters were derived from the X-ray\ndi\u000braction intensities I(001) and I(002), (( xyz) denot-\ning the plane of di\u000braction), through the relationship1,4\ns\u0018p\nI(001) =I(002) and normalizing sto unity for sam-\nple no. 3. We refer to the experimentally obtained chemi-\ncal order parameters as sefor distinction from the chemi-\ncal order parameters sobtained later by \ftting calculated\nMAE-values to experiment.\nThe aim of the present work is to investigate in de-\ntail the e\u000bect of lattice distortion and chemical order on\nthe MAE of FePt by means of the relativistic Korringa-\nKohn-Rostoker5{7method as combined with the coherent\npotential approximation8,9(KKR-CPA). In order to dif-\nferentiate between the two main properties characterizing\nthe samples, namely, the lattice distortion and the chem-\nical disorder, we perform calculations with and without\nthe inclusion of chemical disorder. We then \ft the calcu-TABLE I: Summary of experimental results by Ding et al.3;\nlattice parameters aandc, chemical order parameter, s, mag-\nnetocrystalline anisotropy energy per formula unit, K, and\ndi\u000braction intensity ratio, I(001)=I(002).\nSample a (\u0017A) c (\u0017A) seK(meV) I(001)=I(002)\n1 3.88673 3.69977 0.709 0.493 1\n2 3.88279 3.69387 0.978 0.696 1.9\n3 3.89752 3.68964 1.000 0.841 1.985\n4 3.89646 3.69175 0.965 0.788 1.85\n5 3.86954 3.71378 0.615 0.271 0.7536\nlated MAE to the experimental values using the chemical\norder parameter, s, as a \ftting parameter and draw con-\nclusions from the results of our calculations. We \fnd that\nchemical disorder of each sample is the more important\nfactor in determining the experimental3MAE.\nAs the relativistic KKR method is well documented in\nthe literature (see e.g. 7), here we merely describe some\ndetails of our calculations. We used Density Functional\nTheory within the Local Spin-Density Approximation\n(LSDA) as parametrised by Vosko et al.10. The e\u000bec-\ntive potentials and \felds were treated within the atomic\nsphere approximation (ASA). As the thin-\flm samples in\nthe experiment had a thickness of approximately 20 nm\n(60 formula units),3surface contributions to the MAE\nshould be negligibly small. We therefore modelled the\nFePt samples as face-centered-tetragonal (fct) bulk lat-\ntices with lattice constants as displayed in Table I. The\nself-consistent calculations were performed by using the\nscalar-relativistic approximation, i.e., by neglecting spin-\norbit coupling11and solving the Kohn-Sham-Dirac equa-\ntion using a spherical wave expansion up to an angular\nmomentum quantum number of `= 3. As in earlier the-\noretical work,2we used the coherent potential approx-arXiv:1108.2572v1  [cond-mat.mtrl-sci]  12 Aug 20112\nimation (CPA) to elucidate long-range chemical disor-\nder e\u000bects in FePt. In combination with KKR, the CPA\nhas proved particularly useful in calculating the physical\nproperties of chemically disordered alloys9. The partially\ndisordered FePt alloy is modelled by a stack of alternat-\ning layers with the chemical compositions of Fe sPt1\u0000s\nand Pt sFe1\u0000s.\nThe MAE is then evaluated using the magnetic force\ntheorem12, which states that the di\u000berence in a system's\ntotal energy for two di\u000berent directions of magnetiza-\ntion can be approximated by the corresponding di\u000berence\nof the band energies, neglecting further self-consistency,\ni.e., keeping the e\u000bective potentials and \felds \fxed. From\nprevious experience we know that for transition metal\nsystems these potentials and \felds can safely be taken\nfrom self-consistent scalar-relativistic calculations7. In\norder to achieve a relative accuracy within 5 % for the\nMAE, the associated energy integration was performed\nby sampling 20 energy points along a semi-circular con-\ntour in the upper complex half-plane. At the energy point\nclosest to the real axis the k-integration was calculated\nusing 5050 k-points in the irreducible segment of the two-\ndimensional Brillouin zone.\nAs the MAE should vanish at the Curie temperature,\nit is a rapidly decreasing function of temperature. Whilst\nthe temperature dependence of the MAE of ordered FePt\nhas been previously calculated in terms of di\u000berent the-\noretical methods13,14, in the present work we do not\nmake an attempt to carry out a similar process, since\nsite-resolved information is currently not available for a\nchemically disordered system. Instead, for an approx-\nimate comparison with experiments at room tempera-\nture, we use the scaling obtained for perfectly ordered\nL10FePt in terms of Langevin dynamics simulations14,\nnamely, KT=293K\u00180:6KT=0K.\nUsing the methods described above we performed sys-\ntematic calculations of the MAE of each of the FePt sam-\nples in Table I. In order to separate the e\u000bects of the\nlattice distortion and the chemical disorder, we split our\nstudy into three stages. In our \frst set of calculations, the\nFePt samples were modelled as perfectly ordered alloys\nwith lattice parameters according to Table I. As can be\ninferred from Fig. 1, our calculated values spread around\n3 meV/f.u. and show a very minor dependence on the\nvariation of the lattice parameters. Moreover, this mod-\nerate variation between the samples is contrary to the\nexperimentally observed trend.\nAlthough high in comparison to experiment, our cal-\nculated MAE values are in good agreement with other\ntheoretical results based on the LSDA or the LSDA+U\napproach15. One obvious reason for the discrepancy\nbetween the theoretical and experimental values is the\nstrong temperature dependence of the MAE. We estimate\nthis contribution by scaling the calculated MAE down\nby an approximate factor of 0.6, as described above. The\ncorresponding MAE-values (also shown in Fig. 1) are still\ntoo high as compared to experiment. Thus we conclude\nthat, even when taking temperature-induced spin \ructu-\n 0 1 2 3 4\n 1  2  3  4  5MAE (meV / f.u.)\nSample NoKKR (ordered) T=0 K\nKKR (ordered) T=298 K (est)\nexperimentalFIG. 1: Crosses (solid line): Calculated MAE per formula unit\nfor each of the FePt samples in Table I modelled as perfectly\nordered alloys. Circles (dashed line): The same values scaled\ndown by a factor of 0.6 in order to account for temperature\ninduced e\u000bects. Stars (dotted line): The experimental values.\nations into account, lattice distortion alone can explain\nneither the size nor the trend of the MAE obtained in\nthe experiment.\nSubsequently, the chemical disorder of each sample as\ngiven in Table I was taken into account using the co-\nherent potential approximation. The corresponding re-\nsults are shown in Fig. 2. In accordance with earlier\nwork2, long-range chemical disorder drastically reduced\nthe MAE; for sample no. 1 ( se= 0:709) we obtained a\nvalue of 0.4 meV/f.u., while for sample no. 5 ( se= 0:615)\nthe MAE almost vanished. In fact, reducing sto 0.5 can\neven cause a change of sign of the MAE. In contrast, for\nsamples no. 2 and 4 with a high degree of chemical order\nthe MAE was reduced by less than 10 %, and for sample\nno. 3 ( se= 1) the MAE remained unchanged with respect\nto our previous calculations. Taking into account again a\nreduction by a factor of 0.6 due to temperature e\u000bects, it\nis obvious that the inclusion of chemical disorder has sig-\nni\fcantly improved the agreement between experiment\nand theory: the trend of the MAE between the di\u000berent\nsamples is now correct and the magnitudes of the MAE\nare closer to the range reported by the experiment.\nAs mentioned above, the chemical order parameters\nin Table I were derived from measured di\u000braction inten-\nsity ratios1,4. However, due to an incomplete rocking\ncurve3, the measured di\u000braction intensities, and thereby\nthe experimentally obtained chemical disorder parame-\nters, can only be considered approximate values. Fur-\nthermore, we note the assumption that the sample with\nhighest MAE, sample no. 3, refers to perfect chemical\norder, se= 1. This seems a reasonable working hypoth-\nesis, but one worth investigating theoretically since it is\ncentral to the interpretation.\nThe above uncertainties motivated us to perform a\nthird set of calculations, in which the theoretical MAE\nwas \ftted to the experimental MAE using the chemical3\n 0 1 2 3 4\n 1  2  3  4  5MAE (meV / f.u.)\nSample NoKKR (disordered) T=0 K\nKKR (disordered) T=293 K (est)\nexperimental\nFIG. 2: Crosses (solid line): Calculated MAE per formula unit\nfor each of the FePt samples in Table I modelled as partially\ndisordered alloys with the degree of disorder given by the\nexperiment. Circles (dashed line): The same values scaled\ndown by a factor of 0.6 in order to account for temperature\ninduced e\u000bects. The experimental values are also displayed\nby stars (dotted line).\n0.200.300.400.500.600.700.800.901.00\n0.70 0.75 0.80 0.85 0.90MAE (meV / f.u.)\nChemical order parameter, s123\n4\n5\nFIG. 3: Magnetic anisotropy energy (MAE) calculated as a\nfunction of chemical order parameter, s, for the FePt samples:\n1 +, 2 \u0003, 3\u0004, 4\u0002, 5\u000f. Solid lines serve as a guide for the\neye. Open circles are placed at the best-\ft chemical order\nparameter for each of the samples. Dashed line: Linear \ft.\norder parameter, s, as a \ftting parameter. In Fig. 3, for\neach of the samples we present the calculated MAE foran appropriate set of chemical order parameters. Firstly,\nfor a given sample, i.e. for \fxed lattice parameters, the\ntheoretical MAE shows a non-linear dependence on s. In\nFig. 3 the circles indicate the intersection of the calcu-\nlations with the experimental values for each sample as\nindicated. This determines the best-\ft order parameter\nthat corresponds to the experimental MAE value. As\ncan be clearly inferred from Fig. 2, for samples no. 2, 3\nand 4, a smaller degree of chemical order was \ftted than\npredicted by the experiment, namely, s'0:836;0:874\nand 0.863, respectively. In contrast, for samples no. 1\nand 5 an increased degree of chemical order, s'0:782\nand 0.720, was obtained. Although for a given sample\nthe theoretical MAE shows a non-linear dependence on\ns, there is a nearly perfect linear correlation between the\nexperimental MAE and the best-\ft chemical order pa-\nrameters as indicated by the dashed line in Fig. 3. Ob-\nviously, this remarkable linear behavior is the result of\na subtle interplay of the dependence of the MAE on the\nlattice distortion and the chemical disorder. This is prob-\nably speci\fc to the data set investigated here rather than\nbeing a general property.\nIn summary, our \frst principles calculations imply that\nlattice distortion in the FePt samples has only a minor\ne\u000bect on the MAE, even opposite to the experimental\ntrend. Calculating the MAE using the highly approx-\nimate experimental chemical order parameters signi\f-\ncantly improves the agreement between theory and ex-\nperiment, in particular with regards to the relative dif-\nferences in the MAE between the samples. This indicates\nthat the substrate-sample lattice mismatch e\u000bect on the\nMAE reported by Ding et al.3is mainly due to the varia-\ntion in chemical disorder. To circumvent the uncertainty\nof the experimental determination of chemical disorder,\nwe, furthermore, determined theoretical chemical order\nparameters that reproduced the experimental MAE val-\nues. Interestingly, a linear correlation between the MAE\nand the best-\ft chemical order parameters is found. It\nshould be mentioned that work is underway to perform\nconstrained Monte-Carlo simulations of K(T) for chem-\nically disordered FePt, since this is clearly an important\nfactor in relation to experimental data.\nFinancial support was provided by the Hungarian Re-\nsearch Foundation (contract no. OTKA K77771) and\nby the New Sz\u0013 echenyi Plan of Hungary (Project ID:\nT\u0013AMOP-4.2.1/B-09/1/KMR-2010-0002). CJA is grate-\nful to EPSRC for the provision of a research studentship.\n1S. Okamoto, N. Kikuchi, O. Kitakami, T. Miyazaki, Y. Shi-\nmada and K. Fukamichi, Phys. Rev. B 66, 024413 (2002).\n2J.B. Staunton, S. Ostanin, S.S.A. Razee, B. Gy-\nor\u000by, L. Szunyogh, B. Ginatempo and E. Bruno,\nJ. Phys.: Cond. Mat. 16, S5623 (2004).\n3Y.F. Ding, J.S. Chen, E. Liu, C.J. Sun and G.M. Chow,\nJ. App. Phys. 97, 10H303 (2005)\n4J.A. Christodoulides, P. Farber, M. Daniil, H. Okumura,G.C. Hadjipanayis, V. Skumryev, A. Simopoulos and\nD. Weller, IEEE Trans. Mag. 37, 1292 (2001).\n5J. Korringa, Physica 13, 392 (1947).\n6W. Kohn and N. Rostoker, Phys. Rev. 94, 1111 (1954).\n7J. Zabloudil, R. Hammerling, L. Szunyogh and P. Wein-\nberger, Electron Scattering in Solid Matter (Berlin,\nSpringer 2005).\n8P. Soven, Phys. Rev. 156, 809 (1967).4\n9B. Gy or\u000by, Phys. Rev. B 5, 2382 (1972).\n10S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58,\n1200 (1980).\n11H. Ebert, H. Freyer and M. Deng, Phys. Rev. B 55, 3100\n(1997).\n12H. J. Jansen, Phys. Rev. B 59, 4699 (1999).\n13J.B. Staunton, S. Ostanin, S.S.A. Razee, B.L. Gyor\u000by, L.Szunyogh, B. Ginatempo, and E. Bruno, Phys. Rev. Lett.\n93, 257204 (2004).\n14O.N Mryasov, U. Nowak, K.Y. Guslienko and\nR.W. Chantrell, Europhysics Letters 69, 805 (2005).\n15A. B. Shick and O. N. Mryasov, Phys. Rev. B 67, 172407\n(2003)."    },    {        "title": "1108.5870v1.Spin_orbit_coupling_effect_in__Ga_Mn_As_films__anisotropic_exchange_interactions_and_magnetocrystalline_anisotropy.pdf",        "content": "arXiv:1108.5870v1  [cond-mat.mtrl-sci]  30 Aug 2011Spin-orbit coupling effect in (Ga,Mn)As films: anisotropic e xchange interactions and\nmagnetocrystalline anisotropy\nS. Mankovsky1, S. Polesya1, S. Bornemann1, J. Min´ ar1, F. Hoffmann2, C. H. Back2, and H. Ebert1\n1Department of Chemistry/Phys. Chemistry, LMU Munich,\nButenandtstrasse 11, D-81377 Munich, Germany and\n2Department of Physics, Universit¨ at Regensburg, 93040 Reg ensburg, Germany\nThe magneto-crystalline anisotropy (MCA) of (Ga,Mn)As film s has been studied on the basis of\nab-initio electronic structure theory by performing magne tic torque calculations. An appreciable\ncontribution to the in-plane uniaxial anisotropy can be att ributed to an extended region adjacent\nto the surface. Calculations of the exchange tensor allow to ascribe a significant part to the MCA\nto the exchange anisotropy, caused either by a tetragonal di stortion of the lattice or by the presence\nof the surface or interface.\nPACS numbers: 75.50.Pp, 75.30.Gw, 73., 75.70.-i\nDiluted magnetic semiconductors (DMS) are a class\nof materials having attractive properties for spintronic\napplications (e.g. see review [1]). Many investigations\nin this field are focussed on the (Ga,Mn)As DMS sys-\ntem with 1 to 10% of Mn atoms which have promising\nfeatures from a physical as well as technological point\nof view. The crucial role of valence states with respect\nto various magnetic properties of (Ga,Mn)As was dis-\ncussed in the literature by many authors [1, 2]. First\nof all, the valence band holes are responsible for ferro-\nmagnetic (FM) order in the system mediating the ex-\nchange interaction between well localized Mn magnetic\nmoments. Spin-orbit coupling of the states at the top\nof valence band, being close to the Fermi level, leads\nto a rather strong cubic magnetocrystalline anisotropy\n(MCA) in bulk (Ga,Mn)As and to an in-plane biaxial\nMCA in the (Ga,Mn)As film on top of a GaAs sub-\nstrate [3]. In the latter case the spin-orbit coupling\n(SOC) makes the valence states close to EFsensitive to\nlattice distortions and in that way responsible for the in-\nplane MCA due to compressive strains originating from\nthe lattice mismatch between the (Ga,Mn)As film and\nGaAs substrate [4–14]. As soon as the spin polariza-\ntion of the valence bands is rather small, the MCA in\n(Ga,Mn)As is discussed in terms of anisotropic exchange\ninteractions of the Mn atoms [2–4]. The strength of the\nMCA depends on the hole concentration introduced by\nthe Mn impurity atoms [3, 8, 15, 16] as well as on the\nvariation of equilibrium lattice parameter of (Ga,Mn)As\nincreasing with the increase of Mn content and resulting\nin a larger lattice mismatch with the GaAs substrate.\nNumerous experimental results evidenced a transition\nfrom the bi-axial to the uni-axial in-plane anisotropy [8–\n15]. So far, however, there is no consensus in the lit-\nerature concerning the origin of the in-plane uniaxial\nanisotropy. Although in some recent theoretical works\ntheoriginoftheuniaxialin-planeanisotropyisattributed\nto a trigonal distortion caused by a uniaxial or shear\nstrain within the film plane [10, 15, 17], this type of dis-\ntortion was not observed experimentally. Stacking faultdefects in the (111) and (11 1) planes have been found re-\ncently in experiment [18] which could be responsible for\nbreaking the equivalence of the [110] and [1 10] directions\nin the (Ga,Mn)As films. However, there is so far no ex-\nperimental evidence nor theoretical description showing\nthat these stacking faults are responsible for the in-plane\nuniaxial anisotropy.\nIn order to obtain a more detailed understanding of\nthe subtle electronic effects which determine the MCA\nproperties of (Ga,Mn)As films, we performed ab-initio\nelectronic structure calculations for tetragonally dis-\ntorted (Ga,Mn)As bulk as well as (Ga,Mn)As films de-\nposited on a GaAs substrate. The ab-initio calculations\nhave been performed within the framework of the lo-\ncal spin density approximation (LSDA) of density func-\ntional theory (DFT) using the fully relativistic Korringa-\nKohn-Rostoker (KKR) multiple scattering band struc-\nture method [19, 20]. For the treatment of the chemi-\ncal disorder in (Ga,Mn)As alloys we applied the coher-\nent potential approximation (CPA). Moreover, for the\nbulk and surface calculations we used a regular /vectork-mesh\nof 63×63×63 points in the full 3D Brillouin Zone (BZ)\nand 63×63points in the full 2DBZ, respectively. For the\nangular momentum expansion of the Green’s function a\ncutoff of ℓmax= 3 was applied.\nThe study of the magneto-crystalline anisotropy\n(MCA) was performed by calculating the magnetic\ntorque/vectorT(ˆei)\ni=−∂E({ˆek})/∂ˆei׈eiacting on the mag-\nnetic moment /vector miof the atomic site i, with a unit vector\nˆei=/vector mi/|/vector mi|pointing along the direction of the magne-\ntization /vectorM. The component of the magnetic torque with\nrespect to the axis ˆ u\nTˆu(θ,φ) =−∂E(/vectorM(θ,φ))/∂θ (1)\nwas calculated from first-principles as described in [21].\nHere, the ˆ uvector specified by the angles θandφ(see\nFig. 1a) lies within the surface plane and is perpendicu-\nlar to the direction of the magnetic moment ˆ eM. For an\nuniaxial anisotropya special geometry can be used which\ngives a simple relationship between the magnetic torque2\nand the energy difference between the in-plane and out-\nof-plane magnetization directions. Setting θ=π/4, the\ntorque component Tˆugives the φdependent energy dif-\nferenceTˆu(θ=π/4,φ) =E||(φ)−E⊥. In the case of an\nin-plane anisotropy these values can also be used to eval-\nuate the anisotropy energy within the plane, comparing\nin particular the directions [110] and [1 10].\nThe exchange coupling tensor Jijused below for the\ndiscussions of the magnetic anisotropy in terms of the\nanisotropic Mn-Mn exchange interactions [2–4] was cal-\nculated as described in Ref. [22]. Here, the effective co-\nefficients of the uniaxial MCA are represented by the fol-\nlowing expression [23, 24]:\n˜Ki=−/summationdisplay\nj(Jzz\nij−Jxx\nij)+2Ki, (2)\nwithKibeing the on-site MCA coefficients [23].\nIn order to study the strain-induced effect in the MCA\nof deposited (Ga,Mn)As films, we consider at first a bulk\nsystem with tetragonal distortion (avoiding surface and\ninterface contributions) which is then characterized by\nthec/aratio. Magnetic torque calculations simulat-\ning the strain-induced effects in the alloy with 5% Mn\nyieldalinearvariationofthe magneticanisotropyenergy,\nE[100]−E[001], from +3 .38 to−3.37µeV per unit cell for\nc/aratiovaryingfrom0.99to1.01, i.e. the magneticeasy\naxis changes from an out-of-plane to an in-plane orien-\ntation, which is in line with corresponding experimental\ndata [25]. As the [100] and [010] directions are equiva-\nlent, this leads to the bi-axial in-plane MCA with [100]\nand [010] being easy magnetization directions and an in-\nplane anisotropy energy E[100]−E[110]≈ −0.1µeV per\nunit cell.\nFor a more detailed analysis of the relationship be-\ntween the MCA and anisotropy of Mn-Mn exchange in-\nteractions, calculations of the exchange coupling tensor\nelements Jαβ\nijhave been performed for (Ga,Mn)As with\n5% Mn both without any distortion as well as with a\ntetragonal distortion of c/a= 1.01. For an undistorted\n(Ga,Mn)As system we find that the sum/summationtext\nj(Jαα\nij−Jββ\nij)\n(α,β=x,y,z)overalllatticesitesintheexpressiongiven\nin Eq. (2) vanishes (see Fig. 1b). This is a consequence\nof the system’s symmetry, in spite of the fact that the in-\ndividual terms ( Jαα\nij−Jββ\nij) withα/ne}ationslash=βare non-zero. In\nthe presence of a tetragonal distortion along the z-axis,\nthe symmetry properties within the xyplane (i.e. (001)\nplane) do not change. Therefore, summation over all lat-\nticesitesup to Rij= 5a(with latticeparameter a)shown\nin Fig. 1c gives/summationtext\nj(Jxx\nij−Jyy\nij) = 0. For more details,\nFig. 1d shows the differences Jxx\nij−Jyy\nijfor/vectorRijtaken\nalong [100] and [010] directions (dashed lines). These\nvalues are finite and equal in magnitude, but they have\nan opposite sign and therefore cancel each other upon\nsummation over all sites. However, due to the tetragonal\ndistortion along z, (Jzz\nij−Jyy\nij) for/vectorRijtaken along [001][001]z\nx[100]y\n[010]θ\nφM\na)\n0 500 1000-0.100.1∆Jij (meV)Jxx - Jyy\nJzz - Jyy\n0 500 1000\nNo. of surrounding atoms-10-50510Σj ∆J0j  (µeV)Σj(Jxx - Jyy)\nΣj(Jzz - Jyy)\n1 2 3 4 5 6 7 8\nRij (units of lattice parameters)-0.0500.05Jαα - Jyy (meV)[001], (Jzz - Jyy)\n[010], (Jzz - Jyy)\n[100], (Jxx - Jyy)\n[010], (Jxx - Jyy)b)\nc)\nd)\nFIG. 1. a) Geometry for the torque calculations; b) Jzz−Jyy\nandJxx−Jyyfor bulk (Ga,Mn)As with 5% Mn, with tetrago-\nnaldistortion c/a=1.01; c)/summationtext\nj(Jzz\nij−Jyy\nij)and/summationtext\nj(Jxx\nij−Jyy\nij)\nover all lattice sites up to Rij≤5a; d)Jxx−Jyyalong [100]\nand [010] directions.\nand [010] directions are not equivalent (see Fig. 1d, solid\nlines)andthusthesum/summationtext\nj(Jzz\nij−Jyy\nij)overalllatticesites\ndoes not vanish anymore. The summation over all lattice\nsites up to Rij≤5ais shown in Fig. 1c which gives the\ncontribution to the uniaxial MCA that originates from\nthe exchange anisotropy being ≈2.5µeV. Because of the\nslow convergence of the sum with increasing distance,\nthis gives only an approximation to the true contribu-\ntion due to the exchange anisotropy. Nevertheless, the\nvalue obtained in this way has the same order of magni-\ntude as the MAE obtained from our torque calculations\nleading to the conclusion that the exchange anisotropy\nhas indeed a significant impact on the total MAE.\nOur present investigations of the in-plane uniaxial\nanisotropy have been performed for a 8 monolayer (ML)\nthick (Ga,Mn)As film deposited on a semi-infinite (001)-\noriented GaAs substrate. In order to distinguish the\nanisotropy behaviour in the vicinity to the interface with\nGaAs as well as in the area adjacent to the surface we\nperformed calculations for an uncovered (Ga,Mn)As film\nas well as one with two additional capping layers of Au.\nDuetosmallamountoffreechargecarriersin(Ga,Mn)As\nthe surface potential decays slowly into bulk leading to3\na potential and a charge density gradient within an ex-\ntended region adjacent to the surface. The existence of\nsuch a potential gradient results in the breaking of the\n4-fold symmetry of the bulk (Ga,Mn)As system, making\nthe [110] and [110] directions inequivalent (for the geom-\netry used here this corresponds to the xandydirections,\nrespectively) and leading effectively to a C2vsymmetry\nnot only within the few surface/interface layers but also\nin a rather extended subsurface regime.\nWe discuss now the surface induced MCA in the\nfilm. Here we focus mainly on the MAE properties of\na (Ga,Mn)As film with a clean Ga terminated surface\ndeposited on GaAs(001). The results for the energy dif-\nferences between different magnetization directions are\nE[110]−E[001]=−80.56µeV and E[110]−E[001]=\n−32.96µeV per film unit cell (8 ML). This gives an uni-\naxial in-plane anisotropy with the energy difference of\nE[110]−E[110]=−47.6µeV per film unit cell.\nFig. 2a presents the layer resolved contributions to\ntheE[110]−E[001]andE[110]−E[001]values, indicated\nby open and filled symbols, respectively. The difference\nbetween these values characterizes the MCA within the\nplane. One should emphasize here that the contribu-\ntion to the MCA from the region close to the surface de-\ncays slowly into the bulk. Therefore the surface-induced\nanisotropy effect in the uniaxial in-plane MCA is deter-\nmined by a rather extended region adjacent to the sur-\nface and not just by two or three subsurface layers as\nit is often observed in metallic systems (e.g. [26]). The\ncorresponding contribution to the energy of the uniaxial\nin-plane anisotropy exceeds by far the energy of the bi-\naxial in-plane anisotropy when normalized to the same\nvolume ( E[100]−E[110]≈ −0.1µeV per unit cell of the\nbulk system). Using these results the MCA of exper-\nimental (Ga,Mn)As films consisting of n+ 8 monolay-\ners can be modelled by combining the contribution of n\nbulk-like layers with the contribution of 8 layers of sur-\nface region. This gives two competing contributions to\nthe MCA: a bi-axial in-plane anisotropy from bulk-like\nlayers of (Ga,Mn)As with a tetragonal distortion and a\nuniaxial in-plane anisotropy from the area adjacent to\nthe surface. Applying our obtained MAE values to a\nunit volume, one can get the MCA of the whole film\nincluding the surface region. Within our consideration,\nthe coefficient of the in-plane bi-axial anisotropy K4does\nnot depend on the film thickness L, while the the coef-\nficient of the in-plane uni-axial anisotropy K||\n2recalcu-\nlated per unit volume should decrease with film thick-\nness as 1 /L. Thus, according to our numerical results, a\nrather strong uniaxial anisotropy should be observed in\nthe case of very thin films, while the increase of the film\nthicknessshouldleadtoacompetitionofbi-axialanduni-\naxialanisotropiesbeginning with acertainfilm thickness.\nThe contribution from the ’surface’ region to the out-of-\nplane uniaxial anisotropy E[110]−E[001]decreases as wellwith the film thickness as 1 /L. This results in a leading\nrole of the in-plane anisotropy contribution caused by\nthe tetragonal lattice distortion discussed above. Note\nthat an increase of the Mn concentration results in an\nincrease of the charge carriers in the film which again\nresults in better screening of the surface potential. This\ncan be seen in Fig. 2a, where the values E[110]−E[001]\nandE[110]−E[001]are shown as a function of the dis-\ntance from the surface for a (Ga,Mn)As film with 11%\nMn. This increase in Mn concentration results in an in-\nplane MAE E[110]−E[110]=−20.8µeV per film unit\ncell, i.e. one obtains a smaller anisotropy energy when\ncompared to the case of 5%Mn.\nSince the uniaxial MCA has its origin in an extended\nsubsurface region one can expect that it is an intrin-\nsic property of the systems and should be observed not\nonly in the case of a clean surface but also in the pres-\nence of overlayers on the top of the (Ga,Mn)As film.\nCorresponding investigations have been performed for a\n(Ga,Mn)As film with 3 capping layers of Au on top of\nthe (Ga,Mn)As film. The resulting layer resolved con-\ntribution to the MCA is shown in Fig. 2b. In spite\nof the differences in the MAE between the Au capped\n(Ga,Mn)As film and the case of uncovered film, the gen-\neral trend in both cases is the same, i.e. one can clearly\nsee that the difference in layer contributions to the MCE\nfor different directions of magnetization, along [110] and\n[110], decays slowly with the distance from the surface or\nAu/(Ga,Mn)As interface, respectively.\nTo investigate also the effect caused by a concentra-\ntion gradient along the surface normal within an un-\ncovered (Ga,Mn)As film we dealt with a corresponding\nfilm where the Mn concentration varies from 5% at the\n(Ga,Mn)As/GaAs interface to 6.6% in the surface layer.\nAs can be seen in Fig. 2a such a gradient does not result\nin a noteworthy change in the MCA.\nTo analyze in more detail the origin of the surface-\ninduced in-plane uniaxial anisotropy the contribution of\nthe exchange interaction anisotropy in the (Ga,Mn)As\nfilm was determined. Fig. 3 shows the difference Jxx\nij−\nJyy\nijcalculated along the [110] and [1 10] directions within\nthe film layers where the xandyaxes are chosen along\n[110] and [110] directions, respectively. As discussed\nabove,forbulk(Ga,Mn)Asthevariationof Jxx\nij−Jyy\nijwith\ndistance |/vectorRij|isthesamefor /vectorRijalongthe [110]and[1 10]\nhowever with different sign. This behaviour is more or\nlessthesameforMnatomsnexttothe(Ga,Mn)As/GaAs\ninterface (see Fig. 3a). For Mn in the fourth layer (chos-\ning the surface layer as the first layer), however, the situ-\nation is changed indicating a pronounced modification of\ntheanisotropicexchangecouplingduetothe brokensym-\nmetry. As a result, the sum over lattice sites in Eq. (2)\ndoes not vanish which leads to a contribution to the in-\nplane uniaxial anisotropy. Fig. 3b shows the correspond-\ning results obtained by summing the terms ( Jxx\nij−Jyy\nij)4\n1: Surface 2 3 4 5 6 7 Interface\nLayer index-0.04-0.0200.020.040.060.08E - E[001] (meV)Mn: x=0.11\nMn: x=0.11\nMn: x=0.05\nMn: x=0.05\nMn: grad x\nMn: grad x[11--0]\n[110]\na)\n1: Surface 2 3 4 5 6 7 Interface\nLayer index-0.04-0.0200.020.04E - E[001] (meV)[11--0]\n[110]\nb)\nFIG. 2. a) Layer resolved contributions to the MCA energy in\nthe uncovered 8ML (Ga,Mn)As film with 5 at.% Mn (circles)\nand11(triangles) at.% Mn, aswell as withMncontentvarying\nfrom5at.%attheinterfaceto6.6at.%insurfacelayer( gradx,\nsquares), for two directions of magnetization: /vectorM||[110] and\n/vectorM||[110]; b) layer resolved contributions to the MCA energy\nin the 8ML (Ga,Mn)As film with 5 at.% Mn, with 3 capping\nlayers of Au.\nover all lattice sites jwithin a sphere of radius 2 .9awith\nitaken within the layers 1 −8 in the (Ga,Mn)As film. As\none can see, the anisotropy of the exchange interaction\ngives indeed a substantial contribution to the anisotropy\nenergyE[110]−E[110]for the layers 3 to 8. For the first\ntwo film layers. i.e. surface and subsurface layers the two\ncurves strongly deviate reflecting the dominating on-site\ncontribution to the MCA [23, 24].\nIn summary our results show that the tetragonal dis-\ntortion (caused by a compressive strain due to lattice\nmismatch of (Ga,Mn)As and GaAs lattices) is respon-\nsible for the bi-axial in-plane anisotropy that is in line\nwith the interpretation given in previous investigations.\nAstronguniaxialin-planeMCAwasfoundin(Ga,Mn)As\nfilm in the area adjacent to the surface or an interface.\nWe conclude that this is a result of the slow decay of the\nsurfacepotential gradientdue to the smallamount offree\nchargecarriers. The contributionto the uniaxial in-plane\nanisotropy decays rather slowly into the bulk and is not\nrestrictedto only a few surface layers. Moreover,a signif-\nicant contribution responsible for the MCA in the films is\ncaused by the anisotropic Mn-Mn exchange interactions\nmediated by holes in the valence band of (Ga,Mn)As.\nAcknowledgements\nFinancial support by the DFG through the SFB 689 is\ngratefully acknowledged.1 1.5 2 2.5 3\nRij (units of lattice parameters)-0.100.1Jxx-Jyy (meV)R || [--110]\nR || [110]\nBulkInterface\na)\n1 1.5 2 2.5 3\nRij (units of lattice parameters)-0.1-0.08-0.06-0.04-0.0200.020.040.06Jxx-Jyy (meV)R || [--110]\nR || [110]Bulk\nSurface-3\nb)\n1: Surface 2 3 4 567Interface\nLayer-0.04-0.03-0.02-0.0100.010.02∆E (meV)\n-Σ(Jyy-Jxx)E[110]  - E[11- 0]\nc)\nFIG. 3. 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L. Gyorffy, S. Ostanin, and L. Udvardi,\nPhys. Rev. B 74, 144411 (2006).\n[22] H. Ebert and S. Mankovsky,\nPhys. Rev. B 79, 045209 (2009).\n[23] L. Udvardi, L. Szunyogh, K. Palot´ as, and P. Weinberger ,\nPhys. Rev. B 68, 104436 (2003).\n[24] S. Mankovsky, S. Bornemann, J. Min´ ar, S. Polesya,\nH. Ebert, J. B. Staunton, and A. I. Lichtenstein,\nPhys. Rev. B 80, 014422 (2009).\n[25] J. Daeubler, S. Schwaiger, M. Glunk, M. Ta-\nbor, W. Schoch, R. Sauer, and W. Limmer,\nPhysica E 40, 1876 (2008).\n[26] M. Koˇ suth, Magnetic properties of transition metal sur-\nfaces and GaAs/Fe heterogeneous systems , Ph.D. thesis,\nUniversity of Munich (2007)."    },    {        "title": "1108.5915v1.Electrically_tunable_quantum_anomalous_Hall_effect_in_5d_transition_metal_adatoms_on_graphene.pdf",        "content": "Electrically tunable quantum anomalous Hall e\u000bect in 5dtransition-metal adatoms on\ngraphene\nHongbin Zhang1,\u0003Cesar Lazo2, Stefan Bl ugel1, Stefan Heinze2, and Yuriy Mokrousov1\n1Peter Gr unberg Institut and Institute for Advanced Simulation,\nForschungszentrum J ulich and JARA, D-52425 J ulich, Germany and\n2Institute of Theoretical Physics and Astrophysics, University of Kiel, D-24098 Kiel, Germany\n(Dated: today)\nThe combination of the unique properties of graphene with spin polarization and magnetism for\nthe design of new spintronic concepts and devices has been hampered by the small Coulomb interac-\ntion and the tiny spin-orbit coupling of carbon in pristine graphene. Such device concepts would take\nadvantage of the control of the spin degree of freedom utilizing the widely available electric \felds\nin electronics or of topological transport mechanisms such as the conjectured quantum anomalous\nHall e\u000bect. Here we show, using \frst-principles methods, that 5 dtransition-metal (TM) adatoms\ndeposited on graphene display remarkable magnetic properties. All considered TM adatoms possess\nsigni\fcant spin moments with colossal magnetocrystalline anisotropy energies as large as 50 meV per\nTM atom. We reveal that the magneto-electric response of deposited TM atoms is extremely strong\nand in some cases o\u000bers even the possibility to switch the spontaneous magnetization direction by\na moderate external electric \feld. We predict that an electrically tunable quantum anomalous Hall\ne\u000bect can be observed in this type of hybrid materials.\nSpin-orbit interaction, which couples the spin degree\nof freedom of electrons to their orbital motion in the\nlattice, leads to many prominent physical phenomena,\nsuch as the antisymmetric exchange interaction [1], the\ncolossal magnetic anisotropy [2], or the anomalous Hall\ne\u000bect [3] in conventional ferromagnets. It is also the\nkey interaction in the newly-found quantum topologi-\ncal phase in topological insulators, where the quantum\nspin Hall e\u000bect was observed experimentally [4], and the\nquantum anomalous Hall e\u000bect (QAHE) was predicted\nto exist [5, 6]. Since the orbital motion can be manip-\nulated with external electric \felds, spin-orbit coupling\n(SOC) opens a route to dissipationless transport and to\nelectrical control of magnetic properties [7{9], playing a\ncrucial role in future spintronics applications.\nOwing to its strong spin-orbit coupling, heavy 4 dand\n5dTMs display fascinating physical properties for desir-\nable spintronic applications, especially when combined\nwith non-vanishing magnetization. However, magnetism\nof 5dTMs proved di\u000ecult to achieve due to their more\ndelocalized valence dwavefunctions and the smaller intra-\natomic exchange integrals compared to 3 dTMs such as\nFe, Co, or Ni. Low-dimensional structures, such as mono-\nlayers [10], atomic wires [11], and clusters [12], were pro-\nposed to facilitate the stability of 5 dmagnetism by reduc-\ning thed-dhybridization. Nevertheless, when deposited\non substrates of noble or late transition metals, such as\nCu, Ag, Au, and Pt, interdi\u000busion at interfaces is in-\nevitable, and strong hybridization between the dstates\nof the substrate and of the adatoms is also destructive\nfor 5dmagnetism. From this point of view, using spsub-\nstrates is more promising. In fact, 4 dferromagnetism\nwas \frst observed in a Ru monolayer deposited on the\ngraphite (0001) surface [13], which is close in its chemi-\ncal and physical properties to graphene.Since the day graphene was isolated and produced as\na two-dimensional material [14], it abruptly altered the\nresearch direction of material science [15] with the aim\nof exploring its fascinating transport properties. In a\nsense, graphene serves as a prototype of topological in-\nsulators [16]. For instance, the Berry phase of \u0019of elec-\ntronic states in graphene induces a half-integer quantum\nHall e\u000bect [17, 18], while the existence of the quantum\nspin Hall e\u000bect was \frst suggested for pure graphene\nwhen SOC is taken into account [19]. One common\nfeature for those transport properties is the non-trivial\ntopological origin, resulting in dissipationless charge or\nspin current carried by edge states with conductivity\nquantized in units of e2=h. However, from the appli-\ncation point of view, a large external magnetic \feld is\nrequired to obtain the quantum Hall e\u000bect, and the spin-\ndegeneracy in the quantum spin Hall e\u000bect makes it hard\nto manipulate the spin degree of freedom by controlling\nexternal \felds. To avoid such constraints while keeping\nthe bene\ft of topologically protected quantized topolog-\nical transport, the long-sought QAHE is a perfect solu-\ntion. The essence of the QAHE lies in the quantization\nof the transverse charge conductivity in a material with\nintrinsic non-vanishing magnetization. The fact that the\nmagnetization in ferromagnets can be much easier han-\ndled experimentally than large magnetic \felds makes the\nQAHE extremely attractive for applications in spintron-\nics and quantum information. However, at present the\nQAHE is merely a generic theoretical concept for mag-\nnetically doped topological insulators [5, 6]. Recently,\nQiao et al. suggested that the QAHE could also occur at\ncomparatively low temperatures in graphene decorated\nwith Fe adatoms [20].\nHere, we demonstrate based on \frst-principles theory\nthat 5dTMs deposited on graphene are strongly mag-arXiv:1108.5915v1  [cond-mat.mtrl-sci]  30 Aug 20112\nnetic, provide colossal magneto-crystalline anisotropy en-\nergies and exhibit topologically non-trivial band gaps due\nto very strong spin-orbit interaction. A generic represen-\ntative of this hybrid class of materials has a magneto-\ncrystalline anisotropy energy of 10 \u000030 meV per TM and\na quantum anomalous Hall gap in its electronic spectrum\nwith the size of 20 \u000080 meV. In connection with a large\nmagneto-electric response of the deposited adatoms, we\npredict that within this class of systems an electrically\ntunable QAHE at room temperature could be achieved\nexperimentally.\nWe have performed \frst-principles calculations of 5 d\nTM (Hf, Ta, W, Re, Os, Ir, and Pt) adatoms on graphene\nin 2\u00022, 3\u00023, and 4\u00024 supercell geometries corresponding\nto the deposition density of 4.7, 2.1 and 1.2 atoms/nm2,\nrespectively (see the Method part for details). Through-\nout this work the TM atoms are placed at the hollow\nsites of graphene. According to previous theoretical stud-\nies [21], it is the most favorable absorption site with the\nexceptions of Pt [22] and Ir [23], for which the bridge\nsite is more preferable. Among all systems studied we\nselected one prototype system, W on graphene, that we\ndiscuss in more detail. In the 4 \u00024 geometry, the opti-\nmized distance between W adatoms on the hollow site\nand the C plane is about 1.74 \u0017A, and the hollow site is\nabout 0.14 eV (0.41 eV) per TM atom lower in energy\nthan the bridge (top) site, in agreement with previous\nobservations [24]. Applying an electric \feld, the relaxed\npositions change by at most 0.01 Bohr radii, and the hol-\nlow site remains the preferred adsorption site.\nTurning now to magnetism, and looking \frst at TM\natoms adsorbed in the 4 \u00024 supercell, we \fnd that most\nof the 5dTM atoms (except for Pt) are magnetic with\nsizable magnetic moments (see Fig. 1a) ranging between\n0.5 and 2\u0016B. This is consistent with the large magnetiza-\ntion energy calculated, de\fned as the total energy di\u000ber-\nence between the nonmagnetic and ferromagnetic state,\nreaching e.g. for Ta and W values as high as 0.56 eV\nand 0.34 eV, respectively. Surprisingly, the magnetic mo-\nments across the series of the rather isolated TM atoms\ndo not follow Hund's \frst rule, i.e. the magnetic mo-\nments do not increase in steps of 1 \u0016Bfrom Hf to Re,\nreaching a maximum at the center of the TM series and\ndecreasing again from Re to Pt. On the contrary we\n\fnd the minimal spin moment for Re. This demonstrates\nthe role of the hybridization of the TM d-orbitals with\ngraphene. Increasing the deposition density studying a\n3\u00023 supercell, the results remain basically unaltered, in-\ndicating that the d-dhybridization between neighboring\nadatoms is week and that the physics is determined by\nthe local TM-graphene bonding. This changes in the\ncase of the 2\u00022 geometry, when the adatoms are in close\nvicinity to each other and the magnetism survives only\nfor Ta, W, Re, and Ir atoms.\nUnique to the 5 dTMs is the property that the SOC and\nthe intra-atomic exchange are of the same magnitude.From this we anticipate a signi\fcant in\ruence of SOC on\nthe electronic properties, leading to strong anisotropies\nof the calculated quantities upon changing the direction\nof magnetization Mrelative to the graphene plane.\nThe most impressive manifestation of SOC in the TM-\ngraphene hybrid materials are the colossal values of the\nmagneto-crystalline anisotropy energy (MAE), de\fned as\nthe total-energy di\u000berence between the magnetic states\nwith spin moments aligned in the plane and out of the\ngraphene plane. The MAE presents one of the most fun-\ndamental quantities of any magnetic system as its sign\nde\fnes the easy magnetization axis and its magnitude\ngives an estimate on the stability of the magnetization\nwith respect to temperature \ructuations \u0000a key issue\nfor magnetic storage materials. A large MAE makes the\nmagnetization very stable but also di\u000ecult to manipu-\nlate.\nThe values of the MAE obtained in 3 \u00023 and 4\u00024 ge-\nometry, presented in Fig. 1b, lie in the range of 10 to\n50 meV per TM. These are orders of magnitude larger\nthan the values for 3 dTMs, irrespective of whether they\nare arranged in bulk, thin \flms or multilayers. Simi-\nlar colossal MAE magnitude has been reported for the\nfree-standing 4 dand 5dTM chains [11] as well as dimers\nof TMs [25]. Characteristic for the colossal MAE [2] is\nthe signi\fcant contribution from the variation of mag-\nnetic moments and corresponding magnetization energies\nwhen changing the direction of M, a contribution that is\nabsent for conventional 3 dmagnets. For instance, for\n4\u00024 W on graphene, this variation of the spin moment\nreaches as much as 0.1 \u0016B(Fig. 2a). By looking at the\nMAE values as the function of the band \flling and de-\nposition density in Fig. 2b, we \fnd that, in contrast to\nthe spin moments, the MAE exhbits a much more ir-\nregular behavior as a function of the band \flling and\ndepends much more sensitively on the adatom density.\nE.g. for W the MAE changes sign depending on the den-\nsity of adatoms, and has a small value in the 4 \u00024 geome-\ntry. Such a situation provides a possibility to manipulate\nthe magnetization direction with weak external perturba-\ntions, which can be particularly important for transport\napplications as we demonstrate below.\nIn search for ability to manipulate the magnetic prop-\nerties of adatoms by external \felds, we show in Fig. 2\nthe dependence of the spin moments and the MAE of\nW adatoms in 4\u00024 geometry on the strength of the elec-\ntric \feldE, applied perpendicularly to the graphene layer\n(along thez-axis). Remarkably, the spin moment of W\ndisplays a strong dependence on the \feld strength, espe-\ncially when the magnetization points out of plane. Qual-\nitatively, this response can be characterized by a magne-\ntoelectric coe\u000ecient \u000b, which relates the change in the\nspin moment to the strength of the E-\feld in the zero-\n\feld situation: \u00160\u0001\u0016S(W) =\u000bE, where\u00160is the vacuum\nmagnetic permeability constant. For 4 \u00024 W on graphene\nand an out-of-plane magnetization, \u000b?amounts to about3\n00.51.01.52.0(a)µS(µB)22\n33\n44\n−40−20020(b)MAE (meV/ u.c.)\nHf Ta W Re Os Ir Pt\nFIG. 1: Magnetic moments and MAE of 5 dadatoms on\ngraphene. (a) Magnetic moments due to electron spin, \u0016S,\ncalculated without SOC and (b) the MAE of 5 dTM adatoms\non graphene in 2 \u00022, 3\u00023 and 4\u00024 superlattice geometry.\nPositive (negative) values of MAE imply an out-of-plane (in-\nplane) easy axis of the spin moments, i.e. perpendicular to\n(in) the graphene plane.\n3\u000110\u000013G\u0001cm2/V, which is one order of magnitude\nlarger than that in Fe thin \flms [8]. For an in-plane\nmagnetization, the magnetoelectric coe\u000ecient \u000bkis only\n6\u000110\u000014G\u0001cm2/V, i.e. one order of magnitude smaller\nthan\u000b?, which underlines the strong anisotropy of this\nquantity. We observe similarly large variations of the\nmagnetoelectric coupling strength in other considered 5 d\nTMs. The variation of orbital moment with varying E-\n\feld is negligible for all systems.\nStriking is the e\u000bect of the electric \feld on the magne-\ntization direction (see Fig. 2b). At zero \feld ( E= 0) the\nmagnetization is in-plane. Applying a negative \feld of\nmagnitude of 0.05 \u00000.4 V/ \u0017A, values typical in graphene\n\feld e\u000bect transistor structures [26], the sign of the\nMAE is changed. This means that the equilibrium direc-\ntion of the magnetization can be switched from in-plane\n(MAE<0, forE= 0) to out-of-plane (MAE >0 for\nE<0). Supposing that the E-\feld is completely screened\nin our system by forming a screening charge \u000eq, the vari-\nation of the MAE with respect to \u000eq,\u000eMAE=\u000eq, reaches\nas much as 28 meV/e for negative electric \felds, which is\nmore than three times larger than that on the surface of\nCoPt slabs [27], and one order of magnitude larger than\nthat of Fe slabs [8]. With a moderate out-of-plane elec-\ntric \feld of\u00060.13 V/ \u0017A switching of the magnetization\ncan be also achieved in 4 \u00024 Hf, and the MAE can be sig-\nni\fcantly altered in 4 \u00024 Os on graphene (by \u001910 meV)\n1.51.61.71.8\n(a)µS(µB)M out−of−plane\nM in−plane\n−12−10−8−6−4−202\n−0.6 −0.4 −0.2 0 0.2(b)MAE (meV/u.c.)\nε(V/A0\n)\nFIG. 2: Dependence of the magnetic moments (a) and mag-\nnetic anisotropy energy MAE (b) of 4 \u00024 W on graphene on\nthe strength of an external electric \feld. (a) The spin mo-\nment of W adatoms \u0016S(in\u0016B) as a function of the strength\nof an external electric \feld E, perpendicular to graphene sur-\nface, for the out-of-plane and in-plane magnetization, respec-\ntively. Negative values of Ecorrespond to the electric \feld\nin +z-direction, i.e. the direction from graphene towards the\nadatoms. (b) Dependence of the MAE on the strength of an\nexternal electric \feld E, sign convention of MAE is consis-\ntent to Fig. 1. Inset displays the di\u000berence of spin densities\n\u0001m(r) for the system without electric \feld and the case with\na negative electric \feld of 0.13 V/ \u0017A.\nand 4\u00024 Ir on graphene (by \u001920 meV). This establishes\n5dTM adatoms on graphene as a class of magnetic hybrid\nmaterials with high susceptibility of magnetic properties\nto the electric \feld, thus making the electrical control of\nmagnetism in these systems possible.\nThe reason for such a strong magnetoelectric response\nof 5dTMs on graphene can be exempli\fed for the case of\nW in 4\u00024 geometry. The local W s- andd-decomposed\ndensity of electron states without SOC, grouped into\n\u00011(s;dz2), \u0001 3(dxz;dyz) and \u0001 4(dxy;dx2\u0000y2) contribu-\ntions, is presented in Fig. 3a. As we can see, the W\nspin moment of about 1.6 \u0016Boriginates from two occu-\npied spin-up and two unoccupied spin-down \u0001 1-states,\nwhile the exchange-split \u0001 3- and \u0001 4-states are situated\nabove and below EF, respectively, and almost do not\ncontribute. Upon including SOC (e.g. for out-of-plane\nspins) a strong hybridization between the \u0001#\n1, and \u0001 3\nand \u0001 4states of both spin occurs around EF(see also\nFig. 3b), which results in a formation of hybrid bands of\nmixed spin and orbital character, while the \u0001\"\n1band re-\nmains mainly non-bonding (see Fig. 3c). When an elec-4\n-1.2 -0.8 -0.4 0 0.4 0.8 1.2\nE - EF (eV)-202DOS (states/eV)∆3\n∆4\n∆1\n(a)\nΓ M K Γ-0.6-0.4-0.200.20.40.6E - EF (eV)\n(b)\nΓ M K Γ-0.6-0.4-0.200.20.4E - EF (eV)\n-0.3 -0.2 -0.1 0 0.1 0.2 0.3\nE - EF (eV)-2-1012σxy (e2/h)\n(c)ΓMK\n(d)\n-1-0.5  0 0.5  1-1-0.5 0 0.5 1\n-1000-900-800-700-600-500-400-300-200-100 0 100\nΓMK\n(d)\n-1-0.5  0 0.5  1-1-0.5 0 0.5 1\nFIG. 3: Electronic structure of 4 \u00024 W on graphene. (a) Density of \u0001 1(s;dz2), \u0001 3(dxz;dyz) and \u0001 4(dxy;dx2\u0000y2) states\nwithout SOC. Up and down arrows mark spin-up and spin-down channels, respectively. (b) Band structure without SOC along\nhigh symmetry lines in the 2D Brillouin zone. Red and blue color of the bands stands for the spin-up and spin-down character,\nrespectively. Circles highlight the points at which the gap will open when SOC is considered. (c) Band structures with SOC\nforMjjz: without electric \feld (solid line), with positive (dashed) and negative (dot-dashed) electric \feld of the magnitude of\n0.13 V/ \u0017A. The inset shows the anomalous Hall conductivity of the system with respect to the position of the Fermi energy EF\nforE= 0 and Mkz. (d) Berry curvature distribution of occupied bands in the momentum space (in units of2\u0019\na, withaas the\nin-plane lattice constant of the 4 \u00024 supercell). The Brillouin zone boundaries are marked with solid lines.\ntric \feld is applied along the z-axis, the states which\nexperience most in\ruence of the corresponding poten-\ntial change are the \u0001\"\n1-states, directed perpendicularly\nto the graphene plane. Corresponding modi\fcation of\nthe band structures due to the electric \felds can be seen\nin Fig. 3c, where the \u0001\"\n1band is shifted downwards (up-\nwards) by negative (positive) applied E\felds, while the\nhybrid bands of mixed character remain almost unaf-\nfected. This causes the redistribution of the electrons\nin the \u0001\"\n1-states, its hybridization with the hybrid band\nbelow the Fermi level and hence the variation of the mag-\nnetic moment. This can be visualized by the plot of the\nspin density di\u000berence \u0001 m(r) for the case with E= 0 and\nE=\u00000.13 V/ \u0017A (inset in Fig. 2b). It is obvious that, upon\napplying an electric \feld, a certain amount of spin den-\nsity is transferred from the \u0001 1state ofdz2character to\nthe \u0001 3states. Owing to the di\u000berence in the hybridiza-tion with the graphene states, seen from the width of the\ncorresponding peaks in the density of states, the \u0001 1and\n\u00013states have di\u000berent localization inside the W atoms\nthus leading to a change in the spin moment.\nAt last we turn to the prediction of a stable QAHE,\na manifestation of the quantization of the transverse\nanomalous Hall conductivity. It is motivated by the\nobservation that for 4 \u00024 W on graphene with an out-\nof-plane magnetization, \u0001 3and \u0001 4bands of opposite\nspin cross (see circles in Fig. 3b) and hybridize under the\npresence of SOC, forming a global band gap across the\nBrillouin zone (BZ). Thus, 4 \u00024 W becomes an insulator\nupon a spin-orbit driven metal-insulator transition. We\ncompute the anomalous Hall conductivity of this system,\ngiven by\u001bxy= (e2=h)C, withCas the Chern number of\nall occupied bands that can be obtained as a k-space in-\ntegralC=1\n2\u0019R\nBZ\n(k)d2k. The integrand, \n( k), is the5\nso-called Berry curvature of all states below the Fermi\nlevel:\n\n(k) =X\nn<E FX\nm6=n2Imh nkjvxj mkih mkjvyj nki\n(\"mk\u0000\"nk)2;(1)\nwhere nkis the spinor Bloch wavefunction of band n\nwith corresponding eigen energy \"nk,viis thei'th Carte-\nsian component of the velocity operator. The Berry cur-\nvature in reciprocal space, presented in Fig. 3d, displays\na comparatively complex pattern, with positive contri-\nbutions concentrated along the BZ boundary, and large\nnegative dips at certain isolated k-points away from the\nhigh-symmetry points. Such non-trivial distribution of\nthe Berry curvature, which plays a role of an e\u000bective\nmagnetic \feld in the k-space, is typical for complex\ntransition-metal compounds [3].\nThe calculated anomalous Hall conductivity as a func-\ntion of the electron \flling expressed as energy Erelative\nto the Fermi level is presented in the inset of Fig. 3c for\nan out-of-plane magnetization, Mjjz, and zeroE-\feld.\nWe indeed \fnd that the Chern number of all occupied\nstates acquires an integer value of +2. According to the\nphysics of the quantum Hall e\u000bect, this will result in two\ndissipationless and topologically protected edge states on\neach side of a \fnite graphene ribbon with W adatoms.\nRemarkably, another 32 meV large QAHE gap with the\nC=\u00002 can be observed at the energy of \u00000.27 eV be-\nlowEF, originating from the SOC-mediated hybridiza-\ntion between the \u0001 3and \u0001 4bands of the same spin-up\ncharacter, see Figs. 3b and 3d. Such a peculiar situa-\ntion suggests that the topological properties of the QAHE\nstate in 4\u00024 W on graphene can be controlled by tuning\nthe position of EFe.g. via deposition on an appropriate\nsubstrate.\nMore importantly, at equilibrium the magnetization\ndirection of 4\u00024 W lies in-plane, rendering \u001bxy= 0 ow-\ning to the antisymmetric nature of the anomalous Hall\nconductivity with respect to the magnetization direction.\nFrom our previous observations it follows, however, that\nthe direction of Mcan be conveniently switched to out-\nof-plane by applying a moderate electric \feld without any\nsigni\fcant modi\fcation of the the electronic structure.\nThe gaps remain intact (cf Fig. 3c), its size unchanged,\nand the value of the anomalous Hall conductivity remains\nquantized at both energies.\nSuch non-trivial topological QAHE states occur also\nfor all other 5 dadatoms (except Pt) on graphene in\n4\u00024 geometry, and several of them in 2 \u00022 geometry,\nwith the QAHE gaps of comparable magnitude, although\npositioned away from the Fermi energy. For example,\ngraphene with Re deposited in 4 \u00024 geometry exhibits\nthreeC= +2 QAHE gaps at \u00001,\u00000.15 and +0.67 eV,\nwith the corresponding gap size of 38, 63 and 98 meV,\nrespectively. Also, 4 \u00024 Ir and Os on graphene show two\nQAHE gaps (C= +2 for Ir andC=\u00062 for Os) at 0.24and 0.47 eV for Ir, and \u00000.38 and 0.15 eV for Os, with\ncorresponding gap size in between 11 and 80 meV. The\nfact that several QAHE gaps with di\u000berent Chern num-\nbers at di\u000berent energies can occur in the same material\nmakes these systems a rich playground for topological\ntransport studies. Characteristic of such systems is the\nleading role of the 5 dorbitals and strong SOC of heavy\nadatoms in determining the topological properties: as we\ncan see from Fig. 3b, the creation of the QAHE gaps can\noccur at the points in the Brillouin zone away from high-\nsymmetry points due to hybridization of bands with a\nstrong TM character, which are signi\fcantly altered by\nthe TMs' SOC. This is in contrast to the cases consid-\nered previously [5, 6, 20], in which the role of the mag-\nnetic adatoms is rather to insert a Rashba spin-orbit or\nexchange \felds on the Dirac states at the Fermi energy\nand high-symmetry points in the Brillouin zone. Most\nimportantly, however, the strong SOC of 5 dTMs leads\nto an increase of the QAHE gap by an order of magni-\ntude, when compared to all QAHE systems suggested be-\nfore, guaranteeing a strong topological protection against\ndefects, structural disorder or thermal \ructuations, and\nthus opening a route to QAHE at room temperature.\nTo summarize, we predict that graphene decorated\nwith 5dtransition-metals is a hybrid material display-\ning remarkable magnetic and transport properties, which\nmay be utilized experimentally for the purposes of topo-\nlogical transport applications at reasonable temperatures\nand conditions. The fact that heavy transition-metals de-\nposited on graphene display very stable magnetism due\nto colossal magneto-crystalline anisotropies with exciting\npossibilities in engineering a robust quantum anomalous\nHall e\u000bect with a large band gap stems from the strong\nspin-orbit interaction of these elements. The extreme\nsensitivity of these hybrid systems to an electric \feld that\nwe predicted is likely to be enhanced when graphene is\ndeposited on a ferroelectric substrate. Thus 5 d-graphene\nhybrid materials provide a unique functionality in tai-\nloring the required magnetic and topological properties,\nnot likely to be achieved in conventional magnetic ma-\nterials based on 3 dtransition-metals. Considering the\nbreadth of this material class including other possible\nheavy adatoms, such as the 4 dTMs or 4f-elements, we\nanticipate that room temperature QAHE may become\npossible in these systems and we encourage intense ex-\nperimental research on these materials.\nMethod\nThe theoretical investigations are based on density\nfunctional theory [28], which is '\frst-principles' in that\nit requires no experimental input other than the nu-\nclear charges. We apply the generalized gradient ap-\nproximation [29] to the exchange-correlation potential\nand use the full-potential linearized augmented plane6\nwave method (FLAPW) as implemented in the FLEUR\ncode [30]. The self-consistent calculations with SOC were\ncarried out with the cut-o\u000b parameter k maxof 4.0 bohr\u00001\nand 144k-points (for the 4 \u00024 cases) in the full two-\ndimensional Brillouin zone. The mu\u000en-tin radius of\n2.7 bohr (1.2 bohr) was used for TM (carbon) atoms.\nTo calculate the transverse anomalous Hall conductivity\naccurately, we applied the Wannier interpolation tech-\nnique, using the FLEUR andWannier90 codes to construct\nthe maximally-localized Wannier functions [31{33]. For\nall systems, the distance between the TM atoms and the\nrigid graphene layer was structurally optimized by total\nenergy minimization neglecting SOC. We con\frmed that\nthe relaxations of the carbon atoms within the graphene\nlayer are indeed very small, and neglecting them does not\nlead to any changes in the calculated quantities. TMs\nbond covalently with graphene [21], thus we suspect that\nthe van der Waals interaction plays a minor role.\nAcknowledgments\nWe acknowledge discussions with Mojtaba Alaei,\nFrank Freimuth, Klaus Koepernik, Pengxiang Xu, Tobias\nBurnus, Gustav Bihlmayer, Marjana Le\u0014 zai\u0013 c and Nicolae\nAtodiresei. This work was supported by the HGF-YIG\nProgramme VH-NG-513 and by the DFG through Re-\nsearch Unit 912 and grant HE3292/7-1. Computational\nresources were provided by J ulich Supercomputing Cen-\ntre.\n\u0003corresp. author: h.zhang@fz-juelich.de\n[1] Bode, M., Heide, M., von Bergmann, K., Ferriani, P.,\nHeinze, S., Bihlmayer, G., Kubetzka, A., Pietzsch, O.,\nBl ugel, S., & Wiesendanger, R. Chiral magnetic order\nat surfaces driven by inversion asymmetry. 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Generalized\nGradient Approximation Made Simple. Phys. Rev. Lett.\n77, 3865{3868 (1996).[30] For the description of the code see www.flapw.de\n[31] Wang, X., Yates, J. R., Souza, I., & Vanderbilt, D. Ab\ninitio calculation of the anomalous Hall conductivity by\nWannier interpolation. Phys. Rev. B 74, 195118 (2006).\n[32] Freimuth, F., Mokrousov, Y., Wortmann, D., Heinze,\nS., & Bl ugel, S. Maximally localized Wannier functions\nwithin the FLAPW formalism. Phys. Rev. B 78, 035120\n(2008).\n[33] Mosto\f, A.A., Yates, J.R., Lee, Y.-S., Souza, I., Vander-\nbilt, D., & Marzari, N. Wannier90: A Tool for Obtaining\nMaximally-Localised Wannier Functions. Comput. Phys.\nCommun. 178, 685 (2008)."    },    {        "title": "1109.5514v1.Electric_field_control_of_magnetic_domain_wall_motion_and_local_magnetization_reversal.pdf",        "content": "Electric -field control of magnetic domain wall motion and \nlocal magnetization reversal  \n Tuomas H. E. Lahtinen, Kévin J. A. Franke and Sebastiaan van Dijken * \nNanoSpin, Department of Applied Physics, Aalto University School of Science, P.O. Box 15100, FI -00076 Aalto, \nFinland . *e-mail: sebastiaan .van.dijken@ aalto.fi   \n \n \nSpintronic device s currently rely on magnetic \nswitching or controlled motion of domain walls  by \nan external magnetic field or spin -polarized \ncurrent. Achiev ing the same degree of magnetic \ncontrollability using an electric field has potential  \nadvantages including enhanced  functionality and \nlow power consumption. Here, we report on a n \napproach to electric ally control local magnetic \nproperties, including the writing and erasure of  \nregular ferro magnetic  domain  patterns  and the \nmotion of magnetic domain wall s, in multiferroic \nCoFe -BaTiO 3 heterostructures . Our method is \nbased on recurrent strain transfer from ferroelastic \ndomains in ferroelectric m edia to continuous \nmagnetostrictive film s with negligible \nmagnetocrystalline anisotropy. Optical p olarization \nmicroscopy of both ferromagnetic and ferroelectric  \ndomain structures  reveal s that domain correlations \nand strong  inter -ferroic domain wall pinning  \npersist in an applied electric field. This leads to an \nunprecedented electric controllability over the \nferro magnetic microstructure, an accomplish ment \nthat produces  giant magnetoelectric coupling \neffects  and open s the way to multiferroic spintronic  \ndevice s.  \nIn the field of spintronics, researchers have \ncontrolled magnetic switching and domain wall  \nmotion using spin -polarized currents and these \nphenomena have formed the basis for magnetic \nrandom access memor y (MRAM)  [1] and magnetic \nnanowire device concepts  [2,3]. Employing  significant \namounts of electrical current, however, is inevitably \naccompanied by energy dissipation , and in this \ncontext, electric -field induced magnetization reversal \nwithout major current flow  would be desirable . \nMultiferroic materials are a promising candidate for \nthe realization of electric -field controlled spintronic  \ndevices  [4,5], but practical requirements such as strong \nmagnetoelectric coupling  and room temperature \noperation pose significant scientific challenge s for \nsingle -phase multiferroic  material s. Cross -linking \nbetween ferromagnetic and ferroelectric order \nparameters via interface effects  in multiferroic heterostructures provides a viable alternative , as it \nallows for an independent optimization of  both ferroic \nphases  [6]. Up to now, several mechanisms for \nmagnetoelectric coupling at ferromagnetic -\nferroelectric interfaces have been explore d including  \ncharge modulation  [7-10], exchange interactions  [11-\n17], and strain  transfer  [18-25]. This has led to great \nadvances in electric -field control o f macroscopic \nmagnetic properties (anisotropy, exchange bias , and \nferromagnetic resonance frequency ) and the \ndemonstration of interface -induced multiferroicity  \n[10]. An extension of this degree of controllability to \nmicromagnetic elements such as  magnetic domain s \nand domain walls would constitute another important \nstep towards the realization of practica l multiferroic \ndevices , as it would enable  electric -field actuation of \nmagnetic functions that up to now have only be \naddress ed by an external  magnetic field or spin -\npolarized current.  Previous s tudies on local control of \nferromagnetism using imaging techniques that are \nsensitive to both ferromagnetic  and ferroelectric  order \nparameters have revealed domain correlation s and \nelectric -field induced magnetic switching of irregular \ndomains in single -phase multif erroics  [26,27] and \nexchange -biased multiferroic heterostructures  [13].  \nHere , we demonstrate that it is possib le to precisely \nwrite and eras e regular ferro magnetic domain  pattern s \nand to control the motion of magnetic domain walls in \nsmall electric field s. Our approach consists of two \nsteps. First, a well -defined ferroelastic domain \nstructure is fully transferred from a t etragonal \nferroelectric to a magnetostrictive medium via \ninterface strain transfer during thin film growth. \nHereafter , an electrical bias  voltage is used to \nreversibly manipulate the original magnetic \nmicrostructure by polarization reversal and domain \nwall motion in the ferroelectric . The results show  that \nmicroscopic features such as ferroelastic domain \nhistory  and ferromagnetic -ferroelect ric domain wall \npinning  critically determine the magnitude of \nmacroscopic magnetoelectric coupling  effects .  \nFor this study, thin CoFe films were directly grown \nonto BaTiO 3 substrates with a regular ferroel astic stripe pattern that is typified by 90  in-plane rotations \nof the ferroelectric polarization and elongated c axis \n(Fig. 1a) . This ferroelastic  a1 – a2 domain structure \nproduces  a lateral strain modulation of 1.1% ( a = b = \n3.992 Å , c = 4.036 Å). CoFe with a n atomic \npercentage of 60% Co and 40% Fe was selected  \nbecause of its  large magnetostriction and small  \nmagneto crystalline anisotropy  [28]. This materials \nchoice ensure d that local magnetoelastic energies \ncompared favorably to other energy scales in the \nferromagnet (exchange, magnetostatic, and \nmagnetocrystalline energies), which is a prerequisite \nfor electric -field control o f ferro magnetic \nmicrostructures. Both ferromagnetic and ferroel ectric \ndomain  patterns  were imaged using optical  \npolarization microscopy  techniques . Real-time \nmeasurements  of domain correlations in an appli ed \nelectric field w ere facilitated by the  use of semi -\ntransparent CoFe films with a thickness of 15 nm . \nMore details on sample fabrication and \ncharacterization procedures are presented in the \nMethods section.   \nFull im printing of ferroel astic BaTiO 3 domain s into \nCoFe during  thin-film growth  is demonstrated i n Fig. \n1b. We find exact correlation s between both ferroic \ndomain patterns  across the entire sample  area. \nAnalysis of the birefringent contrast in FE and the \nmagneto -optical Kerr contrast in FM indicates  that the \nferroelectric polarization and magnetization directions \nare collinear in zero magnetic field  (black and white \narrows) . Precise  reproduction of the ferroelastic \ndomain structure  in the CoFe film is explained by  strain transfer at the heterostructure interface , which \ninduces  local modulations in the uniaxial \nmagnetoelastic anisotrop y via inverse \nmagnetostriction . Next, we analyze the micromagnetic \nresponse to an out-of-plane electric field (E = 10 \nkV/cm ) across the BaTiO 3 substrate.  Figure 1d \ncompares the ferroelectric and ferromagnetic domain \npattern after reverting back to electrical remanence ( E \n= 0 kV/cm). The images provide clear evidence of \nhighly correlated changes in both ferroic  domain \nstructures. In the BaTiO 3 substrate, the electric field \ntransform s the ferroelectric microstructure into \nalternating a1 and c domain s as manifested by a 45  \nrotation of the ferroelectric domain boundaries. Th ese \nferro elastic modifications are  accompanied by the \ntransfer of local  strain s to the CoFe film , which , in \nturn, result s in the erasure (on top of the a1 domains) \nand conservation  (on top of c domains) of the original \nmagnetic stripe pattern. The schematic diagrams in \nFig. 1 illustrate  the correlations between the two \nferroic microstructure s in the as -deposited and \nelectric -field induced states.  \nTo interpret the evolution of the ferro magnetic \ndomain pattern, we need to first unravel  the strain -\nmediated relationship between local magnetic \nanisotropies  and the underlying ferroelectric domain \nstructure. For this purpose, we utilized our polarization \nmicroscope to measure local magnet ic hysteresis \ncurves on single domain  areas. Data  as a function of \nin-plane magnetic field angle were  used to determine \nthe orientation of the uniaxial  magnetic easy ax is for \neach domain type. Moreover, the magnitude of the \nFigure 1  Growth - and electric -field control of ferromagnetic domain patterns . a-d, Schematic  diagrams of the ferroelectric \n(FE) and ferromagnetic (FM) microstructure after CoFe film growth  on BaTiO 3 (a) and the application of an out -of-plane \nelectric field of 10 kV/cm (c) and corresponding p olarization microscopy images (b,d). The polarization direction and lattice \nelongation of the BaTiO 3 substrate (black rectangles with arrows) , the orientation of the strain -induced magnetic easy axis \n(white rectangles with double -headed arrows) , and the magnetization direction in zero applied magnetic field (white arrows) are \nindicated.  \n magnetoelastic anisotropy ( Kme) was extracted by \nfitting the slope of hard -axis magnetization curves to \nMs/2K me, where Ms is the saturation magnetization of \nCoFe (1.7  106 A/m). Figure 2 summarizes  the results  \nfor all ferromagnetic -ferroelectric domain \ncombinations . \nAfter growth , the ferro magnetic microstructure  \nconsists of alternating a1 and a2 domains  with \nhorizontal and vertical magnetic easy axes  and Kme = \n1.7 × 104 J/m3. This anisotropy value can be put in to \nperspective by comparing it to a theoretical estimat ion. \nFor polycrystalline films , Kme = 3/2, where  is the \nmagnetostricti on constant and  is the stress, which is \nproportional to the strain  via Young’s modulus Y. \nUsing  = 1.1% for full strain transfer from the BaTiO 3 \nsubstrate to the CoFe film and Y = 2.5   1011 J/m3 [29] \nand   = 6.8  10-5 [28] for CoFe , gives Kme, max = 2.8 × \n105 J/m3. This calculation suggest s that less than 10 % \nof the BaTiO 3 lattice strain is transferred during CoFe \nfilm growth . Yet, our experiments clearly demonstrate \nthat such limited strain effect is sufficient to fully \nimprint ferroelastic domains  into magnetostrictive \nfilms w hen othe r anisotro py contributions are small . \nFigure 1d shows that the magnetic microstructure \nconsists of four  different domain states  after the \napplication of an out -of-plane electric field.  On top of \nthe ferroelectric a1 domains, new compressive and \ntensile strains in the original a2 stripes of the CoFe \nfilm force d the uniaxial magnetic easy axis to rotate by \n90. As a result, the easy magnetization directions  of \nneighboring magnetic stripes align parallel and , thus, \nthe growth -induced domain pattern is electrically \nerased . The magnitude of the uniaxial magnetic \nanisotropy in the original a2 stripes is found to in crease \nsignificantly to Kme = 1.5 × 105 J/m3. This value \ncompares more favorably to the theoretical Kme, max and \nit clearly indicates that  strain  transfer  is more efficient  when the CoFe film is clamped to the BaTiO 3 \nsubstrate  (as compared to the initial stages of film \ngrowth) . For the magnetic a1 stripes, the underlying  \nferroelectric lattice symmetry d id not change , hence, \nKme remain s small (2.1 × 104 J/m3).  \nSimilar arguments hold for the development of \nmagnetic anisotropies on top of the out -of-plane \nferroelectric c domains. In this case, the ferroelectric \ndomain exhibits cubic symmetry in the substrate plane. \nConsequently, the CoFe film is compressed along t he \nsame axis along which it obtained a small tensile strain \nduring deposition. As the electric -field controlled \nlattice compression in the CoFe film exceeds the \ngrowth -induced strain, the uniaxial magnetic easy axis \nrotate s by 90 in both stripe domains an d the strength \nof the magnetic anisotropy is found to increase to Kme \n= 1.3 × 105 J/m3. The original magnetic a1 and a2 \ndomains thus transform to a2 and a1 stripes, \nrespectively, but the pattern  remain s the same . \nElectric -field induced formation of ferroelectric c \ndomains th us results in the conservation and \nstabilization of the original magnetic domain structure .  \nWe now demonstrate that strain -mediated ferroic \ndomain correlations in CoFe -BaTiO 3 can be used to \nwrite and eras e regular ferro magnetic domain patterns \nin small electric field s. An example is shown  in Fig. 3 , \nwhich depicts  the evolution of the magnetic domain \nstructure as a function of out -of-plane bias voltage. \nFirst, the ferroelectric polarization of the BaTiO 3 \nsubstrate is saturated by a bias voltage of 120 V  (Fig. \n3a). As previously discussed, th is uniform c domain  \nproduces a regular magnetic stripe pattern in the CoFe \nfilm. Reverting back to zero bias  causes  the \nferroelectric microstructure to relax  into alternating a1 \nand c domain s (Fig. 3b) . Consequently, t he local strain \nstate in the CoFe film and, thereby, the magnetoelastic \nanisotropies alter in accordance with Fig. 2 . This \nresults in the erasure of magnetic stripes on top of the \nFigure 2. Strai n-mediated r elation ship between local magnetic  anisotrop ies in the CoFe film and the underlying \nferroelectric domain  structure  of the BaTiO 3 substrate . The displayed hysteresis curves are measured o n single magnetic \ndomain  areas  with the in -plane magnetic field along the vertical axis of all images in  this paper.  The solid red lines are linear fits \nto the hard -axis magnetization curves.  ferroelect ric a1 domains . Subsequent application of a  \nsmall  bias voltage causes the ferroelectric c domains to \ngrow at the expense of  a1 domains  by sideways motion \nof the  ferroelastic  a1 – c domain boundaries. This \nagain modifies  the local strain state s in the CoFe film  \nand the regular magnetic stripe pattern is electrically \nrewritten . The images of Fig. 3 indicate that electric \nwriting and erasure of magnetic patterns  based on \nstrain -mediated correlations between local magnetic \nanisotropies and underlyi ng ferroel astic domain s is \nreversible. We note that this unprecedented  electric -\nfield control of ferro magnetic microstructure s was \nobtained at room temperature without the application \nof a magnetic field.  \nMacroscopically, interlinking between \nferromagnetic and ferroelectric order parameters is \noften quantified by the converse magnetoelectric \ncoupling coefficient, α = µ0ΔM/ΔE. In our s tudy, this \ncoefficient is estimated from the electric -field induced \nchange in the sample area that is covered by  a1 \ndomain s (Δa1). At 0 V, the a1 domains cover about \n65% of the CoFe film and this reduces to 0% at 120 V  \n(E = 2.4 kV/cm) . By also taking into account that half \nof the magnetization rot ates by 90° when a1 domains \nare replaced by  c domains, we find  α = 0. 5µ0MsΔa1/ΔE \n= 3 × 10-6 s/m, which  is about one order of magnitude \nlarger than previously reported results  [19,24]. Our \nexperiments  thus clearly indicate  that giant  \nmagnetoelectric coupling effects can be realized by \ncareful optimization of the multiferroic microstructure.  \nAnother  intriguing observation in our study  is \nelectric -field control  of magnetic domain wall  motion . \nIn this paragraph we analyze the main effects  and the  \nunderlying driving  mechanism. Figure 4 a shows an \nenlarged image of the ferromagnetic domain pattern on \ntop of a ferroelectric c and a1 domain. Two different \ntypes of magnetic domain walls are distinguished. The \nwalls labeled as (1) , appear on top of the ferroelectric c \ndomain  and their location is determined by the  \nposition of a1 – a2 boundar ies in the BaTiO 3 substrate \nduring CoFe film growth.  Thus, although this type of \nmagnetic wall is not directly pinned onto a n existing \nferroelastric boundary, its physical properties \n(location, width, and pinning strength) are determined \nby local magnetic anisotropies that result from a \nprevious ferroelastic domain configuration. Th is \nobservation underpins the importance of ferroelastic \ndomain history on magnetoelectric coupling effects.  In \ncontrast, the magnetic domain wall s labeled as (2)  in \nFig. 4 a are pinned on top of an existing ferroelastic a1 \n– c domain boundary . Sideways motion of such \nferroelastic boundaries in  an out -of-plane electric field \nresults in the reappearance of type -1 walls at pre -\ndefined locations, while type -2 walls are dragged \nalong by the ir ferroel ectric counterpart  (Fig. 3). To \nanalyze the properties of magnetic domain walls in more detail, we pe rformed micromagnetic simulations \nusing the local magnetic anisotropies of Fig. 2 as input \nparameter s (Fig. 4b). From these simulations , the spin \nrotation within the domain wall and the pinning energy \nare extracted. The r esults indicate that type -1 and type -\n2 walls are strongly pinned  onto previous and existing  \nferroelastic boundaries, respectively  (Fig. 4c -d). The \npinning energies  provide robust wall localization and \naccurate electric -field control of magnetic domain wall  \nmotion . We note that despite these  promising results, \nour observations are still early in our understanding of \nthe details of ferromagne tic-ferroelectric domain wall \npinning. Future experimental and theoretical studies \nare necessary to carefully examine the dynami cs of \nsuch interlinked ferroic walls  in an applied electric \nfield and the ability to transfer these  phenomena to \nwell-defined nanostructures.      \nIn summary, we have shown that strain transfer \nfrom ferroelastic domains in ferroelectric medi a to \nmagnetostrictive films can be used to write \nferromagnetic domain patterns and control magnetic \ndomain wall motion by the application of an elect ric \nFigure 3. Electric -field controlled writing and erasure of \nferro magnetic domain patterns. a -b, Polarization \nmicroscopy images of  the ferroelectric (FE) and \nferromagnetic ( FM) domain structure  during the application \nof an out -of-plane bias voltage.  Correlations between the \nferromagnetic microstructure and the underlying \nferroelectric lattice are evident.     \n field at room temperature . The realization of th ese \nlocal magnetoelectric  effect s is an important step \ntowards low -power spintronic devices .   \nMethods  \nThin  CoFe films with an atomic percentage of 60% \nCo and 40% Fe w ere directly grown onto ferroelectric \nBaTiO 3 substrates by electron -beam evaporation at \nroom temperature . Each film was covered by a 3 nm \nthick Au capping layer to pre vent oxidation  under \natmospheric conditions . The thickness of the CoFe \nfilms was 15  nm and x -ray diffraction , atomic force \nmicroscopy,  and transmission electron microscopy \nmeasurements confirmed the growth of smooth \npolycrystalline layers  with a small root mean square \nroughness of <0.5 nm . The 0.5 mm thick BaTiO 3 \nsubstrates contained  a regular a1 – a2 domain strip e \npattern  during CoFe film growth. The width of the \nferroelastic stripes varied from 3 – 7 µm.  \nThe ferromagnetic and ferroelectric domains were \nimaged independently using polarization microscop y techniques. The microscope consisted of polarizing \noptics, an adjustable diaphragm, a 100× objective, a \nCCD  camera and an electromagnet for the application \nof in -plane magnetic fields. The ferroelectric BaTiO 3 \ndomains were imaged using birefringent contrast with \nthe polarizer and analyzer in cross -configuration. \nDuring these measurements the CoFe film was \nsaturated by an in -plane magnetic field to eliminate  \nmagneto -optical contrast from overlying magnetic  \ndomai ns. The magnetic domain patterns were \nmeasured using the magneto -optical Kerr effect.  In \nthis case, t he polariz ing optics was set to a few degrees \nfrom extinction  (to optimize the magneto -optical \ncontrast) and a background subtraction method was \nused to ze ro-out contrast from structural defects and \nferroelectric domains. Local magnetic hysteresis \ncurves were constructed from the evolution of \nmagnetic contrast in single domain areas during \nmagnetization reversal. All ferromagnetic domain \nimages in this paper  (including those recorded as a \nfunction of out -of-plane bias voltage) were obtained at \nroom temperature and zero applied magnetic field.  \nElectric  fields were generated by using the CoFe /Au \nfilm as top electrode and silver paint on the back  of the \nBaTiO 3 substrates as bottom electrode . The electric \nbias voltage was ramped at 10 – 20 V/minute  and a \n20× objective was used to allow for wiring space \nduring the poling experiment (Fig. 3) . \nThe micromagnetic simulations (Fig. 4) were \nconducted using object orien ted micromagnetic \nframework (OOMMF) software  [30]. To closely \nmimic the experiments, the local magnetoelastic \nanisotrop ies of Fig. 2 were used for the different \nmagnetic domains. Other input parameters included a \nsaturation magnetization of Ms = 1.7  106 A/m and a \nuniform exchange constant of Kex = 2.1  10-11 J/m. \nThe domain wall pinning potential s were calculated \nfrom the orientation of magnetic s pins along a line \nperpendicular to the walls using   ( ) \n  ∫   ( )    ( ( )  ( )) \n     , where Kme(x),  \n(x), and (x) are the strain -induced magnetoeleastic \nanisotropy, the angle  of the spin moment, and the \norientation of the uniaxial magnetic easy axis a s a \nfunction of position , respectively . The parameters l \nand t indicate  the length of the domain wall  and the \nfilm thickness . In Figs. 4c and 4d, the pinning \npotential is calculated per µm domain wall length, i.e. l \n= 1 µm.  \nAcknowledgements  \nThe work was supported by the academy of \nFinland under contract no. 127731. T. H.E.L. and \nK.J.A.F. also like to acknowledge support from the \nNational Doctoral Program in Materials Physics and \nFigure 4. Electric -field  induced  motion of magnetic \ndomain walls. a , Polarization microscopy image of the \nferromagnetic domain structure on top of ferroelectric  c \n(left) and a1 (right)  domain s. Two types of magnetic \ndomain walls (numbered by (1) and (2)) and the underlying \na1 – c domain boundary are indicated by solid and dashed \nlines, respectively. b, Micromagnetic simulation of the \nmagnetic microstructure in a. c-d, Spin rotation within the \nmagnetic domain walls and pinning energ ies indicating \nstrong clamping of ferro magnetic walls onto ferroelastic \nboundaries (or magnetic anisotropy distributions that are the \nresults of such boundaries).  The pinning energy is \ncalculated per µm domain wall length .  The asymmetry in \nthe wall profile and pinning energy  in d is due to different \nmagnetic anisotropy strengths on the left ( Kme = 1.3 × 105 \nJ/m3) and right ( Kme = 2.1 × 104 J/m3) side of the \nferroelastic a1 – c domain boundary.  \n the Finnish Doctoral Program in Computational \nSciences . \nReferences  \n[1] Chappert, C., Fert, A. & Nguyen van Dau, F. The \nemergence of spin electronics in data storage. 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IEEE Trans. \nMagn.  40, 2335 -2337 (2004).  \n[30] http://math.nist.gov/oommf  \n "    },    {        "title": "1109.6932v1.An_ultrafast_image_recovery_and_recognition_system_implemented_with_nanomagnets_possessing_biaxial_magnetocrystalline_anisotropy.pdf",        "content": "APPLIED PHYSICS LETTERS xx (2011)\nAn ultrafast image recovery and recognition system implemented with\nnanomagnets possessing biaxial magnetocrystalline anisotropy\nNoel D'Souza,1,a)Jayasimha Atulasimha,1and Supriyo Bandyopadhyay2\n1)Department of Mechanical and Nuclear Engr., Virginia Commonwealth University, Richmond, VA 23284,\nUSA.\n2)Department of Electrical and Computer Engr., Virginia Commonwealth University, Richmond, VA 23284,\nUSA.\n(Dated: 21 May 2018)\nA circular magnetic disk with biaxial magnetocrystalline anisotropy has four stable magnetization states\nwhich can be used to encode a pixel's shade in a black/gray/white image. By solving the Landau-Lifshitz-\nGilbert equation, we show that if moderate noise de\rects the magnetization slightly from a stable state, it\nalways returns to the original state, thereby automatically de-noising the corrupted image. The same system\ncan compare a noisy input image with a stored image and make a matching decision using magneto-tunneling\njunctions. These tasks are executed at ultrahigh speeds ( \u00182 ns for a 512\u0002512 pixel image).\nConsider a single crystal single-domain nanomagnet in\nthe shape of a circular disk that has no in-plane shape\nanisotropy but has biaxial magnetocrystalline anisotropy.\nThis magnet has four stable magnetization directions\n(`up', `right', `down', `left') as shown in Fig. 1 (a)1.\nAn array of such magnets can store a black-gray-white\nimage by storing the shade of every pixel in one of the\nfour stable states of a magnet. If corrupted by moderate\nnoise, the magnetization dynamics automatically recover\nthe stored image, so that there is built-in error correction.\nWhen integrated with magneto-tunneling junctions, the\nsame array can recognize images by comparing them with\nstored images pixel by pixel. These tasks are executed\nwith ultrahigh speed since no software is needed.\nFIG. 1. (a) A circular disk-shaped single-crystal Ni layer with\nbiaxial anisotropy, creating four possible magnetization di-\nrections or `easy' axes { `up', `down', `left' and `right'. The\nenergy landscape is depicted by the saddle-shaped curve, re-\nsulting in the four energy minima along the axes. (b) The\nspherical coordinate system used to de\fne the magnetization\nvector and the color scheme used to designate quadrants. (c)\nColor plot of the magnetization vector's \fnal state for all al-\nlowed values of the initial state ( \u0012initial;\u001einitial ).\na)Electronic mail: dsouzanm@vcu.edu.The four states of the magnet encode black, gray, white\nand gray. We could have encoded two di\u000berent shades of\ngray in the four states, but the 3-shade scheme is sim-\npler. Noise can corrupt the stored image by perturbing\nthe magnetization vector and de\recting it away from the\ninitial stable state. The ensuing magnetization dynam-\nics is studied using the Landau-Lifshitz-Gilbert (LLG)\nequation to determine the \fnal state of every perturbed\nmagnet (or pixel). For this purpose, we assume that the\nmagnets are spaced far enough apart that dipole inter-\naction between them can be ignored. Since the useful\npixel density in any optical image is limited by the wave-\nlength of light (0.5 - 1.0 \u0016m for visible light), we can\nspace the magnets apart by 1 \u0016m, which makes the inter-\nmagnet dipole interaction negligible. Using knowledge of\nthe magnetization dynamics of such a 4-state magnet, we\ndemonstrate image recovery in a 512 \u0002512 pixel black-\nwhite-gray image in \u00182 ns while conventional \feld pro-\ngrammable gate array (FPGA)- and CMOS-based image\n\flters would have taken several \u0016s2.\nIn order to demonstrate image recovery, we consider\nan isolated circular nanomagnet of nickel with diameter\n100 nm and thickness 10 nm. Nickel has an `easy' axis\nof magnetization along the h111idirection, a `medium'\naxis along theh110idirection and a `hard' axis along the\nh100idirection. We assume the nickel layer to be in the\n(001) plane. In this plane, the `easy' axis is the h110idi-\nrection (`medium' for the entire crystal) while the `hard'\naxis is along the h100idirection. Because the thickness\nof the magnet is ten times smaller than the diameter,\nout-of-plane excursion of the magnetization vector is en-\nergetically costly, albeit not impossible.\nAs illustrated in Fig. 1 (a), the easy axes of a single-\ncrystal Ni nanomagnet in the x-y plane (magnet's plane)\nare along the [110], [ 110], [ 110] and [1 10] directions, in\nMiller notation. The saddle-shaped curve and arrows de-\npict the potential energy landscape and stable directions\nof the magnetization vectors in the four (degenerate)\nminimum energy locations. The coordinate axis system\nis rotated by 45\u000ein the magnet's plane (i.e. \u001e!\u001e\u0000\u0019=4),\nwhere\u001eis the angle subtended by the in-plane compo-arXiv:1109.6932v1  [cond-mat.mes-hall]  30 Sep 2011APPLIED PHYSICS LETTERS xx (2011) 2\nnent of the magnetization vector with the +x-axis. This\nrotational transformation ensures that the energy min-\nima are along the \u0006x-axes (\u001e= 0\u000eor\u0006180\u000e) and\u0006y-axes\n(\u001e=\u000690\u000e). The angle subtended by the magnetization\nvector with the z-axis is \u0012. When\u0012= 90\u000e, the magneti-\nzation vector is in the plane of the magnet.\nIf noise de\rects the magnetization vector of a mag-\nnet away from its initial stable state to a new state, the\nlatter evolves with time in accordance with the Landau-\nLifshitz-Gilbert (LLG) equation. We take the new state\nas the initial condition and solve the LLG equation to de-\ntermine the \fnal state. We \fnd that as long as the noise\namplitude is not large enough to de\rect the magnetiza-\ntion vector closer to another stable orientation or cause\nlarge out-of-plane excursion, the vector always returns to\nthe initial stable orientation. This happens relatively fast\nin a few ns (for realistic parameters). Thus, the system\nacts like associative memory and could be used in that\nrole as well. However, here we study image recovery.\nThe total magnetic energy of the single-domain nano-\nmagnet per unit voulme is\nEtotal(t) =Emagnetocrystalline (t) +Eshape (t)\n=K1\n4sin22\u0012(t) +\u0014K1\n4sin4\u0012(t) +K2\n16sin2\u0012(t)sin22\u0012(t)\u0015\n+\u00160\n2M2\ns\u0002\nNzcos2\u0012(t) +Nxsin2\u0012(t)\u0003\n;(1)\nwhereK1andK2are the \frst and second order magne-\ntocrystalline anisotropy constants, \u00160is the permeability\nof free space, Msis the saturation magnetization and\nNx;Ny;Nzare the components of the demagnetization\nfactors along the x-, y-, and z-axes, respectively. Since we\nconsider a circular nanomagnet, Nx=Ny. These demag-\nnetizing factors are determined from the oblate spheroid\nestimation3.\nSince the magnet is single-domain, the magnetization\nMhas a constant magnitude. This allows us to de\fne\na unit vector nm=M=jMj=erin the radial direction.\nThe magnetization at any instant is therefore uniquely\nspeci\fed by \u0012(t) and\u001e(t).\nIn any arbitrary orientation [ \u0012(t);\u001e(t)], the magnetiza-\ntion vector experiences a torque per unit volume due to\nmagnetocrystalline and shape anisotropy. The latter can\nbe written as\nTE(t) =\u0000nm\u0002rEtotal(t) =\u0000er\u0002rEtotal(\u0012(t);\u001e(t));\n(2)\nwhereEtotal(t) is given by Equation (1). It can be veri\fed\nthat the torque vanishes when \u0012=\u0019=2 and\u001e=n\u0019=2 [n\n= 0, 1, 2, 3], indicating that these are the four stable ori-\nentations. Whenever the magnetization is de\rected from\na stable orientation by noise, it experiences the above\nrestoring torque that attempts to return the magnetiza-\ntion to the initial state.\nIn order to study the restoring dynamics, we solve the\nLandau-Lifshitz-Gilbert (LLG) equation\ndnm\ndt+\u000b\u0012\nnm(t)\u0002dnm(t)\ndt\u0013\n=2\u0016B\n~Ms\nTE(t);(3)where\u000bis the Gilbert damping coe\u000ecient (0.045 for Ni),\n\n is magnet's volume, and \u0016B= Bohr magneton.\nFrom the last three equations, we derive two coupled\nequations that describe the time-evolution of \u0012(t) and\n\u001e(t) (see the accompanying supplementary material4).\nAfter noise de\rects the magnetization to a new orien-\ntation, we take that new orientation as the initial ori-\nentation [\u0012initial;\u001einitial ] and solve the above coupled\nequations numerically to determine the \fnal state of the\nmagnetization vector [ \u0012final;\u001efinal]. The \fnal state is\nreached when the orientation does not change further.\nIn our simulations, the initial conditions span the space\nof\u001einitial from 0\u000eto 360\u000e(with four narrow excluded\nregions; see later) and \u0012initial from 85\u000eto 95\u000e. The an-\ngular resolution is 1\u000e. The results outside the interval\n[85\u000e\u0014\u0012initial\u001495\u000e] are not considered since they corre-\nspond to large out-of-plane excursions of the magnetiza-\ntion vector that can only occur when the noise amplitude\nis extremely high. Such large perturbations induce non-\nlinear e\u000bects and will be discussed elsewhere.\nThe narrow regions excluded from the \u001einitial space\nare the following. The states \u001e= 45\u000e;135\u000e;225\u000eand\n315\u000eare the highest energy states when \u0012= 90\u000e. If the\nmagnetization is driven to any of these states by noise\n(unlikely since these are the highest energy states), it\nhas equal probability of decaying to either one of its two\nneighboring minimum energy (stable) states. Therefore,\nwe exclude a region of 1\u000earound these critical points.\nIn order to convey the results of magnetization dynam-\nics obtained by solving the LLG equation, we use the fol-\nlowing color scheme (see Fig. 1(b)): blue corresponds to\n\u001e= 0\u000e,redto\u001e= 90\u000e,yellow to\u001e= 180\u000e, andgreen\nto\u001e= 270\u000e. Fig. 1(c) illustrates the magnetization\nvector's \fnal orientation for all allowed values of initial\n(perturbed) orientations \u0012initial and\u001einitial . If\u0012initial\nand\u001einitial are in the blue region, then the \fnal state is\nalways\u001efinal = 0\u000eand\u0012final = 90\u000e. Similarly, if \u0012initial\nand\u001einitial are in the red region, then the \fnal state is al-\nways\u001efinal = 90\u000eand\u0012final = 90\u000e, etc. This plot shows\nthat if\u0012initial = 90\u000e, then the \fnal state for the range\nof initial states\u000044\u000e<\u001einitial<44\u000eis\u001efinal = 0\u000eand\n\u0012final = 90\u000e. Similarly, the \fnal state for \u0012initial = 90\u000e\nand the range of initial states 46\u000e< \u001einitial<134\u000eis\n\u001efinal = 90\u000eand\u0012final = 90\u000e, and so on. This means\nthat if noise de\rects the magnetization vector from a sta-\nble state while still staying in-plane, the vector always re-\nturns to the original state as long as the angular in-plane\nde\rection is within \u000644\u000e.\nHowever, when \u0012initial = 89\u000e, i.e. noise causes -1\u000e\nout-of-plane excursion of the magnetization vector, the\nregions shift by -5\u000e. Now, if the magnetization's ini-\ntial state was in the interval [-50\u000e\u0014\u001einitial\u001440\u000e], the\nthe \fnal state will be \u001efinal = 0\u000e, and so on. This -\n5\u000eshift happens because the out-of-plane excursion of\nthe magnetization vector generates an additional pre-\ncessional torque acting on the magnetization vector ow-\ning to the coupled \u0012\u0000\u001edynamics. Similarly, when\n\u0012initial = 91\u000e, a shift of +5\u000etakes place. As long asAPPLIED PHYSICS LETTERS xx (2011) 3\nFIG. 2. (a) A pixel's shade is given a value of 0 (black), 0.5\n(gray) or 1 (white). (b) Original 512 \u0002512 pixel image. Each\npixel's shade is encoded into the magnetization state of the\ncorresponding magnet. (c) The eye of the bird is blown up for\nclarity. (d) Noise corrupts the image at time t= 0 by de\rect-\ning the magnetization vectors from their initial states. The\nout-of-plane excursion is restricted to 1\u000e. (e) The partially\nrecovered image at t= 1 ns. (f) The fully recovered image at\nt= 2 ns.\nthe out-of-plane excursion \u0001 \u0012initial (=\u0012initial\u000090\u000e) is\nsmall, the shift is linearly proportional to \u0001 \u0012initial . For\n\u0001\u0012initial>5\u000e, the shift increases non-linearly. There-\nfore, out-of-plane excursion is particularly harmful. As\nlong as the magnetization vector is not de\rected too far\nout of the magnet's plane, and does not stray into the\nambit of another stable state, it always returns to the\noriginal state and recovers. This self-correcting behavior\nlends itself to image recovery.\nFig. 2 (a) shows the scheme for encoding pixel shades\nin a black/gray/white image. The magnetization's orien-\ntation encodes three di\u000berent shades as follows: \u001e= 0\u000e\n(black),\u001e= 90\u000e(gray),\u001e= 270\u000e(gray) and \u001e= 180\u000e\n(white). The shades can also be assigned numerical val-\nues: black = 0, gray = 0.5 and white = 1.\nA 512\u0002512 pixel image is encoded by the above scheme\nand is shown in Fig. 2 (b). Next, the numerical value\nof each pixel is changed randomly to simulate the e\u000bect\nof noise. We restrict the random out-of-plane de\rectionof the magnetization vector \u0001 \u0012to\u00061\u000e, which then re-\nstricts the in-plane de\rection \u0001 \u001eto\u000640\u000esince that is\nthe maximum in-plane de\rection that can be corrected\nwhenj\u0001\u0012j\u00141\u000e. The choice of\u00061\u000eout-of-plane de\rec-\ntion is dictated by the fact that this allows a reasonably\nlarge azimuthal de\rection. This visually distorts the im-\nage by a large degree in Fig. 2(d). In accordance with\nthis choice, a pixel with intensity value = 0 is randomly\nassigned a value between 0 and 0.222. If the intensity\nvalue = 0.5, it is changed to something between 0.278\nand 0.722. Similarly, for pixels with intensity value =\n1, the value is changed to something between 0.778 and\n1. These distortions restrict the azimuthal de\rections to\n\u000640\u000e, i.e.j\u0001\u001ej\u001440\u000e.\nFig. 2 (c) shows the corrupted image at t= 0 ns for a\ncritical region (the bird's eye). By converting each pixel\nof the corrupted image to its equivalent \u001e-value and then\nsolving the coupled equations for \u0012\u0000\u001edynamics4, we\ncan determine the \fnal state (black, gray or white) of\nthe pixel. Figs. 2 (d), (e) and (f) illustrate the image\nrecovery process. At t= 1 ns, the noise in the image has\nbeen greatly reduced, with most pixels settling back into\ntheir original states (Fig. 2 (e)). Steady state is achieved\nin 2 ns (Fig. 2 (f)) since the images at 2 ns and at 3\nns (the latter not shown here) are identical. The \fnal\nsteady state image is identical with the original image\npixel by pixel, showing 100% recovery.\nOne could extend this scheme to pattern recognition as\nwell. This would be achieved by integrating a magnetic\ntunnel junction (MTJ) vertically underneath each mag-\nnet. The magnetization orientations of the lower hard\nmagnetic layers store the image to be compared against.\nThe input image is written in the upper soft magnetic lay-\ners (Ni). If the input matches the stored image pixel by\npixel, then the magnetizations of the soft and hard mag-\nnets will be all parallel. This will result in the maximum\ntunnel current \rowing through each MTJ. By setting an\nappropriate current threshold (say X% of the maximum),\nwe can determine if the two images match with probabil-\nity of X% and thus recognize the input image.\nThis work is supported by the US National Science\nFoundation under the Nanoelectronics Beyond the Year\n2020 grant ECCS-1124714.\n1N. D'Souza, J. Atulasimha and S. Bandyopadhyay, J. Phys. D:\nAppl. Phys., 44, 265001 (2011).\n2N. Kawai and S. Kawahito, IEEE Trans. Elec. Dev., 51, 185\n(2004).\n3B. D. Cullity and C. D. Graham, Introduction to Magnetic Ma-\nterials , (Wiley, New York, 2009).\n4Supplementary material located at ....."    },    {        "title": "1111.4043v1.Theory_of_magnetization_precession_induced_by_a_picosecond_strain_pulse_in_ferromagnetic_semiconductor__Ga_Mn_As.pdf",        "content": " \n 1Theory of magnetization precession indu ced by a picosecond strain pulse  \nin ferromagnetic semiconductor (Ga,Mn)As  \n \nT. L. Linnik1, A. V. Scherbakov2, D. R. Yakovlev2,3, X. Liu4, J. K. Furdyna4, and M. Bayer2,3 \n 1\n Department of Theoretical Physics , V. E.  Lashkaryov Institute of Semiconductor Physics, \nNational Academy of Sciences of Ukraine, 03028 Kyiv, Ukraine \n2Ioffe Physical-Technical Institute, Russian Ac ademy of Sciences, 194021 St. Petersburg, Russia \n3Experimentelle Physik 2, Technische Un iversität Dortmund, D-44227 Dortmund, Germany \n4Department of Physics, University of Notre Dame, Notre Dame, Indiana 46556, USA \n \n \nABSTRACT \nA theoretical model of the coherent precession of magnetization excited by a picosecond \nacoustic pulse in a ferromagnetic semiconductor layer of (Ga,Mn)As is developed. The short \nstrain pulse injected into the ferromagnetic layer modifies the magnetocrystalline anisotropy resulting in a tilt of the equilibrium orientatio n of magnetization and subsequent magnetization \nprecession. We derive a quantitative model of this effect using the Landau-Lifshitz equation for the magnetization that is precessing in the time-dependent effective magnetic field. After \ndeveloping the general formalism, we then provide a numerical analysis for a certain structure \nand two typical experimental geometries in whic h an external magnetic field is applied either \nalong the hard or the easy magnetization axis. As a result we identify three main factors, which determine the precession amplitude: the magnetocrystalline anisotropy of the ferromagnetic \nlayer, its thickness, and the strain pulse parameters. \n   \n 21. INTRODUCTION \nUltrafast control of magnetic order is one of the key problems of modern magnetism. The performance of magnetic storage devices, which lags much behind their exponentially increasing \ncapacity, is a bottleneck of current electronics. During the last decade various concepts to \nmanipulate magnetization on a short time scale ut ilizing picosecond magnetic field pulses [1,2] \nor femtosecond optical excitation [3] have been e xplored for magnetic materials. In materials \nwith strong magnetocrystalline anisotropy (MCA) acoustic pulses may be also an effective tool to manipulate magnetization on ultrashort time scales [4,5]. The methods of picosecond laser \nultrasonics allow the generation of ultrashort strain pulses in solids [6]. These strain pulses have \npicosecond duration and amplitude up to 10\n-3. They have a fast and local impact, which may lead \nto a considerable response of the material’s  magnetization, whose magnetic properties are \nsensitive to strain.  Ferromagnetic semiconductors (FMSs), like (Ga,Mn)As, belong to the class of \nferromagnets with strong MCA due to the hole-mediated origin of ferromagnetism [7,8]. In FMS \nepitaxial layers mainly strain determines the directions of the easy magnetization axes. The compressive (tensile) epitaxial strain from lattice mismatch between buffer and FMS layers results in in-plane (out-of-plane) orientation of the easy axes of magnetization for a wide range \nof FMS parameters [9,10,11]. Several ways to control the magnetization in FMS by strain have \nbeen developed recently: (i) the desired direction of the easy magnetization axis may be achieved \nby adjusting the composition of a buffer layer dur ing growth [9]; (ii) after-growth patterning \nallows directing the in-plane magnetization [12]; and (iii) in layered multiferroic structures with the FMS layer grown on piezoelectric material an electric field applied to the piezoelectric layer \ngoverns the in-plane unidirectional strain and a llows manipulation of the magnetization direction \n[13-15].  \n 3 Until very recently the strain-control of magnetization in FMSs has remained static. First \ntime-resolved experiments with strain pulses in FMS epitaxial layers were reported by Thevenard et al . [16] and Scherbakov et al . [5] in 2010. The studies in Ref. [16] focused on \nelasto-optical effects induced by a strain pulse propagating in a magnetized FMS layer, while the \neffect of the strain pulse on the magnetizati on and the strain-induced temporal evolution of \nmagnetization were studied in Ref. [5]. It was demonstrated in a magnetic field normal to the ferromagnetic layer that the strain pulse indu ces a pronounced tilt of magnetization out of its \nequilibrium orientation and subsequently cohere nt magnetization precession. In Ref. [5], for \ndescribing the experimental results the authors considered the simplest model of \nmagnetocrystalline anisotropy of a FMS layer. The proposed model cannot explain a number of effects observed in the later experiments, such as strain pulse induced magnetization precession \nalso for in-plane magnetic fields and even without external field [17]. This observation has stimulated the present theoretical studies, which are aimed at carrying out a comprehensive \nanalysis of the effect of strain pulses on the magnetization in ferromagnetic (Ga,Mn)As. The \nmain goal is to examine how the amplitude of the strain-pulse-induced precession depends on the parameters of the FMS structure, the magnetic fiel d strength and direction and the parameters of \nthe strain pulse. We examine the cases of magnetic field direction normal to the ferromagnetic layer as in Ref. [5] and also parallel to it as well as without magnetic field. The underlying \nanisotropy parameters of the FMS structure have been obtained using the microscopic model for \nhole-mediated ferromagnetism proposed by Dietl et al.  [18]. \n The paper is organized as follows. In Section 2 we briefly describe the considered experiments with picosecond strain pulses hitting FMS layers, introduce the parameters of the \nstrain pulse and qualitatively discuss the effect of  the strain pulse on the magnetization. Section 3 \ndescribes the formalism, which is used later to calculate quantitatively the effect of the strain pulse. In Section 4 we present the results of numerical calculations for a particular FMS structure  \n 4subject to two different orientat ions of external magnetic field. Finally, we summarize and \nconclude the obtained results and discuss the perspectives for controlling magnetization by \npicosecond acoustics.  \n2. EXPERIMENTS WITH PICOSECOND STRAIN PULSES \nIN EPITAXIAL (Ga,Mn)As LAYERS \nFigure 1(a) shows the schematic of experiments w ith picosecond strain pulses applied to a FMS \nlayer. The sample consists of a single \nAs Mn Ga\nMn Mn1 x x−  FMS layer grown on a semi-insulating \nGaAs substrate [5]. The typical content of Mn atoms in the FMS layer is 1.001.0Mn ÷ =x . A thin \nmetal film deposited on the back side of the GaAs substrates serves as optoelastic transducer, \nwhich rapidly expands due to the heating under femtosecond laser excitation [6]. Figure 1(b) \ndemonstrates the bipolar strain pulse δεzz(t) injected into the substrate as result of the thermal \nexpansion of the metal film [19,20]. Pulse duration τ and amplitude max\nzzεdepend on the \ntransducer material and the parameters of optical  excitation, and have typical values of ~10 ps \nand ~10-4÷10-3, respectively. It is important to note, that in high symmetry GaAs substrates \n(typically (001) oriented) the strain pulse contains only longitudinal components for lattice distortions  along the propagation direction perpendicular to the substrate interface. At liquid \nhelium temperatures such a strain pulse propa gates through GaAs over millimeter distances \nwithout scattering [21]. \nIn order to describe the re sponse of the magnetization M of the FMS layer on the strain \npulse we use the standard Landau-Lifshitz appr oach in which the magnetization is precessing \nabout the time-dependent effective magnetic field B\neff [22]. This effective field is the sum of the \nexternal magnetic field B and the intrinsic magnetic anisotropy field, which is determined by the \nparameters of the FMS layer. In equilibrium the magnetization M is parallel to Beff. As an  \n 5example, Fig. 2(a) shows the experimental geometry reported in Ref. [5] when B is applied \nnormal to the (Ga,Mn)As layer with in-plane easy axes. In such a layer the anisotropy field holds \nM in the layer plane, while the external magnetic field turns M out of the layer, so that the \nresulting field Beff has a tilted orientation between in-plane and normal-to-it. When reaching the \nFMS layer, the strain pulse changes the layer properties, namely the zzε static strain component, \nmodifies the magnetic anisotropy field, and  tilts B eff, which is then no more parallel to M. As a \nresult M starts to precess around Beff. After the strain pulse has left the FMS layer, Beff returns to \nits equilibrium orientation, while M remains at some angle relative to Beff. Thus, the precession \ncontinues until relaxation drives M back to equilibrium [Fig. 2( b)]. In the Landau-Lifshitz \napproach value and direction of Beff are determined by the free energy density [23]. The free \nenergy density includes magneto-elastic terms, which provide the direct relation between the strain components and the orientation of B\neff. Thus, one can model the response of Beff and the \nmagnetization on the strain pulse, as shown in the next Section. \n \n3. MAGNETIZATION PRECESSION INDUCED BY A STRAIN PULSE \nIn our theoretical analysis we consider a thin FMS (Ga,Mn)As layer with a typical Mn ion content that is epitaxially strained, at liquid helium temperatures. Figure 2(a) shows the assumed coordinate system, in which the x and y axes lie in the layer plane along the [100] and [010] \ncrystallographic directions, respectively, and the z-axis is perpendicular to the layer growth \ndirection, which is the [001] crystallographic direction. Far below the Curie temperature the \nmagnetization of the FMS layer is close to the saturation value \nmax 0 SNg MMn Bμ= , where g=2 is \nthe Mn Lande factor, Bμ is the Bohr magneton, 2/5max=S  is the maximal total spin of the Mn \natom and 3\n0 Mn/ 4 a x NMn=  is the concentration of Mn atoms (0ais the lattice constant). Assuming \nthat the perturbation induced by the strain pulse is weak and does not affect the absolute value of  \n 6M, and neglecting also damping we may use the Landau-Lifshitz equation to describe the \ndynamics of magnetization in the time-dependent effective field Beff(t) [22]:  \n),( ),( ),,( t F t tdtd\nMm mB mBmm\nm eff eff −∇= ×⋅−=γ ,    (1) \nwhere 0/MMm=  is the normalized magnetization and =/Bgμ γ=  is the gyromagnetic ratio. \nThe effective field Beff acting on m is determined by the gradient of the normalized free energy \ndensity of the FMS layer0/MF FM= . \n Generally, the free energy density FM consists of isotropic and anisotropic parts. The \nisotropic part does not depend on the direction of m and does not contribute to the vector product \nin Eq. (1). Therefore, we have to consider only the anisotropic part of FM, which includes the \nZeeman term, the demagnetization energy, and the MCA terms related to the crystal symmetry. In a thin (Ga,Mn)As layer grown by low-temperature molecular beam epitaxy the cubic symmetry is tetragonally distorted by the epitaxial strain originating from the lattice mismatch between the buffer and the (Ga,Mn)As layers. Most of experiments also indicate the presence of \nan in-plane uniaxial anisotropy in the (Ga,Mn)As films [10,11,24]. The origin of this anisotropy \nis still under discussion, but phenomenologically it can be modeled by a weak shear \nstrain\nxyε[24]. Thus, we write the general expression fo r the anisotropic part of the free energy \ndensity of a thin cubic FMS layer distorted by strain [23,25,26] in the form:  \n ()\nyx xyxy z zz y yy x xx y x zz zx yy zy xxz zz y yy x xx z y x c z d M\nmm A m m m A mm mm mm Am m m A m m mB mB F\nε ε ε ε ε ε εε ε ε\nε εε\n24 4 4 )2(\n422 22 22 )1(\n42 2 2\n24 4 4 2\n) ( ) ( 2) ( ) ( )(\n+ + + + + + ++ + + + + + + +⋅−= Bm m\n, (2) \nwhere mx, my and mz are the projections of m onto the coordinate axes and εij (i,j=x,y,z) are the \nstrain components. The first term in Eq. (2) is the Zeeman energy of m in the external magnetic \nfield B, the second term is the demagnetization energy of the thin ferromagnetic film with \n2/0M Bo dμ=  [27,28], and the five following terms describe the MCA of the strained cubic  \n 7FMS layer. The cubic anisotropy field Bc and the magnetoelastic coefficients \nxyA AAA2)2(\n4)1(\n4 2  and , , ,ε ε ε are parameters of the FMS film, which depend on lattice temperature, \nhole concentration p and Mn content Mnx[9-11,24,29]. The equilibrium orientation of m is \ngiven by the minimum of FM and depends on the balance between Zeeman, demagnetization and \nMCA energies. \n In the unstrained FMS layer the MCA part of FM in Eq. (2) consists of the cubic term \nproportional to Bc only. For the experimentally relevant ranges of p and Mnx at low temperatures \nthe value of Bc may be both negative or positiv e [18,29]. We consider the case Bc<0 when the six \nequivalent easy magnetization axes lie alo ng the [100], [010] and [001] crystallographic \ndirections. This equivalence is destroyed by th e static epitaxial strain with components:  \n11 12 0 / 2 ,/) ( CC aaaxx zz yy xx ⋅ −= − = = ε ε ε ε ,          (3) \nwhere 0a and a are the non-distorted lattice constants of the (Ga,Mn)As and GaAs layers, \nrespectively. C11 and C12 are the elastic modules of (Ga,Mn)As. As a result the in-plane [100] \nand [010] and the out of plane [001] orientations of m become nonequivalent. At low \ntemperatures, for sufficiently high hole con centrations in-plane compressive strain 0< =yy xxε ε  \nis found in (Ga,Mn)As layers grown on GaAs, leading to in-plane orientation of the easy axes \n[10,11,18,29]. Further, the in-plane uniaxial anisotropy determined by the last term of Eq. (2) \nleads to a tilt of the easy magnetization axis from the [100]/[010] crystallographic directions \ntoward [1 10]/ [ 110] for positive xyε. This means that the coefficients A2xy and A2ε must be \npositive. The cubic magnetoelastic coefficients )2(\n4)1(\n4  and  ε ε A A  are one order of magnitude smaller \nthan A2ε and, consequently, do not affect the orientation of the easy magnetization axis. Finally, \nthe demagnetization energy support s the in-plane orientation of m.   \n 8 In the microscopic model used for the calculating the anisotropy coefficients the relation \n)2(\n4)1(\n4  ε εA A=  is fulfilled (see Appendix A) so that we will apply this approximation throughout the \nrest of the paper using the notation ε ε 4)1(\n4  A A≡ . Since also εxx=εyy  for epitaxial strain we may \nsimplify Eq. (2) and rewrite it in spherical coordinates: \n[ ]\n[] [ ]\n. cos sin sin cos sin 2sin sin21)4cos3(41sin) ( cos) (2cos) )(2 ( ),(\n2\n24\n44\n42\n4 2\nθ ϕ θ ϕ θ ϕ θ εϕ θ ε ε θ ε εθ ε ε ϕθ\nε εε ε\nz y x xyxyxx zz c xx zz cxx zz d M\nB B B AA B A BA A B F\n− − − ++ + ⋅ − − + − + ++ − − + =\n  (4) \nThis expression provides a direct relation between  the magnetic anisotropy fields, which are \ntypically used to describe MCA in most publ ications on FMS (Ga,Mn)As, and the strain \ncomponents. The values ) )( 2 (4 2 xx zz A A ε εε ε − − , ) (24 xx zz c A B ε εε − + , ) (4 xx zz cA B ε εε − −  and \nxy xyAε2 are usually defined as perpendicular uniaxial, perpendicular cubic, in-plane cubic and in-\nplane uniaxial anisotropy fields, respectively. \n In the frame of the single-domain model with constant magnetization it is convenient to \nrewrite also Eq.(1) in spherical coordinates [30]: \n   .sin,sin ϕθγ θ\nθθγ ϕ\n∂∂−=∂∂\n∂∂=∂∂M M F\ntF\nt     ( 5 )  \nAssuming that the changes δϕ and δθ of the angles ϕ and θ induced by the strain-pulse δεzz are \nsmall, we can write in linear approximation: \n[]\n[] ),(sin,),(sin\nzt F F Ftzt F F Ft\nzz\nozz\no\nzzzz\nδε δθ δϕθγ θδε δϕ δθθγ ϕ\nϕε ϕθ ϕϕθε θϕ θθ\n+ + −=∂∂+ + =∂∂\n     ( 6 )   \n 9where the jiFFM\nij∂∂∂=2\n (i,j=ϕ,θ,εzz) are calculated at equilibrium orientation )(),(0 0 B Bϕ θ , \ncorresponding to the static orientation of m at a given B. \n Here we introduce the effective rates of strain-induced precession: \n.4sin sinsin), cos 4)1 4(cos sin 2( cossin\n0 03\n4\n002\n4 0 02\n4 2 0\nϕ θ γθγθ ϕ θ θ γθγ\nε ϕε θε ε ε θε ϕ\nA F fA A A F f\nzzzz\no\n−= −=+− + ⋅ −= =\n  (7) \nThe values of fθ and fϕ determine the amplitude and the direction of the tilt of Beff induced by \n),(ztzzδε  for a specific static orientation of m. If both rates are zero, the strain pulse does not tilt \nBeff and, thus, does not induce any magnetiz ation dynamics. One sees that if m lies in the layer \nplane, fϕ =0. In addition, there are specific in-plane directions corresponding to the \ncrystallographic directions [100] , [010], and the diagonals, where 0 =θf , and a tilt of effB by \n),(ztzzδε  is impossible. This means that in a FMS layer with no shear strain ( εxy=0) the strain \npulse ),(ztzzδε  may induce a magnetization precession onl y when applying an external magnetic \nfield, which rotates m out of the easy magnetization axis. Howe ver, the presence of shear strain \n(εxy ≠0) allows launching of a magnetization precession by ),(ztzzδε , even at zero B. So the \npresence of at least one of these factors, either an external magnetic field or an in-plane shear \nstrain, is crucially necessary to induce a magnetization precession by ),(ztzzδε . \nThe precession frequency ω0 is determined by the standard expression for the \nferromagnetic resonance frequency and depends on the static orientation of m [30-32]: \n  2\n00sinθϕ ϕϕ θθθγω F FF − =        ( 8 )   \n 10 It is worth to note, that Eqs. (5-8) cannot be applied, when the equilibrium m is parallel to \nthe [001] axis, where a mathematical singularity ap pears [30,31]. However, it is easy to see that \nfor this orientation of m any perturbation ),(ztzzδε  cannot turn the magnetization out of the \nequilibrium direction. Thus this orientation is not of our interest and we use Eqs (5-8) throughout \nthe rest of the paper.   The developed formalism is well suited for st rain pulses of arbitrary shape but we restrict \nthe numerical calculations to spat ial and temporal dependencies of \nδεzz(t,z) typical for ultrafast \nacoustic experiments. In the (Ga,Mn)As film the strain has a complex shape compared to the one \ninjected into the substrate, as result of interfe rence of the incident and reflected components of \nthe pulse. The spatial-temporal evolution of the strain pulse which propagates with the \nlongitudinal sound velocity νl along the z-axis through the FMS layer with thickness d can be \nmodeled as [19,20]:  \n⎟⎟\n⎠⎞\n⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ −+− −+−⎜⎜\n⎝⎛−⎟⎟\n⎠⎞\n⎜⎜\n⎝⎛ −− − =\n2222 (max)\n2)/)2((exp)/)2((2)/ (exp)/ ( ),(\nττ τεδε\nl\nll\nlzz\nzz\nvd ztvd ztvztvztezt\n   ,  (9) \nwhere e is the base of the natural logarithm. Time t=0 in Eq. (9) corresponds to the moment, \nwhen the center of the bipolar strain pulse reaches the GaAs/(Ga,Mn)As interface ( z=0). The first \nterm in Eq. (9) describes the evolution of the st rain pulse propagating toward the open surface of \nthe FMS layer and the second term describes the strain pulse reflected at the open surface with a \nπ-phase shift and subsequently propagating back toward the substrate. The parameters of the \nstrain pulse that we use for the further calculations are as follows:  max\nzzε=10-4, τ=7ps, and \nskml /5=ν . These values are typical for ultrafast acoustic experiments and correspond to the \nvalues reported in Ref. [5]. Here we do not take into account nonlinear effects, which modify the  \n 11shape of the strain pulse during its propagation through the GaAs substrate. These effects are \ninsignificant for the chosen strain pulse amplitude.  \n Figure 3(a) shows the time evolutions δεzz(t,z) for three different positions inside the 200-\nnm-thick magnetic layer: z=0, 100 nm and 190 nm, which correspond to the GaAs/(Ga,Mn)As \ninterface, the centre of the FMS layer, and the coordinate 10 nm before the open surface, \nrespectively. It is clearly seen that the δεzz(t,z) are not the same for the different coordinates. \nThus, the strain-induced perturbation  of FM is spatially nonuniform and Eq. (6) must be solved at \neach coordinate z inside the FMS layer. Because of this nonuniformity of the perturbation, one \nalso should add the exchange term to the expression for Beff in Eq. (1) [22,25]. Basically \nexchange would lead to two effects. First, it gives rise to a frequency splitting of the magnon \nmodes in a finite-width film [33]. This splitting can manifest itself by a beating due to \ninterference of the split modes contributing to the strain-induced magnetization precession. For \nrealistic (Ga,Mn)As parameters, however, the mentioned splitting is relatively small [34,35]. It is worth to mention also that since the exchange terms are proportional to the spatial derivatives of magnetization, proper boundary conditions must be introduced for the magnetization at the ferromagnetic film interfaces. It is known, however, that this affects the magnetization mainly in \nthe quite thin regions near the interfaces [33]. Le aving these specific effects for further studies \nwe proceed with the analysis of the case without exchange. \nIn the actual experiment probing of the magnetization at a certain coordinate z is \nimpossible. The experimental signal (i.e. the magneto-optical Kerr rotation) reflects the time \nevolution of the magnetization averaged over the layer thickness. Thus, we introduce the mean \nangles: \n∫=d\ndztzdt\n0),(1)( δθ δθ ,  ∫=d\ndztzdt\n0),(1)( δϕ δϕ .   (10)  \n 12Then, Eq. (6) may be rewritten for relating )(tδϕ  and )(tδθ  with the averaged strain-induced \ntemporal perturbation, shown by the thick red line in Fig. 3(a): \n∫=d\nzz dztz t\n0.),( )( δε δε       ( 1 1 )  \nIn the next section we solve Eq. (6) numerically for both the magnetization and the averaged \nmagnetization as function of coordinate z. \n \n4. NUMERICAL ANALYSIS OF STRAIN-INDUCED PRECESSION \nWe examine two characteristic orientations of th e external magnetic field: perpendicular to the \nlayer plane, ),0,0( B =B , and in the layer plane along the [100] crystallographic direction \n)0,0,(B=B . We present the results of a numerical an alysis for certain parameters of the FMS \nlayer. First, we analyze the static orientatio n of magnetization as function of the external \nmagnetic field, calculate the field depend encies of the effective precession rates )()(B fθϕ  and the \nprecession frequency ) (0Bω , and then model the time evolution of the magnetization induced by \nthe strain pulse of chosen shape. We use the following parameters for the structure, which are \ntypical for a thin (Ga,Mn)As layer:  d=200 nm, xMn=0.045, p=4×1020 cm-3, and μ0M0=60 mT. \nThe corresponding values of Bc= −35 mT, A2ε=25 T, A2xy=152 T and T A 5.04=ε  were calculated \nin the frame of the Dietl model, for details see Appendix A. The calculations are limited to the \ncase of compressive epitaxial strain: εxx=εyy<0; εzz>0 and, thus, in-plane orientation of the easy \nmagnetization axes. The factor 2C12/C11=0.89 in Eq. (3) is taken from Ref. [36]. The calculations \nare carried out for several values of the static strain components: εzz=(1÷3) ×10-3 and \nεxy=(0÷2) ×10-4. In the frame of the single domain model we assume that at zero external  \n 13magnetic field m lies along the [100] direction if εxy=0 and along the easy magnetization axis \nthat is closest to the [100] direction if εxy>0. \n \nA. Perpendicular magnetic field \nAn external magnetic field applied perpendicular to the FMS layer rotates the magnetization out \nof the layer plane toward the z-axis. In this case the strain pulse induces a magnetization \nprecession even at 0 =xyε . Since also xyε is typically at least one order of magnitude smaller \nthen the epitaxial strain we first restrict our co nsideration to the case of zero shear strain, and \nthereafter numerically analyze the effect of nonzero xyε.  \n For zero shear strain ( 0 =xyε ) the orientation of m is characterized by 00=ϕ  for any \nvalue of B and we may simplify the expression (4) for FM to: \n[ ]\n[] [ ] . sin) ( cos) (2cos) )(2 ( cos )(\n4\n44\n42\n4 2\nθ ε ε θ ε εθ ε ε θ θ\nε εε ε\nxx zz c xx zz cxx zz d M\nA B A BA A B B F\n− − + − + ++ − − + + −=\n   (12) \nFigure 4(a) shows the angle dependence FM(θ) calculated for 3102−×=zzε  at different B. With B \nincreasing from zero the minimum of the free energy density shifts from θ0 =π/2 toward smaller \nvalues, and m gradually turns toward the field direction as Fig. 4(b) shows. At some magnetic \nfield a second minimum at θ =0 appears, so that FM has two minima separated by a barrier. With \nfurther increasing B the first minimum close to π/2 becomes shallower, while the second \nminimum becomes deeper. Finally, at B=B* the first minimum disappears and the magnetization \nrapidly changes its direction, becoming parallel to B [see Fig. 4(b)]. In realistic structures the \nswitching between the two minima occurs at lower B values smaller than B* due to the finite \ntemperature and the presence of fluctuations [37], but in the present analysis we consider that the  \n 14orientation of m corresponds to the first minimum of FM until B=B*. This corresponds to an \nexperiment at zero temperature with a gradual magnetic field increase starting from zero. \n The equilibrium orientation of magne tization determines the response of effB on the \nstrain pulse. As one sees from Eq. (7), at zero shear strain when 00=ϕ  the rate 0 =θf  and the \ntilt of effB is determined by the value of ϕf. Figure 5(a) shows the field dependence of the \nabsolute value | fϕ(Β)| for εzz= 3101−× , 3102−×  and 3103−× . Since ε ε 2 4 A A<<  the following \napproximation can be made: 0 2cos 2|)(| θ γε ϕ A Bf ≈ , which follows from the field dependence of \n)(Bmz . Therefore | fϕ(Β)| almost linearly increases with B until the jump at B=B*, as clearly seen \nfrom the comparison of Figs. 4(b) and 5(a). The switching field B* is an increasing function of εzz \nand equals to 117 mT, 180 mT and 243 mT (shown by the vertical dashed lines) for εzz=3101−× , \n3102−×  and 3103−× , respectively. Thus, the stronger the magnetization is turned away from the \nin-plane easy axis by the external magnetic field, the larger is | fϕ| and the stronger is the response \nof effB on the perturbation induced by the strain pulse zzδε. \n While fϕ determines the tilt of Beff, the subsequent time evolution of m depends \nsignificantly on the precession frequency. Fig. 5(b) shows the field dependence of ω0(B) for \nseveral values of static strain components zzε. The value of ω0 decreases with increasing B until \nit becomes zero at B=B*. The stronger the static epitaxial strain zzε is, the larger is ω0. \n The precession rate fϕ   and the precession frequency  ω0 at a certain external magnetic \nfield are the parameters of the FMS layer, which do not depend on the shape of the strain pulse. However the spatial-temporal evolution of the magnetization is induced by \n),(tzzzδε . We \ncalculate the magnetization evolution at  three coordinates in the FMS layer: z=0, 100 nm and \n190 nm. Fig. 3(b) shows the corresponding numerical solution for the component \nδmz(t)= δθ(t)sinθ0. We see that the precession starts upon arrival of the strain pulse at the  \n 15corresponding coordinate in the FMS layer. While the strain pulse propagates forwards and \nbackwards the precession trajectory is complicated. When the reflected strain pulse completely has left the layer (\nt=110 ps shown by the vertical line) the magnetization continues to precess \nwithout decay as long as damping does not occur. \n In the considered case of zero shear strain th e simple analytical solutions of Eq.(6) for the \nafter-pulse, free magnetization precession can be written as harmonic oscillations with frequency \n0ω which are shifted in phase by 2/π  relative to each other :  \n()\n() . sin2sin 2 ),(, cos2sin 2),(\n0000\ndd\ntttSfa tztttSf tz\n−⎟\n⎠⎞⎜\n⎝⎛Δ−=− ⎟\n⎠⎞⎜\n⎝⎛Δ=\n⊥ ωωδθωωδϕ\nωϕωϕ\n,     (13) \nwhere Δt=2(d-z)/vl is the travel time of the strain pulse from the coordinate z toward the surface, \nand back; l d vdt /=  is the travel time of the strain pulse through the magnetic layer and \n   )2/ exp(222\n02\n0maxτω π τωεω − = e Szz      ( 1 4 )  \nis the absolute value of the spectral density of the incident strain pulse at frequency0ω. For the \nchosen parameters of the strain pulse Sω is an increasing function of frequency in the considered \nrange around 0ω. The parameter 03\n4\n0sin)) ( ( 4 θ ε εωγ\nε xx zz cA B a − − −=⊥  depends on magnetic \nfield and has values between 0.5 and 1, increasi ng with increasing magnetic field. The presence \nof this parameter shows that the precession trajectory of m is elliptical with one main axis \nparallel to the layer plane. \nTo summarize this part of analysis, the ampl itude of precession is determined by three \nmain factors. The first one is the precession rate ϕf, which describes how sensitive the tilt of \neffective magnetic field Beff to the strain-pulse induced modulation is. The second one is the  \n 16spectral density of the incident strain pulse at the precession frequency0ω. The third one is the \noscillating factor sin( ω0Δt/2), which describes the efficiency of interference between incident \nand reflected parts of the strain pulse at a given coordinate z. The maximum amplitude is \nobtained at a coordinate, where tΔ is equal to half of the precession period. For mT B40=  and \n3102−×=zzε  shown in Fig. 3(b), GHz2.6 2/0 =π ω , and maximum amplitude is reached at \nps t80=Δ  corresponding to z=0. The dependence of the components0 cosθ δϕ δ=ym , which is \nalmost twice larger than zmδ, and 0 cosθ δθ δ=xm , are very similar to zmδ, and therefore, we do \nnot plot them separately. \nWe also solve the dynamical equations for the averaged values )(tδϕ and )(tδθ, which \nare as well harmonic oscillations shifted by 2/π  relative to each other: \n()\n()). ( sin2/ sin4 )(), ( cos2/ sin4)(\n0\n0020\n002\nd\nddd\ndd\nttttSfa tttttSf t\n− −=− =\n⊥ ωωωδθωωωδϕ\nωϕωϕ\n   (15) \nThe precession amplitude of the averaged magnetization is also proportional to ϕf and ωS, but \ndepends on the layer thickness through the oscillating factor ()d d τω τω0 02/2/ sin  with the first \nmaximum at GHz10 2/0 ≈π ω . The thick red line in Figs. 3(a) and 3(b) shows the evolution of \nthe averaged functions )(tδε and )(tmzδ . \n Figure 5(c) shows the field dependence of )(maxB mzδ , the amplitude of the after-pulse \noscillations )(tmzδ . max\nzmδ  was calculated for several values of epitaxial strain =zzε3101−× , \n3102−×  and 3103−× . These dependences reflect the competition between the sensitivity of Beff to \nthe strain pulse that increases with  magnetic field and the response of m that decreases with B \ndue to the decrease of ω0. As a result )(maxB mzδ  has a pronounced maximum 3 max10−≈zmδ  at an  \n 17optimal intermediate magnetic field. A str onger static epitaxial strain at a given B leads to an \nincrease of both )(Bfϕ  and )(0Bω  and, thus, the maximum of )(maxB mzδ  also shifts to higher \nmagnetic fields. In general, the field dependen ce of the precession amplitude, as well as its \nmaximum value of 10-3, is in good agreement with the experimental results [5].  \nWe also numerically analyze the influence of nonzero positive shear strain εxy. At finite \nεxy the precession rate fθ is nonzero even at B=0, but it rapidly decreases and becomes negligible \nwith increasing B, see Eq. (7). As a result, for almost the whole range of B the response of Beff \non the strain-induced perturbation is determined mainly by fϕ and is not affected substantially by \nthe presence of shear strain. In Figs. 5(b) and 5(c) we see the decrease of the precession \nfrequency and the precession amplitude over the whole range of B in presence of shear strain. \nThe calculated field dependencies )(maxB mzδ  for 4102−×=xyε  are shown in Fig. 5(c) by the \ndash-dotted lines. \n \nB. In-plane magnetic field \nIf an external magnetic field is applied along th e [100] crystallographic direction and the shear \nstrain is zero, m is oriented along the [100] axis for any value of B and strain-pulse-induced \nmagnetization precession is impo ssible. Therefore, the presence of shear strain is a key \nrequirement for this geometry. Below we examine the case of nonzero, but small positive xyε, \nfor which m is slightly turned in the film plane toward the ]011[  direction. In this case the free \nenergy density depends only on ϕ and we may simplify expression (4) for MF to: \n[] . cos 2sin21)4cos3() (41)(2 4 ϕ ϕ ε ϕ ε ε ϕε B A A B Fxyxy xx zz c M − + + − − =\n   (16)  \n 18Figure 6(a) shows the dependence )(ϕMF  for 4102−×=xyε  for different B. At zero magnetic \nfield MF has the minimum at finite 00<ϕ . With increasing B the minimum gradually shifts \ntoward the [100] axis. Figure 6(b) show s the field dependence of the projection mx=cos ϕ0 for \ntwo values of εxy. In contrast to the case of perpendicular magnetic field, where we observe a \nrapid step-like turn of m toward the field direction at some threshold, here m continuously is \nrotated with increasing magnetic field.  \n For this geometry the tilt of Beff is determined only by θf, since 0=ϕf . So the strain \npulse ),(ztzzδε  tilts Beff maintaining, however, its in-plane orientation. Figure 7(a) shows the \nfield dependence of | fθ| for two values of =xyε4101−× and 4102−× . The value \nof0 4 4sin || ϕ γε θ A f −=  decreases with increasing B since m is tilted closer to the [100] \ncrystallographic direction. The larger xyε is the stronger is the response of Beff on zzδε, while the \nstatic epitaxial strain zzεdoes not influence the value of fθ substantially. One sees that fθ is two \norders of magnitude smaller than fϕ due to the significant difference in the values of the \nmagnetoelastic coefficients A4ε and A2ε. Obviously, the strain pulse ),(ztzzδε affects the in-plane \norientation of Beff much weaker. \n Figure 7(b) shows the field dependencies of the precession frequency ) (0Bω  for several \nvalues of zzε and xyε. Contrary to the case of a perpendicular magnetic field, here 0ω \ncontinuously increases with increasing B. However, the dependence of 0ω on the static strain \ncomponents is the same: 0ω is larger for stronger epitaxial strain zzε and becomes smaller with \nincreasingxyε. \nWe also give simple analytical expressi ons for the after-pulse  free precession of m:   \n 19()\n() . cos2sin 2),(, sin2sin2),(\n0000\n||\ndd\ntttSf tztttSfatz\n− ⎟\n⎠⎞⎜\n⎝⎛Δ=−⎟\n⎠⎞⎜\n⎝⎛Δ=\nωωδθωωδϕ\nωθωθ\n,     (17) \nand for the corresponding averaged values \n()\n()). ( cos2/ sin4)(), ( sin2/ sin 4)(\n0\n0020\n002\n||\nd\nddd\ndd\nt Sf tt Sfat\nτ ωτωτωδθτ ωτωτωδϕ\nωθωθ\n− =− =\n,    (18) \nwhere []0 0 2 0 4\n0|| cos 2sin 2 4cos)) ( (4 ϕ ϕ ε ϕ ε εωγ\nε B A A B axy xy xx zz c − + − − −=  has a value between \n0.5 and 1 and increases with external magnetic field. \n Figure 7(c) demonstrates the field dependence of )(maxB mzδ , which looks similar to the \npreceding geometry. The main differences are: (i) the smaller value of )(maxB mzδ  due to the \nsignificantly smaller value ofθf and (ii) a different dependence of the precession amplitude on \nthe static strain component zzε. For this magnetic field direction the shear strain affects the \nprecession rateθf significantly, which increases with increasing xyε, but changes only slightly \nthe precession frequency. As a result max\nzmδ is much larger for stronger shear strain. In contrast, \nthe static epitaxial strain zzε does not change the precession rate θf substantially, but 0ω is still \nhigher for larger zzε. As a result, at low magnetic fields this leads to an increase of max\nzmδ  and to \na shift of the maximum to lower B with increasingzzε. At high magnetic fields, however, max\nzmδ  \nis reduced for stronger epitaxial strain zzε. The crossing occurs at a ma gnetic field [shown by the \ndashed line in Fig. 7(c)] at which GHz12 2/0 =π ω , corresponding to the maximum of the  \n 20function ()d d S τω τωω 0 022 sin . Thus, at stronger magnetic fields the increase of the precession \nfrequency leads to a decrease of the precession amplitude as seen in Fig. 7(c).  \n \n5. SUMMARY \nTo summarize the developed analysis, we have elaborated three important factors that \ndetermine the efficiency of the strain pulse-induced magnetization precession. The first one is \nhow strong the distraction of the magnetizatio n direction from equilibrium by the dynamical \nstrain is for a given orientation and strength of the external magnetic field. The distraction is \ndetermined by the magneto-crystalline anisotropy of the FMS layer, which depends on a number of parameters, including the holes and magnetic spins concentrations, the lattice temperature, the \ngrowth direction, as well as the presence of a shear static deformation. The MCA characterizes \nthe sensitivity of the magnetic system to the strain  pulse but does not vary with the specific shape \nof the pulse. \n The second factor arises from the spectral prop erties of the strain pulse. The cumulative \neffect of the pulse is the excitation of precessi on at the frequency of the ferromagnetic resonance. \nNaturally, the amplitude of precession is proporti onal to the spectral density of the strain pulse \ncomponents at this frequency. For the assumed pulse shape, it is determined by the value of S\nω. It \nis worth to mention here that for typical strain pulses the spectrum is quite broad, being extended \nup to a few hundred GHz.  \n Finally, the third factor appears because of the interference of the incident and reflected \nparts of the strain pulse. As a result, the pre cession amplitude averaged over the layer thickness \nis given by the oscillating function of the ratio of the travel time of the strain pulse through the \nfilm and the period of the magnetization preces sion. Thus for a given ferromagnetic resonance  \n 21frequency it is possible to predict at which film thickness the excitation of precession is most \nefficient. \n The maximal amplitude achieved for perpendicular orientation of the external magnetic \nfield is 10-3 and depends on the three factors summarized above. Experimentally strain pulses \nwith 10 times larger amplitudes may be injected into the FMS layer. If in addition the pulse duration, the layer thickness and the precession fre quency are perfectly adjusted to each other, \nthe maximal estimated amplitude of precession is 5\n×10-2. For in-plane orientation of the \nmagnetic field the effect of the strain pulse is  much weaker due to the much smaller anisotropy \ncoefficients. However, in recent experiments on a variety of (Ga,Mn)As layers the precession \namplitude was just twice less for this experimental geometry  compared to the case of a \nperpendicular magnetic field [17]. The differen ce between the experimental observation and the \nresults of our analysis may arise from the uncertainty of the value of A4ε, which is hard to obtain \nby steady-state measurements or to calculate accurately in the frame of a microscopic model. For a larger value of the cubic magnetoelastic coefficient A\n4ε  than assumed here we estimate \ncomparable maximal precession amplitudes for the in-plane field geometry and most importantly \nfor the case of zero magnetic field.  \n Nevertheless, this value is not enough for strain-induced switching of magnetization \nbetween the in-plane easy axes. A much stronger e ffect may be achieved fo r a shear strain pulse \ndue to the much larger value of the in-plane uniaxial magnetoelastic coefficient A2xy. A strain \npulse δεxy of amplitude 4 ×10-4 may rotate Beff completely toward the ]011[ direction for the \nchosen FMS layer parameters. In this case the magnetization will precess between the [100] and \n[010] directions and if the strain pulse is  properly shaped precessional switching of the \nmagnetization in analogy to the experiments with pulsed magnetic fields [2] becomes possible. The idea of precessional switching by modulat ing the MCA of a (Ga,Mn)As layer has been  \n 22discussed recently [38] and has also been demons trated [4], although in a nother material and at \nlower frequencies. The analysis of the strain-pulse induced magnetization precession for a shear strain pulse may be done in the same way.  \n    To conclude we have carried out a comprehensive analysis of the magnetization \nprecession induced by a strain pulse in a thin FMS layer. We have chosen a strain pulse shape that is typical for ultrafast acoustic experiments, and modeled numerically the strain-pulse-\ninduced spatial-temporal evoluti on of magnetization. Solution of the Landau-Lifshitz equation in \nlinear approximation has lead to simple analytical expressions for the amplitude of the strain-pulse-induced precession both for any point in the FMS layer as well as averaged over the whole \nlayer. We have found that strain-pulse-induced precession becomes possible when in equilibrium \nthe magnetization is not parallel to the main cr ystallographic axes and in-plane diagonals. This \ncondition is fulfilled in presence of shear strain in-plane anisotropy or in an external magnetic \nfield, which turns the magnetization out of the easy axis. We have numerically examined two alternative directions of the magnetic field a nd analyzed the dependence of the precession \namplitude on the field strength and the static strain components. The value of epitaxial strain \nmainly influences the precession frequency and in  that way slightly affects the precession \namplitude. The shear strain becomes crucially important for in-plane magnetic fields and mainly determines the precession amplitude in this geometry.  \n \nACKNOWLEDGEMENTS \nWe thank A.V. Akimov, R.V. Pisarev and B. A. Glavin for valuable discussions. The work was \nsupported by the Deutsche Forschungsgemeinschaft (Grant No. BA1549/14-1), the Russian Foundation for Basic Research (11-02-00802), the Russian Academy of Sciences, and the National Science Foundation (grant DMR10-05851).   \n 23APPENDIX A: ANISOTROPY COEFFICIENTS CALCULATION  \nThe anisotropy coefficients )2(\n4)1(\n4 2 ,,,ε ε ε AAABc  and xyA2 for a particular structure, which determine \nthe response of the MCA to the strain pulse, can be obtained experimentally, e.g., from \nferromagnetic resonance or magneto-transport m easurements, or calculated using a microscopic \ntheory. A thorough comparison of experimental and theoretical data may be found in Ref. [29]. \nTheoretical approaches to the ferromagnetism of (Ga,Mn)As are largely based on the Zener mechanism originally proposed for metals [39,40] and assume that the ferromagnetic coupling between the Mn spins is mediated by free holes [7,8,18]. The free energy density can be calculated using the effective mass Hamiltonian which, in addition to the six-band \npk⋅ \nLuttinger-Kohn and the strain terms, includes the p-d exchange interaction of the holes and the \nMn spins in the molecular-field approximation [18] . According to this model the mechanism of \nthe strain-pulse induced precession is that the pulse changes the hole spectrum, giving rise to a \nhole redistribution among the energy bands. This, in turn, results in a change of the \nmagnetization orientation according to the minimum of the free energy . Using this model we \ncalculate the intrinsic anisotropy parameters. Th e hole spectrum calculations are done in the \nlimits of 0=T  and 0=B , in accordance to Refs. [9,18,29]. The parameters of the Hamiltonian \nare chosen like in Ref [18] with the only difference that the shear deformation component is \ntaken into account according to Refs. [41,42]. Since the hydrostatic strain ) (zz yy xx ε ε ε = =  does \nnot affect the magnetic anisotropy in this model the additional relation )2(\n4)1(\n4 ε εA A= between the \nmagnetoelastic constants is fulfilled. \nThe p-d exchange interaction is described in Ref. [18] by the parameter which is \nproportional to the number of Mn spins MnN. Since the presence of Mn interstitial defects \nreduces the number of active Mn spins the real nu mber of Mn spins interacting with the holes is  \n 24smaller than the one introduced by the nominal doping 05.0=Mnx  [7,8]. To account for that, we \ncalculate the anisotropy parameters as  function of saturation magnetization 0 0Mμ  for a range of \nMn ion concentrations 05 .003.0 ÷ =Mnx  and for a range of hole concentrations  \n3 2010)51(−×÷= cm p . The best agreement with the experiment reported in Ref. [5] has been \nobtained for the following parameters: xMn=0.045; p=4×1020 cm-3 and μ0M0=60 mT. 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Balestriere,  T. Devolder, J. Wunderlich, and C. Chappert, Appl. Phys. Lett. 96, 142504 \n(2010). \n[39]  C. Zener, Phys. Rev. 81, 440 (1951). \n[40]  C. Zener, Phys. Rev. 83, 299 (1951). \n[41]  G. L. Bir and G. E. Pikus, Symmetry and strain-induced effects in semiconductors  (Wiley, \nNew York, 1974). \n[42]  R. Winkler, Spin-orbit coupling effects in two-di mensional electron and hole systems \n(Springer, Berlin, 2003). \n  \n 28FIGURE CAPTIONS \n \nFigure 1. (a) Schematic of experiments with picosecond acoustic pulses in ferromagnetic \nepitaxial layers. (b) Temporal profile of the strain pulse injected into the GaAs substrate from the \nmetal film.  \nFigure 2. (a) Equilibrium orientation of the effective field Beff and the magnetization M in \nperpendicular external magnetic field B, and coordinate system orientations used in the article.  \n(b) Magnetization precession after the st rain pulse has left the FMS layer. \nFigure 3. (a) Temporal evolution of the strain pulse ),(ztzzδε  at three positions in the FMS layer \n(black lines) and the relative modulation of the layer thickness )(tδε  (thick red line). \n(b) Strain-pulse-induced temporal evolu tions of the magnetization projection ),(ztmzδ  at three \npositions in the FMS layer (thin black lines) and the value averaged across the layer )(tmzδ  \n(thick red lines). The evol utions are calculated at B=40 mT applied perpendicular to the layer \nplane under static strain zzε=2×10-3 and xyε=2×10-4. Time 0=t  corresponds to the moment \nwhen the center of the incident strain pulse crosses the GaAs /(Ga,Mn)As interface. The vertical \ndot-dashed line shows the time moment at which the strain pulse leaves the FMS layer.   \nFigure 4. (a) Normalized free energy density )( )(x M M M mF F F − = Δ m  as function of angle θ \nfor different values of th e external magnetic field B applied perpendicular to the layer. (b) Field \ndependence of the magnetization projection 0 cosθ =zm  onto the direction of magnetic field for \nthree values of the static epitaxial strain. The vertical dashed lines show the values of B* when m \nrapidly turns toward the external field directi on (see text). The calculations are done for 0 =xyε . \n \nFigure 5 . Magnetic field dependencies of the absolute  value of the effective precession rate | |ϕf \n(a), the precession frequency π ω2/0  (b), and the averaged precession amplitude max\nzmδ (c) for B \nperpendicular to the layer plane, calculated for different values of the static strain components \nzzε and xyε. \n  \n 29Figure 6. (a) Normalized free energy density )( )(x M M M mF F F − = Δ m  as function of the \nequilibrium angle ϕ for different values of B applied along the [100] direction in presence of \nshear strain. (b) Field dependence of the magnetization projection 0 cosϕ =xm  onto the direction \nof the magnetic field for two values of shear strain xyε.  \n \nFigure 7. Magnetic field dependencies of the absolute  value of the effective precession rate ||θf \n(a), the precession frequency π ω2/0  (b) and the averaged precession amplitude max\nzmδ (c) for  \nB ||[100] calculated for different valu es of the static strain components zzε and xyε. The dashed \nlines show the frequency (horizontal) and the corresponding value of magnetic field (vertical) \ndemarking the field-frequency range in which a hi gher precession frequency results in a larger \nprecession amplitude.  \n  \n 30 \n \n \n \n  \n \n  \n  \n \n  \n  \n \nFigure 1. \nT. L. Linnik et al. “Theory of magnetization precession…”  (Ga,Mn)As layerGaAs \nsubstrate\nmetal film\n(~100 nm)laser pulse\n~ 100 fs\nstrain pulsez\n-ττ2τ\nTime, t Strain pulse, δεzz(t)\n  \n0εzzmax(a)\n(b) \n 31 \n \n \n \n \n  \n \n \n \n \n \n  \n \n \n \n  \n \n \n \n  \n \n \nFigure 2. \nT. L. Linnik et al. “Theory of magnetization precession…” . xyz\nM\n[100][010][001]\nϕθ\n[110]Beff\n(a) B\nxyzM\n(b)\nBeff B \n 32 \n \n  \n \n \n  \n \n \n \n  \n \n \n \n  \n \n \n \n \n \n  \n \nFigure 3. \nT. L. Linnik et al. “Theory of magnetization precession…” \n-101\n δε(t)\n z=0 nm\n z=100 nm\n z=190 nm\n  δεzz  (10-4)(a)\n0 50 100 150 200 250 300-505\n Time (ps)(b)\n δmz(t) δmz  (10-4)\n  \n 33 \n \n \n \n \n \n  \n  \n \n  \n \n \n  \n  \n  \n \n \n  \nFigure 4. T. L. Linnik et al. “Theory of magnetization precession…”   \n-0.10.00.1\n210 mT180 mT120 mT80 mT\nπ/2 3π/8 π/4 π/8\n  ΔFM(θ) (mT)\nθ(rad)0B||[001]\nB=0 mT εzz=2x10-3(a)\n0 50 100 150 200 2500.00.51.0\n(b)\n3x10-32x10-3\n243 mT180 mT\n  mz\nB (mT)B*=117 mT\nεzz=1x10-3 \n 34 \n \n \n     \n \n \n \n \n \n \n \n \n \n \n \n \n  \n Figure 5\n. T. L. Linnik et al. “Theory of magnetization precession…”   \n0246 (a) 3x10-3\n2x10-3\nεzz=1x10-3\nB||[001]\n  |fϕ| (ps-1)\n0246\n(c)(b)\n3x10-3 2x10-3εzz=1x10-3\n ω0/2π  (GHz)\n 2x10-4    0εxy\n0 50 100 150 200 2500510\n    εzz=1x10-3\n2x10-3\n3x10-3\n \n B (mT)δmmax\nz  (10-4)   \n 35 \n \n \n \n \n \n \n \n  \n \n \n \n  \n \n \n \n  \n \n \n \n \n \nFigure 6. \nT. L. Linnik et al. “Theory of magnetization precession…”   \n-0.0050.0000.0050.010\nεzz=2x10-3 30 mT\n1020\n0 -π/8\n  ΔFM(ϕ) (mT)\nϕ (rad)-π/4B||[100]\n0εxy=2x10-4\n(a)\n0 50 100 150 200 2500.970.980.991.00\nεzz=2x10-3\n(b)εxy=2x10-4\n  mx\nB (mT)εxy=1x10-4 \n 36 \n \n \n \n \n \n  \n \n \n \n  \n \n \n \n  \n \n \n \n  \n \n \n \n \nFigure 7. \nT. L. Linnik et al. “Theory of magnetization precession…”  \n \n \n  \n0.020.040.060.08(a)\nεxy=1x10-4εxy=2x10-4εzz=2x10-3\nεxy=1x10-4B||[100]\n  \n051015(b)\n3x10-3\n2x10-3\nεzz=1x10-3\n \n 2x10-4  1x10-4εxy\n0 50 100 150 200 250 3000.00.20.40.60.81.0(c)\n  1x10-3\n  2x10-3\n  3x10-3εxy=2x10-4\nεxy=1x10-4\n \n B (mT)\n εzz|fθ| (ps-1)ω0/2π  (GHz) δmmax\nz (10-4)   "    },    {        "title": "1111.5453v1.Magnetic_and_structural_characterization_of_nanosized_BaCo_xZn__2_x_Fe__16_O__27__hexaferrite_in_the_vicinity_of_spin_reorientation_transition.pdf",        "content": " \n \n \n  \n \nMagnetic and structural characterization of nanosized \nBaCo xZn 2-xFe16O27 hexaferrite in the vicinity of spin \nreorientation transition \nA Pasko1, F Mazaleyrat1, M LoBue1, V Loyau1, V Basso2, M Küpferling2, C P \nSasso2 and L Bessais3 \n1 SATIE, ENS Cachan, CNRS, UniverSud, 61  av President Wilson, F-94235 Cachan, \nFrance \n2 INRIM, Strada delle Cacce 91, I-10135 Torino, Italy \n3 ICMPE, CNRS, Université Paris 12, 2-8 rue Henri Dunant, F-94320 Thiais, France \nE-mail: pasko@satie.ens-cachan.fr \nAbstract . Numerous applications of hexagonal ferrites are related to their easy axis or easy \nplane magnetocrystalline anisotropy configurations. Certain W-type ferrites undergo spin \nreorientation transitions (SRT) between differe nt anisotropy states on magnetic field or \ntemperature variation. The transition point can be tuned by modifying the chemical composition, which suggests a potential application of hexaferrites in room temperature \nmagnetic refrigeration. Here we present the resu lts of structural and magnetic characterization \nof BaCo\nxZn2–xFe16O27 (0.7 ≤ x ≤ 2) doped barium ferrites. Fine powders were prepared using a \nsol-gel citrate precursor method. Crystal stru ctures and particle size distributions were \nexamined by X-ray diffracti on and transmission electron microscopy. The optimal synthesis \ntemperature ensuring complete formation of single W-phase with limited grain growth has \nbeen determined. Spin reorientation transitions were revealed by thermomagnetic analysis and \nAC susceptibility measurements. \n1.  Introduction \nHexaferrites have been the subject of intensive studies due to an appealing combination of good \nmagnetic properties and low cost. This large family of oxides with hexagonal crystal structure contains \nferrimagnetic compounds with easy axis of magnetization ( e.g. M-type ferrites) and easy plane of \nmagnetization ( e.g. Y-type ferrites). Hence, hexaferrites have  been widely adopted in two distinct \nfields: permanent magnets and microwave technolog y components [1]. On the other hand, W-type \nferrites Ba M2Fe16O27 (M = Mg, Mn, Fe, Co, Ni, Cu, Zn) can unde rgo spin reorientation transitions \n(SRT) between different anisotropy configurations (easy plane ↔  easy cone ↔ easy axis) induced by \nchange of temperature or applied magnetic field [2–9]. The transition temperatures can be tuned by \nmodifying the chemical compositi on (substitution of bivalent metal M ). Moreover, some SRT are \nexpected to be of the first order, which suggests a potential application of W-type ferrites in room \ntemperature magnetic refrigeration [10]. \nConventional ceramic method of hexaferrite synthesis is efficient, but requires elevated \ntemperatures for solid state reaction to occur between premixed  powders. Alternative production \nroutes (aerosol pyrolysis, chemic al co-precipitation, glass crystallization, hydrothermal synthesis, etc.)  \n \n \n  \n \nare intended to improve mixing of initial component s down to atomic level and thereby to facilitate \ndiffusion. In particular, a sol-gel technique enabl es to obtain sufficiently homogeneous precursors for \nlow-temperature synthesis of nanosized simple (s pinel) ferrites. However, hexagonal ferrites with \ncomplex layered crystal structures still require re latively high temperatures to form because of \nthermodynamic stability conditions. Trying to avoid rapi d growth of grains, we have used this soft-\nchemistry approach for production of W-type ferrit es. An important goal was to determine the heat \ntreatment regimes ensuring complete transformation of a precursor into the smallest particles of single \nW-phase. In this paper the results of structural and magnetic characterization of BaCo xZn2-xFe16O27 \npowders are presented. The chemical compositions were chosen so that SRT occur near room \ntemperature ( x = 0.7, 0.75, 0.8) or much higher (x  = 2). \n2.  Experimental \nHexaferrite powders have been prepared by a so l-gel citrate precursor method [11–13]. High purity \niron(III) nitrate, barium hydroxide, cobalt(II) oxide, zi nc oxide and citric acid were used as starting \nmaterials with the molar ratio of citrate to metal io ns 2:1. Iron(III) nitrate was dissolved in deionized \nwater and quantitatively precipitated with excess of ammonia solution as iron(III) hydroxide. The \nprecipitate was filtered and washed with water until neutrality. Then the obtained iron(III) hydroxide \nwas dissolved in a vigorously stirred citric acid solution at 60–70 °C. Barium hydroxide and other \nmetal sources were added according to stoichiometr y. At this stage 3 samples of each composition \nwere separated: (A) pH value of the solution was adjusted to 6 for better chelation of cations [13]; (B) \nno modification was done [12]; (C) ethylene glycol was added to increase viscosity by \npolycondensation reaction [11]. Water was slowly ev aporated at 80–90 °C with continuing stirring \nuntil a highly viscous residue is formed. The gel was dried at 150–170 °C and heat treated for 2 h at \n450 °C for total elimination of organic matter. Fi nally, the inorganic precursor with homogeneous \ncationic distribution was calcined at temperatures up to 1300 °C for 2 h with heating/cooling rate \n200 K/h to synthesize a hexaferrite phase. Details of the procedure will be discussed elsewhere. \nCrystal structures were examined by PANalytical X’Pert Pro X-ray diffractometer (XRD) in Co-K α \nradiation with X’Celerator detector for rapid data  acquisition. Magnetization curves were recorded on \nLake Shore vibrating sample magnetometer (VSM). Direct observations of powder particles were \ncarried out by FEI Tecnai G2 F20 transmission electron microscope (TEM) operating at 200 keV. \nThermogravimetric analyzer (TGA) PerkinElmer Pyris 6 equipped with a permanent magnet was used \nfor thermomagnetic measurements above room temperature. AC magnetic susceptibility at low \ntemperatures was studied in Quantum Design PPMS. Ri etveld analysis of XRD spectra was performed \nusing MAUD software [14]. \n3.  Results and discussion \nCharacterization had a twofold purpose: to find a correlation between the calcination temperature and \nthe final product properties (phase composition, mean grain size, coercivity); to confirm the existence \nof SRT at expected temperatures in fine hexaferrite powders. \n3.1.  Structural characterization and phase analysis \nFigure 1 represents XRD patterns from BaCo 2Fe16O27 powder synthesized at different temperatures. \nW-ferrite becomes the major phase at 1300 °C, while lower calcination temperatures lead to a mixture \nof W-ferrite, M-ferrite (both are hexagonal) and S-ferr ite (with spinel structure) in the final product. \nTraces of α-Fe 2O3 are sensitive to the method of preparation and dissolve completely with increase of \ncalcination temperature; however, other secondary phases may appear together with W-ferrite. The \nresults of quantitative phase analysis based on Rietveld method of XRD full profile fitting are shown \nin figure 2. At 900 °C only M-ferrite can be fo rmed, with excess of cobalt giving also S-ferrite. \nIncrease of temperature creates favorable conditions for W-ferrite synthesis, directly from the precursor components (nanocrystalline or amorphous  oxides) or through the intermediate reaction \nBaFe\n12O19 + 2 CoFe 2O4 → BaCo 2Fe16O27  \n \n \n  \n \nThe proportion between 3 ferrite phases depends on calcination temperature and production route \n(as the precursor composition is affected, for inst ance, by pH value of the solution). However, at \n1300 °C M-ferrite and S-ferrite almost completely disappear. \n \nFe2O3 (hematite)CoFe2O4 (spinel)M-type Ba ferriteW-type Ba ferrite\nHT 1300 °C\nHT 1250 °C\nHT 1200 °C\nHT 900 °C\n20 25 30 35 40 45 50 55\nDiffraction angle 2 θ [°]Intensity [arb. unit]\n \nFigure 1. XRD patterns of BaCo 2Fe16O27 powder A taken after different heat treatments. \n \nCBA\n0102030405060708090100\n850 950 1050 1150 1250 1350\nTemperature [°C]Mass Fraction [%]\n CBA\n0102030405060708090100\n850 950 1050 1150 1250 1350\nTemperature [°C]Mass Fraction [%]\n M-phase W-phase \n(a) (b) \nFigure 2. Phase composition of BaCo 2Fe16O27 powders A, B, C (from Rietveld fitting of XRD spectra) \nvs. calcination temperature: (a) M-ferrite fraction; (b) W-ferrite fraction.  \n \n \n  \n \n3.2.  Morphology and average size of particles \nThe size of diffracting crystallites in W-ferrite powde rs is derived from XRD data by analysis of line \nbroadening using Rietveld refinement as shown in  figure 3(a). It gives a minimum estimate for \naverage particle size (a particle can contain several crystallites). The mixture of M-ferrite and S-ferrite \nformed at 900 °C have a mean size of crystallites ~ 50 nm. With increase of calcination temperature \nthe calculated values slightly decrease, this can reflect contribution to line broadening caused by \ngeneration of defects in the transforming mixture of phases. At 1300 °C, when W-ferrite phase rapidly \ngrows and becomes dominant, the apparent size of  crystallites significantly increases. In addition, \nXRD spectra of the powders synthesi zed at this temperature exhibit a specific texture characteristic for \nplate-like particles pressed in a holder. \nMagnetic measurements give an illustration to th ese phase transformations. Figure 3(b) shows that \nafter calcination at 900 °C coercivity is high due to the presence of M-ferrite. When W-ferrite forms, \nthe coercivity decreases and reaches ~ 15 mT at 1300 °C, a typical for this phase value. \n \nCBA\n20406080100120140160180200\n850 950 1050 1150 1250 1350\nTemperature [°C]Crystallite Size [nm]\n A\nBC\n0306090120150180210240270300\n850 950 1050 1150 1250 1350\nTemperature [°C]Coercivity [mT]\n W-phase M-phase \nM-phase \nW-phase \n(a) (b) \nFigure 3. BaCo 2Fe16O27 powders A, B, C calcined at different temperatures: (a) mean size of \ncrystallites (from XRD line broadening); (b) coercivity (from VSM measurements). \n \nDirect observations of W-ferrite particles were  performed by TEM. Powders calcined at 900 °C \nconsist of aggregates of clean and almost round particles not exceeding ~ 100 nm, an example is shown in figure 4(a). The shape of crystallites re flects their hexagonal crystal structure. Powders \ncalcined at temperatures high enough to form W-fe rrite exhibit absolutely different morphology as \nshown in figure 4(b). The particles have a wide  size distribution from 50 nm to 2000 nm and are \nmostly plate-like. This morphology is observed not only for 1300 °C when W-ferrite is almost single \nphase, but also for lower temperatures when W-ferrite  appears in the precursor. The average particle \nsize of BaCo\nxZn2-xFe16O27 (x = 0.75) powder estimated by laser light scattering analysis is ~ 700 nm, \nwhich agrees with TEM observations.  \n \n \n  \n \n \n  (a) (b) \nFigure 4. Typical TEM images of powder particles: (a) BaCo 2Fe16O27 precursor after calcination at \n900 °C (mixture of ferrites); (b) W-type BaCo xZn2–xFe16O27 (x = 0.75) ferrite synthesized at 1275 °C. \n \n3.3.  Spin reorientation transitions \nThermomagnetic measurements we re performed on loose powders in a weak (1–10 mT) magnetic \nfield with cycling from room temperature up to 600 °C. An example of magnetization as a function of \ntemperature for W-type BaCo xZn2-xFe16O27 ferrite with x  = 0.8 is shown in figure 5(a). Heating and \ncooling segments do not match (typical hysteretic be haviour), and the first heating differs from others \nbecause of particular magnetization history. Magne tic susceptibility is higher in the easy plane state \nthan in the easy axis one, therefore both spin reorientation (at ~ 55 °C) and ferrimagnetic-\nparamagnetic (at ~ 430 °C) transitions are well reso lved. Small amplitude of the anomalies observed \nabove W-ferrite Curie point confirms that frac tions of magnetic secondary phases are negligible. \n \n2nd cycle1st cycle← Curie pointSRT →\nheating →← cooling\n0 100 200 300 400 500\nTemperature [°C]Magnetization [arb. unit]\n × 10–4\nχ′\nχ″\n← SRT →× 10–6\n2.533.544.555.566.5\n-250 -200 -150 -100 -50 0 50\nTemperature [°C]Susceptibility (Re) [m3·kg–1]\n01234567\nSusceptibility (Im) [m3·kg–1]\n (a) (b) \nFigure 5. Magnetic transitions in W-type BaCo xZn2–xFe16O27 ferrites: (a) magnetization measurements \nfor x = 0.8 (higher temperatures); (b) AC susceptibility measurements for x  = 0.7 (lower temperatures). \n  \n \n \n  \n \nOn the first heating, a sharp drop of magnetization related to SRT is preceded by a distinct peak. \nThe spin reorientation temperature of polycrystallin e powder is also characterized by inflection point \nof magnetization dependence [15]. Another reference can be the point where subsequent cooling and \nheating curves converge (clearly seen in figure 5(a) on the left of magnetization hump). Unlike [6], the \nlatent heat of SRT has not been reliably detected  by differential scanning calorimetry (DSC), while \nCurie point is well visible. \nAC magnetic susceptibility as a function of temperature for W-type BaCo\nxZn2-xFe16O27 ferrite with \nx = 0.7 is shown in Fig. 5(b). The real part of susceptibility has a maximum at ~ 6 °C followed by a \nsharp drop attributed to SRT. Its imaginary part has a minimum at ~ 27 °C which can also be taken as \nthe spin reorientation temperature. Moreover, the mi nimum of imaginary part turns out to approach the \ninflection point of the real part. \n4.  Conclusions \nSingle-phase W-type ferrite submicron powders with different magnetocrystalline anisotropy \nconfigurations are synthesized through a sol-gel ci trate precursor route. The effect of calcination \ntemperature and other production parameters on th e phase composition, average particle size and \nmagnetic properties of the powders is established. Chemical compositions of  substituted hexaferrites \nundergoing spin reorientation transitions near room temperature are determined. \nAcknowledgements \nThis work is supported by EC Seventh Framework Programme fundi ng (project SSEEC, contract FP7-\nNMP-214864). The authors are grateful to P. Au debert (ENS Cachan) for assistance in sol-gel \ntechnique and G. Wang (ICMPE) for TEM observations. \nReferences \n[1] Smit J and Wijn H P J 1959 Ferrites  (Eindhoven: Philips Technical Library) \n[2] Asti G, Bolzoni F, Licci F and Canali M 1978 IEEE Trans. Magn.  14 883–5 \n[3] Albanese G, Calabrese E, Deriu A and Licci F 1986 Hyperfine Interact.  28 487–9 \n[4] Paoluzi A, Licci F, Moze O, Turilli G, Deriu A, Albanese G and Calabrese E 1988 J. Appl. \nPhys.  63 5074–80 \n[5] Samaras D, Collomb A, Hadjivasiliou S, Achille os C, Tsoukalas J, Pannetier J and Rodriguez J \n1989 J. Magn. Magn. Mater. 79 193–201 \n[6] Naiden E P, Maltsev V I and Ryabtsev G I 1990 Phys. Status Solidi A 120 209–20 \n[7] Naiden E P and Ryabtsev G I 1990 Russ. Phys. J.  33 318–21 \n[8] Sürig C, Hempel K A, Müller R and Görnert P 1995 J. Magn. Magn. Mater.  150 270–6 \n[9] Turilli G and Asti G 1996 J. Magn. Magn. Mater.  157–158  371–2 \n[10] Naiden E P and Zhilyakov S M 1997 Russ. Phys. J.  40 869–74 \n[11] Licci F and Besagni T 1984 IEEE Trans. Magn.  20 1639–41 \n[12] Srivastava A, Singh P and Gupta M P 1987 J. Mater. Sci.  22 1489–94 \n[13] Zhang H, Liu Z, Yao X, Zhang L and Wu M 2003 J. Sol-Gel Sci. Technol.  27 277–85 \n[14] Lutterotti L 2000 Acta Crystallogr.  A 56 s54 \n[15] Boltich E B, Pedziwiatr A T and Wallace W E 1987 J. Magn. Magn. Mater.  66 317–22 "    },    {        "title": "1111.6267v1.Large_Coercivity_in_Nanostructured_Rare_earth_free_MnxGa_Films.pdf",        "content": "1 \n Large  Coercivity in Nanostructured Rare -earth -free Mn xGa Films  \n \nT.J. Nummy,1 S.P Bennett,2 T. Cardinal,1 and D. Heiman1 \n1Department of Physics, Northeastern University, Boston, MA 02115  \n2Department of Mechanical Engineering, Northeastern University, Boston, MA 02115  \n \nAbstract  \n \nThe magnetic hysteresis  of Mn xGa films exhibit remarkably large  coercive fields \nas high  as oHC = 2.5 T when fabricated with nanoscale particles of a suitable size \nand orientation.  This coercivity is an order of magnitude larger than in well-\nordered epitaxial film counterparts  and bulk materials . The enhanced  coercivity \nis attributed to  the combinat ion of large magnetocrystalline anisotropy and ~ 50 \nnm size nano particles . The large coercivity  is also replicated  in the electrical \nproperties through the anomalous Hall effect. The magnitude of the coercivity \napproaches that found in rare -earth magnets , mak ing them attractive for rare -\nearth -free magnet applications.  \n \nRare -earth -based magnets provide the backbone of many products, from computers \nand mobile phones to electric cars and wind -powered generators.1 But b ecause of the high cost \nand limited availa bility of rare-earth and precious elements , which are expensive to mine and \nprocess, there is a growing interest in developing new magnetic materials  without these \nelements .2-4 This critical need has fostered innovative research aimed at the discovery of novel \ncompounds and nanoscale composites that are free of these  elements.  One of the key \nproperties of  super strong magnets is their large magnetocrystalline anisotropy.  With this in \nmind, it is ad vantageous to search for rare -earth -free ferromagnets that have large \nanisotropies, such as MnAl and Mn xGa. However, in order to take advantage of the large \nanisotropy and make them suitable for applications they must be synthesized with an \nappropriate com position and structure at the nanoscale level. We find that when the Heusler \ncompound Mn xGa is synthesized with the proper  nanostructuring, a remarkably high  coercive \nfield is produced . These results suggest  that Mn xGa is a good candidate for producing materials \nwith enhanced coercive fields aimed at replacing some rare-earth -based magnets in use today.  \n \nHeusler compounds  possess  a rich variety of fascinating and useful properties .5 \nMagnetic shape -memory, thermoelectrics,  semi and superconductivity, topological insulators , \nand half -metallicity  are a few examples of the special characteristics unfolding from this \nappealing niche of materials.  The binary ferrimagnet Mn xGa (x  = 2 - 3) is one of the simplest \nHeusler materials , yet has remarkable magnetic properties. Its magnetism is  tunable, where the  \nsaturation moment is predicted to be  variable  by as much as a factor of 4 by varying  the \nstoichiometry from x = 2 to x = 3.6 The structural and magnetic properties of Mn xGa have been \ninvestigated  in bulk materials ,6-9 and thin films grown on various substrates including Si,10,11 \nGaAs,12,13 GaN,14 GaSb ,11 Al2O3,11 MgO,15,16 and Cr -MgO17. When g rown on lattice -matched \nsubstrates the resulting epitaxial films exhibit an easy -axis perpendicular to the film \nplane .12,16,17 These oriented films have  led t o measurements of large anisotropy fields   \n2 \n extrapolating to oHA = 10 T or higher .16,17 An anisotropy constant of K ~ 107 erg/cm3 was \ndetermined from  HA = 2K/M S18,19 and the saturation magnetization  MS. In practice, the coercive \nfield is typically limited to H C/HA ~ 0.3 , which could lead to a coercive field of at least oHC ~ 3 T  \nin this material .  \n \nIn the  present work we show that Mn xGa films can be synthesized with nanoparticles  \nhaving ~ 50-100 nm dimensions. This is shown to be an appropriate size for generating \nremarkably large coercive fields,  as large as oHC = 2.5 T , which is nearly an order of magnitude \nlarger than for well -ordered epitaxial films  and bulk samples . The high coercive fields are \nattributed to the combination of specific nanostructuring and a large intrinsic \nmagneto crystal line anisotropy . These results point out a new opportunity for developing rare -\nearth -free magnetic materials.  \n \nThin films of Mn xGa in the range x = 2 to 3 were grown by molecular beam epitaxy \n(MBE ) on Si (001) substrates having an amorphous native oxide surface layer. A view of the \nsurface  of a 20 nm thick Mn xGa film is shown in the  scanning electron microscop e (SEM) image \nin Fig. 1 (a). The particle -like structures are seen to have lateral dimensions on the order of ~ 50-\n100 nm – a size that is crucial for generating high coercivity . Annealing the films produced some \nalignment of the crystalline axes, as shown in the r eflection high energy electron diffraction \n(RHEED) images of Figs. 1(b -c). Before annealing,  Fig. 1(b) shows a predominant random \nalignment displayed  by the polycrystalline -like ring pattern . The nano crystal orientation \nincreased when  the films were anneal ed at high temperatures. After annealing at 400 °C, a \nfraction of the ring intensity coalesced into elongated spots as seen in Fig. 1(c).  The annealed \nparticles were found to have a  distinct D0 22 crystal structure6,12 as determined from θ-2θ x-ray \ndiffraction (XRD) shown in Fig. 1(d). The lattice  constants of the D0 22 structure , illustrated  in Fig. \n1(e), were found to be  a=0.389 nm and c=0.708 nm , indicating a ~ ½ % contraction in lattice \nconstants compared to bulk values6. Magnetization measurements were made in a \nsuperconducting quantum interference device (SQUID) magnetometer from Quantum Design.  It \nwas found that  annealing led to  increase s in the saturation  moment by as much as 102, up to \nvalues a s high  as MS = 130 emu/cm3.  \n \nFigure 2 illustrates the large coercivity in the magnetization that arises from the \nnanostructuring. This figure compares the room temperature magnetization , M(H) , of a 20 nm \nthick nanostructured Mn xGa film to that of a highly -ordered epitaxial film grown on GaAs . In \ncontrast to the epitaxial sample, M(H) of  the nanostructured  film has a wide  “s”-shaped major \nhysteresis  (irreversibility)  loop . The coercive field for the nanostructured film is exceptionally  \nlarge, oHC = 2.5 T. This is  much larger than  the coercive field  for the epitaxial film, oHC = 0.3 6 \nT, which is  typical of other high -quality epitaxial films11-13,15 and bulk samples6. The magnetism  \nof nanostructured film s is robust with respect temperature. The inset of Fig. 2 plots the \nremanent magnetization , M R(H=0),  as a function of temperature, M R(T). The moment remains \nhigh at T = 400 K demonstrating a Curie temperature well above room temperature6,8. The inset  \nalso shows the temperature dependence of the coercive field, H C(T), which appears to have a \ndependence similar to the remanence.  To conclude, we note that the  observed 2.5 T coercive  \n3 \n field is on the order of those  found in some important rare-earth permanent magnets, such as \nNd 2Fe14B where oHC ~ 2.6 T20, and SmCo 5 where oHC ~ 4 T.21 \n \nThe large  coercivity  of nanostructured Mn xGa films  is also present in the electronic \nproperties through the anomalous Hall effect  (AHE) . The Hall effect in magnetic materials \ncontains two main contributions,  ρH (H) = R o H + R 1 M(H,T), where R o = -1/ne is the ordinary Hall \neffect (OHE) coefficient , n the carrier concentration, R 1 the AHE coefficient, and M(H,T) the field \nand temperature -dependent magnetization.  For magnetoconductivity measurements, van der \nPauw squares and lithographically fabricated Hall bars with Ti -Au contacts were measured  using  \na 14 T Cryogenics Limited Cryo -free magnet . Despite  the particle -like growth, all of the films had \nmetallic conductivity with longitudinal resistivities in the range ρ = 200 to 300 ·cm, very \nsimilar  to that measured in epitaxial films.11,15 Figure 3 shows results of room temperature Hall \neffect measurements of a 20 nm thick nanostructured  Mn xGa film. In Fig. 3(a) the raw Hall \nresistivity is plotted as a function of H applied per pendicular to the film. The AHE gives rise to \nthe irrevers ible open -loop hysteresis, which closes up at oH = 5 T . At higher fields the \nresistivity is reversible and linear in field . The dashed line in Fig. 3(a) represents the  total  \nreversible  component of the resistivity , which may contain some residual nonsaturating \nmoment . From th e slope of the  reversible component  we obtain a  lower limit  for the effective  \ncarrier concentration, n ≥ 8 x 1021 cm-3, which  corresponds to one electron per unit ce ll. In \nFigure 3(b) we compare the hysteresis in the AHE with the magnetization. There is excellent  \nagreement between data from the two measurements, including the magnitude of the  coercive \nfield  for that sample . The AHE has been observed previously in Mn xGa, but in those studies the \nmagnitude of the coercive field was limited to < 1 T.11,12,15 At high fields we find that the AHE \nresistivity reaches a saturation value of ρ AHE = 1.5 μΩ·c m, which is a factor of 2 larger than \nobserved  for the precious -metal perpendicular moment material L10-FePt (0.88 μΩ·cm).22 \n \nThe dimensions of the n anoscale particle s are important for achieving large coercive \nfields , as particles that are too large support multidomains , while particle that are too small \nbecome superparamagnetic. Several mechanisms hinder the field -induced magnetic alignment \nthat lead s to  hysteresis . In relatively large particles the hysteresis arises from magnetic \ndomains , which are  characterized by pinning of domain walls and nucleation of reversed \ndomains. On the other hand, w hen particles are too small to support multiple domains , the \nhysteresis arises from coherent reversal of single magnetic domains  that are hindered by \nanisotropy . Single -domain particles must be small enough to prohibit the formation of domain \nwalls . Energy minimization requires that single domain  particles must  be smaller  than several \ntimes (≥ 3) the domain wall width. The width of our domain walls is δ w ~ 20-30 nm, using δ w ~ \n2π (A/K 1)1/2 and A ~ 2 x 10-6 erg/cm as the exchange stiffness obtained from the magnetization \nand Curie temperature19. A sizeable fraction of th e Mn xGa particles are single domain since the \ndomain wall width is a significant fraction of the particle size. For a s ingle domain particle with \nuniaxial anisotropy the coercive field depends on the relative orientation between the unique \n(easy) magnetic axis and the applied field. For a field applied along the unique axis (φ = 0) the \ncoercive field is maximum and is equal to the anisotropy field , H C/HA = 1, and the remanent \nmagnetization is equal to the saturation magnetization, M R/M S = 1, shown by the dashed curve \nin Fig. 3(c ). As the angle between the applied field and the unique axis increases, the coercive  \n4 \n field and remanent magnetization collapse , both reach ing zero for a field applied at right angles \nto the unique axis. The Stoner -Wohlfarth (SW) model23 treats  small noninteracting single -\ndomain particles possessing uniaxial anisotropy. Although it models particle shapes consisting \nof ellipsoids  of revolution having an easy axis along the semi -major axis of the ellipsoid , it is \nnevertheless useful for qualitative comparisons to real systems of particles.  Results of the SW \nmodel predict that for a completely random distribution of easy axis directions  of prolate \nspheroids  (oblate spheroids have H C = 0), the remanent magnetization is one -half the saturation \nvalue , M R/M S = 0.50 , and the coercive field is approximately one -half the anisotropy field , HC/HA \n= 0.48 , shown by the solid curve in  Fig. 3( c). However, the hysteresis data of the nanostructured  \nfilm in Fig. 2 shows a larger remanence ratio of M R/M S = 0.78.  (Note that this measured ratio is \nan upper limit as the saturation moment may be larger due to incomplete saturation at the \naccessible fields.)  The larger observed M R/M S ratio may  be assigned to partial  nonrandom \nalignment of the easy axes, which is confi rmed by the spotty RHEED images.  This remanence  \nvalue is equal to that of a SW particle with its unique axis oriented at φ = 40 deg to the applied \nfield , where  MR/M S = 0.7 7 and H C/HA = 0.50, and shown by the dotted curve in  Fig. 3( c). Using \nan upper limit value  of HC/HA = 0.5 and our observed value  of oHC = 2.5 T, we estimate  a lower \nlimit  for the anisotropy field of oHA ≥ 5 T for the nanostructured  film, consistent with  \nextrapolated measurements16,17. \n \nAs a final point  we consider other contributions to the coercive field. The uniaxial \nanisotropy is not limited to magnetocrystalline  anisotropy, K 1, arising from spin -orbit \ninteractions, but can also have contributions from shape , stress , and surface  uniaxial magnetic \nanisotropies. We estimate  that the shape contribution18,19 to the coercive field is negligible (~ \n0.05 T) and the surface contribution19 is small (≤ 0.4 T). For the strain contribution , we applied  \nthe Williamson -Hall model of XRD line broadening  to the data in Fig. 1(d)24,25 to obtain a value \nof RMS strain26 ε = 0.5 ± 0.1 %.  Although the magnitude of the magnetostriction is not known, \nthe strain could add noticeably  to the coercive field (~ 100 T). In conclusion , the dominant \nanisotropy  leading to the high  coercive field s appears to be  the magnetocrystalline anisotropy, \nbut strain and surface contributions could play a  smaller  role.  \n \nIn summary , we show that when Mn xGa is synthesized with an appropriate nanoscale \nstructure the coercive field is increased by nearly an  order of magnitude  over that found in \nwell-ordered epitaxial films  and bulk samples . Coercive field s as large as  oHC = 2.5  T were \nobtained for films grown on Si substrates. The  nanostructured films have strained particle -like \nfeatures ~ 50-100 nm in size . The remarkably  large coercivity  is attributed to the combination  of \nlarge  intrinsic magnetocrystalline anisotropy  and suitable nanostructuring . In addition , the large \ncoercivity is also present in the electrical conductivity through the anomalous Hall effect. These \nmagnetic and magnetotransport properties  could find applications in  mechanical and electrical \ndevices  that require high coercive fields , and understanding their structures could provide a \npathway for developing new rare -earth -free magnetic materials . \n \nThis work was supported by the National Science Foundation grant DMR -0907007.  We \nthan k L. Lewis , Y. Chen  and W. Nowak  for useful discussions and  W. Fowle for assistance with  \n5 \n the electron microscopy studies.  Added note: a fter this  work was submitted, an article was \npublished describing a  coercive field of 2.05 T in Mn 1.5Ga.27 \n \n \nREFERENCES  \n                                                \n1 Nicola Jones, Nature 472, 22 (2011).  \n2 Editorial, Nature Mater. 10, 157 (2011).  \n3 Eiichi Nakamura and Kentaro Sato, Nature Mater. 10, 158 (2011).  \n4 David Kramer, Phys. Today, Issues and Events , 22 (May 2010).  \n5 Tanja Graf, Claudia Felser, and Stuart S.  P. Parkin, Prog. Sol. State Chem. 39, 1, (2011).  \n6 Jürgen Winterlik, Benjamin Balke, Gerhard H. Fecher, and Claudia Felser, Maria C.M. Alves, \nFabiano Bernardi, and Jonder Morais, Phys. Rev. B 77, 054406 (2008).  \n7 T. A. Bither and W.  H. Cloud, J. Appl. Phys. 36, 1501 -1502 (1965).  \n8 H. Niida, T. Hori, H. Onodera, Y. Yamaguchi, and Y. Nakagawa, J. Appl. Phys. 79, 5946 (1996).  \n9 Benjamin Balke, Gerhard H. Fecher, Jürgen Winterlik, and Claudia Felser , Appl. Phys. Lett. 90, \n152504 (2007).  \n10 C. L. Zha, R.  K. Dumas, J. Persson, S.  M. Mohseni, J. Nogués, and Johan Åkerman, IEEE Mag. \nLett. 1, 2500104 (2010).  \n11 Wuwei Feng , Duong Van Thiet , Dang Duc Dung , Yooleemi Shin , and Sunglae Cho, J. Appl. \nPhys. 108, 113903 (2010).  \n12 M. Tanaka, J.  P. Harbison, J. DeBoeck, T. Sands, B. Philips, T.L. Cheeks, and V.G. Keramidas, \nAppl. Phys. Lett. 62, 1565 (1993).  \n13 W. Van Roy, H. Akinaga, S. Miyanishi, K. Tanaka, and L.  H. Kuo, Appl. Phys. Lett. 69 711 \n(1996).  \n14 E. Lu, D.  C. Ingram, A.  R. Smith, J.  W. Knepper, and F.  Y. Yang, Phys. Rev. Lett. 97, 146101 \n(2006).  \n15 Feng Wu , E. P. Sajitha , Shigemi Mizukami , Daisuke Watanabe , Terunobu Miyazaki , Hiroshi \nNaganuma , Mikihiko Oogane , and Yasuo Ando, Appl. Phys. Lett. 96, 042505 (2010).  \n16 H. Kurt, K. Rode, M. Venkatesan, P. Stamenov and J.  M. D. Coey, Phys. Rev. B 83, 020405(R) \n(2011).  \n17 Feng Wu,  Shigemi Mizukami, Daisuke Watanabe, Hiroshi Naganuma, Mikihiko Oogane, Yasuo \nAndo, and Terunobu Miyazaki,  Appl. Phys. Lett. 94, 122503 (2009).  \n18 B. D. Cullity and C.  D. Graham, Introduction to Magnetic Materials , 2nd ed. (IEEE Press, Wiley \nand Sons, 2009).  \n19  J. M.  D. Coey, Magnetism and Magnetic Materials  (Cambridge University Press, 2010).  \n20 H. Jiang, J. Evans, M.  J. O’Shea, and Jianhua Du, J. Magn. Mag. Matl. 224, 233 (2001).  \n21 M. F. de Campos, F.  J. G. Landgraf, N.  H. Saito, S.  A. Romero, A.  C. Neiva, F.  P. Missell, E. de \nMorais, S. Gama, E.  V. Obrucheva, and B.  V. Jalnin, J. Appl. Phys. 84, 368 (1998).  \n22 J. Yu, U. Ruediger, A. Kent, R.  F. C. Farrow, R.  F. Marks, D. Weller, L. Folks, and S.  S. P. Parkin, \nJ. Appl. Phys. 87, 6854 (2000).  \n23 E. C. Stoner and E.  P. Wohlfarth, Phil. Trans. R. Soc. Lond. A  240, 599 (1948).  \n24 G. K. Williamson and W.  H. Hall, Acta Mettall. 1, 22 (1953).   \n6 \n                                                                                                                                                        \n25 L. H. Lewis, A. R. Moodenbaugh, D.  O. Welch, and V. Panchanathan, J. Phys. D: Appl. Phys. 34, \n744 (2001).  \n26 A. J. Ying, C. E. Murray and I. C. Noyan, J. Appl. Cryst. 42, 401 (2009).  \n27 C. L. Zha, R. K. Dumas, J. W. Lau, S. M. Mohseni, Sohrab R. Sani, I. V. Golosovsky, A . F. \nMonsen, J. Nogués, and Johan Åkerman , J. Appl. Phys. 110, 093902 (2011).  \n \n 7 \n FIG. 1.  (Color online) Images and x -ray diffraction of a nanostructured Mn xGa, x = 2.7, film \ngrown on Si substrate. (a) Nanostructures of island -\nlike particles are pictured in the scanning electron \nmicroscope (SEM) image of the surface, where the \nbar is 200 nm. Reflection high energy electron \ndiffraction (RHEED) images: (b) as -grown at 90 °C; \nand (c) annealed at 400 °C for 60 minutes showing \ndistinct quasi -alignment of crystallites. (d) X -ray θ -2θ \ndiffraction intensity showing Mn xGa(112), \nMn xGa(200) and Mn xGa(224) diffraction peaks  of the \nD0 22 structure . (e) Illustration of crysta l structure of \nMn 3Ga, where Ga atoms are the small grey spheres \nat the corners and the center.6  \n \nFIG. 2. (Color online) Magnetic properties of a nanostructured Mn xGa, x = 2.7, film illustrating \nthe large coercivity at room temperature . Magnetization (scaled and diamagnetism subtracted) \nof a nanostructured film grown on Si (blue solid circles) compared to that of a highly -ordered \nfilm grown epitaxia lly on GaAs (red open squares). \nThe solid curves are guides for the eye. The \nnanostructured film has an order of magnitude \nlarger coercive field, oHC = 2.5 T. The inset shows \nthe temperature dependence of the remanent \nmagnetization, M R(H=0), of the nanost ructured film, \nMR(T) / M R(0), (black crosses), and the temperature \ndependence of the coercive field, H C(T) / H C(0), (blue \nopen circles). The solid curves are empirical fits to (1 \n- T/T’)α, where T’ = 710 K and α = 0.45 for M R(T), \nwhile T’ = 530 K and α = 0.40 for H C(T).  \n \nFIG. 3.  (Color online) Large coercivity shown in the anomalous Hall effect (AHE) at room \ntemperature for a nanostructured Mn xGa, x = 2.7, \nfilm. (a) Solid curve (black) shows the total Hall \nresistivity, arising from the anomalous Hall ef fect \n(AHE) plus reversible resistivity, and the dashed line \n(red) illustrates the reversible resistivity component. \n(b) The solid curve shows the AHE resistivity \ncomponent, and the open (blue) circles show the \nclosely matched magnetization. The magnitudes are \nscaled by the saturation values to provide \ncomparison. (c) Magnetization of Stoner -Wohlfarth \nparticles for easy -axis aligned with field  φ = 0 (black \ndashed curve), a φ = 40 deg angle between easy axis \nand field (red dotted curve), and random alignment \nof easy axes (blue solid curve).  \n"    },    {        "title": "1201.3505v4.Spin_orbit_field_switching_of_magnetization_in_ferromagnetic_films_with_perpendicular_anisotropy.pdf",        "content": "arXiv:1201.3505v4  [cond-mat.mes-hall]  7 Jun 2012Spin-orbit field switching of magnetization in ferromagnet ic films with\nperpendicular anisotropy\nD. Wang1,a)\nDepartment of Physics, National University of Defense Tech nology, Changsha 410073, Hunan,\nChina\n(Dated: 24 June 2018)\nAs analternativeto conventionalmagnetic field, the effectivespin- orbitfield in transitionmetals, derivedfrom\nthe Rashba field experienced by itinerant electrons confined in a spa tial inversion asymmetric plane through\nthes-dexchange interaction, is proposed for the manipulation of magnetiz ation. Magnetization switching\nin ferromagnetic thin films with perpendicular magnetocrystalline anis otropy can be achieved by current\ninduced spin-orbit field, with small in-plane applied magnetic field. Spin- orbit field induced by current pulses\nas short as 10 ps can initiate ultrafast magnetization switching effec tively, with experimentally achievable\ncurrent densities. The whole switching process completes in about 1 00 ps.\nUltrafast manipulation of magnetization is currently\nunder intense investigation, partly driven by the ever in-\ncreasing demand in information industry, partly inspired\nby the intriguing physics involved. Traditional methods\nuse pulsed magnetic field to realize ultrafast switching\nof magnetization, through the spiral motion of magneti-\nzation in a magnetic field, applied in the inverse direc-\ntion of the magnetization. However, due to domain wall\ninstability1, ultrashort field pulses bring about stochas-\ntic behavior, thus imposing limitations on the ultimate\nswitching speed2. In practice, the limitation on this\nswitching scheme is related to the difficulty in the gener-\nation of picosecond, strong magnetic field pulses, which\nentails the use of relativistic electron bunches nowadays.\nPrecessional switching scheme, in which the magnetic\nfield is applied perpendicular to the initial magnetiza-\ntion direction, circumvents this problem by maximizing\nthe precessiontorqueexperienced by the magnetization3.\nThe deficit of precessional switching is manifested by the\nneeded precise control of the pulse duration, on the time\nscale of the magnetization’s precession period. Instead\nof the conventional magnetic field, alternative means,\nsuch as light4, electric field5and electric current6, can\nbe used to manipulate magnetization. Recently, the ef-\nfective spin-orbit field acting on the magnetization at-\ntracts much attention because of its potential applica-\ntions. This spin-orbit field in transition metals results\nfrom the Rashba field7experienced by itinerant elec-\ntrons confined in a spatial inversion asymmetric poten-\ntial through the s-dexchange interaction8. Reversible\nswitching of magnetization in perpendicularly magne-\ntized Co nanodots was already demonstrated9, making\nthe speculation of employing the spin-orbit field to con-\ntrol magnetization in ferromagnetic metals more than\nmere imagination, although the underlying mechanism\nresponsibleforthe observedswitchingis still elusive. It is\nproposed, as will be shown in the following by macrospin\nsimulation, that the pure spin-orbit field, in combination\na)Electronic mail: dwwang@nudt.edu.cnwith the precessional motion induced by it, can explain\nqualitatively the observed experimental results . In addi-\ntion, the feasibility of precessional switching utilizing the\nspin-orbit field will be addressed as well. It is found that,\ndue to the large anisotropy and spin-orbit fields, both\nderived from the large spin-orbit coupling characteristic\nof systems with large perpendicular magnetocrystalline\nanisotropy (PMA), the switching time can be as short as\n100 ps.\nThe prototype material system considered here is a\ntrilayer Pt/Co 6 ˚A/AlO xnanodot, which is a represen-\ntative of thin ferromagnetic metallic nanostructures with\nPMA. The strong perpendicular anisotropy results from\nthe 3d-5dhybridization at the Pt/Co interface and the\n3d-2phybridization at the Co/AlO xinterface10. The\nasymmetry of the top and bottom materials introduces\na spin-orbit field for 3 delectrons confined in the thin Co\nlayer. If current flows along the xdirection (c.f. Fig. 2\nfor the coordinate system used), and the trilayer struc-\nturelies in the xyplane, then the spin-orbitfield is Bso=\n−αso(ˆz×j), where ˆzis a unit vector along the zaxis,j\nis the current density, and αsois the spin-orbit field con-\nstant, which is proportional to the spin-orbit coupling in\nCo. For the optimized thickness of Co considered here,\nαsocould be very large, αso= 10−12T m2/A11. The\ninduced large spin-orbit field by injecting high density\ncurrent into the sample could have profound effects on\nthe magnetization dynamics.\nIn the macrospin approximation, the uniform magne-\ntizationMis treated as a macroscopic spin, whose dy-\nnamicsisgovernedbytheLandau-Lifshitz-Gilbert(LLG)\nequation12\ndm\ndt=−γ/parenleftBig\n(m×B)+αm×(m×B)/parenrightBig\n,(1)\nwherem=M/Msis the normalized magnetization vec-\ntor (Msis the magnitude of M),γ= 1.76×1011Hz/T\nis the free-electron gyromagnetic ratio, and αis the\nphenomenological Gilbert damping constant. The to-\ntal magnetic field B=Ba+Bapp+Bsois a sum of\nthe anisotropy ( Ba), applied ( Bapp) and spin-orbit ( Bso)\nfields. In the simulation, the current flow in Co is along2\nFIG. 1. (a)-(c) Influence of current pulses on magnetization .\nIn (b) and (c), blue up (down) triangles denote mzafter injec-\ntion of a positive (negative) current pulse during the posit ive\nto negative (+B → −B) field sweep, while the corresponding\nmzfor the negative to positive ( −B→+B) half is repre-\nsented by red squares (circles). The asymmetry between the\nhysteresis loops for the positive to negative and the nega-\ntive to positive field sweeps is caused by the slightly differe nt\npaths followed by the magnetization during the time evolu-\ntion to equilibrium. Insets in (a) schematically show the ti me\nsequence of the applied magnetic field and current pulses wit h\nboth polarities. Current densities are given in units of 1012\nA/m2. (d) Dependence of the coercivity (filled squares) and\nthe maximum field (open circles), against which current in-\nduces magnetization switching, on current density. Typica l\nhysteretic behavior of regions I, II and III is given in (a), ( b)\nand (c), respectively.\nthexaxis, while the magnetic field is applied in the\nxzplane, 3◦tilted away from the xaxis. The Gilbert\ndamping is chosen to be α= 0.313. The perpendicular\nanisotropy field has the form Ba=BKmzˆz, withBK=\n0.92 T11. To stabilize the perpendicular magnetization\nconfiguration, an external field Bz=±5 mT is added\nto the total field, depending on the initial magnetization\norientation.\nTo investigate the effect of the spin-orbit field on the\nswitching behavior, the time sequence for the applied\nfield and current pulses, as shown in the insets of Fig.\n1(a), is considered. Essentially, a hysteresis loop is sim-\nulated. But at each field value, current pulses with\nboth polarities, positive and negative, are applied con-\nsecutively. The equilibrium magnetizationdirection after\neach pulse is then recorded. The current pulse is mod-\nelled by a 10 ns square wave with infinitely sharp rising\nand falling edges. The spin dynamics under the influence\nof current is dictated by the LLG equation, Eq. (1). At\nthe rising edge, due to the fact that the length of the\ncurrent pulse is longer than the characteristic time scale\nof the magnetization dynamics triggered by the sudden\napplication of current, the magnetization stops precess-\nFIG. 2. Three dimensional motion of the normalized magne-\ntization under the influence of positive and negative curren t\npulses, with Bapp= 0.3 T and j= 1.5×1012A/m2. The red\n(blue) arrow on the unit sphere represents theequilibrium o ri-\nentationofthemagnetization after anegative (positive)p ulse,\nwhereas the yellow (green) arrow defines the stable directio n\nof the magnetization when the current pulse is present. The\nyellow (green) curve shows the path of motion for the magne-\ntization after the negative (positive) current pulse is tur ned\noff.\ning far before the current pulse is terminated. Once the\nfalling edge of the current pulse is reached, the magne-\ntization vector will start precessing again, damping to a\ndifferent equilibrium position, depending on the polarity\nof the current pulse. The zcomponent of the normalized\nmagnetization, mz, after positive and negative current\npulses, as a function of the applied field, is shown in Figs.\n1(a), 1(b) and 1(c). The current, or the corresponding\nspin-orbit field, effect can be clearly observed: When the\ncurrent density is lower than 5 ×1011A/m2(Bso= 0.5\nT), only the coercivity is decreased (Fig. 1(a)). By in-\ncreasingthe current density to well above 1 ×1012A/m2\n(Bso= 1 T), projection of the magnetization onto the z\naxis is completely determined by the polarity of the cur-\nrent (Fig. 1(c)). Deterministic switching controlled by\nthepolarityofcurrentoccurs. Intheintermediateregion,\ncurrent controlled switching is effective only for a narrow\nfield interval (Fig. 1(b)). Fig. 1(d) gives an overview\nof the different switching behavior of the magnetization,\nfor current density ranging from 0 to 2 ×1012A/m2.\nIt can be seen that the coercivity decreases to zero with\nincreasing current, while the maximum field for current\ninduced switching remainsalmost constant, in agreement\nwith experiment9.\nThe physical mechanism responsible for the reversible,\ncurrent induced switching can be understood by track-\ning the magnetization precession in time. In Fig. 2, two\ntypical precession traces corresponding to j=±1.5×\n1012A/m2andBapp= 0.3 T are shown. The influence3\nof the polarity of the current is obvious. It determines\nwhether the magnetization will spiral upward or down-\nward, initially. If the applied magnetic field is not too\nlarge, which means that the magnetization orientation\npointing up or down is well separated, this initial dis-\ncrepancy will lead to the difference in the final equilib-\nrium position, i.e. whether the magnetization is point-\ning up or down. Effectively, the final orientation of the\nmagnetization is defined by both the spin-orbit field and\nthe applied field, through the cross product Bso×M14,\nwhich is nothing but the initial torque experienced by\nthe magnetization when the current is turned off. This is\nconsistent with the symmetry required by the perpendic-\nular switching scheme9. In the intermediate region (Fig.\n1(b)), the spin-orbit torque is not large enough to induce\nswitching by itself, applied field is required to overcome\nthe action of anisotropy. If the applied field is not large\nenough, the equilibrium mzstays finite even in the pres-\nence of the spin-orbit field, because of the large PMA.\nWhen the current is removed, the magnetization never\ngoes across the xyplane, and switching could not occur.\nThis explains why the spin-orbit torque induced switch-\ning is effective with large applied field, while there is no\nswitching if the field is smaller than a critical value.\nWhen the applied field is rotated away from the xaxis\nby an angle ϕ >0, the hysteretic behavior of the mag-\nnetization under the influence of current becomes asym-\nmetric, because of the non-zero ycomponent of the ap-\nplied magnetic field, Bappsinϕ. For a positive current\npulse to switch the magnetization, it has to overcome\nthis positive yfield, making the current effect less ef-\nficient. The angular dependence of the maximum field\n(not shown) against which a positive current pulse can\ninduce reversible switching supports this intuitive pic-\nture. But, in contrast to the experimental, linear rela-\ntionship, theoretically, the dependence is determined by\nan almost quadratic relation, B∝cos2ϕ. Nevertheless,\nthe overall decrease in the switching efficiency when the\napplied field is rotated away from the current direction\nis observed unanimously.\nFor the case of current flowing along the xaxis, the\nspin-orbit field is parallel to the yaxis. If the applied\nfield is in the xzplane, the magnetization is also in the\nxzplane prior to the application of current pulses. This\nperpendicular configuration between the spin-orbit field\nand the magnetization maximizes the precession torque,\nthus facilitating precessional switching. Using a square-\nwave shaped current pulse, complete switching can be\nachieved in about 100 ps, as shown in Fig. 3. The length\nof the current pulse used in the simulation is 10 ps, which\nis about one half of the precession period corresponding\nto the spin-orbit field induced by the current pulse, with\nthe current density j= 1.5×1012A/m2. The applied\nfield isBapp= 0.2 T. Due to the large spin-orbit field,\nthe time needed to realize precessional switching is solely\ndetermined by the current density, whose direct conse-\nquence is the fact that a very short current pulse can\neffectively initiate the desired magnetization switching.FIG. 3. Time evolution of the normalized magnetization, at\nBapp= 0.2 T, excited by a 10 ps square-wave current pulse,\nwhose amplitude is 1.5 ×1012A/m2. The shaded area signi-\nfies the time interval where the current is present.\nIn the macrospin simulation, domain nucleation and\nthe consequent domain wall motion, which is crucial for\nthe actual determination of the coercivity, are neglected.\nHence the simulated results are only of qualitative sig-\nnificance. However, as can be seen in Fig. 1, the quali-\ntative agreement between the macrospin simulation and\nthe experiment9is satisfactory. Nevertheless, a detailed\nmicromagnetic study, including finite temperature and\nfinite size effects, is needed to gain further insight into\nthe physics involved in the spin-orbit field induced re-\nversible switching of magnetization in perpendicularly\nmagnetized thin films. Experimentally, a thorough in-\nvestigation of the magnetization dynamics following cur-\nrentexcitationinsuchsystemswillprovetobe important\nto clarify the role played by the spin-orbit field in ma-\nnipulating the macroscopic state of magnetization. In\nPt/Co/AlO xor similar systems, this can be achieved by\ntime resolved magneto optical Kerr effect, which is al-\nready demonstrated to be a powerful technique for the\nstudy of magnetization dynamics in thin metallic mag-\nnetic films15.\nIn summary, the spin-orbit field acting on the mag-\nnetization, mediated by the Rashba field experienced by\nitinerant electrons confined in a spatial inversion asym-\nmetric plane, through the s-dexchange coupling, is pro-\nposed for the manipulation of magnetization. Perpen-\ndicular switching of magnetization in Pt/Co/AlO xnan-\nodots, with in-plane applied field, can be realized using\nonly the spin-orbit field, without the need of any extra\nfields. This simplifies the explanation for the experimen-\ntal observation9. Ultrafast switching, on the time scale\nof 100 ps, is made possible by the large magnitude of\nthe spin-orbit field in systems with large PMA, such as\nPt/Co/AlO x. Forperspectives,thespin-orbitfield, prop-4\nerly tailored, can be used to coherently control spin oscil-\nlation and domain wall motion, in conjunction with the\nmore familiar spin transfer torques, thus providing more\nfreedomoverthecontrolofmagnetizationdynamics. The\nmost recent experimental advance on this respect is the\nenhancement of domain wall velocity in perpendicularly\nmagnetizedPt/Co/AlO xnanowires16. Stimulated bythe\nimpetus from information technology, more advances are\nto be expected.\nACKNOWLEDGMENTS\nD.W. thanks group Physics of Nanostructures (FNA),\nEindhoven University of Technology for hospitality.\nEnlightening discussions with Elena Mure and Sjors\nSchellekens are acknowledged. Constructive comments\non the manuscript from Zengxiu Zhao and Jianmin Yuan\nare gratefully appreciated.\n1A. Kashuba, Phys. Rev. Lett. 96, 047601 (2006).\n2I. Tudosa, C. Stamm, A. B. Kashuba, F. King, H. C. Siegmann,\nJ. St¨ ohr, G. Ju, B. Lu, and D. Weller, Nature 428, 831 (2004).\n3C. H. Back, D. Weller, J. Heidmann, D. Mauri, D. Guarisco,\nE. L. Garwin, and H. C. Siegmann, Phys. Rev. Lett. 81, 3251\n(1998); C. H. Back, R. Allenspach, W. Weber, S. S. P. Parkin,\nD. Weller, E. L. Garwin, and H. C. Siegmann, Science 285, 864\n(1999).\n4A. Kirilyuk, A. V. Kimel, and Th. Rasing, Rev. Mod. Phys. 82,\n2731 (2010).\n5H. Ohno, D. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl,\nY. Ohno, and K. Ohtani, Nature 408, 944 (2000).6L. Berger, Phys. Rev. B 54, 9353 (1996); J. C. Slonczewski, J.\nMagn. Magn. Mater. 159, L1 (1996).\n7Yu. A. Bychkov and E. I. Rashba, J. Exp. Theor. Phys. Lett. 39,\n78 (1984).\n8A. Manchon, and S. Zhang, Phys. Rev. B 78, 212405 (2008);\nA. Manchon, and S. Zhang, ibid79, 094422 (2009); A. Matos-\nAbiague, and R. L. Rodriguez-Suarez, ibid80, 094424 (2009); I.\nGarate, and A. H. MacDonald, ibid80, 134403 (2009).\n9I. M. Miron, K. Garello, G. Gaudin, P.-J. Zermatten, M. V.\nCostache, S. Auffret, S. Bandiera, B. Rodmacq, A. Schuhl, and\nP. Gambardella, Nature 476, 189 (2011).\n10B. Rodmacq, A. Manchon, C. Ducruet, S. Auffret, and B. Dieny,\nPhys. Rev. B 79, 024423 (2009).\n11I. M. Miron, G. Gaudin, S. Auffret, B. Rodmacq, A. Schuhl,\nS. Pizzini, J. Vogel, and P. Gambardella, Nature Mater. 9, 230\n(2010); P. Gambardella, and I. M. Miron, Phil. Trans. R. Soc. A\n369, 3175 (2011).\n12L. D. Landau, E. M. Lifshitz, and L. P. Pitaevski, Statistical\nPhysics, Part 2, 3rd ed. (Pergamon, Oxford), 1980; T. L. Gilbert,\nIEEE Trans. Mag. 40, 3443 (2004).\n13α= 0.3 is close to the experimenally determined Gilbert dampi ng\nconstant in a similar Pt/Co/AlO xsample. The detailed determi-\nnation of the intrinsic αwill be published elsewhere.\n14Confusion can arise by simple reference to M. Mironet al.9refer\nto the equilibrium magnetization without current in their e x-\npression, while Mspecifies the magnetization just prior to the\nremoval of current here. For the sense of switching, the sign of the\nzcomponent of the cross product is of significance. Given that\nthe two expressions, in which Mhas different meanings, produce\nthe same sign for the zcomponent, we make no discrimination\nbetween them. They are both proportional to Bso×Bapp, if\nonly the zcomponent is considered.\n15M. van Kampen, C. Jozsa, J. T. Kohlhepp, P. LeClair, L. Lagae,\nW. J. M. de Jonge, and B. Koopmans, Phys. Rev. Lett. 88,\n227201 (2002).\n16I. M. Miron, T. Moore, H. Szambolics, L. D. Buda-Prejbeanu, S .\nAuffret, B. Rodmacq, S. Pizzini, J. Vogel, M. Bonfim, A. Schuhl ,\nand G. Gaudin, Nature Mater. 10, 419 (2011)."    },    {        "title": "1202.3145v1.Magnetic_anisotropies_of_quantum_dots.pdf",        "content": "arXiv:1202.3145v1  [cond-mat.mes-hall]  14 Feb 2012Magnetic anisotropies of quantum dots\nKarel V´ yborn´ y,1,2J. E. Han,1Rafa/suppress l Oszwa/suppress ldowski,1Igor ˇZuti´ c,1and A. G. Petukhov3\n1Department of Physics, University at Buffalo–SUNY, Buffalo, Ne w York 14260, USA\n2Institute of Physics, ASCR, v. v. i., Cukrovarnick´ a 10, CZ-16253 Praha 6, Czech Republic\n3South Dakota School of Mines and Technology, Rapid City, Sou th Dakota 57701, USA\n(Dated: Feb14, 2012)\nMagnetic anisotropies in quantum dots (QDs) doped with magn etic ions are discussed in terms of\ntwo frameworks: anisotropic g-factors and magnetocrystalline anisotropy energy. It is s hown that\neven a simple model of zinc-blende p-doped QDs displays a ric h diagram of magnetic anisotropies\nin the QD parameter space. Tuning the confinement allows to co ntrol magnetic easy axes in QDs\nin ways not available for the better-studied bulk.\nPACS numbers: 73.21.La, 75.75.-c, 75.30.Gw, 75.50.Pp\nI. INTRODUCTION\nOnce the origin of magnetic ordering in a specific ma-\nterial is understood, it is often important to determine\nits magnetic anisotropy (MA) and hard and easy mag-\nnetic axes in particular. A shift of focus towards MA has\nalready occurred for the studies of bulk dilute magnetic\nsemiconductors (DMS),1,2but not yet fully for magnetic\nquantum dots (QDs) where it could play certain role, for\nexample, in context of transport phenomena,3the forma-\ntion of robust magnetic polarons,4–7control of magnetic\nordering,8–12nonvolatile memory,13and quantum bits.14\nIn epilayers of (Ga,Mn)As, a prototypical DMS, the\nmagnetocrystalline anisotropy energy (MAE) has been\nfound to be a significant and often dominant source of\nMA15–17caused by a strong spin-orbit (SO) coupling. It\nturns out that the easy axis direction depends on hole\nconcentration, magnetic doping level as well as on other\nparameters. For example, when (Ga,Mn)As was used\nas a spin injector, the effects of strain (by altering the\nchoice of a substrate) were responsible for changing the\nin-plane to out-plane easy axis.18While the strong SO\ncoupling19is also present in p-type QD of zinc-blende ma-\nterials doped with Mn, its effect on magnetic anisotropies\nwill be significantly modified by the confinement. The en-\nergy levels in such ‘nanomagnets,’20–23where the Mn-Mn\ninteraction is mediated by carriers, depend on the magne-\ntization direction eM= (nx,ny,nz). It is often assumed\nthat the interaction of magnetic moments with holes in\nquantum wells (QWs) or, equivalently in flat QDs, is ef-\nfectively Ising-like.14,24Here we quantify this assumption\nand explore MA using two frameworks: (i) an effective\ntwo-level Hamiltonian with a carrier g–tensor,25which\nis widely employed also in theory of electron spin reso-\nnance, and (ii) MAE, which is commonly used to study\nbulk magnets.\nWhile previous studies focused on specific nonmagnetic\nQDs26and properties sensitive to system details (such as\nprecise position of magnetic ions22,27), we explore more\ngeneric magnetic QD models, which can also serve as\na starting point for more elaborate work. We consider\na Hamiltonian comprising non-magnetic and magneticparts,\nˆH=ˆHQD+ˆHex. (1)\nThe former encodes both QD confinement and SO inter-\naction, which is prerequisite for magnetic anisotropies,\nthe latter expresses the kinetic-exchange coupling be-\ntween holes and localized magnetic moments. For\ntransparency, we disregard the magnetostatic shape\nanisotropy28and assume that the QD contains a fixed\nnumber of carriers. We mostly focus on the case of\na single hole; realistically, such system can be a II-VI\ncolloidal5or epitaxial6QD with a photoinduced carrier.\nMagnetic moments of the Mn atoms are taken to be per-\nfectly ordered (collinear) and are treated at a mean-field\nlevel. The magnetic easy axis is then the direction eMfor\nwhich the zero-temperature free energy F(eM) is mini-\nmized. In this article, we take two different points of view\nonF(eM). On one hand, we discuss the lowest terms of\nF(eM) expanded in powers of the direction cosines of\nmagnetization ( n2\nx+n2\ny+n2\nz= 1), inspired by the stan-\ndard ‘bulk MAE phenomenology’ and pay special atten-\ntion to the case of perfectly cubic QDs, F(eM) =F0(eM).\nThe anisotropies in F0stem purely from the crystalline\nzinc-blende lattice. On the other hand, F(eM) acquires\nadditional terms in systems with less symmetric confine-\nment. We therefore discuss the anisotropic g-factors as a\nuseful framework to handle such systems, e.g. cuboid\nQDs (orthogonal parallelepiped; extremal cases are a\ncube and an infinitely thin slab, i.e., a QW) and show\nhow the expansion\nF(eM) =F0(eM) +AF1(eM) +A2F2(eM) +... (2)\ncan be constructed using powers of Awhich reflects the\nanisotropy in g-factors. We begin by discussing this lat-\nter topic in Section II (quantity Ais defined by Eq. (8)\nat the end of Sec. IIA), then proceed to the phenomeno-\nlogic (symmetry-based) expansions of F0in Section III\nand conclude that Section with calculations of F1in sit-\nuations that are beyond the applicability of the g-factor\nframework.2\nII. EFFECTIVE TWO-LEVEL HAMILTONIAN\nSince ˆHQDis invariant upon time reversal, its spec-\ntrum consists of Kramers doublets.29To study the\nground-state energy in the presence of magnetic mo-\nments, we examine how these doublets are split by\nˆHex(eM) where eMis treated as an external parame-\nter (related to classical magnetization; single-Mn doped\nQDs where the Mn magnetic moment behaves quantum-\nmechanically30require different treatment) and represent\nthem by an effective two-level Hamiltonian of Eq. (6). We\nconsider two example systems: a simple four-level one\nwhere completely analytical treatment is possible, and a\nmore realistic envelope-function based model of a cuboid\nQD.\nA. Four level model\nRelated to the Kohn-Luttinger Hamiltonian of a\nQW,23,31the arguably simplest non-trivial model de-\nscribing anisotropy of a flat QD is\nˆH1=aˆJ2\nz+1\n3heM·ˆJ (3)\nrepresenting hole levels in a zinc-blende structure\nwhose confinement anisotropy and exchange splitting are\nparametrized by aandh, respectively (the term aˆJ2\nz\nimplies that the strongest confinement is along the z-\ndirection and this term also encodes information about\nthe SO coupling). ˆJx,y,z are 4×4 spin-3\n2matrices. In\nterms of Eq. (1), we now choose ˆH=ˆH1and the first\n(second) term in Eq. (3) plays the role of ˆHQD(ˆHex).\nAnisotropic behavior of eigenvalues of ˆH1, to linear or-\nder inh/a, is illustrated in Fig. 1(a). It can be extracted\nfrom the exact eigenvalues,\nE±\nhh(h) =5\n4a±1\n6h+/radicalbigg\na2+1\n9h2∓1\n3ah (4)\nE±\nlh(h) =5\n4a±1\n6h−/radicalbigg\na2+1\n9h2∓1\n3ah (5)\nin the case nx= 1 (orny= 1), shown in Fig. 1(b), which\nclearly differ from the case nz= 1 where the eigenvalues\nare strictly linear functions of h(E±\nhh= 9a/4±h/2 and\nE±\nlh=a/4±h/6); subscripts refer to the E±\nhh(0) = 9a/4\n(‘heavy-hole’, HH) and E±\nlh(0) =a/4 (‘light-hole’, LH)\ndoublets, respectively. In the limit of weak exchange,\nh/a≪1, splitting of each of the Kramers doublets is\nsymmetric and it can be characterized by three param-\neters|∂E/∂ (hnp)|,p=x,y,z , forh→0 as depicted\nin Fig. 1(a). These parameters can be plausibly called,\nby analogy with the Zeeman effect, the anisotropic g-\nfactorsgp. From Eqs. (4),(5), we straightforwardly ob-\ntain (gx,gy,gz) = (0,0,1/2) and (1 /3,1/3,1/6) for the\nHH and LH doublet of the Hamiltonian ˆH1, respectively.\nThis result is known from the context of QWs.31,32WeFIG. 1. (Color online) Splitting of levels E(h) in a flat QD\ndescribed by Eq. (3). (a) For the particular Kramers doublet ,\nE(h) depends on eMand the g-factors (by convention non-\nnegative) are ∂E/∂/vectorh= (gx,gy,gz). (b) Beyond the linear\nregime in h/a,∂E/∂ (hnx) will be different for the upper and\nlower level of the split doublet, it will depend on hand may\neven change sign, indicating that the ˆHeffof Eq. (6) based on\nparameters gx,y,z fails.\nemphasize that these g-factors of the model specified by\nEq. (3) are independent of the parameters a,h(except\nfor the requirement h≪awhich represents the h→0\nlimit).\nIf we focus on one particular Kramers doublet, it is\nstraigtforward to show that ˆH1projects to\nˆHeff=h[nxgxˆτx+nygyˆτy+nzgzˆτz] (6)\nfor a suitably chosen basis |K1∝an}bracketri}ht,|K2∝an}bracketri}htof the doublet. Here\nˆτiare Pauli matrices and we have mapped two eigenstates\nof the original Hamiltonian ˆHQDon a pseudospin |/vector τ|=\n1/2 doublet |+∝an}bracketri}ht,|−∝an}bracketri}ht, where ˆτz|±∝an}bracketri}ht=±|±∝an}bracketri}ht . For ˆH=ˆH1,\nthe eigenstates are only four-dimensional (spanned by\nthe|Jz= 3/2∝an}bracketri}ht,|Jz=−1/2∝an}bracketri}ht,|Jz= 1/2∝an}bracketri}ht,|Jz=−3/2∝an}bracketri}ht\nbasis). We present another example of ˆHin Sec. IIB\nwhere advantage of the projection becomes more appar-\nent. The choice of basis |+∝an}bracketri}ht,|−∝an}bracketri}htis crucial to obtain\nˆHeffin the simple form (6); considering the HH dou-\nblet:|+∝an}bracketri}ht=|Jz= 3/2∝an}bracketri}ht,|−∝an}bracketri}ht=|Jz=−3/2∝an}bracketri}htleads\nto Eq. (6) while for other basis choices the mapping\nˆHex= (h/3)eM·ˆJ∝ma√sto→ˆHeff=heM·g·ˆτmay lead31to3\nnon-symmetric tensor g=gij,i,j∈ {x,y,z}. In general,\nif the mapping is to produce gij= diag (gx,gy,gz) the\n‘suitable choice of the basis |K1∝an}bracketri}ht,|K2∝an}bracketri}ht’ where|K1∝an}bracketri}ht ∝ma√sto→ | +∝an}bracketri}ht\nis such that ∝an}bracketle{tK1|ˆJx,y|K1∝an}bracketri}ht= 0,∝an}bracketle{tK1|ˆJz|K1∝an}bracketri}ht ≥0 (and|K2∝an}bracketri}ht\nis the time-reversed image of |K1∝an}bracketri}htwhich is mapped to\n|−∝an}bracketri}ht).\nLet us now consider a general system described by\nEq. (1). Assuming that the downfolding of ˆHinto ˆHeffis\npossible for given |K1∝an}bracketri}ht,|K2∝an}bracketri}ht(this assumption is discussed\nin Appendix A), the anisotropic g-factors can readily be\ndetermined as ∂E/∂h for the particular Kramers doublet\nlevelE. This is equivalent to perturbatively evaluating\nthe effect of ˆHexon two degenerate levels to the first\norder of has follows: (i) specify the Kramers doublet\nof interest, and find any basis |K1∝an}bracketri}ht,|K2∝an}bracketri}htof this dou-\nblet, (ii) extract the operators ˆtx,y,z from ˆHexby taking\nˆtp=∂ˆHex/∂(nph) (for example, ˆtx=ˆJx/3 for ˆHeffap-\npearing in ˆH1), (iii) evaluate their matrices\n˜tx,y,z =/parenleftbigg\n∝an}bracketle{tK1|ˆtx,y,z|K1∝an}bracketri}ht ∝an}bracketle{tK1|ˆtx,y,z|K2∝an}bracketri}ht\n∝an}bracketle{tK2|ˆtx,y,z|K1∝an}bracketri}ht ∝an}bracketle{tK2|ˆtx,y,z|K2∝an}bracketri}ht/parenrightbigg\n(7)\nin the two-dimensional space spanned by |K1∝an}bracketri}ht,|K2∝an}bracketri}ht,\nand (iv) the non-negative eigenvalue of ˜tpequalsgp\n(p=x,y,z ). We emphasize that while gpdepends on\nsystem parameters in ˆHQDand ˆHex, it also depends on\nwhich Kramers doublet we choose. Higher doublets be-\ncome relevant for QDs containing higher (odd) number\nof holes, for example.\nThe effective Hamiltonian in Eq. (6) can be used for\nvarious purposes, e.g., for studies of fluctuations of mag-\nnetization in magnetic QDs33, spin-selective tunneling\nthrough non-magnetic QDs34or excitons in single-Mn\ndoped QDs.35If the magnetic easy axis is of interest,\ntheg-factors immediately provide the answer: F(eM)\nbased on Eq. (6) is minimized for eMin the direction\nof the largest gp(e.g. for the HH doublet in Fig. 1(a),\nit isnz= 1 because gz> gx,gy). If the full form of\nF(eM) is needed (e.g, for ferromagnetic resonance2), it\ncan be straightforwardly obtained by diagonalizing the\n2×2 matrix of ˆHeff. Assuming gx=gy, the (modulus of\nthe) eigenvalue can be expanded in terms of parameters\nAandkas derived in Appendix B. It is meaningful to\ncall\nA= (g2\nz−g2\nx)/(g2\nz+g2\nx) (8)\nthe asymmetry parameter since it vanishes in a perfectly\ncubic QD ( gx=gy=gz) and it is with respect to this\nparameter that we can identify\nAF1(eM) =−Akn2\nz (9)\nA2F2(eM) = +1\n8A2k(2n2\nz−1)2(10)\nin Eq. (2) to linear order of k∝h.B. A cuboid quantum dot model\nWith this general scheme at hand, we take one step\nin the hierarchy of models towards a more realistic de-\nscription of magnetic QDs. We consider a zinc-blende\nstructure p-doped semiconductor shaped into a cuboid\nof sizeLx×Ly×Lzsuch as can be described by four-\nband Kohn-Luttinger Hamiltonian.23Also in this system,\nˆH=ˆH2is a sum of ˆHexand ˆHQDbut this time, ˆHQD\ncomprises of blocks ∝an}bracketle{tmxmymz|ˆHKL|m′\nxm′\nym′\nz∝an}bracketri}htwith\nˆHKL=/planckover2pi12\n2m0{(γ1+5\n2γ2)p2−2γ2[ˆJ2\nxˆp2\nx+ˆJ2\nyˆp2\ny+ˆJ2\nzˆp2\nz]\n−2γ3[(ˆJxˆJy+ˆJyˆJx)ˆpxˆpy+ c.p.] (11)\nHere,|mxmymz∝an}bracketri}htdenotes the basis of envelope functions,\nγ1,2,3the Luttinger parameters, m0the electron vacuum\nmass, ˆpx,y,z the momentum operators and c.p. denotes\nthe cyclic permutation (see Appendix C for details). The\nenvelope function is conveniently developed into har-\nmonic functions with mp−1 nodes in the p=x,y,z\ndirection:\n∝an}bracketle{t/vector r|mxmymz∝an}bracketri}ht=Nsinmxπx\nλxLsinmyπy\nλyLsinmzπz\nL.(12)\nWe have introduced the dimensionless aspect ratios\nλx,y=Lx,y/Land the normalization factor N. Our sys-\ntem can be viewed as an infinitely deep potential well\nwithV(x,y,z ) = 0 for 0 < x < L x, 0< y < L yand\n0< z < L z≡Land infinite otherwise.\nFor fixed material parameters (Luttinger parameters\nin ratios γ2/γ1,γ3/γ2) and QD shape ( λx,λy), all matrix\nelements of all blocks ∝an}bracketle{tmxmymz|ˆHKL|m′\nxm′\nym′\nz∝an}bracketri}htscale\nas 1/L2. The spectrum, consisting of Kramers dou-\nblets which occasionally combine into larger multiplets,\nis specified by a sequence of dimensionless numbers E/E0\nwhere\nE0=/planckover2pi12π2γ1/(2m0L2). (13)\nFor a cubic QD [ λx=λy= 1; see Fig. 2(a)] the s-like\nstate shown in the inset of Fig. 2(a) forms a quadruplet,\nand depending on the value of γ2/γ1(and to somehow\nlesser extent also of γ3/γ2) this state competes with the\nnext doublet for having the lowest energy. The critical\nvalue (see Appendix C)\ncR= (2 + 128 /9π2)−1≈0.29 (14)\ncan be taken to distinguish materials with small ( γ2/γ1<\ncR, ground state quadruplet) and large ( γ2/γ1> cR,\nground state doublet) splitting between light and heavy\nholes in the bulk; these can be ZnSe and CdTe, respec-\ntively, their values of ¯ γ2/γ1based on approximating γ2\nandγ3by their average ¯ γ2= (γ2+γ3)/2 are indicated\nin Fig. 2(a). By numerical diagonalization we have de-\ntermined the lowest 7 Kramers doublets in slightly de-\nformed QDs ( λx=λy≡λ= 1.01) in these materi-\nals (γ1/2/3= 4.8/0.67/1.53 for ZnSe and 4 .1/1.1/1.6 for4\nCdTe)36and executed the procedure (i)-(iv) above to ob-\ntain theg-factors which are listed in the table on the right\nof Fig. 2 ( gx=gydue toλx=λy). To avoid confusion,\nwe remark that in (i), |K1∝an}bracketri}ht,|K2∝an}bracketri}htare vectors of dimension\n864 in the basis |mxmymz∝an}bracketri}ht⊗|Jz∝an}bracketri}ht(see the discussion of\ncut-off in Appendix C) and in (ii), ˆtx= (1/3)ˆJx⊗1 1xyz,\nwhere 1 1 xyzis the identity operator in the space of the\nenvelope functions given by Eq. (12). Evaluation and di-\nagonalization of the 2 ×2 matrices in Eq. (7) requested in\n(iii,iv) is performed numerically. The possibility to map\nthe action of ˆHex= (h/3)eM·ˆJ⊗1 1xyzon the Kramers\ndoublets |K1∝an}bracketri}ht,|K2∝an}bracketri}htimplied by ˆHQDof a cuboid p-doped\nQD is discussed in Appendix A.\nThe slight deformation of the QD makes the quadru-\nplet split into two doublets (with energies 71 .7 and\n71.9 meV for ZnSe) whose g-factors approach (0 ,0,1/2)\nand (1/3,1/3,1/6). Similar situation occurs for the dou-\nblet pair with energies 52 .8 and 53.0 meV for CdTe. The\nactual ground state in this material is, however, a dou-\nblet of different orbital character than the quadruplet\n(we stress that this is due to the confinement, see Ap-\npendix C); it evolves from the E= 6E0level ofγ2/γ1= 0\nas shown by the solid line in Fig. 2(a) and its g-factors\nare isotropic, (1 /6,1/6,1/6) in the limit λ→1. This\ndoublet, however, remains the ground state only in rather\nsymmetric QDs ( λ≈1.25 in CdTe) and for more strongly\ndeformed QDs, the lower doublet of the E= 3E0(at\nγ2/γ1= 0) quadruplet becomes the ground state just\nas it is the case for ZnSe for arbitrarily small deforma-\ntionsλ > 1. In Fig. 2(b), we show how the g-factors\nof the CdTe QD ground state depend on λbeyond the\nmentioned value ≈1.25. These results, including the g-\nfactors, are independent of the QD size L, except for the\nenergies which scale as 1 /L2as mentioned above.\nFrom Fig. 2, one may conclude that the Ising-like\nHamiltonian is often an excellent approximation ( gx=\ngy= 0, as others assume14,24,33–35) for the lowest\nKramers doublet. To be more specific, we now discuss\nmaterials with small and large HH/LH splitting sepa-\nrately. For γ2/γ1< cR, the out-of-plane g-factor (gz)\noverwhelmingly exceeds the in-plane one ( gx=gy) even\nfor minute deformation of the QD; this can be seen from\nthe numeric ZnSe data in Fig. 2. We find gz= 0.464\nandgx=gy= 0.012 forλ−1 as small as 0 .01. For\nCdTe, which represents the other class ( γ2/γ1> cR), we\nfind similar values ( gz= 0.418) for the second Kramers\ndoublet while the lowest doublet remains rather isotropic\n(gx=gy= 0.166 and gz= 0.164). As we make the QD\ndeformation larger, these two doublets cross, so that the\nground state doublet is Ising like while the second lowest\ndoublet remains more isotropic. This crossing occurs for\nλ≈1.25 in CdTe and data in Fig. 2(b) are only shown\nforλ >1.25.\nWe now elaborate on the properties of the low-energy\nsector of ˆH2(ath= 0). Coupling between blocks of dif-\nferent|mxmymz∝an}bracketri}htvanishes when γ3/γ1,γ2/γ1→0, and\nEq. (3) becomes in this limit the exact effective Hamilto-\nnian of the lowest four levels ( mp= 1 for all p=x,y,z ).\ncR\n.\n 0.42 0.44 0.46 0.48 0.5\n 1 1.5 2 2.5 3 3.5 0 0.02 0.04\nλ(b)gz\ngxE[meV]gx=gygz\n71.7 0.012 0.464\n71.9 0.305 0.171\n92.1 0.167 0.160\n126.1 0.274 0.237\n129.6 0.076 0.069\n130.0 0.082 0.045\n141.0 0.205 0.212\n49.9 0.166 0.164\n52.8 0.027 0.418\n53.0 0.269 0.176\n78.5 0.162 0.169\n84.1 0.010 0.129\n84.4 0.064 0.004\n85.6 0.203 0.279ZnSe CdTe\nFIG. 2. (Color online) (a) Levels in a cubic dot (with γ3=γ2)\nin units of E0defined by Eq. (13). Solid lines indicate ana-\nlytic result obtained when mixing between remote levels is\ndisregarded. Note that their crossing (which we use to dis-\ncern the weak and strong HH/LH splitting materials, dashed\nline) is very close to the actual crossing when level mixing i s\ntaken into account. Values representing ZnSe (¯ γ2/γ1≈0.23)\nand CdTe (¯ γ2/γ1≈0.33) QDs are indicated. Inset: squared\nwavefunction modulus of the ZnSe QD ground state in the\nz=L/2 section. (b) Dependence of the g-factors associated\nwith the ground state Kramers doublet in a CdTe QD on\nits shape ( λx=λy≡λ).Right: Energies and g-factors in\nslightly deformed QDs ( λ= 1.01) for the lowest 7 Kramers\ndoublets for ZnSe and CdTe, where E0≈28 meV and 24 meV,\nrespectively, for L= 8 nm.\nThey form a quadruplet for λ= 1, which splits into two\ndoublets upon deformation of the QD; we can see it by\nwriting\n∝an}bracketle{t111|ˆHKL|111∝an}bracketri}ht= 3E0/bracketleftbigg\n1 14f(λ)−ˆJ2\nz(1−λ−2)2\n3γ2\nγ1/bracketrightbigg\n(15)\nwhere 1 1 4is a unit 4 ×4 matrix and f(λ) is a certain\nfunction with lim λ→1f(λ) = 1. The lower doublet of this\n4×4 effective Hamiltonian has gz= 1/2 (whenλ >1 and\nγ2>0) and therefore the values of gzdeviating from 0 .5\n(appearing in Fig. 2) occur only due to admixtures from\nhigher-orbital ( mp>1) states of LH character. Indeed,\ngoing from ZnSe to CdTe, the mixing becomes stronger\nandgzof the HH-like level drops from 0 .464 to 0 .418\n(λ= 1.01, numerical data in Fig. 2). While Eq. (3)\nmay remain the effective Hamiltonian of the two doublets\noriginating from |mxmymz∝an}bracketri}ht=|111∝an}bracketri}hteven for γ2/γ1> cR\n(CdTe levels of 52 .8 and 53 .0 meV in Fig. 2), for λclose\nto 1, there is the more isotropic doublet on the stage\n(49.9 meV in Fig. 2). Nevertheless, if λis sufficiently\nlarge, the ˆJ2\nzterm in Eq. (11) will eventually dominate,\nit will suppress all mixing between HH and LH states\nand the lowest doublet will again approach ( gx,gy,gz) =\n(0,0,0.5) as it is shown in Fig. 2(b).5\nIII. MAGNETOCRYSTALLINE ANISOTROPY\nENERGY\nIn analogy to the bulk systems, even cubic QDs retain\nanisotropies. However, these cannot be described within\nthe previous framework: for instance, gx,gy,gzare all\nequal to 1 /6 in the cubic CdTe QD ground state hence\nA= 0 in Eq. (8). One could replace gijby a higher\nrank tensor to capture these effects, but MAE formalism\nof bulk magnets seems more customary and informative.\nUnlike the g-factors, MAE analysis does not invoke the\nconcept of Kramers doublets. The zero-temperature free\nenergyF(eM) of a magnetic QD with a single hole is\nnow simply the lowest eigenvalue of Eq. (1) and it can be\nexpanded in powers of nj. The lowest terms compatible\nwith cubic symmetry are39\nF0=Kc(n4\nx+n4\ny+n4\nz) + 27Kc2n2\nxn2\nyn2\nz. (16)\nFor data calculated by numerically diagonalizing ˆH=ˆH2\n(model described in Sec. IIB) it turns out that Eq. (16)\nsuffices to obtain good fits; for instance, lower solid line\nin Fig. 3(a) corresponds to Kc= 0.83 meV and Kc2=\n0.075 meV with easy axis along [111]. There we have\nchosen Cd 1−xMnxTe as the material, L= 16 nm and\nh= 50 meV which corresponds to h=JpdNMnSMn\nwithx≈2.3% (we take36|Jpd|= 60 meV ·nm3,SMn=\n5/2 andNMn= 4x/a2\nlwith CdTe lattice constant al=\n0.648 nm). Results in Fig. 3 are again subject to scaling,\nsimilar to the non-magnetic spectra in Fig. 2(a). When\nthe material parameters (specifically, γ2/γ1andγ3/γ2)\nare fixed, the spectrum of ˆH2, expressed in the units of\nE0, depends on a single dimensionless parameter\nˇZ =h/E0≡2m0hL2/(γ1π2/planckover2pi12). (17)\nThis scaling relates the spectra of e.g. cubic dots of dif-\nferent sizes and Mn contents (if their respective values\nofˇZ are equal). Data in Fig. 3 therefore apply both to\nx= 2.3% atL= 16 nm (if left as they are) and x= 9.2%\natL= 8 nm (if scaled by a factor of 4). It turns out that\ntheg-factor analysis presented in the previous section is\nmeaningful for ˇZ/lessorsimilar0.1 while now we have stepped out of\nthis limit. When the exchange field hbecomes stronger,\nlevels cross and cease to depend linearly on has required\nby Eq. (6); for ˆH=ˆH1, this is illustrated in Fig. 1(b).\nThis limit was determined for CdTe cubic QDs but it will\ntypically not be too different for other materials and/or\naspect ratios λunless accidental (quasi)degeneracies oc-\ncur at ˇZ = 0.\nMAE shown in Fig. 3 describe systems well beyond\nthis limit of small ˇZ (linear regime). We first focus on a\nperfectly cubic CdTe QD where there are no anisotropies\nin the linear regime. As already mentioned, the lowest\nenergy hole state in Fig. 3(a) exhibits a [111] easy axis\nwithKc= 0.83 meV at L= 16 nm and h= 50 meV,\ni.e.ˇZ≈2.8, (this corresponds to a realistic x≈2.3%\nMn doping). In bulk DMSs, [111] would be an unusual\nmagnetic easy axis direction15and we surmise that thecubic deformed\nZnSe CdTe ZnSe CdTe\nh[meV] ˇZKcKcKcKuKcKu\n10 0.55 0.11 0.24 0.23 -3.79 0.35 -3.90\n20 1.1 0.20 0.43 0.36 -5.52 0.62 -6.21\n30 1.7 0.28 0.59 0.44 -6.29 0.83 -7.59\n40 2.2 0.35 0.71 0.50 -6.72 1.01 -8.56\n50 2.8 0.41 0.83 0.56 -7.01 1.16 -9.30\nTABLE I. Magnetic anisotropy constants (in meV) for a 16 ×\n16×16 nm3(cubic) and 16 ×16×8 nm3(deformed) ZnSe\nand CdTe magnetic QD as a function of exchange splitting\n(or dimensionless parameter ˇZ as for CdTe).\nreason for this is that for instance in (Ga,Mn)As grown\non a GaAs substrate, there is a sizable compressive strain\nwhich prefers either parallel or perpendicular orientation\nofeMwith respect to the growth axis.\nWe note that in a QD containing two holes (closed-\nshell system11) the anisotropies will also be present and\nthey will be different from the single-hole case. Free en-\nergy, taken as a sum, F0(eM) =E1+E2, of the lowest\ntwo single-hole states [shown e.g. in Fig. 3(a)], is not a\nconstant independent of eMas one could naively expect.\nThis intuition reflects ˆHeffin Eq. (6) where the two hole\nstates have opposite spin (hence their energies add up to\nzero). Once we leave the linear regime ( ˇZ/greaterorsimilar0.1),ˆHeff\nceases to be a good approximation. Qualitatively, the\nsame behaviour is found for ZnSe (not shown), a smaller\nvalue of Kc= 0.41 meV is accounted for by the smaller\nHH/LH splitting. The value of this constant is a com-\nplicated function of system parameters and it can even\nchange sign as shown in Fig. 3(c) where Kc=−0.63 meV.\nParameters used in this figure ( γ1/γ2/γ3= 4.0/1.5/1.6\nandh= 20 meV) do not strictly correspond to published\nvalues of any semiconductor but they can be viewed as\nreasonable given the uncertainty in experimental deter-\nmination of the Luttinger parameters. Dependence of\nthe anisotropy constants for ZnSe and CdTe QDs on his\nsummarized in Tab. I.\nLet us now return to non-cubic QDs. As already\nexplained, the sizable g-factor anisotropies shown in\nFig. 2(b), relevant to the case of weak magnetism ( ˇZ≪\n1), translate into an additional term AF1=Kun2\nzin the\nfree energy of Eq. (2) where Ku=−kAup to linear or-\nder in ˇZ∝k. Typically, KuexceedsKcalready for small\nQD deformation ( λslightly over one) and the data in\nFig. 3(b) imply Kualmost an order of magnitude larger\nthanKcforλ= 2 (see also data in Fig. 2 where gz≫gx).\nRegardless of the contributions to Kuof higher order in\nˇZ, data in Tab. I imply an out-of-plane easy axis (in the\n[001] direction) as it is the case in QWs. However, upon\ndeforming of a QD the easy axis does not abruptly jump\nfrom [111] to [001] but smoothly interpolates between\nthese two directions. Similar effect, easy axis shifting as\na function of some system parameter, is also known in\nbulk DMS [(Ga,Mn)As epilayers in particular, see Fig. 86\n-2024681012141618\n[100] [110] [001] [100]E [meV]2.73.2\n[100] [110] [001] [100]E [meV](a) (b) CdTe CdTe\n1 1.1 1.2 1.300.10.2\n(1)(2)(3)(c)\n(d)\nz\n[111]E1E2\nE1E2\n-10-9-8-7-6-5-4\n[100] [110] [001] [100]E [meV]\n[010][001]\n[100]\n[100]M\nFIG. 3. (Color online) Magnetocrystalline energy as a funct ion of magnetization direction ( E1); the data labelled E2are\nexplained in the text. CdTe QD with 2 .3% Mn (a) 16 ×16×16 nm3, (b) 16 ×16×8 nm3. (c) Fictitious material with\nparameters described in the text; note that the sign of Kcimplied by Eq. (16) has changed compared to (a,b). (d) Color- coded\neasy axis positions for CdTe QDs as a function of aspect ratio (λ) and effective exchange splitting ˇZ. Black squares (1) indicate\neasy plane perpendicular to z-direction, hollow squares denote an isotropic magnet; whi te region (2) corresponds to easy-axis\n[001]; red squares (3) denote systems with [111] easy axis wh ich gradually shifts towards [001] with increasing λ. This plot is\nuniversal as far as Lis concerned.\nin Ref. 15]. Easy axes as a function of QD shape (oblate\ndots,λ >1) and effective exchange splitting ˇZ are sum-\nmarized in Fig. 3(d) and the mentioned gradual shift of\neasy axis is indicated by shading between regions (3) and\n(2) (easy axes [111] and [001], respectively). On the other\nhand, the easy axis position changes abruptly between\n(1) and (3) or (1) and (2); region (1) corresponds to easy\naxis in the plane perpendicular to [001] (with anisotropies\nwithin this plane being very small). The abrupt changes\nreflect ground state crossings, such as the one with λde-\nscribed below Eq. (14), while the gradual ones stem from\nlevel mixing caused by ˆHex.\nFinally, we comment on MA in QDs occupied by more\nthan one hole. As already mentioned above, one pos-\nsible approach is to discuss open-shell and closed-shell\nsystems separately. This notion is based on the concept\nof the QD being an artificial atom whose levels are or-\nganized into shells comprising of spin-up and spin-down\norbitals. Whenever a shell is completely filled (closed),\nthe numbers of spin-up and spin-down carriers are equal\nhence their total spin is zero. If the QD is magnetically\ndoped, no magnetic ordering is expected and also no MA.\nHowever, strong SO coupling puts this concept into ques-\ntion since it mixes different shells and also invalidates\nthe spin-up and down labels of individual orbitals. The\nMA as a function of particle number Npstrongly varies,\nboth quantitatively and qualitatively. By comparing the\nNp= 1 and Np= 2 cases of a cubic CdTe QD, that isF0(eM) =E1andF0(eM) =E1+E2of Fig. 3(a), we\nfind that while the easy axis [111] in the former case is\nrelatively ‘soft’ (energy difference between eM||[111] and\n[110] is ‘only’ ≈0.1 meV), the QD with two holes has a\n‘robust’ easy axis [110] and the corresponding minimum\ninF0(eM) is as deep as 0 .3 meV. MA as a function of\nNpdisplays rich behavior and one can therefore envision\ncontrol of nanomagnetism by electrostatic gating, illumi-\nnation (used to photoinduce carriers) and possibly also\ntemperature, known to alter the magnetic ordering in the\nbulk-like structures.18,40\nIV. CONCLUSIONS\nWe have discussed two approaches to magnetic\nanisotropies in quantum dots (QDs) described by a\ngeneric model in Eq. (1). An effective Hamiltonian for\nindividual Kramers doublets allows to express the ener-\ngetics of a magnetically doped QD in terms of only three\nparameters (anisotropic g-factor) if the exchange split-\nting due to the magnetic ions is relatively small. On\nthe other hand, if the exchange splitting is large or the\nQD’s symmetry is too high, the symmetry-based expan-\nsion of the magnetocrystalline energy in powers of the\ndirection cosines of magnetization may in principle con-\ntain infinitely many terms (each of them quantified by\none parameter). Focusing on manganese-doped semi-7\nconductor QDs, we find that only first few terms are\nappreciable, present their values and show in Fig. 3(d)\na diagram of magnetic anisotropies in the QD param-\neter space. While we focus on a relatively small pa-\nrameter range in that diagram, and the barriers be-\ntween individual free energy minima are relatively low,\nit demonstrates that the QDs may have rich magnetic\nanisotropies. In spintronics,18,19,41these systems could\nthus enable confinement-controlled multi-level logic. Our\nresults provide a starting point for further studies of\nnanoscale magnetism in QDs. Such studies could relax\nthe mean-field approximation, include multiple-carrier\nstates,22,42or the effect of strain.\nACKNOWLEDGEMENTS\nThis work is supported by DOE-BES de-sc0004890,\nnsf-dmr 0907150, AFOSR-DCT FA9550-09-1-0493, U.S.\nONR N0000140610123, and nsf-eccs 1102092.\nAPPENDIX A\nThe downfolding of ˆHQD+ˆHextoˆHeffis indeed possi-\nble for the two example systems discussed in Sec. IIA\nand IIB. To prove this, we first transform the basis\n|K1∝an}bracketri}ht,|K2∝an}bracketri}htto|K′\n1∝an}bracketri}ht,|K′\n2∝an}bracketri}htwhere ˜tzof Eq. (7) is diagonal\nand then verify that the diagonal elements of ˜txand˜ty\nvanish. This procedure has to be applied to each Kramers\ndoublet of interest. In the case of ˆH1in Eq. (3), this is\ndone simply by construction (e.g. |K′\n1∝an}bracketri}ht,|K′\n2∝an}bracketri}htfor the up-\nper doublet in Fig. 1(a) is just |Jz= 3/2∝an}bracketri}ht,|Jz=−3/2∝an}bracketri}ht).\nIn the model described by ˆH2, one can split the Hilbert\nspace into two disjunct subspaces H1,H2and the above\nassertion can be shown to hold if |K′\n1∝an}bracketri}ht ∈H1and|K′\n2∝an}bracketri}ht ∈\nH2. (The decomposition H1⊕H1relies on ˆHexbeing\nindependent of space coordinates; relaxing the mean-field\ntreatment of Mn magnetic moments thus introduces cor-\nrections to ˆHeff.) Finally, one adjusts the relative phase\nbetween |K′\n1∝an}bracketri}htand|K′\n2∝an}bracketri}ht, so that the matrix ˜txis real and\n˜typurely imaginary.\nAPPENDIX B\nThis Appendix explains the relation between the\nanisotropic g-factors and Eq. (2). The eigenvalues of ˆHeff\nare two numbers of equal magnitude and opposite sign,\nthe lower of which is\n−h/radicalBig\nn2xg2x+n2yg2y+n2zg2z. (18)\nLet us consider for example single hole in a cuboid QD of\ndimensions λL×λL×L(such as it corresponds to data\nin Fig. 2) so that gx=gy. Expression (18) which nowequalsF(eM) can be rewritten as\n−h/radicalbig\ng2x+g2z√\n2/radicalBigg\n1 +g2z−g2x\ng2z+g2x(n2z−n2x−n2y) (19)\nand developped in terms of a small parameter A= (g2\nz−\ng2\nx)/(g2\nz+g2\nx) which quantifies the QD asymmetry as\n−k(1−1\n2A)−Akn2\nz+1\n8A2k(2n2\nz−1)2+... (20)\nwherek=h/radicalbig\n(g2x+g2z)/2. The first term does not de-\npend on the magnetization direction, hence it can be dis-\nregarded for the purposes of magnetic anisotropy analy-\nsis.\nAPPENDIX C\nWe derive Eq. (14) in this Appendix and discuss the\ndetails of the model considered in Sec. IIB. Energies E/E0\nin Fig. 2(a) are calculated by numerical diagonalization\nofˆH2withh= 0, a matrix constructed of 4 ×4 blocks\n∝an}bracketle{tmxmymz|ˆHKL|m′\nxm′\nym′\nz∝an}bracketri}ht/E0introduced at the begin-\nning of Sec. IIB. The basis of ˆHQDconsists thus of direct\nproduct states |mxmymz∝an}bracketri}ht⊗|Jz∝an}bracketri}htwhere|Jz∝an}bracketri}htare the four-\nspinors of total angular momentum J= 3/2 which are\neigenstates to ˆJz. For practical purposes, we cut-off the\nbasis by mx,my,mz≤6, resulting in ˆHQDof dimension\n864. Eigenvalues are typically converged to better than\n0.1 meV for this cut-off.\nThe matrix ˆHQD/E0is block-diagonal for γ2=γ3=\n0 and the block mx,my,mzhas a four-fold degenerate\neigenvalue\n(mx/λx)2+ (my/λy)2+m2\nz. (21)\nDimensionless energies on the left of Fig. 2(a) correspond\ntoλx=λy= 1 and are hence integers. The lowest\nlevelE/E0= 3 belongs to ( mx,my,mz) = (1,1,1) while\nthe first excited state E/E0= 6 entails an additional\nthreefold geometric degeneracy corresponding to orbital\nstates (1 ,1,2), (1,2,1) and (2 ,1,1); theE/E0= 6 level\nforγ2=γ3= 0 is thus twelve-fold degenerate.\nNext, we can treat the HH-LH splitting as a perturba-\ntion when γ2andγ3are turned on. In the lowest order,\nmixing between different ( mx,my,mz) blocks can be ne-\nglected except for the case when their energies were equal\natγ2=γ3= 0 as in the case of the three blocks of the\nE/E0= 6 level. With coupling to the remote levels dis-\nregarded, we are left with a 12 ×12 matrix in this case\nwhich can be diagonalized analytically. It turns out to\nhave two four-fold degenerate eigenvalues\nE±\n4/E0= 6 +64\n3π2γ2\nγ1/parenleftBigg\ns±/radicalbigg\ns2+81π4\n1024/parenrightBigg\n(22)\nand two two-fold degenerate ones\nE±\n2/E0= 6−128\n3π2γ2\nγ1/parenleftbigg\ns∓9π2\n64/parenrightbigg\n. (23)8\nThe lowest of these four energies is E−\n2and it is shown\nin Fig. 2(a) for s≡γ3/γ2= 1 as a solid line which\ncrosses the horizontal line E/E0= 3 corresponding to the\n(mx,my,mz) = (1,1,1) quadruplet which does not shift\nin energy to the first order of this perturbation analysis.Eq. (14) is the solution of E−\n2= 3E0forγ2/γ1under the\nassumption s= 1. Such level crossing (as a function of\nγ2/γ1) is genuinely due to the confinement and no level\ncrossings occur in in bulk as long as 0 < γ2/γ1<1/2.\n1U. Welp, V. K. Vlasko-Vlasov, X. Liu, J. K. Furdyna, and\nT. Wojtowicz, Phys. Rev. Lett. 90, 167206 (2003).\n2X. Liu, Y. Sasaki and J. K. Furdyna, Phys. Rev. B 67,\n205204 (2003).\n3J. Fern´ andez-Rossier and R. Aguado in Handbook of Spin\nTransport and Magnetism, edited by E. Y. Tsymbal and I.\nˇZuti´ c (CRC Press, New York, 2012).\n4D. R. Yakovlev and W. Ossau in Introduction to the\nPhysics of Diluted Magnetic Semiconductors, edited by J.\nKossut and J.A. Gaj (Springer, New York, 2010).\n5R. Beaulac, L. Schneider, P. I. Archer, G. Bacher, and\nD. R. Gamelin, Science 325, 973 (2009).\n6I. R. Sellers, R. Oszwa/suppress ldowski, V. R. Whiteside, M. Egin-\nligil, A. Petrou, I. ˇZuti´ c, W.-C. Chou, W. C. Fan, A. G.\nPetukhov, S. J. Kim, A. N. Cartwright, and B. D. Mc-\nCombe, Phys. Rev. B 82, 195320 (2010).\n7F. Henneberger and J. 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B 76, 075305 (2007).\n28Magnetostatic shape anisotropy will compete with MAE\nas soon as the QD is not perfectly cubic. To estimate\nthe shape anisotropy energy, we use parameters similar\nto those discussed in Sec. III. Taking ˇZ = 0.05 of Eq. (17)\nandλ= 1.1 as a reference point (in-plane magnetization in\naL= 8 nm QD is preferred to eM||ˆzby∼0.1 meV), mag-\nnetostatic shape anisotropy15does not reach even 1 µeV.\n29See Sec. III.A of Ref. 19 for an introduction to Kramers\ndoublets. Note that multiple doublets may sometimes be\ndegenerate either because of symmetry or by accident.\n30L. Besombes, Y. L´ eger, L. Maingault, D. Ferrand, H. Ma-\nriette, and J. Cibert, Phys. Rev. Lett. 93, 207403 (2004);\nL. Besombes, Y. L´ eger, L. Maingault, D. Ferrand, H. Ma-\nriette, and J. Cibert, Phys. Rev. B 71, 161307 (2005).\n31I. A. Merkulov and A. V. Rodina in Introduction to the\nPhysics of Diluted Magnetic Semiconductors, edited by J.\nKossut and J. A. Gaj (Springer, New York, 2010).\n32A. 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(12) in Ref. 15\nwith aesthetically more pleasing n4\nx+n4\ny+n4\nz= 1−2(n2\nxn2\ny+\nn2\nyn2\nz+n2\nzn2\nx).KcequalsBc(Eq. (2) of Ref. 17) or Bx4=\nBz4(Eq. (21) of Ref. 16).\n40T. Jungwirth, Jairo Sinova, J. Maˇ sek, J. Kuˇ cera, and A.\nH. MacDonald, Rev. Mod. Phys. 78, 809 (2006); H. Ohno,\nD. Chiba, F. Matsukura, T. Omiya, E. Abe, T. Dietl, Y.\nOhno, and K. Ohtani, Nature 408, 944 (2000); S. Koshi-\nhara, A. Oiwa, M. Hirasawa, S. Katsumoto, Y. Iye, C.\nUrano, H. Takagi, and H. Munekata, Phys. Rev. Lett.\n78, 4617 (1997); A. G. Petukhov, I. ˇZuti´ c and S. C. Er-\nwin, Phys. Rev. Lett. 99, 257202 (2007); M. J. Calder´ on\nand S. Das Sarma, Phys. Rev. B 75, 235203 (2007); V.\nKrivoruchko, V. Tarenkov, D. Varyukhin, A. D’yachenko,9\nO. Pashkova, and V. Ivanov, J. Magn. Magn. Mater 322,\n915 (2010).\n41Concepts in Spin Electronics , edited by S. Maekawa (Ox-\nford University Press, Oxford, 2006).42N. T. T. Nguyen and F. M. Peeters Phys. Rev. B 80,\n115335 (2009); F. Qu and P. Hawrylak, Phys. Rev. Lett.\n96, 157201 (2006)."    },    {        "title": "1205.5422v1.Enhancing_magnetocrystalline_anisotropy_of_the_Fe70Pd30_magnetic_shape_memory_alloy_by_adding_Cu.pdf",        "content": " 1 Enhancing magnetocrystalline anisotropy of the Fe 70 Pd 30  \nmagnetic shape memory alloy by adding Cu \n \nS Kauffmann-Weiss 1,2 , S Hamann 3, M E Gruner 4, L Schultz 1,2 , A Ludwig 3 and S Fähler 1 \n \n1 IFW Dresden, P.O. Box 270116, 01171 Dresden, Germa ny \n2 Dresden University of Technology, Institute for Mat erials Science, 01062 Dresden, \nGermany \n3 Ruhr-Universität Bochum, Institute for Materials, 44780 Bochum, Germany   \n4 University of Duisburg-Essen, Theoretical Physics,  47048 Duisburg, Germany \n \nE-mail: s.weiss@ifw-dresden.de \n \nAbstract. Strained epitaxial growth provides the opportunity to understand the dependence of \nintrinsic and extrinsic properties of functional ma terials at frozen intermediate stages of a \nphase transformation. In this study, a combination of thin film experiments and first-\nprinciples calculations yields the binding energy a nd magnetic properties of tetragonal \nFe 70 Pd 30-xCu x ferromagnetic shape memory thin films with x = 0, 3, 7 and structures ranging \nfrom bcc  to beyond fcc  (1.07 < c/a bct  < 1.57). \nWe find that Cu enhances the quality of epitaxial g rowth, while spontaneous polarisation and \nCurie temperature are only moderately lowered as ex pected from our calculations. Beyond \nc/a bct > 1.41 the samples undergo structural relaxations t hrough adaptive nanotwinning. For all \ntetragonal structures, we observe a significant inc rease of the magnetocrystalline anisotropy \nconstant K1, which reaches a maximum of K1 ≈ -2.4*10 5 Jm -3 at room temperature around \nc/a bct  = 1.33 and is thus even larger than for binary Fe 70 Pd 30  and the prototype Ni-Mn-Ga \nmagnetic shape memory system. Since K1 represents the driving force for variant reorienta tion \nin magnetic shape memory systems, we conclude that Fe-Pd-Cu alloys offer a promising route \ntowards microactuators applications with significan tly improved work output. \n \nContents \n1. Introduction \n2. Experimental and theoretical methods \n3. Structure and epitaxial relationship  2 4. Remanence, coercivity and saturation field \n5. Change of Curie temperature and spontaneous pola risation \n6. Change of magnetocrystalline anisotropy energy \n7. Conclusions \nAcknowledgments \nReferences \n \n1. Introduction \n \nDue to huge strains up to 10% [1] magnetic shape me mory (MSM) alloys are of particular \ninterest for microactuators [2]. Most research on b ulk and thin films focuses on the Ni-Mn-Ga \nprototype system [3], but in particular for microsy stems Fe 70 Pd 30  as the second system \ndiscovered [4] shows several advantages. While in t he Ni-Mn-Ga system oxidation can result \nin functional degradation [5] the high content of t he noble element Pd in Fe-Pd makes this \nMSM alloy even biocompatible [6]. Spontaneous magne tic polarisation JS and Curie \ntemperature TC are considerably higher [7,8] compared to Ni-Mn-Ga . Whereas the high \nmaterial cost hinder bulk applications of Fe-Pd, mi crosystems require only small masses of \nactive material, which makes the material costs neg ligible compared to the increased process \ncosts of film preparation. \nIn this paper we quantify the influence of Cu alloy ing on the magnetic properties of \nFe 70 Pd 30-xCu x. Recent combinatorial experiments revealed that Cu  can substantially increase \nthe martensitic transformation temperature [9] whic h marks the upper limit of the working \nrange for MSM applications. After a thorough crysta llographic characterisation of the films \nwe will focus on magnetic aspects, in particular on  the magnetocrystalline anisotropy energy. \nAs an intrinsic material property, the anisotropy c onstant K represents the maximum energy \ndensity which can be supplied by an external magnet ic field [10]. In case of highly mobile \ntwin boundaries this energy input limits the mechan ical work which can be provided by an \nMSM element. K as the key intrinsic property depends on structure , composition, and \nchemical order. In addition the following intrinsic  magnetic properties have to be considered \nfor MSM applications. A high JS is favourable, since it allows using low magnetic fields to \nobtain the maximum energy input at the anisotropy f ield S A JK H 2= . Finally, the Curie \ntemperature should be well above application temper ature to allow the MSM effect within an \nincreased temperature regime. Thus, the influence o f Cu on spontaneous magnetic  3 polarisation JS and Curie temperature TC is investigated using both, experimental as well a s \ntheoretical approaches.  \nTo comprehensively characterise materials regarding  the above mentioned properties, a high \nsample quality is required. Single crystals are sui table since finite size, texture and stress \neffects do not affect the measurement results, but they are difficult to produce and often \nexpensive as large amounts of the material is neede d. Epitaxial films represent the thin film \ncounterpart of single crystals and allow to measure  the anisotropic magnetic properties \ndirectly by measuring the magnetisation curves alon g different crystallographic directions. \nThis allows easily to determine the magnetocrystall ine anisotropy constant K since the films \nare attached to a rigid substrate that blocks the r e-orientation of structural variants. \nFurthermore, coherent epitaxial growth on cubic see d buffers with different lattice parameters \ncan be used to adjust the tetragonal distortion of the martensite. This approach allows \ncontrolling the c/a bct  - ratio along the Bain transformation path between  almost body centred \ncubic bcc  (c/a bct  = 1) and face centred cubic fcc  (c/a bct  = 1.41) [11]. By this kind of epitaxial \nfilm growth artificial single variant states are re alised [12] allowing to measure K along the \ndifferent crystallographic directions ( K1 and K3).  \nRecently the authors reported on the straining of e pitaxial Fe 70 Pd 30  films beyond the Bain path \nto c/a bct  >1.41 [13]. For this type of growth, adaptive nano twinning was found as the \nrelaxation mechanism. By using a simple geometric r elation one can still control K by using \nappropriate buffer lattice constants, but additiona lly introduces twin boundaries at the \nnanoscale which can be used as pinning centres e. g . for percolated magnetic recording media. \nThis type of application also requires a high K and benefits from high JS and TC. In contrast to \nMSM microactuators, which require a release from th e substrate [12] this is not necessary for \nrecording applications. \nThis paper is arranged as follows: After a short de scription of sample preparation and \nmethods used for analysis we characterise in detail  the structure of the epitaxial films. \nSupported by extensive first-principles calculation s of the elastic energy associated with \nstraining of Fe-Pd and Fe-Pd-Cu films we develop an  explanation for the significantly \nimproved growth behaviour of Fe-Pd-Cu. As a second focus we probe the influence of Cu on \nextrinsic magnetic properties - for example saturat ion field µ 0HS - as well as the intrinsic \nmagnetic properties like spontaneous polarisation JS and Curie temperature TC. Finally, \nrounding up the beneficial impact of Cu on the func tional properties of Fe-Pd alloys, we \nreport a significant enhancement of the important m agnetocrystalline anisotropy constants K1 \nand K3 upon addition of Cu.  4  \n2. Experimental and theoretical methods   \n \nEpitaxial thin films were deposited at room tempera ture using DC magnetron sputtering in a \nUHV chamber (base pressure <10 −8  mbar, Argon 6N purity). The Cu content was varied by \nco-sputtering from a 4 inch Fe 70 Pd 30  alloy target (purity 99.99 %,) and a 2 inch Cu tar get \n(purity 99.99 %,) and changing the sputter power of  Cu (55… 60W) and Fe 70 Pd 30  (60… 55W) \ntargets. With a sputter rate of ~ 0.3 nm s-1, we prepared 40 nm thick Fe 70 Pd 30-xCu x films. \nEpitaxial metallic layers (Cr, Au, Ir, Rh, and Cu) used as buffers for the strained growth of \nFe 70 Pd 30-xCu x films were deposited at 300°C (and at room tempera ture for Cu) onto \nMgO(100) single crystal substrates. The nominal fil m architecture is: MgO/ 50 nm buffer/ \n40 nm Fe 70 Pd 30-xCu x. When metals with fcc  crystal structure are used as buffer (Au, Ir, Rh, \nCu), prior to the deposition, a layer of 5 nm Cr wa s grown on MgO to enhance adhesion. \nX-ray diffraction (XRD, Philips X’Pert) in Bragg-Br entano geometry was performed at room \ntemperature in a two circle setup using Co K α radiation. Pole figures were measured in a four \ncircle setup with Cu K α radiation.  The composition of all samples was determined by us ing a \nJeol JSM 5800 LV scanning electron microscope (20 k V acceleration voltage) equipped with \nan Oxford Inca energy dispersive X-ray (EDX) analys is system (132 eV resolution, every \nspectrum contained more than 200.000 counts) and a Fe 70 Pd 30  standard. EDX was used to \nprobe if the actual composition meets the intended film composition, which had been obtained \nwith a standard deviation of 1 %. In order to deter mine the thin film thickness, the Thin Film \nID  software from Oxford Instruments was used. This so ftware is based on the StrataGem  \nalgorithm that allows calculating film thickness fr om EDX measurements combined with \nMonte Carlo simulations. After simulation of the el ectron trajectories within the layers the \noptimum acceleration voltage is determined and an E DX spectrum is acquired. Further Monte \nCarlo simulations were separately performed by usin g Casino software to calculate film \nthickness [14].  \nTo determine magnetic properties, temperature and m agnetic field dependent measurements \nwere performed by using a Physical Property Measure ment System (PPMS, Quantum Design) \nwith vibrating sample magnetometer (VSM) add-on. Th e Curie temperature TC was \ndetermined using Kuz’min’s fit [15,16] from tempera ture-dependent in-plane magnetisation \nmeasurements JS(T) between 50 K and 400 K (sweep rate of 2 K min -1) in an applied \nmagnetic field of 1 T:  5  ( )\n( )( )β\n\n\n\n\n\n\n\n−−\n\n\n−==2523\n01 10C CS\nTTsTTsK TJTJ (1) \nFor Fe-Pd as a metallic ferromagnetic material we f ixed the critical exponent at β = 1/3 [15]. \nThe other variables – the Curie temperature TC, the saturation polarisation J0, defined as JS at \n0 K, and the shape parameter s are additional variables within this equation and considered as \nfree parameters. For all samples the values for J0 = 1 ± 0.02 and s = 1 ± 0.2 do not vary \nsignificantly compared to the maximum range of 0 < s < 2.5. This allows to focus on TC in \ndependence of tetragonal deformation and compositio n. MgO single crystals used as \nsubstrates contain superparamagnetic impurities tha t influence the magnetic measurements at \nlow temperatures, with the consequence that J0 was regarded as a free parameter. While for \nmost samples it was sufficient to exclude the tempe rature range below 50 K for analysis, two \nsamples (Fe 70 Pd 23 Cu 7 with c/a bct of 1.07 and 1.09) contained impurities which contri buted up \nto 100 K. This reduces the available temperature ra nge significantly; hence these samples had \nbeen excluded from the TC analysis. \nThe magnetocrystalline anisotropy constants K1 and K3 were calculated from hysteresis \nmeasurements in [100], [110] and [001] direction. M agnetic hysteresis loops were measured \nin the range from -4.5 T up to +4.5 T at room tempe rature. \nCoercivity HC was determined from the intersection with the fiel d axis, remanence JR from \nthe intersection with the polarisation axis. Sponta neous polarisation JS was determined as the \nmaximum polarisation from the in-plane measurements , where the saturation could be \nobtained in low fields, and from the out-of-plane m easurements in high fields. For all samples \nboth values are identical within an accuracy of 3 % . The saturation field HS was determined \nby the minimum of the second derivative of the J(H) curves. All values are averaged from \npositive and negative fields and the difference is about twice the symbol size (when no error \nbar is shown). This also holds for the systematic e rror for the field values given which is \nlimited to about ± 2 mT due to the continuous field  sweep. The accuracy for the polarisation \nvalues is lower and limited by the accuracy of abou t 5 % for the determination of sample \nvolume. \nFirst-principles calculations in the framework of d ensity functional theory (DFT) allow the \naccurate and parameter-free determination of struct ural and electronic properties on the \nelectronic level. In order to obtain information on  the binding energy as a function of \ntetragonality, we performed a full optimisation of the atomic positions without symmetry \nconstraints using the Vienna ab-initio  simulation package (VASP) [17]. We use the exchang e- 6 correlation functional according to Perdew and Wang  [18] in combination with the spin \ninterpolation formula of Vosko, Wilk and Nusair [19 ].The employed projector augmented \nwave potentials (PAW) describe explicitly the 3 d and 4 s valence electrons of Fe and Cu and \nthe 4 d and 5 s of Pd [20]. The cut-off for the plane wave basis w as chosen as 342 eV. The \nBrillouin zone integration was carried out using a 2×2×2 Monkhorst-Pack k-mesh with \nMethfessel-Paxton Fermi-level smearing with a broad ening parameter of 0.1 eV. A disordered \narrangement was realized using a 500 atom supercell  and a random distribution of the 340 Fe \natoms and 160 Pd atoms, or respectively, 135 Pd and  25 Cu atoms. In order to avoid statistical \nuncertainties which hamper a direct comparison, 25 randomly chosen Pd atoms were \nexchanged by Cu to model a comparable ternary distr ibution, while the remaining elements \nwere left untouched. Further computational details can be found in [13,21]. \nIn addition, we investigated composition dependent and finite temperature magnetic \nproperties using the Korringa-Kohn-Rostoker (KKR) a pproach as implemented in the Munich \nSPR-KKR code (version 5.4) [22, 23]. The exchange-c orrelation functional was treated within \nthe generalised gradient approximation (GGA) accord ing to Perdew, Burke and Ernzerhof \n[24] based on a scalar relativistic description of the Hamiltonian in combination with the \natomic sphere approximation (ASA). We used 64 point s for the energy integration contour in \nthe complex plane while taking into account f states for the angular momentum expansion. \nWe used a dense k-mesh of up to 40×40×40 in the ful l Brillouin zone, corresponding to 4500 \npoints in the irreducible zone. Here, the disordere d nature of the Fe-Pd alloy was modelled in \nterms of averaging the electronic scattering proper ties within the coherent potential \napproximation (CPA). The CPA enables the economic u se of small cells but does not allow \nfor structural relaxations. We determined the magne tic exchange parameters for use within a \nHeisenberg model following the approach by Liechten stein et al. [25], starting from the \nferromagnetic state. The corresponding Curie temper atures were estimated within the mean-\nfield approximation. \n \n3. Structure and epitaxial relationship \n \nThere are several crystallographic structures repor ted in quenched Fe 70 Pd 30  bulk. These \ndifferent structures that appear around room temper ature can be described by a phase \nsequence that changes with increasing Pd content as  follows: body-centred-cubic ( bcc ), body-\ncentred-tetragonal ( bct ), face-centred-tetragonal ( fct ) and face-centred-cubic ( fcc ) [7,4,26,27]. \nThe transformation of a crystal lattice from a bcc  to a fcc  structure can be described in terms  7 of a tetragonal distortion, following the so-called  Bain path [28].In order to describe the \ndegree of this tetragonal lattice distortion, we wi ll use the c and abct  lattice parameters \ncorresponding to the bcc  elemental cell. Thus, the ratio c/a bct  ranges from 1 (for a perfect bcc  \nlattice) over 1.33 (corresponding to a fct  structure) to 1.41 (for a fcc  structure). Frequently, the \ndegree of tetragonal distortion is expressed using the austenitic fcc  elemental cell as basis. \nThen, the c/a fcc  ratio ranges from 1 for the austenite ( fcc  lattice) over 0.94 for the fct  \nmartensite to 0.71 for bcc  lattice. The fcc  description is easily converted to the bcc  description \nby multiplying with a conversion factor of √2. Within this paper the c/a fcc  ratio of the bcc  \nelemental cell is used to describe the lattice defo rmations, since bulk experiments and DFT \ncalculations indicate that the bcc  phase defines the ground state [13,27].  \nThe c/a bct  ratio in epitaxial films is determined by dividing  the values of the out-of-plane c-\naxis by the in-plane a-axis of the bct  cell. Within this chapter the route to determine t he c/a  \nratio is described on the basis of the bcc  unit cell. The c/a bct  ratio is used from the beginning \nto identify the samples and to distinguish between the different tetragonal distortions.  \nBy coherent epitaxial growth the lattice constants of the buffer adjust the in-plane lattice \nparameters of Fe-Pd-X films and thus c/a bct . For analysing the crystal structure of Fe 70 Pd 27 Cu 3 \nfilms grown on different buffer layers, XRD scans i n Bragg Brentano geometry were \nperformed (Figure 1). However, this method is only suitable to reveal the out-of-plane lattice \nparameter. When changing the buffer materials from Au to Cu, the position of the (002) \nreflection of fcc  buffers (red circles) increases towards higher 2 θ values. By using a bcc  buffer \nlike Cr the peak position of the (002) reflection i s at considerable higher 2 θ values. This \nreflects the shifting of the XRD peak in dependence  of the lattice parameter for several \nbuffers layers with different structures. When vary ing the buffer material, the position of the \n(002) reflection of Fe 70 Pd 27 Cu 3 (blue pentagons) shifts from 2 θ = 70.5°  ( c/a bct  = 1.07) to a \nlower value at 2 θ = 60.6°  ( c/a bct  = 1.31). For a film on the Cu buffer ( c/a bct  = 1.57) the (002) \nreflection shifts to 2 θ = 52.6°. From these XRD measurements only the (002 ) diffraction peak \nis observed for the Fe 70 Pd 27 Cu 3 films, which indicates the presence of a highly te xtured \nmaterial. The absence of any further diffraction pe aks suggests a coherent epitaxial growth of \nthe films, where the constraint of an almost consta nt unit cell volume results in an increase of \nthe out-of-plane lattice parameter when the in-plan e lattice parameters shrink. \nTo probe this in detail we measured the in-plane la ttice parameters by performing 2 θ scans of \nthe (101) bct  lattice planes in tilted conditions at a tilt angl e Ψ. As sketched in Figure 2b a \nsimple geometric relation allows to use this lattic e plane (together with the measurements in \nBragg-Brentano geometry) to calculate the in-plane lattice parameter. It was found that the in- 8 plane lattice parameters of the Fe 70 Pd 27 Cu 3 films are identical to the buffer lattice spacings \ndbuffer  (Figure 3a). This indicates a coherent growth in a ll thin films deposited on the different \nbuffers and reveals the following relation: dbuffer  = abct . The c/a bct  ratio for every thin film \nsample was then calculated by using Bragg-Brentano as well as tilted XRD measurements. \nThis is concluded in Figure 3b, depicting the depen dency of the tetragonal deformation ( c/a bct  \nratio) for all buffer materials used. This deformat ion behaviour is quite similar to previous \ninvestigations for thin binary Fe 70 Pd 30  films which are added for comparison [11,13]. In \ncontrast to binary Fe-Pd films the c/a bct  ratio of Fe 70 Pd 27 Cu 3 is found to be slightly smaller, \nsuggesting that the addition of smaller atoms like Cu reduce the volume of the Fe 70 Pd 30  unit \ncell. \nThin film samples with varying c/a bct  ratio enable us to investigate the change of intri nsic \nproperties as function of the tetragonal deformatio n and therefore at different structures. In the \nfollowing chapters we use the variation of the c/a bct  ratio in addition to the composition as a \ncontrol parameter to adjust the magnetic properties . For a detailed understanding of the \nanisotropic magnetic properties, however, it is fir st necessary to know how the unit cell is \noriented on the buffer. \n \nFigure 1: Bragg-Brentano scans of Fe 70 Pd 27 Cu 3 films on different buffer layers. Red circles mark  (002) \nreflections of the Fe-Pd-Cu films and blue pentagon s (002) reflections of the cubic buffers. With incr easing c/a bct  \nratio a shift of (002) Fe-Pd-Cu  reflections to lower angles is observed. Red dotte d lines indicate the boundaries of the \nBain transformation path. \n \nTo determine the structural quality of the thin fil m samples (101) bct  pole figure measurements \nwere performed. Due to the four-fold surface symmet ry of the (001) oriented MgO substrate \nonly one quadrant of the pole figure measurement fo r Fe 70 Pd 27 Cu 3 films on different buffers \nare depicted in Figure 2a. Within this drawing the MgO [100] edges are oriented parallel to  9 the edges of the figure. This reveals an orientatio n relationship of the bct  unit cell of the film \nand MgO(001) substrate as follows:  \nFe 70 Pd 30-xCu x(001)[110]|| fcc -buffer(001)[100]||Cr(001)[110]||MgO(001)[100] or \nFe 70 Pd 30-xCu x(001)[110] ||bcc -buffer(001)[110]||MgO(001)[100]. \nThe corresponding pole figures (Figure 2, illustrat e with [29]) indicate a high quality epitaxial \ngrowth of the films since well-defined and sharp pe aks were observed. The unit cells are \nrotated by 45° with respect to each other - shift i n Φ direction of about 45°- to reduce misfits \nbetween different layer materials. \nThese pole figure measurements can be also used to determine the c/a bct  ratio, since the tilt \nangle Ψ of the (101) bct  plane is connected with the tetragonal distortion by \nbct ac=ψtan (Figure 2b). This relation predicts a tilt angle  of the (101) pole at Ψ ≈ 45° for a \nbcc  structure ( c/a bct = 1) and at Ψ ≈ 54° for a fcc  structure ( c/a bct = 1.41).  \nWhen the buffer material is varied, the Ψ shift of the poles reflects the structural changes \nfrom bcc  towards fcc . For a film on Cr buffer the (101) pole is at a ti lt angle Ψ ≈ 47°. This \nconfirms an almost cubic structure with c/a bct = 1.07. On the other side an Ir buffer layer shifts  \nΨ to 52°, that gives a c/a bct ratio of 1.31 corresponding to a tetragonal fct  structure. The pole-\nfigure measurement therefore is a second independen t measurement confirming the c/a bct \nratios obtained from θ− 2θ scans with an accuracy of 1 %. \n \nFigure 2: a) (101) bct  pole figures of Fe 70 Pd 27 Cu 3 films on different buffer layers. With increasing c/a bct  ratio a \nshift of the (101) pole towards higher Ψ angles is observed. b) Sketch illustrating the geo metric relation between \nthe tetragonal distortion ( c/a bct  ratio) and the tilt angle Ψ within a bct  unit cell. \n \nA further increased tetragonal distortion is visibl e for a film on a Cu buffer ( c/a bct  = 1.57). In \nBragg-Brentano scans (Figure 1) the (002) reflectio n of the Fe 70 Pd 27 Cu 3 film is observed \nbeyond the limits of the Bain path. This tetragonal  distortion agrees well with c/a bct  = 1.57 \nderived from the position of the pole in the (101) bct  pole figure at Ψ ≈ 57°. The intensity in  10 this measurement further exhibits a substantial bro adening and splitting in Φ direction. This \nobservation agrees well with the authors’ previous report on highly-strained binary Fe 70 Pd 30  \nfilms on Cu buffer layers [13]. The broadening can be ascribed to adaptive nanotwinning that \nreduces the huge elastic energy induced by the cohe rent epitaxial strain through the formation \nof twin boundaries. The rotation of the tetragonal variants which form (101) fct twin boundaries \nleads to the small observed additional intensities.  \nThe same set of XRD measurements was performed for a series of films with increased Cu \ncontent. These measurements are not shown here, sin ce they are practically identical to the \nones obtained for the lower Cu content. From these measurements we conclude that even at a \nCu content of 7 at.% no macro-precipitations occur in epitaxial films. This is different to \npolycrystalline annealed Fe-Pd-Cu thin films which decompose at Cu contents above 4 at.% \n[9]. We attribute this to the difference in fabrica tion process routes. Epitaxial films are \ndeposited at room temperature without any post anne aling which avoids macro-precipitates \ncompared to annealed films.  \nThe lattice parameters determined for Fe 70 Pd 23 Cu 7 are equal to those of Fe 70 Pd 27 Cu 3. No \nchange in tetragonal deformation with increased Cu content is observed. Accordingly we do \nnot distinguish between 3 and 7 at.% Cu in Figure 3 . Experiments on polycrystalline annealed \nfilms have shown that with increasing Cu content th e lattice constant decreases [9], since the \natomic radius for Cu (0.128 nm) is smaller than for  Pd (0.137 nm). However, it is known that \nalloys, like Fe-Pd, exhibiting the Invar effect [30 ,31,32], deviate from the rule of mixture of \nVegard’s law [33,34]. This must be considered for t he Fe-Pd and the second quasi binary Cu-\nPd systems [35]. \n \nFigure 3: a) In-plane lattice parameters abct  for Fe 70 Pd 30 (rectangle) and Fe 70 Pd 30-xCu x (triangle) are fixed by the \nsubstrate’s lattice spacing d of different buffers (as marked on the top). b) Th e change of tetragonal deformation \n(c/a bct ) dependent on buffer material is smaller for the F e 70 Pd 30-xCu x system (triangle) than for the binary Fe 70 Pd 30 \n(rectangle [11,13]). The curves illustrate the expe cted change in deformation at constant volume of th e unit cell. \n  11 The improved growth behaviour can be motivated in t erms of the structural variation of the \ntotal energy. We obtain this energy from first-prin ciples within a supercell description \ninvolving a full relaxation of the atomic position which minimises the interatomic forces for \neach tetragonal stage. According to Figure 4a, only  a small variation of elastic energy E is \ninvolved if we vary the c/a bct  ratio along the entire Bain transformation path. I n addition to the \nglobal minimum at bcc  we observed a second local minimum beyond the Bain  path \n(c/a bct  > 1.41). Such a local minimum can be attributed to  the formation of a nanotwinned \nmicrostructure in the simulation cell [13] and is f urther confirmed by our pole figure \nmeasurements depicted previously in this chapter (F igure 2). This minimum is observed for \nthe binary Fe 68 Pd 32  solid solution, but also for ternary Fe 68 Pd 27 Cu 5, where the feature is \ncomparatively shallow. Replacing 16% of Pd atoms by  Cu increases the overall valence \nelectron number e/a  by 0.05, which by trend stabilises the fcc  austenite [36,31,32]. This is \nreflected in our calculations by the noticeably dec reased energy difference between the bcc  \nground state at c/a bct  = 1 and the fcc  austenite at c/a bct  = 1.41. Interestingly, the energy profile \ndoes not change significantly at low energies for c/a bct  < 1.3. Changes appear mainly in the \nvicinity of the local fcc  maximum, which is now embedded in flatter energy l andscape. A flat \nprofile in the vicinity of the martensitic phase, h owever, can be considered beneficial for the \nformation of the metastable fct  phase, which is required for the MSM behaviour. Th e decrease \nof the fcc -bcc  energy difference implies a decreased temperature for the onset of the \nmartensitic transformation, but this can be compens ated by the simultaneous replacement of \nPd by Fe, as shown in a previous study [9]. The sim ultaneous substitution of Pd by Cu and Fe \ntherefore opens a way to specifically design the pr ofile of the binding surface around \n2=bct ac and thus the stability range of the fct  phase. \nThe total difference in elastic energy between bcc  and fcc  state ∆E is reduced from \n∆E = 15 meV/atom to ∆E = 12 meV/atom for Fe 68 Pd 32 and Fe 68 Pd 27 Cu 5, respectively. This \nreduces the driving force for the transformation an d all associated relaxation processes and is \nthus favourable for strained epitaxial growth. Inde ed, to avoid unfavourable relaxation \nmechanism like (111) fcc  deformation twinning [37], very low deposition rat es of 0.024 nm s-1 \nwere required for epitaxial growth of binary Fe 70 Pd 30  films [12]. The present Fe-Pd-Cu films, \nhowever, were grown with a deposition rate one orde r of magnitude higher (0.3 nm s-1). Still \nthe poles in the (101) pole figure and the diffract ion peaks in the Bragg-Brentano \nmeasurement reveal a small full-width-at-half-maxim um (FWHM). By using a modified \nScherrer equation [38] the determined coherence len gth is in the range of the overall thickness  12 of the Fe-Pd-Cu films, confirming a high quality ep itaxial growth. This allows producing \nthick epitaxial Fe 68 Pd 30-xCu x films, as required for microsystems, in a shorter time.  \n \nFigure 4: a) Variation of the total energy per atom  as a function of the tetragonal distortion c/a bct  obtained from \nab-initio  calculations of a Fe 68 Pd 32  (black squares) and Fe 68 Pd 27 Cu 5 supercell (red triangles). The calculations \nwere carried out at constant volume of 13.1 Å3/atom and include a full geometric optimisation of the atomic \npositions. Apart from the bcc  ground state minimum at c/a bct = 1 (which defines the energy reference for each \ncomposition), in both cases a second local minimum around c/a bct  =1.5 (beyond fcc ) is obtained which \ncorresponds to the appearance of a finely twinned a daptive superstructure in the 500 atom simulation c ell. b) The \nvariation of the average ground state magnetic mome nt (normalised per atom) for both configurations as  a \nfunction of c/a bct . \n \n4. Remanence, coercivity and saturation field \n \nThe hard magnetic axis in a tetragonal lattice of a  bulk Fe 70 Pd 30  single crystal is aligned along \nthe c-axis while the magnetic easy directions lie within  the basal plane [7,39]. Within the easy \nplane, not all directions are equivalent and a slig ht anisotropy is observed, which favours both \n[110] bct  easy axes. \nTo investigate structure-dependent magnetic propert ies of epitaxial Fe 70 Pd 27 Cu 3 films, in-\nplane hysteresis curves for thin film samples with different tetragonal deformation were \nmeasured at room temperature (Figure 5a-e). The mag netisation behaviour along the different \ncrystallographic in-plane directions - [100] bct  and [110] bct  – differ significantly as known also \nfor binary Fe 70 Pd 30  films. The differences are most evident for c/a bct  = 1.31, which is close to \nthe middle of the Bain transformation path. Here, b oth crystallographic directions show \ndifferent values for remanent polarisations JR and saturation fields HS, while the coercivity \nfields HC do not vary significantly.   13 \n \nFigure 5: Probing magnetocrystalline anisotropy wit hin the basal plane by measuring hysteresis curves for \ndifferent tetragonal deformation ratios measured at  300 K (a - e). Solid lines show the measurements a long \n[110] bct  direction and broken lines along [100] bct  direction. f) Comparison of hysteresis loops measu red out-of-\nplane along [001] bct  at 300 K. The inset shows a zoom around µ0H = 0 T. From these measurements, the \nmagnetic constants JR, JS and HC were determined. \n \nIn dependence of the c/a bct  ratio the shape of magnetisation curves and thus t he magnetic \ncharacteristics change. At low tetragonal deformati on ( c/a bct  = 1.07) the hysteresis measured \nin [100] bct direction shows a step like switching behaviour. In  the [110] bct direction a similar \nswitching process occurs, but additionally a nearly  linear increased magnetisation at higher \nfields is observed. This behaviour originates from magnetisation rotation, indicating that the \n[110] bct direction is the harder axis within the basal plane . Accordingly, JR is reduced in this \ndirection. When increasing the c/a bct  ratio to 1.31 and above, the opposite behaviour is  \nobserved. This can be correlated to a change in sig n of the magnetic anisotropy within the \nbasal plane. The extracted values for the coercivit y field HC and saturation field HS along \n[110] bct  direction are summarised in Figure 6a and b.   14 \n \nFigure 6: c/a bct  dependency of in-plane coercivity field µ0HC ║[100] bct , (a) and saturation field µ0HS ║[100] bct  (b) \nas well as the out-of-plane saturation field µ0HS ║[001] bct  (c) extracted from the hysteresis curves measureme nts \nin Figure 5. Black rectangles represent the results  for Fe 70 Pd 30  films, red circles Fe 70 Pd 27 Cu 3 films and blue \ntriangles Fe 70 Pd 23 Cu 7 films. All lines are guides for the eye. \n \nThe trend of µ0HC and µ0HS within the film plane for different tetragonal dis tortions is similar \nto epitaxial Fe 70 Pd 30 films (black symbols in Figure 6) and previous work  on epitaxial \nFe 1-xPd x films deposited on MgO(100) substrates [40,41]. Du e to high crystal symmetry, films \nclose to the bcc  structure exhibit a low HC and HS while films with a bct  structure have a \nslightly increased HC and slightly reduced HS. Fct  films exhibit a different loop shape with \nhighest HC and HS due to the large lattice deformation. Close to the fcc  phase HC and HS are \nreduced because of the high crystal symmetry of the  fcc  structure. Fe 70 Pd 27 Cu 3 films also \nfollow this behaviour. Both values HC and HS approach zero when the tetragonal distortions is \nclose to the cubic structures bcc  ( c/a bct = 1) and fcc  ( c/a bct = 1.41). For the tetragonal structures \nboth values show a maximum.  \nFor a huge tetragonal deformation of Fe 70 Pd 30  and  Fe 70 Pd 30-xCu x beyond the Bain path \n(c/a bct  = 1.57, Figure 5e), HC decreases again and only a slight differences betw een both in-\nplane curves are observed. This behaviour will be d iscussed together with the magnetic \nanisotropy constants K1 and K3 in chapter 6. \nTo obtain the magnetic anisotropy constant K1, out of plane hysteresis curves along [001] bct  \ndirection were measured (Figure 5f). Along in-plane  directions, all films are fully saturated at \na magnetic field of 0.1 T, while for out-of-plane m easurements substantially higher fields \nabove 1 T are required. Shape anisotropy dominates the magnetisation behaviour for an ideal \nthin film (demagnetisation factor N = 1). When neglecting magnetocrystalline anisotrop y the  15 magnetic saturation is expected at µ 0HS = JS. The difference between µ 0HS and JS will be later \nused to calculate the magnetic anisotropy constant K1. When increasing the tetragonal \ndeformation to c/a bct  ≤ 1.41, HS decreases (Figure 6a).  \nFor all measurements a small hysteresis is observed  (inset in Figure 5f). We attribute this \nswitching process to a slightly angular misalignmen t of the sample normal with respect to the \nfield. For a pinning controlled coercivity mechanis m the coercivity exhibits a Kondorsky like \nstrong increase when the field direction approaches  hard direction [42]. At the same time the \nremanence to saturation ratio decreases. As a detai led analysis requires a more accurate \ncontrol of the tilt angle, we do not give HC from the out-of-plane direction here. \nWe extracted values for HC and HS along [110] bct  and [001] bct  direction also from the in-plane \nand out-of-plane hysteresis curves for Fe 70 Pd 23 Cu 7 films, which follow the same trend as \nfilms with 3 at.% Cu (see blue triangles in Figure 6). \n \n5. Change of Curie temperature and spontaneous pola risation \n \nPrevious experiments have shown that a tetragonal d istortion of the lattice also significantly \naffects the Curie temperature TC [11]. Temperature dependent magnetisation curves w ere \nmeasured in-plane along the [110] bct direction. In this direction an applied field of 1 T is \nsufficient to saturate the sample. Unfortunately, t he magnetisation curves can only measured \nup to 400 K, which is below TC, to avoid the destructive thermal decomposition of  the \nmetastable alloy [43]. Therefore, we were forced to  determine TC of our films by an \nextrapolation of the magnetisation curves using Kuz ’min’s empirical fit [15,16] as explained \nin detail in Section 2. The results are shown in Fi gure 7a.  \nBlack rectangles represent the values for Fe 70 Pd 30 , red circles Fe 70 Pd 27 Cu 3 and blue triangles \nFe 70 Pd 23 Cu 7. The Fe 70 Pd 30  as well as the Fe 70 Pd 30-xCu x systems show a similar behaviour. \nIncreasing c/a bct  ratio up to 1.41 TC decreases monotonously. The value of TC = 652 K at \nc/a bct  = 1.39 approaches the literature value of 600 K re ported for the bulk fcc  phase \n(c/a bct  = 1.41) [8,44]. The change of TC by addition of Cu is within the experimental error  \nrange. Thus, compared to other MSM systems like Ni- Mn-Ga ( TC < 370 K), TC is still much \nhigher [45]. \nAn independent access to the compositional and stru ctural trends concerning TC and ground \nstate JS is provided by our ab-initio  calculations which corroborate the above experimen ts. TC \nis obtained from a classical Heisenberg model which  is parameterised with first principles \nmagnetic exchange constants. In the present case, t he mean-field approximation can already  16 give a reasonable estimate of the structural and co mpositional trends governing the magnetic \ntransformation. Nevertheless, one must be aware tha t this simple approach systematically \noverestimates critical temperatures (typically by 2 0...30%) due to the neglect of spin-\nfluctuations. Furthermore, also the induced nature of the Pd moments, which vary in \nmagnitude according to the field of the surrounding  localised Fe moments, is not taken into \naccount, either [46]. But a comparison with a numer ically exact Monte-Carlo treatment of the \nHeisenberg model [47] demonstrates that for a heuri stic prediction these shortcomings can be \ncompensated sufficiently by correcting TC systematically with an empirical factor of 0.75.  \nThe almost quantitative agreement which we obtain a long the Bain path between bcc  and fcc  \n(Figure 7b) nicely corroborates the experimental pr ocedure, whereas we find no significant \nvariation of TC with the Cu-content.  \nFor c/a bct  > 1.41 - beyond the Bain path - the experimental TC increases again notably. A TC of \n822  K at c/a bct  = 1.56 is similar to values for a structure with c/a bct  = 1.11. In contrast, our \ncalculations assuming a single variant with c/a bct  = c/a box  > 1.41 according to the substrate \nconstraint exhibit only a small increase in TC (solid line in Figure 7b). The agreement is much \nbetter, if we assume an energetically more favourab le nanotwinned fct  configuration, where \nthe individual twins are rather described by config urations with c/a bct  < 1.41. In a recent work, \nwe derived a simple approximate relation between th e epitaxial constraint from the substrate \nc/a box  resulting and the tetragonality of the twins c/a twin  [13]: \n 2\n21−\n\n\n+ =\ntwin box ac\nac. (2) \nUsing this relation to extrapolate our values for c/a bct  > 1.41 beyond fcc , we obtain a \nreasonable agreement with experiment concerning the  Cu buffer (dashed line in Figure 7b).   17 \n \nFigure 7: Experimental (a) and calculated (b) Curie  temperature TC in dependence of c/a bct  and composition. \nWith increasing c/a bct  < 1.41 TC decreases. For c/a bct  = 1.57 similar values as at c/a bct  = 1.11 are observed. The \ntheoretical Curie temperatures are obtained using t he mean-field approximation to the Heisenberg model  and \nconsistently corrected by a factor 0.75. The dashed  lines in b) represent results for the adaptive nan otwinning \nconcept. These data agree well with the experimenta l values. \n \nAnother crucial intrinsic parameter for the MSM eff ect is a high spontaneous magnetic \npolarisation  JS. Figure 8a shows JS for the different Cu contents in dependence of the  \ntetragonal deformation. We extracted these values f rom the deformation dependent  changes in \nthe hysteresis curves in Figure 5f. Similar to the structural variation of TC, we observe a \nminimum of JS = 1.19 T at fcc (c/a bct  = 1.41). Close to bcc  ( c/a bct  = 1.09) and at huge strains, \nJS reaches values of 1.76 T. \nBoth, Fe 70 Pd 27 Cu 3 and Fe 70 Pd 23 Cu 7 films, reach just 80 % of the JS of Fe 70 Pd 30 , which is in \nagreement with previous experiments on polycrystall ine samples [9]. Nevertheless, the values \nfor JS are still much higher than 0.76 T obtained for the  MSM prototype system Ni-Mn-Ga \n[48]. \nWhile the main trends are identical - a decrease of  JS with increasing Cu content and a \nminimum at fcc  - the experimental values obtained at ambient cond itions (300 K) close to TC \nexhibit a significantly stronger variation with bot h, composition and c/a bct  ratio, than the \nresults from the ab-initio  calculations at T = 0 K (Figure 8b). Here, the ground state magnetic  \nmoments vary by a few percent region while the chan ges in experiment are almost one order \nof magnitude larger. A larger reduction at fcc  is expected at ambient conditions, since fcc  \nexhibit the lowest TC. It may also be speculated whether longitudinal sp in fluctuations, which \ncan occur in Invar materials as fcc  Fe 70 Pd 30  [31,32], reduces TC in Austenite further.   18 \n \nFigure 8: a) Spontaneous polarisation JS determined for different c/a bct  ratios and compositions at 300 K. JS for \nFe 70 Pd 30-xCu x is approximately 80 %  of the value for Fe 70 Pd 30 . All lines are guides for the eye. b) Calculated \nground state total (spin + orbital) magnetic moment s in dependence of c/a bct  and composition at 0 K. The dashed \nlines represent results for the adaptive nanotwinni ng concept. \n \n6. Change of magnetocrystalline anisotropy energy \n \nThe key intrinsic magnetic property of MSM alloys i s their magnetocrystalline anisotropy \nenergy ( MAE ). During the MSM effect the MAE  represents the maximum energy input \npossible by a applying an external magnetic field. Hence the MAE  limits the energy available \nto move twin boundaries or conduct external work. MAE  of a tetragonal lattice can be \ndescribed by the following equation [49]:  \n ß K K K MAE 4cos sin sin sin 4\n34\n22\n1 α α α + + = . (3) \nKi are the anisotropy constants, α is the angle between magnetisation direction and c-axis \nand β is the angle to the a-axes within the basal plane of a tetragonal lattic e. In most \ncompounds without rare-earth elements higher order terms ( K2) can be neglected. For the \nFe 70 Pd 30  system this was shown by Cui et  al.  [7]. Hence it is sufficient to consider here only \nK1 which describes the work required to magnetise the  sample along the hard magnetisation c-\naxis and K3 which describes the four-fold anisotropy within th e basal plane.  \nIn this paper the c/a bct  ratio was controlled by strained epitaxial film gr owth. This allows to \ndetermine Ki for all distortions at one temperature (here: room  temperature), which is not \npossible in bulk. Due to the martensitic transforma tion and additional strong temperature \ndependency of the c/a  ratio, it is not possible to separate between the influence of T and c/a  in \nbulk. From hysteresis measurements (Figure 5) we kn ow, that in our epitaxial films the  19 [001] bct direction is the hard magnetisation axis and [100] bct  and [110] bct  directions form the \neasy plane.  \nFrom the present thin film experiments K1 can be extracted from the measurements of HS \nalong the hard [001] bct  direction. The shape anisotropy was considered by using a \ndemagnetisation factor N = 1 of an ideal infinite film and the anisotropy f ield HA can be \ncalculated by S S A NJ H H − =0 0 µ µ . The anisotropy field is then converted in 21S AJ HK⋅−= . \nIn Figure 9a K1 is plotted as a function of the c/a bct  ratio and Cu content. The tetragonal \ndeformation of a Fe-Pd unit cell results in the for mation of an easy plane, i.e., K1 < 0. The \nmaximum effect is observed around c/a bct  = 1.33 corresponding to the fct  structure, which \nexhibits the MSM effect in bulk. For binary fct  Fe 70 Pd 30  we find K1 = -1.6*10 5 Jm -3. This \nagrees well with literature values reported for fct  single crystals (open rectangles in Figure 9a) \nand DFT calculations (stars in Figure 9a) [50,40]. In contrast, Fe 70 Pd 30-xCu x films exhibit an \nincreased absolute value of magnetocrystalline anis otropy of K1 ~ -2.4*10 5 Jm -3.  \nThe absolute values for K1 of Fe 70 Pd 27 Cu 3 and Fe 70 Pd 23 Cu 7 even exceed the \nmagnetocrystalline anisotropy constants reported fo r the Ni-Mn-Ga system at room \ntemperature: K1 = 1.65*10 5 Jm -3 for a 10M single variant single crystal and K1 = 1.7*10 5 Jm -3 \nfor the 14M structure [45]. A larger MAE of similar  magnitude is only observed for \nNi-Mn-Ga at significantly lower temperatures [48, 5 1] according to the considerable variation \nof the MAE with temperature in uniaxial magnets [52 ]. \nWhen changing the tetragonal deformation close to h ighly symmetric cubic structures ( bcc  \nand fcc ,) K1 is reduced for all compositions. For c/a bct  larger than 1.41, magnetocrystalline \nanisotropy increases again, but does not reach the values of fct . This is again consistent with \nthe presence of adaptive nanotwins in the film [13] . For a very small variant width it is not \nanymore possible to form a complete 90° domain wall  at twin boundaries since exchange \nenergy favours a parallel alignment of magnetisatio n. According to the random anisotropy \nmodel of Herzer the critical length for this is the  magnetic exchange length lexch  [53].  For fct  \nFe-Pd lexch  is in the order of 18-85 nm (depending on composit ions) [40], which far exceeds \nthe width of adaptive nanotwins [13].  \nDue to the fourfold symmetry of the basal plane a d eviation from an idealised easy plane \nbehaviour can be observed in films (Figure 5) and b ulk samples [7,39]. K3 is the measure of \nthe anisotropy within this basal plane and defines the work which is necessary to magnetise \nalong the respective directions of the bct  unit cell: [] []bct bct W W K 110 100 23 − = . It can be  20 extracted from magnetisation curves (Figure  5) from the area enclosed by the hysteresis \ncurves measured along both, [110] bct  and [100] bct  directions. \nThe values of K3 are two orders of magnitude smaller than for K1 (Figure 9b) and change sign \nwithin the Bain path. For c/a bct  values close to bcc  positive values are observed. Films with \nc/a bct  ratios close to fcc  have negative K3 values. In between these values no significant \ndifferences were observed for Fe 70 Pd 30  and Fe 70 Pd 30-xCu x films. As for 1K,3K has a \nmaximum with  1.8*10 3 Jm -3 at the fct  structure ( c/a bct  = 1.33).  \n \n \nFigure 9: Magnetocrystalline anisotropy constants K1 (a) and K3 (b) as a function of c/a bct  for various Cu contents \n(solid symbols, T = 300 K). Also shown are the values for a bulk Fe 68.8 Pd 31.2 single crystal (open rectangle [50]) \nand calculations for disordered Fe 70 Pd 30  (stars, T = 0 K [40]). The errors for K1 are shown by error bars and for \nK3 they are in the range of the symbol size. All line s are guides for the eye. \n \n7. Conclusions \n \nOur detailed analysis of structure and magnetism in  ternary Fe-Pd-Cu epitaxial thin films \nsuggests that addition of small amounts of Cu is su itable to significantly enhance the \nfunctional properties of the Fe 70 Pd 30  magnetic shape memory alloy. We selected Cu since \nrecent experiments on polycrystalline films and spl ats show that it can increase the maximum \nsolvable Fe-content and thus the martensitic transf ormation temperature [9]. Our combined \nexperimental and computational approach yields comp lementary insight into the frozen stages \nof the martensitic transformation process within th e limits of the Bain path, which is enforced \nby the epitaxial relation to carefully selected buf fer materials, This is in particular favourable \nfor a detailed analysis of the anisotropic magnetic  properties since the measurements are  21 neither affected by a magnetically induced reorient ation nor the continuous variation of the \ntetragonal distortion with temperature which are bo th occur in bulk samples. \nOne central result is that Cu flattens the energy l andscape, which suppresses common \nrelaxation mechanisms and thus allows for a much be tter film quality in combination with \nfaster growth. More important, we observe an increa se of the magnetocrystalline anisotropy \nconstant K1 by 40 %, which is a substantial improvement of the  key intrinsic property for the \nMSM effect. The values obtained for the fct  structure even exceed those reported for the \nprototype Ni-Mn-Ga system, which makes the Fe-Pd-Cu  system of particular interest for \nmicrosystems with a high energy density. As a minor  drawback, Cu alloying reduces the room \ntemperature spontaneous polarisation by about 15-20  %.  \nThe XRD measurements indicate that the use of room temperature deposition techniques \nprevents macroscopic demixing, which must be expect ed for more than 5 at.% Cu, otherwise \n[9]. However, upon increasing the Cu content from 3  at.% to 7 at.% we do not observe a  \nvariation of the magnetic properties. This might in  turn be taken as an indication for the \npresence of structural or compositional inhomogenei ties at the nanoscale (e.g., a tendency \ntowards a slight L1 0 short range order with antiphase boundaries). Rece nt Mößbauer \nexperiments on Fe-Pd-Cu splats with < 1 at % Cu sug gest some chemical short-range order \n[54].  Moreover, first principles calculations of Fe-rich Fe-Pd on the other hand predict a \ncertain preference for forming a layered type of or der (L1 0 or Z1) [55,56], while the cubic \nFe 3Pt-type L1 2 order appears to be ruled out for energetic reason s [21]. In the martensitic \nstate, aging behaviour resulting to symmetry-confor ming short-range order at this length scale \ndoes not necessarily inhibit shape memory applicati ons [57] and could tentatively induce a \nkind of beneficial two-way behaviour. This aspect, however, cannot be probed at the films \ntested with our methods available and requires furt her investigation. \n \nAcknowledgments \n \nThe authors thank P. Entel, J. Buschbeck, A. Backen , A. Diestel and R. Niemann for \ndiscussions and experimental support and the DFG fo r funding via the Priority Program SPP \n1239. The authors would like to thank the John von Neumann Institute for Computing (NIC), \nJülich Supercomputing Center (JSC), both of Forschu ngszentrum Jülich, as well as the Center \nfor Computational Sciences and Simulation (CCSS) of  the University of Duisburg-Essen for \ncomputing time and support. \n  22 References \n                                                 \n[1]  Sozinov A, Likhachev A A, Lanska N and Ullako K 2002 App. Phys. Lett.  80  1746 \n[2]  Kohl M, Yeduru S R, Khelfaoui F, Krevet B, Bac ken A, Fähler S, Eichhorn T, Jakob \nG and Mecklenburg A 2010 Mat. Sci. 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B  80 , 220201(R) \n[56]  Chepulskii R V, Barabash S V and Zunger A, 20 12 Phys. Rev. B  85  144201  \n[57]  Ren X and Otsuka K 1997 Nature  389  579 "    },    {        "title": "1206.6200v1.Effect_of_Pt_impurities_on_the_magnetocrystalline_anisotropy_of_hcp_Co__a_first_principles_study.pdf",        "content": "E\u000bect of Pt impurities on the magnetocrystalline\nanisotropy of hcp Co: a \frst-principles study\nC.J. Aas1, K. Palot\u0013 as2, L. Szunyogh2;3, R.W. Chantrell1\n1Department of Physics, University of York, York YO10 5DD, United Kingdom\n2Department of Theoretical Physics, Budapest University of Technology and\nEconomics, Budafoki \u0013 ut 8. H1111 Budapest, Hungary\n3Condensed Matter Research Group of Hungarian Academy of Sciences, Budapest\nUniversity of Technology and Economics, Budafoki \u0013 ut 8., H-1111 Budapest, Hungary\nAbstract. In terms of the fully relativistic screened Korringa-Kohn-Rostoker method\nwe investigate the variation in the magnetocrystalline anisotropy energy (MAE) of\nhexagonal close-packed cobalt with the addition of platinum impurities. In particular,\nwe perform calculations on a bulk cobalt system in which one of the atomic layers\ncontains a fractional, substitutional platinum impurity. Our calculations show that at\nsmall concentrations of platinum the MAE is reduced, while at larger concentrations\nthe MAE is enhanced. This change of the MAE can be attributed to an interplay\nbetween on-site Pt MAE contributions and induced MAE contributions on the Co\nsites. The latter ones are subject to pronounced, long-ranged Friedel-oscillations that\ncan lead to signi\fcant size e\u000bects in the experimental determination of the MAE of\nnano-sized samples.arXiv:1206.6200v1  [cond-mat.mtrl-sci]  27 Jun 2012E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 2\n1. Introduction\nCobalt alloys, such as CoPt or CoPd, are ubiquitous in the \feld of magnetic record-\ning and of particular interest to the \feld of ultrafast magneto-optics [1]. In terms of\nmagnetic recording, increasing areal densities require decreased grain size, which in\nturn requires increasing values of magnetocrystalline anisotropy energy (MAE) to en-\nsure thermal stability of written information [2]. Currently this is achieved using CoPt\nalloys with perpendicular anisotropy. Consequently an understanding of the origin of\nthe MAE in CoPt is an important practical problem. Since the magnetic properties of\nthese alloys are highly sensitive to the amount and the spatial distribution of the Pt\ncontent, understanding the e\u000bects of alloying is an important issue. The e\u000bects on the\nmagnetic properties of CoPt as functions of the platinum content have been studied\nextensively, both theoretically [3] and experimentally [4, 5]. Moreover, in recent ex-\nperimental work [6] it was demonstrated that the magnetocrystalline anisotropy energy\n(MAE) of cobalt can be tuned by letting platinum impurities migrate into the cobalt\nsystem. Generally it is agreed that the addition of platinum to a magnetic material,\nsuch as Fe or Co, in\ruences the magnetic properties, in particular, the MAE of the\nmaterial primarily through the strong spin-orbit coupling of Pt [7].\nThe aim of the present work is to elucidate from \frst principles the e\u000bect on the MAE\nof bulk hcp Co by the addition of platinum. To this end, we use the fully relativistic\nscreened Korringa-Kohn-Rostoker (SKKR) method as combined with the coherent-\npotential approximation (CPA), which is well suited to describing substitutional alloys\n[8]. Our model focuses on Pt alloying in a (0001) atomic plane of a hcp Co bulk\nsystem, from the case of an impurity to the case of a complete \flling of the layer by\nPt. After brie\ry discussing the computational methods we present the calculated MAE\nas a function of the Pt concentration and analyze the results in terms of layer- and\nspecies-resolved contributions to the MAE. We note that recording media are complex\nalloy systems, often containing Cr to promote grain boundary separation. It is often\nfound that the maximum MAE as a function of Pt concentration is limited by, for\nexample, the formation of new phases [9] or the presence of stacking faults [10, 11, 12].\nHere we are concerned only with the intrinsic enhancement of the MAE introduced by\nthe Pt impurities. Remarkably, this analysis highlights the role of long-ranged Friedel\noscillations in forming the MAE of the system. Speci\fcally, we demonstrate a layer\ndependence of the valence charge which makes the e\u000bect of the Pt impurities long-\nranged. This might have signi\fcant impact on the determination of the MAE of thin\n\flm samples corresponding to the systems studied in this work. In particular it might be\nexpected to give rise to \fnite size e\u000bects in the MAE of granular thin \flms for magnetic\nrecording which would become more signi\fcant as the grain size is reduced.E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 3\n2. Computational details\nThe central feature of the SKKR method is the evaluation of the electronic Green's\nfunction of a layered system. Here, a layered system refers to a system exhibiting two-\ndimensional translational symmetry in each (in\fnite) atomic plane, but in which there\nare no symmetry requirements along the third axis. From the Green's function one\ncan then determine a number of physical quantities of interest, such as site-projected\ncharges, spin- and orbital moments and the total energy of the system. As the method is\nwell documented elsewhere [13, 14], here we present only some details of our calculations.\nThe calculations were performed within the local spin-density approximation (LSDA)\nof density-functional theory (DFT) as parametrised by Vosko et al. [15] The e\u000bective\npotentials and \felds were treated in the framework of the atomic sphere approximation\n(ASA). The substitutional Pt alloying was treated within the coherent potential ap-\nproximation (CPA) [8, 16]. As the LSDA fails in predicting the orbital moment and the\nMAE for hcp Co correctly, we employed a heuristic extension of the relativistic electron\ntheory by the orbital polarisation (OP) correction [17, 18, 19], as implemented within\nthe KKR method by Ebert and Battocletti [20]. The corresponding Kohn-Sham-Dirac\nequations were solved using a spherical wave expansion up to an angular momentum\nnumber of`= 3, although it should be noted that the OP correction was applied only\nfor the`= 2 orbitals.\nThe magnetocrystalline anisotropy energy was evaluated within the magnetic force\ntheorem [21], in which the total energy of the system can be replaced by the single-\nparticle (band) energy. Moreover, we employed the torque method [22], making use of\nthe fact that, for a uniaxial system, the MAE, K, can be calculated up to second order\nin spin-orbit coupling as\nK=E(\u0012= 90\u000e)\u0000E(\u0012= 0\u000e) =dE\nd\u0012\f\f\f\f\n\u0012=45\u000e; (1)\nwhere, in the case of hcp geometry, \u0012denotes the angle of the spin-polarisation with\nrespect to the (0001) direction, i.e., the direction perpendicular to the hexagonal planes.\nNote that the ^ z-axis of the (global) frame of reference in our calculations is de\fned to\nbe parallel to the (0001) direction. Within the KKR formalism, Kcan be decomposed\ninto site- and species-resolved contributions,\nK=X\ni;\u000bs\u000b\niD(\u000b)\ni; (2)\nwheres\u000b\nidenotes the concentration of species \u000bat siteiandD(\u000b)\nidenotes the\ncorresponding derivative of the band energy. Using Lloyd's formula [23], D(\u000b)\nican be\ncalculated as [24]\nD(\u000b)\ni=\u00001\n\u0019ImZ\u000f(^n)\nF\nd\u000fTr \n@t(\u000b;^ n)\ni(\u000f)\u00001\n@\u0012\u001c(\u000b;^ n)\nii(\u000f)!\n; (3)E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 4\nwhere\u000f(^n)\nFis the Fermi energy and, in case species \u000boccupies site i,t(\u000b;^ n)\ni(\u000f) and\u001c(\u000b;^ n)\nii(\u000f)\nstand for the angular momentum matrices of the single-site toperator and the site-\ndiagonal scattering path operator, respectively. All these quantities are calculated at\nthe direction of the magnetisation ^ n=\u0010\n1p\n2;0;1p\n2\u0011\n, corresponding to \u0012= 45\u000ein Eq. (1).\nThe derivative of the t-matrix is evaluated as described in [25]. The energy integral in\nEq. (3) can be accurately performed by sampling 20 energy points on an asymmetric\nmesh along a semi-circle contour in the upper complex semi-plane. In order to achieve\nan accuracy within 5 % for the MAE, a su\u000eciently dense mesh in the two-dimensional\nBrillouin zone (2D-BZ) was used to evaluate \u001c(\u000b;^ n)\nii(\u000f): at the energy point closest to\nthe Fermi energy, we used 5764 k-points in the irreducible wedge of the 2D-BZ, corre-\nsponding to more than 34 000 k-points in the full 2D-BZ. Due to the two-dimensional\ntranslational symmetry of the system, the MAE should be related to a 2D unit cell,\ntherefore, in the following the index iin Eq. (2) is used to label atomic layers.\nThe SKKR method as applied to layered systems requires the system to be divided\ninto a middle region wedged between two semi-in\fnite bulk regions. Adhering to this\nrequirement, the e\u000bect of platinum alloying in a single atomic layer of bulk hcp Co was\ninvestigated by considering a layered system as shown in Fig. 1. Each atomic layer\nin the semi-in\fnite bulk regions corresponds to pure hcp Co bulk. Since the middle\nregion needs to contain an integer number of unit cells and since each unit cell spans\ntwo atomic layers, this region consists of 2 NLhexagonal Co layers stacked along the\n(0001) direction. In one of the two central layers of the middle region, namely, in the\none indexed by 0 in Fig. 1, a fraction sof the Co atoms are replaced by Pt atoms. From\nhere on, this layer will be referred to as the impurity layer . It should also be mentioned\nthat in this work no attempts are made to trace any structural relaxation e\u000bects of the\nhcp Co lattice caused by Pt impurities.\nTo take into account relaxation of the e\u000bective potentials and \felds, we performed\nself-consistent calculations with NL= 14, i.e. for 28 layers in total. One important\nconsequence of the geometrical construction shown in Fig. 1 is that the calculation of K\nin Eq. (2) is con\fned to layers within the middle region, i.e., for \u0000(NL\u00001)\u0014i\u0014NL.\nThis means that the long-ranged Friedel oscillations that arise due to the presence of\nPt impurities are necessarily truncated. In order to safeguard against any numerical\nartefacts caused by this truncation, we increased the number of atomic layers in the\nmiddle region until the layer-resolved MAE converged to within about 1 % accuracy to\nthe bulk Co MAE at the outer edges of the middle region. According to our calculations\n(see below), this condition requires NL= 40, i.e., 80 atomic layers in total. We\nperformed these calculations of the MAE by appending the perfect bulk potential of\nhcp Co to the layers \u000039\u0014i\u0014\u000014 and 15\u0014i\u001440, i.e., neglecting self-consistency\ne\u000bects for these atomic layers. To check the accuracy of this approach, we compared\nD(Co)\nifor atomic layer no. 14 (with relaxed self-consistent potential) with that for atomic\nlayer no.\u000014 (with appended Co bulk potential) and obtained that the two values agreeE\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 5\nto within 0:02 %.\n0123N -2N -1N\n-1\n-2\n-3\n-(N -3)\n-(N -2)\n-(N -1)semi-infinite \nbulk cobalt\nsemi-infinite \nbulk cobaltz\nx y\n...... ...\n...\nCo\nCo   Pt1-s sLLL\nL\nL\nL\nFigure 1. Sketch of the geometry of the system containing NLhcp unit\ncells, i.e., 2 NLatomic layers wedged between two perfect semi-in\fnite bulk Co\nsystems. In the zero-indexed layer (black circles) a random substitutional alloy\nwith platinum, Co 1\u0000sPts, is considered. Note that the ^ z-axis is de\fned to be\nparallel to the (0001) direction of the hcp crystal.\n3. Results and Discussion\nTo test our computational method, we \frst determined the MAE of bulk hcp Co. Ex-\ncluding the OP correction we obtained an easy-plane magnetisation and a MAE of 6.7\n\u0016eV/Co atom, while including the OP correction we instead obtained an easy axis per-\npendicular to the hexagonal Co planes and a MAE of 84.4 \u0016eV/Co. The latter result\nis in good agreement with the experimental value of 65.5 \u0016eV [26] and with the exper-\nimental easy axis being along the (0001) direction. Our result also compares well with\nthat of Trygg et al. [27], who calculated K= 110\u0016eV for hcp Co using a full-potential\nLMTO method including OP correction.\nAs described in Section 2, we performed calculations of the MAE of a bulk Co system in\nwhich a single layer has been substitutionally alloyed by Pt in a fraction of 0 <s\u00141.E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 6\nThe layer-resolved Co contributions to the MAE, D(Co)\ni, see Eqs. (2) and (3), are shown\nin Fig. 2 for s= 0:01 ands= 0:02. Remarkably, even such small amounts of Pt induce\nlarge \ructuations in D(Co)\ni: in the impurity layer ( i= 0) and in the Co layers near the\nimpurity layer (1 \u0014jij\u00144) the relative changes of D(Co)\niwith respect to the bulk Co\nMAE reach 10 %. In particular, the Co contribution from layers i=\u00061 is enhanced\nto nearly 94 \u0016eV, while that those from layers i=\u00062 are reduced to nearly 76 \u0016eV\nfors= 0:02. For layers further away from the impurity layer ( jij\u00155), oscillations in\nD(Co)\niwith rapidly decreasing amplitude can be seen. Reassuringly, the layer-resolved\nCo contributions approach the bulk Co MAE towards the outer edges of the middle\nregion chosen in our calculations ( i!NL= 40 andi!\u0000 (NL\u00001) =\u000039). The mirror\nsymmetry around the impurity layer, D(Co)\ni=D(Co)\n\u0000i, is also ful\flled with a high accuracy.\nFigure 2. (Color online) Calculated layer-resolved Co contributions, D(Co)\ni, to the\nMAE, see Eq. (3), across the system shown in Fig. 1 for s= 0:01 (red +) and s= 0:02\n(black\u000f). The MAE of bulk Co is indicated by the solid black horizontal line. Solid\nlines connecting the symbols serve as guides for the eye.\nOur earlier studies of the MAE of impurities [28, 29], justi\fed that the MAE is extremely\nsensitive to the presence of Friedel oscillations in the charge density. It is, therefore,\ntempting to relate Fig. 2 to the change in the valence charge on the Co atoms with\nrespect to the distance from the impurity layer. From Fig. 3(a) we can see that the Co\natoms in the impurity layer and, in particular, the Co atoms in layers adjacent the im-\npurity layer gain some extra charge, while the charge transfer to more distant Co layers\ndrops rapidly. The energy shift of the layer-resolved Co valence band position is well\ndescribed by the layer-resolved change in the Madelung potential. (Note that within\nthe ASA the Madelung potential in each atomic plane is a constant.) As is obvious\nfrom Fig. 3(b), an enhanced (reduced) charge at the Co sites is accompanied with aE\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 7\ndownward (upward) shift of the valence states. Comparing with Fig. 2, this shift of the\nvalence states correlates directly with the MAE contributions of the Co layers adjacent\nthe impurity layer, but, clearly enough, the changes in the MAE contributions from\nmore distant Co layers are also subject to \fne details of the valence states in\ruenced\nby the Pt alloying.\nFigure 3. (a) Calculated valence charge on the Co atoms, Q, and (b) relative shift of\nthe Madelung potentials with respect to the bulk case, \u0001 VMad, for layers\u000013\u0014i\u001413\nand for a Pt concentration, s= 0:02. Solid lines serve as guides for the eye.\nThe e\u000bect on the species-resolved MAE by alloying with Pt is demonstrated for the\nwhole range of sin Fig. 4, showing D(Co)\nifor layers 0\u0014i\u00144, together with the direct\ncontribution of Pt, D(Pt)\n0. In the impurity layer i= 0, see Fig. 4(a), the Co contribution\nD(Co)\n0is reduced by the addition of Pt for concentrations up to about s= 0:15 and\nthen enhanced for concentrations 0 :15<s< 0:50. For concentrations s>0:50,D(Co)\n0isE\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 8\nagain reduced with increasing sand ats\u00190:85D(Co)\n0even becomes negative. Note that\nD(Co)\n0fors!1 (not calculated here) would correspond to the contribution of a single\nCo atom in a pure Pt layer which, in general, would di\u000ber from zero. The on-site plat-\ninum contribution, D(Pt)\n0, approaches the very small value of 0.01 meV as s!0, rapidly\nincreases up to 0.30 meV at s\u00190:5 and then saturates at D(Pt)\n0\u00190:35 meV for larger s.\nIt can be inferred from Fig. 4(b), that Pt alloying most dramatically in\ruences the Co\ncontribution at the layers adjacent to the impurity layer: D(Co)\n1increases almost linearly\nfrom the bulk MAE at s= 0 to about 0.7 meV at s= 1. As already seen in Fig. 2, the\nCo contributions D(Co)\nifrom layers further out (2 \u0014i\u00144) decrease with increasing s\nand even becomes negative at s\u00190:25 fori= 2 and 3. For s>0:5 these contributions\nshow a modest increase, but D(Co)\n2still remains negative.\nIt is worth investigating the change of valence states projected onto the Co atoms in\nlayer 1. From Fig. 5(a) it is obvious that the valence charge at this Co atom increases\nalmost linearly with sfrom 9.00eto 9.16e. This increase in the valence charge is\nnecessarily accompanied by a downshift of the corresponding valence states, as charac-\nterized by the change in the Madelung potential, which is also linear s, see Fig. 5(b).\nThe large enhancement of the MAE contribution from layer 1, D(Co)\n1, can therefore be\nrelated directly to the monotonic shift of the corresponding valence states.\nWhile the species- and layer-resolved contributions to the MAE are very illuminating for\na microscopic description of the variations in the MAE, from an experimental point of\nview only the MAE of the whole system can be accessed. Here, this means considering\nthe MAE of the entire middle region illustrated in Fig. 1 for NL= 40. In order to\nextract the change in this MAE induced by the Pt impurities, we de\fne the excess\nMAE, \u0001K(s), by subtracting the MAE of the 'unperturbed' cobalt bulk layers,\n\u0001K(s) =sD(Pt)\n0+ (1\u0000s)D(Co)\n0+ 240X\ni=1D(Co)\ni\u000081KCo; (4)\nwhereKCois the calculated MAE of hcp bulk Co (84.4 \u0016eV). Note that we have\ntaken into account the o\u000b-centre positioning of the impurity layer by doubling the Co\ncontributions D(Co)\nifori2[1;40], thus, in total, a system of 81 layers is considered.\n\u0001Kis shown as a function of sin Fig. 6, demonstrating that for small concentrations of\nplatinum (s<0:24) the addition of platinum to bulk cobalt actually reduces the total\nMAE of the system by about 80 \u0016eV. This is in strong contrast to the on-site contribu-\ntion of Pt, D(Pt)\n0, being positive for all values of sas seen in Fig. 4(a). The reduction\nin MAE for low s, therefore, stems from the decrease in the cobalt contributions D(Co)\ni\nwith increasing s, in particular, for i= 0;2;3 and 4, see Fig. 4. \u0001 Kbecomes positive\nfors>0:24 as the increasing on-site contribution, D(Pt)\n0, gets larger weight (note that\nit is multiplied by s) and due to the large enhancement of D(Co)\n1. Ats= 1, \u0001K= 1:4\nmeV, which is approximately four times the on-site platinum contribution, D(Pt)\n0\u00190:35E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 9\nFigure 4. (Color online) Calculated species-resolved MAE contributions (a) for the\nimpurity layer: D(Pt)\n0(black\u000f) and D(Co)\n0(blueN) and (b) for Co layers: D(Co)\n1(black\n\u000f),D(Co)\n2(red +), D(Co)\n3(green\u0002) and D(Co)\n4(purple\u0004). Solid lines connecting the\nsymbols serve as guides for the eye.\nmeV fors= 1.\nFor nano-sized systems, it might be of interest to consider the change in the MAE per\nplatinum atom in the system ,KPt, de\fned by\nKPt(s) =\u0001K(s)\ns; (5)\nand also, the change in the MAE per platinum atom added to the system ,KPt, obviously\ngiven by\nKPt(s) =d(\u0001K(s))\nds: (6)E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 10\nFigure 5. (a) Calculated valence charge on the Co atom, Q, and (b) relative shift of\nthe Madelung potential with respect to the bulk case, \u0001 VMad, for layer 1 as a function\nof the Pt concentration, s. Solid lines serve as guides for the eye.\nWe obtained KPt(s) by \ftting a fourth-order polynomial to the function \u0001 K(s) in Fig. 6\nand then \fnding the derivative of this function analytically. As apparent from Fig. 7,\nbothKPtandKPtare monotonically increasing with increasing s, starting with the\nsame value of about \u00001 meV ats= 0 (see later). It follows directly from Fig. 6, that\nKPtcrosses zero at s\u00190:24, whileKPtcrosses zero at s\u00190:11 (i.e. where the function\n\u0001K(s) reaches its minimum). For a complete platinum layer immersed in bulk cobalt,\ni.e. fors= 1,KPt= \u0001K\u00191:4 meV. A comparison with Fig. 4 shows that about\n25 % of this value arises from the direct contribution of Pt, D(Pt)\n1, and the rest from\nthe induced contributions at the Co atoms. Interestingly, the change in the MAE by\naddition of a Pt atom to the system, KPt, exhibits a surprisingly large value of about\n2.5 meV at s= 1. From Fig. 4(a) it can be inferred that the on-site Pt contributionE\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 11\nFigure 6. The change in the MAE, \u0001 K, of a system of 81 atomic layers, see Eq. (4),\nas a function of the Pt concentration, s. The solid line connecting the symbols serves\nas a guide for the eye.\nD(Pt)\n0has nearly zero slope in this region of s, thus, this large value of KPtstems mainly\nfrom an increase in D(Co)\nifor 1\u0014jij\u00144 nears= 1.\nFigure 7. (Color online) Black circles: calculated change in the MAE per Pt atom,\nKPt, see Eq. (5), as a function of the Pt concentration, s. The black solid line\nconnecting the symbols serves as a guide for the eye. Red solid line: the change\nin the MAE per Pt atom added, KPt, see Eq. (6), as calculated from a polynomial \ft\nof \u0001K(s) in Fig. 6.E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 12\nIn the limit s!0, corresponding to the case of a single Pt impurity in bulk Co, KPt\nandKPtshould be identical, since for small sthe function \u0001 K(s) exhibits, in principle,\na linear dependence. This is fairly well con\frmed by our calculations. KPt(0) can then\nbe expressed as\nKPt(0) =D(Pt)\n0(0)\u0000KCo+40X\ni=\u000040dD(Co)\ni(s)\nds\f\f\f\f\f\ns=0: (7)\nThe physical meaning of the above equation is that adding a Pt impurity to bulk Co has\ntwo e\u000bects on the MAE of the system: the \frst two terms, D(Pt)\n0(0)\u0000KCo, represent the\ndirect contribution of a Co atom being replaced by a Pt atom, whereas the the last term\nof Eq. (7) quanti\fes the induced change in the MAE contributions from the Co atoms\nthat are not being replaced by Pt. Since the direct contribution is about -0.07 meV,\nsee also Fig. 4(a), the value of KPt(0) =\u00001 meV can again only be explained by the\ninduced Co contributions.\nFigure 8. Calculated derivatives of the layer-resolved Co contributions to the\nMAE, Eq. (8), for the Pt concentration s= 0:02. The solid line serves as a\nguide for the eye.\nIn Fig. 8 we show the approximate layer-resolved derivatives calculated as\ndD(Co)\ni(s)\nds\f\f\f\f\f\nsj=D(Co)\ni(sj+1)\u0000D(Co)\ni(sj\u00001)\nsj+1\u0000sj\u00001; (8)\nforsj= 0:02 withjindexing the Pt concentrations in ascending order. Clearly, this\n\fgure is closely related to Figs. 2 and 4: for small s, the Co contributions fD(Co)\nigshow\nan increasing tendency with increasing sforjij= 1 andjij\u00157, while they decreaseE\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 13\nfor 2\u0014jij\u00145. Apparently, this latter e\u000bect overcomes the former one, leading to the\nrelatively large value of KPt(0) =\u00001 meV.\nWe also investigated possible e\u000bects of the Friedel oscillations on the MAE by truncating\nthe sum in Eq. (4) and considering the variation in KPtagainst the number Nof Co\nplanes included in the sum,\nKPt(s;N) =1\ns \nsD(Pt)\n0+ (1\u0000s)D(Co)\n0+ 2NX\ni=1D(Co)\ni\u0000(2N+ 1)KCo!\n;(9)\nfor 1\u0014N\u001440. The function KPt(N) fors= 0:01, 0:05, 0:10 and 0:20 is shown in\nFig. 9. For all cases, the maximum of KPt(N) occurs at 2 N+ 1 = 3 planes, i.e. includ-\ning only one Co layer on each side of the impurity layer. This is because the induced\ne\u000bect onD(Co)\n1by the addition of platinum is strongly positive for all s, see Figs. 2\nand 4(b). There is a signi\fcant minimum in the calculated KPt(N) at 2N+ 1\u001911\nplanes. Comparing with Fig. 2, it is obvious that this minimum is due to the reduction\nofD(Co)\nifor 3\u0014jij\u00145 caused by the addition of Pt. KPt(N) then exhibits a local\nmaximum at 2 N+ 1\u001931, mostly due to the Co contributions in layers 8 \u0014jij\u001415\ncounterbalancing the Co contributions of opposite sign in layers 3 \u0014jij\u00145. Concerning\nthe overall accuracy of the calculated MAE, the e\u000bects of the Friedel oscillations remain\nsigni\fcant for about 2 N+ 1<70 planes, i.e. for 35 layers on either side of the impurity\nlayer. In general, the variation in KPtwithNspans more than 1.5 meV for all values\nofs. This of course has signi\fcant implications for measuring the MAE in thin \flm\nsamples, as up to about 2 N+ 1<40 planes, i.e. for \flm thicknesses d < 8 nm, the\nchange in the MAE induced by the Pt impurities located at the centre of the sample, is\nexpected to be extremely sensitive on the \flm thickness. Note that this \fnite-size e\u000bect\nis superimposed on and, most likely, ampli\fed by quantum interferences arising from\nthe boundaries of the \fnite \flm sample.\nIt should be noted that, being a mean-\feld approach, the CPA completely neglects\nboth structural and electronic short-range order e\u000bects. Such short-range order e\u000bects\nare likely to be most strongly pronounced for small Pt concentrations s. Therefore, our\nresults in the low- slimit should be tested against another method. By employing a fully\nrelativistic real-space embedded cluster Green's function technique as combined with\nthe SKKR method [30] we made an attempt to test the e\u000bect of electronic relaxations\naround a Pt impurity placed in hcp bulk Co. We performed self-consistent calculations\nfor a cluster containing a Pt impurity and the neighboring Co atoms up to three nearest\nneighbor (NN) distances of the hcp lattice ( aNN). Note that our embedded cluster\nincluded 158 Co atoms around the central Pt atom, sorted out geometrically as follows:\n(i)36, 2\u000230, 2\u000219 and 2\u000212 Co atoms in layers 0, \u00061,\u00062 and\u00063, and (ii)12, 56\nand 158 Co atoms within the spheres centered around the Pt site having the radii of aNN,\n2aNNand 3aNN, respectively. The MAE for this cluster was again calculated using the\nmagnetic force theorem. By summing up all the site-resolved contributions of the MAEE\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 14\nFigure 9. (Color online) Calculated excess MAE per platinum atom, KPt, as a\nfunction of the number of planes Nincluded in the sum in Eq. (9) for s= 0:01 (red\n+),s= 0:05 (green\u0002),s= 0:10 (blueN) and s= 0:20 (black\u000f).\nin the cluster within the spheres as mentioned above, we obtained the values for KPt,\n0.67 meV, -0.31 meV and -0.38 meV, respectively. Obviously, a direct comparison of\nthese values with those for the layered system is hardly possible, since, according to the\ngeometrical classi\fcation (i), theKPtfor the cluster refers to an incomplete summation\nover sites in the respective layers. Nevertheless, comparing with the values in Fig. 9\nrelated to 2 N+ 1 = 3;5 and 7 and for the concentration of s= 0:01 being closest the\ncase of an impurity, we can conclude that both the trend and the magnitude of KPtare\nin satisfactory agreement between the CPA and the real-space calculations.\n4. Summary and Conclusions\nUsing the fully relativistic screened Korringa-Kohn-Rostoker method combined with\nthe coherent potential approximation, we have studied the MAE of a bulk hcp Co\nsystem in which one of the (0001) atomic plane has been alloyed with Pt for the whole\nconcentration range, 0 < s\u00141. We conclude that low concentrations of platinum\nreduce the overall MAE of this system and that the origin of this reduction are the\ninduced changes in the MAE contributions from the Co atoms. In the limit s!0 (bulk\nCo), the change in MAE per platinum atom added is approximately \u00001 meV. At larger\nconcentrations, the direct MAE contribution of the platinum, which is positive for all Pt\nconcentrations, starts to increase, but the overall change in MAE due to the addition of\nplatinum is still dominated by induced Co contributions. Interestingly, in the limit of a\ncompletely \flled Pt layer, addition of one Pt atom increases the MAE of the system byE\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 15\nabout 2.5 meV. We also investigated the e\u000bect of long-ranged Friedel oscillations and\nestablished a large sensitivity of the MAE on the number of Co layers included in the\ncalculations. This might have a signi\fcant impact on the experimental determination\nof the MAE in thin \flm samples of this type of system.\nAcknowledgments\nCJA is grateful to EPSRC and to Seagate Technology for the provision of a research\nstudentship. Financial support was in part provided by the New Sz\u0013 echenyi Plan of\nHungary Project ID. T \u0013AMOP-4.2.2.B-10/1{2010-0009 and the Hungarian Scienti\fc Re-\nsearch Fund (contracts OTKA PD83353, K77771). KP bene\fted the Bolyai Research\nGrant of the Hungarian Academy of Sciences. Support of the HAS Wigner Research\nCentre of Physics through the usage of its computational facilities is also kindly ac-\nknowledged.\n[1] B. Lu, T. Klemmer, K. Wierman, G.P. Ju, D. Weller, A.G. Roy, D.E. Laughlin, C.H. Chang and\nR. Ranjan, J. App. Phys. 91, 8025 (2002).\n[2] D. Weller and A. Moser, IEEE Trans. Magn., 36, 10 (1999).\n[3] S.S.A. Razee, J.B. Staunton and F.J. Pinski , Phys. Rev. B 56, 8082 (1997).\n[4] J. A. Aboaf, S. R. Herd and E. Klokholm, IEEE Trans. Mag. 4, 1514 (1983).\n[5] M. S. Miller, A. E. Schultz, Y. M. Chow and L. A. Heuer, Surface and Coatings Technology 68-69 ,\n696 (1994).\n[6] J. Jaworowicz, A. Maziewski, P. Mazalski, M. Kisielewski, I. Sveklo, M. Tekielak, V. Zablotskii,\nJ. Ferr\u0013 e, N. Vernier, A. Mougin, A. Henschke and J. Fassbender, App. Phys. Lett. 95, 022502\n(2009).\n[7] M. Abes, J. Venuat, A. Carvalho, J. Arabski, D. Muller, G. Schmerber, E. Beaurepaire, P. Panissod,\nA. Dinia and V. Pierron-Bohnes, J. Mag. Mag. Mat. 286, 297 (2005).\n[8] B. Gyor\u000by, Phys. Rev. B 5, 2382 (1972).\n[9] M.F. Toney, E.E. Marinero, M.F. Doerner, and P.M. Rice, J. Appl. Phys. 94, 4018 (2003).\n[10] A. Ishikawa and R. Sinclair, IEEE Trans. Magn. 32, 3605 (1996).\n[11] A. Ishikawa and R. Sinclair, J. Magn. Magn. Mater. 152, 265 (1996).\n[12] V Sokalski, D.E. Laughlin, and J-G Zhu, J. Appl. Phys. 110, 093919 (2011).\n[13] J. Zabloudil, R. Hammerling, L. Szunyogh and P. Weinberger, Electron Scattering in Solid Matter ,\n(Berlin, Springer 2005).\n[14] H. Ebert, D. K odderitzsch, and J. Min\u0013 ar, Rep. Prog. Phys. 74, 096501 (2011).\n[15] S. H. Vosko, L. Wilk and M. Nusair, Can. J. Phys. 58, 1200 (1980).\n[16] P. Soven, Phys. Rev. 156, 809 (1967).\n[17] M.S.S. Brooks, Physica B 130, 6 (1985); 0. Eriksson, B. Johansson, and M. S. S. Brooks, J. Phys.:\nCondensed Matter 1, 4005 (1989); 0. Eriksson, M. S. S. Brooks, and B. Johansson, Phys. Rev.\nB41, 7311 (1990).\n[18] H. Eschrig, The Fundamentals of Density Functional Theory (Leipzig, Teubner 1996).\n[19] H. Eschrig, M. Sargolzaei, K. Koepernik and M. Richter, Europhys. Lett. 72, 611 (2005).\n[20] H. Ebert and M. Battocletti, Solid State Comm. 98, 785 (1996).\n[21] H. J. F. Jansen, Phys. Rev. B 59, 4699 (1999).\n[22] X. Wang, R. Wu, D. Wang and A.J. Freeman, Phys. Rev. B 54, 61 (1996).\n[23] P. Lloyd, Proc. Phys. Soc. 90, 207 (1967).\n[24] J.B. Staunton, L. Szunyogh, A. Buruzs, B.L. Gyor\u000by, S. Ostanin and L. Udvardi, Phys. Rev. B\n74, 144411 (2006).E\u000bect of Pt impurities on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 16\n[25] L. Udvardi, L. Szunyogh, K. Palot\u0013 as and P. Weinberger, Phys. Rev. B 68, 104436 (2003).\n[26] M.B. Stearns, in 3d, 4d, and 5d Elements, Alloys and Compounds , (Berlin, Springer 1986).\n[27] J. Trygg, B. Johansson, O. Eriksson, and J.M. Wills, Phys. Rev. Lett. 75, 2871 (1995).\n[28] L. Szunyogh and B. L. Gyor\u000by, Phys. Rev. Lett. 78, 3765-3768 (1997).\n[29] A. Szilva, S. Gallego, M. C. Munoz, B. L. Gyor\u000by, G. Zarand and L. Szunyogh, Phys. Rev. B 78,\n195418 (2008).\n[30] B. Lazarovits, L. Szunyogh, and P. Weinberger, Phys. Rev. B 65, 104441 (2002)."    },    {        "title": "1208.4403v2.High_temperature_structural_and_magnetic_properties_of_cobalt_nanowires.pdf",        "content": "High temperature structural and magnetic properties of cobalt nanorods.\nKahina Ait Atmanea, Fatih Zighemb, Yaghoub Soumarea, Mona Ibrahimc, Rym\nBoubekric, Thomas Maurerb, Jérémie Margueritatb, Jean-Yves Piquemala, Frédéric Ottb,\nGrégory Chaboussantb, Frédéric SchoensteindNoureddine Jouinidand Guillaume Viauc\naITODYS, CNRS/Université Paris Diderot, Sorbonne Paris Cité, 75205 Paris, France\nbLLB, CEA/CNRS, IRAMIS, 91191 Gif sur Yvette, France\ncLPCNO, CNRS/INSA-Université de Toulouse, 31077 Toulouse, France and\ndLSPM, CNRS/Université Paris XIII, 93430 Villetaneuse, France\u0003\nI. INTRODUCTION\nThe last decade has seen numerous investigations on\nanisotropic inorganic nanoparticles such as nanowires\n(NWs) and nanorods (NRs) for their peculiar physical\nproperties and new applications in optics, magnetism\nor electronics [1, 2]. Several physical and chemical\nmethods have been developed to grow ferromagnetic\nnanowires. The most used consists in the electrochemical\nreduction of Fe, Co or Ni salts within the unixial pores\nof an alumina or a polycarbonate membrane to confine\nthe metal growth [3]. In order to grow NWs with a\ndiameter below 10 nm, other solid hosts have been\nemployed such as carbon nanotubes [4] or mesoporous\nsilica [5]. Recently, cobalt NWs with diameters below\n5 nm embedded in epitaxial CeO 2layers were obtained\nby pulsed-laser deposition [6]. Iron-boride NWs with\ndiameter in the range 5-50 nm were obtained by chemical\nvapour deposition [7]. Wet chemistry methods, for which\nthe adsorption of long chain molecules at the metal\nparticle surface is the driving force of the anisotropic\ngrowth, were developed for the synthesis of cobalt [8]\nand cobalt-nickel NRs and NWs [9].\nOne interest of high aspect ratio ferromagnetic\nnanoparticles is that they can present high coercivity\nproperties due to their large shape anisotropy. Magnetic\nNWshave beenrecentlyproposedas potentialcandidates\nfor high-density magnetic recording [10, 11], nanowire-\nbased motors [11] and the bottom-up fabrication of per-\nmanent magnets [12]. Indeed, very high coercivity val-\nues were observed on assemblies of cobalt NRs or NWs\nobtained using wet chemistry, either by the polyol pro-\ncess [13] or by organometallic chemistry [14]. Compos-\nite materials of well aligned cobalt rods with a volume\nfraction of 50% could exhibit higher (BH)max. values\nthan AlNiCo or ferrite-based magnets [11]. The inter-\nest of these liquid phase processes for the synthesis of\nNRs with hard magnetic properties lies in the possibil-\nity to produce wires combining very good crystallinity\nand small diameter. The cobalt NRs and NWs synthe-\nsized by these chemical methods crystallize with a hcp\n\u0003Electronic address: jean-yves.piquemal@univ-paris-diderot.fr (J.-\nY.P.); gviau@insa-toulouse.fr (G.V.)structure with the c axis along the long axis [8, 15]. The\ncoincidence of the shape anisotropy easy axis and the\nmagneto-crystalline anisotropy easy axis reinforces the\nwhole magnetic anisotropy [12, 14].\nSeveral parameters govern the coercivity of an assem-\nbly of anisotropic magnetic particles: (i) the distribution\nof the particle easy axis orientation with respect to the\nmagnetic field and (ii) the particles shapes. For cobalt\nwires we showed that the coercivity increases when\nthey are aligned parallel to the applied magnetic field\nin good agreement with the Stoner-Wohlfarth model\n[13]. The coercivity is the sum of the magnetocrystalline\nand the anisotropy contributions. The contribution due\nto the shape anisotropy actually strongly depends on\nthe detailed geometry of the wires and especially of\nthe wire tips. This was investigated by micro-magnetic\nsimulations [16]. On the one hand, while the ellipsoid\nshape is the most favourable, a cylindrical shape also\nprovide a good shape anisotropy. On the other hand,\nenlarged tips create nucleation points for the mag-\nnetization reversal which can significantly reduce the\ncoercivity by up to 30. Surface effects related to the thin\nsuperficial CoO layer were extensively studied [18]. The\nmeasured Néel temperature of the CoO shell was 230 K.\nSignificant modifications of the magnetic behaviour take\nplace below this temperature. When the temperature\ndecreases, a coercivity drop is actually observed below\n150 K [18].\nBut even if progresses have been made in the synthesis\nof ferromagnetic NRs and NWs exhibiting large coer-\ncivities at room temperature and in the understanding\nof their magnetic properties [12, 13, 16–18], there is\nstill a lack of information about the stability of both\ntheir structural and magnetic properties above room\ntemperature which is a key information for any practical\nuse at high temperatures of these materials e.g. for\nthe fabrication of rare-earth free permanent magnets.\nAt high temperatures, metal wires generally undergo\nan irreversible transformation to chains of spheres,\ndescribed as the Rayleigh instability[19, 20]. The high\naspect ratio of compacted wires may also be altered by\nsintering. A first study showed that the thermal stability\nof “organometallic” cobalt nanowires was dependent on\nthe atmosphere under which the wires were annealed\n[21]. Fragmentation of cobalt NWs into chains of cobalt\nparticles due to the Rayleigh instability was avoidedarXiv:1208.4403v2  [cond-mat.mtrl-sci]  28 Aug 20122\nwhen the cobalt NWs were coated by a thin carbon shell\n[21].\nIn this paper we present the high temperature struc-\ntural and magnetic properties of cobalt nanorods pre-\npared by the polyol process using in-situ characteriza-\ntions. The scope of this communication is to determine\nwhether the anisotropic structure and texture are modi-\nfied at high temperature and consequently whether their\nmagnetic properties, and especially their large coercivi-\nties, are preserved. The temperature range of stability\nof these NRs and the temperature dependence of their\nintrinsic magnetic properties up to 623 K under different\natmospheres are described.\nII. MATERIALS AND METHODS.\nCoCl 2\u00016H2O (Alfa Aesar, 99.9 %), RuCl 3\u0001xH2O\n(Aldrich, 99.98 %), NaOH (Acros), 1,2- butanediol\n(Fluka, \u001598%), methanol (VWR, Normapur) and\nsodiumlaurate,Na(C 11H23COO)(Acros,98 %)wereused\nwithout any further purification.\nA. Syntheses of Co nanowires.\nThe cobalt laurate precursor, CoII(C11H23COO) 2was\nfirst prepared. In 100 mL of distilled water at 333 K,\nwas dissolved sodium laurate (75 mmol; 16.7 g) and to\nthis solution, was added an aqueous solution (38 mL) of\ncobalt(II) chloride (40 mmol, 10,0 g) pre-heated at 333K\nunder vigorous stirring. This resulted in the formation\nof a purple precipitate which was vigorously stirred at\n333K for 15 min. The Co(II) solid phase was washed\ntwice with distilled water (100 mL) then with methanol\n(100 mL) and finally dried in an oven at 323 K overnight.\nYield: 94 %(based on Co).\nThe synthesis of the cobalt NRs was realized accord-\ning to a procedure previously described [13]. To 75.0\nmLof1,2-butanediolwereaddedCo(C 11H23COO) 2(2.75\ng, 0.08 M), RuCl 3.xH 2O (3.2 10\u00002g) and NaOH (0.225\ng, 0.075 M). The mixture was heated to 448 K with a\nramping rate of 13 K.min\u00001for 20 min until the color\nof the solution turned black, indicating the reduction of\nCo(II) into metallic cobalt. After cooling to room tem-\nperature, the Co NRs were recovered by centrifugation at\n8500 r.p.m. for 15 min, washed with 50 mL of absolute\nethanol (3 times), and finally dried in an oven at 323 K.\nYield: 92%(based on Co). Elemental analyses revealed\nC and H amounts of 3.8 and 0.7 wt. %, respectively.\nB. Preparation of the sample for magnetic\nmeasurements.\nTwo sets of samples were prepared for magnetic mea-\nsurements. Sample (A): a few drops of a Co NRs sus-pension in toluene were deposited on an aluminum foil\nand the toluene was removed by evaporation under the\napplication of an external magnetic field of 1 T. Sample\n(B): a pellet of magnetic NRs was prepared using an in-\nfrared KBr die (internal diameter 13 mm) and applying\na pressure of about 6 tons delivered by a hydraulic press.\nThe mass and the apparent density (including porosity)\nof the pellet were respectively 0.26 g and 2.13 g cm–3.\nThe true density of the powder, obtained at 298 K using\na helium Accupyc 1330 pycnometer from Micromeritics,\nwas found to be 6:76\u00060:41g cm–3. The packing fac-\ntor, defined as the ratio of the volume of the particles by\nthe volume of the pellet was about 30%. Co nanowires\nare randomly oriented inside the pellet since no magnetic\nfield was applied during its preparation. The small den-\nsity of the powder is accounted for by the fact that part\nof the metallic wires are oxidized and that some organic\nmaterial remain in the pellets.\nC. Characterization techniques.\n1. Room temperature characterizations\nTransmission electron microscopy (TEM) characteri-\nzations were performed using a Jeol 100-CX II micro-\nscopeoperatingat100kV.Infraredspectrawererecorded\non a nitrogen purged Nicolet 6700 FT-IR spectrometer\nequipped with a VariGATR accessory (Harrick Scientific\nProducts Inc., NY) fixing the incident angle at 62\u000e. A\ndrop of the colloidal solution was deposited on the Ge\nwafer and the spectrum was recorded when the solvent\n(absolute ethanol or toluene) was fully evaporated. XRD\npatterns obtained using Co K radiation ( \u0015= 1:7889\n) were recorded on a PANalytical X’Pert Pro diffrac-\ntometer equipped with an X’celerator detector in the\nrange 20\u000080\u000ewith a 0:067\u000estep size and 150 s per\nstep. The size of coherent diffraction domains, Lhkl,\nwere determined using MAUD software which is based\non the Rietveld method combined with Fourier analysis,\nwell adapted for broadened diffraction peaks. Magnetic\nmeasurements were performed using a Quantum Design\nMPMS-5S SQUID magnetometer.\n2. Thermal treatments and high temperature\ncharacterizations.\nIn order to follow the structural evolution of the cobalt\nnanowires with temperature, insitu thermal treatments\nwere realized in the range 300-673 K, using a HTK 1200N\nhigh-temperature X-ray diffraction chamber from Anton\nPaar. Twotypesofexperimentswereperformed, depend-\ning on the oxygen content of the nitrogen gas used: in\nthe first one, the powder was heated under a high-purity\nnitrogen atmosphere(O 2< 2 ppm; N 2Alphagaz 1) while\nin the second one, the oxygen content was substantially\nhigher (O 2< 0.1 ppm; N 2Alphagaz 2). The samples3\nwere heated inside the high temperature X-ray diffrac-\ntion chamber from room temperature to the final tem-\nperature using a 5 K \u0001min\u00001rate and maintained for 2 h\nat the final temperature before the acquisition of a X-ray\ndiffraction pattern. Magnetization curves at high tem-\nperature of samples (A) and (B) were measured with a\nSQUID equipped with an oven. In this procedure the\nparticles were heated in a reduced helium pressure.\nIII. RESULTS AND DISCUSSIONS\nA. Morphology and chemical analysis of the cobalt\nnanorods.\nTEM observations on the particles prepared by\nthe polyol process showed Co NRs with a mean\ndiameterdm= 13nm and a mean length Lm= 130nm\n(Figure 1a). The standard deviation of the diameter is\nvery small(<15%of the mean length) as it was observed\nbefore [13]. For the Co NRs deposited on an Al sub-\nstrate, the application of an external magnetic field re-\nsults in the alignment of the cobalt anisotropic nanopar-\nticles while the Co NRs are randomly oriented within\nthe pellet (see Figure 1b and 1c). Moreover Figure 1c\nshows also that the compression was not detrimental to\nthe Co NRs. The Co NRs were further characterized us-\ning TG-DT analyses under pure N 2(O2< 0.1 ppm). The\nthermogram (see Figure 2) showed a 7%weight loss at\nabout 575 K associated with a sharp endothermic peak.\nThis weight loss is attributed to the elimination of or-\nganic matter adsorbed on the particle surface. Infrared\nspectroscopy was thus performed on the cobalt particles\nto characterize the organic matter remaining after the\nsynthesis and the washing procedure at room tempera-\nture. The cobalt NRs infrared spectrum washed twice\nwith ethanol (Figure 3) exhibits a large band centered\nat 3300 cm\u00001corresponding to the O-H stretching vi-\nbration. The OH groups can belong to a surface cobalt\nhydroxide and/or to adsorbed ethanol. At 2850 and 2920\ncm\u00001the symmetric and asymmetric C-H stretching vi-\nbration are respectively observed, which is attributed to\ntheCH 2groupsofthelaurateions. Intheregionbetween\n1400 and 1550 cm\u00001the intense bands are attributed to\nthe asymmetric and symmetric C-O stretching vibration\nof carboxylate groups, indicating that laurate ions re-\nmain at the particle surface. The intensity of all these\nbands decreases with the successive washings (Figure 3)\nshowing that the amount of organic ligands at the par-\nticle surface is strongly dependent on the way they have\nbeen washed.\nB. High temperature structural and chemical\nmodification.\nInordertofollowthestructuralmodificationsoftheCo\nNRs when they are heated up, in-situ X-ray diffraction\nFigure1: (a)TEMimagesofCorods( Lm= 130nm,dm= 13\nnm); SEM images of (b) sample (A), (c) sample (B) and (d)\nmagnetization curves at 300 K of sample (A) (circles) and (B)\n(squares).\nFigure 2: Thermogravimetric and differential thermal analy-\nses realized on Co nanowires under N 2(O2<0.1 ppm).\nhave been performed for different temperatures ranging\nfrom 298 K up to 623 K. The structural and chemical\nmodifications were found to depend on the atmosphere\nunder which the cobalt wires are annealed. The in-situ\nX-ray diffraction patterns of the Co NRs annealed un-\nder a N 2atmosphere with O 2concentration <2 ppm\nat different temperatures showed a progressive oxidation\nemphasized by the growth of the CoO (111) peak around\n42\u000ein spite of the small O 2concentration. With increas-\ning temperature, the very broad peaks of the oxide be-\ncomes thinner indicating grain growth processes and/or\ncrystallization of a pre-existing amorphous phase. Pre-\nvious HRTEM studies performed on Co nanowires have\nevidenced the presence of a CoO layer composed of dis-\noriented crystallites of various sizes [18], indicating that4\nFigure 3: Infrared absorption spectra of cobalt wires washed\ntwice with ethanol (a); washed twice with ethanol and once\n(b), twice (c) and three times (d) with toluene.\nthe former case is the more probable one. No structural\nmodificationwasobservedonthehcpnonoxidizedcobalt\nphase. For Co samples treated under N 2with very low\ndioxygen content ( <0.1 ppm) the Xray diffraction data\n(Fig. 4) show that there is also development of the CoO\noxide between 298K and 473 K. Then, between 473 and\n523 K, CoO vanished at the benefit of metallic Co as in-\ndicated by the sharpening and the increase of intensity\nof the Co peaks. For higher temperatures, 573 K and\n623 K, CoO is detected again. Given the high temper-\nature applied, the very low dioxygen content is never-\ntheless sufficient to induce the formation of this oxide.\nFor the as-synthesized sample, TG-DT analyses realized\nunder N 2(O2content < 0.1 ppm) have shown a weight\nloss of about 7 % at 573 K associated with an endother-\nmic signal (see Figure 2). This weight loss is explained by\nthe decomposition of the remaining metal-organic species\nadsorbed at the particle surface. Indeed, IR-ATR experi-\nments(Figure3)haveclearlyshownvibrationsattributed\nto adsorbed laurate species (see above). Thus, the com-\nparison of the HT XRD patterns with the IR and TGA\nresults suggest that the organic molecules remaining at\nthe particle surface reduce the CoO shell in the temper-\nature range corresponding to their decomposition. This\neffect can be evidenced only when the oxygen concentra-\ntion in the atmosphere is small enough.\nC. High temperature texture modification.\nThe XRD diffractograms showed that the hexagonal\nclose-packed structure of the Co core is preserved for\ntemperatures up to 623 K, whatever the annealing at-\nmosphere. Nevertheless, a close examination of these\npatterns shows that the line broadening is modified dur-\ning the thermal treatment. The mean crystallite sizes\nFigure 4: In-situ X-ray diffraction patterns of Co nanorods\nthermally treated under N 2(O2<0.1 ppm).\nfor the (10.0) and (00.2) reflexions have been determined\nand plotted as a function of the temperature treatment\nfor samples annealed in N2 with the lowest oxygen con-\ntent (Figure 5). At room temperature, the mean crys-\ntallite size for the (10.0) reflexion is always found much\nsmallerthanthemeancrystallitesizeforthe(00.2)reflex-\nion. This observation confirms that the long axis of the\ncobalt wires is the c axis of the hcp structure. The data\nshow that up to 500 K, the (10.0) and (00.2) mean crys-\ntallite sizes are more or less constant, indicating that the\nanisotropic shape of the crystallites is preserved. Major\ntexture modifications are observed above 525K. Indeed,\nat this temperature a considerable increasing of both the\nL10:0and L 00:2mean crystallite sizes is observed and the\ncrystallites lose their anisotropy. At 525 K particles start\nto undergo sintering. In order to probe a possible dete-\nrioration of the Co nanowire morphology, TEM was per-\nformed ex-situ for samples annealed at different temper-\natures. Figures 6a and 6b indicate that the shape of the\nnanowires is kept up to 523 K. On the other hand, Figure\n6c shows that at 573 K and above, the nanowires start to\nsinter so that their shape anisotropy starts to fade away\nat these temperatures. At 623 K, the anisotropic shape is\nlost and only large aggregates are observed (Figure 6d).\nThe sintering of the particles is probably enhanced by\nthe fact that the CoO layer which likely acts as a pro-5\nFigure 5: Variation of the L 10:0and L 00:2mean crystallite\nsizes with increasing temperature for Co samples treated un-\nder N 2(O2<0.1 ppm).\nFigure 6: TEM images of Co wires (a) after drying at 330 K\nand after thermal treatment under N 2(O2< 0.1 ppm) for 2\nhours at 523 K (b), at 573 K (c) and 2 hours at 623 K (d).\ntective layer preventing sintering is reduced to cobalt by\nthe organic molecules at about 523 K as seen by XRD\n(Figure 4).\nD. Magnetic properties at high temperature.\nThe cobalt NRs are ferromagnetic at room tempera-\nture with coercivities of 530 and 230 mT and Mr=MS\nvalues of 0.67 and 0.46 for the samples (A) and (B), re-\nspectively. The difference of HCandMrare essentially\ndue to the high degree of orientation of the rods in thesample (A) which increases both coercivity and rema-\nnence. The saturation magnetization MSof sample (B)\nis 113 emu.g\u00001(1.410–4T.m3.kg\u00001). Such a low value\n(compared to the bulk value of 165 emu.g\u00001) is due to\nthe surface oxidation of the wires and by the remaining\norganic matter in the pellet. The density of the pel-\nlet, determined using He pycnometry, was found to be\n6.76\u00060.41 g.cm\u00003. Taking into account that the organic\nmatter in the particles corresponds to about 7 wt. %\nand that the CoO layer thickness is 1.2 nm for a nanorod\nwith a mean diameter of 15 nm and a mean length of\n130 nm [18], a ratio of 73 wt. %of metallic Co to the\ntotal mass of the solid (including CoO and the organic\nmatter) can be estimated. This value is in good agree-\nment with that determined using the apparent MSvalue\n(113/165 = 68 %). The temperature dependence of the\nsaturation magnetization is presented on Figure 7a. Two\nvery different behaviours are observed. In the case of\nSample (B) (pressed pellet), the saturation magnetiza-\ntionMSis rather stable up to 550 K; MSdecreases by\nlessthan1 %from300Kto550K.At550K,ajumpofthe\nmagnetizationisobservedcorrespondingtoanincreaseof\n25%withrespecttotheroomtemperaturemagnetization\nvalue. At higher temperatures (up to 800 K), the satu-\nration magnetization decreases only moderately (10 %).\nThe magnetization jump at 550 K is irreversible and can\nbe explained by the decomposition of remaining metal-\norganic at the surface of the NRs that reduces the cobalt\noxide layer and causes the apparition of added metallic\nCo. This phenomenon was clearly evidenced by XRD\n(Figure 4). For the sample (A), the saturation magneti-\nzation of the cobalt NRs deposited on an aluminium foil\nvaries only slightly up to 500 K but decreases strongly\nabove this temperature. In this case, the decrease of the\nmagnetization is explained by the oxidation of the cobalt\nNRs into CoO as was inferred by XRD. The successive\nwashings have removed most of the organic matter (Fig-\nure 3) and no reduction can occur. The helium reduced\npressure in the oven of the SQUID was not sufficient to\nprevent the powders from oxidation at high temperature\nsince traces of O 2were probably present.\nThe temperature dependence of the coercivities are\npresented in Figure 7b. When normalized to the to the\nroom temperature coercivity, HC(T)=HC(300K), the co-\nercivity of both samples follows the same behaviour over\nthe temperature range 300-500K. The coercivity \u00160HC\nfollows a linear dependence:\n\u00160HC\n\u00160HC(300K)= 1–a(T–300)\nwhereais 2.4\u000210–3K–1. This dependence can\nbe accounted for by the temperature dependence\nof the magneto-crystalline anisotropy of hcp cobalt.\nThe magneto-crystalline anisotropy of mono-crystalline\ncobalt is usually described by an anisotropy energy of the\nformEa=Ku1sin2\u0012+Ku2sin4\u0012withKu1= 4:1\u0002105\nJ.m\u00003andKu2= 1:4\u0002105J.m\u00003at 300 K and is \u00126\nthe angle of the magnetization with respect to the c axis.\nTherespectivetemperaturevariationsoftheseanisotropy\nconstants is however non-trivial: while the value of Ku2\nmonotonouslydecreasesdownto 0:4\u0002105J.m\u00003at600K,\nthe value of Ku1changes sign at 520 K [22]. The spin re-\norientation (from parallel to perpendicular to the c-axis)\nis mainly driven by the change of sign of Ku1at 520 K,\nthe effect of Ku2being to “smear” the transition so that\nthe reorientation from 0\u000eto90\u000etakes place over a rather\nwide temperature range (520-600 K). At 550 K, the spin\nreorientation is about 40\u000e[23] with a magneto-crystalline\nenergy as low as 0:1\u0002105J.m\u00003, that is less than a 1/50\nof the room temperature anisotropy so that the Co NR\ncan be considered to have no more magneto-crystalline\nanisotropy.\nIn the case of sample B, sintering of the wires inter-\nferes with the intrinsic magnetic properties of the Co\nNRs. On the other hand, in the case of sample A, the\nwires are structurally stable in shape at least up to 600\nK and the wires are rather well aligned with a disper-\nsion in their directions smaller than 7\u000e[17]. We can thus\nconsider sample A as consisting of individual particles\nbehaving as Stoner-Wohlfarth particles. At 550K when\nthe magneto-crystalline anisotropy vanishes, the coer-\ncivity of elongated ellipsoidal particles with their long\naxis aligned with the applied field can be expressed as\n\u00160HC= 2Kshape=MS[24] where Kshapecorresponds to\nthe effective anisotropy constant related to the shape\nanisotropy of the particles. Note that this formula is\nstrictly valid only if the magnetization rotation is co-\nherent, which requires that the particle diameter is very\nsmall ( \u00198 nm for Co). Our NRs dimensions are close\nto this limit. The value of the coercivity at 550K is\n\u00160HC\u0019210mT which thus corresponds to the con-\ntribution of the shape anisotropy. This represents only\nhalf of the theoretical value for an aspect ratio of 5\n(2Kshape=MS\u0019527mT) . The fact that the measured\nvalue is significantly smaller than the maximum theoret-\nical value for equivalent ellipsoid can be easily accounted\nfor by the fact that: (i) the NRs assembly is not well\naligned which dramatically reduces the coercivity related\nto the shape anisotropy, (ii) for an equivalent aspect ra-\ntio, ellipsoids have a higher coercivity compared to cylin-\nders because of domain nucleation [16] and (iii) we are\nnot working at 0 K so that thermally activated reversal\nplays a significant role.\nFrom a point of view of using Co NRs as a basis for\nthe fabrication of permanent magnetic materials, one im-\nportant aspect was to evaluate magnetic dipolar mag-\nnetic interactions between nanowires in dense magnetic\naggregates. This has been very recently addressed and\nmicromagnetic simulations have shown that dipolar in-\nteraction between wires are not detrimental to the high\ncoercivity properties, even for very dense aggregates [17].\nWith respect to high temperature operation, the present\nstudy shows that the temperature dependence of the Co\nmagneto-crystalline anisotropy is a real limitation. KMC\nnot only vanishes around 550 K but becomes negative\nFigure 7: MS(T)=MS(300K)(a) and HC(T)=HC(300K)(b)\nas a function of temperature as a function of temperature for\nsample (A) and sample (B). The solid line is a linear fit to\nthe experimental data corresponding to sample (A), which is\ngiven by\u00160HC\n\u00160HC(300K)= 1–a(T–300)(see text for details).\nabovethistemperature, which, asaconsequence, leadsto\nmagnetic properties which are very sensitive to the tem-\nperature. Nevertheless, the materials are rather resilient\nto high temperatures (up to 550 K) and are competi-\ntive with the other existing types of permanent magnetic\nmaterials. The magnetic performances of Co nanowires\nare of the order of (BH)max\u001912-15 MG Oe at room\ntemperature [12] which rank them in-between ferrites ( \u0019\n3 MG Oe) and AlNiCo ( \u00195 MG Oe) and RE-magnets\n(NdFeB \u001940 MG Oe; SmCo \u001920-30 MG Oe). A general\ndrawback of rare earth based materials is their relatively\nlower Curie temperature (580 K for NdFeB and 1023 K\nfor SmCo) which leads to a significant dependence of the\nmagnetization at high temperatures which is not the case\nfor pure Co systems ( TC\u00191300 K). NdFeB magnets can-\nnot be operated at temperatures above 400 K because\nof irreversible losses. SmCo magnets can be operated at\nelevated temperatures (up to 523 K typically) which is\nequivalent to our materials. They have a similar temper-\nature coefficient for the coercive field ( \u0019 \u00002\u000210\u00003K–1)\nas Co nanowires.7\nIV. CONCLUSION\nThe interest of the Co NRs prepared by the polyol\nprocess lies in their well-controlled morphology, diam-\neter smaller than 15 nm, high aspect ratio and shape\nhomogeneity, and, consequently, in their high coerciv-\nity at room temperature. In this paper, we have pre-\nsented a study on structural and magnetic properties of\nthese particles at high temperatures. In the tempera-\nture range 300-550 K, we show that the coercivity de-\ncreases linearly and that this variation is reversible. This\ncan be accounted for by the temperature dependence of\nthe magneto-crystalline anisotropy of cobalt. At 550 K,\n40%of the room temperature coercivity is maintained.\nThis value corresponds to the contribution of the shape\nanisotropy to the global anisotropy since at 550K the\nmagnetocrystalline contribution has vanished. Above\n525 K, the magnetic properties are irreversibly altered\neither by sintering or by oxidation. In absence of oxygen\nthe decomposition of metal-organic matter remaining at\nthe particle surface reduces the native cobalt oxide layer\nand provokes coalescence. The coercivity is irreversiblyaltered by the loss of shape anisotropy and by the par-\nticle growth that may induce a transition from mono- to\nmulti- domains magnetic particles. In slightly oxidative\nconditions, the growth of a thin oxide layer at the wire\nsurfacepreventsfromsinteringbutdecreasessignificantly\nthe saturation magnetization. In terms of temperature\nstability some further work is necessary to prevent both\ncoalescence and oxidation of the wires at high tempera-\ntures(>550K).Theuseofapassivationlayerisapossible\nroute to that.\nAcknowledgments\nThe authors gratefully acknowledge the Agence Na-\ntionale de la Recherche for their financial support\n(project 07-NANO-009 MAGAFIL). We thank F. 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Lett. 2009, 102, 245504.\n[21] Ciuculescu, D.; Dumestre, F.; Comesaña-Hermo, M.;\nChaudret, B.; Spasova, M.; Farle, M.; Amiens, C., Chem.Mater. 2009, 21, 3987.\n[22] Ono, F. J. Phys. Soc. Jap. 1981, 50, 2564.\n[23] Y. Barnier, R. Pauthenet, G. Rimet, C. R. Hebd. Acad.\nSci. 253 (1961) 400.\n[24] R. C. O’Handley, Modern Magnetic Materials, Principles\nand Applications, Wiley, New York, 2000, p. 319."    },    {        "title": "1209.2702v1.Spin_configurations_in_Co2FeAl0_4Si0_6_Heusler_alloy_thin_film_elements.pdf",        "content": "Spin con\fgurations in Co 2FeAl 0:4Si0:6Heusler alloy thin \flm elements\nC. A. F. Vaz,1,a)J. Rhensius,1, 2, 3, 4J. Heidler,1, 2P. Wohlh uter,1, 2, 4A. Bisig,1, 2, 4H. S. K orner,1, 2, 4T. O.\nMentes,5A. Locatelli,5L. Le Guyader,6F. Nolting,6T. Graf,7C. Felser,7L. J. Heyderman,3and M. Kl aui1, 2, 4, b)\n1)SwissFEL, Paul Scherrer Institut, 5232 Villigen PSI, Switzerland\n2)Laboratory for Nanomagnetism and Spin Dynamics, Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne (EPFL),\n1015 Lausanne, Switzerland\n3)Laboratory for Micro- and Nanotechnology, Paul Scherrer Institut, 5232 Villigen PSI,\nSwitzerland\n4)Fachbereich Physik, Universit at Konstanz, Universit atsstra\u0019e 10, 78457 Konstanz,\nGermany\n5)Sincrotrone Trieste, 34149 Basovizza-Trieste, Italy\n6)Swiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland\n7)Institute for Analytical and Inorganic Chemistry, Johannes Gutenberg-Universt at, 55099 Mainz,\nGermany\n(Dated: 9 November 2021)\nWe determine experimentally the spin structure of half-metallic Co 2FeAl 0:4Si0:6Heusler alloy elements using\nmagnetic microscopy. Following magnetic saturation, the dominant magnetic states consist of quasi-uniform\ncon\fgurations, where a strong in\ruence from the magnetocrystalline anisotropy is visible. Heating experi-\nments show the stability of the spin con\fguration of domain walls in con\fned geometries up to 800 K. The\nswitching temperature for the transition from transverse to vortex walls in ring elements is found to increase\nwith ring width, an e\u000bect attributed to structural changes and consequent changes in magnetic anisotropy,\nwhich start to occur in the narrower elements at lower temperatures.\nHeusler alloys are materials systems characterized by\nrich electronic and magnetic properties, such as half-\nmetallic behavior, large magneto-optic constants, and\nshape memory.1,2The combination of high magnetic\ncritical temperatures, large magnetization, and half-\nmetallicity, makes the Heusler alloys particularly inter-\nesting candidates for spintronics devices, for example,\nas spin injectors in spin polarized \feld e\u000bect transistors\n(spin-FET), magnetic layers in tunnel junctions, or as\nspin polarized materials for spintronics applications,1,3{5\nwhile recently a magnetoelectric coupling6has been pre-\ndicted in a Heusler/ferroelectric junction.7In half-metals,\none of the spin sub-bands is semiconducting, rendering\nelectron transport fully spin-polarized; for a robust half-\nmetallic behavior, a large separation between the Fermi\nenergy and the edges of the semiconducting sub-band is\ndesirable, to avoid thermal excitation and spin-\rip scat-\ntering of electrons to the conduction sub-band. Such\ntuning of the Fermi level position is possible by adjust-\ning the doping level xin half-metallic Co 2FeAl 1\u0000xSix,\nwhere it is found that for x\u00180:5, the Fermi Level lies in\nthe middle of the semiconducting spin sub-band.8,9An-\nother advantage of this composition is that the system\nbecomes less sensitive to atomic disorder, whose e\u000bect is\nto introduce edge states at the band gap of the semicon-\nducting sub-band and to reduce the spin polarization.8,9\nThese factors explain the large tunneling magnetoresis-\ntance (up to 160%)10,11and the e\u000ecient spin-transfer\nswitching in spin-valve nanopillar structures using this\na)Corresponding author. Email: carlos.vaz@cantab.net\nb)Current address: Institut f ur Physik, Johannes Gutenberg-\nUniverst at, 55099 Mainz, Germanyparticular composition.12\nTo utilize these unique features, a detailed knowledge\nof the spin structure in these materials at the nanoscale is\nnecessary. In this work, we study the magnetic spin con-\n\fguration of patterned Heusler alloy Co 2FeAl 0:4Si0:6thin\n\flms using x-ray magnetic circular dichroism photoemis-\nsion electron microscopy (XMCD-PEEM). We \fnd that\nthe spin con\fgurations in small elements (below 2 \u0016m)\nare quasi-uniform, while well de\fned domain walls are\nformed in narrow structures. The magnetic con\fgura-\ntions are found to be stable up to 800 K, making it par-\nticularly suitable for room temperature device applica-\ntions.\nThe structures investigated in this study consist of ar-\nrays of discs, rings, squares, rectangles, triangles, and\nzig-zag wires of various sizes, fabricated by d.c. mag-\nnetron sputtering of [1.5 nm] Ru/ [ t] Co 2FeAl 0:4Si0:6\n\flms, where t= 15;30 nm, on a polymethyl methacrylate\n(PMMA) resist spincoated on a Cr [10 nm]/MgO(001),\nde\fned using e-beam lithography, and subsequent lift-o\u000b.\nThe \flms were deposited at room temperature and the\nsamples were afterwards annealed at 500oC in a nitrogen\natmosphere for 30 min. X-ray di\u000braction measurements\non control samples grown under the same conditions indi-\ncate that the as-deposited Co 2FeAl 0:4Si0:6\flms grow epi-\ntaxially on Cr/MgO(001) with B2-type order, while after\nannealing L21-type order was observed.13The continuous\nCr layer prevents sample charging during magnetic imag-\ning using XMCD-PEEM. In this technique, a photoemis-\nsion electron microscope is used to image the local di\u000ber-\nence in light absorption at resonance for left and right cir-\ncularly polarized light. The magnetic contrast depends\non the angle between the helicity vector and magneti-\nzation, being maximum when they are parallel or anti-arXiv:1209.2702v1  [cond-mat.mtrl-sci]  12 Sep 20122\nparallel. For the XMCD measurements presented here,\nthe x-ray light was tuned to the Co Ledge, which displays\nlarger intensity than the Fe Ledge for Co 2FeAl 0:4Si0:6.\nDue to the grazing incidence of the photon beam, our\nmeasurements are mainly sensitive to the in-plane com-\nponent of the magnetization. Magnetic force microscopy\nimaging of selected elements (not shown) con\frms the\nin-plane orientation of the magnetization. 2D micromag-\nnetic simulations were carried out with the Object Ori-\nented MicroMagnetic Framework (OOMMF)14package\nusing the materials parameters: Ms= 1000 emu/cm3,\nA= 2:3\u000210\u00006erg/cm,K1=\u00009\u0002104erg/cm3, and 5\nnm in-plane cell size.3,13\nA survey of the magnetic con\fguration of the pat-\nterned elements after magnetic saturation (with a mag-\nnetic \feld of about 1 kOe) shows that high remanent\nstates are favored, although low remanent states, such\nas vortex states in discs and rings, and Landau states\nin bar elements, are also observed. Representative mag-\nnetic states of disc elements with varying diameter, fol-\nlowing saturation, are displayed in Fig. 1. For the 15\nnm thick elements, the dominant state is the so-called\ntriangle (T) state,15although vortex (V) states are also\nfound, as can be observed in Fig. 1(a). For the thicker\nelements, the triangle [Fig. 1(d) and (f)] and the S-state\n(S) [Fig. 1(c) and (e)], predicted numerically in Ref. 15,\nprevail. The magnetic contrast perpendicular to the av-\nerage magnetization, Fig. 1(b), shows that these states\nexhibit a signi\fcant modulation of the transverse com-\nponent of the magnetization, in agreement with the re-\nsults of micromagnetic simulations, Fig. 1(e,f). The dom-\ninance of high remanent states points to the important\nrole played by the magnetocrystalline anisotropy, which\nhelps to stabilize more uniform spin con\fgurations.15Our\nresults show that, for Co 2FeAl 0:4Si0:6alloy elements be-\nlow 2\u0016m in size, well de\fned magnetic states are ob-\nserved, determined by the interplay between exchange\nenergy and magnetocrystalline and shape anisotropies.\nThe existence and control of domain walls is a\nprerequisite for domain wall based applications and\nexperiments.16To study domain walls in Co 2FeAl 0:4Si0:6\nHeusler alloy elements, rings and zig-zag shaped wires\nwere imaged. The magnetic con\fguration of the ring el-\nements, after magnetic saturation, consists of so-called\n`onion states,' with domain walls that divide two quasi\nuniform domains in each half of the ring.17A strong\ntendency for ripple domains (which are usually associ-\nated with \ructuations in the direction of the magnetic\nanisotropy18) is found in the wire and ring elements. The\nspin con\fguration of the onion states is characterized by\nthe presence of local domains, as also observed in fcc\nCo and Fe 3O4ring elements due the magnetocrystalline\nanisotropy.19,20For the narrower elements we \fnd better\nde\fned domain wall structures, as illustrated in Fig. 2,\nwhich is explained by the stronger in\ruence of the shape\nanisotropy. While transverse domain walls are dominant,\nwe also observe vortex walls, showing that both spin con-\n\fgurations are stable at room temperature.16The do-\n(b) 30 nm  H 500 nm H \nH (c) (d) \n(e) (f) \n1 µm (a) 15 nm  \n[110]  [110]  - V T \nS V S \nT T S \nS T \nT \nS T H T \n1 µm \nS T S T S T \nS T \nS FIG. 1. XMCD-PEEM images of disc elements with varying\nsize, for (a) 15 nm and (b) 30 nm Co 2FeAl 0:4Si0:6\flms, with\nthe initial magnetic \feld Happlied along the [110] and [ \u0016110]\ndirections, respectively (300 K). Images are representative of\nthe states found across the larger arrays; examples of the\nvortex (V), triangle (T) and the S-state are shown, as labeled.\n(c-f) Detailed view of the triangle and S states for two 30\nnm Co 2FeAl 0:4Si0:6discs and corresponding micromagnetic\nsimulations.\nmain wall structure in the zig-zag wires is found to be well\nde\fned, as shown in the bottom panels of Fig. 2. The\nshape anisotropy leads to head-to-head and tail-to-tail\nspin con\fgurations at the wire bend, so that domain walls\nare formed, mostly transverse. These results show that\ndomain walls in Heusler alloys can be generated repro-\nducibly for elements with typical widths around 500 nm,\ndetermined mostly by the shape anisotropy. The do-\nmain wall spin con\fguration becomes more complicated\nfor wider elements, which are more strongly a\u000bected by\nthe magnetocrystalline anisotropy and magnetic ripple\ndomains. This suggests that there is room for optimiza-\ntion of the \flm quality and/or of the patterning method\nto reduce local pinning.\nWe investigate next the thermal stability of the do-\nmain walls in ring geometries with varying widths, from\n280 to 560 nm. After saturation, transverse domain walls\nare present in the rings, which transform progressively\nto vortex walls with increasing temperature (correspond-\ning the lowest energy state, according to the results of\nmicromagnetic simulations). Examples of transverse to\nvortex wall transitions with increasing temperature are\npresented in Fig. 3, showing images before and after the\ntransition for three ring widths taken during the heating\ncycle.\nThe transition temperature is plotted as a function of\nring width in Fig. 4, determined from the temperature de-\npendence of the number of domain walls that underwent\na transition (out of about six to eight), shown in the inset\nto Fig. 4. A clear dependence on the ring width is observ-3\n(a) 15 nm  \n500 nm  500 nm  \n500 nm  1 µm 250 nm  \n500 nm  250 nm  \nH [110]  [110]  - (b) 30 nm  \nH [110]  [110]  - \nFIG. 2. XMCD-PEEM image of the spin con\fguration in\nrings and curved wires with varying width, initially magne-\ntized along the [110] direction for (a) 15 nm and (b) 30 nm\nCo2FeAl 0:4Si0:6\flms (300 K). The arrow in (a) points to a\nregion in the ring element showing changes in the local trans-\nverse magnetization component that resemble ripple domains\nin continuous \flms. Note the di\u000berent magnetic contrast di-\nrection in the bottom right panel.\n(b) 340 nm  \n (a) 280 nm  (c) 560 nm  693 K  715 K  818 K  842 K  784 K  847 K  \nFIG. 3. Examples of domain wall transitions with increasing\ntemperature (as labeled), from transverse to vortex, in 30 nm\nthick Heusler alloy rings with widths from 280 nm to 560 nm\n(a-c). For each panel, the left image shows a transverse wall\nat a temperature below the domain wall transition, while the\nimage to the right shows a vortex wall at a temperature above\nthe transition.\nable showing surprisingly that wider elements switch at\nhigher temperatures, which is opposite to the expected\nbehavior, since transverse walls tend to be favored in nar-\nrower elements.21{24At temperatures above 890 K, re-\ncrystallization occurs due to Cr interdi\u000busion,10,25lead-\ning to irreversible structural and morphological damage\nto the structures (not shown). The temperature of this\npermanent destruction depends on the element size, with\nnarrow elements degrading \frst. Therefore, possible ex-\nplanations for the increase in the transition tempera-\nture with increasing ring width include changes in mag-\nnetic anisotropy with temperature10,25due to the onset\nof thermally activated Cr interdi\u000busion and crystalliza-\ntion processes, where smaller elements are a\u000bected at\nlower temperatures. Nevertheless, the heating experi-\nments show the stability of the spin con\fguration up to\n800 K, which marks the onset of irreversible magnetic\nchanges in these elements. Another observation is the\nnon-monotonic variation of the switching temperature\ndistribution with the ring width (Fig. 4, inset, and er-\nror bars in main graph), being particularly wide for the\n340 nm wide rings. One possible explanation for this be-havior could be a strong competition between the mag-\nnetocrystalline and shape anisotropy at this ring width,\nleading to a spread in the energy barrier height distri-\nbution that separates the vortex and transverse domain\nwall con\fgurations.\nFIG. 4. Average transition temperature ( T0) as a function of\nring width. Error bars correspond to the width of the tran-\nsition distribution based on sampling of about six transitions\nper ring width, as shown in the inset (lines are Lorentzian \fts\nto the data). For ease of display, the temperature axis in the\n\fgure inset has been o\u000bset by T0.\nIn conclusion, the feasible control of the spin structure\nin patterned Heusler alloys (Co 2FeAl 0:4Si0:6) is demon-\nstrated. In con\fned structures with sizes of around\n500 nm, the spin structure is well de\fned and is deter-\nmined mainly by the element shape. The high spin polar-\nization in Co 2FeAl 0:4Si0:6elements and the resistance to\nthermally activated changes make this material an inter-\nesting candidate for future applications and experiments.\nThis work was funded by the German Ministry for\nEducation and Science (BMBF), project \\Multimag\"\n(13N9911), the Graduate School \\Material Science in\nMainz\" (DFG/GSC 266), EU's 7th Framework Pro-\ngramme IFOX (NMP3-LA-2010 246102), MAGWIRE\n(FP7-ICT-2009-5 257707), the European Research Coun-\ncil through the Starting Independent Researcher Grant\nMASPIC (ERC-2007-StG 208162), the Swiss National\nScience Foundation, and the DFG. We also thank the\nteam of Sensitec GmbH, Mainz for technical support dur-\ning the \flm growth. Part of this work was performed at\nthe Swiss Light Source, Paul Scherrer Institut, Villigen,\nSwitzerland.\n1T. Graf, C. Felser, and S. S. P. Parkin, Progr. Solid State Chem.\n39, 1 (2011).\n2K. H. J. Buschow, P. G. van Engen, and R. Jongebreur, J. Magn.\nMagn. 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Inomata, and K. Hono,\nPhys. Rev. Lett. 102, 246601 (2009).\n12H. Sukegawa, S. Kasai, T. Furubayashi, S. Mitani, and K. Ino-\nmata, Appl. Phys. Lett. 96, 042508 (2010).\n13T. Graf, F. Casper, T. Gasi, V. Ksenofontov, A. Ruiz-Calaforra,\nA. Conca, B. Leven, G. Jakob, B. Hillebrands, and C. Felser,\n(2011), unpublished.\n14Available at http://math.nist.gov/oommf.\n15C. A. F. Vaz, M. Kl aui, L. J. Heyderman, C. David, F. Nolting,\nand J. A. C. Bland, Phys. Rev. B 72, 224426 (2005).\n16M. Kl aui, J. Phys.: Condens. Matter 20, 313001 (2008).17J. Rothman, M. Kl aui, L. Lopez-Diaz, C. A. F. Vaz, A. Bleloch,\nJ. A. C. Bland, Z. Cui, and R. Speaks, Phys. Rev. Lett. 86, 1098\n(2001).\n18K. J. Harte, J. Appl. Phys. 39, 1503 (1968).\n19M. Kl aui, C. A. F. Vaz, J. A. C. Bland, T. L. Monchesky, J. Un-\nguris, E. Bauer, S. Cheri\f, S. Heun, A. Locatelli, L. J. Heyder-\nman, and Z. Cui, Phys. Rev. B 68, 134426 (2003).\n20M. Fonin, C. Hartung, U. R udiger, D. Backes, L. Heyderman,\nF. Nolting, A. F. Rodr\u0013 \u0010guez, and M. Kl aui, J. Appl. Phys. 109,\n07D315 (2011).\n21R. D. McMichael and M. J. Donahue, IEEE Trans. Magn. 33,\n4167 (1997).\n22M. Kl aui, C. A. F. Vaz, J. A. C. Bland, L. J. Heyderman, F. Nolt-\ning, A. Pavlovska, E. Bauer, S. Cheri\f, S. Heun, and A. Locatelli,\nAppl. Phys. Lett. 85, 5637 (2004).\n23Y. Nakatani, A. Thiaville, and J. Miltat, J. Magn. Magn. Mater.\n290-291 , 750 (2005).\n24M. Laufenberg, D. Backes, W. B uhrer, D. Bedau, M. Kl aui,\nU. R udiger, C. A. F. Vaz, J. A. C. Bland, L. J. Heyderman,\nF. Nolting, S. Cheri\f, A. Locatelli, R. Belkhou, S. Heun, and\nE. Bauer, Appl. Phys. Lett. 88, 052507 (2006).\n25W. Wang, H. Sukegawa, R. Shan, T. Furubayashi, and K. Ino-\nmata, Appl. Phys. Lett. 92, 221912 (2008)."    },    {        "title": "1210.4386v1.Degree_of_order_dependence_on_magnetocrystalline_anisotropy_in_bct_FeCo_alloys.pdf",        "content": "arXiv:1210.4386v1  [cond-mat.mtrl-sci]  16 Oct 2012Applied Physics Express\nDegree of Order Dependence on Magnetocrystalline Anisotro py\nin Bct FeCo Alloys∗\nYohei Kota†and Akimasa Sakuma\nDepartment of Applied Physics, Tohoku University, Sendai 9 80-8579, Japan\nWe investigatethe magnetocrystalline anisotropy(MCA) en ergy of tetragonal distorted FeCo alloys depend-\ning on the degree of order by first-principles electronic str ucture calculation combined with the coherent\npotentialapproximation.Theobtained results indicateth at the MCA energyofFeCo alloys stronglydepends\non the degree of order under optimal conditions, where the ax ial ratio of the bct structure is 1.25 and the\ncomposition is Fe 0.5Co0.5. We find that the modification of the electronic structure res ulting from electron\nscattering by chemical disorder has a considerable influenc e on the MCA under these conditions.\nIn recent elemental strategies, it has been essential for hard ma gnetic materials\nappliedtomagneticdevices andpermanent magnetstobefreefrom rare-metalandrare-\nearth elements. Given this situation, Burkert et al.proposed that the FeCo alloy can be\nexpected to exhibit a giant magnetocrystalline anisotropy (MCA) en ergy under certain\nconditions by the first-principles calculation using the virtual cryst al approximation\n(VCA) to treat the binary alloy system.1In their paper, tetragonal distorted Fe 1−xCox\nalloys exhibit a large uniaxial magnetic anisotropy constant Ku, as high as about 700\n– 800µeV/atom, which is sufficiently comparable to the Kuvalue of a series of L10-\ntype alloys such as FePt, CoPt, and FePd,2under the conditions that the axial ratio\nbetween the aandcaxes,c/a,is 1.20–1.25 and that the composition xis 0.5–0.6. This\nresult has stimulated considerable expectation for their use as nov el materials with\nhigh coercivity; thus, many experimental trials have been perform ed to realize such a\nproperty.3–7Under the optimal conditions, the magnetic easy axis coincides with t he\nc-direction in agreement with the theoretical prediction; however, satisfactory results\nhave not yet been obtained at a quantitative level.\nRecently, Neise et al.8have performed a first-principles investigation on the MCA\nenergy of FeCo alloys with a chemical disordered structure by emplo ying both the VCA\n∗Submitted 19 September 2012; Accepted 15 October 2012 in App lied Physics Express.\n†E-mail address: kota@solid.apph.tohoku.ac.jp\n1/9Appl. Phys. Express\nand the supercell method. When using the VCA, they obtained resu lts similar to those\nof Burkert et al.;1however, for the supercell structure reflecting chemical disord er,\nthey obtained values of the MCA energy smaller by factors of 1.5–3.0 . Therefore, we\ncan consider that one of the reasons for the disagreement with th e experiments may\nbe the chemical disorder. Since the VCA is generally the lowest order approximation\nin terms of a random configuration of atoms, it may be inappropriate to calculate\nthe MCA energy in FeCo owing to the insufficient treatment of the che mical disorder.9\nHowever, the physical explanation for this discrepancy and the infl uence of the chemical\ndisorder on the MCA in FeCo are not yet understood. In this work, w e investigate the\nMCA in disordered FeCo alloys by means of the coherent potential ap proximation\n(CPA),10,11which provides us with a physical picture of the effects of chemical d isorder\nin a transparent manner. Employing the CPA has the advantage of r eflecting electron\nscattering by the chemical disorder, which has not been considere d in the VCA and the\nsupercell approach. In addition, it allows us to continuously change the order parameter\nSof the crystal,12–14a parameter that is measurable in experiments.\nElectronic structure calculations are performed by using the tight -binding linear\nmuffin-tin orbital (TB-LMTO) method under the local spin-density f unctional approx-\nimation.15–17Chemical disorder is taken into account by means of both the CPA an d\nthe lowest order VCA to compare the obtained results. In the form er approximation,\nthe averaged potential function /angbracketleftP/angbracketright, which is used in the TB-LMTO method and re-\nflects the random distribution of Fe and Co atoms, is determined via t he iterative CPA\ncondition for single-site t-matrices. In the latter, /angbracketleftP/angbracketrightis determined only by the sim-\nple weighted average of the potential functions of both atoms in ac cordance with the\ncomposition ratio,16so there is no electron scattering. In both methods, we take adva n-\ntage of the Green’s function technique, which needs an infinitesimal imaginary value\nreflecting causality; we set this value to 2.0 mRy in this work. The MCA e nergies are\ncalculated based on the force theorem with inclusion of the spin-orb it interaction after\na self-consistent calculation. The lattice constants are determine d by the volumes of\nthe unit cell of the experimental data of the bulk Fe-Co alloys18and the optional axial\nratioc/a. The number of k-points used for MCA energy calculation is about 503in the\nfull Brillouin zone.\nFirst, we examine the Kuvalue of completely disordered Fe 0.5Co0.5(S= 0.0) as a\nfunction of the axial ratio c/ain Fig. 1. In both the CPA and the VCA, we can measure\nthe peak of Kuaroundc/a= 1.25. We note that Kuis zero for c/a= 1.0 and√\n2\n2/9Appl. Phys. Express\nowing to the cubic symmetry. This behavior is qualitatively consistent with the results\nof Burkert et al.,1although the value of Kucalculated using the VCA is almost half\nat the peak position. This discrepancy may be attributed to the diffe rence in the band\ncalculation method; Burkert et al.1used the full-potential LMTO method, which can\nprovide more accurate results than the present method if one calc ulates the electronic\nstructure of a perfectly ordered crystal, i.e.,S= 1.0. Here, it should be emphasized\nthat the value of Kucalculated using the CPA is further reduced compared with that\ncalculated using the VCA, especially around c/a= 1.25. This behavior is consistent\nwith the results obtained by Neise et al.using the supercell method.8\nNext, let us lookat the degree of order dependence of Kuforc/a= 1.25 in Fe 0.5Co0.5\nin Fig. 2. Here, we note that there are two equivalent sites in the prim itive cell of the\ntetragonal lattice. We set the composition at the body center site and the corner one in\nthe primitive cell to Fe 1−ηCoηand Fe ηCo1−η(0< η <0.5), respectively; the long-range\norder parameter of the crystal is defined by S= 1−2η(0.0< S <1.0).14ForS= 1.0,\nthe lattice is a perfectly ordered structure; therefore, the res ults obtained using the\nVCA and CPA are invariant. In Fig. 2, there is a large difference betwe en these results\nforS <0.8; therefore, we can consider that the values obtained using the lo west order\nVCA possibly overestimate the uniaxial anisotropy constant of FeC o compared with the\nvalues obtained using the CPA, which is a more proper way of dealing wit h a disordered\nalloy system.11Therefore, this result reveals that the giant MCA expected from F eCo\nis not present even if there is a slight disorder in the crystal.\nTo gain insight into the physical feature giving rise to the different be havior of\nthe degree of order dependence of the MCA calculated using the VC A and CPA, we\nfocus on the partial Bloch spectral functions of the dx2−y2anddxyorbital components\nin the minority-spin states at the Γ-point plotted in Fig. 3. According to Burkert et\nal., these two orbital components have a major contribution to the M CA energy of\ntetragonal distorted FeCo alloys such that the magnetic easy axis coincides with the\nc-direction. This can be understood by expressing the energy varia tion arising within\nthe second-order perturbation in terms of the spin-orbit interac tionHSO,19,20\nδE=occ/summationdisplay\nnunocc/summationdisplay\nn′|/angbracketleftkn↓|HSO|kn′\n↓/angbracketright|2\nεkn↓−εkn′↓/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nk=Γ, (1)\nwhereεkn↓denotes the eigenvalue of the nonperturbative eigenstate |kn↓/angbracketrightin the\nminority-spin state. Thus, the projected components /angbracketleftdxy|kn↓/angbracketright·|dxy/angbracketrightand/angbracketleftdx2−y2|kn↓/angbracketright·\n3/9Appl. Phys. Express\n|dx2−y2/angbracketrightmainly contribute to δEand MCA energy. The restriction of the summation\nis applied to the occupied (unoccupied) state below (above) the Fer mi level; therefore,\neq. (1) indicates that the MCA energy becomes large when the Ferm i level lies be-\ntweendx2−y2anddxystates. As shown in Fig. 3, the partial Bloch spectral function\nof FeCo ( x= 0.5,c/a= 1.25) for (a) ordered structure ( S= 1.0) and (b) disordered\none (S= 0.0) calculated using the VCA certainly supports this picture. The pea ks of\nthe spectra of the occupied dx2−y2and unoccupied dxystates are coupled to each other\nthrough the spin-orbit interaction, thereby inducing a large MCA. T he MCA energy is\nmore enhanced for S= 1.0, since the energy separation between the occupied dx2−y2\nand unoccupied dxystates is smaller than that for S= 0.0 calculated using the VCA.\nThis is one of the reasons why the uniaxial anisotropy constants inc rease with Sin Fig.\n2.\nHowever, the spectra for S= 0.0 obtained via the CPA shown in Fig. 3(c) also seem\nto meet this condition, although it should be noted here that in Fig. 3( c), the shapes\nof these spectra around the Fermi level are modified owing to elect ron scattering by\nthe random arrangement of atoms. The width of the spectra incre ases compared with\nthat obtained using the VCA as shown in Fig. 3(b). Some portions of t he spectrum\nlie in the occupied state and others lie in the unoccupied states becau se of the wide\nenergy distribution of each spectrum around the Fermi level. When we evaluate the\nenergy variation δEthat results from mixing dxyanddx2−y2states in the minority-spin\nstate through the spin-orbit interaction by using the nonperturb ative Green’s function\nobtained using the CPA and VCA based on the approach of Solovyev et al.,21we\nconfirm that δEcalculated using the CPA is about one-half of that found using the\nVCA;22therefore, this modification of the spectra contributes to decre asing the MCA\nenergy in the bct FeCo alloy in Figs. 1 and 2. Since in the CPA calculation, the energy\nwidths of these two states increase with decreasing S, theKuvalues from the CPA\ndecrease much faster than those from the VCA when Sdecreases. Therefore, the giant\nMCA energy of the bct FeCo alloy can be realized under the quite stric t condition\nconcerning the dxyanddx2−y2states at the Γ-point.\nIn summary, we investigated the dependence of MCA energy on the degree of order\nof FeCo alloys by means of a first-principles calculation combined with t he CPA. The\nresults indicate that the MCA energy of FeCo alloys strongly depend s on the degree\nof order under the optimal condition of the axial ratio c/a= 1.25. This is because the\nmodification of the spectral function resulting from electron scat tering in the chemical\n4/9Appl. Phys. Express\ndisordered structure has a considerable influence on the MCA ener gy under this condi-\ntion, and so the Kuvalue of the FeCo alloy obtained using the CPA is about one-half\nof that obtained using the VCA; therefore, electron scattering h as a significant influ-\nence on the MCA energy, which gives rise to the large difference betw een the VCA and\nCPA calculations in the small- Sregion. Finally, we should stress that the high degree\nof order, in addition to the large distortion of the lattice, is a necess ary condition for\nobtaining the large MCA from FeCo in experiments.\nAcknowledgments\nOne of the authors (Y.K.) was supported by a Grant-in-Aid for JSPS Fellows (22-\n6092). This work was supported by JST under Collaborative Resear ch Based on Indus-\ntrial Demand “High Performance Magnets: Towards Innovative De velopment of Next\nGeneration Magnets”.\n5/9Appl. Phys. Express\n 0100200300400500\n1.01.11.21.31.4 0204060Ku   [µeV/atom]\nKu   [Merg/cc]\nc/aFe0.5    Co0.5     (S=0.0)\nCPA\nVCA\nFig. 1. Uniaxial anisotropy constant Kuof FeCo for S= 0.0 as a function of the axial ratio c/a.\nThe results obtained from the CPA and VCA are denoted by the open circles and filled squares,\nrespectively.\n 0100200300400500\n0.00.20.40.60.81.0 0204060Ku   [µeV/atom]\nKu   [Merg/cc]\nSFe0.5    Co0.5     (c/a=1.25)\nCPA\nVCA\nFig. 2. Uniaxial anisotropy constant Kuas a function of degree of order Sof FeCo with c/a= 1.25.\nThe open circles and filled squares represent the results from the C PA and VCA, respectively.\n6/9Appl. Phys. Express\n  0 50100150200\n-0.4 -0.2  0.0  0.2pBSF at k= Γ  [states/Ry]\nε - εF    [Ry]FeCo (c/a=1.25,S=1.0)\nd(xy)       \nd(x2 -y2  )\n  0 50100150200\n-0.4 -0.2  0.0  0.2pBSF at k= Γ  [states/Ry]\nε - εF    [Ry]FeCo (c/a=1.25,S=0.0,VCA)\nd(xy)       \nd(x2 -y2  )\n  0 50100150200\n-0.4 -0.2  0.0  0.2pBSF at k= Γ  [states/Ry]\nε - εF    [Ry]FeCo (c/a=1.25,S=0.0,CPA)\nd(xy)       \nd(x2 -y2  )\nFig. 3. Partial Bloch spectral function of dx2−y2anddxystates at the Γ point in FeCo with\nx= 0.5 andc/a= 1.25: (a) results for the ordered structure ( S= 1.0); (b) and (c) results for the\ndisordered structure ( S= 0.0) from the VCA and CPA, respectively.\n7/9Appl. Phys. Express\nReferences\n1)T. Burkert, L. Nordstr¨ om, O. Erikssonm, and O. Heinonen: Phys . Rev. Lett. 93\n(2004) 027203.\n2)T. Klemmer, D. Hoydick, H. Okumura, B. Zhang, and W. A. Soffa: Scr ipta Met.\nMater.33(1995) 1792.\n3)G. Andersson, T. Burkert, P. Warnicke, M. Bj¨ orck, B. Sanyal, C . Chacon, C.\nZlotea, L. Nordst¨ orm, P. Nordblad, and O. Eriksson: Phys. Rev. Lett.96(2006)\n037205.\n4)A. Winkelmann, M. Przybylski, F. Luo, Y. Shi, and J. Barthel: Phys. R ev. Lett.\n96(2006) 257205.\n5)P. Warnicke, G. Andersson, M. Bj¨ orck, J. Ferre´ e, and P. Nord blad: J. Phys.:\nCondens. Matter 19(2007) 226218.\n6)F. Luo, X. L. Fu, A. Winkelmann, and M. Przbylski: Appl. Phys. Lett. 91(2007)\n262512.\n7)F. Yildiz, M. Przybylski, X.-D. Ma, and J. Kirschner: Phys. Rev. B 80(2009)\n064415.\n8)C. Neise, S. Sch¨ onecker, M. Richterm K. Koepernik, and H. Eschr ig: Phys. Status\nSolidi B248(2011) 2398.\n9)G. Hautier, A. Jain, and S. P. Ong: J. Mater. Sci. 47(2012) 7317.\n10)B. Velick´ y, S. Kirkpatrick, and H. Ehrenreich: Phys. Rev. 175(1968) 747.\n11)R. J. Elliott, J. A. Krumhansl, and P. L. Leath: Rev. Mod. Phys. 46(1974) 465.\n12)J. B. Staunton, S. Ostanin, S. S. A. Razee, B. Gyorffy, L. Szunyo gh, B.\nGinatempo, and E. Bruno: J. Phys. C: Condens. Matter 16(2004) S5623.\n13)Y. Kota and A. Sakuma: J. Appl. Phys. 111(2012) 07A310.\n14)Y. Kota and A. Sakuma: J. Phys. Soc. Jpn. 81(2012) 084705.\n15)H. L. Skriver: The LMTO Method (Springer, Berlin, 1984).\n16)I. Turek, V. Drchal, J. Kudrnovsk´ y, M. Sˇ ob, and P. Weinberger :Electronic\nStructure of Disordered Alloys, Surface and Interfaces (Kluwer, Boston, 1997).\n17)I. Turek, V. Drchal, and J. Kudrnovsk´ y: Philos. Mag. 88(2008) 2787.\n18)R. M. Bozorth: Ferromagnetism (van Nostrand, New York, 1951).\n19)P. Bruno: Phys. Rev. B 39(1989) 865.\n20)D. Wang, R. Wu, and A. J. Freeman: Phys. Rev. B 47(1993) 14932.\n21)I. V. Solovyev, P. H. Dederichs, and I. Mertig: Phys. Rev. B 52(1995) 13419.\n8/9Appl. Phys. Express\n22)Y. Kota and A. Sakuma: in preparation for publication.\n9/9"    },    {        "title": "1212.3144v1.First_principles_study_of_the_structural_stability_of_Mn3Z__Z_Ga__Sn_and_Ge__Heusler_compounds.pdf",        "content": "1 First-principles study of the structural stability of Mn3Z (Z=Ga, Sn and Ge) Heusler compounds Delin Zhang 1, Binghai Yan 1,2,3, Shu-Chun Wu 1, Jürgen Kübler 4, Guido Kreiner1, Stuart S. P. Parkin 4, Claudia Felser 1,2  1 Max-Planck-Institute für Chemische Physik fester Stoffe, Nöthnitzer Str. 40, 01187 Dresden, Germany 2 Johannes Gutenberg-Universität Mainz, Staudingerweg 9, 55128 Mainz, Germany 3 Max Planck Institute for the Physics of Complex Systems, Nöthnitzer Straße 38 01187 Dresden, Germany 4 Institut für Festkörperphysik, Technische Universität Darmstadt, 64289 Darmstadt, Germany 5 IBM Almaden Research Center San Jose, CA 95120, USA.  Abstract  We investigate the structural stability and magnetic properties of cubic, tetragonal and hexagonal phases of Mn3Z (Z=Ga, Sn and Ge) Heusler compounds using first-principles density-functional theory. We propose that the cubic phase plays an important role as an intermediate state in the phase transition from the hexagonal to the tetragonal phases. Consequently, Mn3Ga and Mn3Ge behave differently from Mn3Sn, because the relative energies of the cubic and hexagonal phases are different.  This result agrees with experimental observations from these three compounds. The weak ferromagnetism of the hexagonal phase and the perpendicular magnetocrystalline anisotropy of the tetragonal phase obtained in our calculations are also consistent with experiment. 2 I. Introduction The Mn3Z (Z=Ga, Sn and Ge) type of Heusler compounds can have three different structural phases, where each phase exhibits different magnetic properties. The hexagonal phase has been known for decades. Mn atoms form a Kagome lattice in a plane with a Z atom in the center of the hexagon. Here, Mn atoms have a triangular antiferromagnetic (AFM) coupling with a weak net magnetic moment [1~8]. The cubic phase is the standard full Heusler structure. In this phase, Mn atoms are present in two unique lattice sites; these different sites have magnetic moments with opposite directions, leading to ferrimagnetic (FiM) order [9, 10]. The cubic phase has a high density of states at the Fermi energy, and hence a Peierls transition could occur [11], giving rise to the third phase, the tetragonal phase. The tetragonal phase can be treated as a cubic phase with a distortion along the z direction. This distortion makes the magnetic moments favor the z axis, meaning the system possesses perpendicular magnetocrystalline anisotropy (PMA) [11], which promises great potential for future high-density spin-transfer torque applications [12,13,14~21]. In experiments, the hexagonal phase of Mn3Z (Z=Ga, Sn and Ge) was synthesized by annealing the samples at high temperatures [1~3, 22], while the tetragonal phase of Mn3Ga and Mn3Ge was realized by low temperature annealing [1, 3, 23, 24]. The cubic phase has not been observed so far, since it may be unstable as mentioned above. On the other hand, the tetragonal phase of Mn3Sn has not yet been reported in the literature, although it is expected to behave similarly to the other two compounds. The possible transition between hexagonal, cubic and tetragonal phases catches the attention of researchers [3, 25], but a comprehensive study is yet to be undertaken.  In this paper, we investigate the structural stability and magnetic properties of the hexagonal, cubic and tetragonal phases for the three compounds Mn3Z (Z=Ga, Sn and Ge) using first-principles calculations. The tetragonal phase of Mn3Sn is found to have the lowest total energy among the three phases, similar to Mn3Ga and Mn3Ge. However, the cubic phase of Mn3Sn has higher energy than its hexagonal phase, in contrast to the Mn3Ga and Mn3Ge compounds. In the case of Mn3Sn, we suggest it is difficult for the 3 hexagonal phase to transform from the cubic phase into the tetragonal one. This might explain the lack of the tetragonal Mn3Sn phase. II Calculation details First-principles calculations were carried out using Vienna ab initio Simulation Package (VASP) [26]. The ions were described using projector augmented wave (PAW) potentials [27]. The generalized gradient approximation (GGA) in the Perdew-Bruke-Ernzerhof (PBE) form [28] was adopted to describe the exchange-correlation interactions between electrons. An energy cutoff of 500 eV was used for the plane wave basis. Spin-orbit coupling was employed in all calculations to describe the non-collinear spin polarization and magnetocrystalline anisotropy. The volume and shape (c/a) of the cubic, tetragonal and hexagonal structures were fully relaxed to get the stable structural configurations with the lowest energy. In addition, the full potential linear augmented plane wave (FLAPW) method [29] was also used to check the validity of the pseudopotential calculations.  III. Results and discussion As shown in Figure 1, the cubic phase belongs to the X2YZ full Heusler structure (Fm-3m). MnII (X position), MnI (Y position) and Z (Z=Ga, Sn and Ge) atoms occupy the (1/4, 1/4, 1/4), (1/2, 1/2, 1/2) and (0, 0, 0) sites, respectively. The magnetic moments of MnI and MnII orient oppositely. The tetragonal phase (I4/mmm) has an elongated c axis and shortened a axis as compared to the cubic lattice. The magnetic order, as in the cubic phase, is ferrimagnetic. Our optimized lattice parameters and magnetic moments (per Mn3Z unit) agree well with previous experiments [1, 3, 21, 24] and calculations [17, 23] (see Tables I and II). The PMA energy is defined as the energy difference between the easy direction (001) and the in-plane direction (010). In the tetragonal phase, the PMA is around 1 meV for all three compounds, consistent with previous calculations on Mn3Ga and Mn3Ge [11, 21].  The Mn atoms in the hexagonal phase exhibit various possible magnetic configurations. On the one hand, in the plane of the triangle lattice, the magnetic moments of Mn atoms may point in-plane or out-of-plane. On the other hand, one finds 4 that the Mn-Mn bonds between neighboring layers in of Mn3Z are a little shorter than the in-plane Mn-Mn bonds. Therefore the interlayer magnetic coupling is also important. As a summary, we display the most important configurations in Fig. 2. The AFM type of in-plane ordering in Fig. 2(d) is found to be the most stable. However, the directions of the magnetic moments are not equally separated by 120°, therefore they do not fully cancel each other, resulting in a weak ferromagnetic phase [30~32]. The calculated magnetic moments also agree with previous experiments for all three materials. For example, the magnetic moments are 0.03 µB (exp. 0.045 µB [1]) for Mn3Ga, 0.01 µB (exp. 0.009 µB [2]) for Mn3Sn and 0.01 µB (exp. 0.06 µB [3]) for Mn3Ge. In Mn3Ge, the variation between the experimental and calculated results originates mostly from the fact that the experimental compositions are off-stoichiometry  [3, 22]. It is important to compare the energetic stability of the three phases for Mn3Z (Z=Ga, Sn and Ge). Figure 3 shows the total energy dependence on the volume of hexagonal, cubic and tetragonal structures for these three compounds. We relaxed all the structure parameters (volume and shape (c/a) of the lattice) to reach the most stable structures. In the tetragonal and hexagonal phases, for example, the c/a-ratio was fully optimized for any given value of the volume. One unambiguously finds in all three compounds that the tetragonal phase is energetically more stable than the cubic and hexagonal phases. For Mn3Ga and Mn3Ge, the hexagonal phase has the highest total energy, while the cubic phase exhibits the highest energy for Mn3Sn. Although the cubic phase is claimed to be unstable in experimental work [23], its relative energy between these three phases is very important in understanding the stability of the tetragonal and hexagonal phases. We propose that the hexagonal phase does not change into the tetragonal phase directly; rather it transitions through the cubic lattice, which is structurally intermediate. It is simple to see that distortion along the c direction of the cubic phase will lead to the tetragonal phase, while the transition between the cubic and hexagonal phases is not as straightforward. If the cubic lattice is projected along the diagonal direction, one obtains a trianglar lattice. In order to recover a hexagonal phase, four atomic layers including three Mn layers and one Z layer should be compressed into one layer. However, this cannot be realized by a simple projection of ABC sites, for the Z atom overlaps with 5 one Mn atom in this way (Figs. 1(d) and 1(e)). In order to host the additional Mn atoms inside a layer, the honeycomb lattice of Mn (Fig. 1(d)) should change into a Kagome lattice (Figs. 1(f) and 1(g)) to create more available sites for Mn atoms. On the one hand, the transition from the cubic to the hexagonal phase requires compressive pressure along the cubic diagonal direction in order to push Mn and Z atoms inside a plane. On the other hand, the transition from the hexagonal to the cubic phase also needs compressive strain in the ab plane of the hexagonal lattice.  For Mn3Ga and Mn3Ge, the cubic phase is energetically between the hexagonal and tetragonal phases. In this case, the hexagonal phase can easily pass through the cubic phase into the tetragonal phase. Therefore, both phases can be realized in experiments [1, 3, 23, 24]. In contrast to the above two compounds, Mn3Sn has a cubic phase whose energy is higher than the hexagonal phase. Consequently, it is difficult to transform from an existing hexagonal phase into a tetragonal one, possibly explaining why tetragonal Mn3Sn has not been synthesized so far.   We can explain the transition between the hexagonal and tetragonal phases using the cubic phase. However, there still remains a puzzle: the tetragonal structure of Mn3Sn has lower energy than the hexagonal one in our calculations, while experiments only observe the hexagonal phase. This may be due to the structural disorder or off-stoichiometry compositions that commonly exist in experiments [2]. In these cases, the hexagonal phase may have lower energy than the tetragonal counterpart. We will investigate these effects in a future work. IV. Conclusions We calculated the structures and compared the stabilities of the cubic, hexagonal and tetragonal phases for Mn3Ga, Mn3Ge and Mn3Sn. The cubic phase plays an important role as an intermediate state in the phase transition from the hexagonal to the tetragonal phase. Consequently, the cubic phase is necessary to analyse the stability of the other two phases and understand the experiments. For Mn3Ga and Mn3Ge, the cubic phase lies between the hexagonal phase with the highest energy and the tetragonal phase with the lowest energy. Consequently, it is possible to transform the hexagonal phase, through the cubic phase, into the tetragonal phase. This is consistent with experimental 6 observations. However, for Mn3Sn, the cubic phase has a higher energy than the hexagonal phase. 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Santi, arXiv:cond-mat/0408200. 9 Figure 1 (D. L. Zhang et al.)  \n  Figure 1 (Colour online) Crystal structure relationships of (a) cubic-to-tetragonal and (b) cubic-to-hexagonal phases. The projection from the cubic to hexagonal structures is shown in (c)-(g). Four atomic layers (three Mn layers and one Z layer) will be projected into a single layer, in order to form the hexagonal structure. The orange arrows in (a)-(b) indicate the direction of structural deformation, and the arrows on the Mn atoms in (a) denote the direction of magnetic moment. In addition, Mn atoms are represented by red or blue balls, and Z (Z=Ga, Sn and Ge) atoms by green balls.  \n10 Figure 2 (D. L. Zhang et al.)           Figure 2 (Colour online) A schematic of possible magnetic configurations in the hexagonal lattice of Mn3Z (Z=Ga, Sn and Ge). The arrows denote the direction of the magnetic moments. The magnetic moments of Mn atoms lie in the hexagonal basal plane in (a)-(h), and out-of-plane in (i)-(j). The blue and red balls represent the Mn atoms in different planes. The magnetic configuration of (d) is found to be the most stable by our calculations. \n11 Figure 3 (D. L. Zhang et al.) \n859095100105110115120125130-30.6-30.3-30.0-29.7-29.4\n95100105110115120125130-30.9-30.6-30.3-30.0-29.7\n859095100105110115120125-32.1-31.8-31.5-31.2ΔE(tetra-hex)=0.27 eV   Etotal (eV)Etotal (eV)Etotal (eV) cubic tetragonal hexagonalMn3Ga(a)\nΔE(tetra-hex)=0.11 eV    cubic tetragonal hexagonalMn3Sn(b)\nΔE(tetra-hex)=0.24 eV  \n Volume (Å3) cubic tetragonal hexagonalMn3Ge(c)\n  Figure 3 (Colour online) The dependence of the total energy per Mn3Z unit on volume for the cubic, tetragonal and hexagonal structures of (a) Mn3Ga, (b) Mn3Sn, and (c) Mn3Ge. For a given volume, the shape of the lattice (c/a) is also fully optimized.   12 Table I.  Optimized lattice parameters of hexagonal, cubic, and tetragonal structures in Mn3Z (Z=Ga, Sn and Ge) Heusler compounds.  Lattice constant (Å) Mn3Ga Mn3Sn Mn3Ge Hexagonal a=5.26, c=4.26 (a=5.400, c= 4.353 exp.[1]) a=5.57, c=4.43 (a=5.665, c= 4.531 exp.[2]) a=5.28, c=4.22 (a=5.360, c=4.320 exp.[3]) Cubic a=5.82 (a=5.823 calc.[10]) a=6.04 a=5.75 Tetragonal a=3.77, c=7.16  (a=3.77, c=7.16 calc.[23] a=3.909, c=7.098 exp.[24]) a=3.93, c=7.47  a=3.75, c=7.12  (a=3.81, c=7.26 exp.[3, 21])  Table II. Total magnetic moments per Mn3Z unit of hexagonal, cubic, and tetragonal structures. For the hexagonal phase, the µMn denotes the magnetic moment of each Mn atom. The energy of perpendicular magnetocrystalline anisotropy (per primitive unit cell, Mn6Z2) of the tetragonal structure is shown.   Magnetic Moment (µB) Mn3Ga Mn3Sn Mn3Ge Hexagonal 0.03 (0.045 exp.[1]) 0.01 (0.009 exp.[2]) 0.01 (0.06 exp.[3]) µMn=2.5 (2.4±0.2 exp.[1]) µMn=2.9 (3.0 exp.[2]) µMn=2.5 (2.4±0.2 exp.[3]) Cubic -0.01(-0.01 calc.[10]) -1.00 -1.00 Tetragonal 1.78 (1.77 calc.[23]) 1.04 0.97(1.00 exp.[21]) PMA=1.0 meV (1.0 meV calc.[11]) PMA=1.1 meV PMA=1.0 meV (0.8 meV calc.[21])   "    },    {        "title": "1301.4164v1.Imaging_the_antiferromagnetic_to_ferromagnetic_first_order_phase_transition_of_FeRh.pdf",        "content": "arXiv:1301.4164v1  [cond-mat.mtrl-sci]  17 Jan 2013Imaging the antiferromagnetic to ferromagnetic first order phase transition of FeRh\nS. O. Mariager,∗L. Le Guyader,†M. Buzzi, G. Ingold, and C. Quitmann\nSwiss Light Source, Paul Scherrer Institut, 5232 Villigen, Switzerland\nThe antiferromagnetic (AFM) to ferromagnetic (FM) first ord er phase transition of an epitaxial\nFeRhthin-filmhas beenstudied with x-raymagnetic circular dichroism usingphotoemission electron\nmicroscopy. The FM phase is magnetized in-plane due to shape anisotropy, but the magnetocrys-\ntalline anisotropy is negligible and there is no preferred i n-plane magnetization direction. When\nheating through the AFM to FM phase transition the nucleatio n of the FM phase occurs at many\nindependent nucleation sites with random domain orientati on. The domains subsequently align to\nform the final FM domain structure. We observe no pinning of th e FM domain structure.\nI. INTRODUCTION\nThe metallic alloy FeRh undergoes an uncommon\nphase transition at T T≈105oC. Here the magnetic or-\nder upon heating changes from antiferromagnetic(AFM)\nto ferromagnetic (FM) while the lattice simultaneously\nexpands by ∼0.7 %.[1] As the transition is first order\nit involves a latent heat and phase coexistence of the\nAFM and FM states. The value of T Tabove room tem-\nperature is attractive for technological applications and\nhas lead to proposed uses in thermally assisted magnetic\nrecording[2], magnetocaloric refrigiration[3] and magne-\ntostrictive transduction[4]. For any application utilizing\nthe phase transition of FeRh an improved understanding\nof the dynamics and the phase coexistence is of inter-\nest. The phase transition also provides an opportunity\nto study the general phenomena of phase coexistence in\na first order magnetic phase transition. Other materials\nwhere a magnetostructural transition plays a crucial role\nfor the properties include the paramagnetic to FM tran-\nsition in MnAs [5, 6] and the interplay between magnetic\nand structural domains in shape memory alloys like the\nHeusler alloy Ni 2MnGa.[7] Phase coexistence also plays\na vital role in strongly correlated materials like Mangan-\nites. [8, 9]\nFeRh has the CsCl crystal structure and the magnetic\nstructure is known from M¨ ossbauer spectroscopy[10],\nneutron diffraction[11] and band-structure calculations\n[12]. The unit cell is sketched in FIG. 1. The AFM phase\nis of type II with nearest neighbor Fe atoms aligned an-\ntiferromagnetically with moments m Fe≈3µBand zero\nmagnetic moment on the Rh atoms. In the FM phase\nthe Fe atoms align ferromagnetically and a moment is\ninduced on the Rh atoms, m Rh≈1µB. In bulk samples\nthe magnetic transition is accompanied by an isotropic\nstructural expansion of ∼0.7 %, while for thin-films\nthe in-plane expansion is restricted by the substrate[13].\nHere the expansion at the phase transition is only along\nthe surface normal as sketched in FIG. 1. While the\n∗Electronic address: simon.mariager@psi.ch\n†Current address: Helmholtz-Zentrum Berlin fur Materialie n und\nEnergie, Bessy II, 12489 Berlin, Germanyelectronic structure is largely unaffected by the transi-\ntion in both bulk crystals[14] and thin films,[15] the heat\ncapacity, entropy and electrical resistance all change at\nTT[16, 17]. The exact transition temperature depends\non both composition[18] and the addition of transition\nmetal impurities[19], which complicates comparison be-\ntween different experiments. Though the AFM to FM\nphase transition has been known since 1939 [1] the phys-\nicalmechanismbehind it is still debated. The firstsimple\nempirical model proposed historically assumed a change\nof sign of the Fe-Fe interaction as a function of the lattice\nexpansionbutfailedtoexplaintheexperimentalwork,es-\npecially the large change in entropy during the transition\n[18]. On the other hand density functional theory (DFT)\ncomputations reproduce the existence of the AFM and\nFMphasesandthe volumedependencyofthelattice con-\nstant [12]. The low energy difference ( ∼0.2 mRy/atom)\nbetween the AFM and FM phases could be explained by\nconsidering the effect of spin waves [20] and recent local\nDFT work indicate that the volume dependence of the\nAFM Fe-Fe exchange interaction combined with the un-\naffected FM Fe-Rh interaction play a crucial role for the\nFIG.1: (Coloronline)SketchoftheFeRhunitcellintheAFM\nand FM phases. Red spheres symbolize Fe atoms and blue\nspheres Rh atoms with the direction of magnetic moments\nindicated by the arrows. The structural change is shown for\nthe thin film case where expansion only occur along the sur-\nface normal. The structural expansion has been enhanced for\nclarity.2\nphysical properties of FeRh. [21]\nThecoexistenceoftheFM andAFM phasesduringthe\nphase transition has been reported in several recent ex-\nperiments. X-ray magnetic circular dichroism (XMCD)\nexperiments were interpreted to conclude that even in\nthe early stages of the transition the microscopic FM\ndomains were in the final FM state. [22, 23] The lat-\ntice expansion was studied by x-ray diffraction which\nboth directly showed the mixed phase and indicated that\nthe transition was initiated at the free surface of the\nsample.[24] Magnetic force microscopy images of poly-\ncrystalline samples showed a mixed phase[25] and mag-\nnetization curves have been interpreted to support an\nAvrami model behavior for the transition.[26] Similarly,\nwhen induced by afs laserpulse wefound that the transi-\ntion proceeded through the nucleation of many indepen-\ndent and initially unaligned FM domains.[27] In order to\nobtain a microscopic image of the coexisting magnetic\nphases a spatially resolved probe is however needed and\nvery recently Baldasseroni et al. [28] used XMCD photo-\nelectron emission microscopy (PEEM) to image the tran-\nsition. This technique is well suited because PEEM pro-\nvides spatially resolved images with a resolution down to\n50 nm, while XMCD gives excellent magnetic contrast in\nthe FM phase.\nIn this paper we present XMCD PEEM images ob-\ntained at various temperatures of an epitaxial FeRh film\nwhich was slowly heated and cooled through the AFM\nto FM phase transition. We first analyze the domain\nstructure and anisotropy of the FM phase and then fo-\ncus on images obtained during the phase transition and\nshow how the phase transition is initiated at many in-\ndependent nucleation sites. We present a quantitative\nanalysis of the nucleation, and finally compare the mag-\nnetic process to the structural change studied by x-ray\ndiffraction. This work complements both our previously\npublished work on the same sample on the ultrafast laser\ninduced AFM to FM transition[27] and the recent find-\nings by Baldasseroni et al. [28].\nII. EXPERIMENTAL DETAILS\nThe FeRh thin film ( d= 47 nm) was grown on MgO\n(001) by co-magnetron sputtering from elemental targets\n[13]. The single crystal film was epitaxial to the sub-\nstrate with a (001) surface and [100] FeRh∝bardbl[110]MgOas\ncharacterized by x-ray diffraction.\nThe XMCD PEEM measurements were done at the\nSIM beamline of the Swiss Light Source[29]. This beam-\nlineprovidescircularlypolarizedlightwithanenergyres-\nolution of E/∆ E≈5000. The x-ray incidence angle was\n16oand all images were recorded at the Fe L 3edge at\n708 eV. The field of view was varied from 10-20 µm and\nimages were recorded with 512 ×512 pixels and a spatial\nresolution of ∼50 nm. The electron microscope accel-\nerates and detects the emitted photoelectrons and the\ndepth of view is around 5 nm, limited by the mean free\na) − 0o4µm\nk\nb) − 90o\nc) − 180o\nd)\n−π0π\nFIG. 2: XMCD PEEM images of the FeRh film taken at the\nFe L3edge at sample rotations of (a) 0o, (b) 90oand (c)\n180oand T = 150oC. The black circles highlight an exam-\nple of inverted contrast after 180orotation. The white circles\nhighlight an example where the contrast change from grey to\nwhite to grey. In (a) the field of view is indicated by the\nscale bar and the direction of the x-rays is given by the ar-\nrow. This direction roughly corresponds to FeRh [100]. (d)\nThe derived in-plane domain structure obtained from (a), (b )\nand (c). The color indicates the in-plane azimuth angle φof\nthe magnetization, while the saturation of the color is pro-\nportional to the magnitude as shown by the color wheel. The\nlower insert shows the azimuthal angle φalong the contour\ncorresponding to the white circle.\npath of the photoelectrons. The XMCD images are ob-\ntained from two images taken with left (-) and right (+)\ncircular polarization and x-ray intensity I. The asymme-\ntry is calculated for each pixel as:\nIXMCD= (I+−I−)/(I++I−)∝K·m(1)\nThe asymmetry is proportional to the magnetization m\nprojectedontothex-raywavevector K.[30]Inthiscaseall\nnon-magnetic contrast mechanisms cancel because they\nare independent of the photon helicity. By measuring\nIXMCDfor three different sample orientations it is pos-\nsible to obtain three projections of mto determine its\n3 vector components.[31] The sample was heated by a\nsmall resistive heater, with rates of 0.85oK/min during\nheating and -0.7oK/min during cooling.\nThe x-ray diffraction (XRD) experiments were per-\nformed at the MicroXAS beamline of the Swiss Light\nSource. We used a 7 keV x-ray beam in a gracing inci-\ndencegeometrywithincidenceangle0.71◦whichmatches\nthe x-ray penetration depth to the film thickness. The\n(101) Bragg reflection was measured with a PILATUS3\n05101520I [arb.]\n  Fe L3\nFe L2(a)IR\nIL\n705 710 715 720 725 730−2024\nE [eV]IR−IL(b)\n  \nWhite\nGrey\nBlack\nFIG. 3: (Color online) (a) Intensity of the emitted photo ele c-\ntrons as a function of x-ray energy for left and right circula rly\npolarized light. The Fe L 3and L 2edges are visible, and both\nexhibit a splitting. (b) XMCD for three different domains.\nThe dashed line (magenta) corresponds to the spectra in (a).\n100K pixel detector in a rocking curve scan where the\nsample was rotated around the surface normal in 60 dis-\ncrete steps in the interval ±2o. By recording the data in\na 1D scan with a 2D detector the Bragg reflection can\nbe mapped in 3D, which allows tracking of both in-plane\nand out-of-plane peak shifts and hence the shift in lattice\nconstant[32]. The XRD measurements were performed\nin air and the temperature was changed in discrete steps\nwhile allowing the temperature to stabilize between each\nmeasurement.\nIII. FERROMAGNETIC DOMAIN STRUCTURE\nIn order to analyze the domain structure of FeRh it\nis necessary to obtain XMCD PEEM images at least at\nthree different rotations of the sample, and in FIG. 2 we\nshow three PEEM images obtained at 0o, 90oand 180o,\nwith the rotation taken around the surface normal. The\nsample temperature was 150oC, well aboveT T. In all the\nimagesFM domains are clearlyvisible as black and white\ncontrast and the lengthscale of the domain structures is\non the order of 1 µm, as seen by comparison to the 4 µm\nscalebar. The net moment in the images is essentially\nzero, as expected when no external field was applied to\nthe sample.\nThe rotation of the sample by 180ofrom FIG. 2a to\nc results in a reversal of contrast from black to white\nand vice versa, as highlighted by the black circles and\nevident throughout the images. This indicates in-plane\nmagnetic domains because out-of-plane domains do not\nchange contrast upon rotation of the sample. Consis-\ntent with this, a 90orotation results in an interchange\nof grey domains ( m⊥H) with white/black domains.00.0500.51\n|m|  [arb.]%(a)\n0π/2π00.511.522.5\nθ  [rad](b)\n0π2π00.20.4\nφ  [rad](c)\nFIG. 4: Histograms obtained from the analysis of the local\nmagnetic moment in FIG. 2d. (a) the magnitude of the mag-\nnetic moment, (b) the out-of-plane angle θand (c) the az-\nimuth angle φ.\nThis is highlighted for a single feature by the white cir-\ncles in FIG. 2a-c. The images thus directly shows that\nthe FeRh film is dominated by in-plane magnetization.\nThis magnetic structure is due to the shape anisotropy\nof the thin film, where the in-plane magnetization min-\nimizes the stray field and therefore the total energy. In\nFIG. 3a we show the absorption spectrum from a single\ndomain for respectively left and right circular polarized\nx-rays. Both the Fe L 3edge at 708eV and the L 2edge at\n721 eV are split in two peaks of which only the peaks at\nlowerenergyshowdichroism. Thisisatypicalsignofsur-\nfaceoxidation[33]. In this experiment the samplesurface\nwas neither capped nor sputtered prior to imaging which\nexplains the surface oxidation. We did subsequently con-\nfirm that sputtering 2 nm off the surface removed the\noxide layer on a second sample (not shown). In FIG. 3b\nwe show the asymmetry for three different domains, cor-\nresponding to a respectively white, black and grey area\nin FIG. 2a. This confirmsthat the patterns in FIG. 2ado\nindeed arise from the magnetic structure of the sample.\nThe three images in FIG. 2 were used to determine the\nmagnetization m(r).[31] In FIG. 2d we showthe azimuth\nangle of the in-plane magnetization, where the color sat-\nuration is a quantitative scale for the magnitude of the\nmoment. FIG. 2d reveals domains with a typical length-\nscale of 1 µm and fairly straight domain walls. It also\nshows the existence of nodes around which the in-plane\nmagnetization rotates 360o. One example is shown in\nthe lower insert in the figure. Here the azimuthal an-\ngle is taken along the contour shown as a white circle in\nFIG. 2d, and the 0 to 360odegree rotation is evident. As\na consequence there must be an out-of-plane divergence\nbutthesevortexcoresaretoosmalltoberesolved. Previ-\nously reported domain patterns for FeRh polycrystalline\nbulksamples[25]werelikelycausedbydipolarinteraction\nbetween the grains, a feature which is not present in our\nepitaxial film.\nThe reconstruction of the three dimensional magne-\ntization vector allows us to determine if the film has\npreferred directions of the local magnetic moments. In\nFIG. 4 we show histograms of the three spherical coordi-4\nFIG. 5: XMCD PEEM images obtained during cooling (upper row) and heating (lower row) through the AFM to FM phase\ntransition in FeRh and labeled by the temperature. The field o f view is 20 µm as indicated on the scale-bar in the first image.\nThe arrow shows the direction of the x-ray wavevector. In ord er to maximize the contrast the gray-scale is not the same for all\nimages. The number in the upper right corner of each image giv es the limits on the interval used for the gray scale. The first\nimage uses a gray scale running from -9% (black) to 9% (white) . The black square in the last image shows the region which is\nmagnified in FIG. 6.\nnates ofm: the moment magnitude |m|, the out-of-plane\ninclination angle θand in-plane azimuth angle φ. Hereθ\nlies in the range from 0 to πwith 0 corresponding to the\nmagneticmomentlyingalongthesurfacenormaland π/2\nbeing in-plane. The magnitude of the moment shows one\nmajor peak, but is significantly broadened towards zero,\nbecause we do not have the resolution to image domain\nwalls. Due to this areasofthe samplewhere the direction\nof the local magnetic moment changes on a scale shorter\nthan the PEEM resolution will then be measured as hav-\ning a smaller magnitude of the moment. The histogram\nof the out-of-plane angle θhas a clear peak at π/2, con-\nfirming the in-plane magnetization which is also visible\nfrom the raw images in FIG. 2. For the azimuth angle\nφwe find a fairly uniform distribution while a strong\nanisotropy in the plane would give well defined domains\nand reveal itself as distinct peaks in the histogram. The\nabsence of such peaks indicates that FeRh has a very low\nmagnetocrystalline anisotropy. We are not aware of a\nmeasurement of the anisotropy constant K for FeRh, but\nour result supports the assumption that K is small for\nFeRh.[34, 35]\nFrom the presence of two peaks in a histogram of\nthe asymmetry of a single XMCD image Baldasseroni\net al. [28] concluded that their FeRh film had a four fold\nanisotropy. However,sincetheasymmetryforanin-plane\nmagnetized surface is proportional to cos( φ) an isotropic\ndistribution of azimuth angles also results in two peaks\nin the histogram of the asymmetry. Based on our az-\nimuthal study of the magnetization vector we conclude\nin contrast to ref. 28 that the crystalline anisotropy is\ntoo weak to give a well defined 4 fold easy axis in FeRh.It has also been suggested that the tetragonal distor-\ntion imposed by the substrate leads to a magnetocrys-\ntalline anisotropy favoring out-of-plane magnetization\nof the FM phase, when the ratio between the lattice\nconstants satisfy c/a >1 as for our sample.[35] The\nshape anisotropy was however estimated to be several\ntimes larger than the magnetocrystalline anisotropy for\na 150nm film and our resultsclearlyshow that the film is\n100%in-planemagnetized. This showsthat the proposed\nmagnetocrystalline anisotropy if present is too weak to\novercome to the shape anisotropy of our 50 nm thin film.\nIV. MAGNETIC PHASE TRANSITION\nHaving established the ability to map FM domain\nstructures we now discuss the AFM to FM phase tran-\nsition and the phase coexistence. In FIG. 5 we show\nimages taken as the sample was cooled (upper row) and\nheated (lower row) through the AFM to FM transition.\nThe images in FIG. 5 show a different region of the sam-\nple than the images in FIG. 2. The feature consisting of\ntwo straight lines which cross in the upper right corner\nof each image appears to be a scratch on the sample sur-\nface, and was used to monitor the drift of the sample.\nIn the following we refer to each image in FIG. 5 by the\ntemperature at which it was obtained.\nWe first focus on the heating process. In the AFM\nphase (T = 82oC and T = 88oC) no domain structure\nis visible, because the AFM phase does not show any\nXMCD. The growth of the FM phase in the AFM ma-\ntrix then proceeds through the nucleation of many in-5\ndependent domains on a sub micron length-scale (T =\n103oC). This directly shows the coexistence of the two\nmagnetic phases expected for a first order phase transi-\ntion at intermediate temperatures. As the temperature\nis increased (T = 107oC) the dominant feature remains\nnucleation of new domains rather than growth of pre-\nviously nucleated domains. Like the final domain pat-\ntern in FIG. 2 the initially nucleated domains have no\npreferred in-plane magnetization directions. As the film\nreaches a purely FM state neighboring domains to some\nextent realign upon contact (T = 116oC). This is evident\nin FIG. 6 where the initial black contrast of the circled\nfeature changes to white. The result is in the final struc-\nture which as already observed is dominated by domains\nof size∼1µm(T = 125oC). Once the transition reaches\nthis state the domain structure is unchanged upon fur-\nther heating. One example of this nucleation process is\nhighlighted in FIG. 6 which shows a magnification of a\n2×2µm2area corresponding to the square in FIG. 5\n(T = 125oC). At T = 103oC several independent FM\ndomains are visible and in several cases positive (white)\nand negative (black) dichroism are paired. This indicates\nthat the independent FM domains form closed magnetic\nloops in orderto minimize the stray field. As the temper-\nature is raised these smaller structures change or grow to\nlargerdomains. FIG. 5 and FIG. 6showhow the AFM to\nFM transition proceeds through the nucleation of many\nindependent unaligned domains, and subsequent partial\nreorientation of neighboring domains. This observation\nagrees with the recent work by Baldasseroni et al. [28]\nand with our previous results where we have found the\nsame mechanism to dominate the phase transition when\nit is induced with a fs laser pulse.[27] We find the same\nnucleation dynamics on both ps and s timescales, sug-\ngesting that also the laser induced phase transition is a\nthermal process.\nBecause we can only image the FM domains the ob-\nserved process during cooling is slightly different from\nheating. InthiscaseweobservehowtheFMdomainsdis-\nappear, rather than how the AFM phase nucleates in the\nFM matrix. In FIG. 5 we see how the FM domains upon\ncooling first start to shrink (T = 97oC) and eventually\nbreak up (T = 95oC) into independent domains. The\nresult is againa processwhere the final coexistence ofthe\nAFM and FM phase consists of many small sub-micron\nsize FM domains (T = 93oC) in an AFM matrix. This\nprocess is similar to heating, but there is less realignment\nof the FM domains. To quantify this difference we define\nrealignment as a change in sign of dichroism and deter-\nmine the number of pixels which change sign at some\npoint during heating (cooling). In this case the fraction\nof the film which realign is 14% during heating and 7%\nduring cooling. We stress that this definition of realign-\nment is a simplification and that the actual fraction of\nthe film which realigns is higher.\nThe images in FIG. 5 taken at T = 109oC and T =\n125oC wereobtainedbeforeandaftercoolingtothe AFM\nphase. When comparing the two images the two domain103oC \n1µm107oC\n116oC 125oC\nFIG. 6: XMCD-PEEM images obtained during heating. The\nimages show a magnified region of the images in the lower row\nofFIG.5. Thewhite circle highlights asmall FMdomain with\nan apparently closed magnetization loop.\nstructures appearto be independent. To quantify this we\nagain sort pixels according to whether they show positive\nor negative dichroism, and compare how many pixels are\npositive in both images. We find a fraction of 0.26, which\nis practically identical to the 0.25 expected if there was\nno correlation between the domains in the two images.\nA comparison of images during the late stages of cool-\ning (T = 93oC) and early stages of heating (T = 103oC)\nhowever reveal a statistical significant correlation of the\nFM regions [46]. That is, on average the regions which\nnucleate first during heating also retain the FM states\nlongest during cooling. The magnetization direction is\nhowever not correlated in the two images which again\nshows that the film has no memory of the previous FM\nphase. This rules out the existence of a single unique\npathway between the AFM and FM domain patterns.\nTo obtain a more quantitative understanding of the\nphase transition the images in FIG. 5 were analyzed as\nfollows. Due to the low magnetocrystalline anisotropy\nit is not possible to identify domains by the strength of\nthe asymmetry. In addition the XMCD contrastis signif-\nicantlyweakeratthe earlystagesofthe nucleation,which\ncomplicates the comparison of domains at different tem-\nperatures. Instead we define a domain as a connected\narea of pixels with either positive or negative dichroism.\nThis is not a precise definition because it depends upon\nsample orientation (FIG. 2), but it allows us to identify\nthe appearing FM nuclei and quantify their number and\nsizes as well as the total FM area. The result of this\nanalysis is shown in FIG. 7. FIG. 7a shows the number6\nFIG. 7: (Color online) (a) Domains per µm2during cooling\n(blue circles) and heating (red squares). A domain is defined\nas a connected area with either positive or negative dichro-\nism. (b) The total area of the FM domains. The insert show\nthe domain structure corresponding to FIG. 5 T = 125C. (c)\nIn-plane (blue triangles) and out-of-plane (red squares) e x-\npansion of the lattice. (d) Volume fractions of the FM (red\nsquares) and AFM (blue triangles) phases obtained during\nheating.\nof individual domains as a function of temperature for\nboth heating and cooling, while FIG. 7b shows the FM\nfraction of the film. The final FM fraction only reaches\n0.8 because we do not measure domains with a magnetic\nmoment perpendicular to the x-ray beam which have no\nmagnetic contrast. In reality the entire film is in the FM\nphase. The quantitative analysis presented in FIG. 7\nsupport the qualitative interpretation of the images in\nFIG. 5. The peak seen at intermediate temperatures in\nFIG. 7a confirms that the nucleation proceeds through\nan initially large number of domains which later realign\nresulting in fewer domains. We note that the definition\nof a domain used in this analysis will underestimate the\nfinal number of domains according to the standard defi-\nnition.\nWe finally note that while a FM surface layer has pre-\nviously been reported below T Tin the AFM phase for\nboth Au, Al and MgO capped FeRh films [28, 36, 37]\nwe clearly observe no such FM surface layer for the un-\ncapped sample studied here. Though we will not discuss\nthe subject further, we have observed FM surface lay-ers on some uncapped films and imaged FeRh films with\n2 nm Pt cap layer without a FM surface. The many dif-\nferent results thus seems to indicate that details in film\ngrowth and stoichiometry plays a bigger role in defining\nthe magnetic surface properties than the choice of cap-\nping.\nV. STRUCTURAL PHASE TRANSITION\nThe AFM to FM phase transition in FeRh has two\nconnected components as the change in magnetic order\ngoes hand in hand with a structural expansion. To com-\nplete the picture of the transition we show in FIG. 7c\nthe lattice expansion as a function of temperature. The\nshift in lattice constant ∆a can to first approximation\nbe found from the shift in Bragg peak position ∆q be-\ncause ∆a/a≈ −∆q/q. In addition to the thermal ex-\npansion, the out-of-plane lattice constant shows a clear\nshift of 0.7 % at T T. The in-plane lattice constant\nonly shows linear thermal expansion through the en-\ntire temperature range. The in-plane thermal expansion\nα/bardbl≈1.1·10−5of the FeRh film is close to the value for\nMgO (αMgO= 1.04·10−5at T = 300 K).[38] The one\ndimensionalexpansionthus appearstobe due to in-plane\npinning to the MgO substrate. This difference from the\nisotropic expansion in bulk crystals has been reported\npreviously.[13]\nIn the diffraction experiment we measure the total\ndiffraction from the two coexisting phases. The Bragg\npeak measured during the phase-transition is a super-\nposition of two peaks originating from the AFM and\nFM phases respectively and can be decomposed into two\npeaks corresponding to the two phases[27]. As the inte-\ngrated intensity is proportional to the scattering volume\nthe volume fraction of the AFM and FM phase as a func-\ntion of temperature can then be determined. In FIG. 7d\nwe show the volume fractions of the AFM and FM phase\nduring the transition. We find agreement between the\nevolution of the FM structure and the magnetic order.\nThe small differences in temperature is expected given\nthe difference in how the temperature was changed and\nmeasured.\nVI. DISCUSSION\nThe high resolution magnetic images obtained of the\nphase transition and the phase coexistence in FeRh al-\nlows us to speculate about which microscopic mecha-\nnisms drive the phase transition. Here three observa-\ntions play a crucial role. First, the many independent\nnucleation sites present on a sub-micron length scale in-\ndicate that the driving mechanism must appear on a\nsimilar or smaller length scale. Second, the correlation\nbetween images taken during cooling and heating show\nthat there exist either pinning defects or local impurity\nfluctuations leading to local variations in transition tem-7\nperature. Third, the magnetization direction in the FM\nphase is not pinned neither for the final structure nor\nduring nucleation.\nConcerning the existence of pinning centres our spatial\nresolution makes it difficult to discern between a transi-\ntion that appears at topographic features such as defects\norstrainfieldsassuggestedfortheAFMtoFMtransition\nin Gd5Ge4[39] or due to a broader but fixed local varia-\ntion of the transition temperature such as suggested for\nRu doped CeFe 2[40] and the Heusler alloy NiMnIn.[41]\nClear topographic features such as the scratches visible\nin FIG. 5 or intentionally produced antidots (not shown)\nhowever do not dominate as nucleation centres. In ad-\ndition rather than observing nucleation only at fixed\nsites we observe nucleation in all regions but at differ-\nent temperatures. These two observations suggest that\nlocal impurity variations lead to changes in the volume\nfree energy large enough to compensate for the interfaces\ncreated during phase coexistence.[42] The resulting het-\nerogenous nucleation was also observed in previous work\non FeRh. [13, 26, 28]\nThe many independent nucleation sites could support\ntherecenttheoreticalsuggestionthatlocalfluctuationsin\nmagnetic moment and/orvolume acts as driving”forces”\nfor the transition [43]. The many independent nucleation\nsitesthen indicatea highattempt frequencyand/oralow\nenergy barrier for the transition. This can also explain\nthe fact that the domain pattern is dominated by many\nsmall domains because the existence of many indepen-\ndent nucleation centers means that even a moderate do-\nmain wall coercivity will result in many small magnetic\ndomains.\nWe did not apply a magnetic field to the sample be-\ncause an external magnetic field distorts the path of the\nphotoelectrons and hereby reduces the resolution of the\nmicroscope. Inanappliedmagneticfielditcouldbestud-\nied if the nucleated FM phase nucleate aligned to the\nfield, align to the field immediately after nucleation or\nonly after a full FM domain pattern is achieved. We\nhave previously indirectly observed the latter case at\nfields of 0.1 T when the phase transition was induced\nby a fs laser[27]. This question of alignment relates to\nthe question of how the AFM and FM domains are re-\nlated to one another, if at all. A theoretical pathway has\nbeen used in calculations, where the Fe moments rotate\n90oduring the transformation from AFM to FM [21]. In\nthis case an AFM domain can give rise to four differ-\nent directions of the magnetic moment in the FM phase.\nFrom our comparison of images taken at the late stages\nof cooling and early stages of heating, we confirm that\nthere is no unique pathway between the AFM and FM\norder. It was also calculated and shown for one film that\nthe tetragonal crystal structure imposed on the film by\nthe substrate leads to a magnetocrystalline anisotropywhich strengthens this 90orotation of the moments.[35]\nDepending on the preferred magnetization direction this\neffect might howeverbe negated by the shape anisotropy,\nas in our experiment. To verify a possible relation be-\ntween the AFM and FM domain structures an experi-\nmental probe of the AFM domain structure is needed.\nThese are scarce but one candidate could be x-ray lin-\near magnetic dichroism[44, 45]. While this technique has\nbeen applied successfully to oxides only few studies of\nmetallic systems exist.\nVII. CONCLUSIONS\nTo summarize we used XMCD PEEM to measure the\nFM domain structure and XRD to confirm the structural\nchanges of an epitaxial FeRh thinfilm. The film was in-\nplane magnetized due to shape anisotropy and with no\npreferred in-plane direction of the magnetization, which\nindicates a low magnetocrystalline anisotropy. The re-\nsulting domain structure had a lengthscale of 1 µmand\nthe magnetic domains were not pinned when the film was\nrepeatedly cycled through the transition. We directly\nobserved the phase coexistence expected for a first or-\nder phase transition in the XMCD PEEM images, which\nare ideally suited to study this phenomenon. We found\nthat the phase transition proceeds by nucleation at many\nsmall and independent sub-micron sized sites. This ef-\nfect dominates over growth of already existing FM do-\nmainsandneighboringdomainsonlysubsequentlyandto\nsomeextent realigntoform the final FM domainpattern.\nThese findings match our previously developed model for\nthelaserinducedphasetransition[27]andtheexistenceof\nthe samedynamics on ps ands timescale suggestthat the\nlaser induced phase transition is also a thermal process.\nA further understanding of the AFM to FM pathway\nand the underlying driving forces could be obtained by\nimaging also the AFM phase or by time-resolved XMCD\nPEEM.\nAcknowledgments\nThePEEMimageswereobtainedattheX11MAbeam-\nline, and the x-ray diffraction experiments were per-\nformed on the X05LA beam line, both at the Swiss Light\nSource, Paul Scherrer Institut, Villigen, Switzerland. We\nthank D. Grolimund and C. Borcaof X05LAfor help and\nP. Derlet for fruitful discussions. We thank E. E. Fuller-\nton of U. C. San Diego for preparing the sample. 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Titov,3 J.E. Wegrowe,4 D. Byrne5 \n1Laboratoire de Mathématiques et Physique , Université de Perpignan Via Domitia, F-66860, \nPerpignan, France \n2Department of Electronic and Electrical Engine ering, Trinity College, Dublin 2, Ireland \n3Kotel’nikov Institute of Radi o Engineering and Electronics of the Russian Academy of \nSciences, Vvedenskii Square 1, Fr yazino, Moscow Region, 141120, Russia \n4Laboratoire des Solides Irradiés, Ecole Po lytechnique, 91128 Palaiseau Cedex, France \n5School of Physics, University College  Dublin, Belfield, Dublin 4, Ireland \n \nAbstract  \nThermal fluctuations of nanomagnets driven by  spin-polarized currents are treated via the \nLandau-Lifshitz-Gilbert equation generalized to include both the random thermal noise field \nand the Slonczewski spin-transfer torque term. By averaging this stochastic (Langevin) \nequation over its realizations, the explicit infinite hierarchy of differ ential-recurre nce relations \nfor statistical moments (averaged sp herical harmonics) is derived for arbitrary demagnetizing \nfactors and magnetocrystalline anisotropy for th e generic nanopillar model of a spin-torque \ndevice comprising two ferromagnetic strata re presenting the free and fixed layers and a \nnonmagnetic conducting spacer all sandwiched betw een two ohmic contacts. The influence of \nthermal fluctuations and spin-transfer torques on re levant switching characteristics, such as the \nstationary magnetization, the magnetization revers al time, etc., is calculated by solving the \nhierarchy for wide ranges of temperature, damping, external magnetic field, and spin-\npolarized current indicating new spin-torque eff ects in the thermally assisted magnetization \nreversal comprising several orde rs of magnitude. In particul ar, a pronounced dependence of \nthe switching characteristics on the directions of the external magnetic field and the spin polarization exists.   PACS numbers: 75.60.Jk, 75.75.Jn, 75.76.+j   2I. INTRODUCTION \nOne of the most significant developments in the thermally assisted magnetization \nreversal in nanomagnets sin ce the seminal work of Néel1 and Brown2,3 on the reversal time of \nthe magnetization of single-domain ferromagnetic nanoparticles due to thermal fluctuations \nhas been the discovery of the sp in-transfer torque (STT) effect.4-6 The phenomenon exists \nbecause an electric current with spin polariz ation in a small (nanoscale) ferromagnet may \ntransfer spin angular momentum betw een the current and the magnetization M giving rise to \nthe macroscopic spin-torque on M4-7 so that the latter may be altered by spin polarized \ncurrents.8 Such effects underpin the relativ ely new subject of spintronics,9 where the carrier of \ninformation is the spin state of a ferromagnetic material. Typical practical applications include very-high-speed current-induced magnetization sw itching by (a) reversing the orientation of \nmagnetic bits in high density memory structures  as opposed to the more conventional Oersted \nfield switching\n7,10,11 and (b) using spin polarized curre nts both to genera te and manipulate \nsteady state microwave oscillations with a fr equency proportional to the spin-polarization \ncurrent12,13 via the steady state magnetization preces sion. Essentially both objectives (a) and \n(b) can be achieved because, depending on its si gn, the spin current either enhances or \ndiminishes the effective damping representing  the microscopic degrees of freedom of a \nferromagnetic film8 (cf. Eq. (1) below). The meaning of this,8 considering a bi stable potential, \nis that during a precessional pe riod in a well, the average rate of change of energy E may be \neither negative, positive, or indeed zero. If E< 0 the magnetization is forced to relax into its \nenergy minimum in the well. On the other hand, if E is equal to zero, we have stable \nprecession at constant energy as if the G ilbert damping were absent. Finally if E>0, resulting \nin very large precessional orbits at energies  in the vicinity of the saddle energy, the \nmagnetization is ultimately forced to switch to a new stable position in the other well of the \npotential (see Fig. 2 below).  \nRegarding objective (b) above, a simple treatm ent of the onset of stable precessional \nstates at zero temperature has here given in Re f. 6. There, since the damping torque is small \nand roughly balances the STT wh ile ignoring thermal noise (r epresented by a stochastic \nmagnetic field), perturbation theory  is used to investigate the ons et of precessional states. This \nis accomplished by studying those relatively low- energy phase-space trajectories on which the \nmagnetization is close to its stable direc tions. Thus the equati on of motion of the \nmagnetization (cf. Eq. (1) below) may be linear ized, as is usual in the theory of small \noscillations,14 about a stable direction. This leads to situations, where precessional motion \nexponentially decays for currents less than the cr itical value for the onset of precessional states \n[not to be confused with the switching current in objective (a)6] and exponentially grows for \ncurrents exceeding that value. The phenomenon constitutes a parametric excitation because  3the STT is a function of the orientation of M, representing a time varying modification of a \nsystem parameter and thus may exhibit instability unlike conventional resonance.15 Indeed the \noverall behavior is more or less analogous to  that of a triode vacuum tube oscillator16 whereby \na coil connected to the grid circuit is coupled via mutual inductance to a lightly damped \noscillatory circuit in the anode circuit. Then while the triode  is in operation, the resulting \nfeedback effect is either to decrease or incr ease the effective resistance of the oscillatory \nanode circuit according to the sense in which the coil in the grid is c onnected. If the damping \nis decreased and the mutual coupling is sufficien t, the former may be reduced to zero. Thus an \noscillation once started will persist and will gr ow until limited by the characteristics of the \ntube. A discussion of this limiting behavior in  the spin-torque case is given in Ref. 6. \nNow regarding objective (a) because6 the STT represents a parametric excitation with \nE>0 then bifurcation phenomena due to parametric amplification at energies in the vicinity of \nthe barrier energy may manifest themselves wh ereby a qualitatively different solution for a \nnonlinear system may appear follo wing the variation of some para meter. In our context, this \nbehavior represents crossing of  the potential barrier causing th e magnetization to evolve into \nmore highly damped precessional  states exhibiting ringing osci llations which decay rapidly \n(see Fig. 2). Thus the magnetization is driven into its new stati onary state, where precession is \nprevented due to the sign of the STT which now enhances  rather than reduces the damping. \nHowever, unlike the value of the cr itical current characterizing the onset  of precessional states, \noriginating in the small oscilla tions about a stationa ry orientation of the magnetization, no \nclosed form6 for the switching current (at which the di rection of precession reverses) can be \nderived from simple perturbation theo ry. This is so because unlike the small  oscillations, the \nswitching has its origin in the large  oscillations about the direction of precession  \ncharacterizing the motion near the saddle (barrie r) energy between two stable states. Here the \nprecession is almost metastable and so may easily  be reversed in direction following a small \nchange in the energy (see Fi g. 2). An essentially similar argument was used by Kramers.17 He \nutilized the concept of large osci llations of Brownian particles in a well (at energies near the \nseparatrix energy between the bounded moti on in the well and the unbounded motion outside \nit) in discussing noise-induced es cape for very weak coupling to a thermal bath as explained in \nRef. 18. In the STT application, any results that are available ut ilize Melnikov’s method13, 19 \nfor weakly perturbed Hamiltonian systems that  are periodic in time, where the unperturbed \ntrajectories in phase space may be derived from the energyscape. One of the major benefits of \nhis method is that it establishes6 a clear distinction between the critical currents for the onset \nof precessional states and those for switching. \nAll of the foregoing discussion pertains to  zero temperature, where, for example,6 the \nprecessional states are characterized in the si ngle macrospin approximation by a frequency  4that is a function of the curr ent density, the external magne tic field, the anisotropy, the \ndamping parameter, etc. However, the therma l fluctuations cannot be  ignored due to the \nnanometric  size of STT devices, e.g., leading to mainly noise-induc ed switching at currents far \nless than the critical sw itching current without noise20 a phenomenon corroborated by \nexperiments (e.g., Ref. 21) which demonstrate that STT near room temperature alters \nthermally activated switching processes. At fin ite temperatures, randomness due to the thermal \nmotion of the surroundings is in troduced into the magnetization trajectories, counteracting the \ndamping and giving rise to fluctuations as compared to the zero temperature limit. Furthermore, large fluctuations can cause transitions between metastable states\n6 of the \nmagnetization at currents less than the zero temperature current essentially in the manner \nenvisaged by Kramers.17,18 Here we study the effects of thermal fluctuations in the presence of \nSTT using an adaptation of the Langevin equa tion for the evolution of a single macrospin \nproposed by Brown.2,3 He showed how the noise-induced  magnetization relaxation, i.e., \nreversal of the direction of  precession by crossing over a potential barrier between two \nequilibrium states could be set firmly within the context of the theory of stochastic processes. \nHowever, it should be recalled throughout that unlike in the original  work of Brown and \nNéel1-3 STT devices, due to the injection of the spin -polarized current, inva riably represent an \nopen system in an out-of-equilibrium steady state.  Such behavior is in marked contrast to the \nconventional steady state of nanostructures characterized by the Boltzmann equilibrium \ndistribution that arises when the STTs are omitt ed. Now, the effect of thermal fluctuations, \nusing a modification of the customary Néel-Brown model,1-3 treated here represent a very \nsignificant feature of their operati on. Fluctuations, for example, lead to mainly noise-induced \nswitching at currents far less th an the critical current in th e absence of noi se as well as \nintroducing randomness into the precessional orb its which now exhibit energy-controlled \ndiffusion. Thus the effect of the noise is ge nerally to reduce the current-induced switching \ntime.20 To facilitate our discussi on we first describe the archetypal schematic model of the \nSTT effect.  \nII. MODEL \nThe archetypal model (Fig. 1) of a STT de vice is a nanostructure consisting of two \nmagnetic strata labeled the free and fixed layers, respectively, and a nonmagnetic conducting \nspacer all sandwiched on a pillar between two ohmic contacts.6,13 The fixed and free layers \ndiffer significantly because the fixed layer is pinned12 along its orientation much more \nstrongly than the free one usually because the form er is of a harder magnetic material so that \nthe latter is much easier to manipulate in a magn etic sense. When an el ectric current is passed \nthrough the fixed layer, it become s polarized. The polarized spin current then encounters the \nfree layer and induces a spin torque altering its magnetization so permitting8 a variety of  5dynamical regimes. This phenomenon can lead, in particular, to two distinct magnetization \ndynamics regimes which have been ex tensively verified experimentally,20 viz., steady-state \nprecession and STT-induced reversal of the dire ction of precession governed by the transition \nrate between precessional states . Consequently, one can introdu ce two distinct time scales \nassociated with the magnetization vector M, namely, a slow one, corresponding to reversal of \nthe magnetization over a potential barrier and a fast one, characterized by the precession \nfrequency of the bounded motion in a pot ential well for constant energy.  \nNow in the model under consideration both ferromagnetic layers are assumed to be \nuniformly magnetized (for small ferromagnets the STT may lead to a rotation of the magnetization as a whole, rather than to an ex citation of spin waves; even though the single-\ndomain or coherent rotation approximati on cannot explain all observations of the \nmagnetization dynamics in spin-torque systems, many qualitative features needed to explain \nexperimental data are satisfactorily reproduced\n6,13). Thus in the presence of thermal \nfluctuations, the current-induced magnetization dynamics of M in the free layer is governed \nby the Slonczewski equation (i.e., the Landau-Lifshitz-Gilbert equation3 including the spin-\ntorque)13 augmented by a random magnetic field h with Gaussian white noise properties and \nso becoming a Langevin equation4,6,13,22 \n  ef ST     uu H u h u u . (1) \nThe Gaussian field h has the usual white noise properties  \n \nX e   u Z\nY M  \n \neasy axisH0 \nfixed layerfree layer Jeep \n \nFIG. 1. Geometry of the problem. A STT device consists of two ferromagnetic strata labelled \nthe free and fixed layers, respectively, a nd a normal conducting spacer all sandwiched on a \npillar between two ohmic contacts.13 The fixed layer has a fixed magnetization along the \ndirection Pe. eJ is the spin-polari zed current density, M is the magnetization of the free \nlayer, and 0H is the applied magnetic field.  6(a) (b) (c) \nFIG. 2. (a) Biaxia l anisotropy potential (,)V , Eq. (3). (b) Current-induced trajectory of the \nmagnetization escape. Solid line: numerical soluti on of the deterministic Eq. (1), i.e., omitting \nthe random field h, for a strong spin-polarized current, damping 0.01 , and typical values \nof other model parameters (see Sec. V). Reve rsal of the magnetization from one metastable \nstate to another typi cally occurs after ma ny precessions about the X axis, over saddle points. \nThus having traversed the potential barrier, th e magnetization decays to a new stable direction \nof precession so that in this reverse directi on the current accelerates the decay by increasing \nthe effective damping, as obvious from Eq. (1). (c ) The same as in Fig. 2(b) for a weak spin-\npolarized current and the same  values of other model para meters. Here, only the damped \nprecessions of the magnetization ab out the stable direction exist. \n 12 , 1 2\n0S2() 0 , ( ) ( ) .ii j i jkTht hth t t tvM   (2) \nwhere the indices ,1 , 2 , 3ij  in Kronecker’s delta ,ij and ih correspond to the Cartesian \naxes X,Y,Z of the laboratory coordinate system OXYZ , and ()t is the Dirac-delta function. \nThe overbar means the statistical average over an  ensemble of moments which all have at time \nt the same sharp value of the magnetization M, the sharp values subsequently being regarded \nas random variables. In Eq. (1), 1\nSMuM  is a unit vector in the direction of M, SM is the \nsaturation magnetization,  is the gyromagnetic-type constant,  is a damping parameter \nrepresenting the effect of all th e microscopic degrees of freedom, STu is the STT, \n1\nef 0 S () /MV  Hu  is the effective magnetic field co mprising the anisotropy and external \nfields, and the operator /u indicates the gradient on the su rface of the unit sphere. Here V, \nconstituting a conservative potentia l, is the normalized free energy per unit volume of the free \nlayer which we write in the standard form of superimposed easy-plane and in-plane easy-axis \nanisotropies plus the Zeeman term, viz.,13,22 \n \n22 2\n02\nS\n12 3( , ) cos sin cos\n2c o s s i n s i n s i n c o shM VD     \n    \n   (3) \n 7as shown in Fig. 2a. Thus the potential create s an energyscape with two minima and two \nsaddle points and forces the magnetization to ali gn in a given direction in one or other of the \nenergy minima in the equatorial or XY plane. Here the Z axis is obviously taken as the hard \naxis while the X axis is the easy one;  and  are angular coordinates describing the \norientation of the moment u in the spherical polar coordinate system, 1cos sin   , \n2sin sin   , and 3cos  are the direction cosines of the applied field 0H, \n0S/( 2 ) hH M D is the external field parameter, and /DD is the biaxiality parameter, \nwhere D and D account for both demagnetizing a nd magnetocrystal line anisotropy \neffects.13 The spin-transfer torque term STu is defined as \nST\n0S1\nM uuu , \nwhere  is the non-conservative pote ntial due to the spin-polar ized current given by [4] \n  2\n0Sln 1Pe\nPP\nPpMbJccJ uu e . (4) \nIn Eq. (4) the unit vector Pe identifies the magnetization direct ion in the fixed layer, cf. Fig.1, \neJ is the current density, taken as  positive when the electrons flow from the free into the fixed \nlayer, while 2\n0  /pSJM e d  (e is the electronic charge,  is Planck’s reduced constant, \nand d is the thickness of the free layer). The coefficients Pb and Pc are model dependent and \nare determined by the sp in-polarization factor  01PP4 \n 3/2\n33 / 24\n3(1 ) 16PPbP P,  \n 3\n33 / 2(1 )\n3(1 ) 16PPcP P \nwhere 01 / 2Pb  and 1/3 1Pc  as P increases from 0 to 1. The typical value of pJ for \na 3 nm- thick layer of cobalt, where 6\nS1.4 10 MA m1, is 91.1 10pJA/cm2 (cf. Ref. 13, \np. 237).  \nIn tandem with Eq. (1), we have13 (see Sec. 1.17 of Ref.  23) the Fokker-Planck \nequation (FPE) for the probability density function (,, )Wt  of orientations of u on the unit \nsphere, viz.,  8 \n 11\nN\n11\n2112s i nsin sin\n1sin .sinVV WW vWkT\nvWkTt\nVV W\n    \n  \n\n\n    \n\n              \n        \n\n  (5) \nHere 0 N1\nS() / ( 2 ) vM k T   is the free diffusion time of the magnetic moment, kT is \nthe thermal energy, and v is the free layer volume.  \nBy way of background to the Langevin equation method, during the last decade, \nvarious analytical and numerical approaches to the calculation of the measurable parameters \nfrom the Langevin and Fokker-Planck equations (1 ) and (5) have been ex tensively developed, \ne.g., generalizations (e.g., Refs. 24-26)  of the Kramers escape rate theory17 and stochastic \ndynamics simulations (e.g., Refs. 24, 27-29). For example, the pronounced time separation \nbetween fast precessional and slow energy changes in lightly damped ( <<1) closed phase \nspace trajectories (called Stoner-Wo hlfarth orbits) at energies near  the barrier energy has been \nexploited in Ref. 12 to formulate a FPE for the en ergy distribution essentially similar to that of \nKramers17 for point particles. Regarding the magnetization reversal time it will become \napparent that the effect of the spin-polarized current may be of several orders of magnitude in \nthe very-low-damping limit ( 1, the only relevant case) since the stationary distribution of \norientations of M is no longer the Boltzmann one as it now depends both on the spin \npolarized current eJ and on . The dependence is all the more obvious when one considers, \njust as in Ref. 12, the stationary soluti on of the Fokker-Planck equation for the axially \nsymmetric V and  which arises for uniaxial anis otropy with the easy axis, the \nmagnetization direction in the fixe d layer, and the external field taken as collinear. In this \nspecial case alone the magnetization dynamics are determined by a simple generalized \npotential yielding the stationary distribution in exact closed form as well as an approximate \nexpression for the reversal time. The effective potential 1V comprises that of the \nconservative external and anisotropy fields as well as the nonconservative  one due to the spin-\npolarized current. In general, the existe nce of a nonconservative  effective potential12,13,29 \nallows one to define a current-dependent potenti al barrier between stationary self-oscillatory \nstates (limit cycles) of the magnetization and to  estimate transition rates between these states. \nNow for axial symmetry the approximate solu tion procedure for the smallest nonvanishing \neigenvalue 1 of the Fokker-Planck equation for axially symmetric potentials and for high \nbarrier heights given by Brown2,3 may be used. Here the asymptotic calculation of 1, thus the \nmagnetization reversal time 11/ , may be effected2,3 via the purely mathematical method  9of approximate minimization (stemming from the calculus of variations) for 1 of the axially \nsymmetric Fokker-Planck equation when c onverted to a Sturm-Liouville equation. \nHowever, for nonaxially symmetric problems, as depicted in Fig. 2, no such simple \nasymptotic solution exists because it is impossi ble, by inspection of the nonaxially symmetric \nFokker-Planck Eq. (5) when the STT is included, to derive a simple analytic equation for an \neffective potential. [Such a potential can be calculated only in numerically via the time-\nindependent distribution function;  see Eq. (27) below.] Neverthe less, it is still possible to \ncalculate 1 in numerical fashion, once again yi elding the magnetization reversal time. \nMoreover, it is also possible to calculate the time-independent in -plane component of M, i.e., \nin the X or easy-axis direction 0Xu  as well as the corresponding magnetic susceptibility \n2\n0 02\nXXuu , where the angular brackets 0 mean stationary statistical averaging. The \nmerit of such calculations is that inter alia  they allow one to accura tely assess approximate \nlow-damping solutions for the reversal time based on energy-controlled diffusion.13,29 These \nsolutions all more or less rest on (noting the separation of time scales referred to above) \ntreating both the effects of the stochastic torque s due to the heat bath and the spin torque as \nperturbations of the pr ecessional dynamics of M in the wells of the anisotropy-Zeeman \nenergy potential. The corresponding clos ed phase-space traj ectories are known8,20 as Stoner-\nWohlfarth orbits and steady precession along such an orbit of constant energy, belonging to a \nsphere of radius equal to the saturation magnetization, occurs if  the spin-torque cancels out the \ndissipative torque [cf. Eq. (1)]. The origin of th e orbits of course arises from the two well \nstructure of the anisotropy potential. One should  at this juncture me ntion the treatment of \nreversal of the magnetisation by Apalkov and Visscher.25 Here, for example, the time \nseparation between the fast precessional and sl ow energy change in a lightly damped Stoner-\nWohlfarth orbit at energies near to the barr ier energy is used to formulate a Fokker-Planck \nequation for the probability distribution of the en ergy near the barrier. This method is again \nessentially similar to the approach used by Kramers17,18 in the problem of the very-low-\ndamping noise-activated escape rate from a pot ential well. Moreover, the derivation of the \nFokker-Planck equation in energy-phase variable s for point particles with separable and \nadditive Hamiltonians has been ex tensively discussed by Stratonovich30 and Risken.31 The \ncorresponding magnetic problem (where the Hamilt onian in the absence of spin torque is non-\nseparable) in energy-precession va riables has been discussed by Dunn et al.8 in relation to \ntheir Eq. (1.22), which, on assumi ng rapid equilibration of the pre cession variable, leads to an \nenergy diffusion equation, their Eq. (1.23).  10In this paper, we shall present results for the stationary magnetization 0Xu  and static \nsusceptibility 2\n0 02\nXXuu  in the easy-axis direction and for the reversal time of the \nmagnetization. These results are obtained by extending the general method (based on the \nrelevant Langevin equation) of constructing recu rrence relations for the e volution of statistical \nmoments for arbitrary n onaxially symmetric free energy gi ven in Refs. 23 and 32 to include \nthe STT term and then specializing them to Eq. (3). The recurrence relations can then be solved by matrix continued fraction me thods just as with the zero STT term.\n23 The answers \nwill then constitute benchmarks for approximate  solutions obtained by other methods. Indeed \nthe procedure is entirely analogous to that involving the numerical solutions3 used to test \nasymptotic solutions, based on the Kramers escape rate theory,3,17 for the reversal time in the \nNéel-Brown model when the STT is absent. Notice that if the STT is included its effect may also be described via a modifi cation of the energy barrier in the Néel-Brown model (for \ndetailed discussions see Refs. 13, 24-26, and 33). \nIII. DIFFERENTIAL-RECURRENCE RELATION FOR THE STATISTICAL MOMENTS  \nOur method23,32 is based on first averaging th e appropriate Langevin equation \nincluding the spin-torque term (re garded as a Stratonovitch stocha stic differential equation) for \nthe magnetization evolution, therefore written in terms of the spherical harmonics, over its \nrealizations in the representati on space of polar angles  in an infinitesimally small time starting \nfrom a set of sharp angles, which subsequently  are also regarded as random variables. The \nresult is then averaged over the distribution of these angles ultimately yielding the desired \nrecurrence relation for the observables which are the statistical moments of the system. Here \nthe relevant Langevin equation is the Landau-Lifshitz-Gilbert equation for the evolution of M \nin the free layer as modified by Slonczewski4,13 to include the STT, Eq. (1). As shown in \nAppendix A, Eq. (1) can be wr itten in an equivalent Landa u-Lifshitz form, where the \nprecessional and alignment terms are now clearly delineated, viz. \n \nN0S\n1\n0S\nN2\n.2vVMkT\nvVMkT\n            \n               uu hu\nuu hu\n (6) \nIn the spherical polar coordina te basis shown in Fig. 1, th e vector Langevin equation (6)\nrepresents two coupled nonlin ear stochastic differential equations for the angles  and , \nviz.,23  11 \n11 0S\nNN\n1() () () [ () , () ,] [ () , () ,]22\n[( ) ,( ) , ] [( ) ,( ) , ] ,sin ( )vM vt= h t h t V t t t t t tkT kT\nVt t t t t tt      \n  \n    \n    \n (7) \n \n1\n1 0S\n2\nNN\n1() () 1( ) [( ) ,( ) , ] [( ) ,( ) , ]2 sin ( ) 2 sin ( )\n[( ) ,( ) , ] [( ) ,( ) , ] ,sin ( )ht ht vM vt= V t t t t t tkT t kT t\nVt t t t t tt        \n  \n\n    \n    \n (8) \nwhere the components ()ht and ()ht of the Gaussian random field ()th in the spherical \nbasis are expressed in terms of the components ()Xht , ()Yht , ()Zht  in the Cartesian basis as23 \n () () c o s () c o s () () c o s () s i n () () s i n ()XY Z ht h t t t ht t t ht t     , \n () () s i n () () c o s () .XY ht =h t t ht t      \nEquations (7) and (8) then yield the desire d Langevin equation for the evolution of the \nspherical harmonics ,,lmY34 comprising the orthonormal basis set from which the \nobservables are ultimat ely obtained, viz., \n \n \n ,, ,\n1\n,, 1 0S\nN\n, 11\nN\n, 11\n2() ()() ()2s i n\n1\n2s i n\n11,sin sinlm lm lm\nlm lm\nlm\nlmdY Y Y dd\ndt dt dt\nht ht YY vMht htkT\nY vVVkT\nYVV\n\n\n   \n   \n  \n\n\n \n     \n       \n          (9) \nwhere ,(, )lmY are defined by34 \n ,(2 1)( )!(, ) ( c o s ) ,4( ) !im m\nlm lll mYe Plm  \n *\n,, (1 )m\nlm l mYY , \n()m\nlPx are the associated Legendre functions,34 and the asterisk denotes the complex \nconjugate. \n By averaging this Langevin equation (9) as explained in Sec. 9.2 of Ref. 23 and \nsummarized in Appendix B, we have the ev olution equation of the statistical moments \n,()lmYt  (expected values of the spherical harmonics ,lmY34) for arbitrary anisotropy rendered \nas the differential-recurrence relation viz.,  12 N, , , , , ,() () .l m l m lm lm lmdYt e Y tdt    (10) \nIn Eq. (10) angular brackets m ean statistical averaging and ,, ,lml me are expressed via the \nClebsch–Gordan coefficients ,\n,, ,LM\nlml mC (Ref. 34) as  \n \n  \n\n\n,,, , , , 0\n, ,0 ,\n,, 0 , , 0 , , ,\n,,\n,0\n,0, ,0\n,\n2( 1) 1 (2 1)(2 1)124\n11 1\n22 1\n(2 1)( )!\n() !\n!\n!rsm\nlml m s ll s\nrs rr s\nrs l l l m l m s\nrs\nrs rs\nr\nL\nll\nLs\nLll l le\nll r r l l BAC C\nr\nir r sABrs\nLsCLs \n\n\n\n\n  \n\n\n\n\n\n    \n      \n\n\n\n  \n1\n,, 1\n,, , , 1 , ,1,() 1Ls Ls\nlml m s lm l m slm lmmC s CLsLs\n            \n (11) \nwhere 0s and ,rsA and ,rsB are, respectively, the coefficients of the Fourier series \nexpansions in terms of spherical harmonics  of the (conservative) free energy density V and \nthe nonconservative potential , viz., \n ,, ,,rs rs rsvVAYkT  (12) \n ,, ,.rs rs rsvBYkT  (13) \nEquation (10) represents the e volution of a typical entry in a set of differential-recurrence \nrelations with ,, ,lm l me given by Eq. (11). Only the Fourie r expansions of both potentials in \nterms of ,rsA and ,rsB are needed. In the stationary  state, when the statistical moments are \nindependent of time, Eq. (10) becomes \n ,, , , , 00l m lm lm lseY    , (14) \nThe same result may be obtained, albeit with more labor and in a less transparent manner, \nfrom the Fokker-Planck equation, Eq. (5), by se eking the surface density of magnetic moment \norientations on the unit sphere as3,23,32 \n   ,\n0,*,, ( )l\nlm\nllm\nmlWt Y Y t  \n\n , (15) \nwhere \n   ,0,2\n0() , , s i nlm lmYt Y W t d d    (16) \nby the orthogonality property of the spherical harmonics, viz.,  \n \n11 2 2 1 2 122\n*\n,, , ,\n00(,) (,) s i n .lm l m ll mmYY d d\n         13Equations (10) and (14) have been obtained under th e assumption that the damping parameter \n is independent of M. However, the results may be also generalized to magnetization-\ndependent damping ()M.8,20 \nBoth Eq. (10) and Eq. (14) are valid for an arbitrary  free energy. Here we specialize \nthem to the particular free energy given by Eq. (3). In terms of spherical harmonics, Eq. (3) \ncan be written as \n 2\n,,\n12(,)3r\nrs rs\nrs rvVAYkT \n   , (17) \nwhere the nonzero expansion coefficients ,rsA are given by \n 1,0 3 43Ah ,  \n 1, 1 1 28()3Ah i   , \n 204(1 2 )45A ,  \n 2, 22\n15A , \nand 2\n0S /( ) vM D k T  is an anisotropy (or inverse temperature) parameter. Now, the \npotential , Eq. (4), is in spherical harmonic notation \n 1\n*\n1, 1,\n14ln 1 ( , ) ( , )3PP\nmm P P\nm Pvb cJY YkT c  \n  , (18) \nwhere P and P are the spherical polar coor dinates of the unit vector pe which is the \nmagnetization direction of the fixed layer, and 2\n0S /( )ep Jv M J k T J  is the dimensionless \nspin-polarized current parameter. Next retain ing only the two leading terms in the Taylor \nseries expansion  \n23ln 1 / 2 / 3 ... xx x x   , \nwhich converges fairly well for typical values of the model parameters, we will then have  \n 2\n,,\n0(,)r\nap\nrs rs\nrs rvBYkT\n  , (19) \nwhere the expansion coefficients ,rsB are defined as  \n 3/2\n*2 * *\n0,0 1,0 1,1 1, 14(,)2(, ) (,)9PP\nPP PP PPbcJBY Y Y      ,  14(a) (b) \nFIG. 3. 3D plot of th e nonconservative potentials , , Eq. (18), (a) and ,ap , Eq. \n(19), (b) for typical values of the model parameters ( 0.3P , 6J, 0P, and /2P ).  \n *\n1, 1,4(, )3p\nmm P PbJBY  ,   m=0,1 \n 3/2\n*2 * *\n2,0 1,0 1,1 1, 18(, ) (, ) (, )\n95PP\nPP PP PPbcJBY Y Y     , \n 3/2\n**\n2, 1 1,0 1, 18(, ) (, )\n31 5PP\nPP PPbcJBY Y  , \n 3/2\n*2\n2, 2 1, 18(, )\n33 0PP\nPPbcJBY . \nWe remark that the approximation Eq. (19) accurately reproduces all features of the \nnonconservative potential , Eq. (18) (see Fig. 3) because 2/3 0 . 1 5PPcue  for P  0.4 (\n0.3 0.4P  are typical values for ferromagnetic metals13) and all , , P, and P. \nMoreover, in the calculation of the statistical moments, relaxation time, etc., the \napproximation Eq. (19) yields an accuracy be tter than 5% in the majority of cases.  \nIV. CALCULATION OF OBSERVABLES \nThe general time-dependent Eq. (10) and the time independent Eq. (14), as specialized \nto Eqs. (17) and (19), now yields the 25-term differential-recurrence rela tion for the statistical \nmoments ,()lmYt  governing the dynamics of the magne tization [see Eq. (42) of Appendix \nC]. Equations (10) and (14) may be solved by extending the general matrix continued fraction \nmethods developed in Refs. 23 and 32 to include  the STT. Indeed, we can always transform \nthe moment system, Eq. (10) constituting a multiterm scalar  differential-recurrence relation, \ngoverning the magnetization relaxation into the tridiagonal vector differential-recurrence \nrelation \n N1 1() () () ()nn n n n n ntt tt\n  CQ C Q CQ C. (20) \n 15Here ()ntC  are the column vectors arranged in an appropriate way from the entries ,()lmYt  \ngiven by Eq. (11) and ,nnQQ  are matrices formed from ,, ,.lml me  The explicit equations for \n()ntC  and ,nnQQ  for the free energy Eq. (3) are given ex plicitly in Appendi x C. As shown in \nRef. 23, Chap. 2, the exact matrix continued fraction solution of Eq. (20) for the Laplace \ntransform of 1()tC  is given by \n 1N 1 1 1\n2 2() () ( 0 ) () ( 0 ) ,n\nkk n\nn kss s\n\n\n      C Δ C ΔQC  (21) \nwhere 110() ()stst e d tCC, ()nsΔ  is the matrix continued fr action defined by the recurrence \nrelation \n 1\nN1 1 () () ,nn n n nss s\n    IQ Q Q ΔΔ  (22) \nand I is the unit matrix. Having determined the column vectors 1() ,sC\n2() ,sC … as described \nin Refs. 23 and 32, we then have the relevant observables. In a similar way, we also have the \nsmallest nonvanishing eigenvalue (yielding th e reversal time) from the matrix equation3,23 \n NIS  (23) \nwhere the matrix S is defined via the matrix  continued fractions as \n 1\n11 2 2 1 2 2 (0) (0)     SQ Q Q ΔΔ IQ Q  (24) \nand the prime designates the derivative of 2()sΔ  with respect to Ns (see Ref. 23, Chap. 2, \nSec. 2.11.2). Thus 1 is the smallest nonvanishing eigenvalue of S. \nNow in order to calculate the stationary  characteristics (distribution function, \nmagnetization, etc.) we may replace ,()lmYt  in Eq. (42) of Appendix C by ,0lmY  and set the \ntime derivative equal to zero. Then by exte nding the general matrix continued fraction \nmethods developed in Refs. 23 and 32 to include the STT, we have \n 2,20\n2,2 10\n2, 20\n11 1 1\n21 , 210\n21 , 220\n21 , 2101(0) (0) (0) , ( 1,2,....).\n4nn\nnn\nnn\nnn n n\nnn\nnn\nnnY\nY\nY\nn\nY\nY\nY\n\n \n\n \n \n\n\n\n\n \n\nΔ QΔ QΔ Q\n\n\n (25)  16The out-of-equilibrium  or time-independent stationary distribution 0, W is thus rendered \nvia the Fourier series \n  \n00,0*\n, ,,l\nlm\nlm llm WY Y \n\n . (26) \nNext, by analogy with the Boltzmann distribution, since we expect  the stationary distribution \nto be formally similar to it,13 we may define the effective potential efV via \n 0 ,l n,efVW . (27) \nFurthermore, having determined ,0lmY , we have both the stati onary magnetization in the X \ndirection 0Xu  and the corresponding susceptibility 2\n0 02\nXXuu , viz., \n  1, 1 1,1 00 002sin cos3XuY Y \n  , (28) \n \n 22 22\n00 00\n2\n2 , 2 2 ,2 2 , 0 1 ,1 1 , 102\n00 0 0sin cos sin cos\n24 2 1.15 45 3 3XXuu\nYY Y Y Y   \n \n     (29) \nWe remark that in some ranges of the m odel parameters, e.g., for very low damping \n<0.001, and/or very high potential barriers, V>100, the continued fraction method may \nnot be applicable18,31 because the matrices involved become ill-conditioned, meaning that \nnumerical inversions are no longer possible. \nV.RESULTS \nThroughout the calculations the anisotropy an d spin-polarization parameters will be \ntaken as 0.034 D , 20 , and 0.3 P  just as in Ref. 13. Moreover, the applied field 0H \nand the unit vector Pe identifying the magnetization direction in the fixed layer are taken to lie \nin the equatorial or XY plane, i.e., /2P  and /2 . Thus the orientations of 0H and \nPe in the XY plane are entirely determined by the azimuthal angles  and P, respectively. \nThe values 0P  correspond to the particular  configuration whereby both 0H and Pe \nare directed along the easy ( X-)axis. For 52.2 1011m A s, 300T K, 24~1 0v 3m, \n6\nS1.4 10 M1A m (cobalt), 710eJ2A cm, 910pJ2A cm, and 0.02 , we have the \nfollowing estimates for the principal model parameters \n 5.9 J , 20.2 , 8\nN4.8 10 s.  17Moreover, instead of the free diffusion time N, it will be more convenient to use as the \nnormalizing time 11\n0S N/[ ( ) ] ( 2 ) MD     . The above numerical values yield \n11\n04.8 10 s. \nOnce we have determined the time-i ndependent stationa ry distribution 0, W via \nthe Fourier series, Eq. (26), we  have the effective potential  efV from Eq. (27). A typical \nexample of such calculations is  shown in Figs. 4 and 5. The effective potential comprises a \ndouble-well structure with non-equi valent wells. This energyscape (Fig. 4), as expected on \nintuitive grounds, strongly depends on damping,  external magnetic field magnitude and \norientation, and spin-polarized current. In pa rticular, by varying the magnitude of the spin-\npolarized current, damping, etc., one may alter s ubstantially the effective barriers and thus the \nreversal time (cf. Fig. 5). The stationary averages 0Xu  and 2 2\n0 0XXuu  are calculated \nfrom Eqs. (28) and (29), respectively. In Fi g. 6, we show the consequent dependence of 0Xu  \nand 2 2\n0 0XXuu  on the spin-polarized current via a family  of curves with the spin current as \nthe independent variable, for various values of the damping, external field magnitude ( h) and \norientation (), and the magnetization direction in the fixed layer (P), which are all \nregarded as parameters. In contrast, in Fig. 7 we illustrate the dependence of the magnetisation \nand the susceptibility on the external field parameter h via a family of curves for various \nvalues of the current J, the external field orientation and magnetization direction in the fixed \nlayer ( and P), and inverse temperature parameter . Clearly the switching current SWJ \n(i.e., the current when 0Xu  changes sign corresponding to re versal of the direction of \nprecession or, equivalently, when 2 2\n0 0XXuu  attains its maximum and subsequently \nvanishes) strongly depend on the model parameters , J, h, , , , and P. In particular, \nas both the damping  and external field parameters h increase the value of SWJ rapidly \nincreases [see Fig. 6(a), 6( b), and 7(a)].  Moreover, SWJ may also vary significantly with both \nthe orientation of the external field and the direction of the magnetization of the fixed layer \n(see Fig. 6c and 6d). The half-width of 2 2\n0 0XXuu  and the onset of the slope in 0Xu  \nlargely depend on the damping parameter  [Fig. 6(a)] and the inve rse temperature parameter \n [Fig. 7(d)]: higher values of  (lower temperatures) and smaller values of  correspond \nto a narrower half-width and a more rapid onset of the slope. \n  18 \nFIG. 4. (a) 3D plot of the effective potential ,efV , Eq. (27), in the vicinity of the minima \nfor 6J, 0.02 , 0.15h , 20 , 20 , 0P, and 0.  \n \n   \n\n4\n3\n2Vef (   = 0.02P =  = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12\nh = 0.1   = 20  P = 0.3   = 20\n1\n\n(b)3\n2\n1J = 6\n1:  = 0.2\n2:  = 0.02\n3:  = 0.005 (a)\n012301020J = 6 = p = 0Vef ()\nP = 0.3    = 20    = 20    = 0.02\n1: h = 0\n2: h = 0.1\n3: h = 0.2\n123\n01230102030(c)\nJ = 6\n123\n(d) \nFIG. 5. /2 ,efV  vs the azimuthal angle  for various spin polar ized current parameter J = \n6, 0, 6, 12 and 0.02  (a); for various damping  = 0.2, 0.02, 0.005 and J = 6 (b); for \nvarious external field parameter h = 0.0, 0.1, 0.2 and J = 6 (c) and J = 6 (d) ( 0.02 , \n0.15h , 20 , 20 , 0P, and 0). \n 19     h = 0.15 = 20 = 20,  P = 0.3\n4 3 2 1:  = 0.01\n2:  = 0.02\n3:  = 0.05\n4:  = 0.1uX\nJ1(a)\n      P =  /2P = 0 =  /2 = 0\n1 4 3 2\nJu2\nXuX\n\n \n   4\n321: h = 0.05\n2: h = 0.15\n3: h = 0.25\n4: h = 0.40uX = 20 = 20   = 0.02   P= 0.3\nJ1(b)\n   P =  /2P = 0 =  /2 = 0\n143 2\nJu2\nXuX\n\n \n    \n5 = 0.02   h = 0.1 = 20 = 20   P = 0.3\n4 3 21: = 0\n2: =  /4\n3:  =  /2\n4: = 3 /4\n5: = uX\nJ1(c)\n    P= /2P = 0 =  /2\n5 1 4 3 2\nJu2\nXuX\n\n \n   = 0.02   h = 0.1 = 20 = 20   P = 0.3\n43 2\n1: P = 0\n2: P =  /4\n3: P = 3 /4\n4: P = uX\nJ1(d)\n  P= /2 = 0 =  /2\n1432\nJu2\nXuX\n\n \nFIG. 6. 0Xu  and 2 2\n0 0XXuu  vs the dimensionless current parameter J for various values \nof damping  (a), external field parameter h (b), external field orientation in the free layer  \n(c), and spin-polarization orientation P (d).  20    = 0.02 = 20 = 20   P = 0.3\n4 3 2\n1: J = 0\n2: J = 10\n3: J = 20\n4: J = 30uX\nh1(a)\n   P =  /2P = 0 =  /2  =0\n1 4 3 2\nhu2\nXuX\n\n \n   \n51: = 0\n2: =  /6\n3: =  /4\n4: =  /3\n5: = 4321 = 0.02   \nJ = 10\n = 20\n = 20   \nP = 0.3\nhuX(b)\n  \n5P= /2P = 0 =  /2 u2\nXuX\n4 321  = 0.02   \nJ = 10\n = 20\n = 20   \nP = 0.3\nh  \n   \n5\n1:P = 0\n2:P =  /6\n3:P =  /4\n4:P =  /3\n5:P = 4\n321\nhuX(c)\n   5P =  /2= 0 =  /2 u2\nXuX\n4 3 21 = 0.02   \nJ = 10\n = 20\n = 20   \nP = 0.3\nh  \n    5\n43\n2\n11: = 10\n2: = \n3: = \n4: = 25\n5: = 30\nhuX(d)\n  5P= /2P = 0 =  /2,  = 0u2\nXuX\n43 2 1 = 0.02   \nJ = 10\n = 20   \nP = 0.3\nh  \nFIG. 7. 0Xu and 2 2\n0 0XXuu  vs the external field parameter h for various values of the \ncurrent parameter J (a), the external field orientation  (b), the spin-polarization orientation \nP (c), and the inverse temperature parameter  (d). \n Moreover, the smallest nonvanishing eigenvalue 1 of the Fokker-Planck operator \n(inverse of the reversal time of the di rection of precession, i.e., that of the X component of M) \nmay be evaluated from the s ecular Eq. (23). By calculating 1, we also have the dependence \nof the magnetization reversal time 11/  on the spin-polarized current, anisotropy \nparameters, damping, external field magnitude  and orientation in the free layer, and the \nmagnetization direction in the fixed layer. In ge neral, a pronounced dependence of the reversal \ntime on these model parameters exists . Examples are shown in Figs. 8 11. Figure 8 illustrates \nthe damping dependence of  for various values of the current J while Fig. 9 shows the  21temperature dependence of  for various values of the current ( J) and external field ( h) \nparameters. We have also shown for comparison in these figures the reversal time calculated \nfor 0J via escape rate theory for bi axial anisotropy which is given by35 \n 12\nIHD IHD\n12 1 2().2 ( )( )( )AS S\nASA S  (30) \nwhere ()Az is called the depopulation factor, viz. \n 21/4\n21() e x p l n12 1/4z dAz e \n \n\n  , (31) \nIHD\n2 is the escape rate from the shallowe r well 2 to a deeper well 1 given by  \n 21\nIHD 2 2 2 2 2\n2 1\n01() 1 ( 1 ) 4( 1 )2( ) ( 1)hehhh h hh     \n\n           , (32) \n12() ( )hh   , and the dimensionless actions 1S and 2S are given by  \n 21/2 1/212 2 1\n1,2 3/24( 1 )(1 )(1 ) arctan (1 )(1 ) .(1 ) 2hhSh h h h               (33) \nClearly by altering J, the ensuing variation of  may be as much as  several orders of \nmagnitude for very low damping, 1  (Fig. 8). Furthermore,  may greatly exceed or, on \nthe other hand, be much less than the value pertaining to J = 0. Moreover, the increase or \ndecrease in  is entirely governed by the direction of the current, i.e., by the sign of the \nparameter J as expected.3 The temperature dependence of  can be understood via the \neffective potential ef, V, Eq. (27). Clearly, at high barriers, 5, the temperature \ndependence  of  has the customary Arrhenius behavior /( )~efvV k Te, i.e., exponentially \nincreasing with decreasing temperature. The slope of 1()T markedly depends on J, h, , etc. \nbecause the barrier height efV  of the shallow well is strongly influenced by those parameters \n(see Fig. 5). In particular, we  observe that the slope of 1()T significantly decreases with \nincreasing h [Fig. 9(b)] due to a decr ease of the barrier height efV  due to the action of the \nexternal field [see Fig. 5(c) and 5(d) ]. At low barriers, the behavior of 1()T may deviate \nconsiderably from Arrhenius behavior. Figure 10 illustrates the dependence of  on the \ncurrent parameter J for various values of the external field parameter h. Clearly, as J increases \nfrom negative values,  exponentially increases attaining a ma ximum at a critical value of the \nspin-polarized current and then smoothl y switches to exponential decrease as J is further \nincreased through positive values. Such a dependence of  on the applied current implies that \nboth kinds of scaling for the switching time suggested in the literature ,24-26,28 namely, \nSWCJ Je  and 2()SWCJ Je  may be realized for 10SWJJ  and 10SWJJ,  22respectively (where C and SWJ are parameters depending, in general, on h, , , , etc.). \nFigure 11 exemplifies the pronounced dependence of  on the azimuthal angles of the applied \nfield  and the magnetization dire ction in the fixed layer P for various values of J which \nmay comprise several orders  of magnitude [note that () ( 2 )      and \n() ( 2 )P P    ]. Invariably strong STT effects on th e magnetization reversal exist only for \nlow damping, 0.1 , because the magnitude of the STT effects in the magnetization reversal \nis governed by the ratio /J.13 Here, the variation of  with J may be of several orders of \nmagnitude. For 1, however, the STT term in Eq. (1 ) does not influence the reversal \nprocess at all because it is negligible comp ared to the damping and random contributions. \n\n432P =  =  /2\nP =  = 0\n1: J = 0\n2: J = 6\n3: J = 6\n4: J = 12/ \nh =0.15\n =20\n = 20\nP = 0.3\n1\n \nFIG. 8. Reversal time 0/ vs the damping parameter  for various values of the current J \n(solid lines). Asterisks: escape rate formula, Eq.(30). \n  \n/  4321P = =  /2\nP = = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12h = 0.1\n = 0.02\n  = 20\nP = 0.3\n(a)\n \n  \n/ \n4321P = =  /2\nP = = 0\n1: h = 0\n2: h = 0.1\n3: h = \n4: h = J = 6\n = 0.02\n  = 20\nP = 0.3\n(b)\n \nFIG. 9. 0/ vs the inverse temperature parameter 1~T for various values of the current \nJ (a) and the external field h (b). Asterisks: escape rate formula, Eq.(30).  23   \n/ \n43 21P = =  /2P = = 0\n1: h = 0\n2: h = 0.05\n3: h = \n4: h =  = 20  = 20\nP = 0.3  = 0.02   \nJ \nFIG. 10. 0/ vs the spin-polarized current parameter J for various values of the external \nfield 0,0.05,0.1,0.2h . \n\n/ \n4321P = =  /2P = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12h = 0.1 = 0.02  \n  = 20     = 20  \n              P  = 0.3\n(a)\n \n\n/ \n4321P = =  /2 = 0\n1: J = 6\n2: J = 0\n3: J = 6\n4: J = 12h = 0.1\n = 0.02  \n = 20\n  = 20   \nP = 0.3\nP(b)\n \nFIG. 11. 0/ vs the azimuthal angles  (a) and P (b) for 6, 0, 6, 12J . \n \nSTT effects in the thermally assisted magneti zation reversal have been treated via the \nevolution equation for the stat istical moments yielded by th e Langevin equation rendering \nstationary and nonstationary charac teristics for wide ranges of te mperature, damping, external \nmagnetic field, and spin-polarized current. Variat ion of the latter may alter the reversal time \nby several orders of magnitude conc urring with expe rimental results.11 The virtue of our \nnumerically exact solutions of th e recurrence relations for the relevant statistical moments is \nthat they hold for the most comprehensive fo rmulation of the generi c nanopillar model (Fig. \n1), i.e., for arbitrary directions  of the external field and spin polarization and for arbitrary free \nenergy  density, yielding  the STT switching characteris tics under conditions otherwise \ninaccessible. Thus our results may serve both as  a basis for theoretical investigations and \ninterpretation of a broad range  of STT experiments. Additiona lly, they are essential for the \nfuture development of both escap e rate theory and stochastic dynamics simulations of the  24magnetization reversal time in STT systems, re presenting rigorous benchmark solutions with \nwhich calculations of that time by any othe r method must comply (the procedure being \nentirely analogous to that used to validate such  complementary approaches to fine particle \nmagnetization by exactly calculating the re versal time via the smallest nonvanishing \neigenvalue of the FPE fo r the Néel-Brown model3,36). Finally, we believe that the moment \nmethod may be useful in related problems such  as magnetization reversal of STT devices \ndriven by ac external fields and currents, etc.37 \nACKNOWLEDGEMENTS \nWe thank P.M. Déjardin for helpful conve rsations. One of us, D.B., acknowledges the \nSimSci Structured Ph.D. Progra mme at the University College Dublin for financial support. \nSimSci is funded under the Programme for Resear ch in Third-level Institutions and co-funded \nunder the European Regional Development Fund.  All calculations were performed by the \nLonsdale cluster maintained by the Trinity Ce ntre for High Performance Computing. This \nCluster was funded through grants from the Science Foundation Ireland. \nAPPENDIX A: REDUCTION OF EQ. (1) TO THE LANDAU-LIFSHITZ FORM \nIn order to obtain an explicit equation for u, we rewrite the implicit Eq. (1) in the \nLandau-Lifshitz form. Transposing the u term, we have \n   ef ST.       uu u u H u h   (34) \nOn cross-multiplying vectorially by u in Eq. (34) and using the triple vector product formula \n  ()    uu u uu u u  , (35) \nwe obtain since () 0 uu  \n    ef ST .      uu u u H u h u   (36) \nSubstituting Eq. (36) into Eq. (34) yields the explicit form \n   2\nef ef ST ST(1+ )                 uu H u h u H u h u    \nor equivalently, using Eq. (6), \n N\n0S\nN0S2\n2vVMkT\nvVMkT\n          \n                uu u huu\nuu u huu\n (37) \nwhich reduces to the more easily visualized Eq. (6) because  25           uuu uuu. \nIn the spherical polar co ordinate system shown in Fig. 1, one has  \n  1, 0, 0u ,   =0 , , s i n u  ,  \n 10, ,sinVV V\nu,   \n 10, ,sin       uu. \nAPPENDIX B: DERIVATION OF THE DIFFERENTIAL-RECURRENCE RELATION FOR \nSTATISTICAL MOMENTS, EQ. (10) \nOwing to Eqs. (7) and (8), we can write the Langevin equation (9) in  vector notation as \n \n 1 0\n,, ,\n11\n,[]2\n[] .2S\nlm lm lm\nN\nlm\nNvMYY YkT\nvVV YkT\n\n    \n         hu\nu\n (38) \nHere  denotes the orientation space gradient operator defined as  \n  uu. \nOn averaging the Langevin equation (38) with mu ltiplicative noise as explained in detail in \nRef. 32 and Sec. 9.2 of Ref. 23, we have afte r some algebra the evolution equation of the \nstatistical moment ,()lmYt , viz., \n     \n 11 1\n,, , , ,\nN\n11\n,,1\n22\n1\nsinlm lm lm lm lm\nlm lmdvYY V Y Y V V Ydt kT\nVV YY \n\n  \n     \n            (39) \nwhere the operator 2  is the angular part of the Laplacian, viz., \n 2\n2211sin .sin sin       \nHere we have used23 \n  1 0\n,, , []S\nlm lm lmvMYY YkT     hu  \nand that for any function ,Ft23 \n  ,, , , 2lm lm lm lm FY F Y F Y Y F       .  26We now indicate using the th eory of angular momentum34 how Eq. (39) may be \nwritten as a differential-recurrence relation for the statistical moments. This is accomplished \nby reducing (details in Refs. 32 and 23, chapters  7 and 9) the terms inside the angular braces \non the right-hand side of Eq. (39) to the calculati on of the Fourier coefficients in the expansion \nof a product of spherical harmonics as a sum of spherical harmonics. We begin by expressing \nthe terms within the angular braces on the righ t-hand side of Eq. (39) as functions of the \nangular momentum operators 2ˆ,Lˆ,ZL and ˆL defined as34  \n 2ˆ , L ˆ ,ZLiˆ cot .iiLe i e \n    (40) \nThus we have from Eqs. (39) and (40) the evolution equation  \n       \n      22 2 2\n,, , , ,\nN\n1\n1,1 , ,\n1\n1, 1 , ,11 1 1ˆˆ ˆ ˆ\n22\n3ˆˆ ˆˆ\n22\nˆˆ ˆˆ ,lm lm lm lm lm\nZl m Z l m\nZl m Z l mdYL Y L V Y V L Y Y L Vdt\niY L VL Y L VL Y\nY L VL Y L VL Y\n  \n\n\n   \n\n                   \n     \n     (41) \nwhere we have used the following repr esentations for th e expansions of /( )vV kT V V \nand /( )vk T     in terms of spherical harmonics, viz., \n , 1 ,1\nrs r rs rs VA Y\n\n  ,  , 10 , rs rsr\nrsVA Y\n  , \n , 1 ,1\nrs r rs rsBY\n\n  ,  , 10 , rs rsr\nrsBY\n  . \nThen (see Refs. 23 and 32 for details) the right-hand side of Eq. (41) may ultimately be written \nas a linear combination of averag es of spherical harmonics ,i.e., Eq. (10), because the action of \nthe operators 2 ˆˆˆ,,ZLLL  on ,lmY is34 \n ,,ˆ ,Zlm lmLYm Y 2\n,,ˆ 1,lm lmLYl l Y,, 1ˆ (1 ) ( 1 ) ,lm lmLY l l mm Y     \nand products of spherical harmonics may always be reduced to a sum of  spherical harmonics \nusing the Clebsch-Gordan series, viz.,34 \n22 11\n11 1\n11 21\n21,0 ,\n, 0 ,, 0 , ,, 1\n,, ,\n2(2 1)(2 1)\n4 21ll m mll\nll l m l m\nlm l m l m m\nll lCC llYY Y\nl \n\n\n . \nAPPENDIX C: EXPLICIT FORM OF ()ntC , ,nnQQ , AND ,, ,lml me \nThe general Eq. (10) as specialized to Eqs. (17) and (19) yields the 25-term \ndifferential-recurrence equation for the statistical moments ,,() ()lm lmct Y t , viz.,  27 N , ,2 , 2 ,2 , 1 ,2 , ,2 , 1 ,2 , 2\n,1 , 2 ,1 , 1 ,1 , ,1 , 1 ,1 , 2\n,, 2 ,() () () () () ()\n() () () () ()\n()n m n m nm n m nm n m nm n m nm n m nm\nnm n m nm n m nm n m nm n m nm n m\nnm nm ndc t vc t vc t vc t vc t vc tdt\nwc t wc t wc t wc t wc t\nxc t x   \n    \n   \n    \n\n \n \n, 1 ,, ,, 1 ,, 2\n,1 , 2 ,1 , 1 ,1 , ,1 , 1 ,1 , 2\n,2 , 2 ,2 , 1 ,2 , ,2 , 1() () () ()\n() () () ()\n() () () (mn m n mn m n mn m n mn m\nnm n m nm n m nm n m nm n m nm n m\nnm n m nm n m nm n m nm n mct x c t x ct x ct\nyc yc t yc t yc t yc t\nzc t zc t zc t zc t  \n \n   \n    \n  \n    \n   \n  ,2 , 2 )( ) ,nm n mzc t\n (42) \nwhere the coefficients ,nmx, etc. are given by \n * 3\n,1 0\n2\n*2 * *\n10 11 1 1(1 )(, )23\n2 (1 ) 3 1(, ) (, ) (, ),\n21 23 2 3nm P P P\nPP\nPP PP PPmh nnxi i m b J Y\nbcJ nn mYY Y\nnn \n    \n  \n         \n   \n ,1 2\n** *\n11 1 0 1112\n2( 1 2 )(, ) (, ) (, ) ,62 1 2 3nm\nPP\nPP P P P P Pihxn m n m i\nbcJ mib J Y Y Ynn\n   \n  \n    \n \n    *2\n, 11121 3(, ) ,(2 1)(2 3) 4PP\nnm PPnm nm nm nm bcJxYnn\n       \n \n22\n*\n,3 1 0\n*2 * *\n10 11 1 1(1 ) 1(, )21 23 2 3\n2(, ) (, ) (, ),3P\nnm PP\nPP\nPP PP PPnmm n b Jyi n h Ynn\nmb c JiY Y Y \n         \n  \n \n \n*\n,1 1\n**\n12 1 0 1 112(, )12 32 6\n2( 2 )(, ) (, ) ,23P\nnm P P\nPP\nPP PPnm nm nb JyYnn\nbcJn m nhii Y Y\n     \n\n     \n \n \n    \n *2\n,1 1123(, ) ,43 1 2 3 2PP\nnm P Pnm nm nmnm bcJyi Ynn\n      \n22\n,3 2\n** 2 * *\n10 10 11 1 11(1 )41 2\n(1 ) 2( ,) ( ,) ( ,)( ,) ,33nm\nPP P\nPP PP PP PPnm i mwn hn\nbJn m bcJYi Y Y Y \n          \n   \n \n  \n*\n,1 1 2\n**\n12 1 0 1 11 (1 )(, )41 6\n2( 1 2 ) 1(, ) (, ) ,23P\nnm P P\nPP\nPP PPnmnm bJnwYn\nbcJn m nhii Y Y\n     \n\n  \n    \n\n  28    *2\n,1 1 221 1(, ) ,43 4 1PP\nnm P Pnm nm nmnm bcJwi Yn\n       \n \n22 22\n,\n*2 * *\n10 11 1 1(1 ) (2 )\n23 21 25\n2 1(, ) (, ) (, ),23nm\nPP\nPP PP PPnm n m nznn n\nbcJYY Y        \n       \n\n 22\n**\n,1 0 1 1(1 ) 2 ( 3 ) 22(, ) (, ) ,32 3 2 1 2 5PP\nn m PP PPn m nm nm bcJ nzY Ynn n \n     \n \n *2\n,1 112 ( 3 ) ( 4 )(, ) ,43 2 3 2 1 2 5PP\nnm P Pn m nm nm nm bcJ nzYnn n\n        \n \n22 22\n,\n*2 * *\n10 11 1 1(1 ) ( ) 11\n21 21 23 2\n2(, ) (, ) (, ),3nm\nPP\nPP PP PPnm n m nvnn n\nbcJYY Y\n         \n  \n\n 22\n**\n,1 0 1 12( 1 ) ( ) 22 1(, ) (, )32 1 2 1 2 3PP\nn m PP PPnm nm n m bcJ nvY Ynn n \n   , \n   \n *2\n,1 1321 ( ) 1(, )43 2 1 2 1 2 3PP\nnm P Pnm nm nm nm bcJ nvYnn n\n    . \nIn order to rewrite Eq. (42) in the form of Eq. (20) explicitly, we define ()ntC  as the \ncolumn vectors arranged in an appropriate way from ,()nmct , viz., \n 0()t C0 ,  2,2\n2,2 1\n2, 2\n21 , 21\n21 , 22\n21 , 21()\n()\n()\n()()\n()\n()nn\nnn\nnn\nn\nnn\nnn\nnnct\nct\nct\ntct\nct\nct\n\n \n \n\n\n\n\n\n\nC\n, (n  1), (43) \nwhile  the matrices ,,nnnQQQ  are defined as \n 22\n21 21nn\nn\nnnXWQYX, 22\n21nn\nn\nn\nZYQ0Z, 2\n21 21n\nn\nnn\nV0QWV. (44) \nIn turn, the matrices ,,nnnQQQ  themselves consist of five submatrices lV, lW, lX, lY, and \nlZ of dimensions (2 1) (2 3)ll  , (2 1) (2 1)ll , (2 1) (2 1)ll ,(2 1) (2 3)ll  , and  29(2 1) (2 5)ll  , respectively. The elements of thes e five-diagonal submatrices, which are \nformed from the coefficients occurring in Eq. (42), are given by \n 4, , 3 3, , 2 2, , 1 1, , , , 1 , l n ml l m n ml l m n ml l m n ml l m n ml l mnmvvv v v    \n             V , \n 3, , 2 2, , 1 1, , , , 1 1, , 2 , l n ml l m n ml l m nml l m n ml l m nml l mnmww w w w     \n              W , \n 2, , 1 1, , , , 1 1, , 2 2, , 3 , l n m l lm n m l lm n m l lm n m l lm n m l lmnmxx xx x     \n              X , \n1 , , , ,1 1 , ,2 2 , ,3 3 , ,4 ,l n mll m n mll m n mll m n mll m n mll mnmyy y y y     \n                Y , \n,, 1 1 ,, 2 2 ,, 3 3 ,, 4 4 ,, 5 , l n mll m n mll m n mll m n mll m n mll mnmzz z z z      \n                 Z .  30REFERENCES \n1 L. Néel, Ann. Géophys. 5, 99 (1949). \n2 W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). \n3 W. T. Coffey and Yu. P. Kalmykov, J. Appl. Phys. 112, 121301 (2012). \n4 J. C. Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). \n5 L. Berger, Phys. Rev. B 54, 9353 (1996). \n6 M. D. Stiles and J. Miltat, in: Spin Dynamics in Confined Magnetic Structures  III, edited by \nB. 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B 72, 180405 (2005). \n26 T. Taniguchi and H. Imamura, Phys. Rev. B 83, 054432 (2011).  3227 D. V. Berkov and N. L. Gorn, Phys. Rev.  B 72, 094401 (2005); K. It o, IEEE Trans. Magn. \n41, 2630 (2005). \n28 D. Pinna, A.D. Kent, and D.L. Stein, J. Appl. Phys. 114, 033901 (2013). \n29 M. d'Aquino, C. Serpico, R. Bonin, G. Be rtotti, and I. D. Ma yergoyz, Phys. Rev. B 84, \n214415 (2011). \n30 R. L. Stratonovich, Topics in the Theory of Random Noise,  Vol. 1 (Gordon and Breach \nNew York, 1963). \n31 H. Risken, The Fokker-Planck Equation , 2nd ed. (Berlin, Springer-Verlag, 1989). \n32 Yu. P. Kalmykov and S. V. Titov, Phys. Rev. Lett. 82, 2967 (1999). \n33 G. D. Chaves-O'Flynn, D. L. Stein, A. D. Kent, and E. Vanden-Eijnden, J. Appl. Phys. 109, \n07C918 (2011). \n34 D. A. Varshalovitch, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular \nMomentum (World Scientific, Singapore, 1988). \n35 B. Ouari and Yu. P. Kalmykov, J. Appl. Phys. 100, 123912 (2006); Yu. P. Kalmykov and \nB. Ouari, Phys. Rev B 71, 094410 (2005) \n36 Y. P. Kalmykov, W. T. Coffey , U. Atxitia, O. Chubykalo-Fesenko, P. M. Déjardin, and R. \nW. Chantrell, Phys. Rev. B 82, 024412 (2010). \n37 A. N. Slavin and V. S. Tiberkevich, Phys. Rev. B  72, 092407 (2005); J.-V. Kim, V. S. \nTiberkevich, and A. N. Slavin, Phys. Rev. Lett. 100, 017207 (2008); Phys. Rev. Lett. 100, \n167201 (2008).  "    },    {        "title": "1303.5262v1.Co_monolayers_and_adatoms_on_Pd_100___Pd_111__and_Pd_110___Anisotropy_of_magnetic_properties.pdf",        "content": "arXiv:1303.5262v1  [cond-mat.mtrl-sci]  21 Mar 2013Co monolayers and adatoms on Pd(100), Pd(111) and Pd(110): A nisotropy of\nmagnetic properties\nO.ˇSipr,1,∗S. Bornemann,2H. Ebert,2S. Mankovsky,2J. Vack´ aˇ r,1and J. Min´ ar2\n1Institute of Physics of the ASCR v. v. i., Cukrovarnick´ a 10, CZ-162 53 Prague, Czech Republic\n2Universit¨ at M¨ unchen, Department Chemie, Butenandtstr. 5-13, D-81377 M¨ unchen, Germany\n(Dated: September 16, 2018)\nWe investigate to what extent the magnetic properties of dep osited nanostructures can be influ-\nenced by selecting as a support different surfaces of the same substrate material. Fully relativistic\nab initio calculations were performed for Co monolayers and adatoms o n Pd(100), Pd(111), and\nPd(110) surfaces. Changing the crystallographic orientat ion of the surface has a moderate effect\non the spin magnetic moment and on the number of holes in the dband, a larger effect on the\norbital magnetic moment but sometimes a dramatic effect on th e magnetocrystalline anisotropy\nenergy (MAE) and on the magnetic dipole term Tα. The dependence of Tαon the magnetization\ndirection αcan lead to a strong apparent anisotropy of the spin magnetic moment as deduced from\nthe X-ray magnetic circular dichroism (XMCD) sum rules. For systems in which the spin-orbit\ncoupling is not very strong, the Tαterm can be understood as arising from the differences betwee n\ncomponents of the spin magnetic moment associated with diffe rent magnetic quantum numbers m.\nPACS numbers: 75.70.Ak,75.30.Gw,78.70.Dm,73.22.Dj\nKeywords: magnetism,anisotropy,nanosystems,XMCD\nI. INTRODUCTION\nThe magnetic properties of surface deposited nanos-\ntructures have been in the ongoing focus of many ex-\nperimental and theoretical investigations as they often\nexhibit interesting and sometimes unexpected phenom-\nena. One of the main features in this context is that the\nlocal magnetic moments and their mutual interaction as\nwell as the magnetocrystalline anisotropy energy (MAE)\nare in general different and often much larger in nanos-\ntructures than in corresponding bulk systems. Various\naspects of the magnetism of many different nanostruc-\ntures were studied in the past to identify the key fac-\ntors which could then be used to tune the properties of\nsuch systems in a desired way. It has been known for\nsome time that one such key factor is the coordination\nnumber, with smaller coordination numbers generally im-\nplying larger magnetic moments.1–3However, coordina-\ntion numbers alone do not fully determine magnetism of\nnanostructures. The chemical composition can play a\nsignificant role as well. An Fe monolayer, for instance,\nhas a larger spin magnetic moment when deposited on\nAu(111) than when deposited on Pt(111), whereas for a\nCo monolayer it is vice versa .3The situation is even more\ndiverse for the MAE where different substrates may lead\nto different properties of systems of otherwise identical\ngeometries. For example, Co 2and Ni 2dimers on Pt(111)\nhave out-of-plane magnetic easy axis but the same dimers\non Au(111) have an in-plane magnetic easy axis.3\nExperimental research on magnetism of nanostructures\nrelies heavily on the X-ray magnetic circular dichroism\n(XMCD) sum rules.4–6The strength of these sum rules\nis that they give access to spin magnetic moments µspin\nand orbital magnetic moments µorbseparately and in a\nchemically specific way.7,8However, the XMCD spin sum\nrule does not provide µspinalone but only its combinationµspin+ 7Tα, whereTαis the magnetic dipole term (for\nthe magnetization Mparallel to the αaxis,α=x,y,z ).7\nFor bulk systems, Tαcan be usually neglected but for\nsurfaces and clusters the Tαterm can have significant\ninfluence, as it has been demonstrated experimentally9,10\nand theoretically.11–13The anisotropy of the magnetic\ndipole term was predicted on general grounds14and some\nestimates concerning the magnitude of this anisotropy in\nnon-cubic bulk systems were given based on atomic-like\nmodel Hamiltonians14or onab initio calculations.15\nMagnetic nanostructures may be prepared by combin-\ning and arranging different magnetic elements on differ-\nent substrates. In this respect one can also address sur-\nfaces of different crystallographic orientations. Thus, it\nis important to know how the magnetic properties can be\ncontrolled by selecting for the substrate crystallographi-\ncally different surfaces of the same material and whether\none can expect different effects for complete monolayers\nand for adatoms. Connected with this is the question\nabout the effects on the Tαterm, because XMCD is per-\nhaps the most frequently used experimental technique in\nthis field and it is desirable to know how Tαcan influ-\nence the values of magnetic moments deduced from the\nXMCD sum rules. For planning and interpreting such\nexperiments, it would be very useful not only to know\ntheTαvalues from ab initio calculations but also to have\na simple intuitive interpretation of the Tαterm.\nIn order to learn more about this, we undertook a sys-\ntematic study of Co monolayers and adatoms on Pd(100),\nPd(111), and Pd(110) surfaces. Fully relativistic ab ini-\ntiocalculations were performed to obtain µspin,µorb, and\nTαfor different magnetization directions. The MAE was\ndetermined for all these systems as well. The accuracy of\nan approximative expression for the Tαterm was checked\nto see whether it captures the essential physics. It is\nshown in the following that monolayers and adatoms on2\ndifferent crystallographic surfaces may have indeed quite\ndifferent magnetic properties, especially as concerns the\nMAE. Moreover, it is also demonstrated how the depen-\ndence of the Tαterm on the magnetization direction leads\nto a surprisingly strong apparent anisotropy of µspinas\ndeduced from the XMCD sum rules.\nII. METHODS\nA. Investigated systems\nWe investigated Co monolayers on Pd(100), Pd(111)\nand Pd(110) and also Co adatoms on the same surfaces.\nThe corresponding structure diagrams are shown in Fig. 1\n(for adatoms, obviously only one Co atom is kept). Two\nhollow adatom positions are possible for the (111) sur-\nface, differing by the position of the adatom with respect\nto the sub-surface layer; we consider the fcc position in\nthis work (unless specified otherwise).\nThe Pd substrate has fcc structure with lattice con-\nstanta=3.89 ˚A. To determine the distances between the\nCo atoms and the substrate, we relied in most cases on\nthe “constant volume approximation”: the vertical Co–\nPd interplanar distance zCo-Pd is taken as an average\nbetween the interlayer distance in bulk Pd and the inter-\nlayer distance in a hypothetical pseudomorphically grown\nfcc Co film compressed vertically in such a way that\nthe atomic volume of Co is the same as in bulk Co.16\nIn addition we took also into account relevant experi-\nmental data and results of ab-initio geometry relaxations\nwhen available. For example, the constant volume ap-\nproximation yields zCo-Pd =1.70 ˚A for a Co monolayer on\nPd(100) while we took zCo-Pd =1.65 ˚A instead, following\nthe surface X-ray diffraction experiment of Meyerheim et\nal.17For the other two surfaces we used the constant vol-\nume approximation distances, namely, zCo-Pd =1.96 ˚A for\nCo on Pd (111) and zCo-Pd =1.20 ˚A for Co on Pd(110).\nIn the case of the (111) surface we can compare our\ndistance with an EXAFS-derived experimental distance\nzCo-Pd =2.02 ˚A (Ref. 18) and with an ab initio equilib-\nrium distance zCo-Pd =1.91 ˚A (Ref. 19). It follows from\nthis comparison that the constant-volume-approximation\nleads to reasonable distances.\nSystems with interplanar distances as given above will\nbe called systems with “optimized geometries”. Apart\nfrom that, we investigate for comparison also systems\nwhere the Co atoms are located in ideal positions of the\nunderlying Pd lattice. For this we use the designation\n“bulk-like geometry”. The interplanar distances are sum-\nmarized in Tab. I.\nFor adatoms we use the same zCo-Pd distances as for\nmonolayers. This is a simplification because the constant\nvolume approximation will work worse for adatoms than\nfor monolayers. For example the ab initiozCo-Pd dis-\ntance for a Co adatom on Pd(111) is 1.66 ˚A (Ref. 20)\nin contrast to our optimized geometry value of 1.96 ˚A.\nHowever, by using identical zCo-Pd distances for mono-TABLE I. Vertical distances zCo-Pd between the plane con-\ntaining Co atoms and plane containing Pd atoms for systems\ninvestigated in this study. The unit is ˚A.\nsurface optimized geometry bulk-like geometry\n(100) 1 .65 1 .95\n(111) 1 .96 2 .25\n(110) 1 .20 1 .38\nlayers and adatoms, the net effect due to the change in\nCo coordination can be studied. It will be shown that\nthe effect of varying the distances is in fact smaller than\nthe effect of monolayer-to-adatom transition.\nB. Computational scheme\nThe calculations were performed within the ab initio\nspin density functional framework, relying on the local\nspin density approximation (LSDA) with the Vosko, Wilk\nand Nusair parametrization for the exchange and corre-\nlation potential.21The electronic structure is described,\nincluding all relativistic effects, by the Dirac equa-\ntion, which is solved using the spin polarized relativis-\ntic multiple-scattering or Korringa-Kohn-Rostoker (SPR-\nKKR) Green’s function formalism22as implemented in\nthespr-tb-kkr code.23The potentials were treated\nwithin the atomic sphere approximation (ASA) and for\nthe multipole expansion of the Green’s function, an an-\ngular momentum cutoff ℓmax=3 was used.\nThe electronic structure of Co monolayers on Pd sur-\nfaces was calculated by means of the tight-binding or\nscreened KKR technique.24The substrate was modeled\nby slabs of 13–14 layers (i.e. a thickness of 17–27 ˚A, de-\npending on the surface orientation), the vacuum was rep-\nresented by 4–5 layers of empty sites. The adatoms were\ntreated as embedded impurities: first the electronic struc-\nture of the host system (clean surface) was calculated and\nthen a Dyson equation for an embedded impurity cluster\nwas solved.25The impurity cluster contains 135 sites if\nnot specified otherwise; this includes a Co atom, 50–60\nPd atoms and the rest are empty sites.\nIt should be stressed that the embedded clusters define\nthe region where the electronic structure and potential\nof the host is allowed to relax due to the presence of the\nadatom and notthe size of the considered system. In this\nrespect the Green’s function approach differs from the of-\nten used supercell approach: there is an unperturbed host\nbeyond the relaxation zone in the former approach while\nin the latter approach, the supercell is terminated either\nby vacuum or by another (interfering) relaxation zone\npertaining to an adjacent adatom. The sizes of the em-\nbedded clusters and the sizes of the supercells thus have\na different meaning and cannot be directly compared.\nThe magnetocrystalline anisotropy energy (MAE) is\ncalculated by means of the torque T(ˆn)\nˆuwhich describes\nthe variation of the energy if the magnetization direction3\n(100)\nxy(111)\nxy(110)\nxy\nFIG. 1. (Color online) Structure diagrams for a Co monolayer on Pd(100), Pd(111) and Pd(110). The blue and yellow circles\nrepresent the Co and Pd atoms, respectively. The orientatio n of the xandycoordinates used throughout this paper is also\nshown.\nˆnis infinitesimally rotated around an axis ˆ u. For uniaxial\nsystems where the total energy can be approximated by\nE(θ) =E0+K2sin2(θ) +K4sin4(θ),\nthe difference E(90◦)−E(0◦) is equal to the torque eval-\nuated forθ= 45◦.26The torque itself was calculated by\nrelying on the magnetic force theorem.27\nApart from the magnetocrystalline anisotropy induced\nby the spin-orbit coupling, the magnetic easy axis is\nalso determined by the so-called shape anisotropy caused\nby magnetic dipole-dipole interactions. The shape\nanisotropy energy is usually evaluated classically by a\nlattice summation over the magnetostatic energy contri-\nbutions of individual magnetic moments, even though\nit can be in principle obtained ab initio via a Breit\nHamiltonian.28In this paper, we always deal only with\nthe magnetocrystalline contribution to the magnetic\nanisotropy unless stated otherwise.\nIII. RESULTS\nA. Magnetic moments and magnetocrystalline\nanisotropy\nTo assess the effect of selecting different crystallo-\ngraphic surfaces and of going from a monolayer to an\nadatom, we calculated magnetic moments, numbers of\nholes in the Co dband and the MAE for all these sys-\ntems. The results are summarized in Tab. II. For each\nsystem, the data are shown first for the optimized geom-\netry and then for the bulk-like geometry (numbers in the\nbrackets). The x,y, andzsuperscripts in the column\nheader labels indicate the direction of the magnetization\nM.\nThe spin magnetic moment µspinand the number of\nholes in the dbandnhare shown only for M/bardblz, because\nthey are practically independent on the magnetization\ndirection: by varying it, µspincan be changed by no more\nthan 0.2 % and nhby no more than 0.1 %. On the\nother hand, for µorbthe differences can be quite large.\nThe second in-plane magnetization direction M/bardblywasinvestigated only for the (110) surface, because there is\nonly very small “intraplanar anisotropy” for the (100)\nand (111) surfaces (this issue is addressed in more detail\nin Sec. III C). For bulk hcp Co we get µspin=1.61µB,\nµorb=0.08µBandnh=2.48.\nChanging the surface orientation has a moderate effect\nonµspinandnh. The differences in µspinwhen going\nfrom one surface to another are at most 9 %. For nh\nthese differences are at most 5 %. However, the situation\nis quite different for µorbwhere the differences are 20–\n50 %. The sensitivity in µorbfinds its counterpart in\nthe sensitivity of the MAE. For example, the magnetic\neasy axis for a Co monolayer is in-plane for the (100) and\n(110) surfaces but out-of-plane for the (111) surface. For\nthe adatom, the easy axis is in-plane for the (110) surface\nbut out-of-plane for the (100) and (111) surfaces. So in\nthis respect the choice of the crystallographic surface can\nhave a dramatic influence.\nAnother finding emerging from Tab. II is that as\nconcernsµspin, the difference between monolayers and\nadatoms is only quantitative in most cases. A surpris-\ningly small difference in this respect is found for the (110)\nsurface. As the same Co–Pd distances have been used\nfor monolayers and adatoms, one observes here the net\neffect of the change in Co coordination. For µorb, the\ndifference between monolayers and adatoms is obviously\nmuch larger than for µspin. For the MAE this difference\ncan again be essential: The magnetic easy axis for a Co\nmonolayer on Pd(100) is in-plane while for a Co adatom\non the same surface it is out-of-plane. Similarly, the mag-\nnetic easy axis for a monolayer on Pd(110) is parallel to\nthey-axis while for an adatom it is parallel to the x-axis.\nChanging the distance between Co atoms and the sur-\nface clearly affects the magnetic properties (cf. the values\nwith and without brackets in Tab. II). However, it is note-\nworthy that the effect of geometry relaxation is smaller\nthan the effect of the transition from the monolayer to\nthe adatom.\nWe calculated also the magnetic shape anisotropy for\nthe monolayers (classically, via a lattice summation,\ntaking into account also moments on Pd atoms). As\nexpected, this contribution favors always an in-plane4\nTABLE II. Magnetic properties of Co monolayers and adatoms o n Pd(100), Pd(111), and Pd(110). The first column specifies\nwhether the values are for a monolayer or for an adatom, the se cond column contains spin magnetic moment for the Co atom\nforM/bardblz(in units of µB), the third column contains number of holes in the dband for M/bardblz. The fourth, fifth and sixth\ncolumns contain orbital magnetic moments for the Co atom for M/bardblz,M/bardblx, andM/bardbly, respectivelly. The last three columns\ncontain the MAE between indicated magnetization direction s (in meV per Co atom). Numbers without brackets stand for\nsystems with optimized Co–Pd distances, numbers in bracket s stand for systems with a bulk-like geometry (see Sec. II A).\nµ(z)\nspin n(z)\nh µ(z)\norb µ(x)\norb µ(y)\norb E(x)−E(z)E(y)−E(z)E(x)−E(y)\nCo on Pd(100)\nmonolayer 2 .09 2 .45 0 .132 0 .203 −0.73\n(2.07) (2 .39) (0 .190) (0 .241) ( −0.69)\nadatom 2 .29 2 .57 0 .299 0 .279 0 .26\n(2.32) (2 .53) (0 .610) (0 .473) (2 .69)\nCo on Pd(111)\nmonolayer 2 .02 2 .43 0 .135 0 .136 0 .36\n(1.99) (2 .41) (0 .154) (0 .176) (0 .21)\nadatom 2 .35 2 .62 0 .605 0 .355 5 .50\n(2.34) (2 .52) (0 .780) (0 .575) (6 .38)\nCo on Pd(110)\nmonolayer 2 .15 2 .50 0 .192 0 .183 0 .210 −0.15 −0.43 0 .28\n(2.18) (2 .54) (0 .215) (0 .220) (0 .289) ( −0.48) ( −0.97) (0 .49)\nadatom 2 .20 2 .49 0 .270 0 .347 0 .201 −1.51 1 .10 −2.61\n(2.25) (2 .47) (0 .349) (0 .472) (0 .255) ( −1.88) (2 .01) ( −3.89)\norientation of the magnetization. For Co monolayers\non Pd(100) and Pd(111), we get E(x)\ndip-dip−E(z)\ndip-dip=\n−0.1 meV. For Co monolayers on Pd(110), there is a\nsmall difference regarding the xandydirections: we get\nE(x)\ndip-dip−E(z)\ndip-dip=−0.07 meV and E(y)\ndip-dip−E(z)\ndip-dip=\n−0.09 meV. By comparing these values with the values\nshown in Tab. II, we see that the shape anisotropy en-\nergy is smaller in magnitude than the magnetocrystalline\nanisotropy energy and thus the shape anisotropy does not\nchange the orientation of the magnetic easy axis as de-\ntermined by the magnetocrystalline anisotropy.\nB. Induced magnetic moments\nPalladium is not magnetic as an element but it is quite\npolarizable.29,30Spin magnetic moments induced in the\nPd substrate by Co monolayers and adatoms are shown\nin Tab. III for all three surface orientations. In the case\nof Co monolayers, the induced moments are shown for\nthe first three atomic layers of Pd below the Co layer\n[denoted as Pd(1), Pd(2) and Pd(3) in Tab. III]. Note\nthat the interlayer distances are 1.95 ˚A, 2.25 ˚A and 1.38 ˚A\nfor the (100), (111) and (110) surfaces, respectively. The\nrelatively large µspinfor the Pd(2) and Pd(3) sites in\nthe case of the (110) surface reflects the relatively small\ninterlayer distance for this crystallographic orientation.\nIn the case of adatoms, the description is formally more\ncomplicated because Pd atoms belonging to the same co-\nordination shell around the Co atom are not all equiva-\nlent: some of them belong to the surface layer, some to\nthe sub-surface layer and so on. In order not be over-TABLE III. Spin magnetic moments for Pd atoms which are\nfirst, second and third nearest neighbors of Co atoms, in unit s\nofµB. As in Tab. II, the numbers without brackets stand for\nsystems with optimized geometry and the numbers in brackets\nstand for systems with bulk-like geometry.\nPd(1) Pd(2) Pd(3)\nCo on Pd(100)\nmonolayer 0 .29 0 .17 0 .11\n(0.25) (0 .16) (0 .10)\nadatom 0 .18 0 .06 0 .04\n(0.15) (0 .06) (0 .04)\nCo on Pd(111)\nmonolayer 0 .32 0 .16 0 .03\n(0.25) (0 .15) (0 .06)\nadatom 0 .16 0 .02 0 .04\n(0.12) (0 .02) (0 .03)\nCo on Pd(110)\nmonolayer 0 .29 0 .22 0 .17\n(0.29) (0 .24) (0 .19)\nadatom 0 .15 0 .04 0 .04\n(0.15) (0 .05) (0 .04)\nwhelmed by too much data, we display here only mo-\nments averaged over all atoms of a given coordination\nshell. Symbols Pd(1), Pd(2), and Pd(3) in Tab. III stand\nnow for the first, second, and third shell of Pd atoms\naround the Co adatom.\nMoreover, we also calculated the orbital magnetic mo-\nments for the Pd atoms in all systems and we found that\nµorbamounts to about 8–17 % of the corresponding µspin.5\nIn this section we deal only with magnetic moments\non those Pd atoms which are close to the Co atoms. The\nissue of more distant Pd atoms and of the total charge\ncontained in the polarization cloud is dealt with in the\nAppendix. Here, we would only like to stress that it fol-\nlows from the analysis outlined in the Appendix that our\nmodel system is clearly adequate to yield reliable values\nof induced magnetic moments for the Pd(1), Pd(2), and\nPd(3) sites.\nC. Azimuthal dependence of MAE\nIn general, the MAE defined as the difference between\ntotal energies for in-plane and out-of-plane orientation of\nthe magnetization will depend on the azimuthal angle φ.\nThis dependence is often ignored but may sometimes be\nsignificant. In our case, the intraplanar MAE E(x)−E(y)\nis quite comparable to E(x)−E(z)orE(y)−E(z)for\nthe (110) surface (see Tab. II). To get a more com-\nplete picture, we inspect the azimuthal dependence of\nE(/bardbl)(φ)−E(z), whereE(/bardbl)(φ) is the total energy if M\nis in the surface plane ( θ=0◦) with the azimuthal angle\nφ. Our results for a Co adatom on all three Pd surfaces\nare shown in Fig. 2. The data reported here were ob-\ntained for the bulk-like geometry but the trends would\nbe similar for any zCo-Pd distance.\nOne can see from Fig. 2 that the E(/bardbl)(φ)−E(z)curves\nfollow the symmetry of the appropriate surface, as ex-\npected. The amplitude of these curves is the most in-\nteresting information here. For high-symmetry surfaces,\nit is almost negligible: 0.008 meV or 3 % of the average\nvalue for Co on Pd(100) and 0.06 meV or 1 % of the\naverage value for Co on Pd(111). For the (110) surface,\nhowever, the amplitude is 2.6 meV and to speak about an\naverage MAE does not make sense in this case, as illus-\ntrated by the fact that the magnetic easy axis is in-plane\nforφ= 0◦and out-of-plane for φ= 90◦.\nD. Relation between magnetic dipole term and\nm-decomposed spin magnetic moment\nThe spin magnetic moment sum rule for the L2,3edge\nXMCD spectra can be written for a sample magnetized\nalong theαdirection as7\n3\nI/integraldisplay\n(∆µL3−2∆µL2) dE=µspin+ 7Tα\nnh,(1)\nwhere ∆µL2,3are the differences ∆ µ=µ(+)−µ(−)be-\ntween absorption coefficients for the left and right circu-larly polarized light propagating along the αdirection,I\nis the integrated isotropic absorption spectrum, µspinis\nthe local spin magnetic moment (only its dcomponent\nenters here), nhis the number of holes in the dband, and\nTαis the magnetic dipole term related to the delectrons.\nTαcan be written as31,32\nTα=−µB\n/planckover2pi1/angbracketleftˆTα/angbracketright,\n=−µB\n/planckover2pi1/angbracketleftBigg/summationdisplay\nβQαβSβ/angbracketrightBigg\n, (2)\nwhere\nQαβ=δαβ−3r0\nαr0\nβ (3)\nis the quadrupole moment operator and Sαis the spin\noperator. If zis the quantization axis, the eigenvalues of\nSzare±(1/2)/planckover2pi1.\nA more transparent expression for Tαcan be obtained\nif the spin-orbit coupling can be neglected. Then one can\nwrite32\nˆTx=/parenleftBig\n−µB\n/planckover2pi1/parenrightBig\nˆQxxˆS¯zforM/bardblx ,\nˆTy=/parenleftBig\n−µB\n/planckover2pi1/parenrightBig\nˆQyyˆS¯zforM/bardbly ,\nˆTz=/parenleftBig\n−µB\n/planckover2pi1/parenrightBig\nˆQzzˆS¯zforM/bardblz ,(4)\nwhere ˆQxx,ˆQyyand ˆQzzare quadrupole moment com-\nponents referred to the crystal (global) reference frame\nand ˆS¯zis the spin component with respect to the local\nreference frame in which ¯ zis identical to the spin quan-\ntization axis. We are interested in the expectation value\nof the ˆTαoperator acting on the dcomponents of the\nwave function in the vicinity of the photoabsorbing site.\nUsing for the sake of clarity a simplified two-component\nformulation instead of the full Dirac approach, the wave\nfunction can be expanded in the angular-momentum ba-\nsis as\nψEk(r) =/summationdisplay\nℓm/summationdisplay\nsa(s)\nEkℓm(r)Yℓm(ˆr)χ(s)(5)\nto obtain\nTα=/parenleftBig\n−µB\n/planckover2pi1/parenrightBig/integraldisplayEF\n−∞dE/integraldisplay\nBZdk/angbracketleftψEk|ˆQααˆS¯z|ψEk/angbracketright.(6)\nRestricting ourselves just to the ℓ= 2 component and\nomitting the corresponding subscript in a(s)\nEkℓm(r), we get6\n0.2550.260.265E(||)()-E(z)[meV]\n0/2 3/2 2\nazimuthal angle(100) 5.465.485.55.52\n0/2 3/2 2\nazimuthal angle(111)\n-2-101\n0/2 3/2 2\nazimuthal angle(110)\nFIG. 2. (Color online) Difference between total energies for in-plane and out-of-plane magnetization for a Co adatom on P d\n(100), (111), and (110) surfaces (bulk-like geometry). Poi nts are results of the calculation, dashed lines are sinusoi dal fits. The\norientation of the xandyaxes is as in Fig. 1.\nTα=/parenleftBig\n−µB\n/planckover2pi1/parenrightBig/integraldisplayEF\n−∞dE/integraldisplay\nBZdk/summationdisplay\nmm′/summationdisplay\nss′/integraldisplay\ndra(s)∗\nEkm(r)Y∗\n2m(ˆr)Qααa(s′)\nm′kE(r)Y2m′(ˆr)/angbracketleftχ(s)|ˆS¯z|χ(s′)/angbracketright\n=/parenleftBig\n−µB\n/planckover2pi1/parenrightBig/integraldisplayEF\n−∞dE/integraldisplay\nBZdk/summationdisplay\nmm′/integraldisplay\nr2dr/bracketleftBig\na↑∗\nEkm(r)a↑\nm′kE(r)−a↓∗\nEkm(r)a↓\nm′kE(r)/bracketrightBig\n×\n/angbracketleftY2m|ˆQαα|Y2m′/angbracketright1\n2/planckover2pi1\n=1\n2(−µB)/summationdisplay\nmm′/bracketleftBig\nN↑\nmm′−N↓\nmm′/bracketrightBig\n/angbracketleftY2m|ˆQαα|Y2m′/angbracketright, (7)\nwhere the spin-dependent number of states matrix N(s)\nmm′\nis defined as\nN(s)\nmm′=/integraldisplayEF\n−∞dE/integraldisplay\nBZdk/integraldisplay\nr2dra(s)∗\nEkm(r)a(s)\nm′kE(r).\nThe difference of the diagonal terms of N(s)\nmmis just\nthe spin magnetic moment decomposed according to the\nmagnetic quantum number m,\n(−µB)/parenleftbig\nN↑\nmm−N↓\nmm/parenrightbig\n=µ(m)\nspin,\nwith the sum of all the mcomponents giving the total\nspin magnetic moment (of the delectrons, in our case).\nTherefore, if it was possible to restrict the sum (7) just\nto the terms diagonal in m, one would have\nTα=1\n2/summationdisplay\nmµ(m)\nspin/angbracketleftY2m|ˆQαα|Y2m/angbracketright. (8)\nThe procedure we employed above is essentially the one\nsuggested by St¨ ohr,31,32but we present it here in a more\nexplicit way.TABLE IV. Diagonal components of the quadrupole opera-\ntor in the basis of real spherical harmonics. Non-diagonal\ncomponents are all zero except for the components given in\nEq. (9).\nQxx Qyy Qzz\n/angbracketleftYxy|ˆQαα|Yxy/angbracketright −2\n7−2\n74\n7\n/angbracketleftYyz|ˆQαα|Yyz/angbracketright4\n7−2\n7−2\n7\n/angbracketleftY3z2−r2|ˆQαα|Y3z2−r2/angbracketright2\n72\n7−4\n7\n/angbracketleftYxz|ˆQαα|Yxz/angbracketright −2\n74\n7−2\n7\n/angbracketleftYx2−y2|ˆQαα|Yx2−y2/angbracketright −2\n7−2\n74\n7\nThe coefficients /angbracketleftY2m|ˆQαα|Y2m′/angbracketrightcan be obtained by\nanalytic integration. If we use the basis of realspherical\nharmonics, the only “cross-terms” which are non-zero are\n/angbracketleftYx2−y2|ˆQxx|Y3z2−r2/angbracketright= (2/7)√\n3,\n/angbracketleftYx2−y2|ˆQyy|Y3z2−r2/angbracketright=−(2/7)√\n3.(9)\nOtherwise, only the diagonal terms /angbracketleftY2m|ˆQαα|Y2m/angbracketrightare\nnon-zero and we list them in Tab. IV (see also Refs. 14\nand 31). Therefore, in the absence of spin-orbit coupling,7\nEq. (8) presents an exact expression for Tzand an ap-\nproximate expression for TxandTy[due to the existence\nof non-diagonal terms (9)]. As argued by St¨ ohr,32the\nnon-diagonal terms drop out of the sum in Eq. (7) for\nhigh symmetry systems.\nEq. (8) together with Tab. IV illustrate the common\nstatement that the magnetic dipole term Tαis related\nto spin anisotropy: if the m-components of µspinare all\nidentical,Tαis zero (in the absence of spin-orbit cou-\npling). It is also evident from Eq. (8) and Tab. IV that\ntheTαterm will generally depend on the magnetization\ndirectionα.\nTo get a more quantitative feeling of how the various\ncontributions add together to generate Tα, we present in\nTab. V the m-decomposed magnetic moment µ(m)\nspinand\nindividual terms of the sum (8) for Co monolayers on\nPd surfaces. One can see that the Tαterm is formed by\na competition between those mcomponents which con-\ntain theαcoordinate and those which do not (they con-\ntribute with an opposite sign, as it can be seen also from\nTab. IV). In fact, this is what is meant by the statement\nthat theTαterm describes the anisotropy of µspin.\nEq. (8) gives an intuitive insight into Tαprovided that\nthe underlying approximations — the neglect of the spin-\norbit coupling and of the non-diagonal terms shown in\nEq. (9) — are not too crude. To check this, we com-\npare the values of Tαcalculated via the exact relation in\nEq. (2) and via the approximative Eq. (8). Special at-\ntention is paid to the differences between the Tαterms\nfor different orientations of M, because the 7( Tα−Tβ)\nquantities determine the apparent anisotropy of µspinas\ndeduced from the XMCD sum rule in Eq. (1). The out-\ncome for both monolayers and adatoms is summarized\nin Tab. VI. Let us recall that for bulk hcp Co, the mag-\nnetic dipole term is very small (we get Tz=−0.002µB).\nNote that all values presented in Tabs. V–VI were ob-\ntained from fully relativistic calculations, including the\nspin-orbit coupling.\nOne can see from our results that the approximative\nexpression for Tαworks quite well for the Co-Pd systems:\nquantitative deviations sometimes occur but the main\ntrend is well maintained. One can expect that for systems\nwith a strong spin-orbit coupling the deviations between\nEqs. (2) and (8) will be larger.\nThe last two columns of Tab. VI contain the values of\n7(Tx−Tz) and, for the case of the (110) surface, also of\n7(Ty−Tz). These values are comparable to µspinwhich\nmeans that even though µspinpractically does not depend\non the magnetization direction at all, its combination\nµspin+ 7Tαprobed by the XMCD sum rule may strongly\ndepend on the magnetization direction.\nIV. DISCUSSION\nWe investigated how the magnetic properties of Co\nadatoms and monolayers can be manipulated by select-\ning different supporting Pd surfaces. We found that thishas a moderate effect on µspinandnh, larger effect on\nµorband dramatic effect on the MAE and on the Tα\nterm. For the adatoms the effect is larger than for the\nmonolayers. Moreover, the transition from monolayers\nto adatoms has a larger effect than a moderate variation\nin the height of the Co layer above the substrate. If the\nspin-orbit coupling is not very strong, the Tαterm can be\nunderstood as arising from a competition between those\nm-decomposed components of µspinwhich are associated\nwith theαcoordinate and those which are not.\nIn the past, the influence of the orientation of super-\nlattices (multilayers) on magnetic properties was already\ninvestigated, however, the focus was mainly on the role\nof defects and interface abruptness.33Here, we deal with\nperfect monolayers and surfaces and investigate how sole\nselection of a different surface can affect various quanti-\nties related to magnetism. Likewise, the importance of\ntheTzterm for an XMCD sum rules analysis has been\nhighlighted before when it was found that the absolute\nvalue of 7Tzamounts to about 20 % of µspinfor some\nlow-dimensional systems12or that for atomic clusters\nµspincan show a different behavior with changing clus-\nter size when compared to µspin+ 7Tz.13In this study\nthe importance of the anisotropy of the magnetic dipole\nterm in nanostructures is stressed for the first time and\nit should be noted that the anisotropy of Tαwhich we\nhighlight here is primarily connected with the breaking\nof the crystal symmetry at the surface and occurs even\nwithout spin-orbit coupling.\nFor the monolayers, the changes in µspinwhen going\nfrom one surface to another reflect the corresponding\nchanges in the coordination numbers: µspinis largest for\nthe (110) monolayer where each Co atom has got only\ntwo nearest neighboring Co atoms, next comes the (100)\nmonolayer with four Co neighbors and the lowest µspin\nis obtained for the (111) monolayer with six Co neigh-\nbors. This complements an analogous trend found ear-\nlier for free1and supported clusters.2,3,34The magnetic\nmoments induced at individual Pd atoms are larger for\nCo monolayers than for Co adatoms, which reflects the\nfact that for monolayers, Pd atoms are polarized by more\nthan one Co atom.\nThe large amount of data gathered here for quite a\ncomplete set of systems allows a comprehensive look at\nthe relation between the MAE and the anisotropy of µorb.\nIn this respect Bruno’s formula35\nE(α)−E(β)=−ξ\n4/bracketleftBig\nµ(α)\norb−µ(β)\norb/bracketrightBig\n(10)\nconnecting the differences of total energies to the dif-\nferences of orbital magnetic moments for two orienta-\ntions of the magnetization, αandβ, proved to be very\nuseful36despite its limitations,37which become more se-\nvere in the case of multicomponent systems with large\nspin-orbit coupling parameter ξfor the non-magnetic\ncomponent.38,39To assess the situation for 3 d-4dalloys,\nwe compare the differences ∆ µorband ∆E, using all\nthe appropriate values given in Tab. II. The outcome is8\nTABLE V. Spin magnetic moment decomposed according to the ma gnetic quantum number mtogether with the corresponding\nT(m)\nα=1\n2µ(m)\nspin/angbracketleftY2m|ˆQαα|Y2m/angbracketrightterms of the decomposition (8) for Co monolayers on Pd (optim ized geometry). The sums of\nthese components are shown in the last row for each system and they correspond to the total µspin,Tz,Tx, andTyof thed\nelectrons [evaluated using the approximative expression ( 8) in the case of Tα].\ncomponent µ(m)\nspin T(m)\nz T(m)\nx T(m)\ny\nCo on Pd(100)\nxy 0.319 0 .092 −0.046 −0.046\nyz 0.465 −0.066 0 .133 −0.066\n3z2−r20.365 −0.104 0 .052 0 .052\nxz 0.465 −0.066 −0.066 0 .133\nx2−y20.449 0 .128 −0.064 −0.064\nsum 2 .062 −0.018 0 .009 0 .009\nCo on Pd(111)\nxy 0.339 0 .097 −0.048 −0.048\nyz 0.428 −0.061 0 .122 −0.061\n3z2−r20.490 −0.140 0 .070 0 .070\nxz 0.428 −0.061 −0.061 0 .122\nx2−y20.339 0 .097 −0.048 −0.048\nsum 2 .023 −0.069 0 .034 0 .034\nCo on Pd(110)\nxy 0.397 0 .113 −0.057 −0.057\nyz 0.346 −0.049 0 .099 −0.049\n3z2−r20.515 −0.147 0 .074 0 .074\nxz 0.527 −0.075 −0.075 0 .151\nx2−y20.343 0 .098 −0.049 −0.049\nsum 2 .128 −0.060 −0.009 0 .069\nTABLE VI. Magnetic dipole term for Co monolayers and adatoms on Pd(100), Pd(111) and Pd(110) (optimized geometries)\nfor different magnetization directions. For each system, th e first line (“exact”) contains values calculated using Eq. ( 2) and the\nsecond line (“approx.”) contains values calculated using E q. (8). The Tyterms were evaluated only for the (110) surface.\nTz Tx Ty 7(Tx−Tz) 7( Ty−Tz)\nCo on Pd(100)\nmonolayer exact −0.017 0 .010 0 .188\napprox. −0.018 0 .009 0 .184\nadatom exact −0.024 0 .015 0 .275\napprox. −0.026 0 .013 0 .276\nCo on Pd(111)\nmonolayer exact −0.066 0 .035 0 .707\napprox. −0.069 0 .034 0 .723\nadatom exact −0.146 0 .080 1 .577\napprox. −0.154 0 .077 1 .618\nCo on Pd(110)\nmonolayer exact −0.057 −0.008 0 .068 0 .339 0 .872\napprox. −0.060 −0.009 0 .069 0 .360 0 .904\nadatom exact −0.112 −0.020 0 .141 0 .644 1 .768\napprox. −0.117 0 .011 0 .106 0 .900 1 .566\nshown in Fig. 3, together with a straight line represent-\ning Eq. (10). Here we take 85 meV for the spin-orbit\ncoupling parameter ξ[which appears to be a rather uni-\nversal value for Co as our calculations yield ξof 85.4 meV,\n84.5 meV, 84.9 meV and 85.1 meV for bulk hcp Co and\nfor a Co monolayer on Pd(100), Pd(111) and Pd(110),\nrespectively]. It follows from Fig. 3 that Bruno’s for-\nmula Eq. (10) works quite well for adatoms (albeit withsome “noise”) but not so well for monolayers, where re-\nlying solely on Eq. (10) might even lead to a wrong sign\nof the MAE. This may be connected with the fact that\nfor monolayers, the MAE is generally not very large and\nhence small absolute deviations from the rule given in\nEq. (10) can lead to large relative errors.\nThe sizable intraplanar anisotropy E(x)−E(y)which\nwe get for a Co monolayer on Pd(110) had to be expected9\n0246MAE [meV]\n0.0 0.1 0.2 0.3\ndifference inorb[B]adatoms\nmonolayers\nFIG. 3. (Color online) Dependence of the MAE for Co mono-\nlayers and adatoms on the difference of orbital magnetic mo-\nments for respective magnetization directions. The dashed\nline represents Bruno’s formula in Eq. (10).\nas this system could be viewed as a set of Co wires which\nare surely anisotropic in this respect. However, we get a\nvery strong azimuthal dependence of the MAE also for\ntheadatom on the (110) surface which is quite surprising\nas this can be only caused by the underlying substrate.\nThe magnetic moments at Pd atoms are not very large\n(Tab. III), neither is the spin-orbit coupling parameter\nξfor Pd in comparison to, say, 5 delements. Thus, this\nseems to be yet another example of the extreme sensi-\ntivity of the MAE. At the same time, let us note that\nthe calculated azimuthal dependence of the MAE can be\naccurately fitted by smooth sinusoidal curves (see Fig.\n2) which indicates a very good numerical stability of the\ncomputational procedure.\nThe intraplanar anisotropy for a Co adatom on the\nPd(111) surface can be compared to similar systems in-\nvestigated in the past. In particular, for a Co adatom\non Pt(111) the amplitude of the E(/bardbl)(φ)−E(z)curve is\nabout 2 % of the average value,40i.e., similar to the cur-\nrent case. For a 2 ×2 surface supercell coverage of Fe on\nPt(111), this amplitude is 10–25 % (depending on the\ngeometry relaxation)41but this situation is already quite\ndistinct from the isolated adatom case.\nAccording to our calculations, a Co monolayer on\nPd(100) has an in-plane magnetic easy axis, a Co mono-\nlayer on Pd(111) has an out-of-plane magnetic easy axis\nand the difference between the respective MAE values is\nabout 1 meV, which can be seen as a measure of how\nmuch the out-of-plane magnetization is preferred by the\nCo/Pd(111) system in comparison with the Co/Pd(100)\nsystem. This is similar to what was calculated for Co/Pd\nmultilayers: both Co 1Pd3(100) and Co 1Pd2(111) mul-\ntilayers have an out-of-plane magnetic easy axis but theMAE per unit cell is by about 0.9 meV larger for the\n(111) multilayer than for the (100) multilayer.42\nThe theoretical values for the anisotropy of Tαshown\nin Tab. VI can be compared with experimental data for\na similar system, namely, a single Co(111) layer sand-\nwiched between two thick Au layers. By extrapolat-\ning results obtained via angle-dependent XMCD mea-\nsurements, Weller et al.43obtained 7Tx= 0.43µBand\n7Tz=−0.86µB. Our values for a Co monolayer on\nPd(111), 7Tx= 0.24µBand 7Tz=−0.46µB(see Tab.\nVI), are fully consistent with this.\nWe expect that our values for µorbwill be systemat-\nically smaller than experimental values because we rely\nin the LSDA which usually underestimates µorb.44,45The\nsame may be also true for the MAE. However, this does\nnot affect our conclusions.\nWe used potentials subject to the ASA which may limit\nthe numerical accuracy of our results, particularly as con-\ncerns the MAE. On the other hand, our results do not\ndiffer too much from results of full-potential calculations,\nespecially in the case of monolayers. For a Co monolayer\non Pd(100), we get an in-plane magnetic easy axis with\nan MAE of -0.73 meV per Co atom while Wu et al.46ob-\ntained for the same zCo-Pd distance (1.65 ˚A) a theoretical\nMAE of -0.75 meV. Magneto-optic Kerr measurements17\nas well as XMCD experiments47showed that the mag-\nnetic easy axis of ultrathin Co films on Pd(100) is indeed\nin-plane (the experiment includes also an in-plane con-\ntribution from the shape anisotropy). Note that the the-\noretical MAE of -0.18 meV given in Ref. 17 was obtained\nfor a partially disordered Co monolayer simulating the\ngrowth conditions, so it cannot be directly compared to\nour results obtained for an ideal monolayer.\nFor a Co monolayer on Pd(111), we get a µspinvalue of\n2.01µBin a Co ASA sphere with a radius of 1.46 ˚A while\nthe full-potential calculations of Wu et al.48led to aµspin\nvalue of 1.88 µBobtained within a Co muffin-tin sphere\nwith a radius of 1.06 ˚A — both calculations thus again\ngive consistent results. For Pd atoms just below the Co\nlayer, we get a µspinvalue of 0.32 µBin a sphere with a\nradius of 1.49 ˚A while the corresponding µspinvalue of\nWuet al.48obtained within a sphere having a radius of\n1.32 ˚A is 0.37µB. In this last case, one has to bear in\nmind that Wu et al.48used a thin slab of only five Pd\nlayers sandwiched between two Co layers which clearly\nfavors a larger Pd polarization in comparison with just a\nsingle Co-Pd interface considered in this work.\nFor adatoms, the ASA may be more severe than for\nmonolayers, nevertheless, the agreement between our cal-\nculations and the results obtained via a full potential cal-\nculation is pretty good (see the end of the Appendix). As\na whole, the accuracy of our calculations is sufficient to\nwarrant the conclusions which rely on comparing a large\nset of data and not only on results for a singular system.\nIt follows from our results that one can change the\nmagnetic easy axis from in-plane to out-of-plane direction\njust by using as a substrate another surface of the same\nelement. This could be used as yet another ingredient10\nfor engineering the MAE of nanostructures, which has\nbecome a great challenge recently.49We also showed that\nthe magnetic dipole Tαterm can mimic a large anisotropy\nofµspinas determined from the XMCD sum rules. Hence,\nthe anisotropy of Tαhas to be taken fully into account\nwhen analyzing XMCD experiments on nanostructures.\nV. CONCLUSIONS\nCo monolayers and adatoms adsorbed on different sur-\nfaces of Pd exhibit quite different magnetic properties.\nThe effect on µspinis moderate, the effect on µorbis larger\nwhile the effect on the MAE and on the magnetic dipole\ntermTαmay be crucial. A surprisingly strong azimuthal\ndependence of the MAE is predicted for a Co adatom on\nPd(110).\nThe dependence of Tαon the direction of the magne-\ntization can lead to an apparent anisotropy of the spin\nmagnetic moment as deduced from the XMCD sum rules.\nFor systems with small spin-orbit coupling, the Tαterm\ncan be related to the differences between components of\nthe spin magnetic moment associated with different mag-\nnetic quantum numbers.\nACKNOWLEDGMENTS\nThis work was supported by the Grant Agency of the\nCzech Republic within the project 108/11/0853, by the\nBundesministerium f¨ ur Bildung und Forschung (BMBF)\nVerbundprojekt R¨ ontgenabsorptionsspektroskopie\n(05K10WMA) and by the Deutsche Forschungsgemein-\nschaft (DFG) via SFB 689. Stimulating discussions with\nP. Gambardella are gratefully acknowledged.\nAppendix: Effect of the size of the relaxation zone\nWhen studying the magnetism of adatoms, one should\naddress the question to which extent the host around the\nadatom has to be allowed to polarize. Zeller showed29\nthat the polarization cloud around a magnetic impurity\nin bulk Pd extends at least up to 1000 atoms. ˇSipret al.50\nshowed that the convergence of the MAE with respect to\nthe slab thickness and/or with respect to the size of the\nsupercell which simulates the adatom is much slower than\nthe convergence of magnetic moments. In view of these\nfacts, it is desirable to explore more deeply the situation\nfor the systems considered in this work.\nAs a test case, we select a Co adatom on Pd(111).\nTo facilitate the comparison with calculations done by\nother methods, we put the Co adatom in an hcp hollow\nsite, with the vertical distance between the Co adatom\nand the Pd surface layer as zCo-Pd =1.64 ˚A. Our system\nis thus similar to the system investigated by B/suppress lo´ nski et\nal.20(the main difference with respect to Ref. 20 is that\nwe do not consider any buckling of the substrate). To2345totalspinin zone [B]\n0 2 4 6 8 10 12\nradius ofzone [A] Ê220 Pd atoms\n133 Pdatoms\n46 Pdatoms\n7 Pd atoms\nFIG. 4. (Color online) Sum of the spin magnetic moments at\nthe Co adatom and at those substrate Pd atoms which are\nenclosed in hemispherical zones of the given radii, for four\nembedded cluster sizes (identified by numbers of Pd atoms\ncontained in them).\ncheck the convergence with respect to the size of the zone\nwhere the electronic structure is relaxed, we probed a\nseries of embedded cluster sizes, starting with relaxing\nthe electronic structure just in three Pd atoms (i.e., up\nto the distance of 2.3 ˚A from the Co adatom) and ending\nwith relaxing it in 220 Pd atoms (up to 11.7 ˚A from the\nCo adatom). To safely accommodate this large embedded\nclusters, we model the Pd substrate by a slab of 19 layers\n[contrary to 13 layers used in other calculations involving\nthe Pd(111) surface in this work]. The largest embedded\ncluster with 220 Pd atoms contains Pd atoms located\nwithin the fith layer below the surface and comprises 329\nsites altogether.\nFirst we investigate the convergence of the spin mag-\nnetic moments. This can be achieved by inspecting the\ntotalµspincontained inside a hemisphere stretching from\nthe adatom up to a certain radius. The dependence of\nthis totalµspinon the radius of the hemisphere forms an\n“integral magnetic profile”. This is presented in Fig. 4\nfor four embedded cluster sizes containing 7, 46, 133, and\n220 Pd atoms, respectively. The total µspinfor a sphere\nwith zero radius is obviously just the µspinvalue of the\nCo adatom. With increasing sphere radius the spin mag-\nnetic moments of enclosed Pd atoms are added to it. If\nthe radius of the hemisphere becomes larger than the ra-\ndius of the embedded cluster, the total µspinobviously\ndoes not change any more because the Pd atoms outside\nthe embedded impurity cluster are nonmagnetic.\nIt follows from Fig. 4 that the spin magnetic moment\nof the adatom as well as magnetic moments induced in\nthe nearest Pd atoms are actually already well described\nby relatively small embedded clusters. However, the total11\n1.751.81.851.91.95E(x)- E(z)[meV]\n0 50 100 150 200\nPd atoms in embeddedcluster\nFIG. 5. The MAE of a Co adatom in an hcp position on\nPd(111) for different sizes of the embedded clusters.\nµspinconverges only very slowly with increasing size of\nthe relaxation zone because even quite distant Pd atoms\nstill contribute with their non-zero µspin. Our results sug-\ngest that the magnetic moments on all the Pd atoms do\nnot arise due to a direct interaction with the Co adatom.\nRather, the adatom induces a magnetization in its near-\nest neighbors, then these further induce magnetization in\nthe next coordination shell and so on. The emerging pic-\nture of how the magnetism spreads through the Pd host\nis thus consistent with the picture suggested by Polesya\net al.30in terms of an exchange-enhanced magnetic sus-\nceptibility (see Fig. 4 of Ref. 30 and the associated text).\nA plot analogous to Fig. 4 could also be drawn for µorb\nexhibiting the same features as seen in Fig. 4.\nOur results on the convergence of the magnetic mo-\nments may raise objections about the convergence of the\nMAE. If embedded clusters containing as much as 220Pd atoms still do not fully account for the host polariza-\ntion, can one get reliable results for the MAE, which is\nsensitive to the way the substrate is treated?50To check\nthis, we calculated the MAE for a series of embedded\ncluster sizes (Fig. 5). One can see that in fact the MAE\nconverges quickly with increasing size of the embedded\ncluster. Already with a relaxation zone including only\n46 Pd atoms, which corresponds to a radius of the hemi-\nsphere of 6.9 ˚A containing Pd from up to the third Pd\nlayer below the surface, the accuracy of the MAE is bet-\nter than 1 %. This means that all the results presented\nin this work are well converged.\nThe data in Fig. 5 demonstrate that it is sufficient\nto include a rather small polarization cloud within the\nPd host in order to get convergence in the MAE values.\nMore distant Pd atoms do not contribute to the MAE,\neven if they are magnetically polarized . This conclusion\nis not in contradiction with an earlier result that reliable\nvalues of the MAE can be obtained only if the host is\nrepresented by slabs of at least ten layers50because that\nresult concerned the total “physical” size of the model\nsystem while in this appendix we focus only on the size\nof the zone where the electronic structure is allowed to\nrelax to the presence of an adatom (or of an adsorbed\nmonolayer).\nTo complete this part, we should compare our results\nwith the results of B/suppress lo´ nski et al.20which were obtained\nby performing a plane-wave projector-augmented wave\n(PAW) calculation for a supercell comprising five-layers\nthick slabs and a 5 ×5 surface unit cell. 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B 82, 174414 (2010)."    },    {        "title": "1304.1353v1.Effect_of_stacking_faults_on_the_magnetocrystalline_anisotropy_of_hcp_Co__a_first_principles_study.pdf",        "content": "E\u000bect of stacking faults on the magnetocrystalline\nanisotropy of hcp Co: a \frst-principles study\nC.J. Aas1, L. Szunyogh2, R.F.L. Evans1, R.W. Chantrell1\n1Department of Physics, University of York, York YO10 5DD, United Kingdom\n2Department of Theoretical Physics and Condensed Matter Research Group of\nHungarian Academy of Sciences, Budapest University of Technology and Economics,\nBudafoki \u0013 ut 8., H1111 Budapest, Hungary\nAbstract. In terms of the fully relativistic screened Korringa-Kohn-Rostoker method\nwe investigate the e\u000bect of stacking faults on the magnetic properties of hexagonal\nclose-packed cobalt. In particular, we consider the formation energy and the e\u000bect\non the magnetocrystalline anisotropy energy (MAE) of four di\u000berent stacking faults\nin hcp cobalt { an intrinsic growth fault, an intrinsic deformation fault, an extrinsic\nfault and a twin-like fault. We \fnd that the intrinsic growth fault has the lowest\nformation energy, in good agreement with previous \frst-principles calculations. With\nthe exception of the intrinsic deformation fault which has a positive impact on the\nMAE, we \fnd that the presence of a stacking fault generally reduces the MAE of bulk\nCo. Finally, we consider a pair of intrinsic growth faults and \fnd that their e\u000bect on\nthe MAE is not additive, but synergic.arXiv:1304.1353v1  [cond-mat.mtrl-sci]  4 Apr 2013E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 2\n1. Introduction\nWithin the magnetic recording industry, cobalt alloys such as CoPt and CoPd are of\ngreat interest due to their large magnetocrystalline anisotropy energies (MAE) [1]. For\nthe purpose of magnetic recording, a large MAE of the recording medium is crucial in\norder to maintain stability of the written information as larger areal information stor-\nage densities require smaller grain sizes [2]. In close-packed metals and alloys, stacking\nfaults are known to form relatively easily [3]. This is one of the contributing factors to\nthe relatively large ductility and malleability that are observed in many such materi-\nals [3]. For a magnetic recording medium, the presence of stacking faults is generally\nconsidered detrimental, as disturbances in the microstructure will generally worsen the\nsignal-to-noise ratio of the medium [4]. Stacking faults may also break the local lattice\nsymmetry and, therefore, drastically in\ruence the MAE.\nExperimentally, the e\u000bects of stacking faults are generally measured in terms of the\nstacking fault density, which can be determined from X-ray di\u000braction spectra (see\ne.g. [1, 4, 5]). There are a large number of experimental studies into stacking fault\nformation energies and the e\u000bect of the stacking fault density on the magnetocrystalline\nanisotropy for various magnetic recording alloys [1, 6]. However, in experiment, the\nreal e\u000bect of a stacking fault might be obscured by other phenomena, such as migration\nof impurities along the stacking fault, synergies of closely spaced stacking faults, etc.\nConsequently, a number of theoretical methods have been developed for determining\nthe properties and e\u000bects of stacking faults, see e.g. [7]. In particular, there is a large\nnumber of \frst-principles studies of the formation energies of given types of stacking\nfaults in metals [3, 8]. It has been suggested that stacking fault formation energies\ndetermined from \frst-principles may be more accurate than experimental measurements\n[3] as theoretical calculations separate the formation energy from any other correlated\ne\u000bects on the total energy. The e\u000bect on the MAE of a particular stacking fault is,\nhowever, less commonly explored. In this work, we aim to determine from \frst principles\nthe e\u000bect on the MAE of four di\u000berent types of stacking faults in hcp cobalt.\n2. The stacking faults\nHexagonal planes can be packed either in an ...ABAB... sequence, yielding a hexagonal\nclose-packed lattice structure, or in an ...ABCABC... sequence, yielding a face-centred\ncubic lattice structure [9]. In the hexagonal close-packed lattice structure, the stacking\ndirection corresponds to the (0001) axis of the lattice, whereas for the face-centred cubic\nlattice structure, the stacking direction is parallel to the (111) axis of the lattice. In a\nhcp lattice, a stacking fault is de\fned as an interruption in the ...ABAB... stacking of the\nhexagonal planes. While there are of course any number of conceivable stacking faults,\ntheir varying degrees of formation energies and formation mechanisms mean they have\ndi\u000berent probabilities of occurrence [8]. In line with previous work [3, 10], we considerE\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 3\nthe following four di\u000berent stacking faults, denoted in standard notation as I 1, I2, E and\nT2[11, 12]:\n\u000fI1(intrinsic):\u0001\u0001\u0001B A B A BC B C B\u0001\u0001\u0001\n\u000fI2(intrinsic):\u0001\u0001\u0001A B A BjC A C A\u0001\u0001\u0001\n\u000fE (extrinsic):\u0001\u0001\u0001A B A B CA B A B\u0001\u0001\u0001\n\u000fT2(twin-like):\u0001\u0001\u0001A B A B CB A B A\u0001\u0001\u0001\nHere the bold face letters or vertical line denote the plane of re\rection symmetry of the\nstacking fault. In an intrinsic stacking fault (I 1and I 2), the stacking fault is simply a\nshift of one lattice parameter and the stacking on either side is correct all the way up\nto the very fault [9]. The stacking fault I 1is a growth fault while the stacking fault I 2\nis a deformation fault [3]. In the extrinsic stacking fault (E), a plane has been inserted\nso that it is incorrectly stacked with respect to the planes on either side of it [9, 13].\nIn a twin-like fault (T 2), the stacking sequence is re\rected in the fault layer [3]. In the\nfollowing, we refer to the centre of re\rection symmetry as the zeroth layer. The two\nlayers adjacent to the centre of re\rection symmetry are then indexed \u00061, and so on.\nNote that in the case of a stacking fault of type I 2, the plane of re\rection symmetry lies\nin between two atomic layers. Therefore, in the following, for type I 2the atomic layers\nwill be labelled by \u00061\n2;\u00063\n2;:::, rather than by 0 ;\u00061;\u00062;:::as for the other types of\nstacking faults.\n3. Computational details\nFor our theoretical study we employed the fully relativistic Screened Korringa-Kohn-\nRostoker (SKKR) method, in which the Kohn-Sham scheme is performed in terms of\nthe Green's function of the system (rather than the wavefunctions) and the treatment\nof extended layered systems is particularly e\u000ecient [14, 15, 16]. We used the local spin\ndensity approximation (LSDA) of density functional theory (DFT) as parametrised by\nVosko and co-workers [17]. The e\u000bective potentials and \felds were treated within the\natomic sphere approximation (ASA) and an angular momentum cut-o\u000b of `max= 2 was\nused. The magnetocrystalline anisotropy energy (MAE) is calculated using the mag-\nnetic force theorem [18], within which the MAE is de\fned as the di\u000berence in the band\nenergy of the system when magnetised along the easy axis (0001) and perpendicular to\nthe easy axis. Alternatively, the uniaxial MAE can be calculated from the derivative\nof the band energy, for more details see Ref. [19]. Only when calculating the MAE, we\nused an angular momentum cut-o\u000b of `max= 3.\nThe LSDA+ASA fails in describing the orbital moment and the MAE of Co accurately.\nSimilar to our previous work [19] we, therefore, employed the orbital polarisation (OP)\ncorrection [20, 21, 22], as implemented within the KKR method by Ebert and Battocletti\n[23]. Note that the OP correction was applied only for the `= 2 orbitals. Excluding\nthe OP correction we obtained an easy-plane magnetisation and a MAE of 6.7 \u0016eV perE\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 4\ncobalt atom, while including the OP correction we obtained an easy axis perpendicular\nto the hexagonal cobalt planes and a MAE of 84.4 \u0016eV per cobalt atom. This is in good\nagreement with the experimental value of 65.5 \u0016eV [24] and with the experimental easy\naxis being parallel to the (0001) direction. Our result also compares well with that of\nTrygg et al. [25], who calculated K= 110\u0016eV for hcp cobalt using a full-potential\nLMTO method including the OP correction.\nIn this study we consider an in\fnite cobalt system, consisting of two semi-in\fnite bulk\ncobalt systems and an internal region (region I). RegionIcontains the stacking fault\nand is positioned in between the semi-in\fnite regions. The combined system is periodic\nand in\fnite in the plane normal to the (0001) direction. Due to the long-ranged nature\nof the stacking fault e\u000bects on the MAE (see section 4.3), the region Iin this study\nhad to be kept at a size of around 80 atomic layers. More speci\fcally, for stacking\nfaults I 1and I 2, systems of 80 atomic layers were required, while for stacking faults\nE and T 2, 74 atomic layers were required. In order to keep the calculations tractable\nwe limited the self-consistent calculations only for a number of atomic layers near the\nstacking fault, and then appended the bulk potentials for the atomic layers further away\nfrom the stacking fault. We found that it was su\u000ecient to treat only the 20 centremost\nlayers self-consistently. In line with previous \frst-principles studies of stacking faults in\nclose-packed metals, we ignored any atomic and volume relaxations (see e.g. [8]). The\ne\u000bects of such relaxations are normally negligible because atoms in the faulted part of\nthe system tend to retain their close-packed coordination numbers despite the presence\nof the fault [3, 13, 26, 27, 28, 29]. Throughout, therefore, we have used the experimental\nlattice parameter for cobalt, a= 2:507\u0017A.\n4. Results\n4.1. Stacking Fault Formation Energies\nBefore exploring how the stacking faults in\ruence the MAE of bulk Co, we would like to\ngain an idea of their formation energy. Within the SKKR-ASA scheme, the LSDA total\nenergy can be cast into contributions related to individual atomic cells, Ei, comprising\nthe kinetic energy, the intracell Hartree energy and the exchange-correlation energy,\nand into the two-cell Madelung (or intercell Hartree) energy, EMad[14]. For a simple\nbulk metal, like hcp Co, EMadis, in practice, negligible, while in the presence of stack-\ning faults it gives a non-negligible contribution due to charge redistributions. However,\nfrom our self-consistent calculations we found that EMadis in the order of 0 :1\u00000:2 meV\nper stacking fault. Since the typical formation energy of stacking faults are by about\ntwo orders higher in magnitude than this value, in the following we consider only the\nlayer-resolved (cell-like) contributions to the the total energy. In order to check these\ncontributions for artefacts of the appending of the bulk potential, we compare Eifor\ni=\u000010 (layer with e\u000bective potential from a self-consistent stacking fault calculation)E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 5\ntoEifori= 10 (layer with appended bulk potential), since due to the mirror symme-\ntry, these two contributions should be identical. Reassuringly enough, they agreed to\nwithin a relative error of 10\u00009, which is well within intrinsic and numerical error of our\ncomputational method.\nThe layer-resolved contributions to the total energy across the systems containing the\nstacking faults I 1, I2, E and T 2is shown in Fig. 1. Herein we observe the expected\nmirror symmetry and that the layer-resolved total energy contributions approach the\nbulk total energy, ECo=\u000037839:459 eV, towards the edges of each system. From this\n\fgure it is obvious that the deviation of EifromECois signi\fcant up to about 15 layers\naway from the centre of stacking fault.\nFigure 1. The layer-resolved contributions to the total energy in four hcp cobalt\nsystems, each exhibiting one of the four di\u000berent types of stacking fault. The label 0\nrefers to the plane of mirror symmetry. Solid lines serve as a guide for the eyes.E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 6\nThe stacking fault formation energy is de\fned as the di\u000berence in the total energy caused\nby the presence of the stacking fault. In order to ensure we include enough atomic layers\non either side of the stacking fault, we consider the cumulative sums:\n\u0001E(I1;E;T2)(N) =NX\n\u0000NEi\u0000(2N+ 1)ECo; (1)\nand\n\u0001EI2(N) =N\u00001\n2X\n\u0000N+1\n2Ei\u00002NE Co: (2)\nThe formation energy of a given stacking fault X= I1;I2;E;T2,E(X)\nform, is then de\fned\nas\nE(X)\nform = lim\nN!1(\u0001EX(N)): (3)\nFigure 2. The cell-like part of the formation energy, \u0001 EX(N), see Eqs. (1) and (2),\nof stacking faults I 1, I2, E and T 2in hcp cobalt, displayed as a function of the number\nof layers,N, considered in the system on either side of the stacking fault. Solid lines\nserve as guides for the eyes.\nThe calculated values of \u0001 EX(N) are shown in Fig. 2. Quite obviously, for all types of\nstacking faults, nearly 30 atomic layers (i.e., 15 layers on either side of the stacking fault)\nare required in order to obtain well-converged stacking fault formation energies. The\nfact that the e\u000bect of the stacking fault is relatively long-ranged could have signi\fcant\nimpact on nano-sized systems as the formation energy and, consequently, the likelihood\nof occurrence of a stacking fault could be di\u000berent depending on its location in relation\nto, e.g., other imperfections as well as surfaces or interfaces in the sample. We obtain\nthe following formation energies, with a possible error of \u00180:1\u00000:2 meV due to theE\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 7\nMadelung energy not being included:\nE(I1)\nform\u001916 meV\u001940 mJ\u0001m\u00002\nE(I2)\nform\u001948 meV\u0019122 mJ\u0001m\u00002\nE(E)\nform\u001962 meV\u0019160 mJ\u0001m\u00002\nE(T2)\nform\u001939 meV\u0019100 mJ\u0001m\u00002:\nAs expected, all stacking faults incur a positive change in the total energy. Of the four\ntypes of stacking faults considered here, the intrinsic stacking fault I 1has the lowest\nformation energy and the stacking fault E exhibits the highest one. While there is no\navailable experiment in literature, the overall results agree well with e.g. Ref. [8]: the\nextrinsic stacking fault formation energy for the close-packed fcc metals in this study is\ngenerally signi\fcantly larger than that of the intrinsic and twin faults. Moreover, our\ncalculated values for the hcp Co growth stacking fault I 1and the hcp Co extrinsic fault\nE are close to those obtained by Crampin and co-workers for Ni (which is next to Co in\nthe periodic table) [8]: 28 mJ \u0001m\u00002for the intrinsic stacking fault and 180 mJ \u0001m\u00002for\nthe extrinsic fault.\n4.2. Layer-Resolved Contributions to the Magnetocrystalline Anisotropy Energy\nBecause it is calculated directly from the band energy, the MAE can naturally be re-\nsolved into layer-dependent contributions, Di, see Ref. [19]. These layer-resolved con-\ntributions are depicted in Fig. 3 for the di\u000berent types of stacking faults. Note that\nthe mirror symmetry is well reproduced in the layer-resolved MAE contributions for\nall stacking faults. Moreover, the MAE approaches the bulk MAE, KCo= 84:4\u0016eV,\ntowards the edges of all four systems. For stacking faults of type I 1, I2and T 2, the MAE\ncontributions become negative at the centre of the fault, favoring thus an in-plane easy\naxis in these layers. This could indicate that these types of stacking faults may act as\npinning sites. For the type E stacking fault, the layer-resolved MAE contributions near\nthe centre are also reduced, retaining, however, very small positive values.\nFurthermore, we note that all stacking faults induce long-ranged oscillations in the\nMAE. For layers of about jij>15, the four stacking faults exhibit very similar trends\nin the layer-resolved MAE contributions. In other words, at about 15 layers away from\nthe stacking fault, the presence of a stacking fault still in\ruences the MAE, while the\nparticular type of the stacking fault is less signi\fcant. This will, however, obviously\ndepend on the size of the sample.\n4.3. Finite Size E\u000bect on the Magnetocrystalline Anisotropy Energy\nThe long-ranged oscillations in the MAE could cause signi\fcant \fnite-size e\u000bects in the\nexperimental determination of the MAE of nano-sized samples. We therefore considerE\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 8\nFigure 3. Calculated layer-resolved contributions to the MAE for stacking faults I 1\nandI2(upper panel) and E and T 2(lower panel). The horizontal line refers to the\nbulk MAE, 84.4 \u0016eV/atom. Solid lines serve as guide for the eyes.\nthe following cumulative sums,\nK(I1;E;T2)(N) =NX\n\u0000NDi\u0000(2N+ 1)KCo; (4)\nand\nKI2(N) =N\u00001\n2X\n\u0000N+1\n2Di\u00002NK Co; (5)\nwhere the MAE of the stacking fault systems of \fnite width is related to the MAE of\nhcp Co.E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 9\nFigure 4. Cumulative sums of layer-resolved contributions to the MAE, KX(N) (X=\nI1, I2, E and T 2), see Eqs. (4) and (5). Solid lines serve as guide for the eyes.\nFig. 4 shows KX(N) for the four di\u000berent stacking faults as a function of N. Surprisingly,\nforN\u00155 the I 2type stacking fault appears to increase the MAE, i.e., to strengthen the\neasy axis (0001) ( positive e\u000bect ). As seen from Fig. 3, this is due to the positive MAE\ncontributions induced by the stacking fault on neighbouring layers. These apparently\noutweigh the strongly negative MAE contributions induced in the centre of stacking\nfault type I 2. This is an unexpected result as stacking faults are typically reported to\nlower the MAE (see e.g. [4]). It should be noted, however, that, of the stacking faults\nstudied here, type I 2has the next highest formation energy and it is therefore less likely\nto occur in an equilibrated sample. For stacking faults of types I 1, E and T 2, the overall\nchange in the MAE with respect to hcp Co is negative ( negative e\u000bect ). As noted earlier,\nin the vicinity of these stacking faults, the easy dirction is rotated normal to the (0001)\naxis. This is consistent with the reduction in the total MAE observed experimentally\nby Sokalski et al. in[4].\nIt is quite a remarkable feature that, as seen from Fig. 4, the layer-resolved MAE\ncontributions do not settle until at about approximately 35 layers on either side of\nthe stacking fault. This long-ranged behaviour could give rise to signi\fcant \fnite-size\ne\u000bects in nano-sized samples. Moreover, this might have consequences for theoretical\ninvestigations into the formation and e\u000bects of stacking faults on magnetic properties.\nTypically, in Monte Carlo simulations of stacking faults, interactions between stacking\nfaults is kept to around three neighbouring planes [4]. In light of our results, this appears\nto be an uncertain assumption.E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 10\n4.4. Magnetocrystalline Anisosotropy of a Composite Stacking Fault\nExperimentally, the presence of stacking faults is normally quanti\fed in terms of the\nstacking fault density, which is partly a measure of how close the stacking faults are\nlocated. As the simplest assumption, the change in the MAE due to the presence of\na number stacking faults in a sample is approximated by the sum of the changes in\nthe MAE due to each individual stacking fault. If this were the case, the e\u000bect of an\nisolated stacking fault on the MAE could quite straightforwardly be transformed into\nthe change in MAE as a function of the stacking fault density. However, the long-ranged\noscillations in the MAE caused by the presence of each stacking fault indicates that the\nsituation is far more complex.\nIn particular, we considered two stacking faults of type I 1, separated by three atomic\nlayers. In other words, the system exhibits the composite stacking fault:\n\u0001\u0001\u0001A B A BC B C BA B A\u0001\u0001\u0001\nNote that by removing one of the two C-B pairs of atomic layers, a twin-like stacking\nfault T 2is obtained. We have chosen three atomic layers between the centres of the\ntwo stacking faults, since in dynamical models it is often used as the distance beyond\nwhich the interaction between stacking faults is neglected (see e.g. Ref. [4]). Moreover,\nwe deal with a pair of I 1type stacking faults because this type of stacking fault has the\nlowest formation energy and is, therefore, expected to occur more commonly than the\nother three types of stacking faults.\nThe di\u000berence between the layer-resolved MAE contributions and the MAE of bulk hcp\nCo,\n\u0001D(I1I1)\ni =D(I1I1)\ni\u0000KCo; (6)\nis shown in Fig. 5 for the composite stacking fault. As a comparison, we also show the\naverage deviations in the layer-resolved MAE contributions from the bulk MAE of two\nindependent type I 1stacking faults,\n\u0001D(I1+I1)\ni =1\n2\u0010\nD(I1)\ni+2+D(I1)\ni\u00002\u0011\n\u0000KCo: (7)\nIf \u0001D(I1I1)\ni and \u0001D(I1+I1)\ni were equal for each atomic layer i, the change of the MAE\ndue to the presence of the composite stacking fault would be exactly twice that of a\nsingle I 1stacking fault. However, as shown in Fig. 5, \u0001 D(I1I1)\ni and \u0001D(I1+I1)\ni deviate\nsigni\fcantly, particularly in the layers jij\u00142, i.e., in the layers between the two stacking\nfaults. Beyondjij>3, the magnitudes of the MAE contributions are similar for \u0001 D(I1I1)\ni\nand \u0001D(I1+I1)\ni , but with a phase shift of approximately one layer.E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 11\nFigure 5.\u000f: Calculated deviations in the layer-resolved MAE contributions, \u0001 D(I1I1)\ni,\nof the composite stacking fault from the bulk Co MAE, see Eq. (6), and + : the\ncorresponding average deviations, \u0001 D(I1+I1)\ni , of two superposed I 1type stacking faults\ncentred on atomic layers i\u00062, see Eq. (7). Solid lines serve as guide for the eyes.\nSimilar to the case of single stacking faults, we calculate the cumulative sum of the\nMAE contributions of the composite stacking fault,\nKI1I1(N) =NX\ni=\u0000ND(I1I1)\ni\u0000(2N+ 1)KCo; (8)\nand plot it in Fig. 6. Apparently, KI1I1(N) converges to approximately \u00001:18 meV\nfor largeN, which is almost three times the change of the MAE of the single type\nI1stacking fault (\u0018\u00000:40 meV, see Fig. 4). Also shown in Fig. 6 is the di\u000berence\nKI1I1(N)\u00002KI1(N), which appears to settle at approximately \u00000:38 meV. In other\nwords, the two stacking faults interact to yield a stronger negative e\u000bect on the total\nMAE as compared to two isolated type I 1stacking faults. This appears to be mainly\ndue to MAE contributions from the atomic layers located in between the two type I 1\nstacking faults. This could have signi\fcant consequences for predicting the resulting\nMAE in dynamical models used to explain experimental data. To draw any de\fnite\nconclusions, a systematic study of the stacking fault types and separations would be\nrequired. We expect that such a study would be computationally extremely intensive as\ninterlayers (or supercells) of up to approximately 160 atomic layers would be required\nin order to reach the limit in which the two stacking faults are far enough apart not to\ninteract.\n5. Summary and Conclusions\nUsing the fully relativistic screened Korringa-Kohn-Rostoker method, we have studied\nthe MAE of bulk hcp cobalt in the vicinity of four di\u000berent types of stacking faults.\nWe \fnd that, in accordance with experiment, most stacking faults have a detrimen-E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 12\nFigure 6. ?: Change in the MAE of hcp Co due the composite stacking fault,\nKI1I1(N), as de\fned in Eq. 8). \u000f: Interaction term of the two stacking faults in the\nMAE,KI1I1(N)\u00002KI1(N), see. Eqs. (4) and (8) Solid lines serve as guide for the eyes.\n.\ntal overall e\u000bect on the MAE. The one exception to this overall conclusion is the type\nI2intrinsic stacking fault, which, however, exhibits a relatively high formation energy\nand which may, consequently, occur relatively infrequently under standard experimen-\ntal conditions. The e\u000bect of a stacking fault on the layer-resolved contributions to the\nMAE is long-ranged, in the order of 15 atomic layers on either side of each stacking\nfault. Motivated by this observation, we investigated a particular composite stacking\nfault and concluded that the MAE of the composite stacking fault is not identical to\nthe sum of the MAE of the two isolated stacking faults. A further challenging study\nis proposed regarding the dependence of the 'interaction' of two stacking faults on the\nseparation between them.\nCJA is grateful to EPSRC and to Seagate Technology for the provision of a research stu-\ndentship. Support of the EU under FP7 contract NMP3-SL-2012-281043 FEMTOSPIN\nis gratefully acknowledged. Financial support was in part provided by the New Sz\u0013 echenyi\nPlan of Hungary (T \u0013AMOP-4.2.2.B-10/1{2010-0009) and the Hungarian Scienti\fc Re-\nsearch Fund (OTKA K77771).\n[1] B. Lu, T. Klemmer, K. Wierman, G. Ju, D. Weller, A. G. Roy, D. E. Laughlin, C. Chang, and\nR. Ranjan, J. Appl. Phys. 91(2002) 8025\n[2] D. Weller, A. Moser, L. Folks, M. Best, W. Lee, M. Toney, M. Schwickert, J.-U. Thiele, and\nM. Doerner, IEEE Trans. Mag. 36(2000) 10\n[3] N. Chetty and M. Weinert, Phys. Rev. B 56(1997) 10844\n[4] V. Sokalski, D. E. Laughlin, and J.-G. Zhu, J. Appl. Phys. 110(2011) 093919\n[5] G. B. Mitra and N. C. Hadler, Acta Crystallographica , vol. 17, pp. 817{822, July 1964.\n[6] S. Saito, A. Hashimoto, D. Hasegawa, and M. Takahashi, Journal of Physics D: Applied Physics ,\nvol. 42, no. 14, p. 145007, 2009.E\u000bect of stacking faults on the magnetocrystalline anisotropy of hcp Co: a \frst-principles study 13\n[7] R. Berliner and S. A. Werner, Phys. Rev. B 34(1986) 3586{3603\n[8] S. Crampin, K. Hampel, D. Vvedensky, and J. MacLaren, Phil. Mag. A 5(1990) 2107\n[9] R. Abbaschian, L. Abbaschian, and R. Reed-Hill, Physical Metallurgy Principles 2E Stamford:\nCengage Learning, 2009.\n[10] R. R. Zope and Y. Mishin, Phys. Rev. B 68(2003) 024102\n[11] J. Hirth and J. Lothe, Theory of Dislocations New York: Wiley Interscience, 1982.\n[12] F. Frank, Phil. Mag. Series 7 42(1951) 809\n[13] B. Hammer, K. W. Jacobsen, V. Milman, and M. C. Payne, J. Phys.: Condens. Mat. 4(1992)\n10453\n[14] J. Zabloudil, R. Hammerling, L. Szunyogh and P. Weinberger, Electron Scattering in Solid Matter -\nA Theoretical and Computational Treatise Springer{Verlag, Berlin{Heidelberg{New York, 2005.\n[15] L. Szunyogh, B. \u0013Ujfalussy, P. Weinberger and J. Koll\u0013 ar Phys. Rev. B 49(1994) 2721\n[16] R. Zeller, P.H. Dederichs, B. \u0013Ujfalussy, L. Szunyogh and P. Weinberger Phys. Rev. B 52(1995)\n8807\n[17] S. H. Vosko, L. Wilk, and M. Nusair, Canadian Journal of Physics 58(1980) 1200\n[18] H. J. F. Jansen, Phys. Rev. 59(1999) 4699\n[19] C. J. Aas, K. Palot\u0013 as, L. Szunyogh, and R. W. Chantrell, J. Phys.: Condens. Matter 24(2012)\n406001\n[20] M.S.S. Brooks, Physica B 130(1985) 6\n[21] H. Eschrig, The Fundamentals of Density Functional Theory Leipzig: Teubner, 1996.\n[22] H. Eschrig, M. Sargolzaei, K. Koepernik, and M. Richter, Europhys. Lett. 72(2005) 611\n[23] H. Ebert and M. Battocletti Solid State Commun. 98(1996) 785\n[24] M.B. Stearns, 3d, 4d, and 5d Elements, Alloys and Compounds Springer{Verlag, Berlin{\nHeidelberg{New York, 1986.\n[25] J. Trygg, B. Johansson, O. Eriksson, and J. M. Wills, Phys. Rev. Lett. 75(1995) 2871\n[26] P. J. H. Denteneer and J. M. Soler, J. Phys.: Condens. Mat. 3(1992) 8777\n[27] J.-h. Xu, W. Lin, and A. J. Freeman, Phys. Rev. B 43(1991) 2018\n[28] S. Schweizer, C. Els asser, K. Hummler, and M. F ahnle, Phys. Rev. B 46(1992) 14270\n[29] A. F. Wright, M. S. Daw, and C. Y. Fong, Phil. Mag. A 66(1992) 387"    },    {        "title": "1304.2428v2.Effects_of_alloying_and_strain_on_the_magnetic_properties_of_Fe___16__N__2_.pdf",        "content": "E\u000bects of alloying and strain on the magnetic properties of Fe 16N2\nLiqin Ke,1Kirill D. Belashchenko,2Mark van Schilfgaarde,3Takao Kotani,4and Vladimir P Antropov1\n1Ames Laboratory US DOE, Ames, Iowa 50011\n2Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska-Lincoln, Lincoln, Nebraska 68588\n3Department of Physics, King's College London, Strand, London WC2R 2LS, United Kingdom\n4Tottori University, Tottori, Japan\n(Dated: March 28, 2021)\nThe electronic structure and magnetic properties of pure and doped Fe 16N2systems have been\nstudied in the local-density (LDA) and quasiparticle self-consistent GW approximations. The GW\nmagnetic moment of pure Fe 16N2is somewhat larger compared to LDA but not anomalously large.\nThe e\u000bects of doping on magnetic moment and exchange coupling were analyzed using the co-\nherent potential approximation. Our lowest estimate of the Curie temperature in pure Fe 16N2is\nsigni\fcantly higher than the measured value, which we mainly attribute to the quality of available\nsamples and the interpretation of experimental results. We found that di\u000berent Fe sites contribute\nvery di\u000berently to the magnetocrystalline anisotropy energy (MAE), which o\u000bers a way to increase\nthe MAE by small site-speci\fc doping of Co or Ti for Fe. The MAE also increases under tetragonal\nstrain.\nI. INTRODUCTION\nOrdered nitrogen martensite \u000b00-Fe16N2was \frst syn-\nthesized in bulk form by quenching of the cubic nitrogen\naustenite\r-FeN with a subsequent annealing.1Quench-\ning initially produces disordered \u000b0-FeN, which then or-\nders during low-temperature annealing to produce \u000b00-\nFe16N2. The latter is a metastable phase with a dis-\ntorted body-centered tetragonal structure, which decom-\nposes into\u000b-Fe and Fe 4N near 500 K.\nInterest in \u000b00-Fe16N2was revived much later when\nit was synthesized in thin \flm form and a very large\nvalue (\u00183\u0016B) for the average Fe magnetic moment was\nreported.2This result was not independently con\frmed\nuntil twenty years later.3Owing to the rapid develop-\nment of the magnetic recording technologies, this con-\n\frmation inspired numerous studies of thin-\flm samples.\nHowever, the existence of the \\giant\" Fe moment remains\ncontroversial as many researchers did not reproduce these\n\fndings, while others con\frmed them.4{6. The lack of\nconsistent and reproducible experimental results may be\nattributed to the di\u000eculties associated with the prepa-\nration of single-crystal Fe 16N2and stabilization of ni-\ntrogen, as well as with the accurate measurement of the\nmagnetization in multi-phase Fe nitride samples. This is-\nsue has recently attracted additional interest due to the\nsearch for new permanent magnetic materials without\nrare-earth elements.7A new way to prepare single-phase\nFe16N2powder was recently reported, along with evi-\ndence of high maximum energy product ( BH max).8\nMost theoretical studies of the magnetization of \u000b00-\nFe16N2were performed using the local density approx-\nimation, generalized gradient approximation (GGA) or\nLDA+U, though recently Sims et al.9applied a hybrid\nfunctional and the GW approximation to this material.\nIn LDA or GGA the magnetic moment of Fe 16N2is\nonly slightly enhanced compared to elemental Fe. Lai\net al.10included electronic correlations within LDA+ Uand found an enhanced magnetization M=2.85\u0016B/Fe.\nWang et al.11{13identi\fed a localized Fe state coexist-\ning with the itinerant states in X-ray magnetic circular\ndichroism (XMCD) measurements. They introduced a\nspeci\fc charge transfer between di\u000berent Fe sites and ob-\ntained a large Min LDA+U. However, the choice of the\ncorrelated orbitals and the associated value of the Hub-\nbardUparameter is not well-de\fned for metallic systems.\nFor example, the on-site interaction parameters obtained\nby Sims et al.9using the constrained random phase ap-\nproximation (RPA) di\u000ber substantially from those pro-\nposed by Wang et al. . The quasiparticle self-consistent\nGW approximation (QS GW)14,15is more reliable and\nprovides a more satisfactory way to determine the ground\nstate density and magnetic moment. In the present paper\nwe apply this method to Fe 16N2.\nStudies of exchange interaction, Curie temperature\n(TC), and MAE of Fe 16N2met with additional di\u000ecul-\nties. In particular, measurements of TCare hampered by\nthe decomposition of the metastable Fe 16N2into Fe 4N\nand Fe, which was reported to occur above 200\u000eC,6in\nthe 230-300\u000eC range,2or at 400\u000eC.3Sugita et al. extrap-\nolated their data to estimate TCat 540\u000eC.3Thermal sta-\nbility of Fe 16N2was reported to increase with addition of\nCo and Ti16,17(up to 700\u000eC in the Ti case). However, no\nexperimental information is presently available about the\nTCof Co or Ti-doped Fe 16N2, or of any other Fe 16N2sam-\nples stabilized at high temperatures. To the best of our\nknowledge, there have been no theoretical studies of the\nexchange interaction and Curie temperature in Fe 16N2.\nSystematic studies of the e\u000bects of doping on MandTC\nin Fe 16N2also appear to be lacking.\nAs for the MAE, only a few experimental values were\nreported, and they are varied and inconclusive. For ex-\nample, Sugita et al.3obtained an in-plane MAE, while\nTakahashi18found a large uniaxial MAE. The only avail-\nable theoretical calculations of MAE used an empirical\ntight binding (TB) approximation.19arXiv:1304.2428v2  [cond-mat.mtrl-sci]  8 Jul 20132\nIn this paper we study the magnetization, Curie tem-\nperature, and magnetocrystalline anisotropy energy of\npure and doped Fe 16N2using several well-tested elec-\ntronic structure techniques and suggest possible routes\nfor improving its properties for permanent magnet appli-\ncations.\nII. COMPUTATIONAL METHODS\nMost LDA, GGA and QS GW calculations were per-\nformed using a full-potential generalization20of the stan-\ndard linear mu\u000en-tin orbital (LMTO) basis set.21This\nscheme employs generalized Hankel functions as the enve-\nlope functions. Calculations of MAE we also performed\nusing the recently-developed mixed-basis full-potential\nmethod,22which employs a combination of augmented\nplane waves and generalized mu\u000en-tin orbitals to repre-\nsent the wave functions. The results of a traditional non-\nself-consistent application of the GW approximation de-\npend on the non-interacting Hamiltonian generating the\nself-energy. This issue can be particularly problematic\nfor metals. In contrast, QS GW method does not su\u000ber\nfrom this limitation: it is more reliable than the standard\nGW. This method gives quasiparticle energies, spin mo-\nments, dielectric functions, and a host of other properties\nin good agreement with experiments for a wide range of\nmaterials, including correlated ones such as NiO. The de-\ntails of QSGW implementation14,15and applications can\nbe found elsewhere.\nThe pair exchange parameters were obtained using two\nlinear response approaches:\n(1) Static linear-response approach23implemented\nwithin the atomic sphere approximation (ASA) to the\nGreen's function (GF) LMTO method.24In addition to\nmaking a spherical approximation for the potential, this\nmethod makes the long-wave approximation (LWA), so\nthat the pair exchange parameter is proportional to the\ncorresponding spin susceptibility \u001fij25. The exchange\nparameters Aijobtained in this method are related to\nthe parameters of the classical Heisenberg model\nH=\u0000X\nijJijSi\u0001Sj; (1)\nby the following renormalization for ferromagnetic(FM)\nand antiferromagnetic(AFM) cases:\nJij=Aij=SiSj (2)\n= 4Aij=mimj=\u001a\n4Aij=mimjFM\n\u00004Aij=mimjAFM\nwhere miis the magnetic moment on site i. With this\nrenormalization all results obtained for the Heisenberg\nmodel Eq.(1) can be used directly. Thus, parameters Aij\nalways stabilize (destabilize) the given magnetic con\fgu-\nration and can be treated as stability parameters. Curietemperature in the spin classical mean \feld approxima-\ntion (MFA) is simply TC=2=3P\nijAij.\n(2) Dynamical linear response approach with the bare\nsusceptibility \u001f(q;!) calculated in the full product basis\nset representation using the LDA or QS GW electronic\nstructure.26. The results are then projected onto the\nfunctions representing local spin densities on each mag-\nnetic site, which gives a matrix \u001fij(q;!) in basis site\nindices.26This projection corresponds to the rigid spin\napproximation. The inversion of this matrix with a sub-\nsequent Fourier transform provides the real-space rep-\nresentation of the inverse susceptibility representing the\ne\u000bective pair exchange parameters:\nJij= lim\n!\u0000!01\n\nBZZ\ndq[\u001f(q;!)]\u00001eiqRij: (3)\nTCis calculated both in the MFA27and the RPA-\nTiablikov28approximations. The actual TCmay usually\nbe expected to lie between the results of these two ap-\nproximations.\nTo address the e\u000bects of doping, we used our im-\nplementation of the coherent potential approximation\n(CPA) within the TB-LMTO code, which follows the\nformulation of Turek et al.29and Kudrnovsk\u0013 y et al. .30\nA coherent interactor matrix \n iis introduced for each\nbasis siteitreated within CPA. At self-consistency gii=\n(Pi\u0000\ni)\u00001, where Piis the coherent potential matrix for\nsitei, andgiiis the on-site block of the average auxiliary\nLMTO GF matrix g= (P\u0000S)\u00001. This on-site block is\nextracted from the Brillouin zone integral of g(k). The\nconditionally averaged GF at site ioccupied by compo-\nnentaisga\nii= (Pa\u0000\n)\u00001, and the CPA self-consistency\ncondition can be written as gii=P\naca\niga\nii; hereca\niis\nthe concentration of component aat sitei. Using this\nequation, at the beginning of each iteration the stored\nmatrices \n iare used to obtain an initial approximation\ntoPi. In turn, Piis used in the calculation of giiby a\nBrillouin zone integral. The next approximation for \n i\nis obtained from \n i=Pi\u0000g\u00001\nii. These output matrices\nare then linearly mixed with the input \n imatrices at the\nend of the iteration.\nWe found that the mixing coe\u000ecient of 0.4 for \n iworks\nwell in most cases. For fastest overall convergence, we\nfound that it is usually desirable to iterate CPA iterations\nuntil the \n imatrices are converged to a small tolerance,\nand only then perform the charge iteration. The conver-\ngence of \n is done separately for each point on the com-\nplex contour to the same tolerance. With this procedure,\nfairly aggressive Broyden mixing can be used for LMTO\ncharge moments. CPA convergence at each charge it-\neration usually takes 10-50 iterations depending on the\nimaginary part of energy and the selected tolerance. At\nthe beginning of the calculation, the \n imatrices are set\nto zero; afterwards they are stored and reused for subse-\nquent iterations. In order to avoid unphysical symmetry-\nbreaking CPA solutions (which otherwise often appear),\nthe coherent potentials and the k-integrated average aux-\niliary GF are explicitly symmetrized using the full space3\ngroup of the crystal. As a result, the use of CPA does not\nimpose any restrictions on the symmetry of the crystal.\nCalculations reported here were performed without using\ncharge screening corrections for the Madelung potentials\nand total energy.\nThe e\u000bective exchange coupling in CPA is calculated\nas\nA0(c) =cAX(c) + (1\u0000c)AY(c) (4)\nwhere the component-speci\fc Ai(c) are obtained using\nthe conditionally averaged GF and the formalism of Ref.\n31.\nFor MAE calculations the self-consistent solutions are\nfound including spin-orbit coupling (SOC) terms of or-\nder 1=c2. The MAE is de\fned below as K=E100\u0000E001,\nwhereE001andE100are the total energies for the mag-\nnetization oriented along the [001] and [100] directions,\nrespectively. Positive (negative) Kcorresponds to uni-\naxial (planar) anisotropy. We used a 24 \u000224\u000224k-point\nmesh for MAE calculations to ensure su\u000ecient conver-\ngence; MAE changed by less than 2% when a denser\n32\u000232\u000232 mesh was employed. All calculations except\nQSGW were performed with both LDA32and GGA33\nexchange-correlation potentials for comparison.\nIII. RESULTS AND DISCUSSION\nThe crystal structure of Fe 16N2is body-center-\ntetragonal (bct) with space group I4=mmm (#139). It may\nbe viewed as a distorted 2 \u00022\u00022 bct-Fe superlattice with\nc=a=1.1. Crystal structure of Fe 16N2was \frst identi-\n\fed by Jack.1Here we use lattice constants a=b=5:72\u0017A\nc=6:29\u0017A and atomic position parameters z4e=0:3125 and\nx8h=0:25 from Jack's work as the experimental structure.\n(see Fig. 1).\nWe also relaxed the structure by minimizing the total\nenergy in LDA and obtained z4e=0:293 andx8h=0:242,\nnearly identical to that obtained by Sawada et al.34. The\nprimitive cell contains one N and eight Fe atoms divided\ninto three groups indicated by Wycko\u000b sites: two 4 e,\nfour 8hand two 4dsites (correspondingly \frst, second\nand third neighbors to N).\nA. Magnetic moments and electronic structure\nTable I shows the atomic spin moment miat the\nthree Fe sites and magnetization M(orbital magnetic mo-\nment is small, hereafter we only include spin magne-\ntization in M). Within the LDA, M=2:38\u0016B/Fe was\nobtained, in good agreement with previously reported\ncalculations4{6. The enhancement relative to elemental\nbcc-Fe has been attributed to the size e\u000bect35. QSGW\ngivesM=2:59\u0016B/Fe, about 9% larger than LDA. While\nit is known that GW enhances spin moments relative\nFIG. 1: (Color online) (a) Crystal structure of Fe 16N2. The\nexperimental atomic positions are shown. Relaxed structure\nhave slightly di\u000berent z4eandx8h. (b) f001gplane with 8 h\nand N atoms. (c) f110gplane with 4 e, 8hand N atoms. (d)\nf110gplane with 4 dand 8hatoms.\nto LDA, to ensure the genuineness of this enhance-\nment of moment, we also carried out the QS GW cal-\nculation of bcc-Fe and found that QS GW enhance the\nLDA magnetic moment in elemental Fe by only \u00182%\n(2.20!2.24\u0016B). Sims et al.9found a similar Min their\nGW calculation while they also obtained a larger mag-\nnetization enhancement in bcc-Fe with M=2.65\u0016B/Fe.\nConsidering Fe 16N2consists of about 87% Fe, the 9% en-\nhancement we \fnd non-trivial. However, it is still well be-\nlowM= 2:85\u0016B/Fe, obtained in LDA+ Uby Lai et al.10.\nThe spin moment on the 4 dsite reaches mi= 3:11\u0016Bin\nQSGW, though we do not observe any obvious charge\ntransfer from 4 dto 4eand 8hsites in QSGW, relative to\nthe LDA. Hence, we can not attribute the enhancement\nofMto the charge transfer as suggested by others11{13.\nDensity of states (DOS) calculated within LDA and\nQSGW are shown in Fig. 2. The LDA result is simi-\nlar to previously reported results. A careful examination\nof the band structure reveals that QS GW signi\fcantly\nmodi\fes the energy bands near EF, relative to LDA. It\nhas a slightly larger on-site exchange, widening the split\nbetween the majority and minority DOS and increasing\nMby about 9%. Both DOS \fgures show hybridization\nbetween N-2 pand Fe-3dstates at around \u00007 eV, indi-\ncating that QS GW does not strongly modify the relative\nalignment of N-2 pand Fe-3dlevels. Comparing the par-\ntial DOS reveals that bands are slightly wider and hy-\nbridization is overestimated in LDA, as is typical since\nLDA tends to overestimate 3 dbandwidths slightly. The\nDOS also show hybridization is stronger in the 4 eand4\nTABLE I: Atomic spin magnetic moment miand spin magne-\ntizationMin Fe 16N2in di\u000berent methods. Calculations are\nin the LDA unless GGA or QS GW is speci\fed.\nMethodmi(\u0016B)ambMc\n4e8h4dN(\u0016B)(\u0016B/Fe) (emu/g)\nASA 2.07 2.40 3.03 -0.06 2.48 2.47 239\nASA-GF 2.10 2.41 2.99 -0.10 2.48 2.47 239\nFP 2.08 2.32 2.84 -0.05 2.39 2.38 231\nFP(GGA) 2.21 2.40 2.86 -0.04 2.47 2.43 236\nQSGW 2.24 2.55 3.12 -0.01 2.62 2.59 251\naSpin moment inside atomic or mu\u000en-tin sphere.\nbAverage of the atomic spin moments of all Fe sites without taking\naccount of interstitial and N sites.\ncAverage spin moments within the cell (with taking account of\ninterstitial and N sites).\n-4-2 0 2 4\n(a) LDA\nFe-4e\nFe-8h\nFe-4d\nN-4-2 0 2 4\n(a) LDA\n-4-2 0 2 4\n-8-6-4-2024DOS ( states ( eV spin atom )-1 )\nE-EF (eV)(b) QSGW\nFe-4e\nFe-8h\nFe-4d\nN-4-2 0 2 4\n-8-6-4-2024DOS ( states ( eV spin atom )-1 )\nE-EF (eV)(b) QSGW\n-4-2 0 2 4\n-8-6-4-2024DOS ( states ( eV spin atom )-1 )\nE-EF (eV)(b) QSGW\nFIG. 2: (Color online) Site and spin-projected densities of\nstates within LDA (a) and QS GW (b).\n8hchannels while weaker in the 4 dchannels, which are\nthe furthest removed from N. Also, as typical with sec-\nond row elements, QS GW pushes the N-2 sbands down\nrelative to LDA, from -16.2 eV to -18 eV. The N-2 salso\nhybridizes with Fe-4 eand Fe-8h. However there is almost\nno hybridization with the furthest Fe-4 dat all because\nthe N-2sorbitals is very localized.\n-5 0 5 10 15 20 25(a) ASA4e-4e\n8h-8h\n4d-4d\n8h-4e\n8h-4d\n4e-4d\n-5 0 5 10 15 20 25Jij (meV)(b) FP\n-5 0 5 10 15 20 25\n0.4 0.6 0.8 1 1.2\nRij/a(c) QSGW\n-6-4-2 0 2 4 6 8\n0 1 2 3 4 5(Rij/a)2.8Jij (meV )\nRij/a(d) ASA4e-4e\n4d-4dFIG. 3: (Color online) Real-space magnetic exchange param-\netersJijin Fe 16N2within ASA-GF (a), FP (b), and QS GW\n(c) as functions of distance Rij=a. (d) (Rij=a)2:8Jijin ASA-\nGF as a functions of distance Rij=a. The in-plane lattice\nconstantain Fe 16N2is as twice large as in bcc-Fe.\nB. Exchange coupling and Curie temperature\nThe Heisenberg model parameters Jijusing LDA-ASA\nin the LWA, FP-LDA and FP-QS GW are plotted in\nFig. 3 and tabulated in Table II. The two LDA results are\nquite similar, con\frming that the ASA and the LWA form\na reasonable approximation. QS GW shows some di\u000ber-5\nTABLE II: Pairwise exchange parameters of the Heisenberg\nmodelJij(meV) andTC(K) calculated with di\u000berent methods.\njRijj=a direction ASA FP GW\n4e-4e0.454 0 0 -0.455 19.37 23.75 16.75\n0.645 0 0 0.645 4.18 14.20 3.76\n0.713 -0.5 -0.5 0.095 3.24 2.33 1.88\n0.895 -0.5 -0.5 -0.550 -1.92 -2.91 -1.44\n8h-8h0.483 0 -0.484 0 2.00 -2.55 1.69\n0.516 0 0.516 0 1.49 1.37 4.84\n0.550 0.016 0.016 0.550 5.74 4.60 4.67\n0.684 -0.484 -0.484 0 7.00 11.07 11.80\n0.707 0.516 -0.484 0 0.20 -0.68 0.13\n0.730 0.516 0.516 0 0.39 -0.01 -2.50\n4d-4d0.550 0 0 0.550 2.76 1.54 3.67\n0.707 -0.5 0.5 0 1.15 1.44 0.78\n0.895 -0.5 -0.5 0.550 -2.37 -4.49 -1.90\n4e-8h0.430 -0.258 -0.258 -0.227 20.55 20.00 14.08\n0.470 0.242 0.242 0.323 4.20 5.88 5.88\n8h-4d0.448 0.258 -0.242 0.275 15.73 22.30 16.18\n0.827 -0.242 -0.742 -0.275 0.58 0.60 0.64\n4d-4e0.502 -0.5 0 0.048 1.56 1.85 4.72\n0.708 0 -0.5 -0.502 0.49 -0.23 -0.16\nTC(MFA) 1552 1621 1840\nTC(RPA) 1118 1065 1374\nences, particularly reducing those interactions which are\nAFM.\nThe structure of Jijis much more complicated in\nFe16N2than in elemental bcc-Fe. The vectors connected\nthe nearest 8 h-4eor 8h-4dsites are nearly along [111]\ndirection, the magnetic interactions between them are\ngenerally large. In comparison, the largest interaction is\nalso between the nearest sites connected by vectors along\nthe [111] direction in bcc-Fe. Jijbetween 8h-4esites is\nvery anisotropic due to the distortion of lattice around\nN atom. Similar anisotropy was also found for the in-\nplane couplings on the 8 hlattice. Interestingly, a large\ncoupling, (Jij=23.75 meV in the FP-LDA calculation),\noccurs between two 4 eatoms along [001]. This pair has\nbeen squeezed together by neighboring N atoms. Since\nexchange coupling is sensitive to the distance between\nthose two sites, we also examined this exchange param-\neter for the experimental atom coordinates, for which\nthe bond length of the 4 e-4epair shrinks from 2.60 to\n2.36 \u0017A, and found that this Jijincreases from 23.75 to\n36.55 meV, indicating signi\fcant exchange-striction ef-\nfect. The second nearest 4 e-4ecoupling(two Fe atoms\nwith a N atom between them along h110idirection) is\n14.2 meV in FP and 4.18 meV in ASA. The relatively\nlarge disagreement may be a consequence of the shape\napproximation used in ASA, considering the presence of\nN atom and strong lattice distortion around this pair of\natoms.\nThe calculated magnetic interactions between di\u000berent\ntypes of atoms have very di\u000berent spatial dependence\nand correspondent asymptotic behavior. To demon-\nstrate it explicitly on Fig. 3(d) we show Jijscaled with\n(Rij=a)2:8. With this renormalization Jijbetween atomson 4e positions (smaller moments) are approximately\nconstant in this range of distances(long-ranged interac-\ntion), while Jijbetween Fe atoms on 4d sites (with the\nlargest moments) decay much faster( short-ranged inter-\naction), corresponding to more localized moment behav-\nior. Such very di\u000berent asymptotic behaviour suggests\nthat these localized and delocalized interactions corre-\nspond to Fermi surface shapes with di\u000berent dimension-\nalities.\nTCis calculated from the exchange parameters and\ntabulated in Table II. RPA values are about 30% smaller\nthan the MFA ones. Typically experimental values fall\nbetween the MFA and RPA results, with the RPA be-\ning closer to the experiment in normal three dimensional\nsystems. In the present case, however, the reported ex-\ntrapolated experimental estimate TC=810K3is smaller\nthan both of MFA and RPA values, and smaller than\nthe one in bcc-Fe24. This is rather unusual. In bcc-Fe,\nwe estimate TCto be\u00181300 K and\u0018900 K in the MFA\nand RPA respectively, which bracket the experimental\nvalue of 1023 K (as is typical). Contrary to experiment,\nour calculated TCfor Fe 16N2ishigher than for bcc Fe\nin all our estimations. Such disagreement between the-\nory and experiment is much larger in Fe 16N2than other\nFe-rich phases. The disagreement may originate from\napproximations to the theory (absence of spin quantum\ne\u000bects, temperature-dependence of exchange, among oth-\ners) that uniquely a\u000bect Fe 16N2; or alternatively from\nthe experimental interpretation of the measured TC. We\ncannot completely discount the former possibility, but\nfor this local-moment system, it is unlikely that the most\nserious errors originate in density functional theory that\ngenerate exchange parameters. For instance, parameters\ngenerated from QS GW also do not improve agreement\nwith the experiment; indeed this increase the discrep-\nancy. On the other hand, as noted in the introduction\nFe16N2decomposes with increasing T; moreover, there\nis a transition to the (nonmagnetic) \rphase at 1185\nK. Since the measured M(T) is not a measurement of\nthe single-phase material, it is still unknown what is the\nexperimental value for the TCin a single-phase Fe 16N2.\nUnfortunately, disentangling the structural and magnetic\ndegrees of freedom is very di\u000ecult, experimentally. Fi-\nnally, ifMincreases in Fe 16N2, as is observed and pre-\ndicted,TCshould increase. Thus we conclude that TCin\nhigh quality samples of pure Fe 16N2is probably larger\nthan what has been reported so far, and larger than in\npure bcc-Fe.\nLet us now discuss the in\ruence on TCwhen other\natoms substitute for Fe or N.\nThe CPA is an elegant, single-site ab initio approach\nto study substitutional alloys. We have implemented the\nCPA, including a MFA estimate for TC, within the ASA.\nAs we have seen by comparing exchange interactions in\nthe ASA to those without this approximation, the ASA\ndoes not seem to be a serious approximation to the LDA\nin this material.\nFig. 4 shows the Mand and the normalized e\u000bec-6\n 2.32 2.34 2.36 2.38 2.4 2.42 2.44 2.46\n0.0 0.1 0.2M (µB/Atom)\nx(a)\nCo\nMn\nB\nC\nP\nAl 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1\n0.0 0.1 0.2Normalized exchange\nx(b)\nCo\nMn\nB\nC\nP\nAl\nFIG. 4: (Color online) Spin magnetization M(a)\nand normalized exchange J0=J0(Fe16N2) (with respect to\npure Fe 16N2) (b) as functions of doping concentration in\nFe16N2. The concentration xof doping element T is de-\n\fned as (Fe 1\u0000xTx)16N2with Fe site doping (T=Co,Mn); and\nFe16(N1\u0000xTx)2with N site doping(T=B,C,P and Al) .\ntive exchange (or MFA estimation of TCin units of pure\nFe16N2) with doping by di\u000berent elements. Both Co and\nMn doping cause TCto decrease. On the other hand,\nwith N site being doped with B,C,P or Al elements, TC\nandMchange slightly and the Fermi surface character\nis barely a\u000bected.\nAs shown in Fig. 4, Co or Mn-doped Fe 16N2decrease\nmoment and exchange coupling. We neglected the pos-\nsible site preference e\u000bect in this calculation, and doped\nall three Fe sites with equal probability. Table III shows\nthe magnetic moment and J0parameters of Fe and sub-\nstitutional components on all three di\u000berent Fe sites. It\nindicates an opportunity to increase TCby using a sepa-\nrate Co-doping on Fe-4 dsites.\nMagnetic moments of the Fe component decrease with\nMn doping and slightly increase with Co doping. With\nCo doping, the Fe moments on 8 hand 4dsites do slightly\nincrease, however this increment is not big enough to\novercome the decrease resulting from Co substituting for\nFe - the system behaves more like localized moments sys-\ntem. We also carried out the FP calculation of Fe 7CoN,\nwith one out of eight Fe atoms being replaced by Co\natom and con\frmed that the magnetization decrease, es-\npecially when Co replace the Fe on 4 esite. This is con-\nsistent with the CPA results. Mn substituent have larger\nmagnetic moments than Co substituent. However, Mn\ndoping decreases moments on Fe sites. Overall, the de-\npendence of the total magnetic moments on substituent\nconcentration are almost exactly same with Co and Mn\n-3-2-1 0 1 2 (a) Co\nx=0.0\nx=0.1\nx=0.2\n-3-2-1 0 1 2 (a) Co\n-1 0 1 2 3 4\n-0.4 0 0.4DOS ( states ( eV spin atom )-1 )\nE-EF (eV)(b) Mn\n-1 0 1 2 3 4\n-0.4 0 0.4DOS ( states ( eV spin atom )-1 )\nE-EF (eV)(b) MnFIG. 5: (Color online) Densities of state of substitu-\ntional component in random alloy (Fe 1\u0000xCox)16N2(a) and\n(Fe1\u0000xMnx)16N2(b).\ndoping. Another interesting observation is that magnetic\nmoments of both substituents decrease with increasing of\ndoping concentration. This can be explained by the par-\ntial density of states as shown in Fig. 5. The magnetic\nmoment of Co slightly decreases with increasing of dop-\ning concentration. As shown in Fig. 5, the unoccupied\nDOS peak right above the Fermi energy ( EF) in the mi-\nnority spin channel moves toward it. More electrons \fll\nin the minority channel and decrease the magnetic mo-\nment as doping increases. With Mn doping on the other\nhand, the peak in the majority channel right below EF\nbecomes less pronounced. It shifts toward EFand de-\ncreases the magnetic moment of Mn component. For the\nFe component DOS, there is no peak structure near the\nEF, and magnetic moment is much less sensitive to the\nsubstitutional concentration.\nC. Magnetic anisotropy\nValues of MAE from previous work are summarized\nin Tables IV. Results of present work are shown in Ta-\nble V. All calculations are carried out within LDA unless\nGGA is speci\fed. For the pure Fe 16N2, both experimen-\ntal and optimized structure are investigated. Note that\ndoping and SOC lower the symmetry, and the degeneracy\noflvaried. Within LDA, a uniaxial magnetic anisotropy\nK=144\u0002105erg/cm3was obtained with experimen-\ntal atomic coordinates. Structural optimization gives\na smaller MAE with K= 103\u0002105erg/cm3. GGA gives\nsmaller MAE than LDA. It is usually non-trivial to ana-7\nTABLE III: Component-resolved atomic spin moments mi( the atomic spin moment of substitutional component are given in\nparentheses), magnetization Mand exchanges J0in Co and Mn-doped Fe 16N2calculated within ASA-GF.\nSubstituent xmi(\u0016B)M(\u0016B/atom)J0(meV )\n4e 8h 4d 4e 8h 4d\nCo0.00 2.10 ( 1.44) 2.41 ( 1.68) 2.99 ( 2.11) 2.47 12.95 (11.37) 15.70 (14.62) 16.96 (19.42)\n0.10 2.09 ( 1.34) 2.43 ( 1.64) 3.01 ( 2.09) 2.40 11.97 ( 9.62) 15.45 (13.56) 16.70 (18.68)\n0.20 2.08 ( 1.27) 2.45 ( 1.62) 3.02 ( 2.08) 2.32 11.24 ( 8.47) 15.19 (12.91) 16.50 (18.07)\nMn0.00 2.10 ( 1.88) 2.41 ( 2.25) 2.99 ( 3.23) 2.47 12.96 ( 7.05) 15.70 ( 6.57) 16.97 ( 1.07)\n0.10 2.07 ( 1.75) 2.35 ( 2.04) 2.96 ( 3.03) 2.40 12.03 ( 5.64) 14.01 ( 4.03) 14.97 (-1.10)\n0.20 2.06 ( 1.64) 2.30 ( 1.85) 2.93 ( 2.90) 2.32 11.15 ( 4.41) 12.59 ( 2.09) 13.22 (-2.69)\nTABLE IV: Previous works on magnetic anisotropy in Fe 16N2.\nMethod K(105erg\ncm3)Easy axis Ref.\nExp. Sugita et al. 4.8 [100]3\nTakahashi et al. 200a[001]18\nTakahashi et al. 97 [001]19\nKita et al. 44 [001]36\nJiet al. 100b[001]37\nTBcUchida et al. 140 [001]19\naValue of ( K1+K2).\nbMeasured in partial-ordering Fe 16N2, author claimed MAE\nshould be much higher for the single-phase sample.\ncTight binding approximation\n-300-200-100 0 100 200 300 400 500\n[001] [100] [110] [001]Anisotropy ( µeV/Fe )\nSpin direction∆i(4e)\n∆i(8h)\n∆i(4d)\n∆\nELDA\nFIG. 6: (Color online) AMAE on 4 e, 8h, 4dFe sites (\u0001 i)\nand their average value (\u0001) and the LDA total energy relative\nto the ground state ( ELDA) as functions of spin quantization\naxis rotation.TABLE V: The MAE K, on-site orbital magnetic moment l\nand the AMAE \u0001 with di\u000berent spin quantization axis direc-\ntion in pure, Co-doped and Ti-doped Fe 16N2. Spin quantiza-\ntion axis direction eare along [001],[100] and [110] directions\nrespectively. With the spin along [100] and [110], \u0001 and K\nvalues(with respect to [001] direction) directions are given.\nTo estimate \u0001, \u0018i=50,70meV had been used for Fe and Co\natoms respectively.\neK l(10\u00003\u0016B) \u0001\n\u0016eV\nFe105erg\ncm3 4e 8h 4d\u0016eV\nFe\nExp.a001 54 45 71\n100116 144 35 49 64\n110116 144 36 58 39 64\nExp.a001 52 44 68\nGGA 100105 131 36 48 61 110\n110105 131 36 57 39 61 110\nTheo.b001 62 46 67\n100 84 103 39 50 63 137\n110 84 103 39 58 41 63 137\nTheo.b001 56 43 62\nGGA 100 52 65 38 47 58\n110 52 65 38 55 40 58\nFe7CoN 001 91d72 47 76\n(4e)c100165 206 49d30 49 61 64 337\n110165 206 49d30 61 36 63 336\n001 63 69d41 41 44 70\n(8h)100 42 52 36 77d46 50 47 67 123\n110 16 20 40 90d38 38 56 67 81\n001 63 51 120d81\n(4d)100138 171 33 38 52 106d70271\n110138 171 36 62 42 106d70271\nFe7TiN 001 11d63 40 69\n(4e)100127 158 10d27 46 65 65 62\n110127 158 10d27 55 38 65 62\n001 55 14d48 48 41 69\n(8h)100 57 71 38 13d48 50 50 65 122\n110 43 53 35 13d41 41 62 68 83\n001 60 40 14d69\n(4d)100102 127 38 38 45 13d6783\n110103 128 38 52 38 13d6783\naExp. the experimental crystal structure was used.\nbTheo. the theoretically optimized crystal structure was used.\ncDoping site of the substitutional atom.\ndOrbital magnetic moments of the substitutional atoms.8\nlyze the origin or site dependence of magnetic anisotropy.\nBelow we de\fne the atomic magnetic anisotropy energy\n(AMAE) \u0001 ias half of the di\u000berence of the SOC ener-\ngies along di\u000berent magnetic \feld directions, that is in\nturn de\fned by the corresponding anisotropy of orbital\nmagnetic moments:\nX\n\u0001i(\u0012= 90\u000e) =X\n\u0018imi(l001\ni\u0000l100\ni)=4 (5)\nwhere\u0018iis a SOC parameter, while miandliare atomic\nspin and orbital magnetic moments correspondingly. The\nsum of \u0001 ican be compared with the total MAE Kob-\ntained using the total energies. This approach takes into\naccount the SOC anisotropy and its renormalization by\ncrystal \feld e\u000bects. We further assume that the spin mo-\nment has very weak anisotropy38and the main change in\n\u0018L\u0001Sproduct comes from the change of orbital magnetic\nmoment (see also Ref. 39). This is the case for Fe 16N2\n(see Table V). For the pure Fe 16N2, when the spin quan-\ntization axis rotates from [100] to [001], ldecreases on 8 h\nsites, but increases on 4 dand 4e. Whileldepends on site,\nthe totallincreases during this rotation, which agrees\nwith the predicted uniaxial character of MAE. When the\nspin quantization axis points along [110], SOC lowers the\nsymmetry, and splits the four equivalent 8 hsites into two\npairs withlincreasing on one pair and decreasing on the\nother.\nAs shown in Fig. 6, there is a strong correlation be-\ntweenKand \u0001 (with respect to magnetic \feld along\n[001] direction, and atomic value \u0018i=50meV is used for all\nthree di\u000berent Fe sites for simplicity), where iindicates\nall atomic sites. Obviously, the atomic 8 hsites make\nnegative contributions to the desired uniaxial MAE, and\nwhile 4eand 4dsites make positive contributions. Thus,\none may hope that doping on 8 hsite, thus eliminating\nnegative (in-plane) contribution to MAE, may improve\nthe uniaxial MAE.\nSince Co and Ti doping had been reported to stabilize\nthe Fe 16N2phase16,17, it seems logical to study prediction\nabove using these dopants. We replaced one out of eight\nFe atoms in the primitive cell with Co or Ti atom and\nrelax the atomic positions within LDA and then study\nthe anisotropy. If we replace one of four 8 hatoms with\nCo atom, we found that Co atom has a larger lthan any\nother Fe sites, however, it does not eliminate the negative\ncontribution from 8 hsites. Instead, it makes Ksmaller.\nAlsoland thenKalong [100] and [110] directions become\nmore anisotropic. Surprisingly, however, with a Co atom\non 4eor 8hsites, theldi\u000berence between out-of-plane\nand in-plane cases become even larger on 4 eand 4dsites\nand smaller on 8 hsites. In other words, it makes the\npositive contribution from 4 eand 4dsites stronger and\nthe negative contribution from 8 hsites smaller. As a re-\nsult, calculated MAE is doubled ( K=206\u0002105erg/cm3)\nwithin LDA with doped Co being on 4 esite. A similar\ne\u000bect had been found with Ti doping. Kincreases when\nTi is substituted on the 4 eand 4dsites and decreases\nwhen substituted on 8 h. Unlike the Co doping, magnetic\norbital moments of Ti atom are small and barely change\n 0 50 100 150 200\n1.02 1.06 1.10 1.14 1.18Anisotropy ( µeV/Fe )\nc/a Exp. ∆\nKFIG. 7: (Color online) Kand the \u0001 as functions of c=a\nin Fe 16N2. The ideal crystal structure without strain has\nc=a=1:1. For each c=a, the atomic positions are relaxed with\nvolume being conserved.\nwith spin rotation. Generally, for the same structure, K\nis always strongly correlated with \u0001. The larger \u0001 is,\nthe largerKis along that speci\fc spin quantization di-\nrection. However, this correlation may not longer hold\ntrue with di\u000berent structures. For example, in Fe 7TiN,\n\u0001 is the largest with Ti doped on 8 hsite, however K\nis much smaller than those with Ti doped on 4 eand 4d\nsites.\nTetragonality is another factor which may a\u000bect the\nanisotropy in a signi\fcant way. Let us compare Fe 16N2\nwith bct-Fe, where even for c=a=1:1 (thec=aratio for\nFe16N2) MAE is still rather tiny40. In Fig. 7 the cal-\nculated MAE in Fe 16N2is shown as a function of c=a.\nThis dependence is much stronger than in bct-Fe and\nwe assume that MAE mostly originates from distortion\nof Fe sublattice around the N atom and the Fe-N hy-\nbridization. Experimentally, the large tetragonality can\nbe obtained in \flms, where it can be tuned by the nitro-\ngen concentration37. However, according to our results\nabove, doping bulk Fe 16N2in a way that increases c=a\nmay lead to MAE increase. The MAE and AMAE are\nwell correlated as shown in Fig. 7. Within this c=arange,\nthe spin magnetization varies within 2%: it is not likely\nto be responsible for the MAE increase. On the other\nhand the anisotropy of orbital moment strongly corre-\nlates with MAE and is probably responsible for its en-\nhancement as tetragonality increases. Orbital magnetic\nmoments can be measured more precisely, so new XMCD\ntype of experiments for this system are desirable.\nIV. CONCLUSION\nIn this study of intrinsic magnetic properties of Fe 16N2,\nour LDA results for magnetization agree with previously9\nreported values while QS GW increases magnetization by\n9%. This enhancement is largely due to on-site exchange\nsplitting between the dminority and majority states { an\ne\u000bect seen in many other magnetic systems such as NiO\nand MnAs26,41. In Fe 16N2in particular, we \fnd no evi-\ndence of localized states or correlations not already found\nin Fe. Taken together all of those factors we expect that\nthe QSGW prediction for Mis not far from what should\nbe observed in the ideal Fe 16N2compound. We \fnd no\nevidence of charge transfer between di\u000berent Fe sites as\nproposed elsewhere. Thus, the theoretical magnetization\npredicted for Fe 16N2does not exceed the maximum on\nSlater-Pauling curve ( \u00182.5\u0016B) and is smaller than corre-\nsponding maximum of magnetization observed in Fe-Co\nalloys, which may still be considered as a record holder\namongdatomic magnets.\nLDA calculations predict TCsigni\fcantly larger than\nthe experimental value; the QS GW result is even larger.\nWe assume that Fe 16N2will have a higher TCif one can\n\fnd a way to stabilize it. E\u000bects of doping by various\nelements on MandTCwere studied in the LMTO-CPA\napproximation. Various dopants a\u000bect MandTCdi\u000ber-\nently; but unfortunately no dopants we considered en-\nhancedMorTC.\nA uniaxial magnetocrystalline anisotropy\nK=103\u0002105erg/cm3was calculated in the LDAwith the theoretically optimized crystal structure. Kis\nstrongly correlated with the atomic magnetic anisotropy\nenergy due to spin-orbit coupling only. We found it\ncan be increased by increasing c=aor by adding small\namount of Co or Ti atoms on 4 eor 4dsites.\nFe16N2is one of the more promising candidates for per-\nmanent magnets that do not contain rare-earth elements.\nWe believe that there is room for improvement and we\nstudied several possible routes to obtain better proper-\nties. A further investigation on increasing the thermal\nstability and/or changing crystal structure tetragonality\nis desired.\nV. ACKNOWLEDGMENTS\nThis work was supported by the U.S. Department of\nEnergy, O\u000ece of Energy E\u000eciency and Renewable En-\nergy (EERE), under its Vehicle Technologies Program,\nthrough the Ames Laboratory. Ames Laboratory is oper-\nated by Iowa State University under contract DE-AC02-\n07CH11358. K. D. B. acknowledges support from NSF\nthrough grants DMR-1005642, EPS-1010674 (Nebraska\nEPSCoR), and DMR-0820521 (Nebraska MRSEC).\n1K. H. Jack, Proc. R. Soc. A 208, 200 (1951).\n2T. K. Kim and M. Takahashi, Appl. Phys. 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B 76, 165126 (2007)."    },    {        "title": "1304.5307v1.First_Principles_Calculation_of_Magnetocrystalline_Anisotropy_Energy_of_MnBi_and_MnB__1_x_Sn_x.pdf",        "content": "1 \n First Principles C alculation of Magnetocrystalline Anisotropy \nEnergy  of MnBi  and MnBi 1-xSnx \n \n                 A. Sakuma , Y. Manabe , and Y . Kota  \n     Department of Applied Physics, Tohoku University , Sendai Japan  \n                \nWe calculated t he magnetic a nisotropy constant Ku of MnBi using  a first \nprinciples approach , and obtained a negative Ku in agreeable with  \nexperimental results.  Furthermore, we also found a band filling \ndependence indicating that a slight decrease in the valence electron number \nwill change Ku from negative to positive.  When some  of the Bi is replaced \nwith  Sn to decrease the valence electron number,  the Ku value of MnBi 1-xSnx \ndrastically  change s to a positive value , Ku~2 MJ/m3, for x > 0.05.  \n \nKEYWORDS: magnetocrystalline anisotropy en ergy, MnBi, permanent \nmagnets, first principles calculation, LMTO, CPA  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 2 \n      From the view point of elemental strategies, much effort  has been \ndevoted  to realiz e strong magnetocrystalline anisotropy in transition metal \nsystem s, because of the shortage of the rare earth elements used in  \npermanent  magnets  for electric vehicles .  Among  transition  metal system s, \nMnBi has been expected  to be a favorable  candidate as a permanent \nmagnet s1-4) or magneto -optical medi um5-7) because of  its large uni axial \nmagnetic anisotropy.  MnBi has the particular  advantage that the  \nmagnetic anisotropy constant (Ku) increases with increasing temperature , \nstarting from a negative value a round  zero temperature and becoming  \npositive at around 100  K.2,3)  This behavio r continues above room \ntemperature and Ku reaches about 2 MJ/m3 at 300 K.  In addition to the \nlarge magnetocrystalline anisotropy, r ecently, the high spin polarization of \nMnBi has attracted attention  both from theoretical and experimental points \nof view.8) Despite these interesting  propert ies of MnBi, most theoretical \nworks ,9-13) except for Refs. 8 ), 14 ), and 15), have focus ed on the \nmagneto -optical propert ies. \n     In the present work, we perform a first principles calculation  of the Ku \nof MnBi at zero tem perature and of  the magnetic moment  M and Curie \ntemperature  Tc.  The calculated Ku is found to be negative , in agreement \nwith  the experimental result s, and the electron number dependence of Ku \nindicates  that a decrease in the valence electron number will c hange  the Ku \nto a positive value .  Based on  this observation , we examine the partial \nreplacement  of Bi atoms with  Sn atoms , whose electron  number is one  lower , \nand we confirm that the Ku value is raised to positive value at zero \ntemperature  in MnBi 1-xSnx for x >0.05.  \n     For the electronic structure  calculation, we employ the linearized \nmuffin -tin orbital (LMTO) method under the local spin density functional \napproximation.  In the magnetic force theorem, t he magnetic anisotropy \nconstant  is given by \nVcEaE Ku /})ˆ{ }ˆ{(  where \nV  is the unit cell volume \nand \n}ˆ{eE is the band energy calculated with the  semi -relativistic LMTO \nHamiltonian including spin-orbit interaction  with the magnetization \npointing in the \neˆ  direction .  The effective exchange constants  are \ncalculated using  the method developed by Leichtenstein et al.16) From the \neffective exchange constants , the Curie temperatures can be  estimated \nunder the molecular field approximation.  The elect ronic structure s and Ku \nvalue s of partially substituted alloys are calculated under  the coherent 3 \n potential approximation (CPA) within the framework  of the tight -binding \nLMTO method.17,18)  The crystal structure of MnBi is a NiAs -type hexagonal \nstructure , as shown in Fig. 1 , where w e also show t he latest lattice constants  \ngiven by Yang et al.4)  In the electronic structure calculation, we introduce  \nempty spheres located at the dashed circles in Fig. 1 .  About 2×105 k-points \nare sampled  in the full B rilloui n zone , to obtain a sufficiently converged Ku \nvalue.  \n     First, we look at the fundamental properties of MnBi , such as its \nmagneti c moment and Curie temperature.  Figure 2 shows the local density \nof states (DOS) of MnBi.  First, we note that t he spectral shape is close to \nthat reported by  Coehoorn et al.14)  The large spin polarization in the local \nMn DOS  can easily be seen .  In fact, t he magnetic moment s calculated on \nthe Mn sites are up to 3. 8\nB, which is also consistent  with the  previous \nexperimental result s.3,19)  Because of  the strong hybridization of the Bi p \nstates with the Mn d states, the Bi moment is polarized  in the direction \nopposite to that of the Mn moment , with a value of －0.08\nB .  Figure 3 \nshows t he effective exchange constant acting on the Mn moment , which is  \ndenoted by \nMnJ  as a function of the Fermi level in the rigid band scheme.  \nThe real Fermi level position of MnBi is located at the origin of the \nhorizontal line , where as the actual value of \nMnJ  is 72 meV.   The Curie \ntemperature can be estimated under  the molecular field approximation from  \nB C k J T 3/ 2Mn\n, from which we have  Tc ~560 K.  This is compa rable to  the \nexperimental result  (700 K) 2) within a reasonable accuracy . \n     Next, we focus on  the magnetocrystal line anisotropy energy of MnBi. \nFigure 4 shows  the anisotropy constant Ku of MnBi as a function of the \nvalence electron  number  in the rigid b and scheme.   Note that the actual \nelectron number of MnBi is 12 per formula unit  (f.u.) .  We find  that t he \nactual Ku value of MnBi is negative , at about －0.5 MJ/m3.  Although the \nabsolute value is considerably larger than the measured value  of －0.2 \nMJ/m3, the magnetic easy axis is consistent with the experimental result s at \nlow temperature s.  We should emphasi ze here that the actual electron \nnumber of MnBi , 12 per f.u. is located just above  the intersection  point \nbetween the Ku curve and the horizontal axis  where the sign of Ku changes \nfrom positive to negative.   This leads us to expect that the sign of Ku may be \nchangeable  depending on the calculation condition and method.  Actually, 4 \n we have confirmed that the Ku varies with the unit cell volume with \nV Ku/\n~ －0.3×10-30 MJ/m6 (= －0.3 (MJ/m3)/ Å3) from which the Ku is \nfound to turn  positive for V < 95.5  Å3 (V = 97.1 Å3 in the present \ncalculation).   The method for calculating the electronic structure may also \nhave a considerable influence on t he Ku value .  However, at least the Ku \nvalue is shown to become  positive upon a  slight decrease in the valence \nelectron  number . \nExploiting this fact, we proceed ed to calculate  the Ku of MnBi 1-xSnx \nwhere Bi atoms are partially replaced  by Sn , whose electron  number is one \nless than that of Bi.  Here , we avoid ed Pb substitution  because of  \nenvironment al consideration .  In Fig. 5, t he calculated  Ku values  of \nMnBi 1-XSnX are plotted as a function of Sn concentration x.  We fix ed the \nlattice constants at those of M nBi for this substitution ; we have confirmed \nthat the lattice constants do not have  a substantial influence on the results.  \nAs expected, the Ku becomes  positive upon  a slight substitution of Sn  for Bi .  \nIn the figure, we also show , by the dashed line, the result for MnBi based on \nthe rigid band scheme  given in Fig. 4 , for comparison.  Below x = 0.01 of Sn \nsubstitution , the slop e of the Ku values against X seems to be a little bit \nlarger than that obtained in  the rigid band model.  For x >0.01, the behavior  \ndeviates significantly from that predicted by the rigid band model , and the \nKu value roughly remains  constant at around 3  MJ/m3.  This may be \nbecause the atomic potentials of Bi and Sn are much different from eac h \nother, invalidating  the rigid band model.   From  a practical viewpoint , the \n10% Sn substitution may be sufficient to achieve a high magnetic anisotropy \nat low temperature.  In Fig. 6, we show t he Sn concentration dependence s of \nthe Curie temperatures  and magnetic moment s.  The data reveal that both  \nMnJ\n and M gradually decrease with x.  However , the variation  is no t so \ndramatic , so the substitution does not have  a serious influence on the \nmagnetic properties other than the magnetic anisotropy.   \nIt should be noted  here that expe rimental work20) on the magnetic \nproperties of MnBi R (R = In, Ge, and Sn) film s showed that only Ge doping \ncan maintain the hexagonal structure and induce both a Kerr rotation angle \nand coercivity, while the structure of a Sn dop ed film is a mixture of the  \nhexagonal and cubic  structures .  This means that the substitution of Sn for 5 \n Bi is not easy to achieve in practic e, because of the differen t atomic radi i of \nBi and Sn.  The experimental object ive is to achieve  uniform replacement  of \nBi atoms while maintain ing the hexagonal structure.  Theoretically, on the \nother hand, the temperature dependence of the magnetic anisotropy is the \nmain subject to be clarified.  Needless to say, this has been  a common \nproblem for decades in the study of magnetism in transition metal systems.  \nIn particular , a theoretical study of the Ku including  spin reorientation  at a \ncertain temperature (in MnBi3); and also MnSb21)) is sincerely  desired as a \nfuture work.  \nIn summary, we have investigate d the magnetic anisotropy constant \nKu tog ether with the magnetic moment M and the Curie temperature Tc of \nMnBi using a  first-principles calculation.  The calculated Ku, M and Tc are \nto some extent consistent  with the experimental results  and previous \ntheoretical works.  The dependence of Ku on th e valence electron number  \nsuggests that a slight decrease in the valence electron number will change  \nKu from negative to positive.  Based on  this result, we have calculated the \nelectronic structure  of MnBi 1-xSnx whose electron number decreases with x \nand c onfirmed that the Ku dramatically  change s to a positive value of ~2 \nMJ/m3 for x >0.05, while the values of M and Tc decreases slight ly.   \n \nAcknowledgments  \nThis work was supported by JST under the Collaborative Research Based on \nIndustrial Demand program “High Performance Magnets: Towards \nInnovative Development of Next Magnets .” \n \nReferences  \n1) J. Guillaud, J. Phys. Radium 12, 143 (1951) . \n2) B. W. Roberts, Phys. Rev. 104, 607 (1956).  \n3) T. Chen and W.  E. Stutius, IEEE. Trans. Magn.  10, 581 (1974) . \n4) J. B. Yang, K. Kamar aju, W. B. Yelon, W. J. James, Q. Cai, A. Bollero, \nAppl. Phys. Lett. 79, 1846  (2001) . \n5) D. Chen and Y. Gondo, J. Appl. Phys. 35 (1964)  1024.  \n6) R. L. Aagard, F. M. Smith, W. Walters, D. Chen, IEEE. Trans. Magn. 7, \n380 (1970) . \n7) G. Q. Di, S. Iwata, S. Tsunashima, S. Uchiyama, J. Magn. Magn. Mater. \n104-107, 1023  (1992) . 6 \n 8) P. Kharel, P. Thapa, P. Lukashev, R. F. Sabirianov, E. Y. Tsymbal, D. J. \nSellmyer, B. Nadgorny, Phys. Rev. B 83, 024415  (2011) . \n9) S. S. Jaswal, J. X. Shen, R. D. Kirby, D. J. Sellmyer, J. Appl. Phys. 75, \n6346  (1994) . \n10) P. M. Oppeneer, V. A. Antropov, T. Kraft, H. Eschrig, A. N. Yaresko, A. Ya \nPerlov, J. Appl. Phys. 80, 1099  (1996) . \n11) J. Köhler and J. Kübler, Physica  B 237-238, 402 (1997) . \n12) P. Ravindran, A. Delin, P. James, B. Johansson, J. M. Wills, R. Ahuja , O. \nEriksson, Phys. Rev. B 59, 15680  (1999) . \n13) Z. Qing -qi, Z. Zhi, L. Wu -yan, L. Zhi -qing, C. Y. Pan, J. Magn. Magn. \nMater. 104-107, 1019  (1992) . \n14) R. Coehoorn and R. A. de Groot, J. Phys. F: Met. Phys. 15, 2135 (1985).  \n15) J. B. Yang, W. B. Yelon, W. J. James, Q . Cai, M. Kornecki, S. Roy, N. Aji, P. \nHeritier, J. Phys.: Condens. Matter 14, 6509  (2002) . \n16) A. I. Leichtenstein,  M. I. Katsunelson, V. P. Antropov, V. A. Gubanov, J. \nMagn. Magn. Mater. 67, 65 (1987) . \n17) Y. Kota  and A. Sakuma, J. Phys. Soc. Jpn. 81, 084705  (2012). \n18) I. Turek, J. Kudrnovsky, K. Carva, Phys. Rev. B 86 (2012)  174430 . \n19) R. R. Heikes, Phys. Rev. 99, 446 (1955) . \n20) L. Jin, N. Ying, M. Tingjun, F. Ruiyi, J. Appl. Phys. 78, 2697  (1995).  \n21) T. Tobita and Y. Makino, J. Phys. Soc. Jpn. 25, 120 (1968) . \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n 7 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 1  Crystal structure of NiAs -type MnBi.   \nThe dashed circles indicate the empty spheres introduced in the electronic \nstructure calculations.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 2  L ocal density of states (DOS) of MnBi .  The Fermi level is located at \nthe origin of the horizontal axis.  \n \n \n 8 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 3  Effective exchange constant JMn acting on the Mn moment as a \nfunction of the Fermi level.  The actual value of JMn is given at the origin of \nthe horizontal axis.  \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 4  Magnetic anisotropy constant Ku of MnBi as a function of valence \nelectron number in the rigid band scheme.  The actual electron number is \nindicated by the arrow.  \n \n \n \n-250-200-150-100-50050100150\n-0.5 -0.4 -0.3 -0.2 -0.1 00.1 0.2 0.3 0.4 0.5MnBi\nEnergy (Ry)JMn (meV)\nJMn(meV) MnBi\nEnergy ( Ry)\n-250-200-150-100-50050100150\n-0.5 -0.4 -0.3 -0.2 -0.1 00.1 0.2 0.3 0.4 0.5MnBi\nEnergy (Ry)JMn (meV)JMn(meV) MnBi\nEnergy ( Ry) 9 \n  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 5  Ku of MnBi 1-xSnx as a function of Sn concentrat ion x. \nThe dashed line indicates the result for MnBi based on the rigid band \nscheme given in Fig. 4.  \n \n \n \n \n \n \n \n \n \n \n \n \n \n \nFig. 6  M and Tc of MnBi 1-xSnx as a function of Sn concentration x. \n \n "    },    {        "title": "1304.5323v1.Effects_of_composition_and_chemical_disorder_on_the_magnetocrystalline_anisotropy_of_Fe__x_Pt__1_x__alloys.pdf",        "content": "E\u000bects of composition and chemical disorder on the magnetocrystalline anisotropy of\nFexPt1\u0000xalloys\nC.J. Aas1, L. Szunyogh2, and R.W. Chantrell1\n1Department of Physics, University of York, York YO10 5DD, United Kingdom and\n2Department of Theoretical Physics and Condensed Matter Research Group of Hungarian Academy of Sciences,\nBudapest University of Technology and Economics, Budafoki \u0013 ut 8. H1111 Budapest, Hungary\n(Dated: November 3, 2018)\nWe perform \frst principles calculations of the magnetocrystalline anisotropy energy (MAE) of the\nL10-like Fe xPt1\u0000xsamples studied experimentally by Barmak and co-workers in [J. Appl. Phys. 98\n(2005) 033904]. The variation of composition and long-range chemical order in the samples was\nstudied in terms of the coherent potential approximation. In accordance with experimental observa-\ntions, we \fnd that, in the presence of long-range chemical disorder, Fe-rich samples exhibit a larger\nMAE than stoichiometric FePt. By considering the site- and species-resolved contributions to the\nMAE, we infer that the MAE is primarily a function of the degree of completeness of the nominal\nFe layers in the L1 0FePt structure.\nDue to its extraordinarily high magnetocrystalline\nanisotropy energy (MAE), L1 0FePt is of considerable\ninterest to the development of ultrahigh density magnetic\nrecording applications, in particular, for heat-assisted\nmagnetic recording (HAMR). The L1 0phase of Fe 50Pt50\nis a layered face-centered tetragonal structure, exhibiting\nalternating Fe and Pt layers along the (001) direction.\nFePt also exhibits stable FePt 3and Fe 3Pt phases as well\nas a chemically disordered, cubic phase.1,2Accordingly,\nFePt exhibits phase transitions with respect to compo-\nsition as well as to chemical order and understanding\nthe related e\u000bects on the magnetic properties is an\nimportant issue. The large e\u000bect of chemical disorder\non the MAE of Fe 50Pt50has already been outlined both\nexperimentally3{5and theoretically.6,7\nThe degree of long-range chemical order is quanti-\n\fed in terms of a chemical order parameter.8,9The\nL10FexPt1\u0000xalloy is modeled by a repeating sequence\nof two atomic layers, characterized by compositions\nFerFePt1\u0000rFeand Fe 1\u0000rPtPtrPt, respectively. The frac-\ntions,rFeandrPt, are related to each other through the\ncondition, 1 + rFe\u0000rPt= 2x. Furthermore, set by the\nrequirement, rFe\u00151\u0000rPt(the case of rFe<1\u0000rPt\ncan simply be obtained by interchanging the two types\nof layers), the range of rFeis con\fned to rFe\u0015x\n(rPt\u00151\u0000x), whereby obviously rFe\u0014min(1;2x)\n(rPt\u0014min(1;2\u00002x)). The chemical order parameter\nsis then de\fned by\ns= 2(rFe\u0000x) = 2(rPt\u00001 +x); (1)\nand ranges from 0 to max(2 \u00002x;2x). Denoting the\ncompositions of the two repeating layers as ( A;B), the\ncase of complete disorder refers to the compositions\n(FexPt1\u0000x, FexPt1\u0000x) and the maximum order to (Fe,\nFe2x\u00001Pt2\u00002x) forx\u00150:5 and to (Fe 2xPt1\u00002x,Pt) for\nx\u00140:5. Note that only in case of x= 0:5 can the order\nparameter reach the value s= 1. In the following, we\nrefer to the two layers as the nominal Fe layer and the\nnominal Pt layer, respectively.TABLE I. Summary of the experimental data obtained for\nfour samples of Fe xPt1\u0000xin Ref. 5.\nSample x(%) a ( \u0017A) c ( \u0017A) c/a s K (meV/atom)\n1 46.2 3.870 3.721 0.961 0.89 0.453\n2 51.1 3.863 3.710 0.960 0.93 0.709\n3 52.0 3.857 3.706 0.961 0.89 0.775\n4 55.4 3.839 3.704 0.965 0.72 N/A\nOur present study was motivated by the work of\nBarmak and co-workers,5who investigated the MAE\nof four Fe xPt1\u0000xsamples di\u000bering in composition and\ndegree of chemical order. Table I summarizes the\nexperimental geometrical and compositional data, as\nwell as the measured MAE values for the FePt samples\nstudied in Ref. 5. Note that for sample no. 4 the MAE\ncould not be determined. One of the main conclusions of\nRef. 5 is that slightly Fe-rich samples may be preferable\nto Fe 50Pt50for obtaining a large MAE. In terms of \frst\nprinciples calculations we aim to explore the origin of\nthis observation, in particular, whether it is it a pure\ne\u000bect of composition or whether it is also related to the\nchemical order of the sample.\nTo this end, we perform fully relativistic \frst-\nprinciples calculations by means of the screened\nKorringa-Kohn-Rostoker (SKKR) method. As the\nmethod is well documented elsewhere in the literature,\nsee e.g. Refs. 10{13, here we describe only the features\nparticularly relevant to this work. We use the local\nspin density approximation (LSDA) of the density\nfunctional theory (DFT) as parameterized by Vosko et\nal.14and treat the potentials within the atomic sphere\napproximation (ASA). In line with previous work,7the\nself-consistent potentials and \felds are calculated from\nscalar-relativistic calculations and the fully relativistic\nKohn-Sham-Dirac equation is then solved to derive\nthe MAE of the system. In all calculations, an overall\nangular-momentum cut-o\u000b of `max= 3 was used.arXiv:1304.5323v1  [cond-mat.mtrl-sci]  19 Apr 20132\nThe chemical disorder according to the model as\ndescribed above was treated in terms of the coherent\npotential approximation (CPA).15,16It should be men-\ntioned that the MAE of Fe xCo1\u0000xalloys was recently\ninvestigated by using the same model of long-range\nchemical order.17,18In the spirit of the magnetic force\ntheorem, the MAE is evaluated as the di\u000berence in\nthe band energy of the system when polarized along\nthe easy axis (001) and perpendicular to the easy axis,\nalong (100). As in Ref. 7 we estimated the e\u000bect of\ntemperature-induced spin-\ructuations by scaling down\nthe MAE by a factor of 0.6 according to the Langevin\ndynamics simulations of Mryasov et al.19\nIn order to verify our method against the experiments,\n\frst we attempt a direct comparison of our SKKR-CPA\ncalculations to the experimental data of Ref. 5, see\nTable I. As shown in Fig. 1, we performed three sets\nof calculations. The \frst set only takes into account\nchanges in the lattice geometry (i.e., the variation in\nthe lattice parameters), while assuming stoichiometric\ncomposition, x= 0:5, and maximum long-range chem-\nical order, s= 1. Even by taking into account the\n'temperature factor' of 0.6, these calculations yield quite\nhigh MAE values. Such magnitude di\u000berences with\nrespect to the experiment are, however, in agreement\nwith previous \frst-principles calculations of the MAE of\nFePt, see e.g. Refs. 20 and 21. Furthermore, in this set\nof calculations only a very moderate change ( <3%) of\nthe MAE is obtained across the samples.\nIn the second set of calculations, we introduce the\ncomposition xas given in experiment, while keeping the\ndegree of chemical order constant at s= 0:89. This\ngreatly improves the trend of the MAE, however, the\nrelative change of the MAE from sample no. 1 to sample\nno. 2 is still underestimated ( <15%) as compared to\nthe experiment ( \u001850%). The overall magnitude of\nthe MAE is signi\fcantly decreased, but it is still by a\nfactor of 2-2.5 larger than the measured one. Finally,\nincluding also the variation of chemical order as given\nin the experiment clearly improves the above mentioned\nrelative change between samples no 2 and 3 ( \u001830%),\nbut, opposite to the experiment, it predicts a slightly\ndecreasing trend from sample no. 2 to 3. Note, however,\nthat these latter changes are within the range of both\ntheoretical and experimental errors.\nHaving con\frmed that the SKKR-CPA calculations\nsatisfactorily reproduce the experimental trends, we next\nconsider the general e\u000bects of the chemical composition\nxand order parameter son the MAE of FePt. To this\nend, we used the lattice parameters measured for sample\nno. 3 in Ref. 5, a= 3:857\u0017A andc= 3:706\u0017A, while we\nindependently varied the chemical order parameter sas\nwell as the composition x. Note that for this theoretical\nstudy we did not scale down the MAE to mimic temper-\n0.40.60.81.01.21.41.61.82.0\n 1  2  3MAE per formula unit (meV)\nsample nos = 0.89, x as exp\ns as exp, x as exp\ns = 1.00, x = 50 %\nBarmak et alFIG. 1. A comparison of the experimental MAE values of\nthe Fe xPt1\u0000xsamples studied by Barmak et al. in Ref. 5\n(open circles) and theoretical MAE values calculated using\nSKKR-CPA as follows, + : using the experimental lattice\nparameters for each sample, but assuming x= 0:5 and s= 1,\n\u000f: using the lattice parameters and the compositions xas\ngiven in the experiment, but keeping sconstant at 0.89, and\n\u0002: using the lattice parameters as well as the values of xand\nsas given in the experiment. Solid lines serve as guide for the\neyes.\nature induced e\u000bects. The results are shown in Fig. 2 for\nthe range of compositions 0 :4\u0014x\u00140:6. (Beyond this\nrange, the L1 0structure becomes unstable with respect\nto other phases.1,2) Our results are in good agreement\nwith the conclusion of Barmak et al5, inasmuch for any\ngiven degree of chemical order sthe MAE increases\nmonotonically with the Fe-content. However, even\nmaximally ordered Fe xPt1\u0000xalloys with x >50 cannot\nachieve the MAE of fully ordered Fe 50Pt50(3.31 meV\nper formula unit).\nIt should be noted that, at s= 0, the MAE becomes\nnegative. This is in contrast to Ref. 22, which reports\na vanishing MAE for completely disordered FePt under\nthe assumption of a cubic unit cell. The 'residual' neg-\native MAE we obtain in the case of complete chemical\ndisorder is, therefore, due to the tetragonality of the\nlattice (a6=c). For real samples, where the lattice\nparameters cannot be frozen while varying the chemical\norder and composition, in the case of complete chemical\ndisorder the unit cell is expected to become cubic,\nremoving thus this 'residual' MAE.\nIn order to elucidate the origin of the variation in the\nMAE with the composition and the chemical disorder,\nwe consider next the species-resolved contributions to the3\n−0.50.00.51.01.52.02.53.03.5\n0.0 0.2 0.4 0.6 0.8 1.0MAE per formula unit (meV)\nchemical order parameter, sFe60Pt40\nFe55Pt45\nFe51Pt49\nFe50Pt50\nFe49Pt51\nFe45Pt55\nFe40Pt60\nFIG. 2. The variation of the MAE of Fe xPt1\u0000xalloys as a\nfunction of the chemical order parameter sand the composi-\ntionx. Solid lines serve as guide for the eyes.\nMAE. The MAE per unit cell can be decomposed as\nK=rFeDFe\nFe+ (1\u0000rFe)DFe\nPt+ (1\u0000rPt)DPt\nFe+rPtDPt\nPt;\n(2)\nwhereD\f\n\r(\f;\r = Fe or Pt) denotes the MAE contri-\nbution from an atom of species \rwhen it is positioned\nin a nominal \flayer, i.e., within a layer, which in a\nperfectly ordered Fe 50Pt50alloy would contain only\natoms of species \f. For the cases of x= 0:4, 0.5 and 0.6,\nin Fig. 3 we show D\f\n\ras a function of the chemical order\nparameter, s. In completely disordered FePt ( s= 0),\nthe nominal Fe layers and the nominal Pt layers are\nidentical. Therefore, at s= 0, the Fe contributions\nin both layers are equal and take a small negative\nvalue fors= 0, which decreases in magnitude with\nincreasingxand practically vanishes at x= 0:6. The\nPt contributions, on the other hand, are nearly zero for\nall compositions xwhens= 0. As the chemical order s\nincreases, the Fe contribution in the nominal Fe layers\nrapidly increases up to about 1.8 meV, 2.0 meV and 2.2\nmeV ats= 0:8 forx= 0:4, 0.5 and 0.5, respectively. For\nthe fully ordered case, x= 0:5 ands= 1,DFe\nFeeven takes\nthe value of about 3.15 meV, close to the total value of\nthe MAE (3.31 meV). In contrast, the Fe contribution in\nnominal Pt layers decreases up to s'0:3, then slightly\nincreases and, for x\u00150:5, reaches a small positive value\n(<0:5 meV) at maximal chemical order. Remarkably,\nthe magnitude of the Pt contributions remain almost\nnegligible (<0:15 meV) over the whole range of chemical\norder.\nAs indicated in Fig. 3, the dominant contribution\nto the MAE is rFeDFe\nFe, see Eq. (2). It is, therefore,\nintuitive to replot Fig. 2 as a function of the fraction,\nrFe. This interpretation of the MAE is shown in Fig. 4\nfor di\u000berent compositions x. SincerFe=x+s\n2, it is\n−0.50.00.51.01.52.0\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8MAE per atom (meV)\nchemical order parameter, sDFeFe\nDPtFe\nDFePt\nDPtPt\n−0.50.00.51.01.52.02.53.03.5\n0.0 0.2 0.4 0.6 0.8 1.0MAE per atom (meV)\nchemical order parameter, sDFeFe\nDPtFe\nDFePt\nDPtPt\n−0.50.00.51.01.52.02.5\n0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8MAE per atom (meV)\nchemical order parameter, sDFeFe\nDPtFe\nDFePt\nDPtPtFIG. 3. Variation in the species-resolved MAE contributions\nagainst the chemical order parameter sfor compositions x=\n0:40 (upper panel), x= 0:50 (middle panel) and x= 0:60\n(lower panel). + : DFe\nFe, contribution of an Fe atom in nominal\nFe layers, \u0003:DFe\nPt, contribution of a Pt atom in nominal Fe\nlayers, \u0002:DPt\nFe, contribution of an Fe atom in nominal Pt\nlayers, and \u000f:DFe\nPt, contribution of a Pt atom in nominal Pt\nlayers. Solid lines serve as guide for the eyes.\nclear that the horizontal range of the curves in Fig. 2\nis halved and, more importantly, they are shifted to\nthe right by x. As a consequence, for a \fxed value\nofrFethe order of the curves with respect to xis\nreversed as compared the order of curves at a given sin\nFig. 2 . This opposite tendency becomes obvious when4\n−0.50.00.51.01.52.02.53.03.5\n0.4 0.5 0.6 0.7 0.8 0.9 1.0MAE per formula unit (meV)\nprobability of Fe occupancy on Fe sites, rFeFe60Pt40\nFe55Pt45\nFe51Pt49\nFe50Pt50\nFe49Pt51\nFe45Pt55\nFe40Pt60\nFIG. 4. The variation of the MAE of Fe xPt1\u0000xalloys as a\nfunction of the Fe concentration in the nominal Fe layers rFe\nand the overall Fe concentration x. Solid lines serve as guide\nfor the eyes.\nconsidering e.g. the case of rFe= 1 { Fe-rich Fe xPt1\u0000x\nwill exhibit completely Fe-\flled nominal Fe layers at a\nsmallersthan Fe 50Pt50, which requires s= 1 in order\nto exhibit completely \flled Fe layers. On the other\nhand, increasing disorder (i.e., decreasing s) drasticallyreduces the Fe contribution to the MAE in the nominal\nFe layer,DFe\nFe, and, consequently, the MAE of the system.\nIn conclusion, our calculations strongly support the\nconclusion of Barmak and co-workers in Ref. 5, showing\nthat, for a given degree of chemical order, the MAE\nincreases with the Fe concentration of Fe xPt1\u0000x, at\nleast within the range 0 :4\u0014x\u00140:6. This is due to\nthe strongly positive e\u000bect on the MAE of the degree of\nFe-\flling of the nominal Fe layers, rFe, which dominates\nthe variation in the MAE when varying the composition\nx, while keeping the chemical chemical disorder scon-\nstant. 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Phys. 93(2003) 453"    },    {        "title": "1304.6515v3.Spin_Resolution_and_Evidence_for_Superexchange_on_NiO_001__observed_by_Force_Microscopy.pdf",        "content": "arXiv:1304.6515v3  [cond-mat.mtrl-sci]  27 Jun 2013Spin Resolution and Evidence for Superexchange on NiO(001) observed by Force\nMicroscopy\nFlorian Pielmeier∗and Franz J. Giessibl\nInstitute of Experimental and Applied Physics, University of Regensburg, D-93053 Regensburg, Germany\n(Dated: March 10, 2021)\nThe spin order of the nickel oxide (001) surface is resolved, employing non-contact atomic force\nmicroscopy at 4.4 K using bulk Fe- and SmCo-tips mounted on a q Plus sensor that oscillates at sub-\n50pm amplitudes. The spin-dependent signal is hardly detec table with Fe-tips. In contrast, SmCo-\ntips yield a height contrast of 1 .35pm for Ni ions with opposite spins. SmCo tips even show a sma ll\nheight contrast on the O atoms of 0 .5pm within the 2x1 spin unit cell, pointing to the observatio n of\nsuperexchange. We attribute the increased signal-to-nois e ratio to the increased magnetocrystalline\nanisotropy energy of SmCo, which stabilizes the magnetic mo ment at the apex. Atomic force\nspectroscopy on the Ni ↑, Ni↓and O lattice site reveals a magnitude of the exchange energy of\nmerely 1meV at the closest accessible distance with an expon ential decay length of λexc= 18pm.\nHigh resolution non-contact Atomic Force Microscopy\n(nc-AFM) detects short-range chemical interactions be-\ntweenthe foremosttipatomsandsampleatoms, enabling\natomic resolution imaging and quantitative force mea-\nsurements [1–3]. By equipping an AFM with a magnetic\nprobe tip, the sample magnetization can be studied [4]\nat a resolution of several tens of nanometers [5]. Wiesen-\ndanger et al. estimated in 1990, that magnetic exchange\ninteractions that occur in spin-polarized scanning tun-\nneling microscopy can amount to about one pN per ˚A2\nof tip area [6]. Several calculations predicted even larger\nmagnitudes of exchange forces [7–11]. Once atomic reso-\nlution by AFM in ultrahigh vacuum (UHV) became fea-\nsible, extended efforts to detect exchange interactions by\nnc-AFM on NiO at T= 4K and 300K were conducted\n[12–15], initially without success. In 2007, Kaiser et al.\nproved the feasibility of Magnetic Exchange Force Mi-\ncroscopy (MExFM) by imaging the (2 ×1) spin pattern\non the antiferromagnetic insulator NiO [16]. The experi-\nment wasconducted atliquid heliumtemperatures, using\nan iron coated silicon cantilever where the magnetization\nof the tip was stabilized by applying a 5T magnetic field\n[16–18]. The exchange interaction between tip and sam-\nple is qualitatively described by the Heisenberg model,\nH=−J12/vectorS1·/vectorS2, where J12is the exchange coupling\nconstant. For 3 dtransition metals a large magnetic mo-\nment of the foremost tip atom is desirable for achieving\na high signal-to-noise-ratio [18].\nIn this Letter, we report on the detection of spin con-\ntrast on the NiO(001) surface without applying an ex-\nternal magnetic field. We analyse the dependence of the\ncontrast for Fe- and SmCo-tips. Both tips reveal the\nantiferromagnetic structure of NiO(001), but SmCo-tips\nyielda3-10timeshigherspincontrastthanFe-tips. With\nthe magnetic moments of µFe= 2.2µB,µCo= 1.7µB\nandµSm= 0.4µB[19], this finding shows that µis\nnot the only parameter that determines spin contrast in\nMExFM. We attribute the increased contrast in case of\nSmCo-tips to the higher magnetocrystalline anisotropy\nenergy (MAE) compared to Fe, which stabilizes the spin\nO\nNi\n[100] [010][001]\nFIG. 1. (Color online). Left: crystal structure and magneti c\nstructure of nickel oxide (see text). Right: Slightly low pa ss\nfiltered [21] MExFM topography image of NiO(001), show-\ning the (2 ×1) unit cell of the surface. Imaging parame-\nters: SmCo-tip, k= 2425N/m, f0= 39.761kHz, A= 36pm,\nQ= 31,000 and bias voltage Ubias= 0.06V.\norientation of the front atom. Furthermore we present\n∆f(z)-curves acquired with a SmCo-tip and evaluate the\nmagnitude of the exchange interaction on NiO. We find\nthat its magnitude is only about 1/50 of the exchange\ninteraction between Fe-tips and an antiferromagnetically\nordered Fe monolayer on W(001) [20].\nForces are measured by frequency modulation atomic\nforce microscopy [22], where the force sensor with stiff-\nnessk, eigenfrequency f0and quality factor Qoscillates\nat a constant amplitude Aand is subject to a frequency\nshift ∆f=f−f0that is directly related to the averaged\ntip-sample force gradient via /an}bracketle{tkts/an}bracketri}ht=2k\nf0∆f[23]. Forces\nhave been derived by deconvolving the frequency shift\n∆fwith the Sader-Jarvis-method [24]. Optimal sensi-\ntivity to short-range forces is ensured by operating the\nqPlus force sensor at amplitudes below 100pm [25–27].\nThe sensor can be equipped with any tip material, in a\nprevious study on NiO, cobalt was used due to its lower\nchemical reactivity [15, 28]. Iron tips were electrochem-\nically etched from a high purity iron wire (99 .99 + %),\nwhereas a sharp piece of a SmCo permanent magnet was\nglued to the qPlus sensor to obtain a SmCo-tip [29]. Be-2\nc)b)\n0.00 0.10\n 0.02 0.03\na)a)\nFT intensity (arb.u.)\ne)\nd)\nf)0.0 0.2 0.4 0.6 0.8-505z (pm)\ndistance along [100] direction (nm)\n0.0 0.5 1.0 1.5 2.0 2.5024z (pm)\ndistance along [110] direction (nm)0.0 0.2 0.4 0.6 0.8-0.50.00.5z (pm)\ndistance along [100] direction (nm)\nFIG. 2. (Color online). MExFM data acquired with Fe\n(left) and SmCo (right) tips. a) Low-pass filtered (2 ×2)\nunit cell averaged topography image (2 ×2nm2) showing the\nrow-wise contrast, for image processing details see [32]. L ine\nprofile in b) shows a height difference between the local max-\nima of 0 .1pm, the average atomic corrugation is 1 .1pm. c)\nFourier spectra of the raw data corresponding to a), in nor-\nmal and high contrast (right). d) Low-pass filtered topog-\nraphy data (2 .7×2.7nm2) acquired with a SmCo-tip. Each\nsecond Ni row appears darker. e) Line profile, revealing a dif -\nference between Ni sites of up to 1 .35pm. The height of the\noxygen sites within one magnetic unit cell varies by 0 .5pm.\nf) Line profile showing the periodicity of the height varia-\ntions on oxygen sites. Parameters for Fe- (SmCo-) sensor:\nk= 1800N/m (2425N/m), f0= 59.369kHz (39 .761kHz),\nA= 50pm (36pm), Q= 1,362,000 (31,000) and Ubias=\n6.8V (0.06V).\nfore the tips where introduced into the UHV system,\nthey were sharpened by focussed ion beam (FIB) etch-\ning. The native oxide layer of bulk metal tips is removed\nby field evaporation [30] in UHV, afterwards the sensors\nare transfered in-situto the microscope within 15 min-\nutes. The measurements were carried out on an Omi-\ncron LT/qPlus system in UHV ( p≤10−10mbar) and at\na temperature of 4 .4K.\nThe structure of the antiferromagnetic insulator nickel\noxide is shown in Fig. 1. NiO exhibits a rock salt\nstructure with a lattice constant of a= 417pm. Nickel\natoms in {111}planes are coupled ferromagnetically and\nneighbouringNi planesarecoupled antiferromagnetically\nvia superexchange mediated by the oxygen atoms. This\nleads to an antiferromagnetic structure at the (001) sur-\nface with alternating spin orientations of nickel atoms\nalong the /an}bracketle{t110/an}bracketri}htdirection. The NiO crystal (SurfaceNet,Rheine, Germany)wascleavedin-situtoobtaincleanand\nflat terraces up to 100nm in width. Cleaved NiO sur-\nfaces exhibit a bulk-terminated orientation of magnetic\nmoments [31]. On the right in Fig. 1 a model of the sur-\nface atomic and magnetic structure is superimposed to a\nhigh-resolution MExFM image aquired with a SmCo-tip,\nshowing alternating rows of oppositely aligned Ni atoms\nalong the [1 ¯10] direction. When imaging with a metallic\ntip, O atoms usually appear as maxima in constant fre-\nquency shift mode [10], and the minima refer to Ni sites.\nThe difference in apparent height between the two nickel\nsites is due to the exchange interaction which adds to the\nchemical interaction depending on the spin alignment of\nthe surfaceNi atoms relativeto the tip moment. Adirect\nexchange mechanism has been predicted for an Fe atom\nprobing the NiO surface [11].\nAs in all successful MExFM experiments on NiO so\nfar [16–18], we used Fe-tips in our initial experiments.\nHere, we measure exchange contrast on NiO using Fe\ntipswithoutan external magnetic field, yielding a very\nweak exchange contrast that extends over a narrow dis-\ntance range of about 10 −20pm [32]. The small width of\nthe distance range where exchange forces are detectable\nindicates that the stability of the spin orientation of the\ntip apex atom is easily altered by increasing tip-sample\ninteraction forces. Locally, the stability of the spin orien-\ntation is governed by the directional dependent magne-\ntocrystalline anisotropy (MA). Hence, the tip cluster ori-\nentation my effect the ontrast in MExFM experiments.\nThe magnetic easy axis of bulk bcc-iron is parallel to\n/an}bracketle{t100/an}bracketri}htdirections [33]. As a next step we use a tip with\na known tip cluster orientation, achieved by probing the\ntip apex with a CO molecule adsorbed on Cu(111) [34].\nAs both Fe and W are bcc materials, we observe the\nsame symmetries for Fe tips [32] as we did for W tips\nin [34]. After the Fe-tip was characterized by the CO-\nmethod, the Cu sample is removed and the cleaved NiO\nsample is introduced into the microscope. After carefully\napproaching the NiO(001) surface the metallic nature of\nthe tip apex was confirmed by ∆ f(U) curves, where the\nabsenceofchargingeffectsortunnelingtolocalizedstates\nis an indication for a metallic tip apex [32, 35]. Electro-\nstatic forces were minimized by applying a bias voltage\nto the sample.\nFigure 2(a) shows a low-pass filtered, unit cell aver-\naged topographic image acquired with a Fe-tip, which is\noriented along a /an}bracketle{t100/an}bracketri}htdirection [32]. The image was ac-\nquired in constant height mode and the frequency shift\n(∆f) was converted to topography, see [32]. A 2 ×2\nunit cell was used to avoid superimposing the data with\nthe expected 2 ×1 magnetic unit cell. The additional\nmodulation of the atomic contrast can be identified, as a\nrow-wise changing apparent height of the maxima. The\ntopographyline profilein2(b) showsadifferencebetween\ntwo local maxima of only 0 .1pm, the averageatomic cor-\nrugation is 1 .1pm. In Fig. 2(c), two Fourier spectra of3\nthe unfiltered raw data corresponding to a) are shown.\nTwo additional peaks (solid white boxes) appear at half\nthe inverse lattice vector along a line from the lower left\nto the upper right corner. There are two possible reasons\nfor the appearance of larger spin modulation on top of\nthe maxima compared to minima, either the Ni sites are\nimaged as maxima, or due to superexchange on O sites\nwhich might be stronger in this distance regime.\nAlthough the spin contrast using an oriented Fe-tip is\nlarger on maxima than on minima in Fig. 2 a), the mag-\nnitude of the spin contrast is in good agreement with our\ninitial experiments with uncharacterized iron tips, where\nit reached up to 0 .4pm on top of a small chemical in-\nteraction causing 1 .6pm corrugation (Figs. 1, 2 in [32]).\nMExFM with Fe-tips only yields a weak spin contrast\noverathin distancerangewherechemicalforcesaresmall\nand the spin-dependent signal is lost when the tip height\ndeviates from the ideal height by more than ±15pm.\nEven though the observation of low spin contrast can\nbe due to an unfavorable alignment of tip and sample\nspins, Fe tips systematically yield low spin contrast as we\nperformed several experiments with different Fe-tips and\ninvestigated different spots of a given NiO sample. The\ninstability of the spin orientation of the apex atoms upon\nincreased chemical bonding forces between tip and sam-\nple indicates that the spin orientation of the apex atoms\nrotates at closer distances to maximize the chemical in-\nteraction and that the MA in Fe is not high enough to\nstabilize the magneticmoment ofthe frontatom. Indeed,\nthe magnetocrystalline anisotropy energy (MAE) for bcc\niron is only 2 .4µeV/atom, whereas hcp Co already has\na MAE of 45 µeV/atom [33]. Materials with even higher\nMAEs are permanent magnets like samarium-cobalt al-\nloys, their MAE is about 20-40 times larger than hcp Co\nand hence about a factor of 500 higher than the MAE of\nbulk bcc iron [36, 37]. Using such high MAE materials\nas tips in MExFM experiments should lead to a higher\nstability of the spin orientation of the tip apex. To test\nthis hypothesis, the MExFM measurements on NiO were\nrepeated with bulk SmCo-tips. The results are shown in\nFig. 2(d)-(f), the additional modulation is clearly appar-\nent in the low-pass filtered topography image d) of the\nNiO(001)surface. Alineprofilefromthelowpassfiltered\nimage is displayed in e), the average atomic corrugation\nis 12.9pm. The difference between the two local minima\ndue to exchange interaction is 1 .35pm (dark blue shaded\nbar). The chemical and spin resolution is independent of\nthe scan direction [32].\nInterestingly, a small height difference of 0 .5pm (light\nblueshadedbar)betweentheoxygensites(localmaxima)\ncan be identified. These height variations show the same\nperiodicity as the height variation on Ni sites, Fig. 2(f).\nAnadditionalmodulationontopoftheoxygenatomshas\nalready been discussed in [17]. There, it was attributed\nto a magnetic double tip, mainly because the line profile\nshowed an asymmetric, wedgelike shape of the atoms.Furthermore a direct exchange mechanism between the\nmagnetic moment of the oxygen and the tip moment is\nunlikely asit is about an orderofmagnitude smallerthan\nthe moment onthe nickelsites [11, 38]. As the line profile\nin Fig. 2(e) has an overall sinusoidal shape, we believe\nthat the height difference on top of the oxygen sites is\nnot due to a magnetic double tip but rather caused by\nan indirect exchangemechanism between the tip moment\nandthesecondlayernickelatomsunderneaththeoxygen.\nTo evaluate the distance dependence of the atomic and\nexchange interactions, ∆ f(z) curves with the SmCo-tip\nfrom Fig. 2(d) were acquired on three different sites,\nwhich are marked in the insets of figures 3a) and c).\nNamely, O and the two different Ni sites, which are de-\nnoted as Ni ↓and Ni↑for the following discussion. The\nvalue of z= 0 indicates the point of closest approach\nin the ∆ f(z) curves in Fig. 3(c), whereas the curves\nin Fig. 3(a) start at z= 10pm. The image in Fig.\n2(d) was also acquired at z= 10pm, marked by the\ndashed red lines in Fig. 3b)+d). The difference in fre-\nquency shift between the O and the average of the Ni\nsites ∆fO−Ni= ∆fO−∆fNi=(Ni↓−Ni↑)/2= 6.5Hz at\nz= 10pm [3(a)]. Open circles in Fig. 3(b) depict the\ncorresponding force values. Fitting an exponentially de-\ncaying function we obtain a value of FO−Ni=−65pN at\nthe imaging distance and a decay length λNiO= 30pm.\nThe difference between Ni ↓and Ni↑atz= 10pm is\n∆fNi↓−Ni↑= ∆fNi↓−∆fNi↑= 0.93Hz [Fig. 3(c)]. As\nthe Ni sites are chemically equivalent, the difference is\npurely due to short range magnetic exchange interac-\ntions. The exchange force is shown in Fig. 3(d), indi-\ncatingFNi↓−Ni↑=−5.4pN atz= 10pm, and a decay\nlength of λExc= 18pm. The difference between chem-\nical and exchange interaction on NiO with SmCo-tips\nis given by the ratio of FO−Ni/FNi↓−Ni↑=−65pN/−\n5.4pN = 12. Due to the different decay lengths λNiO\nandλExcfor the chemical and exchange interactions the\ndifference in energy is even larger, obtaining a factor of\nEO−Ni/ENi↓−Ni↑=−12meV/−0.6meV = 20. Obvi-\nously, the main challenge in obtaining spin resolution\non NiO is to discriminate the exchange from the chem-\nical interactions. Theoretical predictions, where an Fe\natom probes the NiO surface, find values of the chemical\nforces in the range of nN and exchange forces on the or-\nder of 0.1nN [11]. The experimental exchange force on\nNiO(001) is about 10pN, an order of magnitude smaller\nand even the chemical forces are below 100pN in the\nexperimental distance range. Note, although the SmCo\ndata is not directly comparable with these Fe calcula-\ntions, the smaller contrast we found for Fe-tips implies\nthat the exchange forces are even smaller in this case.\nAs the exchangeforceand energy decreasemonotonically\nwith decreasing tip-sample distance, there is no indica-\ntion for a change in the magnetic coupling, as predicted\nfor Fe-tips, within the resolution of our measurements\n[11]. NiO is a strongly correlated electron system, which4\n0 100 200 300 400 500-70-60-50-40-30-20-100FO - Ni(pN)\nz (pm)-12-10-8-6-4-20EO - Ni(meV)0 20 40 60 80 100 120-1.0-0.8-0.6-0.4-0.20.0/c68fNi/c175- Ni/c173(Hz)\nz (pm)-40-30-20/c68f (Hz)/c68fNi/c173\n/c68fNi/c175a)\nb)c)\nd)\nλNiO= 30 pmλExc= 18 pm\n0 20 40 60 80 100 120-10-50FExc(pN)\nz (pm)-1.0-0.50.0EExc(meV)Ni Ni/c175 /c173O\n0 100 200 300 400 500-7-6-5-4-3-2-10/c68fO - Ni(Hz)\nz (pm)-40-30-20-100/c68f (Hz)/c68fNi = (Ni /c175/c43Ni/c173)/2\n/c68fO\nNi Ni/c175 /c173\nFIG. 3. (Color online). a) ∆ f(z)-spectra on Ni and O sites (inset) and difference between the curves (blue). Starting positions\nof spectra are indicated by arrows in a) and c), while the dash ed red lines in b) and d) indicate the distance z= 10pm where\nFig. 2d) was imaged. b) Force difference between the tip and Ni /O sites, including an exponential fit ∝exp(−z/λ) with\nλ= 30pm (dashed blue line). Integration of the forces yields t he energy difference between Ni and O sites, which reaches\n12meV (inset). c) ∆ f(z)-curves on Ni ↓and Ni↑sites (inset) and difference (green). The resulting exchang e forceFExcin d)\nhas an even smaller decay length than the chemical interacti on of only λExc= 18pm.\nmakes it in general challenging for ab-initio calculations.\nTherefore our measurement of the short range exchange\ninteraction on NiO(001) can serve as input for future cal-\nculations.\nWe conclude that the main challenge of obtaining\nMExFMonNiOismagnetictipstability. Withoutapply-\ning a magnetic field, the magnitude of the exchange con-\ntrast on NiO using Fe-tips is much smaller (100 −400fm)\nthan when applying a field of 5T [16–18]. However, con-\ntrast with a similar magnitude (1 .35pm) can be achieved\nwhen using SmCo-tips, suggesting that the increased\nMAE of SmCo helps to stabilize the spin at the tip apex.\nThe MAE of SmCo is approximately1meV per atom, al-\nmost equalto the Zeemanenergy EZ=gµBB= 0.6meV\nfor a g-factor of 2 .2 for Fe and B= 5T [16]. Our study\nis a step towards a more detailed understanding of the\ninteraction mechanism in magnetic exchange force mi-\ncroscopy on insulating surfaces. Based on these find-\nings, we propose materials with high MAE to be bestsuited for MExFM studies. This is of particular import\nfor the study of antiferromagnetic pinning layers in ex-\nchange bias coupled systems.\nThe authors thank F. Oberhuber for FIB tip etch-\ning, G. 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H1111 Budapest, Hungary and\n3Condensed Matter Research Group of Hungarian Academy of Sciences,\nBudapest University of Technology and Economics, Budafoki \u0013 ut 8., H-1111 Budapest, Hungary\n(Dated: July 17, 2018)\nWe perform fully relativistic \frst principles calculations of the exchange interactions and the mag-\nnetocrystalline anisotropy energy (MAE) in an Fe/FePt/Fe sandwich system in order to elucidate\nhow the presence of Fe/FePt (soft/hard magnetic) interfaces impacts on the magnetic properties\nof Fe/FePt/Fe multilayers. Throughout our study we make comparisons between a geometrically\nunrelaxed system and a geometrically relaxed system. We observe that the Fe layer at the Fe/FePt\ninterface plays a crucial role inasmuch its (isotropic) exchange coupling to the soft (Fe) phase of the\nsystem is substantially reduced. Moreover, this interfacial Fe layer has a substantial impact on the\nMAE of the system. We show that the MAE of the FePt slab, including the contribution from the\nFe/FePt interface, is dominated by anisotropic inter-site exchange interactions. Our calculations\nindicate that the change in the MAE of the FePt slab with respect to the corresponding bulk value\nis negative, i.e., the presence of Fe/FePt interfaces appears to reduce the perpendicular MAE of the\nFe/FePt/Fe system. However, for the relaxed system, this reduction is marginal. It is also shown\nthat the relaxed system exhibits a reduced interfacial exchange. Using a simple linear chain model\nwe demonstrate that the reduced exchange leads to a discontinuity in the magnetisation structure\nat the interface.\nPACS numbers: 75.30.Gw 75.50.Ss 71.15.Mb 71.15.Rf\nI. INTRODUCTION\nExchange-coupled soft/hard composite magnetic sys-\ntems are of signi\fcant interest for their potential ap-\nplication in many di\u000berent \felds of technology such as\nmagnetic recording media [1], permanent magnets [2]\nand magnetic microactuators [3]. A wealth of di\u000berent\nsoft/hard materials has been investigated in the liter-\nature (see, e.g., [4{10]). Experimentally, the Fe/FePt\nsystem is a highly suitable system for studying the fun-\ndamental properties of nano-composite magnetic systems\nas the properties are relatively easy to control [11]. More-\nover, due to the high magnetisation of the saturated\n\u000b-Fe phase and the large magnetocrystalline anisotropy\nenergy (MAE) of the FePt L10phase, Fe/FePt bilay-\ners are considered an ideal structure for exchange spring\nbehaviour [12] and for application in exchange-coupled\n(ECC) magnetic recording media. For ECC applications,\nthe (soft) Fe phase, through its exchange interaction with\nthe (hard) FePt phase, would act as a \\lever\", reducing\nthe write \feld. Meanwhile, the thermal stability of the\nwritten information would be ensured by the large MAE\nofL10FePt. Thus, in order to realise such devices, the\nMAE of FePt needs to be maintained (if possible, en-\nhanced). The e\u000bect of the Fe/FePt interface on the FePt\nMAE is, therefore, a very important aspect.\nThe aim of the present work is to investigate in detail\nthe e\u000bect of the Fe/FePt interface on the exchange cou-\npling and the MAE of an Fe/FePt/Fe system by means\nof \frst-principles calculations. We compare the results of\na geometrically relaxed Fe/FePt/Fe system to the corre-\nsponding results for an unrelaxed such system. The latteris similar to the system studied by Sabiryanov and Jaswal\n[13]. We use CASTEP [14{16] to obtain the relaxed ionic\ncoordinates of an Fe/FePt/Fe system. We then employ\nthe fully relativistic screened Korringa-Kohn-Rostoker\n(SKKR) method [17] to calculate tensorial exchange in-\nteractions and the layer-resolved contributions to the\nMAE of the relaxed and unrelaxed Fe/FePt/Fe struc-\ntures. Moreover, we evaluate the change in the FePt\nMAE induced by the presence of the Fe/FePt interfaces.\nSuch a study is not only important from the point\nof view of understanding the properties of this nano-\ncomposite, but the site-resolved information is also\ncentral to the development and parameterisation of\nlocalised-spin models. This strategy has recently been re-\nalized to study an IrMn 3/Co(111) interface. [18] The ex-\nchange interactions and the on-site magnetic anisotropy\nconstants have been calculated in the antiferromagnetic\nand ferromagnetic parts of the system, as well as at\nthe interface between them, and then used in atomistic\nspin-dynamics simulations to investigate the exchange\nbias e\u000bect. [19] In particular, it was found that the ex-\nchange bias e\u000bect in this system is mainly governed by\nlarge Dzyaloshinskii-Moriya (DM) interactions [20, 21]\nbetween the Mn and Co atoms at the interface.\nWe thus consider the implications of our calculated\nab-initio parameters in a multiscale modelling approach.\nHere one maps ab-initio information onto a \fxed-spin\natomistic model to allow calculations of thermodynamic\nquantities and magnetization dynamics. Using a simple\nmapping onto a linear chain we study domain structures\nat the FePt/Fe interface using the ab-initio parameters.\nImportantly, the ab-initio calculations show that the re-arXiv:1306.3642v1  [cond-mat.mtrl-sci]  16 Jun 20132\nlaxed system exhibits a reduced interfacial exchange cou-\npling. Mapping this information onto the linear chain\nmodel shows that this reduction gives rise to a disconti-\nnuity in the magnetization at the FePt/Fe interface. The\nimplications for the exchange spring e\u000bect are considered.\nThe functionality of magnetic materials increasingly re-\nlies on structural design at the nanoscale, the exchange\nspring phenomenon, its use in permanent magnets and\nrecording media being an excellent example. Mesoscopic\ncalculations often assume bulk exchange coupling across\ninterfaces, which clearly may not be the case, and will cer-\ntainly depend on the material properties in a way which\ncan only be elucidated by electronic structure calcula-\ntions.\nII. DETAILS OF THE CALCULATIONS\nFirst we describe the geometric structure of the\nFe/FePt/Fe sandwich system that we have chosen for\nour investigation. The SKKR method requires the\nsystem to be considered in terms of an interlayer region\n(regionI) positioned between two semi-in\fnite bulk\nregions. For region I, we considered the following\nsequence of atomic layers (AL) as shown in Table I: 7 Fe\nAL + 17 Pt/Fe/\u0001\u0001\u0001/Fe/Pt AL + 7 Fe AL, enclosed by\ntwo semi-in\fnite bulk Fe systems.\nUsing the layer sequence in Table I, we investigated\nthe following two systems:\nAA geometrically unrelaxed system with an over-\nall two-dimensional lattice parameter, a2D=\naFePt=p\n2\u00192:723\u0017A, whereaFePt = 3:852\u0017A is\nthe experimental in-plane lattice parameter of the\nL10lattice of FePt. Note that 2 :723\u0017A is within\n5 % of the experimental lattice parameter of bcc\nFe,a(exp)\nFe = 2:87\u0017A. For the FePt part of the sys-\ntem we used the experimentally measured ratio of\ncFePt=aFePt = 0:964, while for the bcc Fe part,\ncFe=a2D. At the interface between the Fe and\nFePt parts of the system, i.e., between AL's \u00009\nand\u000010, as well as between AL's 9 and 10 (see\nTable I), we set the interlayer separation to\ncinterface =cFe+cFePt\n4:\nBThe interlayer separations were relaxed using\nCASTEP, keeping a2D=a(LDA)\nFe , wherea(LDA)\nFe is\nthe bulk lattice parameter for Fe obtained using\nLocal Density Approximation (LDA) in CASTEP,\n2:659\u0017A. The resulting relaxed structure was then\nisotropically scaled up to the experimental FePt\nlattice parameter a2D= 2:723\u0017A in order to en-\nable a direct comparison with the results for sys-\ntemA. It should be noted here that the CASTEP\ngeometry relaxation yields an Fe region which isslightly tetragonal (rather than cubic), with the ra-\ntiocFe=aFe\u00191:06. As this tetragonalisation must\nbe due to the presence of the FePt slab, the relaxed\ngeometry of system Bcorresponds more closely to\na repeated multilayer structure.\nWe should mention that we also investigated an\nunrelaxed sandwich system with a layout of (1 \u00022)\nFe/Pt AL + 9 Fe AL + 17 Pt/Fe/ \u0001\u0001\u0001/Fe/Pt AL +\n9 Fe AL + (1\u00022) Pt/Fe AL enclosed by FePt bulk\n(system C). Since the considered Fe and FePt layers are\nquite thick, as expected, the magnetic properties (spin\nmoments, exchange interactions and MAE) of system C\nturned out very similar to those of system A. In this\nstudy, we used system Conly for calibrating the change\nin the MAE of the FePt slab with respect to the MAE\nof bulk FePt.\nFor system B, after specifying the vertical coordinate\nziof each atomic layer ifrom the CASTEP geometry\nrelaxation, we needed to determine the corresponding\natomic volumes, Vi. Here the only strict requirement is\nthat the sum of atomic volumes within region Ishould\nbe equal to the total lattice volume of region I, while\nthe choice of the individual atomic volumes is somewhat\narbitrary. As a simple choice, we related the atomic\nvolume of each atom in layer i,Vi, to the layer positions\nfzigasVi=a2\n2D(zi+1\u0000zi\u00001)=2. For Fe or FePt bulk,\nthis construction trivially retains the corresponding bulk\natomic volumes. For the case of system B, in Fig. 1 the\ninterlayer distances, \u0001 zi=zi+1\u0000zi, and the radii of\nthe atomic spheres, Si(de\fned through Vi=4\u0019\n3S3\ni), are\ndepicted according to the above construction.\nFor each of the systems AandB, we performed self-\nconsistent calculations by means of the SKKR method.\nWe used the Local Spin-Density Approximation (LSDA)\nof the Density Functional Theory (DFT) as parameter-\nized by Vosko et al. [22], with e\u000bective potentials and\n\felds treated within the atomic sphere approximation\n(ASA). The self-consistent calculations were performed\nwithin the scalar-relativistic approximation and an\nangular momentum cut-o\u000b `max= 3.\nThe magnetocrystalline anisotropy energy (MAE) was\nthen evaluated in terms of the fully relativistic SKKR\nmethod within the magnetic force theorem [23], in which\nthe total energy of the system can be replaced by the\nsingle-particle (band) energy. Moreover, we employed\nthe torque method [24], making use of the fact that, for\na uniaxial system, the MAE, K, can be calculated up to\nsecond order in spin-orbit coupling as\nK=E(\u0012= 90\u000e)\u0000E(\u0012= 0\u000e) =dE\nd\u0012\f\f\f\f\n\u0012=45\u000e;(1)\nwhere\u0012denotes the angle of the spin-polarization with\nrespect to the [001] direction of the FePt lattice. Within3\n\u0001\u0001\u0001-10 -9 -8 -7 -6 \u0001\u0001\u0001 -1 0 1\u0001\u0001\u0001 6 7 8 9 10 \u0001\u0001\u0001\n\u0001\u0001\u0001 FeFePt Fe Pt\u0001\u0001\u0001Fe Pt Fe\u0001\u0001\u0001Pt Fe Pt FeFe\u0001\u0001\u0001\nTABLE I. The layout of the Fe/FePt/Fe structure enclosed by Fe bulk. Underlined chemical symbols refer to the Fe (soft\nmagnet) part and bold face chemical symbols refer to the FePt (hard magnet) part of the system.\n1.351.401.451.501.551.601.651.701.751.801.85\n-25 -20 -15 -10 -5 0 5 10 15 20 25∆zi(˚A)\natomic layers, i\nSKKR\nCASTEP\n0.500.510.520.530.540.55\n-25 -20 -15 -10 -5 0 5 10 15 20 25Si/a2D\natomic layers, i\nFIG. 1. Top: Interlayer spacings, \u0001 zi, and bottom: radii of\natomic spheres, Sias used for system B. Red + represent\nthe interlayer spacings obtained from CASTEP, while black\n\u000frepresent the interlayer separations and the radii of atomic\nspheres as used in the SKKR calculations. Note that for the\nSKKR calculations the corresponding quantities are constant\nfor layersjij>16. Solid lines serve as guides for the eye.\nthe KKR formalism, Kcan be decomposed into layer-\nresolved contributions, Ki,\nK=X\niKi: (2)\nFor more details on the torque method within the\nKKR method see Ref. [25]. We note that due to the\ntwo-dimensional translational symmetry of the systems,\nthe MAE should be related to a 2D unit cell, therefore,\nin the following the index iin Eq. (2) is used to labelatomic layers.\nHaving evaluated the layer-resolved contributions to\nthe MAE for each Fe/FePt/Fe system, we considered\nnext the potential mapping of such contributions to a\nlocalised-spin model. Supposing that the electronic en-\nergy of a uniaxial magnetic system can be mapped into\na generalised Heisenberg model,\nH=\u00001\n2X\ni6=j~SiJij~Sj\u0000X\nidi\u0010\n~Si\u0001~ e\u00112\n; (3)\nwhere~Sirepresents a classical spin, i.e., a unit vector\nalong the direction of the magnetic moment at site i.\nThe \frst term stands for the exchange contribution to the\nenergy, with Jijdenoting the tensorial exchange interac-\ntion, and the second term denotes the on-site anisotropy,\nwith the anisotropy constant diand the easy magnetic\ndirection~ e. The exchange interaction matrix Jijcan\nfurther be decomposed into three terms [26],\nJij=JijI+JS\nij+JA\nij; (4)\nwithJij=1\n3TrJijthe isotropic exchange interaction,\nJS\nij=1\n2(Jij+JT\nij)\u0000Jiso\nijIthe traceless symmetric\npart of the exchange tensor and JA\nij=1\n2(Jij\u0000JT\nij) the\nantisymmetric part of the exchange tensor.\nWithin the spin model, Eq. (3), the MAE of a uniaxial\nferromagnetic system can be cast into on-site and inter-\nsite parts,\nK=Kon\u0000site+Kinter\u0000site; (5)\nwhere\nKon\u0000site=X\nidi; (6)\nand\nKinter\u0000site=\u00001\n2X\ni6=j\u0000\nJxx\nij\u0000Jzz\nij\u0001\n: (7)\nDe\fning, thus, the layer-resolved inter-site anisotropy as\nKi;inter\u0000site=\u00001\n2X\nj(6=i)\u0000\nJxx\nij\u0000Jzz\nij\u0001\n; (8)4\nthe layer-resolved MAE in Eq. (2) can be compared\nwithin the spin model to\nKi=di+Ki;inter\u0000site: (9)\nThe exchange interaction matrices are calculated using\nthe relativistic torque method as described in Ref. [26].\nThe sum in Eq. (9) over jcan be cast into sums over\natomic layers and over sites within atomic layers. In par-\nticular, the latter one su\u000bers from convergence problems\nsince Jijdecays, at best, as 1 =R3\nij, whereRijdenotes\nthe distance between atoms iandj. For this reason the\ncorresponding sum was transformed into an integral in\nk-space, the convergence of which could easily be con-\ntrolled, for details see Ref. [26].\nIII. RESULTS\nA. Local spin moments\nThe calculated atomic spin moments are plotted in\nFig. 2, displaying a fairly similar picture for the two\nsystems AandB. In the interior of the FePt slab the\nmoments,m(FePt)\nFe = 2:86\u0016Bandm(FePt)\nPt = 0:32\u0016B,\nare close to their bulk values in FePt. Moreover,\nthe Fe moments approach their bulk Fe value at the\nedges of region I. Nevertheless, we observe that the\nbulk Fe spin moment is slightly enhanced in system\nB,m(Fe;B)\nFe = 2:07\u0016B, as compared to system A,\nm(Fe;A)\nFe = 1:97\u0016B. This di\u000berence is, most likely, due\nto the slight tetragonality of the Fe unit cell along the\nzdirection (and the associated increase in volume) as\ndiscussed above for system B. An apparent di\u000berence\nbetween the spin moments of systems AandBoccurs at\nthe interface: although the Fe moments at the interface,\nm\u00069, are slightly increased in system A, for system B\nthe enhancement of these moments is more pronounced.\nMoreover, unlike in system A, in system Bthe spin\nmoments in layer 10, m\u000610, are also enhanced. This\nmeans that, as expected, the transition of the moments\nfrom bulk Fe to bulk FePt is smoother in the relaxed case.\nB. E\u000bective exchange parameters\nIn order to characterise the strength of the isotropic\nexchange interactions in a magnetic system, one often\nde\fnes a site-resolved e\u000bective exchange parameter, Ji,\nde\fned for a given site ias\nJi=X\nj(6=i)Jij: (10)\nFor a 2D translationally invariant system, Jimust\nof course be identical for each site in a given layer.\nTherefore, in the following, idenotes the layer index.\n0.00.51.01.52.02.53.0\n-15 -10 -5 0 5 10 15spin moment ( µB)\natomic layers, iA\nFe\nPt\n0.00.51.01.52.02.53.0\n-15 -10 -5 0 5 10 15spin moment ( µB)\natomic layer, iB\nFe\nPtFIG. 2. Calculated layer-resolved spin moments (green \u0003: Fe,\nblue\u0004: Pt) for systems AandB. Solid lines serve as guides\nfor the eye.\nFor systems AandB, we calculated Jiby considering\nall neighbours within a distance of seven a2D, which en-\nsured a reliable convergence of the sum in Eq. (10). The\ncalculated layer-resolved e\u000bective exchange parameters\nare plotted in Fig. 3 for systems AandB.\nWithin the FePt slab, the e\u000bective exchange in-\nteractions of systems AandBexhibit very similar\nlayer-resolved behaviours. The value of Jifor the Fe\nlayers is about 150 meV in the centre of the FePt slab\nand is slightly enhanced at the edges of the slab (i.e.,\ntowards layers i=\u00067). This is mainly a consequence of\nan enhancement of the ferromagnetic, nearest-neighbour\n(NN) intra-layer Fe-Fe interaction towards the outer\nlayers of the FePt slab. The e\u000bective exchange parameter\nof about 40 meV observed in the Pt layers stems mainly\nfrom the strongly ferromagnetic nearest-neighbour Fe-Pt\ninteractions.\nThe (soft) Fe part of the system is characterised by\nmuch larger e\u000bective exchange, Ji\u0018260 meV. This\nhigh value of the e\u000bective exchange parameter is a\nhighly important property of the soft magnet part in5\n050100150200250300\n-15 -10 -5 0 5 10 15Ji(meV)\natomic layers, iA\nFe\nPt\n050100150200250300\n-15 -10 -5 0 5 10 15Ji(meV)\natomic layers, iB\nFe\nPt\nFIG. 3. Calculated layer-resolved e\u000bective isotropic exchange\nconstants, Ji, see Eq. (10). The blue \u0004and green\u0003symbols\nrepresent Jifor Pt and Fe layers, respectively. Solid lines\nserve as guides for the eye.\nexchange-coupled magnetic recording media as it enables\nthe \\lever\" e\u000bect in switching the magnetisation of the\nhard (FePt) phase. It should be noted that the e\u000bective\nexchange parameters calculated for the interior of the\nFePt slab and the Fe bulk part of the system correspond\nto mean-\feld Curie temperatures of TMF\nC\u0018700Kand\nTFe\nC\u00181000K, in good agreement with the corresponding\nexperimental values [27, 28].\nApproaching the Fe/FePt interface from region I, the\ne\u000bective exchange of the Fe layers drops rapidly and the\ninterfacial Fe layers i=\u00069 exhibit an e\u000bective exchange\nof merely\u001840 meV, almost identical to the e\u000bective\nexchange of the Pt layers. This reduction in Jioriginates\nin the weak interlayer couplings in layers i=\u00069 and the\nrelatively weak exchange of this layer with the soft layers\njij \u0015 10. In other words, what remains is essentially\nthe ferromagnetic NN Fe-Pt interaction, thus giving Fe\nlayersi=\u00069 approximately the same e\u000bective exchange\nas the Pt layers. Our results for the e\u000bective exchange\nparameters in the unrelaxed system Aare in satisfactory\nagreement with those in [13], although the magnitudesofJiare signi\fcantly smaller in [13].\nInterestingly, in system A, the e\u000bective exchange of\nthe Fe layer i=\u000610,J\u000610\u0018230 meV, i.e., it almost\nrecovers the bulk value. In contrast, in system B,\nJ\u000610remains remarkably small ( \u0018160 meV). Also, in\nsystem Bthe e\u000bective exchange exhibits relatively large\n\ructuations throughout the Fe layers jij \u001510. These\ndi\u000berences could be attributed to the fact that the\ngeometry of the Fe bulk is di\u000berent for the two systems\n(see Section II). Although the oscillations can be seen\nfor all the interactions of these Fe layers, the strongest\ncontribution to the oscillatory behaviour comes from the\nferromagnetic NN out-of-plane interactions. This is to be\nexpected as the variation in the interlayer distance (due\nto the geometrical relaxation) mostly a\u000bect the hybridis-\nation between orbitals centered at adjacent atomic layers.\nC. Magnetocrystalline anisotropy energy\nFig. 4 shows the layer-resolved MAE contributions,\nKi, see Eq. 2, for systems AandB. The MAE contribu-\ntions of the Fe layers in the FePt slab oscillate between\nabout 2.5 meV and 3 meV. The frequency and the\nmagnitude of these oscillations is di\u000berent for the two\nsystems, most probably, due to the di\u000berent boundary\nconditions. In system B, due to the evidently smaller\namplitude of oscillation, KFe\nisettles more quickly around\nthe bulk value of 3 meV. In both systems, Kiof all the\nPt layers is very small, \u00180:2 meV. In the Fe parts of the\nsystemKFe\niquickly approaches (practically) zero, since\nthe MAE of Fe bulk is on the order of \u0016eV.\nAs a remarkable di\u000berence between the systems A\nandB, in system Bthe Fe layers at the interface (layers\n\u00069) exhibit a contribution of about 1 meV to the MAE,\nwhile in system Athis contribution is even negative\n(\u0018\u00000:25 meV). In order to gain an understanding of\nthis di\u000berence, we tried applying Bruno's arguments in\nterms of second order perturbation theory [29, 30]. To\nthis end, we performed self-consistent scalar-relativistic\ncalculations (i.e., calculations in which the spin-orbit\ncoupling is excluded) and calculated the local partial\ndensities of states (LPDOS). In Fig. 5 we plotted the\nd-like spin- and orbital-resolved LPDOS at layer 9. The\nLPDOS clearly shows a strong spin-polarisation (almost\n\flled majority-spin band) in this layer, which is a nec-\nessary condition to apply Bruno's theory. Apparently,\nthedz2-LPDOS is nearly insensitive to the geometry\nrelaxation, while upon relaxation a considerable weight\nof thedxz;yz-LPDOS (and, to some extent, also of the\ndxy;x2\u0000y2-LPDOS) is shifted towards the Fermi level\nin the occupied regime in both spin-channels. The\nunoccupied part of the minority-spin channel of these\norbital-resolved states is also a\u000bected by the geometrical\nrelaxation. However, in the vicinity of the Fermi level6\n-0.50.00.51.01.52.02.53.03.54.0\n-15 -10 -5 0 5 10 15Ki(meV)\natomic layers, iA\nFe\nPt\n-0.50.00.51.01.52.02.53.03.54.0\n-15 -10 -5 0 5 10 15Ki(meV)\natomic layers, iB\nFe\nPt\nFIG. 4. Calculated layer-resolved contributions to the MAE,\nKi, (blue\u0004: Pt, green\u0003: Fe) for systems AandB. Solid\nlines serve as guides for the eye.\nonly a small increase in the unoccupied minority-spin\nLPDOS occurs. (The majority-spin LPDOS clearly\ndecreases at the Fermi level, but it is not relevant in\nour present theoretical estimation.) Since the spin-orbit\ninteraction gives rise to couplings between the dxz\nanddyzstates, inducing a perpendicular MAE, as\nwell as to couplings between the dxz;yz anddz2states,\ninducing an in-plane MAE [30], it is hardly possible\nto identify a well-established di\u000berence in the speci\fc\nlocal contribution to the MAE regarding the two systems.\nIn terms of the localised-spin model as described\nin Section II, the MAE can be cast into on-site and\ninter-site contributions as per Eq. (8). Although the\nmicroscopic model to construct an anisotropic spin\nmodel di\u000bers from the one used in this work, the\nresults of Mryasov et al. [31] strongly suggest that the\nMAE of the FePt systems arise mainly from e\u000bective\nFe-Fe inter-site interactions, Eq. (7), mediated by the\nspin-orbit coupling on the Pt atoms. Mapping our\nresults on the MAE to the spin-model, see Eqs. (5)\n{ (9), provides a unique opportunity to check the\nassertion of Ref. [31]. Using the methods introduced in\n-8-6-4-202468\n-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2LPDOS (states ·atom−1·mRyd−1)\nenergy (Ry)dz2\nA\nB\n-4-3-2-101234\n-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2LPDOS (states ·atom−1·mRyd−1)\nenergy (Ry)dxz,yz\nA\nB\n-6-4-20246\n-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2LPDOS (states ·atom−1·mRyd−1)\nenergy (Ry)dxy,x2−y2\nA\nBFIG. 5. Calculated d-like spin- and orbital-resolved local par-\ntial densities of states (LPDOS) for layer 9. Upper: dz2\n(m= 0), middle: dxzanddyz(m=\u00061), lower: dxyand\ndx2\u0000y2(m=\u00062). Positive/negative values stand for the\nminority/majority spin-channels. The zero of the energy is\nshifted to the Fermi energy.\nRef. [26], we calculated the on-site and inter-site parts of\nthe layer-resolved MAE related to an extended Heisen-\nberg spin model and plotted these contributions in Fig. 6.7\n-0.50.00.51.01.52.02.53.03.54.0\n-15 -10 -5 0 5 10 15layer-resolved MAE (meV)\natomic layers, iA\non-site\ninter-site\n-0.50.00.51.01.52.02.53.03.54.0\n-15 -10 -5 0 5 10 15layer-resolved MAE (meV)\natomic layers, iB\non-site\ninter-site\nFIG. 6. Calculated layer-resolved on-site ( \u000f) and inter-site\n(\u000e) anisotropies for systems AandB, see Eq. (9). Solid lines\nserve as guides for the eye.\nInspecting Fig. 6 it is indeed obvious that approx-\nimately 90 % of the MAE of FePt is associated with\nanisotropic inter-site interactions between Fe atoms.\nThe shape of the Ki;inter\u0000siteacross the atomic layers\nicoincides reasonably well with that of Kipresented\nin Fig. 4. The on-site anisotropies are quite stable on\nthe Fe sites within the FePt slab and practically vanish\nat the Pt sites. Furthermore, the increase in Kiat\nthe Fe/FePt interface (i.e., in layers i=\u00069) due to\nrelaxation stems mostly from the inter-site anisotropy.\nUsing the arguments of Mryasov et al. [31], this can be\nexplained in terms of the electron scattering between\nthese Fe atoms and the Pt atoms in the adjacent layer\ni=\u00068, experiencing thus large spin-orbit coupling.\nInterestingly, in system Athis induced anisotropy e\u000bect\nis suppressed and the small negative contribution to the\nMAE in these layers is of on-site origin. Remarkably,\nEq. (9) is satis\fed with a good accuracy if Kiis taken\nfrom the direct calculation via the torque method.\nThis lends substantial credit to the use of the tensorial\nexchange interactions in spin-dynamics simulations.As a \fnal step in our ab initio calculations, we would\nlike to address the question of how much the MAE of the\n\fnite FePt slab is changed with respect to the bulk MAE\nrelated to an FePt layer of the same size. Clearly, this\npoint has a crucial technological impact, namely, whether\nthe perpendicular MAE can be increased by forming an\nFe/FePt multilayer sequence. Regarding Fig. 4, it is obvi-\nous that for the chosen width of the FePt layer the e\u000bect\nof a single interface can hardly be separated, since the\noscillations of Kiindicate strong interaction between the\ntwo Fe/FePt interfaces (quantum interference e\u000bects).\nThus we de\fne the excess MAE generated by the entire\nFePt slab as\n\u0001KFePt\u0000slab=15X\ni=\u000015Ki\u00009KFePt; (11)\nwhere in the sum we also include layers from the Fe\npart of the system. Note that the MAE of Fe bulk\n(on the order of \u0016eV/atom) is neglected and KFePt is\nthe MAE per formula unit (f.u.) for bulk FePt. We\ncalculated KFePt = 3:37 meV/f.u., which, although\nhigh in comparison to experiment, is in good agreement\nwith other theoretical results based on the LSDA or the\nLSDA+U approach [32, 33].\nWhen evaluating \u0001 KFePt\u0000slabwe needed to consider\nthat for the systems AandBthe Fermi level of bulk Fe\nis used instead of the Fermi level of bulk FePt, which\nslightly a\u000bects the value of KFePt calculated within\nthe SKKR-ASA approach. In order to calibrate KFePt\nwe used system C(an unrelaxed Fe/FePt/Fe trilayer\nimmersed in FePt bulk, see Section II). Indeed, in the\nFePt slab of system Cwe obtained a very similar shape\nofKiacross the atomic layers as for system A, while for\nthe innermost FePt layers the bulk MAE, KFePt, was\nretained to within less than 1 % numerical accuracy.\nSince the corresponding MAE contributions in system\nAare by 0.33 meV/f.u. smaller, in Eq. (11) we used a\ncorrected value of 3.04 meV/f.u for KFePt.\nFor system A, we obtain a reduction in the total MAE,\n\u0001KA\nFePt\u0000slab\u0019\u00004:2 meV. This reduction stems primar-\nily from the interfacial layers i=\u00069, see Fig. 4. From\nthis \fgure it is obvious that the contribution of one Fe\nlayer to the MAE of the FePt slab is 'missing', since\nthis Fe layer becomes much rather an interfacial Fe layer.\nUpon relaxation, i.e. in system B, the interfacial Fe lay-\ners have remarkably enhanced contributions to the MAE,\nsee Fig. 4, as these Fe layers seem to belong rather to the\nFePt slab. Moreover, in this case the Fe layers in the FePt\nslab have contributions to the MAE closer to that in bulk\nFePt as compared to system A. Consequently, for sys-\ntemBthe MAE of bulk FePt is almost entirely retained,\n\u0001KB\nFePt\u0000slab\u0019\u00000:4 meV. Comparing this value to the\ntotal MAE of the FePt slab immersed in Fe, 26.9 meV,\nwe conclude that the MAE of a realistic (Fe m/(FePt)k)n\n(m&10,k&9) multilayer sequence is approximately\nequal to the MAE of n\u0001kFePt bulk layers.8\nIV. IMPLICATIONS FOR MESOSCOPIC SPIN\nSTRUCTURES\nThe detailed mapping of the ab-initio information onto\na spin model, which will allow, for example, calculations\nof the temperature dependence of the MAE values, is be-\nyond the scope of the current work. Here we give a simple\nillustration of the implications of the ab initio results and\ntheir e\u000bect on magnetic spin structures and indeed the\nexchange spring phenomenon. Speci\fcally, we consider\nthe e\u000bect of an abrupt change in magnetic properties, es-\npecially the MAE, which is known to give rise to pinning\nof a domain wall at the interface. Kronm uller and Goll\n[34] developed a micromagnetic model of magnetization\nreversal in a material consisting of two coupled phases\nwith di\u000berent magnetic properties. It was found that a\ndomain wall (DW) could be pinned at the interface be-\ntween the di\u000berent layers, the pinning being overcome by\na critical \feld\nHc=2KII\nMIIs1\u0000\u000fK\u000fA\n(1 +p\u000fM\u000fA)2; (12)\nwhere the superscript IIrefers to the properties of\nthe hard phase and \u000fK;\u000fA;\u000fMrefer to the ratio of the\nanisotropy constant, the micromagnetic exchange con-\nstant and saturation magnetization respectively in the\nsoft and hard phases. The coercivity of the hard phase\nis invariably reduced by all combinations of material pa-\nrameters.\nHowever, Eq. (12) was derived under the assumption\nof bulk exchange coupling across the interface, and it\nhas been shown by Guslienko et al. [35] that the ex-\nchange spring e\u000bect is strongly dependent on the degree\nof coupling at the interface. In Ref. [35] the interface\ncoupling was taken as a variable, but the ab-initio calcu-\nlations presented here allow to study the exchange spring\nphenomenon with no \ftting parameters. We use a sim-\nple spin-chain model as in Ref. [35], treating the low-\nexchange layer as an interface providing a weakened ex-\nchange between the FePt and Fe layers. Within each\nlayer we can write down the following spin Hamiltonian\nH=\u0000JX\ni;j(nn)~Si\u0001~Sj+X\niK(Sz\ni)2\u0000X\ni\u0016~H\u0001~Si;(13)\nwhereJis the intralayer nearest-neighbor exchange\ncoupling, ~Sithe unit vector representing the spin\ndirection,Kthe anisotropy constant, \u0016the atomic spin\nin the given layer and ~Hthe applied \feld.\nWe allow for reduced exchange coupling at the inter-\nfaces by writing the exchange energy between interface\nspins as\nHint=\u0000JintX\ni;j~Si\u0001~Sj; (14)\nwhere the spins i;jare in separate layers. Similar to the\ncase of IrMn 3/Co (111) system in Ref. [18], our calcula-\ntions show sizeable DM interactions near the interface.Due to the C4v symmetry, however, the DM energy can-\ncels for the N\u0013 eel walls to be investigated in the following.\nFor that reason, we neglected the DM interactions in our\nmodel of Eq. (14). The equilibrium state of the spin\nsystem is determined by integrating the Landau-Lifshitz\nequation, without the precession term\nd~Si\ndt=\u0000\u000b~Si\u0002\u0010\n~Si\u0002~Hi\u0011\n; (15)\nwith~Hibeing the e\u000bective \feld acting on spin i.\n-1.0-0.50.00.51.0\n80 85 90 95 100 105 110 115 120mz\nDW coordinate\nFIG. 7. Calculated Domain Wall structures prior to mag-\nnetization reversal for fully exchange coupled layers (black \u000f)\nand layers coupled with ab-initio exchange parameters for the\nrelaxed system B(blue\u0004). The relaxed system shows a dis-\ncontinuous DW structure due to the weak interlayer exchange\ncoupling.\nFig. 7 shows calculated Domain Wall (DW) structures\nprior to magnetization reversal for fully exchange cou-\npled layers and layers coupled with ab-initio exchange\nparameter for the relaxed system. In the latter case the\ninterlayer exchange is approximately 20% of the bulk ex-\nchange of FePt. The associated coercivities are 5.9 T and\n7.24 T respectively, reduced from the bulk coercivity of\n14 T due to the exchange spring e\u000bect. For the bulk in-\nterlayer exchange case the DW width in the Fe layer is\nconsiderably smaller than the usual expectation due to\nthe presence of the large applied \feld. Nonetheless the\nDW is continuous across the interface, in contrast to the\ncase of the reduced exchange of the relaxed microstruc-\nture. The e\u000bect of the reduced exchange on the coerciv-\nity is relatively weak, consistent with the results given\nin Ref. [35]. However, even relatively small changes can\nbe signi\fcant for the design and operation of practical\nrecording media.\nV. CONCLUSION\nWe have presented \frst principles calculations of\nthe exchange interactions and the magnetocrystalline9\nanisotropy energy (MAE) in an Fe/FePt/Fe sandwich\nsystem. In particular, we investigated how the geo-\nmetrical relaxation in\ruences the calculated magnetic\nproperties of the system. In accordance with previous\nwork on an unrelaxed Fe/FePt/Fe system [13], we\nfound a dramatic reduction in the exchange coupling\nbetween the Fe layers at the Fe/FePt (soft/hard)\ninterface. Moreover, in the relaxed system, these layers\nadd a remarkable positive contribution to the MAE.\nFrom the tensorial exchange interactions evaluated by\nmeans of the relativistic torque method [26] we have\nshown that the MAE of the FePt slab and the interface\nMAE are dominated by anisotropic inter-site exchange\ninteractions. Moreover, our calculations indicate that\nthe formation of an Fe/FePt layer sequence reduces the\nperpendicular MAE. In the case of a relaxed geometry,\nwhich we consider to be relevant to a multilayer system,\nthis reduction is slight, on the order of \u00180:4 meV per\nFePt slab.\nWe also show that the ab-initio parameters will have\na bearing on the mesoscopic spin structures predicted by\natomistic and micromagnetic models. Speci\fcally, the\nreduced exchange coupling between FePt and Fe layers\ngives rise to a discontinuous spin structure across the\nFePt/Fe interface. Although the exchange spring e\u000bectstill gives rise to a large coercivity reduction it is likely\nthat the discontinuous spin structure could a\u000bect the\nmagnetization dynamics. The reduced exchange could\nalso a\u000bect the temperature dependence of the interface\nMAE values, which could also have a signi\fcant bearing\non the magnetic properties. Both factors require the de-\nvelopment of a detailed mapping onto an atomistic spin\nmodel. Further, each material combination in a mag-\nnetic nanostructure will have interface properties which\nmay di\u000ber signi\fcantly from the bulk, making further ab-\ninitio studies of interface properties important in terms\nof the understanding of the underlying physics of static\nand dynamic magnetic properties.\nVI. ACKNOWLEDGMENTS\nFinancial support was provided by the Hungarian\nNational Research Foundation (under contracts OTKA\n77771 and 84078) and by the New Sz\u0013 echenyi Plan of\nHungary (Project ID: T \u0013AMOP-4.2.2.B-10/1{2010-0009).\nSupport of the EU under FP7 contract NMP3-SL-2012-\n281043 FEMTOSPIN is gratefully acknowledged. CJA\nis grateful to EPSRC for the provision of a research\nstudentship.\n[1] D. Suess, J. Lee, J. Fidler and T. Schre\r,\nJ. Magn. Magn. Mater. 321(2009) 545.\n[2] R. Skomski and J. M. D. Coey, Phys. Rev. B 48(1993)\n15812.\n[3] C. T. Pan and S. C. Shen, J. Magn. Magn. Mater. 285\n(2005) 422.\n[4] F. Casoli, F. Albertini, L. Nasi, S. 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The tested average  perpendicular  magnetostricti on λ⊥ is -886 ppm along the \nmelt-spun ribbon  direction  in the Fe82.89Ga16.88Tb0.23 alloy . The calculated parallel  \nmagnetostriction  λ∥ is 1772 ppm, more than  4 times as large as that of binary Fe83Ga17 \nalloy . The enhanced magnetostriction  should be attributed to a small  amount of Tb \nsolution  into the A2 matrix phase  during rapid solidification . The local ized strong \nmagnet ocrystalline  anisotropy  of Tb element is suggested to  cause the giant \nmagnetostriction . \n \n \n \n \n \n \n \n Giant m agnetostrictive materials have the potential applications  in actuators , \nsensors  and energy harvesting  devices . 1-3 The substitution of nonmagnetic Ga into \ncommon body centered cubic Fe enhanced the magnetostriction λ over tenfold, which \nhas drawn lots of attentions . However, the reported magentostriction  of Fe -Ga alloys  \nis 400 ppm , just one fifth of  that of TbDyFe  alloys . 4-9 Focusing on  improv ing the \nmagnetostricti on of Fe -Ga alloys, many efforts have been made by dop ing the third \nelement s. 10-17 Doping the interstitial elements B , C and N  slightly increas es the \nmagnetostriction of Fe -Ga alloys . 11, 12  ,17 Alloying with the 3d and 4 d transition \nelements , including Ni, V , Cr, Mn, Co , Mo, Rh, etc  drastical ly decrease s the \nmagnetostrictive strain of Fe -Ga al loys. 10, 13 -15 Doping the one main group elements \nSi, Ge and Sn also cannot  enhence  the magnetostriction . 16 Up to now , it is not obvious \nto improv e the magnetostriction of Fe -Ga alloys by doping the above third elements . \nThe magnetostriction of ferromagnetic materials originates from the spin -orbit \ncoupling, with the same origin of the magnetocrystalline  anisotropy. 18, 19 The \nrelatively low magnetostriction of Fe -Ga alloys stems from the low \nmagnetocrystalline anisotropy . The giant  magnetostriction  2600  ppm for TbFe 2 alloy \noriginates  from  its strong magnetocrystalline anisotropy , which is two orders higher \nthan that of Fe-Ga alloys . 4-9 We believe that the  localized strong magnetocrystalline \nanisotropy  possibly  induce s a giant magnetostriction . Based on this idea, the third \nelement Tb is selec ted to dope into the Fe 83Ga17 alloy  to form the localized strong \nmagnetocrystalline anisotropy . However, there is no solid solution of Tb in the A2 \nphase of Fe83Ga17 alloy. M elt spinning  is expected  to increase  the solid solution of Tb into the A2 matrix of Fe83Ga17 alloy  by the rapid solidification . In this letter, slight \namount of Tb element is doped  in Fe83Ga17 alloys by melt spinning.  The tested \naverage  perpendicular  magnetostriction λ⊥ is -886 ppm along the melt-spun ribbon \ndirection in the Fe82.89Ga16.88Tb0.23 alloy . The calculated parallel magnetostriction λ∥ \nis 1772 ppm, more than  4 times as large as that of binary Fe83Ga17 alloy . \nHigh -purity starting elements  iron, gallium and,  terbium  with the purity of 99.99%  \nwere  arc melte d under argon  atmosphere  for four times  with the nominal \ncompositions of Fe83Ga17, Fe82.97Ga16.97Tb0.06, Fe82.89Ga16.88Tb0.23 and \nFe82.77Ga16.76Tb0.47, named as Tb0, Tb0.06, Tb0.23 and Tb0.47 respectively . Ribbon s \nwere prepared  by melt  spinning with the  copper wheel velocity  of 20  m/s. The crystal \nstructur e was characterized by  using x-ray diffraction (XRD) on a Rigaku  X-ray \ndiffractometer  with Cu Kα radiation . The composition and morphology  of the ribbons \nwere determined using a JEOL JXA-8100  electron probe micro -analyzer (EPMA)  and \na JEOL JEM -2100  transmission electron microscope  (TEM) . Initial m agnetization \ncurves  (M-H) were measured on physical properties measurement system (PPMS). \nThe Curie  temperature was det ected  by a NETZSCH TSA449  thermogravimetry  \nanalyzer (TG) with the  20 ℃/min heating rate from room temperature to 1000 ℃. The \nmagnetostrict ion was tested along the ribbon direction by standard resistance strain \ngauge technique . The ribbon was pended vertically and the  magnetic field was applied \nperpendi cular to the ribbon plane . The upper part of the ribbon was fixed with clamp \nand the bottom  part was load ed with a mass block  in order to avoid the bending of \nribbons during the magnetization processes. The magnetostriction was measured three times for each sample with the strain gauge stuck on both sides of the ribbons . And \nthen, rotating the ribbons by 180 ˚, the magnetostriction was measured for another \nthree times .  \nFigure 1 shows the back scattered electron  (BSE)  and bright -field image s of the \nribbon samples . All the ribbons exhibit columnar crystal morphology  with the average \ngrain size of 10 μm in Fig. 1 (a) . Single phase is observed for the Tb0, Tb0.06  and \nTb0.23 ribbons , and dual-phase micro structure exists for  Tb0.47 ribbon as shown in \nFig. 1(b). The second  globular  phase precipitate s in the Tb0.47 ribbon . EDS tests \nconfirm that these second phase s are Tb-rich phase s, with the composition of \nFe60Ga30Tb10, indicat ing the solubility limit of Tb  atoms  in Fe83Ga17 alloy  lower than \n0.47 at%  under  the rapid  solidification.  \nX-ray diffraction s patterns for the Tb -doped Fe83Ga17 ribbons are shown  in Fig . 2. \nAll the detected  diffraction  peaks are indexed as (110), (200) , (211)  and (220)  of the \nbody -centered cubic (bcc) structure , same as A2 phase in the pure Fe 83Ga17 alloy . The \nsmall quantity  of the  precipitate d phase can not be detected by  x-ray diffraction .  \nThe lattice parameters a of the  ribbons  are calculated according to the  XRD data , \nas plotted  in Fig. 3(a). With increasing Tb content, the lattice parameter s first increase  \nfrom a = 0.2900 nm for Tb0 to a = 0.2904 nm for  Tb0.23, and then trend to saturation \nwith a = 0.2904 nm for  Tb0.47. The initial increase of the lattice parameter  a should  \nbe associated with the doping of the big Tb atoms into A2 matrix, and the  saturation \nof the  lattice parameter  a should be attributed to the precipitation of the second \nTb-rich phase. This result is consistent with the x -ray diffraction patterns \nmeasurement, as shown in Fig.2 . \nFigure 3 (b) shows  that the saturation magnetization s MS of the ribbon samples  \nfirst increase  as the Tb content increase  from  180.0 emu/g for Tb0 to 183.6 emu/g for  Tb0.23  and then trend to saturation  with 183.9  emu/g for  Tb0.47. The result shows the \nferromagnetic coupling  of Tb element doped into A2 matrix, different from the usual \nferrimagnetic  coupling  between Fe and Tb. As can be seen from Fig. 3 (c), the Curie \ntempera tures TC of the ribbon increase s first and then slightly decrease with the \nincrease  of Tb content. The initial increase of the Curie temperature further confirm s \nthe ferromagnetic coupling interactions  of the Tb doping into the A2 matrix . The \nprecipitation of the Tb -rich phase results in the decrease of the Curie temperature. The \nenlargement of the lattice parameter, the  enhance ment  of saturation magnetization , \nand the  increase  of the Curie temperature of the Tb -doped Fe -Ga alloys reflect the Tb \nelement  solution  in the A2 matrix .  \nThe room temperature perpendicular  magnetostrictions ( λ⊥) are measured along \nthe length direction of the ribbon samples  and also normal to the  direction of the \napplied field . Each ribbon sample has been tested for twelve  times, as mentioned \nabove, the tested data, accompanied with the average value of the magnetostrictions, \nare plotted  in Fig. 4 (a). One of the λ-H curves for each ribbon  is also presented in Fig. \n4 (b). For the binary  Fe83Ga17 (Tb0) ribbon, the average magnetostriction value λ⊥ is \n-176 ppm, consistent with the previously reported value of -163ppm for Fe 81Ga19 \nribbon  20, 21. The average  magnetostriction values of the  ribbon sample obviously \nincrease from -176 ppm for Tb0, -447 ppm for Tb0.06 to -886 ppm for Tb0.23 and \nslightly decease to -862 ppm for Tb0.47. The results indicate that the giant \nmagnetostriction is achieved by slight amount of Tb element dop ing in the Fe83Ga17 \nribbon.  \n The parallel magnetostrictions  λ// along the direction of the applied field, \nvertical to the plane of the ribbon s, cannot be measured directly since the ribbons are \ntoo thin. Therefore, the parallel magnetostrictions λ// have been calculated . The relationship of  the measuring magnetostriction and the saturation magnetostriction  λs \nfollows  the Equation (1) , which is based on the  isotrop ic grains.  22  \n𝜆  =3\n2𝜆𝑠 cos2θ−1\n3                      (1) \nThere, θ is the included angle between the measuring direction and magnetizing \ndirection as shown in Fig. 4 (a) . The columnar  grains, normal to the plane of the \nribbons,  are anisotrop ic, but they are isotrop ic in the plane of the ribbons.  The tested \nperpendicular  magnetostriction  λ⊥ along the length direction of the ribbon stem s from \nthe in -plane isotropy . It is reasonable to calculate the parallel  magnetostrictions λ// \nbased on the following Equation (1).  Here  the saturate magnetostriction  λs equals to  \nthe parallel  magnetostriction λ// and the tested average magnetostriction  𝜆  equals to  \nthe perpendicular  magnetostriction λ⊥. The angle θ between the measuring direction \nand the magnetizing direction is 90˚ as shown in Fig. 4(a). If we substitute  θ=90˚ into \nEq. (1)  \n                           𝜆∥= −2𝜆⊥                        (2) \nA giant magnetostriction of 1772 ppm  is achieved in the Tb0.23, which  is more \nthan 4 times larger than  the reported value of 400 ppm in  the Fe -Ga alloys , and \ncomparable to that of the  giant rare earth alloys  Terfenol -D, where the amount of rare \nearth (Tb, Dy) is more than 60 % .2, 23  \nThe magnetostriction and the magnetocrystalline anisotropy  of Fe-Ga alloys  \ncommonly originate  from the spin -orbit coupling . Due to their relatively low \nmagnetocrystalline anisotropy , the Fe-Ga alloys  exhibit a moderate  magnetostriction  \nof about 400 ppm . The rare earth element  Tb with high magnetocrystalline anisotropy \nis doped into Fe -Ga ribbons, forming the localized  strong magnetocrystalline \nanisotropy . A giant magnetostriction of 1772 ppm is achieved in the Tb 0.23 ribbon . Therefore, it is believable that  the localized  strong magnetocrystalline anisotropy can \nalso induce a giant magnetostriction.  \n \nThis work is supported by National Basic Research Program  (2012CB619404) ， \nNatural Science Foundation of China （50925101, 51221163, 91016006 ）.  \n \n1 D. C. Jiles, Acta Mater . 51, 5907 (2003). \n2 J. X. Zhang, L.  Q. Chen, Acta Mater . 53, 2845 (2005). \n3 H. Basumatary, M . Palit, J. A . Chelvane, S. Pandian,  M. M . Raja and V . \nChandrasekaran, Scr. Mater. 59, 878 (2008).  \n4 A. E. Clark, K. B. Hathaway, M. Wun -Fogle, J. B. Restorff, T. A. Lograsso, J. R. \nCullen, IEEE. Trans. Magn. 37, 2678(2001).  \n5 O. Ikeda, R. Ka inuma, and I. Ohnuma, J. Alloys . Comp. 347, 198 (2002).  \n6 H. Wang, Y . N. Zhang, T. Yang, Z. D. Zhang, L. Z. Sun, and R. Q. Wu, Appl. Phys. \nLett. 97, 262505 (2010).  \n7 A. Mahadevan, P. G. Evans, and M. J. Dapino,Appl. Phys. Lett.  96, 012502 (2010).  \n8 M. Barturen, B. Rache Salles, P. Schio, J. Milano, A. Butera, S . Bustingorry, C. \nRamos, A. J. A. de Oliveira, M. Eddrief, E. Lacaze, F. Gendron, V . H. Etgens, \nand M. Marangolo, Appl. Phys. Lett. 101, 092404 (2012).  \n9 C. Bormio -Nunes, M. A. Tirelli, R. S. Turtelli, R. Grö ssinger, H. Mü ller, G. \nWiesinger, H. Sassik , and M. Reissner, J. Appl. Phys. 97,033901  (2005).  \n10 A. E. Clark, M. Wun -Fogle, J. B. Restorff, T. A. Lograsso and G. Petculescu, J. \nAppl. Phys.  95, 6942 -6943  (2004).  \n11 S. M. Na and A. B. Flatau, J. Appl. Phys. 103, 07D304  (2008).  \n12 A. E. Clark, J. B. Restorff and M. Wun -Fogle, K. B. Hathaway, T. A. Lograsso and M. Huang, E. Summers, J. Appl. Phys. 101, 09C507  (2007).  \n13 P. Mungsantisuk, R. P. Corson, and S. Guruswamy, J. Appl. Phys. 98, 123907 \n(2005).  \n14. J. B. Restorff , M. Wun -Fogle , A. E. Clark, T. A. Lograsso, A. R. Ross, and D. L. \nSchlagel, J. Appl. Phys. 91, 8225  (2002).  \n15 C. Bormio -Nunes, R. Sato Turtelli, H. Mueller, R. Gr össinger, H. Sassik, M.  A. \nTirelli, J Magn. Magn. Mater. 290-291, 820-822 (2005).  \n16 F. Gao , C. B. Jiang, J. H.  Liu, Acta. Meta. Sini.  43, 683-687 (2007).  \n17 Y . Gong, C. B. Jiang, H. B. Xu, Acta. Meta. Sini.  42, 830-834 (2006).  \n18 A. E. Clark,  Ferromagnetic Materials  (North -Holland, Amsterdam, 1980) , p531  \n19 J. Cullen, P. Zhao, and M. Wuttig, J. Appl. Phys.  101, 123922  (2007).  \n20 N.Mehmood , G. Vlas ák, F. Kubel, R. Sato Turtelli, R. Gr össinger, M. Kriegisch, H. \nSassik, P. Svec,  IEEE  Trans. Magn. 45, 4128 -4131  (2009). \n21 J. J. Zhang, T.  Y . Ma, M . Yan, Phys. B. 404, 4155 -4258 (2002).  \n22 B. D. Cullity , C. D. Graham,  Introduction to Magnetic  Materials  (IEEE Press , \nPiscataway , 2008 ), p.255 . \n23 X. G. Zhao , G. H. Wu, J. H. Wang, K. C. Jia, W. S. Zhan, J. Appl. Phys. 79, 6225  \n(1996 ). \n \n \n \n \n \n \n  \n \nFig.1 (a) BSE images  of the nominal  compositions of  Tb0, Tb0.06, Tb0.23  and \nTb0.47 ribbon samples . (b) Bright -field images  of the Tb0.23 ribbon with single \nphase micro structure  and Tb0.47 ribbon  with the g lobular  precipitate  phase s. \nFig.2 XRD patterns of  melt-spun ribbons  for Tb0, Tb 0.06, Tb0.23  and Tb0.47.  \nFig.3 Composition dependen ce of the lattice constant , saturation m agnetization Ms \nand Curie temperature TC for the Tb-doped Fe 83Ga17 ribbons.  \nFig.4 (a) Magnetostrict ion of Fe Tb0, Tb 0.06, Tb0.23  and Tb0.47 ribbons. The inset \nshows the measuring direction and magnetizing direction.  (b) One of the λ-H curves  \nfor each ribbon  samples .  \n \n \n \n \n 204 06 08 01 00  \nTb0Tb0.06Tb0.23Tb0.472 20211200110Intensity(a.u.)2\ntheta(deg.) \n 0.00.10.20.30.40.5970980990180.0181.5183.0184.50.29000.29020.2904(\nc)T\nb content (at%) \na (nm)(\nb) T c (K)M s (emu/g) \n  \n(a)"    },    {        "title": "1307.2784v1.Magnetocrystalline_Anisotropy_and_the_Magnetocaloric_Effect_in_Fe2P.pdf",        "content": "arXiv:1307.2784v1  [cond-mat.mtrl-sci]  10 Jul 2013Magnetocrystalline Anisotropy and the Magnetocaloric Effe ct in Fe 2P\nL. Caron,1,2,∗M. Hudl,2V. H¨ oglin,3N. H. Dung,1C. P. Gomez,3\nM. Sahlberg,3E. Br¨ uck,1Y. Andersson,3and P. Nordblad2\n1Fundamental Aspects of Materials and Energy, Faculty of App lied Sciences, TUDelft\nMekelweg 15, 2629 JB Delft, The Netherlands\n2Department of Engineering Sciences, Uppsala University, B ox 534, SE-751 21 Uppsala, Sweden\n3Department of Materials Chemistry, Uppsala University, P. O. Box 538, SE-751 21 Uppsala, Sweden\n(Dated: June 26, 2021)\nMagnetic and magnetocaloric properties of high-purity, gi ant magnetocaloric polycrystalline and\nsingle-crystalline Fe 2P are investigated. Fe 2P displays a moderate magnetic entropy change which\nspans over 70 K and the presence of strong magnetization anis otropy proves this system is not fully\nitinerant but displays a mix of itinerant and localized magn etism. The properties of pure Fe 2P\nare compared to those of giant magnetocaloric (Fe,Mn) 2(P,A) compounds helping understand the\nexceptional characteristics shown by the latter which are s o promising for heat pump and energy\nconversion applications.\nPACS numbers: 75.30.Cr,75.30.Sg,75.50.Cc,75.80.+q,65. 40.De\nI. INTRODUCTION\nRecently, magnetic refrigeration based on the magne-\ntocaloric effect has been regarded as a more efficient and\nenvironmentally friendly alternative to gas compression-\nbased refrigeration. Amongst the most promising work-\ning materials for magnetic refrigeration are those based\non Fe2P such as (Fe,Mn) 2(P,A) where A = As, Ge, Si1–3.\nThe promise to magnetic refrigeration these materials\nshow lies in the combination of the properties they retain\nfrom the parent compound with the easy tailoring of its\nmagnetic properties due to stoichiometric changes. The\nformer, a first order magnetoelastic phase transition, en-\nsures high magnetic entropy and adiabatic temperature\nchangeswhile the latter guaranteesgood workingmateri-\nals overa large temperature span. However, as well char-\nacterized as the (Fe,Mn) 2(P,A) compounds have been in\nthe pastdecade1–3, the magnetocaloricproperties ofpure\nFe2P have received very little attention.\nFe2P crystallizes in the so-called Fe 2P-type structure\n(space group P¯62m) where two chemical elements oc-\ncupy four different crystallographic sites. In the hexag-\nonal structure, Fe occupies two different metal sites, the\ntetragonal (Fe I) 3f site, and the pyramidal (Fe II) 3g site,\nwhile P occupies the two dissimilar sites, 2c and 1b4.\nSuch distribution of Fe and P atoms in the crystal lattice\ncreates two magnetic sublattices with different interac-\ntions as well as magnetic moments: Fe Iand Fe IIbeing\nthe low and high moment sites, respectively, with a total\nmoment of ∼2.9µB/f.u.5,6. Below its Curie tempera-\nture (T C), around 214 K, the moments are aligned in\na ferromagnetic arrangement along the c-axis7. At T Ca\nfirst ordermagneticphase transition to the paramagnetic\nstate occurs.\nEarly works about Fe 2P strongly disagree on its T C\nand saturation magnetization, and even the nature of\nthe transition was not clear7,8since the only example\nknown at the time of a ferromagnetic to paramagnetic\nfirst order magnetic phase transition was that of MnAs,described by Bean and Rodbell9. By determining the\npurity of their samples prior to further characterization,\nLundgrenetal.8madeit clearthatthe propertiesofFe 2P\nareverysensitive to stoichiometricdeviations, explaining\nthe differences in saturation moment values. The spread\nin TCis two fold: it is stoichiometry dependent and ex-\ntremely sensitive to the applied magnetic field7. The\nfirst order nature of the transition was first proposed by\nW¨ appling et al.10due to magnetoelastic effects observed\nin M¨ ossbauer measurements. However it was only af-\nter careful measurements that thermal hysteresis8, the\ndiscontinuity of the lattice parameters7,8and a consid-\nerable latent heat contribution11at TCwere observed,\ndetermining once and for all the first order character of\nthe phase transition in question.\nTherefore, Fe 2P undergoes a first ordermagnetoelastic\nphase transition which is accompanied by a discontinu-\nity in lattice parameters and a small decrease in volume\nof about 0.04% (on heating), but no change in crystal\nsymmetry.\nIt has been recently suggested from first principles cal-\nculations that the origin of the metamagnetic transition\nin Fe2P lies in the nature of the Fe Isublattice and its\ninteraction with the Fe IIsublattice. Below T C, while the\nintra layer interactions in the Fe IIlattice are strongly\nferromagnetic, the Fe Ilattice is essentially non-magnetic\nand only acquires moment due to the exchange field gen-\nerated by the ordering of the Fe IIsublattice12. Thus,\nthe Fe Isublattice provides a strong coupling to the crys-\ntal lattice while the interaction between the Fe Iand Fe II\nsublattices generates a large magnetization jump. This\nresult has been independently obtained for Fe 2P-based\ncompounds3as well.\nIn this work we have characterized high purity poly-\nand single-crystalline Fe 2P not only as a magnetocaloric\nmaterial in itself but also to better understand the out-\nstanding properties shown by Fe 2P-based compounds.\nAsFe2Ppresentshighmagneticanisotropy,weemphasize\nthat the anisotropic character of the magnetic response2\nneeds to be taken into account for the correct determi-\nnation of the magnetocaloric effects.\nII. EXPERIMENTAL TECHNIQUES\nThe polycrystalline sample studied was prepared us-\ning the drop synthesis technique13. The single crystalline\nneedle used in this study was prepared using the tin-melt\ntechnique14,15. Since the mass of the needle measured is\nbelow the precision of most balances, its mass was de-\ntermined by estimating its volume under an optical mi-\ncroscope and calculating it from the known density of\nthis compound. In this manner also the aspect ratio of\nthe crystal was determined to be about 1/15 with the\nlong direction being the crystallographic c-axis. For the\nmagnetic measurements the single crystalline needle was\nfixed to a silicon slab for easy handling.\nThe crystallographic properties of both samples were\ninvestigated using X-ray diffraction analysis. For the\npolycrystalline sample the lattice parameters obtained at\n296 K using Cu K α1 (λ= 1.540598 ˚A) radiation are a =\n5.8661(2) ˚A and c = 3.4585(3) ˚A and were refined using\nthe software unitcell16.\nFor the needle, X-ray single crystal diffraction intensi-\nties were recorded at 100 K on a Bruker diffractometer\nequipped with an APEX2 CCD detector and a graphite\nmonochromator. The used radiation was Mo K α(λ=\n0.71073˚A), and the diffractometer was operated at 50\nkV, 40 mA. The initial data collection and reduction\nwas performed using the Bruker APEX software. Crys-\ntal structure refinements were performed using the soft-\nwareJANA200617. The composition was refined to be\nFe1.995(2)P, where only the Fe IIsite (0.592290 0.000000\n0.000000) was fully occupied and the lattice parameters\nobtained were a = 5.8955(0) ˚A and c = 3.4493(0) ˚A.\nComparing the lattice parameters obtained in this\nworkforpolycrystallineFe 2Pand those from the workby\nCarlsson et al.13, it can be concluded that, within error,\nboth the single- and poly-crystalline samples have the\nsame chemical composition. This is further supported\nby the Curie temperatures of both samples which differ\nby only one kelvin (see Fig. 1, 2 and 4).\nThe magnetic measurements were performed in Quan-\ntum Design’s MPMS5XL, MPMS7 and PPMS9 systems.\nThe magnetic entropy change was calculated from the\nisothermal data using Maxwell’s relation. Notice that\nMaxwell’srelationsare, in principle, only validforsecond\norder phase transitions. However, they can be used as a\ngood approximation for first order phase transitions if\nthe magnetization change with temperature and/or field\nis sufficiently smooth.\nIII. RESULTS\nThe temperature dependent magnetic properties of\npolycrystalline and single crystalline Fe 2P are shown in 0 20 40 60 80 100\n 100  150  200  250  300  350  400M (Am2/kg)\nT (K)0.05 T0.5 T\n1 T7 Tpolycrystal\nFIG. 1. Temperature dependence of magnetization at dif-\nferent magnetic fields for polycrystalline Fe 2P. The magnetic\nfield intervals between 1 T and 7 T measurements are of 1T.\nfigures 1 to 3. For both polycrystal and single crystal in\nthe c-direction the characteristic sharp first order phase\ntransition can be observed for low fields. At µ0H = 0.01\nT the transition presents a small thermal hysteresis of\nabout 1 K that quickly decreases with increasing field\nand can no longer be observed for µ0H≥0.1 T. With\nincreasing magnetic field the transition quickly shifts to\nhigher temperatures and broadens, assuming the charac-\nteristics which at first caused Fe 2P to be considered to\nhave a second order phase transition.\nWhen the single crystalline needle is measured with its\nhard magnetization axis perpendicular to the magnetic\nfield, i.e. with c ⊥µ0H, the magnetization direction will\nbe a function of temperature, field and the competition\nbetween field driven alignment and magnetocrystalline\nanisotropy. Thus, the total magnetization M totalcan be\nrepresented as a vector which makes an angle θwith the\nc axis. The component parallel and perpendicular to\nthe applied magnetic field are given by M /bardbl= Mtotal.sinθ\nand M ⊥= Mtotal.cosθ, respectively, so that the total\nmagnetization is given by M total2= M/bardbl2+ M⊥2.\nFigure 3 shows the magnetization component paral-\nlel to the magnetic field when the crystal is measured\nwith c⊥µ0H. M/bardblis deliberately left uncorrected for\ndemagnetizing field. The competition between the tem-\nperature and exchange driven spin alignment along the\nc-direction and the alignment promoted by the applied\nmagnetic field can be clearly observed. Above T Cthe\nmagnetic behavior is dominated by the demagnetization\nfactor of the sample, that is the shape anisotropy. Be-\nlow TCboth magnetocrystalline and shape anisotropies\ncompete with the field driven alignment. Up to 0.5 T\nonly one feature can be observed in the temperature de-\npendence of magnetization as a peak in magnetization.\nAbove 0.5 T both this peak and a broad change in cur-\nvature at temperatures higher than that of the peak are\nobserved.\nThe field dependence of the critical temperatures for\npoly- and single-crystalline Fe 2P is shown in figure 4. All3\n 0 20 40 60 80 100\n 150  200  250  300  350M (Am2/kg)\nT (K)0.01 T0.05 T\n0.1 T\n0.5 T1 T7 T\nc || µ0H\nFIG. 2. Temperature dependence of magnetization at differ-\nent magnetic fields for single crystalline Fe 2P with the applied\nfield parallel to the c direction. The magnetic field interval s\nbetween 1 T and 7 T measurements are of 1T. [Note that in\nthis measurement the magnetization at low temperatures for\nfields above 1 T is actually lower than for lower fields (see\nfigure 2). This is due to the diamagnetic contribution to the\nmagnetization arising from the Si slab where the single crys -\ntalline needle was mounted. In the absence of the Si slab the\nexpected behavior would be very similar to that observed in\npolycrystalline Fe 2P.]\n 0 20 40 60 80 100 120 140\n 0 50 100 150 200 250 300 350M (Am2/kg)\nT (K)7 T\n6 T\n5 T4 T\n3 T2 T\n1 T\n0.5 Tc ⊥ µ0H\n 0 2 4 6\n 150 200 250 300 3500.1 T\n0.05 T\n0.01 T\nFIG. 3. The component parallel to the applied magnetic field\nof the temperature dependence of magnetization at different\nmagnetic fields for single crystalline Fe 2P with the applied\nfield perpendicular to the c direction.\ntransition temperatures were taken as the peak observed\nin the first derivative of the temperature dependence of\nmagnetization. For both polycrystal and single crystal\nwith c/bardblµ0H the results are the same. The field de-\npendence of the apparent transition temperature, here\nreferred loosely as T C, deviates from a linear behavior\nfor fields below 3 T where it is best fit by a third de-\ngree polynomial ( T C= 217.7(2) K + 30.7(6) K/T µ0H\n- 7.9(5) K/T2(µ0H)2+ 1.0(1) K/T3(µ0H)3). Such be-\nhavior is in line with previous observations by Fujii et\nal.7who recorded a shift of 12 K/T for fields below 0.2\nT. However, in this work the δTC/δB observed is much 0 50 100 150 200 250 300\n 0 1 2 3 4 5 6 7T (K)\nµ0H (T)Fe2P\nTC c || µ0H\nTC c ⊥ µ0H\nHAN c ⊥ µ0H\nTC polycrystal\nFIG. 4. Field dependenceof the Curie temperature for severa l\nfields for polycrystalline and single crystalline Fe 2P with the\napplied field parallel to the a and c directions.\nhigher than the values obtained by Fujii in the given field\ninterval, reaching approximately 30 K/T. Above 3 T, the\nincrease of T Cis close to linear with a δTC/δB value of\n7.8(1) K/T.\nFor measurements performed with c ⊥µ0H two curves\nare presented in figure 4. The temperature at which the\npeak is observed represents the compensation point of\nthe competition between magnetocrystalline anisotropy\nand field driven alignment of the spins in the material,\nwhich shifts to lower temperatures with increasing field.\nInotherwords,it representsthe temperatureevolutionof\nthe anisotropyfield and assuch is denoted asafield H AN.\nAn applied magnetic field of approximately 7 T is neces-\nsary to overcome the magnetocrystalline anisotropy at 5\nK.Thesecondcurvepresentedisthederivativemaximum\nof the higher temperature broad change in inflection and\nit follows the trend of T Cobserved when the external\nmagnetic field is applied parallel to the easy magnetiza-\ntion direction, but is shifted about 15 K to lower temper-\natures. The lower T Cobserved when measuring with c\n⊥µ0H arises from the reduction of the effective field in-\nside the sample caused by the demagnetizing field. This\nreduction is proportional to the component of the mag-\nnetization parallel to the applied magnetic field which\nis then given by µ0H’ =µ0H - NM /bardbl, where N = 1/2\nis the demagnetization factor when the field is applied\nperpendicular to a long needle’s axis. As a result of the\nreduction caused by the demagnetizing field, T Cremains\nunchanged for µ0H/lessorapproxeql0.5 T, and shows a response equiv-\nalent to a lowereffective field for higher applied magnetic\nfields.\nThe temperature dependence of H ANdirectly reflects\nthe magnetocrystalline anisotropy, which can be more\ndirectly evaluated calculating the anisotropy constants.\nA ferromagnetic hexagonal single crystal in the shape\nof a needle presents, at least, two contributions to\nthe anisotropy energy: magnetocrystalline and shape\nanisotropies. The magnetocrystalline anisotropy energy4\nis given by:\nE=K1sin2θ\nwhere K 1is the first order anisotropy constant and\nθis the polar angle between the c-axis and the\nmagnetization7. This energy is the magnetic work which\nmust be done by the applied magnetic field to bring the\nmagnetization from the easy direction to that imposed\nby the applied field. This energy can be calculated as\nthe subtraction of the areas under the M Tvsµ0H and\nM/bardblvsµ0H magnetization curves (see figures 5 and 6) or\ndirectly from the extrapolated anisotropy field H AN, at a\ngiven temperature. Since when M /bardbl= Mtotal→sinθ=\n1, then:\nW=/integraldisplay∞\n0/bracketleftbig\nMtotal(H)−M/bardbl(H)/bracketrightbig\nµ0dH\nW=1\n2µ0HANMtotal=K1\nIn figure 7 the temperature dependence of K 1calculated\nusingbothH ANandthedifferenceofthe areasareshown.\nNoticethatK 1calculatedusingthedifferenceintheareas\nis slightly underestimated when compared to K 1calcu-\nlated from the anisotropy field. The curve obtained by\nFujii et al7using the Sucksmith-Thompson18method is\nincluded for comparison. For the calculation of K 1and\nthe entropy change when c ⊥µ0H the applied field was\ncorrected taking into account the shape anisotropy of a\nneedle. All other measurements are presented without\ncorrections. Isothermal measurements show that poly-\nand single-crystalline Fe 2P measured with the magnetic\nfield applied parallel to the easy magnetization direction\nsaturate below 0.1 T. In a very narrow temperature in-\nterval around the phase transition a small magnetic hys-\nteresis can be observed and is presented in figure 8 for\nsingle crystalline Fe 2P with the magnetic field applied\nparallel to the c-direction. Notice that a sharp meta-\nmagnetic transition can only be observed in the same\nrange where magnetic hysteresis is present. The mag-\nnetic entropy change for both single crystalline and poly-\ncrystallineFe 2P wascalculated fromisothermalmeasure-\nments using the Maxwell relations. As expected, the re-\nsults for polycrystalline and single crystalline Fe 2P with\nc/bardblµ0H are very similar (see figures 9 and 10). The mag-\nnetic entropy change for the single crystal being slightly\nhigher than that of the polycrystal, due to a higher satu-\nration magnetization presented by the former (see figure\n12). The magnitude of the maximum magnetic entropy\nchange for a 1 T field change, 2 and 2.2 J/kgK for poly-\nand single-crystalline Fe 2P, respectively, is found to be\nslightly higher than the 1.8 J/kgK observed by Fruchart\net al.19. Note that the given magnetic entropy values\nused here for comparison do not take into consideration\nthe sharp peak observed in the low temperature region of\nthe curve, which is found in all measurements presented\nin both Fruchart’s and this work. 0 20 40 60 80 100 120 140\n 0  1  2  3  4  5M (Am2/kg)\nµ0H (T)c ⊥ µ0HM ⊥ µ0H\nM || µ0H\nFIG. 5. Parallel and perpendicular components of the magne-\ntization as a function of applied field measured with c ⊥µ0H\nfrom 10 K to 360 K. The isotherms were measured using dif-\nferent temperature intervals. Away from the transition, fr om\n10 K to 200 K and from 240 K and 360 K, a 10 K step was\nused. Closer to the transition region, from 203 K to 212 K\nand from 221 K to 230 K, 3 k steps were used. Finally, around\nTC, from 215 K to 218 K, the isotherms were measured taking\n1 K steps. The isotherm corresponding to T Cis highlighted\nin red.\n 0 20 40 60 80 100 120 140\n 0  1  2  3  4  5  6Mtotal (Am2/kg)\nµ0H (T)c ⊥ µ0H10 K\n360 K\nFIG. 6. Total magnetization as a function of the applied field\nmeasured with c ⊥µ0H at different temperatures calculated\nfrom the data presented in figure 5. The large area indicated\ninbluearises from theinteraction betweenmagnetocrystal line\nanisotropy and field induced transitions at T Cand the sudden\nabsence of the latter above T C.\nThe use of the Maxwell relation to calculate the entropy\nchange from isothermal measurements in the case where\nc⊥µ0H requires caution. The Maxwell relations are de-\nrived from the Gibbs (or Helmholtz)free energy, where\nthe magnetic interaction is included in the form of an en-\nergy (or work) which is given by the integral of M·δH,\nwhereHis the effective field and Mthe total magnetiza-\ntion. Here the effective field, hereon denoted as Heff, is\nwritten as Heff=H+Hd+HW, whereHis the applied\nmagnetic field, Hdis the demagnetizing field and HW\nis the field due to the exchange interaction with neigh-5\n 0 1 2 3\n 0  50  100  150  200  250 0 1 2 3\nK1 (T2)\nT (K)K1 (106 J/m3)\nc ⊥ µ0H\nHAN\narea\nFujii\nFIG. 7. K 1calculated from data obtained measuring magne-\ntization parallel and perpendicular to the applied magneti c\nfield while keeping the hard magnetization direction parall el\nto the latter. Open blue squares represents data calculated\nusing the anisotropy field H AN, closed blue squares that from\nthe difference of the areas for isothermal curves and red open\ncircles tothevalues obtainedbyFujii etal.7(redopen circles).\n 0 10 20 30 40 50 60 70\n 0 0.025  0.05  0.075  0.1 0.125  0.15M (Am2/kg)\nµ0H (T)c || µ0H216 K\n217 K\n218 K\n219 K\n220 K222 K\nincreasing µ0H (T)\ndecreasing µ0H (T)\nFIG. 8. Field dependence of the magnetization at several\ntemperatures around T Cfor single crystalline Fe 2P with the\napplied field parallel to the c direction.\n 0 1 2 3 4 5\n 150  200  250  300  350-∆S (J/kgK)\nT (K)polycrystal\n0 - 1 T\n0 - 2 T\n0 - 3 T\n0 - 4 T\n0 - 5 T\nFIG.9. Magnetic entropychangeas afunctionoftemperature\nfor different applied magnetic fields in polycrystalline Fe 2P. 0 1 2 3 4 5\n 150  200  250  300  350-∆S (J/kgK)\nT (K)c || µ0H\n0 - 1 T\n0 - 2 T\n0 - 3 T\n0 - 4 T\n0 - 5 T\nFIG. 10. Magnetic entropy change as a function of tempera-\nture for different applied magnetic fields in single crystall ine\nFe2P with the field applied parallel to the easy magnetization\ndirection.\nboring moments, i.e. the Weiss field20. For an isotropic\nsystem,oranyanisotropicsystemwheretheappliedmag-\nnetic field is parallel to the easy magnetization direction\n(and to the moment), a variation in the effective field\nis equivalent to a change in the applied magnetic field\nonce corrections for shape anisotropy are made, since\nthe field due to the exchange interaction with neighbor-\ning moments points in the same direction as the applied\nmagnetic field. However, due to the magnetocrystalline\nanisotropy, this is not true when the applied magnetic\nfield and the easy magnetization direction are no longer\nparallel21.\nIn the magnetization process of a single crystal aligned\nwith its easy axis perpendicular to the applied magnetic\nfield, the moment or field due to the exchange interac-\ntion with neighboring moments - and the effective field -\nis not parallel to the applied magnetic field. In this case,\nconsideringthat the demagnetizing field is accounted for,\na change in the effective field felt by the single crystal re-\nsults from changes in two components: the applied field\nand the field due to the exchange interaction with neigh-\nboring moments. In order for the magnetization change\nto reflect a change in both these components it is not\nenough to consider only the component in the magne-\ntization along the applied field direction, and the total\nmagnetization needs to be considered. In this manner\nthe magnetic entropy change will reflect the change in\nconfigurational entropy of the microscopic magnetic mo-\nment. Thus, the total magnetization i.e. the magnitude\nof the magnetization vector, should be used as input of\nthe Maxwell relation. To makeour datacomparablewith\nthe literature our entropy change is calculated with re-\nspect to a field change in applied field instead of the ef-\nfective field.\nTherefore, the components of the magnetization par-\nallel and perpendicular to the field (see figure 5) must\nbe measured and vectorially added resulting in the total\nmagnetization, presented in figure 6. The total entropy6\nchange calculated from the computed total magnetiza-\ntionis shownin figure11. Notice that theentropychange\ncurves show a pronounced peak reaching values twice as\nhigh as the one observed in the c /bardblµ0H case (see figure\n10). Numerically, this peak is the direct result of the\nlarge area observed at low fields in the isothermal data\naround the first order phase transition, indicated in blue\nin figure6. In turn, this largeareaspansfromthemagne-\ntocrystalline anisotropy and its interaction with the first\norder phase transition in Fe 2P.\nTo understand the reason for this peculiar behavior\na more detailed analysis of the magnetization process is\nrequired. First we look into the separate components of\nthe magnetization when the crystal is aligned with its\neasy axis perpendicular to the applied magnetic field. In\nfigure 5 one can see that as the applied magnetic field is\nincreasedthecomponentofthemagnetizationperpendic-\nular to the magnetic field presents an initial increase, due\nto the alignment of domains. Subsequently, the magni-\ntude of the magnetization response decreases as the mo-\nment rotates towards the magnetic field direction, and\nthis decrease becomes sharper as temperature increases\nand magnetocrystallineanisotropyis reduced. The coun-\nterpart of this can be observed in the component of the\nmagnetization parallel to the magnetic field which in-\ncreasesasthe perpendicularcomponentdecreases. When\nthe components of the magnetization parallel and per-\npendicular to the field are added a different scenariothan\nthat observed when c /bardblµ0H is obtained (see figure 6). In\nall curves above T Can initial increase of the magneti-\nzation is observed, followed by a small decrease which is\nthen overcome so that the magnetization keeps increas-\ning and saturates. Whereas above T Cthe magnetization\nincreases monotonically.\nThe different behaviors below and above T Care easily\nunderstood considering that the anisotropy field disap-\npears above T Cas magnetic ordering is lost. However,\nbelow T Cthe influence of magnetocrystalline anisotropy\ncanbeclearlyobservedastheslightdipinthemagnetiza-\ntion curves which becomes more pronounced around the\nfirst order phase transition. As can be observed in figure\n8 a field induced transition can only be observed at very\nlow fields and at a narrow temperature interval around\nthe first order phase transition when c /bardblµ0H. Since the\nmagnetic moments in Fe 2P are aligned along the crys-\ntallographic c-axis it is straightforward to assume that a\nfield induced transition can only be observed along this\naxis. This is supported by the measured data in the c ⊥\nµ0H case, since no field induced transition is observed in\nthe component of the magnetization parallel to the ap-\nplied field, i.e. along the hard direction. However, the\nlarge area present at low field around T Cobserved in the\ntotal magnetization measured with c ⊥µ0H (blue area\nin figure 6) can only be explained considering the inter-\naction of the magnetocrystalline anisotropy and the field\ninduced transition.\nAround T Cas the applied magnetic field is increased a\nfield induced transition develops in the direction perpen- 0 1 2 3 4 5 6 7 8 9\n 0 50 100  150  200  250  300-∆S (J/kgK)\nT (K)c ⊥ µ0H\n0 - 1 T\n0 - 2 T\n0 - 3 T\n0 - 4 T\n0 - 5 T 0 3 6 9\n 0 100  200  3000 - 5 Tc ⊥ µ0H\nc || µ0H\nFIG. 11. Magnetic entropy change as a function of tempera-\nture for different applied magnetic fields in single crystall ine\nFe2P with the field applied parallel to the hard magnetization\ndirection. The inset shows the entropy change as a function\nof temperature for a 0 - 5 T field change measured with the c-\naxis parallel and perpendicular to the applied magnetic fiel d.\ndicular to the magnetic field resulting in a sharp increase\nof magnetization. Notice that, because the effective field\nalong the easy direction when c ⊥µ0H is lower, the field\ninduced phase transition can be observed at apparent\nfields higher than in the case where c /bardblµ0H. However,\ncompeting with that increase is the rotation of the mo-\nment in the direction of the applied field and the absence\nof a field induced transition at higher fields, which effec-\ntively results in a decrease of the magnetization above a\ncertain applied field. Once T Cis crossed both field in-\nduced transition and magnetocrystalline anisotropy are\nabsent, resulting in a monotonous increase of the magne-\ntization with increasing applied field. Thus the different\nbehaviors below and above T Care responsible for the\nlarge peak in the entropy change measured in the hard\ndirection. It is worth noticing that the total entropy, i.e.\nthe area under the entropy change vs. temperature curve\nmeasured with c /bardblµ0H and c ⊥µ0H are, within error,\nthe same.\nFor single crystalline Fe 2P the low temperature mag-\nnetization as a function of the applied magnetic field was\nmeasured up to 9 T (see figure 13). Since a 7 T field is\nenoughtoovercomethe magnetocrystallineanisotropyat\n5 K the turn of the curve from a non-saturating behavior\nto a fully saturated ferromagnetic behavior can be ob-\nserved. Surprisingly, Fe 2P displays strong magnetization\nanisotropy: the saturation magnetization when the field\nis applied perpendicular to the c-direction and enough\nto overcome magnetocrystalline anisotropy is about 9%\nbelow the easy axis saturation magnetization values.\nIV. DISCUSSION\nSubstituting Mn on the Fe site and As, Ge or Si on\nthe P site, the crystalline structure and first order mag-7\n 0 20 40 60 80 100 120\n 0  1  2  3  4  5M (Am2/kg)\nµ0H (T)T = 10 Kc || µ0H\nc ⊥ µ0H\npolycrystal\nFIG. 12. Saturation magnetization at 10 K in polycrystallin e\nand single crystalline Fe 2P with the field applied parallel and\nperpendicular tothe c-direction in thelatter case. Notice that\nthe saturation magnetization of the polycrystal is very clo se\nto that of the single crystal measured with c /bardblµ0H, meaning\nthat our polycrystal is likely to be a collection of well alig ned\ncrystallites.\n-1.2-0.8-0.4 0 0.4 0.8 1.2\n-10-8-6-4-2 0 2 4 6 8 10M (T)\nµ0H (T)T = 5 Kc || µ0H\nc ⊥ µ0H\nFIG. 13. Field dependence of the magnetization at 5 K in\nsingle crystalline Fe 2P with the field applied parallel and per-\npendicular to the c-direction.\nnetoelastic phase transition of pure Fe 2P are retained.\nHowever, tuning the magnetic properties of Fe 2P is not\nas simple as substituting similar elements on one of its\nsites. The substitution or doping on the P site quickly\nshifts T Cup, but also leads to the loss of the first order\nmagnetoelastic coupling. Substituting minute amounts\nof Mn on the Fe site is enough to induce antiferromag-\nnetism and change the crystallographic structure22.\nThis reflects the very delicate balance found in the\nmagnetoelastic coupling of Fe 2P. Thermal and magnetic\nhysteresesareboth quite small, and can only be observed\natverylowfields(see figures2and8). Moreover,increas-\ningappliedmagneticfieldquicklybroadensthefirstorder\nphase transition and effectively shifts T Cto higher tem-\nperatures. Such behavior suggests that the energy bar-\nrier needed to be overcome to go between paramagneticand ferromagnetic states is quite low. The high δTC/δB\ncombined with a low magnetic entropy change imply a\nweak magnetoelastic coupling which is easily affected by\nan external magnetic field. From the Clausis-Clapeyron\nequation it is straightforward to conclude that a large\nδTC/δB should result in a low entropy change ∆S:\n∆Stotal(T,∆H) =−∆M/parenleftbiggδTC\nδB/parenrightbigg−1\nwhere ∆M is the change in magnetization due to the\ntransition. Consequently, a low adiabatic temperature\nchange ∆T adis also expected, since it is proportional to\nthe entropy change itself. In this sense the behavior of\nFe2P is very similar to that of the MnCoSi compound\nreported by Sandeman et al.23. MnCoSi shows an even\nlarger sensitivity of the magnetic phase transition to the\napplied magnetic field, reaching values of -50 K/T. Ac-\ncordingly, it also displays low entropy changes, even if\nthe metamagnetic transition survives to very high fields,\nunlike Fe 2P. It is worth noticing that the high peak in\nthe entropy change for Fe 2P when measured with c ⊥\nµ0H is directly reflected in the low field δTC/δB. For\nfields below 0.5 T, due to magnetocrystalline anisotropy,\nTCremains virtually unchanged at 218 K, resulting in a\nlowerδTC/δB and a much higher ∆S than in the case\nwhere c/bardblµ0H.\nThese properties are in stark contrast with most\n(Fe,Mn) 2(P,A) compounds, where A = As, Ge or Si.\nIn (Fe,Mn) 2(P,A) compounds, while thermal hysteresis\ncan often be reduced by the correct synthesis process-\ning methods, it can always be observed up to very high\nmagnetic fields, around 5 T (see figure 14). The tran-\nsition is also hardly broadened by field when compared\nto pure Fe 2P. This becomes particularly evident when\nthe field dependence of the Curie temperatures for pure\nFe2P and (Fe,Mn) 2(P,A) materials, δTC/δB are com-\npared. The first order phase transition in pure Fe 2P is\nextremely sensitive to the applied magnetic field, which\ncauses it not only to broaden but also to be shifted to\nhigher temperatures very quickly. In fact the field de-\npendence of the Curie or transition temperature of Fe 2P\nis not linear and can be as high as 30 K/T for low fields.\nIn (Fe,Mn) 2(P,A) compounds the situation is quite dif-\nferent. The transition is not so easily affected by the\napplied magnetic field, keeping its first order character-\nistics up to 5 T and higher. The observed δTC/δB is\nlinear for (Fe,Mn) 2(P,A) materials, as well as compara-\ntively moderate, reaching maximum values of 4 K/T24.\nTherefore, (Fe,Mn) 2(P,A) materials yield much higher\n∆SMand ∆T adthan Fe 2P (see figure 15). These differ-\nences suggest that the energy barrier associated with the\nfirst order phase transition in (Fe,Mn) 2(P,A) compounds\nis much higher than in the parent compound. This is\nalso reflected in the size of the lattice parameters change\ndue to the transition in the two cases. The jump in the\nlattice parameters in (Fe,Mn) 2(P,A) compounds is much\nlarger than in Fe 2P8. This has intricate consequences\nwhich arise from the nature and change of the magne-8\n 0 25 50 75 100 125\n 175 200 225 250 275 300 325 350M (Am2/kg)\nT (K)Fe2PMn1.30Fe0.65P0.5Si0.5\nFIG. 14. Temperature dependence of the magnetization at\ndifferent applied magnetic fields for polycrystalline Fe 2P and\nMn1.30Fe0.65P0.5Si0.5. Notice that the magnetic fields used\nhere are the same as in figure 1.\n 0 4 8 12 16\n 200  240  280  320  360-∆S (J/kgK)\nT (K)Fe2PMn1.30Fe0.65P0.5Si0.5\n0 - 1 T\n0 - 2 T\nFIG. 15. Temperature dependence of the entropy change for\npolycrystalline Fe 2P and Mn 1.30Fe0.65P0.5Si0.5.\ntoelastic coupling in both Fe 2P and (Fe,Mn) 2(P,A) com-\npounds. The key to understanding these materials lies\nin the coupling of the two different magnetic sublattices.\nThis becomes clear when the interatomic distances are\nchanged. Relatively low pressures are enough to induce\nantiferromagnetism in Fe 2P25. Since the a-direction is\nthe most compressible one26, it is straightforward to as-\nsume that pressure decreases Fe I-FeIand Fe II-FeIImore\nsignificantly than Fe I-FeIIinteratomic distances. Mn\nsubstitution in the Fe site increases the lattice param-\neters and thus interatomic distances, but since Mn has a\nhigher magnetic moment than Fe the exchange interac-\ntions are also larger. Therefore the Mn-Mn interatomic\ndistances are below the critical distance Mn needs to be\nable to order ferromagnetically27, resulting in antiferro-\nmagnetic ordering instead. Thus the insertion of a larger\nnon-magnetic atom, which acts very much as a spacer,\nis needed to increase Mn-Mn distances above the critical\nvalue where it should order ferromagnetically28. This is\nachieved by partially substituting P by As, Ge or Si, en-abling not only ferromagnetic order but also recovering\nthe first order character of the transition found in pure\nFe2P.\nSince the Fe Isublattice is mainly paramagnetic and\nacquires moment due to the interaction with the higher\nmoment Fe II/MnIIsublattice, larger lattice parameters\nmean that a larger change in the phase transition is nec-\nessary to bring the system from the paramagnetic to the\nferromagneticstateandviceversa. Thisresultsinamuch\nlarger change in electronic configuration than in pure\nFe2P, as well as a latent heat contribution at least one\norder of magnitude higher11,29. The larger magnetic mo-\nmentofMnconsiderablyenhancestheexchangefieldgen-\nerated by the Mn II/FeIIsublattice which in turn causes\na much sharper and marked change in the Fe Isublat-\ntice. This is in agreement with first principle calculation\nresults obtained by Delczeg-Czirjak et al.12which show\nthat the structural effects in doped and substituted Fe 2P\ncompounds strengthen the magnetic interactions relative\nto pure Fe 2P.\nSimilarcalculationson (Fe,Mn) 2(P,Si) compoundsalso\npoint to a strongermagnetoelastic coupling and to a sim-\nilar interaction between the magnetic sublattices. As in\npure Fe 2P the Mn II/FeIIsublattice generates a large ex-\nchange field which induces order in the weakly param-\nagnetic Fe Isublattice3. In terms of the coupling of each\nmagneticsublattice to the crystallattice, this means that\ntwo distinct behaviors are present. The fact that the Fe I\nsublattice is mostly non-magnetic above T Cmeans that\nthe valence electrons contribute to the bond and do not\ngenerate moment, having an itinerant character and pro-\nviding strong coupling to the crystal lattice. The situ-\nation is quite different for the Mn II/FeIIsublattice. In\nthe latter, the valence electrons generate high moments\nwhich are not lost in the paramagnetic state. This may\npoint at a more localized character, or that a mix of lo-\ncalized and itinerant characters is present in such site.\nThis essentially means that a previously believed itiner-\nant electron system in fact presents a mix of itinerant\nand localized magnetisms.\nThe observation of magnetization anisotropy in Fe 2P\npresents the first experimental evidence to support this\nlast assumption. Let us first consider a purely itiner-\nant system. In such a system all the valence electrons\nshould be located in the conduction band and thus be\nfree to move. Therefore, in a single crystal, once magne-\ntocrystalline anisotropy is overcome by the applied mag-\nnetic field, it should not matter in which direction (easy\nor hard magnetization) the field is applied, the response\nshould be the same. However, if some of the electrons\nare actually localized, a difference should arise depend-\ning on which direction the magnetic field is applied. This\nis exactly the case for Fe 2P (see figure 13). Moreover,the\nMnII/FeIIsublattice presents a higher moment than that\nof the Fe Isublattice, whereas first principle calculations\npredict the latter to lose its moment above T C. Thus,\nit is most likely that the localized character lies in the\nMnII/FeIIsublattice.9\nV. CONCLUSION\nThe magnetic and magnetocaloric properties of high-\npurity poly- and single-crystalline Fe 2P have been stud-\nied. To the authorsknowledgethis is the first time that a\ncomplete magnetocaloriccharacterizationofpure Fe 2P is\ncarriedout. Alow but broadentropychangepeak as well\nas a strong field dependence of the first order phase tran-\nsitionareobserved. Auniqueinteractionbetweenmagne-\ntocrystalline anisotropy and the first order phase transi-\ntionwasobservedwhilemeasuringsinglecrystallineFe 2P\nwith its hard direction parallel to the applied magnetic\nfield, confirming that not only the moments are aligned\nin the c direction but also that the first order phase tran-\nsition is tied to the c-axis.\nComparison with the known properties of\n(Fe,Mn) 2(P,A) compounds provided considerable\ninsight on the nature of the coupling and thus the origin\nof the magnetocaloric properties of these compounds.\nPure Fe 2P is found to have a weaker magnetoelastic\ncoupling than (Fe,Mn) 2(P,A) compounds, clearly visible\nin stronger first order characteristics such as thermal\nhysteresis and larger volume changes found in the latter.\nThis is in good agreement with the first principle calcu-\nlations of Delczeg-Czirjak et al.12which also conclude\nthat dopings and substitutions strengthen the magnetic\ninteractions.\nMagnetization anisotropy was found to occur in this\nsystem, experimentally showing that a previously be-\nlieved fully itinerant electron system actually displays a\nmix of localized and itinerant characters. Further analy-\nsisstronglysuggeststhatsuchlocalizedcharacterisprob-\nably present in the Mn II/FeIIsublattice.\nACKNOWLEDGMENTS\nFinancial support from the Swedish Energy Agency\n(STEM) and the Swedish Research Council (VR) is ac-\nknowledged. This work is part of an Industrial Part-\nnership Programme IPP I28 of the Stichting voor Fun-\ndamenteel Onderzoek der Materie (FOM) which is fi-\nnancially supported by the Nederlandse Organisatievoor\nWetenschappelijk Onderzoek (NWO) and co-financed by\nBASF Future Business. The authors would like to thank\nDr. Niels van Dijk from the Delft University of Technol-\nogy (TUDelft) for valuable suggestions and discussions.\nAppendix: MCE and Magnetic Anisotropy\nAn internal magnetic field lifts the degeneracy of the\nenergy levels of the spin (angular momentum) states.\nThis is at the basis of the magnetocaloric effect and\nfrom this we immediately also see the applicability of\nthe Maxwell relations, because only the projection of the\nmagnetic moments with respect to the internal field is\nimportant to characterize the occupancy of the differentenergy levels as depicted in figure 16. As described by\nWeiss and Piccard20, the internal field is composed of the\napplied magnetic field and the field generated by neigh-\nboring moments. In a soft ferromagnet these two fields\nare parallel and we don’t need to worry about the mo-\nment direction. In a single crystal of a hard magnet this\nis not the case. Below we give some considerations and\nexperimentalevidencefortheeffectofmagnetocrystalline\nanisotropy.\nIn the absence of an applied field in the ferromagnetic\nstate the moments are all aligned along the easy axis\nand due to demagnetizing effects no net magnetic mo-\nment is observed. If a perfect crystal is placed with its\nhard axis exactly parallel to the direction of an abso-\nlutely homogeneous applied field, no net magnetization\nshould be observed in the easy axis direction. This can\nbe verified by a simple symmetry argument. However,\nthis ideal situation is hardly ever achieved experimen-\ntally, and a moment is always induced along the easy\ndirection. Therefore, to properly evaluate the dynam-\nics of the transition and the magnetocaloric effect of a\nsingle crystal under such conditions both components of\nthe magnetization should always be measured, even if it\nis solely to confirm that your crystal is perfectly aligned!\nHeff+2 \n+1 \n0\n-1 \n-2 ml\nHα\nHeffH\n/g301α\nθ\nFIG. 16. [left] Representation of the Fe 2P needle depicting\nboth applied H and effective H efffields in relation to the c-\naxis of the needle and the corresponding angles. [right] Vec -\ntor model of the atom applied to the situation where l = 2\nin/planckover2pi1/radicalbig\nl(l+1) and non zero applied field at an angle αwith\nrespect to the effective field.\nHere, the fact that we measure a moment in the di-\nrection perpendicular to the applied magnetic field when\nthe hard direction is aligned parallel is due to a slight\nmisalignment. Such misalignment can be estimated from\nthe demagnetization factors calculated when measuring\nthe crystal with its easy axis perpendicular and parallel\nto the applied magnetic field to be around 3◦. Although\nthis value is within the accuracy of the measurement it\nalso carries the error due to the alignment in two differ-\nent measurements and therefore must be considered with\ncare.\nThismeansthattheangle θisnot90◦but90◦±3◦. As\na consequence a moment is induced along the easy axis\ncausing the total magnetization and the effective field to\npoint at an angle θaway from the easy magnetization di-10\n 0 15 30 45 60 75 90\n 0  50  100  150  200  250α\nT (K)c ⊥ µ0H0.4 T\n1 T\nFIG. 17. Temperature dependence of the angle between the\neffective field (or the total magnetization) and the applied\nfield when the latter is applied perpendicular to the easy mag -\nnetization direction.rection or at the complementary angle αaway from the\napplied magnetic field direction. This is represented in\nfigure 16. As expected, with increasing applied field the\ntotalmagnetizationrotatestowardsthedirectionperpen-\ndicular to the c-axis and parallel to the applied magnetic\nfield. Temperaturehasasimilareffect due tothe temper-\nature dependence of the magnetocrystalline anisotropy\nshown in figure 7.\nThis can be clearly observed plotting angle isofields as\none would do for magnetization. In figure 17 the temper-\nature dependence of the angle αat selected applied fields\nis shown. For low fields (0.4 T) the angle only changes\nsignificantly around the first order ferro-paramagnetic\ntransition, at which magnetocrystalline anisotropy dis-\nappears. For higher fields the change is more gradual\nsince the magnitude of the magnetic field is comparable\nto that of the anisotropy field.\n∗L.Caron@tudelft.nl\n1O. Tegus, E. Br¨ uck, K. H. J. Buschow, and F. R. de Boer,\nNature.415, 150 (2002).\n2N. T. Trung, Z. Q. Ou, T. J. Gortenmulder, O. Tegus,\nK. H. J. 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When a\nlow field is applied perpendicular to c we see at 5 K a very\nlow magnetic response, though the magnetic moments in\nthe needle are aligned along the c axis and thus the sample\nis fully magnetized.\n22H. Fujii, T. Hokabe, K. Eguchi, H. Fujiwara, and\nT. Okamoto, Journal of the Physics Society Japan 51, 414\n(1982).\n23K. Sandeman, R. Daou, S. Ozcan, J. Durrell, N. Mathur,\nand D. Fray, Phys. Rev. B 74(2006), 10.1103/Phys-\nRevB.74.224436.\n24L. Caron, Z. Ou, T. Nguyen, D. Cam Thanh, O. Tegus,\nand E. Br¨ uck, J. Magn. Magn. Mater. 321, 3559 (2009).\n25H. Fujiwara, H. Kadomatsu, K. Tohma, H. Fujii, and\nT. Okamoto, Journal of Magnetism and Magnetic Materi-\nals21, 262 (1980).\n26H. Fujiwara, M. Nomura, H. Kadomatsu, N. Nakagiri,\nT. Nishizaka, Y. Yamamoto, H. Fujii, and T. Okamoto,\nJournal of the Physical Society of Japan 50, 3533 (1981).\n27T. Suzuki, Y. Yamaguchi, H. Y. Watanabe, and Hiroshi,\nJournal of the Physical Society of Japan 34, 911 (1973).\n28D. Liu, Q. Huang, M. Yue, J. Lynn, L. Liu, Y. Chen,\nZ. Wu, and J. Zhang, Phys. Rev. B 80(2009),\n10.1103/PhysRevB.80.174415.\n29N. T. Trung, Ph.D. thesis, Delft University of Technology\n(2010)."    },    {        "title": "1307.5961v2.Mechanism_of_uniaxial_magnetocrystalline_anisotropy_in_transition_metal_alloys.pdf",        "content": "arXiv:1307.5961v2  [cond-mat.mtrl-sci]  3 Mar 2014Mechanism of uniaxial magnetocrystalline anisotropy in tr ansition metal alloys\nYohei Kota1,2,∗and Akimasa Sakuma1\n1Department of Applied Physics, Tohoku University, Sendai 9 80-8579, Japan\n2Spintronics Research Center, AIST, Tsukuba, Ibaraki 305-8 568, Japan\n(Dated: March 4, 2014)\nMagnetocrystalline anisotropy in transition metal alloys (FePt, CoPt, FePd, MnAl, MnGa, and\nFeCo) was studied using first-principles calculations to el ucidate its specific mechanism. The tight-\nbinding linear muffin-tin orbital method in the local spin-de nsity approximation was employed to\ncalculate the electronic structure of each compound, and th e anisotropy energy was evaluated using\nthe magnetic force theorem and the second-order perturbati on theory in terms of spin-orbit interac-\ntions. We systematically describe the mechanism of uniaxia l magnetocrystalline anisotropy in real\nmaterials and present the conditions under which the anisot ropy energy can be increased. The large\nmagnetocrystalline anisotropy energy in FePt and CoPt aris es from the strong spin-orbit interaction\nof Pt. In contrast, even though the spin-orbit interaction i n MnAl, MnGa, and FeCo is weak, the\nanisotropy energies of these compounds are comparable to th at of FePd,. We found that MnAl,\nMnGa, and FeCo have an electronic structure that is efficient i n inducing the magnetocrystalline\nanisotropy in terms of the selection rule of spin-orbit inte raction.\nI. INTRODUCTION\nUniaxial magnetic anisotropy is an essential property\nof hard magnetic materials, which are widely utilized\nas permanent magnets and perpendicularly magnetized\nfilms. Magnetic anisotropy has several origins, with the\nmain one being magnetocrystalline anisotropy, which is\ninduced by spin-orbit interactions in addition to the\nanisotropic crystal field.1Since spin-orbit interactions\nare a relativistic effect on electron motion, a large mag-\nnetocrystalline anisotropy has been observed from com-\npounds with heavy elements such as rare-metal and rare-\nearth elements.\nThe transition metal systems FePt, CoPt, and FePd,\nwith a tetragonal crystal structure and heavy 4 dor 5d\nelements, show superior performance as hard magnetic\nmaterials.2–9However, since Pt and Pd are categorized\nas typical rare-metals, there has been a strong demand\nfor rare-element-free magnets such as MnAl, MnGa, and\nFeCo in recent years. Both MnAl and MnGa are fer-\nromagnetic and show uniaxial magnetic anisotropy,10,11\nand a large anisotropy constant has been observed in\ntheir film samples.12–17In contrast to MnAl and MnGa,\na giant magnetocrystalline anisotropy in FeCo was first\npredicted theoretically using a first-principles calculation\nunder optimal conditions in terms of Co concentration\nand tetragonal distortion.18–21Experimentally, uniaxial\nmagneticanisotropywasconfirmedinartificiallystrained\nfilm samples.22–26\nFirst-principles calculations of magnetocrystalline\nanisotropy in transition metal systems have been carried\nout for bulk crystals, monolayers, and multilayers.27–41\nThese studies clarified the experimental results of the di-\nrection of the magnetic easy axis and the relative magni-\ntude of magnetocrystalline anisotropy energy in real ma-\nterials on a quantitative level, although the calculation\nrequiresahighaccuracybecauseofthesmallenergyscale\nof the spin-orbit interaction compared with the energy\nscales of the crystal and exchange fields. Furthermore,a perturbation analysis of the spin-orbit interaction was\nperformed for the magnetocrystalline anisotropy energy\nof ordered FeNi,42and the physical origin of the perpen-\ndicular magnetic anisotropy was quantified.\nMagnetocrystalline anisotropy originates from the\nspin-orbit interaction and the anisotropy energy can\nbe evaluated quantitatively from first-principles calcula-\ntions; however, a qualitative description of the charac-\nteristics in real materials remains an issue. In particular,\nit is important to clarify the mechanism of magnetocrys-\ntalline anisotropy in compounds with no heavy elements.\nIn this paper, we present the characteristicsof the uniax-\nial magnetocrystalline anisotropy of the transition metal\nalloys FePt, CoPt, FePd, MnAl, MnGa, and FeCo with\nordered structures, by using first-principles calculations.\nWe analyze the magnetocrystalline anisotropy energy of\nthese compounds using the magnetic force theorem and\nthe second-order perturbation theory in terms of spin-\norbit interactions to elucidate its specific mechanism.\nThis paper is organized as follows. Details of the cal-\nculation method are given in Sec. 2, and the consistency\nof the perturbation is confirmed in Sec. 3 through a com-\nparison of the magnetocrystalline anisotropy energy cal-\nculatedusingtheforcetheoremandtheperturbationthe-\nory. In Sec. 4, we discuss the mechanism of the uniax-\nial magnetocrystalline anisotropy in FePt, CoPt, FePd,\nMnAl, MnGa, and FeCo. A conclusion is provided in\nSec. 5.\nII. METHODOLOGY\nA. Electronic structure calculation\nFePt, CoPt, FePd, MnAl, and MnGa ordered alloys\nhave the so-called L10-type structure, which can be re-\nduced to the tetragonally distorted B2-type structure\nthat corresponds to a primitive cell.35Tetragonal FeCo\nordered alloy also has the distorted B2-type structure.2\nac\nxyz\nFIG. 1. Crystal lattice of the distorted B2-type structure.\nThe unit cell consists of two sublattices that are located at\nthe corner site (open circles) and the body-center site (clo sed\ncircles).\nThus, we used the unit cell of a crystal lattice compris-\ning two atoms, which are located at the corner and body-\ncenteredsite, asshowninFig.1. Theradiusoftheatomic\nsphere, which determines the volume of the cell, and the\naxial ratio c/awere set to the values used in previous\nstudies.20,28,34,39We considered the ferromagnetic state\nof these alloys.\nTo calculate the electronic structure, we employed\nthe tight-binding linear muffin-tin orbital (TB-LMTO)\nmethod in the atomic sphere approximation.43,44In this\nmethod, the matrix representation of the Kohn-Sham\nHamiltonian without the spin-orbit interaction term H0\nis given by\nH0=C+√\n∆S(1−γS)−1√\n∆, (1)\nwhereC={CRℓσ}, ∆ ={∆Rℓσ}, andγ={γRℓσ}\nare the matrices of the potential parameters in scalar\nrelativistic form, and S={SRL,R′L′}is the matrix of\nthe canonical structure constant.45,46Note that the sub-\nscriptsR,L= (ℓ,m), andσdenote the atomic site, or-\nbital, and spin index, respectively. The potential param-\neters are self-consistently determined in the local spin-\ndensity approximation.\nB. Evaluation of magnetocrystalline anisotropy\nenergy\nFor the evaluation of the magnetocrystalline\nanisotropy energy, we need to take into accountspin-orbit interactions,33,47\nHSO=ξ\n2U(θ,φ)(ℓ·σ)U†(θ,φ), (2)\nin addition to H0. Note that ξ={ξRℓ}is the matrix of\nthe spin-orbit coupling constant, which is estimated by\nsolving the relativistic Dirac equation in a single atomic\nsphere. Here, ℓis the orbital angular momentum op-\nerator,σis the Pauli matrix, and U(θ,φ) is the SU(2)\nrotation matrix of the spin-1/2 system,48\nU(θ,φ) =/parenleftbigg\nexp(iφ\n2)cos(θ\n2) exp(−iφ\n2)sin(θ\n2)\n−exp(iφ\n2)sin(θ\n2) exp(−iφ\n2)cos(θ\n2)/parenrightbigg\n,(3)\nwhereθandφare the angles of the direction of mag-\nnetization when the unit vector is defined by n=\n(sinθcosφ,sinθsinφ,cosθ). Hereafter, we set φ= 0 to\nfocus on the uniaxial anisotropy.\nThe magnetocrystalline anisotropy energy ∆ Ecan be\nevaluated from the energy difference when the magneti-\nzation is aligned in the θ= 0 and θ=π/2 directions.\nFrom the magnetic force theorem,49,50∆Eis given by\nthe band energy difference as\n∆E=occ/summationdisplay\nn/summationdisplay\nkεkn|θ=π\n2−occ/summationdisplay\nn/summationdisplay\nkεkn|θ=0,(4)\nwhereεknis the eigenvalue of the Hamiltonian with the\nspin-orbit interaction H0+HSO, labeled by the wavevec-\ntorkin the first Brillouin zone and the band index n.\nUsing Eq. (4) is a common way of evaluating the mag-\nnetocrystalline anisotropy energy; however, the physical\npicture and mechanism is difficult to comprehend.\nTo address this difficulty, we adopt the expression\nof the magnetocrystalline anisotropy energy within the\nsecond-order perturbation theory in terms of spin-orbit\ninteractions. An expression has been derived in previous\nworks;31,51,52however, here we present a reformulation\nof this expression under the same concept: the energy\nvariation δEdue to spin-orbit interaction is written as\nδE=−1\n4occ/summationdisplay\nnunocc/summationdisplay\nn′/summationdisplay\nk/summationdisplay\n{R}/summationdisplay\n{L}/summationdisplay\n{σ}ξRℓξR′ℓ′′·ρknσ\nR′L′′′,RLρkn′σ′\nRL′,R′L′′\nεkn′σ′−εknσ·/an}bracketle{tLσ|U(ℓ·σ)U†|L′σ′/an}bracketri}ht/an}bracketle{tL′′σ′|U(ℓ·σ)U†|L′′′σ/an}bracketri}ht,(5)3\nand the anisotropy energy ∆ E=δE|θ=π\n2−δE|θ=0is given by\n∆E=/summationdisplay\n{R}/summationdisplay\n{L}/summationdisplay\n{σ}∆ER,R′(L′′′Lσ;L′L′′σ′) (6a)\n∆ER,R′(L′′′Lσ;L′L′′σ′) =1\n4occ/summationdisplay\nnunocc/summationdisplay\nn′/summationdisplay\nkξRℓξR′ℓ′′·ρknσ\nR′L′′′,RLρkn′σ′\nRL′,R′L′′\nεkn′σ′−εknσ·C (6b)\nC=τσ,σ′·[/an}bracketle{tL|ℓz|L′/an}bracketri}ht/an}bracketle{tL′′|ℓz|L′′′/an}bracketri}ht−/an}bracketle{tL|ℓx|L′/an}bracketri}ht/an}bracketle{tL′′|ℓx|L′′′/an}bracketri}ht]. (6c)\nNote that εknσis the eigenvalue of the nonperturbative\nstate, and ρknσ\nR′L′,RL= (cknσ\nRL)∗cknσ\nR′L′are products of the\nexpansion coefficients of the eigenstates on an atomic or-\nbital basis, i.e.,|knσ/an}bracketri}ht=/summationtext\nRLcknσ\nRL|RLσ/an}bracketri}ht. We can ob-\ntain the coefficient cknσ\nRLby solving the secular equation\ndet(ε− H0) = 0 directly, since the TB-LMTO is an or-\nthonormalbasisin R,L, andσ. The factor τσ,σ′gives+1\nfor the same-spin ( σ=σ′) case and −1 for the opposite-\nspin (σ=−σ′) case. In Eqs.(5) and (6), the summations\nover terms in curly brackets denote double or quadru-\nple sums, i.e.,{R}=R,R′,{L}=L,L′,L′′,L′′′, and\n{σ}=σ,σ′. The summation over n(n′) is restricted\nto the occupied (unoccupied) states below (above) the\nFermi level εF, which are determined by the condition\n/integraldisplayεF\n−∞/summationdisplay\nknσδ(ε−εknσ)dε=Nq, (7)\nwhereNqis the number of valence electrons of each com-\npound. Approximately 503k-points in the full Brillouin\nzone were employed for the calculation of the magne-\ntocrystalline anisotropy energy to achieve sufficient ac-\ncuracy in the following numerical results.\nFrom Eq. (6), we are able to conceive a physical pic-\nture in which magnetocrystallineanisotropyis attributed\nto the hybridization between the occupied and unoccu-\npied states through spin-orbit interaction. In particular,\nsuch interaction includes the selection rule with respect\nto the spin and orbital states due to the matrix element\nof the operators. In addition, the degree of hybridization\nis dominated by the strength of spin-orbit interaction, ξ,\nand the inverse of the energy difference between the two\nstates,|εocc−εunocc|−1, on both sides of the Fermi level\nεF.\nIII. CONFIRMATION OF RESULT FROM\nPERTURBATION THEORY\nWe began by confirming that the calculation of\nthe magnetocrystalline anisotropy energy based on the\nsecond-order perturbation theory [Eq. (6)] is consistent\nwith the calculationbased on the magnetic force theorem\n[Eq. (4)]. Figure 2 shows the ∆ Ecurve of FePt calcu-\nlated by both approaches as a function of the number\nof valence electrons Nq. Note that the observed mag-\nnetocrystalline anisotropy energy corresponds to that atNq= 18, which is the actual number of valence elec-\ntrons in FePt. The Nqdependence was evaluated within\nthe rigid-bandschemeusingthe intrinsicelectronicstruc-\nture of FePt ( Nq= 18) as the self-consistent solution.\nAn investigation of the Nqdependence is useful for dis-\ncussing the relationship between the physical quantities\nand electronic structure as well as the consistency of the\ncalculation.\nIn the comparison shown in Fig. 2, the absolute val-\nues and trends of the two ∆ Ecurves are in good agree-\nment with each other. There is very little difference\nbetween the results apart from the disagreement in the\n5< Nq<10 region where the hypothetical Fermi\nlevel is located at the center of the d-orbital bands of Pt,\nwhich has strong spin-orbit interaction. Table I shows\nthe magnetocrystalline anisotropy energy calculated us-\ningtheforcetheoreminEq.(4)andtheperturbationthe-\nory in Eq. (6) for each compound. In FePt, CoPt, FePd,\nMnAl, and MnGa, the ∆ Evalues obtained from the per-\nturbation theory are quantitatively consistent with those\nobtained from the force theorem, since the magnitude\nrelation in these materials is the same and the numeri-\ncal deviation between the two results is less than 20%.\nFurthermore, the calculated results reproduce the exper-\nimental results with respect to the relative magnitude of\n−15 −10  −5  0  5 10 15\n 0  5 10 15 20 ∆E   [meV/f.u.] \nNq  Force theorem\n  Perturbation theory\nFIG. 2. Magnetocrystalline anisotropy energy ∆ Ein FePt as\na function of the number of valence electrons Nq. The solid\nand dashed lines are the results calculated using the magnet ic\nforce theorem in Eq. (4) and the second-order perturbation\ntheory in Eq. (6), respectively. The actual electron number\nin FePt is 18.4\nTABLE I. Magnetocrystalline anisotropy energies ∆ Ein ordered FePt, CoPt, FePd, MnAl, MnGa, and FeCo evaluated u sing\nthe force theorem (FT) in Eq. (4) and perturbation theory (PT ) in Eq. (6). c/adenotes the axial ratio of tetragonal lattice\nused in the calculation. Experimental data are also shown fo r comparison. The number in parentheses shown in the ∆ EExp\ncolumn of FeCo is c/ain experimental samples.\nCompound c/a∆EFT[meV/f.u.] ∆ EPT[meV/f.u.] ∆ EFT[MJ/m3] ∆EPT[MJ/m3] ∆ EExp[MJ/m3]\nFePt 1.36 1.90 2.41 11.01 14.00 ∼103, 6.85, 5.06, 4.17, 5.59\nCoPt 1.38 0.68 0.77 4.12 4.66 2.15, 3.09\nFePd 1.36 0.29 0.33 1.70 1.93 1.54, 2.18\nMnAl 1.28 0.34 0.37 1.98 2.15 1.013, 1.414\nMnGa 1.32 0.37 0.41 2.33 2.55 1.616, 2.217\nFeCo 1.15 0.23 0.20 1.61 1.39\nFeCo 1.25 0.88 1.30 6.09 9.01 2.9 (1.18)22,>0 (1.24)26\nFeCo 1.35 0.34 0.38 2.36 2.63\nthe magnetocrystalline anisotropy energy in these ma-\nterials, although most of the experimental results are\nsmaller than the calculation results. One of the reasons\nfor this is that magnetocrystalline anisotropy energy is\nsensitive to the axial ratio and the degree of order.53,54\nWe adopted a constant c/aand a perfectly orderedstruc-\nture in the calculations; however, the axial ratio and the\ndegree of order depend on the sample preparation condi-\ntions in experiments.\nIn a similar manner, the ∆ Evalues obtained from the\nforce theorem and the perturbation theory are consistent\nwith each other in FeCo for c/a= 1.15 and 1.35; how-\never, there is a large disagreement between the results\nin FeCo for c/a= 1.25, for which a giant magnetocrys-\ntalline anisotropyhas been predicted theoretically.18,20,21\nThe deviation between the results obtained by the two\napproaches is more than 30%, even though the strength\nof the spin-orbit interaction in Fe and Co is one order of\nmagnitude smaller than that in Pt. When we looked at\nthe two ∆ Evalues as a function of the axial ratio, the\ndisagreement becomes significant as c/aapproaches1.25.\nThis large disagreement is a result of the perturbation\nfailing at approximately c/a= 1.25,i.e., the assump-\ntion|εocc−εunocc| ≫ξin Eqs. (5) and (6) is inconsistent\nin somek-areas, as discussed in the next section.\nIV. UNIAXIAL MAGNETOCRYSTALLINE\nANISOTROPY MECHANISM\nWe will discuss the characteristics of magnetocrys-\ntalline anisotropy in FePt, CoPt, and FePd in Sec. 4.1,\nMnAl and MnGa in Sec. 4.2, and FeCo in Sec. 4.3. Also\nwewilloverviewthemagnetocrystallineanisotropymech-\nanism in each compound briefly before presentingthe nu-\nmerical results.\nThe magnetocrystalline anisotropy energy in FePt and\nCoPt originates from the heavy Pt atom, whose spin-\norbit interaction is much larger than those of the other\natoms, and thus, the large coupling constant ξin Eq. (6)\nresults in a larger ∆ E. In contrast, MnAl, MnGa, andFeCo for c/a= 1.25 show a large magnetocrystalline\nanisotropy energy that is comparable to that of FePd in\nspite of the absence of a heavy element, because the elec-\ntronic band structure of these compounds is efficient in\nincreasing the magnetocrystalline anisotropy energy. In\nMnAl and MnGa, ∆ Eis dominated by the hybridiza-\ntion between the occupied and unoccupied states that\nare located near the Fermi level through spin-orbit in-\nteraction, specifically, the occupied d↓\nx2−y2and unoc-\ncupiedd↓\nxystates, the occupied d↑\n3z2−r2and unoccu-\npiedd↓\nyzstates, and the occupied d↑\nyzand unoccupied\nd↓\n3z2−r2states. [The notations ↑and↓are used to de-\nnote the majority- and minority-spin states, respectively,\nandL={dxy,dyz,dxz,dx2−y2,d3r2−z2}is thed-orbital\nstate (ℓ= 2)]. These combinations result in large,\npositive contributions to ∆ Eowing to the coefficient C\nthat determines the selection rule of spin-orbit interac-\ntion in Eq. (6). Furthermore, the electronic structure of\nFeCo for c/a= 1.25 is a special case. The energies\nof two particular bands that mainly consist of the d↓\nxy\nandd↓\nx2−y2states on the upper and lower sides of the\nFermi level around the Γ-point are coincidentally close.\nIn this area, ∆ Eis significantly high, since the energy\ndifference of these states, |εocc−εunocc|, is quite small\n(≪ξ). This implies that the perturbation assumption is\nno longer valid.\nFor the numerical calculations, we employed the per-\nturbationtheoryofthemagnetocrystallineanisotropyen-\nergy given by Eq. (6) for FePt, CoPt, FePd, MnAl, and\nMnGa. We focused on the contributions of several com-\nponents suchas the atomic site R, the orbital L= (ℓ,m),\nand the spin σ. For FeCo, we used the magnetic force\ntheorem given by Eq. (4).\nA. FePt, CoPt, FePd\nThe ∆Evalues of FePt, CoPt, and FePd were decom-\nposed into site Rcontributions because the strength of\nthe spin-orbit interaction varies depending on the atomic5\n−15 −10 −5 0  5 10 15 \n 0  5 10 15 20 ∆ER   [meV/atom] \nNq(b) CoPt\nCo \nPt −15 −10 −5 0  5 10 15 \n 0  5 10 15 20 ∆ER   [meV/atom] \nNq(a) FePt\nFe \nPt \n−3−2−1  0  1  2  3\n 0  5 10 15 20 ∆ER   [meV/atom] \nNq(c) FePd\nFe \nPd \nFIG. 3. Site-decomposed magnetocrystalline anisotropy en -\nergy ∆ERas a function of the number of valence electrons\nNqin (a) FePt, (b) CoPt, and (c) FePd. The actual electron\nnumber are 18 in FePt and FePd, and 19 in CoPt.\nspecies. The spin-orbit coupling constants of Fe, Co, Pd,\nand Pt are ξFe,3d= 54 meV, ξCo,3d= 71 meV,\nξPd,4d= 189 meV, and ξPt,5d= 554 meV,\nrespectively.54We defined the expression of the site-\ndecomposed magnetocrystalline anisotropy energy as\n∆ER=/summationdisplay\nR′/summationdisplay\n{L}/summationdisplay\n{σ}∆ER,R′(L′′′Lσ;L′L′′σ′),\nwhich includes the interference effect with other sites ( R\n/ne}ationslash=R′) via the R′summation. Figure 3 shows the site-\ndecomposed ∆ ERvalues of FePt, CoPt, and FePd as\na function of the number of valence electrons Nq. The∆ERcontributions of Pt and Pd are larger than those of\nFe and Co with respect to the amplitude because of the\nstronger spin-orbit interaction in Pt and Pd.\nFocusing on the values at the actual electron number\n(Nq= 18 for FePt and FePd, and Nq= 19 for CoPt),\nthe ∆ERcontributions of Pt in FePt and CoPt and that\nof Fe in FePd are predominant in Fig. 3. We also con-\nfirmed that the calculated results of ∆ ERat the actual\nelectron number are consistent with those of a previous\nstudy.55In FePt and CoPt, the Pt-based state is still lo-\ncated close to the Fermi level (the local densities of states\nin FePt, CoPt, and FePd are shown in Refs. 35 and 41);\ntherefore, the Pt-based state contributes to an increase\nin the magnetocrystalline anisotropy of FePt and CoPt\nbecause of the large spin-orbit interaction of Pt, which is\nabout ten times as large as those of Fe and Co. The roles\nof the Fe and Co atoms are to induce exchange splitting\nin the Pt atom via orbital hybridization,55even though\nPtisnonmagnetic. Consequently,themagnetocrystalline\nanisotropymechanisminFePtandCoPtcanbedescribed\nby the synergistic effect between the large spin-orbit in-\nteraction in Pt and the large exchangesplitting in Fe and\nCo.\nIn contrast, the Pd contribution to the magnetocrys-\ntalline anisotropy of FePd is not as large, as can be ob-\nserved from the fact that the ∆ ERcurve for Pd crosses\nzero near Nq= 18. Thus, the magnetocrystalline\nanisotropy mainly originates from Fe, and the Pd con-\ntribution appears to be very small.\nB. MnAl, MnGa\nFigure 4 shows the ∆ ERvalues of MnAl and MnGa\nas a function Nq. The actual electron number of both\nMnAl and MnGa is Nq= 10, and the spin-orbit cou-\npling constants of Mn, Al, and Ga are ξMn,3d= 40 meV,\nξAl,3p= 20 meV, and ξGa,4p= 155 meV, respec-\ntively. The ∆ Ekcontribution of Mn is large compared\nwith those of Al and Ga at Nq= 10, although the\nMn and Ga contributions in MnGa are comparable for\nNq<6, for which the Ga p-orbital state with a large\nspin-orbit interaction lies around the hypothetical Fermi\nlevel. Therefore, the magnetocrystalline anisotropy of\nboth MnAl and MnGa is dominated by the Mn contri-\nbution.\nFurthermore, a favorable situation that increases the\nmagnetocrystalline anisotropy of MnAl and MnGa is re-\nalized,sincethereisalargepositive∆ ERpeakatapprox-\nimatelyNq= 10 in Fig. 4 despite the small amplitude\nof the ∆ERcurves compared with those of FePt, CoPt,\nand FePd, as shown in Fig. 3. For a detailed analysis of\nthe ∆Evalues in MnAl and MnGa, we investigated the\nspin- and orbital-resolved magnetocrystalline anisotropy\nenergy of Mn: ∆ ELσ;L′σ′\nR= ∆ER,R(LLσ;L′L′σ′). Here-\nafter, the Rsubscript denotes the Mn site. The en-\nergy ∆ELσ;L′σ′\nRcorresponds to the partial contribution\nof the magnetocrystalline anisotropy energy originating6\nTABLE II. Partial contributions of the magnetocrystalline anisotropy due to the hybridization between the occupied Lσand\nunoccupied L′σ′states, ∆ ELσ;L′σ′\nR, in MnAl and MnGa. The unit of energy is meV/Mn atom. The first g roup (rows 3–6)\nand second group (rows 7–12) of the table correspond to the co ntribution from the nonvanishing matrix elements /angbracketleftL|ℓz|L′/angbracketrightand\n/angbracketleftL|ℓx|L′/angbracketright, respectively.\noccupied unoccupied MnAl MnGa\nL L′∆EL↑;L′↑\nR∆EL↑;L′↓\nR∆EL↓;L′↑\nR∆EL↓;L′↓\nR∆EL↑;L′↑\nR∆EL↑;L′↓\nR∆EL↓;L′↑\nR∆EL↓;L′↓\nR\ndyz dzx 0.01 −0.08 −0.01 0.07 0.02 −0.07 −0.01 0.05\ndzx dyz 0.01 −0.08 −0.01 0.07 0.02 −0.07 −0.01 0.05\ndxydx2−y20.16 −0.22 −0.06 0.11 0.14 −0.21 −0.06 0.08\ndx2−y2dxy 0.02 −0.36 −0.01 0.64 0.02 −0.29 −0.01 0.47\ndzx dxy −0.00 0.13 0.00 −0.07 −0.00 0.10 0.00 −0.04\ndxy dzx −0.02 0.13 0.01 −0.02 −0.02 0.10 0.00 −0.02\ndx2−y2dyz −0.01 0.07 0.02 −0.12 −0.01 0.06 0.01 −0.09\ndyzdx2−y2−0.03 0.05 0.03 −0.04 −0.03 0.05 0.03 −0.03\nd3z2−r2dyz −0.05 0.29 0.02 −0.19 −0.05 0.25 0.03 −0.13\ndyzd3z2−r2−0.07 0.30 0.05 −0.21 −0.05 0.26 0.03 −0.12\n−1.0−0.5 0.0 0.5 1.0\n 0  5 10 ∆ER   [meV/atom] \nNq(a) MnAl\n Mn\n Al\n−1.0−0.5 0.0 0.5 1.0\n 0  5 10 ∆ER   [meV/atom] \nNq(b) MnGa\n Mn\n Ga\nFIG. 4. Site-decomposed magnetocrystalline anisotropy en -\nergy ∆ERas a function of the number of valence electrons\nNqin (a) MnAl and (b) MnGa. The actual electron number\nin both MnAl and MnGa is 10.\nfrom the hybridization between the occupied Lσand un-\noccupied L′σ′states through the spin-orbit interaction\nat the Mn site.\nTable II shows the numerical results of the partial con-\ntributions ∆ ELσ;L′σ′\nRof thed-state of Mn in MnAl andMnGa. Here, we adopted real spherical harmonics as a\nbasis for the orbital Lcomponents, i.e.,dxy,dyz,dzx,\ndx2−y2, andd3z2−r2. TheL=L′′′andL′=L′′contri-\nbutions, for which the coefficient in Eq. (6c) is written\nasC=τσ,σ′·[|/an}bracketle{tL|ℓz|L′/an}bracketri}ht|2−|/an}bracketle{tL|ℓx|L′/an}bracketri}ht|2], are considered\nin Table II with respect to the nonvanishing matrix ele-\nments of /an}bracketle{tL|ℓz|L′/an}bracketri}htand/an}bracketle{tL|ℓx|L′/an}bracketri}ht.52Because of the coeffi-\ncientC, the hybridization between same-spin states with\nanonvanishingmatrixelementof ℓz(ℓx) givesacontribu-\ntion to the decrease of the energy in the system when the\nmagnetization is aligned in the z-direction ( x-direction).\nThe hybridization between different-spin states gives the\ncomplementary contribution. The partial contributions\nofL=L′′andL′=L′′′, and the others are omitted be-\ncausethe values arecloseto zeroorquite small compared\nwith those of L=L′′′andL′=L′′.\nIn Table II, the largest ∆ ELσ;L′σ′\nRis the partial con-\ntribution of hybridization between the occupied d↓\nxyand\nunoccupied d↓\nx2−y2states through spin-orbit interaction:\n0.64and0.47meV/MnatominMnAl andMnGa, respec-\ntively. In addition, the second (third) largest contribu-\ntion with a positive ∆ ELσ;L′σ′\nRis that of the hybridiza-\ntion between the occupied d↑\nyzand unoccupied d↓\n3z2−r2\nstates (between the occupied d↑\n3z2−r2and unoccupied d↓\nyz\nstates). Thus, these contributions increase the uniaxial\nmagnetocrystalline anisotropy of MnAl and MnGa.\nAs described in Sec. 2, the selection rule of spin-orbit\ninteraction is characterized by Cin Eq. (6c) and takes\nvalues of C=±4,±3, and±1 when we consider the\nhybridization between the d-orbital ( ℓ= 2) states.52\nIn particular, the hybridization between the same-spin\ndxyanddx2−y2states and the hybridization between the\nopposite-spin dyzandd3z2−r2states give Cvalues of +4\nand +3, respectively, which are the largest and second-\nlargestpositive Cvalues for ℓ= 2. When these occupied\nand unoccupied states are located near the Fermi level,\nthe magnetocrystalline anisotropy energy is expected to7\n−0.8−0.4 0.0 0.4 0.8\n−6 −4 −2 0 2 4(c) Mn d(x2−y2)\n MnAl\n MnGa\nε − εF   [eV] PDOS   [states/eV/atom] −0.8−0.4 0.0 0.4 0.8\n−6 −4 −2 0 2 4(d) Mn d(3z2−r2)\n MnAl\n MnGaPDOS   [states/eV/atom] \nε − εF   [eV] −0.8−0.4 0.0 0.4 0.8\n−6 −4 −2 0 2 4(b) Mn d(yz), d(zx)\n MnAl\n MnGaPDOS   [states/eV/atom] \nε − εF   [eV] −0.8−0.4 0.0 0.4 0.8\n−6 −4 −2 0 2 4(a) Mn d(xy)\n MnAl\n MnGaPDOS   [states/eV/atom] \nε − εF   [eV] \nFIG. 5. Partial densities of states of the Mn d-orbitals, (a) dxy, (b)dyz,dzx, (c)dx2−y2, and (d) d3z2−r2, in MnAl (solid\nline) and MnGa (dashed line). Positive and negative values o n the vertical axis indicate the densities in the majority- a nd\nminority-spin states, respectively.\nincrease because the strength of the hybridization is pro-\nportional to the inverse of the energy difference of the\ntwo states, |εocc−εunocc|−1, as expressed by Eq. (6b).\nThese are the basic conditions necessary for increasing\n∆E, even though the spin-orbit coupling constant ξis\nsmall.\nTo confirm the numerical results in Table II and the\ncorresponding discussion, the partial density of states\n(PDOS), ρRLσ(ε) =/summationtext\nknρknσ\nRL,RLδ(ε−εknσ), is shown\nin Fig. 5 for each d-orbital component in Mn. The shapes\nof the PDOS of Mn in MnAl and MnGa are similar, al-\nthough there is a small difference in the peak position.\nWe find that the PDOSs of the occupied d↓\nxyand unoccu-\npiedd↓\nx2−y2states of both MnAl and MnGa are located\nnear the Fermi level. There is a high probability that\nthese two states will hybridize through spin-orbit inter-\naction owing to the small |εocc−εunocc|; and therefore,\nthe value of ∆ ELσ;L′σ′\nRis significantly large, as presented\nin Table II. Figure5alsoincludes the PDOSsofthe occu-\npiedd↑\nyzand unoccupied d↓\n3z2−r2states and the occupied\nd↑\n3z2−r2andunoccupied d↓\nyzstatesaroundtheFermilevel,\nand these states additionally contribute to the total ∆ E.\nThe large positive ∆ Epeak at approximately\nNq= 10 in Fig. 4 is mainly attributed to the hybridiza-\ntion between the occupied d↓\nxyand unoccupied d↓\nx2−y2states, the occupied d↑\nyzand unoccupied d↓\n3z2−r2states,\nand the occupied d↑\n3z2−r2and unoccupied d↓\nyzstates\nthrough spin-orbit interaction. Therefore, the mag-\nnetocrystalline anisotropy energy of MnAl and MnGa,\nwhich is comparable to that of FePd, originates from the\nfavorable electronic structure in accordance with the se-\nlection rule, although the spin-orbit coupling constant of\nMn is small.\nC. FeCo\nA giant magnetocrystalline anisotropy has been pre-\ndicted from tetragonal Fe 1−xCoxdisordered alloys un-\nder conditions of an axial ratio of c/a∼1.25 and\nCo concentration of x∼0.5.18Subsequent works\nhave revealed that the magnetocrystalline anisotropy en-\nergy increases further in ordered FeCo ( x= 0.5) for\nc/a∼1.25;19–21thus, we consider ordered FeCo for\nc/a= 1.15, 1.25, and 1.35 to quantify the giant magne-\ntocrystallineanisotropy. Figure6showsthe∆ Evaluesof\norderedFeCo for c/a= 1.15, 1.25 and 1.35as a function\nofNqcalculated using the magnetic force theorem given\nby Eq. (4), where the actual electron number of FeCo is\nNq= 17. In FeCo for c/a= 1.25 [Fig. 6(b)], a sharp\n∆Epeak is located just at Nq= 17; in contrast, ∆ E8\n−1.0−0.5 0.0 0.5 1.0\n 0  5 10 15 20 ∆ER   [meV/f.u.] \nNq(b) FeCo  (c/a=1.25)−1.0−0.5 0.0 0.5 1.0\n 0  5 10 15 20 ∆ER   [meV/f.u.] \nNq(a) FeCo  (c/a=1.15)\n−1.0−0.5 0.0 0.5 1.0\n 0  5 10 15 20 ∆ER   [meV/f.u.] \nNq(c) FeCo  (c/a=1.35)\nFIG. 6. Magnetocrystalline anisotropy energy ∆ Eas a func-\ntion of the number of valence electrons Nqin FeCo for c/aof\n(a) 1.15, (b) 1.25, and (c) 1.35. The actual electron number\nis 17.\natNq= 17 is not so large for c/a= 1.15 [Fig. 6(a)]\nand 1.35 [Fig. 6(c)].\nThe origin of the large magnetocrystalline anisotropy\nin tetragonal Fe-Co alloys has been explained in Ref. 18.\nThe large ∆ EatNq= 17 for c/a= 1.25 in Fig. 6(b)\nis attributed to the closing of the two particular bands\nthat mainly consist of the d↓\nxyandd↓\nx2−y2states near the\nFermi level around the Γ-point.\nInabody-centeredcubic crystal( c/a= 1), the energy\nbands that mainly consist of the t2g(dxy,dyz,dzx) or-\nbital states are triply degenerate, and the energy bands\nthat mainly consist of the eg(dx2−y2,d3z2−r2) orbitalstatesaredoubly degenerateat the Γ-point. The t2g-and\neg-based states are located below and above the Fermi\nlevel, respectively, in the minority-spin states of Fe-Co\nalloys. In contrast, in a body-centered tetragonal crystal\n(c/a > 1), the triple degeneracy in the t2gstates and\nthe double degeneracy in the egstates are resolved, and\nthe energy level of the dxy-based (dx2−y2-based) states\nshifts upward (downward) with increasing c/a.18\nFigure 7 shows the band dispersion εkof FeCo calcu-\nlated without the spin-orbit interaction in the minority-\nspin states. For c/a= 1.15 [Fig. 7(a)], there are two\nbands that mainly consist of the d↓\nxyandd↓\nx2−y2states\n(represented by 1 and 2, respectively), which are located\non the lower and upper sides of the Fermi level around\nthe Γ-point. Moreover, when c/aincreases, the d↓\nxy- and\nd↓\nx2−y2-based bands move up and down, respectively. For\nc/a= 1.25 [Fig. 7(b)], the energy levels of these two\nbands move closer to each other, and the Fermi level\nof FeCo ( Nq= 17) is located just at the interme-\ndiate energy between the d↓\nxy- andd↓\nx2−y2-based bands\naround the Γ-point. This situation is very favorable in\nterms of the conditions for enhancing the uniaxial mag-\nnetocrystalline anisotropy because the d↓\nxyandd↓\nx2−y2\nstates satisfy selection rule of the spin-orbit interaction.\nForc/a= 1.35 [Fig. 7(c)], the energy levels of the d↓\nxy-\nandd↓\nx2−y2-based band are reversed.\nFor the numerical analysis of the magnetocrystalline\nanisotropy energy of tetragonal FeCo, we looked at the\nk-resolved magnetocrystalline anisotropy energy:\n∆Ek=occ/summationdisplay\nnεkn|θ=π\n2−occ/summationdisplay\nnεkn|θ=0.\nFigure 8 shows the ∆ Ekvalues of FeCo for c/a= 1.15,\n1.25, and 1.35 along the X–Γ–Mdirection. The ∆ Ek\nvalues around the Γ-point for c/a= 1.25 [Fig. 8(b)] are\nextraordinarylargein comparisonwith thosein the other\nk-regions and those for c/a= 1.15 [Fig. 8(a)] and 1.35\n[Fig. 8(c)]. As shown in Fig. 8(b), the d↓\nxy- andd↓\nx2−y2-\nbased bands overlap around the Γ-point near the Fermi\nlevel; therefore, the large magnetocrystalline anisotropy\nis induced by the hybridization between these states via\nspin-orbit interaction. In this region, the difference be-\ntween the energies of the two bands |εocc−εunocc|is sig-\nnificantly smaller than the energy scale of the spin-orbit\ncoupling constant ξ; thus, the perturbation assumption\nis no longer valid.\nThe large ∆ Eof FeCo for c/a= 1.25 mostly orig-\ninates from the ∆ Ekcontribution around the Γ-point,\neven though this large ∆ Ekis enhanced in the small k-\nregion in the first Brillouin zone. Therefore, the sharp\npeak at Nq= 17 appears in FeCo for c/a= 1.25,\nas shown in Fig. 6(b). The ∆ Ekforc/a= 1.35\nin Fig. 8(c) also increases around (0 .54π/a,0,0) and\n(0.29π/a,0.29π/a,0) because some bands cross the\nFermi level. However, the range of large ∆ Ekvalues\nis narrow, and the contribution to the total ∆ Eis small.9\n(a) FeCo (c/a=1.15) (b) FeCo (c/a=1.25) (c) FeCo (c/a=1.35) \nkX Γ M\nkX Γ M\nkX Γ M−2−1 0  1  2 εk − εF   [eV] \n−2−1 0  1  2 \n−2−1 0  1  2 εk − εF   [eV] \nεk − εF   [eV] 1\n212\n12\nFIG. 7. Band dispersion εkin the minority-spin states of FeCo for c/aof (a) 1.15, (b) 1.25, and (c) 1.35 without spin-orbit\ninteractions along the X–Γ–Mdirection; Γ : (0 ,0,0),X: (π/a,0,0), andM: (π/a,π/a, 0). The notations 1 and 2 denote the\nbands that mainly consist of dxyanddx2−y2orbitals, respectively.\nkX Γ M\nkX Γ M\nkX Γ M−2 0  2  4 ∆Ek   [eV]\n−2 0  2  4 ∆Ek   [eV]\n−2 0  2  4 ∆Ek   [eV](a) FeCo (c/a=1.15) (b) FeCo (c/a=1.25) (c) FeCo (c/a=1.35) \nFIG. 8. k-resolved magnetocrystalline anisotropy energies ∆ Ekof FeCo for c/aof (a) 1.15, (b) 1.25, and (c) 1.35 along the\nX–Γ–Mdirection; Γ : (0 ,0,0),X: (π/a,0,0), andM: (π/a,π/a, 0).\nConsequently, the mechanism of the giant magne-\ntocrystalline anisotropy in ordered FeCo is the closing\nof the energy level between the two bands that mainly\nconsist of the d↓\nxyandd↓\nx2−y2states near the Fermi\nlevel around the Γ-point for c/a= 1.25. This sit-\nuation is distinguished from the special cases of MnAl\nand MnGa; however, the obtained results imply that\nthis particular band structure in FeCo for c/a= 1.25\nis strictly dependent on the axial ratio and position of\nthe Fermi level, i.e., the number of valence electrons.\nIn addition, the magnetocrystalline anisotropy in FeCo\nforc/a= 1.25 is strongly influenced by the finite life-\ntime of electron scattering caused by chemical disorder,\naccording to recent studies using the coherent potential\napproximation.20,21If the electron lifetime τsatisfies the\ncondition /planckover2pi1/2τ≫ |εocc−εunocc|, then ∆Edecreases con-\nsiderably as a result of the Bloch spectral function being\nsmeared.56\nV. CONCLUSIONS\nWe have examined the uniaxial magnetocrystalline\nanisotropy in FePt, CoPt, FePd, MnAl, MnGa, andFeCo and characterized thespecific mechanisms using\nfirst-principles calculations. In our evaluation of the\nmagnetocrystalline anisotropy energy, the numerical re-\nsults obtained from the second-orderperturbation theory\nin terms of spin-orbit interactions were in quantitative\nagreement with those obtained from the force theorem\nas long as the perturbation assumption was valid. We\nelucidated the mechanism systematically and presented\nthe conditions necessary for increasing the uniaxial mag-\nnetocrystalline anisotropy in real materials.\nThe magnetocrystalline anisotropy energy of FePt and\nCoPt was shown to originate from Pt, which has a strong\nspin-orbit interaction. In contrast, a large magnetocrys-\ntalline anisotropy compared with that of FePd was ob-\nserved in MnAl, MnGa, and FeCo, even though the spin-\norbit interaction is weak. The mechanism of the uni-\naxial anisotropy in MnAl and MnGa was described by\nthe electronic structure that the occupied d↓\nxyand unoc-\ncupiedd↓\nx2−y2states, the occupied d↑\nyzand unoccupied\nd↓\n3z2−r2states, and the occupied d↑\n3z2−r2and unoccupied\nd↓\nyzstates are located near the Fermi level. This situ-\nation is efficient in inducing the uniaxial magnetocrys-\ntalline anisotropy in terms of the selection rule for the\nhybridization of these states through spin-orbit interac-10\ntion. Furthermore, the electronic structure of FeCo for\nc/a= 1.25 is a special case of the electronic structure\nin MnAl and MnGa. The mechanism of the uniaxial\nanisotropy in FeCo for c/a= 1.25 involves a decrease in\nthe energy difference of the two bands based on the d↓\nxy\nandd↓\nx2−y2states on both sides of the Fermi level around\nthe Γ-point. 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Jpn. 37, 17 (2013)."    },    {        "title": "1308.1597v3.Magnetocrystalline_anisotropy_energy_for_adatoms_and_monolayers_on_non_magnetic_substrates__where_does_it_comes_from_.pdf",        "content": "arXiv:1308.1597v3  [cond-mat.mtrl-sci]  5 Mar 2014Magnetocrystalline anisotropy energy for adatoms and mono layers on non-magnetic\nsubstrates: where does it comes from?\nO.ˇSipr,1,∗S. Bornemann,2H. Ebert,2and J. Min´ ar2\n1Institute of Physics of the ASCR v. v. i., Cukrovarnick´ a 10, CZ-162 53 Prague, Czech Republic\n2Universit¨ at M¨ unchen, Department Chemie, Butenandtstr. 5-13, D-81377 M¨ unchen, Germany\nThe substrate contribution to the magnetic anisotropy ener gy (MAE) of supported nanostructures\ncan be assessed by a site-selective manipulation of the spin -orbit coupling (SOC) and of the effective\nexchange field Bex. A systematic study of Co adatoms and Co monolayers on the (11 1) surfaces\nof Cu, Ag, Au, Pd and Pt is performed to study common trends in t his class of materials. It is\nfound that for adatoms, the influence of the substrate SOC and Bexis relatively small (10–30% of\nthe MAE) while for monolayers, this influence can be substant ial. The influence of the substrate\nSOC is much more important than the influence of the substrate Bex, except for highly polarizable\nsubstrates with a strong SOC (such as Pt). The substrate alwa ys promotes the tendency to an\nout-of-plane orientation of the easy magnetic axis for all t he investigated systems.\nI. INTRODUCTION\nOne of the areas of materials research where a lot of\neffort has concentrated lately is artificially prepared sys-\ntems composed of magnetic and non-magnetic elements.\nThis includes multilayers,thin films, monolayersandvar-\nious nanostructures supported by substrates. One of\nthe properties in focus here is the magnetocrystalline\nanisotropy, i.e., the tendency of the system to orient its\nmagnetizationpreferentiallyinonedirectionwith respect\nto the crystal lattice. Such a property is very appealing\nfor technology as it finds its application in device and\ninformation technology and in the whole area of what\nis now called spintronics. However, magnetocrystalline\nanisotropy presents also an interesting topic for funda-\nmental physics.\nAb initio calculations of the magnetic anisotropy en-\nergy (MAE) are especially challenging. First, there are\nbig technical difficulties as evaluating the MAE means\nthat, at least in principle, one has to subtract two large\nnumbers—totalenergiesfortwoorientationsofthemag-\nnetization — with a great accuracy. Second, the cal-\nculated value of the MAE can be affected by various\nfactors that are hard to control, such as many-body ef-\nfects beyond the local density approximation (LDA)1–3\nor “boundary effects” due to the finite sizes of the super-\ncells used for describing the nanostructure4. However,\napart from calculating the MAE as accurately as pos-\nsible one also wants to get an intuitive feeling how it\narises. One direction on this front is to try to link the\nMAE to changes of specific electron states5–7. Another\ndirection, pursued stronglywhen studying composedsys-\ntems, is to try to answer the question: “where does the\nMAE come from”? Which atoms contribute to the MAE\nto which extent? What is the role of the substrate for\nthe MAE of adatoms and monolayers? What is the role\nof non-magnetic atoms in layered compounds such CoPd\nor FePt?A. Assigning the MAE to individual atoms\nThere are several ways how one could try to assign\nMAE contributions to individual atoms. Probably the\nonethatcomesmosthandyistomakeuseofthe factthat\nthe expression for the MAE usually contains a sum over\natomic sites in one way or another. If the MAE is cal-\nculated directly as a difference between total energies for\ntwo orientations of the magnetization, one can subdivide\nthe spatial integral which has to be evaluated in this case\ninto parts coming from different regions. This approach\nwas adopted by Tsujikawa and Oda8to study the spa-\ntial variations of the MAE for Pt/Fe/Pt(001) and FePt.\nA similar philosophy can be applied also if the MAE is\ncalculated simply by subtracting the band energies, rely-\ning on the magnetic force theorem. There again one can\nidentify contributions from different atoms because the\nexpression contains site-projected densities of states. In\nthis way the localization of the MAE was studied, e.g.,\nforYCo 49, FeandCothinfilmsonCu10,11, Comonolayer\non Au12, Co/Pd structures13,14, FePt and Fe 1−xMnxPt\nalloys15, Fe and Co wires on Pt16or Fe and Co adatoms\non Rh and Pd17.\nAnother appealing possibility is to make use of the\ntorque formula (which again relies on the magnetic force\ntheorem). Here one formally adds contributions origi-\nnating from the torque exerted on individual localized\nmagnetic moments, so one can presume that these quan-\ntities correspond to real physical quantities (even though\nthese moments are not independent, because the torque\nformula assumes that allthe moments are infinitesimally\nrotated at the same time). Materials which were investi-\ngated in this way recently include Fe, Co and Ni adatoms\nand monolayers on Ir, Pt and Au4,18, or a diluted Pt\nmonolayer inside Co19.\nApart from making use of formal sums over atomic\nsites which occur in the expressions used for evaluating\nthe MAE, one can turn to various models and try to\nget an insight from there. In this respect the so-called\nBruno formula, linking the MAE to the anisotropy of\nthe orbital magnetic moment µorb, comes into mind: one2\ncan define that for a multicomponent system, the rela-\ntive importance of different atomic types is proportional\nto the anisotropies of µorbfor those types. This ap-\nproach was applied, e.g., to metallic films20or Co/Ni\n(111) superlattices7.\nThere is yet another possibility to use a model-based\napproach to identify localized contributions to the MAE.\nNamely, as the magnetocrystalline anisotropy cannot\narise without the spin-orbit coupling (SOC), one can\nthink ofintroducing a non-zeroSOC only at some atomic\nsites while keeping it zero at the remaining ones. The\nMAE of such a model system could then be seen as the\nMAE due to those atoms where the SOC has been kept.\nSuch an approach was used, e.g., by Wang et al.5for a\nPd/Co/Pd sandwich, ´Ujfalussy et al.21for Fe/Cu thin\nfilm overlayers, Baud et al.22for Co wires on Pt or Sub-\nkow and F¨ ahnle23for Fe films. A similar approach can\nbe adopted in another formulation, where the torque is\nevaluated as a sum of terms due to the SOC at different\natomic types24. In this way contributions to MAE were\nanalyzed,e.g., foraMnmonolayeronW(001)25orforan-\ntiferromagnetic MnX layered materials (X = Ni, Pd, Rh,\nIr)24. Also other authors relied on SOC manipulation to\ndemonstrate the importance of non-magnetic atoms for\nthe MAE of layered compounds of 3 dand 4d/5dnoble\nmetals6,15,26–28, even though they did not perform a full\nanalysis of the various contributions.\nBy analyzing the results obtained for various systems\nup to now (mostly but not exclusively via the torque\ndecomposition scheme), a prevailing pattern concerning\nthe importance of non-magnetic element for the MAE\nemerges: (i) For 3 dadatoms on non-magnetic substrates,\nthe contribution from the magnetic atom clearly domi-\nnates, the contribution from the non-magnetic atoms can\nbe neglected4,18,29. (ii) For monolayers or wires of 3 d\natoms on non-magnetic substrates as well as for layered\nsystems such as L1 0compounds, the role of the substrate\nis important, sometimes even dominant4,15,16,18,22,24,25.\nB. Problems with assigning MAE to individual\natoms\nDespite the common trends that can be extracted\nfrom the results obtained via various methods, there\nare clearly also problems with attempts to attribute the\nMAEtoindividualatoms. First,oneshouldmentionthat\neven technically, the decomposition schemes may be am-\nbiguous, asit wasdemonstratedby SubkowandF¨ ahnle30\nfor a particular implementation of the decomposition via\nthe sum overthe band energies. Second, ifdifferent MAE\ndecomposition schemes are applied to the same system,\ndifferent results are obtained. For example, Burkert et\nal.15investigated FePt and found that Fe atoms are re-\nsponsible for about 70% of the MAE if estimated from\nthe sum of the band energies but only for 14% of the\nMAE if estimated by manipulating with the SOC. This\ncontroversy can be reinforced by observing that if theMAE is decomposed via subdivision of the spatial inte-\ngralin the total energyformula, the contribution from Fe\natoms is more important than the contribution from Pt\natoms8while if the decomposition via real-space calcu-\nlation of the torque is employed, the MAE is attributed\nalmost entirely to the Pt sublattice26.\nWhile one might be able to reconcile these differences\nby one way or another, there is a deep internal problem\nwith the effort to assign MAE contributions to individ-\nual atoms. Namely, total energy of an inhomogeneous\nsystem is not an extensive quantity and so one cannot\ndecompose the MAE uniquely into sums of contributions\ncoming from various spatial regions. Assigning one part\nof the MAE to one atom and another part of the MAE to\na different atom can be always only intuitive and in prin-\nciple ambiguous. It is only the final sum that provides a\nwell-defined quantity.\nThis does not necessarilymean that there can never be\nany physical content in the division of the MAE among\natoms or layers. E.g., it was found that if the MAE for\na Co layer buried in Pt is decomposed via the torque\nformula, the layer-resolved MAE obtained in this way\ncan be related to the shifts of the valence states in a\ngivenlayerinvokedbyFriedel oscillationsofthe charge19,\nwhich clearly is a well defined physical concept. How-\never, in general, whenever one tries to decompose the\nMAE into a sum of spatially located parts, one has to be\nprepared for ambiguities and inconsistencies.\nC. Method for assessing the role of individual sites\nfor the MAE\nDespite all this, one would still like to know what is\nthe role of different atoms for the MAE of complex sys-\ntems, even though such a question has formally no ex-\nact meaning; the intuitive meaning of such a question is\nclear enough. One just needs to reformulate the question\nabout the localization of the MAE in such a way that\nit is formally well-defined and yet it embraces the vague\nbut illuminating concept of “where does the MAE come\nfrom”.\nTo find such a formulation, one should take into ac-\ncount what is the mechanism that leads to the magne-\ntocrystalline anisotropy— in particular, in systems com-\nprising magnetic and non-magnetic atoms alike. Let us\nrecallthatShick et al.25, suggestedthatin3 d–5dbimetal-\nlic systems such as CoPt and FePt, the contribution of\nthe 5datoms to the MAE originates from (i) strong SOC\nat the 5datoms, (ii) exchange splitting at the 5 datoms\ninduced by the magnetic 3 datoms, and (iii) Stoner en-\nhancement of the local spin susceptibility at 5 datoms.\nItems (i) and (iii) can be tested by a computer exper-\niment: one can selectively switch off the SOC at non-\nmagnetic atoms via various schemes and one can also\nsuppress the effective exchange field Bexat non-magnetic\natoms by simply forcing the spin-up and spin-down po-\ntentials to be equal. In this way one gets a well-defined3\nmodel system for which a uniquely defined MAE can be\ncalculated. As concerns the exchange splitting induced\nby the magnetic atoms at the originally non-magnetic\natoms [the point (ii) mentioned above], this cannot be\ncompletely eliminated: by suppressing the Bexfield for\nnon-magneticatoms, onedoesnotremovethe magnetiza-\ntion of the respective atoms completely but allows them\nto be polarized only by hybridization with neighbouring\nmagnetic atoms. The local Pauli or paramagnetic sus-\nceptibility can be seen as a measure for the effectiveness\nof this mechanism.\nThe MAE obtained when suppressing the SOC and\nBexin the region occupied by originally non-magnetic\natoms can be viewed as that part of the MAE of the\nsystem which comes only from the magnetic atoms —\nbecause the SOC and Bexat the non-magnetic atoms\nhave been suppressed. Promoting this picture further,\none can think of the difference between the MAE for\nthe system with full SOC and Bexand the MAE for the\nsystem with SOC and Bexsuppressed in the substrate as\nof the “contribution of the substrate” to the total MAE\nof the system.\nObviously, this is only an intuitive concept that can-\nnot be taken too literally. Due to the non-locality of\nthe MAE, any attempt to decompose it is in principle\nambiguous. E.g., using the same philosophy as above\nbut along a different path, one could alternatively define\nthe contribution of the substrate as the MAE calculated\nwith the SOC suppressed at the magnetic adatoms. Such\na quantity would clearly differ from the difference dis-\ncussed above (see also the results in Tabs. VI and VII\nin Sec. IIIB). There is no formally exact way of say-\ning which approach is better than another one. Still,\nsome approaches may be preferable in the intuitive way,\nby illustrating certain physical aspects. Our concept re-\nspects the fact that there would be no MAE without\nthe magnetic adatoms and contains an aspect of spa-\ncial localization via site-related SOC and Bex. At the\nsame time, unlike some other approaches, is it techni-\ncallyunambiguous because it always involves calculating\nthe MAE for the whole system. One can also view the\nresults presented here disregarding any discussion about\n“localization”, simply as a comprehensive study of sep-\narate effects of SOC and Bexon the MAE of adatoms\nand monolayers. One should also have in mind that we\nfocus here specifically on the role of the substrate SOC\nandBex. However, even without any SOC and Bexthe\nsubstrateaffects the electronstates viahybridization and\nthus has an influence on the MAE. This aspect was thor-\noughly explored, e.g., when comparing the MAE for a\nfree-standing Co monolayer and for Co/Cu, Co/Ag, and\nCo/Pd multilayers5,6or for a Co monolayer on Pt31. In\nparticular, it was noted that the position of the dstates\nof the magnetic atoms with respect to the dstates of the\nsubstrate and to the Fermi energy is important5. In this\nwork we mean by contribution of the substrate to the\nMAE only the contribution of the substrate SOC and\nBex, which can be localized within the limitations andFIG. 1. (Color online) Structure diagrams for a Co adatom\nand a Co monolayer on a (111) surface of an fcc crystal.\nThe Co atoms are represented by blue (dark) circles, various\nshades of orange (grey) represent substrate atoms in differe nt\nlayers.\nambiguities mentioned above.\nOur approach towards assessing the role of non-\nmagnetic atoms for the MAE is in line with earlier\nworks where the SOC was manipulated in a similar\nway5,6,15,22,23,26–28. It should be noted, however,that de-\nspite the numerous works where the SOC manipulation\nwas used to analyze the MAE, the results obtained so far\nare quite sparse and scattered among different systems\nand it is hardly possible to draw systematic conclusions\nconcerning the influence of the SOC at non-magnetic\natoms. The role of the effective exchange field at orig-\ninally non-magnetic atoms has not been investigated in\nthis respect before, to the best of our knowledge.\nIn this work we want to focus on a series of systems\ncomprising magnetic adatoms and monolayers on non-\nmagnetic noble metal substrates and to assess the role\nof the substrate for the MAE. We focus selectively on\nthe role of the SOC and of the effective exchange field\nat the substrate atoms. We will show that while for the\nadatoms the contribution of the substrate SOC and Bex\nis relatively small, for monolayers it can substantial. We\nwill also show that the contribution due to the SOC is\nmore important than the contribution due to Bexand\nthat for highly polarized substrates with large SOC, the\neffect of both factors is non-additive.\nII. COMPUTATIONAL FRAMEWORK\nWe study the MAE forCoadatomsand Comonolayers\non (111) surfaces of noble metals Cu, Ag, Au, Pd, Pt.\nIn this way we include in our study substrates which\nare hard to polarize (Cu, Ag, Au) and substrates that\nare easy to polarize (Pd, Pt) as well as substrates with\nweak SOC (Cu), with moderate SOC (Pd, Ag), and with\nstrong SOC (Pt, Au). The geometry of the systems is\nschematically shown in figure 1, some properties of the\nsubstrates are summarized in table I.\nTheelectronicstructureis calculatedwithin the ab ini-\ntiospindensityfunctionalframework,relyingonthelocal\nspindensityapproximation(LSDA)withtheVosko,Wilk\nand Nusair parametrization for the exchange and corre-\nlation potential32. The electronic structure is described,4\nTABLE I. Comparison of properties of noble metals used as\nsubstrates: lattice constant a, the SOC parameter ξ, mag-\nnetic susceptibility χmand Stoner enhancement factor Sxc.\nThe SOC parameters ξwere calculated for bulk crystals by a\nmethod described by Davenport et al.35, the remaining prop-\nerties were taken from the literature.\na[˚A]ξ[meV] χm[10−6cm3mol−1]Sxc\nCu 3.615 137 -5a1.1,b1.1c\nAg 4.085 299 -20a1.1,b1.2c\nAu 4.078 845 -28a1.1,b1.1c\nPd 3.891 237 540a9.9,d12.1e\nPt 3.911 712 193a3.7,d4.2e\naLide36\nbMacDonald et al.37\ncSmelyansky et al.38\ndS¨ anger and Voitl¨ ander39\nePovzner et al.40\nincluding all relativistic effects, by the Dirac equa-\ntion, which is solved using the spin polarized relativis-\nticmultiple-scatteringorKorringa-Kohn-Rostoker(SPR-\nKKR) Green’s function formalism33as implemented in\nthespr-tb-kkr code34. The potentials were treated\nwithin the atomic sphere approximation (ASA). For the\nmultipole expansion of the Green’s function, an angular\nmomentum cutoff ℓmax=3wasused. The energyintegrals\nwere evaluated by contour integration on a semicircular\npath within the complex energy plane using a logarith-\nmic mesh of 32 points. The integration over the kpoints\nwas done on a regular mesh, using 10000 points in the\nfull surface Brillouin zone. The convergence of the MAE\nwith respect to the kspace integration grid is checked\nin A.\nThe electronic structure of Co monolayers on sur-\nfaces was calculated by means of the tight-binding KKR\ntechnique41. The substrate was modelled by a slab of 16\nlayers, the vacuum was represented by 4 layers of empty\nsites. The adatoms weretreated as embedded impurities:\nfirsttheelectronicstructureofthehostsystem(substrate\nwith a clean surface) was calculated and then a Dyson\nequation for an embedded impurity cluster was solved42.\nThe impurity cluster contains 131 sites; this includes a\nCoatom, 70substrateatomsandtherestareemptysites.\nWe assume that all the atoms are located on ideal lat-\ntice sites of the underlying bulk fcc lattice, no structural\noptimization was attempted. While this affects the com-\nparison of our data with experiment, we do not expect\nthis to have significant influence on our conclusions con-\ncerning the relative role of the substrate for the MAE\nof adatoms and monolayers (see also the discussion in\nSec. IV).\nThe MAE is calculated by means of the torque Tˆu(ˆn)\nwhich describes the variation of the energy if the magne-\ntization direction ˆ nis infinitesimally rotated around an\naxis ˆu. If the expansion of the total energy is restrictedto the second order in directions cosines as43\nE(θ,φ) =E0+K2,1cos2θ\n+K2,2(1−cos2θ)cos2φ\n+K2,3(1−cos2θ)sin2φ\n+K2,4sin2θcosφ\n+K2,5sin2θsinφ , (1)\nwhereθis the angle between the surface normal and the\nmagnetization direction and φis the azimuthal angle,\nthe difference in energy between the in-plane and out-of-\nplane magnetizations can be obtained just by evaluating\nthe torque for θ= 45◦44:\nE(90◦,φ)−E(0◦,φ) =−2K2,1+2K2,2cos2φ\n+2K2,3sin2φ\n=∂E(θ,φ)\n∂θ/vextendsingle/vextendsingle/vextendsingle\nθ=45◦.(2)\nThe torque itself was calculated by relying on the mag-\nnetic force theorem45. We define the MAE as\nEMAE≡E(x)−E(z), (3)\nwhereE(α)is the total energy of a system if the magne-\ntization is parallel to the αaxis. A positive EMAEthus\nimplies an out-of-plane magnetic easy axis.\nIt should be noted that by evaluating the MAE ac-\ncording to equation (2) which relies on the expansion\n(1), we restrict ourselves to the uniaxial contribution to\nthe MAE, neglecting the higher order terms. As our aim\nis to investigate the basic trends concerning the influence\nof the substrate SOC and Bexon the MAE, this simpli-\nfication does not affect our conclusions. A comparison\nof results obtained via equation (2) with results obtained\nvia a full torque integration is presented in B.\nApartfromthemagneto-crystallinecontributiontothe\nMAE which we focus on, there is also a dipole-dipole\ncontribution to the MAE due to the Breit interaction\n(shapeanisotropy)46. Theshape anisotropyenergyisnot\nconsidered here; it’s value for a Co monolayer on noble\nmetal (111) surfaces is about -0.09 meV18.\nIf the spin-orbit coupling is included in the calcula-\ntion implicitly via the Dirac equation, it is not possible\nto study the relation between the SOC strength and a\nselected physical quantity in a direct way — contrary\nto schemes where a SOC term can be identified in the\napproximate Hamiltonian. Rather, one can vary the\nspeed of light cwhich, however, modifies all relativistic\neffects. It is nevertheless possible to isolate the bare ef-\nfect of the SOC by using an approximate two-component\nscheme47where the SOC-related term is identified via re-\nlying on a set of approximate radial Dirac equations. In\nsome respect this approach is an extension of the scheme\nworked out by Koelling and Harmon48and implemented\nby MacLarren and Victora49. This scheme was used in\nthe past to investigate the influence of the SOC on var-\nious properties50–53. In this work we use this scheme\nto suppress the SOC selectively either at the substrate5\nTABLE II. Spin magnetic moments µspin (inµB) inside\natomic spheres around Co adatoms on noble metals, with\nmagnetization perpendicular to the surface. The first data\ncolumn shows the results with the SOC included both at the\nCo atoms and at the substrate, the second column shows the\nresults with the SOC only at Co atoms, and the last col-\numn shows the results with the SOC considered only at the\nsubstrate atoms. Additionally, the first line for each subst rate\nshows the results obtained if no restrictions are laid on the ex-\nchange field Bexwhile the second line shows results obtained\nifBexis suppressed in the substrate.\nξCo/negationslash= 0 ξCo/negationslash= 0 ξCo= 0\nsubstrate ξsub/negationslash= 0 ξsub= 0 ξsub/negationslash= 0\nCu Bex/negationslash= 0 2.11 2.11 2.11\nBex= 0 2.11 2.11 2.11\nAg Bex/negationslash= 0 2.22 2.22 2.22\nBex= 0 2.22 2.22 2.22\nAu Bex/negationslash= 0 2.28 2.28 2.28\nBex= 0 2.28 2.28 2.28\nPd Bex/negationslash= 0 2.33 2.33 2.33\nBex= 0 2.34 2.33 2.34\nPt Bex/negationslash= 0 2.37 2.38 2.37\nBex= 0 2.37 2.38 2.37\natoms or at the Co atoms while retaining all other rela-\ntivistic effects. We checked that if the SOC is included\nat all sites within the approximative scheme47, it yields\npractically identical results to those obtained if the full\nDirac equation is solved (for example, the MAE obtained\nfor the monolayers using a full Dirac equation is by 0.01–\n0.02 meV per Co atom larger than the MAE obtained\nusing the approximative scheme of Ebert et al.47).\nTo assess the role of the Stoner enhancement of the\nlocal spin susceptibility at the substrate atoms, we made\nyet another series of calculations, with the effective ex-\nchange field Bexset to zero for the substrate atoms dur-\ning the self-consistent cycle. In this way the substrate\natoms will be spin-polarized only due to the unenhanced\nPauli susceptibility. Suppressing Bexwould presumably\nhave little effect for Cu, Ag, or Au substrates where the\nStoner enhancement factor Sxcis small. However, for\nPd and Pt, which are close to the ferromagnetic insta-\nbility and the Sxcfactor is relatively large, suppressing\nBexcould affect the outcome significantly (see Polesya et\nal.54for a comparison of enhanced and unenhanced spin\nmagnetic moments in Pd).\nIII. RESULTS\nA. Influence of SOC and Bexon magnetic moments\nWe start by investigating how the spin magnetic mo-\nmentµspinandorbitalmagneticmoment µorbareaffected\nby manipulations with the SOC parameter for Co atoms\nand for the substrate and with Bexfield for the host. InTABLE III. Orbital magnetic moments µorb(inµB) inside\natomic spheres around Co adatoms on noble metals, with\nmagnetization perpendicular to the surface. Otherwise, as\nfor table II.\nξCo/negationslash= 0 ξCo/negationslash= 0 ξCo= 0\nsubstrate ξsub/negationslash= 0 ξsub= 0 ξsub/negationslash= 0\nCu Bex/negationslash= 0 1.092 1.095 -0.010\nBex= 0 1.096 1.099 -0.010\nAg Bex/negationslash= 0 1.619 1.611 -0.017\nBex= 0 1.619 1.612 -0.017\nAu Bex/negationslash= 0 1.388 1.392 -0.048\nBex= 0 1.389 1.393 -0.049\nPd Bex/negationslash= 0 0.788 0.794 -0.018\nBex= 0 0.795 0.803 -0.022\nPt Bex/negationslash= 0 0.748 0.759 -0.054\nBex= 0 0.750 0.762 -0.056\nTABLE IV. Spin magnetic moments µspin inside atomic\nspheres around Co atoms in a monolayer on a noble metal.\nAs for table II, just with additional results for free-stand ing\nCo monolayers.\nξCo/negationslash= 0ξCo/negationslash= 0 free ξCo= 0\nsubstrate ξsub/negationslash= 0ξsub= 0 standing ξsub/negationslash= 0\nCuBex/negationslash= 0 1.68 1.67 1.67\nBex= 0 1.68 1.67 1.91 1.67\nAgBex/negationslash= 0 1.92 1.92 1.92\nBex= 0 1.92 1.92 2.06 1.92\nAuBex/negationslash= 0 1.93 1.93 1.93\nBex= 0 1.93 1.93 2.06 1.93\nPdBex/negationslash= 0 1.99 1.99 1.99\nBex= 0 2.00 2.00 2.02 2.00\nPtBex/negationslash= 0 1.97 1.98 1.97\nBex= 0 1.98 1.99 2.02 1.97\nparticular, we calculated µspinandµorb(i) if the SOC is\nfully accounted for, (ii) if the SOC is included only on Co\natoms, and (iii) if the SOC is included only on the sub-\nstrate atoms. For all three cases, we further distinguish\nthe situations when the effective exchange field in the\nsubstrate is retained ( Bex/ne}ationslash= 0) and when it is suppressed\n(Bex= 0). Moreover, we performed also calculations for\nfree-standing Co monolayers with the same geometry as\nthe (111) layer of the respective substrate. The results\nfor the adatoms are shown in table II and III, the results\nfor the monolayers are shown in table IV and V. For the\npurpose of investigating magnetic moments, we restrict\nourselves only to the case when the magnetization is ori-\nented perpendicular to the surface.\nIt follows from our results that the magnetic moments\nare in many respects quite inert with respect to a ma-\nnipulation of SOC and Bexof the substrate. By sup-\npressing the SOC and/or the exchange field B exin the\nsubstrate, µspinin the Co atomic spheres changes by no\nmore than 0.5%. Likewise, suppressing B exin the sub-\nstratechanges µorbinCoatomicspheresby nomorethan6\nTABLE V. Orbital magnetic moments µorbinside atomic\nspheres around Co atoms in a monolayer on a noble metal.\nAs for table III, just with additional results for free-stan ding\nCo monolayers.\nξCo/negationslash= 0ξCo/negationslash= 0 free ξCo= 0\nsubstrate ξsub/negationslash= 0ξsub= 0 standing ξsub/negationslash= 0\nCuBex/negationslash= 0 0.097 0.101 -0.004\nBex= 0 0.097 0.101 0.108 -0.004\nAgBex/negationslash= 0 0.190 0.193 -0.004\nBex= 0 0.192 0.193 0.204 -0.004\nAuBex/negationslash= 0 0.171 0.191 -0.021\nBex= 0 0.171 0.191 0.202 -0.021\nPdBex/negationslash= 0 0.155 0.167 -0.010\nBex= 0 0.155 0.165 0.160 -0.010\nPtBex/negationslash= 0 0.141 0.162 -0.019\nBex= 0 0.141 0.162 0.164 -0.018\n1%. This applies both for the adatoms and for the mono-\nlayers. If the SOC of the substrate is suppressed, µorbin\nCo atomic spheres always increases (with the exception\nof a Co adatom on Ag); this increase is at most 15%. In-\nterestingly, if the SOC is included only for the substrate\natoms, the µorbat Co atoms arising via hybridization\nwith substrate SOC-split states is always negative.\nThe variation in µspinfor free-standing Co monolayers\nreflects the variation in the lattice constants of the sub-\nstrates to which the geometries are adjusted (cf. table I).\nThe largest decrease of µspindue to the hybridization\nbetween Co and noble metal states is for the Cu sub-\nstrate (about 10%), the smallest decrease is for the Pd\nand Pt substrates (about 2%). The change in µorbin-\nduced by the Co-substrate hybridization is similar as for\nµspin(less than 10%). This reflects the fact that we are\ndealing with perpendicular magnetization here, meaning\nthat evenforsupported monolayersthe quenchingof µorb\nis mainly due to the hybridization with states associated\nwith Co atoms.\nB. Influence of SOC and Bexon the MAE\nOur main focus is the MAE, which we calculated for\nthe same manifold of SOC and Bexoptions for which\nwe calculated µspinandµorbabove. The results are pre-\nsented in table VI for the adatoms and in table VII for\nthe monolayers. Let us recall that results obtained with\nfull SOC are shown in the first column of numbers, re-\nsults obtained if the SOC at the substrate is suppressed\narein the secondcolumn. So the roleofthe substratecan\nbe assessed by comparing the numbers in the first and in\nthe second column and, to account also for the influence\nof the exchange field, the numbers in the first column\nshould be taken for Bex/ne}ationslash= 0 while the numbers in the\nsecond column should be taken for Bex= 0. Table VII\nshows also the MAE for a free-standing Co monolayer\nwith the geometry of the respective substrate. By com-TABLE VI. EMAE =E(x)−E(z)(in meV) for Co adatoms\non noble metals calculated for different ways of inluding the\nSOC and Bex. Similarly to table II, the results in first colum\nwere obtained with the SOC included at the Co atoms as well\nas at the substrate, results in the the second column are for\nthe SOC included only at Co atoms, and the results in the\nthird column are for the SOC included only at the substrate.\nFor each substrate, the first line shows results obtained whe n\nno restrictions were laid on the exchange field Bexwhile the\nsecond line shows results obtained when Bexwas suppressed\nin the substrate.\nξCo/negationslash= 0 ξCo/negationslash= 0 ξCo= 0\nsubstrate ξsub/negationslash= 0 ξsub= 0 ξsub/negationslash= 0\nCu Bex/negationslash= 0 13.17 12.43 -0.03\nBex= 0 13.23 12.50 -0.03\nAg Bex/negationslash= 0 15.87 14.54 -0.05\nBex= 0 15.88 14.54 -0.05\nAu Bex/negationslash= 0 14.72 10.13 -0.45\nBex= 0 14.73 10.14 -0.45\nPd Bex/negationslash= 0 6.58 4.65 0.01\nBex= 0 6.49 4.56 0.03\nPt Bex/negationslash= 0 8.72 5.74 1.19\nBex= 0 8.70 5.69 1.19\nTABLE VII. EMAE =E(x)−E(z)(in meV) for Co monolayers\non noble metals calculated for different ways of inluding the\nSOC and Bex. As for table VI, just with additional results\nfor free-standing Co monolayers.\nξCo/negationslash= 0ξCo/negationslash= 0 free ξCo= 0\nsubstrate ξsub/negationslash= 0ξsub= 0 standing ξsub/negationslash= 0\nCuBex/negationslash= 0 -0.68 -0.83 -0.02\nBex= 0 -0.69 -0.83 -1.20 -0.02\nAgBex/negationslash= 0 -1.59 -1.90 -0.01\nBex= 0 -1.62 -1.90 -2.60 -0.01\nAuBex/negationslash= 0 -0.62 -1.51 -0.25\nBex= 0 -0.63 -1.51 -2.56 -0.26\nPdBex/negationslash= 0 0.20 -0.27 -0.10\nBex= 0 0.15 -0.34 -1.83 -0.18\nPtBex/negationslash= 0 0.08 -0.21 -1.03\nBex= 0 -0.24 -0.26 -1.90 -1.43\nparing this value with the number to the left of it we get\nan idea how the MAE is influenced solely by hybridiza-\ntion of Co states with noble metal states, without any\ncontribution from the substrate SOC or Bex. The num-\nbers in the last column are less important but still inter-\nesting: they represent something that could be viewed as\na “bare” influence of the substrate, if there is no SOC at\nthe Co atoms.\nBy inspecting these tables, one can recognize several\ngeneral trends. First, one can see that for the adatoms\nthe contribution of the substrate SOC and Bexis rela-\ntively small, while for the monolayers this contribution\ncan be sometimes truly substantial. To be more specific,\nthe situation for the adatoms is such that the magnetic7\neasy axis is always out-of-plane, no matter whether the\nsubstrate SOC and Bexis included or not; the effect of\nswitching on the substrate is just that the numerical val-\nues for the EMAEincrease (by 5% for the Cu substrate,\nby 35% for the Pt substrate). For the monolayers, on the\notherhand, includingthe substrateSOCand/or Bexmay\nreorient the magnetic easy axis: it is in-plane if the sub-\nstrate contribution is suppressed but it is rotated to the\nout-of-plane direction if the substrate SOC and Bexis in-\ncludedforthePdandPtsubstrates. (ForCu, Ag,andAu\nsubstrates the easy magnetic axis of a Co monolayer re-\nmainsin-planeifthe substrateSOCand Bexareswitched\non but the absolute value of the EMAEdecreases.)\nIt follows also from Tabs. VI–VII that the contribu-\ntion due to the substrate SOC is practically always more\nimportant than the contribution due to the substrate ex-\nchangefield Bex. Inparticular,suppressingsubstrate Bex\nhas practically no effect for the adatoms. For the mono-\nlayers,Bexhas got a negligible influence in case of Cu,\nAg, and Au substrates, a significant influence in case of\nthe Pd substrate and a crucial influence in case of the\nthe Pt substrate (indeed, it is the substrate exchange\nfield that switches the magnetic easy axis from in-plane\nto out-of-plane).\nEventhoughourfocusis onSOCand Bex, it is instruc-\ntive to have a look at the changes in the MAE caused by\ndepositing a free-standing Co monolayer on a substrate\nwithξsub=0andBex=0. Thiscouldbeviewedasthepure\neffect of Co-substrate hybridization. The strength of this\neffect appears to be significantly larger for the Pd and Pt\nsubstrates than for the Cu, Ag, and Au substrates. This\nseems to reflect the fact that the overlap between Co and\nnoble metal valence bands is larger for the Co/Pd and\nCo/Pt interfaces on the one hand than for the Co/Cu,\nCo/Ag, and Co/Au interfaces on the other hand5,6,55.\nDifferent roles of hybridization in this respect were dis-\ncussed in detail by Wang et al.5and by Daalderop et\nal.6.\nLet us note finally that by comparing the EMAEvalues\nin the first, the second and the last column in Tabs. VI–\nVII, one sees immediately that the effect of the SOC at\ndifferent sites is not additive: the true MAE is clearly\nnot a sum of the MAE obtained if the SOC is included\nonly at the Co atoms with the MAE obtained if the SOC\nis included only at the substrate — not even in case of\nsubstrates with weak or moderate SOC.\nC. Comparing SOC and Bexmanipulation with\ndecomposition of EMAEby means of the torque\ncontributions\nAs it was mentioned in the introduction, the torque\nmethod has often been used to resolve the MAE into\nlocalized contributions. It is thus instructive to compare\nquantitatively the role of the substrate as deduced from\nthe SOC and Bexmanipulation and as provided by a\nmechanical assignment of individual terms in the sum ofTABLE VIII. Role of the substrate in generating the MAE\nfor Co adatoms on noble metals as assessed by SOC and Bex\nmanipulation and as assessed by comparing individual terms\nin the torque evaluation.\nvia SOC and Bex via comparing\nsubstrate manipulation Titerms\nCu 5.1% 0.02%\nAg 8.4% 0.00%\nAu 31.1% 0.01%\nPd 30.7% 0.67%\nPt 34.8% 0.14%\nTABLE IX. As for table VIII, however, for Co monolayers\ninstead of Co adatoms.\nvia SOC and Bex via comparing\nsubstrate manipulation Titerms\nCu -20.4% -0.10%\nAg -19.3% -0.06%\nAu -142.5% -0.39%\nPd 265.5% 24.7%\nPt 422.5% 190.9%\nthe torque contributions to individual atoms,\nEMAE=/summationdisplay\njT(θ=45◦)\nj. (4)\nIn particular, using the method employed in this work,\na quantitative measure of the role of the substrate for\nEMAEcan be obtained by subtracting and dividing ap-\npropriate values in Tabs. VI–VII, which can be symboli-\ncally written as\nw(SOC,Bex)\nsub=EMAE({Co,sub})−EMAE({Co})\nEMAE({Co,sub})(5)\nwith\n{Co,sub}∼={ξCo/ne}ationslash= 0, ξsub/ne}ationslash= 0, Bex/ne}ationslash= 0},\n{Co}∼={ξCo/ne}ationslash= 0, ξsub= 0, Bex= 0}.\nTo get an analogous quantity by relying on resolving the\nsum of the torque contributions, one can apply a proce-\ndure that can be symbolically denoted as\nw(Tj)\nsub=/summationtext\nsubT(θ=45◦)\nj/summationtext\nCo,subT(θ=45◦)\nj. (6)\nOneshould, however,keepin mind that proceedingalong\nthis second scheme is not internally consistent as one im-\nplicitlymakesanassumptionthatthe energyis“spatially\nadditive”. Use of equation (5), on the other hand, is free\nof such issues because now we always evaluate the en-\nergy of the whole model system — we only change its\nproperties by manipulating the SOC and Bex.8\nThe relative importance of the substrate evaluated via\nprocedures outlined in Eqs. (5)–(6) is presented in ta-\nble VIII for Co adatoms and in table IX for Co monolay-\ners. One sees immediately that there are clear differences\nbetween both procedures. The more physical approach\nbased on the SOC and Bexmanipulation reveals that\nthe role of the substrate is significantly larger than what\nwould follow from the mechanistic decomposition of the\ntorquesum. Insomecasesthis differenceis striking(such\nas, e.g., for Co adatoms in Au, Pd, and Pt or for a Co\nmonolayer on Au).\nIV. DISCUSSION\nOur goal was to study the localization of the MAE\nin complex systems, with focus on the question whether\nthe MAE of adatoms and monolayers adsorbed on non-\nmagnetic supports resides mostly in the adsorbed atoms\norinthe substrate. Wenotedthatthis questioninprinci-\nple cannotbe answered, oratleast cannotbe answeredin\nan unambiguous way, because the energy of an inhomo-\ngeneous system is not an extensive quantity and thus the\nenergy of a composed system cannot be split into ener-\ngies residing in sub-parts of the system. At the same\ntime, the simple question “where does the anisotropy\ncome from” follows naturally from the effort to under-\nstand the MAE in simple terms. Therefore, it isdesirable\nto re-formulate it in such a way that it does not suffer\nfrom inconsistencies and still reflects the intuitive ques-\ntion about the role of the adsorbates and the substrates\nin generatingthe magnetocrystallineanisotropy. The ap-\nproach we adopted, namely, comparing the MAE calcu-\nlated for the original system with the MAE calculated\nfor a model system where the key factors contributing to\nthe magnetocrystalline anisotropy (such as the SOC and\nBex) are selectively suppressed satisfies this requirement.\nBefore we proceed further, let us compare our results\nwith earlier theoretical and experimental results for the\nsame systems we explore here. This is done in table X.\nWhen analyzing the theoretical results, one should have\nin mind that comparingtheoreticalMAE values obtained\nby different studies is not always straightforward. First,\nthe MAE is sensitive to the adatom-substrate geometry\nrelaxation58,66, soquantitativedifferencesbetween differ-\nent works may be due to different interatomic distances\nused. However, the MAE of adatoms and monolayers is\nalso sensitive to the way the substrate is accounted for\n(i.e, how many layers have been used to model the semi-\ninfinite half-crystal) and to whether the adatoms are al-\nlowed to interact with each other or not (i.e., what is the\nlateralsizeofthesupercellwhichsimulatestheadatom)4.\nAlso technical parameters such as angular momentum\ncutoffℓmaxare important. To analyze the differences be-\ntween all the various theoretical calculations would thus\nbe quite complicated and beyond our scope. Still we\nshould address two probably most serious simplifications\nof our treatment, which is the use of the ASA and the ne-TABLE X. EMAE (in meV) calculated in this work compared\nwith other ab initio calculations and with experiment. The\nsystems include free-standing monolayers Co ∞with geome-\ntries of Pd(111) and Pt(111) (first two lines) and adatoms\nCo1and monolayers Co ∞supported by (111) surfaces of fcc\nsubstrates. The experimental values for monolayers includ e\nalso a dipole-dipole contribution of about -0.09 meV18.\nsystem this work other theory experiment\nCo∞as Pd -1.8 -2a—\nCo∞as Pt -1.9 -1.6b—\nCo1/Pd 6.6 1.9c∼3d\nCo1/Pt 8.7 8.1,e3.1,f5.0,g5.9h9.3,i10f\nCo∞/Cu -0.7 -0.5j<0k\nCo∞/Au -0.6 -0.6l<0m\nCo∞/Pt 0.1 0.1,h1.1n0.15,i0.12,o>0p\naDaalderop et al.6\nbLehnert et al.56\ncB/suppress lo´ nskiet al.17(for a relaxed geometry)\ndClaude57\neB/suppress lo´ nski & Hafner58(value for a bulk-like geometry is\nshown)\nfBalashov et al.59(calculations for a relaxed geometry)\ngEtzet al.60\nhLazarovits et al.61\niGambardella et al.62\njHammerling et al.11\nkHuanget al.63\nl´Ujfalussy et al.12\nmPadovani et al.64\nnLehnert et al.56(for a monolayer in an hcp position with a\nrelaxed geometry)\noMeieret al.65\npMoulaset al.31\nglect of the geometry relaxation. For open systems such\nas adatomsand, to a lesser degree, supported monolayers\nthe use of the ASA will certainly affect the values of the\nMAE. However, it follows from the comparison between\nASA and full potential calculations for identical systems\nthat the effect of the ASA should not be crucial. E.g.,\nfor a Co monolayer on Pd(100) with a bulk-like geom-\netry, one gets MAE of -0.73 meV using the ASA67and\n-0.75 meV using a full potential68. For a Co adatom on\nPt(111)with abulk-like geometry, the MAE is8.7meV if\nobtained using the ASA (this work), 9.2 meV if obtained\nusing a full-potential for a 4 ×4 supercell on a 4-layers\nthick slab66and 8.1 meV if obtained using a full poten-\ntial for a 5 ×5 supercell on a 5-layers thick slab58. For a\nCoadatomonPt(111)in anadsorptionhcp positionwith\na relaxed geometry, the ASA yields MAE of 1.90 meV67\nand a full potential yields MAE of 0.72 meV (for a 5 ×5\nsupercell on a 5-layers thick slab)17.\nOur use of bulk-like geometrieswill probably affect the\ncalculated MAE more than the ASA does. To be quan-\ntitative, relaxing the geometry for a Co adatom in an fcc\nadsorption position on Pt(111) changes the MAE from\n9.2 meV to 4.8 meV66. Using an optimized geometry for\na Co monolayer on Pd(111) instead of a bulk geometry9\nchanges the MAE from 0.21 meV to 0.36 meV67. Relax-\ning a Fe monolayer on Pt(111) changes the MAE from\n-0.66 meV to -0.47 meV69. Similar deviations have to be\nexpected for our systems. Therefore, one has to take our\nvalues of the MAE with care when interpreting experi-\nments on real materials. This is especially true for the\nPt and Au substrates: atomic volumes of 3 delements\nare significantly smaller than atomic volumes of 5 del-\nements, which will result in shorter Co–Pt and Co–Au\ndistances and, consequently, smaller magnetic moments\nand smaller MAE in comparison with the values we ob-\ntained here for the bulk-like distances. One could argue\nthat acautiousattitude shouldbe applied to allLDAcal-\nculations of MAE for adatoms anyway, because of pos-\nsible orbital polarization effects62,70which are hard to\ndescribe within conventional ab initio procedures. The\nimportant thing is that our focus here is not on the par-\nticular value of the MAE but on the general trends over\na large set systems, each of them being treated with the\nsame technical parameters. As it follows from table X,\nour calculations yield results in the same range of val-\nues as other ab initio calculations and also as provided\nby experiment, which gives us confidence that we can use\nthem to draw reliable conclusionsconcerning the effect of\nthe substrate SOC and Bexwhen going from adatoms to\nmonolayersand when goingthroughsubstratesofvarious\nproperites.\nWe found that, generally, the substrate is more impor-\ntant when dealing with monolayers than when dealing\nwith adatoms. A similar observation could be made also\non the basis of several earlier studies performed via the\ntorque decomposion (see end of Sec. IA). On the one\nhand, this is surprising, because the ratio of the num-\nber of participating substrate atoms to the number of\nadsorbed atoms is much larger for the adatoms than\nfor the monolayers so one would expect that as a con-\nsequence of this, the substrate should be more impor-\ntant for the adatoms than for the monolayers. On the\nother hand, one could argue that the electronic struc-\nture of the (originally) non-magnetic substrate is more\naltered by the presence of monolayers than by the pres-\nence of adatoms, suggesting that the involvement of the\nsubstrate in the magnetocrystalline anisotropy will be\nhigher for the monolayers than for the adatoms. The\nresults demonstrate that the second trend prevails.\nThe exchange field Bexin the substrate has practically\nno effect on the MAE in case of adatoms. This is not sur-\nprising for the Cu, Ag, and Au substrates because they\nhave the enhancement Sxcfactor close to unity. How-\never, this holds also for the Pd and Pt substrates which\nisquitesurprisingbecausetheseelementshavequitelarge\nSxcand, moreover, a 3 dadatom or impurity induces in\nthese materials an extended polarization cloud, the mag-\nnetic moment of which may be larger than the moment\nof the inducing 3 datom71–73.\nTurning to the role of the substrate Bexfield for Co\nmonolayers, it is unimportant in the case of Cu, Ag, and\nAu substrates. However, it is significant in the case ofa Co monolayer on the Pd substrate and crucial in the\ncase of a Co monolayer on the Pt substrate (cf. corre-\nsponding lines in table VII labelled by Bex/ne}ationslash= 0 and by\nBex= 0). Interestingly, the role of the Bexfield is larger\nfor Pt than for Pd even though the Sxcfactor for Pd is\nabout three times larger than for Pt (table I). Another\nintriguing feature is that for a Co monolayer on Pt, the\nimportance of the substrate Bexfield strongly depends\non whether the SOC is fully included or whether it is in-\ncluded only on one type of atoms (either on Co atoms or\non Pt atoms): in the former case, the role of Bexis signif-\nicantly more important than in the latter case. We could\nsummarize this point by saying that the effects of B ex\nand SOC are intertwined in this case and both factors\ncontribute to the MAE in an non-additive way. While\nthe effect of the SOC was explored for some layered sys-\ntems already5,6,15,22,23,26–28, the role of the substrate Bex\nhas been investigated here for the first time.\nLet us recall again that it is in principle not possible to\ndecomposetheMAE intoasum ofsite-relatedquantities.\nThis can be illustrated also by analysis of Tabs. VI–VII,\nbecause the values in the “ ξCo/ne}ationslash= 0,ξsub/ne}ationslash= 0” column\nclearly differ from the sum of the values in the “ ξCo/ne}ationslash= 0,\nξsub= 0” and in the “ ξCo= 0,ξsub/ne}ationslash= 0” columns. The\nfact that one cannot decompose the MAE into a sum of\ncontributions corresponding to situations where the SOC\nis included only onone atomictype at a time waspointed\nout already in some earlier works, e.g., by Wang et al.5\nfor a Pd/Co/Pd sandwich or by Subkow and F¨ ahnle23\nfor a Fe-Au interface.\nAnother interesting point in this respect is that a\nstronger SOC for a substrate does not necessarily mean\nthat it has got a higher relative importance concerning\nthe MAE. In particular, the SOC for Ag is about twice\nas strong as for Cu and yet the relative role of these\nsubstrates for the MAE of a Co monolayer supported\nby them is the same — about 20% (table IX). The Bex\nfield does not interfere here because its role is negligible\nboth for Cu and for Ag (table VII). The relatively small\nrole of the SOC for the Ag substrate reminds a simi-\nlar situation for Co/Ag multilayers: Daalderop et al.6\nfound that even though the SOC strength is similar for\nPd and Ag, its role is more significantly important for\nthe Co 1Pd2multilayers than for the Co 1Ag2multilayers.\nIn other comparisons, however, it appears that stronger\nSOC indeed implies a bigger role of the substrate for the\nMAE (cf. Cu, Ag, and Au substrates for a Co adatom,\ntable VIII). So it seems that there is no unique pattern\nin this respect.\nIt follows from our results that if one analyzes the ef-\nfects of site-related SOC and Bexfor the MAE, the role\nof the substrate is much more important than what one\ngets from comparing individual site-related terms in the\ntorque evaluation. Especially this is true for Co adatoms\non Au, Pd, and Pt and for a Co monolayer on Au, where\nthe differences are two orders of magnitude. So while\nevaluating the torqueis a convenientway to calculate the\nMAE of a system, it should not be used for assessing the10\nroles of various constituents for the magnetocrystalline\nanisotropy of a compound or a nanostructure.\nAlthough we have not explicitly tested for our systems\nthe decomposition scheme based on site-projected den-\nsities of states, we expect that the outcome concerning\nthe role of the substrate would be similar as with the\ntorque formula, among others because both approaches\nare based on the magnetic force theorem. This view is\nbased also on the analysis of the results of works which\nemployed this scheme: By decomposing the MAE for\nsurfaces and multilayers into layer-resolvedcontributions\nvia site-projected DOS it was found that the main con-\ntribution comes from surfaces and interfaces, with only\na small part coming from non-magnetic substrates or\nspacers10–13. Such an outcome clearly differs from the\npicture obtained via site-related SOC and Bexanalysis\nin the present work.\nAllthesubstratematerialsweinvestigatedhadtheten-\ndency to orientthe magnetic easyaxisin the out-of-plane\ndirection: by switching on the SOC and Bexin the sub-\nstrate, eitherthe out-of-planeorientationofthe magnetic\neasy axis was reinforced (in the case of adatoms, see ta-\nble VI), or the preference of the magnetic easy axis for\nthein-planeorientationgotweaker(incaseofmonolayers\non Cu, Ag, and Au, see table VII), or the magnetic easy\naxis was re-oriented from the in-plane direction to the\nout-of-plane direction (in case of monolayers on Pd and\nPt). It would be interesting to check for other adsorbates\nhow general this tendency is. Finally, it should be noted\nthat the same approach we used here could be applied\nalso to layered systems such as CoPt or FePd to assess\nthe role of the non-magnetic element in these systems.\nV. CONCLUSIONS\nThe role of the substrate for generating the magne-\ntocrystalline anisotropy of supported nanostructures can\nbe assessed by comparing the MAE calculated for the\nreal system with the MAE calculated for a model system\nwhere the spin orbit coupling and the effective exchange\nfieldBexis suppressed at the substrate atoms. For Co\nadatoms on noble metals (Cu, Ag, Au, Pd, Pt), the con-\ntribution of the substrate SOC and Bexto the MAE is\nrelatively small while for Co monolayers it can be sub-\nstantial. For all five substrates we explored, we found\nthat their contribution to the MAE is out-of-plane.\nThe role of the substrate SOC is more important than\nthe role of the substrate exchange field Bex. For Co\nadatoms on Cu, Ag, Au, Pd, or Pt, the substrate Bex\nfield has practically no effect on the MAE. For Co mono-\nlayers, the substrate Bexfield is unimportant for sub-\nstrates which are hard to polarize (Cu, Ag, Au) but it\nis significant for highly polarizable substrates (Pd, Pt).\nGenerally, the effects of the SOC and of the Bexfield\neffect are non-additive. The same is true for the effect\nof SOC if it is selectively switched on either only for the\nadsorbed atoms or only for the substrate atoms.TABLE XI. Dependence of the MAE for a Co monolayer on\nCu(111) and on Pt(111) on the number Nkof integration\npoints in the surface Brillouin zone. The MAE was calcu-\nlated both by evaluating the torque according to equation (2 )\n(columns labelled by “ ∂E(θ)/∂θ”) and directly by subtract-\ning the total energies for two orientations of the magnetiza tion\n(columns labelled by “∆ Etot”). The MAE is in meV’s.\nCu Cu Pt Pt\nNk∂E(θ)\n∂θ∆Etot∂E(θ)\n∂θ∆Etot\n3600 -0.684 -0.720 0.115 0.012\n6400 -0.650 -0.696 0.198 0.202\n10000 -0.685 -0.714 0.080 0.100\n22500 -0.688 -0.732 0.128 0.112\n40000 -0.687 -0.726 0.113 0.120\nTABLE XII. Dependence of the MAE for a Co adatom on\nCu(111) and on Pt(111) on the number Nkof integration\npoints in the surface Brillouin zone used when calculating t he\nelectronic structure and the Green’s function of the host. T he\nMAE is in meV’s.\nCu Pt\nNk∂E(θ)\n∂θ∂E(θ)\n∂θ\n3600 12.046 8.474\n6400 12.395 8.692\n10000 13.170 8.719\n22500 12.909 8.689\n40000 13.181 8.763\nACKNOWLEDGMENTS\nThis work was supported by the Grant Agency of the\nCzech Republic within the project 108/11/0853, by the\nBundesministerium f¨ ur Bildung und Forschung (BMBF)\nproject 05K13WMA, and by the Deutsche Forschungsge-\nmeinschaft (DFG) within the SFB 689.\nAppendix A: Convergence of the MAE with respect\nto the k space integration grid\nThe integration within the 2-dimensional kspace was\ndone on a regular grid. We used 10000 points in the full\nsurfaceBrillouinzonetodeterminetheself-consistentpo-\ntentials as well as to evaluate the MAE via equation (2).\nAccording to our experience, this grid density is in the\nregime where the results are already converged enough\nto provide the MAE with accuracy of about 5% (for very\nsmall absolute values of the MAE, the relative accuracy\nmay be somewhat worse). We demonstrate this in ta-\nbles XI–XII where we explore the k-grid dependence of\nthe MAE for two extreme cases of the substrate mate-\nrial, namely, for low-SOC low-polarizable Cu and high-\nSOC highly-polarizable Pt. Apart from results obtained\nby evaluating the torque according to equation (2), we\npresent for the monolayers also results obtained by a di-11\n051015T(,) [meV/ radian]\n0 20 40 60 80\nmagnetizationangleadatom\nonCu\n=90\nuniaxial\n0510T(,) [meV/ radian]\n0 20 40 60 80\nmagnetizationangleadatom\nonPt\n=90\nuniaxial\n-0.8-0.6-0.4-0.20.0T(,) [meV/ radian]\n0 20 40 60 80\nmagnetizationanglemonolayer\non Cu\n=0\n=90\nuniaxial\n0.00.020.040.060.080.10.12T(,) [meV /radian]\n0 20 40 60 80\nmagnetizationanglemonolayer\non Pt\n=0\n=90\nuniaxial\nFIG. 2. (Color online) Torque T(θ,φ) as a function of θfor Co adatoms and monolayers on Cu(111) and Pt(111). Marker s\nrepresent calculated points, thick lines are guides for an e ye, thin lines are uniaxial contributions [proportional to sin(2θ)] which\nwould yield the same MAE as provided by equation (2).\nrect subtraction of the total energies for two orientations\nof the magnetization. This demonstrates that relying on\nthe magnetic force theorem together with equation (2) is\njustified for our purpose.\nAppendix B: Higher-order contributions to the\nMAE\nWhen using equation (2), we assume a simple uniaxial\ndependence ofthe torqueonthe polarangle θ, i.e., we ne-\nglect higher-ordercontributions. These can be accounted\nforifweevaluatethetorque T(θ,φ)forthewhole θrange.\nThe anisotropy then can be evaluated by performing the\nintegral\nEMAE=/integraldisplayπ/2\n0dθT(θ,φ). (B1)\nTo verify that it is sufficient for our purpose to restrict\nourselves to uniaxial contributions, we present here the\nfullθscan of the torque T(θ,φ) for Co adatoms and\nmonolayers on Cu(111) and on Pt(111). For the mono-layers we probe the azimuthal dependence as well, i.e.,\nwe perform the θscans for the magnetization direction\nconfined either to the xzplane (φ= 0◦, horizontal di-\nrection in figure 1) or to the yzplane (φ= 90◦, vertical\ndirection in figure 1). The azimuthal dependence of the\nMAE for the adatoms on fcc (111) surfaces is expected\nto be very weak (cf. data for a Co adatom on Pd(111)67)\nso it is not explored here — just data for φ= 90◦are\npresented in this case.\nThe dependence of the torque on θis shown in figure 2.\nThe comparison of the MAE obtained via equation (2)\nand via equation (B1) is shown in table XIII. One can see\nfromfigure2that therearecleardeviationsfromthe sim-\nple uniaxial behaviour of T(θ,φ) both for adatoms and\nfor monolayers. These deviations are especially evident\nfor the Co monolayer on Pt(111), which is probably con-\nnected with the large role of the substrate in this case\n(see section IIIB). 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Bl¨ ugel1\n1)Peter Gr¨ unberg Institut and Institute for Advanced Simula tion, Forschungszentrum J¨ ulich and JARA, 52425 J¨ ulich,\nGermany\n2)Department of Physics and Astronomy, Uppsala University, B ox 516, 75120 Uppsala,\nSweden\nCombining density-functional theory calculations with a classical Mo nte Carlo method, we show that for\nB2-type FeCo compounds tetragonal distortion gives rise to a str ong reduction of the Curie temperature TC.\nTheTCmonotonically decreases from 1575 K (for c/a= 1) to 940 K (for c/a=√\n2). We find that the nearest\nneighbor Fe-Co exchange interaction is sufficient to explain the c/abehavior of the TC. Combination of high\nmagnetocrystalline anisotropy energy with a moderate TCvalue suggests tetragonal FeCo grown on the Rh\nsubstrate with c/a= 1.24 to be a promising material for heat-assisted magnetic recording applications.\nPACS numbers: 75.47.Np, 75.30.Kz, 75.50.Bb, 75.10.Jm\nThe recording density in a commercial hard disk drive\n(HDD), i.e., theamountofinformationthatcanbestored\nper square inch, has increased by more than 7 orders of\nmagnitude since its first introduction in 1956.1Such an\nincrease has been achieved by a simple scaling of the di-\nmensions of the bits recorded in storage medium. How-\never, the recording density has an upper limit due to the\nsuperparamagnetic effect and limited write field of the\nwriting head. This limit is around 1 Tbit per square inch\nforcurrent perpendicularmagnetic recording.2–5In order\nto further increasethe recordingdensity in future record-\ning media new materials with the following properties\nare sought: (i) they should have large uniaxial magneto-\ncrystalline anisotropy energy (MAE) Ku, (ii) large satu-\nration magnetization, (iii) fast magnetic response to ex-\nternal applied fields, and (iv) moderate Curie tempera-\ntures above the room temperature.\nRetainingthe magnetizationofthe medium overalong\nperiod of time despite thermal fluctuations is one of the\nmajor problems in designing magnetic storage media. If\nthe ratio of the magnetic energy per grain KuV, whereV\nis the grain volume, to the thermal energy kBTbecomes\nsufficiently small, the thermal fluctuations can reverse\nthe magnetization in a region of the medium destroying\nthe information stored there.3,6To further increase the\nrecordingdensityhigh- Kumaterialswithlargesaturation\nmagnetization Msare needed. The large Msis beneficial\nto reduce the write field. In addition to large Kuand\nMsvalues, another important issue in magnetic record-\ning applications is the magnetic switching time, which\nimposes physical limits on areal recording densities and\ndata rates.5In current devices the switching speeds have\nreached a point where dynamical effects are becoming\nimportant.7–13Collective magnonexcitationsplay anim-\nportant role in fast precessional magnetic switching pro-\ncesses because they serve as a heat bath for dissipation of\nthe Zeeman energy and thus contribute to the relaxation\nof magnetization and switching time.\na)Electronic mail: e.sasioglu@fz-juelich.deOn the other hand, the large- Kumaterialsrequirevery\nhigh magnetic fields for writing the information to the\nrecordingmedia. As the bit size gets smallerand smaller,\nat some point the magnetic field required for switching\nthe magnetization direction exceeds the maximal avail-\nable magnetic writing fields and thus data can no longer\nbe written to the disk. To solve this problem the heat-\nassisted magnetic recording (HAMR) was proposed as a\npromising approach, which enables large increases in the\nstorage density of HDD.5,14,15In HAMR a laser is used\nto momentarily and locally heat the recordingarea of the\nmedium to reduce its coercivity. It has been suggested\nthat magnetic recording close to or above the Curie tem-\nperature is required to achieve the highest areal den-\nsity advantage of HAMR, making the TCan important\nparameter for applications and choice of materials.46,47\nWith increasing temperature the Kuof the medium de-\ncreases and above TCKuvanishes, making it possible to\nwrite the information with available head fields. Thus,\ntheTCis an importantparameterin the design ofHAMR\nmedia.\nMaterials that combine most of necessary conditions\nfor HAMR applications are B2-type tetragonal FeCo\ncompounds. The large values of Ku, reaching 600 µeV,\nandMsin these compounds were first predicted by\nfirst-principles calculations16and then confirmed by\nexperiments.17–20Experimentally, FeCocompoundshave\nbeen grown on the Pd, Ir, and Rh substrates in B2-\ntype structure, in which the in-plane lattice constant a\nis enforced by the substrate and the out-of-plane lattice\nconstant cchanges so as to keep the volume constant.\nIn particular, Yildiz et al.19,20found in agreement with\ntheoretical predictions that the perpendicular magnetic\nanisotropy is very sensitive to the tetragonal distortion\nand increases with increasing c/aratio, which allows to\ntune the perpendicular anisotropy value by growing the\nalloys on different substrates. Yildiz et al.19,20have also\nshown that the structure remains stable for film thick-\nnessesofup to15monolayers. Notealsothat microscopic\natomic order in B2-type FeCo compounds is crucial to\nachieve high Kuvalues.34,42\nAt low temperatures the ordered cubic FeCo takes the2\nCsCl (B2) structure, at around 1000 K it undergoes an\norder-disorder transition and at around 1230 K a bcc-fcc\ntransformation accompanied by a magnetic-nonmagnetic\ntransition.21–23In a recent paper S ¸a¸ sıo˘ glu et al.25have\nstudied the effect of the tetragonal distortion on magnon\nspectra of the B2-type FeCo compounds by employing\nthe many-body perturbation theory. The authors have\nshown that tetragonal distortion gives rise to a signifi-\ncant magnon softening, which indicates a strong reduc-\ntion of the Curie temperature. Ab-initio calculations by\nLeˇ zai´ cet al.24on cubic ordered and disordered FeCo al-\nloys have shown that the calculated TCagrees well with\nexperiment.\nThe aim of this Letter is to study the effect of the\ntetragonaldistortiononthe Curietemperatureofthe B2-\ntype FeCo compounds from first principles. It is shown\nthat tetragonal distortion gives rise to a strong reduction\nofTC, that decreases from 1575 K (for c/a= 1) to 940\nK (forc/a=√\n2). Combination of moderate TCval-\nues together with large Kusuggests B2-type tetragonal\nFeCo grown on the Rh substrate with c/a= 1.24 to be a\npromising material for HAMR applications.\nWe calculate the Curie temperature using an estab-\nlished approach: the adiabaticapproximationfor the cal-\nculation of magnon spectra.26–28Ab initio total-energy\nresults, calculated within the frozen-magnon approxima-\ntion, are mapped to the classical Heisenberg model,\nH=−1\n2/summationdisplay\ni,j(i/negationslash=j)Jijei·ej, (1)\nwhereJijare the exchange constants between the mag-\nnetic moments at sites iandjandeiis a unit vector\nalong the moment of atom i. Accounting for the inter-\nactions up to the 12th nearest neighbor, TCis calculated\nwithin this model by a Monte Carlo method by locat-\ning the crossing point of the fourth-order cumulants29\nfor 5488 and 8192-atom supercells in the bulk limit and\nneglecting the anisotropy, which is a good approximation\nfor a film thickness of 15 monolayers.32,33\nThe ab initio results are calculated within the general-\nized gradient approximation31to density-functional the-\nory. We employ the full-potential linearized augmented\nplane-wave (FLAPW) method as implemented in the\nFLEURcode.30The muffin-tin radii of Fe and Co are cho-\nsen to be 1.21 ˚A. A dense 16 ×16×16k-point grid is\nused. Keeping the volume of the B2-type unit cell con-\nstant (V= 23.766˚A3) we vary the c/aratio from 1 to√\n2. Note that if both atoms in the unit cell were iden-\ntical we would get a bcc lattice for c/a= 1 and fcc for\nc/a=√\n2. As FeCo with c/a= 1 crystallizes in ordered\nB2 structure21,22we assume the same type of structure\nwith additional tetragonal distortion in our calculations.\nThe mechanism behind the giant uniaxial MAE observed\nin tetragonal FeCo compounds has been discussed in de-\ntail in Ref.16 and will not be analyzed here. Indeed, our\ncalculated values of uniaxial MAE (results not shown)\nare very similar to those reported by Burkert et al.16We begin the discussion of our results by presenting\nthe exchange interactions. Figures 1(a) and (b) show the\ncalculated sizeable intra-sublattice Co-Co and Fe-Fe ex-\nchange parameters, respectively, as a function of tetrag-\nonal distortion, i.e., from c/a= 1 toc/a=√\n2 (more\ndistant coupling parameters are included in the TCcal-\nculation but not shown here). Figure1(c) shows the c/a\ndependence of the nearest neighbor inter-sublattice Fe-\nCo exchange parameters as well as the calculated Curie\ntemperature of the compounds. Due to the strong ferro-\nmagnetic nature of FeCo compounds (very low majority-\nspin DOS, see Fig.2) the absolute value of the exchange\nparameters decays quickly with increasing interatomic\ndistance35and the main contribution to TCcomes from\nthe interaction between atoms lying in a distance of a\nfew first neighboring shells. The importance of each in-\nteraction ( J1,J2, etc.) should be judged taking into\naccount the number of neighbours in the corresponding\ncoordination sphere, given in parentheses in Fig. 1. At\nc/a= 1eachFe(Co)atomhas8nearestneighborCo(Fe)\natoms and 6 next nearest neighbor Fe (Co) atoms, etc.\nWith tetragonal distortion the distances between Fe or\nCo atoms in the atomic plane become different compared\nto the adjacent planes in the direction of the c/adistor-\ntion and as a consequence the intra-sublattice Fe-Fe and\nCo-Co exchange parameters split into two components,\nwhich are denoted as J1,J3,J5andJ2,J4for in-plane\nand neighboring-plane parameters, respectively. As seen\nfrom Fig.1 the nearest ( J1) and next-nearest neighbor\n(J2) Fe-Fe (Co-Co) parameters stand out and are much\naffected by the distortion showing in part variations be-\ntween ferromagnetic and antiferromagnetic values. A\nsimilar behavior is observed in the case of Ni 2MnGa and\nhcp Gd under distortion.36\nThe most decisive role for TCis played by the nearest\nneighbor Fe-Co exchange as can be seen from Fig.1(c):\nfirstly, their value is significantly larger than the value of\nthe Co-Co or Fe-Fe parameters shown in Fig.1(a,b) [note\nthe different scale in Fig.1(c) compared to Fig.1(a,b)];\nsecondly, we witness that they closely follow the mono-\ntonical reduction of TCwith increasing distortion, except\nfor a flattening-out of TCclose toc/a=√\n2 which is not\nfollowed by the Fe-Co interaction and which we com-\nment on later. The c/abehavior of the exchange inter-\nactions and resulting reduction of TCcan be attributed\nto the complex exchange coupling mechanisms and will\nbe briefly discussed below.\nMany-body model Hamiltonian approaches relevant to\nthe problem provideuseful insight into the qualitativein-\nterpretation of the density-functional results although a\nquantitative analysis of exchange parameters Jin terms\nof different contributions is frequently not possible. It\nis meaningful to separate the interaction in two terms,\nJ=Jdirect+Jindirect.Jdirectstems from the overlap of\n3dwavefunctions of neighboring atoms and practically\nvanishes for distances larger than second-nearest neigh-\nbors. For FeCo it is ferromagnetic because of the double-\nexchange mechanism37, i.e., energy gain by broadening3\n1 1.1 1.2 1.3 1.4\nc/a-606121824JCo-Co  (meV)J1 (4)\nJ2 (2)\nJ3 (4)\nJ4 (8)\n1 1.1 1.2 1.3 1.4\nc/a-12-606121824JFe-Fe  (meV)J1 (4)\nJ2 (2)\nJ3 (4)\nJ4 (8)\nJ5 (8)\n1 1.1 1.2 1.3 1.4\nc/a2030405060JFe-Co  (meV)J1 (8)\n1 1.1 1.2 1.3 1.4\nc/a600900120015001800\nCurie temperature  (K)TC\n(a)                                                            (b)                                                             (c)\nFIG. 1. (Color online) (a) First four nearest neighbor intra -sublattice Co-Co exchange parameters as a function of tetr agonal\ndistortion in B2-type FeCo. (b) the same for Fe sublattice fo r the first five shells. (c) Nearest neighbor inter-sublattic e Fe-Co\nexchange parameters and estimated Curie temperature of the FeCo compounds as a function of tetragonal distortion. In ea ch\npanel for each exchange parameter the number of atoms in the c orresponding coordination sphere is given. Positive excha nge\nparameters correspond to ferromagnetic coupling, negativ e to antiferromagnetic.\nof the half-filled minority dstates due to hybridization\nif the moments are parallel-aligned. Jindirectis mediated\nby the Fermi sea, concerns interatomic distances from\nsecond-nearest neighbors and beyond, and is analyzed\nhere in terms of the Anderson s-dmixing model because\nof the localized nature of magnetic moments in these sys-\ntems.\nWe proceed with a qualitative analysis of the calcu-\nlated exchange parameters. The monotonous reduction\nof the nearest neighbor Fe-Co exchange interaction [see\nFig.1(c)] with tetragonal distortion can be attributed to\ndecreaseofthe directcoupling Jdirectcausedbythree fac-\ntors. (i) The energetic distance of the minority dbands\nto the Fermi level decreases with increasing c/a, from 1\neV to 0 .3 eV, as seen in Fig. 2, leading to a strength-\nening of the antiferromagnetic kinetic-exchange37contri-\nbution to Jdirect. The latter mechanism is related to a\nrepulsion of the occupied majority-spin with unoccupied\nminority-spinlevelsofneighboringatomsthatstemsfrom\nhybridization if the moments are antiparallel-alignedand\nresults in energy gain as the occupied levels move lower\nin energy. (ii) The inter-atomic Fe-Co distance increases\nfrom 2.49 ˚A to 2.56 ˚A asc/aincreases from 1 to√\n2 re-\nsulting in a weakening of the overlap of neighboring 3 d\nwave functions. (iii) The magnetic moment amplitudes,\nthat are included in the values of Jijin Eq. (1), decrease\nfrom 2.86 µBto 2.65µBfor Fe and from 1.82 µBto 1.65\nµBfor Co.\nConcerning the indirect coupling, within the Ander-\nsons-dmixing model Jindirectcan be separated into two\ncontributions (see, e.g., Ref.40): Jindirect=JRKKY+JS.\nHere the first term is an oscillating Ruderman-Kittel-\nKasuya-Yosida (RKKY)-like, which stems from a spin\npolarization of the conduction electron sea by the lo-\ncal moments. The second “superexchange” term, JS, is\nantiferromagnetic, decays exponentially with spatial dis-\ntance, and stems from virtual excitations in which elec-\ntronsfromlocal dstatesofFeandCoarepromotedabove\nthe Fermi sea. JSdepends mostly on the distance of the\nunoccupied Fe(Co) 3 dpeaksfromthe Fermienergy. The\ncloser the peaks to the Fermi level, the stronger becomes-6 -4 -2 0 2-4-2024\n-6 -4 -2 0 2\nE − EF  (eV)-4-2024DOS (states/eV)Fe\nCo\nTotal\nc/a = 1                                                  c/a = 1.29\nc/a = 1.124                                           c/a = 1.414mFe = 2.86 µΒmCo = 1.82 µΒmFe = 2.73 µΒmCo = 1.67 µΒ\nmFe = 2.82 µΒmCo = 1.78 µΒmFe = 2.65 µΒmCo = 1.65 µΒ\nFIG. 2. (Color online) Total and atom-resolved density of\nstates of B2-type FeCo compounds for four different c/ara-\ntios. In each panel we include atom-resolved magnetic mo-\nments. The positive (negative) DOS axis corresponds to the\nmajority-spin (minority-spin) channel.\nJS.\nThe intra-sublattice, i.e. Fe-Fe and Co-Co, exchange\ninteractions depend strongly on Jindirect, showing in part\nstronger variations with tetragonal distortion compared\nto the Fe-Co interaction. However, as the nearest neigh-\nbor Fe-Fe and Co-Co pairs are relatively close (the in-\nplane distance decreases from 2.87 ˚A atc/a= 1 to 2.56 ˚A\natc/a=√\n2), the variation of J1also has a direct-\nexchange contribution. In both sublattices we observe\nthatJ1is sizeableand changessigninthe c/ainterval. In\na large section of the interval the various intra-sublattice\ninteractionspartlycompensate each otherdue to sizeable\nantiferromagnetic J1andJ5(for Co-Co) terms. Only\nclose to the end of the interval at c/a=√\n2 do the Co-\nCo and Fe-Fe interactions contribute towards a stronger\nferromagnetic coupling, which results in a flattening-off\nofthe curve of TCclosetoc/a=√\n2 that is not witnessed\nin the Fe-Co coupling [see Fig. 1(c)].4\nHaving established the possibility of tuning TCvia the\nc/aratio, we should note that an additonal parameter\nthat may be used for the tuning is the film thickness. It\nis known that two-dimensionalHeisenberg magnetswith-\nout anisotropy have TC= 043. However, in the presence\nof uniaxial MAE, TC>0 and it grows with increasing\nfilm thickness, coming close to the bulk value already at\n15-20 atomic layers33,44,45(depending of course on the\nmagnitude of Ku). Since in HAMR applications one\ncould conceivably wish a lower TCthan the bulk limit\nshown here, this can be achieved by reducing the film\nthickness. Of course the functionality will be also deter-\nmined by the thickness dependence of Kuwhich we do\nnot study here, however, for thin films Kuis expected to\nbe appreciable because of the reduced symmetry even if\nit differs from the value of 600 µeV that was found for\nc/a= 1.24.\nIn conclusion, combiningdensity-functionaltheorycal-\nculations with a classical Monte Carlo method, we show\nthat for B2-type FeCo compounds a tetragonal distor-\ntion with 1 < c/a <√\n2 gives rise to a strong reduc-\ntion of the Curie temperature TC. In this interval the\nTCdecreases monotonically from 1575 K to 940 K. 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Jpn. 25, 313 (2003)."    },    {        "title": "1309.4578v1.Magnetocrystalline_anisotropy_energy_of_Fe__001____Fe__110___slabs_and_nanoclusters__a_detailed_local_analysis_within_a_tight_binding_model.pdf",        "content": "Magnetocrystalline anisotropy energy of Fe (001) , Fe(110) slabs and\nnanoclusters: a detailed local analysis within a tight-binding model\nDongzhe Li,1Alexander Smogunov,1Cyrille\nBarreteau,1,\u0003Franc ¸ois Ducastelle,2and Daniel Spanjaard3\n1IRAMIS, SPCSI, CEA Saclay, 91191 Gif-sur-Yvette Cedex, France\n2Laboratoire dEtude des Microstructures,\nONERA-CNRS, BP 72, 92322 Chtillon Cedex, France\n3Laboratoire de Physique des Solides, Universit Paris Sud,\nBatiment 510, F-91405 Orsay, France\n(Dated: September 19, 2013)\nAbstract\nWe report tight-binding (TB) calculations of magnetocrystalline anisotropy energy (MAE) of Iron slabs\nand nanoclusters with a particuler focus on local analysis. After clarifying various concepts and formu-\nlations for the determination of MAE, we apply our realistic TB model to the analysis of the magnetic\nanisotropy of Fe (001) , Fe(110) slabs and of two large Fe clusters with (001) and(110) facets only: a trun-\ncated pyramid and a truncated bipyramid containg 620 and 1096 atoms, respectively. It is shown that the\nMAE of slabs originates mainly from outer layers, a small contribution from the bulk gives rise, however,\nto an oscillatory behavior for large thicknesses. Interestingly, the MAE of the nanoclusters considered is\nalmost solely due to (001) facets and the base perimeter of the pyramid. We believe that this fact could be\nused to efficiently control the anisotropy of Iron nanoparticles and could also have consequences on their\nspin dynamics.\nPACS numbers: 73.20.At, 71.15.Mb, 75.10.Lp, 75.50.Ee, 75.70.Ak\nKeywords:\n1arXiv:1309.4578v1  [cond-mat.mtrl-sci]  18 Sep 2013I. INTRODUCTION\nThe magnetic anisotropy which is characterized by the dependence of the energy of a magnetic\nsystem on the orientation of its magnetization is a quantity of central importance. The orienta-\ntion corresponding to the minimum of energy (so called easy axis) determines the magnetization\ndirection at low temperature. The width of a domain wall directely related to the strength of the\nanisotropy. The spin dynamics is also very much influenced by the shape of the magnetic energy\nlandscape. For example the thermal stability of small nanoparticles with respect to magnetization\nreversal is controled by the height of the energy barrier to overcome during the switching process\nof the magnetization. The development of materials with large uniaxial anisotropy is very useful\nfor technological applications such as high density magnetic recording or memory devices. For\nexample the storage unit can be made up by metallic grains and higher storage densities is achieved\nby reducing the magnetic grains down to nanoscale.\nThe origin of magnetic anisotropy is twofold: magnetostatic interaction and spin-orbit\ncoupling1. The first one gives rise of the so-called shape anisotropy since it depends on the shape\nof the sample while the second is responsible for the overall magnetocrystalline anisotropy energy\n(MAE). The shape anisotropy of classical origin needs not to be included in electronic structure\ncalculation and can be added a posteriori by summing all pairs of magnetic dipole-dipole interac-\ntion energies. In thin films it favors in plane magnetization and is proportional to the thickness of\nthe film and generally dominates for thick enough films. It will not be considered hereafter since\nit behaves almost linearly with the film thickness and cannot be at the origin of any MAE oscil-\nlations. The MAE on the contrary is a purely quantum effect and has a more complex behaviour.\nIts value per atom is usually extremely small in bulk ( \u0016eV) but can get larger in ultrathin films,\nmultilayers or nanostructures.\nDue to the smallness of the energy differences in play, the determination of MAE still remains\nnumerically delicate. However, there now exists a vast body of reasearch devoted to the calcula-\ntions of MAE in monolayers1,2, multilayers3–6, thin films7, clusters8,9or nanowires10systems with\nab-initio as well as tight-binding electronic structure methods. Technically several approaches\nhave been developed for the determination of the MAE. The brute force method consists in per-\nforming self consistent calculations for various orientations of the magnetization. This approach\nalthough straightforward is the most computationnally demanding since it usually necessitates a\nlong self-consistent loop that implies the diagonalization of large matrices. Rather early it was rec-\n2ognized that small changes of the total energy could be related to the changes of the eigenvalues\nof the Hamiltonian. This is the so-called Force Theorem11–13that is very well suited to the cal-\nculation of MAE and has been used extensively in the past. Besides its computational efficiency\nit is also very stable numerically. Several works are also based on a perturbative treatment that\nconsists in writing to second order the energy correction due to the spin-orbit Hamiltonian treated\nas a perturbation1,7. Finally to get a basic understanding of the underlying physical phenomena it\nis very convenient to write the total MAE as a sum of contributions arising from atoms with bulk-\nlike environment and from atoms with lower local symmetry such as surface/interfaces atoms. In\nline with the various methods exposed above there are also many different ways to decompose the\ntotal MAE into atomic contributions and it is not always clear whether or not they are all valid.\nIn this paper we wish to propose a comprehensive overview that will clarify the different\naproaches to calculate the magnetocrystalline anisotropy with a particular emphasis on the atomic\nsite decomposition of MAE and application to Iron slabs and clusters. We will first describe in\nSec.II our tight-binding method used thoughout the paper and then present in detail two alterna-\ntive versions of the Force Theorem (FT) widely used in the litterature. We will basically show\nthat even though these two versions of the FT are equivalent in terms of total energy variatons, the\nso-called grand-canonical FT is the most suited to define local quantities. In Sec. III we illustrate\ntheses concepts on the case of Fe (001) and Fe (110) slabs that behave very differently. In particular\n(001) surface favors out of plane magnetization while the easy axis of (110) slabs is in-plane and\na with a smaller anisotropy. This has important consequences on the MAE of nanoclusters that are\ninvestigated in the Sec. IV. Finally in Sec.V we will draw the main conclusions of this work.\nII. METHOD AND COMPUTATIONAL DETAILS\nA. Magnetic Tight-Binding model\nIn the following most of our calculations are based on an efficient tight-binding (TB) model\nincluding magnetism and spin-orbit interaction that has been described in several publications10,14.\nWe will just briefly recall its main ingredients. The Hamiltonian in a non-orthogonal s;p;d pseudo\natomic basis is written as a sum of 4 terms H=HTB+HLCN+HStoner+HSO. WhereHTBis a\nstandard ”non-magnetic” TB hamiltonian which form is very similar to the one introduced by Mehl\nand Papaconstantopoulos15,HLCNis a term ensuring local charge neutrality, HStoner a Stoner-like\n3contribution that controls the spin magnetization and HSOcoresponds to spin-orbit interaction.\nWithin this model the total energy of the system should be corrected by the so-called double\ncounting terms arising from electron-electron interaction introduced by local charge neutrality and\nStoner terms. The total energy then takes the form as follows:\nEtot=Eb\u0000Edc (1)\nwhereEb=P\n\u000bf\u000b\u000f\u000bis the band energy ( f\u000b=f(\u000f\u000b)being the Fermi-Dirac occupation of\nstate\u000band corresponding eigenvalue \u000f\u000b). The expression of the double counting term is given by:\nEdc=U\n2X\ni[n2\ni\u0000(n0\ni)2]\u00001\n4X\ni;\u0015I\u0015m2\ni\u0015 (2)\nniandmiare respectively the charge and the spin moment of site i,n0\nithe valence charge, Uis\nthe Coulomb integral and I\u0015the Stoner parameter of orbital \u0015(\u0015=s;p;d etc..). In transition met-\nalsdorbitals are the one bearing the magnetism and the amplitude of magnetization is controled\nby the amplitude of Id(the exact value of IsandIphas a minor effect on the total magnetization\nbut in practice we took Is=Ip=Id=10).\nThe hopping and overlap integrals as well as onsite terms of HTBare fitted on ab initio datas\n(bandstructure and total energy). Local charge neutrality is controled by the amplitude of the\nCoulomb energy Uwhich in practice is taken equal to 20 eV. The value of the Stoner parameter\nIdis determined by reproducing ab-initio datas of the spin magnetization of bulk systems as a\nfunction of the lattice constants. The optimal Idvalue is the one that compares the best to ab-initio\ncalculations. In the following we took Id= 0.88eV. The spin-orbit constant \u0018dis also determined\nby comparison with ab-initio bandstructure and we found that 60 meV is a very good estimate for\nIron.\nB. Force Theorem: FT\nThe Force Theorem11has been used in various contexts. In studies of magnetic materials it has\nmainly been used for the calculation of magnetocrystalline anisotropy12or for the determination\nof exchange coupling in magnetic multilayers16. In this section we will illustrate its principle in\na simple magnetic pure dband orthogonal TB model of a monatomic system. The total energy\nreads:\n4Etot=X\n\u000bf\u000bh\u000bjHj\u000bi\u0000U\n2X\ni(n2\ni\u0000(n0\ni)2) +I\n4X\nim2\ni (3)\nwhere the Hamiltonian His made of two terms:\nH=HTB+X\ni;\u001b;\u0015ji;\u0015;\u001bi\u0002\nU(nin0\ni)\u0000I\n2mi\u001b\u0003\nhi;\u0015;\u001bj (4)\nThe total energy obtained from this formula is caclulated by self-consistent loop on the charge\nand magnetic moment. Indeed the onsite terms of the Hamiltonian are renormalized by a quantity\n\"i;\u0015;\u001b=U(ni\u0000n0\ni)ni\u0000I\n2mi\u001bwhich depends itself on the local charge and magnetic moment.\nLet us now consider the effect of a perturbative external potential \u000eVextwhich in our case will\nbe the spin-orbit coupling. This external potential will induce a total potential variation \u000eV=\n\u000eVext+\u000eVindwhere\u000eVindis the potential variation provoked by the modification of on site levels\nin the perturbed system. Within our model \u000eVindis simply related to the variation \u000eniand\u000emiof\nthe charge and magnetic moment thus,\n\u000eVind=X\ni;\u001b;\u0015ji;\u001b;\u0015i\u0002\nU\u000eni\u0000I\n2\u000emi\u0003\nhi;\u001b;\u0015j (5)\nThe variation of the band energy due to \u000eVindcan be straighforwardly calculated from first\norder pertubration expansion17:\nX\n\u000bf\u000bh\u000bj\u000eVindj\u000bi=UX\nini\u000eni\u0000I\n2X\nimi\u000emi (6)\nThis variation is exactly compensated (to linear order) by the one of the double counting term\nand therefore the change of the total energy is equal to the change of band energy induced by the\nexternal potential only, leading to the so-called force theorem:\n\u0001Etot\u0019\u0001EFT\nb= \u0001hX\n\u000bf\u000b\u000f\u000bi\n(7)\nWhere \u0001EFT\nbis the variation of the non-self-consistent band energy. The great advantage of\nthis formulation is obviously that self-consistency effects can (and should) be ignored. In this\ncontext, the total energy variation induced by a change of the external potential from \u000eV1\nextto\n\u000eV2\next(corresponding for instance to a change of the spin-orbit coupling matrices between two spin\n5orientations 1and2) can be written:\n\u0001EFT\nb\u0019ZE1\nF\nEn1(E)dE\u0000ZE2\nF\nEn2(E)dE (8)\nn1(E)andn2(E)being the density of states and E1\nF,E2\nFthe Fermi levels of the configurations\n1and2respectively. The Fermi levels are determined by the condition on the total number of\nelectronsNin the system:\nN=ZE1\nF\nn1(E)dE=ZE2\nF\nn2(E)dE (9)\nC. Grand canonical Force Theorem: FT gc\nIn the previous derivation of the Force Theorem it is important to note that the band energy is a\nsummation of the eigenvalues over the occupied states (at fixed number of electrons) . Therefore\na small variation of Fermi energy is expected with respect to the non perturbed system as follows:\nE1\nF=EF+\u000e1andE2\nF=EF+\u000e2 (10)\nAt linear order the variation of band energy can be written\n\u0001EFT\nb=ZEF\nE\u0001n(E)dE+EF(\u000e1n1(EF)\u0000\u000e2n2(EF)) (11)\nusing the conservation of the total number of electrons it comes that\n\u0001EFT\nb\u0019\u0001EFTgc\nb=ZEF\n(E\u0000EF)\u0001n(E)dE (12)\nWe will denote FT gcthis alternative formulation of the Force Theorem in the rest of the pa-\nper. The FT gcformulation seems very similar to the standard FT formulation, but it leads to very\ndifferent ”space” partition of the energy. The underlying reason is to be found in the type of sta-\ntistical ensemble: canonical for FT and grand-canonical for FT gc. The grand-canonical ensemble\nfor which the ”good” variable is the Fermi energy (and not the total number of electrons) is better\nsuited for a spatial partition of the energy18. For example the Gibbs construction19to define prop-\nerly surface quantities is based on a grand-canonical ensemble. Within this approach the suitable\npotential is the so-called grand-potential \n =Eb\u0000EFN. This formalism can be generalized at fi-\n6nite temperature18,20,21. Since the first-order variation of the Helmholtz Free Energy F=Eb\u0000TSe\nat constant electron-number is equal to the fisrt order variation of the grand-potential at constant\nchemical potential the FT and FT gcformulation are equivalent in terms of variation of total energy.\nHowever the spatial repartition of energy could be very different within these two approaches.\nIII. MAGNETOCRYSTALLINE ANISOTROPY ENERGY OF FE (001) AND FE (110) SLABS\nIn this section we will present results on the MAE of thin layers of Iron. In the first part (III A)\nwe will discuss the validity of the various approximations presented in the methodological section.\nIn particular we will justify the Force Theorem. We will also compare our results with ab-initio\ncalculations proving the quality of our TB model. Sec. III B will be devoted to the comparison of\nFT and FT gcformulations with respect to the layer resolved MAE and we will analyze the surface\nanisotropy energy. Finally the Bruno formula will be discussed.\nA. MAE of Fe (001) slabs: validity of the Force Theorem\nThe MAE is defined as the change of total energy Etotassociated to a change in the direction of\nthe magnetizationh~Sifor a fixed position of atom. In the case of a full self-consistent calculation\nincluding spin-orbit coupling, the MAE is defined as the energy difference \u0001E=E?\ntot\u0000Ek\ntot,\nwhere?andkrefer to a magnetization where all atomic spins are pointing in a direction perpen-\ndicular or parallel to the surface respectively. The MAE is therefore the result of two independant\nself-consistent calculations which have to fulfill an extremely stringent condition for convergency\nsince MAE are typically below meV . In systems containing ”light” atoms like Iron for which the\nspin-orbit coupling constant \u0018is modest (60meV) it is expected that the Force Theorem should\napply very well. Within this approximation the MAE is given by the difference of the band en-\nergy but ignoring any self-consistent effect. This type of calculation is performed in three steps:\ni) Collinear self-consistent calculation without SOC for which the density matrix is diagonal in\nspin space ii) Global rotation of the density matrix to ”prepare” it in the right spin direction iii)\nNon-collinear non-self-consistent calculation including SOC. We have performed a series of cal-\nculations for ultrathin (001) Iron layers of various thicknesses ranging from one to twenty atomic\nlayers, within the full-scf and FT approaches. The lattice parameter of a= 2:85˚A was every-\nwhere used and no atomic relaxation was considered. The convergency of the calculations have\n7been carefully checked, we found that 2500 kkand 4900kkpoints in the first Brillouin zone for cal-\nculations without and with SOC respectively were sufficient to obtain a precision below 10\u00005eV .\nThe MAE obtained by these two methods differ by less than 10\u00005eV proving the validity of FT\napproach which will be used systematically in the rest of the paper. It should be noted that FT\napproach leads to a considerable savings in the computational cost since no self-consistency is\nneeded, therefore only one diagonalization of the full Hamiltonian including SOC is sufficient.\nIn order to check the accuracy of our tight-binding model we have also performed ab-initio\ncalculations using the Quantum-ESPRESSO (QE) package22based on Density Functional Theory\n(DFT). Since no FT approach is yet implemented in QE all the calculations are self-consistent and\nspin-orbit coupling is included via fully-relativistic ultrasoft pseudopotentials. The generalized\ngradient approximation (GGA) for exchange-correlation potential in the Perdew, Burke, and Ernz-\nerhof parametrization was employed. To describe thin films we have used the so-called super-cell\ngeometry separating the adjacent slabs by about 8 ˚A in thezdirection (orthogonal to the surface)\nin order to avoid their unphysical interaction. Since the MAE is usually a tiny quantity, ranging\nfrom\u0016eV tomeV , it requires a very precise determination of total energy, and the total energy\ndifference among various spin directions is very sensitive to the convergence of computational\nparameters. We found that 40\u000240k-point mesh in the two-dimensional Brillouin zone was suf-\nficient to obtain a well-converged MAE for (001) Iron slabs . A Methfessel Paxton broadening\nscheme with 0.05 eVbroadening width was used with plane wave kinetic energy cut-offs of 30 Ry\nand 300 Ry for the wave functions and for the charge density, respectively. Fig. 1 shows the total\nMAE as a function of the number layers of Fe (001) slabs. A good agreement is obtained between\nTB and ab initio calculations which proves once again the efficiency and quality of our TB model.\nB. Layer-resolved MAE: FT versus FT gc\nIn section III A we have only considered variations of total energies but it is also very instruc-\ntive to investigate the local density of energy. Let us write the MAE as a sum of atomic-like\ncontribution within FT and FT gcapproaches:\n\u0001EFT\nb=X\ni\"ZE?\nF\nEminEni\n?(E)dE\u0000ZEk\nF\nEminEni\nk(E)dE#\n(13)\n8\u0001EFTgc\nb=X\ni\"ZEF\nEmin(E\u0000EF)\u0001ni(E)dE#\n(14)\nwhereni\n?(E)andni\nk(E)are the density of states on atom ifor perpendicular or in-plane mag-\nnetization direction, respectively, and \u0001ni(E) =ni\n?(E)\u0000ni\nk(E).E?\nFEk\nFare the corresponding\nFermi energies and EFis the Fermi level of the collinear self-consistent calculation without SOC.\nThe layer-resolved MAE calculated by FT and FT gcmethods for Fe (001) slab of 100 layers is\nshown in Fig.2b. The most striking result is the very large oscillating behaviour which persists very\ndeeply into the bullk for the FT method. In addition, the local MAE obviously does not converge\ntoward the expected bulk value which in this case should be exactly zero (since the three cubic axis\nare equivalent). In contrast, the layer resolved MAE obtained from the FT gcmethod corresponds\nto the behaviour expected from a proper local quantity, namely a dominant variation in the vicinity\nof the surface that attenuates rapidly when penetrating in the bulk. This is indeed the case since\nonly the surface atomic layer is strongly perturbed. In fact there are slight oscillations over the five\nfirst outer layers and an almost perfect convergence towards the bulk value for deeper layers. It is\nthen clear that FT gcis the appropriate method to define a layer resolved MAE. Note, however, that\nthe total MAE are almost strictly indentical for FT and FT gc. Finally, it is very interesting to point\nout a striking analogy that exists with the simple one-dimensional free-electron model discusses\nin the next section III C.\nIt is also useful to note the relation between Eq. 13 and Eq. 14 in order to understand the\ndifference between the two methods:\n\u0001EFTgc\nb,i= \u0001EFT\nb,i\u0000EF(Ni\n?\u0000Ni\nk) (15)\nwhereNi\n?andNi\nkare the Muliken charges on atom ifor perpendicular or in-plane magnetization,\nrespectively. When summed over all the atoms of the system the additionnal term, EF(N?\u0000Nk),\ndisappears since the total number of electrons is preserved and we recover the equivalence be-\ntween FT and FT gcfor total energy differences. This formula is quite instructive since it shows\nthat the difference between FT and FT gcis related to the slight charge redistribution between the\ntwo magnetic configurations. At the first sight it seems that FT and FT gcshould lead to very\nsimilar decomposition of the energy since the local charge neutrality term is supposed to avoid\ncharge transfers and therefore \u0001Ni=Ni\n?\u0000Ni\nk\u00190, but one should bear in mind that the force\ntheorem applies only if self-consistency effects are ignored and therefore larger charge redistribu-\n9tions may appear. They produce irrelevant (to magnetic anisotropy) contributions EF\u0001Nito the\nlocal anisotropy energy which should be substracted as it is accomplished in the FT gcapproach.\nIn Fig.2a we show \u0001Niwhich indeed looks very similar in shape to the FT layer resolved MAE\nand, when substracted, leads thus to well behaved FT gclayer resolved MAE curve.\nThese arguments show that the local variation of band energy should be the same after a self-\nconsistent calculation provided that the local charge neutality is achieved. To check this point we\nhave determined the layer-resolved MAE for a (001) slab of 20 Fe layer with full SCF calculation\nand FT gcmethod. Note that in the case of the full-scf approach one should consider the variation\nof the total energy wich includes band energy as well as double counting terms. In our TB scheme\nthe double counting terms can easily be decomposed as a sum of atomic contributions and will\nparticipate to the local MAE. In Fig. 3 the layer-resolved MAE obtained from the two methods\nare presented and an excellent agreement between them is indeed found.\nFinally let us point out an argument which was originally discussed by Daalderop et. al12: If\na common Fermi energy is used for the two direction of magnetization within the FT formulation\nthen an additional term EF\u0001Nis erroneously contributing to the total MAE.\nC. Didactic example: one-dimensional quantum well\nTo illustrate the difference between FT and FT gclet us consider one of the simplest models,\na one-dimensional free-electron gas bounded within a length Lby infinite barriers (Fig. 4). The\nnormalized wave functions and the corresponding discretized eigenvalues are (atomic units in\nwhich ~2= 2m= 1are used):\n k(z) =r\n2\nLsinkz \u000fk=k2withk=p\u0019\nL(16)\nwhereptakes only positive integer values. For the unbounded electron-gas with periodic Born-\nV on Karman (BVK) boundary conditions:\n BVK\nk(z) =r\n1\nLeikz\u000fk=k2withk= 2n\u0019\nL(17)\nIn that case ntake any postive or negative integer values including 0. In the continuum limit\nthe excess energy due to the creation of two surfaces is given by:\n10\u0001E= 2\u0002L\n\u0019hZkF+\u000ekF\n0\u000fkdk\u0000ZkF\n0\u000fkdki\n; (18)\nwhere the factor 2is due to the spin degeneracy and kF=\u0019N\n2L(Nis the total number of\nelectrons in the box of the length L) is the Fermi wave vector of the unbounded homogeous gas.\nSince an electron at k= 0is not allowed in the case of quantum well, it should be instead placed\non the next free level, which leads to \u000ekF=\u0019\n2Land thus \u0001E=k2\nF=EF. Local decomposition\nof\u0001Eis naturally achieved by weighting each energy eigenvalue in (18) by the squared modulus\nof the corresponding wave function which results in:\n\u0001E(z) =\u00002\n\u0019ZkF\n0k2cos(2kz)dk+2k2\nF\nLsin2(kFz) (19)\nEquivalently, a grand-canonical formulation gives:\n\u0001Egc(z) =\u00002\n\u0019ZkF\n0(k2\u0000k2\nF) cos(2kz)dki\n(20)\nSimple integration leads to exact expressions for \u0001E(z)and\u0001Egc(z):\n\u0001Egc(z) =1\n\u0019\u0010sin(2kFz)\n2(kFz)3\u0000cos(2kFz)\n(kFz)2\u0011\nEFkF (21)\n\u0001E(z) = \u0001Egc(z)\u0000sin(2kFz)\n\u0019zEF+2 sin2(kFz)\nLEF (22)\n(23)\nThese expressions, illustrated in Fig. 4, are quite instructive. Within the FT gcformulation the\ndensity of surface energy behaves like 1=z2for largez. The case of the FT formulation is more\ntricky: it contains, in addition, a term slowly decaying as 1=zand a term which does not decay\n(for a given L) but tends to zero as Lgoes to infinity. In fact, these two last terms are simply\nproportional to the surface excess electronic density:\n\u0001\u001a(z) =\u0000sin(2kFz)\n\u0019z+2 sin2(kFz)\nL(24)\nso that \u0001E(z) = \u0001Egc(z) +EF\u0001\u001a(z). Therefore, we conclude that long-range Friedel oscil-\n11lations in \u0001\u001a(z)are at the origin of slow convergence with zobserved for the FT \u0001E(z)which is\nperfectly in line with our previous analysis of layer-resolved magnetic anisotropies as illustrated\nby the striking similiraties between Fig. 2 and Fig. 4.\nD. MAE: surface and bulk contributions\nFrom the discussion above it is natural to define the surface magnetic anisotropy energy as the\nsum of contributions from five outer layers (from both sides of the slab) obtained using the FT gc\nformulation. The contributions from other layers sum up to what we call a bulk MAE. In Fig.\n5 we plot the evolution of the surface, bulk and total MAE for both Fe (001) and Fe (110) slabs\nwith respect to the total number of layers N(from 15 to 100). Note that the bulk MAE value\nper atom can be obtained by dividing the total bulk value by N\u000010bulk-like layers. Also the\ntrue surface MAE should be obtained by dividing the surface contribution presented in Fig. 5\nby two since the slabs contain two surfaces. Our calculations show that (001) and (110) Fe sur-\nfaces have very different qualitative behaviour, the total MAE is negative for Fe (001) indicating\nan out-of-plane easy axis while it is in-plane for Fe (110) since its MAE is positive. More interest-\ningly, in the case of Fe (110) , additional calculations have shown that the magnitude of the in-plane\nanisotropy is almost as large as the one obtained between in-plane and out-of-plane orientations.\nIt is also important to mention that the amplitude of the oscillations, though do not change the\nsign of the MAE, can however be as large as 0.2meV for Fe (001) and 0.1meV for Fe (110) at\nleast up toN\u001840. In addition, the total MAE is essentially dominated by the surface contribu-\ntion. However, the oscillatory behaviour at large thicknesses, particularly pronounced for Fe (001) ,\nclearly originates from the bulk. This kind of oscillatory behaviour of the MAE has been observed\nexperimentally23,24and was interpreted in terms of quantum well states. The latter are formed in\nthe ferromagnetic films from occupied and unoccupied electronic states close to the Fermi level\nthat contribute significantly to the MAE.\nE. Bruno formula\nTo gain better understanding of MAE beyond bare numbers, investigating related quantities\nis helpful. The orbital moment is a quantity essentially related to the SOC and to the MAE in\nmagnetic systems. It is well known that the easy axis always corresponds to the direction where\n12the orbital moment is the largest. These arguments can be made more quantitative. Patrick Bruno1\nhas derived an interesting relation using second order pertubation theory (since the first order\nterm vanishes) with respect to the SOC parameter25. Provided that the exchange splitting is large\nenough compared to the d-electron bandwidth, the MAE can be made proportional to the variation\nof the orbital moments. More precisely:\nEb;?\u0000Eb;k=\u0000\u0018\n4(hMorb\n?i\u0000(hMorb\nki) (25)\nThis formula is based on a perturbative expansion (and an additionnal approximation concern-\ning spin-flip transitions) for which the reference system and also the Fermi level are those of the\nunperturbed system without SOC. It can be shown that this approach is compatible with a grand\ncanonical ensemble description (see Ref.20 for a detailed discussion about statistical ensemble and\nsecond order corrections in the context of magnetic anisotropy). This relation can be generalized\nto systems with several atoms per unit cells7and also be used to extract a layer resolved MAE26.\nIn Fig. 6 the layer resolved MAE calculated by Eq. 25 and by the Force Theorem are plotted, we\nfound that only the surface layers have a significant contribution, while contribution from inner\nlayers rapidly converges to the bulk (zero) value within the two approaches. However, note that\nthe Bruno’s model results in quite different total MAE compared to the FT approximation in the\nvicinity of the surface. One can say that there is a rather good qualitative agreement between the\ntwo approaches, however the Bruno’s formula can significantly (and quantitatively) differ from the\nFTgcresults.\nIV . ISOLATED FE NANOCLUSTERS\nOnce having properly defined the atomically resolved MAE and analyzed in detail (001) and\n(110) Iron surfaces, it is interesting to study the case of clusters. There exists a vast body of\nresearch on the theoretical investigation of combined structural and magnetic properties of unsup-\nported transition metal clusters, relatively fewer are devoted to the determination of their magnetic\nanisotropy. Moreover most of them are dealing with small particles containing few atoms8,9,27,\nthe case of large clusters is generally treated with empirical Neel-like models of anisotropy28. In\nthis section we will present TB calculation of two large nanoclusters with facets of orientations\n(001) and(110) . We will more specifically consider the case of a truncated pyramid of nanometer\n13size (see inset on Fig.7). This geometry was chosen since such nanocrystals can be obtained by\nepitaxial growth on SrTiO 3(001)29substrate. In a second part we will consider the corresponding\ntruncated bipyramid made of two truncated pyramids joined at their bases.\nA. Truncated pyramid\nThe particular cluster that we investigated is made of 620 atoms, with 12x12 atom lower base\nand 5x5 atom upper face and contains 8 atomic layers. Its length-to-height ratio, 1.14 is close to\nthe experimental value of 1:20\u00060:12. In Fig. 7 we present the variation of the grand-canonical\nband energy with respect to the Euler polar angle \u0012between the magnetization direction and the\nzaxis choosen to be perpendicular to its ”roof” and base of (001) orientation (see inset). The\nazimuthal angle \u001eis kept zero so that the magnetization remains in the xzplane. The easy axis\nis evidently along the zand the magneto-crystalline anisotropy is of the order of 110 meV . We\nalso checked the azimuthal anisiotropy but found an extremely flat energy landscape in the xy\nplane with an amplitude of 3 meV , the hard axis being along the diagonal of the base. To get more\ninsight into the origin of the anisotropy we have decomposed the band energy per atomic sites and\nanalyzed different contributions: total surface, (001) facets, perimeter of the base, etc. Summing\nlocal MAE over atomic sites in the outer shell of the nanocluster (dashed line), we almost recover\nthe total magnetocrystalline anisotropy proving that only the outer shell (so called surface atoms)\nis participating to the overall anisotropy. A more detailed analysis showed only two significative\ncontributions: i) low coordinated perimeter atoms of the base (red line) and ii) two (001) facets,\nexcluding perimeter atoms (blue line).\nIntrestingly, the perimeter atoms have the strongest anisotropy while, on the contrary, the con-\ntribution from (110) side facets is almost negligible (and, moreover, cancel each other because of\ntheir opposite orientations).\nBy counting the number of ”implied” atoms (109 (001) atoms and 44 perimeter atoms) it is\npossible to extract an average anisotropy per (001) surface atom and per perimeter atom. One finds\n0:56meV/atom and 0:90meV/atom for (001) and perimeter atoms, respectively. This coresponds\nquite well to the expected anisotropy found for the Fe (001) slabs.\n14B. Truncated bipyramid\nWe then consider another type of cluster: a truncated bipyramid (lower inset in Fig. 8) made\nof 1096 atoms and obtained by attaching symmetrically to the previous truncated pyramid another\none (with removed base plane) from below. In Fig. 8 we have compared the total MAE of the\ntwo nanoclusters. Although the truncated bipyramid contains more atoms its anisotropy (15meV)\nis much lower than in previous case. The explanation is quite straightforward from the previous\nanalysis: the surface of the (001) facets has been strongly reduced and, moreover, the perimeter\natoms of the base have now more neigbours and no longer contribute so strongly to the total\nanisotropy. The latter comes from two small (001) facets only. This argument works rather well:\nindeed, the number of atoms in (001) facets is now 18 which gives an anisotropy of 18\u00020:56 =\n10meV , the value slightly smaller then the overall MAE, 15meV , with the missing contribution\ncoming from perimeter atoms which were not taken into account.\nV . CONCLUSION\nA comprehensive TB study of magnetocrystalline anisotropy energy of Fe (001) and Fe (110)\nslabs and nanoclusters has been presented. Due to small spin-orbit coupling constant, the Force\nTheorem is valid for Fe-based systems studied in this work. We have shown that a proper way\nto define the layer-resolved MAE should use the grand-canonical FT formulation instead of the\nstandard FT, while the two approaches are equivalent for the total MAE. The prefered orientations\nfor Fe (001) and Fe (110) slabs are out of plane and in-plane, respectively. For both slabs, the total\nMAE is dominated by surface contribution as expected. However, surface contribution converges\nmore rapidly than bulk one with respect to the number of atomic layers of the slabs. The study\nof nanoclusters showed that the dominating MAE originates form the (001) facets and especially\nfrom low coordinated perimeter atoms of the base of the pyramid. On the contrary, the contri-\nbution from (110) side facets is almost negligible. In view of the results presented here, a study\nof the magnetic properties of nanoclusters deposited on a substrate (SrTiO 3(001), Au or Cu etc\n...) is rather promising since depending on the bonding between the substrate and the (001) facets\none could imagine to tune the magnetic anisotropy of these nanoclusters. Finally Skomski et al30\nshowed that the shape of surface anisotropy could have consequences on the magnetization rever-\nsal of nanoparticles. Therefore, it is very likely that a detailed investgation of the spin dynamics\n15of nanometer size iron clusters could reveal such surface effects in the anisotropy.\nAcknowledgement\nThe research leading to these results has received funding from the European Research Council\nunder the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agree-\nment n\u000e259297. This work was performed using HPC resources from GENCI-CINES (Grant Nos.\nx2013096813).\n\u0003Electronic address: cyrille.barreteau@cea.fr\n1P. Bruno, Physical Review B 39, 865 (1989), ISSN 0163-1829, URL http://link.aps.org/\ndoi/10.1103/PhysRevB.39.865 .\n2J. Gay and R. Richter, Physical Review Letters 56, 2728 (1986), ISSN 0031-9007, URL http://\nlink.aps.org/doi/10.1103/PhysRevLett.56.2728 .\n3L. Szunyogh, B. ´Ujfalussy, and P. Weinberger, Physical Review B 51, 9552 (1995), ISSN 0163-1829,\nURLhttp://link.aps.org/doi/10.1103/PhysRevB.51.9552 .\n4L. Szunyogh, B. ´Ujfalussy, C. Blaas, U. Pustogowa, C. Sommers, and P. Weinberger, Physical Review B\n56, 14036 (1997), ISSN 0163-1829, URL http://link.aps.org/doi/10.1103/PhysRevB.\n56.14036 .\n5B.´Ujfalussy, L. Szunyogh, and P. Weinberger, Physical Review B 54, 9883 (1996), ISSN 0163-1829,\nURLhttp://link.aps.org/doi/10.1103/PhysRevB.54.9883 .\n6B.´Ujfalussy, L. Szunyogh, P. Bruno, and P. Weinberger, Physical Review Letters 77, 1805 (1996), ISSN\n0031-9007, URL http://link.aps.org/doi/10.1103/PhysRevLett.77.1805 .\n7M. Cinal, D. Edwards, and J. Mathon, Physical Review B 50, 3754 (1994), ISSN 0163-1829, URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.50.3754 .\n8G. Pastor, J. Dorantes-D ´avila, S. Pick, and H. Dreyss ´e, Physical Review Letters 75, 326 (1995), ISSN\n0031-9007, URL http://link.aps.org/doi/10.1103/PhysRevLett.75.326 .\n9G. Nicolas, J. Dorantes-D ´avila, and G. Pastor, Physical Review B 74, 014415 (2006), ISSN 1098-0121,\nURLhttp://link.aps.org/doi/10.1103/PhysRevB.74.014415 .\n10G. Aut `es, C. Barreteau, D. Spanjaard, and M.-C. Desjonqu `eres, Journal of Physics: Condensed Matter\n18, 6785 (2006), ISSN 0953-8984, URL http://stacks.iop.org/0953-8984/18/i=29/a=\n018.\n1611M. Weinert, R. Watson, and J. Davenport, Physical Review B 32, 2115 (1985), ISSN 0163-1829, URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.32.2115 .\n12G. Daalderop, P. Kelly, and M. Schuurmans, Physical Review B 41, 11919 (1990), ISSN 0163-1829,\nURLhttp://link.aps.org/doi/10.1103/PhysRevB.41.11919 .\n13X. Wang, D.-s. Wang, R. Wu, and A. Freeman, Journal of Magnetism and Magnetic Materi-\nals159, 337 (1996), URL http://www.sciencedirect.com/science/article/pii/\n0304885395009361 .\n14C. Barreteau and D. Spanjaard, Journal of physics. Condensed matter : an Institute of Physics journal 24,\n406004 (2012), ISSN 1361-648X, URL http://stacks.iop.org/0953-8984/24/i=40/a=\n406004 .\n15M. Mehl and D. Papaconstantopoulos, Physical Review B 54, 4519 (1996), ISSN 0163-1829, URL\nhttp://link.aps.org/doi/10.1103/PhysRevB.54.4519 .\n16J. Mathon, M. Villeret, A. Umerski, R. Muniz, J. dAlbuquerque e Castro, and D. Edwards, Physical\nReview B 56, 11797 (1997), ISSN 0163-1829, URL http://link.aps.org/doi/10.1103/\nPhysRevB.56.11797 .\n17The first order variation of an eigenvalue induced by a perturbative potential \u000eVis given by the formula\n\u000e\"\u000b=h\u000bj\u000eVj\u000bi. The band energy of the perturbed system is thereforeX\n\u000bf(\"\u000b+\u000e\"\u000b)(\"\u000b+\u000e\"\u000b).\nUsing the conservation of the total number of electrons it comes that the band energy variation is equal\ntoX\n\u000bf(\"\u000b)\u000e\"\u000b.\n18F. Ducastelle, Order and Phase Stability in Alloys (North Holland, Amsterdam, 1991).\n19M. C. Desjonqu `eres and D. Spanjaard, Concept in Surface Physics (Springer Verlag, Berlin, 1995).\n20M. Cinal and D. M. Edwards, Physical Review B 55, 3636 (1997), ISSN 0163-1829, URL http:\n//link.aps.org/doi/10.1103/PhysRevB.55.3636 .\n21The grand potential at finite temperature Tand chemical potential \u0016can be written in two alternative\nways (thanks to an integration by parts) : \n(T) =Z\nL(E)n(E)dE=\u0000Z\nf(E)N(E)dE. With\nL(E) =\u0000kBTlogf1 + exp[\u0000(E\u0000\u0016)=kBT]gandN(E) =ZE\nn(E0)dE0. Note that the derivative of\nL(E)is the Fermi function f(E) = [1 + exp( E\u0000\u0016)=kBT].\n22P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli, G. L. Chiarotti,\nM. Cococcioni, I. Dabo, et al., Journal of physics. Condensed matter : an Institute of Physics journal 21,\n395502 (2009), ISSN 1361-648X, URL http://iopscience.iop.org/0953-8984/21/39/\n17395502 .\n23M. Przybylski, M. Dabrowski, U. Bauer, M. Cinal, and J. Kirschner, Journal of Applied Physics\n111, 07C102 (2012), ISSN 00218979, URL http://link.aip.org/link/?JAPIAU/111/\n07C102/1 .\n24S. Manna, P. L. Gastelois, M. Dbrowski, P. Ku ´swik, M. Cinal, M. Przybylski, and J. Kirschner, Physical\nReview B 87, 134401 (2013), ISSN 1098-0121, URL http://link.aps.org/doi/10.1103/\nPhysRevB.87.134401 .\n25The second order perturbation expansion at finite temperature is given by the well-known formula \u000e\n =\nX\n\u000b;\fjh\u000bjHSOj\fij2f\u000b(1\u0000f\f)\n\"\u000b\u0000\"\f.\n26F. Gimbert and L. Calmels, Physical Review B 86, 184407 (2012), ISSN 1098-0121, URL http:\n//link.aps.org/doi/10.1103/PhysRevB.86.184407 .\n27M.-C. Desjonqu `eres, C. Barreteau, G. Aut `es, and D. Spanjaard, Physical Review B 76, 024412 (2007),\nISSN 1098-0121, URL http://link.aps.org/doi/10.1103/PhysRevB.76.024412 .\n28M. Jamet, W. Wernsdorfer, C. Thirion, V . Dupuis, P. M ´elinon, A. P ´erez, and D. Mailly, Physical\nReview B 69, 024401 (2004), ISSN 1098-0121, URL http://link.aps.org/doi/10.1103/\nPhysRevB.69.024401 .\n29F. Silly and M. R. Castell, Applied Physics Letters 87, 063106 (2005), ISSN 00036951, URL http:\n//link.aip.org/link/?APPLAB/87/063106/1 .\n30R. Skomski, X.-H. Wei, and D. J. Sellmyer, IEEE Transactions on Magnetics 43, 2890 (2007),\nISSN 0018-9464, URL http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?\narnumber=4202921 .\n180123456789101112−4−3.5−3−2.5−2−1.5−1−0.500.511.5\nNumber of layes NMAE [meV]\n  \nDFT−GGA\nTBFIG. 1: Total MAE versus Fe film thickness Nfor Febcc (001) slabs. TB calculation (in green) are\ncompared with ab-initio DFT-GGA calculations (in red).\n190 10 20 30 40 50 60 70 80 90 100-6-5-4-3-2-101\n0 10 2000 20 40 6080 100-0,0010\nFTFTgc\nAtomic siteAtomic siteMAE  [meV]\nAtomic siteMAE  [meV] ∆N(a)\n(b)FIG. 2: a) Variation of the charge difference \u0001Ni= (Ni\n?\u0000Ni\nk)between out of plane and in plane\nmagnetic configurations (obtained after one diagonalization) on successive atomic layers of a Fe (001) slab\ncontainingN= 100 layers. b) Layer resolved MAE of the Fe (001) slab calculated with two different\nmethods: canonical FT (black lines) and grand canonical FT gc(red lines). The zoom over the first 20 layers\nis shown in the inset.\n200 2 4 6 8 10 12 14 16 18 20−0.5−0.4−0.3−0.2−0.100.1\nAtomic siteLayer resolved MAE [meV]Full SCF calculation\nFTgcFIG. 3: Layer-resolved MAE of the Fe (001) slab withN= 20 layers calculated using the TB fully self-\nconsistent calculation and FT gcapproximation. Very good agreement confirms a proper local decomposition\nof MAE provided by the grand canonical formulation.\n210 20 40 60 80 100-0,200,20,40,6\n0 1000,40 20 40 6080 100-0,4-0,20\nFTFTgc\nz  [length]z [length] ∆E [length -3]\nz ∆E ∆ρ [length-1](a)\n(b)FIG. 4: Graphical representation of the functions \u0001E(z),\u0001Egc(z), and \u0001\u001a(z)for a one-dimensional\nelectron gas confined by infinite barriers in the box of the length L. The discretized calculations were done\nwith the parameters N= 70 (total number of electrons) and L= 100 .\n220 10 20 30 40 50 60 70 80 90 100−1.4−1.2−1−0.8−0.6−0.4−0.200.2\nNumber of layers NMAE [meV]Bulk\nSurface\nTotal(a)\nFe (001)\n0 10 20 30 40 50 60 70 80 90 100−0.1−0.0500.050.10.150.20.250.30.350.40.4\nNumber of layers NMAE [meV]Surface\nTotal\nBulkFe (110)(b)FIG. 5: (Color online) Surface , bulk (see definitions in the text), and total MAE for Fe (001) and Fe (110)\nslabs as a function of the film thickness. The surface contribution is obtained by summing the layer resolved\nMAE over the 5 outer layers on each side of the slab. Consequently the true surface MAE can be obtained\nby dividing by 2 this quantity. Positive (negative) MAE values mean in (out of) plane easy axis direction.\nThe two different slab orientations have magnetic anisotropies of opposite sign.\n230 2 4 6 8 10 12 14 16 18 20−0.5−0.4−0.3−0.2−0.100.1\nAtomic siteLayer resolved MAE [meV]Bruno formula\nFTgcFIG. 6: Layer-resolved MAE of the Fe (001) slab withN= 20 layers obtained from FT gc(red line)\nand the Bruno formula (blue line). The layer-resolved spin-projected orbital moments are obtained by\nself-consistent calculations.\n240 30 60 90 120 150 180020406080100120\nθ [degree] E (θ) − E (0) [meV]Total\nsurface\n(001) surface\nPerimeter of base\nFIG. 7: (Color online) Magnetocrystalline anisotropy of a truncted pyramid with N= 620 atoms, as a\nfunction of the angle \u0012between the zaxis ( [001] direction) and the direction of the spin. Contributions\nfrom atoms of the two (001) facets (excluding perimeter atoms) and from perimeter atoms of the base are\nshown in blue and red lines, respectively. The total MAE and the contribution from atoms of the outer shell\n(surface) are represented in full and dashed black lines which are almost superposed. E(\u0012= 0) is taken as\nthe zero of energy. Note that in all calculations the azimuthal angle \u001eis equal to zero.\n250 30 60 90 120 150 180020406080100120\nθ (degree)E (θ) − E (0) [meV]\nFIG. 8: Total MAE as a function of angle \u0012for a truncated pyramid ( N= 620 ) and a truncated bipyramid\n(N= 1096 ). For the latter, the MAE is strongly reduced because of much smaller area of (001) facets and\nstrongly reduced anisotropy from perimeter atoms.\n26"    },    {        "title": "1310.6204v1.Magnetoelastic_coupling_induced_magnetic_anisotropy_in_Co__2__Fe_Mn_Si_thin_films.pdf",        "content": "arXiv:1310.6204v1  [cond-mat.mtrl-sci]  23 Oct 2013Magnetoelastic coupling induced magnetic anisotropy in\nCo2(Fe/Mn)Si thin films\nHimanshu Pandey,1P. K. Rout,1Anupam,1P. C. Joshi,1Z. Hossain,1and R. C. Budhani1,2∗\n1Condensed Matter-Low Dimensional Systems Laboratory, Dep artment of Physics,\nIndian Institute of Technology Kanpur, Kanpur-208016, Ind ia\n2CSIR-National Physical Laboratory, New Delhi-110012, Ind ia∗\n(Dated: October 24, 2013)\nAbstract\nThe influence of epitaxial strain on uniaxial magnetic aniso tropy of Co 2FeSi (CFS) and Co 2MnSi\n(CMS) Heusler alloy thin films grown on (001) SrTiO 3(STO) and MgO is reported. The in-plane\nbiaxial strain is susceptible to tune by varying the thickne ss of the films on STO, while on MgO the\nfilms show in-plane easy axis for magnetization (− →M) irrespective of their thickness. A variational\nanalysis of magnetic free energy functional within the Ston er-Wohlfarth coherent rotation model\nwith out-of-plane uniaxial anisotropy for the films on STO sh owed the presence of magnetoelastic\nanisotropy with magnetostriction constant ≈(12.22±0.07)×10−6and (2.02 ±0.06)×10−6, in addi-\ntion to intrinsic magnetocrystalline anisotropy ≈-1.72×106erg/cm3and -3.94 ×106erg/cm3for\nCFS and CMS, respectively. The single-domain phase diagram reveals a gradual transition from\nin-plane to out-of-plane orientation of magnetization wit h the decreasing film thickness. A max-\nimum canting angle of 41.5◦with respect to film plane is predicted for the magnetization of the\nthinnest (12 nm) CFS film on STO. The distinct behaviour of− →Min the films with lower thickness\non STO is attributed to strain-induced tetragonal distorti on.\n1I. INTRODUCTION\nThe Heusler alloys have taken the center stage as spintronics mate rials due to their\nhigh degree of spin polarization, high Curie temperature, and low mag netic damping.1,2\nBy tuning the magnetic parameters such as coercivity, anisotropy , exchange interactions\nand damping processes, one can suitably tailor these materials for m agnetic random access\nmemory, magnetic logics, spin-transistors, and related potential applications. However, in\nmost of such applications the magnetic alloy has to be in a thin film form in which its\nmagnetic characteristics can be significantly different due to film thic kness, crystallographic\norientation, growth related strains and interfacial reactions. On e such characteristics is\nmagnetic anisotropy, which should be large for magnetic storage ap plications, and which\nalso determines the magnetization reversal processes in magnetic switching devices. Till\nnow, a large number of full-Heusler alloy thin films have been grown on v arious substrates.\nSome examples of this are Co 2MnGe on GaAs3and Al 2O3,4Co2MnSi on GaAs,5MgO,6–8\nand Al 2O3,9Co2FeAl0.5Si0.5on MgO,10Co2FeSi on GaAs,11,12Al2O3,13and MgO13as well\nas on SrTiO 3(STO).14–16While the substrate lattice parameter, growth, thermal annealing\ncondition and film thickness in these cases vary significantly, the effe ct of such condition on\nmagnetic anisotropy of the films is seldom addressed. In Heusler alloy s films, one expects\na four-fold anisotropy due to the cubic symmetry of the unit cell, wh ile in-plane uniaxial\nanisotropy has also been observed for the case of Co 2FeSi grown on GaAs.11The presence\nof additional uniaxial anisotropy has resulted in multistep magnetiza tion switching in some\nHeusler alloy films.3,5Moreover, Gabor et al.have shown that Co 2FeAl films can have three\ntypes of in-plane anisotropies, namely biaxial (fourfold cubic anisot ropy) and two uniaxial\nanisotropies parallel to the biaxial easy and hard axes.17In some cases, stripe domains\nhave also been seen due to magnetic frustration between two ener getically equivalent easy\naxis.18The interface between the film and substrate also affects the orien tation of magne-\ntization significantly. For example, the out-of-plane magnetic easy axis in Co 2FeAl films\non Cr-buffered MgO substrate seemed to be induced by the interfa cial anisotropy which\nappears after annealing the films in the presence of magnetic field ap plied along out-of-plane\ndirection.19\n2The magnetic anisotropy in thin films originates from fundamental fa ctors such as the\nspin-orbit interaction in the material which controls magnetocryst alline anisotropy and/or\ndue to growth related strain. Any change in the lattice via strain will c hange the distances\nbetween the magnetic atoms and alter the interaction energy, whic h decides the magnetoe-\nlastic anisotropy. The strain therefore becomes a tuning paramet er for magnetic anisotropy\nand can be varied by a choice of substrates of different lattice para meter or films of var-\nied thickness. A consequence of the strain related anisotropy is th e rotation of magnetic\neasy axes from in-plane to out-of-plane configuration or vice vers a. While a strain depen-\ndence of in-plane anisotropy has been reported for Co 2FeAl/MgO thin films,17to the best of\nour knowledge, strain driven out-of-plane anisotropy has not bee n reported for Heusler alloy\nfilms. Herewereportadetailedstudyofthemagneticanisotropyof Co2(Fe/Mn)Si [CF(M)S]\nfilms of various thickness deposited on (001) MgO and (001) STO cry stals. The in-plane bi-\naxial strain was gradually varied from compressive (for the films on S TO) to tensile (for the\nfilms onMgO) bydepositing thefilms of different thickness. Wehave sp ecifically focussed on\nthe strain dependence of out-of-plane uniaxial magnetic anisotro py in CF(M)S/STO films\nand established how the strain induced magnetic anisotropy affects the direction of mag-\nnetization (− →M). It is seen that the tuning of magnetoelastic coupling by varying th e film\nthickness results in the rotation of the magnetization vector towa rds out-of-plane direction\nas the film thickness is lowered.\nII. EXPERIMENTAL DETAILS\nWe have previously demonstrated that the CF(M)S films on SrTiO 3and MgO pro-\ncessed at 600◦C have better crystalline quality as compared to those annealed at lo wer\ntemperatures.6,7,14Therefore, for studies of anisotropy reported here, we mainly co ncen-\ntrate on the films processed at 600◦C. The cubic lattice parameter ( abulk≈0.5656 nm) of\nCF(M)S matches quite well with the face diagonal (√\n2asub) of (001) STO and MgO. The\nlattice misfit [ ǫ= (a−√\n2asub)/√\n2asub] of CM(F)S with STO and MgO lies within 6%.\nTaking advantage of close matching of the lattice parameters, we h ave prepared a series of\nCF(M)S thin films of various thickness ( t= 5-100 nm) epitaxially on (001) STO and MgO\nusing pulsed laser deposition technique. The details of thin film prepar ation are described\n3/s51/s53 /s52/s48 /s52/s53 /s54/s52 /s54/s54 /s54/s56 /s55/s48 /s48 /s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s48/s46/s53/s54/s51/s48/s46/s53/s54/s52/s48/s46/s53/s54/s53/s48/s46/s53/s54/s54/s48/s46/s53/s54/s55/s83/s32/s32\n/s49/s50/s110/s109/s40/s98/s41/s83/s84/s79\n/s98/s117/s108/s107/s32/s40/s48/s48/s52/s41\n/s50/s48/s110/s109/s32\n/s52/s48/s110/s109/s32\n/s83\n/s49/s48/s48/s110/s109/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s77/s103/s79/s55/s48/s110/s109/s32\n/s52/s48/s110/s109/s32\n/s50 /s40/s100/s101/s103/s41/s83/s84/s79\n/s32/s67/s70/s83\n/s32/s67/s77/s83\n/s32/s32/s97 /s32/s40/s110/s109/s41\n/s116/s32/s40/s110/s109/s41/s66/s117/s108/s107/s40/s99/s41\n/s77/s103/s79/s83/s83\n/s83/s40/s48/s48/s52/s41/s32/s32\n/s83/s84/s79\n/s32/s32\n/s40/s48/s48/s50/s41/s32/s40/s97/s41\n/s50/s48 /s52/s48 /s54/s48 /s56/s48 /s49/s48/s48/s83\n/s77/s103/s79\n/s32/s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s41\n/s50 /s40/s100/s101/s103/s41/s83\nFIG. 1. (Color online) (a) The θ−2θX-ray diffraction profiles of 40 nm thick CMS films grown\non STO and MgO. (b) The small range θ−2θscan about (004) peak of CMS films on STO and\nMgO with thickness. The Bragg reflections from (002) planes o f the substrates are also shown.\nThe dashed line shows the position of 2 θvalue corresponding to (004) peak of bulk CMS. (c) The\nin-plane ( a) lattice parameter as a function of film thickness tfor CMS (filled symbols) and CFS\n(empty symbols). The cubic lattice parameter of bulk CMS is m arkrd by the dotted line.\nin our earlier reports.6,7,14,15The structural characterization of the films has been done by\nX-ray diffraction (PANalytical X′Pert PRO X-ray diffractometer) in θ-2θ,ω,ϕ, and grazing\nincidence X-ray diffraction (GIXRD) modes. The magnetic measurem ents were performed\nin a vibrating sample magnetometer (EV7 VSM) at room temperature .\nIII. RESULTS AND DISCUSSION\nA. Structural characterization\nTheθ−2θX-ray diffraction reveals (00 l) oriented growth of CF(M)S films on STO and\nMgO [Fig. 1(a)]. Further evidence of (00 l) texturing is provided by the rocking curves\nabout (004) reflection. The full width at half maximum of these films a re less than 1.9◦,\nwhich corresponds to a crystallite size of 5 nm.15Moreover, the ϕscans confirm the epitaxial\ngrowth of the films with the relation [100] CF(M)S ∝bardbl[110] STO or MgO. The presence of\n(111) superlattice, which governs the ordering of the Mn(or Fe) a nd Si sublattices, and (022)\n4fundamentaldiffractionline, whichconfirmsthepresenceof L21orderinginthefilms, aretwo\nimportant indicators of the structural ordering in the films. From G IXRD measurements,\nwe infer the degree of ordering in the films to be more than 85%. Figur e 1(b) shows the\nθ−2θscan about (004) peak for films of various thickness on MgO and STO substrates. A\nclear shift of the Bragg reflections towards higher (lower) scatte ring angle (2 θ) is seen for the\nfilms grown on MgO (STO) as the thickness is reduced. The out-of-p lane lattice parameter\n(c) obtained from these scans decreases (increases) for the films g rown on STO (MgO) with\nthe increasing t. This can be understood in terms of the strain induced in the films due to\nlattice misfit. The positive misfit value for STO ( ǫ= 2.4%) results in in-plane compressive\nstrains, which decreases the in-plane lattice parameter ( a) as verified by off-axis θ−2θ\nscans about (022) peak. Assuming the volume ( a2c) preserving distortion, we expect an\nincrease in cwith decreasing tfor the films grown on STO. The films with lower thickness\nexperience a relatively strong tetragonal distortion. As the film th ickness increases, the\ndistortion relax by formation of misfit dislocations. With increase in th ickness, the in-plane\nstrainǫxx[=(afilm−abulk)/abulk] approaches zero as seen in Fig. 1(c). We observe that the\nthinnest film ( t= 5 nm) on STO is under highest biaxial compressive strain of ǫxx=ǫyy=\n-0.44 % while the thicker films undergo partial strain relaxation with 10 0 nm film attaining\nbulk values. Similarly, the tensile strain in the Heusler alloy films on MgO dis appears on\nincreasing their thickness.\nB. Magnetization\nWe first discuss the behavior of magnetic hysteresis loops [ M(H)] for in-plane (along\n[110]) and out-of-plane (along [001]) field configurations [See Fig. 2(a -c)]. The hysteresis\nloops of the films on MgO clearly show an in-plane easy axis for magnetiz ation (− →M) as\nrevealed by the squareness of the loop in Fig. 2(a). This result is the same for thicker CFS\nfilms on STO [Fig. 2(b)]. However, for our thinnest film on STO, we obse rve a significantly\nhigher out-of-plane magnetization, which suggests the possibility o f tilted− →Mwith respect to\nthe film plane. Figure 2(d) shows the remanent magnetization ( Mr) at different angles ( θ)\nof the field with respect to the film plane, which looks like a dumbbell with two lobes almost\nseparated from each other for CMS(40 nm)/STO film. Clearly, the Mr/Mr(0) is maximum\n5/s45/s51 /s48 /s51/s45/s49/s48/s49\n/s45/s51 /s48 /s51 /s45/s51 /s48 /s51\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48/s48/s49/s48/s48/s50/s48/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56\n/s48/s57/s48\n/s49/s56/s48\n/s50/s55/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48\n/s48/s57/s48\n/s49/s56/s48\n/s50/s55/s48/s48/s46/s48\n/s48/s46/s53\n/s49/s46/s48/s91/s49/s49/s48/s93\n/s40/s100/s41\n/s32/s32/s77/s32/s47/s32/s77\n/s83\n/s72 /s32/s40/s107/s79/s101/s41/s40/s97/s41/s32/s116/s61/s49/s49/s46/s53/s32/s110/s109\n/s32/s32/s32/s32/s32/s32/s32/s32/s77/s103/s79/s40/s98/s41/s32/s32/s116/s61/s54/s56/s32/s110/s109\n/s32/s32/s32/s32/s32/s32/s32/s32/s83/s84/s79/s32/s32\n/s72 /s32/s40/s107/s79/s101/s41/s91/s48/s48/s49/s93/s40/s99/s41/s32/s32/s116/s61/s49/s50/s32/s110/s109\n/s32/s32/s32/s32/s32/s32/s32/s32/s83/s84/s79/s32\n/s32/s32\n/s72 /s32/s40/s107/s79/s101/s41/s72\n/s67/s32/s40/s79/s101/s41\n/s116/s32/s40/s110/s109/s41/s40/s101/s41\n/s32/s67/s70/s83/s47/s83/s84/s79\n/s32/s67/s77/s83/s47/s83/s84/s79\n/s32/s67/s70/s83/s47/s77/s103/s79\n/s32/s67/s77/s83/s47/s77/s103/s79\n/s32/s32/s77\n/s114/s32/s47/s32/s77\n/s83\n/s116/s61/s53/s32/s110/s109/s32\n/s32/s32/s77\n/s114/s32/s47/s32/s77\n/s114/s40/s48/s41\n/s116/s61/s52/s48/s32/s110/s109\nFIG. 2. (Color online) The magnetic hysteresis loops measur ed along [110] and [001] directions at\nroom temperature of (a) CFS (11.5 nm)/MgO, as well as (b) 68 nm and (c) 12 nm thick CFS/STO\nfilms. (d) Polar plot of Mr/Mr(0) for 5 nm and 40 nm thick CMS/STO film at a step of 5◦. Here\nMr(0) is the Mratθ= 0◦. (e) The upper panel shows the thickness dependence of Mr/MS, where\nMSis the saturation moments. The Slater-Pauling formula pred icts aMSof 5µBand 6µBfor\nCMS and CFS films, respectively.20We have used the experimental values of MS, which are in\nreasonable agreement with the theory.15The lower panel shows the HCas a function of talong\nwith the fits (solid line) according to the relation: HC∝t−n.\nforθ= 0◦and 180◦(in-plane directions) while it is almost zero at 90◦and 270◦(out-of-plane\ndirections). This observation confirms the presence of in-plane ea sy axis for thicker films on\nSTO. However, in the case of CMS(5 nm)/STO film, two lobes are joine d and thus the Mr\nis substantially higher for θ= 90◦and 270◦. This suggests a canted easy axis instead of an\nin-plane one as observed in thicker films. We believe that the substra te-film interface plays\nan important role in tilting the magnetization away from the film plane. T he upper panel\nin Fig. 2(e) shows the thickness dependence of the squareness ( Mr/MS) of magnetization\nextracted from in-plane M(H) loops. In case of films on MgO, it remain s almost constant\nwhereas, for the films on STO, we notice a gradual decline in Mr/MSwith decreasing thick-\nness, which indicates the deviation of easy axis from the film plane. Alt hough the lowest\nobserved value ( ≈0.2 for 5 nm film) does not point towards a distinct out-of-plane easy\naxis, it certainly indicates some canting of− →Maway from the film plane.\n6The coercivity of a material is the principal property related to the rate of change of\nmagnetic relaxation between the remanent and demagnetized stat es. At absolute zero, it\nmeasures the barrier height that is required by magnetic moments t o overcome the demag-\nnetized state. The variation of the coercivity ( HC) of the films with thickness is plotted\nin the lower panel of Fig. 2(e). We observe that HCdecreases gradually with increasing\nthickness in all cases. This may be attributed to a lowering of defect concentration due to\nenhancing crystalline quality or due to lowering of strain in thicker films . Moreover, the\nreduction of HCcan also be due to the changes in the grain size and the surface roug hness\nof the film with its thickness or related to the fact that the film thickn ess decreases to a\npoint where the domain wall thickness becomes comparable to the film thickness. The HC\nfollows a power law type dependence on tof the form: HC∝t−nwithn= 0.50±0.02\nand 0.41±0.17 for CMS and CFS films on STO, respectively. The value of ndepends on\nthe deposition conditions and the choice of ferromagnet, and can h ave values from -0.3 to\n-1.5.21–23\nWe have carried out an analysis of the hard axis magnetization loops in the framework of\nStoner-Wohlfarth formalism.24The total magnetic free energy ( E) of the film in tetragonal\nsymmetry can be expressed as:\nE=K1m2\nz+K2m4\nz+K3m2\nxm2\ny−− →M·− →H+2πM2\nsm2\nz (1)\nwhereK1andK2are second and fourth order uniaxial anisotropy constants, res pectively\nwhileK3is in-plane biaxial anisotropy constant. The mx,y,zare the direction cosines of\nthe magnetization vector− →M. The fourth term of Eq. (1) is the Zeeman energy and the\nlast term represents the thin film demagnetization energy. For out -of-plane field hysteresis\nloopi.e. when− →His applied along [001],− →Mwill rotate from the [110] (in-plane easy axis)\nto [001] direction and thus the term K3mx2my2is always zero. The minimization of total\nmagnetic free energy for an out-of-plane field yields the equilibrium m agnetization Min the\nfield direction given by the relation:\nH=/bracketleftbigg2K1\nM2\nS+4π/bracketrightbigg\nM+4K2\nM4\nSM3(2)\n7The values of K1andK2can be obtained by fitting Eq. (2) to the hysteresis loops.\nThe inset of Fig. 3(a) shows the plot of H/Mvs.M2for 68 nm thick CFS/STO film.\nThe intercept and slope of the linear fit yield K1andK2, respectively. The deviation in\nupper part of the curve from the linearity occurs as Mapproaches saturation, while the\ndeviation at lower Mcan be attributed to magnetic domain effects.25All the films on MgO\nshow in-plane easy axis without any substantial change in M(H) loops with thickness. So\nthe determination of anisotropy coefficients for these films will not b e reliable. While we\nobserved clear change in M(H) loops for the films on STO with varying thickness. Hence\nwe will only focuss on the later films in order to gain further insight of t he magnetic state.\n/s45/s48/s46/s52 /s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48\n/s40/s98/s41/s75\n/s50/s32/s40/s120/s49/s48/s54\n/s32/s101/s114/s103/s47/s99/s109/s51\n/s41\n/s120/s120/s45/s48/s46/s52 /s45/s48/s46/s51 /s45/s48/s46/s50 /s45/s48/s46/s49 /s48/s46/s48/s45/s49/s48/s45/s56/s45/s54/s45/s52/s45/s50/s48\n/s69\n/s68/s32/s102/s111/s114/s32/s67/s77/s83\n/s32/s67/s77/s83\n/s32/s67/s70/s83/s32/s75\n/s49/s32/s40/s120/s49/s48/s54\n/s32/s101/s114/s103/s47/s99/s109/s51\n/s41\n/s120/s120/s40/s97/s41\n/s69\n/s68/s32/s102/s111/s114/s32/s67/s70/s83/s48 /s51 /s54 /s57 /s49/s50/s48/s52/s56/s49/s50/s32\n/s77/s50\n/s120/s49/s48/s53\n/s32/s40/s101/s109/s117/s50\n/s46/s99/s109/s45/s54\n/s41\n/s72/s47/s77 /s32/s40/s79/s101/s46/s99/s109/s51\n/s46/s101/s109/s117/s45/s49\n/s41\nFIG. 3. (Color online) (a) The second order uniaxial anisotr opy constant ( K1) as a function strain\n(ǫxx) with the linear fits (solid lines). The ǫxxhas been calculated using the values of amentioned\nin Fig. 1(c). The dotted lines show demagnetization energy ( ED) for CF(M)S. The inset shows\nthe plot of H/MvsM2for CFS (68 nm)/STO film along with the linear fit (solid line) g iven by\nEq. (2).(b) The fourth order uniaxial anisotropy constant ( K2) as a function of ǫxxwith the linear\nfits (solid lines).\nFigure3 shows thevalues of K1andK2deduced fromEq. (2) for CF(M)S/STO films as a\nfunction of ǫxx. We clearly observe a monotonic increase in anisotropies with the incr easing\nstrain. Moreover, the values of K1are quite similar to previously reported values.6TheK1is\nconnectedto ǫxxthroughthemagnetoelasticcouplingparametersandcanbeexpre ssed asK1\n=Kmc+3λσxx/2.26The first term represents the strain independent magnetic anisot ropy,\ncommonly known as ”magnetocrystalline anisotropy”, which originat es from the inherent\ncrystal structure of ferromagnet.26The linear fits to K1(ǫxx) data yield Kmc≈-1.72×106\nerg/cm3and -3.94 ×106erg/cm3for CFS and CMS, respectively [See Fig. 3(a)]. The second\n8term is purely related to the strain induced anisotropy, which depen ds linearly on stress\nand the magnetostriction constant λ. The stress can be represented as σxx=Yǫxx, where\nthe Youngs modulus ( Y) can be expressed in terms of elastic stiffness constants ( C11and\nC12) as follows: Y= (C11−C12)(C11+2C12)/(C11+C12).27Assuming theoretical values of\nC’s,28we findY≈93 GPa for CFS and 192 GPa for CMS. Using these values, the linear\nfits toK1(ǫxx) data yield λ≈(12.22±0.07)×10−6and (2.02 ±0.06)×10−6for CFS and CMS,\nrespectively. To the best of our knowledge, we are unaware of any values of λandKmcfor\nthese compounds reported in literature. The values of λare comparable to the reported\nvalue of∼15×10−6for another Heusler alloy Co 2MnAl29whileλis of the order of 10−5for\nhalf metallic manganites.30,31\n-2.0 -1.5 -1.0 -0.50.00.20.4\nReg. IIIReg. I\nReg. II\nCFS\nCMS90oK2/ED\nK1/ED0o\nFIG. 4. (Color online) The single-domain magnetic phase dia gram demonstrating different stable\nmagnetic states, namely in-plane(Region I), canted (Regio n II), and out-of-plane (Region III)state\nof magnetization. The symbols are the experimental data.\nOur expression for K1in case of biaxial stress ( σxx=σyy,σzz= 0) is same as the expres-\nsion for uniaxial stress ( σxx∝negationslash= 0,σyy=σzz= 0) induced anisotropy, i.e. K= 3λσxx/2. But\nthese two cases are fundamentally different. In the former scena rio, a uniaxial anisotropy is\ninducedperpendicular totheplane(along z-axis) whileforlattercasetheuniaxialanisotropy\nis along the direction of applied stress (along x-axis). The other anisotropy constant K2also\nshows a linear dependence with ǫxxas shown in Fig. 3(b). Such linear relation has been\npredicted for a cubic system under biaxial strain and experimentally verified for Cu-Ni\nsystems.32Similar to the case for K1, we observe a substantial contribution to K2coming\nfrom magnetocrystalline origin in addition to the magnetoelastic coup lings.\n9The direction of magnetic easy axis depends sensitively on anisotrop y energy dependent\nonK1andK2and the demagnetization energy ED(=2πM2\ns). Only consideration of second-\norder angular term gives an out-of-plane magnetization state for K1/ED<-1 while− →M\nbecomes in-plane for -1 < K1/ED. However, the fourth order anisotropy term introduces\nthe canting states of− →Mallowing a gradual transition between the in-plane and out-of-plane\nstates.32,33Figure 4 shows the general single-domain magnetic phase diagram fo r a system\nwith free energy given by Eq. (1) in zero magnetic field assuming a coh erent rotation of\nmagnetization. The films whose anisotropy data lie in Region II have ca nted magnetization\nstates, where the canting angle θc(the angle between− →Mand film plane) can be obtained\nfrom the relation:32sin2θc=−(K1+ED)/2K2. The CFS (12 nm)/STO film has θc=\n41.5◦while the angles for 5 nm and 10 nm thick CMS/STO films are 31.8◦and 17.9◦,\nrespectively. The data for thicker films fall into Region I, which sugg ests that easy axis of\nmagnetization is in-plane. Clearly, it can be inferred that easy axis ch anges from in-plane\nto canted orientation with increasing compressive strain. Hence, t here is a possibility to\nget the perpendicular magnetic anisotropy in case of films with higher strain. This can be\nachieved either by lowering the film thickness or choosing a substrat e with a larger positive\nmisfit.\nIV. SUMMARY\nWe have presented a study to correlate the crystallographic stru cture and the magnetic\nstate of Co 2FeSi and Co 2MnSi films on (001) STO and MgO substrates. The films on\nSTO are under in-plane biaxial compressive strain while a tensile strain is observed in the\nfilms on MgO. The strain gradually relaxes with increasing film thickness . The hysteresis\nloops clearly show an in-plane easy axis for all the films on MgO, howeve r, for the films on\nSTO, the out-of-planecomponent of magnetization increases with decreasing thickness. The\nanalysis of magnetic free energy functional within the Stoner-Woh lfarth coherent rotation\nmodel with out-of-plane uniaxial anisotropy predicts a canted mag netization state for the\nfilms on STO, which gradually moves towards in-plane state with increa sing thickness in\na single-domain magnetic phase space. The uniaxial anisotropy term s have two distinct\ncontributions; firstoneisintrinsicmagnetocrystallineanisotropy, whichisstrainindependent\n10and the other one is magnetoelastic anisotropy. We have extracte d various anisotropy terms\n(∼106erg/cm3) and magnetostriction constants ∼10−6of Co2FeSi and Co 2MnSi for the\nfirst time. We also predict maximum canting angles of 41.5◦and 31.8◦for Co 2FeSi (12 nm)\nand Co 2MnSi (5 nm) on STO, respectively. 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Lett.101, 022411 (2012).\n13"    },    {        "title": "1311.0238v1.Strain_Tunable_Magnetocrystalline_Anisotropy_in_Epitaxial_Y3Fe5O12_Thin_Films.pdf",        "content": "1 \n Strain-Tunable Magnetocrystalli ne Anisotropy in Epitaxial \nY3Fe5O12 Thin Films \nHailong Wang†, Chunhui Du†, Chris Hammel* and Fengyuan Yang* \nDepartment of Physics, The Ohio State University, Columbus, OH, 43210, USA \n†These authors made equal c ontributions to this work \n*Emails: hammel@physics.osu.edu; fyyang@physics.osu.edu \n \nAbstract \nWe demonstrate strain-tuning of magnetocrystal line anisotropy over a range  o f  m o re  t h a n  o n e  \nthousand Gauss in epitaxial Y 3Fe5O12 films of excellent crystalline quality grown on lattice-\nmismatched Y 3Al5O12 substrates. Ferromagnetic resonance (F MR) measurements reveal a linear \ndependence of both out-of-plane and in-pla ne uniaxial anisotropy on the strain-induced \ntetragonal distortion of Y 3Fe5O12. Importantly, we find th e spin mixing conductance ܩ \ndetermined from inverse spin Hall effect and FMR linewidth broadening remains large: ܩ = \n3.33  1014 -1m-2 in Pt/Y 3Fe5O12/Y3Al5O12 heterostructures, quite comparable to the value \nfound in Pt/Y 3Fe5O12 grown on lattice-matched Gd 3Ga5O12 substrates. \n \n  2 \n Magnetocrystalline anisotropy [1-6] plays an  essential role in permanent magnets, \nmagnetic data storage, energy generation and tr ansformation, magnetic resonance, and there is \nintense interest in understanding the role of magneto elastic coupling in phonon-magnon \ninteractions in thermal spintronics. With the growing applications of magnetic films, it is \nimportant to understand magnetocrystalline anis otropy in the presence of lattice distortion \ninduced by epitaxial strain and the underlying magnetization-lattice coup ling. Tunable magnetic \nanisotropy was observed in GaMnAs films at low temperatures using epit axial strain [3], in \nGaMnAsP films by varying the phosphorous content [4] and in Sr 2FeMoO 6 epitaxial films with \nvarious strains grown on a selected set of single-crystal s ubstrates and buffer layers [6]. \nFerrimagnetic insulating Y 3Fe5O12 (YIG) is widely used in FMR and microwave applications as \nwell as spin dynamics studies [7-10] due to its  exceptionally low magne tic damping. Most YIG \nepitaxial films and single crystals are produced by liquid-phase epitaxy (LPE) with thicknesses \nfrom 100 nm to millimeters [11]. Pulsed-laser de position (PLD) has also been used to grow \nepitaxial YIG thin films [12-14]. However, a systematic study of strain-dependence of \nmagnetocrystalline anisotropy is lacking, largely due to the challenges inhere nt in controlling the \nepitaxial strain while maintaining sufficiently high crystalline quality. Strain control in high \nquality YIG films will allow tuning of magnetocrys talline anisotropy, which in turn determines \nthe static and dynamic magneti zation of the YIG films. \nMost reported YIG epitaxial fi lm fabrication has employed Gd 3Ga5O12 (GGG) substrates \nwhich has a lattice mismatch  = (as – af)/af  100% of 0.057% with YIG, where as = 12.383 Å \nand af = 12.376 Å are the lattice constants of  the GGG substrate and unstrained YIG, \nrespectively. In order to probe the magnetocrys talline anisotropy in epitaxial YIG films in \nresponse to lattice distortion, we re port in this letter th e growth of YIG epit axial thin films on 3 \n (001)-oriented Y 3Al5O12 (YAG) substrate [11, 15, 16] with a = 12.003 Å (  = -3.0%). The larger \nlattice mismatch results in th ickness-controlled stra in-induced tetragonal distortion in the YIG \nfilms, which leads to variation in their out-of-plane and in-plane magnetocrystalline anisotropy \nas discussed below. \nWe grow epitaxial YIG films with thicknesses t ranging from 9.8 to 72.7 nm using a new \nsputtering technique [10, 17, 18] on YAG (001) substrates and determ ine their crystalline quality \nby triple-axis x-ray diffracti on (XRD). Figure 1a shows 2 θ- XRD scans of the YIG films of \nseven different thicknesses on YAG (001). The pronounced Laue oscillations observed in the \n37.9-nm and 72.7-nm films indicate smooth surfaces and sharp YIG/YAG interfaces. The gradual \nshift of the YIG (004) peak posi tion clearly reflects strain relaxa tion as the thickness of the YIG \nfilms increases from 9.8 to 72.7  nm. The lattice mismatch ( \t= -3.0%, compressive) elongates \nthe out-of-plane lattice constant c, resulting in a tetragonal dist ortion. To obtain the in-plane \nlattice constant a, we assume conservation of the unit cel l volume of YIG duri ng stain relaxation, \nܽൌටሺ12.376\tÅሻଷ/ܿ .Figure 1b and Ta ble I show both a and c for the YIG films of 9.8  t  \n72.7 nm, which exhibit a clea r strain relaxation as t increases, while the strain-induced tetragonal \ndistortion  = (c – a)/a of the YIG films decreases from 2.05% to 0.073%. \nWe determine the magnetic anisotropy of our YIG films using FMR spectroscopy at \nradio-frequency (rf) f = 9.60 GHz. A magnetic field H is applied at an angle H with respect to \nthe film normal (see inset to Fig. 2a). Figure 2a  shows four representative FMR spectra for the \n72.7-nm YIG film at H = 0, 30, 50 and 90. The resonance field Hres is defined as the field at \nwhich the derivative of the FMR absorption crosses zero. Figure 2b shows the angular \ndependence of the resonan ce field from out-of-plane ( H = 0) to in-plane ( H = 90) for the 9.8, 4 \n 15.0, 29.3 and 72.7-nm YIG films as the tetragonal distortion  varies from 2.05% to 0.073%. \nThe magnetization can be quantitatively charact erized from the total free energy density F for the \nYIG films with tetragonal symmetry [19, 20],  \nܨൌെ ࡴࡹଵ\nଶܯቄ4ܯߨ ୣ\tcosଶߠെଵଶܪସୄcosସߠെ\tଵ଼ܪସ||ሺ3cos4߶ሻsinସߠെ\nܪଶ||sinଶߠsinଶሺ߶െగ\nସ\tሻቅ,        (1) \nwhere  and  are angles describing the orienta tion of the equilibrium magnetization ( M) (see \ninset to Fig. 2a). The first term in Eq. (1) is the Zeeman energy and the second term is the \neffective demagnetizing energy 4ܯߨୣൌ4 ܯߨ ୱെܪଶୄ which includes the shape anisotropy \n(4Ms) and out-of-plane uniaxial anisotropy ܪଶୄ. The remaining terms are out-of-plane cubic \nanisotropy ( ܪସୄሻ, in-plane cubic anisotropy ( ܪସ||) and in-plane uniaxial anisotropy ( ܪଶ||). We \nmeasure the magnetic hysteresis loops of the YIG films using a vibrating sample magnetometer \n(VSM) to obtain the saturation magnetization Ms. The values of 4 Ms vary from 1590 to 1850 \nOe, which lie in the range of reported magne tization in YIG samples grown by LPE and PLD \n[11-14, 21]. The inset to Fig. 2b sh ows representative in-plane and out-of-plane hysteresis loops \nfor the 37.9-nm YIG film, indicatin g clear magnetic shape anisot ropy. Due to strain relaxation, \nthe coercivity of our YIG films on YAG (001) ranges from 30 to 80 Oe for different thicknesses, \nmuch larger than the values of YIG films on lattice-matched GGG [13]. \nThe equilibrium orientation ( , ) of magnetization can be obtained by minimizing the \nfree energy, and the FMR resonance frequency  in equilibrium is given by [19, 20, 22] \nቀఠ\nఊቁଶ\nൌଵ\nெమୱ୧୬మఏడమி\nడఏమ\tడమி\nడథమെቀడమி\nడఏడథቁଶ\n൨,      ( 2 )  \nwhere ߛൌߤ݃ / is the gyromagnetic ratio. We use a numerical procedure to obtain the \nequilibrium angles at resonan ce condition [23, 24] and fit the Hres vs. H data to determine 5 \n 4ܯߨୣ, ܪସୄ, ܪସ||, ܪଶ||, and g factor. In Figure 2b, the fitting curves agree well with the \nexperimental data which reveal  a systematic variation of 4ܯߨୣ for YIG films of different \nthicknesses. For the 9.8-nm film, 4ܯߨୣ = 3103 Oe while for the 72.7-nm film, 4ܯߨୣ = 1639 \nOe, indicating that the strain induces substa ntial out-of-plane anisotropy. The out-of-plane \nuniaxial anisotropy ܪଶୄ can be calculated from the values of ܯୱ and 4ܯߨୣ. Figure 3a shows \nܪଶୄ as a function of tetragonality ߪ for all the YIG films on YAG; ܪଶୄ varies linearly with strain. \nThis tunability of magnetocrystalline anisotropy through lattice symmetry  highlights the central \nresult of our study: the proportionality of ܪଶୄ to the tetragonal distor tion of the YIG lattice over \na broad range (-2.05% ൏ሺ ܿെܽ ሻ / ܽ൏  -0.073%), \n\tܪଶୄൌ ሺ12േ64ሻെሺ55.8േ5.3ሻ 10ଷሾሺܿ െܽሻ/ܽሿ  (Oe). \nFigure 3a demonstrates a fundamental relation ship between magnetocrys talline anisotropy and \nlattice symmetry which is expected but ha s not been seen before in YIG films.  \nWe also find the in-plane uniaxial anisotropy ܪଶ|| increases with tetragonality of the YIG \nlattice. Figures 2c and 2d show both the experi mental data and fits to the in-plane angular \ndependence of Hres for the 9.8 and 72.7 nm YIG films on Y AG. Clear four-fold symmetry is \nobserved in the 72.7-nm YIG film while superpos ition of two- and four-fold symmetry appears \nin the 9.8-nm YIG film. Based on Eqs. (1) and (2), when the in-plane anisotropy is small, the in-\nplane resonance condition can be expressed by [19, 20]:  \n൬߱\nߛ൰ଶ\nൌቄ ܪܪ ସ||cos4߶െܪ ଶ||cosቀ2߶െߨ\n2\tቁቅ \nൈቄܪ4ܯߨ ୣுర||ሺଷାୡ୭ୱସథሻ\nସܪଶ||sinଶሺ߶െగ\nସሻቅ.  (3) 6 \n Figure 3b plots the dependence of ܪଶ|| on tetragonality ( c-a)/a, where ܪଶ|| can be tuned from 1 to \n60 Oe with the magnitude of tetragonality vary ing from 0.073% to 2.05%. A linear fit to Fig. 3b \ngives \n\tܪଶ||ൌሺ2േ6ሻሺ31.1േ4.6ሻ 10ଶሾሺܿ െܽሻ/ܽሿ  (Oe). \nThe strain-induced anisotropy arises from the magnetizati on-lattice coupli ng [25, 26] in \nwhich a change in inter-atomic distances alters the magnetic properties through spin-orbit \ncoupling. Since \tܪଶୄ\tis more than one order of magnitude larger than \tܪଶ|| in the YIG films on \nYAG, here we focus on the strain-induced \tܪଶୄ\t. The magnetoelastic energy density is given \nby\tܨ ൌ െܾߪ  when M is along the [001] direction, where b and \tߪ are the magnetoelastic constant \nand tetragonality ( c-a)/a, respectively. Figure 3c shows the linear dependence of anisotropy \nenergy, \nܧୟ୬୧ൌെଵ\nଶܪ\tܯଶୄ,         ( 4 )  \non tetragonality for all the YIG films, fr om which a least squares fit gives, \nEani =ሺെ7.0േ54.2 ሻ10ଶሺ40.4േ4.4 ሻ10ହሾሺܿെܽሻ/ܽሿ  (erg/cm3),  \nfrom which we obtain – b = (40.4  4.4)  105 erg/cm3. The negative value of b implies that the \nmagnetic easy axis is parallel to  a short axis of  the tetragonal la ttice. The magnetoelastic constant \nof YIG is somewhat smaller th an but of the same order as  that in double perovskite Sr 2FeMoO 6 \nfilms with – b = (92.9  4.5)  105 erg/cm3 [6]. The similarity may arise because both Y 3Fe5O12 \nand Sr 2FeMoO 6 are Fe3+-based ferrimagnetic oxides, while the presence of 4 d transition metal \nMo5+ in Sr 2FeMoO 6 enhances the spin-orb it coupling and, conseque ntly, the magnetoelastic \ncoupling. This result demonstrates the ability to  tune magnetocrystalline anisotropy in thin YIG \nepitaxial films by substrate lattice mismatch and film thickness. 7 \n YIG is an excellent material for microwave application and spin pumping [7-10] due to \nits narrow linewidth and insulating nature. One fundamentally interesting question is how the \nstrain-induced FMR linewidth broadening in YI G/YAG films will affect the spin transfer \ncapability at YIG/Pt interface [9, 10]. It is belie ved that the FMR linewidth largely determines \nthe quality of YIG films and in terfacial spin mixing conductance ܩ in YIG/normal-metal \nbilayers. Here, we report cavity FMR spin pumping measurements in Pt/YIG/YAG . The FMR \npeak to peak linewidth ∆ܪ is 83.9Oe for the 72.7 nm YIG film  on YAG. Figure 4a shows the \nVISHE vs. H spectra for Pt( 5 nm)/YIG(72.7 nm)/ YAG with an in-plane DC field ࡴ at Prf = 200 \nmW. The ISHE signal is 123 V which, although smaller than our previously reported mV VISHE \nfor Pt/YIG on GGG [10], is still large for Pt/YIG system. Figur e 4b shows the FMR derivative \nabsorption spectra of a single 72.7-nm YIG film and a Pt(5 nm )/YIG(72.7 nm) bilayer on YAG. \nThe real part of interfacial spin mixing conductance ܩ can be determined from [27, 28], \nܩൌ݁2\n݄2√3ܯߨsݐߛF\nߤ݃B߱൫ΔܪPt/YIGെΔܪYIG൯     ( 5 )  \nwhere , ݃ ,ߤ and ݐி are the gyromagnetic ratio, ݃ factor, Bohr magnetron  and thickness of YIG \nfilm, respectively. Using Eq. (5) and the linew idths from Fig. 4b, we obtain the spin mixing \nconductance (3.33 േ 0.15)  1014 -1m-2 for Pt/YIG on YAG, which is slightly smaller but \ncomparable to the values of 3.73  1014 and 4.56  1014 -1m-2 for Pt/YIG bilayers on GGG [10]. \nThis indicates that the larger FMR linewidth for YIG films grown on YAG essentially does not \nchange the effective spin angular momentum tran sfer capability across the Pt/YIG interface. One \npossible explanation is that the strain-induced i nhomogeneity mostly exists in the bulk of the \nYIG film and the Pt/YIG in terface remains high quality. \nThe tunable magnetocrystalline anisotropy in st rained YIG thin films with a clear linear 8 \n dependence on the tetragonal distortion of YIG la ttice allows for fundame ntal understanding of \nmagnetization-lattice coupling in this importan t magnetic material and enables potential \nmicrowave and spin-electronic applications via control of the lattice symmetry. This behavior \npoints towards potential strain en gineering of YIG epitaxial films, for example, with lateral \nmodulation of strain to  tune the magnetic resonance charact eristics and to design microwave \nheterostructures for novel applications. \nAcknowledgements \nThis work is supported by the Center for Emerge nt Materials at the Ohio State University, \na NSF Materials Research Science and Engi neering Center (DMR-0820414) (HLW and FYY) \nand by the Department of Energy through gran t DE-FG02-03ER46054 (PCH). Partial support is \nprovided by Lake Shore Cryogenics Inc. (CHD) and the NanoSystems Laboratory at the Ohio \nState University \n  9 \n References  \n1. .B. Heinrich, K. B. Urquhart, A. S. Arrott, J.  F. Cochran, K. Myrtle, and S. T. Purcell , Phys. \nRev. Lett. 59, 1756 (1987). \n2. M. Kowalewski, C. M. Schneider and B. Heinrich, Phys. Rev. B 47, 8748 (1993). \n3. S. H. Kim, H. K. Lee, T. H.  Yoo, S. Y. Lee, S. H. Lee, X. Liu, and J. K. Furdyna, J. 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Lett.  110, \n127201 (2013). \n  12 \n Figure Captions: \nFigure 1.  (a) Semi-log 2 θ- XRD scans of YIG films of thickness t = 9.8, 12.4, 15.0, 19.5, 29.3, \n37.9 and 72.7 nm grown epitaxially on YAG (001) subs trates. The arrows i ndicate the positions \nof the YIG (004) peak. The sate llite peaks in the s cans of 37.9 and 72.7 nm YIG films are the \nLaue oscillations. (b) Thickness de pendence of the in-plane (blue open squares) lattice constant a \nand out-of-plane (red solid circles) lattice constant c of the YIG films on YAG. The horizontal \ndashed line represents the bulk lattice constant a = 12.376 Å of YIG.  \nFigure 2.  (a) Room-temperature FMR derivative sp ectra for a 72.7 nm YIG film on YAG (001) \nat H = 0, 30, 50, and 90. Inset: coordinate system used for FMR measurements and analysis. \n(b) Out-of-plane angular dependence ( H) of the resonance fields ( Hres) for the 9.8, 15.0, 29.3, \nand 72.7 nm YIG films. The fitting (solid curves) was performed using Eqs. (1) and (2) to obtain \n4πMୣ, from which Hଶୄ\t\twas determined for each film. Inset:  in-plane (blue) and out-of-plane \n(red) magnetic hysteresis loops of a 37.9-nm th ick YIG film. In-plane angular dependence ( H) \nof Hres for the (c) 9.8 nm and (d) 72.7 nm YIG films. \nFigure 3 . (a) Out-of-plane uni axial anisotropy field H2, (b) in-plane anisotropy field H2|| and (c) \n(c) out-of-plane anisotropy energy Eani as a function of the tetragonal distortion ( c–a)/a of the \nYIG films.  \nFigure 4 . (a) VISHE vs. H spectra at θH = 90 and 270 using Prf = 200 mW for a Pt(5 \nnm)/YIG(72.7 nm) bilayer. Inset: FMR spin pumping experimental geometry. (b) FMR \nderivative absorption spectra of the 72.7-nm thick YIG film before (red) and after (blue) the \ndeposition of a 5-nm Pt layer.   \n  13 \n Table I . Structural and magnetic parameters of  YIG epitaxial films with thickness 9.8  t  72.7 \nnm on YAG (001). \nt (nm) a (Å) c (Å) ( c-a)/a H2 (Oe)  Eani (erg/cm3) H 2|| (Oe)  H 4|| (Oe)  \n9.8 12.293 12.545 2.05% -1.25  103 9.22  104 60.4 42.0 \n12.4 12.308 12.513 1.66% -902 6.30  104 48.7 58.6 \n15.0 12.318 12.493 1.43% -701 4.57  104 52.1 66.8 \n19.5 12.334 12.460 1.03% -543 3.77  104 17.9 17.9 \n29.3 12.354 12.420 0.53% -445 2.91  104 23.9 18.0 \n37.9 12.363 12.402 0.31% -139 1.00  104 2.75 25.9 \n72.7 12.373 12.382 0.073% -49 3.10  103 0.941 25.6 \n \n  14 \n  \nFigure 1.   \n  10-11011031051071091011\n27 28 29 30Intensity(c/s)\n2(deg)72.7 nm\n37.9 nm\n29.3 nm\n19.5 nm\n15.0 nm\n12.4 nm\n9.8 nm(a)\nthickness\nYAG(004)YIG\n(004)\n12.312.412.5\n0 1 02 03 04 05 06 07 0Lattice constant (A)\nt (nm)in-plane aout-of-plane c (b)15 \n \nFigure 2.   \n  260026202640\n0 90 180 270 360Hres (Oe)\nH (deg)(d) t = 72.7 nm\n21002140218022202260\n0 90 180 270 360Hres (Oe)\nH (deg)t = 9.8 nm (c)\n-2-101\n2000 4000 6000dIFMR/dH (arb. unit)\nH (Oe)H= 90o\nH= 50oH= 30o\nH= 0o(a)\n20003000400050006000\n0 3 06 09 0Hres (Oe)\nH (deg)9.8 nm\n15.0 nm\n29.3 nm\n72.7 nm(b)-101\n-4000 0 4000M/MS\nH (Oe)H // film\nH film\n37.9 nm16 \n  \n \nFigure 3 .  \n \n  02468\n00 . 511 . 52Eani (104 erg/cm3)\n(c)\n(c-a)/a (%)0204060H2|| (Oe)\n(b)-1200-800-4000H2 (Oe)(a)17 \n  \nFigure 4.  \n-150-100-50050100\n-600 -400 -200 0 200VISHE  (V)\nH-Hres (Oe)H\nH(a)\n-505\n-150 -100 -50 0 50 100 150dIFMR/dH (a.u.)\nH-Hres (Oe)YIG/YAG:\nH = 83.9 Oe\nPt/YIG/YAG:\nH = 88.9Oe(b)"    },    {        "title": "1312.3791v3.Micromagnetics_of_shape_anisotropy_based_permanent_magnets.pdf",        "content": "arXiv:1312.3791v3  [cond-mat.mtrl-sci]  26 Mar 2014Micromagnetics ofshapeanisotropybasedpermanentmagnet s\nSimonBancea, JohannFischbachera,ThomasSchrefla,IngaZinsb,GotthardRiegerb,CarolineCassignolb\naSt. P¨ olten University of Applied Sciences, Matthias Corvi nus Str. 15, 3100 St. P¨ olten, Austria\nbSiemens AG,Corporate Technology, Otto-Hahn-Ring 6,81739 Munich, Germany\nAbstract\nIn the search for rare-earth free permanent magnets, variou s ideas related to shape anisotropy are being pursued. In thi s work we\nassessthelimitsofshapecontributionstothereversalsta bilityusingmicromagneticsimulations. Inafirstseriesof testswealtered\nthe aspect ratio of single phase prolate spheroids from 1 to 1 6. Starting with a sphere of radius 4 .3 times the exchange length\nLexwe kept the total magnetic volume constant as the aspect rati o was modified. For a ferromagnetwith zero magnetocrystalli ne\nanisotropy the maximum coercive field reached up to 0 .5 times the magnetization Ms. Therefore, in materials with moderate\nuniaxial magnetocrystallineanisotropy,the addition of s hape anisotropycould even double the coercive field. Intere stinglydue to\nnon-uniformmagnetizationreversalthereisnosignificant increaseofthecoercivefieldforanaspectratiogreatertha n5. Asimilar\nlimit of the maximumaspect ratio was observedin cylinders. The coercivefield dependson the wire diameter. By decreasin g the\nwire diameterfrom8 .7Lexto 2.2Lexthe coercivefield increasedby 40%. In the cylindersnucleat ionof a reverseddomainstarts at\nthe cornersatthe end. Smoothingtheedgescanimprovetheco ercivefieldbyabout10%.\nIn further simulations we compacted soft magnetic cylinder s into a bulk-like arrangement. Misalignment and magnetost atic\ninteractionscauseaspreadof0 .1Msintheswitchingfieldsoftherods. Comparingthevolumeaver agedhysteresisloopscomputed\nfor isolated rods and the hysteresis loop computed for inter acting rods, we conclude that magnetostatic interactions r educe the\ncoercivefieldbyupto20%.\nKeywords: Permanentmagnets,magnetizationprocesses,simulationa ndnumericalmodelling\n2010MSC: 82D40,81T80\n1. Introduction\nRare earth permanentmagnetsexhibitthe highest maximum\nenergy product ( B·H)maxof all known magnetic materials; a\ncombinationofhighmagneticmomentandhighcoercivity[1] .\nForefficientelectricalgeneratorsandmotorsthisiscrucial;the\npermanentmagnetsthatare containedineitherthe rotoror s ta-\ntor must provide as strong magnetic fields as possible, with-\nout themselves being demagnetized. Recently much of the re-\nsearch into permanent magnets has focussed on reducing thei r\ndependence on rare earth elements. A number of approaches\nto finding new permanent magnets are being pursued, includ-\ning new hardmagneticcompoundsandnanostructuring,where\na soft phase contributes large magnetization and a hard phas e\ncontributeshighcoercivity[2,3,4].\nIn this paperwe assess the limits of shape anisotropye ffects\non the improvement of coercivity in magnets made of a single\nsoft magnetic phase. The coercive field of an ideal magnetic\nparticle is dependent on size and shape, with variousmodeso f\nreversal such as coherent rotation, curling and nucleation [5].\nAt small sizes, where internal magnetization is homogeneou s,\nreversalproceedsbycoherentrotationandisdescribedby\nHc=2K1\nµ0Ms+Ms\n2(1−3N) (1)\nEmailaddress: s.g.bance@gmail.com (Simon Bance)\nURL:http://academic.bancey.com (Simon Bance)whereK1is the uniaxial magnetocrystalline anisotropy con-\nstant,µ0isthevaccuumpermeability, MSisthesaturationmag-\nnetizationand Nisthedemagnetizingfactorparallelwith the c\naxis. The coercive field is dependent on the angle θbetween\nthe field and the caxis of the sample, so that an angle-adjusted\ncoercivefield H∗\ncisgivenby[6]\nH∗\nc=Hc\n(cos2/3θ+sin2/3θ)3/2. (2)\nFor a prolate spheroid (also known as an “ellipsoid of rota-\ntion”),whichhasequaldimensionsalongtwoaxes,thedemag -\nnetizingfactorscanbecalculatedfollowingtheworkofOsb orn\n[7].\nPreviouswork[8, 9,10]showedthat,abovea certainlength,\nthe reversal of columnar grains was no longer dependent on\ngrain length but on the nucleation of a reversal volume. The\nnucleation field then depends on the diameter only, where be-\nlow a certain diameter related to the reversal volume size th e\nnucleation field increases as an inverse function of diamete r.\nThisisaconsequenceofthespatialconfinementoftherevers al\nvolume and the increased exchange energy contribution when\nforminga domainwall.\n2. Method\nThe finite element method is used to numerically solve the\nLandau-Lifschitz-Gilbert(LLG)equation. Ateachtimeste pwe\nPreprint submitted to Journal of Magnetism and Magnetic Mat erials November 5,2018applyahybridfiniteelement /boundaryelementmethodtocom-\npute the magnetic scalar potential [11]. Finite element mod els\narecreatedwithtetrahedralvolumeelementsandtriangula rsur-\nfaceelements. Prolatespheroidsarepreparedbydefiningin ner\naxesain thex−yplaneand balongz, withb≥adefiningthe\nlongaxis(Fig. 2b). Thedimensionsaremodifiedto changethe\naspect ratio, keeping the total volume V=4\n3πa2bconstant at\n1824L3\nex, which correspondsto a sphere with a=b=4.35Lex\nand an aspect ratio of 1.0. Lexis the exchange length of the\nmaterial. Inordertoassessthee ffectsofshapealonewereduce\nthe magnetocrystallineanisotropytozero.\nSingle phase rods are modelled as regular cylinders with di-\nameterDand length L. For comparison,roundedrods are cre-\nated where the total length remains the same but both ends\nare rounded spherically. For these rods a moderate magneto-\ncrystalline anisotropyis used. Reversal loopsare calcula ted by\napplying a slowly-increasing external field opposing the in itial\nmagnetization, where the ramp speed is much slower than the\nLarmor precession, so that dynamic e ffects may be safely ig-\nnored [12]. A small field angle of 2◦is used. The nucleation\nfieldHnucis definedas the field strength requiredto reduce Mz\nto 0.9Ms. The coercive field Hcis defined as the field required\nto reduce Mzto zero. For all simulations the caxis is oriented\nparallel to the cartesian zaxis, except for particle ensembles\nwhere the caxis follows the rotation of the geometry, staying\nwith theshape-definedlongaxis.\n3. Results& Discussion\na\nb(a) (b)\n(c)\n-0.5 -0.4 -0.3 -0.2 -0.1 0.0\nH/Ms-1.0-0.50.00.51.0Mz/Msb/a=\n1.0\n1.95\n4.63\n15.6\n0 5 10 15 20\nAspect ratio b/a0.00.10.20.30.40.5Hnuc/Ms\nSimulation\nTheory\nFigure 1: (a) Computed reversal curves for prolate spheroid s with zero mag-\nnetocrystalline anisotropy and varying aspect ratio betwe en 1.0 and 15.6. (b)\nSchematic ofthespheroid geometry, with abeingtheradius inthe xyplaneand\nbalong the long zaxis. (c) The respective nucleation fields for di fferent aspect\nratios, with the theoretical plot for coherent reversal.\nNumericallycalculatedhysteresisloopsforprolatespher oids\nwith identicalvolumesbut aspect ratiosrangingfrom1.0up to\n15.6 are presented in Fig. 1a. The theoretical values of Hc\nare calculated with Eq. (1) and adjusted for the 2◦field angle\nwith Eq. (2). The curling mode of reversal is not considered,\n(a) \nb/a = 1.95(b) \nb/a = 15.6\nMz/Ms\n0.0 1.0 -1.0xyz\nFigure 2: Visualizations of the computed magnetization dat a during reversal\nof soft prolate spheroids for aspect ratios (a) 1.95 and (b) 1 5.6, showing the\ndifferent reversal modes.\nand images of the changing magnetization configurations dur -\ning reversal for aspect ratios of 1.95 (Fig. 2a) and 15.6 (Fig .\n2b)showthat curlingisnotpresentin thissize regime.\nHnucapproacheszeroforan aspect ratioof 1.0,whichcorre-\nsponds to the theoretical coherent mode anisotropy field HA=\n2K/Ms=0, where for spheres with N=1\n3[13]. For larger as-\npectratiosreversalbeginswithnucleationofa reversaldo main\nat the ends of the wire, followed by domain wall propagation\nuntil the whole wire is switched. The value of Hnucconverges\nat a value of b/a=5, consistent with previous investigations\n[8, 9, 10]. For an aspect ratio of 15.6 the nucleation field is\n0.42Ms.\nThe simulation results match closely with the theoretical\nmodel, even at largeaspect ratios where reversal beginsby n u-\ncleation. This is consistent with the idea of nucleation of a re-\nversal volume at the ends of the sample, since the reversalvo l-\nume forms through localized rotation from the fully saturat ed\nsample (Fig. 2b), therefore the Stoner-Wohlfarth model re-\nmainsvalidforpredictingthenucleationfields. Thiskindo fbe-\nhaviourhasbeen observedbeforein the switchingof pattern ed\nelements in recording media, where non-unform switching is\nobserved but the switching field follows the Stoner-Wohlfar h\nangledependence[14].\nSimilarresultsarepresentedforcylindricalandroundedr ods\n(Fig.3), where for a discrete set of rod diameters Dthe length\nLis modified. The nucleation field Hnucis plotted as a func-\ntion ofL/D, where each symbol represents a di fferent diam-\neterD.Hnucvalues reach a plateau when the aspect ratio is\ngreater than 5. In this regime switching occurs by nucleatio n\nof a reversed domain and wall motion and the nucleation field\n2increases with decreasing D. For each D, the coercive field at\nthe plateau is higher for rounded rods. The smoothness of the\nends reducessurface chargesand the resulting lower magnet o-\nstatic energy stabilises against nucleation. This di fference be-\ncomes larger as the rod diameter increases, since for smalle r\ndiameters the stiffness caused by the exchangeinteractionlim-\nitstheinhomogeneityofthemagnetizationatthesampleedg es.\nFor aspect ratios smaller than 2 we approach the coherent ro-\ntation or curling regime. For the rounded ends with an aspect\nratioL/D=1.0 (a sphere), Hnuc=0.33Msfor allD, which\ncorresponds to the theoretical Stoner-Wohlfarth predicti on for\na small sphere of HA=0.37Ms(when adjusted for the small\nfield angle) [13]. The cylindrical wire with aspect ratio 1.0\nhas a slightly higher nucleation field, due to the higher shap e\nanisotropy contribution along the parallel axis for cylind rical\ngeometries with respect to spheres [15, 16]. In this case all of\ntheroundedparticlesarebelowthetheoreticalcoherencer adius\nfor a sphere Rsphere\ncoh=5.099Lex, and all but the largest cylinder\nare below the cylindricalcoherenceradius Rcylinder\ncoh=3.655Lex,\nmeaningthatreversaliscoherent[17]. For largerdiameter swe\nexpect the nucleation field to reduce, as curling becomes mor e\nsignificant[5].\n0 2 4 6 8 10 12 14 16\nAspect ratio, L/D0.250.30.350.40.450.50.550.60.650.7Hnuc/MsCylinderRounded\nD/Lex\nRounded\nCylinder2.2 4.4 6.5 8.7\nFigure 3: Computed nucleation fields for cylindrical (black markers) and\nrounded (red markers) rods of varying diameter D, as a function of wire as-\npect ratio L/D.\nExchange-decoupled particle ensembles were created by\npacking thirty eight of the rounded rods with dimensions D=\n8.7LexandL=87.1Lexintoaboundingboxofsize65 .3×65.3×\n217.7L3\nex, using the open-source YADE framework (Fig.4b)\n[18]. Using thismethodwe were able to achievevolumepack-\ning densities of between 0.20 and 0.29. With this variation, a\nspreadinthecoercivefieldcalculatedfromthehysteresisl oops\nofaround0.1Mswasfound. Thiscorrespondstoachangeofup\nto30%andcanbeattributedtotheangulardependenceofcoer -\ncivity as well as the changingdistances, orientationsand o ver-\nlap between the individual rodsand the accompanyingmagne-\ntostatic interactions.\nIn order to seperate the e ffect of magnetostatic interactions\non the reversal of rod esembles we perform the following ex-periment: Fig. 4comparestwocomputedreversalsofthe same\nrod arrangement, with and without magnetostatic interacti ons.\nTherodshaveanaveragetiltangleof21 .8◦andusingthesame\narrangementinbothcasesexcludestheinfluenceofangulard is-\ntribution. First, the full model is simulated at once to incl ude\nthemagnetostaticinteractionsbetweentherods. Second,t here-\nversal curvesfor individual rodsare computedseparately, then\nthe combinedvolume-averagereversalcurveis calculated. Ex-\nchange interactions are not included in order to isolate the in-\nfluence of magnetostatic interactions. The full model rever sal\nexhibitsa 0.08Msreductionin coercivity;a 20% reductiondue\ntomagnetostaticinteractionsalone.\nWeexpectthatusingalargernumberofrodswouldbetterre-\nproducethebehaviourofbulkmagnetsandsmooththefeature s\nofthereversalcurve,howeverinthisstudyfiniteelementmo del\nsizeswerelimitedtoaround38rodsbytheavailablecomputi ng\nresources.\n-0.7 -0.6 -0.5 -0.4 -0.3 -0.2 -0.1 0\nH/Ms00.20.40.60.81Mz/MsVolume average\nFull model\nMz/Ms\n0.0 +1.0 -1.0\nFigure 4: The computed reversal curve for the full model rod a rrangement\nshowsa20%reduction incoercivity compared totheisolated volume-averaged\nresults, due to magnetostatic interactions. Inset: visual ization of the packed\nrods model.\n4. Conclusions\nBy changing the shape of soft magnetic particles from\nspheres to long wires we were able to increase the nucleation\nfield by a factor of 15. Above an aspect ratio of five any\nimprovements plateau. The maximum coercive field reaches\n0.65Msfor an aspect ratio of 10 and a diameter of the cylin-\nder ofD=2.2Lex. Similar results were obtained for cylin-\ndrical rods, where smoothing the ends was found to improve\nthe coercivity by reducing surface charges. Compacted rod a r-\nrangementsshowa variationin coercivityofaround30% sinc e\ndifferentpackingarrangementsresultindi fferentangulardistri-\nbutions and different amounts of magnetostatic interaction be-\ntween the rods. Magnetostatic interactions between the rod s\nwereshowntoreducecoercivitybyaround20%.\n3References\n[1] H. Kirchmayr, Permanent magnets and hard magnetic mater ials, Journal\nof Physics D:Applied Physics 29 (11) (1996) 2763.\n[2] E.Kneller, R. Hawig, Theexchange-spring magnet: a new m aterial prin-\nciple for permanent magnets, Magnetics, IEEE Transactions on 27 (4)\n(1991) 3588–3560.\n[3] J. Jiang, J. Pearson, Z. Y. Liu, B. Kabius, S. Trasobares, D. Miller,\nS.Bader,D.Lee,D.Haskel,G.Srajer, J.Liu,Improvingexch ange-spring\nnanocomposite permanent magnets, Applied Physics Letters 85 (22)\n(2004) 5293–5295.\n[4] R. Skomski, P. Manchanda, P. Kumar, B. Balamurugan, A. Ka shyap,\nD. Sellmyer, Predicting the future of permanent-magnet mat erials, Mag-\nnetics, IEEETransactions on 49 (7) (2013) 3215–3220.\n[5] J.M.D.Coey,MagnetismandMagneticMaterials, Cambrid geUniversity\nPress,2004.\n[6] E.Stoner,E.Wohlfarth,Amechanismofmagnetichystere sisinheteroge-\nneousalloys, Philosophical Transactions oftheRoyal Soci ety ofLondon.\nSeries A.Mathematical and Physical Sciences (1948) 599–64 2.\n[7] J. A. Osborn, Demagnetizing factors of the general ellip soid, Phys. Rev.\n67 (1945) 351–357.\n[8] H. Forster, N. Bertram, X. Wang, R. Dittrich, T. Schrefl, E nergy barrier\nandeffectivethermalreversalvolumeincolumnargrains,Journal ofMag-\nnetism and Magnetic Materials 267 (1) (2003) 69 – 79.\n[9] R.Hertel, Computational micromagnetism of magnetizat ion processes in\nnickelnanowires, JournalofMagnetism andMagnetic Materi als 249(12)\n(2002) 251 –256, international Workshopon Magnetic Wires.\n[10] J. Escrig, R. Lavn, J. L. Palma, J. C. Denardin, D. Altbir , A. Corts,\nH.Gmez,Geometrydependenceofcoercivity inninanowirear rays,Nan-\notechnology 19 (7) (2008) 075713.\n[11] T. Schrefl, G. Hrkac, S. Bance, D. Suess, O. Ertl, J. Fidle r, Numerical\nmethods in micromagnetics (finite element method), Handboo k of mag-\nnetism and advanced magnetic materials.\n[12] C. Serpico, G. Bertotti, I. D. Mayergoyz, M. d’Aquino, N onlinear mag-\nnetization dynamics in nanomagnets, Handbook of Magnetism and Ad-\nvanced Magnetic Materials.\n[13] H. Kronm¨ uller, S. Parkin, Handbook of magnetism and ad vanced mag-\nnetic materials vols 1–5 (2007).\n[14] R. Dittrich, G. Hu, T. Schrefl, T. Thomson, D. Suess, B. Te rris, J. Fi-\ndler, Angular dependence of the switching field in patterned magnetic\nelements, Journal of Applied Physics 97 (10).\n[15] H. Kronm¨ uller, M. F¨ ahnle, Micromagnetism and the mic rostructure of\nferromagnetic solids, Cambridge university press, 2003.\n[16] D.-X.Chen,J.Brug,R.B.Goldfarb,Demagnetizing fact orsforcylinders,\nMagnetics, IEEETransactions on 27 (4) (1991) 3601–3619.\n[17] R. Skomski, J. Zhou, Nanomagnetic models, in: Advanced Magnetic\nNanostructures, Springer, 2006, pp. 41–90.\n[18] V.ˇSmilauer,A.Gladky,J.Kozicki,C.Modenese,J.Str´ ansk` y ,YadeUsing\nand Programming, in: V. ˇSmilauer (Ed.), Yade Documentation, 1st Edi-\ntion, The Yade Project, 2010, http: //yade-dem.org/doc/[Accessed 30th\nOctober 2013].\n4"    },    {        "title": "1312.7665v3.Ground_state_search__hysteretic_behaviour__and_reversal_mechanism_of_skyrmionic_textures_in_confined_helimagnetic_nanostructures.pdf",        "content": "Ground state search, hysteretic behaviour, and reversal mechanism of skyrmionic\ntextures in con\fned helimagnetic nanostructures\nMarijan Beg,1,\u0003Rebecca Carey,1Weiwei Wang,1David Cort\u0013 es-Ortu~ no,1Mark Vousden,1Marc-Antonio Bisotti,1\nMaximilian Albert,1Dmitri Chernyshenko,1Ondrej Hovorka,1Robert L. Stamps,2and Hans Fangohr1,y\n1Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, United Kingdom\n2SUPA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom\nMagnetic skyrmions have the potential to provide solutions for low-power, high-density data\nstorage and processing. One of the major challenges in developing skyrmion-based devices is the\nskyrmions' magnetic stability in con\fned helimagnetic nanostructures. Through a systematic study\nof equilibrium states, using a full three-dimensional micromagnetic model including demagnetisation\ne\u000bects, we demonstrate that skyrmionic textures are the lowest energy states in helimagnetic thin\n\flm nanostructures at zero external magnetic \feld and in absence of magnetocrystalline anisotropy.\nWe also report the regions of metastability for non-ground state equilibrium con\fgurations. We show\nthat bistable skyrmionic textures undergo hysteretic behaviour between two energetically equivalent\nskyrmionic states with di\u000berent core orientation, even in absence of both magnetocrystalline and\ndemagnetisation-based shape anisotropies, suggesting the existence of Dzyaloshinskii-Moriya-based\nshape anisotropy. Finally, we show that the skyrmionic texture core reversal dynamics is facilitated\nby the Bloch point occurrence and propagation.\nAn ever increasing need for data storage creates great\nchallenges for the development of high-capacity storage\ndevices that are cheap, fast, reliable, and robust. Nowa-\ndays, hard disk drive technology uses magnetic grains\npointing up or down to encode binary data (0 or 1)\nin so-called perpendicular recording media. Practical\nlimitations are well understood and dubbed the \\mag-\nnetic recording trilemma\" [1]. It de\fnes a trade-o\u000b be-\ntween three con\ricting requirements: signal-to-noise ra-\ntio, thermal stability of the stored data, and the ability\nto imprint information. Because of these fundamental\nconstraints, further progress requires radically di\u000berent\napproaches.\nRecent research demonstrated that topologically sta-\nble magnetic skyrmions have the potential for the devel-\nopment of future data storage and information process-\ning devices. For instance, a skyrmion lattice formed in\na monoatomic Fe layer grown on a Ir(111) surface [2]\nrevealed skyrmions with diameters as small as a few\natom spacings. In addition, it has been demonstrated\nthat skyrmions can be easily manipulated using spin-\npolarised currents of the 106A m\u00002order [3, 4] which\nis a factor 105to 106smaller than the current densi-\nties required in conventional magneto-electronics. These\nunique skyrmion properties point to an opportunity for\nthe realisation of ambitious novel high-density, power-\ne\u000ecient storage [5, 6] and logic [7] devices.\nSkyrmionic textures emerge as a consequence of chi-\nral interactions, also called the Dzyaloshinskii-Moriya In-\nteractions (DMI), that appear when there is no inver-\nsion symmetry in the magnetic system structure. The\nlack of inversion symmetry can be either due to a non-\ncentrosymmetric crystal lattice structure [8, 9] in so-\ncalled helimagnetic materials, or at interfaces between\ndi\u000berent materials that inherently lack inversion symme-\ntry [10, 11]. According to this, the Dzyaloshinskii-Moriyainteraction can be classi\fed either as bulk or interfacial,\nrespectively. Skyrmions, after being predicted [12{14],\nwere later experimentally observed in magnetic systems\nwith both bulk [15{19] and interfacial [2, 20] types of\nDMI.\nSo far, a major challenge obstructing the development\nof skyrmion-based devices has been their thermal and\nmagnetic stability [21]. Only recently, skyrmions were\nobserved at the room temperature in magnetic systems\nwith bulk [22] and interfacial [23{25] DMI. However, the\nmagnetic stability of skyrmions in absence of external\nmagnetic \feld was reported only for magnetic systems\nwith interfacial DMI in one-atom layer thin \flms [2, 26],\nwhere the skyrmion state is stabilised in the presence of\nmagnetocrystalline anisotropy.\nThe focus of this work is on the zero-\feld stability\nof skyrmionic textures in con\fned geometries of bulk\nDMI materials. Zero-\feld stability is a crucial require-\nment for the development of skyrmion-based devices: de-\nvices that require external magnetic \felds to be stabilised\nare volatile, harder to engineer and consume more en-\nergy. We address the following questions that are rele-\nvant for the skyrmion-based data storage and processing\nnanotechnology. Can skyrmionic textures be the ground\nstate (i.e. have the lowest energy) in helimagnetic ma-\nterials at zero external magnetic \feld, and if they can,\nwhat is the mechanism responsible for this stability? Do\nthe demagnetisation energy and magnetisation variation\nalong the out-of-\flm direction [27] have important con-\ntribution to the stability of skyrmionic textures? Is the\nmagnetocrystalline anisotropy an essential stabilisation\nmechanism? Are there any other equilibrium states that\nemerge in con\fned helimagnetic nanostructures? How\nrobust are skyrmionic textures against varying geometry?\nDo skyrmionic textures undergo hysteretic behaviour in\nthe presence of an external magnetic \feld (crucial forarXiv:1312.7665v3  [cond-mat.mtrl-sci]  25 Nov 20152\ndata imprint), and if they do, what is the skyrmionic\ntexture reversal mechanism?\nTo resolve these unknowns, we use a full three-\ndimensional simulation model that makes no assump-\ntion about translational invariance of magnetisation in\nthe out-of-\flm direction and takes full account of the\ndemagnetisation energy. We demonstrate, using this full\nmodel, that DMI-induced skyrmionic textures in con\fned\nthin \flm helimagnetic nanostructures are the lowest en-\nergy states in the absence of both the stabilising external\nmagnetic \feld and the magnetocrystalline anisotropy and\nare able to adapt their size to hosting nanostructures,\nproviding the robustness for their practical use. We\ndemonstrate that both the demagnetisation energy and\nthe magnetisation variation in the out-of-\flm direction\nplay an important role for the stability of skyrmionic tex-\ntures. In addition, we report the parameter space regions\nwhere other magnetisation con\fgurations are in equilib-\nrium. Moreover, we demonstrate that these zero-\feld\nstable skyrmionic textures undergo hysteretic behaviour\nwhen their core orientation is changed using an external\nmagnetic \feld, which is crucial for data imprint. The\nhysteretic behaviour remains present even in the absence\nof all relevant magnetic anisotropies (magnetocrystalline\nand demagnetisation-based shape anisotropies), suggest-\ning the existence of a novel Dzyaloshinskii-Moriya-based\nshape anisotropy. We conclude the study by showing that\nthe skyrmionic texture core orientation reversal is facil-\nitated by the Bloch point occurrence and propagation,\nwhere the Bloch point may propagate in either of the\ntwo possible directions. This work is based on the spe-\nci\fc cubic helimagnetic material, FeGe with 70 nm helical\nperiod, in order to encourage the experimental veri\fca-\ntion of our predictions. Other materials could allow ei-\nther to reduce the helical period [15, 19] and therefore\nthe hosting nanostructure size or increase the operating\ntemperature [22].\nSome stability properties of DMI-induced isolated\nskyrmions in two-dimensional con\fned systems have\nbeen studied analytically [28{30] and using simula-\ntions [26, 31]. However, in all these studies, either mag-\nnetocrystalline anisotropy or an external magnetic \feld\n(or both) are crucial for the stabilisation of skyrmionic\ntextures. In addition, an alternative approach to the\nsimilar problem, in absence of chiral interactions, where\nskyrmionic textures can be stabilised at zero exter-\nnal magnetic \feld and at room temperature using a\nstrong perpendicular anisotropy, has been studied ana-\nlytically [32], experimentally [33, 34], as well as using\nsimulations [35]. Our new results, and in particular the\nzero-\feld skyrmionic ground state in isotropic helimag-\nnetic materials, can only be obtained by allowing the\nchiral modulation of magnetisation direction along the\n\flm normal, which has recently been shown to radically\nchange the skyrmion energetics [27].RESULTS\nEquilibrium states. In order to identify the low-\nest energy magnetisation state in con\fned helimag-\nnetic nanostructures, \frstly, all equilibrium magnetisa-\ntion states (local energy minima) must be identi\fed,\nand secondly, their energies compared. In this sec-\ntion, we focus on the \frst step { identifying the equi-\nlibrium magnetisation states. We compute them by solv-\ning a full three-dimensional model using a \fnite element\nbased micromagnetic simulator. In particular, we sim-\nulate a thin \flm helimagnetic FeGe disk nanostructure\nwith thickness t= 10 nm and diameter d, as shown\nin Fig. 1 inset. The \fnite element mesh discretisation\nis such that the maximum spacing between two neigh-\nbouring mesh nodes is below 3 nm. The material pa-\nrameters are Ms= 384 kA m\u00001,A= 8:78 pJ m\u00001, and\nD= 1:58 mJ m\u00002. We apply a uniform external mag-\nnetic \feld perpendicular to the thin \flm sample, i.e. in\nthe positive z-direction. The Methods section contains\nthe details about the model, FeGe material parameters\nestimation, as well as the simulator software.\nIn this section, we determine what magnetisation con-\n\fgurations emerge as the equilibrium states at di\u000berent\nd{Hparameter space points. In order to do that, we\nsystematically explore the parameter space by varying\nthe disk sample diameter dfrom 40 nm to 180 nm and\nthe external magnetic \feld \u00160Hfrom 0 T to 1 :2 T in\nsteps of \u0001d= 4 nm and \u00160\u0001H= 20 mT, respectively.\nAt every point in the parameter space, we minimise the\nenergy for a set of di\u000berent initial magnetisation con-\n\fgurations: (i) \fve di\u000berent skyrmionic con\fgurations,\n(ii) three helical-like con\fgurations with di\u000berent helical\nperiod, (iii) the uniform out-of-plane con\fguration, and\n(iv) three random magnetisation con\fgurations. We use\nthe random magnetisation con\fgurations in order to cap-\nture other equilibrium states not obtained by relaxing the\nwell-de\fned initial magnetisation con\fgurations. The de-\ntails on how we de\fne and generate initial magnetisation\ncon\fgurations are provided in the Supplementary Section\nS1.\nThe equilibrium states to which di\u000berent initial mag-\nnetisation con\fgurations relax in the energy minimisa-\ntion process (at every d{Hparameter space point) we\npresent in the Supplementary Section S2 as a set of \\re-\nlaxation diagrams\". We summarise these relaxation dia-\ngrams and determine the phase space regions where dif-\nferent magnetisation states are in equilibrium, and show\nthem in Fig. 1. Among the eight computed equilibrium\nstates, three are radially symmetric and we label them as\niSk, Sk, and T, whereas the other states, marked as H2,\nH3, H4, 2Sk, and 3Sk, are not. Subsequently, we discuss\nthe meaning of the chosen labels.\nNow, we focus on the analysis of radially symmetric\nskyrmionic equilibrium states, supported by computing3\nFIG. 1. The metastability phase diagram and magnetisation con\fgurations of all identi\fed equilibrium states.\nThe phase diagram with regions where di\u000berent states are in equilibrium together with magnetisation con\fgurations and out-\nof-plane magnetisation component mz(x) along the horizontal symmetry line corresponding to di\u000berent regions in the phase\ndiagram.\nthe skyrmion number Sand scalar value Saas de\fned in\nthe Methods section. In the \frst con\fguration, marked\nin Fig. 1 as iSk, the out-of-plane magnetisation compo-\nnentmz(x) pro\fle along the horizontal symmetry line\ndoes not cover the entire [ \u00001;1] range, as would be the\ncase for a skyrmion con\fguration (where the magneti-\nsation vector \feld mneeds to cover the whole sphere).\nAccordingly, the scalar value Sa(Eq. (6) in the Meth-\nods section, and plotted in Supplementary Fig. 2 (b) for\na range of con\fgurations), is smaller than 1. For these\nreasons we refer to this skyrmionic equilibrium state as\nthe incomplete Skyrmion (iSk) state. A similar magneti-\nsation con\fguration has been predicted and observed in\nother works for the case of two-dimensional systems in\nthe presence of magnetocrystalline anisotropy where it\nis called either the quasi-ferromagnetic [26, 28] or edged\nvortex state [29, 31]. Because the iSk equilibrium state\nclearly di\u000bers from the ferromagnetic con\fguration and\nusing the word vortex implies the topological charge of\n1=2, we prefer calling this state the incomplete skyrmion\nstate. The incomplete Skyrmion (iSk) state emerges as\nan equilibrium state in the entire simulated d{Hparame-\nter space range. In the second equilibrium state, markedas Sk in Fig. 1, mz(x) covers the entire [ \u00001;1] range,\nthe magnetisation covers the sphere at least once and,\nconsequently, the skyrmion con\fguration is present in\nthe simulated sample. Although the skyrmion number\nvalue (Eq. (5) in the Methods section) for this solution\nisjSj<1 due to the additional magnetisation tilting at\nthe disk boundary [28], which makes it indistinguishable\nfrom the previously described iSk equilibrium state, the\nscalar value is 1 < S a<2. This state is referred to\nas the isolated Skyrmion or just Skyrmion (Sk), in two-\ndimensional systems [26, 28], and we use the same name\nsubsequently in this work. We \fnd that the Sk state\nis not in equilibrium for sample diameters smaller than\n56 nm and external magnetic \feld values larger than ap-\nproximately 1 :14 T. Finally, the equilibrium magnetisa-\ntion state marked as T in Fig. 1 covers the sphere at least\ntwice. In other works, this state together with all other\npredicted higher-order solutions (not observed in this\nwork) are called the \\target states\" [30], and we use the\nsame Target (T) state name. The analytic model, used\nfor generating initial states, also predicts the existence\nof higher-order target states (Supplementary Fig. 2 (c)).\nThe T magnetisation con\fguration emerges as an equi-4\nlibrium state for samples with diameter d\u0015144 nm and\n\feld values \u00160H\u00140:24 T.\nThe equilibrium states lacking radial symmetry can\nbe classi\fed into two groups: helical-like (marked as H2,\nH3, and H4) and multiple skyrmion (marked as 2Sk and\n3Sk) states. The di\u000berence between the three helical-like\nstates is in their helical period. More precisely, in the\nstudied range of disk sample diameter values, either\n2, 3, or 4 helical half-periods, including the additional\nmagnetisation tilting at the disk sample edge due to\nthe speci\fc boundary conditions [28], \ft in the sample\ndiameter. Consequently, we refer to these states, that\noccur as an equilibrium state for samples larger than\n88 nm and \feld values lower than 0 :2 T, as H2, H3, and\nH4 . The other two radially non-symmetric equilibrium\nstates are the multiple skyrmion con\fgurations with 2\nor 3 skyrmions present in the sample and we call these\nequilibrium states 2Sk and 3Sk, respectively. These\ncon\fgurations emerge as equilibrium states for samples\nwithd\u0015132 nm and external magnetic \feld values\nbetween 0:28 T\u0014\u00160H\u00141:06 T.\nGround state. After we identi\fed all observed equi-\nlibrium states in con\fned helimagnetic nanostructures,\nin this section we focus on \fnding the equilibrium state\nwith the lowest energy at all d{Hparameter space points.\nFor every parameter space point ( d,H), after we com-\npute and compare the energies of all found equilibrium\nstates, we determine the lowest energy state, and refer to\nit, in this context, as the ground state. For the identi\fed\nground state, we compute the scalar value Saand use it\nfor plotting a d{Hphase diagram shown in Fig. 2 (a).\nDiscontinuous changes in the scalar value Sade\fne the\nboundaries between regions where di\u000berent magnetisa-\ntion con\fgurations are the ground state. In the studied\nphase space, two di\u000berent ground states emerge in the\ncon\fned helimagnetic FeGe thin \flm disk samples: one\nwithSa<1 and the other with 1 < S a<2. The pre-\nvious discussion of the Savalue suggests that these two\nregions correspond to the incomplete Skyrmion (iSk) and\nthe isolated Skyrmion (Sk) states. We con\frm this by vi-\nsually inspecting two identi\fed ground states, taken from\nthe two phase space points (marked with circle and tri-\nangle symbols) in di\u000berent regions, and show them in\nFig. 2 (b) together with their out-of-plane magnetisation\ncomponent mz(x) along the horizontal symmetry line.\nA key result of this study is that both incomplete\nSkyrmion (iSk) and isolated Skyrmion (Sk) are the\nground states at zero external magnetic \feld for di\u000berent\ndisk sample diameters. More precisely, iSk is the ground\nstate for samples with diameter d<140 nm and Sk is the\nground state for d\u0015140 nm. The Sk changes to the iSk\nground state for large values of external magnetic \feld.\nThe phase diagram in Fig. 2 shows the phase space\nregions where iSk and Sk are the ground states, which\nmeans that all other previously identi\fed equilibrium\nFIG. 2. Thin \flm disk ground state phase diagram\nand corresponding magnetisation states. (a) The scalar\nvalueSafor the thin \flm disk sample with thickness t= 10 nm\nas a function of disk diameter dand external out-of-plane\nmagnetic \feld H(as shown in an inset). ( b) Two identi-\n\fed ground states: incomplete Skyrmion (iSk) and isolated\nSkyrmion (Sk) magnetisation con\fgurations at single phase\ndiagram points together with their out-of-plane magnetisa-\ntion component mz(x) pro\fles along the horizontal symmetry\nline.\nstates are metastable. Now, we focus on computing the\nenergies of metastable states relative to the identi\fed\nground state. Firstly, we compute the energy density\nE=V for all equilibrium states, where Eis the total en-\nergy of the system and Vis the disk sample volume,\nand then subtract the ground state energy density corre-\nsponding to that phase space point. We show the com-\nputed energy density di\u000berences \u0001 E=V when the disk\nsample diameter is changed in steps of \u0001 d= 2 nm at\nzero external magnetic \feld in Fig. 3 (a). Similarly,\nthe case when the disk sample diameter is d= 160 nm\nand the external magnetic \feld is changed in steps of\n\u00160\u0001H= 20 mT is shown in Fig. 3 (b). The magneti-\nsation con\fgurations are the equilibrium states in the d\norHvalues range where the line is shown and collapse\notherwise.5\nFIG. 3. The energy density di\u000berence between identi-\n\fed equilibrium states and the corresponding ground\nstate. Energy density di\u000berences \u0001 E=V at (a) zero \feld\nfor di\u000berent sample diameters dand for ( b) sample diame-\nterd= 160 nm and di\u000berent external magnetic \feld values.\nCon\fgurations are in equilibrium where the line is shown and\ncollapse for other diameter or external magnetic \feld values.\nFor the practical use of ground state skyrmionic tex-\ntures in helimagnetic nanostructures, their robustness is\nof great signi\fcance due to the unavoidable variations\nin the patterning process. Because of that, in Fig. 4 (a)\nwe plot the out-of-plane magnetisation component mz(x)\nalong the horizontal symmetry line for the iSk and the\nSk ground state at zero external magnetic \feld for six\ndi\u000berent diameters dof the hosting disk nanostructure:\nthree iSk pro\fles for d\u0014120 nm, and three Sk pro-\n\fles ford\u0015140 nm. The pro\fles show that the two\nskyrmionic ground states have the opposite core orien-\ntations. In the case of the Sk states, the magnetisation\nat the core is antiparallel and at the outskirt parallel to\nthe external magnetic \feld. This reduces the Zeeman en-\nergyEz=\u0000\u00160R\nH\u0001Md3rbecause the majority of the\nmagnetisation in the isolated skyrmion outskirts points\nin the same direction as the external magnetic \feld H.\nOnce the disk diameter is su\u000eciently small that less than\na complete spin rotation \fts into the sample, this orien-\ntation is not energetically favourable anymore and the\niSk state emerges. In this iSk state, the core magnetisa-\ntion points in the same direction as the external magnetic\nFIG. 4. Themz(x)pro\fles and skyrmionic texture\nsizessfor di\u000berent sizes of hosting nanostructures\nat zero external magnetic \feld. (a) Pro\fles of the out-\nof-plane magnetisation component mz(x) along the horizon-\ntal symmetry line for di\u000berent thin \flm disk sample diam-\neters with thickness t= 10 nm at zero external magnetic\n\feld\u00160H= 0 T. The curves for d\u0014120 nm represent in-\ncomplete skyrmion ( \u000e) states, and for d\u0015140 nm repre-\nsent isolated skyrmion ( \u0002) states. ( b) The skyrmionic tex-\nture sizes= 2\u0019=k (that can be interpreted as the length\nalong which the full magnetisation rotation occurs) as a func-\ntion of the hosting nanostructure size, obtained by \ftting\nmz(x) =\u0006cos(kx) to the simulated pro\fle. ( c) The ratio\nof skyrmionic texture size to disk sample diameter ( s=d) as a\nfunction of hosting nanostructure size d.\n\feld in order to minimise the Zeeman energy. We com-\npute and plot the skyrmionic texture size s= 2\u0019=kas a\nfunction of the disk sample diameter din Fig. 4 (b). We\nobtain the size s, that can be interpreted as the length\nalong which the full magnetisation rotation occurs, by\n\fttingkin thef(x) =\u0006cos(kx) function to the simu-\nlated iSk and Sk mz(x) pro\fles. In Fig. 4 (c), we show\nhow the ratio of skyrmionic texture size to disk sam-\nple diameter ( s=d) depends on the hosting nanostructure6\nsize. Although this ratio is constant ( s=d\u00190:6) for the\nSk state, in the iSk case, it is larger for smaller samples\nand decreases to s=d\u00191:5 in larger nanostructures. In\nagreement with related \fndings for two-dimensional disk\nsamples [29] we \fnd that both iSk and Sk are able to\nchange their size sin order to accommodate the size of\nhosting nanostructure, which provides robustness for the\ntechnological use.\nThe emergence of skyrmionic texture ground state in\nhelimagnetic nanostructures at zero external magnetic\n\feld and in absence of magnetocrystalline anisotropy is\nunexpected [21]. Now, we discuss the possible mecha-\nnisms, apart from the geometrical con\fnement, respon-\nsible for this stability, in particular (i) the demagneti-\nsation energy contribution, and (ii) the magnetisation\nvariation along the out-of-\flm direction which can rad-\nically change the skyrmion energetics in in\fnitely large\nhelimagnetic thin \flms [27]. We repeat the simulations\nusing the same method and model as above but ignor-\ning the demagnetisation energy contribution (i.e. setting\nthe demagnetisation energy density wdin Eq. (1) arti\f-\ncially to zero). We then carry out the calculations (i) on\na three-dimensional (3d) mesh (i.e. with spatial resolu-\ntion inz-direction) and (ii) on a two-dimensional (2d)\nmesh (i.e. with no spatial resolution in z-direction, and\nthus not allowing a variation of the magnetisation along\nthez-direction). The disk sample diameter dis changed\nbetween 40 nm and 180 nm in steps of \u0001 d= 5 nm and\nthe external magnetic \feld \u00160His changed systemati-\ncally between 0 T and 0 :5 T in steps of \u00160\u0001H= 25 mT.\nThe two resulting phase diagrams are shown in Fig. 5,\nwhere subplots (a) and (c) show Saas a function of d\nandH. Because the scalar value Sadoes not provide\nenough contrast to determine the boundaries of the new\nHelical (H) ground state region, the skyrmion number S\nis plotted for the relevant phase diagram areas and shown\nin Fig. 5 (b) and Fig. 5 (d).\nWe demonstrate the importance of including demag-\nnetisation e\u000bects into the model by comparing Fig. 5 (a)\n(without demagnetisation energy) and Fig. 2 (a) (with\ndemagnetisation energy). In the absence of the demag-\nnetisation energy, the isolated Skyrmion (Sk) con\fgura-\ntion is not found as the ground state at zero applied \feld;\ninstead, Helical (H) con\fgurations have lower energies.\nAt the same time, the external magnetic \feld at which\nthe skyrmion con\fguration ground state disappears is re-\nduced from about 0 :7 T to about 0 :44 T.\nBy comparing Fig. 5 (a) computed on a 3d mesh and\nFig. 5 (c) computed on a 2d mesh, we can see the impor-\ntance of spatial resolution in the out-of-plane direction\nof the thin \flm, and how it contributes to the stabilisa-\ntion of isolated Skyrmion (Sk) state. In the 2d model,\nthe \feld range over which skyrmions can be observed\nas the ground state is further reduced to approximately\n[0:05 T, 0:28 T]. In the 3d mesh model the Sk con\fgura-\ntion can reduce its energy by twisting the magnetisationat the top of the disk relative to the bottom of the disk\nso that along the z-direction the magnetisation starts to\nexhibit (a part of) the helix that arises from the com-\npetition between symmetric exchange and DMI energy\nterms, similar to Ref. [27]. A similar twist provides no\nenergetic advantage to the helix con\fguration, thus the\nSk state region in Fig. 5 (a) is signi\fcantly larger than\nthe Sk state region in Fig. 5 (c) where the 2d mesh does\nnot allow any variation of the magnetisation along the\nz-direction and thus the partial helix cannot form.\nWhile the isolated Skyrmion (Sk) con\fguration\nat zero \feld is a metastable state in the absence of\ndemagnetisation energy, or in 2d models, it is not\nthe ground state anymore as there are Helical (H)\nequilibrium con\fgurations that have lower total energy.\nThe demagnetisation energy appears to suppress these\nhelical con\fgurations which have a lower energy than\nthe skyrmion. The variation of the magnetisation along\nthez-direction stabilises the skyrmion con\fguration\nsubstantially. These \fndings demonstrate the subtle\nnature of competition between symmetric exchange,\nDM and demagnetisation interactions, and show that\nignoring the demagnetisation energy or approximating\nthe thin \flm helimagnetic samples using two-dimensional\nmodels is not generally justi\fed.\nHysteretic behaviour. The phase diagram in\nFig. 2 (a) shows the regions in which incomplete\nSkyrmion (iSk) and isolated Skyrmion (Sk) con\fgura-\ntions are the ground states. Intuitively, one can assume\nthat for every sample diameter dat zero external mag-\nnetic \feld, there are two possible skyrmionic magnetisa-\ntion con\fgurations of equivalent energy: core pointing up\nor core pointing down, suggesting that these textures can\nbe used for an information bit (0 or 1) encoding. We now\ninvestigate this hypothesis and study whether an exter-\nnal magnetic \feld can be used to switch the skyrmionic\nstate orientation (crucial for data imprint) by simulat-\ning the hysteretic behaviour of ground state skyrmionic\ntextures.\nWe obtain the hysteresis loops in the usual way by\nevolving the system to an equilibrium state after chang-\ning the external magnetic \feld, and then using the re-\nsulting state as the starting point for a new evolution.\nIn this way, a magnetisation loop takes into account\nthe history of the magnetisation con\fguration. The ex-\nternal magnetic \feld \u00160His applied in the positive z-\ndirection and changed between \u00000:5 T and 0:5 T in steps\nof\u00160\u0001H= 5 mT. The hysteresis loops are represented\nas the dependence of the average out-of-plane magneti-\nsation component hmzion the external magnetic \feld H.\nThe hysteresis loop for a 10 nm thin \flm disk sample with\nd= 80 nm diameter in which the incomplete Skyrmion\n(iSk) is the ground state is shown in Fig. 6 (a) as a solid\nline. Similarly, a solid line in Fig. 6 (b) shows the cor-\nresponding hysteresis loop for a larger disk sample with7\nFIG. 5. The ground state phase diagram in absence of demagnetisation energy contribution. The scalar value\nSaas a function of disk sample diameter dand external magnetic \feld Hcomputed for the ground state at every phase\nspace point in absence of demagnetisation energy contribution for ( a) a 3d mesh and ( c) for a 2d mesh. In order to better\nresolve the boundaries of the Helical (H) state region, the skyrmion number Sis shown in ( b) and ( d). (e) The magnetisation\ncon\fgurations of three identi\fed ground states as well as the out-of-plane magnetisation component mz(x) along the horizontal\nsymmetry line.\nd= 150 nm diameter in which the isolated Skyrmion (Sk)\nis the ground state. The hysteresis between two energeti-\ncally equivalent skyrmionic magnetisation states with the\nopposite core orientation at zero external magnetic \feld,\nshown in Fig. 6 (c), is evident. Moreover, the system does\nnot relax to any other equilibrium state at any point inthe hysteresis loop, which demonstrates the bistability of\nskyrmionic textures in studied system. The area of the\nopen loop in the hysteresis curve is a measure of the work\nneeded to reverse the core orientation by overcoming the\nenergy barrier separating the two skyrmionic states with\nopposite core orientation.8\nFIG. 6. Hysteresis loops and obtained zero-\feld skyrmionic states with di\u000berent orientations. The average\nout-of-plane magnetisation component hmzihysteretic dependence on the external out-of-plane magnetic \feld Hfor 10 nm thin\n\flm disk samples for ( a) incomplete Skyrmion (iSk) magnetisation con\fguration in d= 80 nm diameter sample and ( b) isolated\nSkyrmion (Sk) magnetisation con\fguration in d= 150 nm diameter sample. ( c) The magnetisation states and mz(x) pro\fles\nalong the horizontal symmetry lines for positive and negative iSk and Sk core orientations from H= 0 in the hysteresis loop,\nboth in presence and in absence of demagnetisation energy (demagnetisation-based shape anisotropy).\nAs throughout this work, it is assumed that the\nsimulated helimagnetic material is isotropic, and thus,\nthe magnetocrystalline anisotropy energy contribution\nis neglected. Due to that, one might expect that\nthe obtained hysteresis loops are the consequence of\ndemagnetisation-based shape anisotropy. To address\nthis, we simulate hysteresis using the same method,\nbut this time in absence of the demagnetisation en-\nergy contribution. More precisely, the minimalistic\nenergy model contains only the symmetric exchange\nand Dzyaloshinskii-Moriya interactions together withZeeman coupling to an external magnetic \feld. We\nshow the obtained hysteresis loops in Fig. 6 (a) and\n(b) as dashed lines. The hysteretic behaviour remains,\nalthough all energy terms that usually give rise to the\nhysteretic behaviour (magnetocrystalline anisotropy and\ndemagnetisation energies) were neglected. This suggests\nthe existence of a new magnetic anisotropy that we refer\nto as the Dzyaloshinskii-Moriya-based shape anisotropy.\nReversal mechanism. The hysteresis loops in Fig. 6\nshow that skyrmionic textures in con\fned thin \flm he-9\nlimagnetic nanostructures undergo hysteretic behaviour\nand that an external magnetic \feld can be used to change\ntheir orientation from core pointing up to core pointing\ndown and vice versa. In this section, we discuss the mech-\nanism by which the skyrmionic texture core orientation\nreversal occurs. We simulate a 150 nm diameter thin \flm\nFeGe disk sample with t= 10 nm thickness. The max-\nimum spacing between two neighbouring \fnite element\nmesh nodes is reduced to 1 :5 nm in order to better re-\nsolve the magnetisation \feld. According to the hysteresis\nloop in Fig. 6 (b), the switching \feld Hsof the isolated\nskyrmion state in this geometry from core orientation\ndown to core orientation up is \u00160Hs\u0019\u0000235 mT. There-\nfore, we \frst relax the system at \u0000210 mT external mag-\nnetic \feld and then decrease it abruptly to \u0000250 mT. We\nsimulate the magnetisation dynamics for 1 ns, governed\nby a dissipative LLG equation [36] with Gilbert damping\n\u000b= 0:3 [26], and record it every \u0001 t= 0:5 ps.\nWe now look at how certain magnetisation con\fgura-\ntion parameters evolve during the reversal process. We\nshow the time-dependent average magnetisation compo-\nnentshmxi,hmyi, andhmziin Fig. 7 (a), and on the\nsame time axis, the skyrmion number S, scalar value Sa\nand total energy Ein Fig. 7 (b). The initial magneti-\nsation con\fguration at t= 0 ns is denoted as A and the\n\fnal relaxed magnetisation at t= 1 ns as F. We show in\nFig. 7 (c) the out-of-plane magnetisation \feld component\nmzin the whole sample, in the xzcross section, as well\nas along the horizontal symmetry line. At approximately\n662 ps the skyrmionic core reversal occurs and Fig. 7 (b)\nshows an abrupt change both in skyrmion number Sand\ntotal energy E. We summarise the reversal process with\nthe help of six snapshots shown in Fig. 7 (c). Firstly,\nin (A-B), the isolated skyrmion core shrinks. At some\npoint the maximum mzvalue lowers from 1 to approxi-\nmately 0:1 (C). After that, the core reverses its direction\n(D) and an isolated skyrmion of di\u000berent orientation is\nformed (E). From that time onwards, the core expands in\norder to accommodate the size of hosting nanostructure,\nuntil the \fnal state (F) is reached. The whole reversal\nprocess is also provided in Supplementary Video 1.\nIn order to better understand the actual reversal of\nthe skyrmionic texture core between t1\u0019661 ps and\nt2\u0019663 ps, we show additional snapshots of the mag-\nnetisation vector \feld and mzcolourmap in the xzcross\nsection in Fig. 7 (d). The location marked by a cir-\ncle in subplots L, M, and N identi\fes a Bloch Point\n(BP): a noncontinuous singularity in the magnetisation\npattern where the magnetisation magnitude vanishes to\nzero [37, 38]. Because micromagnetic models assume con-\nstant magnetisation magnitude, the precise magnetisa-\ntion con\fguration at the BP cannot be obtained using\nmicromagnetic simulations [39]. However, it is known\nhow to identify the signature of the BP in such situations:\nthe magnetisation direction covers any su\u000eciently small\nclosed surface surrounding the BP exactly once [40, 41].We illustrate this property in Fig. 7 (e) using a vector plot\ntogether with mx,my, andmzcolour plots that show the\nstructure of a Bloch point. We conclude that the isolated\nskyrmion core reversal occurs via Bloch Point (BP) oc-\ncurrence and propagation. Firstly, at t\u0019661:5 ps the\nBP enters the sample at the bottom boundary and prop-\nagates upwards until t\u0019663 ps when it leaves the sample\nat the top boundary. In the Supplementary Video 2 the\nisolated skyrmion core reversal dynamics is shown.\nWe note that the Bloch point moves upwards in\nFig. 7 (d) but one may ask whether an opposite propaga-\ntion direction can occur and how the Bloch point struc-\nture is going to change. We demonstrate that which of\nthese two propagation directions will occur in the reversal\nprocess depends on the simulation parameters. The re-\nversal mechanism simulation was repeated with increased\nGilbert damping ( \u000b= 0:35 instead of \u000b= 0:3) and the\nresults showing the downwards propagation are shown in\nthe Supplementary Section S3. We hypothesise that both\nreversal paths (Bloch point moving upwards or down-\nwards) exhibit the same energy barriers and that the\nchoice of path is a stochastic process. By analysing the\nresults from Fig. 7 (d) and (e) and Supplementary Fig. 6,\nwe also observe that the change in the BP propagation\ndirection implies the change of the BP structure since the\nout-of-plane magnetisation component mz\feld reverses\nin the vicinity of BP.\nDISCUSSION\nThrough systematic micromagnetic study of equilib-\nrium states in helimagnetic con\fned nanostructures,\nwe identi\fed the ground states and reported the\n(meta)stability regions of other equilibrium states. We\ndemonstrated in Fig. 2 that skyrmionic textures in the\nform of incomplete Skyrmion (iSk) and isolated Skyrmion\n(Sk) con\fgurations are the ground states in disk nanos-\ntructures, and that this occurs in a wide d{Hparameter\nspace range. We have carried out similar studies for a\nsquare geometry and obtain qualitatively similar results.\nOf particular importance is that iSk and Sk states are the\nground states at zero external magnetic \feld which is in\ncontrast to in\fnite thin \flm and bulk helimagnetic sam-\nples. We note that neither an external magnetic \feld is\nnecessary nor magnetocrystalline anisotropy is required\nfor this stability. We also note in Fig. 4 (c) that there is\nsigni\fcant \rexibility in the skyrmionic texture size which\nprovides robustness for technology built on skyrmions,\nwhere fabrication of nanostructures and devices intro-\nduces unavoidable variation in geometries.\nWe have established that including the demagnetisa-\ntion interaction is crucial for the system investigated\nhere, i.e. in the absence of demagnetisation e\u000bects,\nthere are other magnetisation con\fgurations with en-\nergies lower than that of the incomplete and isolated10\nFIG. 7. The isolated skyrmion orientation reversal in con\fned three-dimensional helimagnetic nanostructure.\n(a) The spatially averaged magnetisation components hmxi,hmyi, andhmziand (b) skyrmion number S, scalar value Sa, and\ntotal energy Etime evolutions in the reversal process over 1 ns. The simulated sample is a 10 nm thin \flm disk with 150 nm\ndiameter. ( c) The magnetisation states at di\u000berent instances of time (points A to F) together with mzcolourmap in the xz\ncross section and mz(x) pro\fles along the horizontal symmetry line. ( d) Themzcolourmap and magnetisation \feld in the\ncentral part of xzcross section as shown in an inset together with the position of Bloch point (BP). ( e) The BP structure\nalong with colourmaps of magnetisation components which shows that the magnetisation covers the closed surface (sphere\nsurrounding the BP) exactly once.\nskyrmion. We also note that the translational variance\nof the magnetisation from the lower side of the thin \flm\n(atz= 0 nm) to the top (at z= 10 nm) is essential for\nthe physics reported here: if we use a two-dimensional\nmicromagnetic simulation (i.e. assuming translational in-\nvariance of the magnetisation min the out-of-plane di-\nrection), the isolated skyrmion con\fguration does not\narise as the ground state. Our interpretation is that for\nskyrmion-like con\fgurations the twist of mbetween topand bottom layer allows the system's energy to reduce\nsigni\fcantly while such a reduction is less bene\fcial for\nother con\fgurations such as helices; inline with recent\npredictions in the case of in\fnite thin \flms [27]. Ac-\ncordingly, we conclude that three-dimensional helimag-\nnetic nanostructure models, where demagnetisation en-\nergy contribution is neglected, or the geometry approx-\nimated using a two-dimensional mesh, are not generally\njusti\fed.11\nBecause of the speci\fc boundary conditions [28] and\nthe importance of including the demagnetisation en-\nergy contribution, our predictions cannot be directly ap-\nplied to other helimagnetic materials without repeating\nthe stability study. For instance, although the size of\nskyrmionic textures in this study was based on cubic\nFeGe helimagnetic material with helical period LD=\n70 nm, in order to encourage the experimental veri\fca-\ntion of our predictions, this study could be repeated\nfor materials with smaller LD. In such materials the\nskyrmionic core size is considerably reduced, which al-\nlows the reduction of hosting nanostructure size and is\nan essential requirement for advancing future informa-\ntion storage technologies. Similarly, the ordering tem-\nperature of simulated FeGe helimagnetic material, TC=\n278:7 K [42], is lower than the room temperature, which\nmeans that a device operating at the room temperature\ncannot be constructed using this material. Because of\nthat, in Supplementary Section S4, we demonstrate that\nour predictions are still valid if the ordering temperature\nof simulated B20 helimagnetic material is arti\fcially in-\ncreased to 350 K.\nWe demonstrate in Fig. 6 that skyrmionic textures in\ncon\fned helimagnetic nanostructures exhibit hysteretic\nbehaviour as a consequence of energy barriers between\nenergetically equivalent stable con\fgurations (skyrmionic\ntexture core pointing up or down). In the absence of\nmagnetocrystalline anisotropy and if the demagnetisa-\ntion energy (demagnetisation-based shape anisotropy) is\nremoved from the system's Hamiltonian, the hysteretic\nbehaviour is still present, demonstrating the existence of\na novel Dzyaloshinskii-Moriya-based shape anisotropy.\nFinally, we show how the reversal of the isolated\nskyrmion core orientation is facilitated by the Bloch point\noccurrence and propagation, and demonstrate that the\nBloch point can propagate in both directions along the\nout-of-plane z-direction.\nAll data obtained by micromagnetic simulations in this\nstudy and used to create \fgures both in the main text\nand in the Supplementary Information are included in\nSupplementary Data.\nMETHODS\nModel. We use an energy model consistent with a non-\ncentrosymmetric cubic B20 (P2 13 space group) crystal\nstructure. This is appropriate for a range of isostructural\ncompounds and pseudo-binary alloys in which skyrmionic\ntextures have been experimentally observed [3, 4, 15{\n18, 43, 44]. The magnetic free energy of the system E\ncontains several contributions and can be written in the\nform:\nE=Z\n[wex+wdmi+wz+wd+wa] d3r: (1)The \frst term is the symmetric exchange energy density\nwex=A\u0002\n(rmx)2+ (rmy)2+ (rmz)2\u0003\nwith exchange\nsti\u000bness material parameter A, wheremx,my, andmz\nare the Cartesian components of the vector m=M=Ms\nthat describes the magnetisation M, withMs=jMjbe-\ning the saturation magnetisation. The second term is\nthe Dzyaloshinskii-Moriya Interaction (DMI) energy den-\nsitywdmi=Dm\u0001(r\u0002m), obtained by constructing the\nallowed Lifshitz invariants for the crystallographic class\nT [12, 45], where Dis the material parameter. The third\nterm is the Zeeman energy density term wz=\u0000\u00160H\u0001M\nwhich de\fnes the coupling of magnetisation to an exter-\nnal magnetic \feld H. Thewdterm represents the de-\nmagnetisation (magnetostatic) energy density. The last\ntermwais the magnetocrystalline anisotropy energy den-\nsity, and because the simulated material is assumed to be\nisotropic, we neglect it throughout this work. Neglecting\nthis term also allows us to determine whether the magne-\ntocrystalline anisotropy is a crucial mechanism allowing\nthe stability of skyrmionic textures in con\fned helimag-\nnetic nanostructures.\nThe Landau-Lifshitz-Gilbert (LLG) equation [36]:\n@m\n@t=\r\u0003m\u0002He\u000b+\u000bm\u0002@m\n@t; (2)\ngoverns the magnetisation dynamics, where\n\r\u0003=\r(1+\u000b2), with\r <0 and\u000bbeing the gyromagnetic\nratio and Gilbert damping, respectively. We compute the\ne\u000bective magnetic \feld using He\u000b=\u0000(\u000ew=\u000em)=(\u00160Ms),\nwherewis the total energy density functional. With this\nmodel, we solve for magnetic con\fgurations musing the\ncondition of minimum torque arrived by integrating a set\nof dissipative, time-dependent equations. We validated\nthe boundary conditions by a series of simulations\nreproducing the results in Ref. [26, 28].\nSimulator. We developed a micromagnetic simulation\nsoftware, inspired by the Nmag simulation tool [46, 47].\nUnlike Nmag, we use the FEniCS project [48] instead of\nthe Nsim multi-physics library [46] for the \fnite element\nlow-level operations. In addition, we use IPython [49, 50]\nand Matplotlib [51, 52] extensively in this work.\nMaterial parameters. We estimate the material\nparameters in our simulations to represent the cu-\nbic B20 FeGe helimagnet with four Fe and four Ge\natoms per unit cell [53] and crystal lattice constant\na= 4:7\u0017A [54]. The local magnetic moments of iron\nand germanium atoms are 1 :16\u0016Band\u00000:086\u0016B[55],\nrespectively, where \u0016Bis the Bohr magneton constant.\nAccordingly, we estimate the saturation magnetisation\nasMs= 4N(1:16\u00000:086)\u0016B= 384 kA m\u00001, with\nN=a\u00003being the number of lattice unit cells in a cubic\nmetre. The spin-wave sti\u000bness is Dsw=a2TC[56], where\nthe FeGe ordering temperature is TC= 278:7 K [42].\nConsequently, the exchange sti\u000bness parameter value12\nisA=DswMs=(2g\u0016B) = 8:78 pJ m\u00001[57], where\ng\u00192 is the Land\u0013 e g-factor. The estimated DMI\nmaterial parameter Dfrom the long-range FeGe helical\nperiodLD= 70 nm [42], using LD= 4\u0019A=jDj[43], is\njDj= 1:58 mJ m\u00002.\nSkyrmion number Sand injective scalar value\nSa.In order to support the discussion of skyrmionic tex-\ntures, the topological skyrmion number [2]\nS2D=1\n4\u0019Z\nm\u0001\u0012@m\n@x\u0002@m\n@y\u0013\nd2r; (3)\ncan be computed for two-dimensional samples hosting\nthe magnetisation con\fguration. However, for con\fned\nsystems, the skyrmion number S2Dis not quantised into\nintegers [26, 31], and therefore, a more suitable name for\nS2Dmay be the \\scalar spin chirality\" (and consequently\nthe expression under an integral would be called the \\spin\nchirality density\"), but we will follow the existing liter-\nature [26, 31] and refer to S2Das the skyrmion number.\nWe show its dependence on di\u000berent skyrmionic textures\nthat can be observed in con\fned helimagnetic nanos-\ntructures in Supplementary Fig. 2 (b), demonstrating\nthat the skyrmion number in con\fned geometries is not\nan injective function since it does not preserve distinct-\nness (one-to-one mapping between skyrmionic textures\nand skyrmion number value S2D). Therefore, for two-\ndimensional samples, we de\fne a di\u000berent scalar value\nS2D\na=1\n4\u0019Z\f\f\f\fm\u0001\u0012@m\n@x\u0002@m\n@y\u0013\f\f\f\fd2r; (4)\nand show its dependence on di\u000berent skyrmionic textures\nin Supplementary Fig. 2 (b). This scalar value is injective\nand provides necessary distinctness between S2D\navalues\nfor di\u000berent skyrmionic states. In terms of the termi-\nnology discussion above regarding S2D, the entity S2D\na\ndescribes the \\scalar absolute spin chirality\". We also\nemphasise that although the skyrmion number S2Dhas\na clear mathematical [58] and physical [59] interpreta-\ntion, we de\fne the arti\fcial injective scalar value Saonly\nto support the classi\fcation and discussion of di\u000berent\nskyrmionic textures observed in this work.\nSkyrmion number S2Dand arti\fcially de\fned scalar\nvalueS2D\na, given by Eq. (3) and Eq. (4), respectively,\nare valid only for the two-dimensional samples hosting\nthe magnetisation con\fguration. However, in this work,\nwe also study three-dimensional samples and, because of\nthat, we now de\fne a new set of expressions taking into\naccount the third dimension. The skyrmion number in\nthree-dimensional samples S3Dwe compute using\nS3D=1\n8\u0019Z\nm\u0001\u0012@m\n@x\u0002@m\n@y\u0013\nd3r; (5)\nas suggested by Lee et al. [60], which results in a value\nproportional to the anomalous Hall conductivity. 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Phys. 8, 301{304 (2012).\n[60] Lee, M., Kang, W., Onose, Y., Tokura, Y. & Ong, N. Un-\nusual Hall e\u000bect anomaly in MnSi under pressure. Phys.\nRev. Lett. 102, 186601 (2009).\nACKNOWLEDGEMENTS\nThis work was \fnancially supported by the EPSRCs\nDoctoral Training Centre (DTC) grant EP/ G03690X/1.\nR.L.S. acknowledges the EPSRCs EP/M024423/1 grant\nsupport. D.C.-O. acknowledges the \fnancial support\nfrom CONICYT Chilean scholarship programme Becas\nChile (72140061). We acknowledge the use of the IRIDIS\nHigh Performance Computing Facility, and associated\nsupport services at the University of Southampton, in the\ncompletion of this work. We also thank Karin Everschor-\nSitte for helpful discussions.\nAUTHOR CONTRIBUTIONS\nM.B. and H.F conceived the study, and M.B. per-\nformed micromagnetic simulations. R.L.S. devised the\nanalytic model and discussed its implications. R.C. con-\ntributed to the simulations and analysis of equilibrium\nstates. D.C., M.-A.B., M.A., W.W., M.B., R.C., M.V.,\nD.C.-O. and H.F. developed the micromagnetic \fnite el-\nement based simulator. M.V. and M.A. enabled running\nsimulations on IRIDIS High Performance Computing Fa-\ncility. M.B., H.F., R.L.S., and O.H. interpreted the data\nand prepared the manuscript.\nCOMPETING FINANCIAL INTERESTS\nThe authors declare no competing \fnancial interests.Supplementary Information: Ground state search, hysteretic behaviour, and reversal\nmechanism of skyrmionic textures in helimagnetic nanostructures\nMarijan Beg,1,∗Rebecca Carey,1Weiwei Wang,1David Cort´ es-Ortu˜ no,1Mark Vousden,1Marc-Antonio Bisotti,1\nMaximilian Albert,1Dmitri Chernyshenko,1Ondrej Hovorka,1Robert L. Stamps,2and Hans Fangohr1,†\n1Faculty of Engineering and the Environment, University of Southampton, Southampton SO17 1BJ, United Kingdom\n2SUPA School of Physics and Astronomy, University of Glasgow, Glasgow G12 8QQ, United Kingdom\nSUPPLEMENTARY SECTION S1: INITIAL MAGNETISATION CONFIGURATIONS\nThe magnetisation configurations that we use as initial states for the full three-dimensional micromagnetic simu-\nlations are shown in Supplementary Fig. 1. For every point in the d–Hparameter space, we relax twelve different\ninitial magnetisation configurations. These are the five skyrmionic configurations (A, B, C, D, and E), three helical\nconfigurations (H2, H3, and H4), the uniform configuration (U), and three random magnetisation configurations (R).\nNow, we introduce an approximate analytic model that helps us generate a range of physically meaningful and\nreproducible initial skyrmionic magnetisation configurations labelled A-E in Supplementary Fig. 1. The used DMI\nenergy density term (see Methods section of the main text) is consistent with the helimagnetic material of crystallo-\ngraphic class T, and one expects a skyrmionic texture configuration with no radial spin component (chiral skyrmion)\nto emerge [1]. Consequently, if we consider a two-dimensional disk sample of radius Rin the plane containing the\nxandyaxes, as shown in Supplementary Fig. 2 (b) inset, an approximation of the chiral skyrmionic magnetisation\ntexture (for D> 0), in cylindrical coordinates ( ρ,ϕ,z), is\nmρ= 0,\nmϕ= sin(kρ),\nmz=−cos(kρ),(1)\nFIG. 1. The magnetisation configurations relaxed using the full three-dimensional micromagnetic simulations.\nThe configurations labelled A-E correspond to the first five solutions of the approximate analytic model, whereas the three\nhelical states with 2, 3, and 4 helical half-periods are labelled as H2, H3, and H4, respectively. The uniform out-of-plane\nmagnetisation state is labelled as U, and an example of random magnetisation state is marked with R.2\nwherek= 2π/sis a measure of the skyrmionic texture size s.\nAn equilibrium configuration requires that the torque exerted on the magnetisation mvanishes at every point in\nsample ( m×Heff= 0), including the boundary. The effective field functional Heff=−(δw/δm)/(µ0Ms), due to only\nsymmetric exchange and DMI energy contributions, in absence of an external magnetic field is\nHeff=2\nµ0Ms/bracketleftbig\nA∇2m−D(∇×m)/bracketrightbig\n, (2)\nfor any magnetisation texture. Computing this expression for the radially symmetric approximate skyrmionic texture\nmodel m= sin(kρ)ˆϕ−cos(kρ)ˆ zresults in\nHeff=2\nµ0Ms/bracketleftbigg/parenleftbigg\nDk−Ak2−A\nρ2/parenrightbigg\nsin(kρ) +Ak\nρcos(kρ)/bracketrightbigg\nˆϕ+\n2\nµ0Ms/bracketleftbigg/parenleftbiggAk\nρ−D\nρ/parenrightbigg\nsin(kρ) +/parenleftbig\nAk2−Dk/parenrightbig\ncos(kρ)/bracketrightbigg\nˆ z.(3)\nConsequently, the torque exerted on the magnetisation mis\nm×Heff=2\nµ0Ms/bracketleftbiggAk\nρ−D\nρsin2(kρ)−A\n2ρ2sin(2kρ)/bracketrightbigg\nˆ r. (4)\nRequiring the torque to vanish at the disk boundary ρ=Rresults in the zero-torque condition:\ng(kR)≡−Psin2(kR)−sin(2kR)\n2kR+ 1 = 0, (5)\nwhereP=D/kA . The analysis of this condition shows that the parameter Pmust satisfy the inequality P > 2/3\nin order for g(kR) to have roots and, thus, a skyrmionic texture core to exist in at least metastable equilibrium.\nIn Supplementary Fig. 2 (a), we plot the zero-torque condition as a function of kRfor different values of P. Since\nP= 2/3 is the boundary case (dotted line in the plot), all plots for P < 2/3 have no solutions, whereas the zero-torque\ncondition has multiple solutions if P > 2/3.\nThe skyrmionic magnetisation configurations used as initial states in the energy minimisation process, corresponding\nto the zero-torque condition solutions (marked A-F) for P= 2, we show in Fig. 2 (c). In order to support the discussion\nof these magnetisation configurations, we compute the skyrmion number S, which for our approximate analytic model\nresults inS= (cos(kR)−1)/2. Its dependence on kR, presented in Fig. 2 (b) as a dashed line, shows that the\nskyrmion number for the skyrmionic textures in confined nanostructures is not an injective function since it does not\npreserve distinctness (one-to-one mapping between kRand skyrmion number value S). Therefore, a different scalar\nvalueSais computed and from its dependence on kR, shown in Fig. 2 (b) as a solid line, we conclude that this scalar\nvalue is injective and provides necessary distinctness between Savalues for different skyrmionic states. The details\nabout skyrmion number Sand scalar value Saare in the Methods section of the main text.\nFor the helical state, which emerges as a consequence of the Dzyaloshinskii-Moriya interaction considered in this\nwork, we expect that the magnetisation vector at any point is perpendicular to the helical propagation direction\n(Bloch-wall-like configuration). Consequently, if both xandyaxis are in the plane of the thin-film sample and the x\naxis is chosen as a propagation direction, the helical magnetisation configuration in Cartesian coordinates is\nmx= 0,\nmy= cos(khx),\nmz= sin(khx),(6)\nwherekh= 2π/λ, withλbeing the helical period [2].\nNow, we investigate whether the helical period λin confined nanostructures is independent on the sample diam-\neter, and if not, what are the helical period values that can occur in our simulated system. After relaxing helical\nconfigurations and varying both the helical period and disk sample diameter (up to 180 nm), we find that all relax to\na limited set of helical states with different helical periods. More precisely, the observed relaxed helical states consist\nof either 2, 3, or 4 helical half-periods along the disk sample diameter, including the characteristic magnetisation\ntilting at the boundary [3]. Thus, we define three different helical magnetisation configurations as initial states with\nhelical periods 2 d/2, 2d/3, or 2d/4, wheredis the disk sample diameter, and are named H2, H3, and H4, respectively.\nMagnetisation configurations of these states, together with their mz(x) profiles along the horizontal symmetry, are\nshown in Supplementary Fig. 1.3\nFIG. 2. Zero-torque condition plots and magnetisation configurations corresponding to its solutions. (a) The\nzero-torque condition g(kR) plots, given by Eq. (5), for different values of P=D/kA . All zero-torque conditions for Psmaller\nthan the boundary case P= 2/3 (dotted line) have no solutions, whereas for P > 2/3, multiple solutions (A-F) exist. ( b) The\nnon-injective dependence of skyrmion number Sand the injective dependence of scalar value SaonkR. (c) The magnetisation\nconfigurations and the out-of-plane magnetisation component mz(x) along the horizontal symmetry line for different solutions\n(A-F) of Eq. (5) for P= 2.\nIn addition to the previously defined eight chiral initial states, we also use the uniform magnetisation configuration,\nwhere the magnetisation at all mesh nodes is in the positive out-of-plane z-direction, as shown in Supplementary Fig. 1\nmarked as U. Finally, in order to capture other equilibrium states that cannot be obtained by relaxing previously\ndescribed well-defined magnetisation configurations, we also use additional three random magnetisation configurations.\nAt every mesh node, we choose three random numbers in the [ −1,1] range for three magnetisation components and\nthen normalise them in order to fulfil the |m|= 1 condition. An example of one random magnetisation configuration\nis shown in Supplementary Fig. 1 and labelled as R.\nSUPPLEMENTARY SECTION S2: RELAXATION DIAGRAMS\nIn this section, we compute the equilibrium states at all points in the d–Hparameter space obtained by relaxing\nwell-defined initial states, introduced in the Supplementary Section S1. More precisely, we compute the equilibrium\nstates (local energy minima) that result from a particular initial condition. This allows us to provide a systematic\noverview of equilibrium states, and gain additional insight about the phase space energy landscape in the studied\nsystem. We vary the sample diameter between 40 nm and 180 nm in steps of ∆ d= 4 nm and the external magnetic field\nbetweenµ0H= 0 T andµ0H= 1.2 T in steps of µ0∆H= 20 mT. Relaxation diagrams are represented by plotting\nthe scalar value Saas a function of disk diameter dand applied field strength H, and Supplementary Fig. 3 shows\nthe relaxation diagrams for skyrmionic initial configurations A-E, and Supplementary Fig. 4 shows the relaxation\ndiagrams for helical H2, H3, H4, and uniform U initial configurations. The phase diagram of equilibrium states,\nshown in Fig. 1 in the main text, summarises the main results from the relaxation diagrams presented and discussed\nhere.\nWe now discuss each of relaxation diagrams in Supplementary Fig. 3 and 4, where each subplot corresponds to one\nparticular initial configuration. The scalar value Sa, as a function of disk sample diameter dand external magnetic field4\nFIG. 3. The relaxation diagrams obtained by relaxing skyrmionic initial state A-E. The initial states correspond to\nthe first five solutions of the analytic model and the phase diagrams are marked A-E accordingly. The relaxation diagrams are\nrepresented as the dependence of scalar value Saon the disk sample diameter dand an external field H(as shown in insets).\nH, computed for the final relaxed equilibrium state, i.e. local or global energy minimum, we show in Supplementary\nFig. 3 (A) for the energy minimisation process that started from the skyrmionic initial configuration A (shown in\nSupplementary Fig. 1). The “iSk ↑” label refers to the incomplete Skyrmion (iSk) magnetisation configuration with\nthe core magnetisation pointing in the positive z-direction. An example of this state, marked with the same label, is\nshown in Supplementary Fig. 5. From the Supplementary Fig. 3 (A), we can see that for all examined diameters and\nexternal field values, the final relaxed configuration is the iSk ↑state. Now, this relaxation diagram is compared with5\nFIG. 4. The relaxation diagrams obtained by relaxing helical and uniform initial state. The initial states correspond\nto the helical states H2, H3, and H4, as well as the uniform state U, and the relaxation diagrams are marked accordingly. The\nrelaxation diagrams are represented as the dependence of scalar value Saon the disk sample diameter dand an external field\nH(as shown in insets).\nthe ground state phase diagram (see Fig. 2 (a) in the main text) and the boundary between two ground state (iSk\nand Sk) regions is shown as the dashed line in the discussed relaxation diagram. We can see that the iSk is indeed\nthe ground state (i.e. the global energy minimum) for d <140 nm, but for larger diameters the isolated Skyrmion\n(Sk) configuration has a lower energy, and thus the iSk configuration is only a local energy minimum.\nSimilarly, if the energy minimisation process is started from the initial configuration B, shown in Supplementary\nFig. 1 (B), the Sa(d,H) for the final relaxed magnetisation state is obtained and shown in Supplementary Fig. 3 (B).\nThe vast majority of the final configurations ( d≥80 nm and µ0H<∼1.1 T), labelled as “Sk ↓”, correspond to the\nisolated skyrmion state with the core pointing in the negative z-direction. An example of this state we show in\nSupplementary Fig. 5, marked with the same label. By comparison with the ground state phase diagram shown in\nFig. 2 (a) in the main text (dashed line), we can see that the Sk is the ground state for large diameters d≥140 nm and\nfield values µ0H < 0.7 T. However, for smaller diameters the isolated skyrmion configuration is only metastable (as\nthe iSk configuration is the ground state). We can see that in the vicinity of d≈60 nm andµ0H≈0.1 T parameter\nspace point, the initial configuration B relaxes to the incomplete skyrmion state with core oriented in the negative z-\ndirection (iSk↓), but for larger field values, configuration B falls into the iSk ↑configuration (see Supplementary Fig. 5\nfor detailed configurations of iSk ↓and iSk↑states). The iSk↓has a higher energy than the iSk ↑as the majority of the\nmagnetisation is pointing in the direction opposite to the applied field. However, the initial configuration B is such\nthat the core is pointing down, and there appears to be a direct energy minimisation path to the iSk ↓configuration\nfor field values smaller than approximately 0 .3 T. For larger fields, the Zeeman energy becomes so important that the\ninitial configuration B leads to the the iSk ↑configuration.\nSupplementary Fig. 3 (C) shows that the initial configuration C suppresses the iSk ↓state completely but is otherwise6\nFIG. 5. The identified equilibrium magnetisation configurations.\nsimilar to Supplementary Fig. 3 (B). We note in particular that the Sk configuration cannot exist for d<60 nm even\nif the relaxation is started from a Sk-like configuration B or C, i.e. there are no metastable isolated skyrmion states\nat the smallest diameters.\nIf the system is relaxed from the initial state D, shown in Supplementary Fig. 1 (D), a qualitative change from\nSupplementary Fig. 3 (B) and (C) is evident as shown in Supplementary Fig. 3 (D). In addition to the iSk ↑and Sk↓\nstates, there are now a number of, according to Fig. 3 in the main text, higher energy metastable states emerging as\n2 or 3 skyrmions in the disk (see Supplementary Fig. 5 for detailed plots of 2Sk and 3Sk states). Furthermore, for\nsmall field values µ0H<∼0.2 T and large diameters d>∼152 nm, the Target (T) equilibrium state with core orientation\nin the negative z-direction (T↓) arises. The T↓state is shown in Supplementary Fig. 5.\nThe scalar value Sa(d,H) computed for final equilibrium configurations in Supplementary Fig. 3 (E) we obtained\nby relaxing the initial configuration E shown in Supplementary Fig. 1. The initial skyrmionic state E does not relax\nto 2 and 3 skyrmion configurations but allows the Sk ↑state to arise for small field values.\nSupplementary Fig. 4 (H2), (H3) and (H4), show the Sa(d,H) for starting configurations H2, H3 and H4, re-\nspectively, as shown in Supplementary Fig. 1. All three initial configurations result in the incomplete Skyrmion\nconfiguration with core pointing up (iSk ↑) for the smallest diameters as well as for largest fields. The H3 initial7\nFIG. 6. The isolated skyrmion orientation reversal in confined three-dimensional helimagnetic nanostructure\nwith downwards Bloch point propagation direction. (a) The mzcolourmap and magnetisation field in the central part\nofxzcross section as shown in an inset together with the position of Bloch Point (BP). ( b) The BP structure along with\ncolourmaps of magnetisation components which shows that the magnetisation covers the closed surface (sphere surrounding\nthe BP) exactly once.\nconfiguration relaxes into a configuration with two Skyrmions in the disk (2Sk) for d > 120 nm and field values\nbetween 0.8 T and 1.1 T. These 2Sk configurations had appeared occasionally when starting from configuration D\n(see Supplementary Fig. 3 (D)). The H4 initial configuration also encourages 3 skyrmions in the disk to arise as a\nmetastable state.\nSupplementary Fig. 5 (U) shows Safor final configurations when the simulation starts from a uniform magnetisation,\npointing up in the positive z-direction. This results mostly in the incomplete Skyrmion configuration with core pointing\nup (iSk↑). However, we also find the Skyrmion with core pointing down Sk ↓and the Target configuration T ↑as the\ndiameter increases and the field decreases. Supplementary Fig. 3 (A) is interesting to compare with Supplementary\nFig. 4 (U): in the former, only the iSk ↑state results, presumably because from the initial state A, only the iSk ↑state\nis accessible in the relaxation. On the contrary, for the uniform configuration, the system finds the energy minimum\nfor the isolated Skyrmion state Sk ↓and the Target T ↑because other energy minima can be accessed from this initial\nstate. Fig. 5 (a) in the main text shows the relative energies of the different metastable states for H= 0.\nSUPPLEMENTARY SECTION S3: DIFFERENT BLOCH POINT PROPAGATION DIRECTION\nIn order to illustrate a different direction of the Bloch Point (BP) propagation, we show a result from another\nskyrmion reversal. The simulation parameters are the same as in Fig. 7 of the main text, except that the Gilbert\ndampingαis increased from 0 .3 to 0.35. We show the results of isolated skyrmion core orientation reversal dynamics\nwith modified Gilbert damping in Supplementary Fig. 6.\nNow, the obtained reversal dynamics is compared with the reversal dynamics shown in Fig. 7 in the main text. From\nthe Supplementary Fig. 6 (a), we can see that the Bloch point enters the sample at the top boundary at approximatelly\n684 ps, then propagates downwards to the bottom boundary, where it leaves the sample at approximatelly 685 ps.\nBecause of the opposite BP propagation direction, the structure of the Bloch Point changes (compare Fig. 7 (e) in\nthe main text with Supplementary Fig. 6 (b)). More precisely, the out-of-plane magnetisation component mzfield in\nthe vicinity of BP is changed so that for the upper half of BP mz>0, whereas in the lower half mz<0.\nSUPPLEMENTARY SECTION S4: HIGHER ORDERING TEMPERATURE MATERIAL\nThe ordering temperature of simulated FeGe material, TC= 278.7 K [4], is lower than the room temperature,\nwhich means that this material cannot be used to fabricate a device operating at room temperature. Therefore, it\nis important to determine how our results regarding the identified lowest energy state would change for the material\nwith higher ordering temperature. Because no high ordering temperature helical B20 material has been reported to\nthis day, the best we can do is to artificially increase the ordering temperature, estimate new material parameters,\nand repeat the study of equilibrium states.8\nFIG. 7. The energy density differences between all identified equilibrium states and corresponding lowest energy\nstate as a function of disk sample diameter. A full set of initial state configurations are relaxed for different disk sample\ndiameter values at zero external magnetic field. The helimagnetic material ordering temperature is artificially increased to\nTC= 350 K (above room temperature).\nWe increase the ordering temperature to TC= 350 K, and calculate new values of exchange and Dzyaloshinskii-\nMoriya energy constants (following the estimation described in Methods section of the main text) and obtain A=\n1.1×10−11J/m andD= 1.98×10−3J/m2. Using these values, we relax the full set of initial magnetisation\nconfigurations at zero external magnetic field for different disk sample diameters. More precisely, we vary the disk\nsample diameter between 40 nm and 180 nm in steps of 4 nm and compute the energy density E/V of all identified\nrelaxed equilibrium states, where Vis the sample volume. After that, from the computed energy density, we subtract\nthe energy density of the corresponding lowest energy state. We plot the calculated energy density differences of all\nequilibrium states as a function of disk sample diameter dand show them in Supplementary Fig. 7.\nBy comparing the Supplementary Fig. 7 for higher ordering temperature material with Fig. 3 in the main text, we\nconclude that the incomplete Skyrmion (iSk) state remains being the lowest energy state for d<140 nm, whereas the\nisolated Skyrmion (Sk) state is the lowest energy state for d≥140 nm. However, the iSk state is not in equilibrium for\ndisk sample diameters larger than 172 nm, which is in contrast to the FeGe material case where iSk is in equilibrium\nfor all examined dvalues. Another difference is that the metastable Target (T) state is the highest energy state for\nthe wholedrange where it is in equilibrium. Finally, we do not observe H4 state (helical state with four half periods)\nfor this high ordering temperature material.\nSUPPLEMENTARY REFERENCES\n∗mb4e10@soton.ac.uk\n†h.fangohr@soton.ac.uk\n[1] R¨ oßler, U. K., Bogdanov, A. N. & Pfleiderer, C. Spontaneous skyrmion ground states in magnetic metals. Nature 442,\n797–801 (2006).\n[2] Bak, P. & Jensen, M. H. Theory of helical magnetic structures and phase transitions in MnSi and FeGe. J. Phys. C: Solid\nSt. Phys. 13, L881-L885 (1980).\n[3] Rohart, S. & Thiaville, A. Skyrmion confinement in ultrathin film nanostructures in the presence of Dzyaloshinskii-Moriya\ninteraction. Phys. Rev. B 88, 184422 (2013).\n[4] Lebech, B., Bernhard, J. & Freltoft, T. Magnetic structures of cubic FeGe studied by small-angle neutron scattering. J.\nPhys.: Condens. Matter 1, 6105-6122 (1989)."    },    {        "title": "1402.2684v1.Weak_spin_interactions_in_Mott_insulating_La2O2Fe2OSe2.pdf",        "content": "Weak spin interactions in Mott insulating La 2O2Fe2OSe 2\nE. E. McCabe,1, 2C. Stock,3E. E. Rodriguez,4A. S. Wills,5J. W. Taylor,6and J. S. O. Evans1\n1Department of Chemistry, Durham University, Durham DH1 3LE, UK\n2School of Physical Sciences, University of Kent, Canterbury, CT2 7NH, UK\n3School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK\n4Department of Chemistry of Biochemistry, University of Maryland, College Park, MD, 20742, U.S.A.\n5Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK\n6ISIS Facility, Rutherford Appleton Labs, Chilton Didcot, OX11 0QX, UK\n(Dated: September 2, 2021)\nIdentifying and characterizing the parent phases of iron-based superconductors is an impor-\ntant step towards understanding the mechanism for their high temperature superconductivity. We\npresent an investigation into the magnetic interactions in the Mott insulator La 2O2Fe2OSe 2. This\niron oxyselenide adopts a 2- kmagnetic structure with low levels of magnetic frustration. This mag-\nnetic ground state is found to be dominated by next-nearest neighbor interactions J2andJ2/primeand the\nmagnetocrystalline anisotropy of the Fe2+site, leading to 2D-Ising-like spin S=2 fluctuations. In\ncontrast to calculations, the values are small and confine the spin excitations below ∼25 meV. This\nis further corroborated by sum rules of neutron scattering. This indicates that superconductivity in\nrelated materials may derive from a weakly coupled and unfrustrated magnetic structure.\nThe discovery of iron-based superconductivity at high\ntemperatures in pnictide [1] and chalcogenide [2] systems\nhighlights the importance of magnetism in high- Tcsu-\nperconductivity. [3] Despite the similar phase diagrams\nand the proximity of magnetism to superconductivity re-\nported for both the cuprate and iron-based superconduc-\ntors, these materials otherwise seem remarkably differ-\nent: the cuprate systems are based on doping a strongly-\ncorrelated Mott insulating state, [4] while the parent\nphases for the iron-based materials are either metallic,\nsemiconducting, or semimetallic. [5–7] However, recent\nwork has revealed electron correlation effects in iron pnic-\ntides suggesting that the iron-based systems may be close\nto the Mott boundary, yet a strongly correlated parent\ncompound has not been clearly identified for chalcogenide\nand pnictide superconductors. [8, 9] Also, the spin state\nof the Fe2+in these systems is not understood with dif-\nferent theories suggesting S=1 or 2 ground states [10–12].\nIn this paper, we investigate the magnetic interactions in\nthe Mott insulating iron oxyselenide La 2O2Fe2OSe 2.\nThis layered material (Fig. 1 a) adopts a tetragonal\ncrystal structure composed of fluorite-like [La 2O2]2+lay-\ners and [Fe 2O]2+sheets that are separated by Se2−an-\nions. The [Fe 2O]2+sheets adopt an unusual anti-CuO 2\narrangement with Fe2+cations coordinated by two in-\nplane oxygens and four Se2−anions above and below the\nplane, leading to layers of face-shared FeO 2Se4trans oc-\ntahedra. [13] The Fe grid is similar to that in LaFeAsO\nand FeSe, has similar ∼90◦Fe-Se-Fe interactions but\ncontains additional in-plane O2−ions.\nLa2O2Fe2OSe 2has been described as a Mott insula-\ntor and theoretical work suggests that it is more strongly\ncorrelated than LaFeAsO. [14] La 2O2Fe2OSe 2orders an-\ntiferromagnetically (AFM) below ∼90 K [15] and two\nmagnetic structures have been discussed for the [Fe 2O]2+\nlayers: a collinear model (Fig. 1 b) similar to that re-ported for Fe 1+xTe; [16, 17] and the 2- kmodel (Fig. 1 c)\nfirst proposed for Nd 2O2Fe2OSe 2. [18] These two models\nare indistinguishable from powder diffraction work and in\nthe absence of single crystals of sufficient size and quality,\nthis ambiguity has not been resolved. We present exper-\nimental results here that favour the 2- kmodel proposed\nby Fuwa et al [18] and hope to resolve this ambiguity.\nThe related pnictide and chalcogenide parent com-\npounds have been the subject of theoretical and experi-\nmental studies. Analogous to the cuprates, the spin ex-\nchange constants and spin-wave dispersions in these par-\nent compounds are large, extending up to energy trans-\nfers of ∼100 meV, reflecting strong Fe-Fe coupling. [19–\n24] Electronic structure calculations for La 2O2Fe2OSe 2\nsuggest similar exchange constants to the pnictides but\nwith considerable electronic band narrowing [14]. Until\nnow, neutron inelastic measurements to corroborate such\npredictions have not been reported for La 2O2Fe2OSe 2.\nWe present a combined study of the magnetic struc-\nture and fluctuations to understand the interactions in\nLa2O2Fe2OSe 2using neutron powder diffraction (NPD)\nand inelastic measurements. Full experimental details\nare provided in the supplementary information.\nWe first discuss the elastic magnetic scattering near\nTN(∼89 K) (Fig. 1 e). A broad, low intensity, asym-\nmetric Warren-like peak develops between 103 and 91 K\ncentered at ∼37◦2θ, characteristic of 2D short-ranged\nordering. [25] Fitting with a Warren function gives a 2D\ncorrelation length of ∼23˚A at 103 K that increases to\n∼90˚A (about 20 times the in-plane cell parameter) just\naboveTN. BelowTN, magnetic Bragg reflections appear\nwith the most intense peak at 2 θ∼38◦, such that any\nremaining diffuse scatter becomes hard to fit.\nMagnetic Bragg reflections appear below TN, to which\nthe 2-k(Fig. 2) and collinear spin models give indistin-\nguishable fits. In contrast to the report on Sr 2F2Fe2OS2,arXiv:1402.2684v1  [cond-mat.str-el]  11 Feb 20142\nb) Collinear c) 2-ka) La2O2Fe2OSe2\na\nb\nd)\nLa2O2Fe2OSe2\nLa2O2Mn2OSe2\ne) 91.2 K\n88.2 K\nFIG. 1. [color online] a) nuclear cell of La 2O2Fe2OSe 2b)\ncollinear model and c) 2- kmodel with the three intrapla-\nnar exchange interactions J1,J2andJ2/primeshown; d) evolution\nof magnetic moment for La 2O2Fe2OSe 2and La 2O2Mn2OSe 2\n(Ref. 26) with with M0Fe= 3.701(8) µB,TN=89.50(3) K\nandβFe=0.122(1); M0Mn= 4.5(2)µB,TN= 168.1(1) K and\nβMn= 0.24(3) and e) shows narrow 2 θrange of raw NPD\ndata for La 2O2Fe2OSe 2collected at 91.2 K and at 88.2 K,\nthe Warren-type peak shown by solid blue line.\nthere is no difference in the magnitude of Fe moments for\nthese two models. [27] The magnetic Bragg reflections ob-\nserved for La 2O2Fe2OSe 2are anisotropically broadened\nsimilar to Sr 2F2Fe2OS2, suggesting that both have simi-\nlar magnetic microstructures. This peak broadening can\nbe described by an expression for antiphase boundaries\nperpendicular to the c-axis [28] (Fig. 2 c) with a magnetic\ncorrelation length ξc(T= 2K)=45(3) ˚A that is essentially\nindependent of temperature ( ξc(T= 88K)=42(6) ˚A ). No\nsuch peak broadening has been reported for the Mn2+\nand Co2+analogues. [26, 29, 30]\nSequential NPD Rietveld refinements indicate a\nsmooth increase in the ordered Fe2+moment on cooling.\nThis magnetic order parameter is shown in Fig. 1 d) with\ncritical exponent βFe=0.122(1), similar to the 2D-Ising\nlike behavior of La 2O2Co2OSe 2and BaFe 2As2. [30, 31].\nThis is in contrast to the Mn analogue with an exponent\nβ=0.24(3) (Fig. 1 d)) reflecting greater 3D-like charac-\nter. [26] The ordered Fe2+moment in La 2O2Fe2OSe 2\ndetermined from our Rietveld refinements (3.50(5) µB\nat 2 K) is larger than that reported previously ( ∼2.8\n2ɽ / ° \n65 60 55 50 45 40 35 30 25 20 15 1080\n60\n40\n20\n0La2O2Fe2OSe2 84.20 %\nStructure 15.80 %\n8\n6\n4\n2\n0La2O2Fe2OSe2 84.20 %\nStructure 15.80 %\n \n50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 168\n6\n4\n2\n0La2O2Fe2OSe2 84.20 %\nmagnetic 15.80 %\n \n2ɽ / ° \nRwp = 6.976%\u0003;&Ƶůů\u0003ƉĂƩĞƌŶͿ\n28 independent parameters  \nRwp = 6.976%\u0003;EŽ\u0003ĚŽŵĂŝŶƐͿ\u0003\n28 independent parameters  \nRwp = 5.375%\u0003;\u0004ŶƟƉŚĂƐĞ\u0003ĚŽŵĂŝŶƐͿ\u0003\n29 independent parameters  a)\nb)\nc) Intensity\n(arbitrary units)FIG. 2. [color online] Rietveld refinements (D20, λ=2.41 ˚A)\nwith the 2- kmodel showing a) wide 2 θrange with both nu-\nclear (blue arrows) and magnetic (black tick marks) phases.\nb) refinement with the same peak shape for both nuclear and\nmagnetic phases; c) refinement including antiphase bound-\naries in the magnetic phase. Observed and calculated (upper)\nand difference (lower, at zero intensity) profiles are shown by\nblue points, red, and grey lines, respectively. The tick marks\ndo not include a refined zero offset of ∼0.4◦.\nµB) [15, 32], due to improved fitting of magnetic Bragg\npeaks (Fig. 2 c); our value is similar to that reported for\nSr2F2Fe2OS2(3.3(1)µB) [27] and in the parent phase of\nsuperconducting K xFe2−ySe2(3.3µB). [33, 34]\nWe now discuss spin excitations characterizing the\nmagnetic interactions shown in Fig. 1. Fig. 3 shows\nthe temperature-dependent, powder-averaged inelastic\nresponse. The spectra at 2 K show the magnetic re-\nsponse is gapped and localized in momentum (Fig. 3 a)\nand softens on warming (Fig. 3 b) until gapless scatter-\ning is observed for T > T N(Fig. 3c). This is further\nillustrated in Fig. 3 dande) (showing Q-integrated en-\nergy scans) and in lower resolution scans f−g. The\nintensity distribution at the gap edge is sensitive to the\ndimensionality of the interactions and can be quantified\nthrough use of the first moment sum rule. Fig. 3 dshows\na comparison of the momentum integrated intensity with\ncalculations based on the single mode approximation for\nan isotropic dispersion in a one-dimensional (1D) chain,\n2D plane or 3D structure. [35–37] The 2D model gives\nthe best description consistent with the 2D-Ising critical\nproperties discussed above.\nScans that probe larger energy transfers are shown in\nFigs. 3f−h. Surprisingly, the magnetic excitations ex-\ntend up to only ∼25 meV. This small band accounts for\nall of the expected spectral weight, confirmed by integrat-\ning the intensity and comparing with the zeroeth sum\nrule ( ˜I=/integraltext\nd2Q/integraltext\ndES(/vectorQ,E)//integraltext\nd3Q=S(S+ 1)). Our\ninelastic data (over energy ranges shown in Fig. 3 e) give3\no oo\no\nFIG. 3. [color online] a-c) powder-averaged spectra measured\non DCS; d) shows the momentum-integrated energy scan at\n2 K (upper) and 150 K (lower), the curves are calculations\nusing a single-mode analysis with a 1D model, a 2D model\nand a 3D model; f-h) plot the powder-averaged temperature\nspectra taken on the MARI spectrometer.\n˜Iinelastic = 3.2(4) for the dynamic response. The elas-\ntic magnetic moment of 3.5 µB(determined from NPD\ndiscussed above) implies an elastic contribution to the\nabove integral of ˜Istatic =2.7(1), giving ˜I= 5.9(4), close\nto theS=2 value of 6. Over this narrow energy range,\nall magnetic spectral weight is accounted for.\nThis analysis demonstrates that the total bandwidth of\nthe spin excitations is only ∼20 meV. This is remarkably\nsmall when compared with Mott insulating La 2CuO 4\nand YBa 2Cu3O6+x(with a bandwidth of over 300 meV)\nand with the parent phases of the pnictides (the top\nof the band in BaFe 2As2is∼100 meV and ∼150 meV\nin CaFe 2As2) or the chalcogenide Fe 1+xTe (where ex-\ncitations extend up to ∼150-200 meV). [19–21, 23, 24]\nThe small bandwidth observed for La 2O2Fe2OSe 2implies\nthat magnetic exchange interactions are about an order\nof magnitude smaller than in the cuprates and pnictides.\nTo estimate these exchanges, calculations were per-\nformed fixing the moment direction with a single-ion\nanisotropy and considering Heisenberg spin exchange.\nThe calculation is sensitive to the signs of the interac-\ntions and the ground state. These calculations were car-\nried out based on both the collinear and 2- kmagnetic\nground states (Fig. 1) and results are shown in Fig. 4.\nThe experimental spectrum can be reproduced reason-\nably well for the 2- kground state with J1= 0.75 meV,\noFIG. 4. [color online] a) MARI scans with E i= 40 meV and\nspin-wave models for b) the 2- kstructure and c) the collinear\nmagnetic structure; d) and e) show the effect of weak AFM\nand FM values of the J 2exchange interaction on simulated\nspectra.\nJ2= -0.10 meV and J2/prime= 1.00 meV (Fig. 4 b) and for\nthe collinear ground state with J1= 0.13 meV, J2=\n0.63 meV and J2/prime= 1.00 meV (Fig. 4 c). The predicted\nΘCW, to be compared with a TN∼90 K, are ∼110 K\nfor the 2-kand∼75 K for the collinear models. These\ntwo models give comparable descriptions of the data and\ndiffer mainly in the sign of the J2/primeinteraction with the\n2-k(collinear) ground state giving a FM (AFM) value.\nWe now compare the collinear and 2- kmodels. The\ncollinear model (Fig. 1 b) is a single- kmodel with /vectork=\n(01\n21\n2). Thisk-vector splits the moments of the Fe\nsite (4csite inI4/mmm ) into two orbits that order un-\nder separate irreducible representations (irreps) with the\nmoments along the baxis. The irreps and basis vec-\ntors involved are labelled N+\n2(B3g) andN−\n1(B2g) accord-\ning to ISODISTORT [38], and Γ 2ψ1and Γ 3ψ2following\nSARAh. [39] In terms of energy, none of the three intra-\nplanar exchange interactions are satisfied in the collinear\nstructure, making it disfavoured on energetic grounds.\nAs the mean fields experienced by the different orbits\nare orthogonal, they would order separately and so this\nmodel would also be disfavoured on entropic arguments.\nThe 2-kmodel (Fig. 1 c) can, to a first approximation,\nbe described by the spin Hamiltonian involving single-ion\nanisotropies and Heisenberg terms with AFM J1andJ2/prime\nand FMJ2, consistent with calculations [27] and with the\nvalues postulated here. The nearest neighbor exchange\nJ1is thought to be AFM in all known Ln2O2M2OSe 2\nmaterials and dominates for La 2O2Mn2OSe 2. [26, 29, 40]4\nHowever, in the 2- kmodel, the J1interactions are unim-\nportant as nearest neighbor moments are perpendicular.\nInstead, it is the next nearest neighbor J2andJ2/primethat\ndominate. DFT calculations predict that J2via Se2−\nis FM forM=Fe, but AFM for M=Mn and Co, while\nJ2/prime(180◦exchange via O2−) is predicted to be AFM for\nallM. [41] The FM J2Fe-Se-Fe interactions, predicted\nby DFT, are consistent with the FM chain structure re-\nported for Ce 2O2FeSe 2. [42] 2D exchange concomitant\nwith magnetocrystalline anisotropy (due to partially un-\nquenched orbital angular momentum) is likely to stabilize\nthe 2-kstructure (and the k= (1\n21\n20) structure reported\nfor La 2O2Co2OSe 2). [26, 30] This agrees with the Ising-\nlike character suggested to constrain M2+moments to lie\nalong perpendicular local axes within the abplane forM\n= Fe, Co (i.e. along Fe-O bonds in La 2O2Fe2OSe 2). This\nanisotropy is not found in the high spin M= Mn2+for\nwhich orbital angular momentum is zero and moments\nare oriented out of the abplane. [26, 29] This anisotropy\noverridesJ1and with FM J2and AFMJ2/prime, favors the\n2-kover the collinear model.\nTo stabilize 2- kstructures, energy terms beyond\nsecond order isotropic or antisymmetric exchange\n(Dzyaloshinski-Moriya) are required. Anisotropic ex-\nchange arising from spin anisotropy is able to introduce\nhigher order terms that can stabilize combining the 2- k\ncomponents. In doing so, the C4rotational symmetry\nthat relates the two kvectors is reintroduced into the\nmagnetic symmetry, constraining the moments of what\nwere two independent orbits in the single- kstructure, to\nbe equal in magnitude and related in-phase. This con-\nstraint causes the magnetic ordering to satisfy entropic\nrequirements and the transition is second order as ob-\nserved here by experiment.\nWhile the 2- kstructure cannot be stabilized by second\norder spin terms alone, it is useful to explore the structure\nin terms of the interactions in Fig. 1, which still embod-\nies the two orbit structure of the single- kmodel. In it,\nwith no net J1nearest neighbour interactions, the 2- k\nmodel can be thought of as two interpenetrating square\nsublattices, each described by one of the two k-vectors.\nWithin each sublattice, J2/primecoupling leads to AFM Fe-O-\nFe stripes which are coupled by FM J2Fe-Se-Fe interac-\ntions. The 2- kmodel (and the k= (1\n21\n20) structure de-\nscribed for La 2O2Co2OSe 2) could result from dominant\nJ2/primeinteractions where J2/prime>>J 1,J2. This exchange sce-\nnario would lead to a network of perpendicular quasi-1D\nAFM Fe-O-Fe chains. However, our experimental results\nindicate 2D-like magnetic exchange interactions making\nthis quasi-1D scenario unlikely.\nThe 2-kmodel can be compared with the magnetic\nordering reported for Fe 1+xTe [43] which is also com-\nposed of two interpenetrating square sublattices. [16, 17]\nFirst, the origin of the anisotropy within each sublat-\ntice in Fe 1+xTe (i.e AFM interactions along aTand FM\ninteractions along bTwhereTsubscript denotes tetrag-onal unit cell) is ascribed to orbital ordering, while in\nLa2O2Fe2OSe 2, the anisotropy within each single- ksub-\nlattice is due to different exchange interactions along\neach direction. Second, the mechanism for coupling the\ntwo sublattices differs, with double exchange interac-\ntions proposed for metallic Fe 1+xTe [44] being less likely\nfor insulating La 2O2Fe2OSe 2. Rather, the strong spin-\nanisotropy observed supports a coupling by high order\nanisotropic exchange terms.\nThe observation of a Warren peak characteristic of\nshort-range magnetic ordering only ∼14 K above TN(in\ncontrast to ∼140 K above TNfor La 2O2Mn2OSe 2) [29]\nfurther supports the assignment of the (less frustrated)\n2-krather than the collinear model. This is because the\n2-kstructure diminishes the effects of J1and avoids frus-\ntration ofJ2andJ2/prime. With both J2andJ2/primesatisfied, the\n2-kstructure involves less frustration than in the Mn ana-\nlogues. The anisotropic broadening of magnetic Bragg\nreflections suggests that there is only a small energy cost\nfor disrupting the magnetic ordering along c(e.g. intro-\nducing stacking faults or antiphase boundaries) giving a\nreduced magnetic correlation length in this direction.\nDFT calculations have supported the notion of large\nexchange constants in this material and related iron-\nbased systems, in contrast with our experimental results.\nGiven that Jis proportional to 4 t2/U, [45] these small\nJvalues determined experimentally suggest a small hop-\nping integral tfor these oxychalcogenides, consistent with\ntheoretical work which describes band narrowing in these\nmaterials. [14] These small Jvalues imply that local\nbonding is more important than in related materials such\nas Fe 1+xTe andLnFeAsO, and that La 2O2Fe2OSe 2is a\nmore correlated system than current DFT work suggests.\nThe integrated intensity over the small band width of\nexcitations recovers the total moment for S=2. While\nthis is consistent with a large ordered moment, it implies\nthat Fe2+is in a weak crystal field favoring, a Hund’s\nrules population of the d-orbitals which contrasts with\nsuggestions of a S=1 ground state from analysis of pnic-\ntide and chalcogenide superconductors. [44, 46] Our anal-\nysis, combined with the large ordered magnetic moments\nreported in K xFe2−ySe2, may indicate that the S=1 par-\nent state may need to be reconsidered.\nIn conclusion, Mott-insulating La 2O2Fe2OSe 2adopts\na multi component 2- kmagnetic structure. This struc-\nture is stabilized by AFM J2/primeand FMJ2interactions\nand the magnetocrystalline anisotropy of the Fe site and\nleads to 2D-Ising like spin fluctuations around the critical\npoint. Surprisingly, the magnetic exchange interactions\nare very small in comparison with related systems and\nalso the Mott-insulating cuprates and an integrated in-\ntensity analysis implies a S=2 ground state. 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Evans, Chem.\nCommun. 47, 1261 (2011).\n[43] S. Li, C. Cruz, Q. Huang, Y. Chen, J. W. Lynn, J. Hu,\nY. L. Huang, F. C. Hsu, K. W. Yeh, M. K. Wu, and\nP. Dai, Rev. Rev. B 79, 054503 (2009).\n[44] A. M. Turner, F. Wang, and A. Vishwanath, Phys. Rev.\nB80, 224504 (2009).\n[45] A. H. MacDonald, S. M. Girvin, and D. Yoshioka, Phys.\nRev. B 37, 9753 (1988).\n[46] K. Haule and G. Kotliar, New Journal of Physics. 11,\n025021 (2009).Supplementary information describing the spin excitations and magnetic structure of\nLa2O2Fe2OSe 2\nE. E. McCabe,1, 2C. Stock,3E. E. Rodriguez,4A. S. Wills,5J. W. Taylor,6and J. S. O. Evans1\n1Department of Chemistry, Durham University, Durham DH1 3LE, UK\n2School of Physical Sciences, University of Kent, Canterbury, CT2 7NH, UK\n3School of Physics and Astronomy, University of Edinburgh, Edinburgh EH9 3JZ, UK\n4Department of Chemistry of Biochemistry, University of Maryland, College Park, MD, 20742, U.S.A.\n5Department of Chemistry, University College London, 20 Gordon Street, London WC1H 0AJ, UK\n6ISIS Facility, Rutherford Appleton Labs, Chilton Didcot, OX11 0QX, UK\n(Dated: September 2, 2021)\nSupplementary information regarding neutron elastic and inelastic scattering on La 2O2Fe2OSe 2\nis presented. Further details of the magnetic and nuclear structural refinements are given along with\nadditional information pertaining to the analysis of the magnetic excitations discussed in the main\ntext. A description of the heuristic spin-wave Hamiltonian and model used to parameterize the spin\nexcitations are given along with the calculated Curie-Weiss constants stated in the main text.\nI. EXPERIMENTAL DETAILS AND SAMPLE\nDETAILS\nA 5.25 g sample of La 2O2Fe2OSe 2was prepared follow-\ning the method described by Free and Evans.1Neutron\npowder diffraction data were collected on the high flux\ndiffractometer D20 at the ILL with neutron wavelength\n2.41 ˚A. The powder was placed in an 8 mm diameter\ncylindrical vanadium can (to a height of 4 cm) and data\nwere collected from 5-130◦in 2θ. A 30 minute scan was\ncarried out at 216 K, then 5 minute scans were collected\non cooling at∼3 K intervals, followed by a 40 minute\nscan at 2.4 K. Rietveld refinements were performed using\nTopasAcademic software. The diffractometer zero point\nand neutron wavelength were initially refined while sam-\nple cell parameters were fixed to values obtained at the\nsame temperature from refinements using the previously\npublished HRPD data.1The values for zero point and\nwavelength were then fixed at these values for subsequent\nrefinements. Typically, a background was refined for each\nrefinement, as well as unit cell parameters, atomic posi-\ntions and a Caglioti peak shape. TopasAcademic permits\nnuclear-only and magnetic-only phases to be included in\nrefinements and the unit cell parameters of the magnetic\nphase were constrained to be integer multiples of those of\nthe nuclear phase. The scale factor scales with the square\nof the unit cell volume; the scale factor for the nuclear\nphase was refined and that for the magnetic phase (with\ncell volume four times that of the nuclear phase) was con-\nstrained to be 0.0625 ×that of the nuclear phase. The\nweb-based ISODISTORT software was used to obtain a\nmagnetic symmetry mode description of the magnetic\nstructure; magnetic symmetry modes were then refined\ncorresponding to the two magnetic structures.\nFor inelastic neutron scattering measurements, the\n5.25 g of La 2O2Fe2OSe 2prepared here were combined\nwith an additional 3 g sample to give ∼8 g of powder.\nThis was packed into an Al foil envelope and placed\nin an Al can. Two experiments were performed using\nthe MARI direct geometry chopper instrument locatedat ISIS and also the Disk Chopper Spectrometer (DCS)\nat the NIST reactor source. On MARI, a Gd chopper\nwas used to obtain an incident energy E i=40 meV and\nmeasurements were carried out between 5 and 350 K. A\nfurther scan with E i= 150 meV was collected at 5 K to\nsearch for any higher energy excitations. To normalize\nthese data on an absolute scale, the elastic incoherent\nscattering from a known Vanadium standard was used.\nOn DCS, an incident energy of 25 meV was used and the\nsample was enclosed in an Aluminum can with helium\nexchange gas.\nII. RIETVELD REFINEMENTS\nFurther details from Rietveld refinements (Ref. 2) us-\ning neutron powder data collected at 216 K and 2 K is\nprovided. Refinements were carried out using a nuclear\nstructure in space group I4/mmm with La on 4 esite (1\n21\n2z), Fe on 4csite, Se on 4 esite (0 0z), O(1) on 4 dsite\nand O(2) on 2 bsite. The nuclear structural parameters\nare shown in Table I.\nTABLE I. Refined Parameters\nX T=216 K T= 2 K\na/˚A 4.0792(2) 4.0736(3)\nc/˚A 18.563 18.515(2)\nvolume / ˚A3308.89(4) 307.25(6)\nLaz 0.1840(1) 0.1838(1)\nSez 0.0964(2) 0.0964(2)\nRwp/ % 5.165 5.383\nThe main text describes Rietveld refinements for the\nmagnetic structure using neutron powder diffraction data\ncollected at 2 K with magnetic Bragg peaks fitted by the\n2-kmagnetic phase. This phase and the collinear phase\ngive equivalent fits and those for the collinear phase arearXiv:1402.2684v1  [cond-mat.str-el]  11 Feb 20142\n \n50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 1680,000\n60,000\n40,000\n20,000\n0\n-20,000La2O2Fe2OSe2 84.21 %\nmagnetic 15.79 % \n70 65 60 55 50 45 40 35 30 25 20 15 101,000,000\n800,000\n600,000\n400,000\n200,000\n0\n-200,000La2O2Fe2OSe2 84.20 %\nmagnetic 15.80 %\n \n50 48 46 44 42 40 38 36 34 32 30 28 26 24 22 20 18 1680,000\n60,000\n40,000\n20,000\n0\n-20,000\n-40,000La2O2Fe2OSe2 84.20 %\nmagnetic 15.80 %Rwp = 6.981%  \n28 independent parameters  a) \nb) \nc) 2θ / ° \n2θ / ° \n2θ / °   \n  Rwp = 5.383%  \n29 independent parameters  Intensity  \n(arbitrary units)  Intensity  \n(arbitrary units)  Intensity  \n(arbitrary units)  Rwp = 6.981%  \n28 independent parameters  \nFIG. S1. [color online] Rietveld refinement profiles (D20 with\nλ=2.41 ˚A) for refinements with the collinear model showing\na) wide 2θrange and b) showing narrow 2 θrange to highlight\nmagnetic Bragg reflections, with the same peak shape used for\nboth magnetic and nuclear phases; c) refinement including an-\ntiphase boundaries applied to the magnetic phase. Observed,\ncalculated and difference profiles are shown in blue and red\n(top) and grey (bottom), respectively. The black tick marks\nindicate positions of possible magnetic Bragg peaks and the\nsolid black line highlighs scattering from the magnetic-only\nphase.\nshown here in Figure S1 a−cfor comparison.\nIII. WARREN LINESHAPE\nNear the critical temperature, TN∼89 K, diffuse, two-\ndimensional, critical scattering was observed and fitted to\nthe Warren lineshape (Ref. 3, Equation 1),\nP2θ=KmF2\nhk1 + cos22θ\n2(sinθ)3/2/parenleftbiggL\nλ√π/parenrightbigg\nF(a) (1)\nwherea=/parenleftBig\n2L√π\nλ/parenrightBig\n(sinθ−sinθ0) andF(a) =\n/integraltext∞\n0exp(−x2−a2)dx,Kis the scale factor, mis the mulit-\nplicity of the reflection ( hk) centered at 2 θ0,Fhkis the\nstructure factor (assumed to be constant over this narrow\n2θrange) and λis the wavelength of radiation used.\nExamples of Warren lineshapes are displayed in Fig.\nS2a) andb) and a plot of the fitted temperature depen-\ndent correlation length as a function of temperature is\ndisplayed in Fig. S2 c). Two dimensional diffuse scat-\ntering was only observed over a narrow range near TN\nreflecting the low levels of frustration in the magnetic\nstructure as discussed in the main text. We note that\nthis correlation length is different from the temperature\nindependent magnetic correlation length corresponding\nto magnetic stacking faults along the c-axis which re-\nsults in a broadening of the magnetic reflections at low\ntemperatures.\nFIG. S2. [color online] a)−b) Examples of the Warren line-\nshape characterizing the two dimensional short-range corre-\nlations in La 2O2Fe2OSe 2. The fit is given by the solid blue\nline.c) shows a plot of the correlation length extracted from\nsimlar fits as a function of temperature.\nIV. CANDIDATE MAGNETIC STRUCTURES\nIN LA 2O2FE2OSE 2AND RELATIVE GROUND\nSTATE ENERGIES\nIn the main text, it was pointed out that there have\nbeen two structures proposed to describe the magnetic\ndiffraction patterns in La 2O2Fe2OSe 2and related ma-\nterials - the 2- kmodel and the collinear model. De-\ntailed illustrations of two proposed magnetic structures\nare shown in Fig. S3 showing a) collinear model with\nb) view down long axis and c) 2-kmodel with d) view\ndown long axis, showing only O2−(red) and Fe2+(blue)\nions for clarity, with Fe2+moments shown by blue ar-\nrows. While the structure corresponds to a 2- kmodel\nwhen based on the crystallographic unit cell, analysis us-\ning ISODISTORT5indicates that it can alternatively be\ndescribed using a smaller C-centred monoclinic unit cell\nwith magnetic space group Ca2/m[number 12.64, with\nthe basis (2, -2, 0) (2, 2, 0) (1\n21\n20) and origin (0, 0,\n0)] with moments within the abplane, illustrated in Fig.\nS4).\nFrom our analysis of the magnetic structure and the\nspin excitations described in the main text, we believe the\n2-kstructure is favored. The structure was first proposed3\na) b) c) d) \na \nb a \nb \nFIG. S3. [color online] Candidate magnetic structures to describe the magnetic ordering in La 2O2Fe2OSe 2.a) andb) illustrate\nthe collinear structure first proposed in Ref. 1. c) andd) show the 2- kstructure first proposed in Ref. 4.\nto describe the magnetic structure in Nd 2O2Fe2OSe 2and\nhas more recently been applied to Sr 2F2Fe2OS2. As dis-\ncussed in the main text, the structure can be viewed in\nterms of two interpenetrating lattices made up of single\nkstructures as illustrated in Fig S5.\nWhile we have focussed the discussion in this paper\naround the collinear and 2- kstructures, several other\nmagnetic structures have been proposed to fit related\ncompounds. Relative energies of the proposed magnetic\nstructures based on experimental values for J1,J2and\nJ/prime\n2derived in the main text for the 2- kstructure are pre-\nsented in Table II. The exchange constants were taken as\nJ1= -0.75 meV, J2= 0.10 meV, J/prime\n2= -1.00 meV and\nFIG. S4. [color online] Illustration of the 2- kmagnetic model\nin the smaller Ca2/mmonoclinic space group with cell param-\netersam=bm=2√\n2an∼11.5 ˚A,cm=/radicalbigg/parenleftBig\nan√\n2/parenrightBig2\n+/parenleftbigcn\n2/parenrightbig2∼\n9.70˚A,β= 90◦+ tan−1/parenleftBig\nan/√\n2\ncn/2/parenrightBig\n∼107.3◦and cell volume∼\n1230 ˚A3.here we use the convention that a negative sign denotes\nAFM interaction to be consistent with Kabbour et al.\nFor descriptions of structures FM, AF1, AF2 and AF3,\nsee Kabbour et al. (Ref. 6); for Wu stripe and Wu plane\nAF see Wu (Ref. 7); /vectork=(1\n2,1\n2,0) structure is that de-\nscribed for La 2O2Co2OSe 2, see Free and Evans (Ref. 1)\nand Fuwa et al. (Ref. 4).\nV. ZEROETH MOMENT SUM RULE\nTo compare our calculations with the experiment and\nto understand the excitation spectrum sensitive to the\nspin interactions, it is important to ensure that all spec-\ntral weight is accounted for and we have used the zeroeth\nmoment sum rule to do this. The zeroeth moment sum\nrule states that the integral of the measured intensity\nover all energy transfer (including the elastic, E=0, con-\ntribution) and momentum transfers is\n˜I(Q) =/integraltext+∞\n−∞dE/integraltextQ\n0d3qS(/vector q,E)\n/integraltextQ\n0d3q=S(S+ 1).(2)\nThe integrated inelastic intensity contribution to the in-\ntegral above gives I=3.2 (4), which is consistent with an\nordered moment of 3.5 µBper Fe2+site. Noting that the4\nTABLE II. Spin exchange energies per chemical unit cell for the following spin states for La 2O2Fe2OSe 2:\nStructure Ground State Energy comment\nFM E FM=4×(-8J1-4J2-4J/prime\n2) =38.4 meV –\nAF1 E AF1=4×(8J1-4J2-4J/prime\n2) =-9.6 meV Stabilized over FM by 48 meV\nAF2 E AF2=4×(8J1-4J2-4J/prime\n2) =27.2 meV Stabilized over FM by 11.2 meV\nAF3 E AF3=4×(-4J1-2J2-4J/prime\n2) =19.2 meV Stabilized over FM by 19.2 meV\nWu Stripe E Wus=4×(0J1+4J2+4J/prime\n2) =-14.4 meV Stabilized over FM by 52.8 meV\nWu Plane AF E Wup=4×(0J1-4J2+4J/prime\n2) =-17.4 meV Stabilized over FM by 56 meV\n/vectork=(1\n2,1\n2,0) E ice=4×(0J1+4J2+4J/prime\n2) =-14.4 meV Stabilized over FM by 52.8 meV\n2-k E 2−k=4×(0J1-4J2+4J/prime\n2) =-17.6 meV Stabilized over FM by 56 meV\nFIG. S5. [color online] Schematic illustration of kvectors\na)/vectork=(0,1\n2,1\n2) and b)/vectork=(1\n2,0,1\n2) that together form the 2- k\nstructure, shown in c). The unit cell of the crystal struc-\nture is shown in blue and the interpenetrating sublattices are\nhighlighted with green and red dashed lines.\nelastic line contributes a factor of g2/angbracketleftSz/angbracketright2(withg= 2)\ngives/angbracketleftSz/angbracketright=2.7. The total integral above is therefore close\nto 6, as expected for S=2.\nThe analysis confirming this is presented in Fig. S6\nbased upon a scan performed with E i=150 meV to obtain\na more complete detector coverage than allowed by lower\n~\noFIG. S6. [color online] An analysis of the zeroeth moment sum\nrule.a) illustrates a background corrected scan taken at with\nan E i=100 meV and b) shows the average integrated intensity\nas a function of a momentum transfer. The dashed line is the\naverage value used for the sum rule analysis described in the\nmain text.\nincident energy data sets discussed in the main text. The\ndata is shown in Fig. S6 a) and the integral in Eqn. 4\nis plotted in Fig. S6 b). The initial rise at small momen-\ntum transfers is due to incomplete detector coverage and\nthe large errorbars at larger momentum transfers are due\nto the decrease in intensity owing to the magnetic form\nfactor. The dashed line is the value of 3.5 quoted in the\nmain text.5\nVI. FIRST MOMENT SUM RULE AND SINGLE\nMODE APPROXIMATION\nAs a first step towards understanding the magnetic\nexcitation spectrum, we utilized the single mode approx-\nimation combined with the first moment sum rule. This\nmethod has been used previously to parametrize the ex-\ncitations in low-dimensional and frustrated magnets.8,9\nIn this framework, the measured structure factor can be\nwritten in terms of a momentum dependent part and a\nDirac delta function in energy,\nS(/vectorQ,E) =S(/vectorQ)δ[E−/epsilon1(/vectorQ)]. (3)\nThe first-moment sum relates S(/vectorQ) to the dispersion\n/epsilon1(/vectorQ),\nS(/vectorQ) =−2\n31\n/epsilon1(/vectorQ)/summationdisplay\n/vectordJd/angbracketleft/vectorS0·/vectorS/vectord/angbracketright[1−cos(/vectorQ·/vectord)].(4)\nHere/vectordis the bond vector connecting nearest neighbor\nspins with a superexchange Jd. While this treatment is\nan approximation for our results on La 2O2Fe2OSe 2, a\ngeneral result is that the powder averaged spectrum at\nthe bottom of the dispersion at the gap edge is sensi-\ntive to the dimensionality of the exchange interaction.\nTherefore, through a comparison of calcualtions with the\nlower edge of the measured powder averaged spectrum,\nthe dimensionality of the spin exchange can be verified.\nWe have parameterized this in the main text of the\npaper in Fig.3 by calculating the neutron scattering in-\ntensity using the formalism above for wave vectors near\nthe magnetic zone center. We have described the disper-\nsion of the single mode at small values of /vector q=(H-1\n2, K) as\nfollows,\n/epsilon1(/vectorQ)2= ∆2+α1(H−1\n2)2+α2K2, (5)\nwith the ratioα1\nα2controlling the dimensionality. The\nthree dimensionally coupled limit (labelled 3D in Fig. 3\nof the main text) was calculated by inserting a third α\nequal toα1,2. The values of αwere chosen so that the\nslope of the powder average excitations near the min-\nimum of the dispersion curve agreed with experiment.\nRegardless of this choice, the results illustrated in Fig. 3\nare general. The model calculations clearly show that the\nspin excitations are better described in terms of strong\ntwo dimensional interactions and is consistent with the\n2D Ising universality class derived from a plot of the\nordered moment as a function of temperature from the\ndiffraction data.VII. HEURISTIC SPIN-WAVE ANALYSIS\nTo parametrize the neutron inelastic data and extract\nexchange parameters that could be compared with elec-\ntronic calculations and also compared to the magnetic\nstructure, we have performed calculations considering a\nspin Hamiltonian consisting of Heisenberg exchange in-\nteractions and single-ion anisotropies. The Hamiltonian\nused is described as,\nHstripe =J1/summationdisplay\ni,j/vectorSi·/vectorSj+J2/summationdisplay\ni,j/vectorSi·/vectorSj+...(6)\nJ2/prime/summationdisplay\ni,j/vectorSi·/vectorSj+D1/summationdisplay\niS2\nz,i+D2/summationdisplay\ni(S2\ny,i−S2\nx,i).\nThe exchange constants J1,2,2/primeare schematically repre-\nsented in Fig. 1 of the main text. J1is the nearest neigh-\nbour exchange and J2andJ/prime\n2are the next-nearest. J2and\nJ/prime\n2differ in that they are mediated by oxide and selenide\nanions, respectively. D1is an out of plane anisotropy\nwhich forces the moment direction to be within the a−b\nplane. The second term D2is input to stabilize a struc-\nture with moments selecting a preferred moment direc-\ntion within the a−bplane. These anisotropy terms\nwere used to stablize the magnetic structures so that\nexchange constants could be extracted. Typically, the\nanisotropies were comparable to the extracted exchange\nconstants which is unphysical. This demonstrates that\nhigher order terms in the Hamiltonian are likely needed\nto stablize the magnetic structures.\nTo calculate the spin excitations and the neutron cross\nsection, we used the Holstein Primakoff operators. The\nabove expression for the spin Hamiltonian can be rewrit-\nten in terms of creation and annihilation operators in\nmatrix form as follows,\nH\n4S=α†Mα\nH\n4S=/bracketleftbig\na†\nqb†\nqa−qb−q/bracketrightbig\nA C D F\nC B F E\nD F A C\nF E C B\n\na−q\nb−q\na†\nq\nb†\nq\n\nCalculations were performed for both the collinear and\n2-kmagnetic ground states as discussed in the main text.\nThe matrix elements for the collinear model are as fol-\nlows:\nA=J2\n2+J/prime\n2cos(/vector q·/vectorb) +D1−D2\nB=J/prime\n2\n2+J2cos(/vector q·/vectorb) +D1−D26\nC=J1\n2cos/parenleftBigg\n/vector q·/vectorb\n2/parenrightBigg\ncos/parenleftbigg/vector q·/vector a\n2/parenrightbigg\nD=−J2cos(/vector q·/vector a)−D1\n2+D2\n2\nE=−J/prime\n2cos(/vector q·/vector a)−D1\n2−D2\n2\nF=J1\n2cos/parenleftBigg\n/vector q·/vectorb\n2/parenrightBigg\ncos/parenleftbigg/vector q·/vector a\n2/parenrightbigg\n.\nThe matrix elements for the 2- kstructure are as follows,\nA=J2cos(/vector q·/vector a)−2J2−2J/prime\n2+D1−D2\nB=J2cos(/vector q·/vectorb)−2J2−2J/prime\n2+D1−D2\nC=J1cos/parenleftBigg\n/vector q·/vectorb\n2/parenrightBigg\ncos/parenleftbigg/vector q·/vector a\n2/parenrightbigg\nD=J/prime\n2cos(/vector q·/vectorb)−D1\n2+D2\n2\nE=J/prime\n2cos(/vector q·/vector a)−D1\n2−D2\n2\nF=−J1cos/parenleftBigg\n/vector q·/vectorb\n2/parenrightBigg\ncos/parenleftbigg/vector q·/vector a\n2/parenrightbigg\n.\nThe matrix is Hermitian and follows several symmetry\nrelations of the ground state illustrated in the two mag-\nnetic structures in question. The energy positions were\ncalculated from the eigenvalues of the matrix above and\nthe neutron intensities were calculated from the eigen-\nvectors.10\nA combined analysis of the spin-waves and the diffrac-\ntion indicates that the 2- kstructure is the most likely\nmodel. The spin-wave analysis suggests antiferromag-\nneticJ1andJ2/primeand ferromagnetic J2in contrast to\nthe collinear structure which would require antiferromag-\nneticJ2to reproduce the neutron inelastic results. While\na ferromagnetic J2is consistent with DFT calculations.\nThe superexchange through the 180◦Fe-O-Fe path is ex-\npected to be strong and antiferromagnetic, while weak\nand ferromagnetic exchange is thought to occur via the\n∼90◦Fe-Se-Fe path.VIII. CURIE-WEISS TEMPERATURE - A\nMEAN FIELD DESCRIPTION\nIn this section we discuss the Curie-Weiss tempera-\nture and compare it to our heuristic model of the spin\nexcitations. This analysis assumes a mean-field descrip-\ntion (Ref. 11) and is likely only an approximation for\nLa2O2Fe2OSe 2as a correct and full description of this\npoint needs to include effects of magnetic frustration and\nlow dimensionality. The issue of dimensionality is partic-\nularly important given the presence of magnetic stacking\nfaults evidenced by the Warren line shape and strong dif-\nfuse scattering observed in the magnetic diffraction pat-\ntern for this compound. Nevertheless, while the discus-\nsion below in terms of mean-field theory is speculative,\nit is interesting in light of the spin-wave calculations and\nestimated values of the exchange constants.\nBased on the exchange constants estimated from the\ncalculations described above, a predicted value for the\nCurie-Weiss temperature can be derived based on the\nfollowing formula where Sis the spin value, J nis the\nexchange interaction energy, and the sum is performed\nover all nearest neighbors,\nkBΘCW=−1\n3S(S+ 1)/summationdisplay\nnJn. (7)\nFor the exchange interactions listed in Fig. 1 of the main\ntext, this formula takes the following form for both the\ncollinear and 2- kmagnetic structures,\nkBΘCW=−1\n3S(S+ 1)(4J1+ 2J2+ 2J2/prime).(8)\nUsing this formula we estimate the expected Θ CWtem-\nperature for the 2- kmagnetic structure to be ∼-110 K\nand∼-75 K for the collinear variant. Magnetic suscep-\ntibility data for La 2O2Fe2OSe 2have been reported up to\n300 K12and further measurements to higher tempera-\ntures are required to investigate this material in the para-\nmagnetic phase. The fact that the Neel ordering temper-\nature for the 2- kstructure in La 2O2Fe2OSe 2is close to\nthe calculated Θ CWpotentially reflects the lack of frus-\ntration in the system. The small value predicted for the\ncollinear structure is unphysical given the large degree\nof frustration in that structure discussed in the main\ntext. Therefore, a comparison of the predicted Curie-\nWeiss constant and the Neel temperature also points\ntowards a 2- kmodel being favored for La 2O2Fe2OSe 2.\nAgain, we emphasize that this discussion is based upon\na mean-field description and does not account for frus-\ntrated exchange interactions or dimensionality. The later\nis particularly important in La 2O2Fe2OSe 2as evidenced\nby stacking faults along the caxis. A high degree of frus-\ntration might be expected to lower T N. The observed T N\n(89.5(3) K) is similar to the calculated Weiss tempera-\nture (∼100 K). Whilst this may reflect the low degree of7\nfrustration in the system, other factors such as the Ising\nanisotropy (which would tend to increase T N) and thequasi-2D nature (which might lower T N) must also be\nconsidered and the relative contributions of these factors\nis not known.\n1D. G. Free and J. S. O. Evans, Phys. Rev. B 81, 214433\n(2010).\n2H. M. Rietveld, J. Appl. Cryst. 2, 65 (1969).\n3B. E. Warren, Phys. Rev. 59, 693 (1941).\n4Y. Fuwa, M. Wakeshima, and Y. Hinatsu, J. Phys. Con-\ndens. Matt. 22, 346003 (2010).\n5B. J. Campbell, H. T. Stokes, D. E. Tanner, and D. M.\nHatch, Appl. Cryst. 29, 607 (2006).\n6H. Kabbour, E.Janod, B. Corraze, M. Danot, C. Lee, M. H.\nWhangbo, and M. H. Cario, J. Am. Chem. Soc. 130, 8261\n(2008).\n7H. Wu, Phys. Rev. B 82, 020410 (2010).8C. Stock, L. Chapon, O. Adamopoulos, A. Lappas,\nM. Giot, J. Taylor, M. Green, C. Brown, and P. Radaelli,\nPhys. Rev. Lett. 103, 077202 (2009).\n9P. Hammar, D. H. Reich, and C. Broholm, Phys. Rev. B\n57, 7846 (1998).\n10E. W. Carlson, D. X. Yao, and D. K. Campbell, Phys.\nRev. B 70, 064505 (2004).\n11J. S. Smart, Effective Field Theories of Magnetism (W. B.\nSaunders, New York, USA, 1966).\n12J. M. Mayer, L. F. Schneemeyer, T. Siegrist, J. V.\nWaszczak, and B. V. Dover, Angew. Chem. Int. Ed. Engl.\n31, 1645 (1992)."    },    {        "title": "1402.5543v2.LDA_DMFT_Approach_to_Magnetocrystalline_Anisotropy_of_Strong_Magnets.pdf",        "content": "arXiv:1402.5543v2  [cond-mat.str-el]  4 Jun 2014LDA+DMFT Approach to Magnetocrystalline Anisotropy of Str ong Magnets\nJian-Xin Zhu,1,∗Marc Janoschek,1Richard Rosenberg,2Filip Ronning,1\nJ. D. Thompson,1Michael A. Torrez,1Eric D. Bauer,1and Cristian D. Batista1,†\n1Los Alamos National Laboratory, Los Alamos, New Mexico 8754 5, USA\n2Advanced Photon Source, Argonne National Laboratory, Argo nne, Illinois 60439, USA\nThe new challenges posed by the need of finding strong rare-ea rth-free magnets demand methods\nthat can predict magnetization and magnetocrystalline ani sotropy energy (MAE). We argue that\ncorrelated electron effects, which are normally underestim ated in band structure calculations, play\na crucial role in the development of the orbital component of the magnetic moments. Because\nmagnetic anisotropy arises from this orbital component, th e ability to include correlation effects has\nprofound consequences on our predictive power of the MAE of s trong magnets. Here we show that\nincorporating the local effects of electronic correlations with dynamical mean-field theory provides\nreliable estimates of the orbital moment, the mass enhancem ent and the MAE of YCo 5.\nPACS numbers: 71.15.Mb, 71.15.Rf, 71.27.+a, 75.30.Gw\nI. INTRODUCTION\nMagnets playa central rolein different types ofdevices\nand motors, which are at the heart of modern technol-\nogy. There is an increasing need of permanent magnetic\nmaterials for energy conversionand power generation [ 1].\nMagnetocrystalline anisotropy (MA) is one of the most\nimportant properties of permanent magnets [ 2]. Large\nMA is achieved in existing strong magnets by using rare-\nearth transition-metal intermetallic compounds, such as\nSmCo5and Nd 2Fe14B, which are of direct technological\nuse. However, the shortage of rare-earth elements has\ntriggered the search for rare-earth-free magnetic materi-\nals harnessing sources of magnetic anisotropy other than\nthat provided by the rare-earth components [ 1]. In order\nto guide this search, it is necessary to develop theoret-\nical methods that can estimate the magnetocrystalline\nanisotropy energy (MAE) of 3 d, 4dand 5dtransition\nmetals, which are the natural candidates for replacing\nrare-earth elements.\nThe contribution of itinerant ferromagnetic electrons\nto MA arises from the spin-orbit (SO) interaction that\ncouples the spin and orbital components of the mag-\nnetic moments [ 3]. MA results from the orbital com-\nponent of the moment, which is sensitive to the lattice\nanisotropy. The very first electronic structure analysis of\nMAE forNi wasconducted by Kondorskiiand Straub [ 4].\nWhile a band picture may provide a MAE of the right\norder of magnitude for certain transition metal ferro-\nmagnets [ 5,6], accurate electronic structure calculations\nof the MAE of 3 dmetals, such as Fe, Co and Ni, give\nnumbers that are in disagreement with experiment [ 7,8].\nMoreover, the wrong easy-axis is obtained for Ni. This\nfailure has been attributed to either the omission of the\norbital correlation induced by the intra-atomic Coulomb\ninteractionbetweenelectrons[ 9]orthe limitationofband\n∗jxzhu@lanl.gov\n†cdb@lanl.govstructurecalculationsforcalculatingenergydifferencesof\nthe order of 0.1 meV [ 7].\nYCo5hasoneofthelargestMAEsamongferromagnets\nthat do not include f-electron (actinide or lanthanide)\nions. The MAE is more than 50 times larger than in\nthe pure cobalt metal. Like SmCo 5, it has an easy-axis\nparallel to the c-axis of its hexagonal lattice structure.\nThe primitive unit cell contains six atoms with two dif-\nferent cobalt sites, Co I(2c) and Co II(3g) [10]. Neutron\nscattering experiments by Schweizer et al.have reported\nunusually large orbital moments on these Co sites [ 11]:\nmorb(Co(2c))=0.46 µBand m orb(Co(3g))=0.28 µB. How-\never, our x-ray magnetic circular dichroism (XMCD)\nmeasurements indicate that the average orbital moment\nof Co is 0 .2µB, in better agreement with the value of\n0.25µBreported by Heidemann et al.[12]. These mea-\nsurements suggest that the rather large orbital magnetic\nmomentsofthe Coatomsarepartiallyresponsibleforthe\nstrong MAE of YCo 5. Consequently, reliable estimates\nof the MAE require an accurate calculation of these or-\nbital moments. This is not only true for YCo 5, but also\nfor any other strong magnet based on transition metals.\nWe will then use YCo 5as a prototype compound for de-\nveloping and testing methods for calculating the orbital\nmoments and the MAE of strong magnets.\nNordstr¨ om et al.[10] have applied the force theorem\nto compute the MAE of YCo 5from first-principles cal-\nculations. It was found that, in the absence of atomic\norbital correlation, the MAE is too small and it even\nhas the incorrect sign, in agreement with Ref. [ 7]. Af-\nter including the orbital polarization (OP) scheme sug-\ngested by Brooks [ 13,14], they were able to obtain a\nMAE that has the correct sign. However, the MAE\nvalue of about 50 µRyd, when extrapolated to the in-\nfinitesimal grid in the momentum space although, is still\ntoo small in comparison with the experimental value of\n292µRyd [15]. A similar improvement is obtained for\nestimations of the orbital magnetic moments of both\nCo sites. In absence of orbital correlation, the result\nis morb(Co(2c))=0.1 µBand m orb(Co(3g))=0.13 µB[16],\nwhile the inclusion of OP leads to m orb(Co(2c))=0.27 µB2\nand m orb(Co(3g))=0.20 µB[10].\nThe OP scheme is taken from the theory of open\nshell atoms within the Russel-Saunders coupling. The\nground state energy gain, that is obtained by maximiz-\ning the orbital angular momentum L, is approximated\nbyEOP=−BL2/2, where Bis the Racah parameter\nfordsates. This effect is just a consequence of the\nCoulombinteractionbetween d-electronsthat occupythe\nsame ion and it must influence the final value of the or-\nbital magnetic moment and the MAE. However, it is well\nknown that the on-site electron-electron Coulomb inter-\naction also renormalize the band states (electrons tend\nto avoid each other), for which heavy fermion behav-\nior inf-electron systems is a prototypical example [ 17].\nTherefore, it is reasonableto expect that this secondcon-\nsequence of the Coulomb interaction will also affect the\nmagnitudeoftheorbitalmagneticmomentandtheMAE.\nHere we propose a method for including these additional\ncorrelations.\nII. ROLE OF COULOMB INTERACTION ON\nLOCAL MOMENT FORMATION\nThe effect of electron-electron interaction is to reduce\nthe bandwidth of the quasi-particles and produce an in-\ncoherent component in their spectral weight. The most\ndramatic effect of this Coulomb repulsion is the emer-\ngence of Mott insulators in half-filled bands via localiza-\ntion of individual electrons in their atomic orbitals. The\nelectroniclocalizationisaccompaniedbythe formationof\na local magnetic moment, whose spin and orbital compo-\nnents can be of the order of a Bohr magneton ( µB) [18].\nIt is clear that Coulomb interaction cannot localize the\nelectronic charge away from half-filling. However, the\nband narrowing effect can be interpreted as a tendency\ntowards localization that favors local moment formation.\nThis simple reasoning suggest that the inclusion of elec-\ntronic correlations should lead to more realistic values of\nthe effective mass of the quasiparticles, orbital magnetic\nmoments and MAE.\nStandard LDA calculations lead to orbital magnetic\nmoments of order 0.1 µB. This result can be understood\nin the following way. The typical bandwidths, W, of\n3dmetals like Fe or Co are of the order of a few elec-\ntron volts. The SO interaction is about λ≃0.05−0.07\neV. In the absence of SO coupling, the ground state has\nzero orbital angular momentum, even if it has a net spin\nmagnetization, because single-particle states with oppo-\nsite values of the orbital magnetic moment are degener-\nate and therefore equally occupied. A finite SO coupling\nterm splits states with opposite values of orbital moment\nby an amount that is oforder λ. This observationimplies\nthat only the electronic states that are within a distance\nλfrom the Fermi level contribute to orbital polarization.\nThe fraction of electrons occupying these states is of or-\nderλ/W≃0.02. Becausethe maximum possiblevalue of\nthe orbital moment per atom is of order 1 µB, this roughestimate indicates that morb/lessorsimilar0.1µBin agreement with\nprevious results from standard band structure calcula-\ntions [10]. However, as pointed out in the introduction,\nthe orbital magnetic moment of strong magnets, such as\nYCo5, can be higher than this rough estimate.\nIt is natural to assume that the discrepancy arises\nfrom the effects of rather strong intra and inter-atomic\nelectronic correlations induced by the Coulomb interac-\ntion. The improvement that is obtained after includ-\ningthe intra-atomicOPeffect providesempiricalsupport\nfor this assumption. However, the most basic and gen-\neral argument in favor of this assumption is that intra-\natomic Coulomb repulsion favors local moment forma-\ntion by suppressing double occupancy of single atomic\norbitals. The importance of correlation effects on the\nmagnetic anisotropy of Fe and Ni was already recognized\nmore than ten years ago by Yang et al.[19]. This prob-\nlem is now timely because of the increasing need of find-\ning strong magnets that are free of rare earth elements.\nTherefore, it is crucial to propose new methods that can\nincorporatethe subtle effects ofcorrelationsin solids (the\nOP effect that we discussed above is already captured at\nthe level of single-atom physics). For this purpose we\npropose a method based on the combination of the dy-\nnamical mean-field theory (DMFT) and the LDA [ 20]. A\nsimilar approachhas been successfully applied to the cal-\nculation of neutron magnetic form factors of actinides by\napplying an external magnetic field [ 21], as well as the\nbulk and surface quasiparticle spectra [ 22] and the or-\nbital magnetism [ 23] in Fe, Co, and Ni metals. The basic\nidea is to treat each Co ion as an effective impurity that\nis embedded into the bath generated by the rest of the\nions. The single-ion interactions (including the OP) are\ncaptured by the single-impurity Hamiltonian. The cor-\nrelations developed via the interplay between the single-\nion terms and the interaction with the bath (solid) are\ncaptured by a self-consistent treatment of the full Hamil-\ntonian that we describe in the next section.\nIII. LOCAL DENSITY APPROXIMATION PLUS\nDYNAMICAL MEAN-FIELD THEORY\nTo study the role of electronic correlations on the\norbital moment of the magnetic 3 dions by com-\nbining the LDA with dynamical mean-field theory\n(LDA+DMFT) [ 20] we start with a generalized many-\nbody Hamiltonian:\nˆH=/summationdisplay\nk,lmlσ,l′m′\nlσ′[H(0)\nk]lmlσ,l′m′\nlσ′c†\nklmlσckl′m′\nlσ′\n+1\n2/summationdisplay\n,l=2(3),mlm′\nl,\nm′′\nlm′′′\nlσσ′Vmlm′\nlm′′\nlm′′′\nlc†\nilmlσc†\nilm′\nlσ′cilm′′′\nlσ′cilm′′\nlσ.(1)\nHerekis a wave vectorof the Brillouin-zone, iis a lattice\nsite index for atoms with correlated orbitals, lis the or-\nbitalangularmomentum, ml=−l,−l+1,...,l−1,l,and3\n!\"# $\"%&'( [ ]( ) 2\n,\n,|LDA\neff h V n\nOαβ α β\nαβ α βχ χ\nχ χ=−∇+\n=k k k\nk k k\n)*+,-*.(\n/.0*12-3(0[ , ]G UΣ   Gloc(iω)= (iω+µ)Ok−hk(LDA )−Σel−el(iω)+Σdc\nk∑−1\n  G0−1=Gloc+Σel−el Σel−el\n*\n, ,( ) ( ) ( ; ) ( ) , 0n\nni\nn\nin T G i eω η\nα αβ β\nω αβχ ω χ η− −= = ∑k k\nkr r k r\"%&'(\nFIG. 1. Schematic description of the DMFT+LDA approach.\nσis the spin projectionquantum number. The field oper-\natorc†\nilmlσ(cilmlσ) creates (annihilates) an electron with\nspinσand orbital indices ( lml) at site i, whilec†\nklmlσ\n(cklmlσ) is the corresponding operator in momentum\nspace. The first term of ˆHcontains the single-particle\ncontribution, which is determined by solving the Kohn-\nSham quasi-particle equations [ 24] within LDA. We note\nthat the SO coupling can be included in a second varia-\ntional way in the LDA Hamiltonian. In the second term\nofˆHwe restrict the Coulomb repulsion to the correlated\norbitals (e.g., open shell Co 3 dorbitals ( l= 2) or Ce 4 f\norbitals ( l= 3)) to reduce the complexity of the prob-\nlem [20]. The Coulomb matrix elements are obtained\nfrom atomic physics:\nVmlm′\nlm′′\nlm′′′\nl=2l/summationdisplay\nk(even)=0ak(ml,m′\nl,m′′\nl,m′′′\nl)Fk,(2)\nwhereFkandakare the Slater integrals and the corre-\nsponding expansion coefficients [ 25]. For solids, we iden-\ntify the atomic Slater integral F0with the screened effec-\ntive Coulomb interaction parameter Uof the correlated\norbitals. As a common practice, higher order Slater inte-\ngrals are reduced by 20% from the atomic Hartree-Fock\ncalculations due to screening effects [ 26].\nWithin DMFT, the lattice problem of Eq. ( 1) is\nmapped onto a multi-orbital quantum single impurity\nproblem subject to the self-consistency condition (see\nFig.1):\nˆG−1(iωn) =ˆG−1\nloc(iωn)+ˆΣ(iωn). (3)\nHereˆG(iωn) is the Weiss function, ˆΣ(iωn) is ak-\nindependent self-energy, and the local Green’s function\nis defined as ˆGloc(iωn) =/summationtext\nkˆGk(iωn)/N, where the lat-\ntice Green’s function reads\nˆGk(iωn) = [(iωn+µ)ˆI−ˆH0(k)−ˆΣ(iωn)]−1.(4)\nˆIis the identity matrix in the complete tight-binding ba-\nsis andµis the chemical potential. Because we haveadded the on-site Coulomb terms to the correlated va-\nlence orbitals only, it is evident that the self-energy ˆΣ\nmatrix has nonzero elements only within the 10 ×10d-d\nblock for the case of valence d-orbitals, or the 14 ×14\nf-fblock for the case of valence f-orbitals. This self-\nenergy matrix is a function of the Matsubara frequency:\nΣdd(ff)\nmlσ,m′\nlσ′(iωn). Correspondingly, the local Green’s func-\ntion for the correlated orbitals has the same structure\nGdd(ff)\nloc,mlσ,m′\nlσ′(iωn). We assume that the dominant contri-\nbutions to the spin and orbital components of the mag-\nnetic moments come from the correlated orbitals (Co 3 d-\norbitals for the case of YCo 5). After obtaining the lo-\ncal Green’s function for the correlated orbitals through\nthe full self-consistency, we can evaluate the spin and or-\nbital moments by computing Ms=/summationtext\nmlσσρmlσ,mlσand\nMorb=/summationtext\nmlσmlρmlσ,mlσ,respectively, in the spherical\nharmonics basis. Here the density matrix is related to\nthe local Green’s function as\nˆρ=ˆGdd(ff)\nloc(τ→0−) =1\nβ/summationdisplay\niωnˆGdd(ff)\nloc(iωn)e−iωn0−,\n(5)\nwhereβ= 1/kBT, withkBandTthe Boltzmann con-\nstant and temperature, respectively.\nIn earlier applications of the LDA+DMFT method,\nit is common use to rotate the local Green’s function\nand the corresponding self-energy into a basis in which\nthe diagonal matrix elements are dominant in order to\nneglect the off-diagonal elements. For example, for ac-\ntinide based materials, the correlated 5 forbitals are ro-\ntated into the J-Jbasis because ofthe dominant SO cou-\npling [27,28]. In contrast, SO coupling is subdominant\nford-electron materials, like transition metal oxides, and\nthe self-energy and local Green’s function matrices are\ndiagonal in the crystal field basis when the SO coupling\nis neglected [ 29–32]. However, the off-diagonal matrix el-\nements cannot be neglected if our goal is to compute the\nMAE (the SO coupling must be included to obtain a fi-\nnite MAE and the orbital magnetic moment has only off-\ndiagonal contributions in this basis). This situation re-\nquiresafurtherdevelopmentofquantumimpuritysolvers\nto meet this challenge and similar challenges posed by\nother correlated electron materials, such as the inclusion\nof crystal field terms in 4 fand 5fcompounds.\nIV. COMPUTATIONAL RESULTS\nHere we use the spin-polarized T-matrix fluctuation-\nexchange approximation technique (SPTF) [ 33] to solve\nthe effective quantum impurity problem. In this for-\nmalism, the self-energy includes Hartree and Fock dia-\ngrams with the bare interaction replaced by the Tma-\ntrix and particle-hole contributions with the bare in-\nteraction replaced by the particle-hole potential fluctu-\nation matrix. The Tmatrix and the particle-hole po-\ntential fluctuation matrix are in turn expressed in terms\nof particle-particle and particle-hole susceptibilities. We4\n11.522.533.54\nU (eV)11.21.41.61.82m*/mb\nFIG. 2. Ratio between the effective electronic mass, m∗, in\npresence of on-site Coulomb interaction U, and the mass mb\nobtained from a LDA calculation.\nuse the charge self-consistent LDA+DMFT(SPTF) ap-\nproach as implemented in an electronic structure code\nbased on a full-potential linear muffin-tin orbital method\n(LMTO) [ 22,34–36]. The LMTO basis sets contain a\ntriple basis for sandpstates and a double basis for\nthedorbitals of YCo 5. The basis of the valence elec-\ntrons is constructed with 4 s, 4p, and 3dstates for the Co\natoms, and 5 s, 5p, 4dfor Y atoms. The Nkk-points are\ndistributed with the conventional Monkhorst-Pack grid,\nand the Brillouin zone integration is carried out with the\nFermi smearing at a temperature of T= 474 K. To ex-\nplore the role of electronic correlationeffects arising from\nthe screenedCoulomb interaction U, wefix the higher or-\nder slater integrals of F2= 7.75 eV, and F4= 4.85 eV\nfromRef. 35. Thesevaluesof F2andF4forthed-orbitals\nresult in a Stoner parameter J= 0.9 eV, which is con-\nsistent with the value used in earlier studies of Co met-\nals [22]. We treat the effect of the Coulomb exchange\ninteraction with F2andF4explicitly.\nWe first explore the relevance of the notion of elec-\ntronic correlation in the ferromagnetic magnetic metals\nby studying the quasiparticle renormalization effect. In\nconnection with the specific heat coefficient as measured\nfrom the thermodynamic experiments, the effective mass\nenhancement is proportional to the ratio of the quasipar-\nticle density of states to band one at the Fermi energy:\nm∗/mb= ˜ρ(EF)/ρb(EF). For the cases where the delec-\ntrons are active carriers, the band density of states has\na predominant d-character: ρb(EF) =/summationtext\ni,αwiρb,α(EF),\nwhereρb,i,αis the partial density of states at the Fermi\nenergy from the 10 spin orbitals for the i-th type of Co\natom. Here αis the spin-orbital index, while wiis the\nnumber of equivalent atoms of a given type. Within a\nrenormalized band theory, we can generalize the quasi-\nparticledensityofstatesattheFermienergyas ˜ ρb(EF) =/summationtext\ni,αwi˜ρb,α(EF). Here the spin-orbital dependent quasi-\nparticle density of states at the Fermi energy is given\nby ˜ρb,i,α(EF) =ρb,α(EF)/zi,α, where the quasiparticle\nweight is zi,α= [1−∂ImΣα,i(iωn)/∂ωn|ωn→0]−1with the00.10.20.3Orbital Moment ( µB)\nCo(2c)\nCo(3g)\n0 1 2 3 4 \nU (eV) 050010001500MAE ( µRy) (a)\n(b)\n0 0.02 0.04 0.06\nNk-2/350010001500\nFIG. 3. On-site Coulomb Udependence of the orbital mag-\nnetic moments on Co sites and magnetocrystalline anisotrop y\nenergy (MAE) per formula unit of YCo 5. All the solid curves\ncorrespond to the results obtained with the LDA+DMFT\nmethod described in the text. The dashed line corresponds\nto the measured valued according to Ref. [ 37]. The inset to\npanel (b) shows the MAE dependence on the number of k\npoints in the Brillouin zone for two representative values o f\nHubbard interaction U= 2.0 eV (red line with diamond sym-\nbols) and 2.5 eV (black line with circle symbols). The k-point\nconvergence is reasonably reached.\nself-energy Σ α,idefined on the Matsubara frequency ωn\naxis.\nFigure2showsthe Udependence ofthe mass enhance-\nment relative to LDA calculations, m∗/mb, obtained by\napplying the LDA+DMFT method to YCo 5. As ex-\npected, the effective mass increases monotonically with\nU. The mass enhancement takes values between 1.5 and\n2 forUvarying between 2 .5 and 4 eV. The Sommerfeld\ncoefficient γof the specific heat is proportional to the ef-\nfective mass of the quasiparticles. Based on our specific\nheat measurements of YCo 5, we obtain a Sommerfeld co-\nefficientγ= 90 mJ /mol·K2·f.u., which is ∼2.7 times\nlarger than the value γb≃33 mJ/mol·K2·f.u. extracted\nfrom pure LDA calculations (see Appendix A). Note that\nsome additional contribution to the electronic renormal-\nization arising from the electron-phonon coupling is not\nincluded in our calculations. Because γ/γbis equal to\nm∗/mb, this ratio indicates that YCo 5is a rather cor-\nrelated metal for Ubetween 2.5 and 4 eV. Values of U5\nin this range have been previously reported in the YCo 5\nliterature [ 22,35].\nFig.IV(a) shows the orbital magnetic moments on the\ntwo inequivalent Co atoms as a function of U. The or-\nbital moment of the Co(2c) atoms is always larger than\nthe moment of the Co(3g) atoms and both depend non-\nmonotonically on U, reaching their maximum values at\nU≃2 eV. The results for U→0 reproduce the val-\nues obtained in previous LDA calculations [ 10,16], while\nthe moments increase by a factor of ∼2 forU≃1−3\neV. This increase is consistent with our XMCD measure-\nments, which indicate that the average orbital magnetic\nmoment on the Co ion is 0.20 µB. This observation con-\nfirms the relevant role of Uon the formation of a strong\norbital moment.\nHowever, the most dramatic effect of the on-site\nCoulomb repulsion Uappears when we compute the\nMAE, as is clear from our LDA+DMFT results shown in\nFig.IV(b). By comparingFigs. IV(a) and (b), we can see\nthatthe MAEandthesizeofthe orbitalmomentsexhibit\nthe same non-monotonic dependence on U. The extrap-\nolated LDA value of the MAE is much lower than the\nmeasured value of K1V= 250µRy shown with a dashed\nline in Fig. IV(b) (V= 0.84×10−22cm3is the volume of\ntheprimitiveunitcelland K1= 2.98×1018Ry/cm3[37]).\nHowever, the MAE increases drastically with Ureaching\nvalues that are more than an order of magnitude higher\nin the range U∼1-3.5 eV. This dramatic increase not\nonly explains the reason why LDA calculations system-\natically underestimate the MAE of strong magnets, but\nalso shows the crucial role played by electronic correla-\ntions in the development of large magnetic coercivity. In\naddition, the MAE obtained from our LDA+DMFT cal-\nculationsfor Ubetween3and3.5eVisingoodagreement\nwith the experimental value.\nV. CONCLUSIONS\nThe fact that the strongest magnets are rare-earth\nbased compounds, suggests that the large magnitude of\nthe SOcouplingplaysacrucialrolein the developmentof\nhigh coercivity. One would then expect that the intrin-\nsic magnetocrystalline anisotropyof Nd 2Fe14B or SmCo 5\noriginates in the crystal field splitting of the rare-earth\n4flevels. There is experimental evidence, however, in-\ndicating that substantial magnetocrystalline anisotropy\nmay be associated with the transitional metal sublattice\nitself. For instance, the coercive field of magnetically\nhardened Gd 2Fe14B is 2.5 kOe [ 38,39], but the Gd3+\nion has no significant contribution from 4 f-electrons to\nthe orbital moment, suggesting an increasing role of Gd\n5d-orbitalelectrons [ 40]. In addition, the MAE ofSmCo 5\nis only three times higher than the MAE of YCo 5and Y\nis non-magnetic. Our results indicate that the MAE of a\nmagnetisdramaticallymodifiedbythepresenceofstrong\non-site Coulomb interaction Uthat tends to localize the\nelectrons. We note that enhanced correlations could alsobe playing a role in rare-earth based compounds (rare-\nearths have large ionic radii). This may explain why\nrare-earth based compounds, in which the rare-earth has\nno orbital moment, still have very high MAE.\nBy a close comparison of our LDA+DMFT calcula-\ntions with different key experimental measurements, we\nhave shown that electronic correlation effects play an es-\nsential role in determining the MAE of YCo 5. These\ncalculations suggest that the figure of merit of strong\nmagnets can be greatly optimized by tuning the electron\nCoulomb repulsion U. Our analysis has natural impli-\ncations for the search of rare-earth free strong magnets.\nWhile it may be important to retain a large SO coupling,\nit is equally or even more important to find strongly cor-\nrelated ferromagnets in order to induce a large enough\norbital moment on the transition metal. Developing pre-\ndictive tools for the MAE of strong magnets is an essen-\ntial precondition for guiding the search for new materi-\nals. Our results indicate that LDA+DMFT techniques\nare very promising because they incorporate the relevant\ninterplaybetween kinetic and Coulomb energies. Further\nimprovements in impurity solvers should allow to obtain\neven more reliable values of the MAE for magnets that\nare in the intermediate or strong coupling regime (that\nis,Ucomparable to or larger than the bandwidth).\nACKNOWLEDGMENTS\nWe are grateful to Tomasz Durakiewicz, O. Gr˚ an¨ as, J.\nSchweizer, F. Tasset, P. Thunstr¨ om, and J. M. Wills for\nhelpful discussions. Work at the LANL was performed\nunder the auspices of the U.S. DOE contract No. DE-\nAC52-06NA25396 through the LDRD program. Part of\nthe theoretical calculations were carried out on a Linux\ncluster in the Center for Integrated Nanotechnologies, a\nDOE Office of Basic Energy Sciences user facility.\nAppendix A: Specific Heat Measurements on YCo 5\nWe perform the specific heat measurements on poly-\ncrystalline samples of YCo 5, which were made by arc-\nmeltingtheconstituentsonawater-cooledcopperhearth.\nIt was measured down to 2 K in zero magnetic field using\na thermal relaxation method implemented in a Quantum\nDesign PPMS-9 device. The data is shown in Fig. 4. The\nSommerfeldcoefficient( γ)wasfoundtobe90mJ /mol·K2\nby fitting C/Tbelow 10 K to the form of γ+βT2+δT4.\nWe attribute γto the electronic contribution to the heat\ncapacity, whilethe lattice andmagnetic contributionsare\naccounted for by the βT2andδT4terms.\nTo obtain the mass enhancement due to strong corre-\nlations we compare the measured Sommerfeld coefficient\nto the bare density of states obtained by our DFT cal-\nculations using the generalized gradient approximation\nwith the Perdew-Burke-Ernzerhof exchange correlation\npotential [ 41]. Both the full-potential linear muffin-tin6\n400\n300\n200\n100\n0C/T (mJ/mol K2)\n4003002001000\n T2 (K2)\nFIG. 4. (Color online) C/TvsT2for YCo 5, from which we\nobtain the Sommerfeld coefficient γin theT→0 limit. The\ndashed arrow is value expected based on the band calcula-\ntions.\norbital method as implemented in the RSPt [ 36] program\nand the full-potential linearized augmented plane wave\nas implemented in Wien2k [ 42] program give the con-\nsistent results. By summing both spin contributions we\nfind the density of states at the Fermi level N(EF) = 14\nstates/eV. From this we obtain a mass enhancement\nm∗/mb=γ/(π2k2\nBN(EF)/3) = 2.7.Appendix B: X-ray Circular Magnetic Dichroism\nmeasurements on YCo 5\nThe XMCD measurements were carried out in a total\nelectron yield detection scheme at the beam line 4-ID-C\nof the Advanced Photon Source, Argonne National Lab-\noratory. The beamline 4-ID-C has the ability to generate\ncircularly polarized x-rays at the resonances of 3 dele-\nments with high degree of circular polarization ( >97%)\nby means of an electromagnetic circularly polarizing un-\ndulator, including the ability to switch polarization state\nwith a 1 Hz frequency. For the XMCD measurements\nthe samples have been ground into fine powder and been\npressed directly into electrically conducting carbon tape\nand placed in contact with a Cu holder. The Cu holder\nwaselectricallyisolatedfromthecoldfingerbyasapphire\ndisk. The samples were placed into a 7 Tesla supercon-\nducting magnet with a variable temperature insert. All\nscans were carried out at a temperature T= 20 K and\nover an energy range of 770 to 810 eV to measure the\nCoL3- andL2-edges (778.1 and 793.2 eV, respectively).\nTotal electron yield data sets µ+andµ−recorded with\nleft- and right-circularly polarized x-rays, respectively,\nwere background subtracted and edge-step normalized\n(edge is normalized to one). Moreover, each measure-\nment was carried out for magnetic fields H= 6 Tesla\ndirected along and opposite to the photon wave vector,\nrespectively, to check for experimental artifacts. 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Lett. 68, 1943 (1992).\n[44] P. Carra, B. T. Thole, M. Altarelli, and X. Wang, X-\nRay Circular Dichroism and Local Magnetic Fields , Phys.\nRev. Lett. 70, 694 (1993)."    },    {        "title": "1403.3987v1.Second_order_Perturbation_Formula_for_Magnetocrystalline_Anisotropy_using_Orbital_Angular_Momentum_Matrix.pdf",        "content": "arXiv:1403.3987v1  [cond-mat.mtrl-sci]  17 Mar 2014Typeset with jpsj3.cls <ver.1.1> Full Paper\nSecond-order Perturbation Formula for Magnetocrystallin e Anisotropy\nusing Orbital Angular Momentum Matrix\nTaichi Kosugi1,2, Takashi Miyake1, and Shoji Ishibashi1\n1Nanosystem Research Institute (NRI) “RICS”, AIST, 1-1-1 Um ezono, Tsukuba 305-8568, Japan\n2RIKEN, Advanced Institute for Computational Science, 7-1- 26, Minatojima-minami-machi,\nChuo-ku, Kobe, Hyogo 650-0047, Japan\nWe derive a second-order perturbation formula for an electronic s ystem subject to spin-\norbit interactions (SOI). The energy correction originates in the s pin-conserving and the\nspin-flip transitions. The former are represented by the orbital a ngular momentum (OAM)\nacquired via the SOI. The latter come from the quantum fluctuation effect. By using our\nformula, we examine the relativistic electronic structures of a dorbital chain and L10alloys.\nThe appearance of OAM in the chain is understood by using a parabolic -bands model and\nthe exact expressions of the single-particle states. The total en ergy is found to be accurately\nreproduced by the formula. The self-consistent fully relativistic fir st-principles calculations\nbased on the density functional theory are performed for five L10alloys. It is found that the\nformula reproduces qualitatively the behavior of their exact magne tocrystalline anisotropy\n(MCA) energies. While the MCA of FePt, CoPt, and FePd originates in t he spin-conserving\ntransitions, that in MnAl and MnGa originates in the spin-flip contribu tions. For FePt,\nCoPt, and FePd, the tendency of the MCA energy with the variation in the lattice constant\nobeys basically that of the spin-flip contributions. These results ind icate that not only the\nanisotropy of OAM, but also that of spin-flip contributions must be t aken into account for\nthe understanding of the MCA of the L10alloys.\nKEYWORDS: magnetocrystalline anisotropy, spin-orbit int eraction, first-principles calculation\n1. Introduction\nThe anisotropy of magnetic properties has been attracting m uch attention via the re-\ncent development of technological applications. The magne tic anisotropy is seen in materials\nof various geometries and dimensionalities such as bulks, n anoparticles, surfaces, and wires.\nAmong them, the anisotropic properties in periodic systems are called the magnetocrystalline\nanisotropy (MCA), which are described by the classical magn etic dipolar interactions and the\nelectron-ion interactions. Such a classical dipolar inter action is known to originate from the\nrelativistic quantum mechanical two-electron interactio n, called the Breit interaction.1Itgives\nrise to not only the effective dipole-dipole interactions, bu t also the quadrupole-quadrupole\nones contributing to the MCA.2–4We do not, however, take them into account in the present\n1/43J. Phys. Soc. Jpn. Full Paper\nstudy since these electron-electron contributions are muc h smaller than the electron-ion inter-\nactions in general. We denote the MCA coming from the electro n-ion interactions simply by\nthe MCA in the present study. It is widely accepted that the ph ysical origin of the MCA is\nthe anisotropy of the orbital angular momentum (OAM) caused by the spin-orbit interaction\n(SOI).2,5\nFor electronic structure calculations based on the density functional theory (DFT),6,7the\nforce theorem8,9ensures that MCA energy can be calculated only from the pertu rbed energy\neigenvalues for different spinconfigurations. Thistheorem h as beenused forthecalculations of\nMCA energy by introducing the SOI as perturbation into the Ko hn-Sham Hamiltonian. The\nstate-tracking method10as a way for elaboration of the calculation of MCA energy usin g the\nforce theorem has been proposed. While the force theorem is u sed primarily for obtaining the\nMCA energy in a perturbative DFT calculation, our formula pr ovided below should be used\nfor an analysis of the results for which the MCA energy has bee n obtained in self-consistent\nfully relativistic (FR) DFT calculations.\nBruno11derived a formula for the energy correction based on the seco nd-order perturba-\ntion theory for an electronic system inthe presenceof SOI.H is formulaexpresses explicitly the\nconnection between the OAM induced by the SOI and the MCA in a f erromagnet. It is often\nused for the analyses of the results obtained in model and firs t-principles calculations.12–15An\nextension of the Bruno’s formula containing the spin-flip co ntributions in an approximated\nway has been proposed.16\nIn the present study, we first derive a second-order perturba tion formula for the correction\nto the energy eigenvalue of a many-body electronic state und er SOI. The formula is reduced\nto the Bruno’s formula in a certain limit. We then examine the appearance of net OAM in\na periodic system via SOI by using a parabolic-bands model. A s applications of the formula,\nwe examine the relativistic electronic structures of two ex amples, adorbital chain by and\nL10alloys. The appearance of the net OAM and the energy correcti on for the dorbital\nchain subject to SOI is analyzed by performing tight-bindin g calculations. We pay particular\nattention to the order of perturbation for the chain. The ori gin and the behavior of the L10\nalloys are examined by performing self-consistent FR DFT ca lculations. We focus on the\ndifference in OAM and MCA energy between the alloys.\n2/43J. Phys. Soc. Jpn. Full Paper\n2. Theory\n2.1 Perturbation Hamiltonian\nLet\nH=H0+HSO (1)\nbe the many-body Hamiltonian of an electronic system. We ass ume that the spatial part\nof the many-body ground state |Ψ0/an}bracketri}htfor the unperturbed Hamiltonian H0is nondegenerate.\nThis assumption ensures that the wave function of the ground state is the same as its complex\nconjugate apart from a phase factor. Since the OAM operator Lin spatial representation is\npurely imaginary, the OAM for the ground state in this case va nishes:/an}bracketle{tΨ0|L|Ψ0/an}bracketri}ht= 0, well\nknown as the quench of OAM. The unperturbed state changes whe n the SOI represented by\nthe perturbation Hamiltonian HSOis turned on. We assume that all the electron spins in\n|Ψ0/an}bracketri}htare collinear. Since the first-order energy correction for t he ground state vanishes due to\nthe quench of OAM, the energy correction to the many-body sta te within the second-order\nperturbation can be calculated by using only the perturbed g round state:\nδE0=1\n2/an}bracketle{tΨ|HSO|Ψ/an}bracketri}ht. (2)\nWhen we adopt the single-particles picture for a periodic sy stem, however, it should be\nnoted that a nondegenerate single-particle wave function w ith a nonzero wave vector kcan\nhave an OAM even when the SOI is absent. It is because that its c omplex conjugate has a\nwave vector −k, which in general does not ensure its coincidence with the wa ve function with\nk. This fact allows each single-particle state to undergo the first-order correction of the energy\neigenvalue due to the SOI.\nWhen the valence electrons in the vicinity of each ion are spi n-polarized, the potentials\nthey feel depend on their spin directions (parallel or antip arallel to the quantization axis n)\ndue to the exchange interactions even if the SOI is absent. Wi th the SOI in the crystal turned\non, its strength thus differ for thespin direction of each elec tron since theSOI originally comes\nfrom the gradient of an electrostatic potential.30To describe such a situation, we assume that\nthe SOI is the sum of the contributions from the individual at oms in the crystal and the\nperturbation Hamiltonian takes the following generic form :\nHSO=/summationdisplay\nµQµLµ·SQµ, (3)\nwhere the hermitian operator\nQµ(n)≡/radicalig\nξ↑\nµP↑+/radicalig\nξ↓\nµP↓ (4)\n3/43J. Phys. Soc. Jpn. Full Paper\nhasbeenintroducedsothattheelectrons withadifferentspin directionfeel adifferentstrength\nof the SOI around the atom µ. The OAM operator Lµis effective only in the vicinity of the\natomµ. The spin operator S=σ/2 is the half of the Pauli matrix. P↑=|n/an}bracketri}ht/an}bracketle{tn|is the spin\nprojection operatorfor thespin-upelectrons forthequant ization axis n,whileP↓=|−n/an}bracketri}ht/an}bracketle{t−n|\nis that for the spin-down electrons. The two-component spin ors|n/an}bracketri}htand|−n/an}bracketri}htrepresent the\nspin-up and the spin-down states, respectively, whose expe ctation values of the spin operator\nare/an}bracketle{t±n|S| ±n/an}bracketri}ht=±n/2.ξ↑\nµ(ξ↓\nµ) is the strength of the SOI for the spin-up (spin-down)\nvalence electrons. Sakuma17,18calculated the strengths of SOI for each direction of electr on\nspins for an analysis of MCA. If we set ξ↑\nµ=ξ↓\nµ≡ξµ, the perturbation Hamiltonian becomes\nof the well known n-independent form, HSO=/summationtext\nµξµLµ·S.\n2.2 Derivation of Second-order Perturbation Formula\nWe decompose the OAM operator around the atom µinto the two parts as\nLµ=L/bardbl\nµ+L⊥\nµ, (5)\nwhere\nL/bardbl\nµ≡(n·Lµ)n (6)\nis the part parallel to nand\nL⊥\nµ≡Lµ−L/bardbl\nµ (7)\nis that perpendicular to n.\nThe two-component spinor for the spin-up and spin-down stat es for an arbitrary quanti-\nzation axis nare given by\n|n/an}bracketri}ht=\ncos(θ/2)\neiφsin(θ/2)\n,|−n/an}bracketri}ht=\nsin(θ/2)\n−eiφcos(θ/2)\n, (8)\nwhereθandφare the polar and the azimuthal angles of n, respectively. It is easily confirmed\nthatn·/an}bracketle{t±n|S|∓n/an}bracketri}ht= 0 and we obtain the relation\nn·S=P↑−P↓\n2. (9)\nUsing this relation, the contribution from the parallel com ponent for the energy correction is\ncalculated from eqs. (4) and (6) as,\n/an}bracketle{tQµL/bardbl\nµ·SQµ/an}bracketri}ht=1\n2n·(ξ↑\nµ/an}bracketle{tL↑\nµ/an}bracketri}ht−ξ↓\nµ/an}bracketle{tL↓\nµ/an}bracketri}ht), (10)\nwhere/an}bracketle{tLσ\nµ/an}bracketri}ht ≡ /an}bracketle{tLµPσ/an}bracketri}ht(σ= +,−) is the OAM acquired via the perturbation by the electrons\nof spinσ./an}bracketle{t·/an}bracketri}htrepresents the expectation value with respect to the pertur bed ground state. It\n4/43J. Phys. Soc. Jpn. Full Paper\ncan also be confirmed for an arbitrary σthat/an}bracketle{tPσQµL/bardbl\nµ·SQµP−σ/an}bracketri}ht= 0, which means that the\ncontribution from the parallel component within the second -order perturbation contains only\nthe spin-conserving transitions.\nUsing the relation\nPσSPσ=σ\n2nPσ, (11)\nthe contribution from the perpendicular component for the e nergy correction is calculated\nfrom eqs. (4) and (7) as,\n/an}bracketle{tQµL⊥\nµ·SQµ/an}bracketri}ht=/radicalig\nξ↑\nµξ↓\nµ/an}bracketle{tLµ·T/an}bracketri}ht, (12)\nwhere we have defined the hermitian operator\nT(n)≡P↑SP↓+P↓SP↑. (13)\nIt can also be confirmed for an arbitrary σthat/an}bracketle{tPσQµL⊥\nµ·SQµPσ/an}bracketri}ht= 0, which means that the\ncontributionfromtheperpendicularcomponentwithinthes econd-orderperturbationcontains\nonly the spin-flip transitions.\nBy substituting eqs. (10) and (12) into eq. (2), we obtain the correction to the energy of\nthe ground state,\nδE0(n) =1\n4/summationdisplay\nµn·(ξ↑\nµ/an}bracketle{tL↑\nµ/an}bracketri}ht−ξ↓\nµ/an}bracketle{tL↓\nµ/an}bracketri}ht)\n+1\n2/summationdisplay\nµ/radicalig\nξ↑\nµξ↓\nµ/an}bracketle{tLµ·T/an}bracketri}ht. (14)\nThis expression is exact within the second-order perturbat ion theory. It is clear that δE0(n)\nconsists of the three kinds of contributions: The spin-cons erving two transitions of the spin-up\nelectrons, those of the spin-down electrons, and the spin-fl ip two transitions of the electrons\nof both spin directions. When n=ez, for example, the spin-conserving contributions in eq.\n(14) symbolically correspond to the quantity ( ξ/2)/an}bracketle{tLzSz/an}bracketri}ht, while the spin-flip contributions to\n(ξ/2)/an}bracketle{tLxSx+LySy/an}bracketri}ht. For an exchange splitting ∆ Eex, thespin-flipcontribution ineq. (14) is on\nthe order of ( ξ/∆Eex)2, expected to be much smaller than the spin-conserving contr ibutions.\nIf it is true, we could neglect the spin-flip contribution. Fu rthermore, when the majority spin\nbands, assumed to be spin-up here, are completely filled and t he exchange splitting are very\nlarge, the net OAM of the perturbed spin-up states vanishes. In such a case, the formula eq.\n(14) reads\nδE0(n)≈ −1\n4/summationdisplay\nµξµn·/an}bracketle{tLµ/an}bracketri}ht, (15)\n5/43J. Phys. Soc. Jpn. Full Paper\nwhich is nothing but the well known Bruno’s formula.11\nSince the spin wave function of each electron is |n/an}bracketri}htor|−n/an}bracketri}htin the unperturbed system,\n/an}bracketle{tT/an}bracketri}ht=O(ξµ). Eqs (4) and (11) thus lead to\n/an}bracketle{tQµSQµ/an}bracketri}ht=1\n2/an}bracketle{tξ↑\nµP↑−ξ↓\nµP↓/an}bracketri}htn+O(ξ2\nµ). (16)\nRemembering that /an}bracketle{tL⊥\nµ/an}bracketri}ht=O(ξµ) since the OAM in the unperturbed system vanishes, we\nobtain/an}bracketle{tL⊥\nµ/an}bracketri}ht·/an}bracketle{tQµSQµ/an}bracketri}ht=O(ξ3\nµ). With this relation, the contribution to the energy correc tion\nfrom the perpendicular component is rewritten as\n/an}bracketle{tQµL⊥\nµ·SQµ/an}bracketri}ht=/an}bracketle{tQµ(L⊥\nµ−/an}bracketle{tL⊥\nµ/an}bracketri}ht)·SQµ/an}bracketri}ht (17)\nwithin the second-order perturbation. Since the operator L⊥\nµappears in this expression as the\ndifference between itself and its expectation value, the cont ribution from the perpendicular\ncomponent comes only from the quantum fluctuation effect. This contribution does not vanish\nin general even when /an}bracketle{tL⊥\nµ/an}bracketri}htvanishes. This result means that the contribution from the s pin-flip\ntransitions to the MCA in a ferromagnet is purely of quantum n ature. On the other hand, it\nis easily confirmed that /an}bracketle{tL/bardbl\nµ/an}bracketri}ht·/an}bracketle{tQµSQµ/an}bracketri}ht=O(ξ2\nµ), indicating that the mean-field effect can be\npresent in the contribution from the spin-conserving trans itions.\n2.3 OAM of single-particle states\nIn the derivation of the second-order perturbation formula above, the system was assumed\nto be described by a single many-body wave function. In solid state physics, however, the\nsingle-particle picture is often employed for a periodic sy stem, in which the system consists\nof the single-particle states whose occupation numbers are determined according to the Fermi\nlevel. The net OAM of the system in such a case is calculated as the sum of the contributions\nfrom the occupied single-particle states. The formula deri ved above does not take into account\nthe variation in the Fermi level via the perturbation. Here w e examine the behavior of the net\nOAM of a periodic system with SOI in detail.\n2.3.1 Major contributions to net OAM\nWe assign each of the perturbed single-particle states to fo ur groups according to its\noccupation before and after the SOI is turned on as follows. W e denote an occupied perturbed\nstate by |ψocc(unocc) →occ\ni /an}bracketri}htif it was an unperturbed occupied (unoccupied) one. We denot e\nan unoccupied perturbed state by |ψocc(unocc) →unocc\ni /an}bracketri}htsimilarly. We write the OAM of the ith\nsingle-particlestateasthesumofthecontributionsofall ordersintheSOIas /an}bracketle{tψa→b\ni|L|ψa→b\ni/an}bracketri}ht ≡\nLa→b\ni≡/summationtext∞\nn=0La→b\ni(n)(a,b= occ,unocc). Thenet OAM of theperturbedsystem is then written\n6/43J. Phys. Soc. Jpn. Full Paper\nas/an}bracketle{tL/an}bracketri}ht=/summationtext\niLunocc→occ\ni +/summationtext\niLocc→occ\ni, while the quench of OAM in the unperturbed system\nis expressed as 0 =/summationtext\niLocc→occ\ni(0)+/summationtext\niLocc→unocc\ni(0). We can thus write /an}bracketle{tL/an}bracketri}ht=/summationtext\niLunocc→occ\ni −\n/summationtext\niLocc→unocc\ni(0)+/summationtext\ni,n/negationslash=0Locc→occ\ni(n). Extracting the lowest-order contributions from each term\non the right hand side of this expression, we can write the net OAM as\n/an}bracketle{tL/an}bracketri}ht ≈/summationdisplay\niLunocc→occ\ni(0) −/summationdisplay\niLocc→unocc\ni(0) +/summationdisplay\niLocc→occ\ni(1). (18)\nThe first (second) summation on the right hand side of this exp ression is roughly proportional\nto the number of occupied (unoccupied) perturbed states whi ch was unoccupied (occupied)\nunperturbed states. These contributions are determined no t only by the perturbed energy\neigenvalues of the single-particle states, but also by the p erturbed Fermi level, which is de-\ntermined by the perturbed energy eigenvalues. The order of p erturbation contributing to the\nFermi level and the net OAM is thus not trivial even when the co rrection to the energy\neigenvalues is of the first order.\nSince each of the unperturbed single-particle states does n ot contain the zeroth order\ncontribution for the expectation value of the operator L·T, the leading contributions to the\noperator come only from the states which are occupied before and after the SOI is turned on:\n/an}bracketle{tL·T/an}bracketri}ht ≈/summationdisplay\ni(L·T)occ→occ\ni(1), (19)\nto be compared with the expression for the net OAM, eq. (18).\n2.3.2 Parabolic-bands model\nTo see the behavior of the net OAM acquired by a periodic syste m via the change in\nits Fermi level, here we examine a model consisting of two par abolic bands whose bottoms\nare close to the Fermi level. We assume that the two bands with out SOI coincide with each\nother. We further assume that each unperturbed single-part icle state in one band has an\nintrinsic OAM mand that in the other band has an OAM with the same magnitude bu t in\nthe opposite direction, −m. Such a situation may not be very special since the net OAM of\na generic system without SOI vanishes, as stated above. For s implicity, we consider a case in\nwhich the band with the OAM mundergoes a rigid shift −bξas the perturbation, first-order\nin the strength ξof the SOI, while the other band with −munderdoes a rigid shift bξ.bis\na dimensionless positive constant. The schematic illustra tion of the model is shown in Fig. 1\n(a). Since the expression of DOS for a parabolic band is known ,19the exact expressions of\nthe Fermi level and the net OAM as functions of the SOI strengt hξcan be derived. They are\nprovided in Appendix. While the leading order of variation i n the Fermi level as ξis changed\n7/43J. Phys. Soc. Jpn. Full Paper\ndepends on the dimensionality of the system [see eqs. (A ·5), (A·12), and (A ·22)], interestingly,\nthat in the net OAM is of the first order in ξregardless of the dimensionality [see eqs. (A ·6),\n(A·13), and (A ·23)]. The Fermi level and the net OAM of the perturbed systems for one-,\ntwo-, and three-dimensional systems within our model are pl otted in Fig. 1 (b) as functions of\nthe relative strength bξ/εF0of SOI with respect to the unperturbed Fermi level εF0measured\nfrom the common bottom of the unperturbed bands. For a fixed va lue of the number neof\nelectrons, there exists the critical strength of SOI above w hich the band having OAM −m\nis empty and thus the net OAM is saturated. It is found that, ev en when the strength ξof\nSOI itself is large, the net OAM is small if the unperturbed Fe rmi level is much larger than ξ.\nSimilar discussion is also applicable to the top of paraboli c bands by considering the number\nof holes.\nIf we believe that the mechanism of the appearance of net OAM f or parabolic bands exam-\nined above is true at least qualitatively also for systems ha ving generic bands, we understand\nthat the states which are close not only to the Fermi level but also to the bottoms or the tops\nof bands can contribute to the appearance of the net OAM. Such contributions are assigned\nto the first and the second terms in eq. (18) and they give the fir st-order contributions in SOI,\nas demonstrated above.\n2.4 Connection between Perturbation Formula and DFT calcula tions\nIn the present study, we perform self-consistent FR DFT calc ulations. The resultant two-\ncomponent Bloch states, which are the energy eigenstates of the FR Kohn-Sham Hamiltonian,\narein general the mixtureof the spin-upand -down states wit h respect to a given quantization\naxisn. To evaluate the right hand side of the second-order perturb ation formula, eq. (14), we\ndefine the OAM matrix of the atom µusing the occupied FR eigenstates as\nLττ′\nµ≡/summationdisplay\ni∈occ./an}bracketle{tψiτ′|Lµ|ψiτ/an}bracketri}ht (20)\nforτ,τ′=α,β. The spin indices αandβused in a DFT calculation does not necessarily\ncorrespond to the eigenstates of spin directions for the qua ntization axis n.Lµis a hermitian\nmatrix with respect to the indices τandτ′. By using the matrix representations of the spin\nprojection operator Pσ(σ=↑,↓) and the spin-flip operator T, we obtain\n/an}bracketle{tLσ\nµ/an}bracketri}ht= Tr(PσLµ), (21)\n/an}bracketle{tLµ·T/an}bracketri}ht= Tr(Lµ·T). (22)\n8/43J. Phys. Soc. Jpn. Full Paper\n012\n0 1 2 3\nbξ/εF0εF/εF0\n< L > /mne1 dim.\n012\n0 1 2 3\nbξ/εF0\n012\n0 1 2 3\nbξ/εF0εF/εF0 < L > /mne\n< L > /mne εF/εF02 dim.\n3 dim.\n2-1/3(b)εεF0D0(ε)\nOAM\nFig. 1. (a) (Color online) Schematic illustration of the density of stat es for the model consisting\nof two parabolic bands. The origin of energy is set to the common bot tom of the unperturbed\nbands. Each state in one of the bands has an OAM m, while each state in the other band has an\nOAM−m. The unperturbed two bands coincide with each other and thus the net OAM vanishes.\nWith the SOI turned on, the band with the OAM mundergoes a rigid shift −bξand the other\nband with −munderdoes a rigid shift bξ. The unperturbed Fermi level εF0is changed to the\nperturbed one εFfor the conservation of the number of electrons. (b) The Fermi le vel and the net\nOAM of the perturbed systems for one-, two-, and three-dimens ional systems within our model as\nfunctions of the relative strength bξ/εF0of SOI with respect to the unperturbed Fermi level. For\neach dimension, dashed vertical line represents the critical value o f the SOI strength above which\nthe band having OAM −mis empty.\nUsing the two-component spinor, eq (8), the matrix represen tation ofPσ’s are written as\nP↑=\ncos2θ\n2e−iφcosθ\n2sinθ\n2\neiφcosθ\n2sinθ\n2sin2θ\n2\n, (23)\nP↓=\nsin2θ\n2−e−iφcosθ\n2sinθ\n2\n−eiφcosθ\n2sinθ\n2cos2θ\n2\n. (24)\n9/43J. Phys. Soc. Jpn. Full Paper\nThe matrix representation of T, defined as eq. (13), are written as\nTx=1\n2\n−1\n2sin2θcosφ1−1\n2sin2θ(1+e−2iφ)\n1−1\n2sin2θ(1+e2iφ)1\n2sin2θcosφ\n (25)\nTy=1\n2\n−1\n2sin2θsinφ−i+i\n2sin2θ(1−e−2iφ)\ni−i\n2sin2θ(1−e2iφ)1\n2sin2θsinφ\n (26)\nTz=1\n2\nsin2θ−1\n2sin2θe−iφ\n−1\n2sin2θeiφ−sin2θ\n (27)\nThe explicit expressions of the OAM induced by the perturbat ion to the spin-up and -down\nwave functions are\n/an}bracketle{tL↑\nµ/an}bracketri}ht=Lαα\nµcos2θ\n2+Lββ\nµsin2θ\n2+Re(Lαβ\nµe−iφ)sinθ, (28)\n/an}bracketle{tL↓\nµ/an}bracketri}ht=Lαα\nµsin2θ\n2+Lββ\nµcos2θ\n2−Re(Lαβ\nµe−iφ)sinθ, (29)\nand hence /an}bracketle{tLµ/an}bracketri}ht=/an}bracketle{tL↑\nµ/an}bracketri}ht+/an}bracketle{tL↓\nµ/an}bracketri}ht= TrLµ.\nThe explicit expressions for the evaluation of OAM matrices in a DFT calculation using\na plane-wave basis set are provided in Appendix.\n3. Applications\n3.1 Tight-binding calculation for a dorbital chain\nAsthefirstexample, weexaminetheelectronic structureofa dorbital chain byperforming\ntight-binding calculations. Each site on the chain is dista nt from its neighbor by ain thez\ndirection.\n3.1.1 Hamiltonian and electronic band structure\nOnly the transfer integrals between neighboring sites are c onsidered here. We set their\nvalues astδ=−0.04,tπ= 0.18, andtσ=−0.25 eV [see Fig. 2 (a)]. The exchange splitting is\nset to ∆ ex= 3 eV. These values are the same as in the tight-binding analy sis done by Wang\net al.20for a diatomic molecule of iron.\nWe denote the diorbital (i=xy,yz,zx,x2−y2,3z2−r2) with its spin direc-\ntionn(−n) byd↑\ni(d↓\ni). We arrange the Bloch sums of these orbitals on the chain\nas{d↑\nxy,d↑\nyz,d↑\nzx,d↑\nx2−y2,d↑\n3z2−r2,d↓\nxy,d↓\nyz,d↓\nzx,d↓\nx2−y2,d↓\n3z2−r2}. With this basis set, the tight-\nbinding Hamiltonian matrix for a wave number kin the one-dimensional Brillouin zone is\n10/43J. Phys. Soc. Jpn. Full Paper\nwritten as\nH(k) =\nV(k)−∆ex\n2+H↑↑\nSOH↑↓\nSO\nH↓↑\nSOV(k)+∆ex\n2+H↓↓\nSO\n, (30)\nwhere\nV(k) = 2coska\ntδ\ntπ\ntπ\ntδ\ntσ\n. (31)\nEach component on the right hand side of eq. (30) is a 5 ×5 matrix. The transfer integral is\nnonzero only between the neighboring same orbitals due to th e symmetry. Hσσ′\nSO(σ,σ′=↑,↓)\nrepresents the matrix elements of the SOI Hamiltonian eq. (3 ) with the strength ξcommon to\nboth directions of spin. The electronic band structure of th e chain is obtained by numerically\ndiagonalizing H(k) for eachk.\nThe nonrelativistic ( ξ= 0) electronic band structure is shown in Fig. 2 (b). The ten b ands\nconsist of two band groups corresponding to the spin-up and - down states, each of which\ncontains five bands. The five bands in each band group are made u p of twoδ, twoπ, and one\nσbands due to the axial symmetry of the chain.\nThe relativistic electronic band structures with ξ= 60 meV are shown in Fig. 2 (c) for\nn=exandez. It is seen that the features of band structures for the differe nt spin directions\nare different from each other due to the SOI.\n3.1.2 OAM\nHere we examine the behavior of the net OAM acquired by the sys tem via the variation\nin the strength ξof SOI and the Fermi level. The density of states for the nonre lativistic band\nstructure of the chain is plotted in Fig. 3 (a). For the number s of electrons per site ne= 3,5,\nand 6, the net OAM and the expectation values of /an}bracketle{tL·T/an}bracketri}htas functions of ξare plotted in Fig.\n3 (b).\nLet us analyze the OAM in detail for the case of the electron sp ins in thezdirection ( n=\nez) since the exact eigenstates of the Hamiltonian in this case can be obtained analytically,\nas provided in Appendix. The lowest-order contribution to t he expectation value of Lzof\nevery energy eigenstate in this case is of the second order in ξ[see eqs. from (B ·47) to (B ·56)].\nDespite that, the behaviors of /an}bracketle{tL↑\nz/an}bracketri}htforne= 3 and/an}bracketle{tL↓\nz/an}bracketri}htforne= 6 are obviously not quadratic,\nas seen in Fig. 3 (b). Those of /an}bracketle{tL↑\nz/an}bracketri}htand/an}bracketle{tL↓\nz/an}bracketri}htforne= 5 are, on the other hand, quadratic.\n11/43J. Phys. Soc. Jpn. Full Paper\n(b)\n(c)\n11.21.41.61.82\nka/π1/2 -1/2 0(eV)ξ = 60 meV, n=ex(a)\nka/π1/2 -1/2 011.21.41.61.82ξ = 60 meV, n=ez(eV)\n-2-1012ξ = 0\nka/π1/2 -1/2 0π \u0001\nπ↑δ\u0000\nδ↑σ\u0002\nσ↑tσ\nd3z -r2 2tπ\ndzxtδ\nz\nyx\ndx -y2 2\n(eV)\nFig. 2. (a) Schematic illustration of the transfer integrals used for thedorbital chain. (b) The non-\nrelativistic ( ξ= 0) electronic band structure of the chain accommodating six elect rons per site.\nThe transfer integrals used are tδ=−0.04,tπ= 0.18, andtσ=−0.25 eV. The exchange splitting\nis set to ∆ ex= 3 eV. (c) Solid curves represent the relativistic electronic band st ructures with\nξ= 60 meV for n=exandezon the upper and the lower panels, respectively. The relativistic\nbands coming from the nonrelativistic spin-down states, whose dire ction is−n, are shown in the\nfigures. The nonrelativistic bands are also shown as the dashed cur ves.\n12/43J. Phys. Soc. Jpn. Full Paper\nThese results clearly indicate that the major contribution s for thene= 3 and 6 cases are from\nthe perturbed states that has moved through the Fermi level w hen the SOI was turned on.\n-1.2-0.8-0.40\n00.40.81.21.6\n<L\n\u0003\nx>\n<L↓\nx>  ne=3, n=ex(b)\n-0.2-0.10\n<L T > .\n-40-30-20-100\nspin-cons. + spin-flipspin-flip\nspin-cons.\nexact(c)-1.6-1.2-0.8-0.40\n00.20.40.60.8\n(meV)0 20 40 60 80 100\nξ (meV)-0.16-0.080\n-60-40-200ne=3, n=ez\n(a.u.)0481216\n-2 -1 0 1 2(eV-1) (a)\nne= 3 \nε (eV)σ \u0004π\u0005δ\u0006\nπ\u0007\nσ \bδ\t\nπ\nπ\u000b\nσ \fσ \rδ\u000e δ\u000f\nspin-flip\nspin-cons.\nspin-cons. + spin-flip\nexact0123\n0123\n-0.3-0.2-0.10\n-16-12-8-40012\n0123\n-0.3-0.2-0.10\n-16-12-8-40ne=5, n=ex ne=5, n=ez\nspin-cons.\nspin-flip\nspin-cons. + spin-flip\nexactspin-cons.\nspin-flip\nspin-cons. + spin-flip\nexact0123\n00.10.20.30.40.5\n-0.3-0.2-0.10\n-30-20-1000.40.6\n00.20.81\n00.040.080.12\n-0.2-0.10\n-16-12-8-40ne=6, n=ex ne=6, n=ez\nspin-flipspin-cons.\nspin-cons. + spin-flipspin-flipspin-cons.\nspin-cons. + spin-flip\nexact exactne= 5 ne= 6 \n<L\n\u0010\nz>\n<L↓\nz>  \n<L\n\u0011\nx><L↓\nx>  < L↓\nz>  \n<L\n\u0012\nz><L↓\nx>  \n<L\n\u0013\nx><L↓\nz>  \n<L\n\u0014\nz>\n<L T > . <L T > . <L T > . <L T > . <L T > .(10-3 a.u.)  (10-3 a.u.)  (10-3 a.u.)  (10-3 a.u.)  (10-3 a.u.)  (10-3 a.u.)  (10-3 a.u.)  (a.u.) (a.u.) (10-3 a.u.)  (a.u.)\n(a.u.) (a.u.) (a.u.) (a.u.) (a.u.) (a.u.)\n0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)\n0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)0 20 40 60 80 100\nξ (meV)\nFig. 3. (a) The density of states for the nonrelativistic band struc ture of the chain. Gaussian broad-\nening was used. The vertical dashed lines represent the Fermi leve ls for the numbers of electrons\nper sitene= 3,5,and 6. For ne= 5, the Fermi level lies in the exchange splitting. (b) The net\nOAM and the expectation values of /an}bracketle{tL·T/an}bracketri}htas functions of ξfor the spin directions along the x\nand thezaxes. (c) The contributions to the second-order perturbation f ormula as functions of ξ,\ntogether with the numerically exact correction to the total energ y.\nWe define the OAM density as\nDL(ε)≡/summationdisplay\nk,i/an}bracketle{tψki|L|ψki/an}bracketri}htδ(ε−εki)f(ε;εF), (32)\ncalculated from the perturbed single-particle states and t he corresponding enregy eigenvalues\nfor a given quantization axis n.f(ε;εF) is the Fermi distribution function with the Fermi level\n13/43J. Phys. Soc. Jpn. Full Paper\nεFof the perturbed system. In addition, we define the accumulat ed OAM density as\nLacc(ε)≡/integraldisplayε\n−∞dε′DL(ε′). (33)\nThe net OAM is clearly the accumulated OAM density up to the Fe rmi level: /an}bracketle{tL/an}bracketri}ht=Lacc(εF).\nFor the operator L·T, we define DL·T(ε) andL·Tacc(ε) similarly to the OAM.\nForn=ezandξ= 60 meV, the OAM densities and their accumulated values as fu nctions\nof energy for ne= 3,5, and 6 are plotted in Fig. 4. The densities and the accumulat ed values\nof the operator L·Tare also shown. It is seen for ne= 3,5, and 6 that DL↑\nz(ε) oscillates\nstrongly around the origin. These large amplitudes come mai nly from the intrinsic OAM of\nthe single-particle states, that is, the nonzero OAM presen t even when the SOI is absent [see\neqs. from (B ·47) to (B ·56)].\nWhen the OAM density is integrated for ne= 5, however, these zeroth contributions\nvanish. Due to the absence of the first-order contributions [ see eqs. from (B ·47) to (B ·56)],\nthe leading contribution to the net OAM of the spin-up states ,/an}bracketle{tL↑\nz/an}bracketri}ht, forne= 5 is of second\norder inξ, leading to the rather small /an}bracketle{tL↑\nz/an}bracketri}ht. The net OAM of the spin-down states, /an}bracketle{tL↓\nz/an}bracketri}ht, is\nalso nonzero and of second order since the lower five bands con tain the spin-down components\nwhen the SOI is present.\nForne= 3,/an}bracketle{tL↑\nz/an}bracketri}htis much larger in magnitude than that in the case of ne= 5. It is because\nthat/an}bracketle{tL↑\nz/an}bracketri}htin this case comes mainly from the variation in the occupatio n numbers of the states\nnear the Fermi level, which corresponds to the first and secon d terms on the right hand side of\neq. (18). As stated above, those terms are of first-order in ξ. Since each of the single-particle\nstates does not contain the first-order contribution to Lz, the magnitude of L↑\nzacc(ε) becomes\nlarger steeply as εapproaches the Fermi level, as seen in Fig. 4 (a). /an}bracketle{tL↓\nz/an}bracketri}htin this case is much\nsmaller in magnitude than /an}bracketle{tL↑\nz/an}bracketri}htsince the spin-down components of the single-particle stat es\nin the lower five bands do not have the intrinsic OAM and thus /an}bracketle{tL↓\nz/an}bracketri}htis of second order in ξ\nas well as for ne= 5.\nForne= 6,/an}bracketle{tL↓\nz/an}bracketri}htis much larger than /an}bracketle{tL↑\nz/an}bracketri}htin contrast to the case of ne= 3. It is because\nthat/an}bracketle{tL↓\nz/an}bracketri}htin this case comes mainly from the variation in the occupatio n numbers of the states\nnear the Fermi level. /an}bracketle{tL↑\nz/an}bracketri}htin this case is much smaller than /an}bracketle{tL↓\nz/an}bracketri}htsince the spin-up components\nof the single-particle states in the higher five bands do not h ave the intrinsic OAM and thus\n/an}bracketle{tL↑\nz/an}bracketri}htis of second order in ξas well as for ne= 5. It is found that the magnitude of /an}bracketle{tL↓\nz/an}bracketri}ht\nforne= 6 is much smaller than that of /an}bracketle{tL↑\nz/an}bracketri}htforne= 3. We can understand this result by\nconsidering the parabolic-bands model discussed above, wh ich indicates that the Fermi level\nclose to a band bottom leads to a larger net OAM. As seen in Fig. 3 (a), the Fermi level for\n14/43J. Phys. Soc. Jpn. Full Paper\nne= 3 is closer to its nearest peak of DOS than that for ne= 6 is.\nThe leading contributions to /an}bracketle{tL·T/an}bracketri}htfrom the energy eigenstates in this case are of first\norder inξ[see eqs. from (B ·57) to (B ·66)]. Since the leading contributions to the net /an}bracketle{tL·T/an}bracketri}ht\ncome from all the states below away the Fermi level [see eq. (1 9)], the variation in L·Tacc(ε)\nwith the increase in εis mild compared to that in Lacc(ε) for all the ne’s, as seen in Fig. 4.\nne=3 (a)\n(a.u. eV-1) (a.u.)\n-10010\n-2-1012\nL\n\u0015\naccL↓\nacc\n-0.4-0.200.20.4\n-0.2-0.100.10.2\nε (eV)-2 -1 0 1 2DL T.\nL Tacc.DL↓\nDL↑ne=5 (b)\n-0.4-0.200.20.4\n-0.2-0.100.10.2\nε (eV)-2 -1 0 1 2L Tacc.-0.002-0.00100.0010.002\n-10010(a.u. eV-1) (a.u.)\nL↑\naccL↓\nacc\nDL↓\nDL↑\nDL T.ne=6 (c)\n-0.4-0.200.20.4\nε (eV)-2 -1 0 1 2L Tacc.\n-0.2-0.100.10.2-0.100.1\n-10010(a.u. eV-1) (a.u.)\nL↓\nacc\nL↑\naccDL↑\nDL↓\nDL T.\nFig. 4. (Coloronline)The OAMdensities(solidcurves)and theiraccum ulatedvalues(dashedcurves)\nin the perturbed system with n=ezandξ= 60 meV as functions of energy for ne= 3,5, and\n6. The densities and the accumulated values of the operator L·Tare also shown. For each of the\nne’s, the vertical line represents the Fermi level of the unperturbe d system.\n3.1.3 Analysis using the perturbation formula\nWe definethe total energy of the system simply as the sumof the energy eigenvalues of the\noccupied states. The total-energy corrections due to the SO I calculated from the second-order\nperturbation formula, eq. (14), are shown in Fig. 3 (c) as fun ctions ofξ. It is seen for ne= 5\nthat the values calculated using the second-order formula a re in excellent agreement with\nthe numerically exact values for ξ’s in the adopted range. The agreement between the values\nobtained by the formula and the exact values for ne= 6 are also rather good. The accurate\nreproduction of the numerically exact values is achieved on ly when the spin-conserving and\nthe spin-flip contributions are incorporated together, whi ch indicates that the the Bruno’s\nformula, eq. (15), does not capture the relativistic physic s accurately in this case.\nThe deviation of the values obtained by the formula from the e xact values is, however,\nfound to be much larger for ne= 3 than for ne= 6. These results can be understood via\nconsideration similar to that of the net OAM. For the case of ne= 5, the Fermi level lies in\nthe exchange gap, which ensures that the SOI as perturbation does not allow the unperturbed\n15/43J. Phys. Soc. Jpn. Full Paper\nstatesneartheFermilevel togothroughitwhentheSOIistur nedon.Theoccupationnumbers\nof the unperturbed states are thus unchanged before and afte r the SOI is turned on, so that\nthe energy correction formula, eq. (14), immediately appli es in such a case and gives the exact\ncorrection to the total energy within the second-order pert urbation. For the cases of ne= 3\nand 6, on the other hand, the Fermi level lies in the bands. The effects of the variation in\nthe occupation numbers of the states near the Fermi level via the SOI are thus present in\nsuch cases. These effects are not taken into account in the form ula at all, as discussed above.\nThe correction to the total energy of the system thus cannot b e explained completely by the\nformula even when the SOI is weak enough to be treated within s econd-order perturbation.\n3.2 Density Functional Theory Calculation for L10Alloys\nAs the second example, we perform first-principles electron ic structure calculations based\non DFT for five L10alloys, FePt, CoPt, FePd, MnAl, and MnGa. We examine their MC A\nsystematically by employing the second-order perturbatio n formula.\n3.2.1 Crystal structure\nThe crystal structure of an L10alloy is depicted in Fig. 5. The basal lattice constants afor\ntheL10alloys are fixed at the experimental values, 3 .8600˚A for FePt,213.81˚A for CoPt,22\n3.89˚A for FePd,233.92˚A for MnAl,24and 3.8974˚A for MnGa,25throughout the present\nstudy.\nac\nxzy\nFig. 5. Crystal structure of an L10alloy. The white and the shaded balls represent atoms of different\nkinds. The structure is regarded as stacked atomic layers spaced byc. The thinner solid lines\nrepresent the conventional cell, while the thicker solid lines represe nt the primitive cell. The two\nkinds of atoms are located at the crystallographically equivalent pos itions. The distance between\nthe nearest neighbor atoms of the same kind is a/√\n2. Thexand thezdirections correspond to\nthe [100] and the [001] directions, respectively.\n16/43J. Phys. Soc. Jpn. Full Paper\n3.2.2 Computational details\nWe adopt the projector augmented-wave (PAW) method26using the QMAS (Quantum\nMAterials Simulator) package27within the local-spin-density approximation (LSDA).28We\nperform fully relativistic calculations for periodic syst ems29using two-component pseudo\nBloch wave functions as\n|ψmk/an}bracketri}ht=\n|ψmkα/an}bracketri}ht\n|ψmkβ/an}bracketri}ht\n, (34)\nwheremandkare a band index and a wave vector, respectively. αandβare spin indices.\nIn the present study, the pseudo wave functions are expanded in plane waves with an energy\ncutoff of 35 Ry. The total energy of the system is calculated as a functional of the 2 ×2 density\nmatrix defined in real space representation as\nρττ′(r) =occ./summationdisplay\nm,kψmkτ(r)∗ψmkτ′(r), (35)\nwhereτ,τ′=α,β. In an FR calculation, noncollinear magnetism and spin-orb it interaction\ncan be naturally introduced.\nWhenwesolvetheDiracequationforanisolatedatom30fortheconstructionofapotential,\nwe can continuously move from the scalar relativistic ( λ= 0) to the fully relativistic ( λ= 1)\nequation by varying the dimensionless parameter λof the SOI.31With turning on or off\nthe SOI of each element for the potentials, we can calculate t he MCA energy only with the\nSOI around the atoms of a specific element. For example, in FeP t,EMCA(λFe= 1,λPt= 0)\ninvolves only the transitions between the states at the Fe at oms caused by the SOI around\nthe Fe atoms. We can extract the MCA energy coming only from th e interspecies transitions\nas\nEMCA(betweenFeandPt) = EMCA(λFe= 1,λPt= 1)\n−EMCA(λFe= 1,λPt= 0)−EMCA(λFe= 0,λPt= 1). (36)\nAlthough each transition in a perturbation process occurs a t an atom due to the localized\neffectiveness of SOI, the interspecies contributions do not v anish in general. It is because that\nthe atomic orbitals of the different species in the L10alloys extend to induce the hybridization\nwith each other, which allows an electron to travel between t he atoms of different species, as\nillustrated in Fig. 6. It is clear that the interspecies cont ributions do not contain the influence\nof the first-order transitions.\nThe strengths ξof the SOI of each atom used for the second-order perturbatio n formula\nare estimated from the energy eigenvalues obtained in the DF T calculation for an isolated\n17/43J. Phys. Soc. Jpn. Full Paper\nεFe\nspin-up spin-down(a)\n(b)\nε εPt Fe\nspin-up spin-down spin-up spin-down\nFig. 6. Schematic illustration of transitions in second-order pertur bation processes for FePt. The\ndomes represent the density of states for each element, whose s haded areas represent the occupied\nstates. For the spin-conserving transition of a spin-up electron in FePt, two kinds of transitions\nexist. (a) The one occurs between the states only at the Fe atoms , (b) while the other occurs\nbetween the Fe and the Pt atoms. Since the atomic orbitals of Fe and Pt are hybridized in the\nBloch states, an electron can travel between the atoms of differe nt species via the second-order\nperturbation.\natom. The low-energy expansion of the Dirac equation for a ce ntral potential Vleads to the\nfollowing SOI Hamiltonian of the well known form:30\nHSO=1\n2m2c21\nrdV\ndrL·S (37)\nwith the mass mof a particle. In this case, an energy eigenstate is characte rized by the\nprincipal quantum number n, the orbital angular momentum lof the large component, the\ntotal angular momentum jand itszcomponent, jz. The first-order correction coming from\nHSOto the unperturbed energy eigenvalue Enljis thus the diagonal matrix element:\n∆Enlj=ξnl\n2/bracketleftigg\nj(j+1)−l(l+1)−3\n4/bracketrightigg\n, (38)\nwhereξnl≡(2m2c2)−1/an}bracketle{tr−1dV/dr/an}bracketri}htnl. The energy splitting between the states with a common\nlis calculated as\n∆Enl= ∆Enll+1/2−∆Enll−1/2=ξnl/parenleftigg\nl+1\n2/parenrightigg\n. (39)\nOur estimated values are as follows: ξ3d= 61 meV for Fe, ξ5d= 570 meV for Pt, ξ3d= 74\nmeV for Co, ξ4d= 240 meV for Pd, ξ3d= 48 meV for Mn, ξ3p= 11 meV for Al, and ξ4p= 81\nmeV for Ga.\n18/43J. Phys. Soc. Jpn. Full Paper\nIt is known that the ordinary DFT functional does not contain terms responsible for the\nHund’s second rule, which requires that not only the spin but also the orbital angular mo-\nmentum of a system be maximized. Jansen32demonstrated that the energy functional must\ncontain the term which depend explicitly on the OAM for the de scription of the Hund’s sec-\nond rule for the accurate reproduction of the measured magne tism. The orbital polarization\ncorrection (OPC) for a DFT calculation was introduced by Bro oks33employing the vector\nmodel.34This prescription has been applied not only to 5 fnarrow-band compounds35but\nalso toL10alloys.12,36There also exist, on the other hand, criticisms stating that the under-\nestimation of OAM in DFT calculations using ordinary functi onals such as LDA and GGA\ncomes from different physics than that assumed in the OPC metho d.37–39We do not take into\naccount the OPC in the present study, since our main purpose i s to examine the validity and\nthe reliability of the second-order perturbation formula d erived above by comparing it with\nthe total energies obtained in self-consistent FR DFT calcu lations.\n3.2.3 MCA energies in DFT total-energy calculations\nWe define the MCA energy of an L10alloy as the difference in total energy between the\nspins of the transition metal atoms along the [100] and the [0 01] direction:\nEMCA≡E[100]−E[001]. (40)\nA positive (negative) EMCAmeans the magnetization easy axis along the [001] (the [100] )\ndirection. In the present study, we calculate the MCA energi es by calculating the differences\nin total energy between the self-consistent FR DFT calculat ions for differenet spin directions.\nThe force theorem8,9cannot be used in our case since its mathematical validity is ensured\nonly for a perturbative DFT calculation in which the charge d ensity is not relaxed.\nFor each of the five L10alloys, we obtained the MCA energy in self-consistent FR DFT\ntotal-energy calculations and show them in Table I, togethe r with the results in the literature.\nThe reasonable agreement between our results and the previo us results is seen. The easy axes\nare in the [001] directions for all the systems. It is noticed that the Pt and the Pd atoms\nhas the significant magnitudes of spins, though they are not m agnetic elements. Their spins\noriginate in the hybridization between the dorbitals at the magnetic and the nonmagnetic\nelements. For each atom in of each system, the magnitude of th e spin is found to be almost\nunchanged when its direction is changed, while that of the OA M exhibits anisotropy for FePt,\nCoPt, and FePd. One could expect that the MCA in these three al loys comes directly from\nthe anisotropy of OAM. MnAl and MnGa exhibit, however, the MC A energy larger than that\n19/43J. Phys. Soc. Jpn. Full Paper\nof FePd despite their much weaker anisotropy of OAM. These ob servations suggest that the\norigin ofL10alloys need to be examined in more detail.\n3.2.4 Analysis using the perturbation formula\nFor each of the five L10alloys, we obtained the MCA energy in the self-consistent FR\nDFT total-energy calculations as a function of c/ais plotted in Fig. 7 (a). The contributions\nto the second-order perturbation formula, eq. (14), calcul ated from the OAM matrices, eq.\n(20), are also plotted. Though the variation in the Fermi lev el caused by the SOI is not taken\ninto account in the formula, it is seen for all the five systems that the formula reproduces\nqualitatively rather well the behavior of the exact values. These results for the L10alloys\nencourage us to use the second-order perturbation formula a s a tractable tool for the analyses\nof MCA described in a self-consistent FR DFT calculation.\nIt is found for each of the systems in Fig. 7 (a) that the spin-c onserving transitions\ncontribute in favor of the spins in the zdirection. The spin-flip transitions for FePt, CoPt,\nand FePd with their experimental lattice constants, howeve r, contribute in favor of the spins\nin thexdirection, while those for MnAl and MnGa in favor of the spins in thezdirection.\nThe magnitudes of the spin-conserving contributions in FeP t and CoPt are larger than the\nspin-flip contributions, while the former contributions ar e much smaller in MnAl and MnGa.\nThemagnitudes ofthespin-conservingandthespin-flipcont ributions arecomparableinFePd.\nThe weak anisotropies of OAM in MnAl and MnGa (see Table I) are reflected in their small\nspin-conserving contributions, which indicate that their MCA come mainly from the spin-flip\ntransitions.\nAsc/aincreases, the MCA energies of the systems except for CoPt te nd to become higher.\nRoughly speaking, these tendencies come from the reduction of dimensionality in the systems\nleading to the more localized valence electrons and the more effective SOI, which reinforce\ntheir MCA. It is observed in Fig. 7 (a) that the tendencies in F ePt and FePd are from those\nof the spin-flip transitions, while the tendencies in MnAl an d MnGa are from those of the\nspin-conserving transitions.\nFig. 7 (b) shows the MCA energies in the self-consistent FR DF T total-energy calculations\nas functions of c/awith and without SOI of each element. Those coming only from t he\ninterspecies transitions are calculated by using eq. (36) a nd also plotted. For FePt and CoPt,\nthe contributions from the transitions between the magneti c atoms have only minor effects\non the MCA energy. In contrast, the MCA energy of MnAl and MnGa comes mainly from\nthe transitions between the Mn atoms. For FePd, the transiti ons among the same species\n20/43J. Phys. Soc. Jpn. Full Paper\ncontribute to the MCA energy comparably to the interspecies transitions. Is is found that the\nlarger MCA of FePt and CoPt than FePd comes from the presence o f the Pt atoms, which\nhas the quite large strength ξof SOI.\nSince the spin-flip contributions in MnAl and MnGa are much la rger than the spin-\nconserving contributions [see Fig. 7 (a)], we understand th at their MCA energies come mainly\nfrom the spin-flip transitions occurring only around the Mn a toms. It is the reason for the\nmuch smaller MCA energy of FePd than those of MnAl and MnGa des pite the stronger\nanisotropy of OAM in FePd (see Table I). Although stronger an isotropy of OAM, /an}bracketle{tL/an}bracketri}ht, itself\nimplies a larger contribution to the spin-conserving terms in the formula, eq. (14), it does not\nnecessarily imply a larger contribution to the spin-flip ter ms,/an}bracketle{tL·T/an}bracketri}ht. These results indicate\nthat the Bruno’s formula is not sufficient for an analysis of th e MCA ofL10alloys.\n3.2.5 OAM of FePt, CoPt, and FePd\nFor each atom in FePt, CoPt, and FePd with the electron spins i n thexand thez\ndirections, the OAM projected in the spin direction possess ed by the spin-up and the spin-\ndown electrons are plotted in Fig. (8) as functions of c/a. In each atom µin all the three\nsystems, the magnitude /an}bracketle{tL↑\nµ/an}bracketri}htof the OAM of the spin-up electrons is smaller than that of\nthe spin-down electrons, /an}bracketle{tL↓\nµ/an}bracketri}ht. It is because that the spin-up bands has the larger occupanc y\nthan the spin-down bands and thus the spin-up electrons havi ng opposite OAM among them\nhave the stronger tendency to cancel their net OAM. The OAM of the electrons in each spin\ndirection for the three systems are in the opposite directio n to the spin, though their larger\n/an}bracketle{tL↓\nµ/an}bracketri}htdominate over their smaller /an}bracketle{tL↑\nµ/an}bracketri}htto give rise to the net OAM in the same direction as\nthe spins. The magnitudes /an}bracketle{tLµ/an}bracketri}htof the net OAM at the atom µbecome larger as c/aincreases,\nexcept for the Pt atom in FePt. These tendencies of increase a re the reflections of those of\n/an}bracketle{tL↓\nµ/an}bracketri}ht, while/an}bracketle{tL↑\nµ/an}bracketri}htdoes not exhibit tendency of increase. Since the spin-down s tates at the Fe\nor the Co atoms hybridize strongly with the states of both spi n directions at the Pt or the Pd\natoms,12the OAM of the spin-down states are influenced sensitively by the variation in the\ndistance between the atomic layers. The OAM of the spin-up st ates, on the other hand, are\ninfluenced less sensitively by the interlayer distance than those of the spin-down states and\nthey do not necessarily exhibit tendency of increase as c/aincreases.\nAs found above, the exact MCA energy of CoPt for 0 .91< c/a < 0.98 exhibits the\ntendency of decrease [see Fig. 7 (a)], in contrast to FePt and FePd. This tendency does not\ncome from that in the spin-conserving contributions, which behave similarly in all the three\nsystems. The similarity of their behaviors is a direct conse quence of that of the anisotropy\n21/43J. Phys. Soc. Jpn. Full Paper\n-10123\n0.92 0.96 1exact\nspin-cons.\nspin-flipspin-cons. + spin-flip(meV)\nc/aFePt\n0.94 0.98 1.02 0.9MCA energy\n-20246\n0.92 0.96 1\nc/a0.94 0.98 1.02 0.9(meV) FePt\nPt SOI only\nFe SOI only\nbetween Fe and Pt onlyboth SOI onMCA energy\n-0.2-0.100.10.2\n0.88 0.9 0.92 0.94 0.96 0.98 1FePd\nexactspin-cons.\nspin-flipspin-cons. + spin-flip\nc/a(meV) MCA energy\n-0.8-0.400.4\n0.88 0.9 0.92 0.94 0.96 0.98 1\nc/aFePd\nboth SOI onPd SOI only\nFe SOI only\nbetween Fe and Pd only(meV)MCA energy-0.500.511.52\n0.92 0.96 1CoPt\n0.94 0.98 1.02exactspin-cons.\nspin-flipspin-cons. + spin-flip\nc/a(meV) MCA energy\n-4-2024\n0.92 0.96 1 0.94 0.98 1.02\nc/a(meV) CoPt\nPt SOI only\nCo SOI only\nbetween Co and Pt onlyMCA energyboth SOI on\n00.10.20.3\n0.84 0.86 0.88 0.9 0.92 0.94 0.96MnAl\nexact\nspin-cons.spin-flipspin-cons. + spin-flip\nc/a(meV) MCA energy\n00.10.20.3\n0.84 0.86 0.88 0.9 0.92 0.94 0.96\nc/aMnAl\nboth SOI on\nMn SOI only\nbetween Mn and Al only\nAl SOI only(meV)MCA energy\n00.10.20.30.4\n0.88 0.9 0.92 0.94 0.96 0.98MnGa\nexact\nspin-cons.spin-flipspin-cons. + spin-flip\nc/a(meV) MCA energy\n00.10.20.30.4\n0.88 0.9 0.92 0.94 0.96 0.98\nc/aMnGa (meV)MCA energyMn SOI onlyboth SOI on\nbetween Mn and Ga only\nGa SOI only(a) (b)\nFig. 7. (Color online) (a) On the left panel for each of the five L10alloys, the exact MCA energy\ncalculated as the difference in self-consistent FR DFT total energy between the electron spins in\nthezand thexdirections as a function of c/ais plotted. The contributions to the second-order\nperturbation formula calculated from the OAM matrices are also plot ted. The vertical dashed line\ncorresponds to the experimental lattice constants for each sys tem. (b) On the right panels, the\nexact MCA energies calculated with and without SOI of each element a re plotted.\n22/43J. Phys. Soc. Jpn. Full Paper\nof OAM in the three systems (see Fig. 8). To identify the origi n for this tendency, the MCA\nenergy of CoPt with the SOI only of the Pt atoms are plotted in F ig. 9. It is seen in the\nfigure that the exact MCA energy decreases monotonically as c/aincreases, which should be\nregarded as the origin of the decrease in the MCA of CoPt seen i n Fig. 7 (a). The spin-flip\ntransitions occurring at the Pt atoms thus explain the decre ase in the MCA energy of CoPt.\n4. Conclusions\nWe derived succinctly a second-order perturbation formula for the correction to the en-\nergy eigenvalue of a many-body electronic system subject to SOI. The energy correction was\ndemonstrated to consist of three kinds of contributions: th e spin-conserving transitions of\nthe spin-up electrons, those of the spin-down electrons, an d the spin-flip transitions of the\nelectrons of both spin directions. The first two kinds of cont ributions are represented by\nthe OAM acquired by the valence electrons via the SOI The othe r kind of contributions was\nfound to come from the quantum fluctuation effect. In the limit o f strong exchange interaction\nwith completely filled majority spin bands, the formula deri ved is reduced to the well known\nBruno’s formula. Since it uses only the wave functions of the perturbed system, it serves as\na tractable tool for the analyses of phenomena in which SOI pl ays important roles. In partic-\nular, our formula provides a reliable way to capture essenti al physics of MCA. By using our\nperturbation formula, we examined the relativistic electr onic structures of two examples, a d\norbital chain and L10alloys.\nThe tight-binding calculations were performed for the dorbital chain. The appearance of\nOAM in the chain was clearly understood by using the paraboli c-bands model and the exact\nexpressions of the single-particle states. The total energ y as the sum of the energy eigenvalues\nof the single-particle states were found to be rather accura tely reproduced by the formula,\nthough the formula does not take into account the variation i n the Fermi level.\nThe first-principles calculations based on DFT were perform ed for the five L10alloys and\ntheir MCA energies were analyzed by using the OAM matrices. W e found that the formula\nreproducesqualitatively the behavior of the exact MCA ener gies of the alloys. While the MCA\nof FePt, CoPt, and FePd was found to originate in the spin-con serving transitions, that in\nMnAl and MnGa was found to originate in the spin-flip contribu tions. For FePt, CoPt, and\nFePd, the tendency of the MCA energy with the variation in c/awas found to obey basically\nthat of the spin-flip contributions. These results indicate that not only the anisotropy of OAM\n/an}bracketle{tL/an}bracketri}htitself, but also that of spin-flip contribution, /an}bracketle{tL·T/an}bracketri}ht, must be taken into account for the\nunderstanding of the MCA of the L10alloys.\n23/43J. Phys. Soc. Jpn. Full Paper\n-0.028-0.024-0.02-0.016Fe\n0.060.070.080.09\n0.050.060.07Pt\n-0.19-0.18-0.17-0.16\n0.2160.220.2240.228\n0.040.0440.0480.0520.056(a) FePt\n< L↑\nz >  < L↑\nx >  \n< L↓\nz >  \n< L↓\nx >  \n< Lx >  \n< Lz >  < L↑\nx >  < L↑\nz >  \n< L↓\nz >  \n< Lz >  \n< Lx >  < L↓\nx >  \n0.92 0.96 1\nc/a0.94 0.98 1.02 0.9 0.92 0.96 1\nc/a0.94 0.98 1.02 0.9(a.u.) (a.u.)\n-0.032-0.028-0.024-0.02Co\n0.080.10.12\n0.040.060.080.1-0.18-0.172-0.164Pt\n0.060.070.08\n0.92 0.96 1 0.94 0.98 1.02\nc/a(b) CoPt\n0.2340.2420.25\n0.92 0.96 1 0.94 0.98 1.02\nc/a(a.u.) (a.u.)\n-0.019-0.017-0.015Fe\n0.0760.080.0840.088\n0.0560.0640.072-0.052-0.05-0.048Pd\n0.0740.0780.082\n0.0240.0260.0280.03(c) FePd\n0.88 0.9 0.92 0.94 0.96 0.98 1\nc/a0.88 0.9 0.92 0.94 0.96 0.98 1\nc/a(a.u.) (a.u.)< L↑\nx >  < L↑\nz >  \n< L↓\nz >  \n< L↓\nx >  \n< Lz >  \n< Lx >  < L↑\nx >  \n< L↑\nz >  \n< L↓\nx >  \n< L↓\nz >  \n< Lx >  \n< Lz >  \n< L↑\nz >  \n< L↑\nx >  \n< L↓\nz >  \n< L↓\nx >  \n< Lz >  \n< Lx >  < L↑\nx >  \n< L↑\nz >  \n< L↓\nx >  \n< L↓\nz >  \n< Lx >  \n< Lz >  \nFig. 8. (Color online) For each atom in (a) FePt, (b) CoPt, and (c) Fe Pd with the electron spins in\nthexand thezdirections, the net OAM projected in the spin direction possessed b y the spin-up\nand the spin-down electrons are plotted as functions of c/a. The sums of the net OAM of the\nelectrons of both spin directions are also plotted. The dashed vert ical lines correspond to the\nexperimental lattice constants.\nSince the implementation of the OAM matrix is straightforwa rd and it requires only the\nFR Bloch wave functions, the second-order perturbation for mula derived in the present study\nis tractable and useful for self-consistent FR DFT calculat ions.\n24/43J. Phys. Soc. Jpn. Full Paper\n012345\n0.92 0.96 1 0.94 0.98 1.02\nc/a(meV)MCA energyexactCoPt with Pt SOI only\nspin-cons.spin-flipspin-cons. + spin-flip\nFig. 9. (Color online) For CoPt with SOI only of Pt atoms, the exact MC A energy calculated as\nthe difference in self-consistent FR DFT total energy between the electron spins in the zand the\nxdirections as a function of c/ais plotted. The contributions to the second-order perturbation\nformula calculated from the OAM matrices are also plotted. The vert ical dashed line corresponds\nto the experimental lattice constants.\nAcknowledgement\nTheauthors thank H. Kino, T. Shimazaki, andT. Nakajima for u seful discussions. Numer-\nical calculation was partly carried out at the Supercompute r Center, ISSP, Univ. of Tokyo.\nThis work was supported by Grant-in-Aid for Scientific Resea rch (No. 22104010 and No.\n24540420), Elements Strategy Initiative Center for Magnet ic Materials under the outsourcing\nproject of MEXT, the Strategic Programs for Innovative Rese arch (SPIRE), MEXT, and the\nComputational Materials Science Initiative (CMSI), Japan .\nAppendix A: Exact Expressions of Fermi Level and OAM with SOI for Parabolic\nBands\nIn this Appendix, we derive the exact expressions of Fermi le vel and OAM with SOI\nturned on for the bottoms of parabolic bands in one-, two-, an d three-dimensional periodic\nsystems. The derivation of the expressions for the tops of pa rabolic bands will also be possible\nby considering the number of holes.\nFor each of thethreecases below, we assumethat thetwo bands withoutSOIcoincide with\neach other. We further assume that all the unperturbedsingl e-particle states in one band have\nthe same OAM mand those in the other band have the OAM with the same magnitud e but\nin the opposite direction, −m. We set the origin of energy to the bottom of the unperturbed\nbands. For simplicity, we consider a case in which the band wi th the OAM mundergoes a\nrigid shift −bξas the perturbation, first-order in the strength ξof the SOI, while the other\nband with −munderdoes a rigid shift bξ.bis a dimensionless positive constant.\n25/43J. Phys. Soc. Jpn. Full Paper\nA.1 One-dimensional system\nHere we consider a one-dimensional periodic system. For an e nergyεabove the bottom of\nthe unperturbed two bands, the DOS is of the form D0(ε) = 2a/√ε, whereais a constant.19\nThe number neof electrons and the unperturbed Fermi level εF0thus satisfy the relation\nne= 4aε1/2\nF0. The total energy of the unperturbed system is calculated as E0= (4a/3)ε3/2\nF0.\nThe DOS of the perturbed system is of the form D(ε) =a/√ε−bξ+a/√ε+bξ, whose first\nterm on the right hand side is for the states with the OAM −mand the second term for m.\nWhenneis smaller than the critical value nc≡2a(2bξ)1/2, the band with OAM −mis\nempty. In such a case, the perturbed Fermi level is given by\nεF=/parenleftigne\n2a/parenrightig2\n−bξ=εF0(4−/tildewideξ), (A·1)\nwhere/tildewideξ≡bξ/εF0is the relative strength of the SOI with respect to the unpert urbed Fermi\nlevel. The net OAM then takes the saturated value\n/an}bracketle{tL/an}bracketri}ht\nmne=1\nne/integraldisplayεF\n−∞dεD(ε) = 1, (A·2)\nindependent of ξ. The perturbed total energy is calculated as\nE=/integraldisplayεF\n−∞dεεD(ε) =E0(4−3/tildewideξ). (A·3)\nIn the limit of ne→nc, the Fermi level converges as εF→2εF0.\nWhenne>nc, on the other hand, the perturbed Fermi level is determined s o that\nne=/integraldisplayεF\n−∞dεD(ε) = 2a[(εF−bξ)1/2+(εF+bξ)1/2]. (A·4)\nBy writing εF≡bξcosh2swiths>0, this condition is written as /tildewidene=es, where/tildewidene≡ne/nc.\nThe perturbed Fermi level is thus given by\nεF=εF0/parenleftigg\n1+/tildewideξ2\n4/parenrightigg\n, (A·5)\nThe net OAM is hence calculated as\n/an}bracketle{tL/an}bracketri}ht\nmne=1\nne/integraldisplayεF\n−∞dε/parenleftbigg\n−a√ε−bξ+a√ε+bξ/parenrightbigg\n=/tildewideξ\n2, (A·6)\nwhich, in the limit of ne→nc(/tildewideξ→2), correctly converges to unity [see eq. (A ·2)]. The Fermi\nlevel given by eq. (A ·5) also converges in this limit correctly as εF→2εF0[see eq. (A ·1)]. The\nperturbed total energy is calculated as\nE=/integraldisplayεF\n−∞dεεD(ε) =E0/parenleftbigg\n1−3\n4/tildewideξ2/parenrightbigg\n. (A·7)\nThe Fermi level and the net OAM of the perturbed system as func tions/tildewideξare plotted in\nFig. 1 (b).\n26/43J. Phys. Soc. Jpn. Full Paper\nA.2 Two-dimensional system\nHere we consider a two-dimensional periodic system. For an e nergyεabove the bottom\nof the unperturbed two bands, the DOS is of the form D0(ε) = 2aθ(ε), whereais a constant\nfor the step function.19The number neof electrons and the unperturbed Fermi level εF0thus\nsatisfy the relation ne= 2aεF0. The total energy of the unperturbed system is calculated as\nE0=aε2\nF0. The DOS of the perturbed system is of the form D(ε) =aθ(ε−bξ)+aθ(ε+bξ),\nwhose first term on the right hand side is for the states with th e OAM−mand the second\nterm form.\nWhenneis smaller than the critical value nc≡a(2bξ), the band with OAM −mis empty.\nIn such a case, the perturbed Fermi level is given by\nεF=ne\na−bξ=εF0(2−/tildewideξ), (A·8)\nwhere/tildewideξ≡bξ/εF0is the relative strength of SOI with respect to the unperturb ed Fermi level.\nThe net OAM then takes the saturated value\n/an}bracketle{tL/an}bracketri}ht\nmne=1\nne/integraldisplayεF\n−∞dεD(ε) = 1, (A·9)\nindependent of ξ. The perturbed total energy is calculated as\nE=/integraldisplayεF\n−∞dεεD(ε) =E0(2−2/tildewideξ). (A·10)\nIn the limit of ne→nc, the Fermi level converges as εF→εF0.\nWhenne>nc, on the other hand, the perturbed Fermi level is determined s o that\nne=/integraldisplayεF\n−∞dεD(ε) = 2aεF. (A·11)\nThe perturbed Fermi level is thus given by\nεF=ne\n2a=εF0. (A·12)\nThe net OAM is calculated as\n/an}bracketle{tL/an}bracketri}ht\nmne=1\nne/integraldisplayεF\n−∞dε[−aθ(ε−bξ)+aθ(ε+bξ)] =/tildewideξ, (A·13)\nwhich, in the limit of ne→nc(/tildewideξ→1), correctly converges to unity [see eq. (A ·9)]. The\nperturbed total energy is calculated as\nE=/integraldisplayεF\n−∞dεεD(ε) =E0(1−/tildewideξ2) (A ·14)\nThe Fermi level and the net OAM of the perturbed system as func tions/tildewideξare plotted in\nFig. 1 (b).\n27/43J. Phys. Soc. Jpn. Full Paper\nA.3 Three-dimensional system\nHere we consider a three-dimensional periodic system. For a n energyεabove the bottom\nof the unperturbedtwo bands, the DOS is of the form D0(ε) = 2a√ε, whereais a constant.19\nThe number neof electrons and the unperturbed Fermi level εF0thus satisfy the relation\nne= (4a/3)ε3/2\nF0. The total energy of the unperturbedsystem is calculated as E0= (4a/5)ε5/2\nF0.\nThe DOS of the perturbed system is of the form D(ε) =a√ε−bξ+a√ε+bξ, whose first\nterm on the right hand side is for the states with the OAM −mand the second term for m.\nWhenneis smaller than the critical value nc≡(2a/3)(2bξ)3/2, the band with OAM −m\nis empty. In such a case, the perturbed Fermi level is given by\nεF=/parenleftbigg3ne\n2a/parenrightbigg2/3\n−bξ=εF0(41/3−/tildewideξ), (A·15)\nwhere/tildewideξ≡bξ/εF0is the relative strength of SOI with respect to the unperturb ed Fermi level.\nThe net OAM then takes the saturated value\n/an}bracketle{tL/an}bracketri}ht\nmne=1\nne/integraldisplayεF\n−∞dεD(ε) = 1, (A·16)\nindependent of ξ. The perturbed total energy is calculated as\nE=/integraldisplayεF\n−∞dεεD(ε) =E0/parenleftbigg\n41/3−5\n3/tildewideξ/parenrightbigg\n. (A·17)\nIn the limit of ne→nc, the Fermi level converges as εF→2−1/3εF0.\nWhenne>nc, on the other hand, the perturbed Fermi level is determined s o that\nne=/integraldisplayεF\n−∞dεD(ε) =a2\n3[(εF−bξ)3/2+(εF+bξ)3/2]. (A·18)\nBy writing\nεF≡bξcosh2s (A·19)\nwiths >0, this condition is written as ne= (nc/4)(e3s+ 3e−s). Via a further variable\ntransformation t≡es, the condition to be satisfied becomes t4−4/tildewidenet+ 3 = 0, where /tildewidene≡\nne/nc. The only appropriate solution of this quartic equation for s>0 ist=u+/radicalbig\n/tildewidene/u−u2,\nwhereu≡/radicalbig\n(h2+h−2)/2 andh≡[/tildewiden2\ne+/radicalbig\n/tildewiden4e−1]1/6. The perturbed Fermi level can be\ncalculated by putting this solution into eq. (A ·19). The net OAM is hence calculated as\n/an}bracketle{tL/an}bracketri}ht\nmne=1\nne/integraldisplayεF\n−∞dε(−a/radicalbig\nε−bξ+a/radicalbig\nε+bξ)\n=1\n4/tildewidene(t−3+3t), (A·20)\nwhich, in the limit of ne→nc(/tildewideξ→2−1/3), correctly converges to unity [see eq. (A ·16)]. The\nFermi level given by eq. (A ·19) also converges in this limit correctly as εF→2−1/3εF0[see eq.\n28/43J. Phys. Soc. Jpn. Full Paper\n(A·15)]. The perturbed total energy is calculated as\nE=/integraldisplayεF\n−∞dεεD(ε)\n=2a\n15/bracketleftigg\n/radicalbig\nεF+bξ(3ε2\nF+bξεF−2b2ξ2)\n+/radicalbig\nεF−bξ(3ε2\nF−bξεF−2b2ξ2)/bracketrightigg\n. (A·21)\nWhen the strength of SOI is small compared to the unperturbed Fermi level measured\nfrom the bottom of the bands, the perturbed Fermi level for ne> ncis expressed, from eq.\n(A·19), as\nεF≈εF0/parenleftigg\n1−/tildewideξ2\n4−/tildewideξ4\n16/parenrightigg\n. (A·22)\nThe net OAM is expressed, from eq. (A ·20), as\n/an}bracketle{tL/an}bracketri}ht\nmne≈3\n2/tildewideξ−1\n4/tildewideξ3. (A·23)\nThe total energy is expressed, from eq. (A ·21), as\nE≈E0/parenleftigg\n1−5/tildewideξ2\n4+5/tildewideξ4\n48/parenrightigg\n. (A·24)\nThe Fermi level and the net OAM of the perturbed system as func tions/tildewideξare plotted in\nFig. 1 (b).\nAppendix B: Exact Expressions of Eigenvectors for dOrbital Chain with Elec-\ntron Spins along zAxis\nIn this Appendix, we provide the exact expressions of the ene rgy eigenvalues and the\neigenvectors for the dorbital chain with electron spins along the zaxis (n=ez).\nSubstituting θ= 0 andφ= 0 into the expressions of the spin wave function, eq. (8), we\nobtain the basis functions in spin space as\n| ↑/an}bracketri}ht=\n1\n0\n,| ↓/an}bracketri}ht=\n0\n−1\n. (B·1)\nWerearrangethebasisfunctionsforthechainas {d↑\nxy,d↑\nx2−y2,d↑\n3z2−r2,d↓\nyz,d↓\nzx,d↓\nxy,d↓\nx2−y2,d↓\n3z2−r2,d↑\nyz,d↑\nzx}.\nThe ten-dimensional Hamiltonian matrix, which is represen ted by eq. (30) for the old arrange-\nment of the basis functions, then becomes block diagonal con sisting of two 5 ×5 matrices.\n29/43J. Phys. Soc. Jpn. Full Paper\nThe Hamiltonian matrix for the former five basis functions re ads\nH1(k) =\n2tδp−∆ex/2iξ 0 −ξ/2 iξ/2\n−iξ 2tδp−∆ex/2 0 −iξ/2 −ξ/2\n0 0 2 tσp−∆ex/2−iξ√\n3/2ξ√\n3/2\n−ξ/2 iξ/2iξ√\n3/2 2tπp+∆ex/2−iξ/2\niξ/2 −ξ/2ξ√\n3/2 iξ/2 2tπp+∆ex/2\n,\n(B·2)\nwhile that for the latter reads\nH2(k) =\n2tδp+∆ex/2−iξ 0 ξ/2 iξ/2\niξ 2tδp+∆ex/2 0 −iξ/2 ξ/2\n0 0 2 tσp+∆ex/2−iξ√\n3/2−ξ√\n3/2\nξ/2 iξ/2iξ√\n3/2 2tπp−∆ex/2iξ/2\n−iξ/2 ξ/2 −ξ√\n3/2 −iξ/2 2tπp−∆ex/2\n,\n(B·3)\nwherep≡coska.Theeigenvalueprobleminthiscasehasreducedtothetwoqu inticequations.\nThese Hamiltonians are analytically diagonalizable.\nWe define the following dimensionless parameters:\nη(±)≡ξ\n4(tσ−tπ)p±2∆ex, (B·4)\nγ(±)≡ξ\n−4(tδ−tπ)p±2∆ex. (B·5)\nThe exact energy eigenvalues of H1(k) are then given by\nε11= (tπ+tσ)p−ξ\n4/bracketleftig\n1+/radicalig\n(1+η−1\n(−))2+24/bracketrightig\n, (B·6)\nε12= (tπ+tσ)p−ξ\n4/bracketleftig\n1−/radicalig\n(1+η−1\n(−))2+24/bracketrightig\n, (B·7)\nε13= (tπ+tδ)p−ξ\n4/bracketleftig\n1+/radicalig\n(3+γ−1\n(+))2+16/bracketrightig\n, (B·8)\nε14= (tπ+tδ)p−ξ\n4/bracketleftig\n1−/radicalig\n(3+γ−1\n(+))2+16/bracketrightig\n, (B·9)\nε15= 2tδp−∆ex\n2+ξ, (B·10)\nwhile those of H2(k) are given by\nε21= (tπ+tσ)p−ξ\n4/bracketleftig\n1+/radicalig\n(1+η−1\n(+))2+24/bracketrightig\n, (B·11)\nε22= (tπ+tσ)p−ξ\n4/bracketleftig\n1−/radicalig\n(1+η−1\n(+))2+24/bracketrightig\n, (B·12)\n30/43J. Phys. Soc. Jpn. Full Paper\nε23= (tπ+tδ)p−ξ\n4/bracketleftig\n1+/radicalig\n(3+γ−1\n(−))2+16/bracketrightig\n, (B·13)\nε24= (tπ+tδ)p−ξ\n4/bracketleftig\n1−/radicalig\n(3+γ−1\n(−))2+16/bracketrightig\n, (B·14)\nε25= 2tδp+∆ex\n2+ξ. (B·15)\nWe confirmed that these expressions give the correct energy e igenvalues by comparing them\nwiththoseobtainedvianumericaldiagonalization. Fromth eexacteigenvaluesdisplayedabove,\ntheir expressions for H1(k) correct up to second order in ξare calculated as\nε11≈2tπp+∆ex\n2−ξ\n2−3ξ2\n4(tσ−tπ)p−2∆ex, (B·16)\nε12≈2tσp−∆ex\n2+3ξ2\n4(tσ−tπ)p−2∆ex, (B·17)\nε13≈2tδp−∆ex\n2−ξ−ξ2\n−2(tδ−tπ)p+∆ex, (B·18)\nε14≈2tπp+∆ex\n2+ξ\n2+ξ2\n−2(tδ−tπ)p+∆ex, (B·19)\nwhile those of H2(k) are given by\n(B·20)\nε21≈2tπp−∆ex\n2−ξ\n2−3ξ2\n4(tσ−tπ)p+2∆ex, (B·21)\nε22≈2tσp+∆ex\n2+3ξ2\n4(tσ−tπ)p+2∆ex, (B·22)\nε23≈2tδp+∆ex\n2−ξ−ξ2\n−2(tδ−tπ)p−∆ex, (B·23)\nε24≈2tπp−∆ex\n2+ξ\n2+ξ2\n−2(tδ−tπ)p−∆ex. (B·24)\nThese expressions allow one to find the correspondence betwe en the perturbed energy eigen-\nvalues and the unperturbed states [see Fig. 2 (b)]: ε13andε15fromδ↑,ε23andε25fromδ↓,\nε21andε24fromπ↑,ε11andε14fromπ↓,ε12fromσ↑,ε22fromσ↓states.\nWe define the following eight functions:\nf±(q)≡√\n3q[±q/radicalbig\nq−2+2q−1+25−1+3q]\n±q/radicalbig\nq−2+2q−1+25(−1+q)+1+11q2, (B·25)\ng±(q)≡√\n3\n12[±q/radicalbig\nq−2+2q−1+25−1−q]/q, (B·26)\nu±(q)≡2q[±q/radicalbig\nq−2+6q−1+25+1 −q]\n±q/radicalbig\nq−2+6q−1+25(1+q)+1+4q+11q2, (B·27)\nv±(q)≡1\n4[±q/radicalbig\nq−2+6q−1+25+1+3 q]/q. (B·28)\n31/43J. Phys. Soc. Jpn. Full Paper\nThe normalized exact eigenvectors of H1(k) corresponding to the energy eigenvalues provided\nabove are then given by\n|ψ11/an}bracketri}ht=1\nN11\n0\n0\n1\n−if+(η(−))\ng−(η(−))\n, (B·29)\n|ψ12/an}bracketri}ht=1\nN12\n0\n0\n1\n−if−(η(−))\ng+(η(−))\n, (B·30)\n|ψ13/an}bracketri}ht=1\nN13\n1\ni\n0\nu+(γ(+))\n−iv−(γ(+))\n, (B·31)\n|ψ14/an}bracketri}ht=1\nN14\n1\ni\n0\nu−(γ(+))\n−iv+(γ(+))\n, (B·32)\n|ψ15/an}bracketri}ht=1√\n2\n1\n−i\n0\n0\n0\n, (B·33)\n32/43J. Phys. Soc. Jpn. Full Paper\nwhile those of H2(k) are given by\n|ψ21/an}bracketri}ht=1\nN21\n0\n0\n1\n−if+(η(+))\n−g−(η(+))\n, (B·34)\n|ψ22/an}bracketri}ht=1\nN22\n0\n0\n1\n−if−(η(+))\n−g+(η(+))\n, (B·35)\n|ψ23/an}bracketri}ht=1\nN23\n1\n−i\n0\n−u+(γ(−))\n−iv−(γ(−))\n, (B·36)\n|ψ24/an}bracketri}ht=1\nN24\n1\n−i\n0\n−u−(γ(−))\n−iv+(γ(−))\n, (B·37)\n|ψ25/an}bracketri}ht=1√\n2\n1\ni\n0\n0\n0\n. (B·38)\nNij’s (i= 1,2,j= 1,...,4) are the normalization constants. From the exact eigenvec tors\n33/43J. Phys. Soc. Jpn. Full Paper\ndisplayed above, their expressions for H1(k) correct up to second order in ξare calculated as\n|ψ11/an}bracketri}ht ≈1√\n2\n0\n0\n2√\n3[−η(−)+η2\n(−)]\ni[1−3η2\n(−)]\n1−3η2\n(−)\n, (B·39)\n|ψ12/an}bracketri}ht ≈\n0\n0\n1−3η2\n(−)\ni√\n3[η(−)−η2\n(−)]\n√\n3[η(−)−η2\n(−)]\n, (B·40)\n|ψ13/an}bracketri}ht ≈1√\n2\n1−2γ2\n(+)\ni[1−2γ2\n(+)]\n0\n2[γ(+)−3γ2\n(+)]\n2i[γ(+)−3γ2\n(+)]\n, (B·41)\n|ψ14/an}bracketri}ht ≈1√\n2\n2[γ(+)−3γ2\n(+)]\n2i[γ(+)−3γ2\n(+)]\n0\n−1+2γ2\n(+)\ni[−1+2γ2\n(+)]\n, (B·42)\nwhile those for H2(k) are calculated as\n|ψ21/an}bracketri}ht ≈1√\n2\n0\n0\n2√\n3[η(+)−η2\n(+)]\n−i[1−3η2\n(+)]\n1−3η2\n(+)\n, (B·43)\n|ψ22/an}bracketri}ht ≈\n0\n0\n1−3η2\n(+)\ni√\n3[η(+)−η2\n(+)]\n−√\n3[η(+)−η2\n(+)]\n, (B·44)\n34/43J. Phys. Soc. Jpn. Full Paper\n|ψ23/an}bracketri}ht ≈1√\n2\n1−2γ2\n(−)\n−i[1−2γ2\n(−)]\n0\n−2[γ(−)−3γ2\n(−)]\n2i[γ(−)−3γ2\n(−)]\n, (B·45)\n|ψ24/an}bracketri}ht ≈1√\n2\n2[γ(−)−3γ2\n(−)]\n−2i[γ(−)−3γ2\n(−)]\n0\n1−2γ2\n(−)\n−i[1−2γ2\n(−)]\n. (B·46)\nBy using the expressions for the single-particle states pro vided above, we obtain the ex-\npectation values of Lzcorrect up to second order in ξas\n/an}bracketle{tψ11|Lz|ψ11/an}bracketri}ht ≈1−6η2\n(−), (B·47)\n/an}bracketle{tψ12|Lz|ψ12/an}bracketri}ht ≈6η2\n(−), (B·48)\n/an}bracketle{tψ13|Lz|ψ13/an}bracketri}ht ≈ −2+4γ2\n(+), (B·49)\n/an}bracketle{tψ14|Lz|ψ14/an}bracketri}ht ≈ −1−4γ2\n(+), (B·50)\n/an}bracketle{tψ15|Lz|ψ15/an}bracketri}ht= 2, (B·51)\n/an}bracketle{tψ21|Lz|ψ21/an}bracketri}ht ≈ −1+6η2\n(+), (B·52)\n/an}bracketle{tψ22|Lz|ψ22/an}bracketri}ht ≈ −6η2\n(+), (B·53)\n/an}bracketle{tψ23|Lz|ψ23/an}bracketri}ht ≈2−4γ2\n(−), (B·54)\n/an}bracketle{tψ24|Lz|ψ24/an}bracketri}ht ≈1+4γ2\n(−), (B·55)\n/an}bracketle{tψ25|Lz|ψ25/an}bracketri}ht=−2, (B·56)\namong which none contains the first-order contribution. It i s also easily confirmed for all the\nsingle-particle states that the first-order contribution t o the expectation values of P↑Lzand\nP↓Lzvanishes separately. On the other hand, the expectation val ues ofL·Tcorrect up to\nsecond order in ξare calculated as\n/an}bracketle{tψ11|L·T|ψ11/an}bracketri}ht ≈ −6η(−)+6η2\n(−), (B·57)\n/an}bracketle{tψ12|L·T|ψ12/an}bracketri}ht ≈6η(−)−6η2\n(−), (B·58)\n/an}bracketle{tψ13|L·T|ψ13/an}bracketri}ht ≈ −4γ(+)+12γ2\n(+), (B·59)\n35/43J. Phys. Soc. Jpn. Full Paper\n/an}bracketle{tψ14|L·T|ψ14/an}bracketri}ht ≈4γ(+)−12γ2\n(+), (B·60)\n/an}bracketle{tψ15|L·T|ψ15/an}bracketri}ht= 0, (B·61)\n/an}bracketle{tψ21|L·T|ψ21/an}bracketri}ht ≈ −6η(+)+6η2\n(+), (B·62)\n/an}bracketle{tψ22|L·T|ψ22/an}bracketri}ht ≈6η(+)−6η2\n(+), (B·63)\n/an}bracketle{tψ23|L·T|ψ23/an}bracketri}ht ≈ −4γ(−)+12γ2\n(−), (B·64)\n/an}bracketle{tψ24|L·T|ψ24/an}bracketri}ht ≈4γ(−)−12γ2\n(−), (B·65)\n/an}bracketle{tψ25|L·T|ψ25/an}bracketri}ht= 0, (B·66)\nwhich can contain the first-order contributions.\nAppendix C: Penalty Functional for DFT Calculations\nC.1 Definition and Expressions\nThe formulation of spin-constrained variational problems for the minimization of total\nenergy in DFT calculations has been done in the literature.44In this Appendix, we describe\ntheexplicitexpressionsofpenalty functionalfordetaile d specificationofthespinsofindividual\natoms.\nThe spin of the atom µin a periodic system is evaluated as the sum of the contributi ons\nfrom the occupied Bloch states:\nSµ=occ./summationdisplay\nm,k/an}bracketle{tψmk|PµSPµ|ψmk/an}bracketri}ht, (C·1)\nwherekis the wave vector and mis the band index for a two-component Bloch state |ψmk/an}bracketri}ht.\nFor detailed specification of the directions and/or magnitu des of the spins of individual\natoms, we define the penalty functional consisting of three p arts asP≡Pdir+Pmag+Pcone,\nwhere\nPdir≡Adir/summationdisplay\nµ/parenleftigg\ncos−1Sµ·d0µ\nSµ−δµ/parenrightigg2\n·\n·θ(cosδµ−Sµ·d0µ/Sµ), (C·2)\nPmag≡Amag/summationdisplay\nµ(|Sµ−M0µ|−∆µ)2·\n·θ(|Sµ−M0µ|−∆µ), (C·3)\nPcone≡Acone/summationdisplay\nµ/parenleftigg\ncos−1Sµ·c0µ\nSµ−θµ/parenrightigg2\n. (C·4)\n36/43J. Phys. Soc. Jpn. Full Paper\nAdir,Amag, andAconeare positive constants. θis the step function. Pdiris used for fixing the\ndirections of the spins. If the direction of Sµdeviates from d0µby an angle larger than δµ,\nPdirhas a positive value. Pmagis used for fixing the magnitudes of the spins. If the magnitud e\nofSµdeviates from M0µby a value larger than ∆ µ,Pmaghas a positive value. Pconeis used\nfor forcing the spins to be on the cones. If Sµdeviates from the cone whose axis is c0µ,Pcone\nhas a positive value.\nThegeneralized energy functional to beminimized in this ca se is thus/tildewideE≡E+P, whereE\nis the ordinary energy functional. The penalty functional a cts as the constraint for the energy\nminimization procedure in a DFT calculation. It is noted, ho wever, that the configuration of\nthe spins does not necessarily minimize Pwhen the SCF calculation is converged, since the\nenergy minimization procedure minimizes not P, butE+P.\nThe equation to be solved is obtained from the stationarity c ondition of the generalized\nenergy functional /tildewideE,\n0 =δE\nδ/an}bracketle{tψmk|+δP\nδ/an}bracketle{tψmk|, (C·5)\nwhere the first term on the right-hand side leads to the ordina ry Kohn-Sham Hamiltonian for\n|ψmk/an}bracketri}ht. If we write the variation of the penalty functional as\nδP\nδ/an}bracketle{tψmk|=/summationdisplay\nµ∂P\n∂Sµ·δSµ\nδ/an}bracketle{tψmk|\n=/summationdisplay\nµPµBpen\nµ·SPµ|ψmk/an}bracketri}ht ≡/summationdisplay\nµPµHpen\nµPµ|ψmk/an}bracketri}ht, (C·6)\nBpen\nµ≡Bdir\nµ+Bmag\nµ+Bcone\nµcan be interpreted as the effective magnetic field acting on the\natomµfor fixing its spin. From eqs. (C ·2)-(C·4), the expressions for Bpen\nµare given by\nBdir\nµi= 2Adirθ(cosδµ−dµ)cos−1dµ−δµ/radicalig\n1−d2µ·\n·1\nSµ/parenleftigg\ndµSµ\nSµ−d0µ/parenrightigg\n(C·7)\nBmag\nµ= 2Amag[Sµ−M0µ−sgn(Sµ−M0µ)∆µ]·\n·θ(|Sµ−M0µ|−∆µ)Sµ\nSµ(C·8)\nBcone\nµ= 2Aconecos−1cµ−θµ/radicalig\n1−c2µ1\nSµ/parenleftigg\ncµSµ\nSµ−c0µ/parenrightigg\n, (C·9)\nwheredµ≡Sµ·d0µ/Sµ,cµ≡Sµ·c0µ/Sµ.\n37/43J. Phys. Soc. Jpn. Full Paper\nC.2 Implementation for PAW Method\nWithin the PAW formalism,26an AE wave function and its corresponding PS wave func-\ntion are related via the transformation operator Tas|ψAE/an}bracketri}ht=T|ψPS/an}bracketri}ht. The physical quantity\nrepresented by an AE operator OAEis evaluated using the expectation value of the PS oper-\nator defined as\nOPS≡T†OAET=OAE\n+/summationdisplay\nµ,i,j[/an}bracketle{tφAE\nµi|OAE|φAE\nµj/an}bracketri}ht−/an}bracketle{tφPS\nµi|OAE|φPS\nµj/an}bracketri}ht]|βµi/an}bracketri}ht/an}bracketle{tβµj|, (C·10)\nwhere|φAE\nµi/an}bracketri}htand|φPS\nµi/an}bracketri}htare theith AE and PS atomic orbitals of the µth atom, respectively.\n|βµi/an}bracketri}htis the corresponding projector. It is noted that the atomic o rbitals and the projectors\nare two-component in our fully relativistic calculations.\nThe constrained minimization procedure of the total energy using a plane-wave basis set\nneeds the matrix elements /an}bracketle{tk+G,τ|Hpen\nµ|ψmk/an}bracketri}htof the penalty Hamiltonian, where\n|k+G,α/an}bracketri}ht ≡\n|k+G/an}bracketri}ht\n0\n,|k+G,β/an}bracketri}ht ≡\n0\n|k+G/an}bracketri}ht\n (C·11)\nare the two-component PS plane waves. From eqs. (C ·6) and (C ·10), we obtain\n/an}bracketle{tk+G,τ|(PµHpen\nµPµ)PS|ψmk/an}bracketri}ht (C·12)\n=/summationdisplay\nτ′/an}bracketle{tk+G|Qµττ′|ψmkτ′/an}bracketri}ht\n+/summationdisplay\ni,jLµij/an}bracketle{tk+G|βµiτ/an}bracketri}ht/summationdisplay\nτ′/an}bracketle{tβµjτ′|ψmkτ′/an}bracketri}ht,\nwhere\nQµττ′≡Hpen\nµττ′Pµ (C·13)\nand\nLµij=/summationdisplay\nτ,τ′Hpen\nµττ′√\n4πqµ,00,ττ′\nij, (C·14)\nqµ,lm,ττ′\nij≡ /an}bracketle{tφAE\nµiτ|Ylm|φAE\nµjτ′/an}bracketri}ht−/an}bracketle{tφPS\nµiτ|Ylm|φPS\nµjτ′/an}bracketri}ht. (C·15)\nYlmis thespherical harmonics. Itis obvious fromeq. (C.2) that theintroduction of thepenalty\nfunctional is realized only by replacing the local potentia l of the atom µas\nVloc\nµττ′(r)→Vloc\nµττ′(r)+Qµττ′(r) (C ·16)\n38/43J. Phys. Soc. Jpn. Full Paper\nand by replacing the coefficients of the nonlocal potential Vnonl\nµ=/summationtext\ni,jDµij|βµi/an}bracketri}ht/an}bracketle{tβµj|as\nDµij→Dµij+Lµij. (C·17)\nAppendix D: Calculation of OAM within PAW Method\nHere we describetheimplementation of theOAM of each atom in a periodicsystem within\nthe PAW method using a plane-wave basis set.\nThe OAM of the atom µin a periodic system is evaluated as the sum of the contributi ons\nfrom the occupied Bloch states:\n/an}bracketle{tLµ/an}bracketri}ht=occ./summationdisplay\nm,k/an}bracketle{tψmk|PµLµPµ|ψmk/an}bracketri}ht, (D·1)\nwherekis the wave vector and mis the band index for a two-component PS Bloch state\n|ψmk/an}bracketri}ht.Lµ= (r−Rµ)×pis the OAM operator effective only in the vicinity of the atom µ,\nlocated at Rµ. Within the PAW formalism,26a physical quantity is calculated by using the\nPS operator, defined via the relation eq. (C ·10), and the PS wave functions, as stated above.\nThe OAM of the atom µis thus calculated as\n/an}bracketle{tLµ/an}bracketri}ht=occ./summationdisplay\nm,k/bracketleftigg\n/an}bracketle{tψmk|PµLµPµ|ψmk/an}bracketri}ht+\n/summationdisplay\ni,j[/an}bracketle{tφAE\nµi|L|φAE\nµj/an}bracketri}ht−/an}bracketle{tφPS\nµi|L|φPS\nµj/an}bracketri}ht]/an}bracketle{tψmk|βµi/an}bracketri}ht/an}bracketle{tβµj|ψmk/an}bracketri}ht/bracketrightigg\n. (D·2)\nThe first term in the summation on the right hand side above is w ritten as\nocc./summationdisplay\nm,k/an}bracketle{tψmk|PµLµPµ|ψmk/an}bracketri}ht\n=occ./summationdisplay\nm,k/integraldisplay\nµd3rψmk(r)†(r−Rµ)×pψmk(r), (D·3)\nwhere the integral is taken over the sphere of the ion radius rµcentered at Rµ. By defining a\ncell-periodic function\nPmk(r)≡ψmk(r)†pψmk(r), (D·4)\nwe rewrite the right hand side of eq. (D ·3) as\nocc./summationdisplay\nm,k/integraldisplay\nµd3r(r−Rµ)×Pmk(r) =−occ./summationdisplay\nm,k/summationdisplay\nGeiG·Rµ·\n·Pmk+G×/parenleftigg\n−i∂\n∂G/parenrightigg/integraldisplayrµ\n0r2dr/integraldisplay\ndΩeiG·r, (D·5)\nwhereGis a reciprocal lattice vector and Pmk+Gis the Fourier coefficient of Pmk(r). The\n39/43J. Phys. Soc. Jpn. Full Paper\nintegral on the right hand side above is performed as\n∂\n∂G/integraldisplayrµ\n0r2dr/integraldisplay\ndΩeiG·r\n= 4πr4\nµGµ\nGµ(G2\nµ−3)sinGµ+3GµcosGµ\nG4µ, (D·6)\nwhereGµ≡Grµ. Equation (D ·3) is thus written as\nocc./summationdisplay\nm,k/an}bracketle{tψmk|PµLµPµ|ψmk/an}bracketri}ht=i4πr4\nµocc./summationdisplay\nm,k/summationdisplay\nGeiG·Rµ·\n·Pmk+G×Gµ\nGµ(G2\nµ−3)sinGµ+3GµcosGµ\nG4µ. (D·7)\nBy using this expression in eq. (D ·2), one can evaluate the OAM of the atom µstraightfor-\nwardly in a PAW calculation using a plane-wave basis set.\nThe OAM matrix, defined in eq. (20), can also be evaluated in a m anner similar to that\ndescribed above.\n40/43J. Phys. Soc. Jpn. Full Paper\nTable I. For each of the five L10alloys comprised of the elements A and B (A = Fe, Co, or Mn,\nand B = Pt, Pd, Al, or Ga), the calculated values of MCA energy in meV a re tabulated. The\nnet OAM and the spin angular momentum in a.u. around each atom proj ected along the spin\ndirection ([100] or [001]) are also tabulated. The [100] and the [001] d irections correspond to the\nxand thezdirections, respectively, in Fig. 5. We tabulate the spin angular mome nta multipled\nby 2 since the literature provides not the spin angular momenta but t he spin magnetic moments\nwith agfactor of 2.\nSystem MCA energy LA LB 2SA 2SB\nFePt Present work 3 .145\n[100] 0 .055 0 .057 2 .79 0 .37\n[001] 0 .060 0 .044 2 .79 0 .37\nDaalderop et al.363.3 0 .08 0 .07 2 .91 0 .34\nSakuma172.8 0 .08 0 .05 2 .93 0 .33\nGalanakis et al.403.90 0 .07 0 .05 2 .88 0 .33\nRavindran et al.122.734\n[100] 0 .061 0 .055 2 .89 0 .355\n[001] 0 .067 0 .042 2 .89 0 .353\nBurkertet al.412.84 0.069,0.078 0.045,0.043 2.923,2.937 0.3615,0.296\nLuet al.422.900\nCoPt Present work 1 .307\n[100] 0 .055 0 .078 1 .78 0 .40\n[001] 0 .089 0 .059 1 .77 0 .40\nDaalderop et al.362.0 0 .12 0 .06 1 .86 0 .32\nSakuma171.5 0 .11 0 .07 1 .91 0 .38\nGalanakis et al.402.20\n[100] 0 .06 0 .08 1 .74 0 .35\n[001] 0 .11 0 .06 1 .74 0 .35\nRavindran et al.121.052\n[100] 0 .057 0 .073 1 .809 0 .398\n[001] 0 .089 0 .056 1 .803 0 .394\nFePd Present work 0 .087\n[100] 0 .060 0 .030 2 .88 0 .38\n[001] 0 .070 0 .027 2 .88 0 .38\nGalanakis et al.400.06\n[100] 0 .06 0 .03 2 .90 0 .35\n[001] 0 .07 0 .03 2 .90 0 .35\nMnAl Present work 0 .312\n[100] 0 .028 0 2 .10 −0.033\n[001] 0 .028 −0.001 2 .10 −0.033\nSakuma180.26 0 .059 −0.003 2 .442 −0.095\nMnGa Present work 0 .395\n[100] 0 .024 0 2 .26 −0.066\n[001] 0 .022 0 2 .26 −0.066\nSakuma430.42 0 .056 0 .005 2 .449 −0.08841/43J. 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B 61(2000) R6459.\n43/43"    },    {        "title": "1403.5157v1.Magnetization_of_densely_packed_interacting_magnetic_nanoparticles_with_cubic_and_uniaxial_anisotropies__A_Monte_Carlo_study.pdf",        "content": "arXiv:1403.5157v1  [cond-mat.mtrl-sci]  20 Mar 2014Magnetization of densely packed interacting magnetic nano particles\nwith cubic and uniaxial anisotropies: a Monte Carlo study.\nV. Russiera), C. de-Montferrandb), Y. Lalatonneb)and L. Motteb).\na)ICMPE, UMR 7182 CNRS and UPEC,\n2-8 rue Henri Dunant 94320 Thiais, France. and\nb)CSPBAT UMR 7244 CNRS and University Paris 13, 93017 Bobigny, France.\n(Dated: August 10, 2021)\nThe magnetization curves of densely packed single domain magnetic n anoparticles (MNP)\nare investigated by Monte Carlo simulations in the framework of an eff ective one spin model.\nTheparticleswhosesizepolydispersityistakenintoaccountarearr angedinsphericalclusters\nand both dipole dipole interactions (DDI) and magnetic anisotropy en ergy (MAE) are in-\ncluded in the total energy. Having in mind the special case of spinel f errites of intrinsic cubic\nsymmetry, combined cubic and uniaxial magnetocrystalline anisotro pies are considered with\ndifferent configurations for the orientations of the cubic and uniax ial axes. It is found that\nthe DDI, together with a marked reduction of the linear susceptibilit y are responsible for a\ndamping of the peculiarities due to the MAE cubic component on the ma gnetization. As an\napplication, we show that the simulated magnetization curves compa re well to experimental\nresults for γ–Fe2O3MNP for small to moderate values of the field.\nI. INTRODUCTION\nMagnetic nanoparticles (MNP) assemblies present a fundame ntal interest in the devel-\nopment of nanoscale magnetism research and are promising ca ndidates in a wide range of\npotential applications going from high density recording t o bio-medicine1–5. Experimen-\ntally MNP can be obtained either as colloidal suspensions wh ere the concentration can be\nvaried at will, embedded in non magnetic material where one c an tune the interparticle\ninteractions or as powder samples where they form densely pa cked systems. The case of\niron oxide particles, which aretypical cubicspinel ferrit es take acentral place especially for\nbio-medical applications because of their biocompatibili ty and suitable superparamagnetic\nproperties6,7.\nThe magnetic behavior of nanostructured materials or syste ms including nanoscale\nmagnetic particles is a multiscale problem since the local m agnetic structure within NP\nat the atomic site scale presents non trivial features8–11and the interactions between\nparticles play an important role. A simplification occurs fo r NP of diameter below some\ncritical value of typically few tens of nanometers since the y then reach the single domain\nregime and can be described through an effective one spin model (EOS) where each NP\nis characterized by its moment and anisotropy energy. Howev er both the moment value2\nand the anisotropy energy function are to be understood as effe ctive quantities which take\ninto account some of the atomic scale characteristics12–15. The EOS type of approach is a\nsimplifying but necessary step for the description of inter acting MNP assemblies. In the\nframeworkoftheEOSmodels, thetotal energyincludesonthe onehandtheNPanisotropy\nenergy through a one-body term and on the other hand the inter particle interactions. It\nis generally assumed for frozen systems of well separated NP that the leading term in the\ninterparticle interactions is the dipolar interaction (DD I) between the macrospins which\nis totally determined once the NP saturation magnetization and the size distribution are\nknown. Conversely modeling the anisotropy energy is not str aightforward since in finite\nsized particles it comes from different origins. The intrinsi c contribution which stems form\nthe bulk material is a prioriknown experimentally without ambiguity. It can be of either\nuniaxialorcubicsymmetryaccordingtothecrystallinestr ucturewithanisotropyconstants\nwhosemagnitudeandevensigndependontemperature. Inthew idelystudiedcaseofoxide\nspinel ferrites at room temperature the intrinsic anisotro py is of cubic symmetry16–20with\nin general a negative constant Kc, leading to the moment preferentially oriented along\nthe{111}directions of the crystallites. Then for NP not strictly sph erical one has to\nadd the shape anisotropy term resulting from demagnetizing effect at the particle scale\nwhich for ellipsoidal NP is uniaxial with a shape anisotropy constant proportional to\nthe NP volume1. Finally the finite size of the NP is the source of surface anis otropy\nresulting from symmetry breaking, surface defects or chemi cal bonding of the coating\nlayer. When modeled by a transverse anisotropy or the N´ eel s urface anisotropy model21,\nthe resulting non collinearities of the surface spins can be represented through a cubic\nterm in the framework of the EOS14,15. Concerning spherical iron oxide nanoparticles, the\ngeneral experimental observation is that the uniaxial anis otropy dominates with however a\nrather large dispersion in the effective annisotropy constan t valueKeff13,22–27. Moreover\na small value of Keffis interpreted as a small amount of crystalline defects with in or\nat the nanoparticle surface24,27–29. In any case the effective uniaxial anisotropy constant\ncannot be compared to the intrinsic, or bulk one, since the la tter corresponds to the\ncubic symmetry and is negative at room temperature. Further more from the particle\nsize dependence it is generally concluded that the uniaxial anisotropy is predominantly a\nsurface anisotropy with a related constant Ks= (d/6)Keff13,22,26.\nAt the atomic scale the well known N´ eel model of pair anisotr opy21is often invoked to\ndeduce surface anisotropy either in thin film geometry or in 3 D NP. In the framework of\nEOSapproach, thedeviationfromthesphericalshapetransl atesintheN´ eel pairanisotropy\nmodel, in addition to the magnetic dipolar term responsible for the shape anisotropy, in a\ncontribution with the same symmetry and proportional to the NP surface because of its\nshort range character. This leads for ellipsoidal NP to a sur face contribution of uniaxial3\nsymmetry. One has to keep in mind however that the N´ eel model although useful in the\nsensethatitreproducesthecorrectdescriptionofthesymm etryofthemagneticanisotropy,\ndoes not provide the physical understanding of the single io n anisotropy30. Thus when\ndealing with spinel ferrites oxides as well as with Fe or Ni si ngle domain nearly spherical\nnanoparticles in the framework of a EOS model, combined unia xial and cubic anisotropies\nshould be taken into account because of the intrinsic cubic a nisotropy on the one hand,\nthe shape (uniaxial) and surface contributions (uniaxial a nd/or cubic) on the other hand.\nIn Ref. [31] the expansion of the linear and non linear suscep tibility for non interacting\nassembly with either uniaxial or cubic anisotropy has been p erformed with the result that\nwhen the 3 axes of the cubic contribution are randomly distri buted both the linear and the\nfirst non linear susceptibilities are anisotropy independe nt. In Ref. [32] the LLG equation\nis considered to calculate the hysteresis curve at vanishin g temperature of non interacting\nNP, with randomly oriented cubic axes. The easy axis of the un iaxial term is either fixed\nat conveniently chosen direction or randomly distributed. In Ref. [33] an assembly of\nweakly interacting NP is considered both from perturbation theory and MC simulations\nwith cubic anisotropy relative to the same cubic axes for all the NP combined with an\nuniaxial anisotropy with a random distribution of easy axes .\nIn the present work, we perform MC simulation of NP assemblie s interacting through\nDDI with cubic and uniaxial contributions to the anisotropy energy. Having in mind the\ncase of strongly interacting powder samples of NP dispersed at zero field, we consider the\ncase of NP with cubic axes randomly distributed. The uniaxia l easy axis on the other\nhand is either randomly distributed independently of the cu bic axes or oriented along a\nparticular crystallographic orientation of the particles for which two cases are considered,\nnamely{100}or{111}. Themainpurposeofthepresent workis toinvestigate wheth erthe\ncubic contribution to the anisotropy leads to an observable deviation to the magnetization\ncurve in the superparamagnetic regime. We also revisit the c onsequences of the DDI in\nthe strong coupling regime, in particular on the linear susc eptibility at low field, through\nthe comparison of simulations performed either with free bo undary conditions on spherical\ncluster or with periodic boundary conditions to simulate an infinite system.\nIn section II, we give the details of the model and explicit th e different energy contribu-\ntions. Section III is devoted to the results and the comparis on with experimental results\nand we briefly conclude in section IV.4\nII. MODEL FOR DENSELY PACKED ASSEMBLIES\nWe consider a EOS model with nanoparticles described as non o verlapping spheres\nbearing at their center a permanent point dipole representi ng the uniform magnetization\nof the particle (macro spin). The moment of each particle is e qual to its volume times\nthe bulk magnetization, Ms, which means that no spin canting effect is explicitly taken\ninto account. The particles are surrounded by a non magnetic layer of thickness ∆/2,\nrepresenting the usual coating by organic surfactant molec ules. The particle diameters,\n{di}are distributed according to a log-normal law defined by the m edian diameter dmand\nthe standard deviation σofln(d),\nf(d) =1\nd√\n2πσexp/parenleftbigg\n−(ln(d/dm))2\n2σ2/parenrightbigg\n(1)\nIn the following, we use dmas the unit of length, and the distribution function in reduc ed\nunit is totally determined by the single parameter σwhich characterizes the system poly-\ndispersity. When dealing with interacting particles, we ma inly have in mind the case of\nlyophilized powders samples or high concentration nanopar ticles assemblies embedded in\nnon magnetic matrix. Accordingly the coated particles are d istributed in densely packed\nclusters whose external shape is spherical in order to avoid the demagnetizing effects due\nto the system shape with the free boundary conditions. We emp hasize that this NP\nconfiguration has an experimental justification since upon d rying the NP are likely to ag-\ngregate in spherical shaped large clusters which has been co nfirmed from simulations34.\nMoreover, we consider mainly the superparamagneticregime , wherewe simulate only equi-\nlibrium magnetization curves corresponding to the static o r infinite time measurements\n(τm→ ∞).\nWe include only the leading terms of the anisotropy energy; t he cubic symmetry con-\ntribution for particle say iof moment v(di)Msˆmican be written as\nE(i)\nc\nv(di)=Kc/parenleftbig\nm2\nxim2\nyi+m2\nyim2\nzi+m2\nxim2\nzi/parenrightbig\n=Kc\n2/parenleftBigg\n1−/summationdisplay\nα=x,y,zm4\nαi/parenrightBigg\n(2)\nwhere we have used the unitarity of ˆ miin the second equality. Here and in the following\nhated letters denote unit vectors. In equation (2) mαirefer to the ˆ micomponents in the\nlocal cubic frame of the particle considered. Let us denote b y{ˆxαi},α= 1,3 this local\ncubic frame; dropping an irrelevant constant, the total cub ic anisotropy of the system can\nbe written\nEc=−Kc\n2/summationdisplay\niv(di)/summationdisplay\nα(ˆxαiˆmi)4(3)5\nThe local axes {ˆxαi}can be oriented in different ways according to the physical sys tem\nunder study; for non textured distributions of particles we have to consider a random\ndistribution of the {ˆxαi}. Most of our simulations are performed in the case. The effect o f\nthe texturation is nevertheless examined by considering th at the [111] directions {ˆx1+ˆx2+\nˆx3}ior the{ˆx3}iaxes are confined in a cone along the ˆ z-axis according to the following\nprobability distribution for polar angles\nP(θ) =Csin(θ)exp(−(θ/σθ)2/2) ; (4)\nThe configuration with the {ˆxαi}fixed parallel to the system frame (ˆ x,ˆy,ˆz) is also consid-\nered. This later case is the same as that Margaris et al.33considered for which a strong\neffect of the cubic term is obtained while Usov and Barandiar´ a n32consider cubic axes\nrandomly distributed. Although we have in mind particles of intrinsic cubic anisotropy,\nwe are aware of a possible surface contribution to Ecas shown in Ref. [14] resulting from\nthe non collinearity of the surface spins; as a result the val ue ofKcmay differ form the\nbulk one. The uniaxial term is proportional either to the vol umev(d) or to the surface\ns(d) of the particle. The volume part stems a priorifrom the shape anisotropy where for\nellipsoidal particles K(u)\nsh, given by J2\ns(1−3Nu)/(4µ0) withJs=µ0MsandNu, the demag-\nnetizing factor along the revolution axis, can be deduced fr om the knowledge of the aspect\nratioξ. Notice that one can imagine easily a situation where the dev iation from sphericity\nis not characterized by the same aspect ratio for all particl es leading to a size dependence\nofKsh. For instance one cannot rule out the situation where the dev iation from sphericity\nfollows from a major axis of the form c= (d/2 +δ) and minor axes a=b=d/2, with\na size independent corrugation δ. Then from the demagnetizing factor in the major axis\ndirection\nNu=1−ǫ2\n2ǫ3/bracketleftbigg\nln/parenleftbigg1+ǫ\n1−ǫ/parenrightbigg\n−2ǫ/bracketrightbigg\n; withǫ= (1−1/ξ2)1/2andξ=c/a= 1+2δ/d ,\nthe shape anisotropy may transform in a surface uniaxial ani sotropy with an anisotropy\nconstant given by K(u)\ns≃J2\nsδ/(30µ0) from an expansion of K(u)\nshat order δ/d.\nTheeasy axes {ˆn}iare either randomlydistributedindependently of theparti cles frame\nor aligned along one specified crystallographic axis of the c rystallite. Different origin for\nsuch a easy axes distribution can be invoked. In the framewor k of the uniaxial anisotropy\noriginatingfromthedeviation tosphericityit correspond stoapreferential crystallographic\norientation for crystallite growth, whilein the framework of the uniaxial surfaceanisotropy\nthis may result from a preferential crystallographic orien tation for chemical bondingat the\nparticle surface. We have considered two possibilities, na mely ˆni={001}ior ˆni={111}i.\nIn the total energy, we include formally both surface and vol ume terms in the uniaxial6\ncontribution, with anisotropy constants K(u)\nvandK(u)\nsrespectively and at most one of\nthese is non zero in the simulations. The total energy thus in cludes the DDI, the one-body\nanisotropy term and the Zeeman term corresponding to the int eraction with the external\napplied field /vectorHa=Haˆh. Let{/vector ri},{v(i)},{/vector mi}and{/vector ni}denote the particles locations,\nvolumes, moments and easy axes respectively. The total ener gy of the cluster reads\nE=µ0\n4π/summationdisplay\ni<jmimjˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\nr3\nij−/summationdisplay\ni(K(u)\nvv(i)+K(u)\nss(i))(ˆniˆmi)2\n−Kc\n2/summationdisplay\niv(i)/summationdisplay\nα(ˆmiˆxαi)4−µ0Ha/summationdisplay\nimiˆmiˆh(5)\nmiare the moment magnitudes, rij=|/vector ri−/vector rj|. It is worth mentioning that the considera-\ntion of the anisotropy term with a fixed easy axes distributio n means that the magnetiza-\ntion relax according to a N´ eel process16,35, namely the particles are considered fixed while\ntheir moment relaxes relative to their easy axis. In the foll owing we usereduced quantities;\nfirst the energy is written in kBT0units,T0being a suitable temperature ( T0= 300Kin\nthe present work) and we introduce a reference diameter, dref. The reference diameter,\ndrefis a length unit independent of the size distribution, usefu l for the energy couplings,\nand can be chosen from a convenient criterion independently of the actual structure of the\nMNP assembly. The reduced total energy is given by\nβ0E=−ǫ(0)\nuv/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\nid∗3\ni(ˆniˆmi)2−ǫ(0)\nus/parenleftbiggdm\ndref/parenrightbigg2/summationdisplay\nid∗2\ni(ˆniˆmi)2\n−ǫ(0)\nc\n2/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\nid∗3\ni/summationdisplay\nα(ˆmiˆxαi)4−ǫ(0)\nd/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\ni<jd∗3\nid∗3\njˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\nr∗3\nij\n−h/summationdisplay\nid∗3\niˆmiˆh , (6)\nwith\nǫ(0)\nuv=β0K(u)\nvv(dref) ;ǫ(0)\nus=β0K(u)\nss(dref) ;ǫ(0)\nc=β0Kcv(dref)\nǫ(0)\nd=β0µ0\n4π(π/6)2M2\nsd3\nref;h=β0µ0Ms(π/6)d3\nmHa≡/parenleftbiggdm\ndref/parenrightbigg3Ha\nHref(7)\nwhereβ0= (kBT0)−1and the stared lengths are in dmunit. The dimensionless dipolar\ncoupling constant is then ǫd= (dm/dref)3ǫ(0)\ndand the dimensionless anisotropy constants\nareǫuv= (dm/dref)3ǫ(0)\nuv,ǫus= (dm/dref)2ǫ(0)\nus, andǫc= (dm/dref)3ǫ(0)\ncfor the volume and\nsurface uniaxial and cubic contributions respectively. Th e reference diameter, drefcan be\nchosen such that ǫd(dm=dref)≡ǫ(0)\nd= 1 ; the reduced external field hcoincides with the\nusual Langevin variable at temperature T0for a monodisperse distribution with d=dm7\nIn equation (7), we also introduce the reference external fie ld,Hreffor convenience.\nConcerning the structure in position, the nanoparticles su rrounded by their coating\nlayer of thickness ∆/2 form an assembly of hard spheres of effec tive diameters {di+∆}\nwhich are arranged in large densely packed clusters with eit her a random or a well ordered\nsimple structure (simple cubic lattice) The clusters are bu ilt as in Ref. [36]. First a\nlarge stacking of the coated spheres is made in a parallepipe dic box with the desired\nstructure, random or well ordered. Once this first step is per formed, we cut within the\nglobal stacking the cluster we want to study by imposing both the external shape and the\nnumberof particles Np, with typically Np≃1000. Because of thecoating layer of thickness\n∆/2 the closest distance of approach between particles i,jis shifted from ( di+dj)/2 to\n(di+dj)/2 + ∆ and we therefore define an effective dipolar constant corre sponding the\nparticles uncoated at contact\nǫeff\nd=ǫd1\n(1+∆/dm)3=ǫdφ\nφmϕ(σ,∆/dm) (8)\nwhereφis the volume fraction and φm=φ(∆ = 0) is the maximum value of φfor a\ngiven structure. The function ϕ(see appendix) in equation (8) is equal to 1 for σ= 0 and\nremains very close to 1 for σ <0.1. Two systems differing by ∆ or σand characterized\nby the same value of ǫeff\ndcorrespond to the same intensity of DDI. Notice that for weak\npolydispersity, ǫeff\ndis related to the parameter ywidely used in the works dealing with\nthe dipolar hard sphere fluid (DHS) which in our notations rea dsy= 8β∗φǫd/3.\nAlthough we do not limit our simulations to a specific experim ental system, we have\nin mind iron oxide NP to guide our choice of the physical param eters entering the model.\nAt room temperature the bulk anisotropy constant is Kc=-11.0 to -13.0 kJm−3and -\n4.70 kJm−3for magnetite and maghemite respectively. The saturation m agnetization\nfor these two materials are quite close and lead to Js∼0.50T. Therefore in this work\nwe useJs= 0.50 T, which corresponds to dref= 10nmwhen this later is fixed from\nǫd(dref) = 1.0. The shape anisotropy constant can be estimated for el lipsoidal NP once\nthe aspect ratio is known; using Js= 0.5Twe getK(u)\nsh≃50(1−3Nl) kJm−3which leads\ntoK(u)\nsh<7.0 kJm−3for NP characterized by an aspect ratio ξ <1.20, and accordingly\nǫ(0)\nuv= 0.95. Concerning the surface anisotropy constant, we conside r the experimental\nvalues ranging from Ks= 5.5 10−6Jm−2toKs= 2.7 10−5Jm−2for maghemite13,22,27;\nthusKs= 2.7 10−5Jm−2is considered somewhat as an upper bond for iron oxide NP.\nNotice that if we consider the deviation from sphericity res ulting from a size independent\ncorrugation δas outlined above, we get K(s)\nsh≃2.70 10−5Jm−2withδ=2nm, which\ncorresponds to an aspect ratio ranging from 1.4 to 1.2 for NP o f diameter ranging from8\n10 to 20 nm. The corresponding values of the reduced parameters for Iro n oxide NP of\nc.a.10 to 20 nmin diameter are summarized in table I. In the following we can consider\nthat a characteristic value for the uniaxial anisotropy is a boutǫuv∼5 while a maximum\nvalue for the cubic anisotropy constant is |kc|= 15.\ndm/dref11.201.331.712.0\nǫca)-1.65-2.85-3.88-8.25-13.20\nǫcb)-0.60-1.05-1.41-3.0-4.8\nǫuvc)0.6251.081.473.135.00\nǫuvd)1.001.732.375.08.00\nǫuse)2.052.953.626.08.20\nǫd1.01.732.375.08.00\nǫeff\ndf)0.601.091.563.66.00\nǫeff\ndg)1.56\nTable I: Reduced values for the parameters of the model corresp onding to iron oxide NP. with Kc\n= - 13 kJm−3a); -4.7 13 kJm−3b); an aspect ratio ξ= 1.135c); or 1.20d); or surface anisotropy\nconstant Ks= 2.70 10−5Jm−2e).ǫeff\ndfrom equ. (8) with ∆ =2 nmf)or 14.5 nmg).\nThe main effect of the DDI on the magnetization curve at low field is a strong reduc-\ntion of the initial slope of M(H) versus H, namely of the linear susceptibility36,37. This\nis directly related to the well known plateau in the FC magnet ization in terms of the\ntemperature for T < T Boccurring in strongly interacting NP and also the the platea u\nin theχ(dm) curve obtained in Ref. [36] for densely packed clusters of N P. Indeed, in the\nabsence of anisotropy, the hamiltonian (6) can be rewritten , withhr=Ha/Href\nβE=β∗β0E=−β∗ǫ(0)\nd/parenleftbiggdm\ndref/parenrightbigg31\n(1+∆/dm)3/summationdisplay\ni<jd∗3\nid∗3\njˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\n(r∗\nij/(1+∆/dm))3\n−hrβ∗/parenleftbiggdm\ndref/parenrightbigg3/summationdisplay\nid∗3\niˆmiˆh (9)\nwhere we have introduced the geometrical sum of the reduced D DI of the most con-\ncentrated cluster (∆ =0) of the structure considered, namel y where particles can get at\ncontact which thus depends neither on dmand ∆. Introducing the dimensionless variable9\nλ=β∗(dm/dref)3we get\nβE=−λǫeff(0)\nd/summationdisplay\ni<jd∗3\nid∗3\njˆmiˆmj−3(ˆmiˆrij)(ˆmjˆrij)\n(r∗\nij/(1+∆/dm))3−hrλ/summationdisplay\nid∗3\niˆmiˆh (10)\nFrom equation (10), we can conclude that, when the field is van ishingly small, the leading\ncontribution to the magnetization linear in hrdepends on hrandβ∗only through λhrand\nλǫeff(0)\nd. More precisely, M/Ms≃λhrf(λǫeff(0)\nd) withfa scaling function. Thus the\nlinear susceptibility, χ=∂M/∂H a≃M/Haat vanishing Hamust be in the form\nχ=Ms\nHrefλf(λǫeff(0)\nd) (11)\nIn the limit of zero coupling f(x= 0) is a finite constant and we recover the Langevin\nresult,χ∝λ= (6/π)(T0/T)v(dm). In the interacting system, the strong coupling limit\nλǫeff(0)\nd>>1 or equivalently y >>1 is obtained through the increase of either the DDI\ncoupling, ǫ(0)\ndorβ∗(decrease of T). In this case, the limiting value of the susceptibility\ncan be obtained. We note that the linear susceptibility we de al with is the external one,\nrelating the magnetization to the external, or applied field Haand since we consider the\nmagnetization per unit magnetic volume, the magnetization per unit volume is Mv=Mφ.\nThus the internal field is related to the external one through Hi=Ha−DhφMwhereDh\nis the demagnetizing factor of the sample in the direction of the field. Hence we can relate\nχto the internal susceptibility, χithrough the usual way16\nχ=χi\n1+Dhφχi(12)\nWe can also introduce the relative permeability, µ= (1+φχi) to get\nφχ=µ−1\n1+Dh(µ−1)(13)\nIn the case of a spherical system as those considered here, Dh= 1/3 and equation (13)\nreads\nφχ=3(µ−1)\nµ+2(14)\nItisworthmentioningthat χisrelated tothemomentfluctuationsthroughthefluctuation -\ndissipation theorem40as already used in [36]. We have in an isotropic system\n∂(M/Ms)\n∂h=χr=β∗N¯v\n3v(dm)\n/angbracketleftBig\n(|Σ/vector mi|)2/angbracketrightBig\n(Σmi)2−|∝an}bracketle{tΣ/vector mi∝an}bracketri}ht|2\n(Σmi)2\n≡β∗¯v g\n3v(dm)(15)10\nwhich introduces the factor gand where ¯ vis the average value of the particle volume over\nthe distribution function. From equation (15) we rewrite (1 4) in the equivalent form\n3(µ−1)\nµ+2=φχ= 8φǫdβ∗¯v\nv(dm)g (16)\nNow in the strong coupling limit we expect the system to reach a ferromagnetic transition\nas is the case for the DHS fluid38,39. In this limit the permeability µ→ ∞and a limiting\nvalue for χand thus a plateau in the FC magnetization when the temperatu re is decreased\nis obtained with, from equation (14)\nχ→3\nφor ˜χ→3\n8ǫ(0)\ndφwith ˜χ=Href\nMsχ (17)\nThis is quite well reproduced by the present simulations (se e section (III)) and in total\nagreement with the behavior of ˜ χin terms of the particle size dmwe obtained in Ref. [36]\nin the quasi monodisperse case where ϕ≃1 which is easily deduced from (17) by writing\nφin terms of ∆ /dm\n˜χ→ϕ(1+∆/dm)3\n8ǫ(0)\ndφm(18)\nIt is important to note that equation (14) is the well known re lation between the dielectric\nconstant and the polarization susceptibility in the DHS flui d in the case of an infinite\nspherical system embedded in vacuum, i.e.surrounded by a medium of dielectric constant\nǫs= 1. Indeed the magnetic permeability plays the role of the di electric constant of the\nDHS and the polarization susceptibility is related to the flu ctuations or the Kirkwood\nfactorgK(ǫs), equivalent to the factor gintroduced above; in the monodisperse case, with\nthe dielectric constant, ǫ, in place of µthe DHS satisfies40,41\nµ−1\nµ+2=ygK(ǫs= 1) ; or µ−1 = 3ygK(ǫs=∞) (19)\nNotice that the second equation (19) is the equivalent of (13 ) written for Dh= 0 and cor-\nrespondsto the case whereeither through the boundarycondi tions (ǫs=∞) or the system\nshape(Dh= 0) the system can be uniformly polarized. Equation (19) is s trictly equivalent\nto (16) since in the present model we have, in the monodispers e case,χr=β∗g/3. The\nDHS undergoes a ferromagnetic transition at which the diele ctric constant diverges and as\na result38,42,43, one expects a limiting value for the Kirkwood factor gK(ǫs= 1)→1/y\nand accordingly χr→β∗/(3y) orχ→3/φin agreement with equation (17).\nThe plateau in the FC magnetization at low temperature and lo w field is a behavior\nobserved in the framework of the FC/ZFC procedure25,29,44–46generally related to a col-11\nlective behavior of the dipoles leading to a frozen state. He re, by analogy with the known\nbehavior of the DHS fluid, we relate this plateau to the approa ch of the onset of the fer-\nromagnetic transition at least for σ <<1 and in the absence of MAE. We emphasize that\nas can be deduced from equation (14), in the case of a spherica l system surrounded by\nvacuum, χbecomes nearly independent of µwhenµincreases beyond a sufficiently high\n(µ∼35) but still finite value. As a result χgets close to its limiting value before the\nferromagnetic transition.\nThe Monte Carlo simulations are performed according to the u sual Metropolis\nscheme40,41,47. The trial move of each moment is performed within a solid ang le cen-\ntered on its old position. Since we seek equilibrium configur ations, the maximum solid\nangle of the move is only restricted by the acceptance ratio, R∼0.35–0.50. Moreover we\nuse a annealing scheme at all values of the field in the range wh ere we expect an hysteresis.\nThe averages are performed on 10 to 30 independent runs (up to 70 runs for low tempera-\nture and/or large DDi couplings) with 3104to 4104thermalisation MC steps followed by\nanother set of 3104to 4104MC steps to compute the averages.\nIII. RESULTS\nNon interacting system\nIn this section we deal with the case free of DDI. We first have c hecked that as h→0\nwith volume uniaxial MAE and a random easy axes distribution the linear susceptibility is\nǫuvindependentwhile with cubicMAE and randomly distributed a xes, both the linear and\nthe first non linear susceptibilities are kcindependent and accordingly we get a nearly kc\nindependent M(h) beyond the very vicinity of h=0. This is shown in figure 1 in terms of\nthe inverse reduced temperature β∗. Moreover we also check in figure 1 that the deviation\nofM(h) relative to the isotropic case is negative whatever the sig n ofkcwith the random\ndistribution of cubic axes. This is no more the case when the c ubic axes of the particles\nare fixed where on the one hand only the linear susceptibility iskcindependent and on\nthe other hand the sign of ( M(h,kc)−M(h,kc= 0)) depends on the sign of kc. The same\nresult holds when ǫd∝ne}ationslash= 0.\nFor randomly distributed cubic axes, the cubic MAE has only a negligible effect on\ntheM(h) curve. On the opposite, as shown in figure 2, when the cubic ax es are fixed\nalong the system frame, the cubic MAE has a strong effect on the M(h) curve. Moreover,\nas noted above in the low field region, the sign of the anisotro py induced deviation of\nM(h) depends on the sign of kc. This is expected since a positive value of kcwill favor\nthe principal frame directions for the moments; for an appli ed field along one of these12\ndirections, say ˆh=ˆz,kc>0 leads to a positive deviation of M(h) andvice versa . The\nresults displayed in figure 2 are in agreement with those of Re f. [33] (notice that our kc\ncorresponds to w/2 of Ref. [33]).\nThe effect of the texturation through the preferential orient ation along the ˆ z-axis of\nthe crystallites [111] direction according to the probabil ity density (4) is shown in figure 3\nfor the polydisperse and monodisperse cases.\nConcerning the uniaxial anisotropy, we note that the surfac e contribution can be very\nwell approached by the volume term with the introduction of a n effective volume uniaxial\nconstant, ǫeff\nuvtaking into account the polydispersity. In equation (6), we rewrite the\nuniaxial energy terms by introducing the reduced n−thorder moments d∗\nnof the diameter\ndistribution function and under the hypothesis that (/summationtextd∗n\ni(ˆniˆmi)2)/d∗\nnis independent of\nnat least for n≤3 we get\nǫeff\nuv=d∗\n2\nd∗\n3ǫus= exp(−5σ2/2)ǫus (20)\nwhere we have used the analytical result for the d∗\nnof the lognormal law. The same\nconclusion holds in presence of DDI; in figure 4 we compare the deviation of M(h) due to\nthe surface uniaxial MAE with that due to the volume uniaxial MAE with ǫuv=ǫeff\nuv\ntaken from (20) in the case of a polydisperse interacting sys tem.\nWe now consider the case of combined uniaxial and cubic aniso tropies. The result is\nshown for a typical set of parameters, ǫuv= 5 and |kc|= 15 in figure 5. As is the case\nwhen only the cubic anisotropy is taken into account, we find t hat the effect on M(h) of\nthe cubic anisotropy with random distributed cubic axes is v ery small when the uniaxial\neasy axes are also randomly distributed and uncorrelated fr om the cubic ones. This is no\nmore the case when, still for a random distribution of cubic a xes, the easy axes {ˆn}iare\nalong a specified crystallographic orientation of the cryst allites. The cubic MAE enhances\nthe uniaxial one when ǫc>0 and{ˆn}i= [001], or when ǫc<0 and{ˆn}i=[111]. This is\nqualitatively expected since then the two components of the MAE tend to favor the same\nlocal orientation for the moment.\nA shoulder in M(h) is clearly observed when {ˆn}i= [001] and ǫc>0 or{ˆn}i=[111]\nandǫc<0. This can be compared to the behavior of the hysteresis curv es determined\nby Usov and Barandiar´ an [32] when the easy axis of the uniaxi al MAE component is fixed\nrelative to the NP frame. This shoulder is enhanced when eith er the inverse temperature β\nincreases or when the polydispersity σincreases (see figure 6). This latter point is simply\nduetothepresenceoflargerparticlesinthedistributionw henσincreases, withaccordingly\nlarger anisotropy energies. We can be interpret this featur e as the coherent contributions\nof uniaxial and cubic terms. In the case ǫc<0 where the favorable orientations are the13\n{111}axes, we find that the cubic contribution remains to enhance t he uniaxial anisotropy\nconstant by a factor of roughly |ǫc|/5 as shown in the inset of figure (5).\nInteracting systems\nMost of our simulations with DDI are performed with free boun dary conditions (FBC)\non large spherical NP clusters of Np∼1000 particles. In order to check the validity of the\nmethod, we have performed simulations with periodic bounda ry conditions (PBC) with\nEwald sums for the DDI in both the conducting or the vacuum ext ernal boundary condi-\ntions40,41. This is done by using either ǫs=1 orǫs=∞for the surrounding permeability\n(or dielectric constant in the electric dipolar case). Here we are interested in the determi-\nnation of the linear susceptibility for the infinite system e mbedded in vacuum, as we seek\nthe magnetic response in terms of the external field. Therefo re, we check that one can get\nχr(ǫs= 1) from simulations on a large spherical NP cluster with FBC , or by using PBC\nwith Ewald sums in either the conducting or the vacuum bounda ry conditions. The value\nofχr(ǫs= 1) can be obtained from a simulation with external conducti ng conditions by\nexploiting in equation (19) the independence of µwith respect of ǫsas it is an intrinsic\nproperty ,\nχr(ǫs= 1) =χr(ǫs=∞)/(1+8φǫdχr(ǫs=∞)). (21)\nThe comparison of χr(ǫs= 1) from the three routes is shown in figure 7 in the absence of\nanisotropy and in the quasi monodisperse case ( σ=0.05). We have used the same initial\ncluster and extracted either a spherical cluster for FBC or a cubic simulation box for PBC\nwith a value of ∆ fitted on the volume fraction φ. Moreover we have checked that for\nmoderate values of the DDI coupling the permeability obtain ed from these three routes\nleads to similar values. These two points show the coherence of our simulations with DDI.\nWhen compared to the results of Klapp and Patey48the curve µ(y) we get at φ=0.385\nlies in between the ones of the frozen model with correlation and of the frozen model with\nquenched disorder, much closer to the former and in fact very close to that of the DHS\nfluid.\nBeside the strong reduction of the initial susceptibility, the DDI reduce also the devi-\nation of the M(h) curves due to MAE, as can be seen in figure 8. As expected the cu bic\nanisotropy has nearly no influence on the M(h) when the easy axes and the cubic axes are\nindependently randomly distributed; on the other hand the c hange in the M(h) curve due\nto the cubic contribution when {ˆn}iare along the crystallites [111] with kc<0 or along\nthe[001] with kc>0is smaller than in the absence of DDI. Nevertheless, the con tribution14\nof the cubic anisotropy may be not negligible under the condi tion of a coherence with the\nuniaxial term. Moreover, we do find that in order for the cubic term to give a noticeable\neffect a rather large value of the cubic anisotropy constant, kcis necessary.\nIn opposite to what we get in the absence of DDI, we do not find an y distinctive\nfeature of either the cubic or the uniaxial symmetry on the M(h) curve if the cubic axes\narerandomlydistributedinthecaseofcombinedoronlyunia xialanisotropy. Thisisshown\nin figure9 wheredifferent combinations of anisotropies leadi ng to comparable M(h) curves\nare considered for ǫeff\nd= 1.\nFinally we consider the comparison with the experimental ma gnetization curves of\nRef. [49] on powder samples of maghemite NP differing by their s ize. These samples are\ncharacterized by a polydispersity σ∼0.27 and the estimated coating layer thickness is\nc.a.2nm. The behavior of the M(Ha) curve being controlled by the DDI and the MAE\nat low and intermediate values of the applied field respectiv ely, we fit the value of ∆ by\nthe slope at Ha∼0 and the anisotropy constants on the behavior of M(Ha) at higher\nvalues of Ha. We find that the region Ha∼0 is well reproduced with ∆ = 2 nmfor\ndm= 10nmand 21nm, and ∆ = 2.4 nmfor 12nm, which does not differ much from\nthe estimated experimental value. Concerning the cubic ani sotropy since the experimental\nsamplesarenottexturedweconsideronlyarandomdistribut ionofcubicaxes. Thevalueof\nthe corresponding anisotropy constant may differ from its kno wn bulk value due to surface\neffects; however, we consider the bulk value as a starting poin t. Inany case, since thecubic\nanisotropy constant for iron oxide is rather small, we expec t only a small effect of the cubic\ncontribution to the MAE and accordingly we consider only the case where the cubic and\nthe uniaxial components of the MAE reinforce each other. Wit hǫc<0, this means that\nwe limit ourselves to a easy axes distribution {ˆn}i= [111] i. For the uniaxial MAE we\nhave to choose either a surface or volume dependent MAE (see e quation (6)); however, we\nhave shown that the surface dependent MAE can be reproduced b y the volume dependent\none through the effective constant of (20). Hence, starting fr om the bulk value for ǫc\nwe are left with ǫuvas the only fitting parameter. We find ǫuv= 4.00 for dm= 10nm\nby fitting M(Ha) in the intermediate field range; then, the same quality of ag reement\nbetween the model and the experimental curves is obtained fo rdm= 12nmand 21nm\nby using a value of ǫuvscaling as d3\nm, namely ǫuv= 6.912 and 32.0 for dm/dref= 1.2 and\n2.0 respectively, i.e.Kv= 31.6 kJm−3(we use the simulated curve for dm/dref= 2 for\ncomparison of the experimental curves of the samples with dm18 and 21 nm; only the\nsecond is presented here). Notice the weak hysteresis cycle for the experimental sample\ncharacterized by dm= 21nm; this is due to the largest particles in the distributi on and\nis not reproduced by the M.C. simulations, since we have chos en to perform equilibrium\n(τm=∞) simulations only. The cubic MAE gives only a small contribu tion toM(Ha)15\nas illustrated by the difference obtained using ǫcdeduced from either the magnetite or\nthe maghemite bulk values given in Table I (see figures 11 and 1 2). Therefore, we find\nthat using the iron oxide bulk value for the cubic MAE constan t the experimental NP of\nRef. [49] can be modeled excepted in the high field region, by N P presenting a volume\ndependent uniaxial anisotropy with Kv= 31.6kJm−3. However, as we have shown, we can\nget similar M(Ha)curveswithdifferent combinations ofcubicanduniaxialMAE especially\nwith the DDI which weaken the peculiar features of the cubic c ontribution. Hence, we\ncan get the same agreement with experiment by using on the one hand a uniaxial MAE\nscaling as d2\nmcorresponding to a surface anisotropy and on the other hand a fitted cubic\ncontribution. Starting from ǫuv= 4 fordm/dref= 1.0 this gives ǫuv= 5.76 for dm/dref1.2\n(which translates to ǫus= 7.03 for σ= 0.28 and Ks= 6.4510−5Jm−2). Thecorresponding\ncubic component is obtained from our finding that an increase of|ǫc|corresponds to an\nincrease of ǫuvof roughly |ǫc|/5, leading to ǫc= -9 and Kc= -41kJm−3. We have also\nconsidered a fitted cubic MAE with a positive ǫc, and ˆni= [001] ifor which we find\nǫc= 5.0(Kc= 25.15 kJm−3). The results is shown in figure 11. Doing this means that the\ncubic anisotropy energy present an anomalous component, na mely/vextendsingle/vextendsingleKc−Kbulk\nc/vextendsingle/vextendsingle, scaling as\nthe NP volume while it should be understood as a surface effect. Hence, although it seems\ndifficult to conclude on the best fit of the experimental set con sidered, it may be better\nto avoid the latter contradiction and consider these NP as pr esenting a volume dependent\nuniaxial MAE; however, we then get a value for the effective ani sotropy constant too large\nto be explained only as a shape anisotropy. It is nevertheles s still in the range of what\nis obtained experimentally from TBfor iron oxide NP. In any case, we have to take such\nconclusions with care given the simplicity of the model. Sim ilarly, the high field range\ncannot be reproduced with the simple OSP model and necessita tes a the inclusion of a\nfield dependent description of the individual NP.\nIV. CONCLUSION\nIn this work, we have performed Monte Carlo simulations of ro om temperature magne-\ntization curves in the superparamagnetic regime, with a par ticular attention paid to the\niron oxide based NP. We focused on the search for a peculiar fe ature of the cubic MAE\ncomponent on the M(Ha) curve since iron oxide and spinel ferrites in general prese nts an\nintrinsic MAE with cubic symmetry while from experiments a u niaxial MAE is generally\nfound. Our result is that a peculiar feature of the cubic comp onent can be obtained only\ni)if the the cubic and the uniaxial components are correlated t hrough the alignment of\nthe NP easy axes on a specified crystallographic orientation of the crystallites; ii)if the\nDDI are negligible viaa small NP volume fraction. Nevertheless a large value of the cubic16\nMAE constant compared the uniaxial one is necessary for the f ormer to give a noticeable\neffect on the room temperature M(Ha).\nV. APPENDIX A\nInthisappendixweexplicitthefunction ϕintroducedinequation (8). Thevolumefraction\nis defined as\nφ=Np1\nV/integraldisplay∞\n0f(d)π\n6d3d(d) =πd3\nm\n6Vd∗\n3 (A.1)\nwhereVis the total volume and d∗\nnis the reduced n−thmoment of f(d). Each particle\nof diameter dis surrounded by a coating layer of thickness ∆ /2; the maximum value of\nthe volume fraction, φmis obtained as the volume fraction of the spheres including b oth\nthe particles and the coating layer, namely by replacing din (A.1) by ( d+ ∆) with the\nsame distribution function. Defining ∆∗= ∆/dmwe get\nφm=Np1\nV/integraldisplay∞\n0f(d)π\n6(d+∆)3d(d)\n=πd3\nm\n6Vd∗\n3(1+∆∗)3/bracketleftbigg1+3∆∗(d∗\n2/d∗\n3)+3∆∗2(d∗\n1/d∗\n3)+∆∗3(1/d∗\n3)\n(1+∆∗)3/bracketrightbigg\n(A.2)\nwhich defines the function ϕas the expression in square brackets. 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Patey, The Journal of Chemical Physics 115, 4718 (2001).\n[49] C. de Montferrand, Y. Lalatonne, D. Bonnin, N. Li` evre, M. L ecouvey, P. Monod, V. Russier,\nand L. Motte, Small 8, 1945 (2012).19\n-0.03-0.02-0.01 0 0.01 0.02 0.03\n 0  0.5  1  1.5  2M/Ms - M/Ms(εc = 0)\nβ*\nFigure 1: Deviation of the reduced magnetization M/Msdue to MAE at h= 0.20 for a non inter-\nacting system with cubic anisotropy. Polydispersity: σ= 0.28. Cubic axes randomly distributed\nandǫc=15, solid circles; ǫc= -15, open circles. Cubic axes fixed and parallel to the system fram e\nwithǫc= 15, solid squares; ǫc= -15, open squares.\n 0 0.2 0.4 0.6 0.8 1\n 0 2 4 6 8 10 12 14M/Ms\nhεc < 0εc > 0\nFigure 2: Magnetization curve for a monodisperse non interacting s ystem with cubic anisotropy.\nThe cubic anisotropy axes are fixed along the system frame with ǫc=±15 long dashed; ±12\ndashed; and ±8 short dasched. The sign of ǫcis as indicated. The case with random distribution\nof the cubic axes is shown for comparison with ǫc= 15, thin solid line; and ǫc=-15, thin dotted\nline. The thick solid line is the reference ǫc= 0 case. β∗= 1.20\n 0 0.2 0.4 0.6 0.8 1\n 0 2 4 6 8 10 12 14 16 0.2 0.4 0.6 0.8 1M/Ms\n  h σ = 0.28\nσ = 0\nFigure 3: Magnetization curve for non interacting system with cubic anisotropy, |ǫc|= 15 and\nβ∗= 1. The [111] direction of the cristallites are prefentially oriented alo ng thezaxis (which is\nalso the direction of the field) with the probability distribution of equa tion (4). Polydisperse case\n(σ= 0.28) with ǫc= -15 and σθ= 0.015, long dash; π/10, short dash; π/2, solid line. Same with\nǫc= 15 and σθ=π/2, dotted line; π/10, short dash dot; 0.015, long dash dot. Monodisperse case\n(σ= 0) with ǫc= 15 and σθ= 0.015, open triangles; π/10, open squares; π/2, open circles.\n 0 0.2 0.4 0.6 0.8 1\n 0  2  4  6  8  10  12M/Ms\nh\nFigure4: M(h)foraninteractingsystemcharacterizedby ǫd=2.37,∆ /dref=0.20,dm/dref=1.33\nandβ∗= 1. Without anisotropy: solid line. In the presence of uniaxial anisot ropy with ǫuv= 5.64\nandǫus= 0.0, solid squares; ǫuv= 0.0 and ǫus= 6.88, open circles. (The value ǫus= 6.88\ncorrespondsto ǫuv(d∗\n3(σ)/d∗\n2(σ)) withǫuv=5.64,d∗\nnisthen-thmomentofthediameterdistribution\nfunction.)21\n 0 0.2 0.4 0.6 0.8 1\n 0  2  4  6  8 10  12  14M/Ms\nhεc = 0\nεc =  15; rand\nεc = -15; rand\nεc =  15; [111]\nεc = -15; [111]\nεc =  15; [001] 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7\n 0 1 2 3 4 5\nFigure 5: Magnetization curve for non interacting system with uniax ial and cubic anisotropieswith\nβ∗= 1,ǫuv= 5 ,ǫus= 0 and |ǫc|= 15. Polydispersity : σ= 0.28. Open circles: case free of\nanisotropy for comparison. ǫcand easy axes distributions as indicated. Inset : comparison of the\nM(h) curves for ǫuv= 5 and ǫc=−15, long dash dotted line and for ǫuv= 8 and ǫc= 0, solid line.\n 0 0.2 0.4 0.6 0.8\n 0 2 4 6 8 10 12 14 16 18M/Ms\nha)\n 0.2 0.4 0.6 0.8\n 0  2  4  6  8  10b)\nFigure 6: Reduced magnetization for a non interacting system with ǫuv= 5.0,ǫc= -15, cubic\naxes randomly distributed, easy axes along the [111] NP cristallogra phic orientations and different\nvalues ofthe reduced inversetemperature β∗.β∗= 0.5, dash dotted line; 0.75, dotted line; 1.0, long\ndashed line; 2.0, solid line; 4.0short dashed line. a)monodisperse system ( σ= 0);b)polydispersity\nσ= 0.28.22\n 0 0.05 0.1 0.15 0.2 0.25\n 0  0.5  1  1.5  2  2.5χr \nβ∗εd = 1.33\nεd = 2.66\nFBC\nPBC (εs = 1)\nPBC (εs = ∞)\nFigure 7: Reduced linear susceptibility, χrversus the inverse reduced temperature β∗in the quasi\nmodisperse case, σ= 0.05 for a volumic fraction φ= 0.385,ǫd= 1.33 and 2.66 ( ǫeff\nd= 1.0 and\n2.0 respectively). Different boundary conditions are considered. I n the PBC with Ewald sums, the\nnumber of particles is Np= 600 while the clusters for the FBC include Np= 1000 particles. Solid\nline :M/Msforh= 1. Solid horizontal lines indicate the limit for y→ ∞, (equation (17). The\nsolid triangle at β∗= 1 indicates the value of χrforǫeff\nd= 1.0 in the polydisperse case σ= 0.28\n(ǫd= 1.73; ∆ /rm= 0.40).\n 0 0.2 0.4 0.6 0.8 1\n 0 2 4 6 8 10 12 14 16 18M/Ms\nh\nFigure 8: Reduced magnetization for a polydisperse interacting sys tem with β∗= 1,ǫd= 2.37,\n∆/dm= 0.15, polydispersity σ= 0.28 and different sets of MAE constants. ǫuv= 0.0 and ǫc= 0,\ndotted line; ǫuv= 5.0 and ǫc= 0, solid line; ǫuv= 5.0,ǫc= 15 and ˆ n= random, open squares;\nǫuv= 5.0,ǫc= -15 and ˆ n= random, solid squares; ǫuv= 5.0,ǫc= 15 and ˆ n= [111], short dashed\nline;ǫuv= 5.0,ǫc= -15 and ˆ n= [111], long dashed line; ǫuv= 5.0,ǫc= -15 and ˆ n= [001], solid\ncircles;ǫuv= 5.0,ǫc= 15 and ˆ n= [001], open circles.23\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  2  4  6  8  10  12M/Ms\nh \nFigure 9: Reduced magnetization for the effective DDI coupling cons tantǫeff\nd= 1.0,β∗= 1,\npolydispersity σ= 0.28, open symbols or σ= 0.05, solid symbols. ǫuv= 6.30 and ǫc= 0, circles;\nǫuv= 4.0,ǫc= -12.0 and ˆ n= [111], squares; ǫuv= 4.0,ǫc= 50, and ˆ n= [001], triangles. ǫuv= 0.0,\nǫc= 0 and σ= 0.28, solid line. The dotted lines are guides to the eyes.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  40  80  120  160  200  240M/Ms\nH(kA/m)\nFigure 10: Comparison of the experimental reduced magnetization curve of a maghemite powder\nsample49withdm= 10nm, open circles with the M.C. simulation, solid line. The parameters\nused in the MC simulation are σ= 0.28,ǫd= 1.0, ∆ /dm= 0.20,ǫuv= 4.0 andǫc=−1.5 with\nˆni= [111]. β∗= 1.24\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  40  80  120  160  200M/Ms\nH (kA/m) 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 50  100  150\nFigure 11: Same as figure 10 for dm= 12nm. Experiments49, open circles. The M.C. simula-\ntions are performed with σ= 0.28,ǫd= 1.733, ∆ /dm= 0.20 and different sets of MAE param-\neters.ǫuv= 6.912,ǫc=−2.85 and ˆni= [111], solid line. ǫuv= 6.912,ǫc=−1.1\nand ˆni= [111], open triangles. Inset: Comparison of the simulated M(Ha)/Mscurves with\nǫuv= 6.912, ˆni= [111] and ǫc=−2.85, solid line; ǫuv= 5.76, ˆni= [111] and ǫc=−9.00,\nopen triangles; ǫuv= 5.76, ˆni= [001] and ǫc= 5.50, open squares.\n 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9\n 0  40  80  120  160  200M/Ms\nH (kA/m) 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  10  20  30  40  50\nFigure 12: Same as figure 10 for dm= 21nm. Experiments49, open circles. The M.C. simulations\nare performed with dm/dref= 2,σ= 0.28,ǫd= 8.0, ∆ /dm= 0.10,ǫuv= 32.00, ˆni= [111] and\nǫc= -13.2, solid squares or ǫc= -5.0, open triangles. The thin solid line is a guide to the eyes."    },    {        "title": "1404.5646v2.First_principles_investigation_of_magnetocrystalline_anisotropy_at_the_L2__1__Full_Heusler_MgO_interfaces_and_tunnel_junctions.pdf",        "content": "First principles investigation of magnetocrystalline anisotropy at the L21Full\nHeuslerjMgO interfaces and tunnel junctions\nRajasekarakumar Vadapoo,1, 2, 3,\u0003Ali Hallal,1, 2, 3Hongxin Yang,1, 2, 3and Mairbek Chshiev1, 2, 3\n1Univ. Grenoble Alpes, INAC-SPINTEC, F-38000 Grenoble, France\n2CNRS, SPINTEC, F-38000 Grenoble, France\n3CEA, INAC-SPINTEC, F-38000 Grenoble, France\n(Dated: March 1, 2022)\nMagnetocrystalline anisotropy at Heusler alloy jMgO interfaces have been studied using \frst prin-\nciples calculations. It is found that Co terminated Co 2FeAljMgO interfaces show perpendicular\nmagnetic anisotropy up to 1.31 mJ/m2, while those with FeAl termination exhibit in-plane magnetic\nanisotropy. Layer resolved analysis indicates that the origin of perpendicular magnetic anisotropy\nin Co 2FeAljMgO interfaces can be attributed to the out-of-plane orbital contributions of interfacial\nCo atoms. At the same time, Co 2MnGe and Co 2MnSi interfaced with MgO tend to favor in-plane\nmagnetic anisotropy for all terminations.\nPACS numbers: 75.30.Gw, 75.70.Cn, 75.70.Tj, 72.25.Mk\nINTRODUCTION\nPerpendicular magnetic anisotropy (PMA) in transi-\ntion metaljinsulator interfaces has been demonstrated\nmore than a decade ago.1,2These interfaces have become\na viable alternative to PMA in fully metallic structures\nbased on heavy non-magnetic elements with strong spin-\norbit coupling (SOC)3{6. Indeed, high PMA values were\nobserved in Co(Fe) jMOx (M=Ta, Mg, Al, Ru, etc.) inter-\nfaces despite their weak SOC1,2. These structures serve\nas main constituents for perpendicular magnetic tunnel\njunctions (p-MTJ) which are very promising for realiz-\ning next generation of high density non volatile memo-\nries and logic devices7{11. One of the most important\nrequirements for the use of p-MTJ in spintronic appli-\ncations including high density spin transfer torque mag-\nnetic random access memory (STT-MRAM) is a com-\nbination of large PMA, high thermal stability and low\ncritical current to switch magnetization of the free layer.\nCoFeBjMgO p-MTJ is one of the most promising can-\ndidates among state-of-the-art structures10. However,\nanother class of ferromagnetic electrode materials with\ndrastically improved characteristics for use in p-MTJ are\nHeusler alloys, since they possess much higher spin po-\nlarization12and signi\fcantly lower Gilbert damping13.\nFull Heusler alloys (X 2YZ)jMgO interfaces with high\ninterfacial PMA and weak spin orbit coupling (SOC)\nhave been gaining interest recently12,14{16. For in-\nstance, MgO-based MTJs with Co 2FeAl(CFA) electrodes\nshow high PMA in most of the experiments. The\nsurface anisotropy energy ( Ks) is found to be around\n1 mJ/m2for PtjCFAjMgO trilayer17and CFAjMgO16,18\ninterfaces. The observed PMA values for these struc-\ntures are comparable to those reported for CoFeB jMgO10\nand tetragonally distorted Mn 2:5Ga \flms grown on Cr\nbu\u000bered MgO14. However, there are reports on ob-\nservation of in-plane magnetic anisotropy (IMA) for\nCFAjMgO interfacial structures in di\u000berent cases19,20.\nThus, these interfaces show PMA with values between0.16-1.04 mJ/m216,21,22as well as IMA with Ks=-\n1.8 mJ/m219. On the other hand, some theoretical stud-\nies have reported PMA values of 1.28 mJ/m2for Co ter-\nminated structures23, IMA of 0.78 mJ/m223and PMA\nof 0.428 mJ/m224for FeAl termination. It has been sug-\ngested that interfacial\nFe atoms are responsible for PMA in these structures21\nbut the microscopic origins of anisotropy remains to be\nclari\fed further.\nIn order to elucidate the origin of PMA in these\ninterfaces, we present a systematic study of magnetic\nanisotropy in Heusler alloy (X 2YZ)jMgO interfaces [with\nX=Co, YZ=FeAl, MnGe and MnSi] using \frst principles\nmethod. We explore the di\u000berent interfacial conditions in\nthese interfaces. In order to understand the microscopic\nmechanism of PMA, we employ the onsite projected and\norbital resolved analysis of magnetocrystalline anisotropy\nenergy (MA) which allows identi\fcation of layer contribu-\ntions along with the corresponding di\u000berent orbital con-\ntributions25,26. We found that the magnetic anisotropy\nis much more complex compared to that in Co(Fe) jMgO\nstructures26and it is strongly dependent on the interface\ntermination and composition.\nMETHODS\nCalculations are performed using Vienna ab initio siu-\nmulation package (VASP)27,28with generalized gradient\napproximation29and projected augmented wave poten-\ntials30,31. We used the kinetic energy cuto\u000b of 600 eV\nand a Monkhorst-Pack k-point grid of 13 \u000213\u00023 where\nthe convergence of MAE is checked with repect to the\nnumber of K-points. Initially the structures were relaxed\nin volume and shape until the force acting on each atom\nfalls below 1 meV/ \u0017A. The Kohn-Sham equations were\nthen solved to \fnd the charge distribution of the ground\nstate system without taking spin-orbit interactions (SOI)\ninto account. Finally, the total energy of the system wasarXiv:1404.5646v2  [cond-mat.mtrl-sci]  16 Sep 20162\nFIG. 1. (Color online) Perspective view of (a) X terminated,\n(b) YZ terminated interface structure of Heusler (X 2YZ)jMgO\nand (c) X terminated, (d) YZ terminated Heusler jVacuum\nslabs with X=Co, YZ=FeAl, MnGe and MnSi. Grey, yellow,\npink, blue and red spheres represent X, Y, Z, Mg and O atoms,\nrespectively.\ncalculated for a given orientation of magnetic moments\nin the presence of spin-orbit coupling using a non-self-\nconsistent calculation. The surface magnetic anisotropy\nenergy,Ksis calculated as ( Ek\u0000E?)=a2, whereais\nthe in-plane lattice constant and E?(Ek) represents en-\nergy for out-of-plane [001](in-plane [100]) magnetization\norientation with respect to the interface. The in-plane\nanisotropy (di\u000berence between [100] and [110]) have been\nchecked and found to be negligible. Positive and neg-\native values of Kscorresponds to out-of-plane and in-\nplane anisotropy respectively. In addition, we de\fne the\ne\u000bective anisotropy Keff=Ks=tCFA\u0000Edemag , where\nEdemag is the demagnetization energy which is the sum\nof all the magnetostatic dipole-dipole interactions upto\nin\fnity. We adopt the dipole-dipole interaction method\nto calculate the Edemag term instead of 2 \u0019M2\ns, whereMs\nis the saturation magnetization; since the latter under-\nestimates this term for thin \flms.26,32,33In VASP the\nspin-orbit term is evaluated using the second-order ap-\nproximation:\nFIG. 2. (Color online) Surface magnetic anisotropy energy\n(Ks) as a function of number of heusler atomic-layers (ML)\nin Co and YZ terminated heusler (X 2YZ)jMgO structures.\nFilled data points represent Co terminated and open data\npoints represent YZ terminated interfaces. Blue triangle rep-\nresent Co 2FeAl (CFA), black square for Co 2MnGe (CMG)\nand red circle for Co 2MnSi (CMS) interfaces. Inset shows the\ne\u000bective anisotropy ( Keff\u0003t) as a function of thickness of\nCFA in Co terminated CFA jMgO interface.\nHSOC =1\n2(meC)21\nrdV\ndr~L:~ s (1)\nwhereVdenotes the spherical part of all-electron\nKohn-Sham potential inside the PAW spheres, while ~L\nand~ srepresent the angular momentum operator and the\nPauli spin matrices, respectively. The spin-orbit coupling\nthen can be calculated for each orbital angular momen-\ntum, and from which one can extract layer- and orbital-\nresolved MAE26,33{35.\nRESULTS\nFull-heusler ( X2YZ) alloys are intermetallic com-\npounds with cubic L21structure and belongs to the space\ngroupFm3m12,36. The magnetocrystalline anisotropy of\nbulk heusler is found to be negligible. The Heusler jMgO\ninterfaces have been setup with the crystallographic\norientation of Heusler(001)[100] kMgO (001)[110]24,37{39.\nThis results in a relatively low lattice mismatch between\nHeusler(001) and MgO(001) with a 45 degrees in-plane\nrotation. The energetically stable X and YZ termina-\ntions at the interface were studied and will be denoted as\nX-HeuslerjMgO and YZ-Heusler jMgO as shown respec-\ntively in Fig. 1(a) and (b). The results of these interfaces\nwill be compared to those of X-Heusler jVacuum and YZ-\nHeuslerjVacuum slabs shown in Fig. 1(c) and (d), respec-\ntively.\nIncreasing the MgO thickness beyond 5 atomic-layers\n(ML) is found to have no e\u000bect on magnetic anisotropy.3\nFIG. 3. (Color online) Atomic layer resolved contributions\nto the anisotropy for (a) X - terminated and (b) YZ - ter-\nminated HeuslerjMgO (solid \flled bars) and Heusler jvacuum\n(light \flled bars) structures shown in Fig. 1(a,c) and (b,d),\nrespectively. The Co 2FeAl (CFA), Co 2MnGe (CMG) and\nCo2MnSi (CMS) cases are represented by pink, blue and green\nbars, respectively. The side view of the corresponding Heusler\nlayer are shown on the left for convenience.\nThe variation of suface magnetic anisotropic energy ( Ks)\nwith the thickness of Heusler layers varying from 3 to 11\nML for the Heusler jMgO interfaces is shown in Fig. 2.\nOne can see that only Co-CFA jMgO structure gives rise\nto very high PMA which weakly depend on CFA thick-\nness, while the FeAl-CFA jMgO and all CMG jMgO as well\nas CMSjMgO show IMA. It is interesting to note that\nthe magnetic anisotropy energy for the CMG jMgO and\nCMSjMgO as a function of thickness follow similar trend\nwhich might be due to the inert nature of Z-element (Ge,\nSi). The in-plane anisotropy contribution in these struc-\ntures increases as a function of thickness and stabilizes\nafter 9 ML. It can be seen that Ksfor Co-CFAjMgO in-\ncreases from 1 :20 mJ/m2to a maximum of 1 :31 mJ/m2\nat 7 ML thickness ( \u00180:8 nm), which is in agreement\nwith experimental \fndings of M. S. Gabor et al.22and\nZ. Wen et al.16. Inset in Fig. 2 shows the corresponding\ne\u000bective anisotropy ( Keff\u0003t) as a function of CFA thick-\nness (tCFA). It shows a decaying behavior and vanishes\naround 11 ML becoming IMA beyond this thickness, in\nreasonable agreement with recent experiments16,22.\nIn order to understand the origin of PMA and ef-\nFIG. 4. (Color online) d-orbital resolved contributions\nto magnetic anisotropy for interfacial atoms in X- and YZ-\nterminated (a) Co 2FeAljMgO and (b) Co 2MnGejMgO struc-\ntures along with their free surface counterparts. Black square,\nred circle, blue triangle and purple star represent contribu-\ntions from orbitals with d-character of two Co (Co1 and Co2\nwithin the same atomic-layer), Fe and Mn atoms, respec-\ntively. XV, XM, YV and YM denote X-Heusler jVacuum, X-\nHeuslerjMgO, YZ-Heusler jVacuum and YZ-Heusler jMgO in-\nterfaces respectively.\nfect of MgO, we examined the on-site projected mag-\nnetic anisotropy for the 11 ML of Heusler jMgO and their\nfree surface counterparts as shown in Fig. 3. As one can\nsee, the major PMA contribution of 0.69 mJ=m2in Co-\nCFAjMgO structure comes from the interfacial Co atoms\nwhile the inner layers show fair amount of in-plane or\nout-of-plane contributions represented by solid pink bars\nin Fig. 3(a). By comparing with CFA jVacuum shown\nby un\flled pink bars in the same \fgure, we can clearly\nidentify that the presence of MgO on top of Co layer\nplays a decisive role in establishing the PMA in Co termi-\nnated CFAjMgO structure. More complicated behavior\nis observed for Co-CMG and Co-CMS structures where\nthe role of MgO in anisotropy varies depending on layer.\nWhile it tends to decrease(increase) the IMA in the 1st\nCo layer for Co-CMG(Co-CMS), it simultaneously \rips\nthe IMA into PMA (PMA into IMA) for 2nd YZ (3rd\nCo) layer.\nSimilar nontrivial picture is observed for YZ termi-\nnated structures shown in Fig. 3(b). By employing the\nsame analysis in order to clarify the role of MgO vs vac-\nuum next to YZ-terminated Heusler alloy, one can see\nthat the MgO has a tendency to improve the IMA for the\ncase of YZ-CFA for all layers. Furthermore, it enhances\nthe PMA(IMA) for the 1st(all Co) layers of YZ-CMS and\nYZ-CMG structures.\nOverall it can be concluded that the presence of\nMgO tends to favor IMA from all Co layers except\nthe interfacial ones in Co-CFA and Co-CMG struc-\ntures. At the same time, the inner YZ layers in pres-\nence of MgO have a tendency for the PMA for Co-\nterminated structures (Fig. 3(a)), while YZ interfacial\nlayer favor the IMA(PMA) in YZ-CFA(YZ-CMS and YZ-4\nFIG. 5. (Color online) Magnetic anisotropy contribution from di\u000berent d-orbital hybridizations at the interfacial atoms of\n(a)[(d)] Co1 (b)[(e)] Co2 and (c)[(f)] Fe for Co-terminated Co 2FeAljMgO interface [ FeAl-terminated interface].\nCMG) (cf. Fig. 3(b)).\nTo further elucidate the microscopic origin of PMA,\nwe carried out the d-orbital resolved magnetic anisotropy\ncontributions for interfacial atoms as shown in Fig. 4.\nOne can see that the switch from IMA to PMA when\nMgO is placed on top of Co terminated CFA mainly arises\nfrom the out-of-plane orbitals ( dxz;yz anddz2) as shown\nby comparison of XV and XM columns in Fig. 4(a). Fur-\nthermore, this switch is assisted by all dorbitals within\nthe 2nd (YZ) layer. At the same time, the MgO-induced\nenhancement of the IMA in the \frst two layers from in-\nterface (FeAl and Co) in case of YZ terminated CFA (see\nFig. 3(b)) is due to increase(decrease) of IMA(PMA) con-\ntribution from dx2\u0000y2(dyzanddz2) orbitals, as seen from\ncomparison of columns YV and YM in Fig. 4(a).\nIn the case of Co terminated CMG, the e\u000bect of MgO\nresults in overall tendency to decrease the IMA with par-\nticipation of in-plane dorbitals (dxyanddx2\u0000y2) in the\n\frst Co layer with a quite interesting opposing contribu-\ntions from out-of-plane dyzanddz2orbitals (see XV and\nXM columns in Fig. 4(b)). At the same time, for the\nsecond layer (MnGe) contribution, the presence of MgO\nhas a clear tendency to switch from IMA into PMA as-\nsisted by all d-orbitals. As for Mn terminated CMG,\nthe presence of MgO has almost no e\u000bect on 1st MnGe\nlayer anisotropy contributions, while it induces the \rip\nfrom PMA to IMA from almost all dorbitals within the\nsecond Co layer (Fig. 4(b)). The orbital contributions\nfor CMS are found to be very similar to CMG orbital\ncontributions.\nTo further elucidate the PMA origin in Co 2FeAljMgO,\nFig. 5 shows magnetic anisotropy contribution originated\nfrom the spin orbit coupling induced hybridizations be-\ntween di\u000berent orbital channels for interfacial atoms atthe Co-terminated and FeAl-terminated interface. In\nall cases out-of-plane orbitals (d xz(yz), dz2) mututal hy-\nbridizations strongly favor PMA contribution. At the\nsame time, hybridization among in-plane orbitals (d xy,\ndx2\u0000y2) gives rise to IMA except for the case of Co1\nand Co2 atoms in the Co-terminated interface where they\nhave a slight PMA contribution. In all cases d x2\u0000y2hy-\nbridization with out-of-plane, mainly d yz, orbitals con-\ntribute to IMA. On the other hand, d xyhybridization\nwith out-of-plane, mainly d xz, orbitals contribute to\nPMA except for the case of Co2 atom in Co-terminated\nstructure. However, the sum of the contribution coming\nfrom Co1 and Co2 atoms is favoring PMA.\nDISCUSSION\nAforementioned analysis shows that Co-CFA jMgO\nstructure favors the high PMA while YZ termination in\nCFAjMgO structure give rise to IMA. This allows us to\nconclude that interfacial Co atoms are responsible for the\nPMA. However, it was claimed recently that the origin\nof PMA could be attributed to Fe atoms at the interface\nin CFAjMgO21using XMCD measurements in combina-\ntion with Bruno's model analysis40. In order to resolve\nthis disagreement, we carried out the orbital momentum\ncalculations for 7 ML structure corresponding to that re-\nported in experiments21with both terminations. We sys-\ntematically found that per layer resolved orbital moment\nanisotropy (OMA) is inconsistent with layer resolved MA\ncontributions for Co layers while it remains in qualitative\nagreement for layers containing Fe. We can therefore con-\nclude that Bruno's model should be used with caution\nand its validity may depend on particular system.5\nIn summary, using \frst principles calculation we in-\nvestigated the magnetic anisotropy of Full Heusler jMgO\ninterfaces and MTJs for all terminations. It is found that\nCo terminated CFA jMgO shows the PMA of 1 :31mJ/m2\ninduced by the presence of MgO in agreement with re-\ncent experiments while FeAl terminated CFA and other\nstructures possess the IMA. We also unveiled the micro-\nscopic mechanisms of PMA in Heusler jMgO structures\nby evaluating the onsite projected and orbital resolved\ncontributions to magnetic anisotropy and found that in-terfacial Co atoms are responsible for high PMA (IMA)\nin CFA (CMG,CMS). Finally, out-of-plane (in-plane) or-\nbitals tend to favor mainly PMA (IMA).\nACKNOWLEDGEMENTS\nThe authors thanks C. Tiusan for fruitful discus-\nsions. This work was partly supported by Eu M-era.net\nHeuMem, French ANR SPINHALL and SOspin projects.\n\u0003Present address: Geophysical Laboratory, Carnegie Insti-\ntution of Washington, Washington, DC 20015, USA\n1S. Monso, et al. Appl. Phys. Lett. 80, 4157 (2002).\n2B. Rodmacq, S. Au\u000bret, B. Dieny, S. Monso, and P. Boyer,\nJ. Appl. Phys. 93, 7513 (2003).\n3N. Nakajima, Phys. Rev. Lett. 81, 5229 (1998).\n4P. F. Carcia, A. D. Meinhaldt, and A. Suna, Appl. Phys.\nLett. 47, 178 (1985).\n5H. J. G. Draaisma, W. J. M. de Jonge, and F. J. A. den\nBroeder, J. Magn. Magn. Mater. 66, 351 (1987).\n6D. Weller, et al. Phys. Rev. B 49, 12888 (1994).\n7G. Kim, et al. Appl. Phys. Lett. 92, 172502 (2008).\n8L. E. Nistor, B. Rodmacq, S. Au\u000bret, and B. Dieny, Appl.\nPhys. Lett. 94, 012512 (2009).\n9L. E. Nistor, et al. Phys. Rev. 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Chshiev, and W. H. Butler,\nJ. Appl. Phys. 103, 07D707 (2008).\n37M. Yamamoto, et al . J. Phys. D: Appl. Phys. 39, 824\n(2006).\n38M. S. Gabor, T. Petrisor, C. Tiusan, M. Hehn, and T.\nPetrisor, Phys. Rev. B 84, 134413 (2011).\n39X. Lu, K. L. Varaha, K. Mukherjee, and N. C. Kar, IEEE\nTransactions on magnetics 49, 3965 (2009).\n40P. Bruno, Phys. Rev. B 39, 865 (1989)."    },    {        "title": "1404.7097v2.Electronic_structure_and_magnetic_properties_of_L1_0_binary_alloys.pdf",        "content": "arXiv:1404.7097v2  [cond-mat.mtrl-sci]  13 Jun 2014Electronic structure and magnetic properties of L1 0binary alloys\nAlexander Edström, Jonathan Chico, Adam Jakobsson, Anders Bergman, and Jan Rusz\nDepartment of Physics and Astronomy, Uppsala University, B ox 516, 75121 Uppsala, Sweden\n(Dated: September 26, 2018)\nWe present a systematic study of the magnetic properties of L 10binary alloys FeNi, CoNi, MnAl\nand MnGa via two different density functional theory approac hes. Our calculations show large\nmagnetocrystalline anisotropies in the order 1 MJ/m3or higher for CoNi, MnAl and MnGa while\nFeNi shows a somewhat lower value in the range 0 .48−0.77 MJ/m3. Saturation magnetization\nvalues of 1 .3 MA/m, 1 .0 MA/m, 0 .8 MA/m and 0 .9 MA/m are obtained for FeNi, CoNi, MnAl\nand MnGa respectively. Curie temperatures are evaluated vi a Monte Carlo simulations and show\nTC= 916 K and TC= 1130 K for FeNi and CoNi respectively. For Mn-based compoun ds Mn-rich\noff-stoichiometric compositions are found to be important f or the stability of a ferro or ferrimagnetic\nground state with TCgreater than 600 K. The effect of substitutional disorder is s tudied and found\nto decrease both magnetocrystalline anisotropies and Curi e temperatures in FeNi and CoNi.\nMaterials exhibiting a large saturation magnetization\n(Ms), high Curie temperature ( TC), as well as large mag-\nnetic anisotropy energy (MAE), are of great technological\nimportance in a wide range of permanent magnet appli-\ncations, from electric motors and generators to magnetic\nstorage devices. L1 0ordering of binary compounds is\nknown to be able to significantly increase MAE relative\nto the disordered state and for certain materials, such\nas FePt, an enormous MAE in the order of 5 MJ/m3\nis observed1–4. Large values for MsandTCcan be ob-\ntained with cheap and abundant materials such as bcc\nFe, while achieving large MAE is a challenge. Typically,\nlarge values of the MAE are obtained for materials con-\ntaining heavy elements, such as platinum or rare-earths,\nproviding strong spin-orbit coupling. Such elements are\noften scarcely available and thus expensive. Finding new\nmaterials, with large MAE, made from cheap and readily\navailable elements is therefore a task of great technologi-\ncal importance. Certain L1 0ordered binary compounds,\nsuch as FeNi1,5–9, CoNi10, MnAl11–14and MnGa15, have\nbeen reported to exhibit large MAE without containing\nplatinum or rare-earths, making them potentially inter-\nesting candidates for permanent magnet materials.\nIn this work, a thorough investigation is done into the\nelectronic structure and magnetic properties of L1 0struc-\ntured binary compounds FeNi, CoNi, MnAl and MnGa.\nTo the best of our knowledge, first principles all-electron\nelectronic structure calculations including full-potent ial\neffects have not been presented in the literature for all\nthese compounds yet. Furthermore, all three of the im-\nportant permanent magnet properties Ms,TCand MAE\nare adressed for all of the compounds. In addition to\nthis, substitutional disorder and off-stoichiometric com-\npositions are investigated.\nThree different computational methods were utilized\nin the calculations behind this work. First, two den-\nsity functional theory (DFT) implementations, namely\nfull-potential all-electron code WIEN2k16with linearized\naugmented plane wave basis functions and the Mu-\nnich spin polarized relativistic Korringa-Kohn-Rostoker\n(SPR-KKR) package17,18were used, both with the\ngeneralized gradient approximation19for the exchange-\nFigure 1: Two different unit cells of the L1 0structure.\na′=a√\n2.\ncorrelation potential, to calculate ground state proper-\nties of the investigated systems. Later, Monte Carlo\n(MC) simulations of the Heisenberg hamiltonian were\nperformed, using the Uppsala Atomistic Spin Dynam-\nics (UppASD)20method, with exchange parameters cal-\nculated, via the method of Liechtenstein et al.21,22, in\nSPR-KKR. Results of these calculations are shown in Ta-\nble I. The L1 0structure can be described by either a bct\nor fct-like unit cell as illustrated in Fig. 1. The smaller\nbct-like unit cell is used as input for calculations, as it\nallows lower computational cost due to a smaller basis,\nwhile Table I contains lattice parameters describing the\nfct-like cell, as it is commonly used and gives a c/a-ratio\nbetter describing the deviation from a cubic structure.\nThe lattice parameters were evaluated by total energy\nminimization in WIEN2k and used as input for all fur-\nther calculations. In the case of MnGa a double mini-\nmum is observed in the total energy as function ofc\naas\nshown in Fig. 2. The data for MnGa shown in Table I\nis for the more stable structure, with largerc\na, which\nshows a rather large uniaxial MAE, in contrast to the\nstructure in the local minimum, which reveals a smaller\nin-plane anisotropy. The MAE was evaluated using the\ntorque method18,23in SPR-KKR and total energy differ-\nence calculations in WIEN2k. 160000 k-vectors and 40\nenergy points were used in SPR-KKR and basis functions\nup to l= 3 were included. In WIEN2k, 20000 or more\nk-vectors were used, the smallest muffin-tin radius times\nmaximum k-vector was set to RMTKmax= 9 or higher\nand Brillouin-zone integration was performed using the2\nmodified tetrahedron method24.\n0.60.70.80.91.01.100.20.4E − Emin (eV)\nc/a\nFigure 2: Difference in total energy and total energy of\nthe equilibrium structure as function ofc\na, varied under\nconstant volume, for MnGa.\nMagnetocrystalline anisotropy is a relativistic phe-\nnomenon due to the spin-orbit coupling (SOC). The two\nDFT methods used differ in the way they take relativistic\neffects, in general, and SOC, in particular, into account.\nWIEN2k does a fully relativistic treatment of the core\nelectrons but a scalar relativistic approximation for the\nvalence electrons with SOC included as a perturbation25.\nThis should be a very accurate method for 3d metals and\nhas been shown to yield good results even for significantly\nheavier elements26,27. The SPR-KKR method, on the\nother hand, deals with relativistic effects in all electrons\nvia a fully relativistic four component Dirac formalism18.\nThe data in Table I show a good agreement between\nSPR-KKR and WIEN2k, although there is some minor\ndisagreement in the MAE where SPR-KKR consistently\nyields a larger value. There are a number of reasons\nwhich can contribute to the difference in the MAE found\nfrom the two methods. One of the main possible rea-\nsons is that we did not take full-potential effects into\naccount in the SPR-KKR calculations. Other reasons in-\nclude that, as mentioned, relativistic effects are treated\ndifferently and also different basis functions are used to\ndescribe the Kohn-Sham orbitals. Furthermore, MAEs\nare typically relatively small energies orders of magni-\ntude smaller than, for example, cohesive energies and\nhence difficult to obtain numerically with high accu-\nracy. In view of this, the agreement between the two\nmethods can be considered very good. The MAE has\npreviously been calculated to 0 .5 MJ/m3, 1.0 MJ/m3,\n1.5 MJ/m3and 2 .6 MJ/m3for FeNi, CoNi, MnAl and\nMnGa, respectively5,12,15,28, consistent with the results\npresented here. Table I also contains experimental val-\nues for MAE, where available, for comparison. For CoNi\nand MnAl we see that the theoretical MAEs, both from\nSPR-KKR and WIEN2k are higher than reported exper-\nimental values. This is expected as experimental samples\ntypically do not have perfect ordering and experiments\nare done at finite temperatures, factors which are known\nto reduce MAE1,2. However, in the case of FeNi theo-\nretical and experimental values are of similar magnitude\neven though perfectly ordered samples have not been syn-\nthesized. This might indicate that the theoretical values\npresented here are too low, possibly because these calcu-\nlations ignore orbital polarisation corrections which hav e\nbeen reported to significantly increase MAE in FeNi3,5.\nExchange parameters, Jij, were calculated in SPR-KKR and Fig. 3 shows how these vary with atomic dis-\ntances for FeNi and CoNi. The Jijcan be seen to de-\ncrease approximately as R−3, as one would expect for\nmetals with RKKY-type exchange interactions. These\nexchange parameters were used to calculate the Curie\ntemperatures, presented in Table I, via mean field the-\nory (MFT) as well as MC simulations. MC Curie tem-\nperatures in the thermodynamic limit were evaluated by\nfinite size scaling using the Binder cumulant method30.\nAs expected, MFT overestimates TCcompared to MC\nby around 20%. Both Curie temperatures of 916 K and\n1130 K for FeNi and CoNi are very high, which is suitable\nfor permanent magnet applications. An MFT estimate of\nTChas previously been done to 1000 ±200 K31for FeNi\nwhich is consistent with results presented here. The Jij\nare particularly large for Fe-Fe and Co-Co interactions,\nindicating that these elements contribute significantly to\nproviding a high TCto the materials.\n2468−200−1000100200\nRij / aRij3 ⋅ Jij (a3 ⋅ meV) \n  \nFe-FeFe-NiNi-Ni\n(a) FeNi2468−50050100\nRij / aRij3 ⋅ Jij (a3 ⋅ meV) \n  \nCo-CoCo-NiNi-Ni\n(b) CoNi\nFigure 3: Atomic distance dependence of exchange\nparameters Jij.\nReal samples of L1 0alloys do not exhibit perfect or-\ndering and, for example, FeNi samples have been re-\nported with long-range chemical order parameter around\nS= 0.488(Sdescribes the fraction of atoms on the cor-\nrect sublattice as P=1\n2(1+S)). Disorder has been found\nto be important and have a negative effect on the MAE\nof FeNi as well as a number of other L1 0materials1and\ncould also significantly affect TC. Table II shows the ef-\nfect of some substitutional disorder on the MAE and TC\nof FeNi and CoNi. Calculations were perfomed on sys-\ntems with one atomic position occupied by X 1−ηNiηand\nthe other one by X ηNi1−η, with X=Fe or Co and ηup to\nη= 10%. Disorder was treated using the coherent poten-\ntial approximation (CPA)32in SPR-KKR. The data show\nhow disorder causes a similar reduction of MAE, also in\nCoNi, as it does in FeNi and other L1 0alloys. Also the\nTCof both FeNi and CoNi show a clear decrease with\nincreasing disorder, although they still remain at high\ntemperatures, well above room temperature.\nIt was recently suggested, based on experimental ob-\nservations, that increasing the Fe-content in FeNi to\nFe1.2Ni0.8can increase MAE by around 30%33. SPR-\nKKR-CPA calculations failed to reproduce this result\nand rather indicated a reduction of MAE by around 10%\nto MAE = 98 µeV/f.u. in such a composition. Similarly,3\nQuantity FeNi CoNi MnAl MnGa Mn 1.14Al0.86Mn 1.2Ga0.8\na (Å) 3.56 3.49 3.89 3.83 3.89 3.83\nc (Å) 3.58 3.60 3.49 3.69 3.49 3.69\nmW2k\nX(µB) 2.69 1.77 2.33 2.56 - -\nmkkr\nX(µB) 2.73 1.75 2.49 2.74 2.54/-3.41 2.69/-3.40\nmW2k\nY(µB) 0.67 0.71 -0.04 -0.08 - -\nmkkr\nY(µB) 0.62 0.68 -0.09 -0.12 -0.10 -0.12\nmW2k\ntot(MA/m) 1.33 1.01 0.82 0.86 - -\nmkkr\ntot(MA/m) 1.37 1.03 0.84 0.90 0.69 0.66\nEW2k\nMAE (µeV/f.u.) 68.7 135.1 275.1 378.2 - -\nEkkr\nMAE (µeV/f.u.) 110.3 184.7 320.8 385.7 360.2 428.8\nEW2k\nMAE (MJ/m3)0.48 0.99 1.67 2.24 - -\nEkkr\nMAE (MJ/m3)0.77 1.35 1.95 2.28 2.18 2.54\nEexp\nMAE (MJ/m3)0.58290.54101.3713- - -\nTMFT\nC (K) 1107 1383 - 107 - -\nTMC\nC(K) 916 1130 - 80 670 690\nTable I: Lattice parameters calculated using WIEN2k, magne tic moments and magnetic anisotropies calculated\nusing WIEN2k and SPR-KKR as well as Curie temperatures calcu lated using mean field theory and UppASD Monte\nCarlo for L1 0binary alloys FeNi, CoNi, MnAl and MnGa.\nη 0% 5% 10%\nFeNiEkkr\nMAE (µeV/f.u.) 110.3 102.0 89.5\nTMC\nC(K) 916 880 860\nCoNiEkkr\nMAE (µeV/f.u.) 184.7 170.3 145.2\nTMC\nC(K) 1130 940 935\nTable II: MAE and TCfor FeNi and CoNi with\nsubstitutional disorder described by η.\nin Co 1.2Ni0.8, the MAE was reduced to 141 µeV/f.u..\nAlso the TCwas reduced to 840 K and 1020 K in\nFe1.2Ni0.8and Co 1.2Ni0.8respectively. This can be un-\nderstood from the exchange coupling parameters where\nthere is a slight reduction in the strong positive parame-\nters as one adds excess Fe or Co (not shown).\nFor stoichiometric and perfectly ordered MnAl, the\nMonte Carlo simulations show that an antiferromagnetic\nordering is prefered over a ferromagnetic order. Com-\npeting antiferromagnetic exchange interactions can some-\ntimes infer complex non-collinear ground states, but for\nMnAl, no such tendency was found from the Monte Carlo\nsimulations. The preference of antiferromagnetism in\nMnAl can be qualitatively understood if one looks at\nthe exchange interactions as a function of the distance\nbetween atoms. Fig. 4a shows that the Mn-Mn inter-\nactions have quite strong antiferromagnetic interactions .\nWhen introducing Mn also in the second sublattice, one\ncan observe reduction of the antiferromagnetic coupling\nbetween Mn atoms in the first sublattice while there is\na strong antiferromagnetic coupling between Mn atoms\nin different sublattices, as seen in Fig. 4b. This stabi-\nlizes a ferrimagnetic state with Mn atoms in different\nsublattices having moments in opposite directions, giv-ing a total magnetic moment reduced to 1 .98µB/f.u., but\na considerable critical temperature of TC= 670 K in\nMn1.14Al0.86. Experimentally it has also been reported\nthat increased Mn content can cause increased TCto, for\nexample, TC= 655 K for Mn 1.08Al0.9234.\nIn MnGa only a weak ferromagnetism with very low TC\naround 80 K was found. Similar behaviour as for MnAl\nis observed in the Jijof MnGa, as shown in Fig. 4c-4d.\nAgain, increased Mn content yields a higher TCand anti-\nferromagnetic coupling between the Mn sublattices yields\na reduced total moment. We find, for Mn 1.20Ga0.80,\nTC= 690 K and the saturation magnetization reduced\nby almost 30% to MS= 0.66 MA/m. Experimentally\nit has been reported that pure 1:1 stoichiometric MnGa\nis not stable, while with 55-60 at.% Mn it is, and in\nthis range TCincreases and MSdecreases with increas-\ning Mn content35, which is consistent with our calcu-\nlations of substitutional disorder. Mn 1.18Ga0.82has ex-\nperimentally been reported to show TC= 646 K and\nMS= 0.39 MA/m at room temperature35. At lower Mn\ncontent, with around 10-12% excess Mn, we find more\ncomplicated magnetic structures from MC at low tem-\nperatures which yields a total moment lowered by about\na factor half. Such drastic decreases of moment have also\nbeen reported experimentally, although for a bit higher\nMn content36. The MAE of Mn 1.20Ga0.80is, according\nto SPR-KKR calculations, as large as 429 µeV/f.u.\nFig. 5 shows spin-polarized density of states (DOS)\naround the Fermi energy, calculated in WIEN2k, for the\nstudied stoichiometric compounds. All the plots display\na behaviour with clear exchange splitting as expected for\nferromagnetic metals and are also in accordance with pre-\nceding results for those cases which have been previously\nstudied5,12,15, i.e. FeNi, MnAl and MnGa. The DOS for4\n1 2 3−50510\nRij / aJij (meV) \n  \nMn-MnMn-AlAl-Al\n(a) MnAl1 2 3−20−10010\nRij / aJij (meV) \n  \nMn1-Mn1Mn1-Mn2Mn2-Mn2\n(b) Mn 1.14Al0.86\n0 2 4−1001020\nRij / aJij (meV) \n  \nMn-MnMn-GaGa-Ga\n(c) MnGa0 2 4−20−10010\nRij / aJij (meV) \n  \nMn1-Mn1Mn1-Mn2Mn2-Mn2\n(d) Mn 1.20Ga0.80\nFigure 4: Atomic distance dependence of exchange\nparameters Jij.\nNi is seen to be very similar in FeNi and CoNi, although,\na small peak just below −1 eV in the spin down DOS of\nNi in FeNi, not present in CoNi, explains a slightly re-\nduced moment of the Ni atom in FeNi compared to that\nin CoNi. The DOS of MnAl and MnGa are very simi-\nlar with a pronounced ferromagnetic exchange splitting\nof just over 2 eV on the Mn atom while Ga and Al ex-\nhibit very flat DOS around EF. One then expects overall\nsimilar magnetic properties of the two compounds but at\nthe same time MnGa shows a considerably larger MAE,\nwhich is likely due to stronger spin-orbit interaction in-\nduced by the Ga atom relative to Al37. Another possible\nreason for increased MAE in MnGa, relative to MnAl,\nis increasedc\nawhich might allow for better localization\nof d-orbitals along the z-axis, but this is not likely the\ncause as there is not a significant difference in the occu-\npation of d-orbitals in the two compounds. No significant\nqualitative changes occur in the DOS when introducing\ndisorder or off stoichiometric compositions.\nIn conclusion, the magnetic properties of L1 0binary\nalloys FeNi, CoNi, MnAl and MnGa have been inves-\ntigated, systematically and comprehensively, using two\ndifferent DFT methods. Furthermore, the Curie temper-\natures have been studied in order to have a complete pic-\nture of the three properties Ms, MAE and TCwhich are\nimportant in permanent magnet applications. Three of\nthe studied compounds, namely CoNi, MnAl and MnGa,\nexhibit MAE in the order of 1 MJ/m3or higher, which\nis impressive for rare-earth and platinum free materials.Furthermore, all the compounds show Curie tempera-\ntures in the order of 600 K or higher, allowing them to\nbe used in permanent magnet applications above room\n−10 −5 0 5 10−4−2024\nE − EF (eV)DOS (states / eV)  \nTotal\nFe\nNi\n(a) FeNi\n−10 −5 0 5 10−4−2024\nE − EF (eV)DOS (states / eV)  \nTotal\nCo\nNi\n(b) CoNi\n−10 −5 0 5 10−2024\nE − EF (eV)DOS (states / eV)  \nTotal\nMn\nAl\n(c) MnAl\n−10 −5 0 5 10−2024\nE − EF (eV)DOS (states / eV)  \nTotal\nMn\nGa\n(d) MnGa\nFigure 5: Spin polarized density of states.\ntemperature, although we have shown that for Mn-based\ncompounds it is of importance to increase the Mn-content\nin order to obtain high Curie temperatures. We have also\nexplained the experimentally observed effect of reduced\nmoment in Mn rich compounds due to antiferromagnetic\ncoupling between Mn atoms in the two sublattices. In\naddition, we have shown that, for FeNi and CoNi, it is\nof great importance to obtain a high degree of chemical\nordering as both MAE and TCare reduced by substitu-\ntional disorder.\nWe thank Olle Eriksson for discussions and feedback.\nWe acknowledge support from EU-project REFREEP-\nERMAG, eSSENCE and Swedish Research Council\n(VR). Swedish National Infrastructure for Computing\n(SNIC) and NSC Matter are acknowledged for provid-\ning computational resources.\n1Y. Kota and A. Sakuma, Journal of the Physical Society\nof Japan 81, 1 (2012).2J. B. Staunton, S. Ostanin, S. S. A. Razee, B. Gyorffy,5\nL. Szunyogh, B. Ginatempo, and E. Bruno, Journal of\nPhysics: Condensed Matter 16, S5623 (2004).\n3P. Ravindran, A. Kjekshus, H. Fjellvåg, P. James, L. Nord-\nström, B. Johansson, and O. Eriksson, Phys. Rev. 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Rhim1,3,∗Noejung Park4, Kohji Nakamura5, and Soon-Cheol Hong1†\n1Department of Physics and Energy Harvest Storage Research C enter,\nUniversity of Ulsan, Ulsan,Korea\n2Department of Physics, Incheon National University,\nIncheon, Korea3Department of Physics and Astronomy,\nNorthwestern University, Evanston,IL,60208\n4Department of Physics, UNIST, Ulsan, Korea\n5Department of Physics Engineering,\nMie University,Tsu, Mie, 514-8507, Japan\n(Dated: September 24, 2018)\nA new metastable phase of the body-centered-tetragonal rut henium (bct–Ru) is identified to exhibit a large\nperpendicular magnetocrystalline anisotropy (PMCA), who se energy, EMCA, is as large as 150 µeV/atom, two\nordersofmagnitude greaterthanthoseof3 dmagneticmetals. Furtherinvestigationovertherangeofte tragonal\ndistortion suggests that the appearance of the magnetism in thebct–Ru is governed by the Jahn-Teller spit eg\norbitals. Moreover, from band analysis, MCA is mainly deter mined by an interplay between two egstates,\ndx2−y2anddz2states,as a result of level reversal associated withtetrag onal distortion.\nPACS numbers: 75.30.Gw,75.50.Cc, 75.70.Tj\nPhysics phenomena originated from spin-orbit interaction ,\nsuch as magneto-crystalline anisotropy (MCA), Rashba-typ e\ninteractions, or topological insulator, have attracted hu ge at-\ntention for its intriguing physics as well as great potentia l\nforspintronicsapplications.1–5Inparticular,MCA,whereone\nparticular direction of the magnetization is energeticall y pre-\nferred, offers opportunities in spintronics such as magnet ic\nrandomaccessmemory(MRAM),spin-transfertorque(STT),\nmagneto-optics, and to list a few. With advances of fabri-\ncation techniques in recent years, search for materials wit h\nlarge MCA, more preferrably perpendicular MCA (PMCA),\nhasbeenveryintensive.\nIn particular, ferromagnetic films that can provide per-\npendicular MCA (PMCA) are indispensable constituents in\nSTT memory that utilizes spin-polarized tunneling current to\nswitchmagnetization.6Forpracticaloperationofhigh-density\nmemorybits, two criteria haveto besatisfied forpracticalu s-\nage of high-density magnetic storage - low switching curren t\n(ISW) and thermal stability. Small volume is favored to lower\nISW, but is detrimental for the thermal stability. However, the\nshortcoming of small volume can be compensated by large\nMCA while still retaining the thermal stability. On the othe r\nhand, low magnetization will offer advantage to reduce stra y\nfieldinrealdevices. Therefore,explorationformaterials with\nhighanisotropyandsmall magnetizationwouldbe onefavor-\nable direction to minimize ISWand at the same time to maxi-\nmizethethermalstability.\nIn the framework of perturbation theory7,EMCAis deter-\nminedbythespin-orbitinteractionbetweenoccupiedandun -\noccupiedstates as,\nEσσ′\nMCA≈ξ2∑\no,u|/angbracketleftoσ|ℓZ|uσ′/angbracketright|2−|/angbracketleftoσ|ℓX|uσ′/angbracketright|2\nεu,σ′−εo,σ,(1)\nwhereoσ(uσ′) andεo,σ(εu,σ′) represent eigenstates and\neigenvalues of occupied (unoccupied) for each spin state,\nσ,σ′=↑,↓, respectively; ξis the strength of spin-orbit cou-\npling(SOC).Asthe electronicstructureof magneticmaterialswith non-\nnegigibleMCA is mainly dominated by delectrons, it would\nbeworthwhiletoseehowenergylevelsof dorbitalsevolvein\ndifferentcrystalsymmetry,asillustratedinFig.1. Forth ebcc-\nRuwithc/a=1,thecubicsymmetrysplitsfive dorbitalsinto\ndoublet (eg) and triplet ( t2g). When the lattice changes from\nhigh-symmetric body-centeredto tetragonal with lower sym -\nmetry, additional Jahn-Teller splitting may offer more fre e-\ndom to provide more energy differences in Eq. (1). More\nspecifically, delectrons in the bccstructure split into doublet\n(eg) and triplet ( t2g). The tetragonal distortion further splits\ntheseegandt2glevelsinto twoirreduciblerepresentations: eg\ninto two singlets a1(dz2) andb1(dx2−y2);t2ginto a singlet\nb2(dxy) and a doublet e(dyz,xz), where their relative order is\ndeterminedby c/a,eitherlargerorsmallerthanunity.\nMetalswith4 dand5dvalenceelectronspossessinherently\nlarger SOC than conventional 3 dmetals. Search for mag-\nnetismin these transitionmetalshavea longhistory. The fa ct\nthatPdandPtbarelymisstheStonercriteriatobecomeferro -\nmagnetic(FM) hasincurredenormouseffortsto realize mag-\nnetism in several multilayers and interfaces of 4 dmetals by\nadjustingvolumesorlattice constants,therebyincreased den-\nsity of states (DOS) at the Fermi level ( EF),N(EF), due to\nnarrowed bandwidth, would meet the Stoner criteria. Hence,\n4dand5dmetalswithlargeSOCaswellasmagnetismwould\nbefavorablecandidatetorealizelargeMCA.\nPrevious theoretical study suggested that ferromagnetism\nin Ru is feasible in body-centered-cubic( bcc) structure when\nlattice is expandedby5%.8Other studiespredictedthat mag-\nnetismcanoccurinRhandPdwithvolumechanges.9,10How-\never, those theoretically proposedmagnetism associated w ith\nvolume changes in 4 dmetals have not been fully confirmed\nexperimentally. Nevertheless, with remarkable advances i n\nrecentfabricationtechniques,varioustypesoflatticesa renow\naccessible with diversechoiceof substrates. In particula r,the\nbct–Rufilm hasbeensuccessfullyfabricatedontheMo(110)\nsubstrate, whose lattice constantsare a=3.24˚A andc/a=0.832\nFIG.1: (coloronline)SchematicpresentationoftheJahn-T ellersplittingof delectrons. Inthecubicsymmetry,suchas bcc(c/a=1),dorbital\nsplits into doublet ( eg) and triplet ( t2g). Tetragonal distortion further splits egintoa1andb1;t2ginto a singlet b2and a doublet e, where their\nrelative order is shown depending on c/ais greater or smaller thanunity.\nasidentifiedbyX-rayelectrondiffraction.11Later,theoretical\ncalculationarguedthat magnetismcanexist in the bct–Ru for\nc/a=0.84withmomentof0.4 µB/atom.12\nIn this paper, we present that in a newly identified\nmetastable phase of the bct–Ru,EMCAcan be as large as\n150µeV/atom, two orders of magnitude greater than those\nin 3dmagnetic metals. The magnetic instability driven by\nthis tetragonal distortion is discussed in connection with the\nStonercriteria. Furthermore,weshowthatmagnetismaswel l\nas MCA are governed mainly by the Jahn-Teller split egor-\nbitals.\nDensity functional calculations were performed using the\nhighlyprecisefull-potentiallinearizedaugmentedplane wave\n(FLAPW) method.13For the exchange–correlationpotential,\ngeneralized gradient approximation(GGA) was employed as\nparametrized by Perdew, Burke and Ernzerhof (PBE).14En-\nergy cutoffs of 16 and 256 Ry were used for wave func-\ntion expansions and potential representations. Charge den -\nsities and potential inside muffin-tin (MT) spheres were ex-\npanded with lattice harmonics ℓ≤8 with MT radius of 2.4\na.u. To obtain reliable values of MCA energy ( EMCA), calcu-\nlations with high precision is indispensable. A 40 ×40×40\nmesh in the irreducible Brillouin zone wedge is used for\nkpoint summation. A self-consistent criteria of 1.0 ×10−5\ne/(a.u.)3was imposed for calculations, where convergence\nwith respect to the numbers of basis functions and kpoints\nwas also seriously checked.15,16For the calculation of EMCA,\ntorque method7,17was employed to reduce computational\ncosts, whose validity and accuracy have been proved in con-\nventionalFMmaterials.18–23\nEquilibrium lattice constants of hexagonal-closed-packe d\n(hcp)-, face-center-cubic ( fcc)-, andbcc-Ru are summarized\ninTableI,whichareingoodagreementwithexperiments,11,24\nand previous work.12Thehcpstructure is the most stable\nphase, as Ru crystallizes in hcp. However, the energy differ-\nencebetween hcpandfcc,0.07eV/atom,isverysmall,which\nreflects the feature of closed packed structures of the two bu t\nwith different stacking sequences. In Fig. 2(a) total energ yofnon-magnetic(NM) bct–Ruasafunctionoftetragonaldis-\ntortion(c/a)isplottedforthefixedvolumeoftheequilibrium\nbcc-structure. OurresultreproducesthatbyWatanabe etal.12:\nThereisaglobalminimumat c/a=1.41correspondingtothe\nfccstructure. There are two other extrema, a local maximum\nand minimum at c/a=1 andc/a=0.84, respectively. In\nparticular,thelocalminimumat c/a=0.84suggeststheexis-\nFIG. 2: (color online) (a) Total energy with respect to fccstructure\n(c/a=1.41) of non-magnetic bct–Ru upon the tetragonal distortion\n(c/a) in fixed volume of the bccstructure. The equilibrium c/afor\nbct,bcc, andfccare denoted. (b) N(EF)of non-spin-polarized cal-\nculations (red squares), and magnetic moment of the bcc–Ru as a\nfunction of the uniform lattice constant a(black circles). The arrow\ndenotestheequilibriumlatticeconstantof bcc-Ru. (c)Energydiffer-\nenceΔE=ENM−EFM(red dotted line), magnetic moments (black\nsolid line) as function of c/a. The tetragonal distortion is classified\ninto two regions, AandB, byc/a<1 or>1. (d)N(EF)of NM\nbct–Ru as function of c/a. TotalN(EF), those from dz2,dx2−y2, and\ntheabsolute value ofthedifference ofthetwo egorbitals,denoted as\nΔeg, are shown in black solid circles, black dotted line, red das hed\nline,and blue solidline, respectively.3\nTABLE I: Calculated equilibrium lattice parameters, aandc/a(in˚A), and total energy difference ΔE(in eV/atom) of hcp-,fcc,-bcc-, and\nbct-Ruwithrespect tothe totalenergy of hcpstructure. Experimental andprevious theoretical results are alsogiven for comparison.\nhcp fcc bcc bct\nPresent ExperimentaPresent PreviousbPresent Previous Present PreviousbExperimenta\na 2.70 2.70 3.84 3.84 3.07 3.06 3.25 3.25 3.24\nc/a1.58 1.58 1.09 1.00 1.00 1.00 0.84 0.83 0.83\nΔE 0.0 0.07 0.13 0.56 0.65 0.48 0.55 -\naShiikietal.11\nbWatanabe et al.12\ntence of metastable phase as discussed in Ref.12. Further cal-\nculationsoftotalenergyofthe bctstructureasfunctionofboth\naandc/aconfirms that the local minimum is at a=3.25˚A\nandc/a=0.84, consistent with the fixed volume calculation\nofthebccstructure.\nIn Fig. 2(b), N(EF)of non-spin-polarized and magnetic\nmoment of spin-polarized calculation are plotted as functi on\nof lattice constant. The onset of magnetism in the bccphase\noccurs at a=3.10˚A, which corresponds to 1.1% expansion\nof lattice constant, or3.3% expansion of volume, as consis-\ntentwithRef.8. Inorderforthemagneticinstabilityinthe bcc\nphase to satisfy the Stoner criteria, I·N(EF)≥1, and from\nthe fact that the Stoner factor Iof a particular atom does not\ndiffersubstantially in differentcrystal structures, we e stimate\nI=0.46eVforRufrom N(EF)=2.18eV−1.\nOn the otherhand,as shownin Fig. 2(c),the energydiffer-\nencebetweenNMandFMstates( ΔE=ENM−EFM)andmag-\nnetic moment reveal almost the same trends as c/achanges.\nΔEof thebcc- andfcc-phases are negligibly small, thus both\nphases are non-magnetic. When c/a<1.1 butc/a/negationslash=1, the\nbct–Ru is magnetic ( ΔE>0), whereas c/a>1.1, it is non-\nmagnetic. In particular, c/a=0.84 givesΔE=35 meV/atom\nwith magneticmoment as high as 0.6 µB, largerthan 0.40 µB\nby Ref.12. Interestingly, the magnetic moment of the bct–Ru\nexhibits a re-entrance behavior for c/a>1, as predicted by\nSch¨ onecker et al.25. In region A(c/a<1), magnetic mo-\nment decreases as c/aincreases, whereas magnetism reap-\npears when c/ajust passes unity, which eventually vanishes\nforc/a>1.1.\nTotal DOS and those from egorbitals at EFas function of\nc/aare plotted in Fig. 2(d) for the NM bct–Ru. Most con-\ntributions come from the Jahn-Teller split egorbitals, whose\ndifferencein DOS is also plotted: It resembles magnetic mo-\nment shown in Fig. 2(c). Moreover, among the Jahn-Teller\nsplitegorbitals,dx2−y2(dz2)dominatestheotherfor c/a<1\n(c/a>1).\nPartial DOS (PDOS) of dorbitals are shown in Fig. 3 for\nthe spin-polarizedcases, where the trivial c/a=1 is omitted.\nProminent peaks at c/a=0.84 are mainly from dx2−y2states\nwithoccupied(unoccupied)peaksinmajority(minority)sp in\nbands,whilepeaksin dz2statesevolveas c/aincreases. Con-\ntributionsfrom t2gstatesareratherfeatureless.\nFor simplicity, we assign the energy difference of peaks\ninegstates,dx2−y2forc/a<1 anddz2forc/a>1, re-\nspectively, as the exchange-splitting. Then, as c/aincreases,\nthe exchange-splittings are 1.02, 1.05, 0.80, and 0.66 eV fo r\nFIG.3: (coloronline)Orbital-decomposedDOSof d-orbitalforspin-\npolarized calculations of bct-Ru atc/a= (a) 0.84, (b) 0.90, (c) 0.96,\nand (d) 1.06, respectively. The dorbital states are shown in differ-\nent colors: red ( dz2), black (dx2−y2), blue (dxy), and green ( dxz,yz),\nrespectively.\nc/a=0.84, 0.90, 0.96, and 1.06, respectively, which qual-\nitatively reflects magnetism of the bct–Ru. From this, the\nexchange-splitting is mainly determined by one of the Jahn-\nTellersplit egorbitals.\nIn addition to magnetism, the bct–Ru exhibits large MCA.\nThe angle-dependent total energy in a tetragonal symmetry\nis expressed in the most general form, Etot(θ,ϕ) =E0+\nk1sin2θ+k2sin4θ+k3sin4θcos4ϕ,whereθandϕare po-\nlar and azimuthal angles, respectively, and k1=100,k2=\n−1, andk3≪1µeV. The small value of k3indicates neg-\nligibleϕdependence. EMCA=Etot(θ=90◦)−Etot(θ=\n0◦)as function of the tetragonal distortion c/ais shown in\nFig.4(a). EMCA=150µeV/atomat c/a=0.80,andEMCA=100\nµeV/atomforthelocalminimum( c/a=0.84),whicharetwo\norders of magnitude greater than conventional 3 dmagnetic\nmetals. As the strength of the tetragonal distortion change s,\nEMCAchanges not only in magnitude but also in sign. In re-\ngionA,EMCAbecomes negative near c/a≈0.9 and reaches\n−100µeV/atom around c/a=0.96, whereas in region B,\nEMCA>0 : PMCA is restored. Hence, the strength of the\ntetragonal distortion, c/a, influences magnetic moments as\nwellasEMCA.4\nFIG. 4: (color online) (a) MCA energy dependence on c/aforbct–\nRu, where AB are defined as in Fig. 2(b). (b) Spin-channel deco m-\nposedandtotal EMCAofbct-Ruforvarious c/a. Blackcirclesdenote\ntotal MCA. Upper (lower) triangles denote ↑↑(↓↓)-channel, squares\ndenote↑↓-channel.\nEMCAis decomposed into different spin-channels follow-\ning Eq. (1), as shown in Fig. 4(b) for the bct–Ru with c/a=\n0.84, 0.90, 0.96 and 1.06, respectively. For σσ′=↑↑or↓↓,\npositive (negative)contributionto EMCAis determined by the\nSOCinteractionbetweenoccupiedandunoccupiedstateswit h\nthe same (different by one) magnetic quantum number ( m)\nthroughthe ℓZ(ℓX) operator. For σσ′=↑↓,Eq.(1) has oppo-\nsitesign,sopositive(negative)contributioncomefromth eℓX\n(ℓZ)coupling.\nFrom the spin-channel decomposition of EMCA, one notes\nthat there is no dominant spin-channel. This feature differ s\nfrom the 3 dtransition metals, where particular spin-channel,\ni.e. the↓↓-channel, dominantly contributes to positive value\nthrough the SOC matrix /angbracketleftx2−y2|ℓZ|xy/angbracketrightwith negligible ones\nfromℓXmatrices.7,26Whenc/a=0.84,the↓↓-channelgives\nthe largest contribution, while those from other channels a re\nsmaller than half of the ↓↓-channel with opposite signs. As\nc/aincreases, the ↓↓-channel contribution is reduced, which\nturns negative for c/a>1. MCA almost vanishes for c/a=\n0.90andbecomesnegativefor c/a=0.96. Ontheotherhand,\nforc/a=1.06,the↑↓-and↓↓-channelscontributealmostthe\nsame magnitudes with opposite signs, so just the ↑↑-channel\ncontributionremains.\nTo obtain more insights, band structure is plotted in Fig. 5\nwithdorbital projection, where size of symbols is propor-\ntional to their weights. All bands along the Γ-Z-Xare highly\ndispersive, whereas those along the X-P-N-Γ-Xare less dis-\npersive with rather flat feature from dx2−y2anddz2states.\nLevel reversals between egstates,dx2−y2anddz2, are well\nmanifested, while t2gstates are relatively rigid with respect\nto tetragonal distortion. It is a formidable task to identif y the\nrole of each individual SOC matrix for each c/a. However,\nfrom the spin-channel decomposed MCA [Fig. 4(b)], each\nspin-channelchangesits sign when c/abecomesgreaterthan\nunity,wherethe levelreversaloccursbetween dx2−y2anddz2.\nForasimple analysis,weexpressthe ↓↓-channelas\nE(↓↓)=|/angbracketleftx2−y2|ℓZ|xy/angbracketright|2\nεx2−y2−εxy−|/angbracketleftx2−y2|ℓX|xz/angbracketright|2\nεx2−y2−εxz−|/angbracketleftz2|ℓX|xz/angbracketright|2\nεz2−εxz.\n(2)\nWe focus along the X-P-N-Γ-X, whereegare unoccupied.\nFIG.5: (color online) Bandstructures of bct–Ruforc/a=0.84, 0.90,\n0.96, and 1.06 for majority and minority spin states. dorbital states\nare shown in different colors: red ( dz2), black ( dx2−y2), blue (dxy),\norange (dxz),and green( dyz),respectively.\nThe/angbracketleftyz|ℓZ|xz/angbracketrightcontributions are neglected due to the rigidity\noft2gstatesaswell astheirsmall contributionto EMCAowing\nto large energy denominator. From the fact that E(↓↓)>0\nwhenc/a=0.84 with the largest value, we can infer that the\nfirstterminEq.(2)shouldbelargerthantheothertwo,where\nthe largest occur along the P-N. [See Supplementary Infor-\nmation(SI)forthe k-resolvedMCA analysis.]\nAsc/aincreases but c/a<1, the empty dz2band moves\ndownwardwhile the empty dx2−y2bandgoesupwardwith re-\nspect toEF. As a result, the third term is enhanced due to\nsmaller energy denominator. Hence, E(↓↓)decreases but re-\nmains positive. When c/a>1, however, the level reversal\nbetweenegstates pushes dz2aboveEFanddx2−y2belowEF\nalong the N-Γ-X. The former provides additional negative\ncontributionwhilethe latterreducespositivecontributi on. As\na consequence, E(↓↓)<0 forc/a>1. The sign behavior of\nthe↑↓-component, E(↑↓),iscompletelyoppositeto E(↓↓),as\nalltermsinEq.(2)takeoppositesigns.7Forthe↑↑-component\nwhenc/a<1, we focus near the P-N. The largest positive\ncontribution in the ↓↓-component is significantly reduced in\nthe↑↑-channelbecausetheempty dx2−y2bandintheminority5\nspin is occupiedin the majority spin, and the empty dz2band\ncontributes negatively. Thus, E(↑↑)<0. When c/a>1, the\noccupancy of egstates are reversed again due to the level re-\nversal,therefore E(↑↑)>0. Wewanttopointoutthatthelevel\nreversal betweenthe dx2−y2and thedz2states not only affects\nthe sign behavior of MCA but also the exchange-splitting in\nDOS. Above argument of the sign behavior is more clearly\nsupported by the k-resolved MCA analysis. [See SI Fig. S1-\nS5.]\nInsummary,anewmetastablephaseofthe bct–Ruhasbeen\nidentified to exhibit a large PMCA, two orders of magnitude\ngreater than conventional magnetic metals. In the context o f\nspintronics application, this large anisotropy along with low\nmagnetization and small volume would be key factors to re-\nalize materials with low switching current and high thermal\nstability. Magnetismof the bct–Ruis mainlygovernedbythe\nJahn-Teller split egstates. As the strength of the tetragonal\ndistortion changes, magnetism of the bct–Ru shows an inter-esting reentrance behavior for 1 <c/a<1.1, The tetragonal\ndistortionaccompaniesMCAchangesinbothmagnitudesand\nsigns,asa resultofthelevelreversalbetween dx2−y2anddz2.\nAcknowledgments\nDO, SHR, and SCH are supported by Basic Research Pro-\ngram (20100008842)and Priority Research Centers Program\n(20090093818) through the National Research Foundation\n(NRF) funded by the Korean Ministry of Education, Sci-\nence and Technology. NP was supported by Basic Science\nResearch Program through the NRF funded by the Korean\nMinistry of Education (2013R1A1A2007910). KN acknowl-\nedges support from a Grant-in-Aid for Scientific Research\n(No. 24540344) from the Japan Society for the Promotion\nofScience. DO andSHRequallycontributedauthors.\n∗Email address: sonny@ulsan.ac.kr\n†Email address: schong@ulsan.ac.kr\n1M. 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We discuss a strategy to enha nce the finite-temperature anisotropy\nfield by hole doping, paving the way for an improvement of the c oercivity near room temperature\nor higher.\nPACS numbers: 75.30.Gw, 75.50.Ww, 75.10.Lp, 71.15.Rf\nI. INTRODUCTION\nFerromagnetism in permanent magnets has been a\nlong-standing problem in solid state physics and mate-\nrials science. The mechanism of coercivity, resistance\nof the spontaneous magnetization against the externally\napplied magnetic field in the reverse-direction, is not yet\nentirely known after almost 70 years since Brown’s para-\ndox1. Development of modern rare-earth magnets like\nNd2Fe14B2and SmCo 53,4triggered material-specific in-\nvestigations of the coercivity mechanism5. They culmi-\nnate in today’s state-of-the-art technological optimiza-\ntion of the microstructure of ferromagnetic materials for\nthe enhanced coercivity6on the phenomenological and\nempirical basis. More fundamental understanding and\ncontrolling on the basis of solid-state physics and statis-\ntical physics is desired to better use these technological\nadvances. Especially the temperature decay of coerciv-\nity is a problem in industrial applications and thus more\nand more demands for the control of finite-temperature\nproperties of permanent magnets are rising.\nThe high-temperature tail of the magnetism is mostly\ncarried by 3 d-electrons7in the intermetallic system be-\ntween transition-metal (TM) and rare-earth (RE) ele-\nments while the low-temperature magnetism is domi-\nnated by 4 f-electrons. Thus developing an understand-\ning and the theoretical control over the temperature de-\ncay of 3d-electron magnetism has the universal utility\nfor the optimization of high-temperature magnetism of\nRE magnets. Thus we focus on YCo 5as our target sys-\ntem, extracting the 3 d-electron part out of the SmCo 5\nfamily. The so-called 1-5 magnets represent the RE\nmagnets with one of the simplest crystal structures and\nSmCo5-based magnets are 2ndin magnetic performance\nonlytothechampion-magnetfamilyincludingNd 2Fe14B;\nSmCo5currently serves for some special applications un-\nder extreme conditions8with its high Curie temperature\nand stability against corrosion. Also the large magneto-\ncrystalline anisotropy energy (MAE) of 5.5 [MJ/m3] at\nroom temperature T= 295 [K] in YCo 59can let itself\nqualify for permanent magnet applications8.Describing the magnetic properties of such a material\npresents significant challenges to ab initio theory. Itin-\nerant electron magnetism in realistic magnetic materials\nhas often been addressedin the groundstate with the ap-\nplication of the density-functional theory (DFT). Incor-\nporation of finite temperature physics and description of\ncorrelated electron behavior can be issues for DFT which\ndoes not always include localized electron effects ade-\nquately into its scope. Both issues require a combination\nof DFT and statistical physics in some way. Here we fol-\nlow the idea of disordered local moments10–13to incorpo-\nrate the physics of thermal fluctuations of well-developed\nmagnetic moments into the spin-polarized density func-\ntional theory (DFT) for ferromagnetic materials14–17. A\nrelativisticversionofthedisorderedlocalmoment(DLM)\napproach has recently been developed18,19and success-\nfully applied to finite-temperature magnetic anisotropy\nin L10-FePt alloys18,19and Co films20where magnetism\nis all carried by d-electrons. Thus the d-electron part\nin the magnetism of rare-earth magnets can be a good\ntarget for DLM.\nIn this work we present ab initio calculations based\non DLM for YCo 5at finite temperatures. From calcu-\nlated temperature dependence of the magnetization M\nand the MAE, which we denote by Ku1as it mostly\ncomes from the uni-axial magneto-crystalline anisotropy\n(MCA) for YCo 5, we address the thermal decay of the\nanisotropy field estimated as HA= 2Ku1/Mwhich gen-\nerallyshowsthe approximateproportionalityto the coer-\ncivity field Hc21. The relation is practically Hc∼0.2HA\nwith the practical hard upper limit being Hc<∼0.5HA\nthat is reached only in some exceptional samples22. As-\nsuming such rough proportionality, we propose a strat-\negy to improve HA(T) at high temperatures in the range\nT∼400−500 [K]. Our main message is encapsulated\nin Fig. 1 which shows that hole-doped YCo 5with the\nnumber of doped holes being 0.24 and 0.35 per unit cell\nhas the stronger anisotropy field than pure YCo 5in the\ntemperature range T>∼450 [K] and T>∼550 [K], respec-\ntively.\nExperimentally, doping-enhanced coercivity has been2\n 3 4 5 6 7 8 9\n 200 250 300 350 400 450 500 550 600HA [T]\nT [K]Pure: Nh=0\nHole-doped: Nh=0.24\n0.36\nFIG. 1. (Color online) Calculated temperature dependence\nof the anisotropy field HAfor pure and hole-doped YCo 5,\nwhere doping-induced enhancement of HAis observed in the\ntemperature range T>∼450 [K] and 550 [K] with the number\nof doped holes being Nh= 0.24 and 0 .36, respectively.\n-2-1 0 1 2 3 4\n 50  52  54  56  58  60  62-4-2 0 2 4 6Ku1 [meV/formula unit]\nKu1 [MJ/m3]\nNeOur calculationExp.a\nExp.b\nExp.c\nExp.d\nPrevious\ncalculation\nat T=0\naKu1= 7.38 [MJ/m3] atT= 4.2 [K] from Ref. 23.\nbKu1= 6.3 [MJ/m3] atT= 77 [K] from Ref. 24.\ncK= 6.03 [MJ/m3] from Ref. 25 at T= 293 [K]. We\nassumeKu1dominates in Kas was shown in Ref. 23.\ndKu1= 5.5 [MJ/m3] at room temperature from Ref. 8.\nFIG. 2. (Color online) Calculated magnetic anisotropy ener gy\nplotted as the function of the valence electron number for\nYCo5near the ground state T= 0 [K]. The numbers on the\nleft-hand side of the vertical axis indicates the results pe r the\nformula unit of YCo 5and those on the right-hand side of\nthe vertical axis gives the same quantities converted to per\nthe unit volume using the experimental lattice constants in\nRef. 26. The previous theoretical result for MAE at T= 0\nis taken from Ref. 27. The number of valence electrons per\nformula unit of YCo 5is 54.\nknown for YCo 5−xNixand other RECo 5−xCuxmagnets\n(RE=Ce, Sm)28. The coercivity is indeed significantly\naffected by doping, which is seen in our calculated re-\nsults on the filling dependence of MAE near the ground\nstate as shown in Fig. 2. On top of such a filling depen-\ndence of coercivity determined by the electronic struc-ture, our idea is to let the electronic states below the\npeak of MAE in Fig. 2 be thermally populated so that\nthe high-temperature MAE is enhanced. Experimentally\nobserved doping-enhanced coercivity had been discussed\nin conjunction with intrinsic pinning28. Our results on\nthe finite-temperature physics together with the electron\nband filling-sensitive nature of MAE point to a new sce-\nnario for doping-enhanced coercivity at high tempera-\ntures.\nThe rest of paper is organized as follows. Our pro-\nposal on the strategy to enhance the high-temperature\ncoercivity is found in Sec. IV. This follows an outline of\nthe DLM methodology underpinning the calculations in\nSec. II and Sec. III. Conclusions are given in Sec. V.\nII. METHODS\nOur computational method, the DLM approach,\nis based on the Korringa-Kohn-Rostoker (KKR)\nmethod29,30and coherent potential approximation\n(CPA)31forab initio electronic structure calculations.\nEspecially to address the temperature dependence of\nMAE which is brought about by spin-orbit interaction\n(SOI), we follow the relativistic formulation18,19. These\naspects in Sec. IIA and Sec. IIB, describe the general\nframework and how to extract MAE, respectively at the\nlevel of sketching the basic ideas. The MAE as our key\nobservable is focused on and for the rest of the observ-\nables that we present in Sec. III, we refer to the original\nworks15,19and a recent review article32for their explicit\nexpressions in terms of KKR-CPA. In Sec IIC we give\nthe details of the application to the case of YCo 5.\nA. Framework\n1. Separation of fast and slow dynamics\nUnder the situation that spin-fluctuation dynamics are\nmuch slower than the characteristic time scales of elec-\ntronic motions15, well developed classical local moments\nˆeare assumed to exist on each site i= 1...Nin the unit\ncell, with Nbeing the number of magnetic atoms in the\nunit cell. The thermal average /an}bracketle{t/an}bracketri}htof the local moment\n/an}bracketle{tˆei/an}bracketri}ht=/integraldisplay\n···/integraldisplay\nˆeiP(ˆn)({ˆe})dˆe1... dˆeN(1)\nisassumedtobealignedwiththemagnetizationdirection\nˆnfor a ferromagnet. Here ˆnis fixed in the calculation\nby hand. We have specified the configuration of the local\nmoments by {ˆe}and statisticalweight P(ˆn)({ˆe}) givenin\nterms of the Boltzmann weights at finite temperatures.\n/an}bracketle{tˆei/an}bracketri}ht=miis a magnetic order parameter. For a ferro-\nmagnet, magnetized along a direction ˆn,mi=miˆn.3\n2. DLM as a realistic mean field theory\nDLM is implemented as a realistic mean-field theory\nfor the local moments embedded in a sea of electrons\ngiven by spin-polarized DFT for the target material. For\na given configuration of the local moments\n{ˆe} ≡ {ˆe1,ˆe2,...,ˆeN},\nthe Boltzmann weight is defined as\nexp/bracketleftBig\n−βΩ(ˆn)({ˆe})/bracketrightBig\n,\nreferring to the grand potential Ω(ˆn)({ˆe}) in the spin-\nresolved density functional theory for a given target fer-\nromagnet ( β= 1/kBT). The partition function is ac-\ncordingly written as\nZ(ˆn)=/integraldisplay\n···/integraldisplay\nexp/bracketleftBig\n−βΩ(ˆn)({ˆe})/bracketrightBig\ndˆe1... dˆeN\nand the probability distribution function (PDF) in\nEq. (1) of the local-moment configuration {ˆe}is\nP(ˆn)=exp/bracketleftbig\n−βΩ(ˆn)({ˆe})/bracketrightbig\nZ(ˆn)\nHere we use the mean-field approximation (MFA)\nΩ(ˆn)\n0({ˆe})MFA=N/summationdisplay\ni=1h(ˆn)\ni·ˆei (2)\nwherethe Weiss field h(ˆn)\ni=h(ˆn)\niˆnis written asfollows15.\nh(ˆn)\ni=/integraldisplay3\n4π(ˆei·ˆn)/angbracketleftBig\nΩ(ˆn)/angbracketrightBig\nˆeidˆei (3)\nfori= 1...Nwhereiruns overthe local moments in the\nunit cell and the thermal averagedenoted by /an}bracketle{t·/an}bracketri}htˆeiimplies\nthe restrictionin the averagingprocesswith the direction\nof the local moment on site ifixed to that specified by\nˆei. (Further terms proportional to, say, ( ˆei·ˆn)2, can be\nadded to Eq. 2 to improve the mean field description41\nbut have a rather small effect.) With the MFA in Eq. (2)\ntheoverallpartitionfunctionfactorizesintocontributions\nfrom each of local moments\nZ(ˆn)=N/productdisplay\ni=1Z(ˆn)\ni\nandthepartitionfunction iswrittenintermsoftheWeiss\nfield\nZ(ˆn)\ni=/integraldisplay\nexp/parenleftBig\n−βh(ˆn)\ni·ˆei/parenrightBig\ndˆei\n=4π\nβh(ˆn)\nisinhβh(ˆn)\ni (4)and the PDF is written as follows:\nP(ˆn)\ni(ˆei) = exp/parenleftBig\n−βh(ˆn)\ni·ˆei/parenrightBig\n/Z(ˆn)\ni\n=βh(ˆn)\ni\n4πsinhβh(ˆn)\niexp/parenleftBig\n−βh(ˆn)\ni·ˆei/parenrightBig\n(5)\nWith this expression for the PDF, the free energy is writ-\nten as follows:\nF(ˆn)=/angbracketleftBig\nΩ(ˆn)/angbracketrightBig\n+1\nβN/summationdisplay\ni=1/integraldisplay\nP(ˆn)\ni(ˆei)logP(ˆn)\ni(ˆei)dˆei(6)\nwhere the first term describesthe internal energyand the\nsecond describes −TSwith the magnetic entropy\nS=−N/summationdisplay\ni=1/integraldisplay\nP(ˆn)\ni(ˆei)logP(ˆn)\ni(ˆei)dˆei.\nThe magnetization is given using the PDF in Eq. (5)\nas follows.\nm(ˆn)\ni=/an}bracketle{tˆei/an}bracketri}ht ≡m(ˆn)\nih(ˆn)\ni (7)\nm(ˆn)\ni=−1\nβh(ˆn)\ni+cothβh(ˆn)\ni=L(βh(ˆn)\ni) (8)\nHereL(x) is the Langevin function.\n3. CPA to embed the thermal disorder into DFT\nCoherent Potential Approximation (CPA)31is an ap-\nproach to deal with the disorder physics on a mean-field\nlevel by averaging over the random potentials to intro-\nduce an effective uniformly distributed potential. Fol-\nlowing the original idea by Hasegawa11and Hubbard12,\nthermal disorder in local moments is embedded into the\ndescription of the electronic states15on the level of CPA.\nThe way that the CPA determines the effective\nmedium is by letting the motion of an electron simulate\nthe motion of an electron on the average. The scatter-\ning problem at the heart of KKR is solved to get the\nsingle-site t-matrixtifor each local moment iwhere the\nunderlined t-matrix is that in orbital and spin angular\nmomentum space. Some more elaboration on that solu-\ntion is given below with Eq. (12). For the system mag-\nnetized along the direction ˆn, the medium is specified by\na set of CPA-imposed t-matrices33\nt(ˆn)\nc≡/parenleftbig\nt1,c,t2,c,...tN,c/parenrightbig\nwhich is determined (for details we refer to Ref. 19) for a\ngivensetofWeissfields h(ˆn)\niasinEq.(3)andcorrespond-\ningP(ˆn)\ni(ˆei) as in Eq. (5), where the finite-temperature\nphysics is encapsulated.\nThet-matrices t(ˆn)\ncthat specify the effective medium\ncomes into the formulation of DFT as follows. By mag-\nnetic force theorem we consider only the single-particle4\nenergy part of the spin-resolved DFT grand potential as\nour effective local-moment Hamiltonian\nΩ(ˆn)({ˆe})≃ −/integraldisplay\ndEfFD(E;µ(ˆn))N(ˆn)(E;{ˆe}),(9)\nwherefFD(E;µ(ˆn)) is the Fermi-Dirac distribution\nfunction with the chemical potential being µ(ˆn)\nandN(ˆn)(E;{ˆe}) is the integrated density of states\n(DOS)19,34with the given direction of magnetization ˆn\nand the given configuration of local moments {ˆe}. Using\nthe Lloyd formula35forN(ˆn)(E;{ˆe}) on the right-hand\nside of Eq. (9), the grand potential is expressed in terms\nof thet-matrices19.\nB. Calculation of magnetic anisotropy energy\n1. Defining the MAE for the uni-axial MCA\nFor uniaxial magnets, the free energy takes the form\nF(ˆn)=Fiso+Ku1sin2θ\nwhereFisois the isotropic part, Ku1>0 is the MAE for\nuni-axial MCA and θis the angle between the direction\nof magnetization and the easy axis. Then\nTθ=−∂F\n∂θ(10)\n=−2Ku1sinθcosθ\nThus in order to extract MAE for the uni-axial MCA we\nfix the direction of magnetization to be ˆn= (1,0,1)/√\n2\nthat is, ˆn= (sinθcosφ,sinθsinφ,cosθ) withθ=π/4\nandφ= 0 and calculate the magnetic torque Tθ=π/4.\nOur target MAE is obtained as\nKu1=−Tθ=π/4. (11)\n2. Spin-Orbit Interaction (SOI) in relativistic KKR\nFor a given set of self-consistent potentials, electronic\ncharge, and local moment magnitudes, the directions of\nlocal moments are described by the unitary transforma-\ntion of the single-site t-matrix for local moment i\nti(E;ˆei) =R(ˆei)ti(E;ˆz)R(ˆei)†(12)\nwheret(E;ˆz)isthet-matrixwiththeeffectivefield point-\ning along the local z-axis for given energy E, andR(ˆei)\nis for the O(3) unitary transformation that rotates the\nz-axis along the direction of ˆei. The scattering problem\nat the heart of KKR is solved incorporating SOI with a\nmagnetic field pointing along the z-axis36and the single-\nsitet-matrixti(E;ˆz) is obtained.\nIn the absence of SOI, an element of the single site\nscattering t-matrix\nti;l,m,σ;l′,m′,σ′=ti;l,σδl,l′δm,m′ (13)wherelis the angular momentum quantum number and\nmthe azimuthal component and\nti(E;ˆei) =t+\ni1+t−\niσ·ei (14)\nwhere an element t+(−)\nl,m;l′,m′=1\n2(ti;l,1\n2+(−)ti;l,−1\n2). When\nSOI is included, the t-matrix is no longer independent of\nmwhich leads to the generation of magnetic anisotropic\neffects.\n3. Torque-based formula in KKR-CPA\nThe uni-axial MAE is obtained as the derivative of the\nfree energy following Eq. (10). Inserting Eq. (6) for the\nfree energy, we get the following19.\nT(ˆn)\nθ=−∂\n∂θ/summationdisplay\ni/integraldisplay\nP(ˆn)\ni(ˆei)/angbracketleftBig\nΩ(ˆn)/angbracketrightBig\nˆeidˆei(15)\nThis is our working formula to produce the results for\nthe finite-temperature MAE of YCo 5which we now go\non to describe.\nC. Specifics with the case of YCo 5\nThe past decade has seen the successful application of\nDLMtheorytoL1 0-FePtalloy18,19whichisauniaxialfer-\nromagnet and the magnetism is carried largely by 3 dand\n5delectrons. Since YCo 5involvesmostly 3 d-electrons for\nits magnetism, success of DLM description on the same\nlevel as was achieved for L1 0-FePt is expected provided\nthatthe assumptionoflocalizedmomentonthe magnetic\natoms works well. Some care must be taken in the ap-\nplication of DLM for Co-based magnets since the trend\namong Fe, Co, and Ni13shows that the assumption of\nlocalized moment works well for Fe, faces challenge for\nNi, and Co sits somewhere in-between. Stability of lo-\ncal moments on the Co sublattices in the present case is\ndiscussed in Sec. IIC2. Indeed we will see below some\ninherent underestimate near the Curie point in our cal-\nculations which might indicate some fragile nature of the\nlocal moments in YCo 5. This problem can be cured by\nextending the calculation to that based on the non-local\nCPA37,38. Fornowwearemostlyconcernedwiththelow-\nest temperature properties and the middle-temperature\nrange of T<∼600 [K] which is well below the Curie tem-\nperature at TCurie= 920 [K]26. So we proceed with the\nsingle-site theory within the scope of the present project.\n1. Crystal structure\nWe take the experimental crystal structure of YCo 5\nwith the space group No. 191 a= 4.948 [˚A] and\nc= 3.975 [˚A]26. The unit cell consists of one formula\nunit: Y, Co(2c), and Co(3g) sublattices. The atomic5\n(a)\n 0 0\n-0.5 0 0.5c-axisCo(2c) atoms\nCo(3g) atoms\nY atom\nCo(2c)-Co(2c) network\nCo(3g)-Co(3g) network\nCo(2c)-Co(3g) network\n(b)\n-1 -0.5  0  0.5  1-1-0.5 0 0.5 1\nt1t2\nCo(2)Co(1)\nCo(3)Co(4) Co(5)\nY\nFIG. 3. (Color online) Crystal structure of YCo 5: (a) bird’s\neye view and (b) top view seen along the c-axis. Nearest-\nneighborpairsintheCo(2c)-Co(3g) networkareonlypartia lly\ndrawn for the illustration. Labels for atoms in (b) follow\nTable I.\nconfiguration in the unit cell is shown in Fig. 3 (a) as a\nbird’s eyeview. The top view alongthe c-axisis shownin\nFig.3(b)toillustratetherelativepositionsoftheCo(2c)-\natoms and the Co(3g)-atoms. Within the layer along the\nab-plane, it is seen that nearest-neighbor Co(3g)-atoms\nform a kagom´ e lattice. Between those kagom´ e layers,\nCo(2c)-atoms sit right on the center of Co(3g)-triangles\nto form a hexagonal lattice on the same ab-plane layer as\nY-atoms. The ratio c/a≃0.80 means that the tetrahe-\ndron formed by the Co(3g)-triangle and the Co(2c)-atomsublattice atom atomic coordinates\nY (0 ,0,0)\nCo(2c) Co(1) (1 /(2√\n3),1/2,0)\nCo(2) (1 /√\n3,0,0)\nCo(3g) Co(3) (√\n3/4,−1/4,1/2)\nCo(4) (√\n3/4,1/4,1/2)\nCo(5) (0 ,1/2,1/2)\nprimitive translation vectors\nt1(√\n3/2,−1/2,0)\nt2 (0,1,0)\nt3 (0,0,1)\nTABLE I. Set-up of the unit cell of YCo 5in the present cal-\nculations.\n 1.4 1.45 1.5 1.55 1.6 1.65 1.7 1.75 1.8 1.85 1.9\n 0  200  400  600  800  1000Total (spin + orbital) moment [ µB]\nT [K]Co(2c): Co(1)\nCo(2)\nCo(3g): Co(3)\nCo(4)\nCo(5)\nFIG. 4. (Color online) Calculated temperature dependence o f\nthe total (including both spin and orbital) local moment for\neach magnetic sublattice in YCo 5.\nis close to the regular one. So the Co(2c)-Co(3g) atomic\ndistance is almost equal to Co(3g)-Co(3g)distance which\nis0.5a. Ontheotherhand, Co(2c)-Co(2c)distancewhich\nisa/√\n3≃0.58aspans the longest distance among the\nnearest-neighbor Co atom pairs. We set the position\nof each atom and the direction of the translation vec-\ntors as shown in Table I in our calculation. The labeling\nscheme for the atoms and the directions is illustrated in\nFig. 3 (b).\n2. Robustness of local moments in YCo 5\nWe inspect the validity ofDLM approachatfinite tem-\nperatures in the present calculation. Calculated temper-\nature dependence of each of local moments in YCo 5are\nshown in Fig. 4. The DLM picture of magnetism in met-\nals at finite temperature is based on the assumption that\nthere are some aspects of the interacting electrons of a\nmaterial that vary slowly in comparison with faster de-\ngrees of freedom. These are captured in terms of the\norientational unit vectors, {ˆe}, the transverse part of6\nthe magnetic fluctuations. The electronic grand poten-\ntial Ω(ˆn)({ˆe}) is in principle available from spin-resolved\nDFT (SDFT) generalised for the non-collinear magnetic\nprofiles which are labelled by {ˆe}15. Within the tenets\nof SDFT the charge and magnetisation densities depend\non these configurations, i.e. ρ(r,{ˆe}),/vectorM(r,{ˆe}). In the\nspace around an atom at a site ithe magnetisation is\nconstrainedtofollowthe orientation ˆeitherebut itsmag-\nnitudeµ(r) can depend on the surrounding orientational\nenvironment, µ(r) =µ(r,{ˆe}). In many of the materials\nwhere the DLM theory has been successfully applied, the\nmagnitudes are found to be rather insensitive to the en-\nvironment and can be modelled accurately as rigid local\nmoments. In other materials, e.g. Ni,39, the magnetism\nis driven by the coupling of the magnitudes, (longitudi-\nnal magnetic fluctuations) with these transverse modes.\nWhen there is short-range aligning of the orientations or\nlong range magnetic order, a finite magnetisation mag-\nnitude can establish; whereas in environments where the\norientations differ significantly over short distances, the\nmagnitudes shrink.\nFigure 4 indicates the extent of this effect in YCo 5.\nWe have started with a SDFT calculation of the charge\nand magnetisation density ρ(r) and/vectorM(r) for the T= 0K\nferromagnetic state, {ˆe=ˆn}and find the magnetisation\nmagnitude per site is 1.9 and 1.6 µBfor the Co(2c) and\nCo(3g) sites respectively. When we use the scf poten-\ntials that these produce in a frozen potential approxima-\ntioninourDLMtheoryforincreasingtemperatureswhen\nthe moment orientations become disordered, the average\nmagnetisation magnitude per site output from this cal-\nculation diminishes. For temperatures up to ∼600 [K],\nthis decrease is slight, thereafter more significant as seen\nin the figures. Up to 600 [K] therefore the DLM theory\nas applied is adequate and we can examine the effects\nof transverse fluctuations on the MAE that it describes.\nFor higher temperatures however, the effects of short-\nranged correlations among the ˆeorientation (via the use\nof the non-local CPA for example37,38) and longitudinal\nfluctuations40need to be addressed for a more complete\npicture. Comprehensive analyses on bcc-Fe, where the\nrobustness of the local moments are established, can be\nfound in a recent work41.\nFurther details about the calculations are described in\nAppendix A.\nIII. RESULTS\nWe start with the calculated MAE near the ground\nstate to demonstrate its sensitivity to the filling of the\nelectrons in Fig. 2. Corresponding data for the magneti-\nzation is shownin Fig. 5. We see that even the sign ofthe\nMAE sensitively changes depending on a fraction in the\nfilling of electrons, which is reasonable in the physics of\nmagneticanisotropywhereadding/removingoneelectron\nin an electronic cloud that points to a particular real-\nspace direction in a given crystal-field environment in- 0 2 4 6 8 10\n 50  52  54  56  58  60  62 0 0.2 0.4 0.6 0.8 1 1.2M [µB/(formula unit)]\nJs [T]\nNeExp.bExp.a\nOur calculation\nAdjusted in DLMCalc. MtotalCalc. MspinCalc. Morbital\naM= 8.33 [µB] at liquid-helium temperature from Ref. 23.\nbM= 7.99 [µB] at room temperature from Ref. 26.\nFIG. 5. (Color online) Calculated magnetization of YCo 5\nplotted as a function of the number of electrons near the\nground state at m= 0.967. Our calculated result 8.03 [ µB]\nat around T= 100 [K] falls onto the experimental numbers\nwithin 2 digits. For the closeness of such data to the ground\nstate, see Fig. 7.\ndeed affect which direction the magnetic moment should\nprefer under the presence of SOI. So the challenge is how\npreciselywe can pin-point the electronnumber right onto\nthe realistic one, on which details are given in Sec. A3.\nFixing the electron number by a fine-tuning of the chemi-\ncalpotentialintheDLMruns,weobtainthetemperature\ndependence of MAE and magnetization. The tempera-\nture dependence of the anisotropy field is deduced from\nthem. These data for the bulk are shown in Sec. IIIA.\nThen we inspect how our calculated MAE scales with\nrespect to the calculated magnetization, resolving into\nsublattices and temperature ranges in Sec. IIIB. Com-\nparison to the single ion anisotropy model analyses by\nCallen and Callen42and more recent ones18,43is carried\nout to help uncover the key mechanisms.\nA. Bulk observables\nCalculated temperature dependence of MAE for YCo 5\nby DLM at the fixed correct valence-electron number is\nshown in Fig. 6. We have a systematic underestimate of\nMAE for the overall temperature range as compared to\nthe experimental results found in the literature. How-\never the qualitative temperature dependence is well re-\nproduced. In the similar manner to MAE, calculated\ntemperature dependence of magnetization at the fixed\nvalence-band filling is shown in Fig. 7. Fitting the cal-\nculated temperature dependence of magnetization as it\ndecreases to zero in the temperature range T >900 [K]\nto the following relation\nM(T) =AM|T−TCurie|β,7\n 0 1 2 3 4 5\n 0 100  200  300  400  500  600 0 1 2 3 4 5 6 7 8 9Ku1 [meV/(formula unit)]\nKu1 [MJ/m3]\nT [K]Previous calculation at T=0Our calculation\nExp.a\nExp.b\nExp.c\naRef. 24 for Ku1.bRef. 23 for Ku1.\ncRef. 25 for K, withKu1dominating.\nFIG. 6. (Color online) Calculated temperature dependence o f\nmagnetic anisotropy energy for YCo 5. Previous calculation\nresult at T= 0 is taken from Ref. 27.\n 0 1 2 3 4 5 6 7 8 9\n 0  200  400  600  800  1000 0 0.2 0.4 0.6 0.8 1 1.2M [µB/(formula unit)]\nJs [T]\nT [K]Exp. TCurieExp.bExp.aCalc. MtotalCalc. MspinCalc. Morbital\naRef. 23bRef. 26\nFIG. 7. (Color online) Calculated temperature dependence o f\nmagnetization for YCo 5. Experimental Curie temperature,\n920 [K], is taken from Ref. 26.\nwe getTCurie= 965 [K], β= 0.50, andAM= 0.33. The\ncritical exponent falls onto the mean-field value β= 1/2\nwithin the numerical accuracy as it should, which gives\na check of the data. Our ab initio result for the Curie\ntemperature compares with the experimental Curie tem-\nperature TCurie,exp= 920 [K]26within the deviation of\n5%. A mean-field result on the transition temperature\ntypically gives an overestimate by O(10)% as was ob-\nserved for L1 0-FePt alloy calculated by exactly the same\nmethod18as the one we utilize here; the apparent ab-\nsence of such an overestimate might rather indicate the\nfragility of the local moments close to TCurie.\nWe obtain the ab initio result for the temperature de-\npendenceoftheanisotropyfield HAasshowninFig.8(a)\nfollowingthe simple coherent magnetizationrotation pic- 0 2 4 6 8 10 12 14\n 0  100  200  300  400  500  600HA [T]\nT [K]Exp. at 293 [K]\nOur calc.\nFIG. 8. (Color online) Temperature dependence of the\nanisotropy field for YCo 5as deduced from the numbers pre-\nsented in Figs. 6 and 7, following Eq. (16). The experimental\ndata at 293 [K] is taken from Ref. 25.\nture,\nKu1=1\n2MHA, (16)\nwithKu1andMbeing the calculated uniaxial MAE in\nFig. 6 and magnetization in Fig. 7, respectively. The\nunderestimate by 30% of Ku1on the left-hand side of\nEq. (16) leads to a similar underestimate of HA.\nB. Scaling MAE and Magnetization\nScaling relation concerning MAE with respect to mag-\nnetization\nKu1(T) =A[M(T)/M(0)]α(17)\nhas been discussed for magnetic materials of which the\nrepresentativeonewasestablishedbyCallenandCallen42\non the basis of single-ion theory. Their exponent α=\nl(l+ 1)/2 at low temperatures, where lis the order of\nthe expansion in terms of the spherical harmonics, has\nbeen challenged in some materials18,43. Callen-Callen ar-\ngument for a ferromagnet with the uni-axial anisotropy\npredicts an exponent α= 3 with l= 2. Here we inspect\nwhat sort of scaling relation holds for our target mag-\nnet YCo 5. YCo 5is an intermetallic compound and the\nsingle-iontheoryisnotdirectlyapplicabletoit. Ontopof\nthat, the particular crystal structure of YCo 5comes with\nmultiple magnetic sublattices in the unit cell. The bulk\nscaling between MAE and magnetization might not be\nexactly the same as sublattice-resolved scaling concern-\ning the fact that each sublattice is exposed to its own\ncrystal-field environment, depending on its own charac-\nteristic temperature dependence.\nWe start with the temperature dependence of local-\nmoment resolved MAE as shown in Fig. 9. Exceptional8\n-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6\n 0  100  200  300  400  500  600Ku1,i [meV]\nT [K]i=Co(2c): Co(1)\nCo(2)\nCo(3g): Co(3)\nCo(4)\nCo(5)\nFIG. 9. (Color online) Calculated temperature dependence o f\nlocal-moment-resolved MAE.\n 0.01 0.1\n 0.2  1Ku1,i(T) [meV]\nM(T)/M(0)i=Co(2c): Co(1)\nCo(2)\nCo(3g): Co(3)\nCo(4)\nCo(5)\n 0.1\n 1\nFIG. 10. (Color online) Calculated MAE plotted as a func-\ntion of the order parameter, m≡M(T)/M(0), for each\nlocal moment i. The inset is a zoomed-in plot in region\nM(T)/M(0)<∼1.\nbehavior seen for Co(5) in the Co(3g)-sublattice origi-\nnates in the effect of SOI with the magnetization di-\nrectionn= (1,0,1). Out of the data in Fig. 9, the\nsublattice-resolved MAE as a function of the order pa-\nrameterm=M(T)/M(0) is plotted in Fig. 10 to which\nwe apply the scaling relation Eq. (17) for each of the con-\ntributions fromthe magnetic sublatticesaswell asforthe\nbulk data. The bulk scaling analysis is shown represen-\ntatively in Fig. 11. Experimental bulk data is taken from\nthe past literature23. For the temperature range where\nthe reliable experimental data set seems to be available,\nwe tabulate the extracted scaling exponents in Table II.\nThescalingexponentsdodepend onthe magneticsublat-\ntices. We find that the bulk scaling exponent is observed\nonly in a phenomenological way after summing up the\ncontribution of eachmagnetic atoms in the unit cell. The\noverall calculated trend from the high-temperature side\nwith larger αto the low-temperature side with smaller α\nis shared by all of the sublattice and the bulk. The bulk 0.5 1 1.5 2 2.5 3\n 0.5  0.6  0.7  0.8  0.9  1 1 2 3 4 5Ku1,bulk [meV/unit cell]\nKu1,bulk [MJ/m3]\nm=M(T)/M(0)(i) fit for 0.975<m\n(ii) fit for 0.83<m<0.95\n(iii) fit for 0.75<m<0.86\nDLM results\nFIG. 11. (Color online) Scaling ofthebulkMAE, Ku1,bulk(T),\nwith respect to the order parameter m=M(T)/M(0). The\nscaling exponent αinKu1,bulk(T) =Amαdepends on the\ntemperature-dependent scaling region: i) α= 3.5 form >\n0.975 which corresponds to 41 [K] < T < 75 [K], ii) α=\n5.6 for 0.83< m < 0.95 which corresponds to 186 [K] <\nT <477 [K], and iii) α= 6.5 for 0.75< m < 0.86 which\ncorresponds to 428 [K] < T <610 [K].\nT= 177 [K] 186 [K] 300 [K]\nCo(1)/Co(2) – α= 5.6\nCo(3)/Co(4) – α= 4.9\nCo(5) α= 7.7\nBulk (Our calc.) – α= 5.6\nBulk (Exp.a)α= 6.1 for 150 [K] ≤T≤300 [K]\naRef. 23\nTABLE II. Focus temperature range and the extracted scal-\ning exponents αinKu1∝mαfor each local moment on the\nmagnetic sublattice in the unit cell of YCo 5and for the bulk.\nscaling exponent around the room temperature is calcu-\nlated to be α∼5.6 which reasonably compares with the\nexperimental exponent α∼6.1 that was extracted in the\ntemperature region including the room temperature.\nWe note that our calculated data in Fig. 11 shifts from\nexperimental data on the maxis when compared at the\nsame temperatures. Our calculated data for m >0.96\nare for below the liquid-nitrogen temperatures while for\nthe experimental data23the room-temperature data is\nalreadyin theparameterrange m >0.96. Thatdeviation\ncould be related to the inherent underestimate in our\ncalculations.\nIV. DISCUSSIONS\nWe discuss the strategy to improve the persistence of\ncoercivity against temperature decay. It has been known\nthat the coercivity is approximately proportional to the\nanisotropy field HA5,21. A significant part of the reduc-9\n 0 0.5 1 1.5 2 2.5\n 0  100  200  300  400  500  600Ku1,bulk [meV/(formula unit)]\nT [K]Cf. ∆Ne=0 (Ne=54)\n∆Ne=-0.13\n-0.24\n-0.35\n-0.46\n-0.57\n-0.69\n-0.80\n-0.93\nFIG. 12. (Color online) Calculated temperature-dependenc e\nof MAE for the hole-doped YCo 5.\ntion factor Hc/HAoriginates from the microstructure\nwhich is beyondthe scope ofthe bulk electronicstructure\nconsiderations. Now our interest lies in how to improve\nHA(T) or equivalently Ku1(T) for high T. We propose a\nscheme to exploit the peak position of MAE in Fig. 2 and\nhow the electrons are thermally excited from the ground\nstate following the Fermi-Dirac distribution. The idea is\nto let the thermally excited electrons populate the tar-\ngeted electronic state which lies below the peak in Fig. 2\ntoenhanceMCA.Forthat, weartificiallylowerthechem-\nical potential to make some place for thermally excited\nelectronstopopulate athigh temperatures. Thenwe sac-\nrifice the optimal values of the MCA at low temperatures\nbut we can extract the optimal anisotropy field over the\noperating temperature ranges.\nThe results for the calculated temperature dependence\nofthe MAE in the hole-doped YCo 5’s is shownin Fig. 12.\nHole doping is implemented by artificially lowering the\nFermi level in DLM runs for YCo 5. Measuring the po-\nsition of the lowered Fermi level, EF,doped, from the un-\ndoped one, EF,undoped, in units of Kelvins helps us to fig-\nure out around which temperature range the thermally\nexcited electrons would reach the peak position in Fig. 2.\nWe found that for the reduction range 150 [K]<∼∆EF<∼\n300 [K], with ∆ EF≡(EF,doped−EF,undoped), some en-\nhancement of the MAE around T∼500 [K] is numeri-\ncallyobserved. Thedatawith ∆ EF= 314[K]and471[K]\nwere extracted with the corresponding reduction in the\nelectron number being ∆ Ne=−0.24 and−0.35, respec-\ntively, as the main message of this paper in Fig. 1, where\nMAE is expressed in terms of the anisotropy field follow-\ning Eq. (16).\nRemarkably, non-monotonic temperature dependence\nofHAisfoundforthecomputational“sample”inthelow-\ncoercivity region with the electron number reduced by\naround 0.9 in Fig. 12. This non-monotonic temperature\ntemperature dependence of MAE is reminiscent of what\nhas been known experimentally for Y 2Fe14B since 1986,\nwhere the temperature-enhancement in MAE in the or-\nder of 0.2-0.3 [MJ/m3] is observed around T∼300 [K]44.ThecommonfactoramongY 2Fe14BandYCo 5isthat3d-\nelectron contribution in the MCA of rare-earth magnets\nis extracted out. The non-monotonicity in our finite-\ntemperature data for the chemical-potential controlled\nYCo5comes from the following two factors: a) thermal\npopulation of the particular electronic states that cor-\nresponds to the peak of MAE in Fig. 2, and b) various\ncontributions at finite temperatures from multiple sub-\nlattices in the unit cell. We speculate that analogous\nphysics may be at work in Y 2Fe14B which would require\nmore extensive computations than those for the present\nproject.\nWenotethattheCurietemperatureshiftsasthe chem-\nical potential is reduced. The temperature dependence\nof magnetization shifts rather monotonically with re-\nspect to the artificially manipulated chemical potential\nwhileKu1(T) behaves non-monotonically, and the high-\ntemperature enhancement seen in Figs. 1 and 12 is to be\nconsidered as a finite-temperature reflection of the non-\nmonotonic ground-state behavior in Fig. 2 rather than a\nreflection of the plain shift of the Curie temperature.\nV. CONCLUSIONS\nTemperature dependence of MAE and magnetization\nfor YCo 5has been calculated from first principles within\nthe assumption of local moments to give the qualitative\nagreement with the experimental data. We have pro-\nposed a way to improve the high-temperature coercivity\nby hole doping as was demonstrated by calculations with\nthe filling controlled by the chemical potential manipu-\nlation. That can be implemented as applying the gate\nvoltage for the case of a thin film sample, or alloying\nCo with some other elements to slightly reduce the va-\nlence electron number in the bulk. Since Co is an expen-\nsive element, replacing it with some cheaper ingredients\nwith the improved coercivity in the practical operation\ntemperature range is quite welcome. Thus we have pro-\nposedawayto improvethepracticalutility ofYCo 5from\nthe perspective of the electronic structure and statistical\nphysics.\nACKNOWLEDGMENTS\nThis work is supported by the Elements Strategy\nInitiative Center for Magnetic Materials (ESICMM)\nunder the outsourcing project of the Ministry of\nEducation, Culture, Sports, Science and Technology\n(MEXT), Japan. Support is acknowledged from the\nEPSRC (UK) grant EP/J006750/1 (J.B.S. and R.B.).\nOverall guidance given by T. Miyake and H. Akai is\ngratefully acknowledged. MM thanks ESICMM peo-\nple and Y. Yamaji for helpful discussions, L. Petit and\nD. Paudyal for fruitful interactions during KKR work-\nshop in Warwick in July 2013, O. N. Mryasov for discus-\nsions during his visit to ESICMM, NIMS in July-August10\n2013, and K. Hono for the overall push to the present\nproject. Numerical computations were performed on\n“Minerva” at University of Warwick and SGI Altix at\nNational Institute for Materials Science.\nAppendix A: Details of the calculations for YCo 5\n1. Adjustment of the valence electron number\nThe overall calculation consists of two steps:\n1. The first stage is to generate the grand poten-\ntials for each of the assumed local moment config-\nurations by a scalar-relativistic KKR calculation.\nWe do a spin-polarized calculation for YCo 5as\nparametrized by Vosko et al.45to reach the ferro-\nmagnetic ground state and yield the potentials for\neach local moment on the magnetic sublattices.\n2. Then the second stage consists of solving the fully\nrelativistic equation of state under the potentials\ngenerated by the electronic structure calculation in\nstep 1 and the direction of magnetization is fixed\nto ben= (1,0,1) with the motivation to address\nMAE using the torque-based formula in Eq. (15).\nSolutions are obtained so that Eqs. (3) and (5) are\nsolved self-consistently to yield the result on MAE,\nmagnetization, and the temperature.\nIn these KKR-based calculations we set the cutoff or-\nder in the expansion of spherical harmonics lmax= 3 all\nthrough the above two steps.\nIn step 1, we follow the atomic-sphere approximation\n(ASA) considering the deformed shape of the unit cell to\navoid the complication caused by overlapping muffin-tin\nspheres or too large a interstitial region. We have con-\nfirmed that this ASA construction is adequate from full\npotential (FP)-linear muffin tin orbital (LMTO) calcula-\ntions where we find the ASA and FP-LMTO densities of\nstates for the T= 0K cases to compare well.\nHaving difference in the treatment of the SOI between\nsteps 1 and 2, adjustment of the Fermi level needs to\nbe done to exactly fix the number of electrons, which is\ndemonstrated in Fig. 13. The adjustment is done in the\norder of O(0.01) [Ry] ∼O(0.001) [eV]. We describe the\npractical procedure for this in Sec. A3.\n2. Achieving the self-consistency in DLM with\nmultiple magnetic sublattices in the unit cell\nFurther iterations within DLM is taken for YCo 5to\nachieve the self-consistency of the calculation to incorpo-\nrate the effects of multiple mean-fields that depend on\nthe magnetic sublattice within the unit cell. The target\nmaterial YCo 5has at least two different magnetic sub-\nlattices, Co(2c) and Co(3g), and the magnetic moments\non them can be different from each other as was found in 45 50 55 60 65\n 0.6 0.61 0.62 0.63 0.64 0.65 0.66 0.67 0.68 0.69 0.7Ne(E)\nE [Ry]Fermi level given by KKRFermi level adjusted in DLM\nFIG. 13. (Color online) Calculated Fermi-level dependence of\nnumber of electrons for pure YCo 5near the ground state at\nm= 0.967 [m=L(βh) withβh= 30 as in Eq. (8)]. Equation\nof state is solved with the artificially given Fermi level to\nadjust the valence electron number exactly to the number as\nset up by the calculation.\nthe neutron scattering experiments26. The Weiss fields\nareobtainedfromEqs.(3)and(5)solvedself-consistently\nfor the sites on the sub-lattice where there are local mo-\nments residing. We proceed as follows.\n1. We start with a run with βhuniformly set over\nall of the local moments in the unit cell so that\nmi=L(βh). We refer to this run the zero-th round\nof the correction process.\n2. With the output of the zero-th round, the input\nβh’s are re-set by using the output Weiss fields pro-\nduced by the calculation from Eq (3).\nβ= (βh)i/hi (A1)\nfor alli’s. We note that the left-hand side, in-\nverse temperature, is sublattice-independent. This\nimposes the self-consistency condition: we fix the\ninputβhfor Co(2c)-sublattice and adjust βhfor\nother local moments that sit on the other sublat-\ntices according to the following relation:\n(βh)Co(3g)\nhCo(3g)≡(βh)Co(2c)\nhCo(2c)(A2)\nUsing the output of the zero-th round for h’s,\nthe input numbers for ( βh)Co(3g)=hCo(3g)×\n(βh)Co(2c)/hCo(2c)is used for the first round.\n3. The process is iterated, making the 2nd,\n3rd,...rounds in the correction process, until the\ninputS1and the output S1’s are identical within\nthe numerical accuracy.\nWe have observed that for ( βh)Co(2c)≥10 the conver-\ngence within 4 digits for h’s are quickly reached typically11\nafter 3 iterations while for ( βh)Co(2c)>∼0.1 the conver-\ngence takes around 10 iterations. The iteration can run\ninto a limit cycle of a few periods beyond the 5th digit\nofβh. At such stage of the iteration, we regard that the\nconvergence was reached within the numerical precision\nof 4 digits for S1which we believe is sufficient.\n3. Determining and varying the chemical\npotential in the DLM\nIn addressing magnetic hardness from first principles,\nmagnetic anisotropy energy (MAE) needs to be calcu-\nlated as precisely as possible. The behavior of the calcu-\nlated MAE as a function of the valence electron number\nfor YCo 5is shown in Fig. 2. This data is obtained in a\nseries of DLM runs by artificially modulating the chem-\nical potential near the ground state. Here the data near\nthe groundstate meansthe orderparameteris fixedto be\nm= 0.967 =L(βh) withβh= 30. MAE has the strong\ndependence on the electron filling and only after fixing\nthe electron number by a fine-tuning of the chemical po-\ntential in the DLM run, we can discuss the temperature\ndependence of MAE and magnetization from first prin-\nciples: here we fix the valence electron number down to\n4 digits by manual tuning the chemical potential. The\nfine-tuning follows the Newton’s method in spirit, where\nwe keep on shifting the chemical potential by ∆ µwith\n∆µ= (Ne,desired−Ne)/n(µ)\nwhereNeis the electron number given as N(ˆn)(µ) and\nn(µ) is the density of states at µ, until the desired elec-\ntron number Ndesired= 54 is numerically reached by iter-\nating the calculations with the shifted chemical potential\n(µ+∆µ). At this determined chemical potential and the\nelectron number Ne= 54.00 near the ground state, we\nfind the low-temperature MAE to be 2.3 [meV/unit cell]\nas shown in Fig. 2. Using the experimental lattice con-\nstants26, this is equivalent to 4.4 [MJ/m3], which com-\npareswiththeexperimentalresultatliquidnitrogentem-\nperature, 6.3 [MJ/m3]24, with an underestimate of 30%.\nThe above fine-tuning of the chemical potential also\ndepends on the temperature. At each data point on the\ntemperature axis, the valence electron number is fixed to\nbe 54.00 within the numerical accuracy which practicallygives a temperature-dependent chemical potential from\n0.63609 [Ry] at the lowest temperatures to 0.63357 [Ry]\nat the calculated highest temperature 965 [K].\n4. Magnetization and local moments\nThe total magnetization, and its spin part, and the or-\nbital part, is calculated to be 8.03 [ µB], 7.44 [µB], and\n0.59 [µB], respectively, as shown in Fig. 5 for low tem-\nperature. The calculated total magnetizationamountsto\nJs= 1.11 [T] using the experimental lattice constants26.\ntotal spin orbital\nCo(2c) (our calc.) 1.88 1.81 0.0667\nCo(2c) (exp.) 1.77 1.31 0.46\nCo(3g) (our calc. for Co(3)/Co(4)) 1.59 1.44 0.150\n(our calc. for Co(5)) 1.60 1.44 0.164\nCo(3g) (exp.) 1.72 1.44 0.28\nTABLE III. Calculated magnitude of local moments in [ µB].\nExperimental numbers are taken from Ref. 26.\nTheseresults agreewith the experimental numbersfound\nin the literature of 7.99 [ µB]26orJs= 1.1 [T]8within\n0.5%.\nHowever we note that this apparent excellent agree-\nment is actually realized by a cancellation of overes-\ntimates and underestimates, which is revealed by in-\nspecting each local moment’s contribution to the total\nmagnetization. At the fixed filling, our results are ex-\ntracted as summarized in Table. III. Our calculated lo-\ncal moments on Co(2c)/Co(3g) sublattice are 1.88 and\n1.59 [µB] which compare with the results from neutron\nexperiments 1.77/1.72 [ µB]26within 6-8%. Those polar-\nizations are given by DLM in the fully relativistic cal-\nculation near the ground state, based on the potentials\ngenerated by the scalar-relativistic KKR calculation at\nT= 0 which gave the polarization as 1.48 and 1.52 for\nCo(2c) and Co(3g), respectively. In the present local mo-\nment approach for YCo 5we have systematic underesti-\nmate of the orbital moments compared with the neutron\nmeasurements. However, the resolution of the neutron\ndatainto spin and orbitalcontributionalsoinvolvessome\nsubtlety46. The accurate ab initio estimation of the or-\nbital contribution to the magnetization is left for future\nprojects.\n1W. F. Brown, Rev. Mod. Phys. 17, 15 (1945).\n2M. Sagawa, S. Fujimura, N. Togawa, H. Yamamoto, and\nY. Matsuura, J. Appl. Phys. 55, 2083 (1984).\n3Y. Tawaraand H.Senno,Jpn. J. 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Balogh, L. Szunyogh, M. dos San-\ntos Dias, and J. B. Staunton, Phys. Rev. B 89, 224401\n(2014)\n.\n42H. B. Callen and E. Callen, J. Phys. Chem. Solids 27, 1271\n(1966).\n43O. N. Mryasov, U. Nowak, K. Y. Guslienko, and R. W.\nChantrell, Europhys. Lett. 69, 805 (2005).\n44S. Hirosawa, Y. Matsuura, H. Yamamoto, S. Fujimura,\nand M. Sagawa, J. Appl. Phys. 59, 873 (1986).\n45S. H. Vosko, L. Wilk, and M. Nusair, Can. J. Phys. 58,\n1200 (1980).\n46K. Saito, private communications."    },    {        "title": "1407.1363v2.Atomistic_modeling_of_magnetization_reversal_modes_in__L1__0___FePt_nanodots_with_magnetically_soft_edges.pdf",        "content": "Atomistic modeling of magnetization reversal modes in L10FePt nanodots with magnetically soft\nedges\nJung-Wei Liao1;2, Unai Atxitia2;3,\u0003Richard F. L. Evans2,†Roy W. Chantrell2, and Chih-Huang Lai1‡\n1Department of Materials Science and Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan\n2Department of Physics, The University of York, York YO10 5DD, United Kingdom and\n3Departamento de Fisica de Materiales, Universidad del Pais Vasco, UPV/EHU, 20018 San Sebastian, Spain\n(Dated: September 27, 2018)\nNanopatterned FePt nano-dots often exhibit low coercivity and a broad switching field distribution, which\ncould arise due to edge damage during the patterning process causing a reduction in the L10ordering required\nfor a high magnetocrystalline anisotropy. Using an atomistic spin model, we study the magnetization reversal\nbehavior of L10FePt nanodots with soft magnetic edges. We show that reversal is initiated by nucleation\nfor the whole range of edge widths studied. For narrow soft edges the individual nucleation events dominate\nreversal; for wider edges, multiple nucleation at the edge creates a circular domain wall at the interface which\nprecedes complete reversal. Our simulations compare well with available analytical theories. The increased\nedge width further reduces and saturates the required nucleation field. The nucleation field and the activation\nvolume manipulate the thermally induced switching field distribution. By control of the properties of dot edges\nusing proper patterning methods, it should be possible to realize exchange spring bit patterned media without\nadditional soft layers.\nI. INTRODUCTION\nContinuing requirements for greater digital data storage ca-\npacity has lead to continued growth in data storage density in\nmagnetic recording media. Future improvements are limited\nby the magnetic recording trilemma, caused by competing re-\nquirements of reduced signal to noise ratio, thermal stability\nof written information, and writability.1Two solutions for the\nmagnetic recording trilemma have been proposed: first heat\nassisted magnetic recording (HAMR),2,3where laser heating\nduring recording is used to lower the anisotropy sufficiently\nto achieve writing; the second solution is bit-patterned media\n(BPM), where each bit is defined by a single dot in a litho-\ngraphically defined array4and the larger magnetic volume re-\nduces the requirement for high anisotropy required for long-\nterm thermal stability.\nBit patterned media can be made using a variety of meth-\nods including patterning5–11and self-assembly of magnetic\nnanoparticles.12Controlling the microstructural properties of\nmagnetic nanoparticles is quite challenging, however litho-\ngraphic patterning techniques allow a continuous L10FePt\nfilm to be patterned into an array of isolated magnetic is-\nlands or ’dots’.5–11However, during lithography ions near the\ndot edge can reduce the L10ordering, resulting in magneti-\ncally soft edges.6,7,9–11The presence of damaged edges in the\ndots could reduce both the coercivity6,10,11and the thermal\nstability.9\nIn addition to decreasing the coercivity, a broad switch-\ning field distribution (SFD) can also lead to write errors in\nneighboring bits during the writing process. The SFD (the\nvariation of switching fields between dots) includes both ex-\ntrinsic and intrinsic components.13–17The extrinsic SFD may\nbe caused by dipolar interaction between dots, and the intrin-\nsic SFD arises from variations of intrinsic magnetic properties\nof dots, including anisotropy K, volume V, and the easy axis\nalignment.14Furthermore, thermal fluctuations also broaden\nthe intrinsic SFD, known as the thermal SFD.15–17Within the\nsimple Stoner-Wohlfarth approximation (monodomain), thethermal SFD is mainly related to the anisotropy energy bar-\nrier, KV, and the measurement time scale. This makes the\nthermal SFD pronounced at high field sweep rates associated\nwith the recording process.16,17\nReversal behavior in relatively large dots with magnetically\nsoft edges of fixed width, associated with ion-damage from\nthe etching process, has been studied previously,18where\nthe presence of soft edges was shown to change the rever-\nsal mode. In addition, small-sized dots with ring-shaped soft\nedges of varied width have been investigated by macrospin\nanalytic models without including the thermal fluctuations.19\nThe increased width of edge is found to reduce the coercivity\nof dots,19suggesting a strong relationship between the edge\nwidth and the reversal mode. Further control of the magnetic\nproperties of edges with fixed width in large sized dots can\nalso be done via soft He+irradiation.20The experimental ob-\nservations can only be explained by the model including the\nthermal fluctuations.20All the reported works indicate that ei-\nther the edge width19or the thermal fluctuations20affects the\nreversal mode and could result in different switching field dis-\ntributions of patterned dots. However, the edge-width depen-\ndence of the reversal mode including thermal fluctuations is\nstill not understood.\nHere we develop a computational model to study magneti-\nzation reversal modes in L10FePt dots with magnetically soft\nedges. We employ an atomistic spin model formalism, which\nprovides detailed information on reversal modes unreachable\nby standard micromagnetic simulations.21,22In particular, soft\nedges of only a few nanometers are tractable, and we can\nfurther study the effect of the reduced exchange coupling at\nthe interface, possibly resulting from the core/edge interface\nroughness. Moreover, thermal effects are consistently taken\ninto account within our model using the Langevin dynam-\nics formalism that allows us to study the relationship between\nthe coercivity, the thermal SFD, and reversal modes. We fur-\nther compare the atomistic spin modeled results with avail-\nable analytic approaches, which were originally presented for\nhard/soft nano wires,23–28to examine the validity of these ap-arXiv:1407.1363v2  [cond-mat.mtrl-sci]  18 Nov 20142\nproaches for the core/shell nanostructure. These simpler ap-\nproaches are capable of highlighting the key physics. Addi-\ntionally, since the atomistic resolution in the simulation makes\nthis method computationally intensive restricting the size of\nthe calculated system to nanometer length scales, these vali-\ndated analytic approaches could be potentially utilized to in-\nvestigate properties of large sized systems, e.g., a dot array.\nII. ATOMISTIC SPIN MODEL\nThe studied nanodots are composed of a magnetically hard\ncore and a magnetically soft edge, as illustrated in Fig. 1. In\nthe case of patterned dots, we hypothesize that the edge region\nloses its L10atomic order due to the patterning process, mak-\ning it magnetically soft. We focus exclusively on the width\nof the edge Wedge, with the fixed core size, rcore, on the mag-\nnetization reversal. We therefore fix the diameter of the core\n2rcore=25 nm and the dot thickness, td=4 nm, while the\nedge width is varied systematically from Wedge=0\u000018 nm.\nWe note that in this approach the different Wedgevaries the to-\ntal volume of dot and therefore affects the corresponding ther-\nmal stability, which is beyond the scope of the present work.\nThe system is constructed from a single face-centered cubic\ncrystal and cut into the shape of a nanodot with the desired\ngeometry.\nThe nano dots are modeled using an atomistic spin model\napproach29with the VAMPIRE software package.30The ener-\ngetics of the system are described by the spin Hamiltonian\nwith the Heisenberg exchange, given by:\nH=Hcore+Hedge (1)\nHcore=\u0000å\ni;jJcoreSi\u0001Sj\u0000å\ni;nJceSi\u0001Sn\u0000\nkcoreå\ni(Sz\ni)2\u0000mcoreå\niHapp\u0001Si (2)\nHedge=\u0000å\nn;dJedgeSn\u0001Sd\u0000å\nn;jJceSn\u0001Sj\u0000\nkedgeå\nn(Sz\nn)2\u0000medgeå\nnHapp\u0001Sn (3)\nwhere S=\u0016=mare spin unit vectors, i;jlabel core sites\nwith moment mcore, and n;dlabel edge sites with moment\nmedge. Here we assume the same moment for both core and\nedge such that mcore=medge=1:5mB, which compares well\nto the saturation magnetization of L10FePt as obtained in\nexperiment.31JcoreandJedgeare the exchange interactions be-\ntween moments of the same type in the core and the edge, re-\nspectively. We consider only nearest neighbor interactions be-\ntween the moments. We select values of the exchange energy\nto give a Curie temperature around 700 K comparable with\nexperiment, namely Jcore=Jedge=3\u000210\u000021J/link. Jcerep-\nresents the interfacial exchange interaction between the core\nand the edge and is varied as a parameter between 0 and Jcore.\nkcore=4:9\u000210\u000023J/atom is the uniaxial anisotropy constant\nof the core spins (with easy axis perpendicular to the film\nplane) and kedge=1\u000210\u000024J/atom is the uniaxial anisotropy\nof the edge spins. Happis the external applied field.\nWedge2rcore0 - 18 nmWedge0 - 18 nm\ntd4 nm25 nmFIG. 1. (Color online) Schematic diagram of the atomistic modeled\ndot. Dark and white gray regions represent the core and the edge\natoms, respectively.\nThe hysteresis loops are calculated dynamically using\nthe stochastic Landau-Lifshitz-Gilbert (LLG) equation at the\natomic level, given by\n¶Si\n¶t=\u0000g\n(1+l2)Si\u0002\u0002\nHi;eff+l(Si\u0002Hi;eff)\u0003\n; (4)\nwhere lis the intrinsic damping parameter, g=1:76\u00021011\nT\u00001s\u00001is the absolute value of the gyromagnetic ratio, and\nHi;effis the effective magnetic field in each spin. The field is\nderived from the spin Hamiltonian and is given by\nHi;eff=\u00001\nmi¶H\n¶Si+Hdemag ;i+Hi;th; (5)\nwhere Hdemag ;iandHi;thare the demagnetization and the ther-\nmal fields, respectively. Since the calculation of the demag-\nnetization field at the atomic level is computationally expen-\nsive, we have instead calculated the demagnetization field by\napplying the approach developed by Boerner et al.32Within\nthis approach, the dot is divided into regular macrocells with\nthe volume Vk= (1:77)3nm3which contains 250 atomic\nspins. The value of spin’s moments within each macrocell\nare then summed to obtain the macrocell magnetic moment,\n\u0016k=åd24kmdSd, where klabels macrocell sites, and dla-\nbels spin sites in each macrocell, 4k. We then calculate the\ndemagnetization field of each macrocell, Hdemag ;k, by using\nthe corresponding magnetic moment and treat it as the de-\nmagnetization field of each spin in the macrocell, Hdemag ;i.\nHdemag ;kis calculated by direct pairwise summation including\nthe macrocell self-demagnetization29\nHdemag ;k=m0\n4på\nk6=l3(\u0016l\u0001^rkl)^rkl\u0000\u0016l\njrklj3\u0000m0\n3\u0016k\nVk(6)\nwhere m0=4p\u000210\u00007T2J\u00001m3is the vacuum permeabil-\nity,rklis the vector between kandlmacrocell sites, and\n^rkl=rkl=jrkljis the corresponding unit vector. This is a com-\nputationally efficient approach since the number of macrocells\nis relatively small and moreover, since the magnetostatic field3\n-101-4 -3 -2 -1 0M/MSHappl(T)0n m 1n m 2n m 3n m 7n ma⃝△▽⃝△▽⃝△▽⃝△▽⃝△▽\n-101b0n m 1n m 2n m 3n m 7n m\n’./Figure2b-1.jpg’ binary filetype=auto\nFIG. 2. (Color online) (a) Simulated out-of-plane hysteresis loops for\ndots with different edge widths. Magnetization is normalized to the\nsaturation magnetization at 0 K. Snapshots of domain configurations\nduring reversal, observed along the dot plane normal direction, are\nshown in (b). Symbols in (a) and on the left in (b) indicate the posi-\ntion of snapshots during the reversal process. The color scale (blue\nto red) represents the magnetization component along the easy axis\ndirection. Black dotted circles denote the position of the core/edge\ninterface.\nvaries rather slowly with time it needs updating only on a\ntimescale of around 1000 time steps.29The thermal fluctua-\ntions are represented using Langevin dynamics,33,34where the\nthermal field Hi;this given by\nHi;th=\u0000(t)s\n2lkBT\ngmiDt; (7)\nwhere kBis the Boltzmann constant, Tis the heat bath tem-\nperature, lis the Gilbert damping parameter, gis the abso-\nlute value of the gyromagnetic ratio, and Dtis the integration\ntime step. The thermal fluctuations are represented by a vec-\ntor Gaussian distribution in space \u0000(t)with a mean of zero\nand generated from a pseudo-random number generator. The\nsimulations in this work are carried out at a heat-bath temper-\nature of T=300 K. We set the damping parameter l=1:0 to\nreduce the computational time required for reaching an equi-\nlibrium state. The LLG equation is integrated using the Heun\nintegration scheme34with an integration time step Dt=1 fs.\nIII. RESULTS\nIn order to study reversal modes we simulate hysteresis\nloops as a function of the width of edges, Wedge. To calcu-\nlate the hysteresis loops we apply an external field in a range\nfrom\u00005 to+5 T, which lies above the anisotropy field in the\ncore, at intervals of 5 mT. The field sweep rate is 5 T/ns. Ini-\ntially we consider that the interfacial core and edge spins are\n-1-0.500.51\n-1 -0.5 0 0.5 1M/M S\nHappl(T)Core\nEdgeCore+EdgeFIG. 3. (Color online) Whole hysteresis loop of the dot with Wedge=\n12 nm and two individual loops of its core and its edge, decomposing\nthe whole loop.\nstrongly coupled by setting Jce=Jcore=Jedge=3:0\u000210\u000021\nJ/link.\nFigure 2(a) shows representative out-of-plane hysteresis\nloops for a range of edge widths. One can observe that by\nincreasing edge width, both the nucleation and coercive fields\ndecrease. Furthermore, the square-like hysteresis loop for nar-\nrow edges turns into a two-step reversal as the edge width\nincreases, indicating a change in the reversal process. Fig-\nure 2(b) illustrates the corresponding snapshots of spin config-\nurations during reversal for various Wedge. The reversal mode\nstrongly depends on the edge width, which will be discussed\nin more detail in the following sections. To obtain detailed in-\nformation on the observed reversal behavior, we also calculate\nhysteresis loops for each edge width for 30 different realiza-\ntions of the random number generator. Therefore, we average\nover 60 statistically independent values to obtain the mean\ncoercivity and standard deviation. Since we are considering\ndots with the same magnetic properties in our simulations, the\ndeviation from the mean arises completely from the thermal\nfluctuations. Thus the standard deviation is a manifestation of\nthe intrinsic SFD resulting from thermal fluctuations.17This is\nan important parameter since it increases with increasing field\nsweep rate and is significant at timescales associated with data\ntransfer in information storage. Additionally, we separately\ncalculate the coercivity fields of both the core, Hcore\nc, and the\nedge, Hedge\nc, shown in Fig. 3.\nTo do so, we calculate the individual reduced magnetization\nof the core and edge as follows,\n\u0016core=jmcorej\nNcoreå\ni2coreSi;\u0016edge=jmedgej\nNedgeå\ni2edgeSi; (8)\nwhere Ncore(edge)denotes the number of atoms in the core\n(edge).\nFigure 4(a) shows the variation of the mean coercivity\nHcore(edge)\nc as a function of Wedge, and the corresponding stan-4\n1234Hc(T)a\nHeff\nKHn\nHp\n0.020.040.060.080.10.120.14\n0 2 4 6 8 10 12 14 16 18σ(T)\nWedge(nm)b\nσK σn\nσpledge\nDWCore\nEdge\nFIG. 4. (Color online) (a) Mean coercivity of both the core and the\nedge as a function of the width of edges. The gray dashed line rep-\nresents the effective anisotropy field calculated by the linear chain\nmodel. Purple solid and black dashed lines indicate the nucleation\nand pinning fields respectively. (b) Standard deviation of coercivity\nof both the core and the edge as a function of the width of edges\ngiving an estimate of the thermal switching field distribution. The\ngray dashed line represents the deviation approached by the effective\nanisotropy field. Purple solid and black dashed lines are the devia-\ntion calculated by the nucleation and the pinning fields. The vertical\ndashed line denotes the domain-wall length in the edge.\ndard deviation, score(edge), is shown in Fig. 4(b). The co-\nercivities and the standard deviation are strongly dependent\non the edge width, which will be discussed in the follow-\ning sections. Furthermore, to understand the reversal mode,\nwe will compare coercive fields obtained from atomistic spin\nmodel simulations with those obtained from different theoret-\nical approaches,23–28given by the lines in Fig. 4. The addi-\ntional models, to be discussed later, are all based on a con-\nventional micromagnetic approach. In such models the tem-\nperature dependence of magnetic properties is not intrinsic to\nthe formalism, as it is in the atomistic approach, and must be\nintroduced explicitly. Consequently, in the micromagnetic-\nbased models we will introduce the effect of temperature\n(T=300 K) by normalizing the micromagnetic parameters\nin the theoretical calculations. For the anisotropy constants,\nKcore(edge), we use the Callen-Callen law35\nKcore(edge)(T=300 K )\u0019Kcore(edge)(T=0 K)m3\ne;(9)\nwhere me=Ms(T=300 K )=Ms(T=0 K) =0:82 is obtained\ndirectly from our computational atomistic spin calculations.\nMsis the saturation magnetization of the core (edge). We\nnote that experimentally the exponent value for the decrease\nin anisotropy constant of L10FePt is close to 2.1,31which\ncan be reproduced using the multiscale atomistic spin model\nsimulations.36However, this multiscale simulation is compu-\ntationally expensive for the calculation of dots with the diam-\neter of 25 nm [Fig. 1]. Therefore, we have used a simplified\natomistic spin model, where Kcore(edge)(T)follows the Callen-\nCallen law. In fact at room temperature the 2.1 scaling lawand the Callen-Callen law give similar length scales, e.g. the\nexchange length or the domain-wall length, which can be es-\ntimated by the ratio m2:1=2\ne =m3=2\ne\u00180:93. We also note that\nboth surface and interface effects can slightly vary the Callen-\nCallen law for Kedge(T).37The exchange stiffness of the core\n(edge), Acore(edge)(T), has been shown to scale with meas38,39\nAcore(edge)(T=300 K )\u0019Acore(edge)(T=0 K)m1:745\ne :(10)\nA. Narrow soft edge: individual nucleation\nThe magnetization reversal in the absence of soft edges, as\nshown in the spin configuration snapshot for Wedge=0 nm in\nFig. 2(b), starts by the nucleation of a small region (red area\nin the snapshot) in the boundary and proceeds with the subse-\nquent expansion to the entire dot. At this point it is worthwhile\nconsidering the physical origin of the nucleated reversal. The\norigin of the incoherent nucleated reversal process lies in the\ncombination of high magnetocrystalline anisotropy and ther-\nmal fluctuations. At applied fields in the vicinity of the coer-\ncivity thermal fluctuations break the symmetry of the dot and\ncause a nucleation event. The narrow domain wall width, aris-\ning from the high magnetocrystalline anisotropy in the core,\nstabilizes the nucleated domain. Following the nucleation the\nlowest energy barrier for switching is then propagation of the\ndomain wall, leading to an incoherent reversal mechanism.\nThe combination of short time scales, high anisotropy, and\nsystem size greater than the domain wall width gives the fun-\ndamental physical origin of the thermal switching field dis-\ntribution. For longer timescales more nucleation attempts are\nmade reducing the effective thermal SFD since the material\nswitches at the same field, while for lower anisotropy materi-\nals the nucleated domain is unstable and so the effect of ther-\nmal fluctuations is also lower.\nFor dots with a narrow soft edge, Wedge=1 or 2 nm, the\nreversal mechanism is the same as for dots with no soft edges,\nalthough due to the low coercivity of the edge the nucleation\nfield is reduced significantly. The thermal SFD also reduces\nrapidly with narrow soft edges due to the reduced stability of\nthe nucleated domain owing to the lower effective anisotropy.\nIn an attempt to quantify the reduction in the coercivity as\na function of the edge width we have developed an atomistic\none-dimensional (1D) linear chain model, details of which are\ngiven in Appendix A. By estimating the coercive field as an\neffective anisotropy field of the nucleated area, Heff\nK, the linear\nchain model predicts a linear decrease in the coercivity given\nby\nHcore(edge)\nc =Heff\nK=Hcore\nK\u0000\n1\u0000bWedge\u0001\n: (11)\nIn Fig. 4(a) we can see that for Wedge\u00142 nm, both the coer-\ncivity of the core and the edge are equal and linearly decrease\nas a function of the edge width. It can be seen that Eq. 11\ngives reasonable agreement with the numerical results.5\nB. Wide soft edge: an incomplete to a complete circular\ndomain wall\nWith a further increase in the edge width we observe the re-\nversed region with a negative curvature, shown by Fig. 2(b)\nforWedge=3 nm with nucleated areas denoted by red re-\ngions. The negative curvature could suggest that more than\none reversed region nucleates during the reversal. The devi-\nation between Hcore(edge)andHeff\nK[Eq. (11)] reflects the re-\nversal dominated by multi-reversed regions. These multiple\nnucleation events also mark an increasing difference between\nHedge\ncandHcore\ncvalues with further increases in the edge width\n[Fig. 4(a)]. From the spin configuration snapshots in Fig. 2(b)\nwe observe this behavior corresponds to an incomplete cir-\ncular domain wall formed at the core/edge interface. In this\nregion we cannot approach Hcore(edge)using Eq. (11) because\nthis is only valid for the reversal dominated by a single re-\nversed region. Instead, we find that Hedge\ncapproaches the do-\nmain wall nucleation field Hn, obtained from the analytical\nexpression derived for the limit of strong hard/soft coupling\nwith a soft layer thicker than the domain-wall width ( \u00195 nm\nin this study) in the hard layer23–25given by\nHn=Hedge\nK+\u0010p\n2\u00112\u0010ledge\nEX\nWedge\u00112\nMedge; (12)\nwhere Hedge\nK=2Kedge=Medge is the anisotropy field of the\nedge, and Medgeis the saturation magnetization of the edge\nandledge\nEX=p\nAedge=Kedgeis the exchange length in the edge.\nFor applied fields larger than Hnbut less than the domain-\nwall pinning field at the edge/core interface, Hp, the increased\nfield compresses the domain wall in the edge and therefore\nreduces the corresponding domain-wall width, ledge\nDW. At even\nwider edge widths [for example, see Fig. 2(b) at Wedge=7\nnm], the nucleation occurs in the entire edge, but the domain\nwall is then pinned at the core/edge interface, showing a cir-\ncular domain wall. As the reversal continues, the domain\nwall propagates inwards until collapse and full magnetization\nreversal. In addition, the propagated domain wall shows a\nnon-circular symmetry [Fig. 2(b) at Wedge=7 nm], in con-\ntrast to the circular symmetry of the domain wall pinned at\nthe core/edge interface. The suggests the depinning of part\nof the circular domain wall during the reversal of spins in the\ncore. However, in contrast to the analytical model of the sin-\ngle nucleation region proposed in Ref. 20, the reversed region\nin the core shows a negative curvature [Fig. 2(b) at Wedge=7\nnm], indicating that the reversal could be dominated by the\nmultinucleation events. On the other hand, Hcore\ncsaturates at\nHpwhen Wedge\u0015ledge\nDW, which reads26–28\nledge\nDW=ps\n2Aedge\nKcore+Kedge: (13)Hpis given by26–28\nHp=1\n42\u0002\nKcore\u0000Kedge\u0003\nMedge: (14)\nFigure 4(a) shows that our simulation results fit perfectly to\ntheHp(black dashed line). Thus it confirms that for soft edges\nwider than ledge\nDWatHp, the reversal mechanism is through de-\npinning of part of a circular domain wall at the edge/core in-\nterface driven by the multiple nucleation events in the core\nwith Hcore\nc=Hp.\nC. Thermally induced switching field distribution\nThe calculated thermal switching field distribution\nscore(edge)from the simulations for different edge widths is\nshown in Fig. 4(b). Similarly to the coercive fields, our simu-\nlations show that for Wedge\u00142 nm, score'sedge, the thermal\nSFD displays a linear decrease with the increasing Wedge.\nIn order to quantify the thermal fluctuations in the coercive\nfield within a micromagnetic framework it is necessary to\nassociate the magnetic moment min Eq. (7) with a volume\ncharacteristic of magnetization reversal. For this we use the\nactivation volume, Vact, which is an equilibrium quantity\nand defined as the volume associated with the magnetization\nchange between positions of minimum and maximum static\nenergy.40Furthermore, we average the thermal fluctuation\nfield over a specific time equal to the inverse of an ”attempt\nfrequency” used in phenomenological models of thermal\nactivation processes. The attempt frequency is generally\ntaken as the natural frequency of oscillation in the local\nminimum, i.e., f0=gHKwith HKthe anisotropy field. This\nleads to a variance in the field components, which we take as\nthe standard deviation of coercivity, score(edge), given by\nscore(edge)=s\n2lkBTHK\nMsVact: (15)\nForWedge\u00142 nm, the single nucleated region dominates the\nreversal. However, the observed nucleation is a nonequilib-\nrim quantity.41ForVactone should estimate the volume of\nthe equilibrium domain change during reversal. Consider-\ning the dot size is smaller than the domain size ( \u001826 nm\nin this study), we can treat the dot as a single domain parti-\ncle and therefore approach Vactto the total volume of the dot,\nVact\u0018p(rcore+Wedge)2td. Taking HK=Heff\nK[Eq. (11)], we\narrive at\nscore(edge)=sK=s\n2lkBTHeff\nK\nMs[p(rcore+Wedge)2td]: (16)\nwhere sKisscore(edge)in this region. It can be seen that\nEq. (16) [indicated by the gray dashed line in Fig. 4(b)] gives\nresults reasonably close to the numerical results.\nForWedge\u00153 nm, the common behavior of spins in the core\nstarts to deviate from that in the edge, as we show in Fig. 4(a).6\n01234Hcore\nc(T)a\n01234\n0 0.2 0.4 0.6 0.8 1Hedge\nc (T)\nJce/Jcoreb1 nm 3 nm 7 nm\nFIG. 5. (Color online) (a) Coercivity of the core as a function of the\nnormalized core/edge exchange coupling strength at varied width of\nedges. The core/edge exchange coupling strength is normalized to\nthe exchange interaction between spins in the core (edge). Dashed\nlines are guided by the eye. (b) Coercivity of the edge as a function\nof the normalized core/edge exchange coupling strength. Dashed\ncurves represent a fitting to the Langevin function [Eq. (19)].\nSimilarly we find that scoredeviates from sedge[Fig. 4(b)]. In\nthis region, the different reversal mode of core spins with that\nof edge spins suggests that Vactin the edge approaches to the\nedge volume, Vact\u0018p\u0002\u0000\nrcore+Wedge\u00012\u0000\u0000\nrcore\u00012\u0003\ntd. Using\nEq. (15) with HK\u0018Hedge\nc=Hn[Eq. (12)] sedgein this region,\nsn, is [purple solid line in Fig. 4(b)]\nsn=s\n2lkBTHn\nMsp\u0002\u0000\nrcore+Wedge\u00012\u0000\u0000\nrcore\u00012\u0003\ntd: (17)\nForWedge\u0015ledge\nDW,Hcore\ncsaturates at Hp[Eq. (14)]. The differ-\nent reversal behavior of core spins to that of edge spins bring\nus to the estimation of the activation volume in the core as\nthe core volume, Vact\u0018p\u0002\u0000\u0000\nrcore\u00012\u0003\ntd. Using Eq. (15) with\nHK\u0018Hcore\nc=Hp[Eq. (14)] we arrive at [black dashed line in\nFig. 4(b)]\nsp=s\n2lkBTHp\nMspr2coretd; (18)\nwhere spisscorein this region. Eqations. (17) and (18) gives\nvalues of score(edge)roughly a factor of 2 different from the\nnumerical results [Fig. 4(b)]. Given the assumptions involved\nthe difference is reasonable agreement.\nD. Effect of interfacial exchange coupling on the reversal\nmodes\nFinally we investigate the effect of core/edge exchange cou-\npling strength, Jce, on the reversal modes in the nanodot. To\ndo so we vary the normalized interfacial exchange couplingstrength ˜Jce=Jce=Jcore(edge)from 0 (no coupling) to 1 which\ncorresponds to the strong coupling studied in detail in the pre-\nvious section. Here, we also perform hysteresis-loop calcula-\ntions for varied edge-width Wedge.\nFigure 5(a) shows the coercivity of the core, Hcore\nc, as a\nfunction of ˜JceforWedge=1;3 and 7 nm to cover the three\nwell-separated regimes of reversal modes observed in our sys-\ntem. We observe that for Wedge=1 nm the core coercive\nfield, Hcore\nc, presents a minimum at a relatively weak cou-\npling strength, similar to results observed in hard/soft struc-\ntures, where the observed minimum is related to the two-\nspin behavior.42Considering that the local minimum of Hcore\nc\nhappens when Jedge\u001dJce(˜Jce\u00141=20) in narrowed edges\n(Wedge\u00142 nm), all spins in the edge might behave as a single\nmacrospin. During the reversal, the single-spin behavior in the\nedge could provide a torque to spins in the core and yield the\nlocal minimum of Hcore\nc, which has been previously observed\nin the two-spin model42as well as in the experiment.43\nAs the coupling increases, the coercive field saturates to\nsome value which has been already discussed in previous sec-\ntions of the present work. For edge widths larger than or equal\nto 3 nm, the minimum of the core coercive field disappears\nand a monotonous decrease in Hcore\ncto a saturation value is\nobserved. For Wedge\u00157 nm [equal to ledge\nDWgiven by Eq. (13)],\nthis saturation value corresponds to the domain-wall depin-\nning field. Therefore, the interface coupling dependence of\nthe core coercive field for Wedge\u00157 nm is similar.\nOn the other hand, the edge coercive field, Hedge\nc, consis-\ntently increases with increasing interfacial exchange coupling\nto a saturation value [see Fig. 5(b)] following a Langevin law\nrepresenting the effective bias field created in the edge by\nthe coupling to the core, in direct analogy to a paramagnet\nmagnetization in the presence of an external field and ther-\nmal fluctuations.44This effective bias field is comparable to\nthe external field applied in the calculation of hysteresis loops\nand can be estimated by\nHedge\nc(˜Jce) =Hedge\nc;1L\u0000\nb medgeHex\u0001\n: (19)\nwhere Hexestimates the average effective bias field in the edge\ninduced by the interfacial coupling, and medge=MedgeVedgeis\nthe saturation magnetization of the edge. The Langevin func-\ntion is L(x) =coth(x)\u00001=x.Hedge\nc;1is a fitting constant and\ncoincides with Hedge\ncat˜Jce=1:0 (calculated in the previous\nsections). We can assume that medgeHex=Vedged˜Jcewhere dis\na parameter that measures the energy transferred from the core\nto the edge via the interfacial coupling. This parameter is ex-\npected to depend on the volume of the edge, Vedge\u0018W2\nedgetd,\nso that as the thickness is fixed for all Wedge, we expect that\nd\u00181=W2\nedgesimilar to that in a soft/hard bilayer structure,\nHexµ1=t2\nsoft.23In Fig. 5(b) we show that indeed this relation\nfits very well to simulations.7\nIV . DISCUSSION AND CONCLUSIONS\nTo summarize, using atomistic spin model simulations, we\nhave investigated reversal modes in patterned L10FePt dots\nwith damaged edges in the presence of thermal fluctuations.\nSpecifically, the calculated dot is composed of a hard mag-\nnetic core, which represents the undamaged part of the dot,\nand the damaged edge with soft magnetic properties. We have\ninvestigated the effects of the extent of damage on the edge\nby varying its width. We observe that the nucleation initiates\nreversal for all width of edges. The increased edge width lin-\nearly decreases and then saturates the required field for nucle-\nation, with the curvature of the initially nucleated region re-\nducing from positive to negative. Furthermore, the increased\nedge width reduces the thermally induced switching field dis-\ntribution, which is found related to both the nucleation field\nand the activation volume. We have further studied rever-\nsal modes in dots with varied core/edge interfacial coupling\nstrength, which could possibly result from the core/edge in-\nterfacial roughness. For dots with narrow edges, the reversal\nbehaves in a similar way with that obtained in the two-spin\nmodel, suggesting that we can treat all spins in the edge as a\nsingle effective macrospin. In addition, we describe the coer-\ncivity of the edge using the Langevin function, representing\nthe competition between the effective field generated from the\ncore/edge coupling strength and the thermal fluctuations.\nWhile the numerical simulation by the atomistic spin model\nis sufficient to explain the magnetization reversal, it is insight-\nful to digest these results by simpler analytic methods so that\nthe key physics can be highlighted. In some cases studied\nhere, the reversal dynamics of the minority spins is mainly\none dimensional. We are thus motivated to employ the lin-\near spin chain model to capture the one-dimensional dynam-\nics. Specifically at narrow edges the linear chain model is\nable to estimate the required field for the nucleation. As the\nedge width increases the nucleation field of core spins fits to\nthe domain-wall pinning field at the core/edge interface. Con-\nsidering the computationally intensive nature of the atomistic\nspin model simulation, these analytic theories can provide a\nglobal sketch for different parameters at minimal costs.\nComparing to previous studies focused on the rever-\nsal modes along the layer-growth direction in the typi-\ncal exchange spring media, here we present detailed two-\ndimensional reversal behaviors on the patterned dot-plane as\nwell as the corresponding thermally induced switching field\ndistribution, both of which in fact dominate properties of typ-\nical patterned dots and cannot be investigated by standard mi-\ncromagnetic calculations. We also note that different mag-\nnetic properties of the edge, which have been assumed con-\nstant values in this study and have not been experimentally\nprobed, only vary the characteristic length of different re-\nversal modes and the corresponding coercivity fields with-\nout affecting the validity of theories. According to our study\nhere, the presence of damaged edges with uniform magnetic\nproperties reduces the thermally induced switching field dis-\ntribution, and the width of the damaged edge significantly\nchanges the coercivity in patterned dots. Therefore, the exper-\nimentally observed broadening of the switching field distribu-\n0.40.60.81\n012Knormeff(T)\nWedge(nm)M/MSAtomic plane1n m2n m\nLinear chain modelLinear fit0.5140 45 50 55FIG. 6. (Color online) Effective value of anisotropy, calculated by the\nlinear chain model, as a function of the width of the edge. The gray\ndashed line is the linear fitted function. Inset shows the calculated\nlayer-resolved magnetization prior to magnetization reversal in the\ncore in the linear chain model. The atomic plane is counted from\nthe center of the core toward the edge, and the vertical dashed line\ndenotes the core/edge interface. The blue area indicates the nucleated\nregion for the edge-width of 2 nm.\ntion in patterned L10FePt dots with damaged edges11should\nbe attributed to extrinsic properties of the nanodots created\nby patterning processes, for example, the variation in either\nthe width or the magnetic properties of the damaged edges.\nAs long as we can precisely control properties of damaged\nedges by applying a proper patterning technique, for exam-\nple, ion implantation,20we could realize exchange spring bit-\npatterned media without additional soft layers.\nACKNOWLEDGMENTS\nThe authors would like to thank H.-H. Lin for insightful\nsuggestions. Fruitful discussions with O. Hovorka, J. Wu, W.\nFan, P. Chureemart, and S. Ruta are also acknowledged. J.-W.\nalso highly appreciates the assistance from J. Barker and T.\nOstler on solving computational issues. This work has been\nsupported by the Ministry of Science and Technology under\nGrant No. MOST 101-2917-I-007-016. U.A. gratefully ac-\nknowledges support from Basque Country Government un-\nder ”Programa Posdoctoral de perfeccionamiento de doctores\ndel DEUI del Gobierno Vasco”. The financial support of the\nAdvanced Storage Technology Consortium (ASTC) and EU\nSeventh Framework Programme under Grant Agreement No.\n281043 FEMTOSPIN is also gratefully acknowledged.\nAppendix A: Linear chain model\nIn order to quantify the variation of the coercivity for nar-\nrow edge thicknesses we have developed a 1-D atomistic\nlinear chain model, simplifying the whole dot into a one-8\ndimensional region started from the center of the core to the\nedge. Each spin in the chain model represents the average\nspin within a given atomic plane, and we can write down the\nfollowing spin Hamiltonian\nHi0=\u0000å\ni0;j0Ji0Si0\u0001Sj0\u0000ki0\u0000\nSz\ni0\u00012\u0000mi0Happ\u0001Si0; (A1)\nwhere i0,j0label different spins with the identical moment\nm.Jis the intra-layer exchange coupling, Sis the unit vector\nrepresenting the spin direction, kis the anisotropy constant,\nandHappis the external applied field. We set m=mcore=\nmedge=1:5mB,k=kcore=4:9\u000210\u000023J/link for spins in the\ncore and k=kedge=1\u000210\u000024J/link for those in the edge. We\nallow reduced exchange coupling at the core/edge interface by\nwriting the exchange energy between interface spins as\nHint=JintSi0\u0001Sn0; (A2)\nwhere Jintis the interface exchange coupling, and n0labels\nspins in separate regions (core or edge) from those labeled by\ni0.\nThe equilibrium state of the spin system is determined by\nsolving the Landau-Lifshitz equation, with no precession term\n¶Si0\n¶t=\u0000g\n(1+l2)Si0\u0002l(Si0\u0002Hi0;eff); (A3)\nwhere lis the intrinsic damping parameter, gis the abso-\nlute value of the gyromagnetic ratio, and Hi0;effis the effectivemagnetic field in each atomic plane, given by\nHi0;eff=\u00001\nmi0¶\u0000\nHi0+Hint\u0001\n¶Si0: (A4)\nIn the inset of Fig. 6, we show the calculated layer-resolved\nmagnetization within the spin chain model with various Wedge,\nafter positively saturating all spins and then applying a corre-\nsponding negative field prior to magnetization reversal in the\ncore. We number the atomic plane from the center of the core\nto the edge, and the vertical dashed line in the inset of Fig. 6\ndenotes the core/edge interface. Increasing Wedgegives rise to\nincreasing penetration of the domain wall into the core. From\nthe energy contributed to the reversal, we estimate a normal-\nized effective value of the anisotropy constant, Knorm\neff, by in-\ntegrating the anisotropy energy over the domain-wall width\nfrom the edge to the core in the nucleated region (see the blue\nregion in the inset of Fig. 6 for Wedge=2 nm) and then nor-\nmalizing to Kcore. This quantifies the reduction in the energy\nbarrier due to the exchange spring. Figure 6 illustrates the\nvariation of Knorm\neffwith Wedge. We observe a linear decrease in\nKnorm\neffwith the increase in Wedge, and we further describe the\nlinear decrease as (gray dashed line in Fig. 6)\nKnorm\neff=1\u0000bWedge; (A5)\nwhere b=0:324 (nm\u00001) obtained from fitting. 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Hirjibehedin1,2,7,* 1London Centre for Nanotechnology, University College London (UCL), London WC1H 0AH, UK. 2Department of Physics & Astronomy, UCL, London WC1E 6BT, UK. 3International Iberian Nanotechnology Laboratory (INL), 4715-330 Braga, Portugal. 4Instituto de Nanociencia de Aragón (INA) and Laboratorio de Microscopías Avanzadas (LMA), Universidad de Zaragoza, 50018 Zaragoza, Spain. 5Departamento de Física de la Materia Condensada, Universidad de Zaragoza, 50009 Zaragoza, Spain. 6Max-Planck-Institut für Mikrostrukturphysik, 06120 Halle, Germany. 7Department of Chemistry, UCL, London WC1H 0AJ, UK. *These authors contributed equally to this work. †present address: Department of Physics, Stanford University, Stanford, CA 94305, USA.  Final version published in Nature Nanotechnology 9, 64 (2014); DOI: 10.1038/nnano.2013.264  The properties of quantum systems interacting with their environment, commonly called open quantum systems, can be strongly affected by this interaction. While this can lead to unwanted consequences, such as causing decoherence in qubits used for quantum computation1, it can also be exploited as a probe of the environment. For example, magnetic resonance imaging is based on the dependence of the spin relaxation times of protons2 in water molecules in a host’s tissue3. Here we show that the excitation energy of a single spin, which is determined by magnetocrystalline anisotropy and controls its stability and suitability for use in magnetic data storage devices4,  can be modified by varying the exchange coupling of the spin to a nearby conductive electrode. Using scanning tunnelling microscopy and spectroscopy, we observe variations up to a factor of two of the spin excitation energies of individual atoms as the strength of the spin’s coupling to the surrounding electronic bath changes. These observations, combined with calculations, show that exchange coupling can strongly modify the magnetic anisotropy.  This system is thus one of the few open quantum systems in which the energy levels, and not just the excited-state lifetimes, can be controllably  2 renormalized. Furthermore, we demonstrate that the magnetocrystalline anisotropy, a property normally determined by the local structure around a spin, can be electronically tuned. These effects may play a significant role in the development of spintronic devices5 in which an individual magnetic atom or molecule is coupled to conducting leads.  In quantum mechanical systems, whenever coupling to the environment induces changes in the lifetimes of states it must also induce a shift (often referred to as “dressing” or renormalization) of the energy levels of the system6. Measuring the shifts, as opposed to the lifetimes, is difficult because it is often not straightforward to extract the bare energy from the dressed value obtained from spectroscopic techniques. Furthermore, the effect of the environment can go far beyond the renormalization of the energy levels. This occurs for instance in Kondo systems7, where a localized spin is exchange coupled to a bath of itinerant electrons, screening the localized spin through the formation of a total spin singlet state together with the itinerant electrons. The structure of the environment also influences open quantum systems. One very important and technologically relevant example of this is magnetic anisotropy. The push to increase data storage capacities to the ultimate limit8 has driven research into understanding magnetic anisotropy at the atomic scale9-14. Tuning magnetic anisotropy normally can be done via structural10,13 or mechanical15 means, though electrical control of anisotropy through the addition and subtraction of discrete units of charge on a molecule has been observed16. The interplay between magnetic anisotropy and Kondo screening at the atomic and molecular scale has also recently received theoretical and experimental attention15,17-19. In our experiments (Methods Summary), Co atoms are deposited on a thin-decoupling layer of copper nitride (Cu2N) created on Cu(001). Cu2N reduces the coupling of magnetic atoms with the underlying metallic substrate11,18. As seen in the scanning tunnelling microscopy (STM) image shown in Fig. 1a, the Cu2N islands used here are significantly larger than those used in some prior experiments11,18. Scanning tunnelling spectroscopy (STS) measurements performed on four representative atoms on this island are shown in Fig. 1b; note that the atoms have negligible  3 differences in their topographic appearance at the voltage at which they were imaged. In these spectra, two distinct features are seen: a peak in the local density of states (LDOS) centred at zero bias and two steps in differential conductance that occur symmetrically at positive and negative voltages. In prior experiments18, the zero-bias peak was found to be a Kondo resonance while the differential conductance steps were inelastic electron tunnelling (IET) transitions at spin excitation energies described by the spin Hamiltonian4 H=gµB!B!!S+DSz2+ESx2\"Sy2()\t\r (1)\t\r where µB is the Bohr magneton, g is the Landé g-factor, B is the magnetic field, D and E are the axial and transverse anisotropy constants, S=3/2 is the total spin, and Sx,y,z are the projections of the spin along the appropriate axes. The most striking result of this work is that, as observed in Fig. 1b, the spectra of the different Co atoms on the Cu2N change dramatically even though the atoms are simply at different positions on the same surface, with no observed changes in the local binding. At the edge of a large (18.6×20.5 nm2) Cu2N island, the STS spectrum of Co closely resembles prior measurements for Co on small (5×5 nm2) Cu2N islands18. However, as the atom’s position shifts towards the centre, two striking changes occur: the relative height of the Kondo peak decreases, and the IET step shifts to significantly higher energy. Because the IET step is a measure of the magnetic anisotropy energy, this suggests that the anisotropy energy is increasing as the Kondo screening is decreasing. A first candidate to account for the observed variations in the magnetocrystalline anisotropy would be a change in the structure of the Cu2N. Both magnetic anisotropy and exchange coupling arise from the overlap of the orbitals of the Co atom with those of the atoms in the surface, primarily the neighbouring nitrogen atoms11. For small crystal deformations, the relative change of the magnetocrystalline anisotropy should be proportional to the strain, with the constant of proportionality of order unity in the elastic regime4,20. Atomically resolved images of the Cu2N islands reveal no change in the lattice constant (with an uncertainty of a few pm), limiting the  4 maximum strain to be a few percent. Measurements of the Cu2N bandgap onset, whose change should be very sensitive to small changes in the lattice constant21, further restrict the maximum size of the strain: as seen in Fig. 1c, we find that the bandgap shifts by a few percent at the centre of the island compared to the edges, suggesting substantially smaller changes in the lattice. Therefore, while structural changes of the Cu2N may account for some portion (less than 1%) of our observed energy shift, they cannot account for the majority of the effect. Detailed and technically demanding calculations of large Cu2N islands may provide a more precise measure of this contribution.  Having ruled out a structural origin of the joint variations of Kondo resonance and the effective magnetic anisotropy, we explore a new physical scenario where changes in the exchange coupling between the Co spin and the conduction electrons, which lead naturally to a change of the Kondo temperature, also affect the spin excitation energies. We do so using both the Kondo and Anderson models, generalized to include single ion magnetic anisotropy (Supplementary Discussion). In the Kondo model, the dimensionless constant ρJ, the product of the density of states of substrate electrons at the Fermi energy ρ and the exchange energy J, controls the influence of the conduction electrons: the spin susceptibility is renormalized to linear order in ρJ while the local spin relaxation rate is proportional to (ρJ)2 22-24. In nuclear magnetic resonance, these phenomena are the well-known Knight shift and Korringa spin relaxation, respectively25. The environmentally induced decay rate necessarily comes together with a renormalization of the associated transition energy6. For the Kondo model with single ion anisotropy, second order perturbation theory yields the following expression for the renormalized excitation energy (Supplementary Discussion):  !=!01\"316!J()2ln2W!kBT#$%&'(\t\r (2)\t\r where Δ0 is the bare excitation energy, corresponding to spin excitations between the levels described by equation (1); kB is Boltzmann’s constant; T is the temperature; and W is the bandwidth of the substrate electrons. The second term in this equation is an exchange-driven shift of the spin  5 excitation energies and is formally similar to the normally overlooked second order contribution to the Knight shift26. Qualitatively equation (2) accounts for our central observation: as ρJ decreases, the Kondo temperature7 TK kBTK!W!Je\"1/!J\t\r (3)\t\r goes down while at the same time the spin excitation energy goes up. Whereas in most systems environmentally induced shifts can not be quantified because it is not possible to determine the bare energy Δ0, here the correlated variations of the Kondo temperature and the excitation energy reveal the significant renormalization of the single ion magnetic anisotropy by Kondo exchange.  Equation (2) is based on a perturbative calculation and, as such, cannot reproduce the full Kondo phenomenology. To overcome this limitation, obtain further evidence for the above scenario, and have a more microscopic understanding of the origin of the variation of ρJ, we have carried out non-perturbative calculations, based upon a multi-orbital Anderson model with three local orbitals holding an anisotropic spin-3/2 (Supplementary Discussion). This model is defined by three parameters: the one-electron energies of the local orbitals Ed, the effective Coulomb repulsion between electrons U, and the single-particle broadening Γ due to tunnelling between the local orbitals and the substrate. Ed and U determine the electron removal and addition energies E0 and U-E0 (Supplementary Discussion); therefore E0, U-E0, and Γ are the relevant energy scales that govern the Kondo physics. We solve the generalized Anderson model with the One Crossing Approximation (OCA)27 and obtain the spectral function, which can be related to the experimentally measured STS spectra28. The observed symmetry upon bias inversion of the experimental dI/dV curves is best reproduced when we consider the electron-hole symmetric case (E0=U/2), as illustrated in Fig. 2; however, our results are robust and also occur in the absence of electron-hole symmetry. In the symmetric limit, the relation between the Anderson and Kondo models leads to a particularly  6 simple linear relationship7: ρJ = 8 Γ/U. Thus, a change in ρJ can arise in general from variations of Γ, U, or Ed.  Figures 1d and 2 highlight the results of our OCA calculations. Increasing Γ, keeping U constant, the charge addition peaks at high energy broaden and shift (Fig. 2c). Moreover, in an energy window of ~10 meV around the Fermi energy, two relevant features are found, in agreement with our experimental observations: a Kondo resonance at the Fermi energy and a step a few meV above and below. Our OCA calculations show that the Kondo peak grows as Γ increases, while at the same time the spin excitation step shifts to lower energy (Supplementary Discussion), in agreement with the perturbative theory. As illustrated in Fig. 2d, this general behaviour in our OCA calculations is not sensitive to the specific choice of D or E. Importantly, the non-perturbative results show that the shift also changes linearly with (Γ/U)2, in qualitative agreement with equation (2). A similar shift of the singlet to triplet excitation energy has been recently obtained from an Anderson model of two exchange coupled spin ½ sites treated in the Non Crossing Approximation29. Renormalization of the magnetic anisotropy can arise in a variety of different scenarios where Γ, U, and Ed change at different locations on the surface. Here, we believe that variations in Γ are the most likely cause of the observed changes of the dI/dV. As seen in Fig. 1c, the gap of Cu2N increases by about 0.1 V as we move from the island edge to the island centre. A larger gap implies a higher tunnelling barrier, leading to a smaller Γ and therefore a smaller ρJ. However, a comparison with results obtained on islands with different sizes, showing that large islands present a variation of the magnetic anisotropy far from the edges despite the apparent constant gap, suggests that the situation may be more complex (Supplementary Discussion). For example, surface states confined under the Cu2N may play a role30. In addition, variations in U and Ed, which have been correlated with substantial changes in Kondo screening for Co on Cu(100)31, may also drive variations in exchange coupling. However, our calculations suggest that these parameters must change by more than 1 eV to account for a significant fraction of the observed shifts.  7 The magnetic field behaviour of the STS also changes as the position of a Co atom varies on the large Cu2N island. As seen in Fig. 3a,b,d,e, the field dependence of the IET step for a Co atom near the edge of the island is well-described by equation (1) with a large D term and E~0, consistent with results obtained at the centre of small Cu2N islands18. However, as seen in Fig. 3c,f, the IET step of a Co atom near the centre of the large Cu2N island can only be properly described by including a large E term. Excellent qualitative agreement between the spectral functions calculated using the OCA for the Anderson model (Fig. 1d) and the experimental spectra (Fig. 1b) are obtained using the values of D and E obtained in Fig. 3f. We note that the Co atom’s environment becomes more isotropic as Γ increases. More precisely, for systems with both axial and transverse anisotropy, all three axes are different4. As Γ increases, exchange will dominate the smaller transverse term, leaving the system with just a smaller axial anisotropy; eventually, for large Γ the system will effectively become isotropic. The first stage of this is precisely what is observed experimentally in Fig. 3. Exchange driven renormalization of magnetic anisotropy should be present in any system in which a magnetic impurity is coupled to an electronic bath, even if no Kondo screening occurs, but normally cannot be observed directly because either the unscreened spin excitations cannot be determined directly or the coupling cannot be controllably varied. Understanding this phenomenon is therefore crucial for future engineering of nanoscale quantum spintronic systems, which often involves placing an atomic or molecular spin in contact with an electronic reservoir5. Magnetic atoms on large Cu2N islands are therefore a special physical system with which we can observe and thereby understand the quantum mechanical “dressing” and “undressing” of a spin. This renormalization also provides an electronically tunable mechanism for controlling the magnetic anisotropy experienced by a quantum spin, which could have significant ramifications for the design and control of magnetic bits at the atomic and molecular scale. Not only does this mechanism enable control of the magnitude of the magnetic anisotropy, but it also can be used to  8 tune the relative strengths of the axial and transverse terms, which can be used to enhance or weaken various charge and spin tunnelling phenomena4,19.   9 Methods Summary The majority of the STM experiments were performed using an Omicron Cryogenic STM operating in ultrahigh-vacuum (chamber pressures below 5×10-10 mbar) at an effective sample temperature of 2.5 K. Superconducting magnets can apply fields of up to 6 T perpendicular to the surface of the sample or up to 2 T perpendicular to the surface of the sample plus up to 1 T in the plane. Additional STM experiments were performed using a SPECS JT-STM, a commercial adaptation of the design described by L. Zhang et al.32, operating in ultrahigh-vacuum with similar chamber pressures and at a base temperature of 5 K. Cu(001) samples (MaTeck single crystal with 99.999% purity) were prepared by repeated cycles of sputtering and annealing with Ar and annealing to 500°C. Cu2N was prepared on top of clean Cu(001) samples by sputtering with N2 and annealing to 350°C. The sample was held below 30 K while Co atoms were evaporated onto the surface. The bias voltage V is always quoted in sample bias convention. Topographic images were obtained in the constant current imaging mode with V and tunnel current I set to V0 and I0 respectively and processed using WSxM33. Differential conductance measurements were obtained using a lock-in amplifier, with AC modulation voltages of 100 µV at approximately 750 Hz added to V; spectra were acquired by initially setting V = V0 and I = I0, holding the tip at a fixed position above the surface, and then sweeping V while recording I and dI/dV. Differential conductance spectra shown in Figs 1b-c and 3a-c and Supplementary Figs 3b-c and 4d taken at zero perpendicular field B⊥ were acquired with an in-plane 1 T field to reduce vibrational noise; no noticeable change in the spectral features was observed compared to B=0.      10 References 1. DiVincenzo, D. P. Quantum Computation. Science 270, 255-261 (1995). 2. Damadian, R. Tumor Detection by Nuclear Magnetic Resonance. Science 171, 1151-1153 (1971). 3. Pykett, I. L. et al. Principles of Nuclear Magnetic Resonance Imaging. Radiology 143, 157-168 (1982). 4. Gatteschi, D., Sessoli, R. & Villain, J. Molecular Nanomagnets.  (Oxford University Press, Oxford, 2006). 5. 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Revealing magnetic interactions from single-atom magnetization curves. Science 320, 82-86 (2008). 13. Rusponi, S. et al. The remarkable difference between surface and step atoms in the magnetic anisotropy of two-dimensional nanostructures. Nature Mater. 2, 546-551 (2003). 14. Tsukahara, N. et al. Adsorption-Induced Switching of Magnetic Anisotropy in a Single Iron(II) Phthalocyanine Molecule on an Oxidized Cu(110) Surface. Phys. Rev. Lett. 102, 167203 (2009). 15. Parks, J. J. et al. Mechanical control of spin states in spin-1 molecules and the underscreened Kondo effect. Science 328, 1370-1373 (2010). 16. Zyazin, A. S. et al. Electric field controlled magnetic anisotropy in a single molecule. Nano Lett. 10, 3307-3311 (2010). 17. Höck, M. & Schnack, J. Numerical Renormalization Group calculations of the magnetization of isotropic and anisotropic Kondo impurities. Phys. Rev. B 87, 184408 (2013). 18. Otte, A. F. et al. The role of magnetic anisotropy in the Kondo effect. Nature Phys. 4, 847-850 (2008). 19. Romeike, C., Wegewijs, M., Hofstetter, W. & Schoeller, H. Quantum-Tunneling-Induced Kondo Effect in Single Molecular Magnets. Phys. Rev. Lett. 96, 196601 (2006). 20. Abragam, A. & Bleaney, B. Electron Paramagnetic Resonance of Transition Ions.  (Oxford University Press, Oxford). 21. Ashcroft, N. W. & Mermin, N. D. Solid State Physics.  (Holt, Rinehart, and Winston, New York, 1976). 22. Delgado, F., Palacios, J. J. & Fernández-Rossier, J. Spin-Transfer Torque on a Single Magnetic Adatom. Phys. Rev. Lett. 104, 026601 (2010). 23. Khajetoorians, A. A. et al. Current-Driven Spin Dynamics of Artificially Constructed Quantum Magnets. Science 339, 55-59 (2013). 24. Loth, S. et al. Controlling the state of quantum spins with electric currents. Nature Phys. 6, 340-344 (2010). 25. Slichter, C. P. Principles of Magnetic Resonance.  (Springer, Berlin, 1978).  11 26. Scalapino, D. Curie Law for Anderson's Model of a Dilute Alloy. Phys. Rev. Lett. 16, 937-939 (1966). 27. Haule, K., Kirchner, S., Kroha, J. & Wölfle, P. Anderson impurity model at finite Coulomb interaction U: Generalized noncrossing approximation. Phys. Rev. B 64, 155111 (2001). 28. Maltseva, M., Dzero, M. & Coleman, P. Electron Cotunneling into a Kondo Lattice. Phys. Rev. Lett. 103, 206402 (2009). 29. Korytár, R., Lorente, N. & Gauyacq, J.-P. Many-body effects in magnetic inelastic electron tunneling spectroscopy. Phys. Rev. B 85, 125434 (2012). 30. Ruggiero, C. D. et al. Emergence of surface states in nanoscale Cu2N islands. Phys. Rev. B 83, 245430 (2011). 31. Vitali, L. et al. Kondo Effect in Single Atom Contacts: The Importance of the Atomic Geometry. Phys. Rev. Lett. 101, 216802 (2008). 32. Zhang, L., Miyamachi, T., Tomanic, T., Dehm, R. & Wulfhekel, W. A compact sub-Kelvin ultrahigh vacuum scanning tunneling microscope with high energy resolution and high stability. Rev. Sci. Instrum. 82, 103702 (2011). 33. Horcas, I. et al. WSXM: a software for scanning probe microscopy and a tool for nanotechnology. Rev. Sci. Instrum. 78, 013705 (2007).   Supplementary Information is linked to the online version of the paper at www.nature.com/nature.  Acknowledgments We acknowledge Benjamin E.M. Bryant, Andrew J. Fisher, Katharina J. Franke, Andreas J. Heinrich, Mark Hybertsen, Sebastian Loth, and Alexander F. Otte for discussions and Benjamin E.M. Bryant for support during the experiments. J.F.R. acknowledges the hospitality of the Max-Planck-Institut für Mikrostrukturphysik Halle. Also, J.F.R. is on leave from Departamento de Física Aplicada, Universidad de Alicante, Spain. This work was supported by the EPSRC (EP/D063604/1 and EP/H002022/1); MEC-Spain (FIS2010-21883-C02-01, MAT2010-19236, CONSOLIDER CSD2007-0010, and Programa de Movilidad Postdoctoral); European Commission FP7 programme (PER-GA-2009-251791); and GV grant Prometeo 2012-11.  Author Contributions J.C.O, M.R.C., and C.F.H. conceived of the experiments. J.C.O. and M.R.C. performed the primary experiments and the data analysis supervised by C.F.H. Additional experiments were performed by J.C.O with the assistance of M.M. and supervised by D.S. and  12 C.F.H. F.D. performed the perturbative calculations of exchange-induced modification of magnetic anisotropy, as proposed by J.F.R. D.J. implemented and performed the Anderson model calculations in the one-crossing approximation as proposed by J.F.R. All authors discussed the results and participated in writing the manuscript.  Author Information Reprints and permissions information is available at www.nature.com/reprints. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to C.F.H. (c.hirjibehedin@ucl.ac.uk).     13  Figure 1 | Spectroscopy of Co on a large (18.6×20.5 nm2) Cu2N island. a, Topographic STM image (setpoint voltage V0=100 mV, setpoint current I0=100 pA) of Co atoms on a Cu2N island. Coloured arcs label the atoms and crosses indicate the location of spectra acquired over nearby bare Cu2N. b, Low-bias dI/dV (V0=15 mV, I0=1 nA) spectroscopy acquired at perpendicular magnetic field B⊥=0 on top of four atoms labelled in panel a; spectra are offset vertically for clarity. c, High-bias differential conductance spectra (V0=1.5 V, I0=50 pA) acquired at B⊥=0 near atoms at locations labelled with a cross in panel a; spectra are offset vertically for clarity. d, Spectral function Ad(ω) obtained from the Anderson model calculations (D=3.5 meV, E=2 meV, T~2.3 K, U=4 eV) with Γ=20 meV (red), 50 meV (green), and 90 meV (black). For consistency with the STM spectra in panel b, Ad(ω) is normalized such that the integrated weight up to 15 mV is constant; spectra are vertically offset for clarity. -15-10-50510150.000.050.100.150.20\n-15-10-50510150000000000\n1.61.82.02.22.42.60246810\n  dI/dV (µS)\nVoltage (mV)\n  Normalized Ad(!) (a.u.)\n! (meV)  dI/dV (nS)\nVoltage (V)-15-10-50510150.000.050.100.150.20\n-15-10-50510150000000000\n1.61.82.02.22.42.60246810\n  dI/dV (µS)\nVoltage (mV)\n  Normalized Ad(!) (a.u.)\n! (meV)  dI/dV (nS)\nVoltage (V)-15-10-50510150.000.050.100.150.20\n-15-10-50510150000000000\n1.61.82.02.22.42.60246810\n  dI/dV (µS)\nVoltage (mV)\n  Normalized Ad(!) (a.u.)\n! (meV)  dI/dV (nS)\nVoltage (V)d b \nc \n!\"!\"!\"\n#$#\"%\"&$'\"%\"!\"!\"#$\"a  14  Figure 2 | Generalized Anderson model of the Co electrons coupled to a bath of conduction electrons. a, Scheme of the many body energy levels for the three charge states of Co. b, Scheme of the S=3/2 multiplet split by the magnetic anisotropy. The lowest (shaded) states form an effective two-level Kondo system. c, Scheme of the generalized Anderson model for small Γ (left panel) and large Γ (right panel), showing the addition energies and the spectral function Ad(ω); the gray shaded area represent the Fermi sea of conduction electrons. Fine structure around the Fermi energy EF is shown in the middle sections, with the blue vertical arrows labelling steps in the spectral function corresponding to spin excitations. d, Spin excitation energies obtained from the OCA for various anisotropy values (blue: D=3 meV, E=0 meV; red: 3 meV, 2 meV; green: 4 meV, 0 meV; black: 4 meV, 2 meV) and values of (Γ/U)2.  c a d \n!!\n\"##$!\n\"#!!\n!\"%!%#$\"%%$\"&&$\"&&$\"&&'()\"\n\"#\n'%(*\"b &&'()\"\n+,-./\")!(*\"\n01,22\")!(3&*\"%(*\"'%(*\"\n01,22\")!+,-./\")!\n\"#0.00000.00020.00040.00060246810\n  ! (meV)(\"/U)2 15  Figure 3 | Magnetic field dependence of differential conductance spectra of Co on 18.6×20.5 nm2 Cu2N island. a-c, Low bias differential conductance (dI/dV) spectra acquired at B⊥=6 T (top), 4 T (middle), and 0 T (bottom) over atoms corresponding to those with similar colour labels in Fig. 1 (V0=15 mV, I0=1 nA); spectra are offset vertically for clarity and dashed vertical lines are a guide to the eye highlighting the change in energy of the IET step. d-f, IET step energy vs. perpendicular magnetic field. Solid dark blue line illustrates the evolution of equation (1) with S=3/2, g=2, E=0, and D=2.5 meV, 3.3 meV, and 5.0 meV (assigned based on the excitation energy at B⊥=0) respectively; solid light blue line is for D=3.5 meV and E=2.0 meV, obtained from a fit of all the data points in panel f.  -15-10-5051015-15-10-5051015\n0123456784681012\n012345678012345678-15-10-50510150.000.050.100.150.20\n  dI/dV (µS)\nVoltage (mV)\n   \nVoltage (mV)\n  \nVoltage (mV)  Energy (meV)\nMagnetic Field B! (T)\n  \nMagnetic Field B! (T)\n   Magnetic Field B! (T)b a c e d f "    },    {        "title": "1407.3605v2.Magnetic___dielectric_anomalies_and_magnetodielectric___magnetoelectric_effects_in_Z__and_W_type_hexaferrites.pdf",        "content": "1 Magnetic / dielectric anomalies  and m agnetodielectric / magnetoelectric effects  \nin Z- and W -type hexaferrite s \nJ. Li,1 H.-F. Zhang,1 G.-Q. Shao,1,a) W. Cai,1 D. Chen,1 G.-G. Zhao,1 B.-L. Wu,2 and S. -X. Ouyang3 \n \n1State Key Laboratory of Advanced Technology f or Materials Synthesis and Processing, Wuhan University of \nTechnology, Wuhan, 430070, China  \n2Key Laboratory of New Processing Technology for Nonferrous Metals and Materials, Guilin University of \nTechnology, Guilin, 541004, China  \n3China Building Materials A cademy, Beijing, 100024, China  \n \nTwo kinds of specimens, with the major phase of Sr3Co2Fe24O41 (Sr 3Co2Z) and SrCo 2Fe16O27 (SrCo 2W) \nhexaferrite s respectively, were fabricated through solid -state reaction. The phase composition , magneti c and \ndielectric properties, magnetodielectric (MD)  effect , magnetoelectric (ME) effect and pyroelectric properties  \nwere studied. Results show  that magnetic and dielectric anomalies  are induce d by the magnetocrystalline \nanisotropy  (MCA)  transition . They ca n be considered as characteristic  properties (e.g., Sr3Co2Z at 370 K ) but \nare not a sufficient conditio n for the MD and ME  coupling.  The T-block structure , exist ing in Sr3Co2Z but \nabsent in SrCo 2W, results in the dielectric response with ferroelectric (FE) and magnetic contributions . \nI. INTRODUCTION  \nSr3Co2Z, belongs to Z -type hexaferrite,  shows ME (magnetically induced ferroelectricity ) and MD \n(magnetically  induced dielectric constant  change)  effect s at a low magnetic field ( H) and room temperature \n(RT) .1-3 It has attracted much attention due to the potential technological applications.  The hexaferrites are \nclassified into six types (M, Y, Z, W, X and U) according t o different stacking sequences of three fundamental  \nblocks  (S-, R- and T -block s) in their crystal structure .1,2,4,5 The Z-type hexaferrite is formed according the \nRSTSR*S*T*S* sequence (the asterisk indicates those 180  rotated around c-axis) and W-type according t he \nRSSR*S*S* sequence. Investigations on Sr 3Co2Z have been done in  some aspects.1,3,6-10 However, the early \nstudies on their dielectric propert ies were mainly focused on high frequencies (>  10 MHz) at RT for application s \nin microwave field .11-14 Only two papers involved ε’(T) (temperature dependence of dielectricity ) at low \nfrequencies and  / or in a magnetic field.1,3 Beside s, all the hexaferrites with MD / ME  coupling effect s are found \nto be on the M –Y line in (Ba, Sr)O –Fe2O3–MeO ternary  diagram  (Y-, Z- and U -type, see Fig. 1 , Me denotes the \ndivalent metal ion ).15,16 Due to the synthesis difficulties,1,6,17,18 the reported samples often co -exist with \n                                                        \na)Author to whom correspondence should be addressed. Electronic mail: gqshao@whut.edu.cn.  2 impurities on the M –S line (W - and / or X -type, Fig. 1 ).15,16 So it is a meaningful work to investigate the \nquantity -known samples which contain simultaneously the compounds on both lines.  Target materials in this \nwork are specimens  with the major phase of Z- and W -type hexaferrite s, respectively . The phase composition , \nmagnetic and dielectric properties, MD effect , ME effect and pyroelectric prope rties were studied. Correlation \nbetween magnetic and dielectric anomalies was investigated considering their crysta l and magnetic structures.  \n \nFIG. 1. The  M–Y and M –S lines in the (Ba, Sr)O –Fe2O3–MeO ternary  diagram.  \n \nII. EXPERIMENT  \nSpecimens of ZM and W M, with the major phase of Sr 3Co2Z and SrCo 2W respectively,  were intentionally \nfabricated through a conventional solid -state reaction.19 The stoichiometric mixtures of SrCO 3 (99.5+ wt.%) , \nCo2O3 (99.5+ wt.%)  and Fe 2O3 (99.5+ wt.%)  were weighed, ground and calcined in air at 1250 C for 4 h. Then \nthe calcined powders were p ulverized, ground, pressed into pellets (Φ12  ×  3 mm ), and sintered to ceramics in \noxygen at 1150 C for 8 h (Z M) and 1250 C for 8 h (W M), respectively.  Phase determination was carried out (5°  \n≤ 2θ ≤ 140 ° ) by X-ray diffraction (XRD) using a D8 Advance X -ray diffra ctometer (Bruker,  Germany) , with a \nCu Kα radiation  (λ = 1.54184 Å, 40 kV, 40 mA) at a scan rate of 0.02 ° /s . The magnetization  (M) was tested in \nthe field of 100 Oe ( temperature changing from top down ) by a  vibrating sample magnetometer (VSM) \n(Chan gchun Yingpu Magneto -Electric Corp., China) . Before the measurement of dielectricity , MD effect , ME \neffect and pyroelectric properties,  the Ag electrodes were pasted on the polished plates  (Φ10 × 1 mm) and fired \nin oxygen at 830  C for 10 min. The dielectr icity was determined by using a Precision Impedance Analyzer \n6500B (Wayne Kerr Electronics  Inc., Britain). The MD effect  was measured in the field of  104 Oe employing a \nhomemade device.  The magnetoelectric and pyroelectric properties were tested using a P hysical Properties \nMeasurement System (PPMS, Quantum Design , Inc. USA ) coupled with an electrometer (Keithley 6517B , \nKeithley Instruments, Inc., USA ). The magnetic field dependence of the electric polarization was obtained by \nmeasur ing the ME current. Befo re this measurement, a magnetic field of 3 × 104 Oe was pre-applied. Then an 3 electric field of 2.5 kV / cm  was applied perpendicular to the magnetic field.  Subsequently, the magnetic field \nwas set to 5  × 103 Oe. After these poling procedures, the electric field was removed . The M E current was then \nobtained while the magnetic field  was sweep ing from 5  × 103 to –3 × 104 Oe at a rate of 50 Oe / s.  Before the \nmeasurement of pyroelectric current, the sample was cooled  down  from RT to 200 K under a poling electri c \nfield of 625 V / cm. The pyroelectric current was then obtained during the heating process  (from 200 to 400 K)  \nwith a constant ramping rate of 2 K / min. For measuring the pyroelectric current under a magnetic field, the \nfield was applied during the poli ng and the subsequent processes. The temperature dependence of the electric \npolarization was obtained by integrating the pyroelectric current.  \nIII. RESULTS AND DISCUSSION  \nA. Phases determination  \nXRD patterns in Fig. 2 show that the major phase for Z M and WM is Sr 3Co2Z [PDF 19 -0097, S.G.: P63/mmc  \n(194)] and SrCo 2W [PDF 54 -0106, S.G.: P63/mmc  (194) ], respectively. CoFe 2O4 is the main minor phase. \nDifficult ies in synthes izing single phase of Z- / W-type hexaferrites  were also stated previously.1,6,17,18 The Z M \nand W M in this work represent the specimens  which contain 70 –90 wt.% Sr 3Co2Z phase and 70 –90 wt.% \nSrCo 2W phase, respectively. The subsequent results have high consistency and reproducibility within these \nphase contents.  \n \nFIG. 2. XRD patterns of Sr3Co2Z (ZM) and SrCo 2W (WM) ceramics. The inset shows the enlargement of XRD patterns from \n29°  to 38° .  \n \nB. Dielectric and magnetic properties  \nFig. 3 shows the temperature dependence of magnetization, dielectricity and resistivity ( M–T, ε’–T, tanδ –T \nand ρ–T), and frequency dependence of dielectricity ( ε’–f and tanδ–f at selected temperatures) for Z M and W M. \nStrong correlation was found between the magnetic and dielectric anomalies . 4 Below the Curie temperature ( Tc,FM–PM = 683 K , FM —ferrima gnetic , PM—paramagnetic ),5,18 Ba3Co2Fe24O41 \n(Ba 3Co2Z) and Ba 1.5Sr1.5Co2Fe24O41 go through major changes in the magnetocrystalline anisotropy  (MCA ) with \ntemperature  increasing : Cone 1 → Plane → Cone 2 → Uniaxial.15,18,20 However, there is  no plan ar MCA for  \nSr3Co2Z: Cone 1 [52.3–58 (θ), 300 K; 57 , 373 K ]18,20 → Cone 2 (50.5 –56, 460 –473 K)7,18 → Uniaxial (25 , \n523 K; 20, 573 K; 16 , 623 K ; 0, 566 –673 K)7,18 (θ is the angle between M and c-axis). The θ is less sensitive \nto temperature  and keeps at 52.3–58 in Cone 1. But it decrease s gradually to ~25  in Cone 2.  The st ructure \ncomplexity of the Sr3Co2Z mainly results from the Sr2+ whose radius is comparable to that of O2– [r(Sr2+) = 1.31 \nÅ (9), 1.44 Å (12); r(O2-) = 1.38 Å (4), the digits in brackets refer to coordination number ].21 The Sr2+ ions \nprefer the oxygen positions rather than the interst itial sites , while other metal ions [r(Fe3+) = 0.49 Å (4), 0.65 Å \n(6); r(Fe2+) = 0.63 Å (4), 0.78 Å (6); r(Co2+) = 0.58 Å (4), 0.75 Å (6)]21 are located in non -equivalent interstitial \nsites.  Sr3Co2Z contains Fe(Co) –O–Fe(Co) bonds in the T -block s. Small -moment layers (S) arise in T -block s \nwhile large one s (L) in other block s. Since Sr i ons are located near the two Fe(Co) sites, the substitution of small \nSr for large Ba [vs. Ba3Co2Z, r (Ba2+) = 1.47 Å (9); 1.61 Å (12)]21 increase s the Fe(Co) –O–Fe(Co) bond angle ( φ) \n(Ba3Co2Z: 116°; Sr3Co2Z: 123°) through  the L / S bound ary.5 The φ = 123° causes the magnetic frustration  (θ = \n~55.6) and  stabilizes a trans verse -conical (T-conical)  magnetic  structure  above RT .7 \nFor the ZM (Sr3Co2Z) specimen , the M–T curve exhibits two clear drops at around 370 K and 505 K . The later \nis similar with the report ed while the former is different.1,3,7,18 The anomaly at 370 K represents  the onset of \nestablishment of magnetic structure with the P63/mmc  symmetry,7 below this temperature Z M exhibits MD / ME  \neffect s.3,7 This means that ZM is in the ferroelectric (FE) side while not in the paraelectric  (PE) side below ~370 \nK.3 The anomaly at 505 K corresponds to a transition from the phase with a cone of easy magnetization into the \nphase where Fe and Co magnetic moments become  parallel to c-axis.1,3,7,18 Thus t he M–T curve  can be divided \ninto four regions . Region I , II, III and IV  cover the  Cone 1  (~55.6, < 370 K ), Cone 2 (~55.6 > θ > 25, \n370–505 K), Uniaxial ( 25 ≥ θ ≥ 0, 505–683 K) and PM phase (> 683 K), respectively . \nAs to the temperature dependence of dielectricity ( ε’—the real part of dielectric constant), a broad dielectric \nrelaxation -peak  ε’(T) appears at the border of Reg ion I and II (Fig. 3a), corresponding to the change of magnetic \nstructure . The tanδ(T)-peak at ~550 K (Fig. 3c) coincides with the change from a cone to a uniaxial anisotropy.  \nTemperature increasing causes ε’(f)-peaks shift towards higher frequencies (Fig.  3e). The tanδ(f) decreases with \nfrequency increasing in the whole temperature range (Fig. 3g). The high loss -factor of tanδ(T) can be attributed \nto the contributions from conductio n loss (108 Ωcm  at RT; 104 Ωcm  at 600 K) (Fig. 3c) and ion jump relaxation,  \nespecially at low frequencies.22 The obtained condu ction activation energy ( Ea (c)) extr acted fr om resist ivity (ρ) 5 is 0.72 eV , higher than  that of the Ba 3Co2Z single crystal (~0.1 eV) ,23 indicat ing the condu ction carriers in the \nZM are from the second ionization of oxygen vacancies.24 In the polycrystalline ceramic of Sr3Co2Z hexaferrite, \nmedium resistance (vs. Ba3Co2Z, low resistance ) grains are separated by highly resistive grain -boundaries . The \npreparation in oxygen can decrease Fe2+ concentration and then reduce the hopping  interchange of electrons \nbetween Fe2+ and Fe3+ ions. Thus the oxygen  sinter ing favors a high resistivit y.1 \nBelow the Curie temperature  (Tc,FM–PM = 763 K ; 728 K)5,25, BaCo 2Fe16O27 (BaCo 2W) goes through th e \nfollowing  major changes in MCA with temperature  increasing : Cone 1 (~70, 2–453 K; 68.5 –70, 300 K)5,25,26 \n→ Cone 2 (70  > θ > 0, 453 –553 K)25 → Uniaxial (0, 553 –763 K)25. For the WM specimen (Sr Co2W), the M–T \ncurve  exhibits one drop at ~470 K  (Fig. 3b). When temperature increa ses, the M decre ases steeply at 470–520 K. \nThus t he M–T curve  can also be divid ed into four regions . Region i, ii, iii and iv  cover the Cone 1  (< 470 K), \nCone 2 ( 470–520 K), Uniaxial ( 520–763 K) and PM phase  (> 763 K) , respectively. The difference is obvious \nbetween the tw o specimens. The θ decreases quickly from 70  to 0 within 100 K  (Region ii)25 and t hen keeps at \n0 till Tc,FM -PM (Region iii ) in the WM, but the θ decreases gradual ly from ~55.6  to 0 spanning the Region II \nand III in the  ZM. Compared with the ZM, there is a similar dielectric anomaly of ε’(T) in the W M at the border of \nRegion i and ii. Other characteristics for the WM are analogous with those of the ZM (Fig. 3d, 3f and 3h). \nThe variation  of θ as a function of temperature is summarized for Y-, Z- and W -type hexaferrites in Fig. \n4.4,7,15,18,20,25,27 Below the critical temperature, Y - and Z -type hexaferrites exhibit ME effect. It is amazing for us \nto find that there exists a golden ratio point (critical θ, 55.6 ≈ 0.618 ×  90 ) which is correlative to MD / ME  \neffects for Sr3Co2Z below 400 K (Fig. 4).4,7,15,18,20,25,27 Besides, for Ba 2Co2Y and Ba 3Co2Z, which exibit ME \neffect, their θs are also approaching to the golden ratio point ( 55.6) below critical temperature . There may be \nsome important physical mechanisms accounting for this phenomenon which need further investigation.  6 \n \nFIG. 3. Temperature dependence of magnetization, dielec tricity and resistivity, and frequency dependence of dielectricity for \nSr3Co2Z (ZM) and SrCo 2W (WM) specimens. (a) M–T and ε’–T for Z M (the inset is the enlargement at 300–550 K); (b) M–T \nand ε’–T for W M (ibid); (c) tanδ–T and ρ–T for Z M; (d) tanδ–T and ρ–T for W M; (e) ε’–f for Z M; (f) ε’–f for W M; (g) tanδ–f \nfor Z M; (h) tanδ–f for W M. \n \n \nFIG. 4. The  golden ratio line  (GRL)  of the magnetocrystalline anisotropy  (MCA) transition  in hexaferrites . \n \nSo the magnetic and dielectric anomalies are all associated wit h the MCA  transitions in both specimens (ZM: \n~370  K; WM: ~470 K). The  ZM (Sr 3Co2Z) exhibits MD / ME  effect s above RT  but W M (SrCo 2W) does not in \nany temperature range.4,5,7 That means the magnetic and dielectric anomalies  are characteristic  properties (e.g. 7 ZM at ~370 K ) but do not satisfy a sufficient conditio n for th e MD and ME  coupling.  \nC. M agnet odielectric properties  \nFig. 5 shows the magnetic -field dependence of dielectric ity, i.e., the MD effect  (measured at 100 kHz at RT). \nThe Sr 3Co2Z (ZM) has a smaller ε’ than that of the SrCo 2W (WM) in the whole magnetic fiel d (H = –104 Oe ~ 104 \nOe) (Fig. 5a), while the tanδ of the Z M is larger than that of the W M (Fig. 5b). For the W M, its ε’ and tanδ are not  \naffected by the H up to 104 Oe. For the Z M, the ε’ shows magnetic -field dependence. It decreases with H \nincreasing and is  field-direction dependent. When the H increases from 0 Oe to 104 Oe, the ε’ shows a maximum \n(~16.36) at 0 Oe and decreases gradually to 16.31 at 7 × 103 Oe, then becomes nearly constant above 7 × 103 Oe. \nThe tanδ is almost magnetic -field independent, simi lar with the previously reported.1 Accordingly, the \ndielec tric-constant -change ratio  \n)]0('/))0(')('()0('/'[   H  increases gradual ly with H increasing \nfrom 0 to ~ 7 × 103 Oe, and then becomes nearly constant ( –0.3%)  above 7 × 103 Oe (Fig. 5c). This ratio is \ncomparable with those obtained by Kitagawa et al. (–3%, 100 kHZ )1 and Zhang et al. (–4%, 50 MHz )3. It ha s \nbeen proposed that Δ ε’ is proportional to the square of M based on the framework of Ginzburg –Landau theory , \ni.e., Δε’ ∝ γM2, where γ is a coupling constant and is negative for Sr 3Co2Z.3,28 In the Z M of this work, the M \nincreases with H increasing, leading to the ε’ decreasing. The ε’ shows a maximum in the vicinity o f PE–FE \ntransition induced by H.29 When M is saturated at ~ 7 × 103 Oe, the ε’ becomes a constant.  The ZM presents a \nnegative MD effect  at RT, because it  is in the FE side.3 Therefore the MD effect in the Z M corresponds  to the \nfact that the PE–FE transition is induced by H and the ε’ decreas es with H. 8 \n \nFIG. 5. Magnetic -field dependence of dielectric constant  (a), loss factor (b) and dielectric -constant -change ratio (c) for \nSr3Co2Z (ZM) and SrCo 2W (WM) specimens (100 kHz; RT). The inset is magnetic phase diagram for Sr 3Co2Z with data from \nthe reference ( solid black dots)7 and this work (hollow dots).  \n \nD. M agnetoelectric properties  \nFig. 6 shows the magnetic field dependence of electric polarization and magnetoelectric current (in the inset)  \nfor Sr3Co2Z (Z M). The magnetoelectric current for Z M shows remarka ble dip and peak structure centered at \naround –500 and –8 × 103 Oe at 100, 200 and 300 K. The magnetic field dependence of the electric polarization, \ncalculated by integrating the megnetoelectric current, reveals the ME coupling. There is almost no spontan eous \npolarization at zero magnetic field . By applying a increasing magnetic field, the electric polarization increases \nrapidly and reaches a maximum at about –4 × 103 Oe. It decreases then and vanishes at around –1.3 × 104 Oe \nwhere the syste m becomes a sim ple ferrimagnet.  \nThe results are some similar with those  from Kitagawa et al .1 and Soda et al. .7 Differences are also in \nexistence . The electric polarization decreases with temperature increasing in Kitagawa et al .’s work . But the \nelectric polarization at 200 K is larger than that at 100 K  in this work . The same trend was observed by Soda et 9 al., in which the polarization  of Sr 3Co2Z at 200 K is larger than that at 10 K.7 Besides, the magnitude of electric \npolarization in this work is smaller than those reported previously.  The difference could be attributed to the \ndifferent synthes is process  of Sr 3Co2Z.7 \n \nFIG.6. Magnetic field dependence of the electric polarization and magnetoelectric current (in the inset) for Sr 3Co2Z (Z M) at \ndifferent temperatures.  \n \nE. Pyroelectric properties  \nFig. 7 shows the pyroelectric current and electric polarizatio n (in the inset) as a function of temperature  for \nSr3Co2Z (ZM) under different magnetic field . There is no pyroelectric current found below 300 K during 0 to 1 × \n104 Oe. With temperature increasing, t he pyroelectric current increases gradually, reaches its  maximum at ~380 \nK and then decreases.  The peak -temperature correspond s well to that where the magnetic and dielectric \nanomalies present.  The intensity of the pyroelectric current increases with magnetic field increasing, leading to \nan increased electric  polarization (the inset of Fig. 7).  This indicates that the electric polarization  is enhanced by \nthe electric field . \nThe pyroelectric current peak centered at 380 K implies an intrinsic and over -room -temperature \nferroelectric ity for Z M, even when the magnet ic field is zero . For comparison , the ferroelectricity of Sr 3Co2Z at \nzero magnetic field was claimed by Zhang et al.3 and Wu et al.10 but there was no direct evidence provided.  The \nmagnetic -field-enhanced elec tric-polarization has been reported in DyMnO 3 thin films  which is understood by \nthe Dy -Mn spin interaction .30 In the case of Sr 3Co2Z, its electric polarization ori ginates from the  T-conical  \nstructure through the inverse  Dzyaloshinskii -Moriya  (DM) mechanism. The spin structure of Sr 3Co2Z could be \nslightly modified by the proper magnetic field , result ing in the increase of electric polarization.  10 \n \nFIG. 7. Temperature dependence of pyroelectric current and electric polarization (in the inset) for Sr 3Co2Z (Z M) under \ndifferent magnetic field.  \n \nF. Discussion  \nBecause of its complex T-conical structure which evolves with temperature, the dielectric response in the \nSr3Co2Z (ZM) includes several contributions. Each of them is related to a particular microscopic mechanism and \nbecomes significant in a particular frequency (ω = 2πf) / temperature interval. The ε’ can be represented as:31 \n),( ),,( ),,( ),('),( ),,( ),,( ),( 1),,( 1),,('\nT MT MT TT MT MT T MT MT\nLF M FELF M FE\nii\n      \n       \n \n     (1) \nwhere ε’∞ ∝ ND (ND is dipole d ensity in material) is related to the polarization at high frequencies , χFE ∝ ω–2 ∝ \n(T–Tc,FE–PE)–1 coincides with the contribution of the FE order,31 ΔχM ∝ γM2 (γ < 0 ) is derived from  the \nframework of Ginzburg –Landau theory  as mentioned before, χLF is the low -frequency susceptibility (where the \nconductivity contribution χσ ∝ T / ω is concluded)32 due to the relaxation  of domain walls in polydomain \ncrystals, mobile charge carriers, and crystal defects, etc.31. \nBecause Sr3Co2Z (Z M) locates in its FE zone (Region I) below 370 K, χFE is dominant at low frequencies (400 \nHz - 1 kHz) (Fig. 3 and Fig. 5). A dielect ric anomaly appears at 370 K which coincides to the magnetic structure  \nchange. The dielectric anomalies become weaker with frequencies increasing (10 kHz  - 1 MHz) because the ω is \nmore sensitive  to the χFE than the  T or M. When a magnetic field (H) exists in Region I (below 370 K),  χM brings \na supplementary contribution to ε’ before M reaches saturation.  \nFor a Z -type hexaferrite, the room -temperature  MD effect is attribut ed to the change  of a T-conical spin \nstructure and sp in-phonon coupling.3 The room -temperature ME effect is understood in terms of the appearance \nof electric polarization  which is induced also by a T-conical spin structure through the inverse DM interaction.7 \nThe above T -conical spin structure  refers to the antiphase arrangement of the magnetic moments between \nneighboring T -blocks, which exis ts in Sr3Co2Z (ZM) but absent in SrCo 2W (WM).1,4 As mentioned before, the 11 Fe(Co) –O–Fe(Co) bond angle ( φ) in Sr3Co2Z arouses the magnetic frustration, stabilizes a noncollinear \nmagnetic  structure  above RT and contributes to the ME  performance.7 Thus the Sr3Co2Z has a ME -response \ndriven by phase competition  (i.e., spin-driven ferroelectrics).5 \nIV. CONCLUSIONS  \nIn summary, dielec tric anomalies near the magnetic phase transition temperatures are reported firstly  for \nZ-type hexaferrite Sr3Co2Fe24O41 (Sr3Co2Z) at ~370 K and SrCo 2Fe16O27 (SrCo 2W) at ~470 K in this work . \nCorrelation between the anomalies was investigated considering th e crystal and magnetic structures.  It is \nconcluded that magnetocrystalline anisotropy  (MCA)  transition  induce s the anomalies , which  are characteristic  \nproperties but do not satisfy a sufficient conditio n for the magnetodielectric  / magnetoelectric coupling . T-block \ncrystal structure  that exists in Sr3Co2Z but absent in SrCo 2W is proposed to contribute to the observed results . \nThe composition with full Sr substitution for Z -type hexaferrite s and high resistivity from oxygen sintering \ncause the magnetic frust ration  and stabilize a transverse -conical (T-conical)  magnetic  structure . \nAn over -room -temperature ferroelectric effect is conformed directly and reported firstly in Sr3Co2Z by the \npyroelectric current measurements, where the peak -temperature ( ~380 K ) is correspond ing well to that where \nthe magnetic and dielectric anomalies present. This work helps understanding the origin of MD / ME coupling  \neffect s at a low magnetic field  and above room temperature.  \nACKNOWLEDGEMENTS  \nThe authors gratefully acknowledge Prof . Z.-G. Sun  and Prof. J. -F. Wang in WUT, Prof. J. Xu in WIT, Prof. \nZ.-C. Xia, G. -F. Fan and Z. -M. Tian in HUST , and Dr. J. Lu in IPCAS. This work was supported by grants from \nState Key Laboratory of Advanced Technology for Materials Synthesis and Processin g (WUT, China ) (Grant \nNos. 2010 -PY-4, 2013 -KF-2, 2014 -KF-6), the Open Research Foundation of Key Laboratory of Nondestructive \nTesting (NHU, China ) (Grant No. Zd201329002), the National Natural Science Foundation of China (Grant No. \n51172049) and the Specia l Prophase Project on National Basic Research Program of China (Grant No. \n2012CB722804).  \nREFERENCES  \n1Y. Kitagawa, Y. Hiraoka, T. Honda, T. Ishikura, H. Nakamura, and T. Kimura, Nature Mater. 9 , 797 (2010).  \n2C. J. Fennie and D. G. Schlom, Nature Mater. 9 , 787 (2010).  \n3X. Zhang, Y. -G. Zhao, Y. -F. Cui, L. -D. Ye, J. -W. Wang, S. Zhang, H. -Y. Zhang, and M. -H. Zhu, Appl. Phys. \nLett. 100 , 032901 (2012).  \n4T. Kimura, Annu. Rev. Condes. Matter Phys. 3 , 93 (2012).  \n5R. C. Pullar, Prog. Mater. Sci. 57 , 1191 (2012).  \n6T. Kikuchi, T. Nakamura, T. Yamasaki, M. Nakanishi, T. Fujii, J. 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B 49 , 7868 (1994).  \n \n "    },    {        "title": "1407.5830v1.Out__versus_in_plane_magnetic_anisotropy_of_free_Fe_and_Co_nanocrystals__tight_binding_and_first_principles_studies.pdf",        "content": "Out- versus in-plane magnetic anisotropy of free Fe and Co nanocrystals: tight-binding and\nfirst-principles studies\nDongzhe Li,1Cyrille Barreteau,1, 2Martin R. Castell,3Fabien Silly,1, 3and Alexander Smogunov1,\u0003\n1CEA, IRAMIS, SPEC, CNRS URA 2464, F-91191 Gif-sur-Yvette Cedex, France\n2DTU NANOTECH, Technical University of Denmark, Ørsteds Plads 344, DK-2800 Kgs. Lyngby, Denmark\n3Department of Materials, University of Oxford, Parks Road, Oxford OX1 3PH, UK.\n(Dated: October 2, 2018)\nWe report tight-binding (TB) and Density Function Theory (DFT) calculations of magnetocrystalline\nanisotropy energy (MAE) of free Fe (body centerd cubic) and Co (face centered cubic) slabs and nanocrystals.\nThe nanocrystals are truncated square pyramids which can be obtained experimentally by deposition of metal\non a SrTiO 3(001) substrate. For both elements our local analysis shows that the total MAE of the nanocrystals is\nlargely dominated by the contribution of (001) facets. However, while the easy axis of Fe (001) is out-of-plane,\nit is in-plane for Co (001) . This has direct consequences on the magnetic reversal mechanism of the nanocrys-\ntals. Indeed, the very high uniaxial anisotropy of Fe nanocrystals makes them a much better potential candidate\nfor magnetic storage devices.\nPACS numbers: 71.15.-m, 75.10.Lp, 75.50.Ss, 75.70.-i, 75.75.Lf, 68.47.Jn\nI. Introduction\nLast decades, higher storage densities were achieved by\nreducing the magnetic grains down to nanoscale. However,\nthe magnetic stability of a nano-object decreases proportion-\nally to its size and the ultimate limit is reached when the\nthermal fluctuation overcomes the energy barrier to swicth\nthe global magnetization of the system. The most crucial\nissue in exploring ultimate density data storage (e.g., high-\ndensity magnetic recording1or spintronic devices) is mag-\nnetic anisotropy energy, which is defined as the change of total\nenergy associated to a change in the direction of magnetiza-\ntion. One of the challenge in this route towards high magnetic\ndensity storage is evidently to be able to synthetize well or-\ndered arrays of magnetic nanocrystals with as large magneti-\nzation and magneto-crystalline anisotropy as possible. The\nmagnetic anisotropy energy of magnetic nanocrystals (e.g.,\nFe, Co and Ni etc) is indeed a subject of intense study both\nexperimentally2–5and theoretically6–11but the ability to grow\nwell defined magnetic crytalline nanostructures is also a ma-\njor issue12–18. This is especially the case for Fe and Co nanos-\ntructures, that can adopt various crystalline bulk structures, in\nparticular the body-centered cubic (bcc) and face-centered cu-\nbic structure (fcc) structure in low dimensions15,16. The close-\npacked and lowest-energy facet for bcc structure is the (110)\nfacet whereas it is the (111) facet for the fcc structure. This is\nthe reason for the (110) facets appearing in bcc nanocrystals\n(in fact for Fe the surface energies of (001) and(110) orien-\ntations are almost the same) and the (111) facets appearing\nin fcc nanocrystals. The nanocrystal magnetic properties will\ntherefore not only depend on the bulk structure but also on the\nfacet structure and area.\nThe magnetic anisotropy contains two different parts: the\nfirst part is long-range magnetic dipole-dipole interaction\nwhich leads to so-called shape anisotropy, while the sec-\nond one is referred to magnetocrystalline anisotropy energy\n(MAE) originates from the spin-orbit coupling (SOC)19. The\nlatter effect is a quantum effect (of relativistic nature) thatbreaks the rotational invariance with respect to the spin quan-\ntization axis. Therefore, if SOC is included, the energy of the\nsystem depends on the orientation of the spin with respect to\nthe crystallographic axis.\nThe value of MAE per atom, is extremely small in bulk\n(some\u0016eV), but can get much larger in nanostructures20,21\n(some meV) due to reduced dimensionality. From the point of\nview of theory, there are two different methods which are used\nextensively in the literatures for MAE calculations: i) fully\nrelativistic self-consistent field (SCF) calculations, ii) Force\nTheorem (FT)22–24. Assessing the MAE for systems contain-\ning hundreds of atoms by the former approach is especially\nchallenging, since it requires a well-converged charge den-\nsity as well as a consuming computational time. In the latter\nmethod, the MAE is given by the band energy difference (in-\nstead of total energy difference) obtained after a one-step di-\nagonalization of the full Hamiltonian including SOC, starting\nfrom the self-consistent scalar relativistic density/potential.\nThis approach is not only computationally efficient but also\nnumerically very stable since the self-consistent effect with\nSOC could be ignored.\nThe objective of this paper is to investigate the MAE of\nFe and Co nanocrystals, that can be grown experimentally by\nepitaxy, using tight-binding (TB) as well as first principles\ncalculations in the Density Functional Theory (DFT) frame-\nwork. The nanocrystals adopt a truncated-pyramid shape on a\nreconstructed SrTiO 3(001) substrate but have however a dif-\nferent bulk structure (bcc and fcc). In our theoretical study a\nparticular emphasis is devoted to the local analysis of MAE.\nIn particular, it has been found that the main contribution to\nthe MAE comes from the basal (001) facet of pyramids. This\nresults in strong out-of-plane (in-plane) anisotropy for Fe (Co)\nnanocrystals, in agreement with the study of thick Fe(001) and\nCo(001) slabs.\nThe paper is organized as follows. In Sec. II we present\nthe experimental and theoretical methods used in this work.\nIn Sec. III, we first present Scanning Tunneling Microscope\n(STM) observation of Co nanocrystals on SrTiO 3(001) sub-\nstrate and illustrate the results of TB and DFT calculations forarXiv:1407.5830v1  [cond-mat.mtrl-sci]  22 Jul 20142\nCo(001) and Co(111) slabs. After that, the MAE of free Fe\nand Co nanocrystals will be discussed. Finally, the conclu-\nsions will be presented in Sec. IV.\nII. Methodolody\nIn the following sections we will first briefly present the\nmain ingredients of the experimental set-up to grow the cobalt\nnanocrystals on the SrTiO 3(001) substrate. Then in the sec-\nond part the theoretical model to calculate the MAE will be\npresented starting by the TB Hamiltonian and then the DFT\napproach. In the case of the DFT formalism we will essen-\ntially concentrate on the implementation of the Force Theo-\nrem which we have incorporated in the Quantum-ESPRESSO\n(QE)25package.\nA. Experimental\nwe use SrTiO 3(001) crystals doped with 0.5% (weight)\nNb. The crystals were epi-polished (001) and supplied by PI-\nKEM, Surrey, UK. We deposited Co from an e-beam evap-\norator (Oxford Applied Research EGN4) using 99.95% pure\nCo rods supplied by Goodfellow, UK. Our STM is manufac-\ntured by JEOL (JSTM 4500s) and operates in ultra high vac-\nuum (10\u00008Pa). We used etched W tips for imaging the sam-\nples at room temperature with a bias voltage applied to the\nsample. SrTiO 3(001)-c(4\u00022) was obtained after Ar+bom-\nbardment and annealing in UHV at 600\u000eCfor 2 hours. STM\nimages were processed and analyzed using the home made\nFabViewer application26.\nB. Theoretical\n1. Magnetic tight-binding model\nIn this section, we briefly describe our magnetic tight-\nbinding model (more details can be found in our previous\npublications27,28). The hamiltonian is written as follows:\nH=HTB+HLCN+HStoner +HSOC (1)\nWhereHTBis a standard ”non-magnetic” TB hamiltonian\nwhich form is very similar to the one introduced by Mehl\nand Papaconstantopoulos29,HLCNis the term ensuring a lo-\ncal charge neutrality, HStoner is the Stoner-like contribution\nthat controls the spin magnetization and HSOCcorresponds to\nspin-orbit coupling that operates on dorbitals only.\nThe total energy should be corrected by a double counting\nterm due to inter-electronic interactions introduced by localcharge neutrality and Stoner interaction, explicitly:\nEtot=Eb\u0000Edc=X\n\u000bf\u000b\u000f\u000b\u0000U\n2X\ni[N2\ni\u0000(N0\ni)2]+\n1\n4X\ni;\u0015I\u0015M2\ni\u0015;(2)\nwhereEb=P\n\u000bf\u000b\u000f\u000bis the band energy, f\u000bis the Fermi-\nDirac occupation of state \u000b,NiandMiare the charge and the\nspin moment of site i, respectively. N0\niis the valence charge,\nUis the parameter imposing the local charge neutrality, and\nI\u0015is the Stoner parameter of the orbital \u0015(\u0015=s;p;d ).\nAll the parameters of TB hamiltonian are fitted on bulk ab\ninitio data: bandstructure, total energy, magnetic moment etc.\nThe value of the Stoner parameter Idis taken equal to 0.88\nmeV for Fe and 1.10 meV for Co. The spin-orbit constant\n\u0018dis also determined by comparison with ab initio bandstruc-\nture and we found that 60 meV and 80 meV are very good\nestimates for Fe and Co, respectively.\nThe MAE, in a very good approximation, is calculated by\nusing the Force Theorem (FT)22–24: first, a self-consistent field\n(SCF) collinear calculation without SOC is done followed by\nthe rotation of the density matrix in the right spin direction;\nnext, a non-SCF non-collinear calculation with SOC is per-\nformed. The MAE is obtained as the difference of band ener-\ngies,E1\nb\u0000E2\nb, between two spin moment directions, 1 and\n2. The correct decomposition of total MAE over different\natomic sites ican be done within so-called ”grand canonical”\nformulation24:\nMAEi=EFZ\n(E\u0000EF)\u0001ni(E)dE (3)\nwhere \u0001ni(E) =n1\ni(E)\u0000n2\ni(E)is the change in the density\nof states at atom ifor different spin moment orientations. The\nFermi energy EFof SCF calculation without SOC must be\nsubtracted from all energies in order to suppress the trivial\ncontribution to the local MAE due to charge redistribution as\ndiscussed in Ref.24.\n2. Density Functional Theory (DFT) calculations\nWe perform ab initio DFT calculations using the plane-\nwave electronic structure package Quantum-ESPRESSO\n(QE).25The spin-orbit coupling (SOC), crucial for mag-\nnetocrystalline anisotropy, is taken into account via fully-\nrelativistic pseudo-potentials (FR-PPs)30, describing the inter-\naction of valence electrons with ions, which are in turn gener-\nated by solving atomic Dirac equations for each atomic type.\nWe have implemented the Force Theorem in QE in the same\ntwo-step way as described above for TB model: i) SCF calcu-\nlation with scalar-relativistic PPs (without SOC) is performed\nto obtain the charge density and the spin moment distribu-\ntions in real space; ii) spin moment is globally rotated to a\ncertain direction followed by a non-SCF calculation with FR-\nPPs (with SOC). The change of band energy between two spin3\nmoment directions gives, as above, the total MAE.\nThe total MAE is decomposed over different atoms iin the\nslightly different way:\nMAEi=E1\nFZ\n(E\u0000E2\nF)n1\ni(E)dE\u0000E2\nFZ\n(E\u0000E2\nF)n2\ni(E)dE;\n(4)\nwhere the Fermi level of one of magnetic configurations (we\nhave chosen the second one), E2\nF, is substracted under inte-\ngrals and exact Fermi levels for two configurations are used as\nthe limits of integration. This way we avoid the reference to\nelectronic levels of a system without SOC, since the PPs with\nand without SOC are not generally correlated and can produce\nan arbitrary shift of levels. Due to total charge conservation in\nthis ”canonical” approach, the sum of MAEiover all atoms\ngives exactly the total MAE while for the ”grand canonical”\nscheme, Eq. 3, it was, in principle, only approximate. The\ndescrepancy between ”grand canonical” and ”canonical” for-\nmulations within TB approach is, however, very tiny since the\neffect of SOC on the Fermi level is negligable in the case of\nFe or Co composed materials.\n[100][010]a\n(111)(100)\nb\n30 45 60 7503\nVolume (nm3)\n length / Height\nFIG. 1: (color online) (a) Co deposition onto a 350\u000eCSrTiO 3(001)-\nc(4\u00022) substrate followed by a 320\u000eC anneal gives rise to trun-\ncated pyramid shaped nanocrystals as shown in the STM image,\n80\u000280 nm2, V s= +1.0 V , I t= 0.1 nA. (b) height to length ratio\nconstant of Co nanocrystals.Since QE gives an access to real space wave-functions it is\nnatural to define also the space-resolved MAE as:\nMAE (r) =ZE1\nF\n(E\u0000E2\nF)n1(r;E)dE\u0000\nZE2\nF\n(E\u0000E2\nF)n2(r;E)dE;(5)\nwhere the LDOS is computed via electron wave-functions in\nthe usual way, n1;2(r;E) =P\n\u000bj\t1;2\n\u000b(r)j2\u000e(E\u0000\"1;2\n\u000b). Once\nagain, the integral of MAE(r) over all the space will give ex-\nactly the total MAE.\nIII. Results and discussion\nIn this section we will first briefly present the structural\ncharacterization of supported Co nanocrystals using STM. We\nthen present the results of our calculations on slabs of fcc Co\nwith orientations (001) and(111) corresponding to the facets\nof the nanocrystals. Next, in the second part we consider Fe\nand Co nanocsrystals in form of truncated pyramids with the\nsame length to height ratio as in the experiments.\nA. STM Observations\nThe SrTiO 3(001)-c(4\u00022) surface31is used for cobalt depo-\nsition. The c(4\u00022) reconstruction was verified by STM and\nLEED before deposition. Fig. 1 shows the topography of the\nSrTiO 3(001)-c(4\u00022) surface following deposition of 3 mono-\nlayers (ML) of Co on a substrate heated to 320\u000eCfollowed\nby a subsequent 50 minute anneal at 350\u000eC. The Co has self-\nassembled into similarly sized nanocrystals. Cobalt usually\nadopt a hcp bulk structure but STM shows that Co nanocrys-\ntals have a fcc structure with the shape of a truncated pyramid\n(a square top surface and a square base). The Co nanocrys-\ntals have. The side facets of the nanocrystals were measured\nat an angle of\u001854\u000ewith respect to the substrate. This shows\nthat Co is cubic packed and the nanocrystals have a (001) top\nfacet and four (111) side facets. The interface is therefore\na (001) plane and the interface crystallography is (001) Cok\n(001) SrTiO 3,[100] Cok[100] SrTiO 3. As a guide to the eye we\nhave shown in Fig. 1 (inset) a schematic illustration of a trun-\ncated pyramid. The ratio of the length ( `) of the top square to\nthe height (h) of the truncated pyramids as a function of vol-\nume is shown in Fig. 1b. The constant ratio of `=h=1.48\u00060.13\nsuggests that these pyramidal nanocrystals have reached their\nequilibrium shape. The error in the ratio denotes the standard\ndeviation of the measurements.\nB. Calculations\nAs has been discussed above, fcc Co nanocrystals (Fig. 1)\nas well as bcc Fe nanocrystals16can be epitaxially grown\non SrTiO 3(001) substrate with a remarkable control of size,4\nshape and structure. These crystals can contain up to several\nhundreds of atoms and have the form of truncated pyramids,\nas shown in Fig. 2, with a rather constant length-to-height\nratio,l=h. They however adopt different bulk structure, i.e.\nthe nanocrystal facets will therefore be different because the\nclose-packed and lowest-energy facet for bcc structure is the\n(110) facet whereas it is the (111) facet for the fcc structure.\nIt is expected that the MAE of such pyramids will be dom-\ninated by the surface composed of (001) and (110) or (001)\nand (111) facets for Fe and Co nanocrystals, respectively. It is\ntherefore essential to estimate first the MAE of the bulk slabs\nof these orientations. We present below the results for fcc Co\n(001) and(111) slabs while similar results for bcc Fe slabs\nhave already been reported recently (Ref.24).\nFe (N = 135)\nl/h = 1.41 l/h = 1.0Co (N = 110)\n(001)\n(110)(001)\n(111)\nFIG. 2: (color online) Examples of truncated-pyramid shaped Fe\nand Co nanocrystals studied in the present work. The crystals are\nmade of bcc Fe and fcc Co with two types of facets: (001) and (110)\nfor Fe and (001) and (111) for Co, respectively. Their possible size\nand shape is controlled by length-to-height ratio, l/h, kept to \u00181.0\n(Fe) and 1.41 (Co) which are close to experimental values, \u00181.20\n(Fe)16and\u00181.48 (Co). The zaxis was chosen to be normal to the\npyramid base and the spin moment is rotated in the xzplane forming\nthe angle\u0012with thezaxis.\n1. MAE of Co fcc (001) and(111) slabs\nThe Co slabs were constructed from fcc Co with a lattice\nparameter of a0= 3:531 ˚A found from ab initio calculations\n(which is close to the experimental value of a0= 3:548 ˚A )\nand no atomic relaxations were performed. Fig. 3 shows\nthickness dependence of the total MAE of N-atom fcc Co\nslabs of (001) and (111) orientations. The results of both TB\n(N = 1\u001820) as well as ab initio (N = 1\u001810) calculations are\npresented. Note that the total MAE is obtained as total energy\ndifference for ~Mperpendicular or parallel to the atomic slabs.\nExplicitely, \u0001E=E?\ntot\u0000Ek\ntot.\nIn the TB model, a mesh of 50\u000250in-planekpoints\nhas been used for SCF calculations without SOC whereas\nthe mesh was incresed to 70\u000270in non-SCF calculations\nwith SOC in order to provide a precision below 10\u00005eV . A\nMarzari-Vanderbilt broadening scheme with smearing param-\neter of 50 meV has been used. Ab initio DFT calculations\nwere carried out with Quantum ESPRESSO package25us-\ning generalized gradient approximation (GGA) for exchange-\ncorrelation potential in the Perdew, Burke, and Ernzerhof\nparametrization32. Full self-consistent calculations were per-\nformed with relativistic ultrasoft pseudo-potentials (no FT\nwas employed here) and cut-off energies were set to 30 and300 Ry for wave-functions and charge density, respectively.\nThe mesh of 40\u000240kpoints was used and the same smear-\ning parameter and technique were employed.\nWe find a relatively good overall agreement between TB\nand DFT calculations. MAE oscillations for both slabs can be\nclearly seen even for quite thick slabs (similar results were\nrecently reported for bcc Fe slabs24). This kind of long-\nrange oscillating behavior has been recently reported by ex-\nperiments in thin ferromagnetic films (Fe and Co), and was\ninterpreted in terms of spin-polarized quantum well sates.33,34.\nWe notice further that for Co(001) slabs both calculations give\nrather similar results: the total MAE clearly favours in-plane\nmagnetization with anisotropy energy around 0.6 meV/cell.\nIn the case of Co(111), the MAE oscillates around zero in TB\nmodel while the DFT calculations predict rather small (com-\npared to the (001) case) out-of-plane magnetic anisotropy.\nNote that our results compare rather well with DFT calcula-\ntions in Ref.35done with LDA approximation for exchange-\ncorrelation functional. We further study the local decomposi-\ntion of MAE of (001) and (111) Co slabs made of 20 atomic\nlayers (Fig. 4). Here, we used the FT in TB as well as in\nDFT approaches as described in the previous section. A qual-\nitatively good agreement between TB and DFT calculations is\nagain found for both slabs with the main discrepancy appear-\ning for the surface layers, which indicates that the TB model\nis presumably less accurate for low coordinated atoms. Inter-\nestingly, for both (001) and (111) slabs these surface layers\npossess in-plane anisotropy. The local MAE site decompo-\nsition then shows damped oscillations converging towards a\ntiny bulk value. However, while the MAE of the (001) slab\nis strongly dominated by the outermost surfaces layer, this is\nnot the case for the (111) slab where sub-surface layers cancel\n(and even overcome in the DFT case) the surface contribution.\nThis leads to the large in-plane and rather small out-of-plane\noverall MAE for the (001) and (111) slabs, respectively, as it\nis reported in Fig. 3.\n0 2 4 6 8 10 12 14 16 18 20−0.500.511.5\nNumber of atomic layers NTotal MAE [meV]DFT−GGATBsquares: (001)\ncircles: (111)\nFIG. 3: (color online) Total MAE per unit cell of N atoms (in meV),\nnamely,E?\ntot-Ek\ntot, versus the Co film thickness N for fcc Co slabs.\nSquares and circles are for (001) and (111) slabs respectively. TB\ncalculations (blue) are compared with ab initio DFT-GGA fully rel-\nativistic calculations (red) until N = 10. Lines are guide for the eyes.5\n2. Free Fe and Co nanocrystals\nThe length-to-height ratio of different size of Fe and Co\nnanocrystals can be written l=h= [2(n\u00001)]=(N\u0000n)and\nl=h= (n\u00001)=[p\n2(N\u0000n)], where N\u0002N and n\u0002n are\nthe number of atoms in the first (bottom) and last (up) layers\nof the truncated pyramids. We then selected different sizes of\nFe and Co nanocrystals with the length-to-height ratio of \u0018\n1.0 (l=h= 1.0 for N =29, 135; 1.20 for N = 271; 1.14 for N =\n620) and 1.41 close to the experimental value of 1.20 \u00060.1216\nand 1.48\u00060.13, respectively. Since the MAE in the xyplane\nwas found to be extremely small, we kept the magnetization\nalways in the xzplane making the angle \u0012with thezaxis.\nThe MAE is defined as the change in the band energy between\nmagnetic solutions with magnetization along the zandxaxis,\nMAE =Ez\u0000Ex. In Fig. 5, we plot the total MAE of Fe and\nCo nanocrystals of growing size calculated with TB approach.\nDifferent sign of MAE means that out-of-plane magnetization\nis favored in Fe while in Co the spin moment will be rotating\nin the easyxyplane which makes thus Fe nanocrystals a better\ncandidate for magnetic storage applications. These results can\n02468101214161820−0.2−0.100.10.20.30.4\nAtomic siteE⊥ − E|| (meV)DFT−FT\n(111)(001)\nin−plane\nout−of−plane(b)02468101214161820−0.2−0.100.10.20.30.4\nAtomic siteE⊥ − E|| (meV)\nout−of−planein−plane(001)\n(111)TB−FT (a)\nFIG. 4: (color online) Layer-resolved MAE per Co atom (in meV)\nof slabs with 20 atomic-layer-thick calculated by Tight-Biding (top)\nand DFT-GGA (bottom) within force theorem approximation. Blue\nsquares and red circles are for (001) and (111) slabs, respectively.\nLines are guide for the eyes.be understood from the local analysis reported in Table I for\nbiggest Fe (N = 620) and Co (N = 615) pyramids.\n0 100 200 300 400 500 600 700−120−90−60−300306090120\nNumber of atoms NEz − Ex (meV)Co nanocrystals\nFe nanocrystalsout−of−planein−plane\nFIG. 5: (color online) TB results: total MAE of Co (red squares) and\nFe (blue circles) nanocrystals vs. the number of atoms. The size of\nnanocrystals was chosen so to keep constant length-to-height ratio,\n1.41 (Co) and \u00181.0 (Fe).\nOne can see that the total MAE mainly originates from the\nlower (001) facet and its perimeter composed of least coor-\ndinated atoms. Therefore, in agreement with the previous\nanalysis of (001) Co and Fe24slabs, this would favor the out-\nof-plane/in-plane anisotropy for Fe/Co nanocrystals, respec-\ntively. We notice, moreover, that since nanocrystals of Co are\nmuch flatter then those of Fe (as Fig. 2 illustrates), which is\na consequence of bigger length-to-height ratio for Co, in the\ncase of Co nanocrystals also the upper (001) facet, containing\nmore atoms, gives noticeable contribution to the overall MAE.\nWe have also checked the total MAE in the xyplane but have\nfound it extremely small, of amplitude about 3 meV and 0.8\nmeV for Fe (N=620) and Co (N=615) nanocrystals, respec-\ntively. As mentioned in Sec. I, another important contribution\nto magnetic anisotropy is the so-called shape anistropy en-\nergy. We have calculated it for biggest Fe (N=620) and Co\n(N=615) nanocrystals and have found rather small values, of\nabout 5 meV and 2 meV for Fe and Co, respectively. Note\nthat for both pyramids, the shape anisotropy favors in-plane\nmagnetization direction.\nWe have next performed a more detailed local analysis of\nMAE for a smaller Co nanocrystal made of 110 atoms (shown\non the right panel of Fig. 2). For such a relatively small crys-\ntal,ab initio DFT calculations within FT approach can be also\ncarried out and compared with TB results. Fig. 7 reports\natom-resolved MAE for such pyramid. The atoms of each\natomic layer are numbered starting from the corner and going\nanticlockwise along the spiral to the centre of the plane, as\nshown in Fig. 6 (a) for the base layer. The other layers are\nnumbered in the same way. Again, a qualitatively good agree-\nment has been found between TB and DFT calculations. In-\nterestingly, we found a sign change of MAE between atomic\nlayers: the MAE favors in-plane magnetization for the first\nand third layers and out-of-plane magnetization for the mid-6\nFe (N=620) Co (N=615)\nM AE (meV) M AE/atom (meV) N atoms M AE (meV) M AE/atom (meV) N atoms\nupper perimeter -4.203 -0.262 16 5.799 0.181 32\nupper (001) -3.541 -0.393 9 18.091 0.369 49\nlower perimeter -37.265 -0.846 44 41.590 0.866 48\nlower (001) -52.839 -0.528 100 41.297 0.341 100\nside surfaces -11.457 -0.063 180 1.232 0.010 120\ntotal -103.473 -0.166 620 112.711 0.183 615\nTABLE I: TB results: Local analysis of MAE for Fe (N=620) and Co (N=615) nanocrystals, note that sign negative (positive) means out-of-\nplane (in-plane) magnetization.\ndle layer of the pyramid. The MAE achieves its highest values\nin the middle of two first layer edges aligned with the xaxis,\nnamely for 7-13 and 19-1 segments, and drops down to zero\nfor two other edges. This asymmetry is due to chosen defi-\n010 20 30 40 50 60 70 80 90100 110−0.500.51\nNumber of atomic siteEz − Ex [meV]010 20 30 40 50 60 70 80 90100 110−0.500.511.5\nNumber of atomic siteEz − Ex [meV]\n(a)\n(c)(b)1st layer 2nd layer 3rd layer\nFIG. 6: (color online) Atom-resolved MAE for Co nanocrystal made\nof 110 atoms: (a) trajectory for numbering the base layer atoms start-\ning from the corner and going along the spiral to the center. The\natoms of other layers are numbered in the similar way. (b) MAE per\natom in meV within TB approach. (c) MAE per atom in meV from\nDFT-GGA calculations.nition ofMAE =Ez\u0000Ex, since for the first pair of edges\nwe compare the energies between orthogonal and parallel to\nthe edge directions while for the second pair – between two\nperpendicular directions. Clearly, in the first case the energy\ndifference will be much bigger. Of course, if one chooses\nanother definition of MAE, e.g., as the energy difference be-\ntween the states with spin moment along the zaxis and along\nthe diagonal of the base plane, one would have more symmet-\nric contributions from all four base edges.\npositive\nnegative(a) (b)\n(c) (d)z\nx yz\ny\nx\nFIG. 7: (color online) DFT calculations: real-space distribution of\nMAE for Co nanocrystal of 110 atoms: (a),(b) side views, two iso-\nsurfaces of positive and negative isovalues are shown in red and blue,\nrespectively; (c) cross-section of MAE by the plane passing through\nthe base layer of the pyramid; (d) same as (c) but on the plane slightly\nbelow (by 0.4 ˚A) the base layer. Note that red (blue) colors represent\nthe regions favoring in-plane (out-of-plane) magnetization orienta-\ntion.\nTo get more insight into the local composition of MAE, we\nhave looked at its distribution in the real space as defined in\nEq. 5. Such a real space representation of MAE for the previ-\nously studied 110 atoms Co pyramid is shown in Fig. 7. Inter-\nestingly, there are regions of both positive as well as negative\nMAE around each atom (Fig. 7 a, b), in relative proportion\nwhich changes from layer to layer. This leads, on average, to\nthe change of sign for atomic MAE vs. the layer observed in\nFig. 6. We notice moreover that positive and negative regions\nof MAE have different spatial localization: while the first one7\nextends out of atomic planes (along the zaxis) the second one\nis mostly localized in the xyplane. This observations can be\nimportant when studying the MAE modification due to depo-\nsition of pyramids on various substrates.\nIV . Conclusion\nWe have presented a combined TB and DFT study of mag-\nnetocrystalline anisotropy of iron (bcc) and cobalt (fcc) slabs\nand nanocrystals. The nanocrystals are in shape of truncated\npyramids with the same length to height ratio as in the exper-\niment. Thanks to the use of the Force Theorem that we have\nrecently implemented in the QE package we have been able to\nperform a careful local analysis of the MAE in these nanos-\ntructures. The TB model is in good agreement with the DFT\ncalculations and gives us confidence in the validity of our TB\nresults for large crystals that cannot be done within the DFT\napproach. We found a large in-plane anisotropy for Co (001)\nand a relatively small out-of-plane for Co (111) due to cance-\nlation from the sub-surface layer in the latter case. This is in\ncontrast with iron surfaces since Fe (001) shows a clear out of\nplane anisotropy. The densest surface shows however a rather\nsmall anisotropy for both elements.These results could have a direct consequence on the mag-\nnetic stability of Fe and Co nanocrystals. Indeed, the total\nMAE is of the same order of magnitude for both Fe and Co\nnanocrystals, but opposite in sign. This means that while the\nspin moment of Fe nanocrystals is fixed along the easy out-\nof-plane axis and needs to overcome the high MAE barrier\nto reverse from positive to negative direction, the magnetic\nmoment of Co nanocrystals is allowed to rotate almost freely\n(with a very low in-plane anisotropy barrier) in the easy basal\nplane. One can thus conclude that Fe nanoclusters should be\nbetter candidates for magnetic storage applications. Our lo-\ncal analysis, however, indicates that the MAE of nanocrystals\ncould be substantially altered, for instance, by their covering\nwith a mono-layer of another chemical element or by their\ndeposition on various substrates (SrTiO 3(001), Cu, Au etc).\nAcknowledgement\nThe research leading to these results has received fund-\ning from the European Research Council under the European\nUnion’s Seventh Framework Programme (FP7/2007-2013) /\nERC grant agreement n\u000e259297. 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Zapf1\n1National High Magnetic Field Laboratory (NHMFL), MS E536,\nLos Alamos National Laboratory, Los Alamos, NM 87545, USA\n2Department of Chemistry, University of Edinburgh,\nEdinburgh, Midlothian EH8 9YL, United Kingdom\n3University of Oxford, Department of Physics,\nThe Clarendon Laboratory, Parks Road,\nOxford, OX1 3PU, United Kingdom\n4Department of Physics, Simon Fraser University, Burnaby, BC, V5A 1S6, Canada\n5Department of Physics, University of Warwick,\nGibbet Hill Road, Coventry, CV4 7AL, United Kingdom\n6RCEM & Dept. of Physics and Astronomy,\nRutgers University, Piscataway, NJ 08854, USA and\n7POSTECH, Pohang University of Science and Technology,\nSan 31 Hyoja-dong, Nam-gu, Pohang-si,\nGyungbuk, 790-784, Republic of Korea\n1arXiv:1408.0758v1  [cond-mat.mtrl-sci]  4 Aug 2014Abstract\nWe have measured extremely large coercive magnetic \felds of up to 55 T in Sr 3NiIrO 6, with\na switched magnetic moment \u00190:8\u0016Bper formula unit. As far as we are aware, this is the\nlargest coercive \feld observed thus far. This extraordinarily hard magnetism has a completely\ndi\u000berent origin from that found in conventional ferromagnets. Instead, it is due to the evolution\nof a frustrated antiferromagnetic state in the presence of strong magnetocrystalline anisotropy due\nto the overlap of spatially-extended Ir4+5dorbitals with oxygen 2 pand Ni2+3dorbitals. This\nwork highlights the unusual physics that can result from combining the extended 5 dorbitals in\nIr4+with the frustrated behaviour of triangular lattice antiferromagnets.\nPACS numbers: 71.70.Ej, 75.30.Gw, 75.50.Vv\n2I. INTRODUCTION\nOxides containing iridium in the Ir4+(5d5) state have received much recent attention be-\ncause the energy scales for spin-orbit interactions (SOIs), Coulomb repulsion, and crystalline-\nelectric \felds are all very similar [1{11]. This unusual situation, in comparison to analogous\n3dsystems, results from a decrease in the strength of correlation e\u000bects and an increase\nin SOIs as one descends the periodic table. Usually, the Coulomb repulsion and SOIs are\nresponsible for Hund's rules that determine the groundstates of magnetic ions; however,\nin these 5d(and some 4 d) systems, the competition between the three similarly-sized en-\nergy scales can result in exotic magnetic states [1{11], leading to proposals for the use of\nIr4+-based systems in quantum information processing and other novel applications (see\nRefs. [2, 9] and references therein). In this paper, we report another manifestation of un-\nusual 5dphysics: extremely large coercive magnetic \felds of up to 55 T in Sr 3NiIrO 6, with\na switched magnetic moment of about 0 :8\u0016Bper formula unit. As far as we are aware,\nthis is the largest coercive \feld measured thus far, and a factor \u00185 times those reported\nrecently in novel, ultra-hard ferromagnets [12]. This exemplary hard magnetism is, how-\never, very di\u000berent from that found in conventional ferromagnets in that it results from\nan unusual, frustrated, antiferromagnetic ground state that incorporates a relatively large\nmagnetocrystalline anisotropy.\nThe salient structural details [13] are shown in Fig. 1. The Ni2+and Ir4+magnetic ions\nin Sr 3NiIrO 6occupy oxygen cages that alternate in chains parallel to the c-axis (Fig. 1(a)).\nThese chains are in turn arranged in a hexagonal pattern in the ab-plane (Fig. 1(b)) [13].\nThe Ni2+is surrounded by a trigonal bipyramid of oxygen atoms, while the Ir4+ion sits\nin an octahedral oxygen cage. Magnetic frustration is intrinsic to this structure, and can\nresult from antiferromagnetic interactions within the triangular lattice in the ab-plane, and\nfrom frustration between nearest-neighbour and next-nearest-neighbour interactions along\nthec-axis chains. Electronic structure calculations have suggested both possibilities for\nSr3NiIrO 6[14{16]; the same calculations indicate that the overlap of spatially-extended Ir4+\n5dorbitals with oxygen 2 pand Ni2+3dorbitals leads to a magnetocrystalline anisotropy\nenergy\u001913:5 meV. Schematic energy level diagrams for Ir4+and Ni2+are shown in Fig. 1(c);\nthe e\u000bective spin of Ni2+isS= 1 and that of Ir4+isS=1\n2[14{16].\n3FIG. 1. Structure and energy levels. The crystal structure of Sr 3NiIrO 6as viewed from the\n(a) [110] and (b) [001] directions. The Ni2+and Ir4+ions are within oxygen trigonal bipyramids\n(blue) and oxygen octahedra (orange), respectively. For clarity, Sr ions are not shown in (a).\n(c) Schematic level diagrams for Ir4+and Ni2+(based on Refs [14{16]). Unperturbed Ir4+levels\nare shown in blue; a combination of antiferromagnetic interactions and spin-orbit coupling splits\nthe e0\ngdoublet to give the con\fguration shown in red. Electron spins are shown as arrows: the\ne\u000bective spin of Ni2+isS= 1 and that of Ir4+isS=1\n2in the opposite direction.\nII. RESULTS\nA. Low-\feld magnetization and dynamics\nOur study involves 10 di\u000berent samples of Sr 3NiIrO 6from several di\u000berent batches en-\ncompassing minor variations in growth conditions. The results for all samples are rather\nconsistent; typical data from four single crystals (labelled S1 to S4) and three polycrystalline\nsamples (P1, P2, P3) are shown in the \fgures that follow. Note that sample S2 is a very\nsmall crystal with exceptionally clean hexagonal faces containing virtually no visible defects;\nas will be seen below, this apparent cleanliness is of relevance to the timescales over which\nthe remanent magnetization persists.\nTypical examples of low-magnetic-\feld DC magnetization data for Sr 3NiIrO 6(samples S1,\nS2 and P1) are shown in Fig. 2. Magnetic order sets in below 85 K, being marked by a broad\nfeature in magnetization ( M) vs. temperature ( T) data (Fig. 2(a)), and additional peaks in\nelastic neutron scattering [17{19]. The neutron di\u000braction data are consistent with either\n40501001502002503000.000.020.040.06\n0204060801000.000.020.040.06DC, ZFC, 0.2 T SNIO-I SNIO poly(a)H || c\n M/H (emu/mole)\n DC, ZFC, 0.2 T 10 Hz 10 kHzH || cSNIO-IM/H (emu/mole)\nTemperature (K)(b) S1 S2       P1   S1 \n0246810121401020304050607080\n  Temperature (K)\nMagnetic Field (T)(a)\n0246810121401020304050607080(c)\n  \n0123456701020304050607080(b)\ndM/d(0H) (µB/Ir T)  Temperature (K)\nMagnetic Field (T)00.08\n0123456701020304050607080(d)\nHysteresis (µB/Ir)  \n-0.01980.196\n0(c) pulsed  \n(d) DC (e) pulsed \n(f) DC (1/µ0)dM/dH (µB/f.u. T) \nHysteresis (µB/f.u.) FIG. 2. Frustration-induced slow dynamics. (a) DC magnetic susceptibility ( \u001f=M=H ) as a\nfunction of temperature Tfor Sr 3NiIrO 6single-crystal samples S1 and S2 and polycrystalline sam-\nple P1. The onset of magnetic order is marked by a broad hump superimposed on the background\nslope and centered on T\u001985 K. (b)T-dependent \u001fdata for Sr 3NiIrO 6single crystal sample S1\nmeasured DC (black) and using an AC susceptometer operated at frequencies of 10 Hz (red) and\n10 kHz (blue). The sharp fall in \u001fshifts to higher Tas the frequency increases. The contour plots\nshow dM=dHversus\u00160HandTfor polycrystalline sample P2 measured (c) in a pulsed magnetic\n\feld with a sweep rate \u00193500 T/s and (d) in a SQUID (DC) magnetometer with a magnetic \feld\nsweep rate of 0.008 T/s. Corresponding plots of the hysteresis between falling and rising magnetic\n\feld sweeps, de\fned as hysteresis = [ M(H;T)falling\u0000M(H;T)rising], are shown in (e) (pulsed \feld)\nand (f) (SQUID). Note that as the \feld sweep rate increases, features in d M=dHand regions of\npronounced hysteresis are pushed to higher T.\na partially-disordered antiferromagnet [20] or an amplitude-modulated antiferromagnetic\nstate at low T. Despite the onset of long-range order at 85 K, the heat capacity is nearly\nfeatureless between 4 and 250 K [18].\nFig. 2(b) shows the DC magnetic susceptibility of Sr 3NiIrO 6single crystal sample S1, in\nan applied \feld of \u00160H= 0:2 T, and the AC susceptibility for an oscillating \feld amplitude\nof 10\u00003T at 10 Hz and 10 kHz. In all three data sets, the sharp drop in Malready seen in\n5Fig. 2(a) is observed as Tis lowered. However, the temperature at which the drop occurs is\nfrequency dependent; the half-height point occurs at 28 K for 10 kHz, 20 K for 10 Hz, and\n13 K for DC measurements. Further evidence of slow dynamics over a similar temperature\nrange is seen in the contour plot in Figs. 2 (c) to (f). Here we show d M(H;T)=dHon the\ninitial \feld upsweep following zero-\feld cooling for polycrystalline sample P2. These data\nwere measured in a 7 T SQUID magnetometer with a \feld-sweep rate \u00190:008 T/s (Fig. 2(d))\nand in a pulsed magnetic \feld with a sweep rate \u00193500 T/s (Fig. 2(c)), where the pulse\nshape is shown in Fig. 6(a). Corresponding plots of the hysteresis in Mbetween rising and\nfalling magnetic \feld sweeps are shown in (e) (pulsed \feld) and (f) (SQUID). As the \feld\nsweep rate increases (and therefore the timescale of the measurement decreases), features in\ndM=dHand regions of pronounced hysteresis are pushed to higher T. Thus, this dependence\nofMon sweep rate is analogous to the frequency dependence seen in the AC susceptibility\ndata (Fig. 2(b)), where decreasing the measurement timescale forces features in Mto higher\nT. Similar phenomena have been seen in previous experiments within the ordered phase; as\nTwas lowered below 30 K, a frequency-dependent crossover occurred to hysteretic magnetic\nbehaviour as a function of TandH, as well as a strong frequency dependence of the AC\nsusceptibility [17{19]. A number of other frustrated antiferromagnets isostructural with\nSr3NiIrO 6show qualitatively similar frequency-dependent magnetic behaviour [21{33].\nB. Large coercive \felds\nLarge coercive \felds are demonstrated in Fig. 3, which shows pulsed-\feld M(H) data\nfor four representative samples of Sr 3NiIrO 6(the \feld-pulse pro\fle is shown in Fig. 6(a)).\nData for Figs. 3(a), (b), and (d) are calibrated against absolute measurements on the same\nsamples in quasistatic magnetic \felds of up to 13 T.\nThough the behaviour of all samples is similar, showing that the large coercive \felds are\nintrinsic to Sr 3NiIrO 6, there are some detail di\u000berences from sample to sample. In Fig. 3(a)\n(single crystal S3), the magnetic hysteresis for Hkcis superimposed on a linear M(H)\nbackground, and shows a large \u00160Hc= 34:1 T and a remanent moment of 0 :45\u0016Bper\nformula unit at zero \feld. The magnetization jump at 34.1 T is \u00190:8\u0016Bper formula unit.\nThe data in Fig. 3(e) (Sample S4) show an even larger \u00160Hc= 42:7 T. As mentioned above,\nsample S2 is a very small single crystal with exceptionally clean faces, and Fig. 3(c)) shows\n6-2.0-1.5-1.0-0.50.00.51.01.52.0\n-2.0-1.5-1.0-0.50.00.51.01.52.0\n-4-3-2-101234\n-2.0-1.5-1.0-0.50.00.51.01.52.0\n-60-40-200204060-0.2-0.10.00.10.2\n-60-40-200204060-0.15-0.10-0.050.000.050.100.15SNIO-N(a)H || c T = 4 K 10 K 25 K M (µB/f.u.)H ⊥ cSNIO-N(b)\nM (µB/f.u.)\n 0.5 K 4 K (1st) 4 K (2nd)M (arb. units)SNIO-I(c)H || c\n 4 Kpoly(d)\nM (µB/f.u.) DC field 1st pulse subsequent\nMagnetic field (T) 4 KM (arb. units)SNIO(e)H || c\n H ⊥ c 4 KSNIO(f)\nM (arb. units)S3       S3       \nS2       P2 \nS4       S4       FIG. 3. Hysteresis loops and large coercive \felds. Magnetization Mas a function of magnetic\n\feld\u00160Hmeasured in a series of pulses using a capacitor-bank-driven 65 T pulsed magnet. Sample\nnumbers, \feld directions and measurement temperatures Tare given in each section of the \fgure;\nthe vertical jumps in Moccur at the coercive \feld, \u00160Hc. In (b), (f) no hysteresis is observed in the\nH?ccon\fgurations. In (a), (b), and (d), absolute magnetization values, measured in commercial\nSQUID and vibrating-sample magnetometers, were used to calibrate the pulsed \feld data.\n7that it exhibits qualitatively di\u000berent behaviour: the coercive \feld is very large (38.4 T),\nas in other samples, but Mrelaxes during the millisecond timescales (see Fig. 6(a)) of the\ndownsweep of the magnetic \feld. All subsequent \feld pulses after the \frst one show jumps\nof similar height, consistent with relaxation of Mback to zero between \feld pulses. On the\nother hand, in Fig. 3(a), (b), (d), (e), and (f) the \frst upsweeps following a zero-\feld cool\nresult in an Mjump about half the height of subsequent jumps, showing that the latter\nrepresent a complete reversal of M. For those samples, two consecutive magnetic \felds\nsweeps in the same direction produce no jump (not shown). In polycrystalline sample P2\n(Fig. 3(d)), the M(H) data also show some curvature at low \felds, but Mdoes not relax\nback to zero between \feld pulses. (As will be shown in Fig. 4(a) below, polycrystalline\nsample P3 shows similar behaviour, but superimposed on a much less curved background.)\nIn the initial up-sweep, Mjumps at a record-high magnetic coercive \feld of \u00160Hc= 55 T\nafter zero-\feld cooling; in subsequent pulses, the Mjump doubles in height, but is observed\nat lower \felds (40 T). Note that all samples exhibit a higher Hcvalue in the initial \feld\nsweep after zero-\feld cooling, compared to those seen in subsequent \feld sweeps; we shall\nreturn to this behaviour below. Finally, we remark that the jumps in Mare sharp, occurring\nover a small range of \feld ( <1 T at 4 K). In particular, the transition width in Fig. 3(c) is\n.0:03 T.\nThere is no systematic dependence of the magnetic hysteresis on details of the growth\nmethod in the samples shown in Fig. 3 or in the three other samples measured, suggesting\nthat all samples have the same Ni/Ir ratio. There does, however, appear to be a pinning\ne\u000bect that stabilizes Min most of the samples; in such cases, the remanent magnetization\nwas found to persist unchanged for at least 24 hours at T= 4 K. By contrast, sample S2\nin Fig. 3(c) shows magnetic relaxation on millisecond timescales ( i.e.similar timescales to\nthe downsweep of the pulsed \feld; see Fig. 6(a)). In addition, none of the samples studied\nappear to show signs of saturation of the magnetization even in \felds of 65 T. The anisotropy\nof the hysteresis is illustrated in Figs. 3(b) and (f); here M(H) of single crystals S3 and S4\nis shown for H?c. For this \feld direction, no jump in Mor hysteresis is observed. In both\nsingle crystals, Mis comparable in size at 65 T for H?candHkc.\n81011021031041020304050\n-60-40-200204060-10-50510\n02468101214161820-35-30-25253035404550 (b)dB/dt (T/s) First SubsequentHc (T) (a) First Subsequent M (arb. units) \nMagnetic field (T)4.0 Kpoly 6.7 - 10 kT/s 360 T/s 150 T/s 70 - 120 T/s 40 - 50 T/s 25 - 30 T/sThermal cycle-Hc+HcCoercive magnetic field (T)\nPulse number(c)\n P3       \nµ0Hc (T)      FIG. 4. Sweep-rate dependence of the coercive \feld. (a)M(H) data for polycrystalline\nsample P3 measured in a capacitor bank-driven pulsed magnet. Black dotted lines denote the\ninitial pulse after zero-\feld cooling whilst the green solid line indicates data for subsequent pulses\nat 4 K. (b) Sweep-rate dependence of coercive magnetic \feld ( \u00160Hc). (c) History dependence of\ncoercive magnetic \feld ( \u00160Hc) under various sweep rates (plotted as a function of pulse number).\nFor sweep rates larger than 360 T/s, a capacitor-bank-driven pulsed magnet was used; other data\nwere taken using the generator-driven 60 T Long-Pulse Magnet (see Fig. 6 for magnetic \feld\npro\fles). Twice during this experiment, the sample was warmed to room temperature (denoted by\nred vertical dashed lines).\n9C. Dynamics, and history- and temperature dependences of the coercive \feld\nIn view of the \feld-sweep-rate dependence of the low-\feld magnetization noted earlier\n(Fig. 2(b)-(f)), it is interesting to see if the hysteresis loops are a\u000bected by similar dynamical\nissues, and so Fig. 4 features M(H) measurements of polycrystalline sample P3 in the Los\nAlamos generator-driven 60 T Long-Pulse Magnet, which provides sweep rates between 25\nand 360 T/s, and in a capacitor-driven 65 T magnet with faster sweep rates (pulse shapes\nare shown in Fig. 6). Fig. 4(a) once again demonstrates that the coercive \feld is higher on\nthe initial \feld sweep after zero-\feld cooling (c.f. Figs. 3(c) and (d)), whilst (b) shows that\nthis trend also occurs at lower \feld-sweep rates. The history dependence of the coercive\n\feld throughout a sequence of \feld pulses of varying sweep rate and a couple of thermal\ncycles to room temperature and back is illustrated in Fig. 4(c). All of these data show that\nthe magnetic hysteresis loop is robust at least down to 25 T/s and illustrate that on magnet\npulses subsequent to the initial pulse following zero-\feld cooling, Hcis slightly smaller in\nlower \feld sweep rates; this was observed in all measured samples.\nTheT-dependences of M(H) forHkcandH?care shown in Figs. 3(a) and (b) for\nsingle crystal S3. For H?c,MforT < 25 K is suppressed compared to the value at\n25 K, consistent with antiferromagnetic exchange interactions [36]. For Hkc, hysteresis is\nobserved at 10 K but has disappeared by 25 K. The T-dependence of Hcis given in more\ndetail in Fig. 5(a), which shows d M=dHdata for Hkc(sample S2); peaks occur at the\ncoercive \feld, Hc. Fig. 5(b) shows that Hcdecreases linearly with increasing T(data from\nsamples S2 and S3), extrapolating to zero at \u001925 K. This is close to the onset of hysteretic\nbehaviour and the frequency-dependent fall in M(see Fig. 2(b-f)), linking the observation\nof \fnite coercive \feld with the low-temperature phase of the antiferromagnetic groundstate.\nThe almost identical variations of Hcwith temperature in the two samples show that the\nvalue of this \feld and its temperature dependence are intrinsic to Sr 3NiIrO 6single crystals,\nwhereas the persistence (or not) of the remanent magnetization depends on extrinsic factors\nsuch as defects or disorder (the same samples are used in Figs. 3(a),(c)).\nFinally, we note that a coercive \feld \u00160Hc\u001922 T was measured using polycrystalline\nsamples of Sr 3NiIrO 6in Ref. [17]. All of our polycrystalline samples exhibit low-temperature\ncoercive \felds \u00160Hc\u001940 T, very similar to those seen in the single crystals (see Figs. 3\nand 4). Therefore, the di\u000berence between polycrystalline and single-crystal samples cannot\n10-40-30-20-100102030400.00.51.01.52.0\n051015202530-40-30-20-10010203040 15.0 K, X50 19.0 K, X50 0.5 K 4.0 K 6.0 K, X5 8.0 K, X5 10.0 K, X5 12.0 K, X5Magnetic field (T)(a)H || cSNIO-I dM/dH (arb. units)\n SNIO-I SNIO-NH || cCoercive magnetic field (T)\nTemperature (K)(b)\n  S2       \nS2 S3      \n-40-30-20-100102030400.00.51.01.52.0\n051015202530-40-30-20-10010203040 15.0 K, X50 19.0 K, X50 0.5 K 4.0 K 6.0 K, X5 8.0 K, X5 10.0 K, X5 12.0 K, X5Magnetic field (T)(a)H || cSNIO-I dM/dH (arb. units)\n SNIO-I SNIO-NH || cCoercive magnetic field (T)\nTemperature (K)(b)\n  S2       \nS2 S3      FIG. 5. Delineating the region of large coercive \feld. (a) dM=dHfor single-crystal sample\nS2 at several TforHkc(c.f. Fig. 3(a,c)). The peaks denote the sharp change in Mat the\ncoercive \feld; note that multiplication factors of 5 and 50 are applied for 6, 8, 10 K and 15, 19 K\ndata, respectively as the peak broadens with increasing T. (b)T-dependence of the coercive \feld\nfor single crystals S2 and S3. Filled symbols indicate the result taken in the initial pulse after\nzero-\feld cooling from room T; these show slightly larger Hcvalues compared those measured in\nsubsequent pulses (see Fig. 4). Lines are linear \fts to the data.\naccount for the much lower values of Hcobserved in Ref. [17] compared to those in this\nwork.\nIII. DISCUSSION\nA. Summary of results\nCombining the H- andT-dependent magnetization studies of Sr 3NiIrO 6, a picture\nemerges of an antiferromagnetic order [19] for T.30 K and H= 0 that evolves in\napplied magnetic \felds. This evolution is manifested most markedly in the sharp jump in\nmagnetization that occurs on rising \felds. The \feld position \u00160Hcof the jump shows a\nsimilar history and \feld-sweep-rate dependence for all samples (Figs. 3, 4, 5), with slightly\nsmaller values of Hcoccurring at lower sweep rates. This strongly suggests that the magne-\ntization jump at Hcis an intrinsic property of Sr 3NiIrO 6, with the sweep-rate dependence\nbeing caused by sluggish kinetics associated with the magnetic frustration intrinsic to this\n11structural family [11]; the same frustration is responsible for the slow magnetic relaxation\n(Figs. 2(c-f)) and the strong variation (13 \u000028 K) of the fall in Mwith frequency (Fig. 2(b)).\nBy contrast, in systems such as (Sm,Sr)MnO 3[34], where the magnetization jumps are not\nintrinsic, but associated with quenched disorder, the smaller the \feld-sweep rate, the larger\nthe \feld needed to realize the transition, the opposite of what we observe in Sr 3NiIrO 6\n(Fig. 4).\nIt is likely that the millisecond-timescale magnetic relaxation observed in the hysteresis\nloops for sample S2 (Fig. 3(c)) is also intrinsic behaviour due to frustration, whilst samples\nS3, S4, P2 (Figs. 3(a), (d), (e)) and P3 (Fig. 4(a)) and all other samples studied exhibit\npinning or freezing of the magnetization or an order-by-disorder mechanism, as seen at lower\n\felds in isostructural Ca 3CoMnO 6[35]. In the latter mechanism, magnetic-site disorder\ndisrupts longer-range interactions, leading to a reduction of frustration [35].\nB. Comparison with ferromagnets and other systems\nMagnets with high coercivity, i.e., \\hard\" magnets, are important for a wide range of ap-\nplications involving magnetic actuation and induction, such as loud speakers, wind turbines\nand electric motors [36]. In rare-earth iron boride magnets, among the hardest commercial\nmagnets,\u00160Hccan be as large as 1.5 T at room temperature [37]. At cryogenic tempera-\ntures, magnetic hysteresis e\u000bects can extend to higher \felds, e.g., up to 10 T in the colossal\nmagnetoresistance manganites, and in Li 2(Li1\u0000xFex)N, Gd 5Ge4, Ga-doped CeFe 2, LuFe 2O4,\nand Fe 1=4TaSe 2[12, 28, 38{42]. Large coercive \felds in such magnets are typically caused\nby magnetocrystalline anisotropy due to SOI [37], whilst magnetic hysteresis results from a\ncombination of domain dynamics and anisotropy [36]. In traditional ferromagnets, domains\nresult from competition between the short-range exchange interactions that prefer parallel\nspin alignment, and the free-energy penalty of maintaining a magnetic \feld in an extended\nregion of space around the sample [36]. The e\u000bect of this competition is that the energy\nscale for switching magnetic domains can be orders of magnitude smaller than those of the\nnearest-neighbor ferromagnetic exchange interactions. Generally, the microscopic order in\na traditional ferromagnet does not change signi\fcantly around the hysteresis loop as the\ndomains change direction and/or the domain walls move [36].\nBy contrast, the conventional phenomenology of ferromagnetic domains is notinvolved\n12in the notable Hcvalues observed for Sr 3NiIrO 3(35\u000055 T, this work) and in the lower, but\nnevertheless impressive, values measured in the isostructural family of frustrated triangular-\nlattice antiferromagnets, Ca 3Co2O6(7 T) [28], and Ca 3CoMnO 6(10 T) [33, 43], Sr 3CoIrO 6\n(\u001920 T) [18], and Ca 3CoRhO 6(\u001930 T) [23]. Sr 3NiIrO 6is initially antiferromagnetic after\nzero-\feld cooling and its magnetic groundstate must evolve to produce a net magnetization\n[19]. The energy scale for the coercive \feld ( \u00160Hc= 34\u000055 T) is roughly of the same order\nof magnitude as the temperature of the onset of magnetic hysteresis (13 \u000028 K), and long-\nrange order (85 K), consistent with evolution of the magnetic order driven by large magnetic\n\felds. Indeed, in compounds isostructural with Sr 3NiIrO 6, the microscopic order has been\nshown to change around the hysteresis loop using elastic neutron di\u000braction measurements\n[43, 44]. We can therefore infer that the magnetic hysteresis in Sr 3NiIrO 6can be attributed\nto the evolution of a microscopic frustrated order with magnetic \feld, rather than domain\ne\u000bects ( c.f.Ref. [45]).\nAs mentioned previously, electronic structure calculations suggest that the magnetocrys-\ntalline anisotropy in Sr 3NiIrO 6results from the con\fguration of overlapping orbitals in\nIr4+-O-Ni2+chain [14{16]. This picture is supported by the prevalence of relatively large\ncoercive magnetic \felds in the isostructural family members that have 4 dor 5dions (with\nrelatively extended orbitals) on the octahedral site and 3 dions on the bipyramidal site:\nCa3CoRhO 6, Sr 3NiIrO 6and Sr 3CoIrO 6[17, 18, 23]. The exceptional coercive magnetic \feld\nin the title compound serves as another example of notable physical e\u000bects resulting from\nthe unusual 5 dorbital physics of Ir4+.\nIV. CONCLUSION\nIn conclusion, we observed coercive magnetic \felds of 34 \u000055 T for Hkcin \rux-grown sin-\ngle crystals and polycrystalline Sr 3NiIrO 6, to our knowledge, a record high coercive magnetic\n\feld for any material. Sr 3NiIrO 6shows signatures of a type of frustrated antiferromagnetic\norder in zero \feld [19], and the hysteresis loop is likely due to evolution of this microscopic\norder. The high coercive magnetic \feld is consistent with the large magnetocrystalline\nanisotropy predicted in ab-initio calculations for this compound, due to the Ir4+5dorbitals\noverlapping via intermediate oxygen with Ni2+3dorbitals.\n13V. METHODS\nA. Sample growth\nPolycrystalline Sr 3NiIrO 6was prepared through solid-state reaction at 1300oC. Single\ncrystals of Sr 3NiIrO 6were grown from either a stoichiometric or 20 % Ni-rich or 20 % Ir-rich\n(in molar ratio) composition using K 2CO3as \rux. The single crystals are hexagonal plates\nwith typical dimensions 2 \u00022\u00020:5 mm3.\nB. Magnetization measurements\nQuasistatic-\feld magnetization ( M(H)) data were measured in a vibrating sample magne-\ntometer in a superconducting magnet (PPMS-14, Quantum Design), or in a SQUID (MPMS-\n7, Quantum Design). AC susceptibility data were measured in a 7 T AC SQUID and an ac\nsusceptometer in a 14 T PPMS (Quantum Design).\nThe pulsed-\feld magnetization experiments used a 1.5 mm bore, 1.5 mm long, 1500-turn\ncompensated-coil susceptometer, constructed from 50 gauge high-purity copper wire [46].\nWhen a sample is within the coil, the signal is V/(dM=dt), wheretis the time. Numerical\nintegration is used to evaluate M. Samples were mounted within a 1.3 mm diameter ampoule\nthat can be moved in and out of the coil. Accurate values of Mare obtained by subtracting\nempty-coil data from that measured under identical conditions with the sample present. The\nsusceptometer is calibrated by scaling low-\feld Mvalues to match those recorded with a\nsample of known mass measured in a commercial SQUID or vibrating-sample magnetometer.\nFields were provided by a 65 T short-pulse magnet energized by a 4 MJ capacitor bank, or\nthe generator-driven 60 T Long-Pulse Magnet at NHMFL Los Alamos; the \feld versus time\npro\fles for these two magnets are shown in Fig. 6. The susceptometer was placed within\na3He cryostat providing Ts down to 0.4 K. \u00160Hwas measured by integrating the voltage\ninduced in a ten-turn coil calibrated by observing the de Haas-van Alphen oscillations of the\nbelly orbits of the copper coils of the susceptometer [46].\nIn measuring hysteresis loops, the initial \feld sweep (up and down) is performed after\nzero-\feld cooling from room temperature. Subsequent pulses are delivered after 40 to 60\nminutes (the cooling time of the magnet in question) while maintaining constant sample\ntemperature.\n140.000.020.040.060.080.100102030405060700\n.00.51.01.52.02.53.03.50102030405060short pulse magnet Magnetic field (T)(a)-\n404812S\nweep rate (kT/s)l\nong pulse magnet(b) T\nime (s)FIG. 6. Time-dependence of magnetic \felds for (a) a capacitor-bank-driven 65 T short-pulse\nmagnet and (b) three examples of the controlled sweep patterns possible with the generator-driven\n60 T Long Pulse Magnet. (a) \u00160dH=dtfor the short-pulse magnet is shown in blue (right axis).\nIn (b), the three stages in each of the \feld-sweep patterns are due to three separate coils that are\nenergized in sequence by the generator.\nVI. REFERENCES\n[1] Kim, B.J., et al. . Phase-Sensitive Observation of a Spin-Orbital Mott State in Sr 2IrO4.Science\n323, 3129 (2009).\n[2] Lovesy, S.W., Khalyavin, D.D., Manuel. P., Chapon, L.C. & Qi, T.F.. 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Introduction to Magnetic Materials. 2nd edition (Wiley, New\nYork, 2009)\n[37] Coey, J.M.D.. Advances in Magnetism - Hard Magnetic Materials: A Perspective. IEEE Trans.\nMagn. 47, 4671 (2011).\n[38] Autret, C., et al. . Magnetization steps in a noncharge-ordered manganite, Pr 0:5Ba0:5MnO 3.\nAppl. Phys. Lett. 82, 4746 (2003)\n[39] Morosan, E., et al. . Sharp switching of the magnetization in Fe 4TaS 2.Phys. Rev. B 75, 104401\n(2007)\n[40] Haldar, Arabinda, Suresh, K. G., & Nigam, A. K.. Magnetism in gallium-doped CeFe 2:\nMartensitic scenario. Phys. Rev. B 78144429 (2008)\n[41] Wu, Weida, et al. . Formation of Pancakelike Ising Domains and Giant Magnetic Coercivity in\nFerrimagnetic LuFe 2.Phys. Rev. Lett. 101, 137203 (2008).\n[42] Ko, K.-T., et al. . RKKY Ferromagnetism with Ising-Like Spin States in Intercalated Fe 4TaS 2.\nPhys. Rev. Lett. 107. 247201 (2011).\n[43] Jo, Y.J., et al. . 3:1 magnetization plateau and suppression of ferroelectric polarization in an\nIsing chain multiferroic. Phys. Rev. B 79, 012407 (2009)\n[44] Fleck, C.L., Lees, M.R., Agrestini, S., McIntyre, G.J., & Petrenko, O.A.. Field-driven mag-\nnetisation steps in Ca 3Co2O6: A single-crystal neutron-di\u000braction study. Europhys. Lett. 90,\n1867006 (2010).\n[45] Yanez-Vilar, S., et al. Multiferroic behavior in the double perovskite Lu 2MnCoO 6.Phys. Rev.\nB84, 134427 (2011).\n[46] Goddard, P.A., et al. . Experimentally determining the exchange parameters of quasi-two-\ndimensional Heisenberg magnets. New. J. Phys. 10, 083025 (2008).\nVII. ACKNOWLEDGMENTS\nThis work is funded by the U.S. Department of Energy, Basic Energy Sciences program\n\\Science at 100 Tesla\". The NHMFL Pulsed Field Facility is funded by the US National\nScience Foundation through Cooperative Grant No. DMR-1157490, the State of Florida,\nand the US Department of Energy. Work in the UK is supported by the EPSRC. PAG and\nJS would like to thank the University of Oxford for the provision of visiting fellowships. The\nwork at Rutgers was supported by the NSF under Grant number NSF-DMREF-1233349,\nand the work at Postech was supported by the Max Planck POSTECH/KOREA Research\nInitiative Program (Grant number 2011-031558) through NRF of Korea funded by MEST.\nVIII. AUTHOR CONTRIBUTIONS\nJ.S., J-W.K. and C.T. carried out the pulsed-\feld magnetization measurements, analyzed\nthe data and prepared the \fgures in this paper. C.T., P.A.G. and J.S. constructed the\npulsed-\feld magnetometer. E.D.M. assisted with pulsed-\feld experiments. The SQUID and\nAC and DC magnetometry data in this paper were measured by S.G., P.A.G. and A.H..\nSamples were grown, characterized and prepared by X.L, Y.S.O. and S-W.C.. The text\nwas written by J.S. and V.Z. with input from all authors. This study is part of a research\nprogram on multiferroics directed by V.Z..\nIX. ADDITIONAL INFORMATION\nThe authors declare no competing \fnancial interests.\n19"    },    {        "title": "1408.4078v2.Unquenched__e_g_1__orbital_moment_in_the_Mott_insulating_antiferromagnet_KOsO4.pdf",        "content": "arXiv:1408.4078v2  [cond-mat.str-el]  9 Dec 2014Unquenched e1\ngorbital moment in the Mott-insulating antiferromagnet KOs O4\nYoung-Joon Song1, Kyo-Hoon Ahn1, Kwan-Woo Lee1,2,∗and Warren E. Pickett3†\n1Department of Applied Physics, Graduate School, Korea Univ ersity, Sejong 339-700, Korea\n2Department of Display and Semiconductor Physics, Korea Uni versity, Sejong 339-700, Korea\n3Department of Physics, University of California, Davis, Ca lifornia 95616, USA\n(Dated: July 3, 2021)\nApplying the correlated electronic structure method based on density functional theory plus\nthe Hubbard Uinteraction, we have investigated the tetragonal scheelit e structure Mott insula-\ntor KOsO 4, whose e1\ngconfiguration should be affected only slightly by spin-orbit couping (SOC).\nThe method reproduces the observed antiferromagnetic Mott insulating state, populating the Os\ndz2majority orbital. The quarter-filled egmanifold is characterized by a symmetry breaking due\nto the tetragonal structure, and the Os ion shows a crystal fie ld splitting ∆ cf= 1.7 eV from the\nt2gcomplex, which is relatively small considering the high for mal oxidation state Os7+. The small\nmagnetocrystalline anisotropy before including correlat ion (i.e., in the metallic state) is increased by\nmore than an order of magnitude in the Mott-insulating state , a result of a strong interplay between\nlarge SOC and a strong correlation. In contrast to conventio nal wisdom that the egcomplex will not\nsupport orbital magnetism, we find that for the easy axis [100 ] direction the substantial Os orbital\nmoment ML≈ −0.2µBcompensates half of the Os spin moment MS= 0.4µB. The origin of the\norbital moment is analyzed and understood in terms of additi onal spin-orbital lowering of symmetry,\nand beyond that due to structural distortion, for magnetiza tion along [100]. Further interpretation\nis assisted by analysis of the spin density and the Wannier fu nction with SOC included.\nPACS numbers: 71.20.Be, 71.27.+a, 71.30.+h\nI. INTRODUCTION\nIn condensedmatter, especiallywhen containingheavy\nions, spin-orbit coupling (SOC) leads to phenomena\nthat are lacking without SOC. Examples of recent in-\nterest include the original topological insulators,1be-\nhavior arising from the Rashba effect, unconventional\nmetal-insulating transitions, compensating spin and or-\nbital moments,2,3and the magnetocrystalline anisotropy\n(MCA) that is so important in spintronics applications.\nWhereas SOC within a t2gmanifold in a MO6octahe-\ndron (M= transition metal) has a long history4–6and\nhas been intensively discussed recently in several spe-\ncific contexts,2,7–11corresponding effects in an egmani-\nfold have rarely been considered due to the conventional\nwisdom that the egsubshell ensures a perfectly quenched\norbital moment. From this viewpoint, heavy transition\nmetal oxides containing MO4tetrahedra are of great in-\nterest, since crystal field splitting leads to partially filled\norbitals in the egmanifold.\nAbout a century ago, monoclinic crystals of two toxic,\nvolatile materials, OsO 4and RuO 4, were synthesized.\nThese are presumably textbook band insulators, albeit\nwith remarkably high (8+) formal charges. Although ex-\nisting data on these crystals are limited, the effects of\nSOC have been investigated from a chemical viewpoint\nsince the 1990’s12–14and have been generally found to be\nminor. In 1985, heptavalent AOsO4(A= alkali metal)\ncompounds were synthesized by Levason et al., who de-\ntermined they formed in the tetragonal scheelite crystal\nstructure.15KOsO 4has been often synthesized from a\nmixture of KO 2and Os metal as a precursor for prepa-\nration of the superconductor KOs 2O6,16but further in-vestigations of its physical properties are still lacking.\nKOsO 4seems to be insulating, though detailed resistiv-\nity data are not yet available.17,18\nRecently, Yamaura and collaborators determined the\ncrystal structure parameters and measured the suscep-\ntibility and specific heat.18The Curie-Weiss moment is\nµeff= 1.44µB, 20% reduced from the spin-only moment,\nand the N´ eel temperature is TN= 37 K. These authors\nsuggested that magnetic frustration in this distorted di-\namond lattice may be necessary to account for observa-\ntions. However, the conventional ratio of Curie-Weiss\nto ordering temperatures |θCW|/TN≈1.8 is small (i.e.,\nthere is little frustration in the bipartite Os sublattice)\nso other factors must be considered.\nIn this paper we study the electronic structure of\nKOsO 4, with special attention given to the interplay be-\ntween strong correlation and SOC. The small ligand field\nsplitting of the egorbitals due to distortion of the OsO 4\ntetrahedron plays an important role in determining the\noccupied orbital in the Mott-insulating state, and may\nbecome active in effects arising from SOC as well. A\nmodestt2g-egcrystal field splitting (∆ cf= 1.7 eV) and\nlarge SOC strength ( ∼0.3 eV) bring in another effect of\ncrystallinity that impacts the effects of SOC. This split-\nting is especially small considering that in another Os7+\ncompound, the double perovskite Ba 2NaOsO 6, ∆cf= 6\neV is extremely large.2Results are analyzed in terms\nof magnetization densities, Wannier functions, and spin-\norbital occupation numbers. Symmetry reduction of the\nelectronic state due to SOC when the spin lies in the\n[100] direction is found to have a great consequence: A\npopulation imbalance of the ml=±2 orbitals leads to an\nunexpectedly large orbital moment, as discussed in Sec.\nV.FIG. 1: (Color online) (a) Scheelite-type crystal structur e\nof KOsO 4. (b)G-type antiferromagnetic (AFM) spin order-\ning, whichis thegroundstate inLSDA+ U+SOCcalculations.\nThe arrows indicate the calculated directions of spins (eas y\naxis).\nII. STRUCTURE AND CALCULATION\nMETHODS\nKOsO 4crystallizesinthescheelitelikestructure(space\ngroup:I41/a, No. 88), shown in Fig. 1. In this tetrago-\nnal structure with two formula units (f.u.) per primitive\ncell, the lattice parameters are a= 5.652 ˚A andc=\n12.664˚A,?leading to a ratio of c/√\n2a= 1.58. The\nOs sublattice forms a substantially elongated diamond\nsublattice; this c/√\n2aratio is unity for the cubic dia-\nmond lattice. The K and Os atoms sit at the 4 bsites\n(0,1\n4,5\n8) and 4asites (0,1\n4,1\n8), respectively. The O atoms\nlie on the 16 fsites (0.1320,0.0160,0.2028). In the OsO 4\ntetrahedron, all Os-O bond lengths are 1.81 ˚A, and the\nO-Os-Obond anglesareeither 114◦or107◦, comparedto\n109.5◦for a regular tetrahedron. A similar distortion is\nobserved in the band insulator OsO 4,19while both RuO 4\nand KRuO 4have nearly ideal tetrahedra.20,21This dif-\nference suggests that the distortion is due to a chemical\ndifference between Os and Ru ions.\nOur calculations were carried out with the local (spin)\ndensity approximation [L(S)DA] and its extensions, as\nimplemented in the accurate all-electron full-potential\ncodewien2k.22Since we are interested in a possible\ncompetition between large SOC and strong correlation\neffects in magnetic systems, we compare all of the LDA,\nLSDA, LSDA+SOC, LSDA+ U, and LSDA+ U+SOCap-\nproaches. An effective on-site Coulomb repulsion Uwas\nused for the LDA+ Ucalculations; since Os7+is ad1\nion which is not occupied by more than one electron, the\nHund’s rulecoupling Jbetweentwoelectronsofthe same\nspin was set to zero. To analyze the partially filled Os\ncomplex, the Wannier function approach implemented\ninfploandwien2khas been used.23,24Calculations of-7-6-5-4 -3 -2 -1 0 1 2\nE - EF (eV)051015DOS ( states/eV - f.u. )Total\nOs\nO\nKt2(6) + e(4)\na1(2)t2(6)t1(6)\ne*(4)t2*(6)\nFIG. 2: (Color online) LDA total and atom-projected densi-\nties of states (DOS) of nonmagnetic KOsO 4in the regimes of\nOs 5dand O 2porbitals. The symbols, which are displayed in\neach manifold, represent the molecular orbitals of the OsO 4\ntetrahedron, following the notations of Ref. [25]. The valu es\nin parentheses indicate the number of bands in each mani-\nfold. The symbol ∗denotes the antibonding state. The DOS\nN(EF) at the Fermi energy EF, which is set to zero, is 4.18\nstates/eV f.u. spin.\nthe Wannier function including SOC are available only\nin the latter. In wien2k, the following muffin-tin radii\nare adopted: 2.02 for Os, 1.4 for O, and 2.2 for K (in\nunits of a.u.). The extent of the basis was determined by\nRmtKmax= 7. The Brillouin zone was sampled with a\nsufficiently dense kmesh (for an insulator) of 13 ×13×6.\nIII. THE UNDERLYING ELECTRONIC\nSTRUCTURE\nFigure 2 displays the LDA total and atom-projected\ndensities of states (DOSs), which demonstrates a strong\np-dhybridization not only in the most relevant Os eg\nbands (denoted as the molecular e∗orbitals) but also in\nmore tightly bound oxygen orbitals around –7 eV. This\nhybridization of the transition metal dcharacter into O\n2pbands is common but is not particularly relevant and\nis little discussed. The narrow bands reflect moderately\nbanding molecular orbitals. Some nearly pure oxygen\nbands lie in the –6 to –3 eV range. The t∗\n2bands centered\naround2eVareamixtureofOs t2g, andallO2 porbitals,\nwhile the e∗set is a mixture of egand mostly pπ.\nBefore considering the complications of spin polariza-\ntion, correlation effects, and SOC, we consider the basic\nunderlying features of the electronic structure. Suppos-\ningformalchargesofK+, Os7+, andO2−ions, thecrystal\nfieldeg–t2gsplitting is expected to be 0.8 eV, about half\nof the calculated splitting ∆ cf= 1.7 eV, which is the full\nligand field splitting.\n2Γ H N Γ P-0.400.40.81.21.622.4E - EF (eV)\ndx2-y2\ndz2\nFIG. 3: (Color online) LDA Os 5 dband structure of nonmag-\nnetic KOsO 4, showing an eg–t2gcrystal field splitting of ∼1.8\neV. The partially filled egbands, which are colored with the\ncorresponding Wannier orbitals, lie on the range of –0.3 to 0 .3\neV. In units of ( π/a,π/a,π/c), the symmetry points shown are\nH= (100), N= (1\n21\n20), andP= (1\n21\n21\n2).\nThe LDA nonmagnetic band structure in the Os 5 d\nband region (ten bands due to 2 f.u. per primitive cell)\nis displayed in Fig. 3. The distortion of the OsO 4tetra-\nhedron leads to the crystalfield splitting of dxyabovethe\ndegenerate pair {dxz,dyz}, as is evident in Fig. 3. No-\ntably, the isolated partially filled egmanifold can be fit\nwell using an effective two-bandmodel with three nearest\nneighbor hopping parameters. The hopping parameters\ncorresponding to the corresponding Wannier functions\nare\nt1=∝an}bracketle{tdx2−y2|ˆH|dx2−y2∝an}bracketri}ht=43 meV ,\nt2= ∝an}bracketle{tdz2|ˆH|dz2∝an}bracketri}ht =56 meV ,\nt′= ∝an}bracketle{tdz2|ˆH|dx2−y2∝an}bracketri}ht= 7 meV .(1)\nThe site energies are 59 meV for dz2and 143 meV for\ndx2−y2relative to EF. It is this ligand field splitting\nof 84 meV that determines that the dz2becomes occu-\npied in the Mott-insulating phase (below). As expected\nfrom the small value of t′, each of the dx2−y2anddz2\nbands can be fit nearly as well along symmetry lines us-\ning two independent single band models. A noticeable\nmixing between the two bands only occurs along the Γ-\nHline. The superexchange coupling parameter is deter-\nmined from J=t2/U∼2 meV, using U= 2 eV (see\nbelow). The magnitude of this exchange coupling is sim-\nilar to the ordering temperature kBTN≈3 meV.\nIV. EFFECTS OF CORRELATION AND SOC\nA primary emphasis in our study of this system is to\nassess the interplay in an egsystem between strong cor-Γ H           N           Γ P           00.511.52E - EF (eV)\nLSDA\nLSDA + SOC\nΓ H           N           Γ         P           00.511.52E - EF (eV)LSDA + U\nLSDA + SOC + U\nFIG. 4: (Color online) AFM band structures of (top) LSDA\nwithout and with SOC, and (bottom) LSDA+ Uwithout and\nwith SOC, for U= 2 eV. In the insulating state, the occupied\nstate is mainly dz2.\nrelations, which prefer full occupation of certain orbitals\nand usually increase spin polarization, and SOC, which\nmixesspinorbitalsandcomplicatesallaspectsoftheelec-\ntronic structure while inducing the orbital moment and\nmagnetocrystalline anisotropy (MCA). It was mentioned\nabove that including correlation effects in the LSDA+ U\nmethod leads to preferred occupation of the dz2orbital,\nwhichhas84meVloweron-site(crystalfield)energythan\ndx2−y2due to the distortion of the OsO 4tetrahedron.\nThe band structures including the lower part of the t2g\ncomplex, and the DOS of the egbands alone for the en-\nergetically preferred AFM state, are displayed in Figs. 4\nand 5, respectively. These figures have been constructed\nto allow identification of the individual effects of Uand\nSOC.\nBefore proceeding with a description of the full elec-\ntronic structure and then the spin density itself, we re-\n3TABLE I: Effect of correlation Uon the relative energies ∆ E(in units of meV/f.u.) and Os orbital moments ML(in units\nofµB) for each of four spin quantization directions and for FM and AFM alignments. MLof Os is antialigned to the spin\nmoment of Os, which is ∼0.4µBfor the insulating states. The spin moments contributed by O ions are 0.24 – 0.32 µB/4O in\nLSDA+SOC, increasing to ∼0.4µB/4O for LSDA+SOC+U. U= 2 eV was used for LSDA+SOC+ Ucalculations.\nAFM FM\n[100] [001] [110] [111] [100] [001] [110] [111]\nLSDA+SOC ∆ E0 4.6 2.3 3.7 1.9 3.9 1.8 3.4\nLSDA+U+SOC ∆ E0 14.4 3.0 10.6 19.3 28.7 19.8 26.2\nLSDA+SOC ML–0.134 –0.014 –0.136 –0.052 –0.135 –0.048 –0.135 –0.073\nLSDA+U+SOCML–0.184 0.007 –0.183 –0.053 –0.176 0.006 –0.172 –0.055\nview the energy differences arising from the various in-\nteractions.\nA. Magnetic Energy Differences\nAs expected from the peak at EFin the DOS (see\nFig. 2) and the well known Stoner instability, ferro-\nmagnetism (FM) is energetically favored over the non-\nmagnetic state, by 26 meV/f.u. Our fixed spin mo-\nment calculations of the interacting susceptibility26lead\ntoIN(EF) = 1.60, well above the Stoner instability cri-\nterion of unity, and N(EF) = 4.09 states/eV f.u. spin\ngives the Stoner parameter I= 0.39 eV, similar to the\nvalue obtained2for Ba 2NaOsO 6. Within metallic LSDA\nwhere exchange coupling might be considered to be some\nmixture of double exchange, Ruderman-Kittel-Kasuya-\nYosida (RKKY), and superexchange, the FM ground\nstate lies 5.5 meV/f.u. below the observed AFM state.\nTo assess the effects of SOC before including cor-\nrelation corrections, we display MCA energies with\nLSDA+SOC with several spin orientations in Table I.\n[100] is the AFM easy axis, however, all spin orienta-\ntions differ little in energy compared to the larger differ-\nences when all interactions are included (see below), so\nthe magnetic anisotropy is predicted to be small at this\nlevel of theory. This tentative conclusion, before includ-\ning correlation, is consistent with conventional wisdom\nthat SOC has little effect in egsystems.\nAfter including correlation with U= 2 eV, the AFM\nMott-insulatingstateisobtained(discussedbelow)asthe\nground state, by 19 meV over FM alignment. This favor-\ning of antiferromagnetism over ferromagnetism is com-\nmon when applying the LDA+ Ufunctional in transition\nmetal oxides. For bipartite AFM (alternating) alignment\ncompared to FM alignment, the AFM magnetic coupling\nisJ≈4.8 meV ≈56 K, consistent in magnitude with\nthe experimental ordering temperature TN= 37 K.\nOtheraspectsoftheinterplaybetweenstrongSOCand\nstrong correlation are apparent in Table I. Most notable\nin the energetics is that strong correlationeffects (i.e., in-\ncludingU) greatly enhance the MCA: Energy differences\nbetween different directions of the spin are more than an\norder of magnitude larger. This is more surprising whenone recalls that SOC effects (which provide the MCA)\nare often supposed to be negligible in egsubshells. In-\ncluding both large Uand SOC, the [100] direction is now\nvery clearly determined as the easy axis.\nB. LSDA+SOC+ Uleads to a Mott-insulating state\nThe lowest and highest bands (Fig. 4) in the egman-\nifold extending over the regime of –0.25 to + 0.5 eV are\nthe Os spin-up and spin-down dz2bands, respectively. In\nthe quarter-filled egmanifold, the dz2-dx2−y2degeneracy\nlifting is 0.2 eV, i.e., the egdegeneracy is already split\n(presumably self-consistently by occupation of the dz2\norbital and the resulting Jahn-Teller distortion). Apply-\ning the on-site Coulomb repulsion Ustarting from small\nvalues leads to a metal-insulator transition (MIT) (gap\nopening) at a critical value Uc≈1.2 eV, which is near\nthe bottom of the range of expected values for Os. As\nshown in the bottom panel of Fig. 4 for U= 2 eV and\nspin along the [100] direction, the top of the occupied\nband has a flat region around the Npoint, giving rise to\na one-dimension-like peak and sharp discontinuity in the\nDOS at the top of the band, evident in Fig. 5. Other\nband maxima at Pand midway between Γ and Nare\n(somewhat accidentally) degenerate with the flat band\natN. The occupied bandwidth is 0.2 eV. As shown by\nthe red dashed lines in the band structure of Fig. 4, in-\nclusion of SOC a has negligible effect on the occupied\nstate (position and dispersion) but lowers the uppermost\negband (primarily minority spin) by 0.15 eV. This shift\ncorresponds to a small decrease in the exchange splitting\nof the unoccupied egorbital.\nC. Effects of SOC on spin and orbital moments\nIn the following text and in Table I we quote atomic\nmomentsfromcontributionswithintheinscribedspheres,\nwhich are somewhat smaller than the full value. We re-\nmind that the occupied “ dz2” orbital that is occupied\nbefore including SOC is strongly hybridized with 2 por-\nbitals of the surrounding O ions, so the spin magnetiza-\ntion of 1µBis distributed over oxygen as well as Os. The\n4-0.2 0 0.2 0.4 0.6 0.8\nE - EF (eV)0369DOS ( states/eV - f.u. - spin )Total\nOs(up)\nOs(down)\nO1\nO2\n-0.2 0 0.2 0.406LSDA\nLSDA + SOC\n-0.2 0 0.2 0.4 0.6 0.8\nE - EF (eV)051015DOS ( states/eV - f.u. - spin )Total\nOs(up)\nOs(down)\nO1\nO2-0.2 0 0.2 0.4 0.6 0.8010LSDA + U\nLSDA + SOC + U\nFIG. 5: (Color online) AFM densities of states of (top) LSDA\nand (bottom) LSDA+ UatU= 2 eV, with atomic contribu-\ntions differentiated. Inset: Comparison of DOS with the case\nincluding SOC, near EF. In the metallic U= 0 state, the\nexchange splitting in the egmanifold is 0.4 eV. In the insu-\nlating state, in terms of the spin-up Os, the plots contain th e\nfilled spin up dz2, the unfilled dx2−y2, and the unfilled spin\ndowndz2bands, from the lower energy. In LSDA, N(EF)≈4\nstates/eV spin f.u., but lies on a very sharp edge. Inclusion\nof SOC reduces N(EF) by 7%.\nmoment values should be considered in conjunction with\nthe spin density isosurfaces pictured in Fig. 6.\nFor all spin orientations we have determined that the\nOs spin moment is MS≈0.4µB. This value is almost in-\ndependent of Uin the range0–5eVthat we havestudied.\nThe O net spin MS= 0.07µB/O aligns parallel to that of\nthe nearest neighbor Os. The sum of the full atom mo-\nments must be unity, so atomic values are around 0.5 µB\nand 0.12µBfor Os and O, respectively, versus the atomic\nsphere values just quoted. Including SOC reduces the\nOs spin moment by 10%, transferring the difference to\nFIG. 6: (Color online) Isosurface plots at 0.042 e/˚A3of spin\ndensities for (a) metallic ( U= 0) and (b) insulating AFM ( U\n= 2 eV) KOsO 4, when including SOC. Os at the center is\nsurrounded by four O ions. Red indicates majority spin, and\nblue denotes minority. The insulating density in (b) reflect s\nthe circular shape around both Os and O that provides the\norbital moment.\nneighboring O MSdue to rehybridization. Nonzero ML\nmust arise from mixing in of t2gcharacter, as we discuss\nin Sec. V. For [100] and [110] spin directions, increasing\nUincreases |ML|from∼1/3MSatU= 0 to 1/2 MSat\nU= 2 eV. For [001] spin orientation MLis essentially\nvanishing for any value of U(Table I).\nD. Behavior of the spin density\nThespin densityisosurfaceplotsdisplayedin Fig. 6for\nmetallic ( U= 0) and AFM insulating ( U= 2 eV) phases\nare instructive. Even in the metallic, uncorrelated case\nboth positive and negative lobes of spin density appear\non both Os and O ions, indicating more complexity than\nstrong (but typically simple) p-dhybridization. Since in\nthis limit the lowerHubbard band is fully polarized (only\nspin-up states), the negative polarization arises from po-\nlarizationwithin the filled nominally O 2 pbands at lower\nenergy.\nThe net spin of O lies in 2 porbitals whose orienta-\ntion reflects a πantibonding character with Os dxz,dyz\norbitals. A small negative spin density is induced in a\nlinear combination of the px,pyorbitals, in the local co-\nordinate system. As expected, the Os spin lies mainly in\nthedz2orbital, with some admixture of dx2−y2account-\ning for the square versus circular symmetry of the spin\ndensity in the equatorial plane of Os. A small but clear\nadmixture of dyzanddxzcharacter appears as a negative\nspin density (blue), and this contribution is necessary to\nprovide the Os orbital moment.\nIn the correlated ( U= 2 eV) insulating state, Os\nstill has mainly a dz2character for spin up. However,\nthe circular symmetry indicating dyz-idxzcharacter for\nspin down shows up much more clearly. Unexpectedly,\nthis same development of px-ipy(in an appropriate lo-\ncal frame) shows up on the O ions in the spin-down re-\n5FIG. 7: (Color online) Isosurface plots of the Wannier func-\ntion density |Wσ(r)|2of the occupied band in AFM insulat-\ning KOsO 4(LSDA+ U+SOCwith U= 2 eV), for the majority\nσ=| ↑/angbracketrightand minority σ=| ↓/angbracketrightspins separately. The surfaces\nare colored with the cosine of the phase of each component\n(real positive, red; real negative, blue; imaginary. green ), as\ndescribed in color legend bar. (a), (b) Spin in the [100] di-\nrection; (c), (d) spin in the [001] direction. (a) and (c) are\nfor the majority spin, shown at isovalue = 2 a.u. (b) and (d)\nare for the minority spin, shown at the smaller isovalue of 0. 3\na.u. While the minority spin component is small, its direc-\ntional dependence is evident, with a much larger imaginary\npart for [001].\ngion, whilethespin-up, local pxcharacterisnearlyundis-\nturbed.\nTo generate the complex-valued, mixed-spin Wannier\nfunction W(r) of the occupied band, we projected from\n|5\n2,1\n2∝an}bracketri}htand|3\n2,1\n2∝an}bracketri}htasa trial function in the wien2wannier\npackage. Figure 7 presents isosurface plots of |W(r)|2\nfor each of the two components of spin. The spin-down\npart is much smaller than the spin-up part, as indicated\nseparately by the spin moment which remains close to\n1µB/f.u. Consistent with Eqs. (2) and (3) below, the\nspin-down parts are dxz-like in the [100] direction and\ndxz-idyz-like in the [001] direction. The spin-up parts are\ndz2-like in both directions, but have a squarish negative\nlobe ratherthan a circularshape. The complex character\nof the spin-up part in the [100] direction is visible only\naround the neck of the positive lobe, since the imaginary\npart is small. However, this complex character leads to a\nsymmetry breaking between ml=±2 orbitals, as will be\ndiscussed below. The spin-up part in the [001] direction\nis purely real.\nV. ANALYSIS OF SPIN-ORBIT COUPLING IN\nTHEe1\ngCASE\nNow we address the effects of SOC, especially the ap-\npearance of a surprisingly large orbital moment in an eg\nsubshell which should not produce an orbital moment,\nthrough analysis of the occupation matrices and the as-sociated Wannier function. SOC effects in the egchannel\ntend to be relegated to the background because egcon-\ntains only orbital ml= 0 and ml=±2dorbitals, which\nare not coupled by the electron spin s=1\n2. Note, how-\never, that this is strictly true only in the spherical (iso-\nlated ion) limit and for orbital moments along the axis\nof quantization, i.e., the direction of the spin. Indeed,\nwe find negligible orbital moments for spin along [001].\nCrystalline effects break this orbital-momentkilling sym-\nmetry.\nFirst, in KOsO 4, the crystal field splitting ∆ cf= 1.7\neV is slightly larger than the SOC strength, so virtual\ninclusion of t2gorbitals may be involved. Second, the\nOsO4tetrahedron is distorted, breaking the twofold eg\nsymmetry, which is related to the Mott-insulating be-\nhavior: occupation of a single egorbital and the accom-\npanying Jahn-Teller distortion. Finally, the higher sym-\nmetry crystalline ˆ zaxis is not the easy axis, so additional\ncomplexities arise. The focus begins with the dz2orbital\nthat is occupied before SOC is included, with a slight ad-\nmixture of other 5 dorbitals due to structural symmetry\nbreaking and hopping.\nSpin along [001]. Applying /vectorL·/vectorSto the spin-up dz2\norbital leads to\n/vectorL·/vectorS|dz2∝an}bracketri}ht| ↑∝an}bracketri}htz∝ −(|dxz∝an}bracketri}ht+i|dyz∝an}bracketri}ht)| ↓∝an}bracketri}htz,(2)\nwhich are nominally unoccupied orbitals. Indeed, we cal-\nculate negligible MLfor this orientation, reflecting negli-\ngible intermixing of dxz±idyzorbitals across the crystal\nfield gap ∆ cf. The main occupation amplitudes (eigen-\nvectors of the occupation matrix) are (in |ml,mS∝an}bracketri}htnota-\ntion), 0.96 for |0,↑∝an}bracketri}htand a down-spin amplitude of –0.21\nfor|+1,↓∝an}bracketri}ht(thus decreasing the spin moment by 4%).\nSpin along [100] . For in-plane [100] spin orientation\nSOC leads to the common picture\n/vectorL·/vectorS|dz2∝an}bracketri}ht| ↑∝an}bracketri}htx∝ −i|dyz∝an}bracketri}ht| ↑∝an}bracketri}htx−|dxz∝an}bracketri}ht| ↓∝an}bracketri}htx.(3)\nAnother way to approach the emergence of an orbital\nmoment is to note that when the dz2orbital is expressed\nin local coordinate system X,Y,Z, withZdirected along\n[100], it is a linear combination of dZ2anddX2−Y2or-\nbitals (i.e., the egorbitals). Breaking of symmetries may\ninduce an asymmetry in the ml=±2 orbitals making up\ndX2−Y2. Indeed, this happens. Table II shows the am-\nplitudes of the occupation matrix eigenstate in the local\ncoordinate system. The imbalance in the ml= +2 and\nml=−2 occupations in the spin-up channel results in a\nsurprisingly large (for an egshell) orbital moment.\nVI. SUMMARY\nMaterials such as KOsO 4with ane1\ngconfiguration are\nexpected to have a negligible orbital moment. Mixing of\nt2gcharacter is required, which is aided by small crys-\ntal field splitting and structural symmetry lifting. We\nhave studied the interplay of strong correlation effects\n6TABLE II: Amplitude coefficients of the occupied orbital, ex-\npressed with respect to complex orbitals and both spin com-\nponents, in the the local coordinate system with spin along\nthe [100] direction.\nml\n0 –1 +1 –2 2\n| ↑/angbracketright–0.47 –0.09 i0.11i0.50 0.70\n| ↓/angbracketright0.01i–0.08 0.07 –0.07 i0.08i\nand large spin-orbit coupling strength, and have found\nthat an additional characteristic is very important: the\nadditional symmetry breaking of the electronic state by\nspin-orbit coupling itself. The spin-direction dependent\norbitalmoment in this Os7+e1\ngsystem has been analyzed\nand understood. The occupied orbital without spin-orbit\ncoupling is dz2| ↑∝an}bracketri}ht. For spin along the [001] axis, indeed\nthere is negligible mixing with ml∝ne}ationslash= 0 orbitals and the\nonly change due to SOC is a few percent reduction in the\nspin moment.For the spin along the in-plane [100] axis, however,\nSOC further breaks x↔ysymmetry, inducing a popu-\nlation imbalance in the ml=−2 andml= +2 orbitals\nrelative to the spin direction, which drives the unexpect-\nedly large orbital moment ML= –0.2µB. This moment\ncancels half of the Os spin moment, and the accompany-\ning magnetocrystalline anisotropy favors this [100] spin\norientation.\nVII. ACKNOWLEDGMENTS\nWe acknowledge J. Yamaura for communications on\nresistivity measurement, and J. Kuneˇ s for useful dis-\ncussions on the calculations of Wannier functions in-\ncluding SOC. This research was supported by Na-\ntional Research Foundation of Korea Grant No. NRF-\n2013R1A1A2A10008946 (K.W.L.) and by U.S. Depart-\nment of Energy Grant No. DE-FG02-04ER46111\n(W.E.P.).\n∗Electronic address: mckwan@korea.ac.kr\n†Electronic address: pickett@physics.ucdavis.edu\n1C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 146802\n(2005).\n2K.-W. Lee and W. E. Pickett, EPL 80, 37008 (2007).\n3O. Nganba Meetei, W. S. Cole, M. Randeria, and N.\nTrivedi, arXiv:1311.2823.\n4K. W. H. Stevens, Proc. R. Soc. London, Ser. A 219, 542\n(1953).\n5J. B. Goodenough, Phys. Rev. 171, 466 (1968).\n6C. Lacroix, J. Phys. C 13, 5125 (1980).\n7H. Jin, H. Jeong, T. Ozaki, and J. Yu, Phys. Rev. B 80,\n075112 (2009).\n8G. Chen and L. Balents, Phys. Rev. B 84, 094420 (2011).\n9T. Dodds, T.-P. Choy, and Y. B. Kim, Phys. Rev. B 84,\n104439 (2011).\n10M.-C. Jung, Y.-J. Song, K.-W. Lee, and W. E. Pickett,\nPhys. Rev. B 87, 115119 (2013).\n11H. Matsuura and K. Miyake, J. Phys. Soc. 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Box 270116, 01171 Dresden, Germany\n2)TU Dresden, Faculty of Mechanical Engineering, Institute of Materials Science,\n01062 Dresden, Germany\n3)Demokritos National Center of Scienti\fc Research, 15310 Athens,\nGreece\n4)Uppsala University, Department of Physics and Astronomy, 75120 Uppsala,\nSweden\n(Dated: 20 October 2018)\nRare earth free alloys are in focus of permanent magnet research since the accessibility\nof the elements needed for nowadays conventional magnets is limited. Tetragonally\nstrained iron-cobalt (Fe-Co) has attracted large interest as promising candidate due to\ntheoretical calculations. In experiments, however, the applied strain quickly relaxes\nwith increasing \flm thickness and hampers stabilization of a strong magnetocrys-\ntalline anisotropy. In our study we show that already 2 at% of carbon substantially\nreduce the lattice relaxation leading to the formation of a spontaneously strained\nphase with 3 % tetragonal distortion. In these strained (Fe 0:4Co0:6)0:98C0:02\flms, a\nmagnetocrystalline anisotropy above 0.4 MJ/m3is observed while the large polariza-\ntion of 2:1 T is maintained. Compared to binary Fe-Co this is a remarkable improve-\nment of the intrinsic magnetic properties. In this paper, we relate our experimental\nwork to theoretical studies of strained Fe-Co-C and \fnd a very good agreement.\nPACS numbers: 61.05.Cp, 61.05.Jh, 68.35.Bd, 68.37.Og, 68.55.jm, 71.15.Mb, 75.30.Gw,\n75.70.Ak, 81.15.Fg\nKeywords: Fe-Co, rare earth free permanent magnet, magnetocrystalline anisotropy,\ntetragonal strain, DFT, RHEED\na)Electronic mail: l.reichel@ifw-dresden.de\n1arXiv:1409.4952v1  [cond-mat.mtrl-sci]  17 Sep 2014I. INTRODUCTION\nPermanent magnets are everywhere in our daily lives and will become more important\nin near future. There is a need for new materials as the abundance of the common rare\nearth based alloys has been questioned within the last years1. However, competitive alter-\nnatives are still missing. The alloy Fe-Co has often been considered a promising candidate\nfor rare earth free permanent magnets2{4. Its very high intrinsic magnetic moment5,6is one\nprerequisite. The second requirement for a powerful permanent magnet is a strong mag-\nnetic anisotropy. However, this was not reported yet in Fe-Co samples of considerable size.\nThe reason is the cubic symmetry of the alloy's unit cell in thermodynamic equilibrium.\nThis causes an only weak cubic magnetic anisotropy7. Typically, the magnetocrystalline\nanisotropy (MCA) is weak in cubic materials since spin-orbit coupling is lower than in\nuniaxial crystal structures which tend to exhibit strong magnetic anisotropies. Already a\ndecade ago, it was proposed to overcome the challenge of magnetic anisotropy by straining\nthe unit cell2. Thec=aratio of the unit cell was used to describe the tetragonal strain\nwherecis the length of the strained axis and athe length of the axes perpendicular to\nthe applied strain. Calculations2,3based on Density Functional Theory (DFT) predicted\nremarkably high MCAs if the unit cell would be tetragonally distorted with c=abetween 1.2\nand 1.25. The most common approach to strain the unit cell experimentally8{11is coherent\nepitaxial growth of thin \flms on suitable substrates that provide appropriate in-plane lattice\nparameters a. Assuming constant volume of the Fe-Co unit cell, a strain parallel to the \flm\nnormal is expected when ais smaller than the equilibrium lattice parameter of Fe-Co. Up\nto now, tetragonally distorted Fe-Co \flms with perpendicular easy axis of magnetization\ncould not be produced with \flm thicknesses exceeding 15 monolayers (ML)9{11. Thicker\n\flms do not exhibit a perpendicular easy axis since lattice relaxation takes place. After\nstrain relaxation, shape anisotropy exceeds MCA and dominates the magnetic performance\ncausing in-plane easy axes. This thickness dependent spin reorientation transition has been\ndescribed recently by Kim and Hong12.\nIn order to maintain the strain in \flms thicker than 15 ML, new concepts have to be\napplied. In a recent study, Delczeg-Czirjak et al.13calculated that small atoms like carbon\nmay stabilize the strain in Fe-Co. Their DFT calculations identify the preferential positions\nof the C atoms being interstitials along the caxis of the Fe-Co lattice (Fig. 1) preferring\n2lattice positions between two Co atoms. This may be the main reason for the resulting strain\nsince an arbitrary contribution of C atoms would result in an isotropically strained lattice\nwith no tetragonal distortion. Structures of Fe-Co-C phases with up to 11 at% carbon have\nbeen studied theoretically. The highest magnetic anisotropy energies MAE are reported13\nfor (Fe xCo1\u0000x)16C i. e. in Fe-Co including 5.9 at% C. For such a structure with c=a= 1:12,\ntheMAE was calculated as total-energy di\u000berence for the two magnetization directions to\n0:75 MJ/m3.\nOur study presents the \frst results on Fe-Co thin \flms alloyed with small amounts of\ncarbon. By measuring the structural \flm properties during growth, we can compare these\nFe-Co-C \flms to binary Fe-Co \flms. We show that already 2 at% C stabilizes a tetragonal\ndistortion of the former cubic unit cell. Films with tetragonal distortion of the lattice also\nshow the presence of a MCA pointing in direction of the strained caxis. Their strength is\nin the range of the predictions from DFT calculations. Since the structural and magnetic\nproperties do not diminish at high \flm thickness, Fe-Co with 2 at% C is considered to be a\nspontaneously strained phase.\nII. EXPERIMENTAL AND THEORETICAL METHODS\nFilm preparation was performed by Pulsed Laser Deposition (PLD) in ultra high vacuum\n(5\u000210\u00009mbar) at room temperature. A KrF excimer laser (Lambda Physik LPX 305i) with\n248 nm wavelength and pulse length of 25 ns was used. The \flm substrates were MgO(100)\nsingle crystals. Prior the Fe-Co-C layer, a 50 nm thick Au-Cu bu\u000ber layer was deposited\non a 3 nm thick Cr seed layer. The composition of the Au xCu1\u0000xlayer was adjusted by\nco-deposition from Au and Cu element targets, i. e. the laser pulses hit the targets in an\nalternating manner. For the Fe-Co-C layers, co-deposition from three targets was applied\nwhich were element targets of Fe and Co and a Fe 80C20composite target. During \flm\ngrowth, in situ re\rection high energy electron di\u000braction (RHEED) was performed to study\nthe lattice properties. The electron energy was 30 keV. The di\u000braction pattern was recorded\nby a CCD camera which was triggered by the PLD software. Each RHEED record could\nbe linked to a certain \flm thickness as the deposition rates were prior measured with a rate\nmonitor. With a user developed analysis software, the lattice parameters aandcof the\ngrown \flm were obtained by \ftting the RHEED re\rections.\n3Film compositions were con\frmed by energy dispersive x-ray (EDX) measurements on a\nBruker EDX at a JEOL JSM6510-NX electron microscope. The carbon content was mea-\nsured by Auger Electron Spectroscopy (AES) on a JEOL JAMP-9500F Field Emission Auger\nMicroprobe device after sputter cleaning of the surface with Ar+ions. As reference material\nfor the C quanti\fcation the Fe 80C20target was used. The surface topography was investi-\ngated by atomic force microscopy on an Asylum Research Cypher AFM. X-ray di\u000braction\n(XRD) was performed on a Bruker D8 Advance di\u000bractometer in Bragg-Brentano geometry\n(Co-K\u000bradiation). Pole \fgure measurements were carried out on an X'pert four circle go-\nniometer (Cu-K \u000bradiation). Transmission electron microscopy (TEM) investigations were\nconducted to con\frm the obtained distortion on a Titan380-300 microscope, equipped with\nan imaging C Scorrector and a Schottky \feld emission electron source. The \flm lamellae\nwere cut with focused ion beam on a FEI microscope (Helios NanoLab 600i). Magnetic mea-\nsurements were performed in a Quantum Design Physical Property Measurement System\nusing a vibrational sample magnetometer (VSM) which operated at 40 Hz and 300 K.\nTwo di\u000berent density functional theory (DFT) methods were utilized in the computa-\ntional part of this work. First, full potential code WIEN2k14with linearized augmented\nplane waves as basis functions was used. This method treats core electrons fully relativisti-\ncally, while valence electrons are treated in the scalar relativistic approximation with spin-\norbit coupling, which is essential for the calculation of MAE , added in a second variational\napproach15. TheMAE was evaluated using the force theorem16and disorder was treated by\nthe virtual crystal approximation (VCA) which is, however, known to overestimate MAE in\nFe-Co based systems, although it reproduces the correct qualitative behavior3,4,13. Hence, a\nsecond DFT approach, namely the spin polarized relativistic Korringa-Kohn-Rostoker (SPR-\nKKR) code17,18was used. In SPR-KKR all electrons are treated fully relativistically, but a\nspherical shape approximation is applied for the potential. Disorder is treated via the more\nrealistic coherent potential approximation (CPA) and MAE is evaluated by total energy\ndi\u000berences for two di\u000berent magnetization directions. In both DFT methods the generalized\ngradient approximation19was used for the exchange-correlation potential.\n4FIG. 1. Schematic (Fe 0:4Co0:6)32C supercell as applied in the DFT calculations. Fe and Co atoms\nare marked in red, the smaller C atoms (black) are aligned along the caxis.\nIII. IN SITU OBSERVATIONS DURING FE-CO(-C) FILM GROWTH\nAccording to theory2,3, the magnetic properties of coherently grown Fe-Co \flms should\ndepend on the applied in-plane lattice parameter of the bu\u000ber layer. As a versatile bu\u000ber\nlayer, we selected Au xCu1\u0000x. In this binary system, composition can be used to tune the\nbu\u000ber lattice parameter in a wide range20. For our study, Au contents xbetween 0.45\nand 0.71 were applied which led to abuffer between 0:272 nm and 0 :281 nm1. This is in\nagreement with studies on sputtered Au xCu1\u0000x\flms by Kau\u000bmann-Weiss et al.20, where a\nsmall positive deviation from Vegard's Law for a mixed Au-Cu crystals was reported.\nIn order to monitor the \flm growth in situ, RHEED patterns for the [110] azimuthal\n1In di\u000berence to all other presented \flms, the binary \flm was deposited on an Ir bu\u000ber layer. The lattice\nparameter of Ir ( abcc=aIr=p\n2 = 0:272 nm) is identical when compared to the Au-Cu bu\u000ber of the\nFe-Co-C \flm in Fig. 2b and c.\n5FIG. 2. a) Exemplary RHEED pattern of a (Fe 0:4Co0:6)0:98C0:02\flm of 2:4 nm thickness recorded\nduring \flm deposition along the FeCo[110] azimuth. The horizontal distance between the re\rections\nwas used to obtain the reciprocal in-plane lattice parameter a, the vertical distance gives the out-\nof-plane lattice parameter c. b) In-plane lattice parameters aof a (Fe 0:4Co0:6)0:98C0:02\flm (open\ncircles) compared to a binary Fe-Co \flm (full squares). c) c=aratios for these \flms.\n6direction of Fe-Co were recorded during \flm deposition. An exemplary RHEED pattern\nfor the Fe-Co-C \flm is shown in Fig. 2a. As measured with atomic force microscopy, the\ngrown \flms have a roughness RMS of approx. 0 :3 nm. The \flm surface thus does not grow\nin layer-by-layer-mode. Accordingly, a dot-like di\u000braction pattern is detected which allows\nfor the determination of both lattice parameters canda. An automated analysis, \ftting\nthe detected intensity pro\fle with peak functions was used. In Fig. 2b, in-plane lattice\nparameters aderived from RHEED measurements are plotted versus the thickness of grown\n\flmdFe\u0000Co. Fig. 2c depicts the c=aratio in dependence on dFe\u0000Co. In the graphs, a binary\nFe0:46Co0:54\flm (full squares) is compared to a ternary Fe 0:4Co0:6\flm containing 2 at% C\n(open circles).\nAt \frst, we will discuss the results for the binary Fe 0:46Co0:54\flm. The progression of the\nin-plane lattice parameter (full squares, Fig. 2b) shows that a\frst keeps a nearly constant\nvalue which is identical to the lattice parameter of the used bu\u000ber a buffer in this crystal-\nlographic direction. Hence, the \frst 2 nm of the \flm grow coherently. This is equivalent\nto 14 ML and con\frms observations of other groups which studied ultrathin epitaxial Fe-\nCo \flms8{11. They reported that only Fe-Co \flms with thicknesses up to 15 ML exhibit a\nstrong magnetocrystalline anisotropy due to tetragonal distortion. In this paper, however,\nwe describe \flm thickness in nm units as we intend to increase the thickness where strain is\nstill present.\nWith further increasing the \flm thickness, the described lattice coherence is lost in the\nFe0:46Co0:54\flm. This limit is called critical thickness dC. In the following, aincreases\ncontinuously and, within less than 2 nm additionally grown \flm, reaches a value at which it\n\fnally remains. The increase of areveals the lattice relaxation which commonly proceeds by\nintroduction of mis\ft dislocations. These dislocations do not form before a critical thickness\ndCis reached because of the required dislocation energy. The described relaxation ends\nwhen a \fnal value of ais reached, where no more changes in the lattice occur. For the\nbinary \flm, this \fnal value is equal to the equilibrium lattice parameter21. The described\nmechanisms are also observed on the c=acurve in Fig. 2c. For the \frst monolayers of grown\n\flm, (c=a)bctis close top\n2 which is the expected value for an fcc structure described by\nthe Bain transformation using a body centered tetragonal (bct) unit cell22. Having reached\ndC, the decrease of ( c=a)bctbecomes stronger which can be attributed to a large number of\noccurring mis\ft dislocations. After 2 nm additional deposited \flm i. e. at dFe\u0000Co= 4 nm, the\n7relaxation is completed. For the Fe 0:46Co0:54\flm (full squares), the \fnal value for ( c=a)bct\nis 1. This is consistent to its bcc equilibrium structure. In additional RHEED studies we\nobserved that an Fe content variation in binary Fe yCo1\u0000ybetweeny= 0:3 andy= 0:6 has\nno in\ruence on its lattice relaxation. This allows a comparison between this \flm and the\nternary (Fe 0:4Co0:6)0:98C0:02\flm which we report in the following.\nFrom the RHEED derived in-plane lattice parameters of the Fe-Co-C \flm (open cir-\ncles, Fig. 2b), we already \fnd important di\u000berences when compared to the binary \flm (full\nsquares). Although lattice relaxation is also observed, the critical thickness where the strain\nstarts to relax is increased to approx. 3 nm and also the thickness at which areaches its\n\fnal value is slightly increased. Further, the \fnal in-plane lattice parameter after relaxation\ngives a striking di\u000berence. The C containing \flm \fnishes its lattice relaxation at a reduced\nvalue ofa= 0:281 nm. Regarding c=aratios of the \flms (Fig. 2c), a peculiar di\u000berence\nis the reduced slope of the curve for the C containing \flm (open circles) when compared\nto the binary Fe 0:46Co0:54\flm (full squares). The relaxation thus is considerably slower,\nreaching the \fnal ( c=a)bctvalue after 7 nm of grown \flm. After relaxation ( c=a)bctadapts\na higher \fnal value for the ternary Fe-Co-C \flm, in contrast to the binary \flm where c=a\nis 1. For the (Fe 0:4Co0:6)0:98C0:02\flm it remains at approx. 1.03. This observation of a\nresidual tetragonallity together with the indication, that the formation of mis\ft dislocations\nhas a lower driving force, already implies that Fe-Co-C exhibits a spontaneous tetragonal\ndistortion compared to the induced distortion in binary Fe-Co. The described mechanisms\nof \flm growth were already reported in-depth on thin \flms of pure Fe e. g. by Roldan Cuenya\net al.23. However, our results aim at the di\u000berences between the two regarded \flms which\nhave very di\u000berent structural properties after lattice relaxation.\nIV. EX-SITU OBSERVED STRUCTURAL PROPERTIES OF THE FILMS\nFor a more detailed examination of the in\ruence of the bu\u000ber lattice parameter on the\n(Fe0:4Co0:6)0:98C0:02\flms, we increased \flm thickness to 20 nm and applied XRD in Bragg-\nBrentano geometry. The detected patterns (Fig. 3) con\frm the change of the caxis length\nsince the 002 re\rections of Fe-Co are shifted to lower 2\u0002 angles. The expected value for\nbinary bcc Fe 0:4Co0:6is marked as a dotted line. When comparing \flms with the same\ncomposition deposited on di\u000berent Au xCu1\u0000xbu\u000bers no variation of the caxis length is ob-\n8FIG. 3. XRD results for (Fe 0:4Co0:6)0:98C0:02\flms on di\u000berent Au xCu1\u0000xbu\u000ber layers. The\nequilibrium 002 re\rection of binary bcc Fe-Co is marked with a dotted line21. The shift of the\n002 re\rection to a lower angle indicates a strained caxis in the deposited \flms. The 002 re\rection\nof the Au-Cu bu\u000ber varies between the samples since a di\u000berent composition implies a changing\nlattice parameter. The position of the Fe-Co-C 002 re\rection is independent on the chosen bu\u000ber\nlattice parameter which varies with the Au-Cu composition.\nserved. Its length is calculated as equal to 2 :91\u0017A which is 2.1 % higher than the equilibrium\nvalue of cubic Fe 0:4Co0:621.\nAs independent probe of the tetragonal distortion, we use pole \fgure measurements.\nFig. 4 depicts the 011 pole \fgure of a 100 nm thick (Fe 0:4Co0:6)0:98C0:02\flm. Epitaxial \flm\ngrowth is con\frmed by the four intensity maxima. In this pole \fgure, the tilt angle  of\nthe sample, where the 011 pole gets its highest intensity, is a measure for the tetragonallity\nof the measured lattice. If it is equal to 45\u000e, the unit cells are cubic with c=aof 1. This is\nthe case for binary Fe-Co \flms (not shown). In the present pole \fgure,  has its maximum\n9FIG. 4. XRD 011 pole \fgure of a 100 nm thick (Fe 0:4Co0:6)0:98C0:02\flm on Au 0:53Cu0:47. Its\nfourfold symmetry proves a square base of the lattice. The positive deviation of  from 45\u000e(inset)\nindicates the tetragonal strain which is 1.03 for this sample. The MgO(100) edges [100] and [010]\nare parallel to the \fgure edges.\nat 45:9\u000eas seen in the inset. With ( c=a)bct= tan the tetragonal distortion of the lattice\nis determined to 1.03. From this result, we conclude that even thicker \flms exhibit the\nsame lattice distortion like 10 nm thin \flms. This residual strain of the lattice is a strong\nindication that Fe-Co-C exhibits a spontaneous strain.\nAtomic resolution TEM investigations con\frm a tetragonal strain in the Fe-Co-C \flms.\nFast Fourier transforms (FFT) of di\u000berent \flm sections allow for the determination of the\nlattice parameters. Fig. 5 shows exemplary high resolution TEM images of Fe-Co-C \flms.\n10In Fig. 5a, the cross section of the complete layer structure is depicted con\frming the con-\ntinuity of all layers. Fig. 5b shows a section of the 100 nm thick (Fe 0:4Co0:6)0:98C0:02\flm.\nIts inset gives the FFT pattern of this image. By measuring the distances between the\nintensity maxima in [001] and [100] direction, ( c=a)bctwas determined to 1.02 \u00060.02. This\nvalue is slightly reduced when compared to the XRD measurements. We attribute this re-\nduction to an additional lattice relaxation which occurs when the \flm is cut to a lamella.\nThis assumption is based on the observation that thinner sections of the lamellae exhibited\nlowerc=aratios than thicker sections. FFT patterns of di\u000berent \flm sections with varying\ndistance to the bu\u000ber interface do hardly reveal any di\u000berence in tetragonal distortion. This\ncon\frms that the lattice relaxation does not proceed with increasing \flm thickness, which\nis in agreement with the XRD measurements. However, the ( c=a)bctratio of the \frst 4 nm of\nthe Fe-Co-C \flm i. e. close to the interface to the Au-Cu bu\u000ber is 1.04 \u00060.03 and thus slightly\nhigher than the ( c=a)bctratio measured within the \flm. At the \frst glance, this appears\nto contradict the RHEED measurements, where a substantially higher strain was observed\nin this thickness range. We attribute this to the fact that RHEED is an in situ method\nwhile TEM studies the \flms ex-situ. We argue that most of the formed mis\ft dislocations\npropagate through the already grown \flm until they reach the interface. This explains, why\nwe do not con\frm the much higher values of ( c=a)bctas observed with RHEED during \flm\ngrowth. However, the slightly higher values of ( c=a)bctat the interface indicate an in\ruence\nof the bu\u000ber in this \flm section. XRD pole \fgure measurements of 5 nm thick \flms also\nreveal (c=a)bctof 1.05\u00060.01 (not shown) and con\frm this tendency.\nAs described in the experimental section, the Fe-Co-C \flms were prepared by co-\ndeposition from di\u000berent targets. Since one of the used targets contained 20 at% C, we\ncould study the in\ruence of the C content on the structural \flm properties. AES mea-\nsurements of additionally prepared \flms reveal that the highest achieved C content in\ncontinuous Fe-Co-C \flms was 5 at%. A further increase of C content leads to a partial\ndelamination of the \flms. However, the comparison of (Fe 0:4Co0:6)0:98C0:02\flms and \flms\nwith up to 5 at% C, does not indicate structural di\u000berences { neither in XRD (not shown)\nnor in TEM. The determined lattice parameters remain unaltered as already described and\n(c=a)bctremains 1.03. The only di\u000berence is a reduced X-ray coherence length in the \flms\nwith higher C content which indicates a reduction of crystal size. Since the lattice does not\nchange with increasing C content, we propose that the maximum solubility of C in Fe-Co\n11FIG. 5. a) TEM image of the complete layers architecture of a 20 nm (Fe 0:4Co0:6)0:95C0:05\flm. b)\nMagni\fcation of a 100 nm (Fe 0:4Co0:6)0:98C0:02\flm oriented in [010] zone axis. The inset depicts\nthe FFT image of this section with the re\rections marked which were taken for determination of\na(yellow) and c(orange).12\flms is 2 at% C. We argue that additional C dissolves and accumulates at grain boundaries\nwhich \fnally, at a C content above 5 at%, destabilizes the \flms. Compared to its maximum\nsolubility in bcc equilibrium Fe-Co of only 0.3 at%24, we can reach signi\fcantly higher C\ncontents in our \flms. We attribute this to the high energy of the deposited ions in PLD25\nwhich allows for a preparation of supersolutions with substantially higher solubility limits\nthan in equilibrium26.\nV. MAGNETIC PROPERTIES OF THE STRAINED FILMS\nIn order to investigate the in\ruence of the tetragonal lattice distortion on the magnetic\nproperties, VSM measurements were performed at room temperature. Fig. 6a depicts the\n\frst quadrants of the out-of-plane (oop) hysteresis curves of (Fe 0:4Co0:6)0:98C0:02\flms with\ndi\u000berent thicknesses on Au 0:5Cu0:5bu\u000bers. The magnetic saturation \u00160MSof all of these\n\flms was determined to be 2 :1\u00060:1 T which agrees with DFT as discussed later. To improve\nthe comparability of the hystereses, the maximum magnetizations were normalized. All \flms\nsaturate at a magnetic \feld \u00160HSwhich is lower than 2 :1 T. This implies the presence of\nan oop component of magnetic anisotropy competing with shape anisotropy for all \flm\nthicknesses. For comparison, a hypothetical curve of a \flm with only shape anisotropy\nand no oop component of magnetic anisotropy (grey broken line) has been added. As\nthe epitaxial growth results in a unique alignment of all tetragonal unit cells with their\ncaxis perpendicular to the substrate, we can attribute this additional anisotropy in the\n(Fe0:4Co0:6)0:98C0:02\flms to the tetragonal distortion of the lattice which makes it a MCA.\nAs seen in Fig. 6a, the \flm thickness has an in\ruence on the slope of the hystereses and,\ntherefore, also on its MCA. The \flm with the lowest thickness, 5 nm, shows the highest\nslope of its hysteresis i. e. the highest MCA. This is consistent to the texture measurements,\nwhere for the thinnest \flm, a slightly higher tetragonal distortion is measured. Also TEM\nrevealed a slightly increased distortion of the \frst nanometers of grown \flm. For thicker\n\flms, the shape of the hysteresis curves is unaltered (Fig. 6a), i. e., there is little change of\nthe magnetic properties. This also matches to the structural observations, where no major\ndi\u000berence of lattice distortion was observed for these \flms.\nFrom the magnetic hysteresis curves, values for the magnetocrystalline anisotropy energy\nMAE of the \flms were estimated. We use the triangle, set in the oop hysteresis curve\n13FIG. 6. a) First quadrant of hysteresis curves of (Fe 0:4Co0:6)0:98C0:02\flms with di\u000berent thicknesses\ndFe\u0000Co. For comparison, the expected out-of-plane curve for a \flm with shape anisotropy only\n(grey broken line) has been added. b) Magnetic anisotropy energies of these \flms (open circles)\ncompared to binary Fe-Co \flms (full squares). The broken horizontal lines mark the range of the\nDFT results of (Fe 0:4Co0:6)32C withc=a= 1:03 which were obtained by applying two di\u000berent\nmethods as described in the text. 14between (0,0), (0, \u00160MS) and (\u00160HS,\u00160MS) as measure for the e\u000bective anisotropy energy\nMA eff. The di\u000berence between the shape anisotropy energy MA shape = 1=2\u0001\u00160M2\nSand\nMA effgives then the MAE of the \flm. The results of this estimation are plotted in Fig. 6b\ncompared to the MAE of binary Fe-Co \flms which were determined similarly from their\nhysteresis curves. For both, the binary and the ternary Fe-Co-C \flms, the highest MAE\nis reached for ultrathin \flms with thicknesses below 10 nm. This may be attributed to the\nlarger contribution of the higher strained Fe-Co lattice at the interface. Fe-Co \flms with\noop magnetic easy axis were already reported for thicknesses up to 15 ML, i. e. , 2 nm9{11.\nHowever, the large MAE in the ultrathin \flms may also originate from surface dipole\nanisotropies that may bene\ft magnetic moments to align in perpendicular direction27. The\nhigher the \flm thickness, the lower is the contribution of both surface related e\u000bects and\nthusMAE decreases. With increasing \flm thickness, the MAE of the (Fe 0:4Co0:6)0:98C0:02\n\flms does not decrease as strongly as it is observed for binary Fe-Co \flms. While in the\nlatter, only the \frst few monolayers (see Fig. 2c) are strained tetragonally, the Fe-Co-C\nlattice exhibits a strain up to \flm thicknesses of at least 100 nm. This is the reason why\ntheMAE of these \flms is maintained above 0 :4 MJ/m3. In the binary \flms, the fraction of\ndistorted Fe-Co becomes negligible at higher \flm thicknesses.\nRegarding \flms with higher C content than 2 at%, no further increase of MAE was\nobserved. Instead, a reduction of \u00160MSbelow 2 T for 5 at% C was measured, which is\na deterioration of magnetic properties compared to the (Fe 0:4Co0:6)0:98C0:02\flms. Theory\nagrees at this point. More C in the lattice not only reduces the magnetic saturation due to\nhigher fraction of diamagnetic C, but also reduces the magnetic moment of the Fe and Co\natoms13.\nOur aim is to compare the described experimental observations to theoretical results based\non DFT. As already discussed in the introduction, the tetragonal distortion was predicted\nby Delczeg-Czirjak et al.13for Fe-Co alloyed with carbon. The present study proves these\n\fndings also for lower C contents of only 2 at%. However, higher C contents did not further\nstrain the unit cell as predicted in the literature13where 6 at% C led to ( c=a)bctratios of 1.12.\nWe suggest a thermodynamic limit of solubility of C as most likely reason. Additional DFT\ncalculations were performed based on the structural observations of our (Fe 0:4Co0:6)0:98C0:02\n\flms. The input parameters were c= 0:290 nm and a= 0:281 nm yielding a ( c=a)bctratio\nof 1.03 as experimentally observed with XRD (Fig. 3 and 4). This ( c=a)bctratio is similar to\n15that of (c=a)bct= 1:033 theoretically predicted13for (Fe 0:4Co0:6)24C. The here applied \flm\ncomposition was (Fe 0:4Co0:6)32C i. e. 3 at% C as depicted in Fig. 1. The internal positions\nof atoms were relaxed in WIEN2k after which the same structure was used as input to\nSPR-KKR. Evaluation of the MCA with 1200 kvectors and smallest mu\u000en-tin radius times\nmaximum kvector used in-plane wave basis functions chosen to RKmax= 8:3 yielded\nMAE = 0:51 MJ/m3and\u00160MS= 2:1 T. When MCA is determined in SPR-KKR, with\nCPA and 2600 kvectors, instead of VCA in Wien2k, it is reduced to MAE = 0:224 MJ/m3,\nbut\u00160MSis still 2:1 T. Comparing the results from both applied DFT approaches we \fnd a\ngood agreement with our experimental data e. g. MAE = 0:44\u00060:14 MJ/m3for the 100 nm\nthick (Fe 0:4Co0:6)0:98C0:02\flm. As seen in Fig. 6b, the two calculated MAE could be taken as\nupper and lower limits for the experimentally obtained anisotropy energy for \flm thicknesses\nfrom 20 nm upwards. However, the experimental MAE seems to be closer to the VCA based\ntheoretical value than to the CPA based one. Already Delczeg et al.13reported, that CPA\nmight underestimate the e\u000bect of disorder on MCA when compared to more accurate special\nquasirandom structures (SQS).\nThe observed good agreement between experimental and theoretical results also indicates\nthat a reduction of C content from 3 to 2 at% does not lower the MCA when the state of\nstrain is kept. As expected, the MCA is more sensitive to the c=aratio than to the C\ncontent. An additional explanation is that a higher amount of C atoms, which is expected\nto result in a higher strain13, may introduce local disorder. This disorder could cause a\nreduction of MAE when compared to a highly symmetric distorted Fe-Co crystal with no\nC atoms13. Hence, a not too high C content is expected to be bene\fcial for structural and\nmagnetic properties.\nVI. CONCLUSION\nIn this study, carbon was alloyed to Fe-Co in order to stabilise the tetragonal distortion of\nthe unit cell. It was shown that already a very low fraction of C can bene\ft the formation of\na phase with tetragonal lattice symmetry. For 2 at% C in Fe 0:4Co0:6the former cubic lattice\nis strained by 3 %. This strain is small when compared to theoretical calculations for Fe-Co\nwith higher C content13. However, it has a remarkable in\ruence on the magnetic properties\nas it establishes a uniaxial MCA. Our experimental results con\frm the applied theoretical\n16model where the C atoms preferentially occupy positions along the caxis of the Fe-Co lattice.\nIf this did not happen, an isotropically strained cubic lattice would be expected. We \fnd\nstrong indications that (Fe 0:4Co0:6)0:98C0:02prepared under the named conditions forms a\nspontaneously strained phase: (1) After an initial lattice relaxation which takes place in the\n\frst grown monolayers, the relaxation stops at a state with a lattice distortion of 1.03 (see\nFig. 2c). (2) This lattice distortion is independent of \flm thickness, as it is observed in \flms\nwith thicknesses between 20 nm and 100 nm. We \fnd no indications for a driving force that\npromotes a further relaxation.\nThe magnetocrystalline anisotropy energy of the described alloy was determined to be\napprox. 0:44 MJ/m3. This experimental result agrees with bounds predicted by DFT cal-\nculations based on two di\u000berent methods of treating the chemical disorder.\nOur study shows that apart from ultrathin \flms, also thicker \flms based on Fe-Co can\nexhibit magnetocrystalline anisotropy. Although their anisotropy is still too low to be com-\npetitive to the common hard magnetic alloys, we consider our results as one step towards\npossible permanent magnet alternatives based on 3d elements. The capability of small atoms\nlike C to strain the former cubic lattice of Fe-Co is a key message of this work. Comparable\nelements (B,N) are expected to act similarly. One important advantage of additions of these\nelements in small amount is that a high magnetic moment is maintained. As a spontaneously\nstrained phase in Fe-Co-C is formed, this approach opens many possibilities that may lead\nto novel hard magnetic materials. These potential new permanent magnets would not be\nrestricted to thin \flms.\nACKNOWLEDGMENTS\nWe acknowledge funding of the EU through FP7-REFREEPERMAG. For experimental\nsupport, we gratefully thank Ruben H uhne, Ste\u000e Kaschube, Christine Damm and Juliane\nScheiter.\nREFERENCES\n1J. M. D. Coey, Scr. Mat. 67, 524 (2012)\n172T. Burkert, L. Nordstr om, O. Eriksson, and O. Heinonen, Phys. Rev. Lett. 93, 027203\n(2004)\n3C. Neise, S. Sch onecker, M. Richter, K. Koepernik, and H. Eschrig, Phys. Status Solidi\nB248, 2398 (2011)\n4I. Turek, J. Kudrnovsky, and K. Carva, Phys. Rev. B 86, 174430 (2012)\n5R. H. Victora and L. M. Falicov, Phys. Rev. B 30, 259 (1984)\n6P. James, O. Eriksson, B. Johansson, and I. A. Abrikosov, Phys. Rev. B 59, 419 (1999)\n7J. W. Shih, Phys. Rev. 46, 139 (1934)\n8G. Andersson, T. Burkert, P. Warnicke, M. Bj orck, B. S. Sanyal, C. Chacon, C. Zlotea, L.\nNordstr om, P. Nordblad, and O. Eriksson, Phys. Rev. Lett. 96, 037205 (2006)\n9A. Winkelmann, M. 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Prog.\nPhys. 59, 1409 (1996)\n19"    },    {        "title": "1409.5806v1.Magnetic_anisotropy_of_Fe_1_yX_yPt_L10__X_Cr_Mn_Co_Ni_Cu__bulk_alloys.pdf",        "content": "Magnetic anisotropy of Fe 1\u0000yXyPt{L1 0[X=Cr,Mn,Co,Ni,Cu] bulk alloys.\nR. Cuadrado,1Timothy J. Klemmer,2and R. W. Chantrell1\n1Department of Physics, University of York, York YO10 5DD, United Kingdom\n2Seagate Technology, Fremont, California, 94538, USA\n(Dated: July 28, 2021)\nWe demonstrate by means of fully relativistic \frst principles calculations that, by substitution of\nFe by Cr, Mn, Co, Ni or Cu in FePt{L1 0bulk alloys, with \fxed Pt content, it is possible to tune\nthe magnetocrystalline anisotropy energy by adjusting the content of the non{magnetic species in\nthe material. The changes in the geometry due to the inclusion of each element induces di\u000berent\nvalues of the tetragonality and hence changes in the magnetic anisotropy and in the net magnetic\nmoment. The site resolved magnetic moments of Fe increase with the X content whilst those of\nPt and X are simultaneously reduced. The calculations are in good quantitative agreement with\nexperimental data and demonstrate that models with \fxed band structure but varying numbers of\nelectrons per unit cell are insu\u000ecient to describe the experimental data for doped FePt{L1 0alloys.\nThe chemically ordered face{centred tetragonal ( fct)\nFePt{L1 0phase has attracted much interest because of\nits large magnetocrystalline anisotropy energy (MAE)\nvalue of 7 \u0002107ergs/cm3and hence its potential appli-\ncations such as ultra{high density magnetic recording\nmedia [1, 2]. From the experimental side there are two\nprincipal challenges to the production of high density\nrecording media. First is the large values of the MAE\nto overcome the superparamagnetic limit so as to avoid\nthe loss of recorded information [3] arising from thermal\ninstability. One solution to this problem is the use of L1 0\nbimetallic alloys as magnetic recording media. This leads\nto a second challenge in that the preparation of such al-\nloys generally leads to the deposition of the disordered fcc\nA1 phase with low anisotropy. Transformation into the\nL10phase with large MAE requires elevated annealing\ntemperature, leading to problems with maintaining the\ngranular structure necessary for high density recording.\nSome studies have proposed using FePt{based ternary\nalloys to lower the kinetic ordering temperature lead-\ning to reduced annealing temperatures thereby improv-\ning the orientation and granular structure [4{6]. How-\never, this has the detrimental e\u000bect of reducing the MAE.\nThis problem was studied in a related theoretical paper\nby Sakuma [15], who used a \fxed band structure corre-\nsponding to FePt and varied the number of electrons/unit\ncellneff, \fnding that the MAE had its maximum value\nforneff= 8, corresponding to pure FePt. These predic-\ntions were veri\fed experimentally by Suzuki et.al. [14].\nMore recent work has studied the e\u000bect of substituting\nFe by Cu [7], Mn [9{11], Ni [12, 13]. Gilbert et. al.[13]\nconclude that Cu doping gives a relatively simple ap-\nproach to achieve high quality L1 0FeCuPt \flms that\nhave greater MAE values than current media and there-\nfore are desirable for future magnetic recording technolo-\ngies. Also, the magnetic properties can be smoothly\ntuned by Cu-substitution into Fe sites of the ordered\nalloy. The experiments are generally supported by the\nrigid band model calculations of Sakuma [15]. However,\nit is debatable as to how realistic such models are, given\nthe chemical and structural changes induced by alloy-\ning. This question can only be answered by a detailedinvestigation taking account of the nature of the alloys\nproduced. This is the aim of the current letter. Our\ncalculations are in good agreement with experiment and\ndemonstrate a doping{species dependence of the MAE\nreduction and also, importantly, variations in the local\nMAE arising from the di\u000berent lattice sites available to\nthe impurity atoms.\nConsistent with the experimental studies we proceed\nby replacing the Fe content in bulk FePt{L1 0by Cr,\nMn, Co, Ni and Cu keeping the Pt concentration \fxed.\nWithin Fe 1\u0000yXyPt bulk alloy, we take the yconcentra-\ntion as 0, 0.25, 0.50, 0.75 and 1, ybeing the amount\nof non{magnetic species, whose introduction changes\nthe e\u000bective number of valence electrons, neff, in the\ncell. The e\u000bective valence electrons are computed asP\nsNs\u0001Zs\nval=Ntot, whereNsis the number of atoms of\neach species, s, andNtotthe total number of atoms in\nthe simulation cell. This systematic Fe replacement of\nsimilar 3delements serves to control neffsince for the\nabove mentioned species, the valence charge Zs\nvalis 6, 7,\n9, 10, 11, respectively. Note that only the 3 dselectrons\nare included in the current neffde\fnition; the 5 dcon-\ntribution from the Pt is constant and is not taken into\naccount here.\nAll the geometric, electronic and magnetic structure\ncalculations of Fe 1\u0000yXyPt{L1 0alloys have been done\nby means of DFT using the SIESTA [16] code. As ex-\nchange correlation (XC) potential we have employed the\ngeneralized gradient approximation (GGA) following the\nPerdew, Burke, and Ernzerhof (PBE) version [19]. To\ndescribe the core electrons we have used fully separa-\nble Kleinmann-Bylander [17] and norm-conserving pseu-\ndopotentials (PP) of the Troulliers-Martins [18] type.\nAs a basis set, we have employed double-zeta po-\nlarized (DZP) strictly localized numerical atomic or-\nbitals (AO). The so{called electronic temperature {kT\nin the Fermi-Dirac distribution{ was set to 50 meV.\nThe magnetic anisotropy energie (MAE) has been ob-\ntained using a recent fully relativistic (FR) implemen-\ntation [20] in the GREEN [21, 22] code employing the\nSIESTA framework. As usual, the MAE is de\fned as\nthe di\u000berence in the total energy between hard and easyarXiv:1409.5806v1  [cond-mat.mtrl-sci]  19 Sep 20142\nmagnetization directions. Convergence of the MAE con-\nvergence is dependent on the sampling kpoints. Within\nthe present work we performed an exhaustive analysis of\nthe MAE convergence in order to achieve a tolerance be-\nlow microelectron volts. We employed more than 5000 k\npoints in the calculations for each geometric con\fgura-\ntion, which was su\u000ecient to achieve the stated accuracy.\nThe binary L1 0alloys are formed by alternating planes\nof two distinct species which generates a vertical dis-\ntortion as a result of which two quantities de\fne the\ngeometric structure: the in{plane lattice constant, a,\nand the out{of{plane parameter, c. Speci\fcally, in\nFePt{L1 0the experimental values are aFePt= 3.86 \u0017A and\nc=a=0.98. What we pursue is to study the variation of\nthe anisotropy of bulk FePt{L1 0via the substitution of\nFe atoms by other 3 dspecies keeping Pt \fxed. In doing\nthis we are able to scan two possible ways to control the\nMAE: on the one hand, the species and on the other,\nthe concentration of the impurity (See Fig. 1). Each one\nof the Cr, Mn, Co, Ni and Cu atoms has di\u000berent num-\nber of valence electrons, so that it gives the possibility to\ncontrol the number of total valence electrons in the cell\ndepending on whether one, two, three or four atoms are\nreplaced on the Fe sites.\nIt is complicated in DFT to deal with this kind of cal-\nculation due to the large cells that one has to use to have\na good approximation of the real material in a computer\nmodel, so we doubled the unit cell in X, Y and Z axis in\norder to be able to substitute the X atoms one by one.\nThe minimum unit cell for a XPt{L1 0bulk is composed\nof two atoms (see Fig. 1{B) and in our case the simulation\ncell has 16. This permits to move X atoms on di\u000berent\nFe    X    Pt - L1 075%25%FeFeXPtPtFe    X    Pt - L1 050%50%\nFe    X    Pt - L1 025%75%  X      Pt - L1 0100%AB\n34FePt - L1\nFePt0\nx(010)(100)(001)yca/p212XCr Mn CoNiCu \nFIG. 1. (Color online) (A) Schematic picture of the FePt{\nL10unit cell and its characteristic lattice values: aandc=a.\nNotice that the in{plane diagonal of the unit cell corresponds\nto the lattice constant whilst the edge is a=p\n2; (B) Fe 1\u0000yXyPt\nunit cells. For yvalues of 0.25, 0.50, 0.75 or 1, the cells\nhave been labeled from 1 to 4, respectively. The green arrows\nrepresent di\u000berent locations for the X species (see text for\nexplanation). For speci\fc alloys the X element will be one of\nMn, Cr, Co, Ni and Cu.\n3.90\n3.883.904.004.023.983.963.943.923.943.984.18\n0.840.864.144.024.104.060.880.920.900.940.960.98\n0.880.920.900.940.960.98c/ac/aLattice parametter [A]Lattice parametter [A]FePt - L10FeMnPt - L10\nFeCuPt - L10FePt - L100%0.25%0.50%0.75%1.00%Concentration [y]FIG. 2. (Color online) Lattice constant parameters, aand\nc=a, blue circles and red triangles, respectively, as a function\nof the yconcentration for Fe 1\u0000y(Mn,Cu) yPt{L1 0bulk phases.\nThe straight lines are guide for the eye.\nin{plane and out{of{plane positions as the green arrows\ndepict in Fig. 1{B. The number of di\u000berent geometric\ncon\fgurations keeping the X content \fxed were: three\nfor B{1 and B{3 and six for B{2. For each con\fgura-\ntion we performed a fully relaxation using the conjugate\ngradient (CG) method without any constraint. Special\nattention is needed for the Fe(Cr,Mn)Pt alloys inasmuch\nas the lower energy con\fguration corresponds to anti-\nferromagnetic (AF) alignment of the Mn atoms between\ndi\u000berent atomic planes [8], so we include this restriction\nin our calculations as a magnetic constraint.\nGeometric, electronic and magnetic properties have\nbeen calculated by means of the mean value of \fxed X:Fe\ncomposition unless explicitly expeci\fed, for example, the\ndensity of states (DOS) for a \fxed X:Fe ratio \u000bis\nDOS\u001b;\u000b(\u000f;X) =PN\u000b\nj=1DOS\u001b;\u000b\nj(\u000f;X)\nN\u000b\nwherejruns up to the total number of con\fgurations\nfor a \fxed X composition, N\u000b,\u001bis the spin{up/{down\nstates and\u000fthe energy, usually shifted to the Fermi level,\n\u000f=E\u0000EF.\nAs was pointed out in the experimental work of Gilbert\net al [7], the substitution of Cu into the bulk FePt{\nL10phases promotes an increase of the in{plane lat-\ntice parameter avalues as we observe in Fig 2 (blue\ndots), which is therefore in agreement with experiment.\nThe out{of{plane cparameter is simultaneously reduced\nwith increasing Cu content. Consequently, the tetrago-\nnal distortion of FeCuPt{L1 0increases with Cu content\nleading to an in{plane (out{of{plane) lattice constants\nof 3.98 \u0017A(3.64 \u0017A). In the case of FeMnPt{L1 0(AF),ain-\ncreases with decreasing c, in agreement with the experi-\nmental results of Meyer et al[10].3\nFig. 3 shows the averaged spin resolved density of\nstates for the Fe 1\u0000y(Mn,Cu)yPt alloys as the concentra-\ntionyof the Fe and (Mn,Cu) changes, left and right\npanel, respectively. From top to bottom is shown the\nevolution of the total (solid black), Fe (solid blue),\nPt (dashed red) and the X(=Mn,Cu) (\flled green) DOS\nas the Fe, Mn, Cu species is varied. As pointed out ear-\nlier, the lower energy con\fguration for the FeMnPt{L1 0\ncorresponds to AF coupling of the Mn atoms on alter-\nnating planes so that the up/down charges are equal and\nthe net magnetic moment (MM) is zero which is re\rected\nin the green DOS curves.\nThe DOS curves aid the interpretation of the behav-\nior of the magnetic moments. In bulk FePt{L1 0, the net\nMM is 3.37 \u0016B=f:u: , mainly dominated by the Fe species\nas depicted the blue line in the upper graphs. Only a\nfraction of this net value is contributed by the Pt sites,\nas has been pointed out in previous work [23]. The Fe{\nPt{Fe indirect interactions promote the polarization of\nthe Pt atoms. In our case, the substitution of the the\nFe by non{magnetic species such as Mn or Cu reduces\nprincipally the Fe down{states tending to leave the Fe\n-8-6-4E - E  [eV]FE - E  [eV]FPDOS [Arb. units]\n-2024-8-6-4-2024FeMnPt - AFFeCuPt\nFe     Mn     Pt25%75%100%\nFe     Cu     Pt25%75%100%\nFe     Mn     Pt75%25%100%\nFe     Cu     Pt50%50%100%\nFe     Cu     Pt75%25%100%FePt\nFePt\nFe     Mn     Pt50%50%100%\nMnPtCuPtFeTotalPtMn,Cu\nFIG. 3. (Color online) Spin resolved density of states of\nFe1\u0000y(Mn,Cu) yPt [y=0,0.25,0.50,0.75 and 1], left and right,\nrespectively. The total DOS (black solid line) has been split-\nted in its Fe (blue solid line), Pt (red dashed line) and Mn,\nCu (green \flled curve) contributions. The \frst two graphs on\ntop, represent the DOS for the pure FePt{L1 0bulk alloy.\n10.50%0.75%0.25%03.153.203.253.453.50-0.150.050.150.25\n-0.05\nFeCuPtFePt - L10\nFePt - L10FeMnPtMeanFe - atomsPt - atomsCu/Mn - atoms3.403.353.30BMM [     /at]  Cu/Mn  Concentration [y]BMM [     /at]  FeCuPtFeMnPtMeanFIG. 4. (Color online) Site resolved magnetic moment for\nFe1\u0000y(Mn,Cu) yPt alloys as a function of the Cu/Mn con-\ncentration y. In each panel has shown separately the MM\nof the non{magnetic species (upper) and the MM of the\nFe (bottom). For the same alloy, \fxing the Cu/Mn concen-\ntration, similar symbols represent all the studied con\fgura-\ntions (see Fig. 1). The dashed lines refer to the mean value,P\njMMi\nj;X/Ni\nconf, where X is Fe, Mn, Cu or Pt and irefers\nsome particular concentration.\natoms embeded in a non{magnetic environment, becom-\ning almost magnetically isolated with increasing Mn/Cu\ncontent. This is the reason behind the increase of MM Fe\nwith Cu/Mn content (see Fig. 4). Simultaneously, the\nMMPtdiminishes due to the reduction in Fe neighbors\nuntil the up and down charges compensate. The MM Cu\nis close to zero independent of the amount present in the\nalloy. On the contrary, a small concentration of Mn gives\na MMMnvalue of -0.12 \u0016B=at, its magnitude reducing to\nzero with increasing Mn content. In summary, both the\n(Mn,Cu)Pt{L1 0bulk alloys have a zero net MM as we\nsee in the bottom panel. The addition of the Fe atoms to\nthese alloys enhances the value of the total MM, partly\nfrom the Fe and partly from the induced Pt polarization.\nIt should be noted that in Fig. 4 the scatter of points for\na given impurity concentration indicates the variation of\nthe magnetic moment across the di\u000berent lattice posi-\ntions hosting the impurity. A similar dispersion is also\nseen in the local MAE values, which are considered next.\nThe e\u000bect on magnetization, M, and magnetic\nanisotropy of X = Cr, Mn, Co, Ni, Cu substitution in\nFePt{L1 0as a function of the neffis shown in Fig. 5.\nThe variations with neffof both the magnetization and\nMAE are in good agreement with the experimental data4\nK [10  erg/cm  ]63M [emu/cm  ]3\nneff30507090110-10106780300600900\n91011FePt-L1 0FePt-L1 0A\nB\nFeMnPt FeNiPt FeCoPt FeCuPt FeCrPt \nFIG. 5. (Color online) Magnetization and magnetic\nanisotropy energy of Fe 1\u0000yXyPt alloys as a function of the\nneff, A and B, respectively. Peaks on both graphs depict the\nmagnetization and MAE for pure FePt{L1 0alloy.\nreported by Gilbert [7] et al although a detailed compari-\nson is di\u000ecult as will be discussed shortly. Consider \frst\nthe magnetization M, which is calculated taking account\nof the variation of the MMs between substitution sites.\nIt can be seen that Mhas a maximum for FePt and falls\no\u000b more rapidly for neff<8 than forneff>8. This is\nconsistent with experimental data. We note also that the\nrate of reduction for neff<8 is faster than predicted by\nSakuma [15] emphasizing the importance of taking spe-\nci\fc account of the AF coupling of the impurity spins.\nThe MAE also has a maximum for FePt as expected and\nfalls o\u000b rapidly on either side of the optimum band \flling.\nConsistent with experiment, the dependence of the MAE\nonneffis slower for Ni and Cu impurities, allowing a\nmore controlled tuning of the anisotropy. Some con\fgu-\nrations change their easy axis from out{of{plane to in{\nplane, speci\fcally in Fe 0:50Mn 0:50Pt, Fe 0:25Mn 0:75Pt and\nCoPt. This is not observed experimentally, where the\nrange ofneffdoes not extend into the region of the pre-dicted in{plane anisotropy. Finally, we note that there\nis an important dependence of the MAE on the species\nof the impurity atoms, which is not predicted by the\n(\fxed band) calculations of ref [15]. The experimental\ndata cannot reliably be used to test this prediction since\nthe results summarised in ref [7] were all made on dif-\nferent samples using di\u000berent measurement techniques.\nFor example, the MAE for FePtMn di\u000ber as much as a\nfactor of 2{3 between laboratories, suggesting that the\ncurrent FePtNi and FePtCu data (again measured in dif-\nferent laboratories) cannot be used to test the species-\ndependence predicted here.\nIn summary, we have developed a theoretical method\nto investigate the MAE of the FePt{L1 0phase follow-\ning gradual substitution of Fe by Cr, Mn, Co, Ni and\nCu keeping the Pt content \fxed. The inclusion of the\ndoping elements changes the in{plane and the out{of{\nplane lattice constants characterising the fctphase. In\ngeneral, aincreases with the reduction of the Fe con-\ntent promoting a decrease of c. The magnetic moment\nof the magnetic and non{magnetic species also changes\nsubstitution. Due to the low Fe{Fe in{plane coordina-\ntion that emerges after replacement of the Fe atoms, the\nindirect polarization of Pt and other species is reduced\nsubstantially, disappearing for large dopant concentra-\ntions. On the other hand, the Fe tends to be magneti-\ncally isolated in a non{magnetic environment and hence\nits MM tends to increases. The predicted variation of\nthe magnetization as well as the MAE with the e\u000bective\nvalence charge is in good general agreement with prior ex-\nperiments. The calculations also predict that the local,\nsite resolved, anisotropy constant has a dispersion aris-\ning from di\u000berences in the local environment of doping\natoms situated at di\u000berent lattice sites. We also predict\na species-dependence of the variation of MAE with band\n\flling, which requires further experimentation to evalu-\nate, but which certainly suggests that \fxed band models\nare insu\u000ecient to study the MAE of FePtX alloys and\nthat a full treatment of the nature of the alloys is neces-\nsary.\nThe authors are grateful to Prof. Chih-Huang Lai and\nProf Kai Liu for helpful discussions. Financial support\nof the EU Seventh Framework Programme under Grant\nNo. 281043, FEMTOSPIN is gratefully acknowledged.\n[1] D. Weller and A. Moser IEEE Trans. Magn., 36, 10\n(1999).\n[2] O. A. Ivanov, L. V. Solina, V. A. Demshina and L. M\nMagat Fiz. Metal Metalloyed 35, 92 (1973).\n[3] M. L. Yan, H. Zeng, N. Powers, and D. J. Sellmyer J.\nAppl. 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Chantrell, Europhysics Letters 69, 805 (2005)."    },    {        "title": "1410.6272v2.Surface_aligned_magnetic_moments_and_hysteresis_of_an_endohedral_single_molecule_magnet_on_a_metal.pdf",        "content": "arXiv:1410.6272v2  [cond-mat.mtrl-sci]  8 Jan 2015Surface aligned magnetic moments and hysteresis of an endoh edral single-molecule\nmagnet on a metal\nRasmus Westerstr¨ om,1,2Anne-Christine Uldry,3Roland Stania,1,3Jan Dreiser,4,3Cinthia\nPiamonteze,3Matthias Muntwiler,3Fumihiko Matsui,1,5Stefano Rusponi,4Harald Brune,4\nShangfeng Yang,6Alexey Popov,7Bernd B¨ uchner,7Bernard Delley,3and Thomas Greber1,∗\n1Physik-Institut, Universit¨ at Z¨ urich, Winterthurerstr asse 190, CH-8057 Z¨ urich, Switzerland\n2Department of Physics and Astronomy, Uppsala University, B ox 516, S-751 20 Uppsala, Sweden\n3Swiss Light Source, Paul Scherrer Institut, CH-5232 Villig en PSI, Switzerland\n4Institute of Condensed Matter Physics, Ecole Polytechniqu e F´ ed´ erale de Lausanne, CH-1015 Lausanne, Switzerland\n5Nara Institute of Science and Technology (NAIST),\n8916-5 Takayama, Ikoma, Nara 630-0192, Japan\n6Hefei National Laboratory for Physical Sciences at Microsc ale,\nDepartment of Materials Science and Engineering,\nUniversity of Science and Technology of China, 96 Jinzhai Ro ad, Hefei 230026, China\n7Leibniz Institute of Solid State and Materials Research, Dr esden, D-01069 Dresden, Germany\n(Dated: September 4, 2021)\nThe interaction between the endohedral unit in the single-m olecule magnet Dy 2ScN@C 80and\na rhodium (111) substrate leads to alignment of the Dy 4 forbitals. The resulting orientation of\nthe Dy 2ScN plane parallel to the surface is inferred from compariso n of the angular anisotropy of\nx-ray absorption spectra and multiplet calculations in the corresponding ligand field. The x-ray\nmagnetic circular dichroism (XMCD) is also angle dependent and signals strong magnetocrystalline\nanisotropy. This directly relates geometric and magnetic s tructure. Element specific magnetization\ncurves from different coverages exhibit hysteresis at a samp le temperature of ∼4 K. From the\nmeasured hysteresis curves we estimate the zero field remane nce life-time during x-ray exposure of\na sub-monolayer to be about 30 seconds.\nThe hollow interior of the fullerene [1] carbon cage\ncan be used to encapsulate paramagnetic systems con-\nsisting of single atoms, as well as small clusters\nof different composition [2]. A fascinating example\nis the dysprosium-scandium based endofullerene-series\nDynSc3−nN@C80(n= 1,2,3) where the different stoi-\nchiometries result in distinct ground-state properties like\ntunnelling of magnetization ( n= 1), remanence ( n= 2),\nor frustration ( n= 3) [3–6]. The strong ligand field,\nmainly due to the central N3−ion, imposes orientation\nof the 4fshell and therefore non-collinear magnetism.\nIn the case of the di-dysprosium compound ( n= 2),\nexchange and dipolar coupling between the two mag-\nnetic moments stabilizes hysteresis and a large rema-\nnence with a relaxation time of one hour at 2 K was\nfound [4]. These endofullerenes are thus single-molecule\nmagnets (SMMs) [7–11], a class of magnetic compounds\nwith potential for application in spintronics, quantum\ncomputing, and high density storage [12, 13]. Single-\nmolecule magnets have been studied extensively in the\nbulk phase for the last two decades, but little is known\nregarding possible modifications to their intrinsic mag-\nnetic properties as the molecules are deposited onto sub-\nstrates or integrated into different device architectures.\nThis gap in knowledge can largely be attributed to the\nfragility of most compounds that have restricted research\nto a few families of molecules. A first proof of principle\nthat molecular nanomagnets can retain their magnetic\nbistability on a surface was demonstrated for a mono-layer (ML) of a Fe 4SMM on a gold surface [14]. At\nsub-Kelvin temperatures the Fe 4SMMs exhibited hys-\nteresis, out-of-plane anisotropy, and quantum tunnelling\nofmagnetization(QTM)[14,15]. Thesuccessofthesepi-\noneering experiments is predominantly due to the chem-\nical modification of the Fe 4complex enabling chemically\nand structurally stable MLs which were prepared ex situ\nfrom a solvent under ambient conditions.\nDepositing SMMs ontoa reactivemetal surfacesuch as\nferromagnetic substrates [16, 17] requires in situprepa-\nration under ultra-high vacuum (UHV). In this context\nthemononucleardouble-deckercomplexTbPc 2[18]isthe\nmost studied compound. It has been demonstrated that\nthe magnetic anisotropy is preserved at sub-ML coverage\non Cu(100) [19] and that the magnetic moments couple\nantiferromagneticallyto thin nickel films on Cu(100) and\nAg(100)[16]. Magnetichysteresiscomparabletothebulk\nphase has been observed in thick molecular films [20].\nOnly recently hysteresis with weak remanence was de-\ntected in the monolayer regime with graphite [21] and Si\n[22] as a substrate, though the influence of the substrate\non the magnetic properties is still poorly understood.\nEndofullerenes that are synthesised with the\nKr¨ atschmer-Huffmann method are thermally very\nstable and can be sublimated onto surfaces under UHV.\nFor magnetic endohedral units the carbon cage acts as\na ”spin-shuttle” that protects the spins from chemical\ninteractions. The robustness of the cages also facilitates\nimaging and manipulation by scanning probes [23–25].2\nθ[111] \nMagnetic fieldLphoton N3- Sc 3+ \nC 6- \n80 Dy 3+ (a) (b)\nFIG. 1. (a) Ball-and-stick model of Dy 2ScN@C 80. (b) Mea-\nsurement geometry with the angular momentum of the x-rays\nLph, parallel or antiparallel to the magnetic field and at an\nangle of θwith respect to the normal of the Rh(111) surface.\n.\nIf the cages have a high symmetry, different orientations\nof the endohedral units are possible. This decreases\nthe average magnetic moment of a system with more\nthan one molecule, and strategies that circumvent this\nissue are needed for macroscopic spin alignment. On\nmetallic surfaces it was shown that also endohedral\nunits may order [26], which opens a door for achieving\nnon-vanishing macrospin in zero field. For the case of\nGd3N@C80on Cu(100), a spin system without 4 fcharge\nanisotropy, direction dependent magnetic susceptibility\nwas observed, though it could not be directly related to\nthe geometry of the endohedral cluster [27].\nHere we present an x-ray absorption spectroscopy\n(XAS) study of Dy 2ScN@C 80on Rh(111). In the first\nlayer the endohedral units and their magnetic moments\nalign with the metal substrate, which is inferred from the\nangle dependence in XAS and x-ray magnetic circular\ndichroism (XMCD). Hysteresis curves demonstrate that\ntheproximityofthemetalsurfacehasapronouncedinflu-\nence on the magnetic bi-stability. Compared to thicker\nfilms, which are representative for the bulk phase, the\nsmaller opening of the hysteresis for the sub-ML indi-\ncates faster magnetic relaxation times.\nThe molecules (see Fig. 1(a)) have been sublimated\nin situonto the clean Rh(111) substrate, following the\nrecipe in Ref. [26]. The sample was cooled in zero mag-\nneticfield andthelayerthicknessisestimated fromthe x-\nray absorption of Dy. The XAS measurements were per-\nformed at the X-Treme beamline [28] of the Swiss Light\nSource. Absorption spectra were acquired by recording\nthe total electron yield in the on-the-fly mode [29], at\nsample temperatures of ∼4 K and with an external mag-\nnetic field applied along the x-ray beam.\nFig. 2 (a) shows x-ray absorption for a sub-ML cov-\nerage of Dy 2ScN@C 80on Rh(111) as a function of the\nangleθbetween surface normal and x-ray beam (see Fig.\n1(b)). The data were recorded over the Dy M5-edge\n(3d5/2→4f) using right ( I+) and left ( I−) circular po-\nlarized x-rays. After background subtraction the XAS\n(I++I−) were normalized to the integrated M5absorp-\n1295 1290 1285 1280 1275 127045 o15 o\n30 o\n60 o\n70 o0oIntensity (arb. u) (a) (b)\nPhoton energy (eV) Photon energy (eV) \nIntensity (arb. u) Dy M5I + + I -\nDy 3+ N3- \nIz Ix \nIz Ix \n1295 1290 1285 1280 1275 1270Dy M5\nFIG. 2. (a) XAS measured at the Dy M5-edge from a sub-ML\nof Dy 2ScN@C 80/Rh(111) T= 4 K,µ0H= 6.5 T, measure-\nment geometry in Fig. 1 (b). Each data set is normalized\nto the integrated intensity. (b) Calculated absorption wit h\nthe x-ray beam and external field oriented parallel Iz, and\nperpendicular Ix, to the magnetic easy-axis (Dy-N bond).\n.\ntion signal in order to compensate for the angular de-\npendence of the total electron yield (TEY). A significant\nchange in shape of the XAS multiplet spectra is observed\nas the sample is rotated from normal θ= 0◦to a larger\nangle of incidence θ= 70◦. This anisotropy in the XAS\nis attributed to an anisotropicdistribution of the 4 felec-\ntron charges due to their interaction [30] with the ligand\nfield of the central nitrogen ion, the neighbor rare earth\nions, and the C 80cage. This effect would not be present\nfor an isotropic distribution of the endohedral Dy 2ScN-\nunits and, therefore, indicates a preferred orientation of\nthe Dy 4 forbitals with respect to the surface.\nThe orientation of the Dy 4 forbitals may be inferred\nfromcomparisonoftheexperimentswithmultiplet calcu-\nlations. The crystal- or ligand-field multiplet theory for\nthecircularlypolarizedx-rayabsorptionisacontinuation\nin a long history of conceptually fairly similar [31], close\nto first principles, calculations with semi-empirical pa-\nrameters to fine tune the fit to experiment. This circum-\nvents unfeasible calculations comprising the coupling to\na huge number of electron states of lesser importance for\nthe appearanceof the spectrum. The endohedral Dy ions\nare trivalent (Dy3+), which leads to a 4 f9groundstate\nconfiguration. The final state in the present absorption\nspectra is consequently 3 d94f10. The ligand field deter-\nmines the easy axis with a two fold degenerate ground\nstate±Jz. The magnetic field lifts this degeneracy, and3\nPhoton energy (eV)(a)\nXAS (arb. u) (b)\n(c)θ = 0\nθ = 60oI + \nI -\nI + \nI -\n1290 1280 1270 1290 1280 1270\nPhoton energy (eV)XAS (arb. u) \nXMCD (arb. u) XMCD (arb. u) \nmultilayersub-ML\nθ (deg)M5\nM5I + - I -\nI + - I -XMCD asymmetry 0.5\n0.4\n0.3\n0.2\n0.1\n0.0\n-90 -60 -30 0 30 60 90\nFIG. 3. Sub-ML of Dy 2ScN@C 80/Rh(111), T= 4 K,µ0H=\n6.5 T, measurement geometry of Fig. 1 (b). The polarization\ndependent XAS spectra (left panel), and the corresponding\nXMCD spectra (right panel), were measured at an incidence\nangle of θ= 0◦(a) and θ= 60◦(b). (c) Angle dependence\nof the integrated XMCD signal normalized to the integrated\nXAS over the Dy M5-edge. The dashed line corresponds to\nthe expected angle dependence for magnetic moments ori-\nented parallel to the surface, whereas the blue line takes in to\naccount a Gaussian distribution, centred in the surface pla ne\nand with a standard deviation of 16◦.\n.\ninduces dichroism [32]. For low magnetic fields we find a\nvery small influence on the XAS. The ligand field, here\nfrom a point charge model of the [Dy3+\n2Sc3+N3−]6+ion,\ndescribes the site symmetry. Fig. 2 (b) displays calcu-\nlated XAS spectra from the Dy M5-edge with the x-ray\nbeam and an external magnetic field of 6.5 T applied\nparallel ( Iz) and perpendicular ( Ix) to the Dy-N bond.\nThe resemblance of the calculated Ixspectrum and data\nmeasured at normal incidence ( θ= 0◦) thus indicates\nthat the endohedral units adopt an orientation parallel\nto the surface.\nThe magnetism of the system is governed by the spin\nand orbital moment of the Dy 4 felectrons that have a\ntotal magneticmoment of10 µBper Dy3+ion[3, 4]. Anyanisotropy in the spin and orbital moments will give rise\nto a polarization dependent absorption at the Dy M5-\nedge and an XMCD spectrum ( I+−I−)M5. The magni-\ntude of the XMCD signal is determined by the projection\nof the corresponding magnetic moment /vector µiof the absorb-\ning dysprosium ion ionto the direction of the impinging\nx-rays/vectork[33]\nIXMCD∝/vector µi·/vectork (1)\nFor an isotropic system where the magnetic moments are\neither randomly distributed, or aligned to the external\nmagnetic field, the resulting XMCD signal in the present\nmeasurement geometry is independent of the incidence\nangle. Any macroscopic magnetic anisotropy is reflected\nin different XMCD spectra as a function of incidence\nangle. Polarization dependent XAS and corresponding\nXMCD spectra are shown in Fig. 3 (a) and (b) for inci-\ndence angles θof 0◦and 60◦, respectively. Comparison\nof the two spectra reveals a significant angle dependence\nwhichindicatesamacroscopicmagneticanisotropyin the\nsub-ML. Electrostatic interaction with the surrounding\nligands, in particular the central N3−ion, results in a\nstrong axial anisotropy which restricts the individual Dy\nmoments /vector µito orient parallel, or anti-parallel, to the cor-\nresponding magnetic easy-axis directed along the Dy i-N\nbonds [3–6, 34]. The observed magnetic ordering is thus\ndirectly related to the strong axial anisotropy of the in-\ndividual Dy ions andthe preferred adsorption geometry\nof theendohedral cluster, which must be imposed by the\nsurface.\nThe magnetic anisotropy is quantified in Fig. 3 (c),\nwhere XMCD angular dependence is shown. This con-\nfirms that the dysprosium moments are predominantly\noriented parallel to the surface. A small out-of-plane\nfraction is inferred from the non-vanishing dichroism at\nθ= 0◦. The observed behaviour can be modelled by\nassuming a Gaussian distribution of the magnetic mo-\nments centered in the surface plane ( θ= 90◦). The\nfit yields a distribution of 90 ±16◦. This implies that\nthe Dy-N bonds are not completely parallel to the sur-\nface, which is in line with a resonant x-ray photoelectron\ndiffraction study performed on a ML of Dy 3N@C80on\nCu(111), where the room temperature data indicated a\ncoexistence of planar endohedral units inclined to the\nsurface, and slightly pyramidal configurations parallel to\nthe surface [26].\nThe square symbols in Fig. 3 (c) correspond to the\nsamemeasurementsperformedonamultilayercontaining\n7 times more molecules. The weak angular anisotropy\nobserved is attributed to the residual influence of the\nsurface.\nThe relaxation time of the magnetic moments in a\ngiven environment is a key property of single-molecule\nmagnets. Relaxation times that are slow compared to\nthe time scale of the measurement will result in mag-4\n-1.0-0.50.00.51.0\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5\n-1.5 -1.0 -0.5 0.0 0.5 1.0 1.5m / m sat \nµ0H(T)\nµ0H(T)multilayer\nsub-ML\n-1.0-0.50.00.51.0m / m sat \nFIG. 4. Hysteresis curves measured from a multilayer and a\nsub-ML of Dy 2ScN@C 80/Rh(111) at a magnetic field sweep\nrate of 2 T/min and a sample temperature of ∼4 K. The x-\nray flux was 5 ×1010photons/mm2/s. The data were recorded\nwith the x-ray beam and the magnetic field at an incidence\nangle of θ= 60◦. The magnetization curves correspond to\nthe average of several independent measurements, where the\nerror bars are the standard deviation at each external mag-\nnetic field. The arrows indicate the ramping direction of the\nmagnetic field, the lines are guides to the eye, and msatis\nthe saturated value at ±6.5 T. The drop in magnetization at\nzero field is a consequence of the time of 30 s it takes for the\nmagnet to switch polarity.\n.\nnetic hysteresis. Fig. 4 shows element specific magne-\ntization curves from the Dy M5-edge at a field sweep\nrate of 2 T/min with the x-rays and the magnetic field\nat an angle of 60◦with respect to the surface normal. A\nsignificanthysteresisisobservedforbothsystemsdemon-\nstrating that the corresponding relaxation times are slow\ncomparedtothemeasurementtime. However,comparing\nthe magnetization curves from the two systems indicates\nthatthemagneticbi-stabilityofDy 2ScN@C 80ismodified\nby the proximity of the rhodium metal surface.\nSingle-ion 4 fmagnets, such as e.g. TbPc 2, exhibit\npoor remanence due to the rapid decay of the magneti-\nzation at low fields through QTM. In contrast, for bulk\nsamples of Dy 2ScN@C 80an exchange and dipole barrier\nof 0.96±0.1 meV suppresses QTM that in turn leadsto a significant remanence and coercive field [4]. This is\nclearlyobserved in the multilayer system, where the drop\nin magnetization at zero field is attributed to the delay\nof 30 s when changing the polarity of the magnet. From\nthe 25% decrease in magnetization during these 30 s, we\nderive a remanence relaxation time of 110 s in the mul-\ntilayer system. Compared to bulk samples in the dark\n[4] this is about four times faster, and mainly related to\nx-ray induced demagnetization [35].\nThe remanence-time of the sub-monolayer system is\nstill shorter because the magnetisation vanishes during\nthe switching of the magnet. From the comparison of\nthe two magnetization curves, we can estimate the re-\nmanence time for the sub-ML. Here we assume that the\nratio of the hysteresis openings, recorded for a fixed tem-\nperature and field sweep rate, is a relative measure of the\nrelaxation times in the two systems. Under this assump-\ntion, we obtain a four times faster relaxation rate in the\nsub-ML and a remanence time under x-ray irradiation\nand 4 K of about 30 s.\nThe shorter remanence time of the sub-ML may be re-\nlated to residual interaction of the Rh Fermi sea across\nthe C80shell. The substrate interaction that orients the\nendohedral clusters and spin fluctuations in the metal\nmight impose demagnetizing noise. Furthermore, since\nthe Dy magnetic dipoles lie in a plane their interaction\nis stronger, which may also accelerate demagnetisation.\nAlso, at sub-ML coverage the total electron yield below\ntheM5edge is about 10% higher than in the multi-\nlayer case and the reduced bi-stability could therefore\nbe demagnetisation due to secondary electrons from the\nsubstrate [35]. However, the opening of the hysteresis\ndemonstrates that the rate at which the magnetization\nrelaxes to its equilibrium value is still slow compared to\nthe measurement time.\nIn summary, angle dependent XAS from a sub-ML of\nDy2ScN@C 80on Rh(111) reveals a one-to-onecorrespon-\ndence between structural and magnetic ordering: The\ncombined effect of the local magnetic easy-axis for the\nencapsulated Dy ions, andthe preferred absorption ge-\nometry of the endohedral cluster, indictes surface aligned\n4fmoments and a macroscopic non-collinear anisotropy.\nAt a sample temperature of ∼4 K we observe a hys-\nteresis in the sub-ML. Although orientational ordering\nof endohedral molecules at surfaces, as well as magnetic\nhysteresis of bulk samples of such molecules have been\nshown, we demonstrate here the orientational structural\nand magnetic ordering of surface adsorbed endohedral\nmolecules creating a stable macrospin for molecular sub-\nmonolayers. 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Dreiser et al., Appl. Phys. Lett. 105, 032411 (2014)."    },    {        "title": "1410.6732v1.Strong_magnetic_coupling_in_the_hexagonal_R5Pb3_compounds__R___Gd_Tm_.pdf",        "content": "1 \n Strong magnetic coupling in the hexagonal R5Pb3 compounds ( R = Gd-Tm)  \nAndrea Marcinkova1, Clarina de la Cruz2, Joshua Yip1, Liang L. Zhao1, Jiakui K. Wang1, E. Svanidze1 \nand E. Morosan1 \n1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA  \n2Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA  \nABSTRACT  \n We have synthesized R5Pb3 (R = Gd -Tm) compounds in polycrystalline form and performed \nstructural analysis , magnetization , and neutron scattering measurements. For a ll R5Pb3 reported here  the \nWeiss temperatures  θW are several times smaller  than the ordering temperatures  TORD, while the latter \nare remarkably high ( TORD up to 275 K  for R = Gd) compared to other known R-M binaries ( M = Si, \nGe, Sn and Sb). The magnetic order cha nges from ferromagnetic in R = Gd, Tb to antiferromagnetic in \nR = Dy-Tm. Below TORD, the magnetization measurements together with neutron powder diffraction \nshow complex magnetic behavior and reveal the existence of up to three additional phase transitions . \nWe believe this to be a result of  crystal  electric field effects  responsible for high magnetocrystalline \nanisotropy . The R5Pb3 magnetic unit cells for R = Tb-Tm can be described with incommensurate \nmagnetic wave vector s with spin modulation either along the c axis in R = Tb, Er and Tm  or within  the \nab-plane  in R = Dy and Ho . \n \nKeywords: rare earth led binary systems, incommensurate magnetic structure, crysta l field effects.  \nI. INTRODUCTION  \n The discovery of intermetallic compounds with the chemical formula  R5M3 (R = rare earth , \nM = Si, Ge, Sn, Sb and Bi) attracted attention in condensed matter physics because of their rich \nstructural and physical properties. R5Si3 compounds ( R = La–Nd) crystallize in the  Cr5B3-type \ntetragonal structure1, 2 with space group (SG) I4/mcm , while the R = Gd–Lu, Y, members of this series \ncrystallize in the Mn 5Si3-type hexagonal structure w ith SG P63/mcm .3, 4 All R5Bi3 compounds \ncrystallize in the orthorhombic Y 5Bi3-type structure with SG Pnma.2, 5, 6 R5M3 (where R = La-Nd, Gd -\nLu and M = Ge, Sn , Sb, Pb) adopt a hexagonal P63/mcm  structure , in which the R atoms ( R1, R2) \noccupy two inequivalent crystallographic sites, with 2 (3) R1(R2)/f.u. in the 4 d(6g) positions. Thus one 2 \n can expect different magnetic coupling between the two different sublattices and, he nce, complex  \nmagnetic ordering.  \n Limited physical properties characterization is reported f or the hexagonal R5M3 (M = Si, Ge, Sn \nand Sb ) family of compounds . Yb5Si3 is reported to form with mixed -valence Yb ions,  showing \nantiferromagnetic  order below the N éel temperature TN = 1.6 K. Ho5Si3 undergoes two magnetic \ntransitions at low temp eratures:  first, antiferromagnetic  ordering at TN = 24 K, and second, a spin \nreorientation transition at T2  8 K. Field -dependent magnetization data reveals a metamagnetic \ntransition at Hc = 2.2 T. Later studies by Canepa et al.7 on R5Si3 samples with R = Y, Ce, Pr, Nd, Sm, \nGd, Tb, Yb, and Lu presented temperature -dependent ( T = 4 - 300 K) resistivity data . However, \ndetailed information on the magnetic properties of the R5Si3 is still lacking.  \n The R5Ge3 compounds with R = Pr, Gd, Tb, Ho, and Er show antiferromagnetic  order  with Néel \ntemperatures between  TN = 10 and  85 K and ferrimagnetic (FI) order  in Nd5Ge3 and Ce5Ge3.8, 9 As \nexpected, La5Ge3 and Y 5Ge3 are Pauli paramagnets . No other magnetic transitions were observed \nbelow  the ordering temperature in  the ma gnetic R5Ge3 series . In 2004, Tsutaoka et al.10 presented the \nmagn etic and transport properties of  Gd5Ge3 and Tb 5Ge3 single crystals. Gd 5Ge3 was found to be \nantiferromagnetic  below  TN = 76 K, wi th a spin reorientation transition at TSR  52 K. Tb 5Ge3 shows  an \nantiferromagnetic  order  below TN = 79 K, with large magnetic anisotropy.11 The antimonides R5Sb3 \nwith R = Pr, Nd, Gd -Tm show  antiferromagnetic  order , with Néel temperatures up to  109 K for \nR = Gd.2, 12 Not surprisingly, La 5Sb3 and Y 5Sb3 display  a temperature -independent susceptibility.2 \n Magnetization measurements carried out on R5M3 (R = rare earth, M = Si, Ge, Sn, and Sb ) \ncompounds revealed the presence of strong  crystal field effects  (CEF). As a r esult, these compounds \npresent  complex magnetic behavior, mainly due to exchange interaction between two inequivalent R \natomic sites . Such  a behavior w as confirmed by neutron powder diffraction (NPD)  in some of the R5M3 \nsystems . The magnetic configuration was found to be conical -spiral for  Tb5Sb3,13 flat-spiral for \nTb5Ge314, and amplitude sine -modulated  for Ho 5Si3,1 Ho5Sb3,15 Dy5Sb3,16 Tb5Sn3,17 Er5Sn318 and \nHo5Sn3.19 To better understand the complex  interplay between the exchange int eractions and CEF in the  \nR5M3 compounds, a detailed study on their anisotropic thermodynamic and transport properties is \nneeded.  \n In this manuscript, we performed such a study of the structural and magnetic properties on the \nR5Pb3 compounds ( R = Gd-Tm), from a combination of  magnetization and NPD measurements. We \ninvestigated the influence of  different rare earth  metals on the crystal structure  and the magnetism . 3 \n Lattice parameters and the unit cell volume decrease monotonically with decreasing rare earth  ionic \nradii, according to the expected lanthanide contraction.20 NPD reveals the anisotropy in the R5Pb3 \nsystems  (R = Tb-Tm) and multiple magnetic transitions in the ordered state, with incommensurate \nmagnetic wave vectors km associated with the ordere d state down to the lowest measured temperature \n(T = 4 K). The multiple  magnetic transitions observed in NPD are further confirmed by m agnetization \nmeasurements . Remarkably, after Gd 5Si4 (TORD = 340 K),21 R5Pb3 systems  have the highest observed \nordering temperatures  TORD compared to other known R-M (M = Si, Ge, Sn, and Sb ) binary compounds . \nFor example, Gd5Pb3 orders at 275  K, which is much higher than  Gd5Ge4 (TORD = 127 K),n2 Gd5Sn4 \n(TORD = 80 K),22 Gd5Sn3 (TORD = 68 K),10 Gd5Si3 (TORD = 55 K),7 Gd5Ge3 (TORD = 52 K).10 A striking \nresult of the high anisotropy in the R5Pb3 compounds  is that the Weiss temperatures θW are up to five \ntimes smaller than the ordering temperatures TORD. This was also the case in the high TORD Gd silicates, \nfor example in GdSi , which has a  ordering temperature ( TORD = 78 K)  four times larger  than the Weiss \ntemperature ( θW = 20 K) .23 \nII. MA TERIALS AND  METHODS  \n Polycrystalline R5Pb3 (R = Gd-Tm) samples were prepared from rare earth  ingots (Ames lab \n99.999%) and Pb (Alfa Aesar; 99.98%) in a molar ratio of 5:3.5 , with excess Pb used to compensate for \nmass losses during heating . Arc m elting of the samples w as performed on a water -cooled Copper hearth \nunder a purified argon atmosphere. To ensure homogeneity, the samples were turned over and re -melted \nseveral times. Initial phase analysis was done using x-ray powder diffraction on a Riga ku D/max \nULTIMA II diffractometer with Cu Kα radiation source. The  R5Pb3 (R = Gd-Tm) turned out to be \nextremely  air-sensitive , so the x -ray powders were covered by a layer of amorphous mineral oil. \nRietveld analysis24 of the powder diffraction data was done using the GSAS/EXP GUI suite of \nprograms.25 A pseudo -Voigt function was used to describe the peak shape for all data. As an example, \nthe Rietveld fit of the x-ray powder diffraction for Ho5Pb3 is shown in Fig. 1.  The presence of the \nmineral oil as a protective layer can be obse rved as a broad peak at low angles (2θ = 10° - 20°), \ntogether with  up to 3 wt% of Pb impurity in all polycrystalline R5Pb3 samples .  \n For the NPD experiments, roughly 5 g of each polycrystalline sample was held in a cylindrical \nvanadium container in a top -loading closed -cycle refrigerator and studied using the HB -2A powder \ndiffractometer at the High Flux Isotope Reactor of Oak Ridge National Laboratory.26 Data from HB -2A \nwere collected with neutron wavelengths λ  = 1.54 Å and λ  = 2.41 Å, by (115) and (113) reflections 4 \n from a vertically -focusing Ge monochromator. The data were acquired in the temperature range of 4  –\n 220 K by scanning the detector array consisting of 44 3He tubes in two segments to cover the angle 2 θ \nrange of 4 °–150° in steps of 0.05 °. Overlapping detectors for  the given steps were used to average the \ncounting efficiency of each detector. The shorter wavelength, which gives a greater intensity and higher \nQ coverage, was used to investigate the crystal structures in this low temperature regime, while the \nlonger w avelength gives lower Q coverage and greater resolution that was important for investigating \nthe magnetic structures of these materials. More details about the HB -2A instrument and data collection \nstrategies can be found in Ref. [27].  The NPD data were ana lyzed using the Rietveld refinement \nprogram FULLPROF27 and the representational analysis software SARAh .28 \n Powder diffraction patterns reveal  that all samples have a hexagonal structure with SG P63/mcm , \nin which  the rar e earth atoms occupy two inequivalent crystallographic 4 d and 6 g sites, located at (1/3, \n2/3, 0) and ( xR, 0, 1/4). The atoms in the 4d position ( R1 in Fig. 2c ) are linked into linear chains along \nthe c axis, while the atoms in  the 6g position ( R2 in Fig. 2 c) form face-sharing octahedra l chains along \nc. This structure is similar to that of the R5M3 analogues ( M = Si, Ge, Sn). Both a (squares, left axis in \nFig. 2a)  and c (triangles , right  axis in Fig. 2a)  lattice parameters, as well as the unit cell volume  (circles , \nright  axis in Fig. 2 b), decrease as expected  as R changes from Gd towards Tm  in the R5Pb3 series . From \nRietveld refinements, the distances between the nearest  (NN, R1-R1, R2-R2) and next -nearest (NNN , R1-\nR2) R neighbors  were calculated. The shortes t R-R distances dR1-R1 = 3.2 -3.4 Å were found between R1 \natoms  situated in the quasi -one dimensional chains, parallel to the c axis. \n Zero field -cooled (ZFC) and field -cooled (FC) DC magnetic susceptibilities were measured using \na Quantum Design (QD) Magne tic Property Measurement System (MPMS). The temperature \ndependence of the susceptibility for all samples was measured in H = 0.1 T, with additional fields up to \nH = 7 T measured for a subset of the samples . The antiferromagnetic ( AFM ) transition temperatur es TN \nand the lower -temperature transitions for the AFM compounds  were determined from the peaks in \nd(MT)/dT.29 In the case of ferromagne tic (FM) order, TC and all other transition temperatures were \ndetermined from susceptibility derivatives dM/dT. To further characterize the magnetic behavior of \nR5Pb3 samples, the field -dependent magnetization measurements M(H) were performed  at T = 2 K. A ll \nR5Pb3 samples show metamagnetic phase transitions, with the critical field values determined from the \nmaxima in dM/dH.  \n III. RESULTS  5 \n 3.1 Gd 5Pb3 \n Fig. 3 a shows the ZFC ( closed symbols)  temperature dependence  of the magnetic susceptibility \nM/H for Gd 5Pb3 in applied magnetic field H = 0.1 T (black circles ), 1 T (red triangles) and 7  T (blue \nsquares ). The Curie -Weiss law is follo wed at temperatures above 300  K, as revealed by the linear \ninverse susceptibility H/ M (open circles, right axis, Fig. 3a). The linear fit above ~300 K yields a Weiss \ntemperature θW = 158.7  K and effective magnetic moment\nmeffexp ≈ 7.46 μB / Gd3+, close to the theoretical  \nvalue  \ntheory\neff≈ 7.94 μB for Gd3+. The positive θW value is indicative of ferromagnetic  coupling , \nconsistent with the shape of the magnetic susceptibility, abrupt ly increasing  below TORD  ~ 275 K. More \ninterestingly, θW is nearly half  of the ordering temperature TORD = 275 K, with TORD determined from \nthe minimum in dM/dT for H = 0.1 T,  as shown in Fig.3b . Below TORD another magnetic transition is \nevident at T2 = 85.5 K, as seen in the high field M(T) data ( inset, Fig. 3a)  and also revealed by the \nmagnetization derivative d/dT (inset, Fig. 3b).29 The transition temperatures TORD and T2 do not \nchange with increasing magnetic field  up to  H = 7 T \n Compared to others R5Pb3 compounds, the M(H)  isotherm for Gd 5Pb3 (Fig. 4 ) shows almost \nlinear behavior in the applied magnetic field of 0.2  T  H  7 T. However, within our field range the \nmaximum magnetization (  1.5 μB/Gd3+) is much smaller than  the Gd3+ saturated moment μsat \n= 7 μB/Gd3+. The steep M(H) increase for fields H < 0. 2 T, is reminiscent of the anisotropic behavior in \nsome G d interm etallic compounds ,30 suggesting that a helical (or more complex) magne tic moment \nconfiguration in Gd5Pb3 may be responsible for both the M(H) shape and the small M values reached \nwith H = 7 T.  No magnetic hysteresis is observed (Fig. 4). \nNo NPD has been collected on Gd 5Pb3 due to its very high absorption cross -section  for ne utrons.  \nFor example, Gd 2O3 is used as a neutron beam stopper .31 \n3.2 Tb5Pb3 \n The ZFC ( full circles ) and FC ( full triangles ) temperature dependence of the magnetic \nsusceptibility  M/H (left axis)  and the  inverse susceptibility H/M (right axis)  is shown in Fig. 5 a and \nreveals tw o magnetic transitions, first at TORD = 215.3  K and second at T2 = 69.4 K. The values of TORD \nand T2 were determined from peaks  in dM/dT . The inverse magnetic suscep tibility shows linear \ntemperature dependence above 300 K, indicative of Curie -Weiss behavio r. The linear fit H/M above ~ \n300 K  yields a Weiss temperature θW = 96.4 K. This is less tha n half of TORD, a likely  consequence of \nstrong crystal field effects, or anisotropic coupling , or both . The positive θW value indicates the 6 \n dominance of the ferroma gnetic coupling between the Tb3+ magnetic ions. The effective magnetic \nmoment is \nmeffexp ≈ 9.66 μB/Tb3+, which is consistent with the theoretical value calculated for the Tb3+ free \nion, \ntheory\neff ≈ 9.72 μB. The  T = 2 K  M(H) isotherm  (black squares, Fig. 5b)  shows small hyst eresis  below \nH ~ 2.2 T. Considering the lack of the secondary phases in the neutron data shown below, the observed \nsmall hysteresis in Tb 5Pb3 measured at T = 2 K can be ascribed to a small ferromagnetic  component.  \nWithin our field range , the maximum magnetization (  2 μB/Tb3+) is much smaller than the Tb3+ \nsaturated moment μsat = 9 μB/Tb3+. The low measured magnetization values M(H) ≤ 2 μB could be a \nconsequence of the crystal field anisotropy.  \n The magnetic order in Tb 5Pb3 is further investigated by the NPD data collec ted at T = 300, 150, \n90, 70, 60, 50  K using a wavelength  = 1.54 Å, and at T = 4 K using  = 2.41 Å (Fig. 6). In the \nparamagnetic state ( T = 300 K bottom curve in Fig.  6), only the nuclear peaks are visible. As the \ntemperature is decreased  through  T = 150 K, extra intensity on the nuclear (2,1,1) Bragg peak develops  \nat Q = 2.29 Å-1 for all lower  temperatures.  Additional magnetic peaks are observed  below T = 70 K, in \nthe range Q = 1 - 2.5 Å-1, with intensities varying as a function of temperature.  Using a l arger \nwavelength  = 2.41 Å, a (0,0, l) magnetic peak with l = 1.52  is revealed at T = 4 K (inset, Fig. 6), \nsuggesti ng that the spins order along c axis at this temperature.  \n A careful comparison of the T = 300 K and 150  K data ( inset, Fig. 7) reveals weak magnetic \ncontribu tions to the nuclear (1,0,2), (1,1,1),  and (2,1,1)  Bragg ref lections. This is indicative of  weak \nferromagnetic  coupling of the R spins, which is consistent with the ferromagnetic ordering  observed \nfrom magnetization around TORD = 215.3 K  (Fig. 5a). All ma gnetic Bragg reflections were  indexed with \na hexagonal cell aM = aN, cM = cN, where the subscript M denotes magnetic and N the nuclear cell . The \ntemperature evolution of the magnetic (2,1,1)+km Bragg reflection  was measured  in the temperature \nrange T = 150  K - 260 K, and its intensity is shown  in Fig. 7 (black squares ). The intensity of this peak \nincreases in the ordered state , resulting in an estimate of the ordering temperature TORD = 211 K, taken \nas the intercept of the two solid lines in Fig. 7. This is  in good agreement with the magnetization data \nshown in Fig.  5a. \n Upon further cooling, the new magnetic peaks  developing  below T = 70 K point to an AFM state \nin this temperature range . Indexing these magnetic peaks suggests  that the magnetic unit cell is almost \ntwice as  large as the nuclear unit cell  with aM = aN and cM = 1.88cN. The corresponding propagation \nmagnetic vector  below  T = 60 K is  km1 = (0, 0, 0.535) , Fig. 8a. At 4  K, the incommensurability  \nchanges , with the corresponding  magnetic vector km2 = (0, 0, 0.520) , Fig. 8b. A detailed analysis of the 7 \n magnetic structure in Tb5Pb3 is underway, to  be published elsewher e.32 The peak at Q = 2.29 Å-1 \npersists th roughout the AFM state (Fi g. 8), making it unclear whether it is intrinsic to the  AFM order or \nif it is associated with a FM component within the AFM state.  \n3.3 Dy5Pb3 \n Fig. 9a shows the ZFC ( circles ) and FC ( triangles ) temperature -dependen t magnetic susceptibility \nM/H (left axis ) and the inverse susceptibility H/M (open symbols, right axis), measured in an applied \nfield H = 0.1 T. By contrast with the R = Gd and Tb compounds, where M(T) diverged at TORD (Figs., \n3a and 5a), in Dy 5Pb3 the magnetic order is indicated by a peak close t o 160 K. The shape of the \nmagnetic susceptibility  and the indistinguishable ZFC/FC M(T) close to this temperature  suggest that \nthe transition corresponds to AFM order , with  the N éel temperature TN determined from d(MT)/dT23 to \nbe TN = 160.6  K. However, the  upturn in the magnetization and the additional peaks at lower \ntemperatures reveal very complex magnetic behavior in Dy 5Pb3. Three local magnetization maxima at \nT2 = 75.3 K, T3 = 43.3 K and T4 = 15.4 K are determined from the d(T)/dT, with small ZFC /FC \nirreversibility observed below T2. The spin reorientation transitions at T2-T4 might also be associated \nwith a small ferromagnetic component at low temperatures.  The inverse magnetic susceptibility \ndisplayed in Fig. 9a (right axis ) is linear above 220  K, indicative of Curie -Weiss behavior.  From the \nlinear fit of H/M above T = 220 K, the resulting  Weiss temperature  is θW = 36.4 K, with the effective \nmoment \nmeffexp ≈ 10.38  μB/Dy3+, consistent with the theoretical value \ntheory\neff ≈ 10.63  μB/Dy3+. The positive \nθW value indicat es FM coupling between Dy3+ ions. It is very likely then that  strong CEF effects result \nin the AFM order , and are responsible for the r emarkably  large  ordering temperature (almost five times ) \ncompared to  θW. \n The M(H) isotherm measured at T = 2 K (Fig. 9b) displays one broad metamagnetic transition  at \nHc = 5.9 ( 6.25) T for increasing (decreasing) applied field , as determined from the peaks position in the \ndM/dH plot (inset, Fig. 9b ). The magnetization in the maximum applied field H = 7 T (5.8 μB / Tb3+) is \nclose to half of  the Dy3+ saturated moment μsat = 10 μB/Dy3+. It is therefore likely that more \nmetamagnetic transitions occur in higher magnetic fields. Small hysteresis can be observed for the \nwhole measured field range, a likely result of the FM component at low tempe ratures, as suggested by \nthe ZFC /FC irreversibi lity in M(T) data (Fig. 9a) .  \n Neutron diffraction measurements  on Dy-containing samples  are inherently difficult given that \nDy is a strong neutron absorber. However, measurements on Dy 5Pb3 did reveal two magnetic peaks 8 \n between T = 30 K and 4 K , marked by the vertical arrows in Fig. 10a. These magnetic peaks were \nidentified by comparison with the measurement in the paramagnetic state T = 250 K (bottom curve, \nFig. 10a).  The order parameter for Dy 5Pb3, measured on the (-3,1,1)+ km magnetic Bragg reflection in  \nthe temperature range T = 4 - 95 K (Fig. 10b) reveals the third magnetic transition at T3 = 39 K. This is \nclose to the value determined from magnetization T3 = 43.3 K (Fig. 9a) . The magnetic peaks are \nconsistent with  a magnetic wavevector  km = (0, 0.301, 0 ), determined from  the Lebail fit of the T = 4 K \nNPD data (Fig. 11). \n3.4 Ho 5Pb3 \n Fig. 12 a shows  the temperature -dependent magnetic susceptibility M/H (black circles , left axis ) \nand the inverse susceptibility  H/M (open circles, right axis ) for Ho 5Pb3 in an applied field H = 0.1 T. \nNo ZFC/ FC irreversibility was registered down to 1.8 K, which, together with the weak peak around \n110 K,  suggest  AFM order  in this compound. T he Néel temperature TN was determined from d(MT)/dT \nto be TN = 111 K. Upon f urther coolin g, two local magnetization maxima at T2 = 23.7 K and \nT3 = 8.03 K are determined from the d(T)/dT. The inverse magnetic susceptibi lity displayed in Fig. \n12 (right axis) is linear above 1 10 K, indicative of Curie -Weiss behavior. From the linear fit of H/M \nabove T = 110 K, the resulting Weiss temperature is θW = 30.4 K, with the effective moment \nmeffexp\n≈ 10.95  μB/Ho3+ close to  the theoretical value \ntheory\neff ≈ 10.61 μB/Ho3. The positive θW value \nindicates FM coupling betwe en Ho3+ ions. However, the evidence for magnetic order from M(T) data \npoints to AFM order, suggesting possible anisotropic spin coupling and strong CEF effects, as well as \nthe remarkably large ordering temperature TORD compared to θW, TORD/θW ~4. \n The T = 2 K M(H) isotherm (Fig. 12b) shows two metamagnetic transitions up to H = 7 T, also \nconsistent with spin -flop transitions in the AFM state. However, within our field range the maximum \nmagnetization ( 3 μB/Ho3+) is less than half the Ho3+ saturated moment μsat(Ho3+) = 10.6 μB. It is \ntherefore likely that more metamagnetic transitions occur in higher magnetic fields, or it could be a \nconsequence of the crystal field anisotropy.  \n  Fig. 1 3 shows the temperature evolution of the (3,-1,0)+km1, (1,0,0)+km2, and ( 0,1,0)+km2+km3 \nas the most intense  Ho5Pb3  magnetic Bragg peaks for the respective  temperature interval s. The order \nparameter  for Ho 5Pb3 measured on the ( 3,-1,0)+km1 magnetic Bragg reflection in  the temperature range \nT = 70 - 120 K reveals the ordering tempe rature T = 110.5  K, taken as the intercept o f the two solid \nlines in Fig. 13a. This is close to the value determined from magnetization TORD = 111 K. The 9 \n (1,0,0)+ km2 and (0,1,0)+km2+km3 magnetic Bragg reflections measured in the temperature range T = 4 -\n 40 K (Fig. 13b) were used to determine the magnetic transition temperatures T2  28.6 K and T3  \n8.7 K. These transition temperatures are close to the values determined from the magnetic susceptibility \nin Fig. 12 a. \n A comparison of the NPD patterns for Ho 5Pb3 collected at T = 150 and 50  K (Fig. 14) for \nQ = 1.9-2.3 Å-1 points to an AFM state in this temperature range. Indexing these magnetic peaks \nsuggests that the magnetic unit cell is more than three times larger than the nuclear unit cell with \naM = 3.46a N and cM = cN. The corresponding propagation magnetic vector determined from the Lebail \nfit to the T = 50 K NPD data (Fig. 14 ) is km1 = (0, 0 .295, 0). The (h,0+k,0) magnetic peak with h = 1 \nand k = 0.295 at T = 50 K (left inset , Fig. 14) suggests that the Ho  spins order within  the ab-plane at \nthis temperature. With the second magnetic transition at T2 = 28.6 K, the incommensurability of the \nmagnetic vector changes to km2 = (0, 0.288, 0) ( orange , Fig. 15a). Below T3 = 8.7 K, the additional \nAFM ordering along t he b axis is indexed using km3 = (0, 0.5, 0) ( blue, Fig 15b). At T = 4 K, all \nmagnetic Bragg reflections are indexed using the same two magnetic propagation vectors km2 and km3, \nwith an additional rotation of the spins towards the basal plane.  \n3.5 Er5Pb3. \n Fig. 16a shows the temperature -dependent magnetic susceptibility M/H (full symbols, left axis ) \nand the inverse susceptibility H/M (open symbols, right axis) for Er 5Pb3 in an applied field H = 0.1 T. \nThe magnetic order is mark ed by a peak in M/H close to 3 6 K and, together with the indistinguishable \nZFC/FC M(T) close to this temperature, suggests that the transition corresponds to AFM order. The \nNéel temperature TN determined from d(MT)/dT is TN = 36.3 K. Upon further cooling, another local \nmagnetization ma ximum at T2 = 11.0 K is observed in d(T)/dT, consistent with a spin reorientation \ntransition. The inverse magnetic susceptibility displayed in Fig. 16 (right axis) is linear above 50  K, \nindicative of Curie -Weiss behavior. From the linear fit of H/M above T = 50 K, the resulting Weiss \ntemperat ure is θW = 21.3 K, with the effective moment \nmeffexp  ≈ 9.62 μB/Er3+. This is consistent with the \ncalculated value \ntheory\neff ≈ 9.58 μB/Er3+. As was also the case for R = Dy and Ho , the positive θW value \nindicates FM cou pling between Er3+ ions, which together with likely strong CEF effects, results in the \nAFM order. As was the case for the other R members of this series, the ordering temperature TORD \nexceeds the Weiss temperature, and in this case TORD/θW ~ 2. \n The T = 2 K M(H) isotherm (Fig. 16 b) displays  a metamagnetic transition with the critical field 10 \n Hc = 1.25 (2.37)  T for increasing (decreasing) applied field, as determined from the peak position in \ndM/dH. The me tamagnetism is likely associated with  spin-flop transition s caused by strong CEF \nanisotropy in this system . It was shown33 that in antiferromagnets with strong uniaxial anisotropy, e.g. \nhexagonal or tetragonal systems with a n easy axis, the transition would be a single -step from a low \nmagnetization state toward a fully saturated state. However,  within our field range the maximum \nmagnetization (  8 μB/Er3+) is less than the Er3+ saturated moment μsat = 9 μB/Er3+. The finite slope of \nM(H) at H = 7 T suggests that this system is approaching saturation at a field slightly higher than our \nmaximum mea sured field.  \n Fig. 17 shows a  comparison of the NPD patterns for Er 5Pb3 collected at T = 150, 60, 20 and 4  K, \nusing wavelength  = 1.54 Å and plotted in the Q = 0.4 - 2.6 Å-1 interval. Fig. 1 8 shows the LeBail \nrefinement of the NPD data for Er 5Pb3 measured  at T = 20  K (a) and  4 K (b) . The insets show NPD \ndata for Q = 0.1 -1.3 Å-1. Below T = 20 K the NPD data (Fig. 1 8a) reveals new magnetic Bragg \nreflections compared to the high temperature  data (T = 150 K , Fig. 1 7) indicative of AFM state. This is \nin good ag reement with the magnetization da ta with AFM ordering below  TN = 36.3 K. Indexing the \nnew magnetic peaks suggests that the magnetic unit cell is more than three times larger than the nuclear \nunit cell with aM = aN, cM = 3.40c N. The corresponding propagatio n magnetic vector determined from \nthe Lebail fit to the T = 20 K NPD data (Fig. 1 8a) is km1 = (0, 0, 0 .294). A second magnetic transition \nobserved in the magnetization data at T2 = 11 K is caused  by an additional spin rotation along  the c axis  \n(Fig. 1 8b). At T = 4 K, intensities of the ( h,h,0)+km magnetic peaks decrease compared to those \nmeasured at T = 20 K, typical of a spin rotation  from an in-plane to an out-of-plane  arrangement.  In our \ncase,  out-of-plane spin  arrangement s lower  the magnetic component  of the system along c axis which \nleads to  the intensity decrease  of the ( h,h,0)+km magnetic peaks.  \n3.6 Tm 5Pb3. \n Fig. 19 a shows the temperature -dependent magnetic susceptibility M/H (black circles, left axis) \nand the inverse susceptibility H/M (open circles,  right axis) for Tm 5Pb3 in an applied field of H = 0.1 T. \nThe magnetic order is marked by a peak in M/H close to 18 K and, together with the indistinguishable \nZFC/FC M(T) close to this temperature, suggests that the transition corresponds to AFM order. T he \nNéel temperature TN determined  is from d(MT)/dT to be TN = 17.8 K. The inverse magnetic \nsusceptibility displayed in Fig. 1 9a (right axis) is linear above 16 K, indicative of Curie -Weiss \nbehavior. From the linear fit of H/M above T = 40 K, the resulting We iss temperature is θW = -1.7 K, 11 \n with the effective moment \nmeffexp ≈ 7.45 μB/Tm3+. This is consistent with the calculated value \ntheory\neff\n≈ 7.56 μB/Tm3+. The negative  θW value is consistent with AFM coupling between Tm3+ ions.  \n The M(H) iso therm meas ured at T = 2 K (Fig. 19 b) displays  two-step metamagnetic transition \nwith a critical field for increasing (decreasing) applied field, as determined from the peak s in the \ndM/dH. 3.5 μB/Tm3+ The magnetization in an applied field H = 7 T corresponds to 7  μB/Tm3+ and is \nconsiste nt with the calculated saturated moment μsat = 7 μB/Tm3+. \n A comparison of the NPD patterns for Tm 5Pb3 collected at T = 150, 50, 30 and 4 K (Fig. 20) \nusing wavelength  = 1.54 Å plotted in the Q = 0.70 - 2.8 Å-1 interval  reveal extra peaks at  T = 4 K, \nsugg esting AFM magnetic order between 30 and 4 K . Indexing these magnetic peaks suggests that the \nmagnetic unit cell is more tha n three times  large r than the nuclear unit cell with aM = aN, cM = 3.60c N. \nThe corresponding propagation magnetic vector determined from the Lebail fit to the T = 4 K NPD \ndata (Fig. 20) is km1 = (0, 0, 0 .275). Such a magnetic vector is similar to the one found in Er 5Pb3 or \nEr5Si3.23 \nIV . CONCLUSIONS  \n We have successfully synthesized polycrystalline R5Pb3 (R = Gd-Tm) samples and perfor med \nstructural and magnetic  analysis , along with  neutron powder diffraction  on the R = Tb – Tm R5Pb3 \ncompounds . The inverse magnetic susceptibilities of all R5Pb3 showed that the ordering temperature \nTORD decreases when going from Gd3+ towards Tm3+ (Fig. 21a), consistent with the expected deGen nes \nscaling[ref] . Below TORD, the behavior is complex and reveals other phase transitions at T2, T3, T4. A fit \nof the linear parts of the H /M to the Curie -Weiss law yields large  differences between Weiss \ntemperatures θW and actual measured TORD (Fig. 21), with the  latter several times larger tha n the former  \nWe believe this to be a result of strong CEF effects . Although the Curie -Weiss fit yields positive θW \nvalue s for R5Pb3 (R = Gd - Er), NPD analysis indicates more complex ordered states. This is plausibly  \nalso caused by different exchange interactions between  the two inequivalent R atomic sites , and is often  \naccompanied  by the competition between magnetic incommensurability and commensurability.13-19 \n The T = 2 K  isotherms  for R5Pb3 with R = Tb - Tm compou nds reveals  metamagnetic behavior  \nassociated with the  crystal field anisotropy . For R5Pb3 (R = Tb  and Dy), the S-shaped  isotherms  and \ntheir smooth variation in the measured field range is typical of a continuous metamagnetic transition, \nwhich is generally of second order . Note that there is a n almost nonexistent  hysteresis  in R5Pb3 (R = Tb 12 \n and Dy) isotherms . The opposite was observed for Er5Pb3. The isotherm  for Er 5Pb3 shows  a step -like \nmetamagnetic transition with  pronounced hysteresis. Such  discontinuous behavior is usually, but not \nalways, associated with a first -order transition.  The same isotherms for  R5Pb3 with R = Ho and Tm \nshow  multiple metamagnetic transitions. Due to the polycrystallinity and air -sensitivity of our samples, \none cannot fully describ e the interionic interactions between R sites. It is evident that the CEF , \nfrustration (hexagonal structure  with two inequivalent crystallographic R sites) , and magnetic coupling \nplay a leading role in the complex magnetic properties observed in R5Pb3. \n The ordering temperature Tord follow the expected  de Gennes  scaling Tord ~ dG, where the \ndeGennes factor dGis given by dG = (gJ - 1)2J(J + 1), in which gJ is the Landé factor and J is the total \nangular momentum of the R3+ ion Hund’s rule ground state  multipl et. The magnetic ordering \ntemperatures ( TORD, T2, T3, T4), Weiss temperatures θW and the ratio f = θW / TORD for R5Pb3 (R = Gd - \nTm) are shown in Fig. 21. The dashed line in Fig 21a shows the expected linear de Gennes scaling.  \nCEF effects often reduce the  Hund’s rule ground state degeneracy, leading to deviations from the \nCurie -Weiss behavior of the temperature -dependent magnetic susceptibility. In our case, such a \nbehavior manifests itself in TORD being much higher than the Weiss temperature θW. The ratio , f = \nθW / TORD, or the  frustration parameter , is usually employed  to quantify the  frustration in magnetic  \nsystem s. With frustration, f is close to 10, or at least larger then unity. This ratio in R5Pb3 (R = Gd – \nTm) is notably up to an order of magnitud e smaller  than one, which implies strong crystalline \nanisotropy in these materials.  \n Like most of the previously studied R5X3 compounds ( X = Si, Ge, Sn, Sb) , compl ex magnetic \nstructures  are also detected in  R5Pb3 (R = Gd - Tm). For the Pb compounds,  NPD da ta reveal the \npresence of multiple magnetic phases , which appear to be incommensurate for all R5Pb3. As quoted \nabove, in a large number of the magnetic materials the periodicity  near TN is incommensurate or long \nperiod  commensurate lea ding to amplitude mod ulated mag netic structures. Only a few  known  \ncompounds exhibit simple  commensurate AFM  structures over the whole temperature  range below TN \n(e.g. ErGa 2).34 The analysis of our NPD  data shows  that magnetic propagation vectors km for R5Pb3 \nwith changes as follows : Tb (1, 1, 1.88), Dy ( 1, 3.333 , 1), Ho (1, 3.46, 1), Er ( 1, 1, 3.40) and Tm ( 1, 1, \n3.63), indicating that the  R = Dy and Ho compounds have magnetic spins aligned in the ab-plane, while \nin the R = Tb, Er, Tm magnetic spins are aligned along the c axis. At low temperatures T = 4 K, a spin \nrotation occurs in the R = Tb, Ho and Er  members of this series, which was observed in other rare earth \nintermetallics .33 In these systems , except in the case of a  non-magnetic CEF sing let ground state, 13 \n amplitude mod ulated magnetic  structures are proved  not to be stable at low tem perature, and additional \nspin rotation  or reorientation is observed.33 \n Based on the increasing c/a ratio ( Table . 2), it is obvious that  the a axis decreases more compared \nto the c axis with heavier R ions. In other words, upon  decreasing  the R ionic size, the in-plane dR2-R2 \ndistance decreases more than the out -of-plane dR2-R2 distance. This possibly  decrease s the frustration \nand allow s magnetic spins to order within the ab-plane in some R5Pb3. Thus by geometry, increasing  \nin-plane anisotropy should be expect ed going from Tb towards Tm. But as one can see from neutron \ndata ( Table  1) only Dy 5Pb3 and Ho 5Pb3 show in -plane magnetic ordering. In the other R5Pb3 (R = Tb, \nEr and Tm)  the magnetic spins are aligned along the c axis. Neutron data and the magnetic k vectors \nreveal that CEF effects  play a very important role. For the smaller members of the series, R = Er and \nTm, correlations along the c axis are much stronger  then for example in the R = Tb compound , which is \nwhy the  former  compounds hav e larger correlation  length s (more that 3 unit cells along the c axis).  \n As one can see from Table s 1 and  2, there is no obvious trend that could  explain the compl ex \nmagnetic behavior observed in R5Pb3; rather the c/a ratio, the temperature ratio  f and the k vectors have \nto be considered  together to describe  the competition of the in -plane and out -of-plane dR2-R2 magnetic \ninteractions. The character of the modulations and detailed analysis of the NPD data together with the \nrepresentational symmetry analysis will be published s eparately.32 Inelastic neutron scattering and \nsingle crystals of the R5Pb3 are n eeded in order to determine the crystal field parameters, which would \ndescribe quantitatively the complex magnetic behavior observed in R5Pb3 compounds.  \nACKNOWLEDGEMENT  \n Work at Rice was partially supported b y NSF Grant No. DMR 0847681  and the DOD PECASE. \nResearch conducted at ORNL's High Flux Isotope Reactor was sponsored by the Scientific User \nFacilities Division, Office of Basic Energy Sciences, US Department of Energy.  14 \n REFERENCES  \n1 P. Hill and L. L. Miller, J.  Appl. Phys.  87, 6034 (2 000).  \n2 J. K. Yakin thos and I. P. Semitelou, J.  Magn . Magn . Mater. 36, 136 (1983).  \n3 J. D. Corbett, E. Garcia, A. M. Guloy, W. -M. Hurng, Y. -U. Kwon, and E. A.  Leon -Escamilla, \nChem. Mat.  10, 2824 (1998).  \n4 W. Jeitschko and E. 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Mater. 30, 111 (1982).  15 \n Table 1 : Measured and calculated magnetic moment, ordering temperatures, Weiss temperature, and \nincommens urate magnetic  structures for  R5Pb3 series.  \nR5Pb3eff ( B per R3+) \nTORD (K) Weiss \ntemperature \nW (K) Magnetic structure  \nmeasured  theoretical  \nGd5Pb3 7.46 7.94 TN = 275  \nT2= 85.5  \nT3 = 4.9  158.7  \nNo NPD data  \nTb5Pb3 9.66 9.72 TN = \n215.3  \nT2 = 69.4  \nT3 = 4 115 \nkm1 = (0,  0, 0.535)  \n \nkm2 = (0, 0, 0.520)  \nDy5Pb3 10.38  10.63  TN = \n160.6  \nT2 = 75.3  \nT3 = 43.3  \nT4 = 15.4  36.4 \n-* \n-* \n-* \nkm = (0, 0.301, 0)  \nHo5Pb3 10.61  10.95  T?= 111  \nT2 = 23.7  \nT3 = 8.0  \nT4 = 4 30.4 km1 = (0, 0.295, 0)  \nkm2 = (0, 0.288, 0)  \nkm2 + km3 = (0, 0.5, 0)  \nkm2 + km3 + spin rotation \ntowards ab-plane \nEr5Pb3 9.62 9.58 TN = 36.6  \nT2 = 11 21.3 km1 = (0, 0, 0.294)  \nkm1 + spin rotation \ntowards along  c-axis \nTm 5Pb3 7.45 7.56 TN = 17.8  -1.7 km1 = (0, 0, 0.275)  \n* see text (3.3)  16 \n Table 2 : Room temperature l attice constants, c/ a ratio, and all R-R bond distance s for R5Pb3 series.  \nR5Pb3 Gd Tb Dy Ho Er Tm \na (Å) 9.0879(4)  9.0104(7)  8.9612(5)  8.9100(6)  8.8724(2)  8.8286(3)  \nc (Å) 6.6487(5)  6.5943(5)  6.5641(4)  6.5326(5)  6.5148(2)  6.49017(2)  \nc/a 0.7316  0.7319  0.7325  0.7332  0.7343  0.7351 \ndR1-R1 \n(Å)+ 3.3182(4)  3.2962(3)  3.2813(3)  3.2628(2)  3.2474(3)  3.2329(4)  \ndR1-R2 \n(Å) 3.8998(4)  3.9058(2)  3.8583(2)  3.8390(2)  3.8338(2)  3.7977(5)  \ndR2-R2 \n(Å) , \nout-of-\nplane* \n 3.9729(4)  3.9066(2)  3.8831(2)  3.8831(2)  3.8544(2)  3.8565(3)  \ndR2-R2 \n(Å), \nin-plane* \n 3.7844(4)  3.6319(3)  3.6880(3)  3.6469(2)  3.5963(3)  3.6417(3)  \n+ dR1-R1 bonds are depicted in Fig. 2c  \n* dR2-R2 out-of-plane (dash light blue line) and dR2-R2 in-plane (solid light blue line)  bonds are depicted in \nFig. 2 d 17 \n FIGURE CAPTIONS  \n \nFig. 1 Rietveld  analysis of powder x-ray diffraction data of Ho 5Pb3 (fit statistics: wR P = 7.9%, \nRP = 6.0% and χ2 = 1.4). Collected data are shown as  black crosses, calculated model as red line, and \nbackground as green line. The Bragg markers are for the Ho 5Pb3 main phase, black asterisk  for a Pb \nimpurity.  \nFig. 2  (a) Left axis: crystallographic a-axis versus r R3+ for R5Pb3. Right axis: the c axis versus r R3+. (b) \nLeft axis: crystallographic c/a-ratio plotted versus r R3+. Right axis: volume plotted versus r R3+. The solid \nlines are the guides to the eye. Depicted c rystal structure of R5Pb3 consists of (c) linear chain s of R1 \natoms along c axis (pink solid arrow highlights  dR1-R1 bond distance);  and (d) distorted R2 hexagonal \nrings and R1 honeycomb layers (dark blue line). Also d epicted are in -plane dR2-R2 (solid light blue line) \nand o ut-of-plane dR2-R2 (dash light blue  line) bond distances.  \nFig. 3: (a) Left axis: t emperature dependence of th e magnetic susceptibility of  Gd5Pb3 measured in \ndifferent magnetic fields H = 0.1 T, 1 T , and 7 T. Right axis: inverse magnetic susceptibility. Solid \nblack line is a Curie -Weiss fit.  The inset shows a magnetic transition observed at low temperatures. (b) \ndM/dT for Gd 5Pb3 measured  at H = 0.1 T . Inset displays a d MT/dT for Gd 5Pb3 measured  at H = 1 T.  \nFig. 4:  The field -dependent magnetization measured at T  = 2 K for Gd 5Pb3. The inset dis plays The \nfield-dependent magnetization measured at low magnetic fields.  \nFig. 5: (a) Left axis:  ZFC and FC temperature dependence of th e magnetic susceptibility of  Tb5Pb3 \nmeasured in magnetic field H = 0.1 T. Right axis: inverse magnetic susceptibility . The solid red lin e is a \nCurie -Weiss fit . (b) The magnetization isotherm M(H) measured at T  = 2 K. The inset shows a \nhysteresis observed at low magnetic fields ( H < 2.2  T). \nFig. 6: Comparison of the NPD patterns  for Tb5Pb3 collected at different temperatures using \nwavelength of  = 1.54  Å. The inset shows the NPD data measured at T = 2 K using wavelength  = \n2.41 Å in the range of Q = 0.2 - 1.8 Å-1 revealing the most intense (0,0,1)+ km2 magnetic peak at Q  \n0.5 Å-1. \nFig. 7 : (a) The temperature evolution of the  (2,1,1)+km magnetic Bragg reflection, as a most intense \nmagnetic peak , revealing the ordering temperature TORD of Tb 5Pb3. An in set depicts the FM \ncontribution  on top of the nuclear peaks (black arrows) by overlaying data collected at T = 300  K \n(black line ) T = 150  K (red line) . 18 \n Fig. 8: LeBail refinement of the NPD data for Tb 5Pb3 measured at (a) T = 60  K and (b) 4 K. The \ndifference between the measured data (black) and LeBail fit (red) is shown as a blue line, the calculated \nBragg positions are indicated b y vertical markers (upper - nuclear phase, lower  - magnetic phase). The \ninsets show the difference in incommensurability of the magnetic structures with a magnetic \npropagation vector (a) km1 = (0,0,0.535) at T = 60 K and (b) km2 = (0,0,0.520) at T = 4 K. \nFig. 9: (a) Left axis: ZFC and FC temperature dependence of th e magnetic susceptibility of  Dy5Pb3 \nmeasured in magnetic field H = 0.1 T. Right axis: dependence of the inv erse magnetic susceptibility  for \nDy5Pb3. The solid red line is a Curie -Weiss fit . (b) The magnetization isotherm  M(H)  measured at \nT = 2 K. The inset shows d M/dH. \nFig. 10: (a) Comparison of the NPD patterns collected at different temperatures for Dy 5Pb3. Black \narrows indicate magnetic Bragg peaks. The inset shows the temperature evolution of t he two magnetic \nBragg peaks indicated by black arrows.  (b) The temperature evolution of the ( -3,1,1)+km Bragg \nreflection, as a most intense magnetic Bragg peak . The solid black line as a polynomial fit and serves as \na guide to the eye.  \nFig. 1 1: LeBail refi nement of the NPD data for Dy 5Pb3 measured at T = 4  K. The difference between \nthe measured data (black) and LeBail fit (red) is shown as a blue line, the calculated Bragg positions \nare indicated by vertical markers (upper - nuclear phase, lower  - magnetic phase). The inset shows the \nleBail fit of the two magnetic Bragg peaks.  \nFig. 1 2: (a) Left axis: ZFC and FC temperature dependence of the magnetic susceptibility of the \nHo5Pb3 measured in magnetic field H = 0.1 T. Right axis: dependence of the inv erse magne tic \nsusceptibility  for Ho 5Pb3. The solid red lin e is a Curie -Weiss fit . (b) The magnetization isotherm M(H) \nmeasured at T  = 2 K. \nFig. 1 3: (a) The temperature evolution of the (3,-1,0)+km1 Bragg reflection, as a most intense magnetic \nBragg peak  revealing th e ordering temperature TORD of Ho 5Pb3. (b) The temperature evolution of the \n(1,0,0)+ km2 Bragg reflection revealing the second magnetic transition T2 of Ho 5Pb3. An inset shows  the \ntemperature evolution of the (0,1,0)+ km2+km3 magnetic Bragg reflection reveal ing the third magnetic  \ntransition  T3 of Ho 5Pb3.  \nFig. 1 4: LeBail refinement of the NPD data for Ho 5Pb3 measured at T = 50  K using magnetic \npropagation vector km = (0,0.296,0). The difference between the measured data (black) and LeBail fit 19 \n (red) is shown a s a blue line, the calculated Bragg positions are indicated by vertical markers (upper - \nnuclear phase, lower  - magnetic phase). The inset shows a c omparison of the NPD patterns for Ho 5Pb3 \ncollected at temperatures T = 150  K (purple)  and 300  K (black)  plotted in Q = 0-0.75 Å-1 and Q = 1.6 -\n2.8 Å-1 interval.  \nFig. 1 5: Temperature evolution of the NPD data  for Ho 5Pb3 collected at temperatures (a) T = 50  K \n(brown ), 15  K (orange ) and (b) T = 15  K (orange ), and 4  K (blue) plotted in (a) Q = 0.6 -1.9 Å-1 and (b) \nQ = 1.5-3.5 Å-1 interval.  \nFig. 1 6: (a) Left axis: ZFC and FC temperature dependence of the magnetic susceptibility of the Er 5Pb3 \nmeasured in magnetic field H = 0.1 T. Right axis: dependence of the inv erse magnetic susceptibility  for \nEr5Pb3. The solid red line  is a Curie -Weiss fit . (b) magnetization isotherm M(H) measured at T  = 2 K. \nFig. 17:  Comparison of the NPD patterns collected at different temperatures for Er 5Pb3. \nFig. 1 8: (a) LeBail refinement of the NPD data for Er 5Pb3 measured at T = 20  K and (b) at T = 4 K \nusing magnetic propagation vector k m = (0,0,0.294). The difference between the measured data (black) \nand LeBail fit (red) is shown as a blue line, the calculated Bragg positions are indicated by vertical \nmarkers (upper - nuclear phase, lower  - magnet ic phase). The insets show NPD data  for Er 5Pb3 plotted \nin an interval Q = 0.1 -1.3 Å-1 at (a) T = 20 K and (b) T = 4 K. Arrows highlight the intensity difference \nof the low Q (h,h,0) magnetic peaks (see text).  \nFig. 1 9: (a) Left axis: ZFC and FC temperature dependence of the magnetic susceptibility of the \nTm 5Pb3 measured in magnetic field H = 0.1 T. Right axis: temperature dependence of the inv erse \nmagnetic susceptibility  for Tm 5Pb3. The solid red lin e is a Curie -Weiss fit . (b) The magnetization \nisotherm M(H)  measured at T  = 2 K. \nFig. 20 : LeBail refinement of the NPD data for Tm 5Pb3 measured at T = 4  K using magnetic \npropagation vector k m = (0,0,0.275). The difference between the measured data (black) and LeBail fit \n(red) is shown as a blue line, the calculate d Bragg positions are indicated by vertical markers (upper - \nnuclear phase, lower  - magnetic phase). The inset shows the comparison of the NPD patterns collected \nat different temperatures for Er 5Pb3 plotted in  Q = 0.7 -2.6 Å-1 interval.  \nFig. 21: (a) Magneti c ordering temperatures (left axis) and Weiss temperatures θW (right axis) . The \ndashed line is a guide to the eye and shows the expected de Gennes scaling. (b) Frustration parameter \nf = W/ TORD. 20 \n  Fig. 1.  \n \n \n \nFig. 2.  \n 21 \n FIG. 3.  \n \n \nFig.4  \n 22 \n FIG. 5. \n \n \nFIG. 6 \n 23 \n FIG. 7 \n \n \nFIG. 8  \n 24 \n FIG. 9 \n \n \nFIG. 10  \n 25 \n FIG. 1 1 \n \n \nFIG. 1 2 \n \n \n 26 \n FIG. 1 3 \n \n \n \nFIG. 1 4 \n 27 \n FIG. 1 5 \n \n \n \nFIG. 1 6 \n \n 28 \n FIG. 1 7 \n \n \nFIG. 18  \n 29 \n  \nFIG. 1 9 \n \n \n \nFIG. 20 \n 30 \n FIG. 21  \n \n "    },    {        "title": "1411.2448v1.Symmetry_lowering_lattice_distortion_at_the_spin_reorientation_in_MnBi_single_crystals.pdf",        "content": "arXiv:1411.2448v1  [cond-mat.mtrl-sci]  10 Nov 2014Symmetry-lowering lattice distortion at the spin-reorien tation in MnBi single crystals\nMichael A. McGuire,1,∗Huibo Cao,2Bryan C. Chakoumakos,2and Brian C. Sales1\n1Materials Science and Technology Division, Oak Ridge Natio nal Laboratory, Oak Ridge, Tennessee 37831, USA\n2Quantum Condensed Matter Division, Oak Ridge National Labo ratory, Oak Ridge, Tennessee 37831, USA\n(Dated: July 15, 2018)\nStructural and physical properties determined by measurem ents on large single crystals of the\nanisotropic ferromagnet MnBi are reported. The findings sup port the importance of magneto-\nelastic effects in this material. X-ray diffraction reveals a structural phase transition at the spin\nreorientation temperature TSR= 90 K. The distortion is driven by magneto-elastic coupling , and\nuponcoolingtransforms thestructurefromhexagonaltoort horhombic. Heatcapacitymeasurements\nshow a thermal anomaly at the crystallographic transition, which is suppressed rapidly by applied\nmagneticfields. Effectsonthetransportandanisotropic mag neticpropertiesofthesinglecrystalsare\nalso presented. Increasing anisotropy of the atomic displa cement parameters for Bi with increasing\ntemperature above TSRis revealed by neutron diffraction measurements. It is likel y that this is\ndirectly related to the anisotropic thermal expansion in Mn Bi, which plays a key role in the spin\nreorientation and magnetocrystalline anisotropy. The ide ntification of the true ground state crystal\nstructure reported here may be important for future experim ental and theoretical studies of this\npermanent magnet material, which have to date been performe d and interpreted using only the high\ntemperature structure.\nPACS numbers:\nI. INTRODUCTION\nDue to its high Curie temperature and strong mag-\nnetic anisotropy,MnBi hasattractedincreasingattention\nin the pursuit of rare-earth-free permanent magnets (see\nfor example the review by Poudyal and Liu [1]). Ad-\nvances in the understanding of the properties and pro-\ncessing of this material continue, despite its relatively\nsimple, NiAs-type crystal structure, and the fact that\nferromagnetism in MnBi was first reported over a cen-\ntury ago [2]. Key early studies were preformed by Guil-\nlaud [3, 4]. In the 1950’s, this material was considered\nas a candidate to replace permanent magnets containing\ncobalt and nickel, and an energy product of 4.3 MGOe\nwas achieved in the resulting “Bismanol” magnets [5, 6].\nSubsequently, energy products as high as 7.7 MGOe have\nbeen reported [7]. MnBi has several promising aspects as\na candidate replacement for rare earth magnets. These\ninclude: (1) relatively inexpensive components [8], (2)\nhigh ordered moment and saturation magnetization of\n0.58 MA m−1or about 3.5 µBper Mn at room tempera-\nture [9], (3) ferromagnetismthat persiststo 630K(about\n40 K higher than Nd 2Fe14B) [3], (4) large and uniaxial\nmagnetocrystalline anisotropy energy near 1 MJ m−3at\nroom temperature (moments along the c-axis), which in-\ncreases upon heating above room temperature [10–13].\nThe observed increase in magnetic anisotropy and coer-\ncivity with increasing temperature is perhaps the most\ninteresting, unique, and potentially important property\nof MnBi.\nThe ferromagnetism in MnBi vanishes abruptly upon\n∗Electronic address: mcguirema@ornl.govheating above 630 K [3, 4]. Importantly, the phase tran-\nsition that occurs near 630 K is not a typical magnetic\ntransition. This is not technically the Curie tempera-\nture, and the transition is not directly driven by mag-\nnetism. This transitions has been identified as a peri-\ntectic decomposition of MnBi. Heikes noted that the\ndecomposition products contain a ferromagnetic phase\nwith a Curie temperature well below the decomposition\ntemperature of MnBi [14]. It has since been determined\nthat near 630 K MnBi decomposes into Mn 1.08Bi and Bi\n[15], both of which are paramagnetic at this tempera-\nture. The Mn rich phase is an orthorhombic variant of\nthe NiAs-structure with the excess Mn occupying trig-\nonal prismatic interstices similar to those occupied by\nthe Bi atoms [16]. Since MnBi is ferromagnetic and the\ndecomposition products are paramagnetic (at the peri-\ntectic temperature), application of strong magnetic field\nhasbeenshowntostabilizeMnBitohighertemperatures,\nup to about 650 K in a 10 Tesla magnetic field [17, 18].\nSynthesis of high quality single phase samples and in\nparticular large single crystals of MnBi is complicated by\nthe peritectic nature of this reaction, the relatively low\ntemperature at which it occurs (630 K), and the nearness\nto the eutectic temperature (535 K). MnBi can be grown\nfrom a melt with excess Bi, as first described by Adams\net al. [5] Bycoolingthe melt in amagneticfield, textured\ncompositeswith coalignedMnBi crystallitesembedded in\na Bi matrix have been obtained and studied [17, 19]. For\npermanent magnet applications, fine-grained, polycrys-\ntalline samples are usually desired. These have been pro-\nduced from melt-spun and/or mechanically milled mate-\nrial [20, 21], and by magnetic separation of MnBi from\nexcess Bi in samples produced by powder-metallurgical\nroutes [7].\nMagnetic measurements show that the magnetocrys-2\ntalline anisotropy of MnBi passes through zero and\nthe coercivity of MnBi powders vanishes near 90 K\n[11, 12, 22, 23]. These observations are associated with\nthe reorientation of the Mn magnetic moment from par-\nallel to perpendicular to the crystallographic c-axis upon\ncooling. The spin reorientation temperature, TSR, de-\nnotes the temperature at which magnetic moment de-\nvelops a c-component upon heating and the coercivity\nbegins to increase. The rotation of the moments has\nbeen oberved in neutron diffraction from MnBi powders\n[7, 9, 24] and also nuclear magnetic resonance (NMR)\nexperiments [25]. The NMR results from powders [25]\nand magnetization measurements on single crystals [12]\nshow the rotation away from the c-axis upon cooling be-\ngins near 140 K, increases gradually upon further cooing\ntoTSR= 90 K, and then abruptly completes, with the\nmomentfloppingdiscontinuouslyintothe ab-plane. How-\never, neutron diffraction from MnBi powder suggests the\nmoments are not fully in-plane below 90 K [7, 9].\nIn the presence of significant magneto-elastic coupling,\naresponsein the crystallatticeto thereorientationofthe\nmagnetic moments is expected. Signatures of this cou-\npling have been reported in the temperature dependence\nof the lattice constants of MnBi [22, 26, 27]. While the\nhexagonal NiAs structure type describes the powder x-\nray diffraction results at all temperatures, anomalies in\nboththeaandclatticeparametersoccurattemperatures\nnear the spin reorientation. Recent first-principles cal-\nculations have reproduced these magneto-elastic effects,\nand linked the spin-reorientation to anisotropic thermal\nexpansion [28], and in particular to resulting changes in\nthe anisotropic pairwise exchange interactions between\nBi p-states [29].\nMost of the experimental studies performed on MnBi\nto date have used polycrystalline material. Single crys-\ntals are better suited for the detailed study of intrinsic\nand anisotropic properties. Here we report the results of\nour detailed investigation of the structural and physical\nproperties of single crystal MnBi, including x-ray and\nneutron diffraction, magnetization, heat capacity, and\nelectrical resistivity measurements. In addition to elu-\ncidating the intrinsic properties of MnBi, we find signifi-\ncant differences between the behavior of the single crys-\ntals and the reported behavior of polycrystalline MnBi\nin the literature, in particular regarding the spin reorien-\ntation. Our x-ray diffraction results reveal a symmetry-\nlowering structural distortion occurs at TSR, and suggest\nthat MnBi adopts an orthorhombicstructure at low tem-\nperature, similar to Mn 1.08Bi but without the excess in-\nterstitial Mn. This transition is extremely sensitive to\nstrain, and is not observed in powders gently ground\nfrom the crystals. The structure change is accompanied\nby a thermal anomaly which responds strongly to ap-\nplied magnetic field, and a sharp decrease in electrical\nresistivity. Examination of the anisotropic atomic dis-\nplacement parameters determined from neutron diffrac-\ntionanalysisshowsthattheBivibrationsalongthe c-axis\nare enhanced with increasing temperature above the spinreorientation, suggesting a relationship with the well-\ndocumented increase in magnetocrystalline anisotropy\nenergy with temperature.\nII. EXPERIMENTAL DETAILS\n012345\n20 40 60 80 100 120 02468(b)(006) (004) intensity (10 3 cts) (002) \na = 4.287 Å \nc = 6.117 Å  (a)\n(400) (200) (300) \n(100)  intensity (10 3 cts) \n2θ (deg) Bi Mn \na bcT = 293 K \n1 cm\nFIG. 1: X-ray diffraction patterns from MnBi single crystal\nfaces at room temperature. (a) Diffraction from a face per-\npendicular to the c-axis. (b) Diffraction from a face perpen-\ndicular to the a-axis. The inset in (a) shows a photograph\nof typical MnBi crystals. The inset in (b) shows the room\ntemperature hexagonal crystal structure of MnBi. The lat-\ntice constants determined from the reflections shown in the\nfigure are listed in (b).\nStarting materials for the crystal growths were Bi shot\n(Cominco American, 99.9999%) and Mn pieces (Alfa Ae-\nsar, 99.95%), and the crystal were grown from a flux\nwith excess Bi based on the published binary phase di-\nagram [30]. The Mn pieces were lightly ground into a\npowder and 0.65 g of this material was then immediately\ncombined with 35 g of Bi and loaded into an alumina\ncrucible. The 10 mL crucible, covered with an inverted\n10 mL “catch” crucible half-filled with quartz wool was\nsealed under vacuum inside a silica ampoule. The am-\npoule was heated to 1000◦C at 1◦C/min, held for 24 h,\ncooled to 440◦C at 1◦C/min, held at 440◦C for 1 h, and\nthen cooled to 275◦C at 0.4◦C/h. At 275◦C, the excess\nBi flux was centrifuged into the catch crucible. Several\nmm-cm sized MnBi singlecrystalswereproduced via this\nmethod, some of which are shown in Fig. 1a. The largest3\ncrystal obtained weighed close to 2 g. When exposed to\nair for hours, crystals developed a visible patina. X-ray\ndiffraction from crystals which had been powdered and\nthen exposed toairfor morethan 12hoursindicated slow\ndecomposition into MnBi, MnO 2, and Bi.\nX-ray diffraction measurements were conducted with a\nPANalytical X’Pert Pro MPD diffractometer, equipped\nwith an incident beam monochromator (Cu K α1radia-\ntion)andanOxfordPheniXclosed-cycleheliumcryostat.\nSingle crystal neutron diffraction was performed at the\nHB-3A four-circle diffractometer at the High Flux Iso-\ntope Reactor at Oak Ridge National Laboratory. A neu-\ntron wavelength of 1.003 ˚A was used from a bent perfect\nSi-331 monochromator [31]. The data were collected at\nthe selected temperatures of 5 K, 50 K, 70 K, 90 K, 110\nK, 130 K, 150 K, 200 K, 300 K, 350 K, 400 K, and 450\nK. At each temperature, more than 330 reflections were\ncollected and used for the refinements. Refinements of\nthe X-ray and neutron diffraction data were performed\nusingFullProf[32]. dcmagnetizationmeasurementswere\nperformed using a Quantum Design Magnetic Properties\nMeasurement System and ac measurements were made\nusing a Quantum Design Physical Properties Measure-\nment System, which was also used for the resistivity and\nheat capacity measurements. Spot-welded Pt leads were\nused for electrical contacts.\nIII. RESULTS AND DISCUSSION\nA. X-ray diffraction\nA photograph of typical crystals used in this study is\nshown in Fig. 1a. Diffraction from powdered crystals at\nroomtemperatureconfirmedthemtobeNiAs-typeMnBi\n(space group P63/mmc). The crystal structure is shown\nin Fig. 1b, along with the lattice constants determined\nfrom this room temperature data. Diffraction patterns,\nlike those shown in Fig. 1, from crystal facets were used\nto identify the hexagonal c-axis (typically a hexagonal\nface) and the a-axis (typically a rectangular face).\nTo examine the lattice response to the spin reorienta-\ntion, diffractionpatternsfromcrystalfaceswerecollected\nat temperatures between 300 and 20 K. At 90 K, the\nsixfold symmetry in the ab-plane is broken, as demon-\nstrated by the splitting of the hexagonal 300 reflection in\nFig. 2a. Such a distortion lowers the unit cell symmetry\nfrom primitive hexagonal to C-centered orthorhombic.\nThe temperature dependence of the lattice parameters\ndetermined from these measurements is shown in Fig. 2c\nand e. The relationship between the two unit cells is\nshown in Fig. 2d. For the orthorhombic structure, half\nof the face diagonal, labeled din the figure, is plotted\nas well in Fig. 2c. The sixfold symmetry in the hexag-\nonal structure constrains these two distances to be the\nsame, and so the divergence of aanddillustrates clearly\nthe distortion. In addition, a change in the temperature\ndependence of the lattice parameters is observed in theao, ah\nbod77.1 77.2 77.3 77.4  2θ  (deg.) (3 0 0) h\n60 70 80 90 100 110 12060.961.061.1(0 0 4)h\nT (K)60 70 80 90 100 110 120\nT (K)\n(c)(a)\n(d) (e)(b)\n0 50 100 150 200 250 3004.2684.2704.2724.2744.2764.2784.2804.2824.2844.286\n ah\nao\nd = (ao2+b o2)1/2  / 2ah, a o, d  (Å)\nT (K) 7.4067.4087.4107.4127.4147.4167.4187.4207.4227.424\nbo b h√3\nbo,b h√3 (Å)\nT = 20 K\nao = 4.271 \nbo = 7.407 \nco = 6.062 \n0 100 200 300 96.0 96.5 97.0 97.5 \nV\n volume  (Å3)\nT (K) 6.06 6.08 6.10 6.12 \nc\nc (Å)\nFIG. 2: Results of temperature dependent x-ray diffraction\nmeasurements from indexed faces of MnBi single crystals.\n(a,b) Contour plots of intensity vs diffraction angle and tem -\nperature for the 300 and 004 reflections (h denotes the hexag-\nonal unit cell). (c) In-plane lattice constants vs temperat ure\nfor the high temperature hexagonal and low temperature or-\nthorhombic (denoted by subscript o) structures. (d) The re-\nlationship between the high temperature hexagonal and low\ntemperature orthorhombic unit cells. (e) c-axis length and\nprimitive unit cell volume vs temperature (the volume of the\nC-centered orthorhombic cell is twice the value plotted).\nhexagonal state near 140 K, where the spin reorientation\nonsets (see below).\nThere is a small expansion of the c-axis upon cool-\ning through 90 K (Fig. 2e). This is also apparent in\nthe contour plot of the 004 reflection in Fig. 2b, which\nshowsanincreaseind-spacing(decreasein2 θ)uponcool-\ning at 90 K, the temperature below which the magnetic\nmoment is confined to the ab-plane. This suggests that\nmagneto-elastic coupling produces a contraction of the\nlattice along the direction that the moment is directed.4\nThis is consistent with the behavior seen in the basal\nplane dimensions as well. Upon cooling through the spin\nreorientation, the basal plane contraction exceeds the c-\naxis expansion, resulting in a sharp decrease in unit cell\nvolume (Fig. 2e). The orthorhombic lattice parameters\ndetermined at 20 K from this data are listed in Fig. 2c.\nThe importance of the coupling between the proper-\nties of the crystal lattice and the magnetism in MnBi has\nbeen recently noted by Zarkevich et al. [28]. They re-\nportacorrelationbetweenlatticeparametersandmagne-\ntocrystallineanisotropyenergyidentifiedbydensityfunc-\ntional theory calculations. In particular, the calculated\nanisotropy is strongly dependent on a. When the value\nofaused in the calculations is increased, the preferred\ndirection of the magnetic moment switches from in the\nab-plane to along the c-axis. This is generally consistent\nwith the lattice parameters reported here, where the ab-\nplane of the orthorhombic phase (moments in the plane)\nis contracted relative to the hexagonal phase (moments\nout of the plane). First principles calculations reported\nby Antropov et al.[29] find that this is due to spin-orbit\ncoupling and an unusual evolution of Bi −Bi interactions\nas the lattice constants change. Note that these calcula-\ntions, and all others for MnBi reported to date, used the\nNiAs-structure at all temperatures.\nStrain masks or suppresses the distortion so that only\nan abnormal temperature dependence of the lattice pa-\nrameters is observed in powders ground from single crys-\ntals, with no detectable symmetry change at low tem-\nperature. This was true even for gently ground, coarse\npowders and for powders gently ground under liquid ni-\ntrogen. The lattice parametersdetermined from powders\nground from the crystals showed behavior similar to pre-\nviously reported data for polycrystalline MnBi [26, 27].\nThe grinding-induced strain in powdersproduced from\nsinglecrystalscouldberelievedtosomedegreebyanneal-\ning, and diffraction patterns from the annealed powders\nare shown and discussed in the Supplemental Material.\nRietveld analysis of data collected at 20 K from annealed\npowder indicate the low temperature structure adopts\nspace group Cmcmwith lattice constants a= 4.269˚A,b\n=7.404˚A,andc=6.062˚A,ingoodagreementwiththose\ndetermined from the single crystal measurements shown\nin Fig. 2c, and Mn at (0,1\n2, 0) and Bi at (0, 0.167(1),1\n4).\nThis structure closely related to the structure adopted\nby Mn 1.08Bi (space group Pmma) [16], but without the\nexcess Mn in the interstices. As discussed in the Supple-\nmental Material, the Cmcmmodel is the simplest way to\ndescribe the distortion (it is a subgroup of the high tem-\nperature space group), and it fits the powder diffraction\ndata well. Thus, it is likely to be the correct low tem-\nperature structure. To examine the structure of the low\ntemperature phase further, single crystal x-ray diffrac-\ntion measurements were performed at 80 and 200 K. The\ndistortion is very small, near the detection limit for lab-\noratory single crystal diffraction measurements, but can\nbe identified with somecertainty. Details can be found in\nthe Supplemental Material. The results were consistent0 100 200 300 400 500 0.00.51.01.52.02.53.0\n Mn\n Bi U11 (10 -2 Å2)\n0 100 200 300 400 5000.0 0.5 1.0 1.5 2.0 2.5 3.0 \n Mn \n Bi \n \nT (K) 0 100 200 300 400 500 0.81.01.21.41.61.8 Mn\n 0 100 200 300 400 5000.8 1.0 1.2 1.4 \n Bi U33 (10 -2  Å 2)U11  / U33 \nU11  / U33 (c)\n(e)(d)\n(f)\nT (K) 0 100 200 300 400 500 0.00.51.01.52.02.53.0 \nUeq (10 -2 Å2)(b) (a)\n0 100 200 300 400 5000.0 1.0 2.0 3.0 4.0 mMn  ( µB)\n m c m a m total \n110 K 450 K \nacΘD(Mn) = 244 K\nΘD(Bi) = 127 K\nΘD(avg) = 159 K\n Mn\n Bi \nFIG. 3: Results of single-crystal neutron diffraction measu re-\nments on MnBi analyzed using the hexagonal NiAs structure\ntype at all temperatures. (a) Refined magnetic moment on\nMn, including the total moment, as well as the projections of\nthemomentalongthehexagonal a-axisand c-axis. (b)Equiv-\nalent isotropic atomic displacement parameters, Ueq. The ele-\nment specific and average Debye temperatures and the linear\nfits used to determine them are shown on the plot. (c, d)\nAnisotropic atomic displacement parameters for both atoms .\nU11measures displacement in the ab-plane, and U33mea-\nsures displacement along the c-axis. (e, f) The ratio U11/U33,\na measure of the compression of the displacement ellipsoids\nalong the crystallographic c-axis. The inset in (f) compare s\nthe relative sizes and shapes of the Bi ellipsoids at 110 and\n450 K.\nwith the powder diffraction results. Orthorhombic space\ngroupsCmcm,Cmc21, andC2cmwere identified as can-\ndidates based on systematic absences. A good structure\nrefinement was achieved in the Cmcm, the highest sym-\nmetry group of the three candidates and a subgroup of\nthe high temperature space group. The refinement gives\nMn at (0,1\n2, 0) and Bi at (0, 0.16611(6),1\n4).5\nB. Neutron diffraction\nResults of our single crystal neutron diffraction mea-\nsurementsaresummarizedinFig. 3. Previouslyreported\nneutron diffraction results are limited to powder diffrac-\ntion[7,9,24]. Thesinglecrystaldataallowustonotonly\ndetermine with greater reliability the intrinsic nature of\nthe spin reorientation transition, but also to extract in-\nformationaboutthelatticevibrationsthroughanalysisof\nthe temperature dependence of the atomic displacement\nparameters.\nThe neutron diffraction experiment did not have suffi-\ncient resolution to detect the small lattice distortion ob-\nserved by X-ray diffraction (Fig. 2). For this reason, the\nneutron data were analyzed using the hexagonal NiAs-\ntype structure at all temperatures. Thus, for T≤90 K,\nthere is potential for systematic effects on the refinement\nresults. Possible effects on the magnetic moment and\nstructural properties derived from the data are discussed\nbelow.\nThe refinement results for the Mn magnetic moment\nare shown in Fig. 3a. At all temperatures, separate re-\nfinements were performed with the moment along c-axis,\nwith the moment in the ab-plane, and with the moment\nhaving projections along both the c-axis and in the ab-\nplane. As expected, the results show moments along the\nc-axis at higher temperatures ( ≥150 K) and in the ab-\nplane at lower temperatures ( ≤70 K). The model with\npartially rotated moments provided a better fit to the\ndata only for the intermediate temperatures. The in-\nplane component of the moment is along the a-axis.\nThe magnetic structure with the in-plane components\nhas a lower symmetry (orthorhombic) than that used to\ndescribe the nuclear structure (hexagonal). Therefore,\nthree equivalent magnetic domains, generated by rota-\ntions of 120 degrees about the c-axis, are expected in\nthe large single crystal used for this study. These do-\nmains, equally weighted, were used in the refinements of\nthe neutron data. Belowthe structuretransition at 90K,\neach magnetic domain can be uniquely related to an or-\nthorhombic structure domain, and the moment along the\n[100]/[010]/[110] directions in the hexagonal description\nalignsalongthe a-axis ofthe orthorhombiclattice ofeach\nstructural domain (see Fig. 2d). Because of this domain\nformation, we expect little effect on the magnetic struc-\nture refinement to arise from use of hexagonal symmetry\nat low temperature.\nPrevious studies have reached conflicting conclusions\nregarding the completeness of the spin rotation at low\ntemperatures, with some finding partial rotation [7, 9],\nand other full rotation [24, 25]. Our results show that in\nMnBi single crystals, the rotation is complete. The ob-\nservation of only partial rotation in powder samples may\nbe related to the sensitivity of this material to strain. As\nnoted above, the structural phase transition observed at\nTSRis suppressed with only a small amount of mechan-\nical stress (gentle grinding). Thus, a strain effect on the\nspin reorientation itself may be expected.We find an ordered moment on Mn of 3.50(2) µBat\n300 K, and 3.90(2) µBat 5 K. These compare well with\nthe values of 3.6 and 4.1 µBdetermined from our magne-\ntization measurements at 300 and 5 K, respectively (see\nbelow). They are in reasonable agreement with pow-\nder diffraction reports as well [7]. Note that assignment\nof nominal oxidation states would give Mn3+Bi3−, and\ntrivalent Mn (3 d4) would be expected to have a moment\nofgS= 4µB.\nAtomic displacement parameters (ADP) determined\nfrom the neutron diffraction data are also shown in Fig.\n3. Theseareameasureofthemeansquareddisplacement\nof atoms from their ideal crystallographic position, and\ninclude contributions from thermal motion and, when\npresent, static displacements (off-centering). The data\nprovides not only information about the temperature de-\npendence of the ADPs, but also their anisotropy. Fig.\n3b shows the equivalent isotropic displacement parame-\ntersUeq, while Figs. 3c,d show the anisotropic values.\nThe displacement in the ab-plane is measured by U11.\nThe displacement along the c-axis is measured by U33.\nThe ratio U11/U33for each atom is shown in Figs. 3e,f.\nThis ratio characterizes the degree of oblateness of the\nellipsoids (squashing along the c-axis).\nInspection of Ueq(Fig. 3b) shows that both atoms,\ndespite their disparate masses, vibrate similarly in the\ncrystal lattice. This indicates potential wells with simi-\nlar force constants are experienced by both Mn and Bi.\nFrom the temperature dependence, element specific De-\nbye temperatures (Θ D) can be extracted [33]. Above the\nDebye temperature, and in the Debye model, the slope\n(S) ofUeqvsTis given by S= 3h2/(4π2mkBΘD), where\nm is the atoms mass, h is Planck’s constant and kBis\nBoltzmann’s constant. For mass in atomic mass units,\nand the slope measured in ˚A2/K, Θ Din K is given by\nΘD=/radicalBig\n146\nm S. The Debye temperatures determined in\nthis way, and the linear fits used to determine them, are\nshown on Fig. 3b). The value determined using average\nmass and average slope is also shown.\nAmong the results for the anisotropic ADPs, the tem-\nperature dependence ofthe oblateness ofthe Bi ellipsoids\nare particularly noteworthy (Fig. 3d). Above about 100\nK,U11/U33for Bi decreases steadily up to the highest\nmeasurement temperature. That is, the Bi ellipsoid be-\ncomes more and more elongated along the c-direction as\ntemperature increases, while no temperature dependence\nin the shape ofthe Mn ellipsoid canbe resolved(Fig. 3e).\nThe Bi ellipsoids are shown in the inset of Fig. 3f for T\n= 110 K, just above the structural distortion, and 450 K,\nthe highest temperature investigated.\nThe anisotropy of the Bi ADPs reflects the anisotropy\nof the restoring force experienced by Bi when it is moved\naway from its ideal position. This restoring force can\narise from both chemical bonding and magnetoelastic ef-\nfects. While the detailed origin of the ADP anisotropy\nwould require a careful theoretical investigation, it seems\nreasonable to conclude that the increase in the Bi vi-6\nbrations along the c-axis as temperature is raised con-\ntributes to the anisotropic thermal expansion in this\nmaterial (Fig. 2c). This anisotropic thermal expan-\nsion has been linked in a recent theoretical study to\nthe magnetic anisotropy and spin reorientation in MnBi\n[28]. The important role played by the nominally non-\nmagnetic Bi atoms in the magnetism of MnBi also has\nbeen recently identified by first principles calculations\n[29], where the fine details of the Bi-Bi exchange inter-\nactions and spin-orbit coupling on Bi are demonstrated\nto influence strongly the magnetic anisotropy. Thus,\natomic displacement parameters, as especially those of\nBi, may provide a fundamental link between the temper-\nature dependence of structural and magnetic properties,\nand could be a fruitful target for further theoretical in-\nvestigations.\nWhileU33(Fig. 3d) for both Mn and Bi varies\nsmoothly throughout the entire temperature range, a\nclear kink in U11(Fig. 3c) is seen for both atoms be-\ntween 130 and 150 K. This is also reflected in Ueqval-\nues of Fig. 3b. It is expected that the orthorhombic\ndistortion would manifest itself here as an inflation of\nthe in-plane displacements determined using the hexag-\nonal structure. However, this should appear for T≤90\nK. The kink in U11near 140 K occurs in the hexagonal\nstructure, and is robust against changes in the magnetic\nstructure model. This is the temperature at which the\nhexagonal lattice constants experience a deviation from\ntheir high temperature behavior (Fig. 2c), and the mo-\nments begin to rotate away from the c-axis (Fig. 3a).\nThis shows that not only the crystal structure and sym-\nmetry, but also the atomic displacements in MnBi are\nclosely coupled to the magnetism.\nC. Physical properties\nEffects of the crystallographic and magnetic phase\ntransitions occurringat TSRon the physical propertiesof\nthe MnBi single crystals were examined. The magnetic\nbehavior, shown in Fig. 4, is consistent with previous\nliterature reports on polycrystalline materials. Detailed\nreports of the heat capacity and anisotropic electrical re-\nsistivity are not available for comparison. Measurements\non the single crystalsreveal sharp anomalies in both heat\ncapacity (Fig. 5) and resistivity (Fig. 6). The following\nwill discuss the individual properties in some detail.\nResults of magnetization measurements on the crys-\ntals are summarized in Fig. 4. The results are in general\nagreement with the previously reported behaviors noted\nin the Introduction. Upon heating above room temper-\nature (Fig. 4a), the magnetization decreases and then\nabruptly drops by orders of magnitude near Tdec= 630\nK, signaling the decomposition of MnBi. A simple model\n[34] is used to fit the temperature dependence up to Tdec,\ngiving an estimate of the hypothetical Curie temperature\nof 680 K.\nIsothermal magnetization curves measured with thefield parallelandperpendicularto the c-axisareshownin\nFig. 4b. This data demonstrates the change in the easy\naxis of magnetization from along cnear room tempera-\nture to perpendicular to cat 5 K. The spin reorientation\nis also clearly seen in the temperature dependent mag-\nnetization results shown in Fig. 4c,d. Upon cooling in\nthe lowest applied fields, the magnetization for H⊥cin-\ncreases below about 150 K, and the magnetization for\nH/bardblcdecreases abruptly below 90 K. This is consistent\nwith previous reports from textured polycrystalline ma-\nterial[17]andwith ourneutrondiffractiondatadiscussed\nabove, which find the moments to rotate awayfrom cbe-\nlow about 140 K and lie fully in the ab-plane below 90\nK. The dc magnetic susceptibility ( M/Hat low field) is\nlow when measured perpendicular to the moment direc-\ntion when the moments are locked into the cdirection\nor theab-plane. At intermediate temperatures, when\nthe moment has components both parallel and perpen-\ndicular to c, the magnetic susceptibility is large in both\ndirections. This is clearly illustrated in in Fig. 4f, which\ncompares ac magnetic susceptibility measured at Hdc=\n0 with the magnetic moment components determined by\nneutron diffraction.\nForH⊥c, a small but sharp decrease in mis observed\nat 90 K for H = 0.1 kOe (also seen in the zero field ac\nmeasurement in Fig. 4f). Such a feature could occur\nfrom misalignment of the crystal during the measure-\nment; however, its absence in the 1 kOe data suggests\nit is an intrinsic and field dependent feature. At H= 1\nkOe in this direction, a sharp increase in mis observed\nnear 100 K. Signatures of the spin reorientation transi-\ntion in the magnetization data are generally broadened\nand moved to higher temperatures as the applied mag-\nnetic field is increased. Taking the point of minimum\nslope in the m(T) data for H⊥cand 1 kOe ≤H≤2\nkOe asTSR, the temperature at which the spin reorien-\ntation is complete, a plot of TSR(H) is shown in Fig. 4e.\nInterpolating this back to H= 0 using a second order\npolynomial gives TSR(0) = 90 K, coincident with sharp\ndrops in χac(Fig. 4f) and mmeasured at H= 0.1 kOe\n(Fig. 4c). A simple linear fit to the data in Fig. 4e gives\na similar value of TSR(0) = 95 K.\nThe measured heat capacity cPof MnBi single crys-\ntals is shown in Fig. 5. Measurements are shown for two\ndifferent crystals. A sharp thermal anomaly is observed\natTSR= 90 K. A broad feature has been reported in the\nheat capacity of polycrystalline MnBi at a significantly\nhigher temperature of 118 K [35]. The integrated en-\ntropy associated with the peak for H= 0 is 0.01 R (mol\nMn)−1. Similar anomalies are seen in the heat capacity\nof Nd2Fe14B and related compounds have been reported,\nand used as a precise way to determine spin reorienta-\ntion temperatures in these materials [36, 37]. The heat\ncapacity anomaly at the spin reorientation in MnBi is\nsuppressed by applied magnetic fields, as shown in the\nupper inset of Fig. 5a, decreasing in magnitude with\nincreasing field while remaining at 90 K.\nThere is a low temperature upturn in cPupon cooling7\n/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48/s48/s49/s50/s51/s52\n/s51/s48/s48 /s52/s48/s48 /s53/s48/s48 /s54/s48/s48 /s55/s48/s48/s48/s50/s48/s52/s48/s54/s48/s56/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48/s49/s54/s48/s49/s56/s48\n/s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s53/s49/s48/s49/s53/s50/s48/s32/s51/s48/s48/s32/s75 /s44/s32/s72 /s99\n/s32/s51/s48/s48/s32/s75 /s44/s32/s72/s32/s124/s124/s32/s99\n/s32/s53/s32/s75 /s44/s32/s72/s32 /s99\n/s32/s53/s32/s75 /s44/s32/s72/s32/s124/s124/s32/s99/s109 /s32/s40\n/s66 /s32/s47/s32/s77/s110/s41\n/s72 /s32/s40/s107/s79/s101 /s41/s32/s109 /s32 /s32/s40 /s84\n/s67 /s45/s84 /s41/s110\n/s84\n/s67 /s32/s61/s32/s54/s56/s48/s32/s75 \n/s110 /s32/s61/s32/s48/s46/s50/s56\n/s32/s77 /s32/s40/s101/s109/s117/s32/s47/s32/s103/s41\n/s84 /s32/s40/s75/s41/s84\n/s100 /s101 /s99/s61/s32/s54/s51/s48/s32/s75 /s72 /s32/s61/s32/s49/s48/s32/s107/s79/s101 \n/s32/s32\n/s32/s53/s48/s32/s107 /s79/s101\n/s32/s50/s48/s32/s107 /s79/s101\n/s32/s49/s48/s32/s107 /s79/s101\n/s32/s53/s32/s107 /s79/s101\n/s32/s49/s32/s107 /s79/s101\n/s32/s48/s46/s49/s32/s107 /s79/s101/s77 /s32/s40/s101/s109/s117/s32/s47/s32/s103/s41\n/s84 /s32/s40/s75/s41/s72 /s32 /s32/s99/s40/s100/s41/s40/s99/s41/s40/s98/s41\n/s72 /s32/s124/s124/s32 /s99/s32\n/s32/s50/s48/s32/s107 /s79/s101\n/s32/s49/s48/s32/s107 /s79/s101\n/s32/s53/s32/s107 /s79/s101\n/s32/s49/s32/s107 /s79/s101\n/s32/s48/s46/s49/s32/s107 /s79/s101/s32/s40/s120 /s53/s41/s77 /s32/s40/s101/s109/s117/s32/s47/s32/s103/s41\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s32/s32/s84 \n/s83 /s82 /s32/s40/s75/s41\n/s72/s32/s40/s107/s79/s101 /s41/s40/s101/s41\n/s40/s102/s41\n/s32/s32/s65 /s67/s39/s32/s40/s99/s109/s51\n/s32/s47/s32/s109/s111/s108/s32/s70/s46/s85/s46/s41\n/s84 /s32/s40/s75/s41/s40/s72 /s32 /s32/s99/s41\n/s40/s72 /s32 /s32/s99 /s41/s32\n/s120 /s51\n/s48/s49/s50/s51/s52\n/s32/s109 \n/s99\n/s32/s109 \n/s97/s98\n/s109\n/s77/s110/s32/s40\n/s66 /s41\nFIG. 4: Magnetic properties of MnBi single crystals. (a) Mag -\nnetization ( M) above room temperature. A simple model,\nshown on the plot, is used to estimate the hypothetical Curie\ntemperature. (b) Magnetic moment ( m) vs applied field at\n5 and 300 K. (c and d) Magnetization vs temperature mea-\nsured at the indicated magnetic fields for Hperpendicular an d\nparallel to the c-axis, respectively. (e) Field dependence of\nthe spin reorientation temperature defined by the minimum\nof dM/dTforH⊥c. The line is a fit using a second order\npolynomial. (f) ac magnetic susceptibility measured at 20 H z\nin zero applied dc field, with magnetic moment components\ndetermined by neutron diffraction shown for comparison.\nbelow about 2.5 K, shown in the lower inset of Fig. 5(a).\nThe origin of this behavior is unclear at this time. No\nprevious reports of the low temperature heat capacity for\nMnBi were found in the literature for comparison. To ex-\ntract a Debye temperature for comparison with those de-\ntermined from the temperature dependence of the ADP\nvalues discussed above (Fig. 3e), linear fits to the low-\ntemperature cP/TvsT2data were performed, excludingthe upturn below 2.5 K. The values of the Debye temper-\nature and Sommerfeld coefficients determined from them\nare listed in the Figure. The Debye temperature deter-\nmined in this way is in very good agreement with the\naverage Debye temperature of 159 K determined from\nthe neutron diffraction data.\nTo examine the behavior ofthe heat capacity near TSR\nin detail, measurement were performed in which a large\nheat pulse is applied to the sample and then removed,\nso that the sample passes through the spin reorientation\nupon heating and then again upon cooling. Analysis of\nthe time dependence of the sample temperature [38] is\nused to derive heat capacity values, using software pro-\nvided by Quantum Design. Fig. 5b shows the heat ca-\npacity determined separately from the heating and cool-\ning curves. A thermal hysteresis in the peak position of\nabout 0.6 K is observed. Since there is expected to be\nsome thermal lag between the sample and the thermome-\nter in this measurement, this can be considered an upper\nbound any the intrinsic thermal hysteresis. Inspection of\nthe temperature vs. time curves (Fig. 5c) used to de-\ntermine the heat capacity reveals no detectable thermal\narrest near TSR, indicating little latent heat is associated\nwith the transition.\nFig. 6 shows results of electrical resistivity measure-\nments on MnBi single crystals. For the current parallel\nand perpendicular to the c-axis, the residual resistivity\nratios [ρ(300 K) / ρ(5 K)] are 88 and 98, respectively. A\ndecrease in resistivity is observed upon cooling through\nTSR, indicated by the arrow on the figure. This is shown\nclearly by the temperature derivative shown in the upper\ninset of Fig. 6. Hihara and Koi observed a small resis-\ntivity anomaly in polycrystalline MnBi occurring near\n115 K [25]. Resistivity measurements on thin films have\nshown anomalies near 50 K [39]. The anomaly is most\nclearly seen in d ρ/dTshown in the inset of Fig. 6. No\nthermal hysteresis is observed within the 1 K resolution\nof the resistivity data. Due to the irregular shape of the\ncrystals, there is some uncertainty associated with the\nabsolute value of the resistivity reported here. However,\nthe data does suggest that there is not a large amount\nof anisotropy in the electrical transport. The material is\na good metal, reaching about 1 µΩcmat 2 K. Measure-\nments in the ab-plane performed on films have shown\nsimilar resistivity values at near room temperature, but\nlarger low temperature resistance than we find in our\ncrystals [39].\nIV. SUMMARY AND CONCLUSIONS\nIn summary, we have identified a symmetry-lowering\nstructural phase transition which occurs at the spin-\nreorientation temperature in MnBi, driven by magne-\ntostriction. Below TSR= 90 K, when the magnetic\nmoment lies in the ab-plane the structure is orthorhom-\nbic. Diffraction data indicates that the low temperature\nstructure can be described in space group Cmcm, a8\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53/s48/s49/s50/s51\n/s48 /s50/s53 /s53/s48 /s55/s53 /s49/s48/s48 /s49/s50/s53/s56/s48/s57/s48/s49/s48/s48/s48 /s50/s48 /s52/s48 /s54/s48/s48/s50/s52/s54\n/s56/s53 /s57/s48 /s57/s53/s50/s46/s50/s50/s46/s51/s50/s46/s52/s56/s54 /s56/s56 /s57/s48 /s57/s50 /s57/s52/s50/s46/s50/s50/s46/s51/s50/s46/s52 /s32/s99/s114/s121 /s115/s116/s97/s108 /s32/s35/s49/s99\n/s80 /s32/s40/s82/s32/s47/s32/s109/s111/s108/s32/s97/s116/s46/s41\n/s84 /s32/s40/s75/s41/s40/s97/s41\n/s84 /s32/s40/s75/s41/s40/s99/s41\n/s99/s114/s121 /s115/s116/s97/s108/s32/s35/s50\n/s116/s105/s109 /s101 /s32/s40/s115/s41/s99/s114/s121 /s115/s116/s97/s108/s32/s35/s49\n/s99/s114/s121 /s115/s116/s97/s108/s32/s35/s50\n/s32/s32/s99\n/s80 /s32/s47/s32/s84 /s32/s40/s49/s48/s45/s51\n/s32/s82/s32/s47/s32/s109/s111/s108/s32/s97/s116/s46/s32/s75/s41\n/s84/s32/s50 \n/s32/s40/s75 /s50 \n/s41/s68 /s32 /s32/s49/s54/s48/s32/s75 \n/s32 /s32/s48/s46/s52/s32/s109/s74/s32/s47/s32/s109/s111/s108/s32/s97/s116/s46/s32/s47/s32/s75 /s50 /s99\n/s80 /s32/s40/s82/s32/s47/s32/s109/s111/s108/s32/s97/s116/s46/s41\n/s32/s32/s32/s99/s114/s121 /s115/s116/s97/s108/s32/s35/s50\n/s32/s32/s32/s72 /s32/s61/s32/s48\n/s32/s104/s101/s97/s116/s105/s110/s103\n/s32/s99/s111/s111/s108/s105/s110/s103/s40/s98/s41/s32\n/s84 /s32/s40/s75/s41/s32/s32/s32/s99/s114/s121 /s115/s116/s97/s108/s32/s35/s50\n/s32/s72 /s32/s61/s32/s48\n/s32/s49/s32/s107 /s79/s101\n/s32/s50/s32/s107 /s79/s101\n/s84 /s32/s40/s75/s41\nFIG. 5: Heat capacity of MnBi single crystals showing an\nanomaly at the spin-reorientation and crystallographic ph ase\ntransition at 90 K. The upper inset shows the effects of a\nmagnetic field near TSRand the lower inset shows the low\ntemperature behavior of cP/TvsT2. Linear fits to the data\njust above the low-temperature upturn are used to estimate\nthe Debye temperature Θ Dand the Sommerfeld coefficient γ.\n(b) Heat capacity derived from temperature vs. time data\ncollected during a large heat pulse which moves the sample\nthrough thephase transition temperature on heatingand sub -\nsequent cooling, showing the intrinsic thermal hysteresis as-\nsociated with the transition is <0.6 K. (c) Temperature vs\ntime data used to determine the cPdata shown in (b).\nsubgroup of P63/mmc, the space group of the hexagonal\nNiAs-type structure adopted at higher temperature\nwhen the moment in aligned along the c-axis. Such\na distortion is allowed to occur via a second order\ntransition. Heat capacity data place an upper bound of\n0.6 K on the thermal hysteresis of the spin reorientation.\nMagnetoelastic coupling results in a shortening of the\naxis along which the Mn moments are directed in\nboth phases. Single crystal neutron diffraction analysis\nreveals an increasing elongation of the Bi displacement\nellipsoid along the c-axis as temperature is increased in\nthe high temperature state. This is likely related to the\nobserved anisotropic thermal expansion, and may playan important role in the magnetocrystalline anisotropy,\nwhich has been attributed to Bi −Bi exchange interac-\ntions [29]. The relationship between atomic vibrations\n/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48 /s50/s53/s48 /s51/s48/s48/s48/s50/s53/s53/s48/s55/s53/s49/s48/s48\n/s54/s48 /s56/s48 /s49/s48/s48 /s49/s50/s48\n/s84\n/s83/s82/s32/s61/s32/s57/s48/s32/s75\n/s32/s73/s32/s124/s124/s32 /s99\n/s32/s73/s32 /s99/s40 /s32/s99/s109/s41\n/s84 /s32/s40/s75/s41/s99/s111/s111/s108/s105/s110/s103/s100 /s32/s47/s32/s100 /s84 /s32/s40/s97/s114/s98/s46/s32/s117/s110/s105/s116/s115/s41\n/s84 /s32/s40/s75/s41/s119/s97/s114/s109/s105/s110/s103\nFIG. 6: Resistivity of MnBi single crystals. Data are shown\nwith current ( I) flowing both parallel and perpendicular to\nthe c-axis. The upper inset shows the temperature derivativ e\nof the resistivity near the TSRmeasured both upon warming\nand cooling. The derivative curves are offset vertically for\nclarity.\nand magnetic anisotropy is also supported by the kink\nobserved in the atomic displacement parameters at\n140 K, a temperature well above the orthorhombic\ndistortion, and at which the moment first begins to cant\naway from the c-axis upon cooling. The crystallographic\nphase transition reported here may prove to be key in\ninterpreting future experimental and theoretical studies\nof the intrinsic properties of this unusual permanent\nmagnet material.\nResearch sponsored by the U. S. Department of En-\nergy, Office of Energy Efficiency and Renewable En-\nergy, Vehicle Technologies Office, Propulsion Materials\nProgram (M.A.M.) and the Critical Materials Institute,\nan Energy Innovation Hub funded by the U.S. Depart-\nment of Energy, Office of Energy Efficiency and Renew-\nable Energy, Advanced Manufacturing Office (B.C.S.).\nNeutron diffraction measurements conducted at ORNL’s\nHigh Flux Isotope Reactor (H.C. and B.C.C.) were spon-\nsored by the Scientific User Facilities Division, Office of\nBasic Energy Sciences, US Department of Energy. The\nauthors thank David S. Parker and David J. Singh for\nhelpful discussions and insight, and for suggesting that\na structural distortion should occur in this material, and\nBayrammurad Saparov and Radu Custelcean for assis-\ntance with transport measurements and single crystal x-\nray diffraction, respectively.9\n[1] N. Poudyal and J. P. Liu, J. Phys. D: Appl. Phys. 46,\n043001 (2013).\n[2] F. Heusler, Z. Angew. 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Winkler1,2\n1Centro Atómico Bariloche , CNEA, 8400 S.C. de Bariloche, Río Negro, Argentina\n2Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina\nAbstract\nIn this work we present a study of the low temperature magnetic phases ofpolycrystalline\nMnCr2O4spinel through dc magnetization and ferromagnetic resonance spectroscopy\n(FMR). Through these experiments we determined the main characteristic temperatures: T C\n~ 41 K and T H~ 18 K corresponding, respectively, to the ferrimagnetic order and to the low\ntemperature helicoidal transitions.  The temperature evolution of the system is describe dby\na phenomenological approach that considers the different terms that contribute to the free\nenergy density. Below the Curie temperature the FMRspectra were modeled by a cubic\nmagnetocrystalline anisotropy to the second order, with K1andK2anisotropy constants that\ndefine the easy magnetization axis along the <110> direction . At lower temperatures, the\nformation of ahelicoidal phase was considered by including uniaxial anisotropy axis along\nthe]011[\npropagation direction o f the spiral arrange, with a Kuanisotropy constant . The\nvalues obtained from the fittings at 5 K are K1=-2.3x104erg/cm3,K2= 6.4x104erg/cm3and\nKu= 7.5x104erg/cm3.\n1. IntroductionThe cubic spinels AB 2O4, where the tetrahedral A -sites are occupied by non -magnetic ions\nand the octahedral B -sites are occupied by Cr ions, are model systems to study magnetic\nfrustration [1, 2, 3]. In these compounds the main magnetic interaction isthestrong J CrCr\nantiferromagnetic direct exchange between the nearest neighbors ions[4, 5]. However, the\ngeometrical arrangement of thesemagnetic ions in a pyrochlore -like array prevents the\nmagnetic order till very low temperature, ascompared to the Curie temperature ,CW[2, 6,\n7]. Several authors have proposed that trough the magnetoelastic coupling the strong\nmagnetic frustration could be released and the system could develop a magnetic transition\n[8,9]; in fact the low temperature magnetic ordered state is usually accompanied by\nstructural distortions. [ 10, 11] Instead, when the tetrahedral A -site is occupied by a\nmagnetic ion, the magnetic frustration is partially relieved by the J ACrsuperexchange\ninteraction. [1 2] In this case the system presents nearly degenera ted ground states and it\ndevelops complex low temperature magnetic order.\nIn particular in the MnCr 2O4the competing Cr -Cr, Cr-Mn and Mn -Mn exchange\ninteractions prevent the development of ferrimagnetic order till to TC~41 K,even\nconsidering the importan t exchange energies observed ( CW/TC>10)[4].Neutron\ndiffraction studies reported that bel ow TCthe system presents long -rangeferrimagnetic\norder with an easy axis parallel to the <1 10> direction [13 -15], when the temperature\ndecreases belowTH~18 K,this magnetic phase coexists with short -rangespiral order. In the\nspiral arrange two positions can be distinguished for the Cr, and the magnetic moment s\ndescribea cone on each sublattice, with helicoidal propagation vector in the ]011[\ndirection.\nThe complex low temperature order, where the spin rotation axis does not coincide with the\nhelicoidal propagation vector, positioned this material as a good candidate to presentmagnetodiel ectric coupling [16 -18]. Recently, Mufti and collaborat ors [19,20] have\nreported that the dielectric and magnetic properties are coupled below THin powder\nMnCr2O4oxide.In addition ,recent FMR results on frustrated spinels [ 21] have related the\nunusual FMR temperature dependence to phase separation. In thiscomplex scenario the\nferromagnetic resonance (FMR) spectroscopy emerges as a suitable technique because it\nprovides microscopic information related to the exchange and magnetic anisotropy and\nallows extending the knowledge of the nature of the long-rangeferrimagnetic order and the\nspiralshort-rangestate. In this context we present a study of the low temperature magnetic\nphasesin a cubic chromium spinel with A=Mn by magnetic and FMR measurements. We\nfollow the temperature evolution of the parameters tha t characterize the FMR spectra in a\npolycrystalline sample. We describe the evolution of the FMR spectra by a\nphenomenological model that takes into account the different terms that contribute to the\nmagnetic anisotropy of the system.\n2. Experimental\nSingle phase polycrystalline samples of MnCr 2O4were fabricated by solid state reaction of\nMnO and Cr 2O3powders, as described elsewhere [4].This system has a normal cubic spinel\nstructure, belonging to the Fd -3mspace group. The magnetic properties were in vestigated\non loosely packed powdered samples in the 5 –90 K temperature range, with applied fields\nup to 5 T, using a commercial superconducting quantum interference device (SQUID,\nQuantum Design MPMS-5S) magnetometer. The temperature dependence of the\nferromagnetic resonance ( FMR) spectra was recorded by a Bruker ESP300 spectrometer\noperating in the conventional absorption mode at 2~24 GHz (K -band), for temperaturesranging from 4 K to 300 K. Magnetic -field scans were performed in the range 0 –15000\nOe.Care was taken in order to avoid cavity detuning effects, as are usually present in\nspectra of strongly magnetic compounds. For that purpose, the MnCr 2O4powder was\nthoroughly milled and mixed with a non -absorbing KCl salt. No noticeable changes in the\nquality factor (Q) of the cavity w ereregistered in the whole set of experiments.\n3. Results and discussion\n3.1 Magnetic properties\nFigure 1 presents the magnetization vs. temperature measurements, M(T), under zero -field-\ncooling (ZFC) and field -cooling(FC) conditions, with an applied field of 50 Oe. Near 41\nK, a sudden jump is observed, consistent with the ferrimagnetic transition ( TC). As the\ntemperature is further lowered, other anomalies are manifested at TH~18 K and Tf~14 K,\ncorresponding, respectively, to the helicoidal order temperature and to the “lock -in”\ntransition at which the spiral becomes fully developed, as it was determined from neutron\ndiffraction experiments [ 13-15]. The inset in figure 1 exhibits the M(T) ZFC -FC curves\nmeasured with an applied field of 8 kOe, where it can be observed that the TCvalue\nincreases and the transition becomes broader. Also, when the applied magnetic field is\nenhanced, both low -temperature anomalies become less defined, as it was previously\nreported b y Mufti et al. [ 19,20].01020304050607080900246810Tf\nTHM (emu/g)\nT (K)TC0102030405060708090051015202530M (emu/g)\nT (K)\nFigure 1. Temperature dependence of the ZFC (solid symbols) and FC (open symbols)\nmagnetization measured in a field of 50 Oe. The arrows signal the ferrimagnetic transition\n(TC),the helicoidal order temperature (TH)and the “lock -in” transition where the spiral\ncomponent is fully developed (Tf).  The inset shows the M(T) ZFC -FC curves measured\nwith an applied field of 8 kOe.\nFigure 2 shows the magnetization as a function of the applied magnetic field acquired at\ndifferent temperatures. As the temperature descends below ~45 K the magnetization\npresents a n important increase that starts near 2.5 kOe . The spontaneous magnetization of\nMnCr2O4at 5 K was estimated to be ~1.1 Bper unit formula in agreement with the value\npreviously reported [19,20,22 ].Noticeable, a linear increase of the high field magnetization\nis clearly observed for temperatures below 30 K .This lineal contribution signals a non -\ncollinear spins arrangement of the MnCr 2O4ferrimagnet .As is stated in references [23,24]in non-collinear configuration the applied magnetic field exerts a torque that could change\nthe angles between the canted magnetic moments; as a result the magnetization increases\nlinearly with the magnetic field. By neu tron diffraction studies non -collinear order was\nfound below T~18 K where short -range spiral arrangement is developed [13,14].In order\nto shed light onto this complex behavior we have performed ferromagnetic resonance\nmeasurements.\n0 10 20 30 40 50051015202530\n30K\n35K\n40K\n45K\n50KM (emu/g)\nH (kOe) 5K\n15K\n20K\n25K\nFigure 2. Magnetizati on versus applied field at different temperatures near and below TC.\n3.2Ferromagnetic resonance\nTheFMRspectroscopy is a very sensitive technique to detect magnetic transitions as well\nas changes in the magnetic ani sotropy of local -moment systems [25, 26],which are usually\ndifficult to measure by other techniques, particularly in polycrystalline samples. Figure 3\n(a) and (b) exhibit representative FMRspectra measured at different temperatures in theT<THandTH<T≤TCranges, respectively. For polycrystalline samples the resonance\nspectrum includes the contribution of the absorption lines of the crystallites oriented in all\nthe possible space directions relative to the magnetic field. The main features observed in\nthetemperature evolution of the spectra can be summarized as follows:\ni) For temperatures above TC(i.e. in the paramagnetic phase) only one symmetric\nabsorption line is observed, centered at an approximate constant value of Hr= 8619 Oe,\ncorresponding to a sp ectroscopic splitting g-factorg=(ħ)/(BHr)=1.991 (6), where ħis the\nPlanck'sconstantdivided by 2 andBis the Bohr 'smagneton.\nii) As the temperature decreases below TCthe absorption line grows up, becomes\nasymmetric and the peak to peak linewidth, H,enhances. Furthermore, Hrshifts to lower\nmagnetic fields.\niii) Below T ~18 K more significant changes are detected: a secondary peak emerge sand\nshiftsto lower field swhen the temperature diminishes.\nThese features could be explained by the presence of internal fields when the system goes\nthrough the magnetic transitions. In order to account forthe temperature evolution of the\nspectrum, we introduce in the next sectio n a phenomenological model that takes into\naccount different terms that contribute to the free energy.68101245K38K34K\n14K11K9K7K5KAbsorption derivative (arb. units)\nH (kOe)3K 20K\n24K\n29K\n78910\nH (kOe)\nFigure 3. FMRabsorption derivative spectra of the MnCr 2O4powder sample ( open circles )\nmeasured at different temperatures: (a) T<THand (b) TH<T≤TC. Thestraightlines\ncorrespond to the fittings with the phenomenological model.\n3.3 Evolution of the effective magnetic anisotropy: phenomenological model\nThe ferromagnetic resonance condition is obtained from the magnetic free energy of the\nsystem fol lowing the Smit and Beljers formalism [2 7,28]. Equation(1)describes the\ndifferent terms that contribute to the magnetic free energy Eof the MnCr 2O4system:\nKu KcubZ EEEE  (1)\nThe first term of Eq. (1) corresponds to the Zeeman energy, described in Eq. (2), where\n H H H H HHH      cos,sinsin,sincos0is the applied magnetic field vector, and      cos,sinsin,sincos0MMis the magnetization vector in the laboratory coordinate\nsystem presented in Figure 4. The second term in E q. (1) accounts forthe second order\ncubic magnetocrystalline anisotropy, Eq. (3), characterized by the K1andK2anisotropy\nconstants, where 1=sincos,2= sin sinand3= cos . Finally, in order to consider the\nformation of the helicoidal phase, we haveincludeda third term accounting for auniaxial\nanisotropy characterized by the Kuparameter, described by Eq. (4). This last term\ndeterminesan easy axis in the ]011[\ndirection, which is the propagation direction re ported\nfor thehelicoidal order [14].\n H H H Z HMHME      sinsin) cos(coscos .00  (2)\n \n             \n2 4 2 2\n24 2 2 2 2\n12\n32\n22\n122\n12\n32\n32\n22\n22\n11\ncossinsincos sinsincossincos K KK KEKcub\n    (3)\n)2sin(1sin22  u\nKuKE (4)\nq\nfqH\nfH[001]\n[100][010]MHFigure 4. Schematic representation of the magnetization ( M)andmagnetic field ( H)\nvectors and the angles involved in the description of the free energy.\nNotice that the formalism applied to calculate the resonance mode of the ferrimagnetic\nMnCr2O4spinel is essentially the same as the ferromagnetic resonance, in the sense that it\nis considered the precession of the spontaneous magnetization as a whole around their\nequilibrium orientation. Additional resonance modes, which depend explicitly on the\nmagnetic sublattice structure, are not considered because they are located at frequencies\nmuch higher than the microwave.  These modes involve the exchange interaction between\nthe different magnetic sublattices which are usually above the infrared part of t he spectrum.\nIn fact,from the Mn -Mn, Cr-Mn and Cr -Cr exchange constants reported for the MnCr2O4\nspinel the exchange resonance modes are above ~ 5 x1011s-1, which is far fromthe\nmicrowave frequency range, and as a consequence these exchange mode scouldnot be\nexcited [23, 29]. Therefore, as it is usually implemented , we only include in the magnetic\nfree energy the Zeeman interaction and the magnetic anisotropy terms. In the case of the\ncubic magnetocrystalline anisotropy, the easy direction of the magneti zation depends on the\nsigns and relative magnitude of K1andK2. In the present case, we set K1<0 and\n9|K1|/4<K2<9|K1|[30]. With this choice it results that the easy,medium andhard\nmagnetization axis are parallel to the <110>, <111> and <100> directions of the crystal,\nrespectively , for all temperatures .It is noteworthy that ,if another relation between K1and\nK2is chosen (resulting in different medium and hard magnetization directions), thisleads to\nqualitatively different spectra features, where secondary absorption peaks are localized in\nthe g<2 higher field region. We also want to remark that the aforementioned choice ofparameters is consistent with the magnetization easy axis direction r eported from neutron\ndiffraction and magnetization studiesperformed on single crystal samples[13,14,31]and\ndiffersfromthe results reported by [32] where different orientation of the easy\nmagnetization direction was found.\nRegarding Ku, this parameter takes into account the propagation dir ection of the helicoidal\norder[14,15], that breaks the cubic symmetry imposed by the crystalline structure. This\nkind of magnetic ordering is observed when several comparable exchange interactions are\npresent and the description in terms of sublattices is interdicted. This is the case, for\nexample, when the step of the spiral is not commensurate with the lattice parameter [33 ].\nFrom the magnetic free energy, equations (1) to (4), the angular derivative s\n) and , (2 2 22 2    E E E , evaluated at the equilibrium angles for the\nmagnetization for each orientation of the magnetic field, were calculated. The FMR\nresonance condition was obtained evaluating the Smit -Beljers equation [ 27,28]:\n.\nsin122\n22\n22\n22\n02\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n     EEE\nM(5)\nHereis the angular frequency  GHz,is the gyromagnetic ratio and M0is the\nsaturation magnetization value measured at kOe (Figure 2). As we measured a\npolycrystalline sample we assume that the absorption line corresponds to thesum of\nLorentzian lineshape resonances with a homogeneous angular distribution of the\nanisotropies ax es related to the magnetic field. For simplicity no angular variation of the\nresonance linewidth was considered. Furthermore ,for temperatures near and be lowTCthe\nlines present an additional asymmetry that could be attribut ed to a dispersive component[34,35]as we are going to discuss later . Consequently, in this range we have also included\nin the simulat ed spectra a dispersive term , determining a linesh ape of the form: (1 -)\nAbsorption + Dispersion, where 0< <1[35-36].We solved the Smit -Beljers equation\n(Eq. 5) in a self -consistent way, with g K1,K2andKuas adjusted parameters, and we have\nobtained a numerical simulation for the FMR resonance absorption at each temperature.\nThe gyromagnetic factor obtained from the fittings in all the T ≤TCrange isg~2.05(2). The\ncalculated spectraarepresented in straightlines in figure 3 , wheregood agreement between\nthe spectral lines and the model is observed in all the studiedtemperature range. Notice that\nthe calculated spectra reproduce well the general features of the lineshape, as the resonant\nfield, the field positions of the satellite peaks and the linewidth, even for T<THwhere the\nexperimental peaks are broader than the fitting. The difference b etween the experimental\nand the calculated spectra could be attributed to the simplifications of our model, as we\nconsidered no angular variation of H on the resonance lines that form the powder\nspectrum and also no distribution of anisotropy values were considered that could account\nfor some degree of crystalline disorder. Nevertheless, this simple model allow s us to extract\nquantitative information of the evolution of the system with temperature.0 10 20 30 40 50-202468TCK1, K2, Ku (104 erg/cm3)\nT (K)K1\nK2\nKuTH\nFigure5.Temperature evolution of the anisotropy constants calculated from the\nphenomenological model. The lines are guides for the eye and the vertical dotted lines\nsignal the ferrimagnetic order temperature TCand the helicoidal temperature TH.\n0 10 20 30 40 500.150.180.210.240.270.300.33dispersive component parameter \nT (K)Figure 6.Temperature evolution of the parameter that corresponds to the proportion of\ndispersive component included in the spectra simulations .\nIn figure 5wehavepresentedthe temperature evolution of the magnetic anisotropy\nconstants obtained from the simulation of FMR spectra. Below TC, themagnitude of the\ncubic magnetocrystalline anisotropy constants, K1andK2, startsto increase smoothly and\nshowsa more important enhancement below TH. Thesecubic anisotropy values are in the\nsame order of magnitude thanthe values reported for similar oxide and chalcogenide spinel\nsystems as MnFe 2O4(K1~-3.3x104erg/cm3[38,39]) and MnCr 2S4(K1~ 4.2x104erg/cm3\nandK2~ 1x105erg/cm3[40]). On the other hand, Kuremains equal to zero down to ~30K,\nwhere it starts to enhance smoothly. Finally, a jump in the Kuvalue is observed below TH,\nfollowed by a lineal increase up to the lowest measured temperature. The value of\nKu~8x104erg/cm3obtained at 4.2 K agreeswith that reported in Ref. [41] of ~2x104\nerg/cm3.The abrupt jump of Kuis reflected in the low field satellite peakthat clearly\nappearsin theFMRspectra below TH[see figure 3(a) ] which is related to the formation of\ntheshort-range helicoidal order. However, Kuis non-zero above TH,till T~30 K. This result\ncould indicate that the short -range helicoidal order still coexists with the ferrimagnetic\norder above the helical transition temperature . Although no evidence of this fact was\ndetected within statistical uncertainties from neutron diffraction experiments for this\nsystem, Tomiyasu and co -workers [1 4] observed that for the CoCr 2O4spinel, the spiral\ncomponent retains the correlation well above TH.\nFurthermore, this result is consistent with the magnetization measurements where the high\nfield lineal contribution is observed from low temperature up to T ~ 30 K. This complexstageis a consequence of exchange and superexchange competing interactions in this\ngeometrically frustrated magnetic material. Although the relevant interaction is the direct\nCr-Cr exchange interaction, in the MnCr 2O4the superexchange interactions between Cr -Mn\nand Mn-Mn present comparable magnitude [4] . Therefore thelong-rangehelicoidal order\ncannot be stabilized, as is calculated for AB2O4spinelwhen the A -A interaction is neglected\n[42,43].These results suggest that, at low temperature, the long -range ferrimagnetism\ncoexists with the spiral order where the transverse compon ent of the Mn and Cr conical\narrange is ordered. However, when the temperature increases the conical arrange preserves\nthe longitudinal order up to T C, while the transverse component is largely disordered above\n~18 K.\nFigure 6 exhibits the parameter that determines the proportion of dispersive component\nincluded in the fittings, as a function of temperature. Notice that this component is larger in\ntheTH<T<TCtemperature range where the magnetization increase s. It is well known that in\na medium which has conductivity ( ) the wave is attenuated as i tprogressesthrough the\nsample [35-37]. Therefore, when the penetration depth of the microwave ( ) is less than the\nthickness of the sample (d) the medium will not be homogeneous and the resonance line\nwill beasymmetric. This effect is more important for materials with high magnetic\npermeability ( ). For conductive materials this resonance correspond to a Dysonian line,\nwhich in the limit 0 ≤ d/≤ 2 results a linear combination of absorption and dispersion\nLorentzian lines [ 36]. In the case of insulat ormaterials, as the MnCr 2O4[5, 15], the\nmicrowave penetration depth result s:\n2,wherecorrespond sto the dielectric\nconstant [ 35]. Therefore, the microwave penetration depth decreases with the magneticpermeability, as a consequence the dispersive component in the magnetic resonance\nincreases (seefigure1). This result is in agreement with the measured temperature\ndependence of the magnetization which is larger in the TH<T<TCtemperature range.\nHowever,the wave propagation in the medium depends on the interplay between the\ncharacteristic ,andparameters. Recently it has been reported that multiferroic spinels\npresent in the magnetic orderedphase important changes in the dielectric properties,\nbesides the increases of the magnetization [ 44,45].Consequently, in order to study this\ntopic in depth, careful measurements of the temperature evolution of these parameters in\nMnCr2O4should be performed.\n4. Conclu sions\nIn summary, we have investigated the low temperature magnetic phases present in a\npolycrystalline sample of the MnCr 2O4spineland quantified the temperature evolution of\nthe magnetic anisotropy constants in a wide temperature range. In themagnetization versus\ntemperature measurements we have observed anomalies consistent with the ferrimagnetic\norder atTC~41K and the formation of the helicoidal spin arrangement at TH~18 K. The\nelectron spin resonance spectra exhibit important changes as a function of the temperature.\nThis behavior could be explained through a phenomenological model consider ingthe\ndifferent terms that contribute to the magnetic free energy of the system. Below TCthe\nFMR spectra can be fitted bya cubic magnetocrystalline a nisotropy term with constants K1\nandK2that increase when the temperature diminishes. Near TH, an additional magnetic\nanisotropy term should be included to account forthe noticeable changes observed in the\nFMR spectra. This anisotropy, accounted by a Kuparameter of the uniaxial anisotropy term,is associated to the breaking of the cubic anisotropy due to the formation of the helicoidal\norder that propagates in the ]011[\ndirection. We remark that the structure observed in the\nFMR spectra can be explained taking into account the change of the magnetic symmetry of\na single magnetic phase as a function of the temperature. The fact that Kuisnon-zero above\nTHcould indicate that the short -range helicoidal order still coexists with the ferrimagnetic\norder above the helical transition .The fact that Ku is non -zero above THand also the lineal\nincrease of the high field magnetization up to 30 K, could indicate th at the conical\narrangement of the spins coexists with the ferrimagnetic order above TH. When the\ntemperature diminishes the transverse component of the Mn and Cr conical arrange orders\nand the spiral order stabilizes. Finally, we want to emphasize the sens itivity of the electron\nspin resonance spectroscopy to detect magnetic transitions and anisotropic interactions\nwhich enable us to obtain fundamental information that complements the magnetic\nmeasurements, and it allows us to calculate the characteristic parameters even in\npolycrystalline samples.\nAcknowledgements\nThe authors thank Francisco Rivadulla for thesamples and valuable sugges tionsand\ndiscussions .The work was supported by PIP 112 -200801-0133 CONICET (Argentina) and\nC011Universidad Nacional de Cuyo (Argentina).\nReferences\n[1] R. Moessner and J. T. Chalker 1998 Phys. Rev. B 5812049[2] S. E. Dutton, Q. Huang, O. Tchernyshyov, C. L. Broholm and R. J. Cava 2011Phys.\nRev. B83064407\n[3] S.-H. Lee, C. Broholm, W. Ratcliff, G. Gasparovic, Q. Huang, T. H. Kim and S. -W.\nCheong2002Nature418856-858\n[4] E. Winkler, S. Blanco Canosa, F. Rivadulla, M. A. López -Quintela, J. Rivas, A.\nCaneiro, M. T. Causa and M. Tovar 2009Phys.Rev. 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Menyuk 1962 Phys. Rev. 126540\n[43] J. B. Goodenough, Magnetism and Chemical Bond (Wiley, New York, 1963).\n[44]Y. Yamasaki, S. Miyasaka, Y. Kaneko, J. -P.He,T. Arima, and Y. Tokura 2006 Phys\nRev. Lett. 96207204\n[45]E. D. Mun, Gia -Wei Chern, V. Pardo, F. Rivadulla, R. Sinclair, H. D. Zhou, V. S.\nZapf, and C. D. Batista 2014 Phys Rev. Lett. 112017207"    },    {        "title": "1412.7454v2.Tuning_the_magnetic_anisotropy_of_single_molecules.pdf",        "content": "Tuning the magnetic anisotropy of single molecules\nBenjamin W. Heinrich,\u0003,†Lukas Braun,†,§Jose I. Pascual,†,‡,¶and Katharina J.\nFranke†\nFachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany, CIC\nnanoGUNE, 20018 Donostia-San Sebastián, Spain, and Ikerbasque, Basque Foundation for\nScience, 48011 Bilbao, Spain\nE-mail: bheinrich@physik.fu-berlin.de\nAbstract\nThe magnetism of single atoms and molecules is governed by the atomic scale environ-\nment. In general, the reduced symmetry of the surrounding splits the dstates and aligns\nthe magnetic moment along certain favorable directions. Here, we show that we can re-\nversibly modify the magnetocrystalline anisotropy by manipulating the environment of sin-\ngle iron(II) porphyrin molecules adsorbed on Pb(111) with the tip of a scanning tunneling\nmicroscope. When we decrease the tip–molecule distance, we first observe a small increase\nfollowed by an exponential decrease of the axial anisotropy on the molecules. This is in con-\ntrast to the monotonous increase observed earlier for the same molecule with an additional\naxial Cl ligand.1We ascribe the changes in the anisotropy of both species to a deformation\nof the molecules in the presence of the attractive force of the tip, which leads to a change in\nthedlevel alignment. These experiments demonstrate the feasibility of a precise tuning of the\nmagnetic anisotropy of an individual molecule by mechanical control.\n\u0003To whom correspondence should be addressed\n†Fachbereich Physik, Freie Universität Berlin, Arnimallee 14, 14195 Berlin, Germany\n‡CIC nanoGUNE, 20018 Donostia-San Sebastián, Spain\n¶Ikerbasque, Basque Foundation for Science, 48011 Bilbao, Spain\n§present address: Fritz-Haber-Institut der Max-Planck-Gesellschaft, 14195 Berlin, Germany\n1arXiv:1412.7454v2  [cond-mat.mes-hall]  8 May 2015KEYWORDS: magnetocrystalline anisotropy, spin excitation, strain-induced tuning, porphyrins,\nmanipulation, scanning tunneling microscopy\nAn important goal in device technology is the construction of nanomagnets and addressability\nof their magnetic states.2,3This requires the understanding of how the atomic-scale environment\ninfluences the spin state and magnetocrystalline anisotropy of single atoms, the building blocks of\nnanomagnets.4–7The spin state of a magnetic atom is governed by the competition of two energy\nscales: the splitting of the dorbitals by the reduced symmetry in their crystal field, which tends to\nlower the total spin, and the tendency of electron spins to avoid spin pairing and align according\nto Hund’s rules. In a metal-organic molecule, the dlevel splitting is adjusted by the organic ligand\naround the paramagnetic metal atom. The magnetocrystalline anisotropy is a consequence of spin-\norbit coupling via the admixture of low-lying excited states as a second order perturbation.8Recent\nworks have shown that the careful choice of the atomic surrounding on surfaces, such as adsorption\nsites,4–7,9or an organic ligand10–12leads to sizable magnetocrystalline anisotropy, up to some tens\nof meV .\nTunable control of the magnetocrystalline anisotropy is difficult to achieve, because it demands\na handle to continuously shift the energy levels.13So far, this has only been achieved in mechani-\ncally controlled break junctions,14which lack the precise control of the molecular geometry and,\nhence, of the type of deformation induced. One way to tune properties of nanoobjects on surfaces\nwith a high degree of control consists in approaching the tip of an STM to the object and modifying\nthe local magnetic, electronic or mechanical properties. This has been used to intentionally shift\nsurface,15or molecular states16in energy, alter the Kondo screening,17or the (anti)ferromagnetic\ncoupling strength,18and change the spin state mixing of individual nanomagnets.19\nHere, we develop a strategy to precisely tune the magnetocrystalline anisotropy in metal-\norganic complexes. We study and manipulate iron(II) porphyrin molecules adsorbed on Pb(111)\nby means of scanning tunneling microscopy and inelastic scanning tunneling spectroscopy (ISTS)\nand compare the results to measurements on the same molecule with an additional axial Cl ligand.\n2FeOEP-ClFeOEP\nSample bias (mV)-10 0\n3 nm\n200\n0200\n0dI/dV (nS)\n1005004∆/e\n5x\n5x\nPb(111)Figure 1: ISTS on Fe-OEP on Pb(111). The dI=dVspectra are acquired above the center of an\nFe-OEP, and an Fe-OEP-Cl molecule, respectively, and on pristine Pb(111), using conventional\nlock-in technique at a frequency of 912Hz with an acbias modulation of 50 mVrms. The insets\nshow a topographic image of a molecular island on Pb(111) ( V=130mV, I=60pA) and a high\nresolution zoom on two molecules. Molecules with an axial Cl ligand appear with a protrusion in\nthe center (Fe-OEP-Cl, left in the zoom), molecules without Cl ligand exhibit a depression in the\ncenter (Fe-OEP, right).\nWe use inelastic spin excitations to monitor changes in the magnetic anisotropy. To be able to\nresolve even small variations, a very good energy resolution is required. Thus, we employ a su-\nperconducting tip and substrate to probe the magnetic excitations in the single molecules. We then\nuse the force exerted by the tip of our microscope to affect the molecules with picometer precision.\nWe gain a handle on the ligand field splitting and, hence, the magnetocrystalline anisotropy. These\nfindings pave the way for a flexible atomic-scale control of spins in nanomagnets.\nWe carry out our experiments in a SPECS JT-STM operating at a base temperature of 1 :2K.\nThe Pb(111) surface was cleaned by repeated sputter/anneal cycles until a clean, superconduct-\n3ing surface was obtained.20To increase the energy resolution beyond the Fermi-Dirac-limit of\nnormal metal tips, we covered the bulk W-tip with superconducting Pb by controlled indentation\ninto the clean surface, until the tip showed a bulk-like superconducting order parameter D. Iron-\noctaethylporphyrin-chloride (Fe-OEP-Cl, structure model as inset in Figure 1) was sublimated\nfrom a crucible at 490K onto the clean Pb(111) surface held at 120K. After annealing to 240K,\nthe porphyrin molecules self-assemble in ordered monolayer islands of quasi-hexagonal structure,\nwith the ethyl-groups clearly visible in the STM images (see insets in Figure 1). About two thirds\nof the molecules exhibit a depression in the center. They have lost the axial Cl ligand upon ad-\nsorption (in the following referred to as Fe-OEP). The second type of molecules shows a bright\nprotrusion in the center, which we identify as the Cl ligand. By means of a voltage pulse, the ligand\ncan controllably be removed (compare to Ref.21,22), and Fe-OEP-Cl is transformed into Fe-OEP.\nTo characterize the spin state of Fe-OEP, we recorded spectra of the differential conductance\n(dI=dV) with the tip placed over the center of the molecule (Figure 1 top). The spectrum exhibits\na gap region around the Fermi level, enclosed by sharp resonances of quasi-particle excitations at\neV=\u00062Das also observed for the pristine Pb(111) surface (Figure 1 bottom).23This characterizes\nthe unperturbed superconductor-superconductor tunneling junction. No inner-gap bound states,\nwhich are ascribed to local screening of the spin through the superconductor’s quasi-particles,25,26\nare observed on the paramagnetic molecule. This shows the absence of any sizable interaction\nwith the substrate, probably due to the eight ethyl groups acting as spacers. Instead, we found\ntwo pairs of new resonances appearing symmetric to zero bias at \u000612:6 and\u000614:0 mV , outside\nthe superconducting gap. On Fe-OEP-Cl, we observed similar peak shapes (see Figure 1 middle)\nat lower energies.1The resonances are signatures of inelastic excitations in a superconductor-\nsuperconductor junction.1The opening of the inelastic tunneling channel at the threshold energy\nejVj=2D+eis reflected in the dI=dVspectra by BCS-like resonances due to the structure of the\nsuperconducting density of states of tip and sample.\nMeasurements on Fe-OEP-Cl in a magnetic field of up to 3 T perpendicular to the sample sur-\nface (Supporting Information, Figure S1) show that the origin of these peaks is magnetic. The ob-\n4served Zeeman shift of the low energy excitation and the appearance of a new transition around the\nFermi level reveals a half integer spin state of either S=3=2 or 5 =2 with an easy-plane anisotropy.\nCurrent-dependent changes of the excitation intensities allowed us to identify the excitation scheme\nof the molecule [illustrated in Figure 2(c)].1The lower-energy excitation corresponds to transitions\nbetween the ground state ( jMSi=j\u00061=2i) and the first excited state ( j\u00063=2i). The second excita-\ntion then proved the S=5=2 state, because it is due to transitions between j\u00063=2iandj\u00065=2i, the\nfirst and second excited state, respectively. The deduced easy-plane anisotropy can be described in\nterms of the phenomenological Spin–Hamiltonian at zero magnetic field: H=DS2\nz+E(S2\nx\u0000S2\ny)\n(here Sx,Sy, and Szare the spin operators of the three Cartesian axis).2The axial anisotropy param-\neterD=0:7meV is close to the bulk value.27We did not detect any additional in-plane distortion\n(E=0).\nThe removal of the Cl ligand causes a drastic change in the excitation spectrum [Figure 2(b)].\nThe two excitations at 12 :6 and 14 :0mV are of equal intensity independent of the tunneling cur-\nrent.28This spectrum agrees with a spin S=1, which is expected for an Fe center reduced from\noxidation state III to II. The intermediate spin state is typical for Fe2+in the square-planar ligand\nfield of a porphyrin ligand.29–31In the S=1 state, easy-plane anisotropy ( D>0) with an additional\nin-plane distortion ( E6=0) gives rise to two possible excitations with equal transition probabilities\n[sketch in Figure 2(c)]. From the spectrum we deduce the anisotropy parameters D=10:6meV\nandE=0:7meV. The Dvalue is similar to those measured for other Fe2+porphyrins in the bulk\nphase.32,33However, we find variations in EandDaround these values [Figure 2(d)] for different\nFe-OEP molecules on the surface without any obvious correlation between the two anisotropy pa-\nrameters (note that no such variations are observed for Fe-OEP-Cl). The fluctuation in EandD\nvalues are probably due to slight variations in the adsorption site within the self-assembled islands.\nTo gain an intuitive picture of the origin of the magnetocrystalline anisotropy of the two\nmolecules, we consider the different ligand field splitting and dlevel occupation of the Fe ion\nin the two complexes (Figure 3). The ligand field experienced by the Fe3+in Fe-OEP-Cl is of\nsquare-pyramidal nature. The axial Cl ligand, which is a weak ligand compared to the pyrole ni-\n5 3/2 (a)\n(d)0.10\nSample bias (mV)3 60300 dI/dV (nS)\n(c)S = 5/2FeOEP-Cl\n2D\n4D\nZFS of\n4D\n2DPb(111)0.15 dI/dV (arb. units)\nSample bias (mV)0 10D+E\nD-EFeOEP\nS = 1(b)\nE = 0D\nE ≠ 02E FeOEP-Cl  FeOEP\n0 1\nD (meV)10 111.4E (meV)\n00.7\nFeOEP\n 1/2  5/2Figure 2: Inelastic excitations. (a) High-resolution ISTS spectrum of Fe-OEP-Cl (25 mVrms). The\ntwo excitations with an energy of 1 :4 and 2 :8meV identify the S=5=2 state with a magnetocrys-\ntalline anisotropy parameter D=0:7meV.1(b) The zoom on the ISTS spectrum of Fe-OEP unveils\ntwo inelastic excitations of equal intensity at 12 :6 and 14 :0mV (50 mVrms). As guide for the eye,\nthe spectrum of pristine Pb(111) is superimposed as dotted line. (c) Scheme of the zero field split-\nting of Fe-OEP and Fe-OEP-Cl. For Fe-OEP we detect, additionally to the main anisotropy axis,\nan in-plan distortion (rhombicity), which gives rise to the parameter E. (d) Distributions of Eand\nDas measured for 71 different Fe-OEP molecules (measured at 4 :5K).\ntrogen, lifts the Fe3+out of the macrocycle plane.29This leads to a small ligand field splitting\nand yields a high-spin S=5=2 ground state with singly occupied dlevels. In this case, the spin-\npairing energy is larger than the energy level difference. The magnetocrystalline anisotropy results\nfrom the admixture of electronically excited states of reduced total spin [ e.g. E1and E 2in Fig-\nure 3(a)]. Furthermore, the lifting of the Fe center out of the pyrole plane leads to a relatively large\nFe–surface distance, which makes the Fe3+insensitive to variations of the adsorption site.\nIn the case of Fe-OEP, the Fe2+lies in the square-planar ligand field of the porphyrin macro-\ncycle. The in-plane position of the Fe2+yields shorter Fe–N distances and a larger ligand field\nsplitting, in particular the energy of the dx2\u0000y2level is strongly increased compared to Fe-OEP-Cl.\nThis results in the intermediate spin ( S=1) ground state E 0as depicted in Figure 3(b). Since the\n6square planar\nπxy\nz2x2-y2\nE0E2E1(b) (a)\nπxyz2x2-y2\nE0E1E2square pyramidalFigure 3: Sketch of the electronic ground (E 0), first (E 1), and second (E 2) excited state in the\ncase of: (a) high-spin ( S=5=2) Fe3+in a square-pyramidal ligand field (as in Fe-OEP-Cl); (b)\nintermediate spin ( S=1) Fe2+in a square-planar ligand field34(as in Fe-OEP). Note that in (a) E 1\n(degenerate) and E 2are of reduced total spin compared to E 0, while in (b) the spin is conserved.\nEnergies not to scale.\nsplitting between the dpanddz2is small in Fe(II) porphyrins,35there is a strong admixture of E 1\nand E 2to the ground state and hence a large magnetocrystalline anisotropy. The admixture of these\nlowest lying states yields contributions to the orbital moments LxxandLyyinxandydirection,\nrespectively, but not to Lzz, the orbital moment in zdirection.37Therefore, the resulting anisotropy\nis easy-plane, i.e.,D>0, as we can write D=\u0000l2\n2(2Lzz\u0000Lxx\u0000Lyy),37with lbeing the spin-\norbit coupling constant. At the same time, the shorter Fe–surface distance compared to Fe-OEP-Cl\nresults in a higher sensitivity to variations in the adsorption site. The atomic environment under-\nneath the Fe2+ion mainly affects the dz2anddplevels, because they extend toward the surface,\nresulting in slight variations of LxxandLyy, and, therefore, DandE[note: E=\u0000l2\n2(Lxx\u0000Lyy)].\nThe interaction with the surface also explains the reduction of the D4hsymmetry evidenced by the\nnon-zero Eparameter.\nThe above shown sensitivity of the zero field splitting to small variations in the environment\nprovides access to a controlled tuning. With this intention, we approach the tip of the STM to the\ncenter of an Fe-OEP molecule. The presence of the tip alters the ligand field, while we simultane-\nously record ISTS curves at varying tip–sample distances [see Figure 4(a)]. First, both excitation\npeaks shift to higher energies and reach a maximum at Dz\u0019\u0000200pm. With a further reduction\nof the distance, both peaks shift to energies lower than the initial values until, at \u0000330pm, the\njunction becomes instable. Figure 4(b) shows the extracted values of D vs.Dz.38From Dz=0 to\n\u0000200pm the axial anisotropy Dslightly increases, before it rapidly decreases for shorter distances.\n7This variation of Dis qualitatively different from the variation observed earlier on Fe-OEP-Cl,1\nwhere Dexponentially increased with decreasing tip–sample distance.\nThese variations in the zero field splitting of the two species can be understood considering the\nchanges in geometry induced by the proximity of the STM tip. In the relaxed adsorption state of\nFe-OEP (in absence of the STM tip), the Fe ion is expected to be attracted toward the surface, thus\nlying slightly below the porphyrin plane. The tip potential then exerts an opposed attractive force\non the Fe atom, which pulls it toward the opposite side of the macrocycle due to a surface trans\neffect.39Passing through the molecular plane causes first a small decrease in Fe–N bond length,\nfollowed by a subsequent Fe–N bond elongation. A larger Fe–N bond length causes the dx2\u0000y2\norbital to shift down in energy, while the concomitant decrease in the tip–Fe distance increases the\nenergy of the dz2orbital [sketched in Figure 4(c)]. The larger energy difference between dpanddz2\nenhances also the energy difference E 1\u0000E0. This reduces the admixture of the excited states and,\nhence, the magnetocrystalline anisotropy. This scenario thus explains the initial slight increase in\nDbetween Dz=0 and\u0000200pm (Fe–surface distance increased and, therefore, lower dz2energy)\nand its subsequent pronounced decrease upon further tip approach.\nIn the case of Fe-OEP-Cl, the tip potential acts (mainly) on the Cl ligand pointing upwards\nand attracts the Cl atom toward the tip. In turn, the Cl–Fe bond weakens and the Fe ion relaxes\ntoward the molecular plane, decreasing the Fe–N bond length. The resulting changes of the ligand\nfield splitting are sketched in Figure 4(c): the dz2(dx2\u0000y2) orbital shifts down (up) in energy as\nthe Fe–Cl (Fe–N) distance increases (decreases). These changes increase the overall ligand field\nsplitting and reduce the total energy difference E 1\u0000E0, which also includes the spin pairing en-\nergy. The smaller energy difference between ground and excited states enhances their admixture\naccording to perturbation theory. This yields larger orbital moments LxxandLyyand, therefore,\nlarger magnetocrystalline anisotropy.\nThe local control of the magnetic properties of nanostructures, such as the spin state or the mag-\nnetocrystalline anisotropy, is a prerequisite for their successful application in spintronic devices. In\nthin metallic films40,41and clusters,42an external electric field can be applied to tune the magnetic\n8far close\n(c)\nfar close∆z (pm)\nSample bias (mV)\n-16 -10 -13-50\n-300-100 pmFe-OEP\n hi low (a) dI/dV\n(b)\nD (meV)\n∆z (pm)0 -150 -30011.0\n10.010.5\nFe-OEPFe-OEP\nFe-OEP-ClFigure 4: Tip-induced changes of the zero field splitting of Fe-OEP. (a) Color plot of the normalized\ndI=dVexcitation spectra of Fe-OEP as a function of Dz(50mVrms, setpoint: 200pA, 50mV). (b)\nAxial anisotropy D vs.Dzas extracted from the spectra shown in (a). (c) Scheme of the changes in\nthe ligand field splitting of Fe-OEP and FeOEP-Cl with the approaching tip.\nanisotropy. For single atoms, so far only static control of the magnetocrystalline anisotropy has\nbeen achieved by the selection of the adsorption site on surfaces.4,5,7,43,44In the chemical approach,\norganic ligands are used to determine the ligand field splitting and magnetocrystalline anisotropy.2\nOur approach provides the flexibility to continuously tune the magnetocrystalline anisotropy by\nmodifying the geometry of the atomic-scale surrounding, even though it is, so far, restricted to\nchanges in the order of 10%. The opposed variations in the anisotropy induced by the tip for\nthe two types of molecules discussed, underline the importance of a smart chemical engineering to\nachieve desired functionality and highlight the degree of freedom that can thereby be reached. Fur-\nthermore, we show how the anisotropy may serve as a highly sensitive probe to identify variations\nin the atomic scale interactions, which are, e.g., induced by the presence of the tip.\n9Acknowledgement\nWe thank O. Peters and N. Hatter for assistance during magnetic-field dependent measurements.\nThis research has been supported by the DFG grant FR2726/4, ERC grant NanoSpin, MINECO\ngrant MAT2013-46593-C6-01, and the focus area \"Nanoscale\" of Freie Universität Berlin.\nSupporting Information Available Magnetic field dependent dI=dVmeasurements on Fe-OEP-\nCl. This material is available free of charge via the Internet at http://pubs.acs.org.\nReferences\n(1) Heinrich, B. W.; Braun, L.; Pascual, J. I.; Franke, K. J. Nature Phys. 2013, 9,765.\n(2) Gatteschi, D.; Sessoli, R.; Villain, J. Molecular Nanomagnets , Oxford Univ. Press: Ox-\nford, 2006.\n(3) Affronte, M. J. Mater. Chem. 2009, 19,1731.\n(4) Gambardella, P.; Rusponi, S.; Veronese, M.; Dhesi, S. S.; Grazioli, C.; Dallmeyer, A.;\nCabria, I.; Zeller, R.; Dederichs, P.H.; Kern, K.; Carbone, C.; and Brune, H. Science\n2003, 300, 1130.\n(5) Bryant, B.; Spinelli, A.; Wagenaar, J. J. T.; Gerrits, M.; and Otte, A. F. Phys. Rev. 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Nano Lett. 2008, 8,1291.\n(17) Choi, D.-J.; Rastei, M. V .; Simon, P.; and Limot, L. Phys. Rev. Lett. 2012, 108, 266803.\n(18) Bork, J.; Zhang, Y .-h.; Diekhöner, L.; Borda, L.; Simon, P.; Kroha, J.; Wahl, P.; and\nKern, K. Nature Phys. 2011, 7,901.\n11(19) Yan, S.; Choi, D.-J.; Burgess, J. A. J.; Rolf-Pissarczyk, S.; Loth, S. Nature Nanotech.\n2014, 10,40.\n(20) Pb is a type I superconductor with a critical temperature of 7 :2K. At 1 :2K its energy gap\nhas a width of 2 D=2:7meV.\n(21) Gopakumar, T. G.; Tang, H.; Morillo, J.; and Berndt, R. J. Am. Chem. Soc. 2012, 134,\n11844.\n(22) Stró ˙zecka, A.; Soriano, M.; Pascual, J. I.; and Palacios, J. J. Phys. Rev. Lett. 2012, 109,\n147202.\n(23) The doubling of the size of the superconducting gap is due to the convolution of the BCS-\nlike (Bardeen–Cooper–Schrieffer) density of states of tip and sample. The shoulder-like\nfeatures visible at\u00066:7 and\u000611:2meV are, again, also present in the spectrum of pris-\ntine Pb(111). These are signatures of the strong electron-phonon coupling in Pb and\ncorrespond to the transverse and longitudinal phonon bands, respectively.24\n(24) Rowell, J. M.; Anderson, P. W.; and Thomas, D. E. Phys. Rev. Lett. 1963, 10,334.\n(25) Shiba, H. Prog. Theor. Phys. 1968, 40,435.\n(26) Franke, K. J.; Schulze, G.; and Pascual, J. I. Science 2011, 332, 940.\n(27) Nishio, T.; Yokoyama, S.; Sato, K.; Shiomi, D.; Ichimura, A. S.; Lin, W. C.; Dolphin,\nD.; McDowell, C. A.; and Takui, T. Synthetic Metals 2001, 121, 1820.\n(28) The absence of any resonances in this energy range in the spectrum of Fe-OEP-Cl ren-\nders a vibrational origin implausible.\n(29) Kadish, K. M.; and Ruoff, R. S. in The Porphyrin Handbook ; Kadish, K. M., Smith, K.\nM., Guilard, R., Eds.; Academic Press: San Diego, 2000; V ol. 3.\n12(30) Wende, H; Bernien, M.; Luo, J.; Sorg, C.; Ponpandian, N.; Kurde, J.; Miguel, J.; Piantek,\nM.; Xu, X.; Eckhold, Ph.; Kuch, W.; Baberschke, K.; Panchmatia, P. M.; Sanyal, B.;\nOppeneer, P. M.; & Eriksson, O. Nature Mater. 2007, 6,516.\n(31) Bernien, M.; Xu, X.; Miguel, J.; Piantek, M.; Eckhold, Ph.; Luo, J.; Kurde, J.; Kuch, W.;\nBaberschke, K.; Wende, H.; and Srivastava, P. Phys. Rev. B 2007, 76,214406.\n(32) Barraclough, C. G.; Martin, R. L.; Mitra, S.; and Sherwood, R. C. J. Chem. Phys. 1970,\n53,1643.\n(33) Boyd, P. D. W.; Buckingham, D. A.; McMeeking, R. F.; and Mitra, S. Inorg. Chem.\n1979, 18,3585.\n(34) Huheey, J.; Keiter, E.; Keiter, R. Anorganische Chemie , 3rd. ed.; Walter de Gruyter:\nNew York, 2003.\n(35) Note that the ground state electronic configuration of Fe(II) porphyrins is under debate\nfor decades. However, it is agreed that the dx2\u0000y2orbital is high up in energy, while\nall other dlevels are close in energy and the actual alignment depends on subtle de-\ntails.36The resulting S=1 and in-plane anisotropy are robust. Despite being simplified,\nour sketch of the electronic configuration allows an intuitive insight and is sufficient\nto explain our experimental observations. Note that also the other low-lying excitation\n(dp!dxy) contributes to an in-plane orbital moment.\n(36) Liao, M.-S.; Kar, T.; Gorun, S. M.; and Scheiner, S. Inorg. Chem. 2004, 43,7151.\n(37) Dai, D.; Xiang, H.; and Whangbo, M. J. Comput. Chem. 2008, 29,2187.\n(38) We repeated this experiment on six Fe-OEP molecules with different DandEparame-\nters. We observe qualitatively the same D vs. Dzdependence. The E vs.Dzdependence\ndiffers, however, qualitatively for different molecules and tips. Eis a measure of dis-\n13tortions in the system and is, therefore, expected to strongly depend on details in the\nadsorption geometry. Ddepends less on these distortions.\n(39) Flechtner, K.; Kretschmann, A.; Steinrück, H.-P.; and Gottfried, J. M. J. Am. Chem. Soc.\n2007, 129, 12110.\n(40) Weisheit, M.; Fähler, S.; Marty, A.; Souche, Y .; Poinsignon, C.; Givord, D. Science\n2007, 315, 349.\n(41) Maruyama, T.; Shiota, Y .; Nozaki, T.; Ohta, K.; Toda, N.; Mizuguchi, M.; Tulapurkar,\nA. A.; Shinjo, T.; Shiraishi, M.; Mizukami, S.; Ando, Y .; Suzuki, Y . Nature Nanotech.\n2009, 4,158.\n(42) Sonntag, A.; Hermenau, J.; Schlenhoff, A.; Friedlein, J. Krause, S.; and Wiesendanger,\nR.Phys. Rev. Lett. 2014, 112, 017204.\n(43) Hirjibehedin, C. F.; Lin, C.-Y .; Otte, A. F.; Ternes, M.; Lutz, C. P.; Jones, B. A.; and\nHeinrich, A. J. Science 2007, 317, 1199.\n(44) Loth, S.; Etzkorn, M.; Lutz, C. P.; Eigler, D. M.; Heinrich, A. J. Science 2010, 329,\n1628.\n14Supporting Information:\nZeeman shift of the spin excitations\nTo prove the magnetic origin of the inelastic excitation detected for Fe-OEP-Cl, we acquired dI=dV\nspectra at different magnetic (B) field strengths aligned perpendicular to the sample surface. Fig-\nure 5(a) shows dI=dVspectra measured on an Fe-OEP-Cl molecule at fields ranging from 0 :5 to\n3 T. The B field quenches the superconducting state of sample and tip. Therefore, the inelastic\nexcitations now appear as steps rather then as resonances at a threshold energy jeVj=ein the\ndI=dVspectra.\n 0.5 T\n 1.0 T\n 1.5 T\n 2.0 T\n 2.5 T\n 3.0 T 0.0 T\n 0.5 T\n 1.0 T\n 1.5 T\n 2.0 T\n 2.5 T\n 3.0 T\nSample bias (mV) Sample bias (mV)-3 0 1 3 2 -2 -1dI/dV (arb. units)\n0.41.0\n0.60.8dI/dV (arb. units)\n0.41.0\n0.60.8\n-3 0 1 3 2 -2 -1(a) (b) (c)\nB field0Energy\n 3/2\n 1/2 5/2\nFigure 5: (a) Magnetic field dependent ISTS spectra of Fe-OEP-Cl (feedback parameters: I=\n50 mV , V=200 pA; dz=\u0000100 pm, Vmod=35meV). (b) Fit of the experimental data in (a) with the\nfollowing parameters: spin S=5=2, axial anisotropy D=0:72\u00060:02 meV , g-factor g=1:8\u00060:2,\neffective temperature Te f f=1:3 K. (c) Scheme of the B field dependent state energies. Inelastic\ntransitions as observed in the experiment are indicated by arrows.\nWith increasing field strength, the excitation with a zero-field energy of 1 :4 meV as detected\nin the superconducting state [compare to Figure 2(a) of the main manuscript], moves to lower\nenergies. Simultaneously, a V-shaped gap is opened around zero bias. The energy of this new\nexcitation increases with the B field strength. The appearance as a dip is due to the overlap of\nsteps at opposite energies, which are Fermi-Dirac broadened at 1 :2 K. We can describe the spin\nexcitations in an anisotropic environment and magnetic field by the following phenomenological\nSpin–Hamiltonian:2\nH=gmB~B\u0001~S+DS2\nz+E(S2\nx\u0000S2\ny): (1)\n15The first term yields the Zeeman splitting, with gbeing the Landé g-factor, mBthe Bohr magne-\nton,~Bthe magnetic field vector and ~S= (Sx;Sy;Sz)the spin operator. DandEare the axial and\ntransverse anisotropy parameters. Setting the spin to S=5=2 and E=0,1we can simulate the\nexperimental spectra assuming the axial anisotropy parallel to the applied B-field axis and an ef-\nfective temperature Te f f=1:3 K. The fit yields D=0:72\u00060:02 meV and g=1:8\u00060:2 and the\nsimulated curves are shown in Figure 5(b).\nFigure 5(c) presents a scheme of the B field-dependent state energies and the observed transi-\ntions. In the superconducting state of the substrate at zero field strength, two excitations can be\nobserved: one ground state excitation (full orange arrow) from MS=j\u00061=2itoj\u00063=2i, and, at\nhigher currents, a second excitation from the first excited state j\u00063=2itoj\u00065=2i(dashed orange\narrow). In magnetic fields above the critical field of Pb (80 mT), only ground state excitations are\nobserved. The violet arrows indicate the transitions from j\u00001=2itoj+1=2i, and fromj\u00001=2ito\nj\u00003=2i, respectively.\nIn the case of Fe-OEP, we were not able to detect unambiguous evidence of a Zeeman shift of\nthe spin excitations in fields of up to 3 T at a temperature of 1 :2 K. This is understood by taking the\nrelevant energies of the system into account. In a S=1 system in zero magnetic field, the transverse\nanisotropy Emixes the pure spin states j+1iandj\u00001i. An applied magnetic field in zdirection\nwill reduce the mixing and restore the pure spin state if gmBBz\u001dE. At intermediate fields, the\nenergy of the two states is given as E\u0006=D=3\u0006p\nE2+ (gmBBz)2. With a transverse anisotropy of\nE=0:7 meV (approximate median of all measured molecules), this results in changes of the exci-\ntation energies of 60 meV at 3 T, which is small compared to the temperature broadening at 1 :2 K\n(3:5kBT=360meV). Yet, the agreement of the deduced axial anisotropy with bulk measurements\nof other Fe2+porphyrins,32,33together with the absence of any excitation in this energy range\nin the spectrum of Fe-OEP-Cl, renders a vibrational origin of the observed inelastic excitations\nimplausible and corroborates our interpretation as spin excitations of a S=1 system.\n16"    },    {        "title": "1501.02154v1.Single_crystal_study_of_layered_U___n__RhIn___3n_2___materials__case_of_the_novel_U___2__RhIn___8___compound.pdf",        "content": "arXiv:1501.02154v1  [cond-mat.str-el]  9 Jan 2015Single crystal study of layered U nRhIn3n+2materials:\ncase of the novel U 2RhIn8compound\nAttila Barthaa,1,∗, M. Kratochv´ ılov´ aa, M. Duˇ sekb, M. Diviˇ sa, J. Custersa, V.\nSechovsk´ ya\naDepartment of Condensed Matter Physics, Charles Universit y, Ke Karlovu 5, 121 16\nPraha 2, Czech Republic\nbDepartment of Structure Analysis, Institute of Physics ASC R, Cukrovarnick´ a 10, 162\n00 Praha 6, Czech Republic\nAbstract\nWe report on the single crystal properties of the novel U 2RhIn8compound\nstudied in the context of parent URhIn 5and UIn 3systems. The compounds\nwere prepared by In self-flux method. U 2RhIn8adopts the Ho 2CoGa8-type\nstructure with lattice parameters a= 4.6056(6)˚A andc= 11.9911(15) ˚A.\nThe behavior of U 2RhIn8strongly resembles that of the related URhIn 5and\nUIn3with respect to magnetization, specific heat and resistivity except for\nmagnetocrystallineanisotropydeveloping withlowering dimensionality inthe\nseries UIn 3vs. U2RhIn8and URhIn 5. U2RhIn8orders antiferromagnetically\nbelowTN= 117 K and exhibits a slightly enhanced Sommerfeld coefficient\nγ= 47 mJ ·mol−1·K−2. Magnetic field leaves the value of N´ eel temperature\nforbothURhIn 5andU 2RhIn8unaffected upto9 T. Ontheother hand, TNis\nincreasing with applying hydrostatic pressure up to 3.2 GPa. The cha racter\n∗Corresponding author\nEmail address: bartha@mag.mff.cuni.cz (Attila Bartha)\n1Address: Charles University, Department of Condensed Matter P hysics, Ke Karlovu\n5, 121 16 Praha 2, Czech Republic; Tel. +420221911456\nPreprint submitted to Journal of magnetism and Magnetic Mat erials March 10, 2022of uranium 5 felectron states of U 2RhIn8was studied by first principles cal-\nculations based on the density functional theory. The overall pha se diagram\nof U2RhIn8is discussed in the context of magnetism in the related URh X5\nand UX3(X= In, Ga) compounds.\nKeywords: single crystal growth, antifgerromagentism, magnetocrystalline\nanisotropy, U 2RhIn8\nPACS:75.30.Gw, 75.50.Ee, 81.120.Du\n1. Introduction\nMagnetism of uranium compounds is characterized by the large spat ial\nextent of the 5 fwave functions which perceive their physical surroundings\nmore intensively compared to the localized behavior of 4 felectrons. Typical\nexample of that is the 5 f-ligand hybridization causing nonmagnetic behavior\nin several compounds characterized by the distance between the nearest U\nions far larger than the Hill limit [1]. When considering the U X3(X=p-\nmetal) materials, the size of the p-atom is a very important parameter. In\nthe case of smaller X-ions (Si, Ge) [2], the p-wave function decays slower\nat the U-site, resulting in strong 5 f-phybridization which leads to lack of\nmagnetic ordering (UGe 3, USi3) [4, 2, 3] while larger X-ions (In, Pb) cause\nthe hybridization to be weaker resulting in magnetic ground state (U In3,\nUPb3) [5, 6].\nThe U nTX3n+2(n= 1, 2;T= transition metal; X= In, Ga) [7, 8, 9, 19,\n24]compoundsadoptthelayered Ho nCoGa3n+2-typestructurewhichconsists\nofnUX3layers alternating with a TX2layer sequentially along the [001] di-\nrection in the tetragonal lattice. They are isostructural with the thoroughly\n2investigated Ce nTX3n+2[10] compounds known for their outstanding phys-\nical properties such as the coexistence of unconventional super conductivity\nand magnetism or non-Fermi liquid behavior. These families of compou nds\nprovide unique opportunity to study the effect of dimensionality on p hysical\nproperties due to their layered tetragonal structure. Adding a la yer ofTX2\npushes the character of the structural dimensionality from 3D to more 2D.\nSince the U 2RhIn8compound has not been reported yet, we focused in\nthispaperonthestructurestudyfollowedbyinvestigationofmagn etic, trans-\nport and thermodynamic properties with respect to applied magnet ic fields\nand hydrostatic pressure. In order to study the evolution of gro und state\nproperties on the structural dimensionality, we also prepared and investi-\ngated single crystals of URhIn 5and UIn 3.\n2. Experimental\nSingle crystals of UIn 3, URhIn 5andU 2RhIn8have been preparedusing In\nself-flux method. High-quality elements U (purified by SSE [11]), Rh (3 N5)\nand In (5N) were used. The starting composition of U:In = 1:10, U:Rh:I n\n= 1:1:25 and U:Rh:In = 2:1:25 were placed in alumina crucibles in order to\nobtain UIn 3, URhIn 5and U 2RhIn8, respectively. The crucibles were further\nsealed in evacuated quartz tubes. The ampoules were then heated up to\n950◦C, kept at this temperature for 10 h to let the mixture homogenize\nproperly and consequently cooled down to 600◦C in 120 h. After decanting,\nplate-like single crystals of U 2RhIn8(URhIn 5) with typical dimensions of\n1×0.5×0.3 mm3(1×1×0.5 mm3) were obtained. In case of UIn 3, however,\nourgrowattemptsledtogrowthofsingle crystalsoftypical masse s<0.1 mg.\n3The single crystal of UIn 3(2×2×2 mm3) suitable for the bulk measurements\nwas obtained as a by-product of the URhIn 5synthesis.\nHomogeneity and chemical composition of the single crystals were co n-\nfirmedbyscanningelectronmicroscope(TescanMIRAILMHSEM)eq uipped\nwith energy dispersive X-ray analyzer (Bruker AXS). The crystal structures\nwere determined by single crystal X-ray diffraction using X-ray diffractome-\nter Gemini, equipped with an Mo lamp, graphite monochromator and an\nMo-enhance collimator producing Mo K αradiation, and a CCD detector At-\nlas. Absorption correction of the strongly absorbing samples ( µ∼50 mm−1)\nwas done by combination of the numerical absorption correction ba sed onthe\ncrystal shapes and empirical absorption correction based on sph erical har-\nmonic functions, using the software of the diffractometer CrysAlis PRO. The\ncrystal structures were solved by SUPERFLIP [13] and refined b y software\nJana2006 [14].\nThe electrical resistivity measurements were done utilizing standar d four-\npoint method down to 2 K in a Physical Property Measurement Syste m\n(PPMS). The specific heat measurements down to 400 mK were carr ied out\nusing the He3 option. Magnetization measurements were performe d in a su-\nperconducting quantum interference device (MPMS) from 2 to 300 K/400 K\nand magnetic fields up to 7 T.\nTo investigate the effect of hydrostatic pressure on the transitio n temper-\natureTN, we measured the temperature dependence of electrical resistiv ity\nusingadouble-layered(CuBe/NiCrAl) piston-cylindertypepressur ecellwith\nDaphne 7373 oil as the pressure-transmitting medium [15, 16]. Pres sures up\nto 3.2 GPa were reached.\n4In order to acquire information about formation of magnetic momen ts in\nU2RhIn8, we applied the theoretical methods based on the density function al\ntheory. The electronic structure and magnetic moments were calc ulated us-\ning the latest version of APW+lo WIEN2k code [17]. The 5 felectrons form\nthe Bloch states with non-integer occupation number. The spin-or bit cou-\npling was included using second-order variational step [18]. Since we f ound\nthesmallervalueofthetotalmagneticmomentthanexpected, wea ppliedthe\nLSDA+U method [17] and tuned the effective U to obtain the required to-\ntal magnetic moment. The electronic structure calculations were p erformed\nat experimental equilibrium. The calculations were ferromagnetic fo r the\nsake of simplicity, since we have no information about the character of the\nantiferromagnetic ground state.\n3. Results and discussion\nThe obtained diffraction patterns revealed the Ho 2CoGa8- (HoCoGa 5)-\ntype structure (P4/mmm) for U 2RhIn8(URhIn 5). Table 1 summarizes the\nlattice parameters, atomic coordinates and the equivalent isotrop ic displace-\nment parameters Ueq. The refinement parameters of the obtained data for\nU2RhIn8equalRint= 0.076,R[F2>3σF2] = 0.035, the largest peak/hole in\ndifferenceFouriermap∆ ρmax= 5.84e˚A−3/∆ρmin=−4.04e˚A−3. ForURhIn 5:\nRint= 0.041,R[F2>3σF2]=0.022,∆ ρmax= 2.55e˚A−3/∆ρmin=−2.01 e˚A−3.\nThe temperature dependence of the specific heat C(T) divided by tem-\nperature for U 2RhIn8and URhIn 5is presented in Fig. 1; a clear λ-shaped\nanomaly at TN= 117 K and TN= 98 K, respectively, indicates a second-\norder phase transition in both materials. Closer observation of the C(T) vs.\n5Table 1: Lattice parameters, fractional atomic coordinates and is otropic or equivalent\nisotropic displacement parameters for U 2RhIn8and URhIn 5.\nU2RhIn 8Atomx y z Uiso*/Ueq\na= 4.6056(6) ˚A U 0.5 0.5 0.30883(7) 0.0059(3)\nc= 11.9911(15) ˚A Rh 0.5 -0.5 0 0.0078(6)\nIn(1) 0.5 0 0.5 0.0080(5)\nIn(2) 0.5 0 0.12263(11) 0.0091(4)\nIn(3) 0 0 0.30916(14) 0.0079(4)\nURhIn 5\na= 4.6210(5) ˚A U 0 0 0 0.00474(19)\nc= 7.4231(7) ˚A Rh 1 0 0.5 0.0059(4)\nIn(1) 0.5 0 0.30179(11) 0.0078(2)\nIn(2) 0.5 0.5 0 0.0076(3)\nTcurveofU 2RhIn8reveals asmallanomalyat T∼100 K, which arisesfrom\na tiny amount of URhIn 5. The magnitude of the phonon contribution to the\nspecific heat for both ternary compounds was determined from a C/T=γ+\nβT2fittothedata(fitinterval 1K <T<10K).FortheU 2RhIn8compound,\nthe value of Sommerfeld coefficient yields γ= 47 mJ·mol−1·U·K−2and the\nβcoefficient equals to 3.4 mJ ·mol−1·K−4which corresponds to a Debye tem-\nperature TD= 150 K. Sommerfeld coefficient of URhIn 5equals to 60.7\nmJ·mol−1·U·K−2while the βcoefficient yields the value of 3.3 mJ ·mol−1·K−4\ncorresponding to the Debye temperature of 165 K. The values for URhIn 5\nare close to those presented recently [24].\nFig. 1 (b) represents data in applied magnetic field of 9 T for U 2RhIn8\n6\u0002\u0001\u0002\u0003\u0004\u0005 \u0006\u0005 \u0007\u0005\u0005 \u0007\u0006\u0005 \b\u0005\u0005\u0003\t\u0002\u0001\u0002\n\u000b\f\r\u000e\u000f\u0007\u000b\u0003\u000f\b\u0004\n\u0005\u0007\b\u0010\u0011\u0006\n\u0002\u0001\u0002\u0003\u0004\u0007\u0005\u0006 \u0007\u0007\u0005 \u0007\u0007\u0006 \u0007\b\u0005\u0003\t\u0002\u0001\u0002\n\u000b\f\r\u000e\u000f\u0007\u000b\u0003\u000f\b\u0004\n\u0010\u000b\u0012\u0010\u000b\u0013\u0011\u000b\b\u0011\u000b\u0006\u0014\b\u0015\u0016\u0017\u0018\u0019\n\u0004\u0001\u001a\u001a\u0001\u001b\u0005\u0005\u0007\u001c \u0002\u0005\u0004\u0001\u001d\u0001\u0005\u0001\u001e\n\u0002\u001f\u0002\u001f\u0014\b\u0015\u0016\u0017\u0018\u0019\n\u0014\u0015\u0016\u0017\u0018\u0006\n\u0002\u0001\u0002\u0003\u0004\u0019\u0019 \u0013\b \u0013\u0012 \u0007\u0005\u0005\u0007\u000b\u0013\b\u000b\u0005\b\u000b\u0007\b\u000b\b\u0003\t\u0002\u0001\u0002\n\u000b\f\r\u000e\u000f\u0007\u000b\u0003\u000f\b\u0004\n\u0014\u0015\u0016\u0017\u0018\u0006\u0005\u0001\u001e\n\u0013\u0001\u001e\n\u0002 \u0004\u0002!\u0004\u0004\u0001\u001a\u001a\u0001\u001b\u0005\u0005\u0007\u001c\u0005\u0001\u001e\n\u0013\u0001\u001e\n\u0002\"\u0004\nFigure 1: Temperature dependence of the specific heat divided by t emperature (a). The\ntransition into the magnetically ordered state at TN= 117 K ( TN= 98 K) for U 2RhIn8\n(URhIn 5) is marked by vertical dashed lines. Comparison of C/Tfor U2RhIn8(b) and\nURhIn 5(c) in zero and 9 T magnetic field, respectively, applied along the [001] axis.\nalong the [001] axis. The direct comparison with the zero field measur ement\nreveals that TNis almost unaffected within experimental uncertainty. Equiv-\nalent behavior isobserved, Fig. 1(c), inthecaseofURhIn 5. Similar response\nto magnetic field is observed in the structurally related Ce-based co mpound,\nCeRhIn 5, where the transition temperature TNtends to be rather insensitive\nto the application of magnetic field along the same direction [25].\nFig. 2 shows the temperature dependence of the χ(T) and 1/ χ(T) of\nU2RhIn8and URhIn 5in magnetic field oriented along [100] and [001] direc-\ntions and [110] and [001] in the case of UIn 3. The analysis of the χ(T) data\nlead to the determination of N´ eel temperatures as proposed by F isher [20];\nthe maximum of the ∂(χ(T))/∂Tcurve. The behavior of the susceptibility\ncurves resembles the one shown for URhIn 5and UIn 3[19, 5], therefore we\n7conclude that the phase transition drives the compound into an ant iferro-\nmagnetic state. The magnetic susceptibility increases in all compoun ds with\ndecreasing temperature and this increase is much pronounced for the [001]\ndirection in the ternary compounds. The maximum value of susceptib ility\nis reached at Tχmax= 130 K for UIn 3(consistently with literature [5]), at\nTχmax= 160 K for URhIn 5[24] and at Tχmax= 150 K for U 2RhIn8. Such be-\nhavior was previously observed in several different uranium compou nds [21]\nand it is generally supposed that this character of the susceptibility curves\nis associated with antiferromagnetic correlations when approachin gTN. The\nvalue ofTχmaxfor both studied ternary compounds is the highest among ura-\nnium compounds up to our knowledge (UPd 2Al3:Tχmax= 30 K; URu 2Si2:\nTχmax= 60 K [22]).\nTable 2: N´ eel temperatures, effective magnetic moments obtaine d from Curie-Weiss fits\nand paramagnetic Curie temperatures for different orientation of magnetic fields for UIn 3\n([001] and [110] orientation), U 2RhIn8and URhIn 5.\nH/bardbl[001]UIn3U2RhIn8URhIn 5\nTN(K) 88 117 98\nµeff(µB/U) 3.16 3.45 3.6\nθp(K)−300−240−400\nH/bardbl[110]\nµeff(µB/U) 3.15 - -\nθp(K)−310 - -\nIn the vicinity of TN= 117 K (98 K, 88 K) for U 2RhIn8(URhIn 5, UIn3),\na sharp drop of the magnetic susceptibility is observed in agreement with\n8literature [5, 24, 19]. This rapid decrease is again more pronounced in the\n[001] direction in the studied ternary compounds, pointing to the fa ct that\nthe magnetic moments probably lie in this direction in the ordered stat e.\nAt low temperatures ( T∼40 K) the susceptibility reaches its minimum\nvalue and rises up again. This increase is negligible in the case of URhIn 5\nand U 2RhIn8; however, it plays a dominant role in case of UIn 3. Previous\nstudies of UIn 3[5] revealed similar behavior to that one presented for ternary\ncompounds. This effect may have intrinsic nature or it could be also co n-\nnected with a non-negligible amount of paramagnetic impurities, requ iring\nfurther investigations.\nFrom the character of the 1/ χ(T) curves above the N´ eel temperature it is\nevident, that the behavior of ternary compounds does not follow t he Curie-\nWeiss law in H/bardbl[100] direction due to magnetocrystalline anisotropy. The\nlinear behavior appears probably above 400 K, as it is shown i.e. for UP tGa5\n[22], whichwewereabletoconfirmexperimentally inthecaseofU 2RhIn8(see\nFig. 2 (a)). Such recovery of Curie-Weiss law at high temperatures indicates\na localized nature of 5 felectrons. Thus, a crossover of the 5 felectrons\nfrom a low-temperature itinerant nature to a high-temperature lo calized one\nis observed. This crossover effect is characteristic for many heav y fermion\ncompounds such as UPt 3, UPd 2Al3and URu 2Si2[22].\nIn accordance with previous arguments, we applied the Curie-Weiss law\nin theH/bardbl[001] direction for U 2RhIn8and URhIn 5, and in [001] and [110]\ndirection in the case of UIn 3. We obtained qualitative values of the effective\nmagnetic moments, summarized in table 2, that we were able to compa re\nwith those previously obtained [24, 19, 5].The large negative values of para-\n9magnetic Curie temperatures reflect the huge uniaxial magnetocr ystalline\nanisotropy in U 2RhIn8and URhIn 5induced by anisotropic 5 f-ligand hy-\nbridization.\nThe magnetic field dependence of magnetization (see Fig. 3) of U 2RhIn8\nwas measured at T= 4 K for magnetic field oriented along the [001] and\n[100] directions. Both magnetization curves reveal linear charact er up to\n7 T; the [100] axis is almost twice higher than the magnetization in the\nother direction, which resembles the behavior of its more 2D counte rpart\n[24, 19]. Using a relation kBTχmax≃µBHc[21], where Tχmax= 150 K defines\nthe position of the maximum of the magnetic susceptibility data, and Hcis\nthe critical magnetic field of metamagnetic transition, we obtain a va lue of\nHc= 220 T for U 2RhIn8. This extremely large value explains the absence of\nmetamagnetic transition in our experimental data.\nFig. 4 (a) shows the temperature dependence of the electrical re sistivity\nof U2RhIn8for electrical current japplied along the [100] and [110] axes.\nThe room temperature resistivity equals 320 µΩ·cm along the [100] direction\nand is only slightly lower for the [110] direction (310 µΩ·cm). The resid-\nual resistivity ratio (RRR) exceeds 500 being a sign of a sample of ver y\nhigh quality. The electrical resistivity shows a monotonous decreas e down\nto the value of the transition temperature TN. Near the transition temper-\natureTN= 117 K, a tiny kink is observed, accompanied by a second-order\nphase transition and a formation of a gap at the Fermi surface. Su bse-\nquently, the resistivity decreases rapidly with decreasing tempera ture. The\nlow-temperature part of the electrical resistivity (2 K <Tfit<30 K) can\nbe fitted well using the equation appropriate for an energy gap (∆) antifer-\n10\u0002\u0001\u0002\u0003\u0004\u0005 \u0006\u0005\u0005 \u0007\u0005\u0005 \b\u0005\u0005 \t\u0005\u0005\u0002\u0001\u0002\u0006\u0005\n\u000b\u0001\f\b\r\f\u000e\u000f\n\u0006\u0004\n\u0005\r\u0007\u0005\r\t\u0005\r\u0010\u0005\r\u0011\u0006\r\u0005\n\u0006\u0012\u0002\u0001\u0002\u0006\u0005\u000b\u0001\f\u000e\u000f\u0006\r\f\n\b\u0004\n\u0006\r\u0007\u0006\r\u0013\u0006\r\u0011\u0007\r\u0006\u0007\r\t\u0007\r\u000b\n\u0003\u0001\u0014\u0014\u0001\u0015\u0005\u0005\u0006\u0016\n\u0003\u0001\u0014\u0014\u0001\u0015\u0006\u0005\u0005\u0016\u0001\u0017\u0007\u0018\u0019\u001a\u001b\u0011\n\u0003\u0005\u0003\u0001\u001c\u0001\u0006\u0001\u001d\n\u0002\u0002\u0006\u0005\n\u0011\u0001\f\u000e\u000f\n\u0006\r\f\b\u0004\n\u0006\r\u0007\u0006\r\u0011\u0007\r\t\b\r\u0005\b\r\u0010\n\u0006\u0012\u0002\u0001\u0002\u0006\u0005\u000b\f\u000e\u000f\u0006\r\f\n\b\u0004\n\u0007\r\u0005\u0007\r\u0013\b\r\u0005\b\r\u0013\t\r\u0005\t\r\u0013\n\u0003\u0001\u0014\u0014\u0001\u0015\u0006\u0005\u0005\u0016\u0003\u0001\u0014\u0014\u0001\u0015\u0005\u0005\u0006\u0016\u0017\u0018\u0019\u001a\u001b\u0013\n\u0003\u0002\u0003\u0001\u001c\u0001\u0007\u0001\u001d\n\u0002\u001e\n\u0002\u0001\u0002\u0003\u0004\u0005 \u0006\u0005\u0005 \u0007\u0005\u0005 \b\u0005\u0005\u0002\u0001\u0002\u0006\u0005\n\u0011\u0001\f\b\r\f\u000e\u000f\n\u0006\u0004\n\u0007\r\t\u0007\r\u0011\b\r\u0007\b\r\u0010\n\u0006\u0012\u0002\u0001\u0002\u0006\u0005\u000b\f\u000e\u000f\u0006\r\f\n\b\u0004\n\u0007\r\u0011\b\r\u0005\b\r\u0007\b\r\t\b\r\u0010\b\r\u0011\u0017\u001a\u001b\b\n\u0003\u0005\u0003\u0001\u001c\u0001\u0006\u0001\u001d\n\u0002\u001e\u0002\u001e\n\u0002\u001f\u0004\n\u0002 \u0004\n\u0002!\u0004\u0003\u0001\u0014\u0014\u0001\u0015\u0005\u0005\u0006\u0016\n\u0003\u0001\u0014\u0014\u0001\u0015\u0006\u0006\u0005\u0016\nFigure 2: Temperature dependence of χ(T) and 1/ χ(T) and Curie-Weiss fit of (a)\nU2RhIn8, (b) URhIn 5and (c) UIn 3for magnetic field oriented along the [100] and [001]\ndirection in the case of U 2RhIn8and URhIn 5along the [100] and [110] direction for UIn 3,\nrespectively. Vertical dashed line marks the transition temperatu reTN.\n11\u0002\u0002\u0002\u0001\u0003\u0004\u0005\u0002 \u0006 \u0007 \b\u0002\u0001\u0003\u0002\t\n\u000b\f\r\f\u0005\n\u0002\f\u0002\u0002\u0002\f\u0002\u000e\u0002\f\u0002\u0006\u0002\f\u0002\u000f\n\u0002\u0001\u0010\u0010\u0001\u0011\u000e\u0002\u0002\u0012\n\u0002\u0001\u0010\u0010\u0001\u0011\u0002\u0002\u000e\u0012\u0013\u0006\u0014\u0015\u0016\u0017\u0018\n\u0003\u0001\u0019\u0001\u0007\u0001\u001a\nFigure3: Magneticfield dependenceofmagnetizationtakenat4Kof U2RhIn8inmagnetic\nfield oriented along the [100] and [001] axes.\nromagnet with an additional Fermi-liquid term, being: ρ(T) =ρ0+AT2+\nDT(1 + 2T/∆)exp(−∆/T) [26]. Optimal fitting of ρ[100]gives a residual\nresistivity value ρ0= 0.56µΩ·cm, an electron-electron scattering coefficient\nofA= 0.006 µΩ·cm·K−2, an electron-magnon and spin-disorder scattering\nprefactor D= 1.1µΩ·cm·K−1and ∆ = 118 K. Our fit yielded similar values\nof the parameters for current applied along the [110] direction.\nFig. 4 (b) summarizes the overall temperature dependence of the elec-\ntrical resistivity of URhIn 5for electrical current japplied along the [100],\n[110] and [001] axes. The room temperature resistivity equals 180 µΩ·cm\nin the basal plane and is only slightly lower for the [001]-axis direction\nρ[001]= 170µΩ·cm. Note that ρ[100](300 K) is almost the same as reported\nby Matsumoto and co-workers [24]. Similar to U 2RhIn8, the RRR ∼200\n12\u0002\u0001\u0002\u0003\u0004\u0002\u0001\u0002\u0003\u0004\u0005\u0005\u0006\u0004\n\u0007\b\u0007\t\u0007\u0007\t\b\u0007\n\u0007\u0007\n\b\u0007\u000b\u0007\u0007\f\n\r\u000e\u000f\u0010\u0011 \r\r\r\u0001\u0012\u0001\b\u000b\u0007\n\r\r\r\u0001\u0012\u0001\b\u0007\u0007\n\u0002\u0013\n\u0002\u0001\u0002\u0003\u0004\u0007 \b\u0007 \t\u0007\u0007 \t\b\u0007 \n\u0007\u0007 \n\b\u0007 \u000b\u0007\u0007\u0002\u0001\u0002\u0003\u0004\u0005\u0005\u0006\u0003\n\u0007\b\u0007\t\u0007\u0007\t\b\u0007\n\u0007\u0007\u0003\u0001\u0004\u0004\u0001\u0014\t\u0007\u0007\u0015\n\u0003\u0001\u0004\u0004\u0001\u0014\t\t\u0007\u0015\n\u0003\u0001\u0004\u0004\u0001\u0014\u0007\u0007\t\u0015\n\u0002\u0013\f\r\u000e\u000f\u0010\b\n\u0005\u0005\u0005\u0001\u0012\u0001\n\u0007\u0007\u0005\u0005\u0005\u0001\u0012\u0001\n\n\u0007\n\u0016\u0007 \t\u0007\u0007 \t\t\u0007 \t\n\u0007\t\u000b\b\t\b\u0007\t\u0017\b\t\u0011\u0007\n\u0002\u0001\u0002\u0003\u0004\u0005\u0005\u0006\u0004\n\u0002\u0001\u0002\u0003\u0004\u0002\u0018\u0004\n\u0002\u0019\u0004\nFigure 4: Temperature dependence ofthe electrical resistivityfo r electrical current applied\nalong [100] and [110] directions is shown for U 2RhIn8(a). The temperature dependence\nof the electrical resistivity for current applied along the [100], [110] and [001] directions\nfor URhIn 5compound (b). The inset in Fig. 4 (b) shows the transition in detail. Th e\narrows mark the onset of the antiferromagnetic transition at TN= 117 K ( TN= 98 K)\nfor U2RhIn8(URhIn 5). One legend applies for both figures.\n13of the URhIn 5single crystals confirms their high quality. The data mani-\nfest distinct anomalies with onset at around T∼100 K for ρ[001]and at a\nslightly lower temperature T= 98 K for ρ[100]andρ[110], respectively, which\nis reminiscent of the N´ eel temperature anomaly for ρ(T) in pure Cr [31] -\na spin-density-wave (SDW) antiferromagnet. Accordingly, the on set marks\nTNand the increase in resistivity results from opening of the SDW gap.\nBelowTN, the resistivity drops down rapidly in all directions. The low-\ntemperature part was fitted according to similar formula as for U 2RhIn8\nwith results ρ0= 1µΩ·cm,A= 0.013µΩ·cm·K−2,D= 0.35µΩ·cm·K−1and\n∆ = 82 K. Our fit yielded similar values of the parameters for current a long\nthe [110] direction. However, in the case of j/bardbl[001] we obtained a somewhat\nhigher value of the gap energy ∆ = 119 K.\nAfter evaluating the data from specific heat and electrical resistiv ity mea-\nsurements of U 2RhIn8and URhIn 5, the Kadowaki-Woods [27] ratio A/γ2\ncould be calculated, where Ais the coefficient of the quadratic term in\nthe temperature dependence of electrical resistivity. The value o f 2·10−6\nµΩcm·(mol·K/mJ)2forU2RhIn8and3.6·10−6µΩcm·(mol·K/mJ)2forURhIn 5\nwasobtained, beingoneorderofmagnitudelower thaninthecommon heavy-\nfermion systems [28].\nTo investigate the effect of hydrostatic pressure, we performed electrical\nresistivity measurement of U 2RhIn8up to 3.2 GPa. The N´ eel temperature\nincreases gradually with hydrostatic pressure, as is shown in the T-pphase\ndiagraminFig. 5. Apossibleexplanationofthepositivepressuredepe ndence\nofTNis given by the spin-fluctuation theory of an itinerant 5 felectron sys-\ntem alongside with the Hubbard model [29, 30]. According to this scen ario,\n14hydrostatic pressure induces an increase in the hybridization betw een 5fand\nconduction electrons, which strengthens the exchange coupling Jbetween U\nions. On the other hand, it also decreases the 5 fmagnetic moment at the\nuranium site. The value of TNchanges from 117 K to 128 K at ambient\npressure and 3.2 GPa, respectively, with the rate of 5.4 ±0.9 K·GPa−1. This\nslope corresponds well to the pressure evolution of TNin URhIn 5[24]. The\ndetail of the resistivity curve near TNfor ambient pressure and for 3.2 GPa\nwith electrical current applied along the [001] axis is shown in the inset of\nFig. 5.\n\u0002\u0003\u0004\u0005\u0006\u0007\n\b\t\n\u000b\f\r\u0006\t\u000e\t\u0001\u000f\u0010\u0011\n\u0012\u0013\u0014\u0001\u0015\u0004\u0001\u0016\u0017\u0001\u0012\u0013\u0014\u0001\u0015\u0005\u0001\n\u0002\u0001\u0018\u000f\u0010\u0011\u0019\u0004 \u0005 \t \u001a\u0003\u001b\u0001\u0018\u001c\u0019\n\u001d\u0004\u0006\u0004\u001e\u0004\u0005\u0004\u0004\u0005\u0005\u0004\u0005\t\u0004\u0005\u001a\u0004\n\b\t\n\u000b\f\r\u0006\n\u0003\u0001\u0018\u001c\u0019\u0005\u0005\u001d \u0005\t\u0015 \u0005\u001a\u001f\u0002\u0001\u0018\u0003\u0004\u0005 !\u0019\n\u0005\u0006\u0004\t\u0004\u0004\t\t\u0004\t\"\u0004\u0011!#$\u0007\r%\u0001&'\u0007\u0017\u0017('\u0007\n\u001a\u000e\t\u0001\u000f\u0010\u0011\u0004\u0001))\u0001*\u0005\u0004\u0004+\nFigure 5: Temperature-pressure phase diagram of U 2RhIn8. Inset shows the tempera-\nture dependence of the electrical resistivity for electrical curre nt applied along the [100]\ndirection at ambient pressure and at 3.2 GPa. Arrows indicate the tr ansition at TN.\nThe ternary compounds U nTX3n+2withT= Rh and X= In represent a\nsystem which offersthepossibility tostudy theeffect ofstructura l dimension-\n15ality on the magnetism of uranium layered structures. Dimensionality devel-\nops from 3D towards 2D when spanning the series: UIn 3(1-0-3)→U2RhIn8\n(2-1-8)→URhIn 5(1-1-5).In theother cases, either onlythecubic compound\n(USn3, UPb3[32]) or the cubic compound (UGa 3[33]) together with its 1-1-5\n(or 2-1-8) parent system ( T= Fe, Co, Ni, Pd, Ir, Pt [34, 35, 9, 36, 37]) are\nknown. The Ga-based compounds with T= Fe, Rh, which form both 1-1-5\nand 2-1-8 are paramagnetic. Since all the known U nRhIn3n+2compounds\norder antiferromagnetically, a TNvs.c/adiagram can be constructed (see\nFig. 6) in order to study the effect of structural dimensionality on t he mag-\nnetic ordering. The evolution of transition temperature does not f ollow the\ndimensionality sequence ”1-0-3 →2-1-8→1-1-5” as discussed by Cornelius\net al.[25]. In contrast to the behavior in cerium compounds [25], the N´ eel\ntemperatures ofU nRhIn3n+2reveal amonotonicevolutionwithrespect tothe\nc/aratio. A possible explanation of this discrepancy is given by the differe nt\ndriving microscopic mechanisms in the compounds. The microscopic me cha-\nnisms in Ce-based compounds are mostly RKKY-type while in the uraniu m\ncompounds the 5 f-ligand hybridization plays a substantial role. Moreover,\nthe cerium compounds order magnetically well below 10 K while the orde r-\ning temperature in the case of U-based compounds is at least an ord er of\nmagnitude higher.\nThe spin-polarized LSDA calculation splits the spin-up and spin-down\nbands with spin magnetic moment at uranium site 2.17 µB. Since strong\nmagnetocrystalline anisotropy is present, the spin-orbit coupling is also in-\ncluded into calculations. The calculated spin magnetic moment at ura-\nnium site decreases to MS= 1.744 µBand the orbital magnetic moment\n16\u0002\u0002\u0003\u0003\u0004\u0005 \u0003\u0004\u0006 \u0007\u0004\u0005 \u0007\u0004\u0006\u0004\b\u0001\t\n\u000b\n\f\u0005\r\u0005\u0003\u0005\u0005\u0003\u0003\u0005\u0003\u0007\u0005\n\u000e\u000f\u0010\u0011\u000e\u0012\u0013\u000f\u0010\u0006\u000e\u0007\u0012\u0013\u000f\u0010\f\nFigure 6: TNvs.c/adiagram depicting the relation of the structural dimensionality and\nmagnetism of uranium layered structures. The points are accompa nied by the structures\nof UIn 3(cubic), URhIn 5(tetragonal) and U 2RhIn8(tetragonal) compounds. Solid black\nline is a guide to the eye.\nML= -2.418 µBis antiparallel. Magnetic moments located at rhodium and\nindium sites are both smaller than 0.1 µB. Total uranium magnetic moment\nis|MT|= 0.674µB. From comparison with URhIn 5we expected larger total\nuraniummomentabout |MT|=1.6µB. Thus, thecorrelationmovement of5 f\nelectrons is not negligible. Therefore we used LSDA+U method to des cribe\ncorrelated movement of 5 felectrons. Tuning effective Hubbard Uwe have\nfound spin magnetic moment MS= 1.738µBand orbital magnetic moment\nML= -3.3µBproviding the total magnetic moment |MT|= 1.592µBfor\nmedium effective U= 1.3 eV. We are fully aware that such calculation loses\nits first-principle character on this level, but on the other hand, we showed\nthat these heuristically derived values of effective Uallow us to obtain valu-\n17able results.\n4. Conclusions\nSingle crystals of UIn 3, URhIn 5and the novel U 2RhIn8phase were syn-\nthesized using the In self-flux method and studied by means of magn etiza-\ntion, thermodynamic and transport measurements. The U 2RhIn8compound\nadoptstheHo 2CoGa8-typestructurewithlatticeparameters a= 4.6056(6) ˚A\nandc= 11.9911(15) ˚A. Measurements of specific heat and magnetization\nrevealed a second-order phase transition into the antiferromagn etic state at\nTN= 117 K. The compounds UIn 3and URhIn 5order antiferromagnetically\nbelowTN= 88K and98 K,respectively, inaccordance with literature[5, 24].\nElectrical resistivity measurement revealed the very high quality of the stud-\nied ternary compounds with RRR exceeding 200 and 500, respective ly. The\ntemperature dependence of magnetic susceptibility for URhIn 5and U 2RhIn8\nreveals strong magnetic anisotropy and suggests that both syst ems undergo\nan itinerant-localized crossover at high temperatures above 300 K , similar to\notheruranium-basedcompounds(UPd 2Al3,URu 2Si2[22]),includingUPtGa 5\n[22] from the same group of compounds. The application of hydrost atic pres-\nsure supports the robustness of the antiferromagnetic phase in both ternary\ncompounds with similar pressure coefficients [24]. Successful synth esis of\nU2RhIn8provides an opportunity among 5 fsystems to study the evolution\nof ground state properties depending on c/aratio. As is shown on Fig. 6\nthe evolution of TNis monotonic with respect to c/aratio. This behavior\nis in contrast with the cerium analogs [25] and will be a subject of fur ther\ninvestigation.\n185. Acknowledgement\nExperiments were performed in MLTL ( http://mltl.eu/ ), which is sup-\nported within the program of Czech Research Infrastructures ( project no.\nLM2011025). Our work was supported by the Grant Agency of the Charles\nUniversity (project no. 362214) and Czech Science Foundation (P roject\nP203/12/1201). Single crystal X-ray diffraction was performed in the De-\npartment of Structure Analysis, Institute of Physics ASCR.\n6. References\nReferences\n[1] H.H.Hill, Plutonium1970andotheractinides, AIME,NewYork(197 0),\np. 2\n[2] Y. Tokiwa, H. Harima, D. Aoki, S. Nojiri, M. Murakawa, K. Miyake,\nN. Watanabe, R. Settai, Y. Inada, H. Sugawara, H. Sato, Y. Haga , E.\nYamamoto and Y. ¯Onuki,J. Phys. Soc. Jpn 69 (2000) 1105-1112\n[3] G. W. Crabtree, B. D. DunlapandD. D. Koelling, Physica B 135 (1985)\n38-40\n[4] A. Hasegawa, J. Phys. Soc. Jpn. 53 (1984) 3929-3937\n[5] Y. Tokiwa, D. Aoki, Y. Haga, E. Yamamoto, S. Ikeda, R. Settai, A .\nNakamura and Y. ¯Onuki,J. Phys. Soc. 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Phys, Soc.\nJpn.71 (2002) 326-331\n[33] D. Kaczorowski, R. Hauser and A. Czopnik, Physica B 230-230 (1997)\n35-38\n[34] Y. Tokiwa, T. Maehira, S. Ikeda, Y. Haga, E. Yamamoto, A. Nak amura,\nY.¯Onuki, M. Higuchi and A. Hasegawa, J. Phys. Soc. Jpn 70 (2001)\n2982-2988\n[35] R. Wawryk, Z. Henkie, T. Cichorek, C. Geibel and F. Steglich, phys.\nstat. sol. (b) 232 (2002) R4-R6\n[36] Y. Tokiwa, S. Ikeda, Y. Haga, T. Okubo, T. Iizuka, K. Sugiyama , A.\nNakamura and Y. ¯Onuki,J. Phys. Soc. Jpn 71 (2002) 845-851\n[37] S. Ikeda, T. ¯Okubo, Y. Inada, Y. Tokiwa, K. Kaneko, T. D. Matsuda,\nE. Yamamoto, Y. Haga and Y. ¯Onuki,J. Phys.: Condens. Matter 15\n(2003) S2015-S2018\n22"    },    {        "title": "1501.03483v2.Origin_of_the_spin_reorientation_transitions_in__Fe___1_x__Co___x______2__B_alloys.pdf",        "content": "arXiv:1501.03483v2  [cond-mat.mtrl-sci]  3 Feb 2015Origin of the spin reorientation transitions in (Fe 1−xCox)2B alloysa)\nKirill D. Belashchenko,1Liqin Ke,2Markus D¨ ane,3Lorin X. Benedict,3Tej Nath Lamichhane,2,4Valentin\nTaufour,2,4Anton Jesche,2,4Sergey L. Bud’ko,2,4Paul C. Canfield,2,4and Vladimir P. Antropov2\n1)Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA\n2)Ames Laboratory, U.S. Department of Energy, Ames, Iowa 5001 1, USA\n3)Lawrence Livermore National Laboratory, Livermore, Calif ornia 94550, USA\n4)Department of Physics and Astronomy, Iowa State University , Ames, Iowa 50011,\nUSA\n(Dated: 5 July 2018)\nLow-temperature measurements of the magnetocrystalline aniso tropy energy Kin (Fe1−xCox)2B alloys are\nreported, and the origin of this anisotropy is elucidated using a first -principles electronic structure analysis.\nThe calculated concentration dependence K(x) with a maximum near x= 0.3 and a minimum near x= 0.8\nis in excellent agreement with experiment. This dependence is traced down to spin-orbital selection rules and\nthe filling of electronic bands with increasing electronic concentratio n. At the optimal Co concentration, K\ndepends strongly on the tetragonality and doubles under a modest 3% increase of the c/aratio, suggesting\nthat the magnetocrystalline anisotropy can be further enhanced using epitaxial or chemical strain.\nMagnetocrystalline anisotropy (MCA) of a magnetic\nmaterial is one of its key properties for practical applica-\ntions, large easy-axis anisotropy being favorable for per-\nmanent magnets.1Intelligent search for new materials\nrequires understanding of the underlying mechanisms of\nMCA. This can be particularly fruitful for substitutional\nalloys whose properties can be tuned by varying the con-\ncentrations of their components. The analysis is often\nrelatively simple in insulators, where MCA is dominated\nby single-ion terms which can be deduced from crystal-\nfield splittings and spin-orbital (SO) selection rules. In\ncontrast, in typical metallic alloys the band width sets\nthe largest energy scale, and MCA depends on the de-\ntails of the electronic structure.\nThe (Fe 1−xCox)2B solid solution2–6(space group\nI4/mcm7) is a remarkable case in point. Fe 2B has a\nfairly strong easy-plane MCA, and Co 2B, at low temper-\natures, a small easy-axis MCA. However, the alloy has a\nsubstantial easy-axis MCA around x= 0.3,2making it\na potentially useful rare-earth-free8permanent magnet.\nAtx≈0.5 the MCA again turns easy-plane, peaks at\nx= 0.8, and then turns easy-axis close to x= 1. These\nthree spin reorientation transitions must be related to\nthe continuous evolution of the electronic structure with\nconcentration. The goal of this Letter is to elucidate the\norigin of this rare phenomenon.\nFirst, we report the results of experimental mea-\nsurements at low temperatures. Single crystals of\n(Fe1−xCox)2B were grown from solution growth out of\nan excess of (Fe,Co) which was decanted in a centrifuge.9\nThe single crystals were grown as tetragonal rods which\nwerecutusingawiresawtogivethemtheshapeofarect-\nangular prism. The demagnetization factor was calcu-\na)This article has been accepted by Applied Physics\nLetters. After it is published, it will be found at\nhttp://scitation.aip.org/content/aip/journal/apl .lated using Ref. 10. Field-dependent magnetization mea-\nsurements were performed in a Quantum Design MPMS\nat 2 K in fields up to 5.5 T. The MCA energy Kwas\ndetermined as the area between the two magnetization\ncurves, with the field parallel and perpendicular to the c\naxis, taken at the same temperature.6The results shown\nin Fig. 1 (blue stars) measured at 2 K confirm the non-\nmonotonic concentration dependence, in good agreement\nwith the measurements at 77 K from Ref. 2.\nFIG. 1. Calculated (black circles) and experimental (blue\nstars) MCA energy Kin (Fe 1−xCox)2B alloys. Gray curve:\nKSO. The other lines show the spin decomposition Kσσ′.\nDensity-functional calculations using several different\nmethods showthatthe choiceofthe exchange-correlation\npotential and other computational details strongly affect\nthe calculated MCA in Fe 2B and Co 2B. We have ascer-\ntained that this sensitivity is largely due to the variation\nof the exchange splitting, which controls the position of\nthe Fermi level relative to the minority-spin bands. The\nsystematic variation of MCA with xis also controlled\nby the Fermi level shift. This continuous variation is,\ntherefore, reliably predicted as long as the end points are\ncorrectly fixed using experimental input.2\nThe results reported below were obtained using the\nGreen’s function-based formulation of the tight-binding\nlinear muffin-tin orbital (GF-LMTO) method.11Substi-\ntutional disorder was treated using our implementation\nof the coherent potential approximation (CPA),12with\nthe SO coupling included perturbatively as described in\nthe supplementary material.13\nThe lattice constants and the internal coordinates are\nlinearly interpolated between the experimental data for\nthe end compounds extrapolated to zero temperature:2\na= 5.109 and 4.997 ˚A,c= 4.249 and 4.213 ˚A, and\nu= 0.166 and 0.168 for Fe 2B and Co 2B, respectively.\nThe exchange and correlationare treated within the gen-\neralized gradient approximation (GGA).14\nThe correct exchange splitting at the end points can\nbe enforced by using the experimental data2–6for the\nmagnetization M: 1.9µB/Fe in Fe 2B and 0.76 µB/Co\nin Co2B. In Fe 2B it is only slightly overestimated, but\nin Co2B it is much too large at about 1.1 µB/Co in all\ndensity-functional calculations. The relatively small spin\nmoment of Co indicates a pronounced itinerant charac-\nter of magnetism in Co 2B, which tends to be sensitive to\nquantum spin fluctuations. Therefore, we introduced a\nscaling factor for the local part of the effective magnetic\nfield in the GGA functional for the Co atoms and ad-\njusted it to match the experimental magnetization. The\nresultingscalingfactorof0.80wasthenusedforCoatoms\nat all concentrations. The spin moments on different\natoms and the total spin magnetization obtained in this\nway are shown in Fig. 2. While the Fe spin moment is\nalmost constant, the Co spin moment declines with x.\nMoreover, this decline accelerates at x>∼0.6, which is in\nexcellent agreement with experimental data.3\nFIG. 2. Spin moments on different atoms and the total spin\nmagnetization Mper transition-metal atom.\nThe MCA energy Kwas obtained by calculating the\nsingle-particle energy difference for in-plane and out-of-\nplane orientationsofthe magnetization while keeping the\nLMTO charges fixed at their self-consistent values found\nwithout SO coupling. A uniform mesh of 303points in\nthe full Brillouinzoneprovidedsufficientaccuracyforthe\nkintegration. We have verified that the values of Kforpure Fe 2B and Co 2B agree very well with Hamiltonian\nLMTO results. The concentration dependence of Kis\nshown in Fig. 1. The agreement with low-temperature\nexperimental data is remarkably good, suggesting that\nthe electronic mechanisms of MCA are correctly cap-\ntured in the calculations. If the spin moment of Co is\nnot corrected by scaling the exchange-correlation field,\nthe downward trend in Kat the Co-rich end continues\ntolargenegativevaluesin disagreementwith experiment.\nIn 3dsystems the SO band shifts are usually well\ndescribed by second-order perturbation theory, except\nperhaps in small regions of the Brillouin zone. Conse-\nquently, when MCA appears in second order in SO cou-\npling (as in the tetragonal system under consideration),\nthe anisotropy of the expectation value of the SO opera-\ntor ∆ESO=/angbracketleftVSO/angbracketrightx−/angbracketleftVSO/angbracketrightzis approximately equal to\n2K.13(Herexorzshows the orientation of the magne-\ntization axis.) We therefore denote KSO= ∆ESO/2 and\nuse the expression 2 /angbracketleftSL/angbracketright=/angbracketleftLz′/angbracketright↑↑−/angbracketleftLz′/angbracketright↓↓+/angbracketleftL+/angbracketright↓↑+\n/angbracketleftL−/angbracketright↑↓toseparatethecontributionsto Kbypairsofspin\nchannels. Here we use Lz′to denote the component of\nLparallel to the magnetization axis, to avoid confusion\nwith the crystallographic zaxis;L±are the usual linear\ncombinationsofthe othertwo (primed) components of L.\nThe contributions Kσσ′toKSOare accumulated as en-\nergy integrals taking into account the energy dependence\nof the SO coupling parameters. The results for KSOand\nKσσ′are shown in Fig. 1. First, we see that KSOis close\ntoK, confirmingthevalidityofthisanalysis. Second, the\nnonmonotonic concentration dependence of Kis almost\nentirely due to K↓↓forx<∼0.7. While K↑↑is sizeable,\nit depends weakly on xin this region. Additional discus-\nsion about the spin decomposition of MCA is provided\nin the supplementary material.13\nLet us now analyzethe electronic structure and the de-\ntails of SO coupling-induced mixing for the minority-spin\nelectrons. To resolvethe minority-spincontribution to K\nby wave vector k, we calculated the minority-spin spec-\ntral function in the presence ofSO coupling and found its\nfirst energy moment at each k. Fig. 3 shows the differ-\nence of these integrals for magnetization along the xand\nzaxes at three key concentrations: pure Fe 2B (x= 0),\nthe maximum of K(x= 0.3) and its minimum ( x= 0.8).\nWe have checked that the Brillouin zone integral of the\nk-resolved MCA energy (summed up over both spins)\nagrees almost exactly with the value of Kcalculated in\nthe usual way.\nFig. 3 shows that the MCA energy accumulates over a\nfairly large part of the Brillouin zone. At x= 0 negative\ncontributions to Kdominate over most of the Brillouin\nzone. At x= 0.3 both positive and negative contribu-\ntions are small. At x= 0.8 there are regions with large\npositive and large negative contributions. Overall, it ap-\npears that the most important contributions come from\nthe vicinity of the ΓXM ( kz= 0) plane and from the\nvicinity of the ΓH ( kx=ky= 0) line.\nThe partial minority-spin spectral functions for the\ntransition-metal site are displayed in Fig. 4 (panels (a)-3\nFIG. 3. Brillouin zone map of the k-resolved minority-spin contribution to Kat: (a)x= 0, (b) x= 0.3, (c)x= 0.8. Half of\nthe Brillouin zone is shown; the top face of the plot is kz= 0. Points H (same as M) and X are shown in panel (b). The color\nintensity indicates the magnitude of negative (blue) and po sitive (red) values.\n(c)) along the important high-symmetry directions for\nthe same three concentrations. The coloring in this fig-\nure resolves the contributions from different 3 dorbitals.\nAtx= 0 the spectral function resolves the conventional\nelectronicbandsofpureFe 2B(animaginarypartof0.004\neV is added to the energy to acquire them). At x= 0.3\nandx= 0.8 substitutional disorder broadens the bands\nby a few tenths of an electronvolt, but their identity is\nin most cases preserved. Thus, we will discuss the SO-\ninduced band mixing in the alloy, bearing in mind that\nband broadening should reduce the values of MCA at\nintermediate x.\nAs we have learned above, the dominant concentration\ndependence of Kcomes from the /angbracketleftLz′/angbracketright↓↓term, where z′\nis the magnetization axis. The electronic states on the\nwhole ΓXM plane can be classified as even or odd under\nreflection z→ −z. Even (odd) states have m= 0,±2\n(m=±1) character and appear red and green (blue) in\nFig. 4. States of different parity do not intermix on this\nplane in the absence of SO coupling, as is clearly seen\nin Fig. 4a. The selection rules for SO coupling of the\nminority-spin states follow from the definite parity of the\ncomponents of ˆLunder reflection. ˆLz(even; relevant for\nM/bardblz) only mixes states of the same parity on the ΓXM\nplane,ormoregenerallyorbitalsofthesame m. Coupling\nbetween states of the m=±2 character (red) is stronger\ncompared to states of the m=±1 character (blue). In\ncontrast, ˆLx(odd; relevant for M/bardblx) couples states of\nthe opposite parity on the ΓXM plane, or more generally\norbitalsmandm±1. All these couplings contribute to\nKonly when the Fermi level lies between the two states\nthat are being coupled. Whenever ˆLzorˆLxcouples such\nstates, there is a negative contribution to the energy of\nthe system with the corresponding direction of M.\nWith the help of Fig. 4 we can now deduce which cou-\nplings contribute to Kat different concentrations. At\nx= 0 the Fermi level lies between the filled even states\n(bands 1-2) and empty odd states (bands 3-4) near Γ.\nCoupling of these states by ˆLxcontributes to negative\nK. Filling of the hole pocket at Γ (bands 3-4) with in-\ncreasing xsuppresses this contribution. At x= 0.3 the\nodd bands 3-4 are filled (Fig. 4b), and the minority-spincontribution to Kis small (Fig. 3b). These two cases are\nsketched in Fig. 4d.\nAtstilllarger xtheoddband5getsgraduallyfilled, ac-\ntivating the negative contribution to Kfrom the mixing\nof band 5 with empty even bands 6-7. This trend contin-\nues till about x= 0.8, where an even pair of bands 6-7\n(degenerate at Γ) begins to fill (Fig. 4c). Mixing of these\nbands by ˆLzleads to an intense positive contribution to\nKnearthe Γpoint(clearlyseeninFig.3c), andthetrend\nreverses again. Thus, the nonmonotonic dependence of\nKin the whole concentration range is explained. Note\nthat if the exchange-correlation field on the Co atoms\nis not scaled down to bring the magnetization in agree-\nment with experiment, the exchange splitting remains\ntoo large, and bands 6-7 remain empty up to x= 1. As a\nresult, without this correctionthe uninterruptednegative\ntrend brings Kto large negative values in disagreement\nwith experiment.\nTo assess the effect of atomic relaxations on K, we op-\ntimized allinequivalentstructureswith twoformulaunits\nper unit cell using the VASP code15and the GGA. The\nvolumes were fixed at the same values as in the CPA\nstudies at the same x, while the cell shape and inter-\nnal coordinates were relaxed. Since all these supercells\npreserve the σzreflection plane, the displacements of Fe\nand Co atoms are confined to the xyplane. All displace-\nments were less than 0.025 ˚A. One of the two structures\natx= 0.5 breaks the C4symmetry. For this structure\nwe took the average of the energies for M/bardblxandM/bardbly\nas the in-plane value in the calculation of K, which cor-\nresponds to the averaging over different orientations of\nthe same local ordering.\nThe changes in the absolute value of Kdue to the re-\nlaxation and its values (meV/f.u.) in the relaxed struc-\ntures were found to be: −26% and −0.11 in Fe 2B,−6%\nand 0.25 in Fe 1.5Co0.5B,−13% and 0.15 in the FeCoB\n[100] superlattice, 6% and 0.12 in the FeCoB [110] su-\nperlattice, −11% and −0.31 in Fe 0.5Co1.5B), and−19%\nand−0.04 in Co 2B. While MCA tends to be larger for\nordered structures, the concentration trend in supercell\ncalculations agrees well with the CPA results for disor-\ndered alloys. Although this set ofunit cells is limited, the4\nFIG. 4. (a-c) Minority-spin partial spectral functions for the transition-metal site in the absence of SO coupling at (a )x= 0,\n(b)x= 0.3, and (c) x= 0.8. Energy is in eV. (d) Level diagram and SO selection rules at the Γ point (bands 1-4). Color\nencodes the orbital character of the states. The intensitie s of the red, blue and green color channels are proportional t o the\nsum ofxyandx2−y2(m=±2), sum of xzandyz(m=±1), andz2(m= 0) character, respectively.\nresults suggest that local relaxations do not qualitatively\nchange the concentration dependence shown in Fig. 1.\nLarger positive values of Kare favorable for perma-\nnent magnet applications. Our electronic structure anal-\nysis shows that the maximum near x≈0.3 corresponds\nto the optimal band filling. Further raising of Kre-\nquires favorable changes in the band structure, which\ncould be induced by epitaxial or chemical strain. We\ntherefore considered the dependence of Katx= 0.3 on\nthe volume-conserving tetragonal distortion. The results\nplotted in Fig. 5 show a very strong effect: Kis doubled\nunder a modest 3% increase in c/adue to the sharply\nincreasing spin-flip contributions. A more detailed anal-\nysis shows that the latter is largely due to the increase in\nthecparameter. On the other hand, the minority-spin\ncontribution increases with decreasing volume. Thus, in-\ncreasing cand decreasing aboth have a positive effect on\nMCA. This enhancement could be achieved through epi-\ntaxial multilayer engineering. The search for a suitable\nalloyingelement toenhancethe c/aratiois aninteresting\nsubject for further investigation.\nFIG. 5. MCA energy Kas a function of the c/aratio at\nx= 0.3 (the value c/a= 1 is assigned to the unstrained\nlattice). KSOandKσσ′are also shown.\nIn conclusion, we have explained how the spin re-\norientation transitions in (Fe 1−xCox)2B alloys originate\nin the SO selection rules and the consecutive filling ofminority-spin electronic bands of particular orbital char-\nacter. Near the optimal 30% concentration of Co, the\nMCA energy is predicted to increasequickly with the c/a\nratio, which could be implemented by epitaxial strain or\na suitable chemical doping.\nThe work at UNL was supported by the National Sci-\nence Foundation through Grant No. DMR-1308751 and\nperformedutilizing the Holland Computing Center ofthe\nUniversity of Nebraska. Work at Ames Lab and LLNL\nwas supported in part by the Critical Materials Insti-\ntute, an Energy Innovation Hub funded by the US DOE\nand by the Office of Basic Energy Science, Division of\nMaterials Science and Engineering. Ames Laboratory is\noperated for the US DOE by Iowa State University un-\nder Contract No. DE-AC02-07CH11358. Lawrence Liv-\nermore National Laboratory is operated for the US DOE\nunder Contract DE-AC52-07NA27344.\n1J. Stohr and H. Siegmann, in Magnetism: From fundamentals to\nNanoscale Dynamics (Springer, Berlin, 2006), p. 805.\n2A. Iga, Jpn. J. Appl. Phys. 9, 415 (1970).\n3M. C. 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B 88, 024404 (2013).\n13See the attached supplementary material for implementatio n de-\ntails, explanation about KSO, and the discussion of orbital mo-\nment anisotropy.\n14J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77,\n3865 (1996); 78, 1396 (1997).\n15G. Kresse and J. Furthm¨ uller, Comput. Mat. Sci. 6, 15 (1996).1\nSupplemental Material:\nOrigin of the spin reorientation transitions in (Fe 1−xCox)2B alloys\nI. IMPLEMENTATION OF SPIN-ORBIT COUPLING IN GREEN’S FUNCTION -BASED LINEAR MUFFIN-TIN\nORBITAL (LMTO) METHOD\nMagnetocrystalline anisotropy (MCA) of substitutional alloys is oft en studied using the fully-relativistic Korringa-\nKohn-Rostockeror Green’s function-based LMTO (GF-LMTO) met hod combined with the coherentpotential approx-\nimation (CPA).1–4Here we employed a perturbative implementation of spin-orbit couplin g in GF-LMTO. Dealing\nwith conventional spinor wave functions, it simplifies the analysis of M CA without a significant loss of accuracy\ncompared to the fully-relativistic approach.\nA. Relation between the Hamiltonian and Green’s function-b ased LMTO formulation\nThe so-called orthogonal Hamiltonian and Green’s function-based L MTO formulations are equivalent to second\norder in ( E−Eν) whereEνis the linearization energy.5,6The Hamiltonian in the orthogonal basis is\nHorth=C+√\n∆S(1−γS)−1√\n∆, (1)\nwhereC, ∆, and γare the diagonal matrices containing the LMTO potential paramete rs, andSis the structure\nconstant matrix. On the other hand, in GF-LMTO the Green’s funct ion\nG(z) =λ(z)+µ(z)[P(z)−S]−1µ(z), (2)\nis constructed using the diagonal matrices containing the potentia l functions\nP(z) =z−C\n∆l+γl(z−C), µ(z) =√\n∆\n∆+γ(z−C), λ(z) =γ\n∆+γ(z−C). (3)\nThese definitions are related through energy derivatives (denote d by an overdot):\nµ2=˙P, λ=−1\n2¨P/˙P. (4)\nTheserelationsguaranteethat G(z)doesnothavepolesatthepointswhere P(z)hassimplepoles. Itisstraightforward\nto show that G(z) = (z−Horth)−1, which means that G(z) in (2) is the exact resolvent of Horth.\nThird-orderparametrizationofthe potential functions5is often considerablymoreaccuratecomparedto the second-\norder one. The accuracy of this parametrization is similar to that of the three-center LMTO Hamiltonian, although\nthey are not exactly equivalent. In the nearly orthogonal LMTO ba sis, the Hamiltonian and overlap matrices are7\nH=Horth+(Horth−Eν)Eνp(Horth−Eν),\nO= 1+(Horth−Eν)p(Horth−Eν) (5)\nwhereEνandpare diagonal matrices containing the linearization energies and the L MTO parameters p=/angbracketleft˙φ2/angbracketright, and\nHorthis the same as in (1). The generalized eigenvalue problem is then writte n as\ndet[(Horth−E)−(Horth−Eν)(E−Eν)p(Horth−Eν)] = 0 (6)\nThe third-order term can be treated as a perturbation, and the v ariational correction formally amounts to the\nsubstitution Eˆ1→Eˆ1+(E−Eν)3pin the second-order eigenvalue equation det( Horth−E) = 0.\nIn the GF-LMTO formulation, third-order accuracy is similarly achiev ed by redefining the potential functions:5,7\n˜P=P(˜z),˜µ=√\n˜z′µ(˜z),˜λ=−1\n2˜z′′\n˜z′+ ˜z′λ(˜z) (7)\nwhere ˜z=z+p(z−Eν)3and ˜z′=d˜z/dz. Thesedefinitions aredesigned to preservethe relations(4). Ift he third-order\nGreen’s function ˜Gis defined similarly to (2) but with the third-order potential function s (7), it can be shown that\n˜G=−1\n2˜z′′\n˜z′+√\n˜z′(˜z−Horth)−1√\n˜z′. (8)2\nA solution of det( Horth−˜z) = 0 corresponds to a pole of ˜G, and the factors√\n˜z′guarantee that the residue at that\npole is equal to 1. Thus, the eigenstates of the three-center LMT O Hamiltonian are correctly represented by ˜Gto\nthird order in E−Eν. The first term in (8) introduces two unphysical poles at z=Eν±i/√3pfor each orbital. It is\na diagonal matrix which is real on the real axis and has no effect on th e spectrum of the physical states.\nB. Spin-orbit coupling in GF-LMTO\nIn the Green’s function formalism the perturbation can be introduc ed through the Dyson equation\nGV=G0+G0ΣGV (9)\nwhereG0andGVare the unperturbed and perturbed Green’s functions and Σ the e nergy-dependent self-energy due\nto the perturbation V. ForVrepresenting the spin-orbit coupling, Σ is local.\nIn the second-order representation G0= (z−Horth)−1, and therefore GV= (z−Horth−Σ)−1. ThisGVcan be\nconstructed from the potential functions by denoting\nCV=C+Σ (10)\nand defining the following nondiagonal matrices:\nPV=√\n∆[∆+(z−CV)γ]−1(z−CV)1√\n∆=/bracketleftBig\nγ+√\n∆(z−CV)−1√\n∆/bracketrightBig−1\nµL\nV= [∆+γ(z−CV)]−1√\n∆ = (z−CV)−1√\n∆PV\nµR\nV=√\n∆[∆+(z−CV)γ]−1=PV√\n∆(z−CV)−1\nλV= [∆+γ(z−CV)]−1γ=γ[∆+(z−CV)γ]−1(11)\nThe relations (4) are then replaced by\n˙PV=µR\nVµL\nV, λV=−1\n2(µR\nV)−1¨P(µL\nV)−1. (12)\nA tedious but straightforward derivation then shows\nGV=λV+µL\nV(PV−S)−1µR\nV. (13)\nIftheunperturbedGreen’sfunctioniscalculatedtothird-ordera ccuracy,wecanstillproceedfromDyson’sequation,\nbut the situation is complicated by the first term κ=−˜z′′/(2˜z′) in Eq. (8) for ˜G0, which modifies the perturbed\nGreen’s function ˜GV. However, for spin-orbit coupling this modification may be neglected , as we now argue. Let us\ndenote˜G0=κ+¯G0and write down the diagrammatic expansion of ˜GVin powers of Σ. It is just a usual expansion\nin which each Green’s function line can be either ¯G0orκ. Now let us resum all diagram insertions with no external\nlines and no ¯G0lines inside. This gives a renormalized self-energy ¯Σ = (1−Σκ)−1Σ. Taking this into account and\nperforming the remaining summations, we find\n˜GV=¯G+κ+κ¯Σκ+κ¯Σ¯G+¯G¯Σκ (14)\nwhere\n¯G= (1−¯G0¯Σ)−1¯G0. (15)\nThe quantity Σ κ∼p(E−Eν)Σ is very small for spin-orbit coupling in transition metals: Σ ∼50 meV, p∼0.01\neV−2,E−Eν<∼5 eV. Since κis analytic on the real axis, ¯Σ has no poles there too. Therefore, the poles of ˜GV\ncoincide with the poles of ¯G. The latter are just the poles of ¯G0which are shifted by the self-energy ¯Σ. But¯Σ differs\nfrom Σ by a factor (1 −Σκ)−1which is equal to 1 up to small corrections of order Σ κ. Up to similar corrections, the\nresidues of the poles of ˜GVare equal to those for ¯G. Thus, neglecting all small terms or order Σ κcompared to 1, we\nfind˜GV≈¯G+κ.\nSubstituting the second term in (8) as ¯G0in (15), we find the perturbed Green’s function:\n˜GV=√\n˜z′/parenleftBig\n˜z−Horth−√\n˜z′Σ√\n˜z′/parenrightBig−1√\n˜z′. (16)3\nwhere we have dropped the inconsequential term κ. This expression for ˜GVis equivalent to\n˜GV=˜λV+ ˜µL\nV(˜PV−S)−1˜µR\nV (17)\nwith redefined potential functions\n˜PV=PV(˜z),˜µL\nV=√\n˜z′µL\nV(˜z),˜µR\nV=µR\nV(˜z)√\n˜z′,˜λV=√\n˜z′λV(˜z)√\n˜z′, (18)\nwhere instead of CVin (10) we should use\n˜CV=C+√\n˜z′Σ√\n˜z′. (19)\nThe self-energy matrix for spin-orbit coupling within the given lsubspace is defined as\nΣSO=ξlσσ′(z)/angbracketleftlmσ|SL|lm′σ′/angbracketright (20)\nwhere the energy dependence of the coupling parameter ξlσσ′(z) comes from the radial wave functions. To second\norder in ( z−Eν), the radial function in the LMTO basis is φνlσ(z) =φνlσ+(z−Eν)˙φνlσ. Therefore, we have\nξlσσ′(z) =/angbracketleftφνlσ(z)|VSO(r)|φνlσ′(z)/angbracketright\n=/bracketleftbig\n/angbracketleftφνlσ|VSO(r)|φνlσ′/angbracketright+(z−Eνlσ)/angbracketleft˙φνlσ|VSO(r)|φνlσ′/angbracketright+(z−Eνlσ′)/angbracketleftφνlσ|VSO(r)|˙φνlσ′/angbracketright\n+(z−Eνlσ)(z−Eνlσ′)/angbracketleft˙φνlσ|VSO(r)|˙φνlσ′/angbracketright/bracketrightbig\n//radicalbig\n[1+plσ(z−Eνlσ)2][1+plσ′(z−Eνlσ′)2] (21)\nwhere the square roots in the denominator come from the normaliza tion ofφνlσ(z).\nAn approximate form can be obtained by neglecting the terms of ord erp(E−Eν)2when they appear in the\nperturbation, i. e. approximating√\n˜z′≈1 in (19) and neglecting the square root in the denominator of (21). We\nfound that for (Fe 1−xCox)2B alloys this approximation captures all the essential physics (see, for example, Fig. 1),\nwhile the p(E−Eν)2terms only change Kby an average of about 10%.\nThe above treatment of spin-orbit coupling is extended to CPA with n o modifications. Some implementation notes\nabout our CPA code can be found in Ref. 8.\nII. RELATION TO SPIN-ORBIT COUPLING ENERGY\nWhen MCA energy Kappears in second order in spin-orbit coupling, the anisotropy of th e expectation value of\nthe SO operator ∆ ESO=/angbracketleftVSO/angbracketrightx−/angbracketleftVSO/angbracketrightzis approximately equal to 2 K.9Here we transcribe this relation in terms\nof Green’s functions.\nThe single-particle energy is by definition\nEsp=−1\nπImTr/integraldisplay\nz˜GVdz (22)\nThe second-order term from perturbation theory is\nE(2)\nsp=−1\nπImTr/integraldisplay\nz˜G0Σ˜G0Σ˜G0dz=−1\nπImTr/integraldisplay\nz˜G2\n0Σ˜G0Σdz (23)\nThe expectation value of VSO, with the radial integral included in Σ, in the same order is\n/angbracketleftVSO/angbracketright=−1\nπImTr/integraldisplay\nΣ˜Gdz≈ −1\nπTr/integraldisplay\nΣ˜G0Σ˜G0dz. (24)\nAn exact quasiparticle Green’s function with simple poles of unit residu e satisfies dG/dz=−G2. For˜G0this identity\nis satisfied approximately up to a real term of second order in ( E−Eν). Thus, setting d˜G0/dz≈ −˜G2\n0, we find\nE(2)\nsp≈1\nπImTr/integraldisplay\nz1\n2d\ndz(˜G0Σ˜G0Σ)dz+((dΣ/dz)) =/angbracketleftVSO/angbracketright\n2+((dΣ/dz)) (25)\nwhere the first term was integrated by parts, and (( dΣ/dz)) denotes the terms with the energy derivative of Σ. For\nsubstitutional alloys treated in CPA, the Green’s function does not have a purely quasiparticle structure, and in\ngeneral the relation dG/dz=−G2is violated due to the energy dependence of the coherent potentia l. This energy\ndependence contributes additional terms to the right side of (25) . However, here we are dealing with relatively weak\nsubstitutional Fe/Co disorder, and the electronic structure larg ely preserves its quasiparticle character. Therefore, we\nmay expect the relation E(2)\nsp≈ /angbracketleftVSO/angbracketright/2 to hold approximately, which is borne out by calculations. Deviations from\nthis relation can also occur near the points of degeneracy close to t he Fermi level, where second-order perturbation\ntheory breaks down. As seen in Fig. 4, this situation appears near x= 0.8 where the Fermi level cuts through a pair\nof flat, nearly degenerate bands. From Fig. 1 we see the largest diff erence between KandKSOat this point.4\nIII. SPIN-RESOLVED ORBITAL MOMENT ANISOTROPY\nOrbital moments and their dependence on the direction of magnetiz ation are often discussed in connection with\nMCA.9–14In (Fe 1−xCox)2B both spin channels contribute comparably to K, and therefore there is no direct relation\nbetween Kand the anisotropy of the total orbital moment. However, the sp in-flip terms are small and change\nslowly when xis not close to 1 (see Fig. 1). Since the energy dependence of the SO parameters is relatively weak,\nthe contributions K↑↑andK↓↓are closely related to the orbital moment anisotropy (OMA) for the states of the\ncorresponding spin. Fig. S1 shows that the concentration-weight ed average of the spin-down OMA behaves similarly\ntoK↓↓. The spin-up OMA is almost constant except for the Co-rich end whe re, as seen in Fig. 1, the spin-flip terms\nalso increase substantially. Note that negative OMA for spin-up sta tes corresponds to positive K↑↑, but the sign of\nOMA for spin-down states is the same as that of K↓↓. This is because /angbracketleftLz/angbracketright↓↓appears in KSOwith a minus sign, as\nreflected in Hund’s third rule.\nFIG. S1. Orbital moment anisotropy (OMA) for majority and mi nority-spin states of Fe and Co. The thick black line shows\nthe concentration-weighted average of Fe and Co contributi ons from the minority-spin states.\nThe concentration dependence of K↑↑and of the majority-spin OMA is weak because the majority-spin 3 dbands\nare fully filled. This feature is typical for Fe, Co, and Ni-based magne tic alloys. Therefore, while Kis almost never\nproportional to the total OMA, their derivatives with respect to t he concentration may be strongly correlated in many\nmagnetic 3 dalloys.\n1S. S. A. Razee, J. B. Staunton, and F. J. Pinski, Phys. Rev. B 56, 8082 (1997).\n2J. Zabloudil, L. Szunyogh, U. Pustogowa, C. Uiberacker, and P. Weinberger, Phys. Rev. B 58, 6316 (1998).\n3I. Turek, J. Kudrnovsk´ y, and K. Carva, Phys. Rev. B 86, 174430 (2012).\n4P. Weinberger, Magnetic anisotropies in nanostructured matter (CRC Press, Boca Raton, 2009).\n5O. Gunnarsson, O. Jepsen, and O. K. Andersen, Phys. Rev. B 27, 7144 (1983).\n6I. Turek, V. Drchal, J. Kudrnovsk´ y, M. ˇSob, and P. Weinberger, Electronic structure of disordered alloys, surfaces and interfaces (Kluwer,\nBoston, 1997).\n7O. K. Andersen, O. Jepsen, and D. Gl¨ otzel, in Highlights of Consensed-Matter Theory , ed. F. Bassani, F. Fumi, and M. P. Tosi\n(North-Holland, Amsterdam, 1985).\n8L. Ke, K. D. Belashchenko, M. van Schilfgaarde, T. Kotani, an d V. P. Antropov, Phys. Rev. B 88, 024404 (2013).\n9V. P. Antropov, L. Ke, and D. Aberg, Solid State Commun. 194, 35 (2014).\n10K. Yosida, A. Okiji, and S. Chikazumi, Progr. Theor. Phys. 33, 559 (1965).\n11R. L. Streever, Phys. Rev. B 19, 2704 (1979).\n12P. Bruno, Phys. Rev. B 39, 865 (1989).\n13G. van der Laan, J. Phys.: Condens. Matter 10, 3239 (1998).\n14J. Stohr and H. Siegmann, in Magnetism. From fundamentals to Nanoscale Dynamics (Springer, Berlin, 2006), p. 805."    },    {        "title": "1502.05139v1.Magnetocrystalline_anisotropic_effect_in_GdCo___1_x__Fe__x_AsO___x___0__0_05__.pdf",        "content": "arXiv:1502.05139v1  [cond-mat.str-el]  18 Feb 2015APS/123-QED\nMagnetocrystalline anisotropic effect in GdCo 1−xFexAsO (x= 0,0.05)\nT. Shang,1,∗Y. H. Chen,1F. Ronning,2N. Cornell,3J. D.\nThompson,2A. Zakhidov,3M. B. Salamon,3and H. Q. Yuan1,4,†\n1Center for Correlated Matter and Department of Physics, Zhej iang University, Hangzhou 310058, China\n2Los Alamos National Laboratory, Los Alamos, New Mexico 8754 5, USA\n3UTD-NanoTech Institute, The University of Texas at Dallas, Richardson, Texas 75083 –0688, USA\n4Collaborative Innovation Center of Advanced Microstructu res, Nanjing 210093, China\n(Dated: September 20, 2018)\nFrom a systematic study of the electrical resistivity ρ(T,H), magnetic susceptibility χ(T,H),\nisothermal magnetization M(H) and the specific heat C(T,H), a temperature-magnetic field ( T-H)\nphase diagram has been established for GdCo 1−xFexAsO (x= 0 and 0 .05) polycrystalline com-\npounds. GdCoAsO undergoes two long-range magnetic transit ions: ferromagnetic (FM) transition\nof Co 3delectrons ( TCo\nC) and antiferromagnetic (AFM) transition of Gd 4 felectrons ( TGd\nN). For\nthe Fe-doped sample ( x= 0.05), an extra magnetic reorientation transition takes plac e belowTGd\nN,\nwhich is likely associated with Co moments. The two magnetic species of Gd and Co are coupled\nantiferromagnetically to give rise to ferrimagnetic (FIM) behavior in the magnetic susceptibility.\nUpon decreasing the temperature ( T < TCo\nC), the magnetocrystalline anisotropy breaks up the FM\norder of Co by aligning the moments with the local easy axes of the various grains, leading to a spin\nreorientation transition at TCo\nR. By applying a magnetic field, TCo\nRmonotonically decreases to lower\ntemperatures, while the TGd\nNis relatively robust against the external field. On the other hand, the\napplied magnetic field pulls the magnetization of grains fro m the local easy direction to the field\ndirection via a first-order reorientation transition, with the transition field ( HM) increasing upon\ncooling the temperature.\nPACS number(s): 74.70.Xa,75.30.Kz,75.30.Gw\nI. INTRODUCTION\nThe interplay of 3 dand 4felectrons in the FeAs-based\ncompounds gives rise to rich physical properties, e.g.,\nquantum phase transition, heavy fermion behavior, reen-\ntrance of superconductivity and complex magnetism.1–4\nIn the ZrCuSiAs-type REFePnO compounds ( RE=\nrare earth, Pn= pnictogen), the 4 f-electrons usually\nform an antiferromagnetic (AFM) order at very low tem-\nperatures, while the occurrence of superconductivity is\nmainly associated with the Fe-3 delectrons.5–9For in-\nstance, CeFeAsO sequentially undergoes two AFM-type\ntransitions upon cooling from room temperature, one\nassociated with Fe ( TFe\nN≈150 K) and the other one\nattributed to Ce ( TCe\nN≈3.4 K).9,10Elemental substi-\ntutions in CeFeAsO, e.g., Fe/Co or As/P, may induce\nsuperconductivity while suppressing the magnetic order\nof Fe-3delectrons.11–15On the other hand, CeFePO is\na paramagnetic (PM) heavy fermion metal.1Evidence\nfor a magnetic quantum phase transition was shown in\nCeFeAs 1−xPxO, where the AFM state is separated from\nthe PM heavy fermion state, and a ferromagnetic order\ndevelops in the intermediate doping region.11–13These\nindicate that the 1111-type iron pnictides may provide\na representative system to study the interplay of 4 fand\n3delectrons, and their emergent properties. A system-\natic study of the interplay between 4 fand 3delectrons\nwould help elucidate the nature of superconductivity and\nmagnetism in iron pnictides.\nIntheCe(Fe,Co)AsOseries,superconductivityappears\nwhile the magnetic order at TFe\nNis quickly suppressed\nupon substituting Fe with Co.14,15The Ce N´ eel tempera-tureslightlyincreasesneartheFe-3 dmagneticinstability,\nbut showsanearlyunchanged value on further increasing\nthe Co concentration,14indicating a strong coupling be-\ntweenCe4 felectronsandFe3 delectronsinCeFeAsO,as\npreviously reported by neutron scattering and muon spin\nrelaxation measurements.16,17On the Co-rich side, the\nferromagnetically ordered Co ions have a strong polar-\nization effect on the AFM order of Ce moments.14,18On\nthe otherhand, in the Gd(Fe,Co)AsO series, which corre-\nspond to the case of a large 4 f-magnetic moment, Fe/Co\nsubstitution also leads to a superconducting dome near\nthe 3d-electron magnetic instability.14On the Co-rich\nside, in contrast to the Ce-case, the large moment of Gd\nions is robust against the FM order of Co moments, and\nmaintains its AFM order with a nearly unchanged N´ eel\ntemperatureoverthe entiredopingrange.14However,the\nAFM coupling between the Gd- and Co-blocks leads to\nferrimagnetic behavior at low fields and a possible mag-\nnetic reorientation below TGd\nN.14,19For the case of RE=\nSm and Nd, both show behavior similar to the Ce com-\npounds on the Fe-rich side.14,20,21However, on the Co-\nend, namely, SmCoAsO and NdCoAsO, complex mag-\nnetic properties were observed, in which the Co moments\nsequentially undergo an FM and an AFM transition, fol-\nlowedbyanAFM transitionof4 felectronsat TSm\nN=5K\nandTNd\nN= 3.5 K.22,23By applying a magnetic field, such\nan FM-AFM transition quickly shifts to lower tempera-\ntures, showing a large magnetoresistance effect up to the\nCurie temperature.24However, in GdCoAsO, the mag-\nnetic susceptibility exhibits a broad maximum at tem-\nperatures below TCo\nC, being different from the FM-AFM\ntransition in the Sm- and Nd-case, and its origin remainsunclear.14,19,24Since the transitions of the Co 3 delec-\ntrons are susceptible to the application of magnetic field\nin these compounds, e.g., SmCoAsO and NdCoAsO,24it\nis also interesting to tune the d-fmagnetic coupling in\nGdCoAsO and investigate the associated phenomena.\nIn this paper, we present the resultant T-Hphase di-\nagram for GdCo 1−xFexAsO (x= 0 and 0.05) polycrys-\ntalline compounds based on systematic measurements of\nthe physical properties. We have studied the properties\nofx= 0, 0.05 and 0.1 (nominal values) under various\nmagnetic fields and allthese samplesdemonstratesimilar\nbehaviors even though the Curie temperature TCo\nCsys-\ntematically decreases with increasing x. For clarity, only\nthe results of x= 0 and 0.05 are going to be presented in\nthe paper. For these compounds, the AFM order of Gd\n4felectrons is relatively robust against magnetic field,\nwhile the magnetic transitions of Co blocks are sensitive\nto the magnetic field. The structure ofthis paper is orga-\nnized as follows: to begin with, we will provide a brief in-\ntroduction in Sec.I. Experimental methods are described\nin Sec.II. In Sec. III, we present the experimental results\nof the electrical resistivity, magnetic properties, and spe-\ncific heat under various magnetic fields. Finally, Sec.IV\nsummarizes the results of this investigation.\nII. EXPERIMENTAL DETAILS\nPolycrystalline samples of GdCo 1−xFexAsO (x= 0.0\nand 0.05) were synthesized by solid-state reaction as\ndescribed elsewhere.14The crystal structure of these\ncompounds was characterized by powder x-ray diffrac-\ntion (XRD) at room temperature using the PANalytical\nX’PertMRDdiffractometerandagraphitemonochroma-\ntor. The chemical compositions of the compounds were\nestimated by using an energy dispersive X-ray spectrom-\neter (EDXS), from which an actual Fe concentration of\nx=0.07and 0.12weredetermined forthe nominal x=0.05\nand 0.1, respectively. We note that the nominal values\nofx= 0 and 0.05 are used in this paper. Measurements\nof magnetic properties and specific heat were performed\nin a Quantum-Design magnetic property measurement\nsystem (MPMS-7T) and a physical property measure-\nment system (PPMS-9T), respectively. Temperature de-\npendence of the electrical resistivity was measured by a\nstandard four-point method for temperatures T= 2-300\nK by using a LR700 resistance bridge combined with the\nPPMS temperature control system.\nIII. RESULTS AND DISCUSSION\nFigure 1 presents the XRD patterns of GdCoAsO\nat room temperature, with GdCo 0.95Fe0.05AsO show-\ning similar results. The XRD patterns were ana-\nlyzed by Rietveld refinement using the GSAS+EXPGUI\nprogram.25,26Both samples exhibit a single phase and\ncrystallize in the tetragonal ZrCuSiAs-type structure/s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48 /s54/s48 /s55/s48 /s56/s48/s79\n/s67/s111\n/s71/s100/s98\n/s97 \n/s51/s49/s49/s51/s49/s48/s49/s48/s54/s50/s49/s52/s50/s50/s48\n/s50/s50/s49/s50/s48/s52/s50/s49/s51/s49/s48/s53/s50/s48/s51/s50/s49/s50/s48/s48/s53/s49/s49/s52/s50/s49/s49/s49/s48/s52/s49/s49/s51/s50/s48/s48/s48/s48/s52/s49/s48/s51/s49/s49/s50/s49/s49/s49/s49/s49/s48\n/s48/s48/s51/s49/s48/s50/s49/s48/s49/s48/s48/s50\n/s32 /s32/s73/s110/s116/s101/s110/s115/s105/s116/s121/s32/s40/s97/s46/s117/s46/s41\n/s50 /s32/s40/s100/s101 /s103/s114/s101 /s101 /s41/s32/s73\n/s79 /s98/s115\n/s32/s73\n/s67 /s97/s108\n/s32/s73\n/s79 /s98/s115/s45/s73\n/s67 /s97/s108\n/s32/s66/s114/s97/s103/s103/s32/s80/s111/s115/s105/s116 /s105/s111/s110/s115/s71/s100\n/s65 /s115\n/s79/s48/s48/s49/s99 \nFIG. 1. (Color online) Room-temperature powder XRD pat-\nterns for GdCoAsO (black open circles). The solid red line is\nthe Rietveld refinement profile, while the blue one represent s\nthe differences between the calculated and the experimental\npatterns. The vertical green bars mark the calculated posi-\ntions of the Bragg peaks. The crystal structure of GdCoAsO\nis shown in the inset.\nwith space group P4/nmm, as shown in the inset of\nFig. 1, which can be viewed as alternating GdO- and\nCo(Fe)As-layers stacking along the c-axis. According to\nthe Rietveld refinement, the derived lattice parameters\narea=b= 3.937˚A,c= 8.204˚A for GdCoAsO, being\nconsistent with previous results.14,24\nFigure 2 plots the electrical resistivity ρ(T) of\nGdCo1−xFexAsO (x= 0 and 0.05) at various magnetic\nfields up to 9 T, showing metallic behavior for both sam-\nples. At zero field, we obtained a residual resistivity\nratio (RRR) of 16 and 5 for x= 0 and 0.05, respec-\ntively. This indicates a good quality of the polycrys-\ntalline compounds. In contrast to the GdFeAsO, where a\nbroadmaximumattributed tothe Fe-AFMorderappears\naround 130 K in the electrical resistivity,7,14no distinct\nresistiveanomalyassociatedwiththeComagnetictransi-\ntions canbe found at zerofield in GdCo 1−xFexAsO.Nev-\nertheless, one may see a slope change of the ρ(T) around\ntheCoCurietemperature TCo\nCof75Kand50Kfor x= 0\nand 0.05, respectively. The TCo\nCcan be tracked from\nthe derivatives of the electrical resistivity with respect to\ntemperature (d ρ/dT) (seex= 0 as an example in the in-\nset of Fig. 2(a)). For x= 0, atµ0H≤0.5 T no obvious\ntransitioncanbefoundin theelectricalresistivityaround\nthe temperature where a broad maximum demonstrated\nin the magnetic susceptibility (see below), which is dis-\ntinct from the analogue compound, NdCoAsO, where a\nstep-like FM-AFM transition of Co moments is clearly\nseen around 14 K both in the electrical resistivity and\nmagnetic susceptibility.23,24However, at µ0H= 1 T an\nobvious transition appears around TCo\nR≈25 K, as in-\n2/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48/s54/s48/s55/s48/s56/s48/s57/s48/s49/s48/s48\n/s84/s67/s111\n/s82\n/s84/s32/s40/s75/s41/s40/s98/s41\n/s120/s32/s61/s32/s48/s46/s48/s53\n/s32/s32/s32/s40 /s99/s109/s41/s84/s71/s100\n/s78/s84/s42\n/s50 /s52 /s54 /s56 /s49/s48/s54/s48/s54/s53/s55/s48/s55/s53/s56/s48/s32/s40 /s99/s109/s41\n/s84/s32/s40/s75/s41/s50/s48/s50/s50/s50/s52/s50/s54/s50/s56/s51/s48\n/s84/s67/s111\n/s82/s120/s32/s61/s32/s48\n/s32/s48/s46/s48/s84\n/s32/s48/s46/s53\n/s32/s49/s46/s48\n/s32/s49/s46/s53\n/s32/s50/s46/s48\n/s32/s51/s46/s48\n/s32/s52/s46/s48\n/s32/s53/s46/s48\n/s32/s55/s46/s48\n/s32/s57/s46/s48\n/s32/s32/s40 /s99/s109/s41/s40/s97/s41\n/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s48/s49/s48/s48/s50/s48/s48/s51/s48/s48\n/s84/s67/s111\n/s67\n/s32/s84/s32/s40/s75/s41/s32 /s40 /s99/s109/s41\n/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s32\n/s100 /s47/s100/s84/s32/s40 /s99/s109/s32/s75/s45/s49\n/s41\nFIG. 2. (Color online) Temperature dependence of the\nelectrical resistivity measured at various magnetic fields for\nGdCo 1−xFexAsO with x= 0 (a) and 0 .05 (b). Inset of (a)\nplots the zero-field electrical resistivity and its derivat ive with\nrespect to temperature for x= 0 in a temperature range of\n2-300 K. Inset of (b) expands the low- Tregion for x= 0.05.\nThe dashed lines are guides to the eye.\ndicated by the arrow in Fig. 2(a), which develops into\na resistive minimum at higher fields ( µ0H >1 T). Such\ntransitionscorrespondtothespinreorientationofComo-\nments, being consistent with the maximum in the suscep-\ntibilityχ(T) in Fig. 3(a). The TCo\nRis suppressedto lower\ntemperatures with increasing the magnetic field, and dis-\nappears at µ0H >7 T. We note that strong negative\nmagnetoresistance persists down to the temperature we\nhave identified as TCo\nR. Below that temperature, the re-\nsistivity returns to the low-temperature limit of the zero-\nfield resistivity. We propose that the FM Co lattice has\nstrong uniaxial anisotropy, with a random orientation of\nhardc-axis in the polycrystalline crystals. In zero field,\nthe grain-to-grainmagnetic orientationis random, giving\nrise to spin-scattering at grain boundaries. In an applied\nfield, the magnetization of neighboring grains is aligned\nwith the magnetic field so long as the Zeeman energy\nexceeds the anisotropy energy. The difference between\nlow-field and high-field resistance, then, is attributed tospin-dependent grain-boundary scattering. As the tem-\nperature decreases, two effects change the balance be-\ntween Zeeman and anisotropy energy: (1) the anisotropy\nenergyincreaseswith decreasingtemperatureand (2) the\nmagnetization of Gd exerts an AFM molecular field on\nCo, reducing the net magnetic field acting on the Co\nmagnetization. Ignoring this latter effect, we note that\nthe grain-boundary resistance is completely suppressed\nat 9 T. Using the estimate of 0.32 µB/Co, we place an\nupperlimit on theanisotropyenergytobe approximately\n3×105J/m3, similar to other Co compounds.27At low\ntemperatures, the Gd-AFM transition is barely visible\nin the electrical resistivity (see Fig. 2(a)), however, we\ncan track it in the magnetic susceptibility and specific\nheat (see below). Upon Co/Fe substitution, the Co mag-\nnetic transitions decrease to lower temperatures.14For\nx= 0.05, theρ(T) curve shows a distinct transition at\nTCo\nR≈25 K in a field of 0.5 T, as the arrow shows in\nFig. 2(b). Upon increasing the magnetic field, TCo\nRis\nalso quickly suppressed to lower temperatures before dis-\nappearing at µ0H >4 T. However, the TGd\nNis robust\nagainst the magnetic field, and the resistive upturn at\nTGd\nNis likely associated with a gap opening.14We note\nthat the TGd\nNofx= 0.05 is hardly seen in the electrical\nresistivity at 9 T, but still can be found in the specific\nheat. In addition, for x= 0.05, a subsequent transition\nwas observed below TGd\nNforµ0H≤4 T (marked as T∗\nin the inset of Fig. 2(b)), which is present in the thermo-\ndynamic properties as well (see below). Since this sub-\nsequent transition disappears at the field where the TCo\nR\nis suppressed, it may arise from a magnetic reorientation\nof Co moments.\nThe dc magnetic susceptibility for GdCo 1−xFexAsO,\nmeasured as a function of temperature at various mag-\nnetic fields up to 7 T, is presented in Fig. 3. As an\nexample, here we take GdCoAsO for a detailed analy-\nsis (Fig. 3(a)). The magnetic susceptibility increases\nsharply at a temperature that decreases with x, indicat-\ning FM nature of this transition. The Curie temperature\nof Co moments TCo\nCcan be obtained from the deriva-\ntives of d χ/dT(see details in Ref. 14). At an extremely\nsmall field, the FM transition is sharp in temperature,\nbut it broadens and slightly shifts to higher tempera-\ntures with increasing magnetic field.19,28Similar effects\nwere reported previously in SmCoAsO and NdCoAsO.24\nAt high temperatures ( T >150 K), the magnetic suscep-\ntibility follows a Curie-Weiss law, with the derived effec-\ntivemoment closeto the freeionmoment ofGd (7.94 µB),\nindicating the negligible contribution of Co moments.14\nFurthermore, the susceptibility can be modeled within\nthe mean field approach by considering the AFM cou-\npling between the Gd and Co sublattice, giving rise to\nFIM behavior in the magnetic susceptibility.14At tem-\nperatures just below TCo\nC, a broad maximum was ob-\nserved at low fields ( µ0H <1 T) that becomes much\nsharper (peak) at large fields ( µ0H≥1 T), as shown\nin Fig. 3, being consistent with the reorientation transi-\ntions observed in M(H) curves in Fig. 4. As discussed\n3/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s48/s46/s48/s49/s50/s48/s46/s48/s49/s54\n/s84/s42\n/s32/s84/s32/s40/s75/s41\n/s32/s32/s40/s101/s109/s117/s47/s79/s101/s32/s99/s109/s51\n/s41/s120/s32/s61/s32/s48/s46/s48/s53/s40/s98/s41\n/s84/s71/s100\n/s78/s84/s67/s111\n/s82\n/s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50/s48/s46/s48/s49/s52\n/s53/s84/s48/s46/s51/s84/s48/s46/s49/s84/s40/s101/s109/s117/s47/s79/s101/s32/s99/s109/s51\n/s41\n/s84/s32/s40/s75/s41/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s48/s46/s48/s48/s48/s48/s46/s48/s48/s52/s48/s46/s48/s48/s56/s48/s46/s48/s49/s50/s48/s46/s48/s49/s54\n/s84/s71/s100\n/s78\n/s32/s32/s40/s101/s109/s117/s47/s79/s101/s32/s99/s109/s51\n/s41\n/s32/s48/s46/s49/s84\n/s32/s48/s46/s51\n/s32/s48/s46/s52\n/s32/s48/s46/s53\n/s32/s48/s46/s55\n/s32/s49\n/s32/s50\n/s32/s51\n/s32/s53\n/s32/s55/s120/s32/s61/s32/s48 /s40/s97/s41 /s84/s67/s111\n/s82\n/s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s56/s48/s46/s48/s49/s48/s48/s46/s48/s49/s50\n/s55/s84/s40/s101/s109/s117/s47/s79/s101/s32/s99/s109/s51\n/s41\n/s84/s32/s40/s75/s41/s48/s46/s49/s84\nFIG. 3. (Color online) Temperature dependence of the dc\nmagnetic susceptibility χ(T) at various magnetic fields for\nGdCo 1−xFexAsO with x= 0 (a) and 0 .05 (b). The insets\nexpand the χ(T) in the low-temperature regions.\nabove, such behavior corresponds to the realignment of\nCo moments awayfrom the field direction due to increas-\ning magnetocrystalline anisotropy at low temperatures.\nRoughly 1/3of the magnetic moment should be involved,\nan estimation that is consistent with the observed de-\ncrease in magnetization. For fields above 7 T (5 T for\nx= 0.05), the local anisotropy is unable to reorient the\nCo magnetization in hard-axis grains and no sharp de-\ncrease can be observed. However, no distinct anomaly\ncan be found in the electrical resistivity and the specific\nheat forµ0H <1 T, implying that the reorientation pro-\ncess is broad in temperature at low fields. At lower tem-\nperatures, the Gd AFM transition ( TGd\nN) can be clearly\nfoundinthemagneticsusceptibility χ(T), whichisnearly\nunchanged upon Co/Fe substitution.14With increasing\nthe magnetic field, the feature at TCo\nCis smeared, but\nTCo\nRis rapidly suppressed to lower temperatures and be-\ncomes barely visible at µ0H= 7 T and 5 T for x= 0\nand 0.05, respectively. On the other hand, TGd\nNis ro-\nbust against the magnetic field, as the up arrows indi-\ncated in the insets of Figs. 3(a) and (b). For x= 0.05,\nthe subsequent transition below TGd\nNalso appears in the\nmagnetic susceptibility, as indicated by T∗in the inset/s48/s49/s50/s51/s52\n/s48 /s49 /s50 /s51 /s52 /s53/s48/s49/s50/s51/s50/s46/s53 /s53/s32/s32 /s49/s48\n/s50/s48/s32 /s51/s48 /s54/s48\n/s49/s48/s48\n/s32/s32/s77 /s40 \n/s66/s70/s46/s85 /s46 /s41/s40/s97/s41\n/s120/s32/s61/s32/s48\n/s120/s32/s61/s32/s48/s46/s48/s53/s40/s98/s41\n/s50/s46/s53/s32 /s53/s32/s32 /s49/s48\n/s50/s48/s32/s32 /s52/s48 /s54/s48\n/s32 /s72/s32/s40/s84/s41\n/s32/s77/s32 /s40 /s70/s46/s85/s46 /s41\nFIG. 4. (Color online) Field dependence of the magnetiza-\ntionM(H) at various representative temperatures for (a) x\n= 0 and (b) 0.05, respectively. The arrows mark the spin\nreorientation transitions.\nof Fig. 3(b)(down arrows). Such a subsequent transition\nis barely visible at µ0H >1 T in the χ(T), but it can be\ntracked up to 4 T in ρ(T) (see inset of Fig. 2(b)).\nWe also performed measurements of the field depen-\ndent magnetization M(H) up to 5 T at various temper-\natures (see Fig. 4). In the PM state, the magnetization\nof both samples is linear in field, e.g., T= 100 K and 60\nK forx= 0 and 0.05, respectively; while at tempera-\ntures below TCo\nC, theM(H) curve changes its slope at\na magnetic field that increases with decreasing temper-\nature, indicating a reorientation transition. The applied\nmagnetic field pulls the magnetization of each grain from\nthe local easy direction to the field direction via a first\norder reorientation transition, i.e., all the moments are\npolarized to the field direction (see polarized phase in the\nphase diagram (see below)). Because the orientations of\nthe grains are random, the transition is spread out and\nbecomes continuous. The reorientation transition field\nHMcan be determined from the derivatives of magneti-\nzation with respect to magnetic field d M/dH. The de-\nrivedHM, marked by arrows in Fig. 4, are summarized\nin theT−Hphase diagram (see below). For x= 0,\nthe derived HMare consistent with previous results on\nGdCoAsO.24,29It is noted that, for T= 2.5 K, the HM\nofx= 0 is larger than 5 T. HMalso decreases upon\nCo/Fe substitution, suggesting that the Fe moments are\nlikely oriented opposite to the Co moments due to in-\nternal fields. A scenario involving disorder within the\nCoAs-layers is also suggested. The Fe atoms acting as\nimpurities in the Co sites, might decrease the Co-Co ex-\nchange interaction, making the reorientation transition\n4/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s50/s52 /s50/s53 /s50/s54/s53/s46/s53/s54/s46/s48/s54/s46/s53/s55/s46/s48/s55/s46/s53/s56/s46/s48/s32/s48/s84\n/s32/s48/s46/s53\n/s32/s49\n/s32/s49/s46/s53\n/s32/s50\n/s32/s51\n/s32/s52\n/s32/s53\n/s32/s55\n/s32/s57/s120/s32/s61/s32/s48/s46/s48/s53\n/s32/s32\n/s40/s98/s41\n/s32/s32/s67/s32/s40/s74/s47/s109/s111/s108/s32/s75/s41\n/s84/s32/s40/s75/s41\n/s48 /s53 /s49/s48 /s49/s53 /s50/s48/s48/s51/s54/s57/s49/s50/s49/s53\n/s32/s32/s67/s32/s40/s74/s47/s109/s111/s108/s32/s75/s41/s120/s32/s61/s32/s48/s40/s97/s41\n/s48 /s51/s48 /s54/s48 /s57/s48/s48/s50/s48/s52/s48/s54/s48\n/s32/s84/s32/s40/s75/s41/s32/s67/s32/s40/s74/s47/s109/s111/s108/s32/s75/s41\n/s48 /s50 /s52 /s54 /s56 /s49/s48\n/s32/s84/s32/s40/s75/s41/s40/s100/s41\n/s120/s32/s61/s32/s48/s46/s48/s53\n/s32/s32\n/s84/s42/s84/s71/s100\n/s78\n/s48 /s50 /s52 /s54 /s56/s48/s52/s56/s49/s50/s49/s54/s50/s48\n/s32/s84/s32/s40/s75/s41/s49/s51\n/s50/s52/s55\n/s53/s57\n/s48/s46/s53/s40/s99/s41\n/s32/s32/s67/s47/s84/s32/s40/s74/s47/s109/s111/s108/s32/s75/s50\n/s41/s120/s32/s61/s32/s48\n/s48/s84\n/s84/s71/s100\n/s78\nFIG.5. (Coloronline)Temperaturedependenceofthespecifi c\nheatC(T) at different magnetic fields for GdCo 1−xFexAsO:\nx= 0 (a) and 0 .05 (b). The inset of (a) plots C(T) up to a\ntemperature of 100 K for x= 0 atµ0H= 0 T and 0.5 T. The\ninset of (b) enlarges the temperature range near TCo\nRforx=\n0 atµ0H= 0 T and 0.5 T. (c) and (d) plot C(T)/Tversus\nTin the low-T regime for x= 0 and 0.05, respectively. The\ncolor of the arrows marks the value of applied magnetic field.\noccur at lower field.\nIn Fig. 5, we present the temperature dependence of\nthe specific heat for GdCo 1−xFexAsO measured under\nvarious magnetic fields up to 9 T. The huge magnetic\nentropy at TGd\nNis attributed to the large spin freedom of\nGd-ions( S≃Rln(2s+1)), while the entropychangeasso-\nciated with Co magnetic transitions is weak, indicating a\nlow-spin configuration of Co-ions. Such behavior agrees\nwith the effective moment derived from the Curie-Weiss\nanalysisofthe high- Tsusceptibility, which is nearlyiden-\ntical to the free-ion moment of Gd, with the contribution\nof Co moments being negligible. Further experiments,\ne.g, electron spin resonance (ESR) measurements, would\nbe useful to confirm the spin configurations of Co. In\norder to demonstrate the Co magnetic transitions, low- T\nspecific heat C(T) is plotted in Figs. 5(a) and (b) for x=\n0 and 0.05, respectively. According to the Landau-type\ntheory, the heat capacity would exhibit a jump at the re-\norientation transition.30Forx= 0, no obvious anomaly\natTCo\nRcan be found in the specific heat for µ0H <1\nT, but the slope of C(T) changes at TCo\nC, as shown in\nthe inset of Fig. 5(a). However, when further increas-ing the magnetic field ( µ0H≥1 T), the C(T) does ex-\nhibit a step-like increase at TCo\nR, as the arrows indicate\nin Fig. 5(a). The reorientation transition temperatures\nTCo\nRin the specific heat are highly consistent with those\nderived from the electrical resistivity ρ(T) and magnetic\nsusceptibility χ(T) data. The TCo\nRis also monotonically\nsuppressed on increasing the magnetic field in the C(T),\nwhich decreases to 5.7 K at µ0H= 5 T before encoun-\ntering the Gd AFM transition ( TGd\nN≈3.5 K). We note\nthat atµ0H= 7 T, it is difficult to determine the TCo\nR\nin the specific heat and magnetic susceptibility due to\nthe proximity to TGd\nN, but it still can be tracked in the\nelectrical resistivity (see Fig. 2(a)). Similar behaviorwas\nalso observed for x= 0.05 except that the TCo\nRis observ-\nable already at µ0H= 0.5 T (see inset of Fig. 5(b)),\nreflecting that Co/Fe substitution decreases the magne-\ntocrystalline anisotropy.\nIn Figs. 5(c) and (d), we plot the specific heat C/T\nbelow 10 K for x= 0 and 0.05 to show the magnetic\nproperties of Gd moments under various magnetic fields.\nA value of 2 J/mol-K2is added as an offset. Again, for\nboth compounds, the AFM order of Gd moments is rela-\ntively robust againstthe magneticfield. Similar behavior\nwas also observed in ρ(T) andχ(T). On the other hand,\nforx= 0.05, a subsequent transition below TGd\nNwas also\nfound in the specific heat at µ0H≤3 T, denoted as T∗\nin Fig. 5(d), being consistent with the ρ(T) andχ(T)\ndata. This subsequent transition is likely attributed to a\nfurther reorientation of Co moments due to the coupling\nbetween Co 3 dand Gd 4 felectrons.14\nFigure6 summarizesall the experimental resultsin the\nform of a T-Hmagnetic phase diagram. The applica-\ntion of a magnetic field leads to rich physical properties,\nwhich are correspondingly associated with Co 3 dand Gd\n4felectrons. At zero field, the Co moments form a long-\nrange FM order with a Curie temperature TCo\nC≈75 K\nandTCo\nC≈50 K for x= 0 and x= 0.05, respectively.\nJust below TCo\nCand at low field, the AFM coupling be-\ntween Co and Gd may be sufficient to polarize the Gd\nmoments opposite to the FM Co and the applied field,\nyielding a narrow FIM regime. As usual, the FM tran-\nsition gradually broadens as the magnetic field increases\n(marked by FMCo). At temperatures below TCo\nC, when\nmagnetocrystalline anisotropy exceeds the Zeeman en-\nergy, it aligns the Co magnetization with the local easy\naxes of various grains, causing a spin reorientation tran-\nsition at TCo\nR(marked by SRCo). With increasing mag-\nnetic field, the “gap ”between FMCoand SRCo(H) grad-\nually increases before the FM order fades out. At very\nlow temperatures, the Gd moments are ordered antifer-\nromagnetically with a N´ eel temperature of TGd\nN≈3.5\nK (marked by AFMGd). TheTGd\nNis nearly unchanged,\nwhilethe TCo\nRgraduallymovesto lowertemperaturewith\nincreasing magnetic field. Since the Gd moments are still\nordered antiferromagnetically at 9 T (see Figs. 5(c) and\n(d)), further study under higher magnetic field is desir-\nable to clarify whether there exists an f-electron mag-\nnetic critical point in these compounds. It is noted that,\n5/s70/s77/s67/s111\n/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s49/s48/s49/s48/s48\n/s32 /s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41/s80/s111/s108/s97/s114/s105/s122/s101/s100/s32/s80/s104/s97/s115/s101\n/s65 /s70/s77/s71/s100/s84 /s67/s111\n/s67/s40 /s41/s32 /s32/s84 /s67/s111\n/s67/s40 /s41/s32 /s32/s84 /s67/s111\n/s67/s40/s67 /s41\n/s84 /s67/s111\n/s82/s40 /s41/s32 /s32/s84 /s67/s111\n/s82/s40 /s41/s32 /s32/s84 /s67/s111\n/s82/s40/s67 /s41/s32/s32\n/s84 /s71 /s100/s32\n/s78 /s40 /s41 /s32/s84 /s71 /s100\n/s78 /s40 /s41/s32 /s32/s84 /s71 /s100\n/s78 /s40/s67 /s41\n/s32/s72 \n/s77 \n/s83/s82/s67/s111/s80/s77/s40/s97/s41 /s32/s32/s32/s120 /s32/s61/s32/s48\n/s80/s111/s108/s97/s114/s105/s122/s101/s100/s32/s80/s104/s97/s115/s101/s40/s98/s41 /s32/s32/s32/s120 /s32/s61/s32/s48/s46/s48/s53\n/s48/s46/s49 /s49 /s49/s48 /s49/s48/s48/s49/s49/s48/s49/s48/s48\n/s32/s32/s84/s101/s109/s112/s101/s114/s97/s116/s117/s114/s101/s32/s40/s75/s41\n/s32 /s32\n/s83/s82/s67/s111/s70/s77/s67/s111\n/s72/s32/s40/s84/s41/s65 /s70/s77/s71/s100/s84 /s67/s111\n/s67/s40 /s41/s32/s32/s32 /s32/s84 /s67/s111\n/s67/s40 /s41/s32/s32/s32 /s72 \n/s77 \n/s32/s84 /s67/s111\n/s82/s40 /s41/s32/s32 /s32/s84 /s67/s111\n/s82/s40 /s41/s32/s32/s32 /s84 /s67/s111\n/s82/s40/s67 /s41/s32/s32/s32\n/s32/s32/s84 /s71 /s100\n/s78 /s40 /s41/s32 /s32/s84 /s71 /s100\n/s78 /s40 /s41/s32/s32/s32 /s84 /s71 /s100\n/s78 /s40/s67 /s41\n/s32/s84 /s40 /s41/s32/s32/s32/s32 /s32/s84 /s40 /s41/s32/s32/s32/s32/s32 /s84 /s40/s67 /s41/s80/s77\nFIG. 6. (Color online) T-Hphase diagram for\nGdCo 1−xFexAsO, (a) x= 0 and (b) x= 0.05. The var-\nious symbols denote different types of magnetic transitions\ndetermined from the ρ(T),χ(T) andC(T). The×symbols\nare taken from Ref. 29. The dashed and solid lines are the\nguides to the eyes.\nforx= 0.05, there is another magnetic transition inside\nthe Gd AFM phase, denoted as T∗in Fig. 6(b). Such a\ntransition disappears around the field where TCo\nRis sup-\npressed, so we attribute it to the magnetic reorientation\nof Co moments. Furthermore, inside the SRCophase, the\napplied magnetic field can pull the magnetization of Co\ngrains from the local easy direction to the field direction\n(marked by Polarized Phase).\nIn GdCo 1−xFexAsO, we propose that the Co layers\nexhibit significant magnetocrystalline anisotropy; each\ngrain will have its own local magnetic orientation and\nthere will be spin-flip scattering among the grains. The\napplied magnetic field can pull the magnetization of each\ngrain from the local easy axes to the field direction via\na first-order reorientation transition. As we discussed\npreviously, considering the AFM coupling between the\nGd and Co sublattice, the effective field on Co moments\ncan be described as Heff=H+λMCo−µMGdwithin\na mean-field approach.14Here the λis the FM Co-Co\ncoupling constant and µis the AFM Co-Gd coupling. As\nwe discussed in the resistivity section, at a fixed externalfield, two factors will modify the reorientation process on\ndecreasing the temperature: (i) the magnetocrystalline\nanisotropy energy increases and (ii) the effective mag-\nnetic field, acting on Co, decreases due to the increased\nAFM coupling to the Gd paramagnetism ( µ). Thus,\nthe magnetic field is unable to suppress the reorientation\ntransition at lower temperatures, causing the Co magne-\ntization to rotate toward the local easy direction in each\ngrain. Consequently, the electrical resistivity approaches\nthe low-field behavior which is dominated by spin-flip\nscattering at grain boundaries and a step-increase in the\nspecificheataccompaniesthereorientation. Atlowfields,\nthe direction of magnetization changes continuously over\na temperature interval of several degrees with no observ-\nable changes in the magnitude of magnetization, but it\nbecomesmuchsharperwithincreasingmagneticfield(see\ndetails in Fig. 3). As an example, for x= 0, the reorien-\ntation transition at low fields ( µ0H <1 T) is very broad,\nthe susceptibility exhibits very broad maximum, and no\nobservable transition can be found in the electrical resis-\ntivity and specific heat. While for x= 0.05, the criti-\ncal field decreases to 0.5 T, as expected since the Co/Fe\nsubstitution in the CoAs-layers weakens the FM Co-Co\ncoupling constant λor increases the magnetocrystalline\nanisotropy. Therefore, both magnetic field and chemical\nsubstitutions caneffectivelytune the magneticproperties\nof these compounds, providing the examples for funda-\nmental research and application.\nIV. CONCLUSION\nIn conclusion, we have synthesized GdCo 1−xFexAsO\n(x= 0 and 0.05) polycrystalline samples and systemat-\nically investigated the field-tuning effects on the electri-\ncal resistivity, magnetization and specific heat. A rich\nT-Hphase diagram has been constructed. It is found\nthat the magnetic field can effectively tune the magnetic\nproperties of Co 3 delectrons, while the Gd 4 felectrons\nare insensitive to the applied magnetic field. Further in-\nvestigations on single crystals are badly needed in order\nto clarify the role of magnetocrystalline anisotropy. To\nachieve that, one still needs to overcome the difficulties\nof growing sizable single crystals. Moreover, it is also\ninteresting to extend the measurements to higher mag-\nnetic fields and lower temperatures to study the possible\nmagnetic critical behaviors.\nACKNOWLEDGMENTS\nWe acknowledge the fruitful discussions with Marcelo\nJaime. Work at Zhejiang University is supported\nby the National Basic Research Program of China\n(2011CBA00103),the National Science Foundation of\nChina (No. 11174245) and the Fundamental Research\nFunds for the Central Universities. Work at Los Alamos\nNational Lab was performed under the auspices of the\n6US DOE. Work at The University of Texas at Dallas is supported by the AFOSR grant (No.FA9550-09-1-0384)\non search for novel superconductors.\n∗Present Address: Key Laboratory of Magnetic Materials\nand Devices & Zhejiang Province Key Laboratory of Mag-\nnetic Materials and Application Technology, Ningbo In-\nstitute of Materials Technology and Engineering, Chinese\nAcademy of Sciences, Ningbo 315201, China\n†hqyuan@zju.edu.cn\n1E. M. 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Fidler\nAdvanced Magnetics Group, Institute of Solid State Physics,\nVienna University of Technology, Wiedner Hauptstraße 8-10, 1040 Vienna, Austria\nWe have explored, computationally and experimentally, the magnetic properties of (Fe 1−xCox)2B\nalloys. Calculations provide a good agreement with experiment in terms of the saturation magne-\ntization and the magnetocrystalline anisotropy energy with some difficulty in describing Co 2B, for\nwhich it is found that both full potential effects and electron correlations treated within dynamical\nmean field theory are of importance for a correct description. The material exhibits a uniaxial\nmagnetic anisotropy for a range of cobalt concentrations between x= 0.1andx= 0.5. A sim-\nple model for the temperature dependence of magnetic anisotropy suggests that the complicated\nnon-monotonous temperature behaviour is mainly due to variations in the band structure as the\nexchange splitting is reduced by temperature. Using density functional theory based calculations we\nhave explored the effect of substitutional doping the transition metal sublattice by the whole range\nof 5dtransition metals and found that doping by Re or W elements should significantly enhance the\nmagnetocrystalline anisotropy energy. Experimentally, W doping did not succeed in enhancing the\nmagnetic anisotropy due to formation of other phases. On the other hand, doping by Ir and Re was\nsuccessful and resulted in magnetic anisotropies that are in agreement with theoretical predictions.\nIn particular, doping by 2.5 at.% of Re on the Fe/Co site shows a magnetocrystalline anisotropy\nenergy which is increased by 50% compared to its parent (Fe 0.7Co0.3)2B compound, making this\nsystem interesting, for example, in the context of permanent magnet replacement materials or in\nother areas where a large magnetic anisotropy is of importance.\nPACS numbers: 75.50.Ww, 75.30.Gw, 75.50.Bb, 71.15.Nc, 71.20.Be\nI. INTRODUCTION\nPermanent magnets are used in a wide range of appli-\ncations, for example in data storage, energy conversion,\nmagnetic refrigeration, wind power generators, to name\njust a few52. The determining properties of the quality of\npermanent magnet materials are the operating tempera-\nture range and, primarily, the energy product. In terms\nof operating temperature range, it depends on particular\napplications, but mostly we demand that the material\nremains magnetic well above room temperature. For the\nenergy product, we generally require it to be as large as\npossible. Energy product is a relatively complex quan-\ntity; it depends on the structure of the material (size\nof crystalline grains, their shape and orientation), but\nfrom the microscopic point of view it depends mainly\non the saturation magnetization Msand magnetocrys-\ntalline anisotropy energy (MAE), which is a measure of\nhow difficult it is to rotate the magnetization direction\nby external magnetic field1–3.\nThe best permanent magnet materials known today\nare typically utilizing rare-earth elements, most notably,\nneodymium in Nd 2Fe14B and samarium in SmCo 5mag-\nnets. Both exhibit high Curie temperatures and large en-ergy products. Recently, however, the availability (and\nthus price) of the rare-earth elements became rather\nvolatile, calling for development of replacement mate-\nrials, which would use less or none of the rare-earth\nelements. Intense research efforts have started world-\nwide, revisiting previously known materials, such as\nFe2P4–6, FeNi7or Fe 16N28, doing computational data\nmining among the large family of Heusler alloys9, ex-\nploringtheeffectsofstrain10–16anddopingbyinterstitial\nelements17,18, multilayerssuchasFe/W-Re19, orasalim-\niting case of multilayers, the L 10family of compounds20,\nor promising Mn-based systems21–24,50,51, among others.\nIn this work we explore the magnetic properties of the\n(Fe1−xCox)2B alloys. In 1970 the magnetic properties\nof these compounds have been experimentally studied\nby Iga25, later by Coene et al.26and very recently by\nKuz’min et al.27. It was found that within a certain\nrange of Co contents the material exhibits uniaxial mag-\nnetocrystalline anisotropy, with a relatively large MAE,\nsignificantly higher than pure iron or even cobalt ele-\nmental metals. To the best of our knowledge, theoretical\nunderstanding of the observations is missing.\nMoreover, motivated by the works of Andersson28and\nBhandary19, we have attempted to further improve the\nMAE of the (Fe 1−xCox)2B materials by doping themarXiv:1502.05916v3  [cond-mat.mtrl-sci]  1 Sep 20152\nwith 5dtransition metal elements. The mentioned works\nhave shown that the hybridization of the exchange-split\n3dbands of Fe or Co with the 5 dbands of heavier tran-\nsition metal elements may lead to a significant enhance-\nment of the MAE, even if the 5 delement itself is non-\nmagnetic or very weakly magnetic. We have thus con-\nsidered low percentage substitutions of Fe or Co by the\nwhole series of 5 dtransition metal elements and compu-\ntationally explored the effect of the substitutions on the\nMAE.\nTogether with the first principles electronic structure\ncalculations we have performed experimental studies of\n(Fe1−xCox)2B alloys for selected Co concentrations and\nat the optimal concentration in terms of maximal MAE,\nwe have synthesized single crystals with 2.5 at.% of se-\nlected 5delements alloyed on the 3 dsite. For these single\ncrystals we have then measured the magnetic properties\nand compared them to previous results25and our calcu-\nlations.\nThe structure of the manuscript is the following: In\nSec. II we present our experimental results of synthesis\nand magnetic characterization of (Fe 1−xCox)2B alloys.\nIn Sec. III we present our results of computational stud-\nies of (Fe 1−xCox)2B alloys using two different approaches\nto the problem of substitutional disorder. Different com-\nputational models are compared and the importance of\ncorrelation effects are discussied. Furthermore, we re-\nlate the MAE to the electronic band structure around\nthe Fermi energy (E F) since it is well known that the\nMAE ind-electron systems is determined by the elec-\ntronic structure close to E F43and in particular by the\nspin-orbit coupling (SOC) between occupied and unoc-\ncupied states with small difference in energy. Sec. IV\nthen presents theoretical and experimental results of the\n(Fe1−xCox)2B alloys doped by 5 dtransition metal ele-\nments.\nII. MEASUREMENT OF MAGNETIC\nPROPERTIES\nA. Experimental Methods\nUnambiguous determination of anisotropy constants\nrequires single crystals. Therefore, several single crystals\nof (Fe 1−xCox)2B were grown as part of this work. The\nfirst stage consisted in melting a mixture of Fe and Co\n(both 99.99% pure) with crystalline B of 99.999% purity.\nThemeltingwasperformedinaluminacruciblesplacedin\nan induction furnace under a gauge pressure of 2 bars of\npure argon. When we proceeded from the stoichiometric\ncomposition, the powder XRD patterns of (Fe 1−xCox)2B\nalloys corresponded well to the CuAl 2-type structure but\nthe ingots also contained some precipitants (3-4 wt.%)\nrich in the 3delements. To avoid the precipitants, the\nstarting mixture was taken with a 1.5 at.% excess of B;\nthe so obtained ingot appeared as single-phase in powder\nx-ray diffraction patterns and scanning electron micro-scope images.\nThe ingots were subsequently remelted. For better ho-\nmogeneity, the melts were held for 5 minutes at 1425◦C\nin an argon atmosphere. Then the temperature was re-\nduced slowly, at a rate of 0.1◦C/min, down to 1100◦C.\nAt that point the power was switched off and the furnace\nquickly (initially at approx. 1◦C/min) cooled down to\nroom temperature. The remelted ingots contained large\n(approx. 2-3 mm) crystalline grains. Individual grains\nwere isolated and polished to give them the shape of\na rectangular prism with edges parallel to the principal\ncrystallographic axes. The crystals studied in more de-\ntail had the dimensions approx. 0.5 ×3×3 mm3with one\nof the longest edges being along [001] and the other one\nalong [100]. X-ray back-scattering Laue patterns were\nused for the orientation and quality control of the crys-\ntals.\nThe XRD measurements were done at room tempera-\nture using a STOE Stadi P diffractometer with Mo K α1\nradiation.\nMagnetization isotherms were measured in static mag-\nnetic fields up to 3 T at temperatures ranging between 10\nand 1000 K using a Physical Property Measurement Sys-\ntem (PPMS-14 of Quantum Design). The magnetic field\nwas applied either in the magnetization direction [001]\nor along [100]. Test measurements were also performed\nfor the direction [110] and the resulting curves proved\nindistinguishable from the corresponding [100] curves.\nB. Magnetization and Magnetocrystalline\nAnisotropy\nFigure 1 displays the magnetization curves measured\nalong [100] and [001] for Co 2B (Fig. 1a) and Fe 2B\n(Fig. 1b) single crystals. All of M(H)curves depicted\nin Fig. 1 are shown after correction on demagnetiza-\ntion factor. The Co 2B compound has uniaxial mag-\nnetic anisotropy at low temperature (magnetic moment\nis parallel to the c-axis), however at Tsrt= 72K the\nspin reorientation transition occurs, see Fig. 3. As a re-\nsult atTsrtthe Co 2B single crystal becomes magneti-\ncally isotropic and above this temperature, till the Curie\ntemperature, the magnetization vector lies in the (001)\nplane (easy plane anisotropy). Contrary, the Fe 2B single\ncrystal has easy plane anisotropy at temperatures below\nTsrt= 593K and by passing through the isotropic state\natTsrtbecomes magnetically uniaxial.\nSpontaneousmagnetizations, Ms, determinedfromthe\neasy axis magnetization curves are plotted versus tem-\nperature in Fig. 2. At low temperatures the Mswas\ndetermined as the ordinate of the crossing-point of the\nlinearly extrapolated high-field portions of the easy-axis\ncurves (as shown on the 10 K curve in Fig. 1). For a more\naccurate determination of Msnear theTC(above 800 K)\nBelov–Arrott plots29,30were used. The continuous line3\n01 2 3 500\n1 2 3 50100150(\nb)H II [100]H\n II [001]H II [001]1 0K  \nM (Am2kg-1)/s61549\n0H (T)300KC\no2BMsH\n II [100](\na)6\n00K10KH\n II [001]H II [100]  \nM (Am2kg-1)/s61549\n0H (T)Fe2BMs\nFigure 1: Magnetization curves of (a) Co 2B and (b)\nFe2B single crystals measured along [001] (dashed lines)\nand [100] (solid lines).\nis a fit to the following expression27:\nMs(T) =Ms(0)/bracketleftBigg\n1−s/parenleftbiggT\nTC/parenrightbigg3\n2\n−(1−s)/parenleftbiggT\nTC/parenrightbigg5\n2/bracketrightBigg1\n3\n(1)\nwhereMs(0)isspontaneousmagnetizationat0Kand TC\nis the Curie temperature. Values of the fitting parameter\nsare shown in Fig. 2.\nThe anisotropy energy Eawas determined as the area\nbetween the magnetization curves along [100] and [001]\ntaken at the same temperature. No demagnetization\ncorrection was necessary since the sample dimensions in\nboth directions had been made practically equal. The so\nobtainedEais presented as a function of temperature in\nFig. 3. Also shown in Fig. 3 are data from an early work\nof Iga25. One can appreciate that Iga’s values of K1are\nnoticeably lower than ours. The brevity of Iga’s paper25\nprecludes us from making any definite statement about\nthe source of his underestimation; Iga’s K1were deduced\nfrom torque curves taken in a field of 1.6 T, which is just\nabove the anisotropy field at low temperatures. In an\n2004006008001000050100150s=0.72s\n=0.7s\n=0.5 \n Ms (Am2kg-1)T\n (K) Fe2B \nFe1.5Co0.5B \nFe1Co1B \nCo2Bs=0.9Figure 2: Temperature dependence of the spontaneous\nmagnetization of (Fe 1−xCox)2B.\noblique orientation, such a field is insufficient for mak-\ning the magnetization vector parallel to the applied field.\nThe remaining misalignment distorts the torque curves\nand has to be corrected for. Unfortunately, there is no\nmention of such a correction in Ref.25. We are led to con-\nclude that (Fe 1−xCox)2B are more strongly anisotropic\nthan thought previously.\n/s48 /s50/s48/s48 /s52/s48/s48 /s54/s48/s48 /s56/s48/s48/s45/s56/s48/s48/s45/s54/s48/s48/s45/s52/s48/s48/s45/s50/s48/s48/s48/s50/s48/s48/s52/s48/s48\n/s40/s70/s101\n/s49/s45/s120/s67/s111\n/s120/s41\n/s50/s66/s69\n/s97/s32/s40/s107/s74/s32/s109/s45/s51\n/s41\n/s84 /s32/s40/s75/s41/s120/s61/s48/s58/s32/s32/s32/s32 /s32/s116/s104/s105/s115/s32/s119/s111/s114/s107\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s119/s111/s114/s107/s32/s111/s102/s32/s73/s103/s97\n/s120/s61/s48/s46/s53/s58/s32 /s32/s116/s104/s105/s115/s32/s119/s111/s114/s107\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s119/s111/s114/s107/s32/s111/s102/s32/s73/s103/s97\n/s120/s61/s49/s58/s32/s32/s32/s32 /s32/s116/s104/s105/s115/s32/s119/s111/s114/s107/s32\n/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32 /s32/s119/s111/s114/s107/s32/s111/s102/s32/s73/s103/s97\nFigure 3: Temperature dependence of the anisotropy\nenergy as determined from the leading anisotropy\nconstantK1of (Fe 1−xCox)2B. Dashed lines are the\nvalues from the work of Iga25\n.4\nIII. DENSITY FUNCTIONAL THEORY\nCALCULATIONS OF (Fe1−xCox)2B ALLOYS\nA. Fe 2B and Co 2B\nTheoretical analysis of (Fe 1−xCox)2B alloys starts\nfrom the study of the boundary compounds Fe 2B and\nCo2B. Both these systems crystallize in the tetragonal\nstructure with the space group I4/mcm. The calcu-\nlated equilibrium lattice parameters are a= 5.05Å,\nc= 4.24ÅforFe 2B(cf. experimentalvalues: a= 5.11Å,\nc= 4.25Å) anda= 4.97Å,c= 4.24Å for Co 2B\n(cf. experimental values: a= 5.02Å,c= 4.22Å).\nThe optimized Wyckoff positions differ only negligibly\nfrom rational numbers (1/6, 2/3 and 1/4). Thus, the\ncalculations are based on the latter ones (see Tab. I).\nStructural optimizations are performed with the full po-\ntential linearized augmented plane wave method (FP-\nLAPW) implemented in the WIEN2k code31. The rel-\nativistic effects are included within the scalar relativistic\napproach with the second variational treatment of spin-\norbit coupling. For the exchange-correlation potential\nthe generalized gradient approximation (GGA) in the\nPerdew, Burke, Ernzerhof form (PBE)32is used. Cal-\nculations are performed with a plane wave cut-off pa-\nrameterRKmax= 8, total energy convergence criterion\n10−8Ry and with 20×20×20k-points which was tested\nto give well converged MAE values.\nTable I: Atomic coordinates for Fe 2B and Co 2B.\nAtomSitex y z\nFe/Co8(h)1/6 2/3 0\nB4(a)0 0 1/4\nThe magnetocrystalline anisotropy energy (MAE) is\nevaluated as the difference between total energies cal-\nculated for [100] and [001] quantization axes. The ini-\ntial results for MAE are -0.052 meV/f.u. (-0.31 MJ/m3)\nfor Fe 2B and -0.168 meV/f.u. (-1.03 MJ/m3) for Co 2B.\nThe experimental values of the anisotropy constant K1\nare -0.80 MJ/m3for Fe 2B and +0.10 MJ/m3for Co 2B,\nwhileK2are insignificant25. Although both theoreti-\ncal and experimental anisotropy constants indicate the\nin-plane anisotropy and have the same order of magni-\ntude for Fe 2B, for Co 2B this comparison appears worse.\nA larger discrepancy between theoretical (-1.03 MJ/m3)\nand experimental (+0.1 MJ/m3) anisotropy constants\nfor Co 2B occurs together with another discrepancy be-\ntween calculated ( 1.09µB) and measured ( 0.8µB33) mag-\nnetic moments on the Co atom. This correlation of vari-\nances motivates us to study the dependence of MAE\non magnetic moment. Fully relativistic fixed spin mo-\nment (FSM) calculations were carried out based on\nthe Full-Potential Local-Orbital Minimum-Basis Scheme\n(FPLO-14)34. These calculations were performed with a\n20×20×20k-mesh, the PBE form32of the exchange-correlation potential and convergence criterion 10−8Ha.\n0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-5-4-3-2-10123MAE [ MJ/m3 ]\nµFe/Co [ µB ] Co2BFe2B\ncalc. µCoexp. µCo\ncalc. µFeexp. µFe\n(a) MAE (µ)\nT (K)0 200 400 600 800 1000MAE (MJ/m3)\n-1-0.500.51\nFe2B\nCo2B\n(b) MAE (T)\nFigure 4: MAE as function of total magnetic moment\non the 3datom for Fe 2B and Co 2B as calculated with\nFSM FPLO in (a). MAE (T)for Fe 2B and Co 2B as\nobtained by the scheme described in the text in (b).\nMAE as a function of magnetic moment (see Fig. 4a)\nindicates, for Co 2B, that between the measured ( 0.8µB)\nand calculated ( 1.09µB) values of magnetic moments, the\nMAE varies substantially in the range between +1 and\n-1 MJ/m3. Furthermore, the slope of MAE as a func-\ntion ofµat the point 1.09µBis twice bigger for Co 2B\ncompared to the slope at 1.98µBfor Fe 2B. In summary,\nFSM calculations show that the MAE of Co 2B is a very\nsensitive function of magnetic moment on Co. Overesti-\nmation of the Co magnetic moment by more than 30%\nin the present calculations leads to inaccuracy of the cal-\nculated MAE for Co 2B.\nNote that the dependence of MAE on magnetic mo-\nment qualitatively can be related with effects of chang-\ning temperature and as such it can give clues about the\nbehavior of MAE as a function of temperature. If we\nassume that the magnetic moment (magnetization) de-\ncreases with T, first slowly and then rather rapidly close\ntoTC, we may relate the calculated MAE( µ) to experi-\nmental MAE( T). For Fe 2B we see that with decreasing5\nofµthe MAE starts from a negative value then increases\nto a positive maximum and after that it decreases to-\nwards zero, in a qualitative agreement with the exper-\nimentalK1(T)curve. For Co 2B, if we start from the\nexperimental µvalue, we see that MAE starts from a\npositive value, decreases to negative minimum, and then\nonce again increases, also in qualitative agreement with\nexperimental K1(T). The analysis above should however\nbe taken with some care since the curves in Fig. 4a are\nfrom collinear configurations in which each atomic mo-\nment decreases, while in the experiments non-collinear\nconfigurations certainly also play role. Still, the observa-\ntion described above gives an indication to the source of\nthe rather complicated temperature dependence of the\nMAE in this system. It appears that it might be ex-\nplained by the variation in the MAE due to changes in\nthe band structure as the exchange splitting is reduced\nwith increasing temperature.\nIn an attempt to quantify the above arguments we\npresent a highly simplified model to map the MAE (µ)\ncurves in Fig. 4a to temperature. To large extent the\ntemperature dependence of the MAE is expected to come\nfrom two sources, namely the reduction in the spin split-\nting of the bands and the fluctuations in the directions of\nthe moments. The magnetism observed here is expected\nto be of a highly itinerant character and since the major-\nityspinchannelcontainsunoccupiedstates, itisitinerant\nferromagnetism of the weak kind. Such a system is ex-\npected to exhibit a magnetic moment which scales with\ntemperature approximately as47\nm2∼1−T2\nT2\nC. (2)\nExperimental values for Curie temperatures allow an ini-\ntial mapping of MAE to temperature from the data in\nFig. 4a via Eq. 2. In the next step we wish to take\ninto account also the directional fluctuations of the spins.\nThe Callen and Callen model48predics that such fluctu-\nations result in a /angbracketleftm(T)/m(0)/angbracketright3behaviour in the MAE.\nA classical mean field model with a Langevin function\nis used to enforce such a scaling. Finally, we choose the\nthe zero-temperature reference values for the moments\nin order to obtain MAE (T= 0)values closer to experi-\nmentandhenceaquantitativelysomewhatmoreaccurate\nmodel than that obtained by using the experimentally or\ncomputationally obtained moments at low temperature.\nIn doing so the zero-temperature moments are chosen\ntomCo(T= 0) = 0.7µBandmFe(T= 0) = 2.1µB.\nThe resulting MAE (T)curves for Fe 2B and Co 2B are\nincluded in Fig. 4b. It appears that this crude and sim-\nplified model at least captures the main features of the\nexperimental MAE (T)curves when comparing to Fig. 3.\nThis further supports the idea that much of the interest-\ning non-monotonous behaviour in the MAE (T)is due to\nvariations in the band structure as temperature reduces\nthe exchange splitting.B. Virtual Crystal Approximation\nTo study the whole range of compositions between\nFe2B and Co 2B the Virtual Crystal Approximation\n(VCA) method was used. VCA imitates a random on-\nsite occupation of two types of atoms by using a vir-\ntual atom with an averaged value of the nuclear charge\nZVCA. In the case of the (Fe 1−xCox)2B alloys it is\nZVCA = (1−x)·ZFe+x·ZCo= 26 +x. The num-\nber of valence electrons on the VCA site is also increased\ncorrespondingly.\nCo concentration x0 0.2 0.4 0.6 0.8 1a(Å)\n4.955.15.25.3\nc (Å)\n4.154.24.25\nExp. c\nTheo. c\nExp. a\nTheo. a\nFigure 5: Lattice parameters as functions of xin\n(Fe1−xCox)2B, from experiment and calculated with\nWIEN2k in the GGA, treating disorder by the VCA.\nIn the first step of investigating the alloy compound,\nthe lattice parameters are studied as a function of alloy\nconcentration using WIEN2k within the GGA. Calcula-\ntions were performed with and without the inclusion of\nSOC which, however, had a negligible effect on lattice\nparameters. Fig. 5 shows experimentally measured and\ntheoretically calculated lattice parameters, aandc, as\nfunctions of xin (Fe 1−xCox)2B.\nOverall, the discrepancy between theory and experi-\nment is small and in general the variation in lattice pa-\nrameters are small, in the order of 1-2% over the whole\nrange. Calculations were also performed within the lo-\ncal density approximation (LDA)36with similar results\nforaandcbut somewhat greater disagreement with ex-\nperiment (not shown). Further results presented in this\nsection are obtained with structures from linear interpo-\nlation between theoretically obtained lattice parameters\nof the end compounds. For a few alloy concentrations\nthe MAE obtained by using the computationally opti-\nmized lattice parameters was compared to that obtained\nby using the lattice parameters from linear interpolation\nand the difference was found to be negligible. It was also\ninvestigated how internal atomic positions vary with x\nand the variation was found to be minute, within numer-\nical and experimental uncertainties, whereby fixed values\nwere assumed as presented in Tab. I.\nFig. 6 reveals an excellent agreement between MAE’s\nas functions of xcalculated with the WIEN2k and FPLO\ncodes. Such consistency reflects a good convergence and6\n0 0.2 0.4 0.6 0.8 1-4-3-2-1012\nFPLO\nWIEN2k\nexp.MAE [ MJ/m3 ]\nCo concentration x\nFigure 6: MAE as a function of xin (Fe 1−xCox)2B\ncalculated with WIEN2k and FPLO, in both cases\ntreating disorder by the VCA.\nhigh numerical accuracy of these results. The func-\ntion MAE (x)is in good qualitative agreement with mea-\nsuredK1(x)at low temperatures25. Both the uniaxial\nanisotropy maximum, closer to Fe-rich side, and the in-\nplane anisotropy minimum, on Co-rich side, are repro-\nduced. The highest uniaxial anisotropy, with MAE = 1.5\nMJ/m3, occurs for the (Fe 0.6Co0.4)2B alloy. Quantita-\ntive overestimation of the MAE for the whole concentra-\ntion range brings reminiscence to the case of tetragonally\nstrained Fe/Co alloys, where VCA treatment of substi-\ntutional disorder10also led to an overestimation of MAE\ncompared to experiments12or more sophisticated treat-\nments of disorder15,16. Therefore, in the next subsection\nwe will also present results based on a more advanced\ndescription of disorder, the coherent potential approxi-\nmation (CPA).\nWe continue first with our analysis of the VCA re-\nsults by exploring the relation of the MAE to the orbital\nangular momentum anisotropy. Calculated orbital mag-\nnetic moments µLvary between 0.008 and 0.014 µBper\nFe/Co atom. As seen from Figs. 6 and 7 the MAE closely\nfollows the orbital moments anisotropy, as suggested by\nBruno’s formula35, although with a slight shift towards\nthe positive values.\nTotal magnetic moment µS+Lper (Fe/Co) virtual\natom calculated as a function of xin (Fe 1−xCox)2B\nis compared with experimental values of mean\nmagnetization33in Fig. 8. Discrepancy between experi-\nment and theory is equal to 0.1 µB/atom for Fe 2B and\n0.3µB/atom for Co 2B. For intermediate concentrations\ndiscrepancies scale almost linearly with x.\nIn Fig. 9 the FSM and VCA dependencies of MAE\nare presented as a 2D color map in superposition with\nthetheoreticalequilibriummagneticmoments µS+L’sde-\nnoted by black dots. The MAE landscape reveals that\nby going from magnetic moment of pure Fe 2B towards\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.0080.0090.0100.0110.0120.0130.014\n001\n100µL [ µB/(Fe/Co) atom ]\nCo concentration x\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-0.005-0.004-0.003-0.002-0.00100.001µL 100  - µL 001 [ µB/(Fe/Co) atom]\nCo concentration xFigure 7: The orbital magnetic moment µLand\ndifference of µL’s for [100] and [001] quantization axis\nas functions of xin (Fe 1−xCox)2B, calculated with\nFPLO treating disorder by the VCA.\nthe much lower magnetic moment of Co 2B the MAE\npath passes across a broad range of positive values and\na narrower and steeper valley of negative values. Par-\nticularly, the final value of MAE on the Co 2B side de-\npends sensitively on the Co magnetic moment. Clearly,\non the Co-rich side the overestimation of Co moment af-\nfects substantially the shape of MAE curve. Focusing on\nthe uniaxial anisotropy, the highest MAE value along the\nequilibrium path is reached for the (Fe 0.6Co0.4)2B alloy.\nWe note that on the calculated MAE landscape, there\nis a region with about three times higher values of MAE\nforµabout 1.25 µB/atom on the Fe-rich side. To ap-\nproach this region, starting, e.g., from the (Fe 0.6Co0.4)2B\ncomposition, both the magnetic moment and the d-\nelectrons number (proportional to x) should be reduced.\nThe desired magnetic moment reduction from 1.75 to\naround 1.25 µB/atom is around 25%. This can be ob-\ntained by alloying Fe 0.6Co0.4with 25% of a suitable non-7\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.60.81.01.21.41.61.82.0\nexp.: mean magnetization\nexp.: Fe magnetization\nexp.: Co magnetization\ntheory: µS+L (Fe/Co) µ [ µB ]\nCo concentration x\nFigure 8: Total and species-resolved magnetic moments\nas functions of xin (Fe 1−xCox)2B. Experimental values\nfrom Ref.33, theoretical moments calculated with FPLO\ntreating disorder by the VCA.\nmagnetic element. The non-magnetic alloying should at\nthe same time decrease x(which may be understood as\nthe number of delectrons beyond those of Fe) by about\n0.2. Such decrease can be obtained by alloying Fe 0.6Co0.4\nwith elements having less of delectrons, i.e., the elements\nfrom columns of the periodic table preceding the column\nwith Fe. Among 3delements which fulfill the latter con-\ndition are Cr or Mn, but these often carry rather large\nmagnetic moments, therefore they would not fulfill the\nfirst condition – reduction of average moment per atom.\nFrom 4dand 5delements, possible candidates are Mo\nand Tc, or W and Re, respectively. Doping with heavier\ntransition metal elements will be explored in Sec. IV.\nC. Coherent Potential Approximation\nAs seen in the previous section, the VCA yields a cor-\nrect qualitative agreement for MAE as a function of xin\n(Fe1−xCox)2B when comparing with experiments. How-\never, as previously discussed, it quantitatively overes-\ntimates the MAE. Hence, calculations have also been\nperformed using the coherent potential approximation\n(CPA) to treat the alloying. These calculations were car-\nried out with the SPRKKR39,40method and GGA32for\nthe exchange-correlation potential. The MAE was eval-\nuated by the torque method41and as much as 160000\nˆk-vectors were used for the numerical integration over\nthe full Brillouin zone in order to obtain MAE values\nwith numerical accuracy within a few percent. Calcula-\ntions were done using the lowest temperature, i.e. 77 K,\nexperimental lattice parameters reported by Iga25.\nFig. 10 shows the MAE (x)and saturation magneti-\nzation,Ms(x), as result of these calculations. As ex-\npected, application of CPA instead of VCA removes the\n 0  0.2  0.4  0.6  0.8  1 0 0.5 1 1.5 2\n-4-3-2-1 0 1 2 3 4MAE [MJ/m3] \n33\n22\n111111\n11\n110000\n00-1-1\n-1-1\n-1-1-1-1\n-2-2-2-2\n-2-2-3-3\n-3-3-4-4\n 0  0.2  0.4  0.6  0.8  1\nCo concentration x 0 0.5 1 1.5 2µS+L (Fe/Co) [ µB/virtual atom]\n 0  0.2  0.4  0.6  0.8  1 0 0.5 1 1.5 2Figure 9: Fully relativistic FPLO calculations of MAE\nas a function of xand total magnetic moment ( µS+L)\non 3d atom for (Fe 1−xCox)2B. Disorder was treated by\nthe VCA and µS+Lwere stabilized with fixed spin\nmoment FSM approach. Equilibrium µS+L’s (see\nFig. 8) are denoted by black dots.\nproblem of significantly overestimating the MAE of the\nalloy, at least on the Fe-rich side of the diagram. In-\nstead the qualitative behavior appears to be inconsistent\nwith the experiments of Iga25, as well as the experimen-\ntal data provided in this paper, on the Co-rich side of\nthe diagrams. This failure is most likely due to the\natomic sphere approximation (ASA) which approximates\nthe potential as being spherically symmetric around each\natom, in contrast to the VCA calculations presented in\nthe previous section, which included full potential (FP)\neffects. This is in contrast with a recent publication42,\nwhere authors suggest that the qualitatively wrong be-\nhavior in the MAE predicted for the Co-rich side is a\nfailure of exchange-correlation functional itself, yielding\ntoo high magnetization. Nevertheless, the CPA descrip-\ntion in ASA coincides with experiments on the Fe-rich\nside. In particular the region around 0.2≤x≤0.4, with\nrather large uniaxial MAE, which is the most interest-\ning region from a practical perspective, is correctly de-8\nscribed. The maximum MAE is found for x= 0.3where\nMAE = 131µeV/f.u. = 0.77MJ/m3.\nCo concentration x0 0.2 0.4 0.6 0.8 1MAE (µeV / f.u.)\n-1200-1000-800-600-400-2000200\nMAE (MJ/m3)\n-6-5-4-3-2-101\nTheo. (MJ/m3)\nExp. (MJ/m3)\nTheo. (µeV / f.u.)\nCo concentration x0 0.2 0.4 0.6 0.8 1moment ( µB)\n01234\nµ0Ms (T)\n00.511.52\nµ0Ms\nmtotal / f.u.\nmFe\nmCo\nExp. mtotal / f.u.\nFigure 10: MAE and saturation magnetization as\nfunctions of xin (Fe 1−xCox)2B as calculated with\nSPRKKR treating disorder by the CPA.\nIn order to further investigate the importance of full\npotential (FP) effects, calculations using both ASA and\nFP in SPRKKR were performed for Fe 2B and Co 2B. The\nband structures around the Fermi energy resulting from\nthese calculations are shown in Fig. 11. In the case of\nFe2B, there are quantitative changes in the bands but\nqualitatively they look similar. The MAE of the system\nchanges from−0.11meV/f.u. =−0.64MJ/m3in ASA\nto−0.22meV/f.u. =−1.28MJ/m3in FP, which can\nbe compared to the experimental value of −0.80MJ/m3.\nThis variation in MAE is certainly of numerical signifi-\ncancebutstillthenumbersqualitativelyagreeonthesign\nand order of magnitude and both are reasonably close to\nexperiment.\nFor the case of Co 2B, the MAE changes from\n−0.89meV/f.u. =−5.4MJ/m3to−0.37meV/f.u. =\n−2.2MJ/m3upon inclusion of FP in the calculations.\nThe later value is significantly closer to the experimen-tal low temperature values, as well as to the results of\nFP calculations presented in previous sections of this pa-\nper and it could potentially allow for reproducing the\ncorrect qualitative behavior for the Co-rich alloys. In\nthe band structure one can observe certain qualitative\nchanges which can be of relevance with respect to the\nMAE. In particular around the high symmetry point X, a\nrelatively flat band which is unoccupied in ASA becomes\noccupied in FP which is relevant to the MAE since bands\ncrossing the Fermi energy tend to dramatically affect the\nMAE44. However, to allow for further analysis of the\nimportance of various bands to the MAE, we have used\nscalar relativistic, spin polarized calculations, neglecting\nspin-orbit interaction, carried out in WIEN2k to identify\nthe orbital and spin characters of some bands which are\nmarked in Fig. 11b. This information can be used in an-\nalyzing how the coupling between the bands contribute\nto the MAE by considering the SOC in second order per-\nturbation theory35, where the only non-zero contribution\ncomes from coupling between occupied and unoccupied\nstates, and by looking at the spin-orbit matrix elements\nwhich are tabulated e.g. in Ref.45. In the bottom part\nof Fig. 11b, the WIEN2k FP-LAPW bands, which are\nvery similar to the SPRKKR-FP bands, are also shown\nfor comparison together with the MAE contribution per\nk-point as obtained by the magnetic force theorem44,46.\nIn the ASA case, the highest occupied band at X shows\nmainly minority spin dz2character while the lowest un-\noccupied is minority spin dxz. These states do not couple\nvia spin-orbit interaction whereby a weak contribution to\nthe MAE is expected in this region. As the minority spin\ndxzband becomes occupied in FP, it will couple to both\nof the unoccupied states with magnetic quantum number\nm=±2which have opposite spin character and cause\nMAE contribution of opposite signs resulting in cancel-\nlation and again a weak effect on the MAE as also seen\nexplicitly in the bottom panel of Fig. 11b. An important\nnegativeMAEcontributioninsteadappearstocomefrom\na region around Γand the coupling of the occupied mi-\nnority spin dxzband with the flat unoccupied minority\nspindxyband (in general, coupling between states with\nthe same spin but different magnetic quantum numbers\ncontributenegativelytotheMAE44,45). Thesetwobands\nare closer to each other in the ASA case allowing for this\nnegative region to be overestimated and contribute to the\ntoo large negative MAE obtained for Co 2B in ASA calcu-\nlations. Typically, flat bands such as that just above the\nFermi energy at Γcan contribute strongly to the MAE\nand small shifts in such bands will be essential and might\npartially explain the sensitivity in the MAE of Co 2B to\nvarious parameters. Including FP effects, the MAE is\nstill more strongly negative both than the experimental\nvalue, which is slightly positive at low temperatures, and\nthat from previous FP (WIEN2k and FPLO) calcula-\ntions, which yield a negative value with approximately\nhalf the magnitude. However, the previous calculations\nwere performed using computationally optimized lattice\nparameters while the SPRKKR calculations used slightly9\nX N Γ\nE-E\nF (eV)-0.500.5\n-0.500.5ASA\nFP\n(a) Fe 2B\nE-E\nF(eV)-0.500.5\nX N ΓASA\n-0.500.5\nE(k) (eV)FPFP ↓,z²↓,xz↑,xy\n↓,xz↓,x²-y²\n↑,xy↑,x²-y²\n↓,xy\nX Γ NE-EF(eV)\nMAE (eV/k-point)\n-0.0200.020.5\n0FP-LAPW\n-0.5\n(b) Co 2B\nFigure 11: Band structure of Fe 2B and Co 2B obtained\nin SPRKKR-ASA and SPRKKR-FP and for Co 2B also\nthat obtained with FP-LAPW calculations, in the lower\npanel,together with the MAE contribution of each\nk-point (solid red line) as obtained by the force\ntheorem. The orbital and spin characters of some bands\nare indicated with majority and minority spin denoted\nby↑and↓, respectively.\nlarger experimental lattice parameters. Calculating the\nMAE for Co 2B with the computationally optimized lat-tice parameters presented in previous sections of this pa-\nper in SPRKKR-FP yields MAE =−0.19meV/f.u. =\n−1.2MJ/m3, in good agreement with the FPLO cal-\nculations (−0.17meV/f.u. =−1.03MJ/m3). Another\nindication that FP effects are of great significance for\nCo2B comes from the density of states at the Fermi en-\nergy. For Co 2B this quantity changes significantly from\n8.1to5.9states/eVuponinclusionofFPeffects, whilefor\nFe2B the change is from 4.4to4.3states/eV. Moreover,\nthe relatively large density of states at the Fermi energy\nfor Co 2B could partially explain why the MAE of this\ncompound appears more sensitive to various parameters\nsince, with many states near the Fermi energy, a precise\ndescription of these states will be essential for a correct\nMAE. In conclusion it is clear that the MAE, which is in\ngeneral a challenging quantity to obtain from first prin-\nciples, is for this system, especially on the Co-rich side,\nparticularly sensitive to various parameters and difficult\nto describe correctly. However, the Fe-rich side of the\ncompound appears to be well described using ASA and\nCPA, which will hence be used in Sec. IV. Since the full\npotential VCA calculations overestimate the MAE but\nproduce the qualitatively correct shape of the MAE (x)\ncurve, while the ASA CPA calculations yield a qualita-\ntively incorrect curve on the Co-rich side but the correct\nbehaviour on the Fe-rich side, one is led to believe that\na combination of FP and CPA might produce an MAE\ncurve which is both qualitatively and quantitatively in\nreasonable agreement with experiment for all alloy con-\ncentrations.\nForfurtherinsightintotheMAEofthealloysystemwe\nhaveplottedtheBlochspectralfunctionsfromSPRKKR-\nCPA calculations together with the band structure from\nFPLO-VCA calculations for a few alloy concentrations\nin Fig. 12. In addition to the smearing in the spec-\ntral functions, which can be seen to increase somewhat\nwithx, certain bands appear slightly shifted in compar-\ning the bands and spectral functions. These shifts are\nexpected to be mainly due to FP effects, which are ab-\nsent in SPRKR-CPA, a conclusion which is supported\nby comparison of Fig. 12 to Fig. 11a, and in general\nthe electronic structure appears very similar in the two\nmodels. This similarity is the reason why the VCA and\nCPA tend to yield qualitatively similar results while the\nquantitative differences appear because the band smear-\ning is absent in the VCA. In Ref.42the filling of states\nwith alloy concentration was investigated and related to\nthe variation of the MAE. It was suggested42that the\nspin-diagonal part of the spin-orbit operator is most im-\nportant with rather strong and nearly constant, positive\nMAE contributions coming from the coupling between\nmajorityspinstateswhilecouplingbetweenminorityspin\nstates yields the variation in the MAE. More specifically,\nit was suggested that minority spin bands yield a nega-\ntive contribution for x≤0.3, which is suppressed as the\ntwo minority spin bands on opposite sides of E Fat the Γ\npoint forx= 0.2become occupied, which they both are\natx= 0.3. Identification of the character of the bands in10\nX Γ NE - EF(eV)\n-1-0.500.51\n↓,xz\n↓,yz\n↓,xy\n↓,x²-y²\n(a)x= 0.1\nX Γ NE - EF (eV)\n-1-0.500.51 (b)x= 0.2\nX Γ NE - EF (eV)\n-1-0.500.51 (c)x= 0.3\nFigure 12: Spectral functions from SPRKKR in the CPA and electronic bands from FPLO in the VCA for\n(Fe1−xCox)2B. In (a) the main spin and d-orbital character of the two lowest unoccupied and two highest occupied\nstates at Γare indicated with ↓denoting the minority spin channel.\nFe2B using a scalar relativistic, spin polarized calculation\nneglecting SOC supports that the two relevant lowest un-\noccupied bands at Γin Fig. 12a are mainly of minority\nspin character with magnetic quantum number m=±1.\nIt is also confirmed that the two highest occupied states\natΓare of minority spin character with magnetic quan-\ntum number m=±2. Since the spin-diagonal part of the\nspin-orbit coupling between states of different magnetic\nquantum numbers yiels a negative MAE contribution,\nthis coincides with the picture given in Ref.42. How-\never, for a more complete picture of the total effect on\nthe MAE, contributions from bands around the Fermi\nenergy in the entire Brillouin zone should be carefully\nconsidered. This is illustrated further by the observation\nthat the qualitative description, just discussed to explain\nwhythereisamaximumintheMAEfor x= 0.3, appears\nto hold also for the FPLO VCA calculations even though\nthese do not show the maximum MAE until x= 0.4. We\nfinalize this discussion by concluding that the MAE is a\ndelicate property sensitive to the electronic structure in\na region near the Fermi energy while major changes tend\nto be traceable to bands crossing the Fermi energy.\nD. Electron Correlations in Fe 2B and Co 2B within\nDMFT\nThe disagreement between the presented DFT calcula-\ntionsandexperimentalmeasurementsregardingthemag-\nnetic moment of Co 2B motivates further investigation to\nresolve this issue and correlation effects beyond standard\nDFTmethodswillthereforebeconsidered. Inordertoin-\nvestigate correlation effects on the magnetic properties of\nthe Fe 2B and Co 2B compounds, we have performed elec-\ntronicstructurecalculationswithindynamicalmean-field\ntheory (DMFT),55,56using the so-called DFT+DMFT\napproach53,54,57,58as implemented in the full-potential\nlinear muffin-tin orbital (FP-LMTO) code “RSPt”.59,60\nThe 3dsubset of correlated orbitals was defined using\nthe “muffin-tin head” (MT) projection. For further de-a)\n00.511.5 Fe DOS (states/eV)GGA\nGGA+DMFT\n-16 -12 -8 -4 0 4 812\nE-EF(eV)0\n0.5\n1\n1.5 b)\n00.511.5 Co DOS (states/eV)GGA\nGGA+DMFT\n-16 -12 -8 -4 0 4 812\nE-EF(eV)0\n0.5\n1\n1.5\nFigure 13: Projected density of states of the 3dstates\nin Fe 2B (fig. a) and Co 2B (fig. b), respectively,\ncalculated within DFT and DMFT (SPTF solver). The\nFermi level is set to zero.\ntails we refer the reader to Refs. 58, 61, and 62. The\nDMFTcalculationsareperformedatafinitetemperature\nset to room temperature in this study. Spin-orbit cou-\npling in the muffin-tin region was included in the calcu-\nlations. The impurity problem occurring within DMFT\nwas solved via the spin-polarized T-matrix fluctuation-\nexchange (SPTF) solver,63,64which is based on pertur-\nbation theory. The Hubbard Uwas set to 1.5 eV and\n2.5 eV for Fe 2B and Co 2B, respectively, and the Hund’s\nexchange Jto 0.9 eV. These values of Uare chosen since\nvalues in the range 1−3eV are commonly used for metal-\nlic Fe and Co while using a somewhat larger value for Co\nthan for Fe has been shown to yield a good description of\nthe orbital magnetism in these elements49. The around\nmean-field approximation was used for the double count-\ning correction.\nThe calculated magnetic moments within DFT and\nDMFT are given in Table II. DMFT brings a decrease in\nthe spin polarization for the transition metal moments in\nboth Co 2B and Fe 2B. The decrease in the Co spin polar-\nizationisconsiderable, ingoodagreementwiththeexper-11\nimentally measured moments. The decrease can be un-\nderstood by investigating the projected density of states\nof the transition metals, shown in Fig. 13. Negligible\ndifferences among DFT and DMFT are found for the mi-\nnority spins for both compounds, as well as the majority\nspins for Fe 2B. As for the Co 2B majority spins, the peak\nbelow the Fermi level is pushed to higher energies, thus\ndecreasing the overall spin polarization. In addition, a\nsatellite-like feature is observed at low energies.\nBasedontheseresultsweconcludethatnotonlyanac-\ncurate description of the electrostatic potential, but also\na treatment of correlation effects more advanced than\nthat offered by the GGA is of importance to obtain a\nsatisfying picture of the magnetic properties of Co 2B.\nFor Fe 2B on the other hand, such effects appear to be of\nless significance.\nIV. SUBSTITUTIONAL DOPING BY 5 d\nTRANSITION ELEMENTS\nVarious ways of enhancing magnetocrystalline\nanisotropy in 3 dmagnets have been discussed by\nKuz’min et al.27. In this work we tried to prove the\nconcept of 3d–5dinteractions in a more complex system\nlike (Fe 0.7Co0.3)2B. As spin-orbit coupling is essential\nfor the MAE, heavier elements with large SOC can\npossibly have significant effects on the MAE. Even\nnon-magnetic elements can be of great importance via\nhybridization19,28. Hence, the possibility of tailoring\nthe MAE by adding small amounts of 5 delements\nsubstituting some of the Fe and Co in the concentration\naround (Fe 0.7Co0.3)2B, where the MAE is large and\npositive, has been explored.\nA. Theory\nUsing the same computational methods as described\nin Sec. IIIC (SPRKKR-ASA in the CPA), the magnetic\nproperties and in particular the MAE, was calculated\nfor various elements Xfrom the 5drow of the peri-\nodic table in the compounds (Fe 0.675Co0.3X0.025)2B and\n(Fe0.675Co0.275X0.05)2B. The result is shown in Fig. 14\nwith theXmarked on the x-axis and a dotted line indi-\nTable II: Calculated magnetic moments within DFT\nand DMFT (SPTF solver).\nCo2B Co spin moment [ µB] Co orbital moment [ µB]\nDFT 1.083 0.036\nDMFT 0.813 0.034\nFe2B Fe spin moment [ µB] Fe orbital moment [ µB]\nDFT 1.938 0.041\nDMFT 1.841 0.041cating the MAE of (Fe 0.7Co0.3)2B for comparison. The\ncalculations indicate that doping with W or Re indeed\nappears to cause a remarkable increase of the MAE,\nwith the cost of a small decrease in Msdue to substi-\ntuting some magnetic elements for non-magnetic ones.\nThe greatest increase in MAE is for Re where there is a\nrise from MAE = 0.77MJ/m3to MAE = 1.14MJ/m3\nor1.38MJ/m3, assuming 2.5 % or 5.0 % substitutions\nper Fe/Co atom, respectively.\nLu Ta W Re Os Ir Pt Au HgMAE (µeV/f.u.)\n100120140160180200220240\n(Fe0.675Co0.3X0.025)2B\n(Fe0.675Co0.275X0.05)2B\n(Fe0.7Co0.3)2B\n(Fe0.675Co0.275X0.05)2B\nwith no SOC on X\nMAE (MJ/m3)\n0.60.70.80.91.01.11.21.31.4\nFigure 14: MAE for various elements Xin\n(Fe0.675Co0.3X0.025)2B and (Fe 0.675Co0.275X0.05)2B.\nThe dotted line indicates the MAE of (Fe 0.7Co0.3)2B for\ncomparison. Crosses indicate the MAE obtained when\nthe spin-orbit coupling on the 5 delements is set to zero.\nThe large increase in MAE only appears to be seen\nwhen doping with W or Re and not with other elements\nstudied, in agreement with qualitative prediction based\non Fig. 9. The MAE is sensitive to the electronic struc-\nture around the Fermi energy and the dopants will not\nonly contribute with stronger SOC but also affect the\ngeneral electronic structure. In some cases this effect will\nbe beneficial for the MAE, such as for W or Re here, and\nin some cases not, such as for Ir or Ag where the MAE\nis reduced. In order to investigate the importance of the\nstrong SOC of the 5 datoms, calculations were performed\nwhere the SOC is set to zero on these atoms and the re-\nsults are included in Fig. 14. There is a small variation in\nthe MAE also in this case where the SOC is zero on the\n5datoms and the trend in this variation is essentially the\nsame as in the case with all SOC included. However, it is\nclear that these variations can be significantly enhanced\ndue to the strong 5 dSOC and that this is what yields\nthe immense MAE for Re or W doping. Other relevant\nchanges to the electronic structure around the Fermi en-\nergycausedbythedopantscouldinprinciplebeanalyzed\nin the spectral functions but it is difficult to observe spe-\ncific changes due to the small amounts of dopants in the\nrather complicated spectral functions for the disordered\nsystems and hence these have not been included. As the\neffect of the dopants is more complex than simply caus-12\ning an enhancement of the MAE, it could be of interest\nto study 5dsubstitution around other values of xthan\nx= 0.3as well.\nB. Experiments on (Fe 0.675Co0.3X0.025)2B\nAs predicted by the theoretical calculations we tried\nto substitute the pure (Fe 0.7Co0.3)2B system with 5 del-\nements of W, Re and Ir. The room temperature XRD\nmeasurements of 5 ddoped samples are shown in Fig. 15.\n102 03 04 05 0(a)(\nb)(\nc) 1\n82 02 2Intensity (a.u.)2\n/s61553 (deg.) \nIntensity (arb.units) \n2\n/s61553 (deg.)\nFigure 15: Room temperature XRD plots of\n(Fe0.675Co0.3X0.025)2B forX=(a) W, (b) Ir and (c)\nRe. Colored lines indicate the Bragg positions of\nFe2B-phase and CoWB-phase in red and blue,\nrespectively. The inset in (a) shows an enlarged region\nof17◦−23◦for W substituted sample.\nThe room temperature XRD pattern (Fig. 15a) of\nW doped sample shows two different phases, main(Fe,Co) 2B phase and a secondary CoWB (Pnma) phase.\nWe tried different synthesis methods to avoid the forma-\ntion of the secondary phase, but in all cases W tends to\ncreate a secondary phase rather than replacing the Fe in\nthe main phase. In contrary to the W substitution, single\nphase materials were obtained for Ir (Fig. 15b) and Re\n(Fig. 15c) substitutions.\n0.00 .51 .01 .52 .0050100150T\n = 300K \n (\nFe0.675Co0.3Ir0.025)2BM (Am2kg-1)/s61549\n0H (T)(Fe0.675Co0.3Re0.025)2B(Fe0.7Co0.3)2BH\n II [100]H II [001]\nFigure 16: Magnetization curves of (Fe 0.7Co0.3)2B and\n(Fe0.675Co0.3X0.025)2B single crystals along [100] and\n[001] (X=Re, Ir).\nThe comparison of the room temperature magnetic\nmeasurements of pure (Fe 0.7Co0.3)2B and 5dsubstituted\n(Fe0.675Co0.3X0.025)2B withX=Re, Ir samples are\nshown in Fig. 16. For the Re substituted sample, the\nsaturation magnetization at 2 T decreases from 143.3\nAm2kg−1to 122.2 Am2kg−1which is expected due to\nreduced amount of magnetic elements and, consequently,\nincreased Fe-Fe interaction distances. Contrary to the\nobserved decrease in the saturation magnetization, quite\na big improvement is observed in the anisotropy field.\nThe Re substitution increased the anisotropy field of the\npure system from 1 T to 1.6 T.\nThe Ir substitution slightly decreases the saturation\nmagnetization at 2 T to 138.2 Am2kg−1(Fig. 16). In\naddition the anisotropy field is also decreased down to\n0.8 T, which leads to a decrease in the leading anisotropy\nconstant.\nMAE values for both pure and 5dsubstituted samples\nare shown in Fig. 17 for different temperatures. The re-\nsults show that for the measured temperature interval an\nincrease in MAE is observed for the Re substitution. The\nroom temperature MAE increases from 0.42 MJm−3to\n0.63 MJm−3for Re substitution (approx. 32% increase).\nThis ratio stays almost constant down to 10 K. This rel-\native increase is similar to the theoretical data shown in\nFig. 14.\nThe MAE for the Ir substituted sample is 0.36 MJm−3\nat room temperature, which is approximately 15% lower\nthan for the pure system. For the high temperature re-13\n02 004 006 008 00100200300400500600700800(\nFe0.675Co0.3Re0.025)2B(\nFe0.675Co0.3Ir0.025)2B \n Ea (kJ/m3)T\n (K)(Fe0.7Co0.3)2B\nFigure 17: Temperature dependence of the leading\nanisotropy constants of (Fe 0.7Co0.3)2B,\n(Fe0.675Co0.3Re0.025)2B and (Fe 0.675Co0.3Ir0.025)2B.\ngion (T > 600K), On the other hand, the MAE shows\nhigher values compared to the pure system.\nV. CONCLUSIONS\nIn summary, we have presented a combined theoreti-\ncal and experimental study of structural and magnetic\nproperties of (Fe 1−xCox)2B alloys with emphasis on the\nmagnetocrystalline anisotropy as this property is essen-\ntial, for example, for replacement material candidates\nfor permanent magnet applications. The qualitative\nshape of the MAE curve as function of Co-concentration\nagrees with experiment for full-potential calculations in\nthe VCA. However, the quantitative discrepancy is se-\nvere as commonly observed when studying the MAE in\nthe VCA15–17. Calculations in the atomic sphere approx-\nimation, treating disorder in the CPA, solves the prob-\nlem of the quantitative disagreement with experiment onthe Fe-rich side of the alloy but instead yields a quali-\ntatively incorrect behavior on the Co-rich side. Further-\nmore, DFT with the generalized gradient approximation\nfor exchange-correlation potential results in an overesti-\nmate, compared to experiment, of the magnetic moment\nof Co 2B, which is shown to strongly affect the MAE.\nIt is shown that a reduction of the exchange splitting\nand the correct magnetic moment of Co 2B is reproduced\nif electron correlations are treated by dynamical mean\nfield theory. This leads us to believe that it would in\nprinciple be possible to correctly describe both the mag-\nnetic moment and the MAE over the whole range of alloy\nconcentrations by taking into account both full potential\neffects and dynamical mean field theory while describing\nthe alloy with the coherent potential approximation. Un-\nfortunately, such calculations are outside of our current\ncapabilities. This provides an alternative description of\nthe observed MAE curve compared to a recent work42,\nwhereauthorsartificiallyreducetheexchangeinteraction\nin order to describe the upturn in MAE on the Co-rich\nside. In addition, we have studied the effect of doping\nthese materials with 5delements on their magnetic prop-\nerties. We have found this to be a viable route to en-\nhancing the MAE of (Fe 1−xCox)2B alloys. Simulations\nsuggest that 5 at.% doping by W or Re should approx-\nimately double their MAE, with only modest reduction\nof saturation magnetization. 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Eriksson, “AnLDA+DMFT study of the orbital magnetism and total\nenergy properties of the late transition metals: conserving\nand non-conserving approximations,” (unpublished)\n."    },    {        "title": "1503.00063v2.Band_filling_effect_on_magnetic_anisotropy_using_a_Green_s_function_method.pdf",        "content": "Band-\flling e\u000bect on magnetic anisotropy using a Green's function method\nLiqin Ke1,\u0003and Mark van Schilfgaarde2\n1Ames Laboratory U.S. Department of Energy, Ames, Iowa 50011, USA\n2Department of Physics, King's College London,\nStrand, London WC2R 2LS, United Kingdom\n(Dated: October 1, 2018)\nAbstract\nWe use an analytical model to describe the magnetocrystalline anisotropy energy (MAE) in solids as a function of band\n\flling. The MAE is evaluated in second-order perturbation theory, which makes it possible to decompose the MAE into a sum\nof transitions between occupied and unoccupied pairs. The model enables us to characterize the MAE as a sum of contributions\nfrom di\u000berent, often competing terms. The nitridometalates Li 2[(Li 1\u0000xTx)N], withT=Mn, Fe, Co, Ni, provide a system where\nthe model is very e\u000bective because atomic like orbital characters are preserved and the decomposition is fairly clean. Model\nresults are also compared against MAE evaluated directly from \frst-principles calculations for this system. Good qualitative\nagreement is found.\nI. INTRODUCTION\nMagnetocrystalline anisotropy is a particularly impor-\ntant intrinsic magnetic property[1]. Materials with per-\npendicular magnetic anisotropy are used in an enormous\nvariety of applications, including permanent magnets,\nmagnetic random access memory, magnetic storage de-\nvices, and other spintronics applications.[2{5]\nModern band theory methods have been widely used\nto investigate the magnetocrystalline anisotropy energy\n(MAE) in many systems[6, 7]. The MAE in a uni-\naxial system can be obtained by calculating the total-\nenergy di\u000berence between di\u000berent spin orientations (out\nof plane and in plane). However, MAE is usually a small\nquantity and a reliable ab initio calculation requires very\nprecise, extensive calculations. Moreover, MAE is, in\ngeneral, harder to interpret from the electronic structure\nthan other properties, such as the magnetization. MAE\noften depends on very delicate details of the electronic\nstructure[8]. Using perturbation theory, the MAE can be\ndecomposed into virtual transitions between di\u000berent or-\nbital pairs. In practice, the dbandwidth is large enough\nthat it is nontrivial to meaningfully resolve the MAE into\norbital components and predict its dependence on band\n\flling.\nThe magnetocrystalline anisotropy originates from\nspin-orbit coupling (SOC)[9] or, more precisely, the\nchange in SOC as the spin-quantization axis rotates. In-\ncluding the relativistic corrections to the Hamiltonian\nlowers the system energy and breaks the rotational in-\nvariance with respect to the spin-quantization axis. Here\nwe refer to the additional energy due to the relativistic\ncorrection as SOC energy or relativistic energy Er. MAE\nis a result of the interplay between SOC and the crystal\n\feld[10]. The MAE and change in orbital moment on\nrotation of the spin-quantization axis are closely related.\nWe describe this below and denote them as KandKL,\n\u0003Corresponding author: liqinke@ameslab.govrespectively. Without the SOC, the orbital moment is\ntotally quenched by the crystal \feld in solids. Except for\nvery heavy elements such as the actinides, SOC usually\nalleviates only a small part of the quenching and induces\na small orbital moment relative to the spin moment. For\n3dtransition metals, SOC is often much smaller than\nthe bandwidth and crystal \feld splitting, and thus can\nbe neglected in a \frst approximation. While the Eris\ngenerally small, its anisotropy with respect to spin rota-\ntion is often even orders of magnitude smaller.\nRecently, it had been found that a very high magnetic\nanisotropy can be obtained in 3 dsystems such as lithium\nnitridoferrate Li 2[(Li 1\u0000xFex)N][11{14], which can be\nviewed as an \u000b-Li3N crystal with Fe impurities. As\nfound both in experiments[15] and calculations[12, 13] us-\ning density functional theory (DFT), the Li 2(Li1\u0000xFex)N\nsystem possesses an extraordinary uniaxial anisotropy\nthat originates from Fe impurities. The linear geom-\netry of Fe-impurity sites results in an atomic like or-\nbital and then a large MAE. As found in both x-ray\nabsorption spectroscopy[11] and DFT calculations[11{\n13], 3dionsThave an unusually low oxidation state\n(+1 ) in Li 2(Li1\u0000xTx)N forT= Mn, Fe, Co, and\nNi. Recently, Jesche et al. [16] developed a single-crystal\ngrowth technique for these systems and directly ob-\nserved that the MAE oscillates when progressing from\nT=Mn!Fe!Co!Ni.[16] Electronic structure calcula-\ntions also show that the atomic like orbital features\nare preserved for di\u000berent Telements. Considering the\nrather large MAE and well-separated density of states\n(DOS) peaks in this system, it provides us with a unique\nplatform to investigate the MAE as a function of band\n\flling.\nLi and N are very light elements with sandpelectrons,\nrespectively. They barely contribute to the MAE in\nLi2[(Li 1\u0000xTx)N]; rather, MAE is dominated by single-ion\nanisotropy from impurity Tatoms, especially for lower T\nconcentration, where T-Tatoms become well separated.\nIn this work, we investigate the magnetic anisotropy with\ndi\u000berentTelements based on second-order perturbation\ntheory by using a Green's function method. Lorentzians\n1arXiv:1503.00063v2  [cond-mat.mtrl-sci]  2 Oct 2016are used to represent local impurity densities of states\nand calculate the MAE as a continuous function of band\n\flling. First-principles calculations of MAE are also per-\nformed to compare with our analytical modeling.\nThe present paper is organized in the following way. In\nSec. II, we overview the general formalism of the single-\nion anisotropy[17, 18] with Green's functions and second-\norder perturbation approach[19{24]. Analytical model-\ning and calculational details are discussed. In Sec. III, we\ndiscuss the scalar-relativistic electronic structure of these\nsystems. The band-\flling e\u000bect on MAE in Li 2[(Li 1\u0000xTx\n)N], withT=Mn, Fe, Co, and Ni, is examined within\nour analytical model and results are compared with \frst-\nprinciples DFT calculations. The results are summarized\nin Sec. IV.\nII. THEORY AND COMPUTATIONAL DETAILS\nA. Perturbation theory of the magnetocrystalline\nanisotropy and orbital moment\nPerturbation theory allows us to calculate magnetic\nanisotropy directly from the unperturbed band structure.\nOrbital moment, SOC energy, and their anisotropies can\nbe written in terms of the susceptibility.[7, 17, 21, 23] The\nrelativistic energy Erdue to the spin-orbit interaction\n\u0001Vso=\u0018L\u0001Scan be written as\nEr=\u00001\n2ZEF\n\u00001dE\n\u0019=(Tr[G(E)\u0001Vso]) (1)\nwhereG(E) is the full Green's function, which includes\nSOC and can be constructed from the non-perturbed\nGreen's function G0. Using second-order perturbation\ntheory (here we consider only systems with a uniaxial\ngeometry), the relativistic energy can be written as\nEr=\u00001\n2=X\nijZEF\n\u00001dE\n\u0019TrfGij\n0(E)\u0001Vj\nsoGji\n0(E)\u0001Vi\nsog\n=\u00001\n2X\ni\u00182\niX\n\u001b=\u00061X\nm;m0jhm\u001bj~l\u0001~ sjm0\u001b0ij2\u001f\u001b\u001b0(i)\nmm0\n+ intersite terms\n(2)\nGreen's functions are represented in a basis of or-\nthonormalized atomic functions ji;m;\u001bi, andilabels\natomic sites, msubbands (in cubic harmonics), and \u001b\nthe spin. The local susceptibility \u001f\u001b\u001b0\nmm0, characterizing\nthe transition between two subbands jm;\u001biandjm0;\u001b0i,is de\fned as\n\u001f\u001b\u001b0\nmm0(EF) =\u001f\u001b0\u001b\nm0m(EF) =ZEF\n\u00001dE\n\u0019=fg\u001b\nmg\u001b0\nm0g;(3)\nwhereg\u001b\nmis the unperturbed on-site Green's func-\ntion. Because we only consider the on-site contribution of\nMAE, only the on-site Green's function or local suscep-\ntibility is needed to investigate MAE. We further assume\nthat on-site Green's functions diagonalize in real har-\nmonic space. The angular dependence and band struc-\nture dependence of relativistic energy Erare decoupled.\nIn the following, we assume that MAE is dominated by\na particular site i, and consider only its contribution.\nWhen the spin-quantization axis is along the 001 direc-\ntion, the spin-parallel (longitudinal) components of SO\ninteraction lzcouple orbitals with the same jmjquantum\nnumber (m=-m0), while the spin-\rip (transverse) ones l\u0006\ncouple orbitals with di\u000berent jmjnumbers (jmj=jmj\u00061).\nHereafter, we refer to those two types of coupling as\nintra-jmjand inter-jmjtypes, respectively. According to\nEq. (2) and absorbing the site index i, the relativistic\nenergy can be written as\nEr\n001=\u0000\u00182\n8X\n\u001b=\u00061X\nm;m0\u0000\nAmm0\u001f\u001b\u001b\nmm0+ 2Bmm0\u001f\u0000\u001b\u001b\nmm0\u0001\n(4)\nPositive-de\fnite coe\u000ecients AandBare just the spin-\nparallel and spin-\rip parts of the jL\u0001Sj2matrix elements.\nThey can be written as\nAmm0=m2\u000em;\u0000m0 (5)\nBmm0=1\n4(l(l+ 1)\u0000m(m\u00061))\u000ejmj;jm0j\u00061:(6)\nAandBcorrespond to intra- jmjand inter-jmjtransi-\ntions, respectively. An interesting property of the coe\u000e-\ncient matrices is\nX\nmm0Bmm0=X\nmm0Amm0 (7)\nFor an arbitrary spin orientation other than the 001\ndirection, one can either obtain the relativistic energy Er\nby rotating G0[7] orVso[25, 26] in spin subspace. Here we\nuse the latter approach and the relativistic energy with\nspin being along the 110 direction can be written as\nEr\n110=\u0000\u00182\n8X\n\u001b=\u00061X\nm;m0\u0000\nBmm0\u001f\u001b\u001b\nmm0+ (Amm0+Bmm0)\u001f\u0000\u001b\u001b\nmm0\u0001\n(8)\n2Notice that spin-parallel coe\u000ecients in Eq. (8) are ex-\nactly half of the spin-\rip coe\u000ecients in Eq. (4). If the\nsusceptibility matrix \u001fis relatively homogeneous with re-\nspect to spin, then according to Eqs. (4), (7), and (8), we\nshould expect the spin-\rip components of the relativistic\nenergyErto be about twice as large as the spin-parallel\ncomponents[27]. This is true for the weakly magnetic\natoms in di\u000berent compounds.Let us de\fne the orbital moment anisotropy (OMA)\nand MAE, respectively, as KL=hLzi001\u0000hLzi110and\nK=Er\n110\u0000Er\n001. In this de\fnition, a positive Kindicates\nthat the system has a uniaxial anisotropy. If KLis also\npositive, then the system has a larger orbital magnetic\nmoment along the easy axis. Using Eq. (4) and Eq. (8),\nthe MAEKcan be written as\nK=\u00182\n8X\nm;m0(Amm0\u0000Bmm0)(\u001f\"\"\nmm0+\u001f##\nmm0\u0000\u001f\"#\nmm0\u0000\u001f#\"\nmm0): (9)\nMAE is resolved into allowed transitions between all\npairs of orbitals jm;\u001bi$jm0;\u001bi, corresponding to the\n\u001f\u001b\u001b0\nmm0terms. Since AandBare positive de\fnite, the\ncoe\u000ecient of \u001f\u001b\u001b0\nmm0is positive when ( m=\u0000m0and\u001b=\u001b0)\nor (jmj=jm0j\u00061 and\u001b=\u0000\u001b0), and is negative when\n(m=\u0000m0and\u001b=\u0000\u001b0) or (jmj=jm0j\u00061 and\u001b=\u001b0). In\ngeneral, the local susceptibility \u001f\u001b\u001b0\nmm0is also positive def-\ninite; hence we have the following simple selection rule\nfor MAE: For intra- jmjorbital pairs, transitions between\nsame (di\u000berent) spin channels promote easy-axis (easy-\nplane) anisotropy; for inter- jmjpairs, the sign is the other\nway around, i.e., transitions between same (di\u000berent)\nspin channels promote easy-plane (easy-axis) anisotropy.\nThis simple rule is illustrated in Fig. 1.\n\tD\n \tE\n\tC\n \tB\nFIG. 1: (Color online) Illustration of the dependence of the\neasy-axis direction on the orbital quantum numbers ( m;m0)\nand the spin quantum numbers ( \u001b;\u001b0) of two subbands. Con-\n\fgurations (a) and (d) favor uniaxial anisotropy, while (b)\nand (c) favor easy-plane anisotropy. The vertical dotted line\ncorresponds to the Fermi energy, EF. The horizontal line\nseparates the majority (up) and minority (down) spin chan-\nnels. Occupied states with di\u000berent jmjnumbers are \flled\nwith di\u000berent colors.Similarly, the OMA KLcan be written as\nKL=\u0018\n2X\nm;m0(Amm0\u0000Bmm0)(\u001f##\nmm0\u0000\u001f\"\"\nmm0) (10)\nHence, OMA originates from the di\u000berence between \"\"\nand##components of each pair susceptibility, while MAE\noriginates from the di\u000berence between the spin-parallel\nand spin-\rip components. If we sum over contributions\nfrom all the spin components from each pair of orbitals\n(m;m0) and de\fne\n\u001f\u000f\nmm0=\u001f\"\"\nmm0+\u001f##\nmm0\u0000\u001f\"#\nmm0\u0000\u001f#\"\nmm0 (11)\n\u001fl\nmm0=\u001f##\nmm0\u0000\u001f\"\"\nmm0; (12)\nthen Eqs. (9), and (10) can be written as\n4\n\u00182K=1\n2X\nm;m0(Amm0\u0000Bmm0)\u001f\u000f\nmm0 (13)\n1\n\u0018KL=1\n2X\nm;m0(Amm0\u0000Bmm0)\u001fl\nmm0 (14)\nObviously, the correlation between OMA and MAE[28]\nonly happens when the susceptibility is dominated only\nby one of the spin-parallel components. If it is dominated\nby\u001f\"\", then the system has a smaller orbital moment\nalong the easy axis[27]. If it is dominated by \u001f##, then\nthe system has a larger orbital moment along the easy\naxis and we have K=\u0018\n4KL.\nEquation (9) is useful to explain the MAE in two ex-\ntreme cases. (i) Nonmagnetic limit: Since the orbitals\nare spin independent, we have \u001f\"\"\nmm0=\u001f\"#\nmm0=\u001f#\"\nmm0=\n\u001f##\nmm0.\u001f\u000f\nmm0vanishes for every pair of subbands mm0be-\ncause the spin-parallel components cancel out the spin-\n\rip ones. (ii) Zero crystal-\feld limit: Since orbitals are\n3degenerate,P\nmm0(Amm0\u0000Bmm0)\u001f\u001b\u001b0\nmm0in Eq. (9) van-\nishes for each of the four spin components \u001b\u001b0. Thus the\ntotal anisotropy vanishes as in a free atom.Using the expressions of coe\u000ecients in Eqs. (5) and\n(6), for ad-orbital system, Eq. (9) can be written as\n4\n\u00182K= 4\u001f\u000f\n\u00002;2+\u001f\u000f\n\u00001;1\u00003\n2\u0000\n\u001f\u000f\n\u00001;0+\u001f\u000f\n0;1\u0001\n\u00001\n2\u0000\n\u001f\u000f\n\u00002;\u00001+\u001f\u000f\n\u00002;1+\u001f\u000f\n\u00001;2+\u001f\u000f\n1;2\u0001\n(15)\nwhere the ordering of the states is j-2i=dxy,j-1i=dyz,\nj0i=dz2,j1i=dxz, andj2i=dx2\u0000z2. Di\u000berent point-group\nsymmetry results in di\u000berent orbital degeneracy on site\ni. By summing up the coe\u000ecients of equivalent orbital\npairs, Eq. (15) can be simpli\fed.\nFor tetragonal, square planar, or square pyramidal ge-\nometries, one pair of orbitals ( dxz;dyz) is degenerate.\nEquation (15) can be written as\n4\n\u00182K= 4\u001f\u000f\n\u000022+\u001f\u000f\n11\u0000\u001f\u000f\n12\u00003\u001f\u000f\n01\u0000\u001f\u000f\n\u00002;1: (16)\nWe recover Eq. (13) in Ref.[21].\nFor linear, trigonal, petagonal bipyramidal, and square\nantiprismatic geometries, besides ( dxz;dyz) orbitals,\n(dx2\u0000y2,dxy) orbitals are also degenerate. Equation (16)\ncan be further simpli\fed as\n4\n\u00182K= 4\u001f\u000f\n22+\u001f\u000f\n11\u00003\u001f\u000f\n01\u00002\u001f\u000f\n12 (17)\nOn the other hand, for tetrahedral and octahedral ge-\nometries, \fve dorbitals split into two groups EgandT2g,\nnamely, (dz2,dx2\u0000y2) and (dxy,dyz,dxz). One can easily\nshow that the right side of Eq. (15) vanishes as expected\nfor cubic geometry.\nSimilarly, with the coe\u000ecient matrices and orbital de-\ngeneracy, one easily recovers the formulas for the orbital\nmoment in the tetragonal system as in Ref.[17] or A1 and\nA2 as in Ref.[7].\nB. band-\flling e\u000bect on MAE in a two-level model\nAs shown in Eq. (9), the MAE and OMA can be re-\nsolved into contributions from allowed transitions be-\ntween all pairs of orbitals. The sign and weight of the\ncontribution are determined by coe\u000ecients Am;m0and\nBm;m0, which only depend on the orbital characters of the\ncorresponding orbital pairs. On the other hand, \u001f\u000f\nmm0,\nor its four components \u001f\u001b\u001b0\nmm0, are determined by the elec-\ntronic structure, namely, the Fermi level (electron occu-\npancy or band \flling), band width, crystal-\feld splitting,\nand spin splitting. Here we investigate the band-\flling\ne\u000bect on the MAE contribution from a single pair of or-\nbitals. For each orbital pair mm0, there are four spin\ncomponents: two spin-parallel ( \"\"and##) terms and two\nspin-\rip terms (\"#and#\"). As assumed in the Anderson\n\tB\n\tC\n\tE\n \tD\nFIG. 2: (Color online) (a) Schematic Lorentzian-shape densi-\nties of states for subbands mandm0. (b)\u001f\u000f\nmm0and its four\nspin components as functions of Fermi energy. The ampli-\ntudes of\u001f\u000f\nmm0with (c) the maximum at \"(1;3)\nF and (d) the\nminimum at \"(2)\nFas functions of spin splitting \u0001 sand crystal-\n\feld splitting \u0001 c.\nmodel, Lorentzians are used to represent the local densi-\nties of state (LDOS) in our analytical model to illustrate\nthe electronic structure dependence of \u001f\u001b\u001b0\nmm0and MAE.\nSimilarly, Ebert et al. [17] used Lorentzians DOS to an-\nalytically investigate the orbital magnetic moment and\nrelate it to the impurity density of states at the Fermi\nlevel. For simplicity, we use the same width for every\nLorentzian orbital, and the on-site Green's function for\n4subbandjmiin one spin channel \u001bis given by\ng\u001b\nm(E) =1\nE\u0000\"\u001bm+iw(18)\nwhere\"\u001b\nmis the band center and wis the half width. The\ncorresponding LDOS for subbands jmiandjm0iin two\nspin channels are shown in Fig. 2(a). For simplicity, wefurther assume that the two subbands have the same spin\nsplitting,\"\u001b\nm\u0000\"\u001b0\nm=\"\u001b\nm0\u0000\"\u001b0\nm0\u0011\u0001s, or, equivalently, have\nthe same crystal-\feld splitting, \"\u001b\nm\u0000\"\u001b\nm0=\"\u001b0\nm\u0000\"\u001b0\nm0\u0011\u0001c,\nin the two spin channels.\nAccording to Eq. (3), the pairwise local susceptibility\nfor orbitalsjm;\u001biandjm0;\u001b0ican be written as\n\u001f\u001b\u001b0\nmm0(EF) =8\n<\n:1\n\u00191\n\"\u001b0\nm0\u0000\"\u001bm(arctan[EF\u0000\"\u001b\nm\nw]\u0000arctan[EF\u0000\"\u001b0\nm0\nw]) if\"\u001b\nm6=\"\u001b0\nm0\nD(EF) =1\n\u0019w\n(EF\u0000\"\u001bm)2+w2 if\"\u001b\nm=\"\u001b0\nm0(19)\n\u001f\u001b\u001b0\nmm0(EF) is a positive-de\fnite function for any EF\nand reaches the maximum at EF=(\"\u001b\nm+\"\u001b0\nm0)=2. The\nmaximum value increases as the two band centers ap-\nproach each other until becoming degenerate, because\nthe energies required to transfer electrons from occupied\nstates to the unoccupied states become smaller. Band\nnarrowing increases \u001f\u001b\u001b0\nmm0quickly (nearly 1 =w) until it\nreaches the atomic limit. When the bandwidth becomes\ncomparable to or smaller than the SOC constant, SOC\ncan lift the orbital degeneracy and shift two states, i.e.,\none above and the other below the Fermi level EFcom-\npletely. On the other hand, if the Fermi level sits between\ntwo well-separated narrow subbands and bandwidth is\nsmall compared to the distance between the Fermi level\nand the two band centers, w\u001cEF\u0000\"\u001b\nmandw\u001c\"\u001b0\u0000EF\nm0,\naccording to Eq. (19), then \u001f\u000f\nmm0=1=(\"\u001b0\nm0-\"\u001b\nm) does not\ndepend on the Fermi energy.\nUsing Eqs. (11) and (19), the dependencies of \u001f\u000f\nmm0\nand its four spin components on the Fermi energy EF\nare shown in Fig.2(b). There is one minimum at \"(2)\nFand\ntwo maxima at \"(1;3)\nF, with\n\"(i)\nF=\"1+\"2+4s\n2+i\u00002\n2p\n(\u0001c)2+ (4s)2+ 4w2\n(20)\nThe two maximum peaks originate from the two spin-\nparallel terms \u001f\"\"\nmm0and\u001f##\nmm0, while the minimum orig-\ninates from the spin-\rip terms \u0000(\u001f\"#\nmm0+\u001f#\"\nmm0). In\nEq. (20), each spin component \u001f\u001b\u001b0\nmm0has its maximum\namplitude when the Fermi level is around the middle\nof the corresponding two band centers. The two spin-\n\rip components have their maximum values at the same\nFermi level \"(2)\nFbecause we assume that the two orbitals\nhave the same spin splittings. Contributions from the two\nspin-\rip components become identical when two states\njmiandjm0iare degenerate.\nAs shown in Eqs. (9) and (13), the MAE coe\u000ecients\nfor intra-jmj(A) and inter-jmjterms (-B) have di\u000berent\nsigns. To have a large uniaxial anisotropy, the Fermi levelshould be around the \"(1)\nFor\"(3)\nFfor intra-jmjorbital pairs\nand\"(2)\nFfor inter-jmjorbital pairs. Two orbitals can ac-\ncommodate four electrons in two spin channels, and \"(i)\nF\nroughly corresponds to band \flling of one, two, and three\nelectrons with i=1, 2, and 3, respectively. Figures 2(c)\nand 2(d) shows the maximum amplitude of \u001f\u000f\nij(EF=\"(i)\nF)\nas functions of crystal splitting \u0001 cand spin splitting 4s.\nForEF=\"(1;3)\nF, it requires4c=0 to align the two sub-\nbands in the same spin channel (two subbands becomes\ndegenerate). For EF=\"(2)\nF, it requires4s=\u00064cto align\nthe two subbands in di\u000berent spin channels.\nC. Crystal structures\nLi2(Li1\u0000xTx)N crystallizes in the \u000b-Li3N structure\ntype, which is hexagonal and with space group P6=mmm\n(no. 191). The unit cell of \u000b-Li3N contains one formula\nunit. There are two crystallographically inequivalent sets\nof Li atoms, Li I(1b) and LiII(2c), with 6=mmm and\n\u00006m2 point-group symmetries, respectively. The Li I\natoms are sandwiched between two N atoms and form\na linear -Li I-N- chain along the axial direction, while\nLiIIsites have twofold multiplicities and form coplanar\nhexagons which are centered at -Li I-N- chains and paral-\nlel to the basal plane. Li IIis more close packed in lateral\ndirections and 3 datoms randomly occupy Li Isites. We\ncarried out DFT calculations for small doping concen-\ntration with x=0.166 and found that all Telements with\nT=Mn, Fe, Co, and Ni indeed prefer to occupy Li Isites.\nTo calculate the electronic structure and MAE, we use a\nsupercell which corresponds to ap\n3\u0002p\n3\u00022 superstruc-\nture of the original \u000b-Li3N unit cell. Details of the super-\ncell construction can be found in Ref.[12]. For x=0.5, as\nshown in Fig. 3, there are three Tatoms in the 24-atom\nsupercell with one on the 1 asite and the other two on\nthe 2dsites. Both T1aandT2dsites are derived from the\n1bsite in the original \u000b-Li3N. They have a linear geome-\ntry and a strong hybridization with neighboring N atoms\n51a\nLi\n2d\nNFIG. 3: (Color online) Schematic representation of the su-\npercell used in the DFT calculation for Li 2[(Li 1\u0000xTx)N] with\nx=0.5. Both T1aandT2dsites are derived from the Li I(1b)\nsite in the original \u000b-Li3N structure, while other Li atoms,\nwhich form coplanar hexagons, correspond to Li II(2c) sites\nin the original \u000b-Li3N structure.\nalong the axial direction. T1ahave six Li neighbors, while\nT2dhave three T2dand three Li neighbors in the T-Li\nplane. This structure (denoted as hex2 in Ref.[12]) is\nof particular interest because two types of Tsites,T1a\nandT2d, possess very di\u000berent local surroundings and\nrepresent di\u000berent local impurity concentrations. Along\nthe in-plane direction, T-Tdistances are rather large, es-\npecially for the 1 asite. Since the T1asite represents a\nrelatively low impurity concentration and dominates the\nuniaxial MAE for T=Fe, most of the results in this work\nare focused on the T1asite in the hex2 supercell. We\nalso consider other concentrations such as x=0.16 and\nx=0.33.\nD. DFT calculational details\nWe carried out \frst principles DFT calculations using\nthe Vienna ab initio simulation package (VASP)[29, 30]\nand a variant of the full-potential linear mu\u000en-tin orbital\n(LMTO) method[31]. We fully relaxed the atomic posi-\ntions and lattice parameters, while preserving the sym-\nmetry using VASP. The nuclei and core electrons were\ndescribed by the projector augmented-wave potential[32]\nand the wave functions of valence electrons were ex-\npanded in a plane-wave basis set with a cuto\u000b energy\nof 520 eV. For relaxation, the generalized gradient ap-\nproximation of Perdew, Burke, and Ernzerhof was used\nfor the correlation and exchange potentials. The spin-\norbit coupling is included using the second-variation\nprocedure[33, 34]. We also calculated the MAE by car-\nrying out all-electron calculations using the full-potentialLMTO (FP-LMTO) method to check our calculational\nresults. For the MAE calculation, the k-point integration\nwas performed using a modi\fed tetrahedron method with\nBl ochl corrections, with 163k-points in the \frst Brillouin\nzone of the 24-atom unit cell. By evaluating the SOC ma-\ntrix elementshVSOiand its anisotropy[27], we resolve the\nanisotropy of orbital moment and MAE into sites, spins,\nand orbital pairs. The correlation e\u000bects are also consid-\nered by using the local-density approximation (LDA)+ U\nmethod. Here we choose the fully localized limit imple-\nmentations of the double counting introduced by Liecht-\nenstein et al. [35] considering it is more appropriate for\nmaterials with electrons localized on speci\fc orbitals.\nIII. RESULTS AND DISCUSSIONS\nA. Electronic structures\nWithout considering SOC, the axial crystal \feld on\nbothT1aandT2dsites splits \fve 3 dorbitals into three\ngroups: degenerate ( dxy,dx2\u0000y2) states, degenerate ( dyz,\ndxz) states, and dz2state. Equivalently, they can be la-\nbeled asm=\u00062,m=\u00061, andm=0 using cubic harmonics.\nThe scalar-relativistic partial densities of states\n(PDOS) projected on the T1asite are shown in Fig. 4.\nForT=Fe, the PDOS obtained is very similar to what\nwas previously reported [12]. The Fe 3 dshell has seven\nelectrons and the majority spin channels of dorbitals are\nfully occupied with \fve electrons.\nThe Fedz2states hybridize with pzstates ofNatoms\nalong the axial direction and mix with on-site 4 sstates,\nwhich causes the dz2orbital to be lower in energy than\nthe otherdorbitals.[12] The dz2states spread out and\nlie below the Fermi level and accommodates one electron\nin the minority spin channel. The last electron occupies\nhalf of the degenerate ( dxy,dx2\u0000y2) states in the minority\nspin channel. These states have a very narrow bandwidth\nand cross the Fermi level.\nThe linear geometry minimizes the in-plane hybridiza-\ntion between the T3dorbitals and the neighboring\natoms, making them atomic like and resulting in nar-\nrower bands. The T2dsite shows a similar PDOS as the\nT1asite; however, the in-plane hybridization with other\nT2dsites results in a much broader bandwidth than the\n1asites.\nFor otherTelements, the DOS peaks are well sep-\narated as in T=Fe. The minority spin channel clearly\nshows a di\u000berent band-\flling pattern with di\u000berent Tel-\nements. The deviation from the rigid-band model is also\nobvious. Spin splitting decreases from Mn to Ni, while\nthe crystal-\feld splitting values (the energy di\u000berence be-\ntweenm=\u00061 andm=\u00062 states) are larger for T=Mn and\nFe than for T=Co and Ni.\nFigure 5(a) shows the schematic Fe PDOS, and how\nthe Fermi level changes with di\u000berent Tin a rigid-band\napproximation (RBA). Di\u000berent Telements correspond\nto di\u000berent integer number of 3 delectrons. Since each\n6−4−2024\n  \nMn (a)\nm=±2\nm=±1\nm=0\n−4−2024Fe (b)DOS ( states ( eV spin atom )−1)\n−4−2024Co (c)\n−5 −4 −3 −2 −1 0 1 2−4−2024Ni (d)\nE(eV)FIG. 4: (Color online) Partial densities of states projected\non the 3dstates of the T1asite in the hex2 structure in\nLi2[(Li 1\u0000xTx)N], wherex=0.5 andTis (a) Mn, (b) Fe, (c) Co,\nand (d) Ni. The vertical dotted line corresponds to the Fermi\nenergy,EF. The horizontal dotted line separates the major-\nity (up) and minority (down) spin channels. Calculation is\nwithin LDA, without spin-orbit coupling included.\ndegenerate state pair can accommodate two electrons in\none spin channel, the Fermi level either intersects the\ndegenerate peaks or sits in the middle of two peaks.\nB. MAE in Li 2[(Li 1\u0000xTx)N] withT=Fe\nMAE in Li 2[(Li 1\u0000xTx)N] withT=Mn, Fe, Co, and Ni\nandx= 0:5 are calculated in DFT and summarized inTABLE I: Lattice constants, total and site-resolved MAE\nLi2[(Li 0:5T0:5)N] withT=Mn, Fe, Co and Ni. The MAE val-\nues forT2dsite are in unit of meV/atom, and there are two\nT2datoms in the supercell.\nLattice parameters K(meV )\nTa(a.u.)c=a cellT1aT2dothers\nMn 12.143 1.202 -1.14 -0.35 -0.38 -0.03\nFe 12.091 1.183 20.83 14.77 3.09 -0.12\nCo 12.144 1.154 -3.69 -0.89 -1.32 -0.15\nNi 12.113 1.156 2.52 1.71 0.37 0.06\nTable I. The system has uniaxial anisotropy with T=Fe or\nNi and easy-plane anisotropy with T=Mn or Co. MAE is\ndominated by the contributions from the 1 asite forT=Fe\nor Ni. Results are in qualitative agreement with previous\ncalculations.[11{13] The extraordinary MAE for T=Fe\noriginates from the unique band structure in this system.\nBecause the well-isolated Fe atoms, such as the Fe 1asite\nin thehex2 supercell, provide the major contribution to\nthe uniaxial anisotropy, we focus on the Fe 1asite.\nAs shown in Fig. 1, the sign of the MAE contribu-\ntion from transitions between a pair of subbands jm;\u001bi\nandjm0;\u001b0iis determined by the spin and orbital char-\nacter of the involved orbitals. Because the dz2orbital is\nspread out relatively further below the Fermi level and\ncontributes negligibly to the MAE, we only consider the\ntransitions between subbands with m=\u00002,\u00001, 1, and 2.\nIntra-jmjtransitionsj1i$j\u0000 1iandj2i$j\u0000 2ipromote\neasy-axis anisotropy when they are within the same spin\nchannel, and easy-plane anisotropy when between dif-\nferent spin channels. For inter- jmjtransitions, it is the\nother way around. Transition j\u00061i$j\u0006 2ipromotes\neasy-plane anisotropy when it is within the same spin\nchannel and easy-axis anisotropy when between di\u000berent\nspin channels. The signs and coe\u000ecients of the MAE\ncontributions from di\u000berent orbital pair transitions are\nindicated in Fig. 5(a). Transitions contribute to MAE\nonly when they cross the Fermi level. The amplitude\nof MAE depends on the orbital characters and also the\nenergy di\u000berence between the two band centers. When\nthe Fermi level intersects the narrow degenerate states,\nthe transition energy required to excite an electron across\nthe Fermi level is very small (between 0 and bandwidth),\nmaking the MAE contribution from this pair of orbitals\nvery large. On the other hand, when the Fermi level\nis between two well-separated DOS peaks, the required\ntransition energy is much larger so the amplitude is much\nsmaller.\nTo elucidate the orbital contributions from the Fe 1a\nsite to the MAE in Li 2[(Li 0:5Fe0:5)N], we approximate\nthe densities of states of j\u00061i(dxz;dyz) andj\u00062i\n(dxy,dx2\u0000y2) subbands with two Lorentzian functions.\nCrystal-\feld splitting \u0001 c=\u000fjmj=1\u0000\u000fjmj=2=1:8eV, spin\nsplitting \u0001 s=2.4 eV, and half width w= 0:06 eV are\nused to represent the DFT-calculated PDOS, as shown in\nFig. 4. The PDOS used in our model is shown in Fig. 5(a)\nand the MAE contribution from the 1 asite and its de-\n7m2\nm1Fe Co Ni Mn\n3 2 1 0 1 26420246\nEeVDOS states eVspinatom1\n4Χ22Ε\nΧ11Ε\n/Minus2Χ12Ε\n4K/Slash1Ξ2\nFe\n/Minus3/Minus2/Minus1 0 1 205101520\nEF/LParen1eV/RParen14K/Slash1Ξ2/LParen11/Slash1eV/RParen1\n/SolidCircle/SolidCircle\n/SolidCircle/SolidCircleMn:/CapDeltac/Equal2/CapDeltas/Equal3 w/Equal0.08\nFe:/CapDeltac/Equal1.7/CapDeltas/Equal2.4 w/Equal0.06\nCo:/CapDeltac/Equal0.9/CapDeltas/Equal1.7 w/Equal0.1\nNi:/CapDeltac/Equal1/CapDeltas/Equal0.7 w/Equal0.12\n/SolidCircleDFTFe\nMnCoNi\n2 4 6 8 10051015\nNeK/LParen1meV/Slash1atom/RParen1FIG. 5: (Color online) (a) Schematic partial densities of states\nprojected on the 3 dstates of Fe 1asites. Orbital transitions\nand the sign of their contributions to the MAE are also shown.\nSolid line indicates positive contribution (easy axis) and the\ndashed line indicates negative contribution (easy plane) to\nthe easy-axis anisotropy. (b) Scaled MAE 4 K=\u00182fromT1a\nsite and its decomposition into orbital susceptibilities as func-\ntions of band \flling. (c) Magnetic anisotropy energy Kfrom\nT1asite as a function of T. Di\u000berent sets of electronic struc-\nture parameters \u0001 s, \u0001c, andware used to represent the\nDFT PDOS on T1asites in Li 2[(Li 0:5T0:5)N] for di\u000berent T\nelements.composition into orbital pair transitions as functions of\nthe Fermi energy are shown in Fig. 5(b). With T=Fe, the\nFermi level intersects the j\u00062;#istates, which results in\na large uniaxial anisotropy. Using Eq. (17), Fe 1ahas a\nMAE contribution which is of the order of 15 meV/Fe.\nAs shown in Fig. 5(b), for T= Fe, nearly all MAE contri-\nbutions are from the transitions j2;#i$j\u0000 2;#i, in other\nwords, between dx2\u0000y2anddxyorbitals in the minority\nspin channel.\nTo compare with the above analytical modeling, MAE\ncalculations were carried out in both VASP and all-\nelectron FP-LMTO. The di\u000berence of MAE values using\ntwo methods is less than 5% for T=Fe. To decompose\nthe MAE, we evaluate the SOC matrix element hVsoiand\nits anisotropy K(hVsoi), which can be easily decomposed\ninto sites, spins, and orbital pairs[27]. We found that\nK\u0019K(hVsoi)=2 for allTcompounds, which suggests\nthat second-order perturbation theory is a good approx-\nimation. As shown in Table I, for T=Fe, the total MAE\nis 20.8meV (per 24-atom cell) and MAE contributions\nfrom 1aand 2dsites are 14.77 and 3.09 meV/Fe, respec-\ntively. The contributions from Li and N atoms are nearly\nzero as expected. Thus, the impurity Fe (especially Fe 1a)\natoms are essentially the only MAE providers. By fur-\nther investigating the matrix element of SOC on the 1 a\nsite, we found that nearly all the MAE contributions\ncame from intra- jmjtransitions ofj2;#i$j\u0000 2;#i. As\nshown in Table II, the 4 \u001f\u000f\n22term (dominated by \u001f##\n22for\nT=Fe ) contributes 15.1 meV /Fe and the \u001f\u000f\n11term has a\nmuch smaller negative value of -0.42 meV /Fe, while other\nterms are negligible. Hence, DFT results agree with our\nmodel very well.\nWith magnetization along the cdirection, the SOC can\nlift the orbital degeneracy and shift two narrow bands\nm=\u00062, one below and the other above the Fermi level\ncompletely, with orbital quantum number mc=\u00062, re-\nspectively, where mcis the orbital quantum number in\nthe complex spherical harmonics. As a result, the density\nof states at the Fermi level becomes very small. Indeed,\nexperiments[15] found this system to be an insulator for\nT=Fe. It had been shown that [11{13, 36] the correlation\ne\u000bect further enhances the separation between occupied\nand unoccupied states. Using the LDA+ Umethod, we\nalso found that correlation can enhance the orbital mo-\nment when the spin is along the axial direction.\nFe concentration and site disordering can signi\fcantly\na\u000bect the MAE. As we have shown, the Fe 2dsites, which\nrepresent a high-doping concentration, have much lower\nanisotropy than the Fe 1asites, which represent a lower-\ndoping concentration. By replacing the Fe 2dsites back\nwith Li atoms in the hex2 supercell, we calculated the\nMAE with a smaller concentration x=0.166 and found\nthat MAE increase to 22 meV/Fe, which is in very good\nagreement with previous calculations.[12]. An interest-\ning concentration is x=0.33. If only one of two 2 dsites is\noccupied by Fe in the hex2 supercell, as shown in Fig. 3,\nthen this con\fguration would correspond to x=0.33 and\nthe supercell has two well-isolated Fe atoms. The DFT\n8calculation shows high MAE with a value of 20 meV/Fe.\nOn the other hand, if the two Fe atoms occupy the 2d\nsites and then are not well separated, the resulting MAE\nis much smaller (2.8 meV/Fe). Even if we assume that\nFe atoms tend to separate, with a concentration beyond\nx=0.33, it is unavoidable to have Fe atoms neighboring\neach other and the hybridization between them causes\nthe MAE (per Fe) to decrease. Furthermore, impurity\nsites are disordered, as found in experiments. At least at\na higher concentration, many Fe atoms would not have\nthe symmetric lateral surroundings as the two Fe sites do\nin thehex2 supercell we used in the calculations. This\nsite disordering may also have an e\u000bect on MAE by lower-\ning the point-group symmetry of Fe impurity sites. And\nthem=\u00062 states on Fe sites are no longer degenerate,\nwhich may decrease MAE per Fe.\nC. MAE in Li 2[(Li 1\u0000xTx)N] withT=Mn, Co and\nNi: The band-\flling e\u000bect\nFigure 5(a) shows how the Fermi level changes with\ndi\u000berentTelements in a simple rigid-band picture. Only\nthose transitions across the Fermi level contribute to\nMAE. With Telements other than Fe, the j\u00062;#i\nstates become either fully occupied or unoccupied. The\nlarge uniaxial anisotropy that originated from transition\nj2;#i$j\u0000 2;#i(term 4\u001f##\n22) vanishes and other transitions\nbecomes important, depending on the position of the\nFermi level. For T=Ni, the Fermi level intersects the de-\ngeneratej\u00061;#istates. Hence anisotropy contributions\nare dominated by the transitions j1;#i$j\u0000 1;#i(term\n\u001f##\n11). This transition promotes the uniaxial anisotropy,\nas 4\u001f##\n22does forT=Fe. ForT= Co, the Fermi level is\nbetweenj\u00062;#iandj\u00061;#ipeaks. The transitions of\nj\u00062;#i$j\u0006 1;#i(term -3\u001f##\n12) andj\u00061;\"i$j\u0007 1;#i\n[term -(\u001f\"#\n11+\u001f#\"\n11)] support easy-plane anisotropy, while\nthe transitionj\u00062;\"i$j\u0006 1;#i(term 3\u001f#\"\n12) promotes\neasy-axis anisotropy. However the two bands involved in\nthe last transition are far away from each other and this\ncontribution is relatively small. Hence, for T=Co, one\nshould expect the system to have easy-plane anisotropy.\nForT=Mn, there are four transitions that contribute to\nthe MAE; all of them are between the two spin channels,\nin which two inter- jmjtransitionsj\u00061;\"i$j\u0006 2;#i(term\n3\u001f\"#\n12) andj\u00062;\"i$j\u0006 1;#isupport easy-axis anisotropy,\nwhile two other intra- jmjtransitionsj\u00061;\"i$j\u0007 1;#iand\nj\u00062;\"i$j\u0007 2;#i[term -4(\u001f\"#\n22+\u001f#\"\n22)] support easy-plane\nanisotropy. The four transitions compete and the sign of\nthe total MAE is not obvious and requires a more quan-\ntitative description.\nThe SOC constant \u0018changes with element. In\nFig. 5(b), we plot the scaled MAE ~K=K=4\u00182and its\norbital-resolved components as functions of the Fermi\nlevel by using parameters of \u0001 s, \u0001c, andwforT=Fe. In\na rigid-band picture, it clearly shows that Ni also has\na uniaxial anisotropy with contributions coming fromthe\u001f##\n11term. Since we are using the same half width\nwof LDOS for m=\u00061 andm=\u00062 subbands, we have\n~KNi\u00191\n4~KFebecause of the intra- jmjtransitions coef-\n\fcientsm2, as shown in Eqs. (5) and (9). Figure 5(c)\nshows the MAE Kas a function of the number of oc-\ncupied electrons by using di\u000berent sets of \u0001 s, \u0001c, and\nwparameters to better present DFT-calculated PDOS\nfor di\u000berent Telements, as shown in Fig. 4. The SOC\nconstant\u0018is interpolated by using DFT-calculated \u0018val-\nues for 3delements. Since \u0018decreases with the atomic\nnumber within a given nlshell,Kquickly decreases with\nsmaller atomic numbers due to the factor \u00182. The DFT\nMAE values are also plotted to compare with the mod-\neling MAE function. As shown in Fig. 5(c), with T=Fe\nparameters, the modeling MAE (Fe rigid-band approx-\nimation) can already correctly describe the MAE trend\nwith di\u000berent Telements.\nAlthough the RBA predicts the correct easy-axis di-\nrection for T=Ni, the di\u000berence between RBA mod-\neling and DFT is rather large. In RBA modeling,\nKNi=KFe=(\u0018Ni=\u0018Fe)2=4\u00190.6, while the DFT value\n(1.71meV /atom) for T=Ni is about one order of mag-\nnitude smaller than for T=Fe. This can be explained\nas follows. First, we use the same band width for all\nDOS peaks in our modeling. In fact, the j\u00061;#ibands\nare much broader than the j\u00062;#ibands. The easy-\naxis anisotropy contribution from the transition between\nj\u00061;#istates decreases with increasing band width. Sec-\nond, the Ni PDOS deviates from the Fe PDOS more than\nMn or Co, so RBA is less appropriate for T=Ni. The\nspin splitting \u0001 sand crystal-\feld splitting \u0001 care much\nsmaller in Ni than in Fe. This causes the amplitudes of\nthe negative contributions from j\u00062;#i$j\u0006 1;#iand\nj\u00061;\"i$j\u0007 1;#ito become larger and decrease the total\nuniaxial anisotropy. As shown in Fig. 5(c), if we use a\nsmaller \u0001s, smaller \u0001c, and larger wto better represent\nthe Ni PDOS calculated from DFT calculations, then\nmuch better agreement between model and DFT values\ncan be reached.\nForT=Co, the model MAE is about twice the DFT\nvalue, probably because of the simpli\fed model DOS.\nThe orbital-resolved T1aMAE calculated in DFT are\nsummarized in Table II. Overall, there is a qualitative\nagreement between DFT and the analytical model for\nthe orbital-resolved MAE values for all Telements. It\nis interesting that with T=Co, the contribution of the\n4\u001f\u000f\n22term is comparable to that of \u00002\u001f\u000f\n12and\u001f\u000f\n11in\nDFT, which is not expected in the model. As shown in\nFig. 4(c), there is a small portion of unoccupied j\u00062;#i\nstates right above the Fermi level in the minority spin\nchannel, which makes the 4 \u001f##\n22terms comparable to oth-\ners. However, this electronic structure detail is not con-\nsidered in the simpli\fed DOS we use in modeling. If we\nneglect the 4 \u001f\u000f\n22terms in DFT, then a better agreement\nbetween modeling and DFT can be achieved for T=Co.\nThus, the contributions from well-separated impurity\nsites withTcan be well understood. For T=Mn and\nCo, the easy-plane anisotropy is a result of competition\n9TABLE II: Orbital-resolved MAE from the T1asite in\nLi2[(Li 0:5T0:5)N] withT=Mn, Fe, Co, and Ni.\nK(meV)\nTerm Orbital Transition Mn Fe Co Ni\n4\u001f\u000f\n22dxy,dx2\u0000y2 -0.86 15.10 0.71 -0.03\n\u001f\u000f\n11dyz,dxz -0.22 -0.42 -0.78 3.68\n-2\u001f\u000f\n12dyz,dxz,dxy,dx2\u0000y20.73 -0.18 -0.81 0.09\n-3\u001f\u000f\n01dz2,dyz,dxz 0.03 0.08 -0.01 -0.25\nbetween di\u000berent transitions, instead of being dominated\nby the intra-jmjtransition, which strongly depends on\nthe bandwidth of the degenerate j\u0006mistates that are\nintersected by the Fermi level. As a result, the band-\nnarrowing e\u000bect on MAE is not as strong as for T=Fe or\nNi. As shown in Table I, the contributions from 2 dsites\nare comparable or even larger than 1 asites forT=Mn\nand Co.\nIV. SUMMARY AND CONCLUSION\nBased on second-order perturbation theory, MAE is\nresolved into contributions from di\u000berent pairs of or-\nbital transitions, more precisely, the di\u000berence between\nspin-parallel and spin-\rip components of the orbital sus-\nceptibilities of the corresponding orbital pair. In the\nLi2[(Li 1\u0000xTx)N] systems, with T=Mn, Fe, Co, and Ni,\nthe linear geometry of the Tsites minimizes the in-plane\nhybridization and results in atomic like orbitals around\nthe Fermi level for all Telements. The MAE oscillates\nwith the atomic number from T=Mn toT=Ni, which is\na result of the competition between contributions fromall allowed orbital transitions. As the Fermi level evolves\nwithT, di\u000berent orbital pair transitions dominate the\ncontribution to MAE. For T=Fe andT=Ni, the intra-jmj\ntransitions within the minority spin channel dominate\nthe MAE contribution and result in a uniaxial anisotropy.\nForT=Mn and Co, the easy-plane anisotropy is a result\nof the competition between contributions from several\ntransitions with di\u000berent signs. Using Lorentzian den-\nsity of states, we investigate the band-\flling e\u000bect on\nMAE in an analytical model based on a Green's function\ntechnique. We show the MAE as a continuous function\nof atomic number. This analytical model can already\ndescribe the correct trend of the MAE obtained using\nDFT, by just using a simple rigid Fe band picture. If we\ntake into account the deviation from the rigid Fe band\nmodel and some details of DFT electronic structure, an\neven better agreement between the model and DFT can\nbe found. To further validate our modeling analysis, we\nalso calculate the orbital-resolved MAE by evaluating the\nSOC matrix element in DFT. Overall, Li 2[(Li 1\u0000xTx)N],\nwithT=Mn, Fe, Co, and Ni, is a unique system which\nclearly shows the band-\flling e\u000bect on MAE and the na-\nture of this e\u000bect can be understood in a very simple\nmodel.\nAcknowledgement\nWe would like to thank A. Jesche, P. Can\feld, V.\nAntropov, A. Chantis, B. Harmon, T. Ho\u000bmann, and\nD. Johnson for helpful discussions. Work at Ames Lab-\noratory was supported by the US Department of En-\nergy, Energy E\u000eciency and Renewable Energy, Vehicles\nTechnology O\u000ece, Advanced Power Electronics and Elec-\ntric Motors program, under Contract No. DE-AC02-\n07CH11358.\n[1] I. G. Rau, S. Baumann, S. Rusponi, F. Donati,\nS. Stepanow, L. Gragnaniello, J. Dreiser, C. Piamonteze,\nF. Nolting, S. Gangopadhyay, et al., Science 344, 988\n(2014).\n[2] R. McCallum, L. Lewis, R. Skomski, M. Kramer, and\nI. Anderson, Annual Review of Materials Research 44,\n451 (2014).\n[3] J. Cirera, E. Ruiz, S. Alvarez, F. Neese, and J. 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Belashchenko1\n1Department of Physics and Astronomy and Nebraska Center for Materials and Nanoscience,\nUniversity of Nebraska-Lincoln, Lincoln, Nebraska 68588, USA\n2Ames Laboratory, U.S. Department of Energy, Ames, Iowa 5001 1, USA\nThe origins of the anomalous temperature dependence of magn etocrystalline anisotropy in\n(Fe1−xCox)2B alloys are elucidated using first-principles calculation s within the disordered local\nmoment model. Excellent agreement with experimental data i s obtained. The anomalies are asso-\nciated with the changes in band occupations due to Stoner-li ke band shifts and with the selective\nsuppression of spin-orbit “hot spots” by thermal spin fluctu ations. Under certain conditions, the\nanisotropy can increase, rather than decrease, with decrea sing magnetization due to these peculiar\nelectronic mechanisms, which contrast starkly with those a ssumed in existing models.\nMagnetocrystallineanisotropy(MCA) isoneofthe key\nproperties of a magnetic material [1]. Understanding of\nits temperature dependence is a challenging theoretical\nproblem with implications for the design of better mate-\nrials for permanent magnets [2], heat-assisted magnetic\nrecording [3], and other applications. While the MCA\nenergyKusually declines monotonically with increasing\ntemperature as predicted by simple models [4], in some\nmagnets it behaves very differently and can even increase\nwith temperature. Such anomalous K(T) dependence\nmakes some materials useful as permanent magnets and\ncan potentially facilitate specialized applications.\nWell-known anomalies in the temperature dependence\nof MCA include spin reorientation transitions (SRT) in\ncobalt [5] and MnBi [6], which have been attributed to\nthermal expansion; an SRT in gadolinium, which may be\ndue to higher-order terms in MCA [7]; SRT in R 2Fe14B\nhard magnets [8] due to the ordering of the rare-earth\nspins at low T; and SRT in thin films [9, 10] associated\nwith the competition between the bulk and surface con-\ntributions to MCA. Competition between single-site and\ntwo-site MCA can also lead to an SRT [11].\nMCA in metallic magnets is rarely dominated by the\nsingle-ionmechanismleadingtothe K∝M3dependence\non the magnetization [4]. For example, two-ion terms in\n3d-5dalloys like FePt modify this dependence to K∝\nM2.1[12, 13]. Clear understanding of the anomalous\ntemperature dependence of MCA has been so far limited\nto the cases when competing contributions to MCA can\nbesortedoutin realspace,suchas, forexample, bulk and\nsurface terms in thin films. In contrast, understanding of\nMCA in itinerant magnets usually requires a reciprocal\nspace analysis [14].\nOne such system is the disordered substitutional\n(Fe1−xCox)2B alloy, which exhibits three concentration-\ndriven SRTs at T= 0, a high-temperature SRT at the\nFe-rich end, and a strongly non-monotonic temperature\ndependence at the Co-rich end with a low-temperature\nSRT [15, 16]. The SRT’s at T= 0 were traced down\nto the variation of the band filling with concentrationcombined with spin-orbital selection rules [16]. Here we\nelucidate the unconventional mechanisms leading to the\nspectacular anomalies in the temperature dependence of\nMCA in this system and show that they stem from the\nchanges in the electronic structure induced by spin fluc-\ntuations. We will see that under certain conditions MCA\ncan increase, rather than decrease, with decreasing mag-\nnetization due to these mechanisms.\nOur calculations employ the Green’s function-based\nlinear muffin-tin orbital method [17] with spin-orbit cou-\npling (SOC) included as a perturbation to the potential\nparameters [16, 18]. Thermal spin fluctuations are in-\ncluded within the disorderedlocal moment (DLM) model\n[19, 20], which treats them within the coherent potential\napproximation (CPA) on the same footing with chemical\ndisorder. The DLM method has been previously used\nto calculate the K(T) dependence in systems like FePt\n[21, 22] and YCo 5[23]. Although K(T) in these metals\ndoes not follow the Callen-Callen model [4] designed for\nmaterials with single-ion MCA, it still decreases mono-\ntonically. In contrast, we will see that the changes in\nthe electronic structure with temperature lead to strong\nanomalies in (Fe 1−xCox)2B. Our implementation of the\nDLM method is described in Ref. 24. (See Supplemental\nMaterial [25] for additional details.)\nApart from the inclusion of spin disorder, the com-\nputational details are similar to Ref. 16. In particular,\nthe large overestimation of the magnetization in density-\nfunctional calculations for Co 2B (1.1µBcompared to\nexperimental 0.76 µBper Co atom) is corrected by scal-\ning the local part of the exchange-correlationfield for Co\natoms by a factor 0.8 at all concentrations. This treat-\nment is consistent with spin-fluctuation theories showing\nthat spin fluctuations tend to reduce the effective Stoner\nparameter [26, 27] and allows us to take into account the\nresulting changes in the electronic structure.\nMagnetism in (Fe 1−xCox)2B alloys is much more itin-\nerant compared to systems like FePt; the spin moments\nof Fe and, especially, Co atoms are not rigid in density-\nfunctional calculations. To implement spin disorder2\nwithin the DLM method, we make a simple assumption\nthat the spin moments of both Fe and Co at finite Tcan\nbe taken from the ferromagnetic state at T= 0. This as-\nsumption is based on the expectation that thermal spin\nfluctuations to a large extent restore the “soft” spin mo-\nments [26]. On the other hand, the variation of the elec-\ntronic structure with Tshould not be very sensitive to\nthe details of the spin fluctuation model. For simplicity,\na similar approach is used for the (Co 1−xNix)2B system,\nincluding the small spin moments on the Ni atoms.\nThe distribution functions for spin orientations are\ntaken in the Weiss form: pν(θ)∝exp(ανcosθ), where\nθis the angle made by the spin with the magnetization\naxis, and νlabels the alloy component. The temperature\ndependence of the coefficients ανis determined using the\ncalculated effective exchange parameters, as explained in\nthe Supplemental Material [25]. Fermi-Dirac smearing\nis neglected, because the effects of spin fluctuations are\noverwhelmingly stronger.\nThe results of K(x,T) calculations shown in Fig. 1,\nwhich were obtained with temperature-independent lat-\ntice parameters, are in excellent agreement with experi-\nmental data [15]. Both the thermal SRT at the Fe-rich\nend and the non-monotonic temperature dependence at\nthe Co-rich end in (Fe 1−xCox)2B alloys are captured (see\nSupplemental Material [25] for a direct comparison). For\n(Co0.9Ni0.1)2B the MCA energy at T= 0 is large and\nnegativein agreementwith experiment [15], although the\ninitial decline of K(T) similar to Co 2B is not observed\nin experiment. The finite slope in the K(T) curves at\nzero temperature is due to the classical treatment of spin\nfluctuations. We have explicitly verified that the effect of\nthermal expansion on K(T) in Fe 2B and Co 2B is almost\nunnoticeable.\nThe effects of spin disorder on the electronic structure\ncan be understood from Fig. 2, which shows the partial\nminority-spinBlochspectralfunction at x= 0.95forT=\n0 andT/TC= 0.7. Here, at the Co-rich end, all bands\nare easily identifiable and relatively weakly broadened at\nT/TC= 0.7. In addition, they are shifted down relative\nto their positions at T= 0, which is a hallmark of an\nitinerant Stoner system. In contrast, at the Fe-rich end\nthe bands are strongly broadened by spin fluctuations,\nso that most bands in the 1 eV window below EFare\nbarelyvisible(seeSupplementalMaterial[25]). Thelarge\ndifference in the degree of band broadening between the\nFe-rich and Co-rich ends is due to the 2.5-fold difference\nin the magnitude of the spin moments. The effect of\nphonon scattering on band broadening in (Fe 1−xCox)2B\nalloys is likely much smaller and is neglected here.\nThe usual expectation is that spin disorder should re-\nduce MCA as a result of averaging over spin directions.\nSuch normal behavior is seen, for example, at x= 0.3\nin Fig. 1. This expectation is violated at many concen-\ntrations: K(x,T) is non-monotonic with respect to Tat\n0≤x≤0.2, 0.5≤x≤0.6, and 0.9≤x≤1; we will\nFIG. 1. Calculated temperature dependencies of MCA energy\nKin (Fe 1−xCox)2B and (Co 0.9Ni0.1)2B alloys.\ncall this behavior anomalous. At x≤0.6 the anomalous\ntemperature dependence of Kat a given xfollows the\nvariation of Kwith increasing xatT= 0. For exam-\nple,K(0.2,0)> K(0.1,0), andK(0.1,T) anomalously\nincreases with T. Atx≥0.9 the anomalous variation is\nopposite to the trend in K(x,0) with increasing x. To\nunderstand this difference, we first need to examine the\neffect of disorder on MCA.\nFig. 3 compares K(x,0) calculated within the vir-\ntual crystal approximation (VCA) with CPA results for\n(Fe1−xCox)2B [16] and (Co 1−xNix)2B systems [28]. Note\nthat in the (Co 1−xNix)2B system the spin moments van-\nish near 40% Ni, in agreement with experiment [29]. In\naddition to the MCA energy K, Fig. 3 also shows its\napproximate spin decomposition Kσσ′obtained from the\nSOC energy [16, 25]. Because the 3 dshell in this system\nis more than half filled, the variation of MCA with xis\nlargelycontrolled by the K↓↓term, i.e., by the LzSzmix-\ning of the minority-spin states. Substitutional disorder\nstrongly suppresses MCA, an effect that was also found\nin tetragonal Fe-Co alloys [30]. The suppression is due\nto band broadening, which reduces the efficiency of spin-\norbital selection rules. Importantly, bands broaden at\ndifferent rates; the contributions to MCA from the bands\nthat lie close to EFand broaden strongly are most effec-\ntively suppressed. The dispersive majority-spin bands\nare weakly broadened, and hence the K↑↑term is almost\nunaffected by disorder; in contrast, K↓↓is strongly re-3\nFIG. 2. Partial minority-spin spectral function for the\ntransition-metal site in (Fe 0.05Co0.95)2B at (a) T= 0, and\n(b)T/TC= 0.7. SOC is included, M/bardblz, and energy is in\neV. Color encodes the orbital character of the states. The\nintensities of the red, blue and green color channels are pro -\nportional to the sum of m=±2 (xyandx2−y2), sum of\nm=±1 (xzandyz), andm= 0 (z2) character, respectively.\nduced. We note that although band broadening (and\nthereby MCA) can depend on chemical short-range or-\nder, the latter is expected to be negligible in the present\nalloy with chemically similar constituents.\nThe strongest suppression of MCA can be expected for\nthe “hot spots” appearing when nearly degenerate bands\natEFare split by SOC [14]. A clear example of such\nbands is seen near the Γ point in Fig. 2a. The effect\nof disorder is further illustrated in Fig. 4 showing the\nspectral function at the Γ point for two orientations of\nthe magnetization at x= 1, 0.9, and 0.8, all at T= 0.\nAtx= 1 there is no disorder, and the sharp bands are\nfully split by SOC for M∝bardblz. With the addition of Fe,\nthe broadeningquickly exceeds the originalSOC-induced\nsplitting, and the effect of SOC is strongly suppressed.\nDisorderhasa similareffect on the mixing ofelectronic\nbands of opposite spin by L+S−andL−S+. Indeed,\nwhile in Fig. 2a for T= 0 the anticrossings with the\nmajority-spin bands are clearly visible, in Fig. 2b, for\nT/TC= 0.7, they are almost completely suppressed.\nWe now return to the analysis of the anomalous tem-FIG. 3. MCA in (Fe 1−xCox)2B and (Co 1−xNix)2B alloys\ncalculated within VCA (empty circles) compared with CPA\n(filled circles). The spin decomposition is given for VCA.\n(a)\n(b)\n-1.5 -1 -0.5 0 0.5\nE, eV(c)\nFIG. 4. Spectral functions at the Γ point at (a) x= 1, (b)\nx= 0.9, (c)x= 0.8. Solid lines: M/bardblz. Dashed lines:\nM/bardblx. A small imaginary part is added to energy to resolve\nthe bands in panel (a).\nperature dependence of K. We expect that these anoma-\nlies come from the effects of thermal spin fluctuations on\nthe electronic structure beyond a simple averaging over\nspin directions. As we saw in Fig. 2, there are two such\neffects in (Fe 1−xCox)2B: reduction of the exchange split-\nting ∆, and band broadening. The reduction of ∆ shifts\nthe minority-spin bands downward relative to EF, just\nas the band filling with increasing xdoes. Band broad-\nening has a stronger effect on the minority-spin states,\nwhereEFlies within the relatively heavy 3 dbands, and\nit is particularly important for nearly degenerate bands\nstraddling the Fermi level, as we saw in Fig. 4.\nTo understand how these effects lead to to the anoma-\nlies inK(T), it is convenient to examine two quantities,4\nK↑andK↓, defined as Kσ=/integraltextE0(E−E0)∆Nσ(E)dE,\nwhereE0is the Fermi energy in the absence of SOC, and\n∆Nσis the difference, between M∝bardblxandM∝bardblz, in\nthe partial density of states for spin σin the global ref-\nerence frame. Their sum K↑+K↓closely approximates\nK, and their analysis can help identify the contributions\nof different bands to K, particularly in combination with\nreciprocal-space resolution [16, 25].\nFig. 5a shows the temperature dependence of Kσin\nFe2B. Since the spin-mixing contribution K↑↓here is\nsmall (Fig. 3), K↑andK↓provide information similar to\nK↑↑andK↓↓atT= 0 while retaining clear meaning at\nfinite temperature [25]. We see that K↓decreasesquickly\nwith increasing T. This happens because the downward\nshift and broadening of the minority-spin bands strongly\nsuppress the negative minority-spin contribution to K.\nIn contrast, the initial increase in K↑mirrors the upward\nslope ofK↑↑(x,0) as a function of x[16], which occurs as\nthe majority-spin bands shift upward relative to EFwith\ndecreasing ∆. At elevated temperatures the majority-\nspin contribution becomes dominant, and Kundergoes\nan anomalous sign change, i.e., a spin-reorientation tran-\nsition.\nFIG. 5. Contributions to Kin (a) Fe 2B and (b)\n(Fe0.05Co0.95)2B from different spins ( K↑andK↓).K+\nσand\nK−\nσin panel (b): total positive and negative contributions to\nKσcoming from different kpoints. (Dotted lines show K+\n↑,\nK−\n↑.)\nAt the Co-rich end the situation is complicated by the\npresenceoflargecontributionsofoppositesign that comefrom the minority-spin states in different regions of the\nBrillouin zone [16]. Near the Γ point there is a large\npositive contribution from the degenerate bands that are\nmixed by Lz. There is also a large negative contribution\nfromthe mixingofminority-spinbandsofoppositeparity\nwithrespectto σzreflection, whichisdistributed overthe\nwholeBrillouinzone. Tohelp resolvethesecontributions,\nFig.5b for(Fe 0.05Co0.95)2B shows, in addition to Kσ, the\ntotal positive ( K+\nσ) and negative ( K−\nσ) contributions to\nKσ, which were sorted by wave vector. Fig. 6 displays\nk-resolved K↓on the ΓMXplane at T= 0 and T/TC=\n0.7. The bright red ring around the Γ point in Fig. 6\nis the hot spot coming from the two nearly-denegerate\nbands that are split by SOC (see Fig. 2a and 4).\nAs seen in Fig. 6, thermal spin disorder strongly sup-\npresses the hot spot observed at T= 0: it is strongly\nwashed out at T/TC= 0.7, while the contributions from\nother regions decline almost homogeneously. This effect\nis similar to that of chemical disorder (Fig. 4). As a re-\nsult,K+\n↓declines faster compared to other contributions\nshown in Fig. 5b, and the negative value of Kgrows\nanomalously with T.\nInterestingly, while in VCA the maximum in K(x,0)\nwith respect to band filling occurs near x= 0.95 (Fig. 3),\nin CPA there is a cusped maximum exactlyin Co2B. The\nlatter is due to the fact that the bands are broadened by\ndisorder with any admixture, reducing the positive con-\ntribution from the hot spots. This dominant effect of\ndisorder explains why, as noted above, the anomalous\nK(T) dependence at x≥0.9 is opposite to the trend\nexpected from increasing x, which holds at other con-\ncentrations. In Co 2B, where the positive contribution is\nat its maximum, both band broadening and decreasing\n∆ contribute to the anomalous decrease in K(T), as the\nnearly degenerate bands broaden and sink below EF.\nFIG. 6. Wave vector-resolved K↓(units of meV a3\n0, wherea0is\nthe Bohr radius) on the ΓMX plane in (Fe 0.05Co0.95)2B alloy\natT= 0 (upper left) and T/TC= 0.7 (lower right).5\nIn conclusion, we found that the anomalous tempera-\nture dependence of MCA in (Fe 1−xCox)2B alloys is due\nto the changesin the electronic structure induced by spin\nfluctuations. Thisunconventionalmechanismcanbehar-\nnessed in applicationswheretemperature-independent or\nincreasing MCA is required.\nThe work at UNL was supported by the National Sci-\nence Foundation through Grant No. DMR-1308751 and\nperformed utilizing the Holland Computing Center ofthe\nUniversity of Nebraska. Work at Ames Lab was sup-\nported in part by the Critical Materials Institute, an En-\nergy Innovation Hub funded by the US DOE and by the\nOffice of Basic Energy Science, Division of Materials Sci-\nence and Engineering. Ames Laboratory is operated for\nthe US DOE by Iowa State University under Contract\nNo. DE-AC02-07CH11358.\n[1] J. Stohr and H. Siegmann, in Magnetism: From Funda-\nmentals to Nanoscale Dynamics (Springer, Berlin, 2006),\np. 805.\n[2] L. H. Lewis and F. Jim´ enez-Villacorta, Metall. Mater.\nTrans. A 44, 2 (2013).\n[3] M. H. Kryder, E. C. Gage, T. W. McDaniel, W. A. Chal-\nlener, R. E. Rottmayer, G. Ju, Y.-T. Hsia, and M. F.\nErden, Proc. IEEE 96, 1810 (2008).\n[4] H. 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Fiz. 63,\n356 (1972) [Sov. Phys. JETP 36, 188 (1973)].\n[15] A. Iga, Jpn. J. Appl. Phys. 9, 415 (1970).\n[16] K. D. Belashchenko, L. Ke, M. D¨ ane, L. X. Benedict, T.\nN. Lamichhane, V. Tarfour, A. Jesche, S. L. Bud’ko, P.\nC. Canfield, and V. P. Antropov, Appl. Phys. Lett. 106,\n062408 (2015).\n[17] I. Turek, V.Drchal, J. Kudrnovsk´ y,M. ˘Sob, andP.Wein-\nberger,Electronic Structure of Disordered Alloys, Sur-\nfaces, and Interfaces (Kluwer, Boston, 1997).\n[18] I. Turek, V. Drchal, and J. Kudrnovsk´ y, Philos. Mag. 88,\n2787 (2008).\n[19] T. Oguchi, K. Terakura, and N. Hamada, J. Phys. F:\nMet. Phys. 13, 145 (1983).\n[20] B. L. Gy¨ orffy, A. J. Pindor, J. B. Staunton, G. M. Stocks,\nand H. Winter, J. Phys. F: Met. Phys. 15, 1337 (1985).\n[21] J. B. Staunton, S. Ostanin, S. S. A. Razee, B. L. Gyorffy,\nL. Szunyogh, B. Ginatempo, and E. Bruno, Phys. Rev.\nLett.93, 257204 (2004).\n[22] J. B. Staunton, L. Szunyogh, A. Buruzs, B. L. Gyorffy,\nS. Ostanin, and L. Udvardi, Phys. Rev. B 74, 144411\n(2006).\n[23] M. Matsumoto, R. Banerjee, and J. B. Staunton, Phys.\nRev. B90, 054421 (2014).\n[24] B. S. Pujari, P. Larson, V. P. Antropov, and K. D. Be-\nlashchenko, Phys. Rev. Lett. 115, 057203 (2015).\n[25] See Supplemental Material for technical details and ad -\nditional figures.\n[26] T. Moriya, Spin Fluctuations in Itinerant Electron Mag-\nnetism(Springer, Berlin, 1985).\n[27] V. P. Antropov and A. Solontsov, J. Appl. Phys. 109,\n07E116 (2011).\n[28] The scaling of the exchange-correlation field in CPA is\nused only for Co. In VCA calculations the scaling factor\nis interpolated, similar to the nuclear charge.\n[29] K. H. J. Buschow, in Boron and Refractory Borides , ed.\nV. I. Matkovich (Springer, Berlin, 1977).\n[30] I. Turek, J. Kudrnovsk´ y, and K. Carva, Phys. Rev. B 86,\n174430 (2012).6\nSUPPLEMENTAL MATERIAL\nDESCRIPTION OF SPIN DISORDER\nAll calculations were performed using our implementa-\ntion[S1]ofthedisorderedlocalmoment(DLM) modelfor\npartially ordered magnetic states. Thermal spin disorder\nwas introduced as follows. Fig. S1 shows the effective\nexchange parameters Jµ\n0=∂2E/∂θ2\nµfor both compo-\nnents (µ= Fe, Co) calculated at all concentrations using\nthe linear response technique [S2]. These parameters de-\nscribe the exchange interaction of an atom of a given\ntype with the rest of the crystal. Within the mean-field\napproximation (MFA), these data predict the Curie tem-\nperatures TCof pure Fe 2B and Co 2B to be 1570 K and\n290 K, which can be compared with experimental values\nof 1013 K and 429 K, respectively [S3].\nFIG. S1. Effective exchange parameters Jµ\n0for Fe and Co\n(dashed lines) and calculated Curie temperature TC(solid\nline) as a function of concentration. Blue triangles: exper i-\nmental data for TC[S3].\nFor Fe 2B theTCis overestimatedby about 35%, which\ncan largely be attributed to the neglect of short-range\norder in MFA. Indeed, linear response calculations show\nthat the spin of a Fe atom is strongly ferromagnetically\ncoupled to seven neighbors, five of which are within the\nlayer. Although the couplings to more distant atoms are\nnot negligible, they alternate in sign and contribute little\nto the effective exchange parameter. Thus, MFA can be\nexpected to overestimate TCby about as much as it does\n(about23%[S4])forthebcclatticewithnearest-neighbor\nexchange(coordination number 8). In contrast, for Co 2B\ntheTCis underestimated by about a factor 1.5. This sit-\nuation appears to be similar to the well-known case of\nfcc Ni, where TCcalculated from J0is underestimated\nby nearly a factor of 2 [S5]. The similarity is due to the\nfact that both Ni and Co 2B are strongly itinerant fer-\nromagnets with relatively small local moments. In such\nsystems the effects of quantum spin fluctuations become\nsignificant, which for Co 2B is also reflected in the large\noverestimationofthe magnetizationin density-functional\ncalculations. In addition, the long-wave approximation\ninherent in the linear-response calculation of J0becomesunreliable [S6].\nFor our present problem, it is important to capture\nthe gradual increase of the spin disorder with increas-\ning temperature leading to decreased exchange splitting\nand band broadening. It is likely that these effects are\nnot sensitive to the moderate uncertainties in the calcu-\nlation of Jµ\n0. The latter only affect the relative degree\nof disordering for Fe and Co, while the most interest-\ning anomalies in K(x,T) occur close to pure Fe 2B and\nCo2B.Therefore, weadoptthe followingschemebasedon\nthe values of Jµ\n0calculated above. The MFA equations\nfor a two-component alloy described within the Heisen-\nberg model contain four component-resolved parameters\nJµνdefined as the exchange interaction of the spin of\natom type µwith atoms of type νeverywhere else in the\ncrystal. Introducing the pair exchange parameters Jij\nµν,\nwe have Jµν=xν/summationtext\njJij\nµν=xν˜JµνandJµ\n0=/summationtext\nνJµν,\nwherexνis the concentration of the component ν. In\nour case this translates to JF\n0= (1−x)˜JFF+x˜JFCand\nJC\n0= (1−x)˜JFC+x˜JCC, where we took into account\nthat˜JFC=˜JCF(F stands for Fe and C for Co). We\nfurther fit the concentration dependence of ˜JFCand˜JCC\nto a linear and ˜JFFto a quadratic polynomial in xso as\nto best approximate the concentration dependence of JF\n0\nandJC\n0in Fig. S1. This fitting gives ˜JFF= 406+220 x2,\n˜JFC= 225−11x, and˜JCC= 116−41xin meV units.\n(The linear term in ˜JFFis negligibly small.)\nThe coefficients αµin the distribution functions for\nspin orientations are determined from the solution of the\nMFA matrix equation T¯α=˜J¯m, where ¯αis a column\nvector with two elements αµ,˜Jis a 2×2 matrix with el-\nements˜Jµν, and ¯misacolumnvector(ofreducedcompo-\nnent magnetizations)with elements xµL(αµ),L(α) being\nthe Langevin function. The Curie temperature TCcor-\nresponds to the vanishing of both αµ. The MCA energy\nKis calculated as a function of xandT/TC. To facili-\ntate the comparisonwith experimental data, the K(x,T)\ncurves in Fig. 1 of the main text are plotted using the ex-\nperimental values of TCfor each concentration [S7].\nIn the (Co 0.9Ni0.1)2B system the spin moments of Ni\natomsarequitesmall, andinprincipletheyshouldnotbe\ntreatedwithin the DLMmodel. However,suchtreatment\ndoes not introduce significant errors, because disorder of\nthese small spin moments has a negligible effect on the\nelectronic structure. Therefore, in this system we treated\nboth Co and Ni within the DLM model with identical\nvalues of αfor Co and Ni. In this way the system auto-\nmatically tends to a paramagnetic state at α→0, and\nthe self-consistent calculation of the local moment of Ni\nat each temperature is avoided.7\nSPIN-ORBIT COUPLING AND\nMAGNETOCRYSTALLINE ANISOTROPY\nThe spin-orbit coupling (SOC) is included as a per-\nturbation of the potential parameters in the Green’s\nfunction-based tight-binding linear muffin-tin orbital\n(LMTO) method [S8, S9], and the resulting band prob-\nlem is solved exactly. This is analogous to the so-called\n“pseudoperturbative” treatment of SOC in conventional\nband-structure methods, in which the eigenvalues of the\nHamiltonian perturbed by SOC are found exactly [S10–\nS12]. The MCA energy is calculated as the difference in\nthe single-particle energy for two directions of the mag-\nnetization, M∝bardblxandM∝bardblz, calculated with the same\npotential parameters and distribution functions for spin\norientations.\nThe coherent-potential equations involve an integra-\ntion over the orientations of the spin local moment on\neach magnetic atom treated within the DLM model. For\ncollinear magnetic orderings in the non-relativistic case\nthe axial spin symmetry with respect to the direction of\nthe magnetic order parameter is retained, and the in-\ntegration over the azimuthal angle can be handled an-\nalytically [S1]. SOC breaks this symmetry, and a full\nintegration over the sphere needs to be taken. Here we\nused an 88-point quadrature for this integration, which\nwas found to provide well-converged results. Thus, for-\nmally we apply the coherent potential approximation to\nan 176-component alloy (88 orientations of the local mo-\nment for Fe and Co). Each such component is described\nby LMTO potential parameter matrices which are first\ncalculated in the reference frame of the local moment\nand then rotated to the prescribed direction by a rota-\ntion operator generated by the total angular momentum\noperator ˆJ[S1].\nIn the analysis of the underlying mechanisms of MCA,\nwe employ two approximate decompositions of K. The\nfirst one utilizes the calculated anisotropy of the SOC\nenergy:\nKSO=−1\n2π∆/summationdisplay\nσσ′ImEF/integraldisplay\nVσσ′Gσ′σdE (1)\nHere and in the following we use the notation ∆ Ato\ndenote the difference in the values of a quantity Afor\nmagnetization oriented along xand along z.Vis the\nperturbing SOC operator, and Gthe Green’s function\ncalculated with SOC. The quantity KSOis naturally sep-\narated in four spin contributions Kσσ′using the identity\n2∝angbracketleftSL∝angbracketright=∝angbracketleftLz′∝angbracketright↑↑−∝angbracketleftLz′∝angbracketright↓↓+∝angbracketleftL+∝angbracketright↓↑+∝angbracketleftL−∝angbracketright↑↓[S8]. The\nanalysis of KSOis useful, because it approximates K\nwell unless second-order perturbation theory is strongly\nviolated [S13]; in (Fe 1−xCox)2B the concentration de-\npendence of KSOagrees very well with that of K. This\ndecomposition, however, loses its utility at finite temper-\natures.The second decomposition is defined as follows. First,\nwe can write the single-particle energy, calculated with\nSOC included, for magnetization direction nas\nEn\nsp=En\nF/integraldisplay\nENn(E)dE=E0\nFQval+En\nF/integraldisplay\n(E−E0\nF)Nn(E)dE\n(2)\nwhereQvalis the total valence charge, Nn(E) the density\nof states (DOS), and E0\nFcan be set to the value of the\nFermi energy calculated without SOC or to En\nFfor some\nspecific orientation of n. The replacement of En\nFbyE0\nF\nin the upper limit of the last term introduces an error\nδEn∼N(EF)(En\nF−E0\nF)2. We now define\nKσ=E0\nF/integraldisplay\n(E−E0\nF)∆Nσ(E)dE, (3)\nwhereNσ(E) =−1\nπImTrG(E) is the DOS of spin σ.\nThe sum K↑+K↓differs from Konly in the term δEx−\nδEz, which we have verified to be negligible.\nThe definition (3) is used explicitly in the calculations\nofKσ. It is, however, useful to observe that in the per-\nturbativeregime there is arelationbetween KσandKσσ′\natT= 0. This can be seen by following the derivation in\nRef. S8 while sorting out the spin-dependent terms. The\nsecond-order correction to the density of states is\nδNσ(E) =−1\nπImTrG0σVG0VG0σ (4)\nwhereVisthe perturbingSOCoperatorand G0thespin-\ndiagonalGreen’sfunction calculatedwithoutSOC.Using\nthe cyclic property of the trace and the relation G2\n0σ≈\n−∂G0σ/∂E, which in CPA is satisfied approximately as\nlong as disorder is not too strong [S8], we obtain\nKσ=/summationdisplay\nσ′Xσσ′, (5)\nwhere\nXσσ′=1\nπ∆ImTrE0\nF/integraldisplay\n(E−E0\nF)∂G0σ\n∂EVG0σ′VdE.(6)\nOn the other hand, inserting the first-ordercorrectionfor\nGσσ′in (1), we have\nKσσ′≈ −1\n2π∆Im/integraldisplay\nVσσ′G0σ′Vσ′σG0σdE.(7)\nNote that, while it can be employed in a perturbative\ncalculation of MCA energy [S12], Eq. (7) is not used ex-\nplicitly here. Using integration by parts and ignoring the\nrelativelysmall energydependence ofthe SOC constants,\nit is now easy to show\nKσσ′=Xσσ′+Xσ′σ\n2. (8)8\nCombining this with (5), we find, in the perturbative\nregime,\nKσ≈Kσσ+Xσ¯σ, K↑↓≈X↑↓+X↓↑\n2,(9)\nwhere ¯σ∝negationslash=σ. The relations (8) hold when the pertur-\nbative approximations are admissible, i.e., as long as the\nquasi-degenerate states near EFdo not dominate in the\nMCA energy [S12, S14].\nIf spin-off-diagonal band mixing, coming from the\nL±S∓terms in the SOC operator, can be neglected, then\nKσ≈Kσσcomes entirely from the mixing of states in\nspin channel σby theLzSzoperator, and Kσσrepresents\na perturbative approximation for Kσ. This correspon-\ndence holds at all concentrations where K↑↓is small, i.e.,\nsufficiently far from x= 1 [S8]. However, even when K↑↓\nis appreciable, the reciprocal-space resolution of Kσcan\nbe used to identify the hot spots in reciprocal space and\ntheir origin. Indeed, mixing of nearly-degenerate bands\nof the same spin σnearEFgenerates hot spots in Kσ,\nwhile strong off-diagonal band mixing should give rise\nto hot spots in both K↑andK↓. The analysis of Kσ\nnaturally extends to finite temperatures.\nCOMPARISON OF K(T) WITH EXPERIMENT\nThe calculated K(T) curves for Fe 2B and Co 2B are\ncompared with experimental data in Fig. S2. The ex-\nperimental temperature dependence is well reproduced\nin the calculations. The differences between different ex-\nperiments are discussed in Ref. S16.\nSPECTRAL FUNCTIONS FOR Fe 2B\nFig. S3 shows the Bloch spectral functions in Fe 2B at\nT= 0 and T/TC= 0.7. These figures can be com-\npared with Fig. 2 of the main text corresponding to the\n(Fe0.05Co0.95)2B composition. At the Co-rich end the\ndominant effect of spin disorder is the Stoner-like re-\nduction of exchange splitting with only moderate band\nbroadening. In contrast, at the Fe-rich end there is very\nstrong band broadening, particularly in the 1 eV energy\nwindow below the Fermi level.\nMAGNETOCRYSTALLINE ANISOTROPY IN\nREAL SPACE\nThe main text presents the analysis of MCA in re-\nciprocal space. In principle, a real-space analysis could\nprovide an alternative description. Fig. S4 compares the\nK↓↓term with the estimated single-site contribution to\nit. The single-site terms for Fe (or Co) were computed by\nsetting the SOC parameters to zero for all atoms except\nFIG. S2. Comparison of theoretical and experimental K(T)\ncurves for (a) Fe 2B, (b) Co 2B. Solid black lines: theory; red\ndotted lines: Ref. S15; green dashed lines: Ref. S16.\nFe (or Co) on one of the four transition-metal sites in the\nunit cell. If MCA were dominated by single-site terms,\nthe concentration-weighted average of these terms would\ncoincide with K↓↓, but Fig. S4 shows not even a corre-\nlation between them. Clearly, reciprocal space analysis\nis preferable to the real-space decomposition of MCA in\nthis itinerant system.\n[S1] B. S. Pujari, P. Larson, V. P. Antropov, and K. D.\nBelashchenko, Phys. Rev. Lett. 115, 057203 (2015).\n[S2] A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov,\nand V. A. Gubanov, J. Magn. Magn. Mater. 67, 65\n(1987).\n[S3] K. H. J. Buschow, in Boron and Refractory Borides , ed.\nV. I. Matkovich (Springer, Berlin, 1977).\n[S4] K. Chen, A. M. Ferrenberg, and D. P. Landau, Phys.\nRev. B48, 3249 (1993).\n[S5] V. P. Antropov, M. I. Katsnelson, A. I. Liechtenstein,\nPhysica B 237-238 , 336 (1997).\n[S6] V. P. Antropov, J. Magn. Magn. Mater. 262, L192\n(2003).\n[S7] L. Takacs, M. C. Cadeville, and I. Vincze, J. Phys. F:\nMet. Phys. 5, 800 (1975).\n[S8] K. D. Belashchenko, L. Ke, M. D¨ ane, L. X. Benedict,\nT. N. Lamichhane, V. Tarfour, A. Jesche, S. L. Bud’ko,9\nFIG. S3. Partial minority-spin spectral function for the\ntransition-metal site in Fe 2B at (a) T= 0, and (b) T/TC=\n0.7. All details are as in Fig. 2 of the main text.FIG. S4. Single-site terms in K↓↓(circles): two Fe sites\n(open squares), two Co sites (open triangles), and their\nconcentration-weighted sum (filled diamonds).\nP. C. Canfield, and V. P. Antropov, Appl. Phys. Lett.\n106, 062408 (2015).\n[S9] I. Turek, V. Drchal, and J. Kudrnovsk´ y, Philos. Mag.\n88, 2787 (2008).\n[S10] O. K. Andersen, Phys. Rev. B 12, 3060 (1975).\n[S11] D. D. Koelling and B. N. Harmon, J. Phys. C 10, 3107\n(1977).\n[S12] I. V. Solovyev, P. H. Dederichs, and I. Mertig, Phys.\nRev. B52, 13419 (1995).\n[S13] V. P. Antropov, L. Ke, and D. Aberg, Solid State Com-\nmun.194, 35 (2014).\n[S14] E. I. Kondorski ˘i and E. Straube, Zh. Eksp. Teor. Fiz.\n63, 356 (1972) [Sov. Phys. JETP 36, 188 (1973)].\n[S15] A. Iga, Jpn. J. Appl. Phys. 9, 415 (1970).\n[S16] A. Edstr¨ om, M. Werwi´ nski, J. Rusz, O. Eriksson, K.\nP. Skokov, I. A. Radulov, S. Ener, M. D. Kuz’min, J.\nHong, M. Fries, D. Yu. Karpenkov, O. Gutfleisch, P.\nToson, and J. Fidler, arXiv:1502.05916."    },    {        "title": "1504.02320v2.Site_occupancy_and_magnetic_properties_of_Al_substituted_M_type_strontium_hexaferrite.pdf",        "content": "Site occupancy and magnetic properties of Al-substituted M-type strontium\nhexaferrite\nVivek Dixit, Chandani N. Nandadasa, and Seong-Gon Kim\u0003\nDepartment of Physics and Astronomy, Mississippi State University, Mississippi State, MS 39762, USA and\nCenter for Computational Sciences, Mississippi State University, Mississippi State, MS 39762, USA\nSungho Kim\nCenter for Computational Sciences, Mississippi State University, Mississippi State, MS 39762, USA\nJihoon Park and Yang-Ki Hong\nDepartment of Electrical and Computer Engineering and MINT Center,\nThe University of Alabama, Tuscaloosa, AL 35487, USA\nLaalitha S. I. Liyanage\nDepartment of Physics, University of North Texas, Denton, TX 76203, USA\nAmitava Moitra\nThematic Unit of Excellence on Computational Materials Science,\nS.N. Bose National Centre for Basic Sciences, Sector-III, Block-JD, Salt Lake, Kolkata-700098, India\n(Dated: April 6, 2015)\nWe use \frst-principles total-energy calculations based on density functional theory to study\nthe site occupancy and magnetic properties of Al-substituted M-type strontium hexaferrite\nSrFe 12\u0000xAlxO19withx= 0:5 andx= 1:0. We \fnd that the non-magnetic Al3+ions preferen-\ntially replace Fe3+ions at two of the majority spin sites, 2 aand 12k, eliminating their positive\ncontribution to the total magnetization causing the saturation magnetization Msto be reduced as\nAl concentration xis increased. Our formation probability analysis further provides the explanation\nfor increased magnetic anisotropy \feld when the fraction of Al is increased. Although Al3+ions\npreferentially occupy the 2 asites at a low temperature, the occupation probability of the 12 ksite\nincreases with the rise of the temperature. At a typical annealing temperature ( >700\u000eC) Al3+\nions are much more likely to occupy the 12 ksite than the 2 asite. Although this causes the mag-\nnetocrystalline anisotropy K1to be reduced slightly, the reduction in Msis much more signi\fcant.\nTheir combined e\u000bect causes the anisotropy \feld Hato increase as the fraction of Al is increased,\nconsistent with recent experimental measurements.\nI. INTRODUCTION\nStrontium hexaferrite, SrFe 12O19(SFO) is one of the\nmost commonly used materials for permanent magnets,\nmagnetic recording and data storage, and components in\nelectrical devices operating at microwave/GHz frequen-\ncies, due to its high Curie temperature, large saturation\nmagnetization, excellent chemical stability and low man-\nufacturing cost [1{5]. However, in comparison with Nd-\nFe-B and magnet, the coercivity of the SFO is low and\npresents a signi\fcant limitation in its application. There-\nfore, enhancement of the coercivity is an important re-\nsearch topic for the strontium hexaferrite.\nIn order to tailor the magnetic properties such as mag-\nnetization and coercivity, various cation substitutions in\nthe M-type hexaferrites have been investigated. For ex-\nample, the substitution of La [6, 7], Sm [8], Pr [9] and\nNd [10] in the SFO increased coercivity moderately while\nthe substitution of Zn-Nb [11], Zn-Sn [12{14] and Sn-Mg\n\u0003Corresponding author: kimsg@ccs.msstate.edu[4] decreased coercivity. However, the coercivity of the\nM-type hexaferrites is not increased signi\fcantly by these\ncation substitutions, and is still much smaller than that\nof Nd-Fe-B magnet [15].\nAl substitution in the M-type hexaferrite has been\nmore e\u000bective in enhancing coercivity [16{20]. Par-\nticularly, Wang et al synthesized Al-doped SFO\nSrFe 12\u0000xAlxO19(SFAO) with Al content of x= 0\u00004\nusing glycinnitrate method and subsequent annealing in\na temperature over 700\u000eC obtaining the largest coer-\ncivity of 17.570 kOe, which is much larger than that of\nSFO (5.356 kOe) and exceeds even the coercivity of the\nNd2Fe17B (15.072 kOe) [1]. Wang and co-workers also\nobserved that the coercivity of the SFAO increases with\nincreasing Al concentration at a \fxed annealing temper-\nature. These results call for a systematic understand-\ning, from \frst principles, of why certain combinations of\ndopants lead to particular results. This theoretical un-\nderstanding will be essential in systematically tailoring\nthe properties of SFO.\nThere have been several previous \frst-principles inves-\ntigations of SFO. Fang et al investigated the electronic\nstructure of SFO using density-functional theory (DFT)arXiv:1504.02320v2  [cond-mat.mtrl-sci]  27 Apr 20152\n[21]. Park et al have calculated the exchange interaction\nof SFO from the di\u000berences of the total energy of dif-\nferent collinear spin con\fgurations [22]. In spite of the\nimportance of substituted SFO, only a few theoretical\ninvestigations have been done. Magnetism in La substi-\ntuted SFO has been studied using DFT [23, 24]. The site\noccupancy and magnetic properties of Zn-Sn substituted\nSFO has been investigated [14].\nIn this work we use \frst-principles total-energy calcu-\nlations to study the site occupation and magnetic prop-\nerties of Al substituted M-type strontium hexaferrite\nSrFe 12\u0000xAlxO19withx= 0:5 andx= 1:0. Based on\nDFT calculations, we determine the the structure of var-\nious con\fgurations of SFAO with di\u000berent Al concentra-\ntions and compute the occupation probabilities for dif-\nferent substitution sites at elevated temperatures. We\nshow that our model predicts an decrease of saturation\nmagnetization as well as a decrease in magnetocrystalline\nanisotropyK1, and the increase of the anisotropy \feld Ha\nas the fraction of Al is increased, consistent with recent\nexperimental measurements.\nII. METHODS\nSFO has a hexagonal magnetoplumbite crystal struc-\nture that belongs to P63=mmc space group. Fig. 1 shows\na unit cell of SFO used in the present work that con-\ntains 64 atoms of two formula units. Magnetism in SFO\narises from Fe3+ions occupying \fve crystallographically\ninequivalent sites in the unit cell, three octahedral sites\n(2a, 12k, and 4f2), one tetrahedral site (4 f1), and one\ntrigonal bipyramidal site (2 b) as represented by the poly-\nhedra in Fig. 1(a). SFO is also a ferrimagnetic material\nthat has 16 Fe3+ions with spins in the majority direction\n(2a, 2b, and 12ksites) and 8 Fe3+ions with spins in the\nminority direction (4 f1and 4f2sites) as indicated by the\narrows in Fig. 1(b).\nTotal energies and forces were calculated using DFT\nwith projector augmented wave (PAW) potentials as im-\nplemented in VASP [25, 26]. All calculations were spin\npolarized according to the ferrimagnetic ordering of Fe\nspins as \frst proposed by Gorter [21, 27]. A plane-wave\nenergy cuto\u000b of 520 eV was used both for pure SFO and\nAl-substituted SFO. Reciprocal space was sampled with\na 7\u00027\u00021 Monkhorst-Pack mesh [28] with a Fermi-\nlevel smearing of 0.2 eV applied through the Methfessel-\nPaxton method [29]. We performed relaxation of the elec-\ntronic degrees of freedom until the change in free energy\nand the band structure energy was less than 10\u00007eV. We\nperformed geometric optimizations to relax the positions\nof ions and cell shape until the change in total energy\nbetween two ionic step was less than 10\u00004eV. Electron\nexchange and correlation was treated with the general-\nized gradient approximation (GGA) as parameterized by\nthe Perdew-Burke-Ernzerhof (PBE) scheme [30]. To im-\nprove the description of localized Fe 3 delectrons, we em-\nployed the GGA+U method in the simpli\fed rotationally\nFIG. 1. (color online) (a) A unit cell of SFO containing\ntwo formula units. Two large gold spheres are Sr atoms and\nsmall gray spheres are O atoms. Colored spheres enclosed\nby polyhedra formed by O atoms represent Fe3+ions in dif-\nferent inequivalent sites: 2 a(blue), 2b(cyan), 12k(purple),\n4f1(green), and 4 f2(red). (b) A schematic diagram of the\nlowest-energy spin con\fguration of Fe3+ions of SFO. The\narrows represent the local magnetic moment at each atomic\nsite. (For interpretation of the references to color in this \fg-\nure caption, the reader is referred to the online version of this\npaper.)\ninvariant approach described by Dudarev et al [31]. This\nmethod requires an e\u000bective Uvalue (Ue\u000b) equal to the\ndi\u000berence between the Hubbard parameter Uand the ex-\nchange parameter J. We chose Ue\u000bequal to 3.7 eV for\nFe based on the previous result [14].\nIII. RESULTS AND DISCUSSION\nThe substitution of Fe3+ions by Al3+ions consider-\nably a\u000bects the unit cell parameters. We have calculated\nthe lattice parameters of pure and Al-substituted SFO by\nrelaxing ionic positions as well as the volume and shape\nof the unit cell. In all cases the \fnal unit cell was found\nto remain hexagonal. In the case of pure SFO, the lat-\ntice parameters aandcwere found to be 5.93 \u0017A and\n23.21 \u0017A in good agreement with the experimental values\nofa= 5:88\u0017A andc= 23:04\u0017A, respectively [19, 32];\nthe deviation between the experimental and the theo-\nretical values is less than 1%. In the case of x= 0:5\nin SrFe 12\u0000xAlxO19the lattice parameters aandcwere\ncalculated to be 5 :92\u0017A and 23:16\u0017A respectively, while\nthe volume of the unit cell was reduced by 0.61%. For3\nFIG. 2. (color online) Comparison of calculated and experi-\nmental (Ref. [19]) volume of the unit cell of SrFe 12\u0000xAlxO19\nas a function of the fraction of Al x.\nx= 1:0,a= 5:91\u0017A andc= 23:04\u0017A were found, and\nreduction in the unit cell volume was 2.51%. Fig. 2 shows\nthat the reduction of unit cell volume predicted by our\nDFT calculation is consistent with the experimental re-\nsults [1, 19].\nWe investigated the site preference of Al substituting\nFe in SrFe 12\u0000xAlxO19for (i)x= 0:5 and, (ii) x= 1:0.\nThex= 0:5 case corresponds to the condition where\none Al atom is substituted in the unit cell, while two Al\natoms were substituted in the case of x= 1:0 as shown\nFig. 3. To determine the site preference of the substituted\nAl atoms, the substitution energy of con\fguration iwas\ncalculated using the following expression:\nEsub(i) =E(SFAO(i))\u0000E(SFO) \u0000X\n\u000bn\u000b\u000f(\u000b) (1)\nwhereE(SFAO(i)) is the total energy per unit cell at\n0 K for SFAO in con\fguration iwhileE(SFO) is the\ntotal energy per unit cell at 0 K for SFO. \u000f(\u000b) is the\ntotal energy per atom for element \u000b(\u000b= Al, Fe) at 0 K\nin its most stable crystal structure. n\u000bis the number\nof atoms of type \u000badded: if two atoms are added then\nn\u000b= +2 while n\u000b=\u00001 when one atom is removed.\nThe con\fguration with the lowest Esubis concluded to\nbe the ground state con\fguration, and the corresponding\nsubstitution site is the preferred site of Al atoms at 0 K.\nTo understand the site preference of the substituted\nAl3+ions at higher temperatures, we compute the for-\nmation probability of con\fguration iusing the Maxwell-\nBoltzmann statistical distribution [33]:\nPi=giexp(\u0000\u0001Gi=kBT)P\njgjexp(\u0000\u0001Gj=kBT)(2)\nFIG. 3. (color online) The structures of SrFe 12\u0000xAlxO19\nwith spins oriented in the easy axis (001): (a) con\fguration\n[2a] forx= 0:5 and (b) con\fguration [2 a;12k]:1 forx= 1:0.\nAl atoms are labeled and other atoms are colored as in Fig. 1.\nwheregiis the multiplicity of con\fguration i(number of\nequivalent con\fgurations) and\n\u0001Gi= \u0001Esub(i) +P\u0001Vi\u0000T\u0001Si (3)\nis the change of the free energy of con\fguration irela-\ntive to that of the ground state con\fguration; \u0001 Esub(i),\n\u0001Vi, and \u0001Siare the substitution energy change, volume\nchange, entropy change for con\fguration i;PandkBare\nthe pressure and Boltzmann constant.\nFor thex= 0:5 concentration, one Al atom is substi-\ntuted at one of the 24 Fe sites of the unit cell as shown in\nFig. 3(a). The application of crystallographic symmetry\noperations shows that many of these Fe sites are equiv-\nalent and leaves only \fve inequivalent structures. We\nlabel these inequivalent con\fgurations using the crystal-\nlographic name of the Fe site: [2 a], [2b], [4f1], [4f2], and\n[12k]. These structures were created by substituting one\nAl atom to the respective Fe site of a SFO unit cell and\nperforming full optimization of the unit cell shape and\nvolume, and ionic positions.\nTable I lists the results of our calculation for all \fve in-\nequivalent con\fgurations in the order of increasing sub-\nstitution energy. The lowest Esubis found for con\fgu-\nration [2a] shown in Fig. 1(a). We can conclude that\nat 0 K the most preferred site for the substituted Al\natom is the 2 asite. We used Eq. (2) to compute the4\nTABLE I. Five inequivalent con\fgurations of\nSrFe 12\u0000xAlxO19withx= 0:5.gis the multiplicity of\nthe con\fguration. Esubis the substitution energy of the\nSFAO. The total magnetic moment ( mtot) and its change\nwith respect to SFO (\u0001 mtot) are also given. All values are\nfor a double formula unit cell containing 64 atoms. Energies\nare in eV while magnetic moments are in \u0016B.\nCon\fg g E sub mtot \u0001mtot\n[2a] 2 -6.04 35 -5\n[12k] 12 -6.00 35 -5\n[4f2] 4 -5.63 45 +5\n[2b] 2 -5.60 35 -5\n[4f1] 4 -5.57 45 +5\nFIG. 4. (color online) Temperature dependence of the forma-\ntion probability of di\u000berent con\fgurations of SrFe 12\u0000xAlxO19\nwithx= 0:5. The con\fgurations with negligible probability\nare not shown.\nprobability to form each con\fguration as a function of\ntemperature. Since the volume change among di\u000berent\ncon\fgurations is very small (less than 0.1 \u0017A3), we can\nsafely regard P\u0001Vterm to be negligible (in the order\nof 10\u00007eV at the standard pressure of 1 atm) com-\npared to the \u0001 Esub(i) term in Eq. (3). The entropy\nchange \u0001Shas a con\fgurational part, \u0001 Sc, and a vi-\nbrational part, \u0001 Svib[34]. For binary substitutional al-\nloys such as the present system, \u0001 Svibis around 0.1-\n0.2kB/atom, and \u0001 Scis 0.1732kB/atom [33]. Therefore,\nwe set \u0001S= 0:3732kB/atom.\nFig. 4 displays the temperature dependence of the\nformation probability of di\u000berent con\fgurations of\nSrFe 12\u0000xAlxO19withx= 0:5. The doped Al3+ions\nmainly replace Fe3+ions from the 2 aand the 12ksites.\nThe formation probabilities of [2 b], [4f1] and [4f2] are\nnegligible and not shown in Fig. 4. The probability that\nthe doped Al3+ion replaces Fe3+ion from the 2 asite is\nmaximum at 0 K and it falls as temperature increases,TABLE II. Ten lowest energy inequivalent con\fgurations of\nSrFe 12\u0000xAlxO19withx= 1:0.gis the multiplicity of the con-\n\fguration. Esubis the substitution energy per Al atom. The\ntotal magnetic moment ( mtot) and its change with respect to\nSFO (\u0001mtot) are also given. All values are for a double for-\nmula unit cell containing 64 atoms. Energies are in eV while\nmoments are in \u0016B.\nCon\fg g E submtot \u0001mtot\n[2a;2a] 1 -6.056 30 -10\n[2a;12k]:1 12 -6.054 30 -10\n[2a;12k]:2 12 -6.041 30 -10\n[12k;12k]:1 6 -6.025 30 -10\n[12k;12k]:2 12 -6.025 30 -10\n[12k;12k]:3 12 -6.027 30 -10\n[12k;12k]:4 12 -6.025 30 -10\n[12k;12k]:5 6 -6.023 30 -10\n[12k;12k]:6 6 -6.017 30 -10\n[12k;12k]:7 12 -6.014 30 -10\nwhile the occupancy of Al3+at the 12ksite rises with\ntemperature. The two curves cross at T\u0018220 K. At\na typical annealing temperature of 1000 K for SFAO [1]\nthe site occupation probability of the site 2 aand 12kis\n0.196 and 0.798, respectively. Thus, during the anneal-\ning process of the synthesis of the SFAO the doped Al3+\nions are more likely to replace Fe3+ions from the 12 ksite\nthan the 2asite despite of higher substitution energy.\nFor thex= 1:0 concentration, two Al atoms are substi-\ntuted at two of the 24 Fe sites of the unit cell as shown in\nFig. 3(b). These Fe sites have more than one equivalent\nsite. Substitution of Al atoms breaks the symmetry of the\nequivalent sites of pure SFO. Out of all C(24;2) = 276\npossible structures, many of the structures are crystal-\nlographically equivalent. On applying crystallographic\nsymmetry operations, the number of inequivalent struc-\ntures reduces to 40. We label these inequivalent con\fgu-\nrations using the convention of [(site for the \frst Al),(site\nfor the 2nd Al)].(unique index). For example, when two\nAl atoms are substituted at the 2 aand 12ksites, there\nare 2 inequivalent con\fgurations, which are labeled as\n[2a;12k]:1 and [2a;12k]:2. These structures are fully op-\ntimized and their substitution energies are calculated us-\ning Eq. (1). When there are more than one inequivalent\ncon\fguration, we assign the unique index in the order of\nincreasingEsub.\nTable II lists the ten lowest energy con\fgurations of\nSrFe 12\u0000xAlxO19withx= 1:0. The con\fguration [2 a;2a]\nwhere two Al3+ions replace Fe3+ions from two 2 asites\nhas the lowest Esub, and it is the most energetically fa-\nvorable con\fguration at 0 K. To investigate the site oc-\ncupation at nonzero temperatures we compute the for-\nmation probability of each con\fguration using Eq. (2).\nSimilar to the previous case the volume change among\ndi\u000berent con\fgurations is very small (less than 0.7 \u0017A3)\nand we can safely ignore the P\u0001Vterm. The entropy\nterm is calculated in the same way as the x= 0:5 case.\nFig. 5 shows the variation of the formation probability5\nFIG. 5. (color online) Temperature dependence of the forma-\ntion probability of di\u000berent con\fgurations of SrFe 12\u0000xAlxO19\nwithx= 1:0. For clarity only the con\fgurations with signi\f-\ncant formation probability are labeled.\nof di\u000berent con\fgurations with temperature. We note\nthat due to low multiplicity of the con\fguration [2 a;2a],\nits formation probability falls rapidly as temperature in-\ncreases. On the other hand, the formation probability of\nthe con\fguration [2 a;12k] (sum of the formation proba-\nbilities for all [2 a;12k]:ncon\fgurations) increases steeply\nand reaches a maximum value at 50 K and then falls with\ntemperature. Fig. 5 shows that the formation probability\nof the [2a;12k] con\fguration becomes larger that that of\n[2a;2a] beyondT\u001810 K, which is a much lower transi-\ntion temperature than in the x= 0:5 case.\nWe can calculate the occupation probability of Al at\nnonzero temperatures for a given site by adding all for-\nmation probabilities of the con\fgurations where at least\none Al3+ion is substituted in that site. At the annealing\ntemperature of 1000 K, the occupation probability of Al\nfor 12ksite is 79.8% for x= 0:5 as given in Table IV.\nThe same probability is increased to 97.7% for x= 1:0\nas calculated by adding the P1000's for all con\fgurations\nthat contain the 12 ksite. This means that the fraction\nof Al3+ions occupying the 12 ksite increases when the\nfraction of Al is increased from x= 0:5 tox= 1:0. This\nconclusion is in agreement with the previously reported\nmeasurements [1, 16, 35].\nIn Table III we compare the contribution of di\u000ber-\nent sublattices to the total magnetic moment in Al-\nsubstituted SFO. To see the e\u000bect of Al3+ions in di\u000ber-\nent substitution sites, we split the entries of sublattices\ncontaining these ions (2 aand 12k). As expected, Al3+\nions carry negligible magnetic moment regardless of their\nsubstitution sites. Consequently, when they replace Fe3+\nions in the minority spin sites (4 f1and 4f2), they elimi-\nnate a negative contribution and hence increase the total\nmagnetic moment. On the other hand, when they re-place Fe3+ions in the majority spin sites (12 k, 2a, and\n2b), they eliminate a positive contribution and hence re-\nduce the total magnetic moment. For the x= 0:5 case,\nthe most probable sites are 12 kand 2a(majority sites)\nand the net magnetic moment of the unit cell is reduced\nby 5\u0016B. For the con\fguration [2 a;12k]:1 of thex= 1:0\ncase, two Al atoms are substituted in the 2 aand 12k\nsites, there is a reduction of 10 \u0016Bin the total magnetic\nmoment per unit cell.\nMagnetic Anisotropy determines the capacity of a\nmagnet to withstand external magnetic and electric\n\felds. This property is of considerable practical in-\nterest, because anisotropy is exploited in the design of\nthe most magnetic materials of commercial importance.\nThe magnetocrystalline anisotropy energy (MAE) is one\nof the main factors that determine the total magnetic\nanisotropy of the material. To investigate the e\u000bect of\nAl substitution on the magnetic anisotropy of SFO, we\ncomputed the MAE and the magnetic anisotropy con-\nstant of SrFe 12\u0000xAlxO19forx= 0;0:5 and 1. The MAE,\nin the present case, is de\fned as the di\u000berence between\nthe two total energies where electron spins are aligned\nalong two di\u000berent directions [36]:\nEMAE =E(100)\u0000E(001) (4)\nwhereE(100) is the total energy with spin quantization\naxis in the magnetically hard plane and E(001) is the to-\ntal energy with spin quantization axis in the magneti-\ncally easy axis. Using the MAE, the uniaxial magnetic\nanisotropy constant K1can be computed [37, 38]\nK1=EMAE\nVsin2\u0012(5)\nwhereVis the equilibrium volume of the unit cell and\n\u0012is the angle between the two spin quantization axis\norientations (90\u000ein the present case). The anisotropy\n\feldHacan be expressed as [39]\nHa=2K1\nMs(6)\nwhereK1is a magnetocrystalline anisotropy constant\nandMsis saturation magnetization.\nThe results for the MAE, the magnetocrystalline\nanisotropy constant K1, and anisotropy \feld Hafor\nSFAO with di\u000berent Al concentration are presented in\nTable IV. To compare with experimental results, we also\ncompute the weighted average of K1andHausing the\nformation probability P1000at a typical annealing tem-\nperature of 1000 K [1]. We note that SFAO considered in\nthe present work loses most of its magnetic properties at\ntypical annealing temperatures (1000 K or higher) that\nare near or above its Curie temperature. The magnetic\nproperties listed in Table IV refer to their ground state\nproperties at the temperature T= 0. We use the for-\nmation probability at 1000 K to compute the weighted\naverages as the crystalline con\fgurations of SFAO will be6\nTABLE III. Contribution of atoms in each sublattice to the total magnetic moment of Al-substituted SFO structures [12 k],\n[2a], and [2a;12k]:1 compared with pure SFO. All magnetic moments are in \u0016B. \u0001mis measured relative to the values for\nthe pure SFO. Note that the total magnetic moment of the unit cell ( mtot) is slightly di\u000berent than the sum of local magnetic\nmoments due to the contribution from the interstitial region.\nsiteSFO [12k] [2a] [2a;12k]:1\natoms m atoms m \u0001m atoms m \u0001m atoms m \u0001m\n2d 2 Sr -0.006 2 Sr -0.006 0.000 2 Sr -0.006 0.000 2 Sr -0.006 0.000\n2a1 Fe 4.156 1 Fe 4.155 -0.001 1Al -0.010 -4.166 1Al -0.010 -4.166\n1 Fe 4.156 1 Fe 4.156 0.000 1 Fe 4.156 0.000 1 Fe 4.156 0.000\n2b 2 Fe 8.098 2 Fe 8.086 -0.012 2 Fe 8.100 0.001 2 Fe 8.090 -0.008\n4f1 4 Fe -16.152 4 Fe -16.189 -0.037 4 Fe -16.268 -0.116 4 Fe -16.304 -0.152\n4f2 4 Fe -16.384 4 Fe -16.420 -0.036 4 Fe -16.382 0.002 4 Fe -16.418 -0.034\n12k1 Fe 4.172 1Al 0.000 -4.172 1 Fe 4.170 -0.002 1Al -0.001 -4.173\n11 Fe 45.884 11 Fe 45.861 -0.023 11 Fe 45.872 -0.012 11 Fe 45.846 -0.038\n4e 4 O 1.416 4 O 1.304 -0.112 4 O 1.424 0.008 4 O 1.316 -0.100\n4f 4 O 0.360 4 O 0.281 -0.079 4 O 0.310 -0.050 4 O 0.230 -0.129\n6h 6 O 0.124 6 O 0.115 0.009 6 O 0.134 0.010 6 O 0.117 -0.007\n12k 12 O 1.016 12 O 0.877 -0.129 12 O 0.548 -0.468 12 O 0.404 -0.612\n12k 12 O 2.140 12 O 1.895 -0.245 12 O 2.088 -0.052 12 O 1.839 -0.301Pm 38.980 34.114 -4.837 34.136 -4.845 29.259 -9.720\nmtot 40 35 -5 35 -5 30 -10\nTABLE IV. The saturation magnetization ( Ms), magnetocrystalline anisotropy energy (MAE), magnetocrystalline anisotropy\nconstant (K1) and anisotropy \feld ( Ha) for SFO and SFAO. xis the Al fraction in SrFe 12\u0000xAlxO19andVis the volume of the\nunit cell in \u0017A3.P1000is the formation probability at 1000 K. The averaged quantities are weighted by P1000.Msis in emu/g,\nMAE in meV, Hain kOe, and K1in kJ\u0001m\u00003.\nx Con\fg Ms MAE V K 1HaP1000hMsi hK1i hHai\n0.0 SFO 110 :19 0.85 707.29 193 7 :35 1 :000 110.19 193 7.35\n0.5[2a] 96 :41 0.95 703.29 216 9 :38 0 :196\n96.49 189 8.18[12k] 96 :41 0.80 703.19 182 7 :90 0 :798\n[2b] 96 :41 0.67 702.82 152 6 :62 0 :003\n[4f1] 123 :96 0.86 704.22 196 6 :61 0 :001\n[4f2] 123 :96 0.83 702.58 189 6 :38 0 :001\n1.0[2a;2a] 82 :64 0.99 698.94 227 11 :41 0 :019\n83.03 184 9.23[2a;12k] 82:64 0.88 699.08 202 10 :13 0 :379\n[12k;12k] 82:64 0.75 698.66 172 8 :64 0 :585\n[12k;4f2] 110:19 0.78 690.64 181 6 :74 0 :007\n[12k;4f1] 110:19 0.80 700.29 183 6 :92 0 :004\n[12k;2b] 82:64 0.62 698.98 142 7 :14 0 :002\n[4f2;4f2] 137:74 0.80 697.96 184 5 :53 0 :000\n[4f2;4f1] 137:74 0.83 699.62 191 5 :74 0 :000\n[4f2;2b] 110:19 0.60 698.82 138 5 :19 0 :000\n[4f2;2a] 110:19 0.90 698.53 206 7 :78 0 :002\n[4f1;4f1] 137:74 0.86 701.38 196 5 :95 0 :000\n[4f1;2b] 110:19 0.65 699.95 149 5 :62 0 :000\n[4f1;2a] 110:19 0.91 700.11 208 7 :87 0 :001\n[2b;2b] 82 :64 0.45 698.86 103 5 :19 0 :000\n[2b;2a] 82 :64 0.74 698.82 170 8 :53 0 :001\ndistributed according to this value during the annealing\nprocess.\nTable IV shows that Msdecreases as the concentration\nof Alxis increased from 0 to 0.5 to 1.0, consistent with\nthe previous experimental results [1, 20, 40, 41]. Our\ncalculation also shows that K1decreases as the concen-\ntration of Al xis increased from 0 to 0.5 to 1.0. At a\nlow temperature Al atoms prefer to occupy the 2 asites,which would have increased K1(seeK1values for [2 a]\nand [2a;2a] in Table IV). However, the formation prob-\nability of the con\fgurations involving 12 ksite (such as\n[12k], [2a;12k] and [12k;12k]) increases signi\fcantly as\nthe temperature rises due to the entropy contribution of\nthe free energy. At the annealing temperature Al3+ions\nare much more likely to occupy the 12 ksite than the 2 a\nsite. This causes the magnetocrystalline anisotropy con-7\nstantK1of Al-substituted SFO to be reduced with the\nincrease of Al fraction x, consistent with the experimen-\ntal measurement reported by Albanes [40]. Despite of\nthis,Msis reduced more signi\fcantly than K1and this\ncauses the anisotropy \feld Hain Eq. (6) to increase as\nthe concentration of Al xis increased from 0 to 0.5 to\n1.0 as shown in Table IV. This result is consistent with\nseveral experimental results [1, 41].\nIV. CONCLUSIONS\nUsing the \frst-principles total energy calculations\nbased on density functional theory, we obtained the\nground state structures and associated formation prob-\nabilities at \fnite temperatures for Al-substituted SFO,\nSrFe 12\u0000xAlxO19withx= 0:5 and 1:0. The structures de-\nrived from our calculations show that the total magnetic\nmoment of the SFO unit cell is reduced as the fraction\nof Al atoms increases. This reduction of magnetization\nis explained by the fact that the non-magnetic Al atoms\nprefer to replace Fe3+ions at two of the majority spin\nsites, 2aand 12k, eliminating their positive contributionto the total magnetization. Our model also explains the\nincrease of the observed anisotropy \feld when the frac-\ntion of Al in SFO is increased. At the annealing temper-\nature Al3+ions are much more likely to occupy the 12 k\nsite than the 2 asite. Although this causes the magne-\ntocrystalline anisotropy to decrease slightly, the reduc-\ntion in the saturation magnetization is larger and their\ncombined e\u000bect causes the magnetic anisotropy \feld of\nAl-substituted SFO to be reduced with increase of Al\nfractionx. Our results are consistent with the available\nexperimental measurement on Al-substituted SFO.\nV. ACKNOWLEDGMENTS\nThis work was supported in part by the U.S. De-\npartment of Energy ARPA-E REACT program under\nAward Number DE-AR0000189 and the Center for Com-\nputational Science (CCS) at Mississippi State University.\nComputer time allocation has been provided by the High\nPerformance Computing Collaboratory (HPC2) at Mis-\nsissippi State University.\n[1] H. Wang, B. Yao, Y. Xu, Q. He, G. Wen, S. Long, J. Fan,\nG. Li, L. Shan, B. Liu, L. Jiang, and L. Gao, Journal of\nAlloys and Compounds 537, 43 (2012).\n[2] R. C. Pullar, Progress in Materials Science 57, 1191\n(2012).\n[3] Z. Pang, X. Zhang, B. Ding, D. 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Solovyev1,2,∗\n1Computational Materials Science Unit,\nNational Institute for Materials Science,\n1-1 Namiki, Tsukuba, Ibaraki 305-0044, Japan\n2Department of Theoretical Physics and Applied Mathematics ,\nUral Federal University, Mira str. 19, 620002 Ekaterinburg , Russia\n(Dated: August 13, 2018)\n1Abstract\nBa2CoGe2O7, crystallizing in the noncentrosymmetric but nonpolar P421mstructure, belongs\nto a special class of multiferroic materials, whose propert ies are predeterminedby theavailability of\nthe rotoinversion symmetry. Unlike inversion, the rotoinv ersion symmetry can be easily destroyed\nby the magnetization. Moreover, due to specific structural p attern in the xyplane, in which the\nmagnetic Co2+ions are separated by the nonmagnetic GeO 4tetraherda, the magnetic structure of\nBa2CoGe2O7is relatively soft. Altogether, this leads to the rich varie ty of multiferroic properties of\nBa2CoGe2O7, where the magnetic structure can be easily deformed by the m agnetic field, inducing\nthe net electric polarization in the direction depending on the magnetic symmetry of the system,\nwhich itself depends on the direction of the magnetic field. I n this paper, we show that all these\nproperties can be successfully explained on the basis of rea listic low-energy model, derived from\nthe first-principles electronic structure calculations fo r the magnetically active Co 3 dbands, and\nthe Berry-phase theory of electronic polarization. Partic ularly, we argue that the magnetization\ninduced electric polarization in Ba 2CoGe2O7is essentially local and expressed via the expectation\nvalues∝an}bracketle{tˆp∝an}bracketri}ht= Tr[ˆpˆD] of some dipole matrices ˆp, calculated in the Wannier basis of the model,\nand the site-diagonal density matrices ˆDof the magnetic Co sites. Thus, the basic aspects of the\nbehavior of Ba 2CoGe2O7can be understood already in the atomic limit, where both mag netic\nanisotropy and magnetoelectric coupling are specified by ˆD. Then, the macroscopic polarization\ncan be found as a superposition of ∝an}bracketle{tˆp∝an}bracketri}htof the individual Co sites. We discuss the behavior of\ninteratomic magnetic interactions, main contributions to the magnetocrystalline anisotropy and\nthe spin canting in the xyplane, as well as the similarities and differences of the propo sed picture\nfrom the phenomenological model of spin-dependent p-dhybridization.\n2I. INTRODUCTION\nMagnetically driven ferroelectricity is one of the major topics in the c ondensed matter\nphysics today. Even though the basic crystallographic symmetry o f a magnetic material\nmay not allow for the existence of a spontaneous electric polarizatio n, this symmetry can be\nfurtherlowered bythemagneticorder, whichinsomecases givesris etotheferroelectric(FE)\nactivity. After discovery of such effect in TbMnO 3,1where the inversion symmetry is broken\nby some complex noncollinear magnetic order, there is a large number of experimental and\ntheoretical proposals pointing at the existence of a similar effect in o ther magnetic materials,\nwhich are called multiferroics.2\nBa2CoGe2O7and related compounds take a special place among multiferroic mate rials.\nThey crystallize in the noncentrosymmetric P421mstructure,3which nonetheless does not\npermit the FE effect because of the fourfold rotoinversion symmet ry. Nevertheless, unlike\ninversion, therotoinversionsymmetrycanberelativelyeasydestr oyedbythemagneticorder,\nby canting the magnetic moments out of the rotoinversion axis. Thu s, one unique aspect of\nBa2CoGe2O7isthattheferroelectricity inthiscompoundcanbeinducedrelatively simpleC-\ntype antiferromagnetic (AFM) order. Furthermore, the magnet ic structure of Ba 2CoGe2O7\nis relatively soft: It consists of the CoO 4tetrahedra, which are interconnected with each\nother via nonmagnetic GeO 4tetrahedra (see Fig. 1a). Since two magnetic Co sites in this\nstructure are separated by the long Co-O-Ge-O-Copaths, the e xchange interactions between\nthem are expected to be small and the magnetic structure can be e asily deformed by the\nexternal magnetic field. The possibilities of easy manipulation by the m agnetic structure\nand switching the FE properties have attracted a great deal of at tention to Ba 2CoGe2O7,\nand today it was demonstrated in many details how the electric polariz ation in Ba 2CoGe2O7\ncan be tuned by the magnetic field (Refs. 4 and 5) as well as the uniax ial stress (Ref. 6).\nThe experimental behavior of the electric polarization in Ba 2CoGe2O7is frequently inter-\npreted basing on the model model of spin-dependent p-dhybridization: if eis the direction\nof the spin magnetic moment at the Co site, located in the origin, and υis the position\nof a ligand oxygen atom, relative to this origin, the bond Co-O contrib utes to the local\nelectric dipole moment as p∝(e·υ)2υ.5,7This simple phenomenological expression is able\nto capture the symmetry properties of the electric polarization an d, thus, explain the be-\nhavior of this polarization on a phenomenological level. It is therefor e not surprising that a\n3/s49/s50\n/s49/s50/s67/s111/s79\n/s52\n/s71/s101/s79\n/s53/s66/s97\n/s120/s120\n/s121/s121\n/s32/s48\n/s32/s50\n/s32/s48\n/s32/s49/s40/s97/s41\n/s40/s98/s41\n/s40/s99/s41 /s40/s100/s41\n/s49\n/s49/s50/s50\nFIG. 1. (Color online) Crystal and spin magnetic structure o f Ba2CoGe2O7: (a) General view\non the crystal structure in the xyplane. The Co and O atoms are indicated by the medium red\nand yellow spheres, respectively, the Ba atoms are indicate d by the big green spheres, and the\nGe atoms are indicated by the small blue spheres. The MnO 4and GeO 4tetrahedra are colored\nred and green, respectively. (b) Details of the rotations of MnO4tetrahedra associated with two\nMn sites in the xyplane. (c) and (d) Two possible spin magnetic structures, re alized without\nexternal magnetic field. The light (cyan) arrows indicate th e directions of spins favored by the\nsingle-ion anisotropy, while the dark (blue) arrows are the joint effect of the single-ion anisotropy\nand interatomic exchange interactions.\nqualitatively similar behavior of the electric polarization was obtained in the first-principles\nelectronic structure calculations,8which generally support the model of spin-dependent p-d\nhybridization.\nOn the other hand, the correct quantum-mechanical definition of the electric polarization\n4is solids should be based on the Berry-phase theory.9This theory provides not only an effi-\ncient computational framework, which is used today in most of the fi rst-principles electronic\nstructure calculations, but also appears to be a good starting poin t for the construction\nof realistic microscopic models, explaining the origin and basic aspects of the behavior of\nelectric polarization in various types of compounds.10–12In this work we will pursue this\nstrategy for the analysis of electric polarization in Ba 2CoGe2O7. First, we will show that,\nin the atomic (Wannier) basis, the electric polarization consists of tw o parts: the local one,\nwhich is expressed via the expectation value ∝an}bracketle{tˆp∝an}bracketri}ht= Tr[ˆpˆD] of some local dipole matrices\nˆpand site-diagonal density matrices ˆD, and the anomalous one, which is solely related to\nthe phases of the coefficients of the expansion of the wavefunctio ns in the basis of atomic\nWannier orbitals. Similar classification holds for the orbital magnetiza tion.13Then, if the\ncrystal structure possesses the inversion symmetry, the local term becomes inactive and the\nelectric polarization, induced by the magnetic inversion symmetry br eaking, is anomalous in\norigin. Such situation occurs, for instance, in multiferroic manganit es with the orthorhombic\nstructure.11,14In Ba2CoGe2O7, however, the situation is exactly the opposite: to a good ap-\nproximation, the electric polarization is expressed as a sum of local e lectric dipole moments\nof individual Co sites, while the anomalous contribution is negligibly small. This explains\nmany experimental details of the behavior of electric polarization in B a2CoGe2O7as well as\nbasic difference of this compound from other multiferroic materials.\nThe rest of the paper is organized as follows. The method is describe d in Sec. II. We\nuse the same strategy as in our previous works, devoted to the an alysis of multiferroic\nproperties of various transition-metal oxides on the basis of effec tive realistic low-energy\nmodels, derived from the first-principles calculations.11,14Therefore, in Sec. IIA we will\nremind the main details of the construction of such effective low-ene rgy model. In Sec. IIB,\nwe will analyze the main contributions to the electric polarization in the case of basis, when\nthe Bloch wavefunction is expanded over some atomic Wannier orbita ls, centered at the\nmagnetic sites. Since we do not consider the magnetostriction effec ts, which can low the\noriginalP421msymmetry of the crystal structure, the considered electric pola rization is in\nfacttheelectronicone.9InSec.IIIwewillpresent resultsofourcalculationsforBa 2CoGe2O7,\nstarting from the atomic limit and, then, consecutively considering t he effect of interatomic\nexchange interactions and the external magnetic field. Finally, in Se c. IV, we will summarize\nour work. In the Appendix we will estimate the ferromagnetic (FM) c ontribution to the\n5interatomic exchange interaction cause by the magnetic polarizatio n of the oxygen 2 pband\nand discuss how such effect can be evaluated starting from the low- energy electron model\nfor the magnetic Co 3 dbands.\nII. METHOD\nA. Effective low-energy model\nIn this section, we briefly remind the reader the main ideas behind the construction of the\neffective low-energy electron model. The details can be found in the r eview article (Ref. 15).\nThe model Hamiltonian,\nˆH=/summationdisplay\nij/summationdisplay\nαβtαβ\nijˆc†\niαˆcjβ+1\n2/summationdisplay\ni/summationdisplay\nαβγδUi\nαβγδˆc†\niαˆc†\niγˆciβˆciδ, (1)\nis formulated in the basis of Wannier orbitals {φiα}, which are constructed for the mag-\nnetically active Co 3 dbands near the Fermi level, starting from the band structure in th e\nlocal-density approximation (LDA) without spin-orbit (SO) coupling ( Fig. 2). Here, each\n/s45/s56 /s45/s54 /s45/s52 /s45/s50 /s48 /s50 /s52 /s54/s48/s50/s52/s54/s56/s49/s48/s49/s50\n/s66/s97/s32/s53 /s32/s100/s32/s67/s111/s32/s51 /s32/s100/s32\n/s32/s32/s68/s101/s110/s115/s105/s116/s121/s32/s111/s102/s32/s83/s116/s97/s116/s101/s115/s32/s40/s49/s47/s101/s86/s32/s102/s46/s117/s46/s41\n/s69/s110/s101/s114/s103/s121 /s32/s40/s101/s86/s41/s79/s32/s50 /s32/s112\nFIG. 2. (Color online) Total and partial densities of states of Ba2CoGe2O7in the local density\napproximation. The shaded light (blue) area shows the contr ibution of the Co 3 dstates. The\npositions of the main bands are indicated by symbols. The Fer mi level is at zero energy (shown by\ndot-dashed line).\n6Greek symbol ( α,β,γ, orδ) stands for the combination of spin ( σα,σβ,σγ, orσδ) and\norbital (a,b,c, ord) indices, for which we adapt the following order: xy,yz, 3z2−r2,zx,\nx2−y2. Each lattice point i(j) is specified by the position τ(τ′) of the atomic site in the\nprimitive cell and the lattice translation R. Hence, the basis orbitals φiα(r)≡φτα(r−R−τ)\nare centered in the lattice point ( R+τ) and labeled by the indices τandα. Moreover, they\nsatisfies the orthonormality condition:\n∝an}bracketle{tφτ′α′(r−R′−τ′)|φτα(r−R−τ)∝an}bracketri}ht=δR′Rδτ′τδα′α. (2)\nIn our case, the Wannier function were calculated using the projec tor-operator method\n(Refs. 15 and 16) and orthonormal linear muffin-tin orbitals (LMTO’s , Ref. 17) as the\ntrial wave functions. Typically such procedure allows us to generat e well localized Wannier\nfunctions, that is guaranteed by the good localization of LMTO’s the mselves. Then, the\none-electron part of the model (1) is identified with the matrix eleme nts of LDA Hamilto-\nnian (HLDA) in the Wannier basis: tαβ\nτ,τ′+R=∝an}bracketle{tφτα(r−τ)|HLDA|φτ′β(r−R−τ′)∝an}bracketri}ht. Since the\nWannier basis is complete in the low-energy part of the spectrum, th e construction is exact\nin the sense that the band structure, obtained from tαβ\nτ,τ′+R, exactly coincides with the one\nof LDA.\nAll Wannier basis and one-electron parameters tαβ\nτ,τ′+Rwere first computed without the\nSO interaction. Then, the SO interaction at each atomic site was inclu ded in the “second-\nvariation step”, in the basis of the nonrelativistic Wannier functions :∝an}bracketle{tφiα(r)|∆HSO|φiβ(r)∝an}bracketri}ht,\nas explained in Ref. 15.\nMatrix elements of screened Coulomb interactions can be also compu ted in the Wannier\nbasis as\nUi\nαβγδ=/integraldisplay\ndr/integraldisplay\ndr′φ∗\niα(r)φiβ(r)vscr(r,r′)φ∗\niγ(r′)φiδ(r′),\nwherevscr(r,r′) is obtained using the constrained RPA technique.18Then,vscr(r,r′) does not\ndepend on spin variables and Ui\nαβγδ=Ui\nabcdδσασβδσγσδ. Since RPA is very time consuming\ntechnique, we employ additional simplifications, which were discussed in Ref. 15. Namely,\nfirst we evaluate the screened Coulomb and exchange interactions between atomic Co 3 d\norbitals, using fast and more suitable for these purposes constra ined LDA technique. After\nthat, we consider additional channel of screening caused by the 3 d→3dtransitions in\nthe polarization function of constrained RPA and projecting this fu nction onto atomic 3 d\n7orbitals. The so obtained parameters of screened Coulomb interac tions are well consistent\nwith results of full-scale constrained RPA calculations without additio nal simplifications.\nAll calculations have been performed for the room temperature P421mstructure re-\nported in Ref. 3. The corresponding parameters of the low-energ y model are summarized in\nsupplemental materials.19\nAfter the construction, the model is solved in the unrestricted Ha rtree-Fock (HF) approx-\nimation, which is well justified for the considered case where the deg eneracy of the ground\nstate is lifted by the crystal distortion.15\nB. Electronic polarization\nThe electronic polarization can be computed in the reciprocal space , using the formula\nof King-Smith and Vanderbilt:9\nP=−ie\n(2π)3/summationdisplay\nn/integraldisplay\nBZdk∝an}bracketle{tunk|∂kunk∝an}bracketri}ht, (3)\nwhereunk(r) =e−ikrψnk(r) is the cell-periodic eigenstate of the Hamiltonian Hk=\ne−ikrHeikr, the summation runs over the occupied bands ( n), thek-space integration goes\nover the first Brillouin zone (BZ), and −e(e>0) is the electron charge. In our case, each\nψnk(r) is expanded in the basis of Wannier orbitals φτα(r−R−τ), used for the construction\nof the low-energy model:\nψnk(r) =1√\nN/summationdisplay\nRταcτα\nnkeik(R+τ)φτα(r−R−τ), (4)\nwhereNis the number of primitive cells. Then, the k-space gradient of unkwill have two\ncontributions:\n∂kunk=−i√\nN/summationdisplay\nRτα(r−R−τ)e−ik(r−R−τ)cτα\nnkφτα(r−R−τ)\n+1√\nN/summationdisplay\nRταe−ik(r−R−τ)∂kcτα\nnkφτα(r−R−τ),(5)\nand the electronic polarization Pwill also include two terms:\nP=/summationdisplay\nn/integraldisplay\nBZdk\nΩ∝an}bracketle{tcnk|ˆpk|cnk∝an}bracketri}ht−ie\n(2π)3/summationdisplay\nn/integraldisplay\nBZdk∝an}bracketle{tcnk|∂kcnk∝an}bracketri}ht. (6)\n8Here,|cnk∝an}bracketri}htdenotes the column vector |cnk∝an}bracketri}ht ≡[cτα\nnk],ˆpkis the matrix ˆpk≡[pτ′a′,τa\nkδσα′σα],\nwhere\npτ′a′,τa\nk=−e\nV/summationdisplay\nR∝an}bracketle{tφτ′α′(r−τ′)|r|φτα(r−R−τ)∝an}bracketri}hteik(R+τ−τ′), (7)\nVis the unit cell volume, and Ω = (2 π)3/Vis the first BZ volume. Moreover, since Pis\nunderstood as the changeof the polarization in the process of adiabatic symmetry lowering\n(Ref. 9), which in our case is driven by the magnetic degrees of free dom, here and below we\ndropallnonmagneticcontributionsto pτ′a′,τa\nk, whichareirrelevanttothemagneticsymmetry\nlowering.\nIf the transition-metal site is located in the inversion center, the fi rst term of Eq. (6) iden-\ntically vanishes. Such situation is realized, for instance, in multiferro ic manganites, crystal-\nlizing in the orthorhombic Pbnmstructure, where the FE activity is entirely “anomalous”\nand associated with the second term of Eq. (6).14However, in Ba 2CoGe2O7the situation\nappears to be exactly the opposite: the first term dominates, while the second contribution\nis negligibly small (about 0 .1µC/m2in the magnetic ground state). Furthermore, due to\nthe orthogonality condition (2), the leading contribution to pτ′a′,τa\nkcomes from the Wannier\nfunctions centered at the same atomic site. Then, one can impose in Eq. (7) the additional\ncondition τ′=R+τ, which yields pτ′a′,τa\nk≈pa′a\nτδτ′τ, wherepa′a\nτdoes not depend on k.\nThus, the electronic polarization in Ba 2CoGe2O7will be given by the sum\nP≈/summationdisplay\nτPτ (8)\nof the local electric dipoles:\nPτ≡ ∝an}bracketle{tˆpτ∝an}bracketri}ht= Tra{ˆDτˆpτ}, (9)\nwhere Tr ais the trace over a. Each such dipole is given by the expectation value ∝an}bracketle{tˆpτ∝an}bracketri}htof the\ndipole matrix ˆpτ= [pa′a\nτ] and the spin-independent part of the density matrix ˆDτ= [Dαα′\nτ]\nat the siteτ:\nDaa′\nτ=/summationdisplay\nσαδσασα′/summationdisplay\nn/integraldisplay\nBZdk\nΩcτα\nnkcτα′∗\nnk.\nIn order to evaluate ˆpτwe use the LMTO method and expand each φτα(r) is the basis\n{χυβ}of linear muffin-tin orbitals, which can be viewed as orthonormalized at omic-like\nWannier orbitals, constructed in the whole region of valence states .17Moreover, we shift\neach siteτto the origin, that does not affect the polarization change caused b y the magnetic\n9symmetry lowering. Then, without SO interaction, we have\nφτa(r) =/summationdisplay\nυbqυb\nτaχυb(r−υ+τ). (10)\nwhere,υ=τcorresponds to the “head” of the Wannier function, centered at the Co-site\nτ, while all other contributions describe the “tails” of the Wannier fun ctions, spreading to\nthe neighboring sites υ. Then, around each such site we identically present the position\noperator as r= (υ−τ)+(r−υ+τ) and assume that the leading contribution to ˆpτcomes\nfrom the first term. This is reasonable because, at each site, ( r−υ+τ) couples the atomic\nstates with different parity, which are typically well separated in ene rgy. Then, the matrix\nelements pαα′\nτcan be easily evaluated as\npa′a\nτ=−e\nV/summationdisplay\nυb(υ−τ)qυb∗\nτa′qυb\nτa.\nThen, by considering the leading contributions of four oxygen sites surrounding each Co\natom, one obtains the following matrices (in µC/m2):\nˆp1,2x=\n0∓3509 0 6357 0\n∓3509 0 −801 0 −3553\n0−801 0 ±303 0\n6357 0 ±303 0 ∓6142\n0−3553 0 ∓6142 0\n, (11)\nˆp1,2y=\n0 6357 0 ±3509 0\n6357 0 ∓303 0 ∓6142\n0∓303 0 −801 0\n±3509 0 −801 0 3553\n0∓6142 0 3553 0\n, (12)\nand\nˆp1,2z=\n0 0 −3682 0 0\n0±3394 0 3524 0\n−3682 0 0 0 ±2870\n0 3524 0 ∓3394 0\n0 0 ±2870 0 0\n, (13)\n10where the upper (lower) signs correspond to the Co-sites 1 (2) in F ig. 1. As expected, the\nmatrices p1,2≡(ˆp1,2x,ˆp1,2y,ˆp1,2z) obey the P421msymmetry. If the density matrix ˆD1,2\nobeys the same symmetry, all local electric dipoles will vanish and the re will be no net\npolarization in the ground state. However, if the symmetry of ˆD1,2is lowered by magnetism,\none can expect the appearance of local electric dipoles with some or der between sites 1 and\n2. If this order is antiferroelectric, there will be no net polarization . Nevertheless, if this\norder permits a FE component, the system will exhibit a finite net pola rization. The details\nof such symmetry lowering will depend on directions of magnetic mome nts at the sites 1 and\n2. For instance, if the spins are parallel to the zaxis, the fourfold rotoinversion around z,\nˆSz\n4, will remain among the symmetry operations of the magnetic space g roup, and there will\nbe no local electric dipoles. However, if the moments lie in the xyplane, the rotoinversion\nˆSz\n4is replaced by the regular twofold rotation, ˆCz\n2, which allows for the existence of local\nelectric dipoles parallel to z. For an arbitrary direction of spin, the symmetry will be further\nlowered, and the local electric dipoles may have all three component s. However, the relative\ndirections of dipoles at the sites 1 and 2 will depend on other symmetr y operations. Below,\nwe will investigate this magnetic symmetry lowering more in details.\nTo conclude this section, we would like to emphasize that Eqs. (8) and (9) are the cor-\nrect quantum-mechanical expressions for the electric polarizatio n in Ba 2CoGe2O7, which\nare based on the very general Berry-phase theory.9These expressions allows us clarify the\nmicroscopic origin of the polarization, which is frequently ascribed to the spin-dependent p-d\nhybridization.8Thep-dhybridization does play a very important role in this material as it\ndefines the matrix elements of ˆpτin Eq. (9). Moreover, the contribution of each Co-O bond\nto the local electric dipole moment is indeed proportional to the vect or (υ−τ), connecting\nthe Co and O sites. However, the matrix elements pa′a\nτthemselves do not depend on the\nspin state. In this sense, the spin dependence of the hybridization is not the most impor-\ntant aspect. The spin dependence of the electric polarization in our picture comes from the\nmatrix elements of the density matrix, which describes the effect of the SO coupling at the\ntransition-metal sites. The local magnetic moment deforms the ele ctron density around the\nCo sites. This deformation, which itself depends on the direction of lo cal magnetic moment,\nspreads to the neighboring oxygen sites via the tails of the Wannier f unctions and produces\nfinite electric moment. This is the basic microscopic picture underlying the formation of the\nlocal electric moments in Ba 2CoGe2O7.\n11III. RESULTS AND DISCUSSIONS\nThe first important question we need to address is the local proper ties developed at each\nof the Co sites. For these purposes we set all transfer integrals in Eq. (1) equal to zero\nand solve the model in the mean-field approximation separately for e ach Co site. The SO\ninteraction in these calculations is treated in the frameworks of the self-consistent linear\nresponse (SCLR) theory.20For each direction of spin, it gives us the self-consistent HF\npotential in the first order of the SO coupling, which can be used aga in as the input of HF\nequations in order to obtain at the output the change of the densit y matrix and the total\nenergy beyond the first order. We would like to emphasize that the c onsidered below effects\nare beyond the first order of the SO coupling.\nThe obtained polarization is in the good agreement with the regular se lf-consistent HF\ncalculations, which can be performed for the high-symmetry points . For the in-plane ro-\ntations of the spin magnetization, the results are summarized in Fig. (3). In this case, Px\n/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s32/s48\n/s32/s49/s80 /s32/s40 /s67/s47/s109/s50\n/s41/s97/s116/s111/s109 /s32/s49\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s67/s111/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s45/s52/s48/s45/s50/s48/s48/s50/s48/s52/s48\n/s32/s48\n/s32/s50\n/s32/s80\n/s120\n/s32/s80\n/s121\n/s32/s80\n/s122/s80 /s32/s40 /s67/s47/s109/s50\n/s41/s97/s116/s111/s109 /s32/s50\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s48/s46/s48/s48/s48/s46/s48/s49/s48/s46/s48/s50/s48/s46/s48/s51/s48/s46/s48/s52/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s67/s111/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s48/s46/s52/s51/s51/s48/s46/s52/s51/s52/s48/s46/s52/s51/s53/s48/s46/s52/s51/s54/s48/s46/s52/s51/s55/s48/s46/s52/s51/s56/s124/s77 /s124/s32/s40\n/s66/s41\n/s48/s46/s52/s51/s51/s48/s46/s52/s51/s52/s48/s46/s52/s51/s53/s48/s46/s52/s51/s54/s48/s46/s52/s51/s55/s48/s46/s52/s51/s56/s124/s77 /s124/s32/s40\n/s66/s41\nFIG. 3. (Color online) Electronic polarization, absolute v alue of orbital magnetization, and total\nenergy depending on the direction e= (cosφ,sinφ,0) of spin magnetization in the xyplane. The\nlinesshow resultsofself-consistent linear responsetheo ryforthespin-orbitcoupling.20Thesymbols\nshow results of unrestricted Hartree-Fock calculations fo r the high-symmetry points.\nandPyidentically vanish, while Pz(φ) exhibits the characteristic cosine-like behavior.5As\n12expected, each Pz(φ) takes its minimum value at the angle φ0\nτ, which specifies the direction\nof the upper O-O bond of the CoO 4tetrahedron (see Fig. 1). Note also the minus sign\nin Eq. (7), which means that the considered polarization is purely elec tronic. Since two\nCo sites in Ba 2CoGe2O7are connected by the symmetry operation {ˆCx\n2|a1/2+a2/2}(the\ntwofold rotation around x,ˆCx\n2, associated with the half of the primitive translations in the\nxyplane), the angles φ0\n2andφ0\n1satisfy the property: φ0\n2=π/2−φ0\n1, where the experimental\nvalue ofφ0\n1is 67◦.3The maximums of Pz(φ) are atφ=φ0\nτ+π/2, which specify the directions\nof the lower O-O bonds of the CoO 4tetrahedra. Correspondingly, Pz(φ) = 0 when the spin\nmoment is aligned in between the upper and lower O-O bonds ( φ=φ0\nτ+π/4), and the\nnegative and positive contributions to the polarization cancel each other.\nVery generally, the single-ion (SI) anisotropy energy in tetragona l systems has the fol-\nlowing form:21\nESI=K1sin2θ+K2sin4θ+K′\n2sin4θcos4(φ−φ0\nτ).\nBy fitting the total energies, obtained in the SCLR calculations, one can find that K1=\n−2.307 meV,K2= 0.072 meV, and K′\n2=−0.018 meV. The obtained value of K1is well con-\nsistent with the experimental estimate K1≈ −2.327 meV, reported in Refs. 22 and 23. The\nparameterK′\n2, controlling the in-plane anisotropy, is small but finite. Anyway, this behavior\nis different from the conventional S= 3/2 spin-only model, where noin-plane anisotropy is\nexpected.22The difference is caused by an appreciable orbital magnetization ( ∼0.434µBper\nCo site), which is unquenched in the xyplane. Moreover, this orbital magnetization exhibits\na sizable anisotropy ( ∼0.005µBper Co site – see Fig. 3), depending on the direction in the\nxyplane. Then, it is reasonable to expect that the magnetocrystalline anisotropy energy\nshould be related to the behavior of the orbital magnetization.24Indeed, we observe a close\ncorrelation between behavior of the SI anisotropy energy and the anisotropy of the orbital\nmagnetization in the xyplane (see Fig. 3). Thus, similar to the polarization, the origin of\nthe in-plane SI anisotropy is related to the orbital degrees of free dom, which become active\ndue to the magnetic symmetry lowering. The SI anisotropy can be fu rther controlled by\napplying the uniaxial stress.6\nBy summarizing the behavior of the SI anisotropy energy, the spin m oments are expected\nto lie in the xyplane and, at each Co site, be parallel to either upper or lower O-O bo nd.\nTherefore, as far as the SI anisotropy energy is concerned, one expects two magnetic config-\nurations, which are depicted in Figs. 1c and 1d. In the first case (Fig . 1c), the spins at the\n13site 1 and 2 are parallel to, respectively, lower and upper O-O bonds and hence this phase\nif antiferroelectric. In the second case (Fig. 1d), all spins are par allel to the upper bonds,\ngiving rise to the FE order. In the atomic limit, these two configuratio ns are degenerate.\nAnother factor, controlling the relative direction of spins, is the int eratomic exchange\ninteractions. The isotropic part of these interactions can be comp uted by considering the\ninfinitesimal rotations of spins, which provides the local mapping of t he total energies onto\nthe spin (Heisenberg) model EH=−/summationtext\ni>jJijei·ej, witheidenoting the direction of spin\nat the site i.25There are six types of nonvanishing exchange interactions, which a re ex-\nplained in Fig. 4. The values of these interactions, obtained in the C-t ype AFM state, are\n/s74\n/s49\n/s74\n/s50/s74\n/s51/s74\n/s52\n/s74\n/s53/s74\n/s54\nFIG. 4. (Color online) Main parameters of isotropic exchang e interactions.\nJ1= 0.012 meV,J2=−2.730 meV,J3=−0.046 meV,J4=−0.071 meV,J5=−0.007\nmeV, andJ6=−0.012 meV. Thus, the exchange interaction J2≡Jstabilizes the AFM\ncoupling in the xyplane, while the FM coupling between the planes is stabilized by the com-\nbination of J1,J3, andJ5. The leading interaction Jis relatively weak (compared to other\ntransition-metal oxides), that is directly related with the crystal structure of Ba 2CoGe2O7,\nwhere neighboring Co sites in the xyplane are separated by relatively long Co-O-Ge-O-Co\npaths. Nevertheless, the value of Jseems to be overestimated in comparison with the exper-\nimental data. For example, the N´ eel temperature TN≈34 K, evaluated using Tyablikov’s\nrandom-phase approximation,26is about five time larger than the experimental TN= 6.7 K.3\nSimilar disagreement is found for the exchange coupling itself: our ca lculations overestimate\nthe experimental J, reported in Refs. 22, 23, and 27, by the same order of magnitude . This\n14seems to be a negative aspect of the low-energy model (1), which w as constructed only for\nthe Co 3dbands and neglects several important contributions to the magne tic properties of\nBa2CoGe2O7, related to the magnetic polarization of the O 2 pband. This is also the main\nreason why we are not able to obtain a good quantitative agreement with the experimental\ndata for the behavior of electric polarization in the magnetic field: alt hough our low-energy\nmodel correctly reproduces the main tendencies, the magnetic fie ld should by additionally\nscaled (by the same factor as the exchange coupling J) to be compared with the experi-\nmental data. In the Appendix, we will evaluate the change of the ma gnetic energy, caused\nby the polarization of the O 2 pband, and show that this mechanism, which favors the FM\nalignment, can indeed reduce the effective AFM coupling J. Nevertheless, this mechanism\noverestimates the FM contribution to Jand alone does not resolve the quantitative dis-\nagreement with the experimental data, which should involve addition al factors, such as the\nexchange striction below TN.\nTheDzyaloshinskii-Moriya (DM)interactionscanbecalculatedbyapp lyingSCLR theory\nfor the SO coupling and considering a mixed perturbation, where the SO interaction is\ncombined with rotations of the spin magnetization.20The DM interactions between nearest\nneighbors along zare forbidden by the symmetry. Therefore, the strongest inter action is\nagaind2≡(dx,dy,dz) , which takes place between nearest neighbors in the xyplane. Forthe\nbond, connecting the sites 1 and 2 in Fig. 1a, this vector is given by d2= (−5,5,−6)µeV.\nSimilar parameters for other bonds can be obtained by applying the s ymmetry operations\nof the space group P421m, which will change the signs of dxanddy. Thus,d2is comparable\nwithK′\n2and can also contribute to the canting of spins.\nThe AFM interaction Jenforces the collinear alignment between spins in the xyplane\n(see Fig. 1), while the SI anisotropy and DM interactions result in a sm all canting of spins,\nwhich is given by:19\nδφ≈K′\n2cos4Φsin4φ0\n1+dz\n2J, (14)\nwhere Φ =1\n2(φ1+φ2) is the average azimuthal angle, formed by the spins 1 and 2. It is\ninteresting that the magnitude of the canting depends on Φ, which c ontributes to the SI\nanisotropy, but not to the DM energy. Indeed, for the antiferro magnetically coupled spins,\nbeing parallel to the [100] and [110] axes in the xyplane, cos4Φ is equal to 1 and −1,\nrespectively. Therefore, in the first case (Fig. 1c), the effects o f the SI anisotropy and DM\ninteractions will partly compensate each other (note that sin4 φ0\n1<0), while in the second\n15case (Fig. 1d), these two terms will collaborate, that leads to large r spin canting. This\nanalysis is totally consistent with results of unrestricted HF calculat ions for the low-energy\nmodel (1).\nThe exchange coupling J, in the combination with the SI anisotropy K′\n2, lifts the de-\ngeneracy between states e||[100] and e||[110]. However, the corresponding energy difference\n(per one formula unit),19\n∆E= 2K′\n2cos4φ0\n1,\nis very small (about 1 µeV), mainly because cos4 φ0\n1is small. This is the main reason why\nthe direction of spins in the xyplane cannot be not easily determined experimentally.22,27\nAs was discussed in Sec. IIB, the finite value of the polarization is due to the magnetic\nsymmetry lowering, which is reflected in the behavior of the density m atricesˆD1,2. The\nlatter can be identically presented as ˆD1,2=ˆD0\n1,2+δˆD1,2, whereˆD0\n1,2is the average density,\nobeying the P421msymmetry, and δˆD1,2is a perturbation, which depends on the direction\nof spine. Straightforward unrestricted HF calculations yield (in 10−3)\nReδˆD1,2=\n0 0 ∓3.92 0 0\n0 3.43 0 ∓1.90 0\n∓3.92 0 0 0 −1.30\n0∓1.90 0 −3.43 0\n0 0 −1.30 0 0\n(15)\nand\nReδˆD1,2=\n0 0 −0.16 0 0\n0±1.85 0 −2.88 0\n−0.16 0 0 0 ∓3.90\n0−2.88 0 ∓1.85 0\n0 0 ∓3.90 0 0\n(16)\nfor the solutions with the spin magnetization being parallel to the axe s [100] and [110],\nrespectively (see Figs. 1b and c), where the upper (lower) signs co rrespond to the Co-sites\n1 (2) in Fig. 1. By combining these matrices with ˆ p1,2z, given by Eq. (13), one obtains the\nfollowing contributions of the sites 1 and 2 to the electric polarization :P1,2z=±31.2µC/m2\nandP1,2z=−28.9µC/m2fore||[100]and e||[110],respectively. Thus, themagneticstructure\nwithe||[100] preserves the symmetry operation {ˆCx\n2|a1/2+a2/2}, connecting the sites 1 and\n162, which results in the antiferroelectric state. On the other hand, the magnetic structure\nwithe||[110] breaks this symmetry, giving rise to the FE order. The total p olarization\nPz=P1z+P2z= 57.8µC/m2is in fair agreement with the experimental data. According to\nEqs. (13), (15), and (16), such a behavior is related to the phase s of the matrix elements of\nReδˆD1,2, which interplay with the phases of ˆ p1,2z: The phases are organized in such a way\nthat fore||[100] and e||[110] their interplay yields P2z=P1zandP2z=−P1z, respectively.\nNote also that the obtained matrices Re δˆD1,2do not couple with ˆ p1,2xand ˆp1,2y, so that the\nxandycomponents of the polarization are identically equal to zero.\nNext, we discuss how the electric polarization can be controlled by th e external magnetic\nfieldH. First, we consider the situation, where the AFM spins are parallel t o the [110] axis\n(Fig. 1d) and apply Halong the perpendicular direction [ ¯110]. The results are summarized\nin Fig. 5, where for an easier comparison with experimental data we p lot−Pz.28In this case,\n/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48\n/s32/s80\n/s122/s32/s40 /s67/s47/s109/s50\n/s41\n/s45/s57/s48 /s45/s54/s48 /s45/s51/s48 /s48 /s51/s48 /s54/s48 /s57/s48/s45/s50/s45/s49/s48/s49/s50/s77 /s32/s124/s124/s32/s91/s49/s49/s48/s93/s32/s40\n/s66/s47/s67/s111/s41\n/s72 /s32/s124 /s124 /s32/s91/s49/s49/s48/s93/s32/s40/s84/s41\nFIG. 5. (Color online) Electric polarization and net spin ma gnetization in the external magnetic\nfield parallel to the [ ¯110] axis, as obtained from the solution of the low-energy el ectron model in\nthe Hartree-Fock approximation.\nHcontrols the magnitude of the spin canting and the direction of spins relative to the upper\n(lower) O-O bonds. When both spins are parallel to the upper O-O bo nds,−Pztakes the\nmaximal value. The corresponding magnetic field can be easily found f rom the analysis of\nthe spin Hamiltonian, which yields\nµBHm=−8J\ngSsin/parenleftig\nφ0\n1−π\n4/parenrightig\n.\n17Using above values of Jandφ0\n1,g≈2 andS= 3/2,Hmcan be evaluated as 47 T, which is\nin the very good agreement with the results of HF calculations for th e low-energy electron\nmodel displayed in Fig. 5. Nevertheless, Hmis overestimated by factor five in comparison\nwith the experimental data,5,6following similar overestimation of J, as was explained above.\nWhen the spins are aligned in between the upper and lower O-O bonds, Pzis equal to\nzero. The maximal value of the polarization at H=HmisPm≈90µC/m2, which is in fair\nagreement with the experimental data (about 120 µC/m2, Ref. 5).\nSince the in-plane anisotropy is small, the spins in the xyplane can be easily rotated by\nthe magnetic field, which couples to the net magnetization. The resu lts of such calculations\nare displayed in Fig. 6. In this case, the value of the magnetic field also plays an important\n/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48/s32/s97/s116/s111/s109 /s32/s49\n/s32/s97/s116/s111/s109 /s32/s50\n/s32/s116/s111/s116/s97/s108/s80\n/s122/s32/s40 /s67/s47/s109/s50\n/s41\n/s72 /s32/s61/s32/s52/s55/s46/s48/s32/s84\n/s48 /s57/s48 /s49/s56/s48 /s50/s55/s48 /s51/s54/s48/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48/s80\n/s122/s32/s40 /s67/s47/s109/s50\n/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s72 /s32/s61/s32/s53/s56/s46/s56/s32/s84\nFIG. 6. (Color online) Behavior of electric polarization (t otal and partial contributions of the\nCo sites 1 and 2) under the rotation of the external magnetic fi eld in the plane xy, as obtained\nfrom the solution of the low-energy electron model in the Har tree-Fock approximation. Here, the\nangleφspecifies the direction of the magnetic field, while the antif erromagnetic component of the\nmagnetization is perpendicular to the filed.\nrole, as it controls the angle between spins at two Co sublattices: if H≈Hm, the individual\ncontributions P1zandP2zchange “in phase”, and for φ=π/4 (modulo π/2)|P1z+P2z|\nreaches its maximal possible value Pm. However, if H∝ne}ationslash=Hm, there is some “dephasing” and\n|P1z+P2z|is smaller than Pm.\nAs was already discussed in Sec. IIB, for an arbitrary direction of s pins, the original\nP421msymmetry is completely destroyed and the polarization Pcan also have an arbitrary\n18direction. Similar to the in-plane rotations (Fig. 3), this behavior can be well understood\nalready in the atomic limit, by considering the change of the density ma trix, induced by\nthe SO interaction at a given Co site, which couples with the electric dip ole matrix ˆpτ. For\nthese purposes we again employ the SCLR method. It is convenient t o start with the AFM\nconfiguration of spins parallel to the [110] axis and rotate them out of thexyplane. The\nresults of such calculations are summarized in Fig. 7. One can clearly s ee that, in addition\n/s45/s54/s48/s45/s51/s48/s48/s51/s48/s54/s48/s80 /s32/s40 /s67/s47/s109/s50\n/s41\n/s32/s61/s32/s52/s53/s111/s97/s116/s111/s109 /s32/s49\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s67/s111/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s115/s41/s45/s54/s48/s45/s51/s48/s48/s51/s48/s54/s48/s32/s80\n/s120\n/s32/s80\n/s121\n/s32/s80\n/s122/s97/s116/s111/s109 /s32/s50/s80 /s32/s40 /s67/s47/s109/s50\n/s41\n/s32/s61/s32/s50/s50/s53/s111\n/s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s69/s110/s101/s114/s103/s121/s32/s40/s109/s101/s86/s47/s67/s111/s41\n/s32/s40/s100/s101/s103/s114/s101/s101/s115/s41\nFIG. 7. (Color online) Electronic polarization and total en ergy depending on the direction e=\n(cosφsinθ,sinφsinθ,cosθ) of the spin magnetization, rotated out of the xyplane for φ=π/4 and\nφ= 5π/4, as obtained in the self-consistent linear response theor y for the spin-orbit coupling.20\nThe azimuthal angles φ=π/4 andφ= 5π/4 at the sites 1 and 2, respectively, were chosen to\nsimulate the antiferromagnetic spin alignment in the xyplane.\ntoPz, there are finite perpendicular components of the polarization, PxandPy, which obey\ncertainsymmetry rulesandreplicatethegeometryoftherotated CoO4tetrahedra. Thetotal\nenergy change in this case is mainly controlled by relatively large anisot ropy parameter K1.\nIn practice, such a situation can be realized by applying the magnetic field along the z\naxis, which cants the spins out of the xyplane (Fig. 8). The angle θbetween the spins and\nthexyplane can be estimated as cos θ=−µBgHS/(8J+2K1). There is a clear similarity\nwith the atomic picture, depicted in Fig. 7: the magnetic field slightly de creases|Pz|and\ninduces two perpendicular components of the the polarization, whic h satisfy the condition\nPy=−Px. According to the atomic calculations (Fig. 7), the components PyandPxare\nmainly induced at the Co sites 1 and 2, respectively, that is closely rela ted to the geometry\nof the rotated CoO 4tetrahedra.\n19/s45/s54/s48 /s45/s52/s48 /s45/s50/s48 /s48 /s50/s48 /s52/s48 /s54/s48/s45/s56/s48/s45/s52/s48/s48/s52/s48/s56/s48\n/s80 /s32/s40 /s67/s47/s109/s50\n/s41\n/s72 /s32/s124 /s124 /s32/s91/s48/s48/s49/s93/s32/s40/s84/s41/s32/s80\n/s120\n/s32/s80\n/s121\n/s32/s80\n/s122\nFIG. 8. (Color online) Electric polarization in the externa l magnetic field parallel to the zaxis, as\nobtained from the solution of the low-energy electron model in the Hartree-Fock approximation.\nThe starting configuration, where antiferromagnetic spins were parallel to the [110] axis, is shown\nin the inset.\nIV. SUMMARY AND CONCLUSIONS\nUsing effective low-energy model, derived from the first-principles e lectronic structure\ncalculations, we have investigated the multiferroic properties of Ba 2CoGe2O7. There are two\nimportant aspects, which make this material especially interesting in the field of multiferroic\napplications: (i)Ba 2CoGe2O7crystallizesinthenoncentrosymmetric butnonpolarstructure.\nThekeysymmetryoperation,whichcontrolsthemultiferroicprope rtiesofthismaterialisthe\nrotoinversion. From the view point of magnetic symmetry breaking, there is a fundamental\ndifference between inversion and rotoinversion. The magnetic patt ern, which can break the\ninversion symmetry, should be rather nontrivial. Typically, such mag netic order arises from\nthe competition of many magnetic interactions and crucially depends on a delicate balance\nbetween these interactions. On the contrary, the rotoinversion symmetry can be broken\nvery easily, simply by caning the magnetic moments out of the rotoinv ersion axis. (ii) The\nmagnetic structure of Ba 2CoGe2O7is very soft and can be easily deformed by the external\nmagnetic field. This property is related to the specific geometry of t he Ba2CoGe2O7lattice,\nwhere the CoO 4tetrahedra are interconnected with each other via the GeO 4octahedra.\nThus, the magnetic Co atoms are separated by the long Co-O-Ge-O -Co paths, resulting\n20in the relatively weak exchange coupling J. Another important ingredient is the weak in-\nplane anisotropy, which is inherent to magnetic compounds with the s pin 3/2.22We propose\nthat this anisotropy has an intraatomic nature and related to the a nisotropy of the orbital\nmagnetization.\nNevertheless, it seems that the magnetic softness of Ba 2CoGe2O7has also one negative\naspect: the N´ eel temperature TN= 6.7 K, below which the multiferroic behavior has been\nobserved, isrelatively small.3This imposes aserious constraint onthepractical realizationof\nthe considered effects: since TNis controlled by the same exchange coupling J, any attempts\nto increase TNwill make it more difficult to deform the magnetic structure and manipu late\nthe properties of Ba 2CoGe2O7by the magnetic field.\nOn the theoretical side, we have shown that the electric polarizatio n of Ba 2CoGe2O7can\nbe presented as the sum of electric dipoles, which are induced at eac h Co site by the local\nexchange field. Each such dipole is given by the expectation value ∝an}bracketle{tˆp∝an}bracketri}ht= Tr[ˆpˆD] of the\ndipole matrix ˆpand the site-diagonal density matrix ˆD. This is rather general property of\nBa2CoGe2O7, whichwasderivedstartingfromtheBerry-phasetheoryofelect ricpolarization\nin the Wannier basis. The local character of polarization in the case o f Ba2CoGe2O7is\ndirectly related to the rotoinversion symmetry.\nThe magnetic state dependence of the electric polarization is fully de scribed by the site-\ndiagonal density matrix ˆD. Any rotation of the local magnetization from the rotoinversion\naxis lowers thesymmetry of thedensity matrix ˆDat theCo site andinduces thelocal electric\ndipole due to the transfer of the weight of the Wannier functions to the neighboring oxygen\nsites. This transfer is possible due to the p-dhybridization. However, the spin dependence of\nthe hybridization itself does not play an important role. The direction of the electric dipole\ndepends, via ˆD, on that of the local magnetization. The total polarization of the c rystal is\nthe macroscopic average over the microscopic electric dipoles. This is the basic microscopic\npicture underlying the multiferroic behavior of Ba 2CoGe2O7.\nAcknowledgements . This work is partly supported by the grant of Russian Science Foun -\ndation (project No. 14-12-00306).\n21Appendix: Polarization of oxygen band and interatomic exchange interactions\nIn this appendix, we evaluate the change of the magnetic energy, c aused by the polariza-\ntion of the O 2 pband.\nAfter the solution of the low-energy model, consisting of the Co 3 dbands, we expand the\nbasis Wannier functions of the model in the original LMTO basis, Eq. ( 10), and construct\nthe spin magnetization density m(r) =n↑(r)−n↓(r), associated with the Co 3 dband. In this\nappendix,n↑andn↓(v↑andv↓) denote the electron densities (potentials) for the majority\n(↑) and minority ( ↓) spin states. m(r) has major contributions at the Co sites as well as\nsomehybridization-induced contributionat theoxygen andother a tomicsites. Following the\nphilosophy of the low-energy model,15the interaction of m(r) with the rest of the electronic\nstates can be described in the frameworks of the local-spin-densit y approximation (LSDA).\nTherefore, our strategy is to evaluate, within LSDA, the exchang e-correlation (xc) field\nb(r) =v↓(r)−v↑(r), which is induced by m(r) and polarizes the O 2 pband, and find the\nself-consistent change of m(r) andb(r), caused by the polarization of the O 2 pband. For\nthese purposes, it is convenient to use the SCLR theory.20For simplicity, let us consider\nthe discrete lattice model and assume that all weights of m(r) are concentrated in the\nlattice points: m(r) =/summationtext\nυmυδ(r−υ), wheremυis the local magnetic moment at the\nsiteυ. Furthermore, we recall that LSDA is conceptually close to the Sto ner model.29\nThen, the magnetic part of the xc energy can be approximated as Exc=−1\n4/summationtext\nυIυm2\nυ. In\npractical calculations, the parameters {Iυ}can be obtained using the values of intraatomic\nspin splitting and local magnetic moments. In LMTO, the intraatomic e xchange splitting\ncanbeconvenientlyexpressed viaC-parametersofthecenterso fgravityoftheCo3 dstates.17\nThen, by introducing the vector notations /vectorb≡[bυ], and the tensors ˆI= [Iυδυυ′] and\nˆR= [Rυυ′], the self-consistent field can be found as\n/vectorb=/bracketleftig\n1+ˆIˆR/bracketrightig−1/vectorb0,\nwhere/vectorb0=ˆI/vector mis the xc field induced by the Co 3 dband, and the response tensor is\nobtained in the first order of the perturbation theory for the wav efunctions, starting from\nthe nonmagnetic band structure in LDA:\nRυυ′=/summationdisplay\nabocc/summationdisplay\nnunocc/summationdisplay\nn′BZ/summationdisplay\nk/braceleftbigg(Cυa\nnk)∗Cυa\nn′k(Cυ′b\nn′k)∗Cυ′b\nnk\nεnk−εn′k+c.c./bracerightbigg\n. (A.1)\n22In these notations, {Cυa\nnk}are the coefficients of the expansion of the LDA wave functions\nover the LMTO basis, {εnk}are the LDA eigenvalues, and kruns over the first Brillouin\nzone (BZ). Moreover, similar to constrained random-phase appro ximation for the screened\nCoulomb interactions (Ref. 18), we have to exclude from the summa tion in Eq. (A.1) the\ncontributions, where both indexes nandn′belong to the Co 3 dband. In the present\nperturbation theory, such terms describe the change of the mag netization, which is caused\nby the LSDA potential in the Co 3 dband. However, these effects are already taken into\naccount in the low-energy model, where the LSDA part is replaced by a more rigorous\nunrestricted HF approximation with the screened Coulomb interact ions. Therefore, such\nterms should be excluded at the level of SCLR calculations for the LS DA part. In practical\ncalculations, nruns over the occupied O 2 pbands and n′runs over the unoccupied Co 3 d\nbands.\nOnce/vectorbis known, the change of /vector m, caused by the polarization of the oxygen band, can\nbe found as\nδ/vector m=−ˆR/vectorb\nand corresponding change of the xc-field is δ/vectorb=ˆIδ/vector m. Since O 2pband is fully occupied, the\nnet change of the magnetic moment is identically equal to zero:/summationtext\nυδmυ= 0, irrespectively\non the type of the magnetic order. Nevertheless, the individual mo mentsδmυcan be finite\nand contribute to the energy. The corresponding correction to t he total energy consists of\ntwo parts:δE=δECo−O+δEO, whereδECo−O=−1\n2δ/vector mTˆI/vector mis the interaction of δmυwith\nthe “external” xc field, created by the Co 3 dband, andδEOis the energy change caused\nbyδ/vector min the O 2pband. It also consists of two parts: δEO=δEsp+δEdc, whereδEspis\nthe single-particle energy, which can be found in the second order o fδ/vectorbasδEsp=1\n4δ/vectorbTˆR/vectorb,20\nδEdc=1\n4δ/vector mTˆIδ/vector mis the double-counting energy, and δ/vector mTis the row vector, corresponding\nto the column vector δ/vector m. Meanwhile, it is assumed that the magnetic energy of the Co 3 d\nband itself is described by the low-energy model in the HF approximat ion.\nδEmay have different values in the case of the FM and AFM alignment of sp ins in the\nxyplane. This difference additionally contribute to interatomic exchang e interactions in the\nplane.\nIn theP421mstructure of Ba 2CoGe2O7, there are three types of oxygen atoms: O1, O2,\nand O3, which are located in the Wyckoff positions 2 c, 4e, and 8f, respectively.3The ob-\ntainedparameters {Iυ}are0.98, 1.92, 1.10,and2.01eVforCo, O1, O2, andO3, respectively.\n23The magnetic moments are listed in Table I and the energies are in Table II. As expected,\nTABLE I. Local magnetic moments mυ, derived from the low-energy model for the Co 3 dband,\nand the moments δmυ, caused by the polarization of the O 2 pband in the ferromagnetic (F) and\nC-type antiferromagnetic (C) state of Ba 2CoGe2O7. All magnetic moments are in µBper site and\nthe number of such sites in the unit cell is given in the parent heses.\nF C\nmυ δmυ mυ δmυ\nCo (×1) 2 .251 0 .304 2 .245 0 .234\nO1 (×1) 0 .005 −0.004 0 0\nO2 (×2) 0 .004 −0.004 0 0\nO3 (×4) 0 .184 −0.073 0 .180 −0.057\nTABLE II. Magnetic contributions to the energy of interacti on between Co 3 dand O 2pbands\n(δECo−O), the single particle energy in the O 2 pband (δEsp), the total energy of the 2 pband\n(δEO), and the total energy ( δE=δECo−O+δEO) as obtained for the ferromagnetic (F) and\nC-type antiferromagnetic (C) states. All energies are in me V per one formula unit.\nF C\nδECo−O −282.98 −217.67\nδEsp −6.46 −3.24\nδEO 27.07 16.91\nδE −255.91 −200.76\nthe moments mυaredistributed mainly between central Co site andits neighboring sit es O3.\nIn the FM state, the total moment is mCo+mO1+2mO2+4mO3= 3µB, which is totally\nconsistent with the value obtained in the Wannier basis. The moments mυandδmυare\nparallel at the Co sites and antiparallel at the oxygen sites. This ten dency is consistent with\nresults of the first-principles calculations and can be explained by th e hybridization between\nCo 3dand O 2pstates.30Therefore, the negative sign of δECo−Ois due to the contributions\nof the Co sites, which are partly compensated by the positive contr ibutions of the oxygen\nsites. The absolute value of δECo−Ois larger for the FM state, mainly because mCoand\n24δmCoare larger than those in the C-type AFM state. Thus, the Co-O inte raction addition-\nally stabilizes the FM order. Then, since the O 2 pband is fully occupied, the contribution\nδEspis relatively small. The total contribution of the O 2 pband to the magnetic energy is\npositive. This is because the fully occupied O 2 pband itself does not favor the magnetism\nand any magnetic polarization of this band increases the energy. Th is also explains why\nδEOis slightly smaller in the C-type AFM state: the magnetic moments δmυare smaller\nand, therefore, the magnetic perturbation of the O 2 pband is also smaller. Anyway, this\neffect is considerably weaker in comparison with the change of δECo−O.\nIn total, the magnetic polarization of the oxygen band favors the F M alignment of spins\nin thexyplane. By mapping the energy change δEonto the spin model, which includes only\nnearest-neighbor interactions in the xyplane, the change of the exchange coupling, caused\nby the polarization of the oxygen band, can be estimated as δJ=1\n4(δE[C]−δE[F])≈\n14 meV. Thus, δJwill indeed compensate the AFM exchange coupling, obtained in the\nlow-energy model for isolated Co 3 dbands. However, despite the correct tendency, the\nobtained change δJis too large (and would lead to the FM alignment in the xyplane).\nThere may be several reasons for it: (i) The SCLR theory may be to crude for treating the\nmagnetic polarization of the oxygen band (in fact, the perturbatio n, which is introduced\nby/vectorb0in the O 2 pband is not small); (ii) Some quantitative estimates may change by\nconsidering the correct crystal structure below TN(which is not available yet). Particularly,\nthe original P421msymmetry can be lowered by the exchange striction effects; (iii) Tp s ome\nextent, the correlation interactions in the Co 3 dband, beyond the HF approximation, will\nadditionally stabilize the C-type AFM order. The corresponding cont ribution to the total\nenergy difference between FM and C-type AFM states can be estima ted using the second-\norder perturbation theory (Ref. 15) as 2 meV per one formula unit . This change alone does\nnot significantly change the FM contribution δJ, caused by the polarization of the O 2 p\nband. However, it can collaborate with other effects, such as the e xchange striction.\n∗SOLOVYEV.Igor@nims.go.jp\n1T. Kimura, T. Goto, H. Shintani, K. Ishizaka, T. Arima, and Y. Tokura, Nature 426, 55 (2003).\n2T. Kimura, Annu. Rev. Mater. Res. 37, 387 (2007); S.-W. Cheong and M. Mostovoy, Nature\n25Materials 6, 13 (2007); D. Khomskii, Physics 2, 20 (2009); Y. Tokura and S. Seki, Adv. Mater.\n22, 1554 (2010).\n3V. Hutanu, A. Sazonov, H. Murakawa, Y. Tokura, B. N´ afr´ adi, and D. Chernyshov, Phys. Rev.\nB84, 212101 (2011).\n4H. T. Yi, Y. J. Choi, S. Lee, and S.-W. Cheong, Appl. Phys. Lett .92, 212904 (2008).\n5H. Murakawa, Y. Onose, S. Miyahara, N. Furukawa, and Y. Tokur a, Phys. Rev. Lett. 105,\n137202 (2010).\n6T. Nakajima, Y. Tokunaga, V. Kocsis, Y. Taguchi, Y. Tokura, a nd T. Arima, Phys. Rev. Lett.\n114, 067201 (2015).\n7T. Arima, J. Phys. Soc. 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Tyablikov, Methods of Quantum Theory of Magnetism (Nauka, Moscow, 1975).\n27V. Hutanu, A. P. Sazonov, M. Meven, G. Roth, A. Gukasov, H. Mur akawa, Y. Tokura, D.\nSzaller, S. Bord´ acs, I. K´ ezsm´ arki, V. K. Guduru, L. C. J. M . Peters, U. Zeitler, J. Romh´ anyi,\nand B. N´ afr´ adi, Phys. Rev. B 89, 064403 (2014).\n28Absolutely the same calculations can be performed by aligni ng the AFM spins parallel to the\n[¯110] axis and applying the magnetic field along the [110] axis , which will lead to the positive\nnet polarization, being in total agreement with the experim ental data.5\n29O. Gunnarsson, J. Phys. F: Met. Phys. 6, 587 (1976).\n30I. Solovyev, J. Phys. Soc. Jpn. 78, 054710 (2009).\n27"    },    {        "title": "1504.04252v1.Heat_capacity_study_of_the_magnetic_phases_in_a_Nd__5_Ge__3__single_crystal.pdf",        "content": "arXiv:1504.04252v1  [cond-mat.str-el]  16 Apr 2015Heat capacity study of the magnetic phases in a Nd 5Ge3single\ncrystal\nD. Villuendas∗and J. M. Hernàndez\nFacultat de Física, Universitat de Barcelona,\nMartí i Franquès 1, 08028 Barcelona, Spain\nT. Tsutaoka\nGraduate School of Education, Hiroshima University,\nHigashi-Hiroshima, Hiroshima 739-5824, Japan\n(Dated: September 22, 2021)\nAbstract\nThe different magnetic phases of the intermetallic compound Nd5Ge3are studied in terms of the\nspecific heat, in a broad range of temperatures (350 mK–140 K) and magnetic fields (up to 40 kOe).\nThe expected T3andT3/2terms are not found in the antiferromagnetic (AFM) and ferro magnetic\n(FM) phases respectively, but a gapped T2contribution that originates from a mixture of AFM\nand FM interactions in different dimensionalities under a la rge magnetocrystalline anisotropy, is\npresent in both. An almost identical Schottky anomaly, that arises from the hyperfine splitting of\nthe nuclear levels of the Nd3+ions, is observed in both phases, which leads us to state that the\nmagnetic-field induced transition AFM →FM that the system experiments below 26K consists in\nthe flip of the magnetic moments of the Nd ions, conserving the average local moment.\n1I. INTRODUCTION\nIn the last years the binary intermetallic compound Nd 5Ge3has been the object of dif-\nferent studies. The interest in this system comes from the fa ct that many of its physical\nproperties present abrupt changes when the transition betw een its two magnetic phases oc-\ncurs. The first investigations of the magnetic properties sh owed that this material orders\nferrimagnetically in zero applied magnetic field at the Néel temperature, TN≈50K, and\npossesses a remanent moment at 4.2K1. Later on, neutron diffraction experiments sug-\ngested that when the system is cooled in zero applied magneti c field an antiferromagnetic\n(AFM) state is established below TN, although the hysteresis loop typical of a hard magnetic\nmaterial was found at 4.2K. This fact indicated a magnetic-field-induced phase trans ition\nto a ferromagnetic (FM) state2. In recent years, the system has regained attention fostere d\nby the exploration of the temperature dependence of the irre versibility of this transition\nbelowTt= 26K3. Some papers have been published on this compound showing ri ch phe-\nnomena in magnetostriction4, electric resistance, specific heat, spontaneous magnetic phase\ntransitions5, and more recently optical properties and electronic struc ture6.\nThe intermetallic alloy Nd 5Ge3belongs to the family R 5Ge3, with R = rare earth, that\ncrystallizes in the hexagonal Mn 5Si3-type structure ( P63/mcm , space group No. 193). The\nstructure contains two formula units per unit cell, in which R atoms occupy the two non-\nequivalent crystallographic sites 4dand6g, while Ge atoms occupy the 6gsite7. Although it\nis still unclear, the most accepted magnetic structure, whi ch has been derived from neutron\ndiffraction experiments2,8,9, consists in a complex double sheet, with the magnetic momen ts\nof the Nd ions located in the 6gposition oriented parallel to the c-axis, and the moments of\nthe Nd ions in the 4dposition oriented along the c-axis with a deviation angle of 31◦and a\npropagation vector k= (0.2500) ; thezcomponent changes sign every two successive (110)\nplanes.\nThe temperature variation of the specific heat in zero field wa s measured in Ref. 3 and\n10 to examine the magnetic phase transitions. Both works obs erved a hump around 50K\nbut no anomaly around 26K, which was attributed to the existence of a spin-glass stat e,\nbecause of the occurrence of similar features in the Cp(T)curve in other well-known spin-\nglass systems. In this paper we will conduct a detailed study of the specific heat, as a\n2function of the temperature and the magnetic field, seeking a better understanding of the\nproperties of the different magnetic phases.\nII. METHODS\nPolycrystalline ingots were prepared by arc-melting the co nstituting 99.9%-pure Nd and\n99.999%-pure Ge elements under high-purity argon atmosphe re. The compounds were found\nto be single-phase by powder X-ray diffraction. Single cryst als were grown by the Czochralski\nmethod from single-phase polycrystalline samples using a t ri-arc furnace. It should be noted\nthat it is difficult to grow a large single crystal of Nd 5Ge3because the grown crystal ingots\ntend to have small single-crystalline grains. The sample wa s cut from one ingot into a\nrectangular shape ( 1×1.5×2mm3) and annealed at 300◦C for 24 h in an evacuated\nquartz tube. The crystal orientation was determined by the b ack-reflection Laue method.\nMeasurements of the specific heat in the temperature range fr om 350 mK to 300 K and\nmagnetic fields up to 40 kOe were made using the heat pulse-rel axation method with the\nheat capacity option of the PPMS/circleRsystem, produced by Quantum Design/circleR.\nIII. RESULTS AND DISCUSSION\nFigure 1 shows the specific heat, C, of our Nd 5Ge3single crystal (triangles) and a poly-\ncrystaline sample of the nonmagnetic isostructural compou nd La5Ge3(diamonds; extracted\nform Joshi et al.11) as a function of temperature. The specific heat of the latter will be used\nas a blank and follows the expected monotonic behavior from t he electronic and phononic\ncontributions12, while the specific heat of Nd 5Ge3presents a hump around TN≃50K and\nthenceforth tends progressively to the Dulong-Petit limit .\nThe dependence of the specific heat of Nd 5Ge3with the temperature was investigated as\nfollows. Following a zero-field cooling process (ZFC) the sp ecific heat was measured from\n140K down to 350mK. During this process we observed the expected hump associ ated\nto the paramagnetic–AFM transition at 50 K. The next step was to measure the system in\nthe FM state. It is known that cooling the Nd 5Ge3compound below 26K with an applied\nmagnetic field larger than 5kOe leaves the system in the saturated FM state3. To ensure\nthis fact we measured the specific heat following a field cooli ng process (FC) in an applied\n3050100150200\n20 40 60 80 100 120 140C(J mol−1K−1)\nT(K)Nd5Ge3\nLa5Ge3\nFIG. 1. (Color online) Zero-field temperature dependence of the specific heat of a single crystal of\nNd5Ge3(triangles) and a polycrystal of La 5Ge3(diamonds; extracted from Joshi et al.11). The lines\njoining the data points are guides to the eye. The vertical do tted line indicates the temperature\n(50 K) at which a hump is observed for Nd 5Ge3. The horizontal solid line indicates the Dulong-Petit\nlimit of the specific heat.\nfield of15kOe, large enough for our purpose. Nonetheless our goal was t o compare the\nmagnetic contribution of the different magnetic phases to th e specific heat, and the applied\nmagnetic field could play an undesired role. Therefore, afte r measuring the FC process we\nset the magnetic field to zero and measured the specific heat of the ferromagnetic remanent\nstate (FM rem) as we increased the temperature. From magnetization exper iments it is known\nthat well below 26K the FM remand the FM FCare magnetically equivalent3. We observed\nthat this equivalence is also present in terms of specific hea t, as it is shown in Fig. 2. In this\nfigure the three data sets are plotted together showing the AF M curve and how the FM rem\nand FM FCcurves superimpose. Consequently, from now on we will renam e the FM remstate\nas FM state in this temperature region.\nThe specific heat can be assumed to be made up of four independe nt contributions,\nC(T) =Cel(T)+Clat(T)+Chyp(T)+Cmag(T). (1)\nThe contribution from phonons, Clat, can be subtracted using the specific heat of the\nnonmagnetic isostructural compound La 5Ge3, taking into account the different molar masses\nof Nd and La via the two-Debye function method13,14,\nCNd5Ge3\nlat(T) =CLa5Ge3\nlat(rT), (2)\n40.13\n1\n0.3 0.5 1 2 3 4 5C(J mol−1K−1)\nT(K)AFM ZFC\nFMrem\nFMFC\nFIG. 2. (Color online) Log-log plot of the low-temperature d ependence of the specific heat of the\nthree magnetic states: the AFM (crosses), the FM rem(squares), and the FM FC(circles).\nwith\nr=/parenleftBigg\n5M3/2\nLa+3M3/2\nGe\n5M3/2\nNd+3M3/2\nGe/parenrightBigg1/3\n= 0.98. (3)\nTherefore we get\nC(T)−CLa5Ge3\nlat(rT) =Cel(T)+Chyp(T)+Cmag(T). (4)\nTo determine the Cmag(T)contribution for each magnetic phase, we can attempt to mode l\nthe experimental data taking into account the different term s:Cel(T) =γTfrom free charge\ncarriers, Chyp(T) =AT−2from the high-temperature limit of the Schottky anomaly due to\nthe hyperfine splitting of the nuclear levels of the Nd3+ions, and Cmag(T)from spin waves.\nThe approach to study the last term was to consider the more ge neral dispersion relations\nfor the long-wavelength spin interactions. We examined the cases of AFM, FM, and type- A\nAFM (ferromagnetic layers antiferromagnetically coupled ) states. The last is one of the\nproposed magnetic structures to occur below TNat zero applied magnetic field2. Because\nof the large magnetic anisotropy and the magnetoelastic effe cts present in this system3,4, we\nalso took into account the possibility of the presence of a ga p in the dispersion relation, ∆.\nIn the low-temperature limit, the specific heat from each dis persion relation is found to be\nCmag(T) =Be−∆\nT\n\nT(12T2+6T∆+∆2) (5a)\nT−1\n2(15T2+12T∆+4∆2) (5b)\n(6T2+4T∆+∆2). (5c)\n50.13\n1\n0.3 0.5 1 2 3 4 5C−CLa5Ge3\nlat (J mol−1K−1)\nT(K)AFM\nFM\nFIG. 3. (Color online) Zero-field low-temperature specific h eat for both the AFM (diamonds) and\nFM (triangles) states, plotted in a log-log scale to remark t he differences between both states. The\nlattice contribution to the specific heat has been subtracte d to the experimental data. The lines\nare the best fits of Eq. (4) to the data.\nTABLE I. Results of the fitting of Eq. (4) to the specific heat dat a, for both the AFM and FM\nstates. The units are mJ /(mol Km+1)wheremis the power of Tcorresponding to each coefficient,\nm= 1forγ,m=−2forA,m= 2forB. The number in parentheses is the statistical uncertainty\nin the last digit from the least-squares fitting procedure.\nstate γ A B ∆(K)\nAFM 115(2) 39.3(4) 22.2(1) 4.34(8)\nFM 75(1) 40.6(3) 23.1(1) 4.75(5)\nEquations (5a), (5b) and (5c) correspond, respectively, to the low-temperature magnetic\ncontribution to the specific heat of the AFM, the FM, and the ty pe-AAFM states. The\nusual expressions ( T3,T3/2andT2, respectively) are recovered when ∆is set to zero. We\nfitted the experimental data to the expressions with and with out gap and found that, nei-\nther in the AFM nor in the FM phase, the “pure” AFM/FM contribu tions gave physically\nreasonable values for the parameters. On the contrary, in bo th phases the gapped type- A\nAFM contribution [Eq. (5c)] was found to fit precisely. Figur e 3 shows the specific heat for\nthe AFM and FM states together with the best fits of Eq. (4) to th e data. Table I lists the\ncoefficients of all contributions.\n6The mixture of interactions and dimensionalities resultin g in aT2contribution to Cmag\nhas been also considered in magnetic structures where FM dro plets are found in an AFM\nphase15. In our system, nevertheless, the magnetization measureme nts indicate a saturated\nFM state, without evidence of AFM interactions. Despite we d o not have a clear explanation\nto the presence of this term in the FM phase, the values for the parameters obtained fitting\nother contributions [Eq. (5a) and (5b)] do not have any physi cal meaning. The larger gap\nobtained in the FM state corresponds with the larger interna l magnetic field in this phase, as\nit will be shown below. The values of γfor both phases are reasonable within the electronic\ncontributions of rare earth intermetallics R 5Ge316and do not need extra considerations. The\nreduction of the value in the FM phase with respect to the AFM c an be attributed to a\ndecrease in the density of states at the Fermi level, that wou ld probably favour one of the\nelectronic spin projections.\nThe hyperfine contribution can not be omitted to fit completel y the measured specific\nheat. The Schottky anomaly consists in a peak originated fro m the (de)population of discrete\nenergy levels. In this case, these correspond to the hyperfin e split nuclear levels of the Nd3+\nions. The Schottky anomaly can be approximated to A/T2in its high-temperature limit,\nwhereAis related to the internal hyperfine magnetic field by the expr ession15\nA= 5NAkB\n3/parenleftbiggI+1\nI/parenrightbigg/parenleftbiggµIHhyp\nkB/parenrightbigg2\n. (6)\nHereIis the nuclear spin, µIis the nuclear magnetic moment, Hhypis the internal mag-\nnetic field at the Nd site, and the factor 5 is the number of mole s of Nd per mole of\nNd5Ge3. Only two isotopes of Nd have nuclear spin different from zero (I= 7/2),143Nd\nand145Nd with the natural abundances of 12.18% and 8.29%, whose nuc lear magnetic mo-\nments are µI=−1.208µNandµI=−0.744µNrespectively17. The hyperfine field values\nobtained are µ0Hhyp(AFM) = 272(2) T andµ0Hhyp(FM) = 276(1) T. The energy splitting\n(∆E=µIHhyp/I) found is ∼2.5µeV for both phases. We can compare this value with\nthe splitting of other Nd compounds studied with neutron sca ttering. Figure 4 plots the\nhyperfine energy splitting versus the saturated ionic magne tic moment of Nd for several\nNd-based compounds18, along with the data point obtained in this work. The value us ed for\nthe magnetic moment of the Nd ion corresponds to the one obser ved in the saturated FM\nstate3,µNd≈2µB. It is remarkable that we have obtained approximately the sa me splitting\nfor both magnetic phases, ∆E(AFM) = 2.50(2)µeV and∆E(FM) = 2.53(1)µeV, which\n700.511.522.533.54\n0 0.5 1 1 .5 2 2 .5 3 3 .5∆E(µeV)\nNd ionic moment ( µB)NdFeO 3Nd2CuO4 NdMnO 3NdGaO 3NdMg 3NdCu2NdNdAl2\nNdCo2\nNd5Ge3\nFIG. 4. (Color online) Energy splitting of inelastic neutro n scattering signals in several Nd-based\ncompounds (circles) as a function of the corresponding satu rated ionic magnetic moment of Nd\nat low temperatures (adapted from Chatterji et al.18). The data point obtained in this work for\nNd5Ge3(square) is also shown.\nmeans that the average local magnetic moment per Nd ion is rou ghly the same in the two\nphases and corresponds to the value in the FM saturated state . Therefore, we may assert\nthat the magnetic-field-induced AFM →FM transition simply flips the magnetic moments\nand preserves the value of µNd.\nWe will now proceed to investigate the temperature dependen ce of the magnetic contri-\nbution to the specific heat. Subtracting to the total measure d specific heat the analytical\nfunctions of the electronic contribution, the Schottky ano maly and the phononic contribu-\ntion from the corrected La 5Ge3data we obtain Cmag(T). Fig. 5 shows the zero-field-cooled\n(ZFC) and remanent (rem) curves, where the latter was acquir ed as the sample was heated\nin zero applied magnetic field after it had been driven to the F M saturated state. The peak\nat 50 K indicates the temperature at which the AFM ordering ta kes place ( TN). The inflec-\ntion point observed around 26 K is in contradiction with the p reviously reported absence of\nany anomaly around this temperature, which was related to th e possibility of the system to\nbe in a spin glass state3,10.\nFromCmag(T)we can compute the magnetic entropy as\nSmag(T) =/integraldisplayT\n0Cmag(T′)\nT′dT′, (7)\nwhere it is assumed that the magnetic entropies of AFM and FM m aterials at zero tem-\n8010203040506070\n0 20 40 60 80 100 120 140Cmag(J mol−1K−1)\nT(K)ZFC\nrem\nFIG. 5. (Color online) Temperature dependence of the magnet ic contribution to the specific heat\nobtained following the ZFC (diamonds) and rem (triangles) p rocesses. A maximum at 50 K and an\ninflection point around 26 K are highlighted with vertical do tted lines. The lines joining the data\npoints are guides to the eye.\n020406080100\n20 40 60 80 100 120 140Smag(J mol−1K−1)\nT(K)ZFC\nrem\nFIG. 6. (Color online) Temperature dependence of the magnet ic entropy obtained for the ZFC (dia-\nmonds) and rem (triangles) processes. The horizontal solid line represents the value of 5Rln(2J+1)\nforJ= 9/2. The lines joining the data points are guides to the eye.\nperature are zero. Fig. 6 shows how Smagattains the value of Rln(2J+ 1)expected for a\nparamagnet19as the temperature grows above TN. In our case the entropy tends to 5Rln10,\ncorresponding to 5 Nd3+free ions with J= 9/2. The actual value at which the obtained\nentropy tends is moderately smaller because the zero-field s plitting due to the anisotropy\ncould play a significant role even in the paramagnetic state.\n90.120.140.160.180.200.22\n0 5 10 15 20 25 30 35 40C(J mol−1K−1)\nH(kOe)1.2 K\nFIG. 7. (Color online) Magnetic field dependence of the speci fic heat at 1.2 K, starting with the\nsystem in the AFM state ( C≈0.2J mol−1K−1) and ending with the system in the FM state\n(C≈0.14J mol−1K−1). Two independent runs are plotted. The lines joining the da ta points are\nguides to the eye.\nFinally, the dependence of the specific heat with the applied magnetic field at fixed tem-\nperature was studied to explore the magnetic-field-induced AFM→FM transition . The\nsystem was prepared following a ZFC process from T≫TNdown to 1.2 K, where the rel-\native difference between the specific heat of the two phases ha s a maximum, and then the\nspecific heat was measured varying the field, from 0 to 40kOe and back to 0Oe. Fig. 7\nshows how a large, abrupt, and irreversible change in C(H), associated with the AFM–FM\ntransition, takes place between 16 and 17 kOe as the magnetic field ncreases. Two inde-\npendent runs are plotted in the figure showing the reproducib ility of this transition. From\nthe magnetic field-temperature phase diagram obtained from magnetization measurements\nin Ref. 3 one expects this transition to happen at much higher fields(H/greaterorsimilar35kOe). One\nexplanation for this reduction of the field at which the spont aneous transition occurs is to\nconsider it of thermally assisted origin. The large differen ce in the thermal bath properties\nbetween the two experimental setups (MPMS/circleRfor magnetic measurements, versus PPMS/circleR\nfor specific heat measurements) can strongly affect how therm ally-assisted abrupt transi-\ntions develop20,21. We also see in the figure that the decreasing dependence of th e specific\nheat with the increasing magnetic field, above H≈10kOe, is consistent with the behaviour\nof aCmagterm with a gap in the dispersion relation of the spin waves pr oportional to the\napplied magnetic field, ∆∝H. Nevertheless, a change in the sign of the slope is observed\n10in both states around 10kOe, for which we do not have an explanation. A more detailed\nstudy considering also magnetoelastic effects could give a b etter description of the exact\nbehaviour of the specific heat as a function of the applied mag netic field.\nIV. CONCLUSIONS\nIn summary, we have performed measurements of the specific he at in the two magnetic\nphases of the system Nd 5Ge3. From the low-temperature data we have modeled the dif-\nferent contributions to the specific heat. A magnetic T2contribution is found in both the\nferromagnetic (FM) and antiferromagnetic (AFM) phases. Th is term can be understood\nas a mixture of FM and AFM interactions in different dimension alities. In the case of the\nAFM phase this T2term can be attributed to a type- AAFM, while in the case of the FM\nphase can be interpreted as a remanence of AFM interactions. The large magnetocrystalline\nanisotropy of Nd 5Ge3is evidenced by a gapped spin-wave spectrum in both phases.\nThe average magnetic moment at low temperature in the two mag netic phases has been\nobtained by means of the specific heat contribution of the hyp erfine splitting of the nuclear\nmoment of the Nd3+ions. The value of this magnitude is approximately 2µBin both\nphases, which corresponds to the saturation value of the FM s tate at low temperature.\nHence, we state that the magnetic-field-induced transition between both states corresponds\nto an irreversible spin-flip transition of the Nd ions.\nFinally, from the magnetic field dependence we observe that t he field at which the sponta-\nneous transition takes place is remarkably smaller than the expected value from the magnetic\nfield-temperature phase diagram. This is most likely due to t he effect of being in an envi-\nronment with a smaller thermal coupling (PPMS/circleRvs MPMS/circleR), leading to a spontaneous\nignition of a thermally assisted transition at smaller field s, probably by means of a magnetic\ndeflagration process.\nACKNOWLEDGMENTS\nThis work was financially supported by Spanish Government pr oject MAT2011-23698.\nAuthors would like to acknowledge the use of Servicio Genera l de Apoyo a la Investigación-\nSAI, Universidad de Zaragoza. D. V. and J. M. H. also thank A. G arcía-Santiago (UB) for\n11useful discussions.\n∗Corresponding author: diego@ubxlab.com\n1K. H. J. Buschow and J. F. Fast, Phys. Stat. Sol. 21, 593 (1967).\n2P. Schobinger-Papamantellos and K. H. J. Buschow, J. Magn. Ma gn. Mater. 49, 349 (1985).\n3T. Tsutaoka, A. Tanaka, Y. Narumi, M. Iwaki, and K. Kindo,\nPhys. B Condens. Matter 405, 180 (2010).\n4M. Doerr, M. Rotter, A. Devishvili, A. Stunault, J. A. A. J. Pe renboom, T. Tsutaoka, A. Tanaka,\nY. Narumi, M. Zschintzsch, and M. Loewenhaupt, J. Phys. Conf . Ser.150, 042025 (2009).\n5B. Maji, K. G. Suresh, and A. K. Nigam, EPL 91, 37007 (2010).\n6Y. Knyazev, A. Lukoyanov, Y. Kuz’min, B. Maji, and K. Suresh,\nJ. Alloys Compd. 588, 725 (2014).\n7L. Zeng and H. F. Franzen, J. Alloys Compd. 313, 75 (2000).\n8R. Nirmala, A. V. Morozkin, A. K. Nigam, J. Lamsal, W. B. Yelon, O. Isnard, S. A. Granovsky,\nK. K. Bharathi, S. Quezado, and S. K. Malik, J. Appl. Phys. 109, 07A716 (2011).\n9A. P. Vokhmyanin, B. Medzhi, A. N. Pirogov, and A. E. Teplykh,\nPhys. Solid State 56, 34 (2014).\n10B. Maji, K. G. Suresh, and A. K. Nigam, J. Phys. Condens. Matter 23, 506002 (2011).\n11D. Joshi, A. Thamizhavel, and S. K. Dhar, Phys. Rev. B 79, 014425 (2009).\n12C. Kittel, Introduction to Solid State Physics , 8th ed. (Wiley, New York, 2005) p. 703.\n13M. Bouvier, P. Lethuillier, and D. Schmitt, Phys. Rev. B 43, 137 (1991).\n14J. Hoffmann, A. Paskin, K. Tauer, and R. Weiss, J. Phys. Chem. S olids1, 45 (1956).\n15C. He, H. Zheng, J. F. Mitchell, M. L. Foo, R. J. Cava, and C. Lei ghton,\nAppl. Phys. Lett. 94, 102514 (2009).\n16N. P. Gorbachuk, Powder Metall. Met. Ceram. 49, 1 (2010).\n17R. K. Harris, Encyclopedia of Nuclear Magnetic Resonance, Vol. 5 , edited by D. M. Granty and\nR. K. Harris (Wiley, New York, 1996).\n18T. Chatterji, G. J. Schneider, L. van Eijck, B. Frick, and D. Bha ttacharya,\nJ. Phys. Condens. Matter 21, 126003 (2009).\n19S. Blundell, Magnetism in Condensed Matter (Oxford University Press, 2001).\n1220C. Webster, O. Kazakova, J. Gallop, P. Josephs-Franks, A. He rnández-Mínguez, and A. Tza-\nlenchuk, Phys. Rev. B 76, 012403 (2007).\n21F. Macià, G. Abril, J. M. Hernandez, and J. Tejada,\nJ. Phys. Condens. Matter 21, 406005 (2009).\n13"    },    {        "title": "1504.05404v2.Tailoring_the_magnetodynamic_properties_of_nanomagnets_using_magnetocrystalline_and_shape_anisotropies.pdf",        "content": "Tailoring the magnetodynamic properties of nanomagnets using magnetocrystalline\nand shape anisotropies\nVegard Flovik,1,∗Ferran Maci` a,2Joan Manel Hern` andez,2Rimantas Bruˇ cas,3Maj Hanson,4and Erik Wahlstr¨ om1\n1Department of Physics, Norwegian University of Science and Technology, N-7491 Trondheim, Norway\n2Grup de Magnetisme, Dept. de F´ ısica Fonamental, Universitat de Barcelona, Spain\n3Department of Engineering Sciences, Uppsala University, SE-751 21 Uppsala, Sweden\n4Department of Applied Physics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden\n(Dated: March 6, 2022)\nMagnetodynamical properties of nanomagnets are affected by the demagnetizing fields created\nby the same nanoelements. In addition, magnetocrystalline anisotropy produces an effective field\nthat also contributes to the spin dynamics. In this article we show how the dimensions of magnetic\nelements can be used to balance crystalline and shape anisotropies, and that this can be used to\ntailor the magnetodynamic properties. We study ferromagnetic ellipses patterned from a 10 nm\nthick epitaxial Fe film with dimensions ranging from 50 ×150 nm to 150 ×450 nm. The study com-\nbines ferromagnetic resonance (FMR) spectroscopy with analytical calculations and micromagnetic\nsimulations, and proves that the dynamical properties can be effectively controlled by changing the\nsize of the nanomagnets. We also show how edge defects in the samples influence the magnetization\ndynamics. Dynamical edge modes localized along the sample edges are strongly influenced by edge\ndefects, and this needs to be taken into account in understanding the full FMR spectrum.\nI. INTRODUCTION\nThe magnetodynamic properties of nanostructures\nhave received extensive attention, from both fundamental\nand applications viewpoints1–5. Nanometer sized mag-\nnetic elements play an important role in advanced mag-\nnetic storage schemes6,7, and their static and most im-\nportantly their dynamic magnetic properties are being\nintensely studied. While technological applications are\nimportant, there is also significant interest in understand-\ning the fundamental behavior of magnetic materials when\nthey are confined to nanoscale dimensions. In confined\nmagnetic elements, there is a complex competition be-\ntween exchange, dipolar and anisotropic magnetic ener-\ngies. Understanding the interplay between the various\nenergy terms is thus of importance when investigating\nthe magnetodynamics of such systems.\nThe magnetization dynamics in patterned magnetic\nstructures has been extensively studied previously8–14.\nThe spin dynamics in elliptical permalloy dots were in-\nvestigated by Gubbiotti et al.9. They studied the various\nexcitation modes as a function of dot eccentricity and in-\nplane orientation of the applied field, showing how the\nshape of the ellipses affects the spectrum of excitable\nmodes and their frequencies.\nHowever, the above mentioned studies of patterned\nmagnetic structures were all performed for systems hav-\ning a negligible magnetocrystalline anisotropy. Mate-\nrial systems with a significant crystalline anisotropy pro-\nduce an effective field which also contributes to the spin\ndynamics. The combination of shape and crystalline\nanisotropy results in a complex energy landscape, where\nthe interplay of these energy terms determines the mag-\nnetodynamic properties of the system.\nThe influence of shape and crystalline anisotropy on\nmagnetic hysteresis and domain structures in submicron-size Fe particles have previously been investigated by M.\nHanson et al.15. However, to the best of our knowledge,\nthe dynamic properties of magnetic structures utilizing\nboth crystalline and shape anisotropies remains unex-\nplored. The goal of this study is thus to investigate a\nsystem where the energy terms from both crystalline and\nshape anisotropy contribute to determine the dynamics\nof the system.\nWe have investigated a system utilizing epitaxial Fe\nas the ferromagnetic (FM) material, patterned to an ar-\nray of elliptical nanomagnets. This results in a system\ncombining the cubic crystalline anisotropy of Fe with the\nshape anisotropy due to the elliptical shape of the con-\nfined magnetic elements.\nThe dynamic properties were investigated by ferromag-\nnetic resonance (FMR) experiments for ellipses with a\nthickness of 10 nm and lateral dimensions of 50 ×150\nnm, 100 ×300 nm and 150 ×450 nm. The experimen-\ntal results are compared with micromagnetic simulations,\nand a macrospin model considering the total free energy\ndensity of a ferromagnetic structure containing both crys-\ntalline and shape anisotropies. The macrospin model is\nthen used to explore the properties of ellipses with lateral\ndimensions ranging from 50 ×150 nm to 500 ×1500 nm,\nshowing how the ellipse size governs the balance between\ncrystalline and shape anisotropy.\nDuring the fabrication of such structures, the magnetic\nproperties may be affected by edge defects and shape\ndistortions16–19. As the size of the magnetic elements are\nreduced, the edge regions become increasingly important.\nUnderstanding how edge defects affect the magnetody-\nnamic properties of the elements is thus of importance in\nnanomagnets, where the edge region covers a significant\namount of the total sample area. We show how this af-\nfects the magnetization dynamics, and that edge defects\nneed to be taken into account in understanding the full\nFMR spectrum.arXiv:1504.05404v2  [cond-mat.mes-hall]  27 Aug 2015ii\nII. EXPERIMENTAL SETUP\nThe samples are based on a single crystalline Fe film\nepitaxially grown on MgO(001) substrates. The ferro-\nmagnetic ellipses were patterned by e-beam lithography\nand ion beam milling from a 10 nm thick Fe layer, and\nhave lateral dimensions of 50 ×150 nm, 100 ×300 nm\nand 150 ×450 nm. The crystalline easy axis [100] and\n[010] of the Fe film are oriented along the long/short axis\nof the ellipses, as indicated in Fig. 1a. Further details\nconcerning sample growth and processing are similar to\nthat described earlier20.\nThe FMR experiments were performed using two com-\nplementary setups. The cavity FMR measurements were\ncarried out in a commercial X-band electron paramag-\nnetic resonance (EPR) setup with a fixed microwave fre-\nquency of 9.4 GHz (Bruker Bio-spin ELEXSYS 500, with\na cylindrical TE-011 microwave cavity). The magnitude\nof the external field is then swept to locate the resonance\nfield,HR. The sample is attached to a quartz rod con-\nnected to a goniometer, allowing to rotate the sample 360\ndegrees in order to accurately resolve the angular depen-\ndence. The FMR measurements were performed with a\nlow amplitude ac modulation of the static field, which\nallows lock-in detection to be used in order to increase\nthe signal to noise ratio.\nFor the broadband FMR measurements, we used a vec-\ntor network analyzer (VNA) FMR setup with a coplanar\nwaveguide (CPW) excitation structure. The static exter-\nnal field,H0, was applied in the sample plane, and per-\npendicular to the microwave field from the CPW. This\nwas used to obtain the standard microwave S parameters\nas a function of frequency for various fixed values of the\nstatic field. This allows for a complete field versus fre-\nquency map of the resonance absorption, not being lim-\nited to a fixed frequency as for the cavity measurements.\nData was then collected in a field range of ±500 mT,\nand a frequency range of 1-25 GHz. Typical absorption\nmaps had a step size of ∆ f= 0.1 GHz and ∆ H0= 5mT.\nIII. MICROMAGNETIC SIMULATIONS\nThe micromagnetic calculations were performed using\nMuMax21. The simulated ellipses have a dimension of\n150×450 nm, with a thickness of 10 nm. In order to have\nmesh independence, the discretization cells should have\nsides of the same order, or less than, the two character-\nistic magnetic length scales of the system. The exchange\nlength,lexch= (A\nK1)1/2and the magnetostatic exchange\nlengthldem= (A\nKd)1/2. HereAis the exchange stiffness\nconstant,K1the first order anisotropy constant, Kdthe\nenergy density of the stray field, and an upper limit for\nKdis given by1\n2µ0Ms2.\nMaterial parameters used in the simulations are stan-\ndard literature values, with a saturation magnetization,\nMs= 1.7×106A/m and a crystalline anisotropy con-\nstant ofK1= 4.3×104J/m3, with the easy axis orientedalong the long and short axis of the ellipse. The exchange\nstiffness was set to a value off A = 21 ×10−12Jm−1, and\nthe damping coefficient to α= 0.01.\nPerforming simulations for a 3d model and a 2d model\nwe obtained the same results, and varying the grid size\nit was found that the results converge at a grid size of\n2×2 nm. To save computation time the simulation model\nwas thus implemented as a 2d model with a grid size of\n2×2 nm, which is well below the characteristic magnetic\nlength scales of the system ( lexch= 21 nm,ldem= 3.5\nnm.)\nSimulations of the FMR spectrums were performed by\nusing a field relaxation process. The system is first initial-\nized at zero applied field. If a static field, H0, is applied,\nthe simulations are run until the system reaches the new\nground state configuration. A 10 mT perturbation field,\nHpis then applied along the z-axis (out of plane), and\nthe simulation is run until it reaches the ground state\nconfiguration for the field H0+Hp. The perturbation\nfield is then switched off, allowing the system to relax.\nThe perturbation causes oscillations of the magnetiza-\ntion around the equilibrium position with a maximum\ndeviation of approx. 1 degrees, avoiding any non-linear\neffects. To obtain the resonance frequencies, we take the\nFourier power spectrum of the mzcomponent the first\n10 ns of the magnetization relaxation. The various exci-\ntation modes of the system will then appear as distinct\npeaks in the Fourier spectrum.8\nSimulations with an ac field of varying frequency as the\nperturbing field were also performed, and we obtained the\nsame results as for the field relaxation procedure. The ac\napproach is however more time consuming, as one has to\nscan the full frequency range for each value of the applied\nstatic field in order to locate all the resonances. To obtain\nthe full field vs. frequency map of the excitation modes\nin the system we thus used the field relaxation process.\nIV. FREE ENERGY DENSITY AND\nTHEORETICAL FMR SPECTRUM\nDue to the size and shape of the ellipses, we consider\nthe individual magnetic elements to be in a single do-\nmain state. This was also confirmed by MFM imaging of\nsimilar samples22, where all particles were found to be in\na single domain state for a thickness of 10 nm. Increas-\ning the thickness makes it energetically favorable to form\nflux closure domains, and already at a thickness of 30 nm\nsome of the particles were found to be in such multi do-\nmain states. This means that to make sure the magnetic\nelements are in single domain states, one has to keep the\nfilm thickness well below 30 nm for ellipses of the di-\nmensions we have investigated. Having a single domain\nstate allows us to use an analytical macrospin model to\ninvestigate the ferromagnetic resonance properties of the\nsystem.\nThe array of ellipses has an inter-particle spacing of\ntwo times the corresponding ellipse dimension in eachiii\ndirection, as illustrated in Fig. 1a. This spacing is suffi-\ncient to significantly reduce the dipolar coupling between\nthe individual elements, and as a first approximation we\nconsider the ellipses as uncoupled magnetic elements.\nWe start by defining the geometry of the system, and\nconsider the free energy density of the individual mag-\nnetic elements. From the sample geometry illustrated in\nFig. 1b (magnetic element in the x-y plane), one gets\nthat:\nxyz\nHM\nAB\n2A2Ba) b)\n[100]\n[010]\nFIG. 1. a) Array of ellipses with dimension A×B, an aspect\nratio ofA/B = 3 and inter-particle spacing of two times the corre-\nsponding ellipse dimension in each direction. The [100] and [010]\ncrystallograpic axis of Fe is oriented along the long/short ellipse\naxis. b) Field geometry of the individual ellipses\nMx=Mssinθcosφ\nMy=Mssinθsinφ\nMz=Mscosθ,(1)\nwhereMsis the saturation magnetization. Assuming\nthe external applied field, H0, is oriented in the sample\nplane,θH=π/2, gives\nHx=H0cosφH\nHy=H0sinφH(2)\nAfter defining the geometry, one can calculate the free\nenergy density of the system by adding up the vari-\nous energy terms. Using a macrospin model, we do\nnot consider the exchange energy. The total free en-\nergy density of the system is then given by, Etot=\nEZeeman +EDemagnetization +EAnisotropy .\nEZ=−/vectorM·/vectorH\n=−MsH0[sinθcosφcosφH+ sinθsinφsinφH]\n=−MsH0sinθcos(φ−φH),(3)EDemag =µ0\n2[NxMx2+NyMy2+NzMz2]\n=µ0Ms2\n2/bracketleftBig\nNxsin2θcos2φ+Nysin2θsin2φ\n+Nzcos2θ/bracketrightBig\n,(4)\nwhereµ0is the vacuum permeability, Niare the de-\nmagnetization factors and Nx+Ny+Nz= 1. The units\nfor the saturation magnetization and magnetic field are\n[Ms] =A/m and [ H] =T respectively. We assume cubic\ncrystalline anisotropy for the epitaxial Fe film, with the\neasy axis oriented parallel to the long/short axis of the\nellipse, as indicated in Fig. 1a. The lowest order term in\nthe crystalline anisotropy energy is then the fourth order\nterm:\nEAnis=K1[α2\nxα2\ny+α2\nyα2\nz+α2\nzα2\nx]\n=K1/bracketleftBig\nsin4θsin2φcos2φ+ sin2θsin2φcos2θ\n+ sin2θcos2φcos2θ/bracketrightBig\n,(5)\nwhereK1is the magnetocrystalline anisotropy con-\nstant andαi=Mi/Ms. After adding the terms, one\ncan write the total free energy density as:\nEtot=−MsH0sinθcos(φ−φH)\n+µ0Ms2\n2/bracketleftBigg\nsin2θcos2φ/parenleftbigg\nNx+2K1\nµ0M2ssin2θsin2φ/parenrightbigg\n+ sin2θsin2φ/parenleftbigg\nNy+2K1\nµ0M2scos2θ/parenrightbigg\n+ cos2θ/parenleftbigg\nNz+2K1\nµ0M2ssin2θcos2φ/parenrightbigg/bracketrightBigg\n.\n(6)\nEquation (6) describes a complex energy landscape,\nwith competing energies from the various terms. It is\nimportant to note that the orientation of the magneti-\nzation, given by φ, might not be parallel to the applied\nfield,φH. Thus, to investigate the resonance conditions\nof the system one must first find the equilibrium orien-\ntation of the magnetization. The equilibrium orientation\nwas found by minimizing the free energy density of the\nsystem given by Eq.(6) for each value of HandφH, and\nwas performed numerically. After obtaining the equilib-\nrium orientation of the magnetization, one can calculate\nthe resonance frequency ω, given by:23\nω=γ\nµ0Mssinθ/radicaltp/radicalvertex/radicalvertex/radicalbt/parenleftBigg\n∂2Etot\n∂θ2∂2Etot\n∂φ2−/parenleftbigg∂2Etot\n∂θ∂φ/parenrightbigg2/parenrightBigg\n.(7)\nBy solving Eq.(7), one can obtain the resonance fre-\nquency as a function of magnitude and direction of theiv\napplied field, ω(H,φ H). Calculating the various terms in\nEq.(7), one obtains:\n∂2Etot\n∂θ2=MsHsinθcos(φ−φH)\n+K1\n4/bracketleftBigg\ncos 2θ/parenleftBig\n1−cos 4φ−4µ0M2\nsNz/K1\n+2µ0M2\ns\nK1(Nx+Ny+ (Nx−Ny) cos 2φ)/parenrightBig\n+ (cos 4φ+ 7) cos 4θ/bracketrightBigg\n,(8)\n∂2Etot\n∂φ2=MsHsinθcos(φ−φH)\n+ 2K1sin2θ/bracketleftBigg\ncos 4φsin2θ\n+µ0Ms2(Ny−Nx)\n2K1cos 2φ/bracketrightBigg\n,(9)\n∂2Etot\n∂θ∂φ=MsHcosθsin(φ−φH)\n+ 8K1sinφcosφsinθcosθ/bracketleftBigg\ncos 2φsin2θ\n+µ0M2\ns(Ny−Nx)\n4K1/bracketrightBigg\n.(10)\nFor thin films, one can simplify these expressions by\nassuming that the magnetization is oriented in the film\nplane,θ=π/2. After introducing the anisotropy field,\nHk= 2K1/Ms, one obtains the resonance frequency\ngiven by Eq.(7):\n/parenleftbiggω\nγ/parenrightbigg2\n=/bracketleftbigg\nHcos(φ−φH) +µ0Ms/parenleftBig\nNz\n−(Nx+Ny+ (Nx−Ny) cos 2φ\n2)/parenrightBig\n+Hk\n4(3 + cos 4φ)/bracketrightbigg\n×/bracketleftbigg\nHcos(φ−φH) +Hkcos 4φ\n+µ0Ms(Ny−Nx) cos 2φ/bracketrightbigg\n.\n(11)\nEquation (11) gives the resonance frequency for the\ngeneral case, with the assumption that the magnetization\nis oriented in the sample plane. Depending on the shape\nand size of the magnetic elements, one can then adjust\nthe demagnetization factors Nito obtain the resonance\nconditions for various samples.In addition to the four-fold symmetry from the cubic\nanisotropy, one notices that in this case there are ad-\nditional terms of two-fold symmetry due to the shape\nanisotropy along the long/short axis of the ellipse. The\nresonance conditions of the system are thus more com-\nplicated, and are determined by the interplay of shape\nand crystalline anisotropies. This brings us to the main\ntopic of the study, to investigate how tuning the various\nenergy terms changes the magnetodynamic properties of\nthe system.\nV. RESULTS AND DISCUSSION\nA. Cavity FMR measurements\nThe experiments to investigate the angular dependence\nwere performed in the X-band cavity FMR setup de-\nscribed in section II. This gives an angular FMR spec-\ntrum for both the continuous film and an array of ellipses\nof dimension 150 ×450 nm, as shown in Fig. 2.\na) b)\nAdditional modes Main modes\nIntensity, [A.U]\nFIG. 2. (Color online) Experimental FMR spectrum for a) con-\ntinuous film and b) ellipses of dimension 150 ×450 nm from the\nX-band cavity FMR setup.\nGoing from a continuous film to a patterned array of el-\nlipses, there is a significant difference. For the continuous\nfilm, the four-fold symmetry due to the cubic crystalline\nanisotropy in Fe is dominating. For the ellipses the situa-\ntion is more complicated, as there are competing energies\nalso from the shape anisotropy.\nTo investigate this, we compare the experimental and\ntheoretical results. By solving Eq.(11) after first mini-\nmizing the free energy density for each value of Hand\nφH, one gets the FMR dispersion relations shown in the\nlower panel of Fig. 3. From Eq.(11), the relevant pa-\nrameters determining the dispersion are the demagneti-\nzation factors Ni, the anisotropy field Hkand the satu-\nration magnetization Ms. In nanometer-dimension mag-\nnetic structures, estimates of the demagnetization factors\nusing an ellipsoidal formulae are considered to represent\nthe anisotropy fields well24,25. The factors Niwere found\nfrom24, and for an ellipse of dimension 10 ×150×450 nm\nthey are:Nx≈0.005,Ny≈0.05 andNz= 1−Nx−Ny.v\nThe anisotropy field Hkwas determined from the ex-\nperimental FMR spectrum in Fig. 2a, and was found\nto be approx. 50 mT. In the calculations, Mswas ad-\njusted to obtain the best fit between the experimental\nand theoretical spectrum, and the best fit was found for\nMs= 1.5×106A/m (a reduction of approx. 10% com-\npared to textbook values of Msfor Fe).\nTo compare the angular dependence of the theoretical\nspectrum with experimental results from the cavity mea-\nsurements shown in Fig. 2, one can inverte the solution.\nThis rather gives the resonance field HR, as a function\nof rotation angle for a fixed excitation frequency of 9.4\nGHz, and the inverted solution is shown in the upper\npanel of Fig. 3. To distinguish the effect of crystalline\nanisotropy and shape anisotropy, the same calculations\nwere also performed assuming polycrystalline Fe, setting\nHk= 0.\nNgN5\nNgL\nNgL5\n3N\nSLN6N\nS4N9N\nS7NLSN\n3NNL5N\n33NL8N N\nResonance2fieldv2[Tesla]v2on2radial2axis\nvsg2rotation2in2the2x−y2planev2[degrees]\nNgN5\nNgL\nNgL5\n3N\nSLN6N\nS4N9N\nS7NLSN\n3NNL5N\n33NL8N N\nResonance2fieldv2[Tesla]v2on2radial2axis\nvsg2rotation2anglev2[degrees]\nThin2film Ellipse aR bR\ncR dR\nN NgL NgS Ng3 Ng4NNg5LLg5SSg5x2LNLNFrequencyv2[Hz]\nApplied2fieldv2[Tesla]\nEasy2axis28N2degreesR\nHard2axis28452degreesR\nNo2anisotropy\nN NgL NgS Ng3 Ng4NNg5LLg5SSg5x2LNLNFrequencyv2[Hz]\nApplied2fieldv2[Tesla]\nLong2axis\nShort2axis\nFrequencyv2[GHz]\n5LNL5SNS5\nFrequencyv2[GHz]\n5LNL5SNS5\nApplied2fieldv2[T] Applied2fieldv2[T]\nFIG. 3. (Color online) Upper figures: Theoretical data for res-\nonance field versus rotation angle for a) continuous film and b)\nellipse of dimension 10 ×150×450 nm, with (red) and without\n(blue) crystalline anisotropy. Lower figures: Dispersion for c) con-\ntinuous film and d) ellipse of dimension 10 ×150×450 nm, with\n(solid lines) and without (dotted lines) crystalline anisotropy.\nComparing theory and experiment in Fig. 2 and 3, one\nnotices that for the continuous film, both show the ex-\npected four-fold cubic symmetry. For the ellipses, the\ntheory replicates the ”heart shape” of the resonance well.\nIn the experimental data in Fig. 2b, there are also some\nadditional weak resonance lines. It is known that regions\nalong the sample edges could lead to a spectrum of addi-\ntional edge modes9,10,16. However, from our experiments\nwe observe that the main mode is dominating, and thus\nfocus on this in the following. The other resonances are\ncharacterized and discussed in detail in section V C.B. Size of the ellipses\nTo investigate the interplay of shape anisotropy and\ncrystalline anisotropy, we studied ellipses of various lat-\neral dimensions but with the same aspect ratio of 1:3.\nChanging the sample size affects the balance between\ncrystalline and shape anisotropy in the free energy den-\nsity. As shown using our macrospin model for the main\nFMR mode, this will in turn change the resonance fre-\nquency. There are two limiting cases worth noticing: in\nthe limit of a very large ellipse, one should expect a be-\nhavior close to that of a continuous film, where crystalline\nanisotropy is dominating. By gradually reducing the size\nof the ellipse, shape anisotropy becomes increasingly im-\nportant. This means that one can use the size of the\nmagnetic elements to tune the ratio between crystalline\nand shape anisotropies, and thus change the magnetody-\nnamic properties of the system.\nChanging the dimensions of the ellipse affects the free\nenergy density of the system, given by Eq.(6). The tran-\nsition from a continuous film to a small ellipse can be\nobserved by considering the energy landscape of the sys-\ntem as a function of the ellipse dimension, as shown in\nFig. 4.\na) b)\nc) d)Continuous film 500x1500 nm Ellipse\n150x450 nm Ellipse 50x150 nm Ellipse\nE(J/m³)E(J/m³)E(J/m³)E(J/m³)\nFIG. 4. (Color online) Free energy density given by Eq.(6) for a)\ncontinuous film, b) 500 ×1500 nm ellipse, c) 150 ×450 nm ellipse,\nd) 50×150 nm ellipse. Film thickness is 10 nm in all cases.\nFigure 4 indicates how the free energy density changes\nwhen one gradually reduces the size of the ellipse from\nthe upper limit of a continuous film, to an ellipse of di-\nmension 50 ×150 nm. As expected, one notices that in\nall cases the magnetization favors an orientation in the\nsample plane ( θ= 90, from sample geometry as defined\nin Fig. 1b). For the continuous film and the largest ellipse\nin Fig. 4a and b, one can clearly see the dominating crys-\ntalline anisotropy, with a four-fold symmetry between thevi\nenergy minima along the φaxis.\nIn the intermediate case for an ellipse of dimension\n150×450 nm, one has two dominating energy minima at\nφ= 0 andφ= 180 (magnetization along the long axis of\nthe ellipse). In addition, there is a quite flat saddle point\natφ= 90 (which corresponds to a magnetization along\nthe short axis of the ellipse). This is not a stable energy\nminimum, but the flatness of the saddle point means that\napplying a small magnetic field along this axis will create\na local energy minimum along this direction.\nFor the smallest ellipse, the energy landscape is dom-\ninated by the two-fold shape anisotropy along the long\naxis of the ellipse. To align the magnetization along the\nshort axis of the ellipse ( φ= 90) will thus require a quite\nlarge external field.\nAs shown in section IV, the FMR frequency given by\nEq.(11) is determined by the free energy density of the\nsystem. Adjusting the lateral dimensions of the ellipse\nis thus an important parameter controlling the FMR fre-\nquency. From Eq.(11), one notices that the resonance\nfrequency is determined by contributions of both two-\nfold and four-fold symmetry. From this expression, the\nrelevant ratio to determine which term will dominate is\ngiven byHK/µ0Ms(Nx−Ny). Changing the ellipse di-\nmensions, and thus the demagnetization factors Ni, af-\nfects the resonance frequency significantly, as shown in\nthe upper panel of Fig. 5.\nTheoreticalvdata:\nExperimentalvdata:\n158x458vnmaH bH\ncH dH\n8985\n8915\n8925\n38\n21868\n24898\n278128\n388158\n338188 8\nResonancevfieldAv[Tesla]Avonvradialvaxis\nvs9vrotationvinvthevx−yvplaneAv[degrees]\neH 188x388vnm 58x158vnm\n8 891 892 893 894 895889511952295xv1818FrequencyAv[Hz]\nAppliedvfieldAv[Tesla]\n58x158:vlongvaxis\n58x158:vshortvaxis\n188x388:vlongvaxis\n188x388:vshortvaxis\n158x458:vlongvaxis\n158x458:vshortvaxis\nResonancevfieldAv[Tesla]Avonvradialvaxisvvs9vrotationvinvthevx6yvplaneAv[degrees]FrequencyAv[GHz]\n518152825\nAppliedvfieldAv[T]\nFIG. 5. (Color online) a) Theoretical dispersion for ellipses of\ndimension 50 ×150 nm (Black), 100 ×300 nm (Blue) and 150 ×450\nnm (Red). b) Angular dependence of same data. c) Experimental\ndata for ellipse of dimension 150 ×450 nm d) 100 ×300 nm and e)\n50×150 nm.\nFigure 5a and b compare the theoretical FMR spec-\ntrum for ellipses of dimension 150 ×450 nm, 100 ×300 nmand 50 ×150 nm. As the dimensions of the ellipse are re-\nduced, the two-fold shape anisotropy tends to dominate\nover the crystalline anisotropy, and the ”heart shape”\nof the spectrum in Fig. 5b due to the cubic crystalline\nanisotropy is suppressed. Comparing the theoretical re-\nsults with the experimental data in the lower panel of\nFig. 5, they follow the same trend. As the size is re-\nduced, the resonance is shifted to slightly higher fields,\nand the ”heart shape” of the resonance gets suppressed.\nInvestigating the opposite limit, one can determine\nwhen the crystalline anisotropy starts to dominate. Com-\nparing the theoretical FMR spectrum for ellipses of di-\nmension 150 ×450 nm, 250 ×750 nm and 500 ×1500 nm\nin Fig. 6, one notices that by increasing the size, the ef-\nfect of shape anisotropy is suppressed compared to that\nof crystalline anisotropy.\nav bv\n0.05\n0.1\n0.15\n30\n21060\n24090\n270120\n300150\n330180 0\nResonance6fieldg6[Tesla]g6on6radial6axis\nvs.6rotation6in6the6x−y6planeg6[degrees]\n0 0.1 0.2 0.3 0.4 0.500.511.522.5x61010Frequencyg6[Hz]\nApplied6fieldg6[Tesla]\n500x1500:6long6axis\n500x1500:6short6axis\n250x750:6long6axis\n250x750:6short6axis\n150x450:6long6axis\n150x450:6short6axis\nFrequencyg6[GHz]510152025\nApplied6fieldg6[T]\nFIG. 6. (Color online) a) Theoretical dispersion for ellipses of\ndimension 500 ×1500 nm (Black), 250 ×750 nm (Blue) and 150 ×450\nnm (Red). b) Angular dependence of same data.\nFor an ellipse of dimension 500 ×1500 nm the dispersion\nstarts to look similar along the long/short axis of the\nellipse, as indicated in Fig. 6a. If the only contribution\nwas from the crystalline anisotropy, the dispersion should\nbe identical along the long/short axis due to the four-\nfold symmetry. Comparing the FMR spectrum for the\nlargest ellipse in Fig. 6b to that of a continuous film in\nFig. 3a, they look very similar. This indicates that as\nthe sample dimensions approach the micrometric scale,\nshape anisotropies play a minor role compared to the\ncrystalline anisotropy.\nTo summarize the size dependence, we have shown\nthat for sample dimensions above approx. 1 µm, crys-\ntalline anisotropy will dominate. In the opposite size\nlimit, shape anisotropy will dominate for sample dimen-\nsions below approx. 50 ×150 nm. In this intermediate\nregime, one can thus effectively use the sample size as a\nparameter to tune the balance between crystalline and\nshape anisotropies.\nC. Broadband FMR measurements and\nmicromagnetic simulations\nThe assumption that the magnetization in the individ-\nual ellipses is uniform is a good approximation at thevii\ncenter of the ellipse, but along the edges the magnetiza-\ntion will be less uniform due to the demagnetizing fields.\nRegions along the sample edges could lead to a spectrum\nof additional edge modes9,16. In addition, there could be\nother spin-wave excitations with non zero wave vectors,\nand correspondingly varying frequencies9,10. To charac-\nterize the various resonances, we thus performed a series\nof broadband FMR measurement in combination with\nmicromagnetic simulations.\nTo obtain a complete field versus frequency map of the\nFMR absorption, we performed experiments using the\nbroadband setup described in section II. The experimen-\ntal FMR absorption peaks were extracted, and are shown\nin the upper panel of Fig. 7. Red dots represent the main\nFMR mode, and the blue squares the additional weaker\nmode. For clarity only a few selected data points are\nincluded, where the uncertainty in determining the ab-\nsorption peak position is of the order of the dot size. The\nexperimental results are then compared with the theoret-\nical FMR spectrum from the macrospin model, shown as\ndotted black lines.\nFrequency, [GHz]\n−400 −200 0 200 400510152025\n−400 −200 0 200 400510152025\n−400 −200 0 200 400510152025Frequency, [GHz]\nField, [mT]\n−400 −200 0 200 400510152025Field, [mT]a) b)\nc) d)\nFIG. 7. (Color online) Upper panel: Experimental FMR spectrum\nfor ellipses with field oriented along long/short axis. Experimental\ndata is shown as red dots/blue squares, and theoretical spectrum\nas dotted black line. a) Field sweep from negative to positive field\nalong long axis, also showing the switching of the magnetization\nat approx. 75mT. b) Similar measurement along the short axis,\nshowing the main mode (red dots) and an additional weak reso-\nnance at lower frequency (blue squares). Lower panel: Simulated\nFMR spectrum for a single ellipse of dimension 150 ×450 nm with\nthe field oriented along the c) long and d) short axis.\nThe agreement between theory and experiment is good\nfor an applied field oriented along the long axis of the\nellipse, as indicated in Fig. 7a. Sweeping the field from\nnegative to positive, one also notices the switching of themagnetization. As the field is swept from negative to\nzero, the FMR frequency decreases as expected. This\ncontinues also for positive fields until the external field\nis strong enough to overcome the anisotropy favoring the\nmagnetization along the long axis of the ellipse. The\nswitching is then observed as an abrubt jump in the FMR\nspectrum.\nWhen applying the field along the short axis there are\ntwo parallel dispersing lines, as shown in Fig. 7b. A high\nfrequency resonance and an additional weaker resonance\nat lower frequency, which corresponds well to the addi-\ntional resonance also seen in the cavity measurements\n(see Fig. 2b). Comparing the measurements along the\nshort axis with the theoretical dispersion, one does not\nobserve the low field resonance in Fig. 7b (the black dot-\nted line below 100 mT). In this field range the magneti-\nzation is not saturated and it is still oriented along the\nlong axis of the ellipse, being parallel to the microwave\n(MW) pumping field from the CPW. However, in the\ncavity measurements we observed both resonances, be-\ncause the pumping field is, in this case, oriented out of\nthe sample plane and thus perpendicular to the magneti-\nzation. The first resonance is observed at a field of ∼50\nmT (see Fig. 2b), and a second one at ∼100 mT, which\nagrees well with the expected resonance fields from the\ntheoretical curves shown in Fig. 7b at a frequency of 9.4\nGHz. At higher fields the magnetization in the ellipse\nsaturates in the direction of the external field, being per-\npendicular to the MW pumping field from the CPW, and\nthus the theoretical spectrum corresponds well with the\nhigh-frequency branch of the experimental data.\nUsing a macrospin model, one accounts only for the\nmain FMR mode. In order to investigate the observed\nlow frequency resonance we performed micromagnetic\nsimulations. The model was implemented as a single el-\nlipse of dimensions 150 ×450 nm with a thickness of 10\nnm, and the simulated FMR spectrums are shown in the\nlower panel of Fig. 7. Comparing the experimental data\nwith the micromagnetic simulations, we notice a few dif-\nferences. Applying the field along the long axis of the\nellipse, the simulated and experimental data both show\na single dispersing resonance. The simulated FMR fre-\nquency is however noticeably higher than the experimen-\ntal results. Applying the field along the short axis of the\nellipse, the differences between the experimental and sim-\nulated FMR spectrum are more significant. The exper-\nimental data show two parallel dispersing lines, whereas\nthe simulated spectrum shows a whole range of various\nexcitation modes.\nA similar splitting of the main mode has been observed\nexperimentally in elliptical permalloy dots, and was at-\ntributed to a hybridization of the main mode with other\nspin-wave modes9,10. A study of the excitation modes in\npermalloy dots as a function of dot eccentricity has been\nperformed by Gubiotti et al.9, where they found a large\nrange of possible modes depending on the orientation of\nthe external field with respect to the axis of the dots. The\nnumber of modes in our system compared to theirs mayviii\nbe smaller because of the different material parameters\nand sample size. The exchange stiffness in Fe is almost\ntwice that of permalloy, and combined with a smaller\nsample size this results in a reduction in the number of\nexcitation modes due to the increased exchange energy.\nThis was also confirmed in our simulations, where the\nmode splitting disappear when reducing the sample size\nor increasing the exchange stiffness.\nThe low frequency branch in Fig. 7d was identified\nby imaging the mzcomponent from the micromagnetic\nsimulations (out of plane component). From the periodic\noscillations of the magnetization, we determined the low\nfrequency resonance to be localized along the edges of\nthe ellipse, as indicated in Fig. 8 for an applied field of\n150 mT along the short axis.\n1.5 2 2.5 3 3.5 4\nx 10−70.20.40.60.811.21.41.61.822.2x 10−7\n1.5 2 2.5 3 3.5 4\nx 10−70.20.40.60.811.21.41.61.822.2x 10−7\n1.5 2 2.5 3 3.5 4\nx 10−70.20.40.60.811.21.41.61.822.2x 10−7\n1.5 2 2.5 3 3.5 4\nx 10−70.20.40.60.811.21.41.61.822.2x 10−7\n1.5 2 2.5 3 3.5 4\nx 10−70.20.40.60.811.21.41.61.822.2x 10−7t=0 ns t=0.06 ns t=0.12 ns t=0.18 ns t=0.24 ns\nOscillation of\nedge modes\n< 0 > 0 \nFIG. 8. (Color online) mzcomponent, showing one oscillation pe-\nriod of the edge mode at H= 150 mT, corresponding to a frequency\nof approx. 4 GHz.\nAfter identifying the excitation modes, one needs to\nconsider why there is a significant difference between the\nsimulated and experimental FMR spectrum. It is known\nthat the fabrication process of nanostructures can lead\nto distortions and defects at the sample edges16–19. To\ninvestigate how this would affect the magnetodynamic\nproperties, the effects of edge defects need to be taken\ninto account in the simulation model.\n1. Edge modes and edge defects\nIn the initial simulations, the edges of the ellipse\nwere treated as ideal. However, the samples most likely\nhave some kind of non-ideal edges which could influ-\nence the FMR spectrum. The effects of non-ideal edges\non the dynamics have been investigated theoretically by\nMcMichael et al.16. It was shown that several cases such\nas edge geometry, reduced edge magnetization and sur-\nface anisotropy on the edge surface all had similar effects.\nThe main effect was to reduce the edge saturation field,\nwhich is the field needed to align the magnetization at\nthe edge nearly parallel to the applied field. A reduced\nedge magnetization will also lead to a smaller effective de-\nmagnetization field along the edges. This would cause a\nsignificant increase of the edge mode resonance frequency\ncompared to that of an ideal edge, and the shift could be\nin the order of several GHz16. Such effects would be lessimportant when the field is oriented along the easy axis of\nthe ellipse, explaining the better agreement between the\nsimulated and experimental spectrum in that geometry.\nTo account for edge defects in the simulations, we made\na model where the material properties were changed\nalong the edges of the ellipse. In a real sample the vari-\nation of the material properties when approaching the\nsample edge should be gradual, but as a first approxi-\nmation the model was defined with two distinct regions.\nThe width of the edge region was set to 10 nm, and is\nwithin the same width range as that investigated theoret-\nically by McMichael et al.16. A schematic of the model\nincluding edge defects is shown in Fig. 9b.\nAs mentioned in section II, the samples were defined\nby ion beam milling. This can affect the magnetic prop-\nerties of the sample19, and a more disordered edge region\ncould lead to an increased damping of the FMR modes.\nTwo kinds of defects have thus been considered in the\nsimulations; increased damping α, and reduced Ms.\n0 100 2000510152025\n0 100 2000510152025Normal]edges\nEdge:]Reduced]MsSplitting]of]\nmain]mode\nEdge]mode\nFrequency,][GHz]Edge]regionCenter]region\n0 100 2000510152025\nEdge:]Increased]damping\nField,][mT] Field,][mT]Frequency,][GHz]a3 b3\nc3 d3\n0]]]]]]]]100]]]]]]]]200]]]]]]]300 0]]]]]]]]100]]]]]]]]200]]]]]]300\nFIG. 9. (Color online) a) Simulation using normal edges, showing\nthe edge mode and splitting of the main mode. b) Schematic of\nsimulation model with a defined edge region. c) Simulation with\nreducedMsin edge region (reduction of 40 %). d) Simulation with\nincreased damping αin edge region (from 0.01 to 0.1).\nIn the initial simulation model with ideal edges, excited\nspin waves would be reflected at the edges of the sample.\nThis explains the multiple excitation modes observed in\nthe simulations, due to a hybridization of the main mode\nwith other spin-wave modes9,10(see Fig. 9a). As a dis-\nordered edge region could lead to increased damping of\nthe FMR modes, we introduced an edge region where theix\ndamping was increased from α= 0.01 toα= 0.1. This\nwould absorb the propagation of spin waves, reducing\nthe spin-wave reflection at the sample edges. As seen in\nFig. 9d, the increased damping lead to a broadening of\nthe FMR modes, and suppress some of the splitting of\nthe main mode. The low frequency edge mode however,\nremains relatively unaffected.\nThe edge magnetization Mswas found to be the most\nimportant parameter, and we made simulation models\nwhere the outer region of the ellipse had a significantly\nreducedMsfromMs= 1.7×106A/m toMs= 1×106\nA/m. Reducing Msin the edge region changes the FMR\nspectrum considerably, as seen when comparing Fig. 9a\nand c. The splitting of the main mode is suppressed,\nand the resonance frequency of the edge mode is shifted\nsignificantly. The resulting spectrum now resembles the\nexperimental data, showing mainly two parallel dispers-\ning modes.\nAnother important effect to consider in arrays of nano-\nmagnets is the dipolar interaction among the individual\nparticles. In order to take this into account, we per-\nformed simulations for arrays of interacting ellipses.\n2. Dipolar interactions\nThe simulations so far have been performed for single\nellipses. However, due to the periodic array of ellipses (as\nshown in Fig. 1a), there will be some degree of dipolar\ninteraction between the individual ellipses. The dipolar\ninteraction in arrays of magnetic particles can have both\nstatic and dynamic contributions. The effects of static\ndipolar interaction on the magnetization reversal of the\nsame samples have been investigated previously, and an\ninteraction field in the order of tens of mT was found22.\nThe dynamic interaction can couple the magnetization\ndynamics of adjacent dots through the stray field gener-\nated by the precessing magnetization, forming collective\nspin excitations in the system4,5.\nInteractions were included in the simulations by using\nperiodic boundary conditions (b.c.), with the same pe-\nriodicity as that indicated in Fig. 1a. In the limit of\nstrong dipolar interaction, one could also expect collec-\ntive modes in the system. A simple model of a single\nellipse with periodic b.c. would not be sufficient to re-\nsolve such modes, as the neighboring ellipses could rotate\neither in phase (acoustic mode) or out of phase (optic\nmode)26,27. To take this into account, we compared the\nsimulation results for a single ellipse with periodic b.c.\nversus arrays of 3 ×3, 5×5 and 10 ×10 ellipses. Compar-\ning the simulated FMR spectrums for the various array\nsizes, we found no indication of such collective modes in\nour system. In the following simulations the dipolar in-\nteraction was thus taken into account by using a simple\nmodel for a single ellipse with periodic b.c.\nComparing the simulated spectrums for a single ellipse\nversus an array of ellipses, we found that the dipolar in-\nteraction changes the effective field felt by the individualellipses. At zero applied field, the magnetization is ori-\nented along the long axis of the ellipses. The overall\ndipolar field caused by the array geometry will then op-\npose the magnetization direction. As seen in Fig. 10a,\nthe dipolar field reduces the resonance frequency at zero\napplied field for the array compared to a single ellipse.\nIncreasing the field along the short axis of the ellipse, the\nmagnetization will reorient itself along the short axis at\nan applied field of approx. 75 mT (seen as a ”dip” in\nthe FMR spectrum in Fig. 9c ). At fields above this\nswitching field, the dipolar interaction acts to increase\nthe effective magnetic field felt by the ellipses, and thus\nincreases the FMR frequency. These shifts can be seen in\nFig. 10a for an applied field between 150 mT - 350 mT,\nand are in the order of 1 GHz. These shifts in the FMR\nfrequencies along the hard/easy axis are similar to those\nobserved by Carlotti et al.28, who studied the effects of\ndipolar interactions in arrays of rectangular permalloy\ndots.\nTo capture all significant effects we thus made a simu-\nlation model with periodic b.c., where edge defects were\nmodelled as a reduced Msat the sample edges. After\nincluding both edge defects and dipolar interactions, one\ncan compare the simulated and experimental spectrums\nin Fig. 10b and c.\n0 100 200 3000510152025\n−300 −200 −100 00510152025\nField,A[mT]\nFrequency,A[GHz]\n0 100 200 3000510152025\n−300 −200 −100 0510152025\nAA-300AAAAA-100AAAAAA100AAAAAA300A\nField,A[mT]25\n20\n15\n10\n525\n20\n15\n10\n5b)\n0 5 10 15 20 25 3000.511.522.5xA10−3\nFrequency,A[GHz]0AmT150AmT250AmT350AmT\n|F(f)|,A[A.U]\nSingleArray\nc)a)\nAAAAAAAAAAA0AAAAAAAAAAAAA5AAAAAAAAAAAAAA10AAAAAAAAAAAA15AAAAAAAAAAAA20AAAAAAAAAAAA25AAAAAAAAAAAA30AAAA\nAA-300AAAAA-100AAAAAA100AAAAAA300A\nFIG. 10. (Color online ) a) FFT spectrum of a single ellipse (black\ndots) vs. an array of ellipses (blue line) using periodic b.c, for an\napplied field oriented along the short axis of the ellipse. b) Left\nhalf: Experimental data and macrospin model as dotted black line.\nRight half: Micromagnetic simulation including edge defects and\ndipolar interactions in an array of ellipses. Data shown for field\noriented along the long axis. c) same data for field oriented along\nthe short axisx\nWe notice that the inclusion of edge defects and dipolar\ninteractions give a better agreement between the simu-\nlated and experimental FMR spectrum. As expected,\nthe edge modes are strongly influenced by edge defects\nin the samples. To accurately capture the behavior of all\nthe FMR modes, it is thus important to take edge defects\ninto account in the simulation model. Due to the large\nspacing between the individual ellipses in the array, the\ndipolar interaction is quite weak. In the simulations we\nobserve a small shift in the FMR frequencies, but not\nany indications of collective modes between neighboring\nellipses.\nThe fact that the amplitude of the main mode domi-\nnates in the experiments, together with the weak dipolar\ncoupling, explains the good agreement between the an-\nalytical macrospin model and experimental data. This\nindicates that in the limit of weak dipolar interaction,\nour macrospin model can be used to estimate the FMR\nfrequency of the main mode in magnetic elements within\nthe investigated size range (e.g. in a single domain state).\nUsing an analytical macrospin model compared to per-\nforming numerical simulations simplifies the analysis con-\nsiderably. The various energy terms contributing to the\nFMR dynamics can then be separated, and their relative\nimportance investigated.\nVI. CONCLUSIONS\nIn this study we have investigated how the com-\nbined interplay between shape anisotropy and crystalline\nanisotropy affects the magnetodynamic properties of con-\nfined magnetic elements. We have shown how the dimen-\nsions of the magnetic elements can be used to balance\ncrystalline and shape anisotropies, and that this can be\nused to tailor the magnetodynamic properties\nWe have shown that a simple macrospin model for\nthe FMR frequency gives good agreement with the ex-\nperimental results for the main FMR mode. Com-\nparing experimental data and model calculations, we\nshow how changing the sample size affects the magne-\ntodynamic properties. For the smallest ellipses, shape\nanisotropy is dominating, whereas for the largest ellipsescrystalline anisotropy is the dominating energy term.\nFrom Eq.(11), the relative contributions to the resonance\nfrequency from crystalline and shape anisotropy is given\nby:Hk/µ0Ms(Nx−Ny), determined by the anisotropy\nfieldHk, the saturation magnetization Msand the de-\nmagnetization factors Ni. This means that for the case\nof a 10 nm thick epitaxial Fe film, one has an intermedi-\nate regime between approximately 50 nm to 1 µm where\none can use the sample size as an additional tuning pa-\nrameter for the dynamic properties. For other materials\nwith a different HkandMs, this regime can be shifted\nto smaller/larger sample sizes.\nThe effects of non ideal sample edges and dipolar in-\nteraction in the array of ellipses were investigated using\nmicromagnetic simulations. We found that edge defects\nin the form of a reduced edge magnetization had to be in-\ncluded in the micromagnetic model, and that this needs\nto be taken into account in understanding the full FMR\nspectrum. The static dipolar interaction in the array was\nfound to shift the FMR frequency in the order of 1 GHz\ncompared to that of a single ellipse. From the simulated\nFMR spectrums we found no indications of collective spin\nexcitations due to the dynamic dipolar interaction be-\ntween neighboring ellipses.\nThe tunability of the relative contributions from crys-\ntalline and shape anisotropies means that by changing\nthe material parameters and sample size one can tai-\nlor the magnetodynamic properties of the magnetic el-\nements, which could be of importance for magnonics ap-\nplications.\nACKNOWLEDGEMENTS\nThis work was supported by the Norwegian Research\nCouncil (NFR), project number 216700. V.F acknowl-\nedge partial funding obtained from the Norwegian PhD\nNetwork on Nanotechnology for Microsystems, which\nis sponsored by the Research Council of Norway, Di-\nvision for Science, under contract no. 221860/F40.\nF.M. acknowledges support from Catalan Government\nCOFUND-FP7. JMH and FM also thank the Spanish\nGovernment (Grant No. MAT2011-23698.)\n∗vegard.flovik@ntnu.no\n1R. L. Stamps et al. J. Phys. D: Appl. Phys. 47 333001\n(2014)\n2J.˚Akerman, Science 308, 508 (2005)\n3S. D. Bader Rev. Mod. Phys. 78, 1 (2006)\n4M. Krawczyk, D. Grundler J. Phys.: Condens. Matter 26\n123202 (2014)\n5V. V. Kruglyak, S. O. Demokritov, D. Grundler J. Phys.\nD. Appl. Phys. 43 264001 (2010)\n6A. Moser, K. Takano, D. T. Margulies, M. Albrecht, Y.\nSonobe, T. Ikeda, S. Sun, E. E. Fullerton, J. Phys. D.\nAppl. Phys. 35 R157, (2002)7S. S. P. Parkin, K. P. Roche, M. G. Samant, P. M. Rice,\nR. B. Beyers, R. E. Scheuerlein, E. J. O‘Sullivan, S. L.\nBrown, J. Bucchigano, D. W. Abraham, Y. Lu, M. 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Kruglak, M. Franchin, H. Fangohr,\nL. Giovannini, F. Montoncello Phys. Rev. B. 87, 174422,\n(2013)\n28G. Carlotti, S. Tacchi, G. Gubbiotti, M. Madami, H. Dey,\nG. Csaba, W. Porod J. Appl. Phys. 117, 17A316, (2015)"    },    {        "title": "1504.05420v2.Anisotropic_interactions_opposing_magnetocrystalline_anisotropy_in_Sr__3_NiIrO__6_.pdf",        "content": "Anisotropic interactions opposing magnetocrystalline anisotropy in Sr 3NiIrO 6\nE. Lefrançois,1, 2, 3,\u0003A.-M. Pradipto,4,yM. Moretti Sala,5L. C. Chapon,1\nV. Simonet,2, 3S. Picozzi,4P. Lejay,2, 3S. Petit,6and R. Ballou2, 3\n1Institut Laue Langevin, CS 20156, 38042 Grenoble Cedex 9, France\n2CNRS, Institut Néel, 38042 Grenoble, France\n3Univ. Grenoble Alpes, Institut Néel, 38042 Grenoble, France\n4Consiglio Nazionale delle Ricerche CNR-SPIN, L’Aquila, Italy\n5European Synchrotron Radiation Facility, CS 40220, 38043 Grenoble Cedex 9, France\n6Laboratoire Léon Brillouin, CEA, CNRS, Univ. Paris Saclay, CEA Saclay, F-91191 Gif-sur-Yvette, France\n(Dated: June 15, 2021)\nWe report our investigation of the electronic and magnetic excitations of Sr 3NiIrO 6by resonant\ninelastic x-ray scattering at the Ir L 3edge. The intra- t2gelectronic transitions are analyzed using\nan atomic model, including spin-orbit coupling and trigonal distortion of the IrO 6octahedron, con-\nfronted to ab initio quantum chemistry calculations. The Ir spin-orbital entanglement is quantified\nand its implication on the magnetic properties, in particular in inducing highly anisotropic magnetic\ninteractions, is highlighted. These are included in the spin-wave model proposed to account for the\ndispersionless magnetic excitation that we observe at 90 meV. By counterbalancing the strong Ni2+\neasy-plane anisotropy that manifests itself at high temperature, the anisotropy of the interactions\nfinally leads to the remarkable easy-axis magnetism reported in this material at low temperature.\nPACS numbers: 71.70.Ej, 78.70.Ck, 75.10.Dg\nSince the discovery of spin-orbit induced Mott insula-\ntor in Sr 2IrO4[1], it has become apparent that electron\ncorrelationeffectscanbeimportantin5 dtransitionmetal\noxides when combined with spin-orbit coupling (SOC),\ndespite rather wide delectronic bands. In a perfect oc-\ntahedral environment of 5d5low-spin systems, e.g. in\nIr4+oxide compounds, SOC splits the t2gband occupied\nby the 5 electrons, into a fully-occupied je\u000b=3\n2band\nand a narrow singly-occupied je\u000b=1\n2band. This band\ncan be gapped by a fair amount of Hubbard repulsion\ndriving the system toward an insulating state. This can\npersist even when the ideal je\u000b=1\n2state is not realized,\ni.e. when it is mixed with the je\u000b=3\n2states due to the\nIr4+non regular octahedral environment (see Fig. 1(b)),\nas it is generally the case in the known iridates [2–5].\nThe influence of SOC is not limited to transport prop-\nerties. Another very interesting perspective in the field\nof iridates is the strong anisotropy of the magnetic inter-\nactions produced by spin-orbital entanglement. This is\nexpected to give rise to novel exotic magnetic states and\nexcitations, still to be discovered [6, 7].\nInthiscontext,thecompoundsofthefamilyA 3MM’O 6\n(A = alkaline-earth metal, M,M’ = transition metal)\nare of interest because the M’ site can be occupied by\nIr4+. Most members of the family crystallize in the\nK4CdCl 6-derived rhombohedral structure in which al-\nternating distorted trigonal prismatic MO 6and nearly\noctahedral M’O 6coordinations constitute chains along\nthe rhombohedral axis (see Fig. 3) arranged on a trian-\ngular lattice. They show complex magnetic behaviors\nattributed mainly to frustration and low-dimensionality,\nenhanced by strong magnetocrystalline anisotropy and\nthe presence of two magnetic species. Sr 3NiIrO 6displays\nparticularly intriguing properties: a complex antiferro-magnetic structure below a Néel temperature T Nof 70 K\nwith slow spin dynamics and two consecutive correlated\nmagnetic regimes when lowering the temperature [8–11].\nThe ordered magnetic moments are aligned along the c-\naxis, but there is currently no explanation for the ori-\ngin of the large anisotropy (single ion effect or exchange-\ndriven mechanism) and its relation with the very high\ncoercive field [12] of this material. The role played by\nSOC and anisotropic exchange is therefore a key issue.\nThree groups have performed ab initio electronic struc-\nture calculations [13–15] confirming Sr 3NiIrO 6Mott in-\nsulating state. Nevertheless, they disagree about the\nnature of the frontiers 5delectronic levels, the sign of\nthe nearest-neighbor Ni-Ir interactions, and the magne-\ntocrystalline anisotropy. The role of SOC has been inves-\ntigated in the related Cu compound by resonant inelastic\nx-ray scattering (RIXS) [16, 17]. However, the distor-\ntion of the Ir4+octahedral environment in Sr 3CuIrO 6is\nmonoclinic, which leads to nonequivalent Ir-O distances\nand O-Ir-O angles. The resulting je\u000b=1\n2andje\u000b=3\n2\nmixedstatesandtheiroverlapwiththeCu 3dorbitalsare\nthen at the origin of an unusual ferromagnetic exchange\nanisotropy arising from antiferromagnetic superexchange\n[17]. Ir4+octahedra in Sr 3NiIrO 6, instead, only experi-\nence a pure elongation along the trigonal c-axis, which\npreserves equivalent Ir-O distances. Sr 3NiIrO 6is accord-\ningly a nice candidate to study the influence of simple\ndistortion on the electronic and magnetic properties.\nHereafter, we report RIXS experiment on Sr 3NiIrO 6\nprobing simultaneously its electronic and magnetic ex-\ncitations. The local atomic effective Hamiltonian of\nSr3NiIrO 6is derived, complemented by quantum chem-\nistry ab initio calculations. The analysis of the magnetic\nexcitations reveals a large Ising anisotropy in the Ni-IrarXiv:1504.05420v2  [cond-mat.str-el]  13 Apr 20162\njeff=±3/2jeff=1/2λSOCTrigonal distortioneg’a1gΔTrigonal distortion + SOCd1d2d3t2g10Dqeg\n−0.200.20.40.60.81050100150200250300350400450\nTransferred energy (eV)Intensity (a.u.)Q = (10 −1 7)300 K150 K100 K20 K10 K\na)b)c)d)\nFIG. 1: (Color online) (a) Incident energy dependence of the Sr 3NiIrO 6RIXS spectra at the L3Ir edge measured at\n300K, showing the resonance of the t2gandeglevels. (b) Splitting of the 5dlevels due to the octahedral\nenvironment (10 Dq), to its trigonal distortion and to SOC. (c) RIXS spectra at the reciprocal space position\nQ= (10 -1 7) between 10 and 300 K. The scans are offset by 75 counts for clarity. The dashed curves on the 300 K\ndata correspond to least-square refinements of the excitations using Pearson-VII functions. The vertical lines on the\n10Kcurve corresponds to the eigenvalues of Hcalculated for \u0015=396 meV,\u0001=294 meV,\u000bx;y=0 and\u000bz= 0.1 eV.\n(d) RIXS map measured at 10 Kalong the (10 -1 L) direction. The dotted lines show the spin-wave excitations\ncalculated with Jxx=Jyy= 20 meV,Jzz=46 meVandD=9meV. The Ir and Ni main contributions are in white\nand grey respectively.\nmagnetic exchange which competes with a sizable single-\nion planar anisotropy for the Ni ions. This competition\nexplains the directional crossover of the magnetization\nobserved at high temperature.\nRIXS measurements were performed at the ID20 beam\nlineoftheEuropeanSynchrotronRadiationFacility. The\nbeam line is equipped with a 2 m arm spectrometer,\nbased on spherical Si(844) diced crystal analyzers. The\ncombination of the Si(111) double-crystal monochroma-\ntor with a channel-cut allows for a flexible choice of the\nenergy-resolution down to 25 meV[18] at the Ir L3edge.\nRIXS has proven an ideal tool in the study of the iri-\ndate electronic structure [3, 16, 19, 20], as it directly\nprobes 5dstates via two successive dipole transitions\n[21, 22]. Measurements were performed between 300 and\n10Kon a single crystal of Sr 3NiIrO 6, grown using the\nflux method [11]. Fig. 1(a) shows a RIXS map obtained\nfrom inelastic scans at different incident photon energies.\nThe resonance of the Ir4+electronic levels is observed\nat 11.214 keVfor thet2gand at 11.218 keVfor theeg.\nFor all measurements presented below, the incident pho-\nton energy was fixed to 11.214 keV, which enhances the\nexcitations within the t2gmanifold.\nThe RIXS spectra of Sr 3NiIrO 6measured at the mo-\nmentum transfer Q= (10 -1 7) of a forbidden Bragg\npeak position are shown Fig. 1(c). We first focus on the\nroom temperature measurement where four features are\nvisible beside the elastic peak, at the energies of 50(5),\n322(20), 568(2) and 728(5) meV, none of them showing\nany detectable dispersion within the lowest instrumen-tal resolution. The most intense peaks at 568(2) and\n728(5) meVare interpreted as d-dexcitations within the\nt2glevels [4, 16, 17], i.e. as transitions of the hole from\nthe ground state to the two lower lying filled states (see\nFig. 1(b)). In addition to the SOC splitting of the t2g,\na trigonal crystal field indeed further splits the je\u000b=3\n2\nground manifold. This effect can be quantified using a\nlocal atomic model for the single hole in the2T2gstates\nwith the Hamiltonian:\nH= \u0001(2\n3ja1giha1gj\u00001\n3je0\u0006\ngihe0\u0006\ngj)+\u0015L·S\u0000X\ni2x;y;z\u000biSi\n(1)\nwhere \u0001describes the t2gorbitals splitting due to the\nIrO6octahedron trigonal distortion ( \u0001>0for an axial\nelongation) and \u0015is the SOC parameter. The symmetry-\nadaptedja1giandje0\u0006\ngiwave-functions for trigonal sym-\nmetry are defined in the Sup. Mat. [23]. The 10Dq\nsplitting between t2gandeglevels is of the order of 4 eV\n(see Fig. 1(a)), i.e. much larger than \u0015and\u0001, and there-\nfore theeglevels can safely be ignored. The last terms,\n\u000biSi=\u00002Jii<SNi\ni>Si(i=x;y;z), are the molecular\nfield components produced on the Ir spin by its two Ni2+\nnearest neighbors. For the analysis of the 300 Kdata, far\nabove TN, we first assume \u000bx;y;z = 0, i.e. weak influence\nof the magnetic interactions.\nThe values of \u0015and\u0001are obtained by constraining the\neigenvalues ofHto the energies of the t2gmanifold exci-\ntations determined from the experiment. Two solutions3\nexist, depending on the sign of \u0001:\u0015=396(1) meVand\n\u0001=294(7) meVfor the first one and \u0015=417(4) meVand\n\u0001=-218(8) meVfor the second one. The wave-functions\n(shown in Fig. 2) for the highest doublet in the2T2gsub-\nspace (occupied by the hole) are:\n8\n>><\n>>:j0;\"i=ipja1g;\"i+ije0+\ng;#i+je0\u0000\ng;#ip\np2+2\nj0;#i=pja1g;#i\u0000je0+\ng;\"i\u0000ije0\u0000\ng;\"ip\np2+2(2)\nwith the evolution of pwith \u0001=\u0015shown in Fig. 2. The\nsign of \u0001can not be determined by the RIXS analysis\nalone. However, the Ir magnetic moment refined in neu-\ntron diffraction [11] seems in better agreement with the z\ncomponent of the total magnetic moment, <Lz+2Sz>,\ncalculated for a positive \u0001(0.25\u0016B) than for a nega-\ntive one (1.35 \u0016B), as shown in Fig. 2. Nevertheless, a\nstrong reduction of the magnetic moment by quantum\nfluctuations, enhanced by low dimensionality or geomet-\nric frustration, is not excluded.\nAnotherargumentinfavorof \u0001>0comesfrom ab ini-\ntiocalculations. For these, we used a quantum chemistry\ninherited computational scheme [3, 4, 16, 24], in which it\nwas shown that the combination of Multireference Con-\nfiguration Interaction method with SOC (MRCI+SOC)\nis able to reproduce the RIXS spectra. The local Ir- 5d\nelectronic structure has been investigated in the basis of\nmulti-configurational wave-function-based methods [23]\nwhich is implemented in MOLCAS 7.8 code [25] to en-\nsure a proper description of the multiplet physics.\nThe 54 spin-orbit Ni-Ir coupled states and their en-\nergies were calculated for the cluster shown in Fig. 3\n[23]. In order to estimate the excitations only within\nthe Ir- 5dorbitals, we performed calculations by replac-\ning the Ni2+with non-magnetic Zn2+. Without SOC,\nwe observe three doublet states, in which the degenerate\n2E0\ngdoublets lie 0.21 eVhigher than the ground state\n(see Table I). This is in perfect agreement with the en-\nergywindowscalculatedwithNi2+, confirmingthatthese\nNo SOC With SOC Exp. values\nConfiguration \u000f0\u000f1\u000f2\u000f0\u000f1\u000f2\u000f0\u000f1\u000f2\n0.00 0.21 0.21 0.00 0.64 0.80 0.00 0.57 0.73\nj 1i=d1\n1d2\n2d2\n31.00 0.46 0.54 0.00 0.58 0.42 0.00\nj 2i=d2\n1d1\n2d2\n31.00 0.27 0.23 0.50 0.21 0.29 0.50\nj 3i=d2\n1d2\n2d1\n3 1.00 0.27 0.23 0.50 0.21 0.29 0.50\nTABLE I: SD-MRCI results for the t2glevel splittings\n(ineV) and weights of the different Ir- t2g\nconfigurations. In the absence of SOC, \u000f1=\u000f2\ncorresponds to the energy of the doubly degenerate E0\ng\nstate. The energies and weights extracted from the\nexperiment for \u0001>0 are shown in the last column.correspond to the excitation energies within the Ir-5 t2g\nmanifold. Furthermore, the ground doublet state corre-\nsponds to the d1\n1d2\n2d2\n3configuration, where d1is thea1g\norbital that can be written as dz2, whiled2andd3cor-\nrespond to the e0\ngorbitals in the trigonal representation.\nOne can hence conclude that the a1gorbital lies higher\nin energy than the e0\ngones, in accordance with a positive\n\u0001.\nTaking into account the SOC, two excitations are cal-\nculated at 0.64 and 0.80 eV. These values are remarkably\nclose to the energies found in the RIXS spectra, confirm-\ning that the associated peaks correspond to the splitting\nof theje\u000b=3\n2quartet into two doublets. The weights\nof different t5\n2gcontributions to the spin-orbit coupled\ndoublet states are shown in Table I. The ground state is\nnot an equal mixture of the t2gorbitals, the weight of\nj 1ibeing larger than the other two degenerate config-\nurations, with a ratio of 0:46 : 0:27 : 0:27(to be com-\npared with 0:33 : 0:33 : 0:33in a perfect cubic environ-\nment). Overall, the room temperature RIXS measure-\nments confronted to the ab initio calculations are fully\nconsistent and allow us to describe quantitatively the\nelectronicspectrumoftheIr4+anditsdeparturefromthe\nperfectje\u000b=1\n2state due to its non-cubic environment.\nAn Ir-5t2gorbitals splitting \u0001ranging between 0.21 and\n0.30eVis obtained, leading to a ratio \u0001=\u0015\u00190:53\u00000:74\n(\u0015=0.396 eVestimated from RIXS). This scheme does\nnot explain the weak signal observed at 322 meV. Its\norigin is unclear, but a feature with similar energy and\nspectral weight was reported in almost all the iridates\n[3, 4, 16], suggesting that it might be an intrinsic char-\nacteristics of this class of materials.\nLowering the temperature down to 10 K, the two\nintense peaks related to the t2gelectronic transitions\nchange shape and acquire a substructure (see Fig. 1(c)).\nThis reflects the increasing influence, as the tempera-\nture decreases, of the Ni-Ir nearest neighbor magnetic\nexchange interactions on the Ir electronic states. The\nmolecular field produces a splitting of the peaks that can\nbe calculated from the last term of Eq. 1. Although the\nfeatures observed in the range 500-900 meVdo not allow\nto univocally determine the values of \u000bx;y;z, an exam-\nple of splitting is shown on the 10 K spectrum of figure\n1(c), in reasonable agreement with the experiment. It is\nobtained with \u000bx;y=0 and\u000bz=0.1 eV, agreeing with the\nstrong uniaxial character of the magnetic structure, and\nwith the value of the Jzzinteraction determined here-\nafter.\nAt 300 K, the excitation at the lowest energy is broad\nwith a position of the maximum at \u001950(5) meV. It\nrises in intensity while narrowing as the temperature is\nlowered (see Fig. 1(c)), and its energy rapidly shifts to\n\u001990(5) meVat 100 K, below which it remains constant.\nThis excitation cannot be accounted for by the local elec-\ntronic level scheme derived previously. It is attributed to\na magnetic excitation whose observation by RIXS is al-4\nFIG. 2: (Color online) Evolution of the zcomponents of\nthe orbital <L z>, spin <S z>, total magnetic moment\n<Lz+ 2Sz>and ofpas a function of\u0001\n\u0015. The grey\nstripe shows the Ir magnetic moment refined from\nneutron diffraction [11]. The amplitudes of the\nprobability of presence of the t2ghole are shown at the\nbottom: for a perfect octahedral environment (\u0001\n\u0015= 0),\nfor a trigonal compression (\u0001\n\u0015=\u00000:52) and for a\ntrigonal elongation (\u0001\n\u0015= 0:73) of the octahedron. The\nlatter two are in agreement with the RIXS results but\nonly the positive\u0001\n\u0015is compatible with quantum\nchemistry calculations. The surfaces of constant\namplitude of probability are colored from red for pure\nja1g;\"istate, to blue for pure ( ije0+\ng;#i+je0\u0000\ng;#i) state,\nthrough white for an equal mix of these states.\nlowed in this system [23]. The absence of any detectable\ndispersion is illustrated by the RIXS measurement along\nthe (10 -1 L) direction (see Fig. 1(d)). Another mag-\nnetic excitation was identified at 35 meVby inelastic\nneutron scattering (INS) on a Sr 3NiIrO 6powder sam-\nple [26]. It was also shown to persist up to 200 K, i.e\nmuch above T N, a behavior attributed to the presence of\nstrongspincorrelationsassociatedtothelow-dimensional\nnature of the magnetism. We succeeded in reproducing\nthe spin-wave modes at 35 and 90 meVobserved by INS\nand RIXS respectively using the Holstein-Primakov for-\nmalism within the linear approximation (see Fig. 1(d))\n[23]. We used a simple model, in line with the reported\nmagnetic structure [11, 23], of isolated chains with (i)\nstrong anisotropic antiferromagnetic exchange interac-\ntions between the nearest-neighbor Ni and Ir ions along\nthechain:Jxx=Jyy=20(2) meV,Jzz=46(2) meVand(ii)\nmagnetocrystalline anisotropy of Ni2+:DS2\nzterm with\nD=9(1) meVcorresponding to an easy-plane single-ion\nanisotropy perpendicular to the chain axis. The effective\ninterchaininteractionwasestimatedfromboththeorder-\ning temperature and the coercive field. It is more than\none order of magnitude smaller than the intrachain in-\nteractions, affecting negligibly the spin excitations. It is\nfurthermore irrelevant to the spin wave dispersion alongthe chains. It can thus be safely ignored [23]. Extract-\ning the ion-dependent neutron spectral weight of the ex-\ncitations clearly demonstrates that Ir(Ni) mainly con-\ntributes to the high(low) mode [23].This explains the ab-\nsence of the INS 35 meVmode in the RIXS data since\nthe measurements at the Ir L3edge are mainly sensi-\ntive to the resonant inelastic signal of Ir. Our model is\nhighly constrained by the energies and dispersion of the\nmodes extracted from the RIXS and INS data [23] and\ncan uniquely account for the experimental results if one\nconsiders this easy-plane single-ion term. This term is\na hallmark difference between the physics of Sr 3NiIrO 6\nand Sr 3CuIrO 6, which otherwise shows rather similar\nanisotropic exchanges [17]. The strength of the exchange\nanisotropy is driven by p, which is related to the rela-\ntive weight of the ja1giandje0\ngiwave functions in Eq.2.\nThose are respectively preserving or not L zwhen flipping\nSz, hencecontributingtotheisotropic(resp. anisotropic)\npart of the interactions [17]. The parameter pvaries with\nthe octahedral distortion \u0001and might be tunable by a\nchemical or external pressure.\n501001502002503003504000.40.60.811.21.41.6\nTemperature (K)Magnetic susceptibility   H ⊥ c H // c5010015020025030000.0050.01\nTemperature (K)M/H (emu/mole.Oe)\nc-axisSrNiIrO\nFIG. 3: (Color online) Magnetic susceptibility of\nSr3NiIrO 6calculated by Monte Carlo simulation using\nthe spin-wave parameters. The blue (red) curve is\nobtained for the magnetic field applied along\n(perpendicular to) the chain direction. Top insert:\nComparison with the measurements reported in Ref.\n[11]. The arrows indicate the crossing of the curves.\nBottom insert: Detail of the Sr 3NiIrO 6structure with\nthe central distorted IrO 6octahedron along the chain\naxis.\nTo further support this model, it is found that the\nNi anisotropy perfectly matches the one reported for\nthe isostructural compound Sr 3NiPtO 6, for which D\u0019\n9 meV[27] and which is even considered as a prototypal\nexample of a \"large- D\" system. Furthermore, the com-\npetition between strong exchange anisotropy ( Jzz=Jxx)\nand single-ion anisotropy ( DS2\nz) on the Ni site produces\na hallmark feature in the magnetization measurement:5\nat high temperature, the susceptibility measured for a\nmagnetic field applied along the c-direction is lower than\nthe one obtained with a field in-plane. Below 200 K, the\ntwo curves cross due to the increasing influence of the\nlarge exchange anisotropy when correlations build-up, ul-\ntimately leading to a uniaxial antiferromagnetic order.\nThe directional crossover in the magnetic susceptibility\nis nicely reproduced by Monte-Carlo simulations using\na one-dimensional chain model and the parameters ex-\ntracted from the spin-wave calculations (see Fig. 3) [23].\nFinally, the large coercive field reported in Sr 3NiIrO 6is\nsimply explained as the minimum field responsible for\nthe coherent reversal of the spin chains, its value agree-\ning with the interchain coupling [23].\nTo summarize, we have determined the 5dIr elec-\ntronic scheme in the Sr 3NiIrO 6chain compound through\nour RIXS measurements and their analysis. The re-\nsulting spin-orbital entangled ground state produces a\nstrong anisotropy of the interactions, which is evidenced\nin our analysis of the spin-wave excitations including the\ntwo gapped modes seen by RIXS and neutron scatter-\ning. This uniaxial anisotropy of the interactions com-\npetes with a strong Ni2+single ion easy-plane anisotropy,\na unique feature of this system in the series. Compar-\ning our magnetic Hamiltonian to the one of a recent nu-\nmerical study of a mixed-spin XXZ chain with single-ion\nanisotropy, it is found that, in the parameter space of ex-\nchangeandsingle-ionanisotropies, Sr 3NiIrO 6liescloseto\nthe boundary separating the observed ordered Ising ferri-\nmagnetic phase from a disordered XY phase [28]. These\nresults demonstrate the potential of SOC in triggering\nemerging physics. In mixed 3d-5d transition metal chain\nsystems, it can allow to probe unexplored regions of their\nphase diagram and quantum phase transitions.\nAcknowledgments\nWe thank A. Al-Zein for this help during the RIXS\nexperiment and A. Hadj-Azzem for his collaboration on\nthe crystal synthesis. A.-M. P. and S. P. thank R. Broer\nforprovidingwiththecomputationalfacilitiesinthelocal\ncomputer cluster at the University of Groningen.\n\u0003Corresponding author: lefrancois@ill.fr\nyCorresponding author: a.m.t.pradipto@gmail.com\n[1] B. J. Kim, H. Jin, S. J. Moon, J.-Y. Kim, B.-G. Park,\nC. S. Leem, J. Yu, T. W. Noh, C. Kim, S.-J. Oh, et al.,\nPhys. Rev. Lett. 101, 076402 (2008).\n[2] M. Moretti Sala, M. Rossi, A. Al-Zein, S. Boseggia, E. C.\nHunter, R. S. Perry, D. Prabhakaran, A. T. Boothroyd,\nN. B. Brookes, D. F. McMorrow, et al., Phys. Rev. B 90,\n085126 (2014).[3] H. Gretarsson, J. P. Clancy, X. Liu, J. P. Hill, E. Bozin,\nY. Singh, S. Manni, P. Gegenwart, J. Kim, A. H. 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Tian, International Jour-\nnal of Modern Physics B 29, 1550070 (2015).1\nSupplemental Materials: Anisotropic interactions opposing magnetocrystalline\nanisotropy in Sr 3NiIrO 6.\nDEFINITIONS OF THE ORBITALS USED IN THE SIMPLE HAMILTONIAN ACCOUNTING FOR THE\nRIXS MEASUREMENTS\n8\n>>>>><\n>>>>>:(xy) =\u0000{p\n2(j2;2i\u0000j2;\u00002i)\n(yz) ={p\n2(j2;1i+j2;\u00001i)\n(zx) =\u00001p\n2(j2;1i\u0000j2;\u00001i)\n(x2\u0000y2) =1p\n2(j2;2i+j2;\u00002i)\n(z2) =j2;0i(E1)\n8\n>>>>>>>><\n>>>>>>>>:ja1gi= (z2)\nje0+\ngi=q\n2\n3(x2\u0000y2)\u0000q\n1\n3(zx)\nje0\u0000\ngi=q\n2\n3(xy) +q\n1\n3(yz)\nje+\ngi=q\n1\n3(x2\u0000y2) +q\n2\n3(zx)\nje\u0000\ngi=q\n1\n3(xy)\u0000q\n2\n3(yz)(E2)\nAB INITIO CALCULATION DETAILS\nThe local Ir- 5delectronic structure has been investigated in the basis of multi-configurational Complete Active\nSpace Self-Consistent Field (CASSCF) [R1] method which is implemented in MOLCAS 7.8 code [R2] to ensure a\nproper description of the multiplet physics. To make such multi-electron approach feasible, a small part representing\nthe crystal in which a finite number of atoms are treated explicitly in the calculation, called a ‘cluster’, has to be\nchosen. In our case, we include an IrO 6octahedra, its neighboring NiO 6trigonal prismatic coordinations and the\nadjacent Sr atoms (see insert of Fig. 2 in the article). This choice is expected to provide an accurate description of\nthe charge distribution in the region of IrO 6octahedra. To model the crystalline environment, we employ a standard\nembedding procedure of the cluster [R3, R4] including (i) a set of optimized point charges [R5], and (ii) the Ab Initio\nEmbedding Model Potentials [R6] to avoid artificial excessive polarization of the cluster electrons towards the point\ncharges. An active orbital space has been chosen to include the t2g-like orbitals of the Ir- 5dshell and the magnetic\norbitals of the Ni atoms to describe the excitations within the Ir- 5t2gmanifold. The chosen active space is enlarged\nto include also the Ir- 5egorbitals when the t2g!egexcitations are to be considered. The dynamical correlation is\nthen taken into account at a Single and Double Multireference Configuration Interaction (SD-MRCI) level using the\nRestricted Active Space Self-Consistent Field [R7] module of MOLCAS, in which a maximum number of 2 electrons\nare allowed to be excited from the 2porbitals of oxygen atoms surrounding the iridium, without relaxing the orbitals\nfrom CASSCF. Further enlarging the external active space in the SD-MRCI step, e.g. to include all doubly occupied\nNi-3dshells or the empty t2gorbitals from the Ir-6 dshell to improve the electron correlation description, does\nnot significantly change the results. The scalar relativistic effects are accounted for with the Douglass-Kroll-Hess\nHamiltonian [R8, R9]. The wavefunctions and energies of the spin-orbit free multiplets obtained at the SD-MRCI\nlevel are then included for the spin-orbit coupling (SOC) calculation using the Spin-Orbit Restricted Active Space\nState Interaction method [R10, R11] as implemented in MOLCAS code. The cluster ‘molecular’ orbitals are expanded\nwith ANO-RCC basis sets [R12] in all calculations, with the contraction schemes ( 8s7p5d2f), (6s5p4d), (4s3p1d),\nand ( 6s5p2d1f) for the Ir, Ni, O, and Sr atoms, respectively.\nAs expected, due to a large crystal field splitting 10 Dqof about 3.7 eV(estimated from Single and Double MRCI\n+ SOC, in good agreement with the experimental value of 4 eV), all 5 electrons of Ir- 5dare arranged in a low-spin\nS=1\n2manner to occupy the t2g-like orbitals. In the spin-orbit free framework, these three doublet states of Ir4+and\ntheS= 1states of the two Ni2+ions can couple to produce three sets of one sextet, two quartet and two doublet\nstates, lying in energy windows of 0\u000033 meVand0:24\u00000:26 eV. This gives an estimate to the Ir-5 t2gsplitting of2\n0:2\u00000:24 eVdue to the non-cubic crystal field. Allowing these multiplets to interact via SOC results in a total of\n54 spin-orbit coupled states lying in energy windows of 0\u000022 meV,0:64\u00000:66 eV, and 0:82\u00000:84 eV. In order\nto get better estimate on the excitations only within the Ir- 5dorbitals, further calculations have been performed by\nreplacing the Ni2+with non-magnetic Zn2+ions, as presented in the main article. One result of these SD-MRCI\ncalculations is that the three lowest doublets are always dominated by t5\n2gconfigurations, weighting more than 98%\nof the total contributions, confirming a minor t2g\u0000egmixing to the ground state wavefunction.\nMAGNETIC EXCITATION AND SPIN WAVES CALCULATIONS\nWe checked beforehand that single magnon excitations in Sr 3NiIrO 6can effectively be detected by RIXS. The RIXS\nscattering intensity is given by the Kramers-Heisenberg formula [R13] and is proportional to the matrix elements of\nthe polarization dependent dipole operator C\u000f=\u000f~ ras:\nI~!!~!0/jh5dfjC\u000f0j3p3\n2ih3p3\n2jC\u000fj5diij2; (E3)\nwhere 5diand5dfstand for the initial and final d states and 3p3\n2for the intermediate states. Using the dipole\nmatrix reported in ref [R14], we can compute the matrix elements displayed in Eq. E3 for the different possible\nintermediate 3p3\n2states. They are gathered in Table I for initial and final dstates in the ground level with respect to\nd-dexcitations ( i.e.j0;\"iandj0;#istates defined in the main text). All of them are not null thereby implying that\nsingle magnons with spin deviations in the ground level do contribute to the scattering.\nj3\n2;\u00063\n2i j3\n2;\u00061\n2i\nj0;\"i\nj0;#ii2p\n2p\n2+p2\u0000ip\n2(2+5p+2p2)\n3(2+p2)\nTABLE I: Computed h0;\"(#)jC\u000f0j3p3\n2ih3p3\n2jC\u000fj0;#(\")iquantities for the different 3p3\n2states.\nTo model the spin dynamics, spin-wave calculations have been performed using the Spinwave software developed at\nLaboratoire Léon Brillouin for inelastic neutron scattering experiments. Based on the Holstein-Primakov approxima-\ntion, the code diagonalises any bilinear spin Hamiltonian and takes into account (isotropic or anisotropic) exchange\ncouplings acting between neighboring spins, as well as single ions anisotropy terms. The initial spin configuration is\nobtained from the mean field solution of the following Hamiltonian:\nH=X\niJxx(Sx\niSx\ni+1+Sy\niSy\ni+1) +JzzSz\niSz\ni+1+X\ni2NiD(Sz\ni)2(E4)\nwhich describes an isolated chain with alternating spin 1 and spin1\n2antiferromagnetically coupled along the\nchains [R15]. The exchange tensor acting between the first neighbors along the caxis is highly anisotropic,\ndefined by JxxandJzz.Halso takes into account the magnetocrystalline anisotropy of the Ni spins, modeled\nbyD(Sz\ni)2, and which corresponds to an easy plane anisotropy perpendicular to the zaxis. The point symmetry\nbetween Ni and Ir site is 3, thus allowing for antisymmetric exchange interactions with a Dzyaloshinskii-Moriya\nD-vector pointing along the c-axis. The upper limit estimate for this interaction, \u00195 meV, shows that it is not\ntheleadingtermintheHamiltonian, andhasthusbeenneglected, togetherwiththeinterchaininteractions(seebelow).\nFigure F1 shows the calculations performed with the parameters: Jxx= 20 meV,Jzz= 46 meVandD=\n9meV, along aQ-direction corresponding to the zone explored during the RIXS experiment (see left Fig. F1). Our\ncalculation shows the presence of two excitations around 35 and 90 meVwith a small dispersion of the order of 10 meV.\nWe also calculated the powder average of the spin wave spectrum in order to compare our model with the powder\nneutron inelastic experiment performed bu Wu et al.[R16] (see right Fig. F1). Because of the integration over the\nmomentum directions, the resulting spectra lack directional information but give a weighted density of states for each\nmagnitude of Q. This results in two band-like excitations, the lower one reproducing fairly well the INS results. The\nvalues of the different parameters are highly constrained by the results from the RIXS and INS experiments. Indeed\nthe position of the maxima of the two modes are equal to Jzz\u0000Dand to 2Jzz, whereas the dispersion is related to\nJxx=Jzz: The amplitude of the dispersion decreases as Jxxdecreases and as Jzzincreases.3\nThe Spinwave software also allows one to identify the contributions arising from the different magnetic species.\nIndeed, since a spin-wave is a collective excitation, each spin within the unit cell contributes with however a different\nweight. A close look at those individual contributions shows that the lower branch is essentially due to the Ni\ncontribution while the upper branch arises mainly from the Ir contribution (see Fig. F2). This explains why we did not\nobserved the low-energy excitation in the RIXS experiment as the energy was fixed at the Ir- L3resonance edge. The\nreason why the high-energy excitation was not seen in INS is probably due to the weakness of the signal in the rather\nhighQ-range investigated. Overall, the calculations give an excellent agreement with both RIXS and INS experiments.\nNote that Figure F2 shows two interlaced branches around 25 and 90 meV. This is due to the fact that the unit cell\ncontainstwoIrandtwoNiatoms. OnlyonebranchactuallycarriessignificantphysicalintensityasshowninFigureF1.\nUsing this program, we also examined the influence of the mean interchain interactions on the spin wave spectrum\nby calculating the response of three chains on a triangular lattice: two of them are antiferromagnetically coupled,\nwhile the third one is unfrozen in orientation. This spin configuration is one of the two that are compatible with\nthe neutron diffraction experiment [R15], the other one involving a modulation of the amplitude of the magnetic\nmoments, which cannot be implemented in the Spinwave software. Note that these two magnetic structures are\nequivalent up to a global phase. The inclusion of the interchain interactions modifies slightly the energy of the\ntwo branches (see Fig. F3) by increasing (decreasing) their energy for ferromagnetic (antiferromagnetic) interchain\ncoupling without affecting the amplitude of the dispersion.\nThe interchain exchange interaction might be estimated from the ordering temperature, but this is not straightfor-\nward for mixed-spin quantum chains where the intrachain exchange anisotropy competes with single-ion anisotropy\nso that the spin dimensionality evolves from almost XY to almost Ising by decreasing the temperature. An upper\nbound can nevertheless be perceived through analytical and numerical estimates from different models [R17–R21].\nIn the case of classical spin-S chains without anisotropy, by treating exactly the effects of the intrachain exchange\nJkand in mean field those of the interchain exchange J?, the ordering temperature is analytically computed in the\nformTN\u0019JkS(S+ 1)((8=3)z?J?=Jk)1=2wherez?is the number of chains with which each chain interacts. With\nS= 1=2,Jk=Jxx=Jyy= 20 meV andTN\u001980 K\u00196:9 meVthis leads to the exchange ratio \u0011=J?=Jk\u00190:02\nand an interchain exchange J?\u00190:4 meV. With the larger exchange Jzza smaller exchange ration \u0011is obtained.\nNéel ordering is inhibited by interchain correlations and quantum fluctuations, which would suggest that J?is larger.\nApproached numerically these effects in the case of the extreme limit of S=1/2 quantum spins lead indeed to an\nincrease of the exchange ratio up to \u0011\u00190:1and therefore of interchain exchange up to J?\u00192 meV. These decrease\nto\u0011\u00190:075andJ?\u00191:5 meVfor quantum spins S=3/2. Néel ordering on the other hand should be favored by the\nreduction of the spin dimensionality caused by the single ion and intra-chain exchange anisotropies that should lead\nto much lower values of the interchain exchange J?, as suggested by the exchange ratio \u0011\u00190:009obtained in the\nIsing limit. It follows that the J?\u00190:4 meVestimated from the classical spin-S chains without anisotropy can be\nconsidered safely as providing an upper bound.\nAs a matter of fact, another independent estimate of J?can be obtained from the coercive field Hcwhich\nwas observed up to 55 Tin the magnetization processes [R12]. We assume that this coercive field corresponds\nto the minimum field of coherent reversal of spin chains. In case of the frustrated magnetic structure with two\nfrozen chains of antiparallel moments over three with the third being paramagnetic, there is a degeneracy of the\npossible configurations. Each frozen chain can be surrounded by (3 + n) chains of opposite moment orientation and\n6-(3+n) chains of same moment orientation, with n = 0, 1, 2, 3. On the other hand, each unfrozen chain is always\nsurrounded by 3 chains of opposite moment orientation and 3 chains of same moment orientation. We can write\n(3 +hni=3)Jperp\u0019C Hcwith C\u00190:058forHcgiven in Tesla and Jperpin meV, which leads to values of Jperp(\u0019\n1.06 meVforhni= 0,\u00190.80 meVforhni= 1,\u00190.64 meVforhni= 2and\u00190.53 meVforhni= 3) of the order of\nthe upper bound extracted from the Néel temperature. Accordingly, the interchain exchange is expected to have a\nminor effect on the spin wave spectrum and confirms the quasi-one dimensional character of Sr 3NiIrO 6.\nWe also calculated the effect of the intrachain Dzyaloshinskii-Moriya interactions, whose component along the c\naxis is allowed. We found that the small changes produced by this term in the energy position and the dispersion of\nthe spin waves cannot be seen in the RIXS and INS experiments, up to values of 5 meV. Up to this upper bound, It\nhas also no effect on the magnetic configuration.4\nMONTE CARLO CALCULATIONS\nMonte-Carlo simulations have been performed by considering a system of chains of alternating classical spin 1/2\nand spin 1 sites. Interchain interactions have been neglected for simplicity. The exchange interactions and single ion\nanisotropies have been fixed to that provided by the spin-waves analysis. The simulations have been performed using\na conventional Metropolis algorithm, cooling the system from 2000K down to 1K. The temperature was decreases\nexponentially (0.95 times the previous temperature). At each step in temperature, 104spin flips per spin were\nconducted for equilibration and a further 104steps for data taking. The magnetic susceptibility was calculated using\nthe fluctuation-dissipation theorem:\n\u001f\u000b=hm2\n\u000bi\u0000hm\u000bi2\nNT(E5)\nwhereNis the number of spins in the simulations and \u000b=x,y,z is the direction. Simulations using supercells of\n3x3x200 up to 3x3x800 have been used. The results presented in the main article correspond to the largest supercell\nwith 20000 spins.\nMAGNETIZATION MEASUREMENTS\nThe easy-plane nature of the magnetocrystalline anisotropy of the compound can also be observed in the magnetic\nfield dependence of the magnetization (see Fig. F4). Indeed, at 300 K, the magnetization measured in the direction\nof the c axis is smaller than in the perpendicular direction. At 150 K, the inverse is observed.\n\u0003Corresponding author: lefrancois@ill.fr\nyCorresponding author: a.m.t.pradipto@gmail.com\n[R1] B. O. Roos, P. R. Taylor, and P. E. M. Siegbahn, Chem. Phys. 48, 157 (1980).\n[R2] F. Aquilante, L. De Vico, N. Ferre, G. Ghigo, P.-A. Malmqvist, P. Neogrady, T. Pedersen, M. Pitonak, M. Reiher,\nB. Roos, et al., J. Comp. Chem. 31, 224 (2010).\n[R3] A. Sadoc, C. de Graaf, and R. Broer, Phys. Rev. B 75, 165116 (2007).\n[R4] R. Maurice, A.-M. Pradipto, C. de Graaf, and R. Broer, Phys. Rev. B 86, 024411 (2012).\n[R5] B. Roos and U. Wahlgren, MADPOT andMADFIT program (1969).\n[R6] Z. Barandiarán and L. Seijo, J. Chem. Phys. 89, 5739 (1988).\n[R7] P.-A. Malmqvist, A. Rendell, and B. O. Roos, J. Phys. Chem. 94, 5477 (1990).\n[R8] M. Douglas and N. M. Kroll, Ann. Phys. 82, 89 (1974).\n[R9] B. A. Hess, Phys. Rev. A 33, 3742 (1986).\n[R10] P.-A. Malmqvist and B. O. Roos, Chem. Phys. Lett. 155, 189 (1989).\n[R11] P.-A. Malmqvist, B. O. Roos, and B. Schimmelpfennig, Chem. Phys. Lett. 357, 230 (2002).\n[R12] B. O. Roos, R. Lindh, P.-A. Malmqvist, V. Veryazov, and P.-O. Widmark, J. Phys. Chem. A 109, 6575 (2005).\n[R13] T. Åberg and B. Crasemann, in X-ray Anomalous (Resonance) Scattering: Theory and Experiment , edited by K. Fisher,\nG. Materlik, and C. Sparks (Elsevier, Amsterdam, 1994).\n[R14] F. M. F. de Groot, P. Kuiper, and G. A. Sawatzky, Phys. Rev. B 57, 14584 (1998).\n[R15] E. Lefrançois, L. C. Chapon, V. Simonet, P. Lejay, D. Khalyavin, S. Rayaprol, E. V. Sampathkumaran, R. Ballou, and\nD. T. Adroja, Phys. Rev. B 90, 014408 (2014).\n[R16] W. Wu, D. T. Adroja, S. Toth, S. Rayaprol, and E. V. Sampathkumaran, ArXiv e-prints (2015), 1501.05735.\n[R17] T. Oguchi, Phys. Rev. 133, A1098 (1964).\n[R18] D. J. Scalapino, Y. Imry, and P. Pincus, Phys. Rev. B 11, 2042 (1975).\n[R19] J. Villain and J. M. Loveluck, 38, 77 (1977).\n[R20] F. Boersma, W. J. M. de Jonge, and K. Kopinga, Phys. Rev. B 23, 186 (1981).\n[R21] C. Yasuda, S. Todo, K. Hukushima, F. Alet, M. Keller, M. Troyer, and H. Takayama, Phys. Rev. Lett. 94, 217201 (2005).\n[R22] J. Singleton, J. W. Kim, C. V. Topping, A. Hansen, E.-D. Mun, S. Ghannadzadeh, P. Goddard, X. Luo, Y. S. Oh, S.-W.\nCheong, et al., ArXiv e-prints (2014), 1408.0758.5\nFIG. F1: (Color online) Calculated spin waves with exchange interactions Jxx=Jyy= 20 meVandJzz= 46 meV\nand magnetocrystalline anisotropy D= 9meVfor a single crystal (left) and for a polycrystalline sample (right).\nThe white zone represents the region which was not accessible in the INS experiment from Fig.4 of ref. [R26] up to\n50 meV.\nFIG. F2: (Color online) Contributions of the Ir ions (left) and Ni ions (right) in the calculated spin waves with\nexchange interactions Jxx= 20 meVandJzz= 46 meVand magnetocrystalline anisotropy D= 9meV. The\ncolorbar is in log scale.6\nFIG. F3: (Color online) Calculated spin waves with intrachain interactions Jxx= 20 meV,Jzz= 46 meV,\nmagnetocrystalline anisotropy D= 9meV, and interchain interaction J?= -1 meV(left), 0 meV(middle) and\n+1meV(right).\n01234500.0050.010.0150.020.0250.030.0350.040.0450.05\nMagnetic Field (T)Magnetic moment (µB/fu)\n01234500.0050.010.0150.020.0250.030.0350.040.0450.05\nTemperature (K)Magnetic moment (µB/fu)\n   H ⊥ c H // c\n01234500.0050.010.0150.020.0250.030.0350.040.0450.05\nTemperature (K)Magnetic moment (µB/fu)\n   H ⊥ c H // c\n01234500.0050.010.0150.020.0250.030.0350.040.0450.05\nTemperature (K)Magnetic moment (µB/fu)\n   H ⊥ c H // c150 K300 K\nFIG. F4: (Color online) Magnetic field dependence of the magnetization measured with the magnetic field applied\nparallel (filled symbol) and perpendicular (empty symbol) to the c-axis at 300 K(in cyan) and 150 K(in red)."    },    {        "title": "1504.05829v1.Structures_and_magnetic_properties_of_Co_Zr_B_magnets_studied_by_first_principles_calculations.pdf",        "content": "Structures and magnetic properties of Co-Zr-B magnet s studied  by \nfirst-principles  calculations  \nXin Zhao, Liqin Ke, Manh Cuong Nguyen, Cai -Zhuang Wang*, and Kai -Ming Ho§ \nAmes Laboratory, US DOE and Department of Physics and Astronomy, Iowa State \nUniversity, Ames, Iowa 50011, USA  \n \nAbstract  \nThe structure s and magnetic properties of the Co-Zr-B alloy s near the Co 5Zr composition  \nwere studied using adaptive genetic algorithm and first-principles calculations  to guide \nfurther experimental effort on optimizing their magnetic performance s. Through  \nextensive  structural searches, we constructed  the contour maps of the energetics and  \nmagnetic moment s of the Co -Zr-B magnet alloys  as a function of composition . We found \nthat the Co -Zr-B system  exhibits the same structural motif as the “Co11Zr2” polymorphs, \nwhich  plays  a key role  in achieving  high coercivity. Boron atom s can either substitute \nselective  cobalt atoms  or occupy  the interstitial site s. First-principles calculation  show s \nthat the magn etocrystalline  anisotropy energies can be significant ly improved  through \nproper boron doping . \n*e-mail: wangcz@ameslab.gov  \n§e-mail: kmh@ameslab.gov  \n  Introduction  \nAs promising candidate s for non-rare-earth  permanent  magnet s, Co xZr alloy s with x near \n5 and related compounds, such as Co -Zr-B, Co -Zr-M-B (M=C, Si, Mo, etc.), have \nattracted considerable  attention s [1-10]. Great effort has been devoted to the \nimprovement of their hard magnetic properties. The reported highest coercivity was 9.7 \nkOe, found in annealed Co74Zr16Mo 4Si3B3 ribbons [8] and the optimal magnetic \nproperties were obtained in Co 80Zr18B2 with intrinsic coercivity Hc=4.1 kOe and  energy \nproduct  (BH) max = 5.1 MGOe  [5]. More recently, cluster beam deposition has been used \nto make Co-Zr/Hf samples and energy products of 16 -20 MGOe were reported [ 9, 10 ]. \nThe Co -Zr/Hf  magnet alloys typically contain multiple ph ases and  identifying the phase \nresponsible for the magnetic hardness  has been one of the research  focuses . Several \nstudies  [11-13] assume d that the hard magnetic phase in  the Co-Zr system is the \nmetastable Co 5Zr phase with the structure of Ni 5Zr. However, Ni5Zr structure  is cubic \nand thus unlikely to provide strong magnetocrystalline  anisotropy energies , which was \nconfirmed by first -principles calculations [14] . Co3ZrB 2 has also been proposed  to be a \ncandidate for the hard magnetic phase  [15], which remains to be  validated .  \n \nDetermining the hard magnetic phase in the above -mentioned  alloys has been  a long -\nstanding issue due to the ambiguity of their crystal structure s. Recently, progress  has \nbeen made  in solving  the crystal  structures of the complex CoxZr alloys . Using adaptive \ngenetic algorithm (AGA)  [16], we studied the crystal structure s of the rhombohedral,  \nhexagonal, and orthorhombic polymorphs close to the Co 11Zr2 intermetallic compound  [14]. The common building block in the structures of these polymorphs  was identified  as \na derivative from the  SmC o5 structure.  Decreas e of the temperature induces a phase \ntransition from high symmetry  rhombohedral/hexagonal phase to low symmet ry \northorhombic/monoclinic phase, along with a slight increase of the Co concentration. The \nexperimental data  from  the x-ray diffraction (XRD) and transmission electron \nmicroscopy  were well explained  by the crystal structure s obtained from AGA  searches . \nThrough  first-principles magnetic properties calculation s, the hard magnetic phase  in the \nCoxZr alloys was identified  to be the high -temperature rhombohedral/hexagonal phase .  \n \nIn this work, we extended the investigation to  the effect of boron doping on the structures  \nand magnetic properties  of the Co xZr alloys . Structure searches by AGA  allowed us to \naccess the preferred position s of boron atoms , thus  energetics and magnetic properties of \ndifferent Co-Zr-B compositions can be  studied by  first-principles calculations . \n \nComputational methods  \nCrystal structures of  Co-Zr-B were  investigated  by AGA  [14, 16]. The structur e searches \nwere performed without any assumption on the Bravais lattice type, atom basis or unit \ncell dimensions.  The size of the unit cell studied  in this work was up to  100 atoms. In the \nAGA search  for this system , embedded -atom method [17] was used as the auxiliary \nclassical potential . The parameters for Co -Co and Zr -Zr interactions were from the \nliterature [ 18]. B-B interaction and the  crossing -pair interactions ( i.e., B-Co, B -Zr and  \nCo-Zr) were modeled by  Morse function  (eq. 1 ), with 3 adjustable parameters each  (D, α, r0). For Co and Zr atoms, parameters of the density function and embedding function \nwere taken  from r ef. [18] as well, a nd for B atom, e xponential decaying function was \nused as the  density function  (eq. 2, with 2 adjustable parameters : α, β) and the form \nproposed by Benerjea and Smith [ 19] was used as the embedding function  (eq. 3, with 2 \nadjustable parameters : F0, γ).  \n𝜙(𝑟𝑖𝑗)=𝐷[𝑒−2𝛼(𝑟𝑖𝑗−𝑟0)−2𝑒−𝛼(𝑟𝑖𝑗−𝑟0)] (eq. 1 ) \n𝜌(𝑟𝑖𝑗)=𝛼exp[−𝛽(𝑟𝑖𝑗−𝑟0)]  (eq. 2) \n𝐹(𝑛)=𝐹0[1−𝛾ln𝑛]𝑛𝛾                 (eq. 3) \nThe total energy of the system then has the following form:  \n𝐸𝑡𝑜𝑡𝑎𝑙 =1\n2∑ 𝜙𝑖𝑗𝑁\n𝑖,𝑗(𝑖≠𝑗) (𝑟𝑖𝑗)+∑𝐹𝑖(𝑛𝑖) 𝑖         (eq. 4) \nWhere  𝑟𝑖𝑗 is the distance between atoms i and j, 𝑛𝑖=∑ 𝜌𝑗(𝑟𝑖𝑗) 𝑗≠𝑖  is the electron density \nat the site occupied by atom i.  \n \nThe potential parameters were adjusted adaptively by fitting to the DFT energies, forces, \nand stresses of selected structures according to the A GA scheme  [16]. The fitting was \nperformed by the force -matching method with stochastic simulated annealing algorithm \nimplemented in the potfit  code [ 20, 21]. First-principles calculations were performed \nusing the  projector augmented wave (PAW) method [ 22] within density  functional theory \n(DFT)  as implemented in VASP code [ 23]. The exchange and correlation  energy is \ntreated within the spin -polarized generalized gradient  approximation (GGA) and parameterized by Perdew -Burke -Ernzerhof  formula (PBE) [ 24]. Wave functions are \nexpanded in plane  waves up to a kinetic energy cut -off of 350 eV. Brillouin -zone \nintegration was performed using the Monkhorst -Pack sampling scheme [ 25] over k-point \nmesh resolution  of 2π×0.0 33 Å-1. The formation energy E F of the alloy is calculated as:  \n𝐸𝐹=[𝐸(𝐶𝑜𝑚𝑍𝑟𝑛𝐵𝑝)−𝑚∙𝐸(𝐶𝑜)−𝑛∙𝐸(𝑍𝑟)−𝑝∙𝐸(𝐵)]/(𝑚+𝑛+𝑝)   (eq. 5) \nWhere  E(Co mZrnBp) is the total energy of  the Co mZrnBp alloy; E(Co), E(Zr) and E(B) are \nthe energy per atom of Co, Zr and B in the reference structures, which are HCP  Co, HCP  \nZr and α-boron , respectively . \n \nIntrinsic magnetic properties of the Co -Zr-B structures, such as magnetic moment and \nmagnetocrystalline anisotropy energy (MAE) were calculated using VASP code.  The \nspin-orbit coupling (SOC) is included using the second -variation procedure  [26]. We also  \ncalculated the MAE of the rhombohedral Co 5Zr structure by carrying out all -electron \ncalculations using the full -potential  (FP) LMTO method to check VASP  calculation \nresults. In addition, b y evaluating the SOC matrix elements, the anisotropy of orbital \nmom ent and MAE was resolve d into sites, spins and orbital pairs [ 27] to identify their \ncontribution to the magnetic properties . Curie temperature ( Tc) is checked for selected \nstructures using mean -field approximation  and more details can be found  in Ref.  [28].  \n \nResults and discussions  To validate the selection of the auxiliary classical potential , we first performed crystal \nstructure search  for the Co3ZrB 2 phase , whose crystal structure was well -characterized.  \nThe ground state structure of Co3ZrB 2 was successfully found  in the AGA search with \nabove  setup  [16]. Further calculat ions on  its magnetic properties  by DFT  showed  this \nphase is non-magnetic with zero  magnetic moment s. Therefore,  this structure  cannot be \nresponsible for the magnetic properties observed in th e Co-Zr-B system.  \n \nIn order to obtain practically useful magnet s, we then performed extensive  AGA searches \nfor Co-Zr-B with Co:Zr ratio around 5 and boron composition  less than  6 at %. The \ncontour map of the ir formation energies is plotted  in Fig. 1 , where  the compositions \nsearched in current  work  are represented by squares . It can be seen that near the Co5Zr \ncomposition  (Co, at  % ~ 83.3%)  there  is a local minimum in the energy landscape, which \nexplains the Co xZr (x ~ 5)  phases  obtained  in experiment s. For certain compositions at \nthe high energy areas , such as Co84Zr15B, and Co46Zr8B2, it is unlikely to synthesize such \ncompounds experiment ally. Among the compositions considered in Fig. 1, the lowest \nformation energy is found around Co 40Zr9B and  Co40Zr8B2, which are consistent with \nexperimental results since  most  of samples produced by experiments we re around these \ncompositions  [4-6]. In the following, structures and magnetic properties  of the Co -Zr-B \nalloys will be discussed respectively .  \nFIG. 1. Contour map of the formation energies in the Co -Zr-B system. Only partial \ncomposition range is considered and the squares represent the compositions  searched by \nAGA in the present work.  \n \nI. Structures  \nSeveral  low-energy boron -doped Co xZr structures obtained from our AGA search es are \nplotted  in Fig. 2 (a-d). Co and Zr atoms form the same building block as discovered in \nCoxZr [1 4], while boron atoms can either  substitut e Co atoms [e.g. Fig. 2( a)] or occupy  \ninterstitial positions  [e.g. Fig. 2(c)] in company with local distortions. In the Co 40Zr8B2 \nstructure  plotted in Fig. 2(b) , boron atoms  can be considered as  interstitial atoms in the \nCo5Zr structure  of high temperature phase , or as substitutional atoms  in the Co5.25Zr \nstructure  of low temperature phase , because  the main difference  between the Co 5Zr and \nCo5.25Zr structures  comes from  the different packing density of  one of the two pure Co \nlayer s [14]. In our previous  study , we also showed the layer -stacking  feature  in CoxZr \npolymorphs is frequently interrupted  to adjust  the strain due to the rippled hexagonal  Co \nlayer  [14]. Figure 2(d) shows a similar structure with boron atoms located  at the \ninterruption site.  \n \nFIG. 2.  Examples of the low -energy structures obtained from the AGA searches with \ncompositions of (a) Co 40Zr8B; (b) Co 40Zr8B2; (c) Co 42Zr8B2; (d) Co 40Zr8B. Unit cell of \neach structure is indicated by black lines. (e) Typical boron -centered clusters extracted \nfrom  the Co -Zr-B crystal structures. The label under each cluster represents the \nneighboring atoms of boron.  \n \nTo give a  better picture  of the local environment s of boron atoms, Fig. 2(e) listed several  \ntypical boron -centered clusters found in the Co -Zr-B struc tures. In general, the nearest \nneighbor distances for the B -Co and B -Zr pairs are about 2.1  Å and 2.6  Å, respectively. \nThe coordination number of the boron atoms is 7 or 8 , and the neighbor ing atoms are \nfound to be  Co or Zr atoms  in most cases . The effect of different boron positions on the \nmagnetic properties will be discussed later.  \n \nThe structure and glass formability in the Co -Zr-B alloy system ha ve been studied \nexperiment ally [29-31]. In the XRD  analysis  [29], the intensity of the crystalline peaks \nbecomes weaker and broader as the boron content increases , indicating the reduction of \nthe crystalline size in the samples. Amorphous and partially crystalline alloys have also  \nbeen observed in this system [ 2, 31]. From the AGA search, we found that all the low-\nenergy structures of Co -Zr-B have low symmetries (triclinic system) due to the \ndistortions induced  by the doping of boron atoms. Moreover , many different structures \nwere found to have closely competitive energies (within a few meV  per atom) , similar to  \nthe Co xZr binary system . Therefore, within the composition range plotted in Fig. 1, \ngrowing single  crystals of  Co-Zr-B alloy is difficult  and the sample is expected to have \nsmall grains and defects.  \n \nII. Magnetic properties  \nA. High -temperature Co5Zr phase revisited  \nIn our previous study  [14], the high -temperature rhombohedral  phase was assigned  to be \nresponsible for the magnetic hardness in the Co xZr alloys . The full potential calculation (GGA) showed it has a magnetic mome nt of around 1.0  μB/atom and MAE of 1.4 MJ/m3. \nThe rhombohedral structure , plotted in Fig. 3(a),  has a space group R32 (#155)  and 4 \ninequivalent  Co sites as indicated by different colors in Fig. 3. Views along c axis of the \ndifferent layers are plotted in Fig. 3(b). Among the four inequivalent  Co sites, two of \nthem  (Co1, Co3)  have nine-fold multipli city and the other two  (Co2, Co4)  have six-fold \nmultiplicity.  \n \nFIG. 3. (a) Crystal structures of the rhombohedral Co 5Zr with different Co sites presented \nby different colors. The lattice parameters of the structure are a=4.66  Å and c=24.0 Å. It \nhas one Zr site: 6c ( 0.0000 , 0.0000 , 0.4314 ) and four Co sites: Co1 9d ( 0.3300 , 0.0000 , \n0.0000 ), Co2 6c (0.0000, 0.0000 , 0.0795 ), Co3 9e ( 0.494 6, 0.0000 , 0.5000 ), and Co4 6c \n(0.0000, 0.0000 , 0.2549 ). (b) Views of layer I, II and III along c axis. In the plot of layer \nIII, all possible interstitial positions are grouped into 3 inequivalent sites based on \nsymmetry.  Unit cells of the crystal structures are indicated by the black boxe s. \nTo examine the contribution of different sites to the magnetic properties  of the \nrhombohedral phase , Fig. 4 shows the variations of orbital magnetic moments and \nrelativistic energy as functions of th e spin rotation. Rhombohedral Co5Zr has uniaxial \nanisotropy. By evaluating the SOC matrix element, we found th e Co3 site ha s in-plane \nmagnetic easy axes while all other Co sites , especially Co 4, support the uniaxial \nanisotropy. As shown in Fig. 4, the correlation between orbital moment and magnetic \nanisotropy is obvious. Co 1, Co2  and Co 4 sites have larger orbital magnetic moments \nalong the z axis while Co 3 has larger orbital magnetic moments when spin is along in -\nplane directions. The MAE calculated in LDA is smaller than the one calculated using \nPBE functional  [14]. By evaluating the SOC matrix elements, we found that this \ndifference mostly comes from the Co 1 site, whose contribution to MAE nearly disappear s \nin LDA. The MAE contributions from all other sites barely depend on the exchange -\ncorrelation functionals used in our calculations.  Above analysis indicates if the Co 3 site \ncan be modified , such as substitutin g Co 3 by other elements , the MAE  of the system  \ncould be improved .  \nFIG. 4. Variations of the orbital moment (a) and relativistic energy (b) as a function of \nspin quantization direction. Orbital moment and relativistic energy values are averaged \nover all atoms which belong to the corresponding inequivalent sites.  \n \nB. Boron doping on m agnetic moments  \nTo map  out the m agnetic moment s of the Co -Zr-B alloys , results  from VASP calculations \nwere collected  for all the compositions  presented in Fig. 1 . The results  of total moments  \nin the system are plotted in Fig. 5(a), and the partial  contributions from Co, Zr , and B \natoms are plotted in Fig. 5(b), (c), and (d) respectively . The total magnetic moment per \natom is calculated  as the moment of the whole system divid ed by the total number of \natoms, while  the moment contribution from atom type M (M=Co, Zr, or B) is c alculated \nas the moment from all the M atoms divided by the number of atom M. Results  plotted  in \nFig. 5 for each composition were averaged over ten lowest -energy structures  from the \nAGA searches . \n \nFIG. 5 (a) Contour map of the total magnetic moment per atom in the composition range \nstudied for Co -Zr-B; (b, c, d) contour plots of the partial contribution s from Co, Zr and B \natoms  to the magnetic moment respectively.  \n \nIt can be seen that the magnetization in the Co -Zr-B system mainly comes from the Co \natoms . Both the Zr and B atoms are an tiferromagnetically coupled to  the Co atoms.  As \nshown i n Fig. 5(a), the magnetic  moment of the system becomes smaller  with the \ndecrease of Co atomic composition , which can be explained by  two reasons. First, Zr  and \nB atoms give negative contribution to the total moment of the system . More Zr and B \natoms will lower the moment of the system.  Second, the Zr and B atoms suppress the \nmoment of Co, which can be seen from Fig. 5(b). The average moment of the Co atoms i s \ndecreased with the increase of the Zr, B compositions. In contrast to the Co moment, t he \nvariation of the moment in  Zr and B atoms  as the function of composition is more \ncomplicated and t here is no clear trend of how the magnetic moment s of Zr and B  chan ge \nwith composition. However, due to the small atomic percentages of the Zr  and B atoms \nand their small moments, total magnetic moment of the system is dominated by Co atoms  \nand varies in the same manner  as that of Co .  \n \nC. Boron doping on MAE  \nThe computational cost of calculating m agnetocrystalline anisotropy energy  can be \nenormous, which makes it infeasible to scan all the low -energy structures from AGA \nsearches, especially when the unit cells contain as many  as 100 atoms. In the following, \nthe effect of boron dopin g on MAE was investigated  based on the rhombohedral Co 5Zr \nstructure  and the knowledge of the preferred sites by boron atoms  from above analysis . \nAll calculations were performed using VASP.  To compare, the MAE of the rhombohedral structure calculated from VASP is about 1.6 MJ/m3, which is very close to \nthe result from FP calculations  (1.4 MJ/m3). \n \nFIG. 6 (a) Effect of boron doping on MAE. The calculations were based on the \nrhombohedral Co 5Zr structure. In the case of substitution, results of both the relaxed \n(solid triangles) and unrelaxed structures (empty triangles) are plotted, while for \ninterstitial positions, only results of the relaxed structures (solid squares) are plotted. The \npositions of doped boron atoms are  discussed in the main text. (b) Volume comparison \nbetween the original Co 5Zr structure (dash line) and boron doped structures after DFT \nrelaxations. The layer III with interstitial boron atom after relaxation is plotted as the \ninset.  \n \nWe have  showed  that the same structure motif found in Co 5Zr polymorphs also exists  in \nthe boron -doped Co xZr alloys , which explain s the origin of the high coercivity observed \nin Co-Zr-B alloys . Referring to the rhombohedral Co 5Zr structure plotted in Fig. 3, t he \nboron atoms appear to prefer substituting Co atoms in layer I or entering layer III as \ninterstitial atoms. Therefore, we scanned all the possibilities of adding up to 3  boron \natoms in to the conventional unit cell of the rhombohedral Co 5Zr structure (Fig. 3 a, 36 \natoms) and selected the one with lowest energy for each scenario to calculate MAE. All \nstructures have uniaxial anisotropy unless noted otherwise.  \n \nIn the substitutional case, all the Co atoms in layer I belong to the same Wyckoff position, \ntherefore the choices of substituting Co by B are limited. We found  while substituting \nmore than one Co atoms, lower energies were obtained with one B atom per layer, i.e. \ntwo B atom s substituting the same layer is not energetically favored for the 1 1 cel l \nstudied in this work . This can be explained  by the fact that  the size of the boron atom  is \nmuch smaller than that of Co atom . Large distortion s would be introduced if the boron \ndensity in one layer is too big. The  calculated  MAE results are plotted in Fig. 6 (a) for \nboth the relaxed and unrelaxed structures. It can be seen that the MAE increases \nsignificantly with the number of boron atoms for the unrelaxed structures, which \nconfirms our speculation  that replacing Co atoms at the Co3 site with other elements \nwithout modifying  the structure can improve the MAE . However, after structure \nrelaxations, MAE values  of the boron substituted structures  become  slightly smaller than the original Co5Zr structure . From volume comparison s plotted in Fig. 6(b), we can see \nthe relaxation changes the structure noticeably. T he changes in the  environments of Co \natoms cause the change of their electronic configuration and contributions to the MAE.  \n \nIn the interstitial case, there are three inequ ivalent positions in each Co1 layer where \nboron atoms can occupy, as indicated in Fig. 3(b). We scanned all the possibilities of \nadding up to 3 boron atoms and plotted  the MAE results in Fig. 6(a). Considering that  \ninterstitial defects usually introduce mu ch larger distortions to the neighboring atoms , we \nonly calculated the MAE of  the relaxed structures.  We note again that adding one boron \natom in to each layer III  gives more competitive  energy  and the relaxed structure of the B -\nCo mixed layer is plotted as  the inset of Fig. 6(b) . The MAE data shows the interstitial \nboron atoms  in the Co1 layer  do not change the MAE much compared with the high-\ntemperature Co5Zr phase . \n \nAlthough a bove calculations were performed on model s that were created based on the \nrhombo hedral Co 5Zr structure , the results are representative due to the consideration of  \nthe preferable positions of boron. In our previous study, we showed in the low \ntemperature Co 5.25Zr phase where extra Co atoms packed in layer III  to form the  \northorhombic phase , the MAE is much lower than the high temperature Co 5Zr \nrhombohedral phase  [14]. However, if the extra atoms are boron atoms instead, such  as \nFig. 2(b) , the MAE is expected to be close to the rhombohedral Co5Zr from above \nanalysis. Meanw hile, when the density of boron substitution to the Co 3 site is much smaller, such as Fig. 2(a), the distortion introduced to the neighboring atoms will be \nsmaller , thus there exists a great chance to increase the anisotropy . Finally , we calculated \nthe Curie temperatures for the model structures discussed above  and it shows that  the \nchange in Curie temperature due to  boron addition is not significant. The calculated Curie \ntemperature  is around 700 K which is  high enough for practical use.  \n \nConclusion  \nIn summary,  we studied the Co -Zr-B system using AGA method and first-principles \ncalculations. We noted that  the Co 3ZrB 2 phase is paramagnetic and cannot be responsible \nfor magnetic hardness. Near the Co 5Zr composition, the Co and Zr atoms in the structures \nof Co-Zr-B share the same structural motif as  discovered in the CoxZr polymorphs , while  \nboron atoms can a ppear both a s substitutions for Co atoms or  in the interstitial positions . \nBased on the AGA results , we constructed  the formation energy and magnetic moment \ncontour maps for partial composition range of the Co -Zr-B system, which can be used as \nguidance to adjust the experiment al processing  to further optimize  the magnetic \nproperties .  \n \nWe believe the h igh coercivity observed in the ternary alloy  system  origin s from the Co -\nZr layer packing  feature , as in the high temperature Co 5Zr rhombohedral phase . Through \nthe MAE calculations on Co -Zr-B model structures, we found both substitutional and \ninterstitial boron atoms give similar magnetic anisotropy energies  as the original rhombohedral Co 5Zr structure. Our calculations provide insight  into the significant  \nimprove ment of  the MAE  in Co -Zr system  through chemical doping.  \n \nAcknowledgement  \nThis work was supported by the US Department of Energy -Energy Efficiency and  \nRenewable Energy, Vehicles Technology Office, PEEM program, and by the US  \nDepartment of Energy, Basic Energy Sciences, Division of Materials Science and \nEngineering. The research was performed at the Ames Laborato ry, which is operated for \nthe U.S. DOE by Iowa State University under Contract No. DE -AC02 -07CH11358. This \nresearch used resources of the Oak Ridge Leadership Computing Facility (OLCF) in Oak \nRidge, TN  and the National Energy Research Scientific Computing Center (NERSC) in \nBerkeley, CA . \n \nReference  \n1. B. Balamurugan, B. Das, W. Y. Zhang, R. Skomski, and D. J. Sellmyer, J. Phys.: \nCondens. Matter 26, 264204 (2014).  \n2. A. Mitra Ghemawat, M. Foldeski, R.  A. Dunlap, and R. C. O’Handley, IEEE \nTrans. 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Alloys Compounds 450, 245 \n(2008).  \n "    },    {        "title": "1504.07349v3.Surface_termination_dependent_magnetism_and_strong_perpendicular_magnetocrystalline_anisotropy_of_a_FeRh__001__thin_film__A_density_functional_study.pdf",        "content": "arXiv:1504.07349v3  [cond-mat.mtrl-sci]  18 May 2015Surface-termination dependent magnetismandstrong perpe ndicular magnetocrystallineanisotropy\nofaFeRh(001)thin film: Adensity-functional study\nSoyoung Jekal1, S. H. Rhim1,∗Soon Cheol Hong1,†Won-joon Son2, and Alexander B. Shick3\n1Department of Physics and Energy Harvest-Storage Research Center,\nUniversity of Ulsan, Ulsan, 680-749, Republic of Korea\n2Samsung Advanced Institute of Technology,\nSuwon, 443-803, Republic of Korea\n3Institute of Physics, ASCR, Na Slovance 2,\nCZ-18221 Prague, Czech Republic\n(Dated: September 30, 2018)\nMagnetismofFeRh(001)filmsstronglydepends onfilmthickne ssandsurfaceterminations. Whilemagnetic\nground state of bulk FeRh is G-type antiferromagnetism, the Rh-terminated films exhibit ferromagnetism with\nstrongperpendicularMCAwhoseenergy+2.1meV/ /squareistwoordersofmagnitudegreaterthanbulk3dmagnetic\nmetals, where /squareis area of two-dimensional unit cell. While Goodenough-Kan amori-Anderson rule on the\nsuperexchange interaction is crucial in determining the ma gnetic ground phases of FeRh bulk and thin films,\nthe magnetic phases are results of interplay and competitio n between three mechanisms the superexchange\ninteraction, the Zener-type direct-interaction,and magn etic energygain.\nPACS numbers: 68.37.Ef,75.70.Tj,75.70.Rf\nFeRh alloys have attracted significantly because of their\nvarious intriguing physics phenomena including magneto-\ncaloric effect and huge magnetoresistance[1–4]. Transi-\ntionbetweenantiferromagnetism(AFM) andferromagnetism\n(FM) occurs above room-temperatureabout 350 K and ultra-\nfast phase transition of magnetic phases in the FeRh alloys i s\ninducedbyfemto-secondlaser,whichhavedrawnmoreatten-\ntion due to possibility of applications for heat-assisted m ag-\nneticrecording(HAMR)[5–7].\nFurthermore, feasibility of room-temperature memories\nbased on the AFM spintronics has been successfully demon-\nstrated in the FeRh alloys very recently utilizing anisotro pic\nmagneto-resistance(AMR)ofthebistableAFMstates[8–10] .\nThisAFMspintronicshassomeadvantagesoverthatbasedon\nFM states[8–16] owing to the absence of stray magnetic field\nfrom the zero net magnetization and the insensitivity to the\nexternal magnetic fields. Regarding FeRh thin films, several\nexperimentshavebeenreported[17–19]. InsteadoftheG-ty pe\nantiferromagnetic (G-AFM) in bulk, some thin films exhibit\nFM statesstabilized at theinterfacewithmetal,while theF M\nstatesareunstableatinterfacewithoxide[17]. Interesti ngly,it\nwasreportedthatanelectricfieldofonlyafewvoltsisneces -\nsarytodrivetheAFM-FMtransitionfortheepitaxiallygrow n\nFeRh films on the ferroelectric BaTiO3 substrate[18]. Also,\nit was revealed that the spin orientation of the FeRh film on\nthe MgO (001) depends on the strength of lattice strain and\nmagneticstate[19].\nInthispaper,magnetismandmagnetocrystallineanisotrop y\n(MCA) of FeRh films are investigated using first-principles\ncalculations. It is found that magnetism and MCA are sig-\nnificantly affected not only by film thickness but also by the\nsurface-terminations. The Rh-terminated films are more sta -\nble in FM state by quite big energy differences relative to\nthe magnetic ground G-AFM state in bulk. Furthermore,\nthe Rh-termination exhibits strong perpendicular MCA of+2.1 meV/ /square, where/squareis two-dimensional (2D) unit cell\narea. The Fe-terminations, on the other hand, are in G-\nAFM states as in bulk. The strikingly different behavior of\nthese two terminations is explained mainly in the framework\nof Goodenough-Kanamori-Anderson (GKA) rule on the su-\nperexchangeinteraction[20,21]withtheZener-typedirec tex-\nchangeinteraction[22] alsotakenintoaccount.\nDensity functional calculations are performed using Vi-\nennaab initio Simulation package (VASP)[23]. Results by\nVASP, particularlyMCA, have been double-checkedwith the\nhighlyprecisefull-potentiallinearizedaugmentedplane wave\n(FLAPW) method[24]. Generalized gradient approximation\n(GGA) within projector augmented-wave(PAW) scheme[25]\nis employed for the exchange-correlation interaction. k\nmeshes of 24 ×24×24 and 24 ×24×1 in Monkhorst-Pack\nscheme are used for bulk and films, respectively. For wave\nfunction expansions, 500 eV is used for cutoff energy. Con-\nvergencewithrespecttocutoffenergyandnumberof kpoints\nare seriously checked. Magnetic structures of the bulk FeRh\nare illustrated schematically in Fig. 1(a)-(c) for A-, C- an d\nG-type AFM, respectively, where arrows represent the direc -\ntion of magnetic moments of Fe atoms. To account for the\nAFM structures, a tetragonal magnetic unit cell, c(2×2)in\nthexyplane and doubled along the zaxis, is taken. To in-\nvestigate thickness and surface-termination dependent ma g-\nnetismoftheFeRh(001)films,twoterminations,Fe-andRh-\nterminationwiththicknessfrom3-to15-MLhavebeentaken\nintoaccount,whereweallowrelaxationalongthe z-axiswhile\nfixingtwo-dimensional(2D)lattice constantasthe calcula ted\nbulkvalue(3.007 ˚A) ofthe G-AFM.\nFirst,magnetismofbulkFeRhispresented. AFMinG-type\nis found to be most stable from total energy calculations by\n48.3, 56.7, and 187 meV/Fe with respect to FM and AFM in\nA- and C-type, respectively. Since the C-AFM has the high-\nest energy, it will be excluded in forthcoming discussion if2\nFIG. 1: Schematic diagrams of magnetic structures of (a) A-, (b) C-\nand (c) G-AFM states of the bulk FeRh, and (d) exchange intera c-\ntion between Fe atoms in the G-AFM state. Red and blue spheres\nat the corners represent Fe atoms with different magnetic st ates, re-\nspectively, while grey spheres represent Rh atom. In (d), cy linder\nconnecting Fe 1and Fe2denote the 180◦superexchange, while red\ncylinders connecting Fe 1-Rh-Fe3do the 90◦superexchange. Blue\ndotted line which connects Fe 1-Fe4denotes the direct exchange in-\nteraction.\nnot necessary. Calculated lattice constant (3.012 ˚A) of the\nbulk FeRh in FM state is a little bit larger than that in the G-\nAFM (3.007 ˚A). Magnetic moments of Fe and Rh atoms in\nthe G-AFM (FM) are 3.158 (3.144) µBand 0.00 (1.041) µB,\nrespectively. Lattice constants and magnetic moments in th is\nwork are reasonably consistent with experiments[26, 27] an d\npreviouscalculation[28].\nIn FeRh (001) films, the interlayer spacing exhibits oscil-\nlatoryfeature: the topmostsurfacemovesdownwardwhereas\nthe subsurface layers do upward, and so on, where the inner-\nmost layerconvergetobulk behavior[See SupplementaryIn-\nformation for fully relaxed interlayer spacings of the Fe- a nd\ntheRh-terminatedfilm inFig. S1(a)and(b),respectively]. 7-\nor9-MLfilmsarethickenoughduetoshortmetallicscreening\nlength.\nTotal energy differences of the FM (A-AFM) and the G-\nAFM, which we denote as ΔE=EFM(EA)−EG, are pre-\nsented in Fig. 2(a) and (b) for the Fe- and Rh-termination,\nrespectively,whereFM, A, andG in subscriptsstand for FM,\nA-AFM, and G-AFM, respectively. As shown in Fig. 2(a),\nthe G-AFM is the most stable for the Fe-termination regard-\nless of thickness. We recall here that magnetism is so deli-\ncate physics phenomenonthat it can behave differentlyin re -\nduced dimension[29, 30]. As film gets thinner, ΔE decreases\nwhileΔE=36.6meV/Fefor9-MLisstill lessthanthatofbulk\nFIG. 2: The thickness dependent total energy differences of the FM\nand A-AFM states from that of the G-AFMstate, ΔE=EFM(EA)−\nEG; (a)the Fe-terminatedand (b) theRh-terminated films.\n(48.3meV/Fe). As shown in Fig. 2(b),magnetismof the Rh-\ntermination is totally different from the bulk. FM is favore d\nover G-AFM up to 15-ML by big energy difference whose\nabsolute value is larger than 70 meV/Fe for 7-ML. Notewor-\nthy,thereisacrossoverfromFMtoG-AFMwhenthenumber\nof layersexceeds17-ML.Furthermore,evenA-AFM is more\nstable than G-AFM for films thinner than 7-ML. Magnetic\nmomentsofFeandRhatomsinbothterminationsarelistedin\nTable 1 and 2 for their respective magnetic ground states, G-\nAFM and FM. Thesurface layershave almost the same mag-\nnetic moments as the bulk unlike other surfaces of magnetic\nelements, Fe, Co, and Ni, where moments are strongly en-\nhanced.\nBefore we discuss magnetism the FeRh films, we provide\nhere a detailed analysis of bulk magnetism. egandt2gare\ndegenerate in G-AFM and FM due to the cubic symmetry,\nwhereas those in A- and C-AFM are no longer degenerate\nowing to the tetragonal magnetic structure (see SI Fig. S2).\nStrong hybridization between Fe and Rh dstates brings in\nalmost fully occupied (unoccupied) majority (minority) sp in\nstates of Fe dorbital, particularly egstates for all magnetic\nphases,whichresults inenhancedmomentsofFe atoms. The\nnearly half-filled band favors AFM as in bulk Mn and Cr. It\nis noteworthy that the majority spin bands of Rh are almost\nfully occupied in the FM similarly to Fe, but featureless in\nAFMstates.\nThe magnetic phase of FeRh alloy can be explained\nby interplay between three mechanisms - superex-\nchange interaction[20, 21], Zener-type direct-exchange\ninteraction[22], and magnetic energy gain. In the frame-\nwork of GKA rule, whether FM or AFM is preferred by\nsuperexchange interaction is explained by magnetic ion-\nligand-magnetic ion angle. Here we view Fe atoms as\nmagneticionsanddelocalized sandporbitalsofRhasligand\norbitals in GKA rule. Fig. 1(d) schematically illustrates\nmagnetic interactions between Fe atoms to be involved in\ndetermining magnetic structure. The GKA superexchange\ninteractions are shown as solid lines and Zener-type direct\ninteraction as a dotted line. In accord with GKA rule on\nmagnetic coupling, Fe 1prefers AFM coupling to Fe 2and\nFM coupling to Fe 3because angles of Fe 1-Rh-Fe 2and3\nFIG. 3: MCA energies, defined as EMCA≡E(→)−E(↑), of the\nFeRh(001)thinfilmsfor(a)Fe-and(b)Rh-terminatedfilms,r espec-\ntively. Solid(dashed) linesdenote MCA of FM(G-AFM)phase.\nFe1-Rh-Fe 3are 180◦and 109.5◦(which is close to 90◦),\nrespectively.\nOn the other hand, the interaction between Fe 1and Fe4is\nmoreorlessdirectsinceRhatomisnotmuchinvolvedinthis\ncoupling. Thehalf-filled egstatesdirectingalongtheprincipal\naxesaremoreinvolvedintheZenerdirectinteractionbetwe en\nFe1and Fe4comparedto the t2gstates, whichresultsin AFM\ncoupling[22]. Since the dstates are highly localized giving\nlittle wave function overlap, the Zener-type direct intera ction\nbetween Fe 1and Fe4must be weak. As a result of combina-\ntionofthesuperexchangeandthedirectinteractionsdiscu ssed\nabove,G-AFMismoststableamongotherAFMstates. Inthe\nFM states, on the other hand, there is magnetic energy gain\ndue to the considerable magnetic moment of Rh atom (1.041\nµB), which reduces total energy to a certain degree, hence a\nFM state is more stable than A-AFM and C-AFM states even\nthoughit isless stablecomparedto thegroundG-AFMstate.\nIn the Rh-termination, the 180◦superexchange interaction\ndisappears because of the absence of Fe layer above the Rh-\nterminatedsurface. The90◦superexchangeinteractionmakes\nthe A-AFM more stable than G-AFM. Instead, the magnetic\nenergy gain of the surface Rh atom plays a key role in sta-\nbilizing FM. Hence, FM is the magnetic ground state in the\nRh-terminated FeRh thin films. In the Fe-termination, on the\nother hand, G-AFM is most stable as in bulk since the 180◦\nsuperexchangeinteractionstill works.\nInFig. 3,calculatedMCAenergies, EMCA≡E(→)−E(↑),\nare presentedas functionof thickness for the Fe- and the Rh-\nterminated films in their for G-AFM and FM states. EMCA’s\nfor other magnetic states are also shown for comparison.\nFrom the definition of EMCA, positive (negative) value im-\npliesperpendicular(parallel)magnetizationtothesurfa cenor-\nmal. Interestingly,theRh-terminationinFMstateshowsqu ite\nstrong persistent perpendicularMCA regardlessof thickne ss,\nwhereas the Fe-termination exhibits in-plane MCA for all\nmagneticstates. In particular, EMCA=+2.1meV/ /squareof the Rh-\ntermination is greater than 3 dmagnetic metals such as bulk\nFe, Co, and Ni by two orders of magnitude and larger than\nmagnetic films by several factors. From the fact that both the\nFe-terminatedFM states and the Rh-terminatedG-AFM state\nshowparallelMCA,themagneticstatesarenotakeyfactorin\nFIG. 4: Projected density of states of dorbitals of 9-ML FeRh film\nin the Fe-termination (upper panels) and the Rh-terminatio n (lower\npanels)forFMandG-AFMstates. SandS-1indicatethetopsurface\nandsubsurface layer, respectively.\ndetermining MCA, but the surface termination is. Moreover,\nthesaturatedfeatureof EMCA=+2.1meV/ /squarewithrespecttothe\nthicknessimpliesthatmagnetismoftheRhsurfaceplaysake y\nroleindeterminingthestrongperpendicularMCA.Todiscus s\nthe role of the thickness and the surface termination on mag-\nnetism as well as MCA, we present density of states (DOS)\nof the Fe- and the Rh-terminated 9-ML film is presented in\nFig. 4, where left and right columns represent DOS’s of Fe\nand Rh atoms for better comparison as in Fig. S2. First and\nsecondrowsarefortheFe-terminatedfilminG-AFMandFM\nstates; third and forth rows are for the Rh-terminated film in\nFMandG-AFMstates. DOSfeaturesdonotdifferverymuch\nfrom the bulk, despite lifted degeneracies in egandt2gstates\natthesurfacesandsomechangesinsurfaceatoms. Inparticu -\nlar,DOS’sofsubsurfaceatomsareessentiallythesameasth e\nbulk. This result confirms that the surface atoms indeed play4\nthe key role in determining magnetism in film geometry, as\nsaturated EMCAwithrespecttothicknessimplies[see Fig.3].\nTo confirm and analyze calculated results on MCA, ad-\nditional calculations have been carried out using FLAPW\nmethodusingGGAfor5-,7-,and9-MLRh-terminatedfilms.\nResults by FLAPW method are quite consistent with VASP\nmethod: i) FM is much more stable than G-AFM states,\nii) magnetic moments of the surface Rh atoms are 1.107,\n1.106, and 1.107 (Compare with SI Table S2) for 5-, 7-, and\n9-ML, respectively, and iii) EMCA=+1.97, +2.41, and 2.10\nmeV//square(cf. Fig. 3), respectively. EMCA’s are further de-\ncomposed into individual atomic contribution and differen t\nspin-channels[31]. It is found that the surface Rh atom with\nstronger spin-orbit coupling[32–34] and the ↑↓channel play\na dominantrole in determiningMCA. Results using FLAPW\nadoptingconventionalvonBarth-Hedinlocalspindensitya p-\nproximation(LSDA)arealso listedin SITableS3.\nThe role of the surface Rh layer in determiningperpendic-\nularMCAiswellmanifestedinDOS:thelifteddegeneracyof\negstatesmakespeakfrom dz2moreprominentintheminority\nspinband,whichisjustabovetheFermilevel. Thispeaked dz2\nstate contributes significantly to perpendicularMCA throu gh\n/angbracketleftyz/zx,↑|LX|z2,↓/angbracketrightmatrix[31] intheRh-termination.\nIn order to elucidate the magnetic behavior when the\nFeRh(001) films are under an applied magnetic field or heat,\nwe present total energy of the Fe-terminated FeRh(001) thin\nfilm as a function of angle between magnetic orientations of\nthe closest Fe atoms.[See SI Fig. S3]. The energy barrier of\nthe AFM-to-FM transition under an applied magnetic field is\nreducedwithdecreasingofthickness. Thisinformationmig ht\nbe useful in designing spintronics devices such as an AFM\nmemory[8–10] andHAMR[5–7].\nIn summary, magnetism of the Fe- and the Rh-terminated\nFeRh (001) are studied for variousfilm thicknesses. The ori-\ngin of stability of G-AFM in bulk is well explained in the\nframework of Goodenough-Kanamori-Andersonrules on the\nsuper-exchange interaction, where subsidiary Zener-type di-\nrect exchange interaction and magnetization energy are als o\ntaken into account. The thickness and the surface termina-\ntion turn out really significant as the two terminations give\ndifferent magnetic ground state. The Fe-termination is sta -\nbilized in G-AFM as in bulk, while the energy difference\nbetween G-AFM and FM is greatly reduced. On the other\nhand, the Rh-terminationstrongly prefersFM when films are\nthinner than 15-ML. Furthermore, the Rh-termination in FM\nstate exhibits quite strong perpendicular MCA, 2.10 meV/ /square\n, which is greater than bulk 3 dconvention magnetic metals\nbytwoordersofmagnitudeandlargerthan3 dmagneticfilms\nby several factors. Utilizing the thickness and the surface -\nterminationdependentmagnetismof FeRh film, highlydesir-\nable materials can be tailored as required by spintronics de -\nvices.\nThis work was supported by the Priority Research Cen-\nters Program (2009-0093818) and the Basic Science Re-\nsearch Program (2010-0008842) through the National Re-\nsearchFoundationfundedbytheMinistryofEducationofKo-rea. ABS acknowledges support from the Czech Rep. grant\n(GACR14-37427G).\n∗Email address: sonny@ulsan.ac.kr\n†Email address: schong@ulsan.ac.kr\n[1] M. Manekar andS.B.Roy, J.Phys.D: Appl. Phys. 41, 192004\n(2008).\n[2] S.A.Nikitin,G.Myalikguliev,M.P.Annaorazov, A.L.Ty urin,\nR. W. Myndyevb, and S. A. Akopyan, Phys. Lett. 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We di scuss in detail the first-order con-\nstantK1and show that the results reproduce previous results. We als o apply our method to\nNd2Fe14B compounds and demonstrate that the temperature dependenc ies of the magnetocrys-\ntalline anisotropy constants K1,K2, andK3are successfully computed.\n1Understanding and controlling magnetic anisotropy is a central issu e in developing mag-\nnetic devices such as spin RAMs, high-density storage, and perman ent magnets. The mag-\nnetic anisotropy at room temperature (or above) is important at a practical level and should\nbe evaluated in consideration of finite-temperature effects for ma terials whose magnetic\nproperties strongly depend on temperature. This letter present s appropriate expressions for\ncomputing temperature-dependent magnetic anisotropy consta nts (MACs) and physically\ninterpreting them.\nAn early study of the temperature dependence of the magnetocr ystalline anisotropy was\nfirst performed by Zener [1], which was followed by Callen and Callen [2, 3 ]. Considering an\ninsulating ferromagnet having localized moments, they phenomenolo gically derived a very\nsimple expression, the so-called Callen–Callen law: Kn(T) =Kn(0)[M(T)/M(0)]n(2n+1),\nwhereKn(T) is thenth-order MAC at the temperature T, andMis the magnetization. This\ntheoryisbasedontheconceptinwhichadecreaseintheMACoriginat esfromthebreakdown\nof the asphericity of the electron cloud of the magnetic ions due to t hermal excitation [4].\nBecause a spherical electron cloud means the absence of a specific direction for the magnetic\npolarization (note that this does not mean the disappearance of th e local moment), the\ntemperature dependence of the MAC can be related to that of the magnetization, as shown\nabove. After this pioneering work, there have been few theoretic al works on the temperature\ndependence of the MACs. Much later, Skomski et al. [5–7] discussed the Callen–Callen law\nby analyzing various magnetic materials andshowed that it can beapp lied to simple systems\nsuch as Fe and Co. However, they also pointed out that complex mag netic materials do not\nobeythislaw, suchasNdCo 5andNd 2Fe14B,whichexhibitaspinreorientation. Thusfar, the\nCallen–Callen theory is not always satisfactory quantitatively for th e range of applicability.\nIn recent years, theoretical trials have been conducted to evalu ate the magnetocrystalline\nanisotropy energy (MAE) at a finite temperature for spin Hamiltonia ns including the mag-\nnetic anisotropy term by a Monte Carlo approach [8, 9]. Further, fr om a technological\nviewpoint, micromagnetic simulations of the Landau–Lifshitz–Gilbert equation including\nthe random field describing thermal noise [8, 10–12] or the Landau– Lifshitz–Bloch equa-\ntion [13, 14] have been performed to observe the thermal stabiliza tion of the magnetization.\nUsing a first-principles approach, the MACs at T= 0 of transition-metal systems and\nrare-earth (RE) compounds have been calculated, and the result s were at a satisfactory\nlevel compared with the experimental data within the numerical acc uracy. For transition-\n2metal systems, considerable effort has been made to clarify the eff ects of practical factors\nsuch as the interfaces [15, 16] and distortion [17] in magnetic multilay ers and the chemical\ndisorder [18, 19] in alloys on the MAE. For the finite-temperature tr eatment within the\nfirst-principles approach, Staunton et al. [20] and Matsumoto et al. [21] have successfully\ndemonstrated the temperature dependencies of the MAE of FePt and YCo 5, respectively,\non the basis of the functional integral method. As for RE compoun ds, the MACs at T= 0\nhave been successfully evaluated by calculating the crystalline elect ric field (CEF) acting\non the 4f electrons in the RE ions using the first-principles approach [22–24]. In the 1970s,\nWijnet al. [25] and Buschow et al. [26] calculated the magnetic stabilization energy related\nto a Sm ion in SmCo 5from the Helmholtz free energy using ligand field theory. Recently,\nwe have theoretically studied the temperature dependence of K1andK2of Nd2Fe14B and\nwell-reproduced the experimental data for Ki(T) using appropriate crystalline field param-\neters [27]. The method employed in this work is to derive K1andK2numerically from the\nfree energy calculated as a function of the magnetization direction .\nAll approaches mentioned above have been rapidly developed in the p ast two decades ow-\ning to the considerable progress in numerical techniques and compu ter performance. How-\never, it is still difficult to measure Kidirectly, especially at a finite temperature, because of\nthe lack of an explicit quantitative expression for Ki. Given this background, we provide a\ngeneral expression for Kiin the present work, starting from the Hamiltonian describing the\nmagnetic system, and describe a method to evaluate Kiquantitatively for realistic ferro-\nmagnetic systems. In particular, for localized electronic systems s uch as RE compounds or\nsystems that can only be described by angular momentum operator s, we demonstrate that\nthe proposed method is useful and convenient for evaluating the t emperature dependence of\nKi(T).\nFirst, we derive the microscopic expression for the first-order MA C,K1(T), of crys-\ntals with N-fold rotational symmetry ( N≥3). Introducing the Helmholtz free energy,\nF(θ,φ) :=−β−1ln/summationtext\nnexp[−βEn(θ,φ)] and comparing F(θ,φ) with the phenomenological\nexpression [28, 29], we have the relation\nK1(T) =1\n2∂2\nθF(θ,φ)/vextendsingle/vextendsingle\nθ=0, (1)\nwhereβ:= 1/(kBT) is the inverse temperature corresponding to T, andEn(θ,φ) is an\nenergy eigenvalue of the Hamiltonian ˆH(θ,φ). Let us consider cases where the Hamiltonian\n3is explicitly written as a function of the polar angles θandφdenoting the direction of the\nmagnetization relative to the caxis (the hat denotes an operator):\nˆH(θ,φ) :=ˆh+m(θ,φ)·ˆD, (2)\nwherem(θ,φ) := (sinθcosφ,sinθsinφ,cosθ),andˆhis the angle-independent part. ˆDis an\noperator with the transformation property of\nˆC†\nNˆD±ˆCN= e±i2π\nNˆD±, (3)\nˆC†\nNˆDzˆCN=ˆDz, (4)\nwhereˆD±:=ˆDx±iˆDy, andˆCNis theN-fold rotational symmetry operator around the\nsymmetrical axis parallel to the caxis. The derivatives of F(θ,φ) are expressed as\n∂θF=/summationdisplay\nne−β(En−F)∂θEn, (5)\n∂2\nθF=/summationdisplay\nne−β(En−F)/bracketleftbig\n∂2\nθEn−β(∂θEn)2/bracketrightbig\n+β(∂θF)2, (6)\nin terms of the derivatives of En(θ,φ). As above in the purely theoretical treatment, the\nMACs can be described in terms of En(θ,φ) in the neighborhood of θ= 0. Introducing\nˆV(θ,φ) :=m(θ,φ)·ˆD−ˆDz, we divide the Hamiltonian ˆH(θ,φ) into the unperturbative part\nˆH0:=ˆh+ˆDzand the perturbative part ˆV(θ,φ). Here, we notice that ˆV(θ,φ)→0 in the\nlimitθ→0 and\n/bracketleftBig\nˆCN,ˆH0/bracketrightBig\n= 0. (7)\nTo obtain K1, we perturbatively calculate Enup to the second order in ˆV. We assume that\nthe unperturbed eigenvalues ǫnand eigenstates |n/angbracketrightare given, which satisfy the unperturbed\nSchr¨ odingerequation, ˆH0|n/angbracketright=ǫn|n/angbracketright. Assuming thatthevaluesof ǫnarenotdegenerate, the\nperturbative energyterms areexpressed as∆(1)\nn=/angbracketleftn|ˆV|n/angbracketright,∆(2)\nn=/angbracketleftn|ˆVˆPnˆV|n/angbracketright, whereˆPn:=\n/summationtext\nk/negationslash=n|k/angbracketright/angbracketleftk|/(ǫn−ǫk), and we have let En(θ,φ)≃ǫn+∆(1)\nn(θ,φ)+∆(2)\nn(θ,φ). Considering\nthe symmetries in Eqs. (3) and (7), we obtain /angbracketleftn|ˆD+|n/angbracketright=/angbracketleftn|ˆD+ˆPnˆD+|n/angbracketright= 0 such that\n∂θEn(θ,φ)|θ=0= 0 and ∂2\nθEn(θ,φ)|θ=0=−/angbracketleftn|ˆDz|n/angbracketright+[/angbracketleftn|ˆD+ˆPnˆD−|n/angbracketright+/angbracketleftn|ˆD−ˆPnˆD+|n/angbracketright]/2.\nThus, from Eqs. (1), (5), and (6), we obtain\nK1(T) =1\n2/summationdisplay\nne−β(ǫn−F0)/bracketleftbigg\n−/angbracketleftn|ˆDz|n/angbracketright+1\n2/angbracketleftn|(ˆD+ˆPnˆD−+ˆD−ˆPnˆD+)|n/angbracketright/bracketrightbigg\n,(8)\n4whereF0:=F(0,φ). This is one of main results used to compute K1(T) later in this letter.\nBefore discussing this expression, let us further transform it. If we find an operator ˆJ\nsatisfying\n[ˆJα,ˆJβ] = iǫαβγˆJγ, (9)\n[ˆJα,ˆDβ] = iǫαβγˆDγ, (10)\nwe can then eliminate the matrix elements of ˆD±with the help of ˆJ. Here,ǫαβγis the\nLevi–Civita tensor, the repeated Greek indices are summed for all c ases (α=x,y,z), the\ncondition in Eq. (9) means that ˆJis an angular momentum operator, and the condition\nin Eq. (10) geometrically means that ˆDis a vector operator for ˆJ. Using the identity\n/angbracketleftn|ˆD±|k/angbracketright=±(ǫn−ǫk)/angbracketleftn|ˆJ±|k/angbracketright±/angbracketleftn|[ˆJ±,ˆh]|k/angbracketright, we obtain the commutator form\nK1(T) =−1\n8/angbracketleft[ˆJ−,[ˆJ+,ˆh]+h.c.]/angbracketright+1\n4/summationdisplay\nne−β(ǫn−F0)/summationdisplay\nk/negationslash=n|/angbracketleftn|[ˆJ+,ˆh]|k/angbracketright|2+|/angbracketleftk|[ˆJ+,ˆh]|n/angbracketright|2\nǫn−ǫk,\n(11)\nwhere/angbracketleft···/angbracketrightdenotes the statistical average in ˆH0. This is another one of the main results\nthat is appropriate for describing physical pictures of the magnet ic anisotropy because the\ncommutators distill the essential features from the Hamiltonian. E quations (8) and (11) are\nequivalent and general for K1.\nFor example, let us apply these formulae to localized spin systems. We consider the\nsingle-spin Hamiltonian\nHspin:=−A(ˆSz)2−2Hex·ˆS, (12)\nwhereˆSis a spin operator; Ais the single-ion MAE; Hexis the exchange field parallel to\nthe magnetization direction, i.e., Hex=Hexm; and we assume the condition A <2Hex.\nHence, we find the correspondences ˆD→ −2HexˆSandˆh→ −A(ˆSz)2. Obviously, ˆSsat-\nisfies the conditions in Eqs. (9) and (10) as ˆJ. Then, the commutators are calculated\nas [ˆS+,−A(ˆSz)2] =A(ˆSzˆS++ˆS+ˆSz) and [ˆS−,[ˆS+,−A(ˆSz)2]] = 2A[ˆS2−3(ˆSz)2]. From\n5Eq. (11),\nKspin\n1(T) =−A\n2/bracketleftBigg\nS(S+1)−3S/summationdisplay\nM=−Se−β(ǫspin\nM−Fspin\n0)M2/bracketrightBigg\n+1\n4S/summationdisplay\nM=−Se−β(ǫspin\nM−Fspin\n0)/parenleftbiggA2(1−2M)2(S+M)(S−M+1)\nA(1−2M)−2Hex\n+A2(1+2M)2(S−M)(S+M+1)\nA(1+2M)+2Hex/parenrightbigg\n, (13)\nwhereǫspin\nM:=−AM2−2HexMis an energy eigenvalue of ˆH0\nspin:=−A(ˆSz)2−2HexˆSz, and\nFspin\n0:=−β−1ln/summationtextS\nM=−Sexp(−βǫspin\nM). At zero temperature, we have\nKspin\n1(0) =AS/parenleftbigg\nS−1\n2/parenrightbigg1\n1+A(S−1/2)/Hex. (14)\nIt is clear that this tends to AS2in the classical limit S→ ∞, as expected from Eq. (12).\nNow, we refer to the relation between Kspin\n1(T)/Kspin\n1(0) andS(T)/S(0). Here, the statis-\ntically averaged spin is defined by S(T) =/summationtextS\nM=−Sexp/bracketleftBig\n−β(ǫspin\nM−Fspin\n0)/bracketrightBig\nM. The Callen–\nCallen law states that Kspin\n1(T)/Kspin\n1(0) is [S(T)/S(0)]3. To consider this power law, we\nintroduce a temperature-dependent power\nα(T) :=ln[Kspin\n1(T)/Kspin\n1(0)]\nln[S(T)/S(0)]. (15)\nThe zero-temperature value is exactly obtained as\nα(0) =6+A(2S−3)/Hex\n2+A(2S−3)/Hex. (16)\nThus, our formula supports the Callen–Callen law: α(0)→3 in the limit A/Hex→0. How-\never, we observe a deviation from the law in the classical limit because α(0) monotonically\ndecreases to 1 with respect to S. Thus, the Callen–Callen law is valid under the condi-\ntionAS/H ex≪1. Figure (1) shows the temperature dependencies of the quantit ies with\nS= 1 andA/Hex= 0.1. We observe that α(T) slightly deviates from the Callen–Callen law\nbecause of corrections from the higher-order terms with respec t toA/Hex.\nThe other sample is a localized 4f-electron system. Let us consider a model in which 4f\nelectrons are in ligand and exchange fields [25, 28–30]:\nˆH4f:=λˆLf·ˆSf+ˆHcry+2Hex·ˆSf, (17)\n6whereˆHcryandHexrepresent, respectively, the CEF and exchange field acting on the\nelectrons; λis the strength of the spin–orbit interaction; and ˆLandˆSare the total orbital\nand total spin angular momenta of the electrons, respectively. As suming that Hex=Hexm,\nthecorrespondences are ˆD→2HexˆSfandˆh→λˆLf·ˆSf+ˆHcry. UsingˆJ→ˆSf, theconditions\nin Eqs. (9) and (10) are satisfied. Therefore,\nKf\n1(T) =−λ\n4/angbracketleftˆLf·ˆSf+ˆLz\nfˆSz\nf/angbracketrightf+λ2\n4/summationdisplay\nne−β(ǫf\nn−Ff\n0)/summationdisplay\nk/negationslash=n/vextendsingle/vextendsingle/vextendsingle/angbracketleftn|(ˆL+\nfˆSz\nf−ˆLz\nfˆS+\nf)|k/angbracketrightf/vextendsingle/vextendsingle/vextendsingle2\n+(n↔k)\nǫfn−ǫf\nk.\n(18)\nAt first glance, we can understand that Kf\n1vanishes in absence of the spin–orbit interaction.\nNow, we purturbatively evaluate Eq. (18) with respect to λto the second order under the\nassumption that the CEF has tetragonal symmetry. The straight forward calculation gives\nKf\n1(T)≃λ2\n4/summationdisplay\nne−β(εn−f)/summationdisplay\nk/negationslash=n4∆Lz\nnk−∆L+\nnk−∆L−\nnk\nεk−εn, (19)\nwhere(ˆHcry+2HexˆSz\nf)|un/angbracketright=εn|un/angbracketright,f:=−β−1ln/summationtext\nnexp(−βεn),and∆Lα\nnk:=|/angbracketleftun|ˆLz\nfˆSα\nf|uk/angbracketright|2−\n|/angbracketleftun|ˆLx\nfˆSα\nf|uk/angbracketright|2. This is a version of a localized spin system for one in itinerant electron ic\nsystems [31–34]. Here, it is noteworthy that we can also regard ˆJf:=ˆLf+ˆSfasˆJbecause\n[ˆLα\nf,ˆDβ] = 0. This choice leads to another form\nKf\n1(T) =−1\n8/angbracketleft[ˆL−\nf,[ˆL+\nf,ˆHcry]+h.c.]/angbracketrightf+1\n4/summationdisplay\nne−β(ǫf\nn−Ff\n0)/summationdisplay\nk/negationslash=n|/angbracketleftn|[ˆL+\nf,ˆHcry]|k/angbracketrightf|2+(n↔k)\nǫfn−ǫf\nk,\n(20)\nwhere we have used [ ˆJf,λˆLf·ˆSf] =0and [ˆSf,ˆHcry] =0. Thus, we can easily confirm that\nKf\n1(T) = 0 when ˆHcry= 0. The commutators can be calculated by representing ˆHcryin\nterms of spherical tensor operators constructed from ˆLfon the basis of the Wigner–Eckart\ntheorem. For simplicity, let us consider the minimum case that\nˆHcry=C0\n2/bracketleftBig\n3ˆLz\nf2−ˆLf2/bracketrightBig\nandHex→ ∞ (21)\nforλ→ ∞, whereC0\n2is a constant encoding a CEF. The commutators are calculated as\n[ˆL−\nf,[ˆL+\nf,ˆHcry]] =−C0\n2[2ˆL2\nf−6ˆLz\nf2], and the second term in Eq. (20) vanishes in the limit\nof a large exchange field such that Kf\n1(T) =3\n2C0\n2/angbracketleftˆL2\nf−3ˆLz\nf2/angbracketright. Then, replacing ˆLfbyˆJf\n7on the basis of the equivalent-operator technique [35, 36] for λ→ ∞, we have Kf\n1(T) =\n3\n2B0\n2/angbracketleftˆJ2\nf−3ˆJz\nf2/angbracketright. Therefore, for light RE elements, our formula reproduces\nKf\n1(T) =−3J(J−1/2)B0\n2, (22)\nwhereB0\n2is a CEF parameter, and Jis a total angular momentum [28]. Note that in the\nlimit ofHex→0, we observe that Kf\n1→0 because the first and second terms in Eq. (20)\ncancel.\nFinally, we demonstrate numerical calculations for the temperatur e-dependent MACs\n(K1,K2, andK3) of Nd 2Fe14B. Following the derivation of Eq. (8) and purturbatively calcu-\nlatingEn(θ,φ)uptothefourthorder, themicroscopic expressions of K2andK3areobtained\nas\nK2=1\n4/summationdisplay\nne−β(ǫn−F0)[−HexC(1)\nn+4Hex2C(3)\nn\n+4Hex3(C(1)\nnC(5)\nn−C(4)\nn)−4Hex4(C(2)\nnC(5)\nn−C(6)\nn)]\n+β\n2/bracketleftBigg\nK12−/summationdisplay\nne−β(ǫn−F0)(HexC(1)\nn−Hex2C(2)\nn2)/bracketrightBigg\n, (23)\nK3= 2Hex4/summationdisplay\nne−β(ǫn−F0)C(7)\nn, (24)\nwherewehavedefinedtheconstantsas C(1)\nn:=/angbracketleftn|ˆSz\nf|n/angbracketright,C(2)\nn:=/angbracketleftn|(ˆS−\nfˆPnˆS+\nf+ˆS+\nfˆPnˆS−\nf)|n/angbracketright,\nC(3)\nn:=/angbracketleftn|ˆSz\nfˆPnˆSz\nf|n/angbracketright,C(4)\nn:=/angbracketleftn|(ˆS−\nfˆPnˆSz\nfˆPnˆS+\nf+ˆS+\nfˆPnˆSz\nfˆPnˆS−\nf+ 2ˆS−\nfˆPnˆS+\nfˆPnˆSz\nf+\n2ˆS+\nfˆPnˆS−\nfˆPnˆSz\nf)|n/angbracketright,C(5)\nn:=/angbracketleftn|(ˆS−\nfˆPn2ˆS+\nf+ˆS+\nfˆPn2ˆS−\nf)|n/angbracketright,C(6)\nn:=/angbracketleftn|(ˆS+\nfˆPnˆS+\nfˆPnˆS−\nfˆPnˆS−\nf+\nˆS−\nfˆPnˆS−\nfˆPnˆS+\nfˆPnˆS+\nf+ˆS+\nfˆPnˆS−\nfˆPnˆS+\nfˆPnˆS−\nf+ˆS−\nfˆPnˆS+\nfˆPnˆS−\nfˆPnˆS+\nf+2ˆS+\nfˆPnˆS−\nfˆPnˆS−\nfˆPnˆS+\nf)|n/angbracketright,\nandC(7)\nn:=/angbracketleftn|ˆS+\nfˆPnˆS+\nfˆPnˆS+\nfˆPnˆS+\nf|n/angbracketright. Figure2 shows thetemperature dependencies of the\nMACs for Nd 2Fe14B calculated using Eqs. (8), (23), and (24). Here, we have not rep eated\nthe discussion of the comparison with experimental results (see Re f [27]). See Ref. [25, 28]\nfor detailed computational methods for solving the eigenproblems. The expressions have\nthe following advantages. (I) There are no calculation parameters . In contrast, a mesh pa-\nrameter is necessary for finite-difference calculations of Eq. (1). Although there are integral\ntechniques using Fourier series, thespherical harmonics expansio n [37], andother expansions\nin terms of various complete sets, mesh parameters are also neede d. (II) The calculation\ndiagonalizing the Hamiltonian matrix only needs to be performed once. If finite-difference\nor integral methods are used, the eigenvalues at the mesh points n eed to be computed.\n8(III) For materials with small MACs, high-precision results can be ex pected because the\ncancellation of significant digits caused by finite-difference calculatio ns is avoided.\nIn summary, we have derived direct expressions for the MACs at a fi nite temperature\nfrom microscopic viewpoints. It is only assumed that the Hamiltonian is a linear function\nwith respect to the magnetization direction. We have discussed in de tail the first-order\nconstant, K1, and shown that the results reproduce previous results. Furthe rmore, we have\nsuccessfully demonstrated the calculation of the temperature-d ependent K1,K2, andK3for\nNd2Fe14B.\nThis work was supported by JST–CREST.\n∗dmiura@solid.apph.tohoku.ac.jp\n[1] C. Zener, Phys. Rev. 96, 1335 (1954).\n[2] E. R. Callen and H. B. Callen, Phys. Rev. 129, 578 (1963).\n[3] H. Callen and E. Callen, J. Phys. Chem. 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Keeler, Phys. Stat. Sol. (b) 64, 357 (1974).\n[38] M. Sagawa, S. Hirosawa, K. Tokuhara, H. Yamamoto, S. Fuj imura, Y. Tsubokawa, and\nR. Shimizu, J. Appl. Phys. 61(1987).\n10 0 1 2 3 4\n 0  1  2  3\nT/Hexα\nσ\nκ1κ1=σα\nFIG. 1. Calculated results of σ(T) :=S(T)/S(0),κ1(T) :=K1(T)/K1(0), and the power α(T) as\na function of the temperature TwithS= 1 and A/Hex= 0.1. The Callen–Callen law predicts\nα(T) = 3.\n-20 0 20 40 60 80\n 0  50  100  150  200\nTemperature [K]Ki[MJ/m3] Nd2Fe14B K2\nK3\nK1\nFIG. 2. Calculated magnetocrystalline anisotropy constan tsK1,K2, andK3as a function of\ntemperature. For Nd 2Fe14B,/angbracketleftˆr2/angbracketright= 1.001 a02,/angbracketleftˆr4/angbracketright= 2.401 a04,/angbracketleftˆr6/angbracketright= 12.396 a06,A0\n2= 295\nK/a02,A0\n4=−12.3 K/a04,A0\n6=−1.84 K/a 06,A4\n6=−15.9 K/a06,Hex= 350 K [28], λ→ ∞, and\nthe lattice constants a = b = 0 .881 nm, c = 1 .221 nm [38].\n11"    },    {        "title": "1506.00505v1.Evaluation_of__BH_max_and_magnetic_anisotropy_of_cobalt_ferrite_nanoparticles_synthesized_in_gelatin.pdf",        "content": "1 \n Accepted to be published in Ceramics International: 10.1016/j.ceramint.2015.05.14 8 \n \nEvaluation of  (BH)max and magnetic anisotropy of cobalt ferrite nano particles  \nsynthesized in gelatin  \n   \nA. C. Lima1, M. A. Morales1*, J. H. Araújo1, J. M. Soares2, D. M. A. Melo3, A. S. Carriço1. \n1Departamento de Física Teórica e Experi mental , UFRN, Natal, RN 59078 -970, Brazil  \n2Departamento de Física , UERN, Mossoró, RN 59610 -210, Brazil  \n3Programa de Pós -Graduação em Ci ência e Engenharia de Materiais , UFRN , Natal, RN 59078 -970, \nBrazil  \n*Corresponding author.  \nE-mail address: Marco.MoralesTorres @gmail.com ( M.A. Morales ) \nAbstract  \nCoFe 2O4 nanoparticles were synthesized using gelatin as a polymerizing  agent. Structural , \nmorpholog ical and magnetic properties  of samples  treated  at different  temperatures were  investigated  by \nX-ray diffraction , scanning electron microscopy, Mössbauer spectroscopy  and magnet ization \nmeasurements . Our results  revealed  that the sample s annealed at 623 K and temperatures above 973 K \nhave a cation distribution s given by  (Co 0.19Fe0.81)[Co 0.81Fe1.19]O4 and (Co 0.06Fe0.94)[Co 0.94Fe1.06]O4, \nrespectively . The particle  sizes varied from 73 to 296 nm  and the magnetocrystalline anisotropy, K 1, has \nvalues ranging from 2.60x106 to 2.71x106 J/m3, as det ermined from the law of approach to s aturation  \napplied to the M xH data at high field . At 5 K, the saturation magnetization , coercive field and (BH) max \nvaried from 76 to 9 5 Am2/kg, 479.9  to 278.5 kA/m  and 9.7 to 20.9 kJ/m3, respectively . The reported \nvalues are in good  agreement with near-stoichiometric cobalt ferrite  samples . \n \nKeywords:   Gelatin ; CoFe2O4 ;  Mössbauer spectroscopy;  Magnetic properties ; nanoparticles .  \n \n \n 2 \n 1. Introduction  \nRecently , there is a considerable interest  in research for n ew methods to synthesize no n  rare \nearth based hard magnetic materials. Accordingly, magnetic ferrites have drawn attention due their wide \ntechnological applications [1-2]. CoFe 2O4 is a well -known magnetic material with  high coercivity and \nremanence, moderate saturation magnetization, good  chemical stabilit y and mechanical hardness [ 3-4]. \nThese characteristics  make  it a candidate in applications such as magnetic recording, permanent magnets , \nmagneto -hyperthermia, magnetic drug delivery  and magnetic resonance imaging [ 5-6]. However , to \nobtain Co ferrites  for these  applications  is important to optimize  their chemical composition, structure and \nmagnetic  properties . These  properties  are sensitive to shape  and particle  size as well as  the occupancy of \ntetrahedral and octahedral  sites by the metal ions [ 7].  \nSeveral  methods to synthesize CoFe 2O4 nano particles have  been extensively studied , such as sol -\ngel process , thermal decomposition, co -precipitation, microemulsions, hydrothermal, and combustion  \nreaction  [8-10]. Among these , solution -phase chemical methods have att racted attention  because allow to \nprepare  nanocrystalline materials with high purity and controlled particle size  at relatively low \ntemperatures.  \nRecent works  have reported  a wet chemical method using gelatin as a polymerizing  agent  [11-\n13]. Gelatin is a protein produced by partial hydrolysis of collagen extracted from bon es, connective \ntissues, organs of animals such as cattle, pigs, and  horses. Because  its solubility in water and ability to \ninteract with metal ions in solution through the amino and carbo xylic groups present in its structure , \ngelatin  can be used as  a binder gel  to synthesize nanometer scale precursor particles . After the burning of \nthis gel, nanof errites phases can be obtained  at lower  temperature compared to  other methods [ 13].  \nCosta et al. [11] have synthesized CuFe 2O4 and CuFeCrO 4 nanoparticles using gelatin  for \napplication as ceramic pigments.  Their results showed that the onset of crystallization of samples \noccurred  at 625 K , but the spinel phase was obtained at temperatures  above 775 K. For the CuFe 2O4 and \nCuFeCrO 4 phases, t he crystallite  sizes ranged  from 50 to 70 nm and 47 to 60 nm, respectively . \nFurthermore, they observed that color of pigments varied as a function of composition and heat treatment . \nPeres et al. [12] prepared NiCo 2O4 nanoparticles  using Ni and Co nitrates and commercial gelatin [12]. \nThe results show ed the formation of NiCo 2O4 and Ni xCo1−xO phases  with sizes ranging from 2 0 to 100 \nnm, when the sample was annealed  from 633 to 1223  K. 3 \n Despite several studies found in the literature, we noted that there is no t any report on the \nsynthesis of  Co ferrites by chemical method using gelatin. Thus,  the aim of this work is to report  the \npreparation  of Co ferrite by this method  and study its chemical composition , structur e and magnetic  \nproperties .  \n  \n2. Experimental  \n Cobalt ferrite  nanoparticles  were synthesized by using Co(NO 3)2.6H 2O (VETEC), \nFe(NO 3)3.9H 2O (VETEC) and gelatin . The masses  of reagents  were 6.202  x 10-3 kg of Co(NO 3)2.6H 2O, \n17.221  x 10-3 kg of Fe(NO 3)3.9H 2O and 5  x 10-3 kg of gelatin . Initially, gelatin was dispersed  in distilled \nwater at 323 K. Then , Co(NO 3)2.6H 2O and Fe(NO 3)3.9H 2O were added to the above  solution  and kept \nunder stirring at 353 K until the formation of a viscous gel.  To obtain the precursor powder, the gel was \nheat treated  at 623 K for 3h , then, the precursor was finely crushed and again annealed  for 2  h at 973, \n1173  and 1273 K . The samples annealed  at 623, 9 73, 1173 and 1273 K were named S1 , S2, S3 and S3, \nrespectively.  \n Structural characterization of s amples  was carried out by  X-ray diffraction (XRD) using a Mini \nFlex II Rigaku  diffractometer and Cu Kα radiation.  XRD data were collected  in the 2θ range between  10° \nand 80° with a scan rate of 5° min-1 and 0.02° step. Crystalline phases were identified using the ICDD  \ndatabase. Relative concentration of phases , lattice  parameters and crystallite size were obtained by using \nthe Rietveld refinement method . The m orphology and particle size distribution  of samples were analyzed \non a TESCAN  MIRA3 field emission scanning electron  microscope  (SEM) . Mössbauer spectra were \nrecorded in transmission mode , at room temperature , using a spectrometer from Wiessel with a 57Co:Rh \nsource and activity of 25 mCi. Isomer shifts values are related to  -Fe. Magnetic measurements were \nperformed as a function of temperature ( 5 to 300 K ) and magnetic field (up to 10T) by using a \ncommercial VSM - Physical Properties Measurement System (PPMS) Dynacool from Quantum Design.   \n \n3. Results and discussion  \n The XRD patterns of CoFe 2O4 samples  treated  at different temperatures are shown in Fig. 1. \nSimilar di ffractograms  were observed for all samples and were  indexed to cubic spinel phase  (109044 -\nICSD ). Besides the spinel phase, a  small amount of CoO  (9865 -ICSD)  was identified as a secondary  4 \n phase for sample S1. We noted  that crystallite size increased with heat treat ment  temperature , this may \nhappen  as result of  coalescence of particles [14]. The l attice parameter s and crystallite  sizes varied  from \n8.373 to 8.387 Å , and 73 to 296 nm, respectively . The lattice parameter s are in good agreement  with those \nreported in  literature for cobalt ferrite [15]. Table 1 shows  the results obtained from the  Rietveld \nrefinement .  \n \nFig. 1. XRD patterns  of CoFe 2O4 samples  (a) S1, (b) S2, (c) S3 and (d) S4. Symbols are related to phases:  \n CoFe 2O4,  CoO  \n \n \n5 \n Table 1.  Refined parameters  of CoFe 2O4 samples  \nSamples  Phases  \n% Dm \nCoFe 2O4 \n(nm)  Lattice parameter  \nCoFe 2O4 \n(Å) \n CoFe 2O4 CoO  \nS1 94 6 73 8.373  \nS2 100 - 128 8.384  \nS3 100 - 189 8.387  \nS4 100 - 296 8.386  \n \nSEM images of samples  S1 and S4 are shown  in Fig. 2. The micrograph for sample S1 (Fig. 2a) \nrevealed  rounded particles with uniform size distribution.  Elongated particles  observed in figure 2b are \nmay be due to  coalescence  of small particles . The sample S4 show  agglomerated and pores , the porous \nstructure is formed by escaping gases during  heat treatment . In fact, ge latin provides a large amount of \norganic matter to the system, which may promotes the appearance of pores [11].  Inset in the top right  \nshow a histogram  indicating the particle size distribution . The distributions were  lognormal with average  \nparticle sizes of 53 and 308 nm. In both cases , the average  particle sizes were  of the same order as the \nones determined from  the XRD analysis . \n \nFigure. 2. SEM images  of CoFe 2O4 samples  (a) S1 and (b) S4. Insets  to the right  show histograms \nrevealing the l ognormal size distribution  \n \n(c) (d) 6 \n  Mössbauer spectra  (MS) recorded  at room temperature are shown in Fig. 3. The MS were fit ted \nto two sextets related to the Zeeman interaction between the hyperfine magnetic field and the nuclear \nmagnetic moment. These subspectra  are assigned to iron ions located in the tetrahedral (Fe-Tetr) and \noctahedral (Fe-Oct) coordination  symmetry . The isomer shift (IS), hyperfine magnetic field (Hhf) and \nquadrople splitting (QS) values are typical for cobalt ferrite samples [16] . In these sites, Fe3+ is \ncoordinated by four and six oxy gens. No doublet or singlet related to superparamagnetic particles or \nparamagnetic phases were  observed. When the annealing temperature  is increase d, we observe d a small \ndecrease in the Fe -Oct relative absorption area (RA ), accompanied by an equivalent increase in the RA  of \nFe-Tetr. The chemical formula  unit (f.u.)  of cobalt ferrites , (Co 1-xFex)[Co xFe2-x]O4 , can be obtained from \ndegree of inversion  parameter (x), which is defined as the  fraction  of tetrahedral sites occupied by Fe3+. In \nthe above formula, cations enclosed  in round and square brackets  are ions in tetrahedral  (A-sites)  and \noctahedral  (B-sites) sites , respectively.  Co ferrite is completely inversed when x=1. The degree of \ninversion  can be calculated from the ratio of sub spectra areas, RA(A)/ RA(B)=f A/fB(x/(2 -x)), where the \nratio of recoilless  fraction between octahedral and tetrahedral sites  at 300K  is f B/fA = 0.94  [16]. Table 2 \nshow the hyperfine parameters determined from the fits . The samples heat treated  at 623 K  and above 973 \nK, have RA(A)/RA(B) ratios of 0.73 and 0.97, respectively. The f.u. determined from these values w ere \nof Co0.19Fe0.81)[Co 0.81Fe1.19]O4 and (Co 0.06Fe0.94)[Co 0.94Fe1.06]O4, respectively . \nThe linewidth attributed to the Fe -Oct is larger  indicating different surroundings for Fe  ions \nlocated at this site.  In fact, the broadening of the B -site line was interpreted as being due to a distribution \nin Hhf caused by several configuration of Co and Fe nearest A -site neighbors [ 17]. Similar results have \nbeen observed in cobalt ferrite samples prepa red by using several cooling rates [17, 18]. 7 \n  \nFigure 3. Mössbauer spectra recorded at 300  K for CoFe 2O4 samples (a) S1, (b) S2 and (c) S4.  \n \nTable  2. Hyperfine  parameters of CoFe 2O4 samples  \nSamples  Fe Sites  Hhf  \n(T ) QS \n(mm/s)  IS  \n(mm/s)  RA \n(%) Linew idth \n(mm/s)  RA(A)/RA(B)  \nS1 Fe-Oct (B) \nFe-Tetr (A) 50.3 \n49.8 0.18 \n-0.07 0.48 \n0.18 58 \n42 0.60 \n0.40 0.73 \nS2 Fe-Oct (B) \nFe-Tetr (A) 49.3 \n48.6 0.16 \n-0.06 0.46 \n0.22 51 \n49 0.65 \n0.40 0.97 \nS4 Fe-Oct (B) \nFe-Tetr (A) 49.5 \n48.8 0.16 \n-0.06 0.45 \n0.22 51 \n49 0.67 \n0.41 0.97 \n \n \nZero f ield cool ed magnetization (ZFCM)  measurements versus temperature under a field of 1 0 T \nshowed a nearly constant magnetization value below 100  K, indicating that samp les have reached the  \nsaturated  regime . Figure 4c shows these measurements for samples S1, S2 and S4.  \nMagnetization versus magnetic field measurements were recorded at 5K and 300 K under a n \napplied  magnetic field of up to 10 T, figures 4a and 4b show  these measurements . Conversion magnetic \n8 \n units from CGS to IS are 1emu/g =1Am2/kg, and 1 Oe = 79.58 A/m.  For all the samples, the upper and \nlower branches of the hy steresis loops approach each other asymptotically at magnetic fields above 6 T. \nThe anisotropy constant was determined at 5 K by fitting the high field regions (H » coercive field ) to the \nLaw of Approach to Saturation (LAS), based on  the assumption that at sufficiently high field only \nrotational processes remain . According to the LAS , the magnetization as a function of magnetic field is \nusually writte n as [19]: \n        \n      \n     \n  \nwhere the numerical coefficient 8/105 h olds for random polycrystalline samples with cubic anisotropy \nand K 1 is the anisotropy constant.  The anisotropy con stant, K1, varied from 2.60 x106  to 2.71 x106 J/m3 in \nagreement with results obtained from magnetic torque measurements [ 20] and by fitting the LAS equation \nto MxH  magnetic measurements [21] in Co ferrite samples . The high K1 values are related to  the strong \nanisotropy of Co ferrite ions and to their presence in octahedral -B sites of the spinel structure  [20]. These \nfindings were confirmed  through the f.u. obtained from  the MS analysis,  which show  that the amount of \nCo2+ increases with annealing temperature . Table 3  shows t he saturation magnetization values obtained \nfrom the LAS fittings . The magnetic moment (M m) per f.u. can be determined from the Ms values . Thus, \nMm=WxMsx10-3 /(Nax9.274 x10-24), where W is the molecular weight of Co ferrite and Na is the \nAvogadro’s number . Therefore, the values 3.19 µB, 3.78  µB and 3.95 µB are the magnetic moments  per \nf.u. for samples S1, S2 and S4, respectively.  The higher magnetic moments for samples S2 and S4 reflects \nthe higher occupancy of Co ions in the octahedral sites.  These values are in agreement with the ones \nreported for Co ferrites  with different  stoichiometries and indicates  the spin and orbital magnetic \ncontribution of Co ions [17, 22]. 9 \n  \nFigure 4  – Magnetization of samples S1, S2 and S4  measured at (a) 5K and (b) 300K. (c) ZFCM  \nmeasurements under a magnetic field of 10 T . (d) Fit to the LAS of the high field of the MxH-2 data for \nsample S4. \n \nThe magnetic parameters obtained from hys teresis loops,  such as  saturation magnetization (M s), \nremanent magnetization (M r), coercivity (H c) and ratio M r/Ms, are shown in Table 3.  The values in \nparenthesis corr espon d to measurements performed at 300 K, while the other s correspond to \nmeasurements performed at 5 K. The crossover from the single domain to multi domain magnetic system  \nis related to the magnetocrystalline anisotropy. In CoFe 2O4, the particle size to form a multidomain \nsystem is about  70 nm [ 23], crystallites exceeding this size will have reduced Hc values. Although, r ecent \nstudies showed that CoFe 2O4 nanoparticles with a partial ly inverse structure ha d a maximum  value of Hc \nat size between 25 -30 nm, and small values above 45 nm [ 24]. Our sample s exhibited  decreasing Hc  \n10 \n values when part icle size increas ed, as expected for particles  with sizes above the critical diameter to \nform a multi domain regime.  \nTable 3 .  Magnetic parameters of Co  ferrite samples  \nSample  Ms  \n(Am2/kg) Mr  \n(Am2/kg) Hc  \n(kA/m ) Mr/Ms K1 \n(J/m3) (BH)max  \n(kJ/m3) \nS1 76 (72 ) 40 (40) 479.9  (167.1 ) 0.53 (0.56)  2.40x106 9.7 \nS2 90 (78)  58 (34)  369.3 (87.5 ) 0.64 (0.45)  2.65x106 18.8 \nS4 95 (82)  65 (33) 278.5 (65.3 ) 0.68 (0.40)  2.71x106 20.9 \nParameters in parenthesis are results from  measurements performed at 300 K. Other values are related to  \nmeasurements  recorded at 5 K. \n \n Figure 5a shows the B=  o (H + M) versus H curve for sample S4 measured at 5  K. An \nimportant  parameter for hard magnetic materials is the (BH)max value , which is the largest area of the \nrectangle that can fit in the demagnetizing B versus H curve at the second quadrant , see figure 5b. Figure \n5c shows the modulus  of the product B*H versus H , where  the maximum value determined from th is plot \nwas 2 0.9 kJ/m3. The (BH)max valu es show a strong increase from sample S1 to S2 and from sample S2 to \nS4 did not change to much . This result is in agreement with the large value  of remanence magnetization \nand high anisotropy constant for samples S2 and S4 . 11 \n  \nFigure 5  – Magnetization of sample S4. (a) B or J  versus magnetic field  H. (b) Second quadrant of the \ndemagnetizing curve. (c) Modulus (BH) versus magnetic field.  \n \n4. Conclusions  \n The synthesis using gelatin is an alternative route, simple and inexpensive to prepare oxide \nnanoparticles.  Cobalt ferrite nanocrystals with diameters ranging from 73 to 296 nm were prepared in \ngelatin. The samples  have diameter compatible wit h multi  domain magnetic particles. Mössbauer \nspectroscopy measurements showed that the annealing process of the cobalt ferrite nanocrystals affect ed \nthe average crystallite size but also the distribution of Fe ions within A - and B -sites in the spinel struc ture. \nWe found a degree of inversion close to 1.0 for samples heat treated  above 973 K . The anisotrop y values \nare in agreement with hard magnetic material s. The saturation magnetization showed magnetic moments \nper f.u. compatible with near-stoichiometric Co ferrite.  \n \n \n12 \n 5. Acknowledgments  \n A.C. Lima thanks to CNPq/PNPD (561256/2010 -1) by the financial support.  \n6- References  \n[1] M.A.G. Soler, E.C.D. Lima, S.W. daSilva, T.F.O. Melo, A.C.M. Pimenta, J.P. Sinnecker, R.B. \nAzevedo, V.K. Garg , A.C. Oliveira, M.A. Novak, P.C. Morais, Aging Investigation of Cobalt Ferrite \nNanoparticles in Low pH Magnetic Fluid, Langmuir 23 (2007) 9611.  \n[2] E. Manova, B. Kunev, D. Paneva, I. Mitov, L. Petrov, C. Estournés, C. D’Orléans, J. -L. Rehspringer,  \nM. Kurmoo,  Mechano -Synthesis, Characterization, and Magnetic Properties of Nanoparticles of Cobalt \nFerrite, CoFe 2O4, Chem.  Mater.  16 (2004)  5689.  \n[3] L.D. Tung, V. Kolesnichenko, D. Caruntu, N.H. Chou, C.J. O’Connor, L. Spinu, Magnetic properties \nof ultrafine  cobalt ferrite particles, J. Appl. Phys. 93 (2003) 7486.  \n[4] L. Ai, J. Jiang , Influence of annealing temperature on the formation, microstructure and magnetic \nproperties of spinel nanocrystalline cobalt ferrites,  Curr. Appl . Phys . 10 (2010 ) 284 . \n[5] D.-H. Kim, D.E. Nikles, D.T. Johnson, C.S. Brazel , Heat generation of aqueously dispersed CoFe 2O4 \nnanoparticles as heating agents for magnetically activated drug delivery and hyperthermia,  J. Magn . \nMagn . Mater . 320 ( 2008 ) 2390 . \n[6] Q. Song, Z.J. Zhang , Shape Control and Associated Magnetic Properties of Spinel Cobalt Ferrite \nNanocrystals,  J. Am. Chem . Soc. 126 ( 2004 ) 6164 . \n [7] S.R. Naik, A.V. Salker, S.M. Yusuf, S.S. Meena, Influence of Co2+ distribution and spin –orbit \ncoupling on the resultant magnetic properties of spinel cobalt ferrite nanocrystals, J. Alloys Compd 566 \n(2013) 54.  \n[8] M. Shi, R. Zuo, Y. Xu, Y. Jiang, G. Yu, H. Su, J. Zhong , Preparation  and  characterization  of \nCoFe 2O4 powders and films via the sol –gel method,  J. Alloys Compd . 512 ( 2012 ) 165 . \n[9] P. Kumar, S.K. Sharma, M. Knobel, M. Singh , Effect of La3+ doping on the electric, dielectric and \nmagnetic properties of cobalt ferrite processed by co -precipitation technique, J. Alloys Compd . 508 \n(2010 ) 115 .  \n [10] X. Zhang, W. Jiang, D. Song, H. Sun, Z. Sun, F. Li, Salt-assisted combustion synthesis of highly \ndispersed superparamagnetic CoFe 2O4 nanoparticles, J. Alloys Compd . 475 ( 2009 ) L34 . 13 \n  [11] A.F. Costa, P.M. Pimentel, F.M. Aquino, D.M.A. Melo , M.A.F. Melo, I.M.G. Santos , Gelatin \nsynthesis of CuFe 2O4 and CuFeCrO 4 ceramic pigments, Mater . Lett. 112 ( 2013 ) 58. \n[12] A.P.S. Peres, A.C. Lima, B.S Barros, D.M.A. Melo , Synthesis and characterization of NiCo2O4 \nspinel using gelatin as an organic precursor,  Mater . Lett. 89 (2012 ) 36. \n[13] M.A. Gabal, Y.M. Al Angari, A.Y. Obaid, A. Qusti , Structural analysis and magnetic properties of \nnanocrystalline NiCuZn ferrites synthesized via a novel gelatin method,  Adv. Powder Techn . 25 ( 2014 ) \n457. \n[14] M. Gharagozlou , Synthesis, characterization and influence of calcination temperature on magnetic \nproperties of nanocrystalline spinel Co -ferrite prepared by polymeric precursor method,  J Alloys Compd \n486 ( 2009 ) 660 .  \n[15] J.D. Baraliya, H.H. Joshi , Spectroscopy investigation of nanometric cobalt ferrite synthesized by  \ndifferent techniques, Vibrational Spectroscopy 74 ( 2014 ) 75. \n[16] G.A. Sawatzky , F. Van der Woude , A.H. Morrish, Recoilless -Fraction Ratios for Fe57 in Octahedral \nand Tetrahedra l Sites of a Spinel and a Garnet,  Phys. Rev. 183 ( 1969 ) 383.  \n[17] G. A. Sawatzky, F . Van der Woude, A. H. Morrish, Cation Distributions in Octahedral and \nTetrahedral Sites of the Ferrimagnetic Spinel CoFe 2O4, J. Appl . Phys.  39 (1968 ) 1204.  \n[18] M. R. De Guire, R.C. O’Handley, G. Kalonji, The cooling rate dependence of cation distributions in \nCoFe 2O4, J. Appl. Phys.  65 (1989) 3167.  \n[19] S. Chikazumi, Physics o f Ferromagnetism, Second ed ., Oxford University Press, Oxford, 1997 . \n[20] H. Shenker , Magnetic Anisotropy of Cobalt Ferrite (Co 1.01Fe2.00O3.62) and Nickel Cobalt Ferrite \n(Ni 0.72Fe0.20Co0.08Fe2O4), Phys. Rev.  107 (1957) 1246 . \n[21] A. Franco , F.L.A. Machado, V.S. Zapf, Magnetic properties of nanoparticles of cobalt ferrite at high \nmagnetic field, J Appl. Phys 110 (2011) 053913 . \n[22]- W. Gorter, Saturation magnetization and crystal chemistry of ferromagnetic oxides,  Philips  Res. \nRep. 9 (1954) 295 . \n[23] A.E. Berkowitz, W.J. Schuelle, Magnetic properties of some ferrite micropowde rs, J. Appl. Phys. 30 \n(1959) 134S . \n[24] K. Maaz, A. Mumtaz, S.K. Hasanain, A. Ceylan, Synthesis and magnetic properties of cobalt ferrite \n(CoFe 2O4) nanoparticles prepared by wet chemical route, J. Magn. Magn. Mater. 308 (2007) 289.  "    },    {        "title": "1506.03735v1.Magnetism_in_tetragonal_manganese_rich_Heusler_compounds.pdf",        "content": "arXiv:1506.03735v1  [cond-mat.mtrl-sci]  11 Jun 2015Magnetism in tetragonal manganese-rich Heusler compounds\nLukas Wollmann,1Stanislav Chadov,1J¨ urgen K¨ ubler,2Claudia Felser1\n1Max-Planck-Institut f¨ ur Chemische Physik fester Stoffe,\nN¨ othnitzer Strasse 40, 01187 Dresden, Germany and\n2Institut f¨ ur Festk¨ orperphysik, Technische Universit¨ a t Darmstadt, 64289 Darmstadt, Germany\nA comprehensive study of the total energy of manganese-rich Heusler compounds using density\nfunctional theory is presented. Starting from a large set of cubic parent systems, the response\nto tetragonal distortions is studied in detail. We single ou t the systems that remain cubic from\nthose that most likely become tetragonal. The driving force of the tetragonal distortion and its\neffect on the magnetic properties, especially where they dev iate from the Slater–Pauling rule, as\nwell as the trends in the Curie temperatures, are highlighte d. By means of partial densities of\nstates, the electronic structural changes reveal the micro scopic origin of the observed trends. We\nfocus our attention on the magnetocrystalline anisotropy a nd find astonishingly high values for\ntetragonal Heusler compounds containing heavy transition metals accompanied by low magnetic\nmoments, which indicates that these materials are promisin g candidates for spin-transfer torque\nmagnetization-switching applications.\nI. INTRODUCTION\nThe spintronics community demands materials with\nuniaxial anisotropy for spin transfer torque–random ac-\ncess memory (STT-RAM) applications, as well as for\nfundamental skyrmion-related research or in the field\nof magnetic shape-memory alloys [ 1]. Such materials\nare needed to improve the functionality and applica-\nbility of modern devices or device concepts within the\nscope of mass production or proof of practicality. Ad-\ndressing this request withing the class of compounds\nwith Heusler and Heusler-like structures, the task is ap-\nproached by means of relatives of familiar systems [ 2,3].\nThese relatives arethe family of Mn 2-based Heusler com-\npounds, the famous pioneering material and ancestor\nof which is Mn 3Ga [4,5]. The uniaxial magnetocrys-\ntalline anisotropy of Mn 3Ga has been calculated [ 6] and\nmeasured [ 7] several times on different occasions. It is\nthought that anisotropic materials such as these could\nconstitute the foundation for magnetic racetrack mem-\nory as proposed by Parkin et al. [8]. The indispensable\nperpendicularmagneticanisotropy(PMA)in(ultra-)thin\nstructures is best controlledby intrinsic propertiesrather\nthan by shape- or strain-induced anisotropy. As a re-\nsult, the perpendicular orientation of magnetization is a\ndesired property of the material. Recently, Mn 2-based\nHeusler systems, Mn 2YZ, were reconsidered as promis-\ning materials. Thus, considerable researchhas been done\non related systems. Mn 2NiGa in particular is a well-\nstudied material as it is directly related to Ni 2MnGa,\nwhichhasbeenthemoststudiedandbestunderstoodfer-\nromagneticshape-memoryalloysinceitsdiscovery[ 9,10].\nMn2NiGa, however, is a ferrimagnetic shape-memory al-\nloy that is theoretically linked to Ni 2MnGa through a\nsubstitution series, with a transition from ferro- to ferri-\nmagnetic ordering due to the increasing manganese con-\ntent. In addition to Mn 2NiGa, other Mn 2-based Heusler\nalloys have been synthesized and characterized or have\nbeen theoretically treated, such as Mn 2CoGa [11–15],\nMn2FeGa [16], and Mn 3Ga [4,7]. In addition to ma-terials in which Yis an atom from period IV (or the\n3dseries), equivalent systems with heavier constituents\nasYspecies have been investigated. Among these were\nMn2RuGa [17] (which has been found to have more or\nless random occupation of sites), Mn 2RhGa [18] (cubic,\ndisordered), Mn 2PtGa [19,20], and Mn 2PtIn [21], in the\ncontext of large exchange bias effects.\nAlthough a detailed study of a single material is an im-\nportant task that results in valuable knowledge, the in-\nclusion of cluster knowledge into a general concept cre-\nates comprehensive insight. The same intention that\nguided our previous work [ 15] motivated us to undertake\na similar approach in the current study, i.e., comprehen-\nsively studying a selected set of systems. We intend to\nunderstand the general trends governing the formation\nand magnetism of tetragonal materials for the aforemen-\ntioned applications.\nIn this paper, we show how the magnetism and\nthe atomic structure change withinthe Mn 2Y(3d)Ga,\nMn2Y(4d)Ga, and Mn 2Y(5d)Ga series. The trigger quan-\ntitiescausingthetetragonaldistortion, aswellasthecon-\nsequences of this distortion, i.e., the magnetocrystalline\nanisotropy(MCA) [ 3], willbe highlighted andplacedinto\nan appropriate context.\nII. CRYSTAL STRUCTURE\nHeusler alloys are nowadays informally divided into\ntwo structure types, the so-called regularandin-\nversetypes, referring to the original Heusler compound\nCu2MnAl as the reference system [ 22]. The materials\nfirst associated with the name ofFritz Heusler were cubic\nphases analogous to Cu 2MnAl, with the stoichiometry\nX2YZ. Fromthenon,similarmaterialswerethuslabeled\n“Heusler compound”, extending the original definition of\nHeusler compounds to the family of Heusler materials\nthat incorporates a variety of similar structures. These\nstructures are derived from the original compound, with\noccupation of the Wyckoff positions, 8 c, 4b, and 4a, in2\na) b) c) d) a\na/√2 }}\n4b 4a \n4c 4d \nFIG. 1. (Color online) The conversion from a (a) cubic Heusle r structure to a (b) tetragonal derivative phase is displaye d in\nterms of a non-displasive transformation for a system with t he general composition of XX′YZ(X,X′,Y- transition metals\nandZ- main-group element, marked as red, orange, blue, and green , respectively) within the fcc lattice. (c) Relationship of\nthe lattice parameters as atet=acub/√\n2.\nthe spacegroup (SG) 225 through the introduction of va-\ncancies or slight structural changes. These modifications\nalter the structure by breaking the inversion symmetry\nwhen going from a regularHeusler phase to an inverseor\nhalf-Heusler structure.\nHeusler SG 4d4c4b4a\nregular L21225Mn Mn YGa\ninverse Xa216Mn YMn Ga\nhalf C1b216Mn Mn /squareGa\nTABLE I. The structural relationship of the regularand\ninverse structure types for Mn 2YGa is shown, where\nX=X′=Mn and Z=Ga, as compared to Fig. 1. Only one\npossible configuration of a half-Heusler type is listed exem-\nplary.\nHeusler materials are generally understood to be in-\ntermetallic compounds, distinguishing them from gen-\neral intermetallics which are forming a broad range of\nsolid solutions, with no preferred but statistical occupa-\ntion of crystallographic sites. They are also set apart\nfrom other ionic or covalent compounds because Heusler\nsystems allow the formation of substitutional series of\nsingle sites. However, they maintain the character of an\nordered compound, and thus they are on the borderline\nbetween alloys and compounds. Some distinct systems\nexhibit a tendency to form alloys, nevertheless. The gen-\neral composition is given by XX′YZ, where the classical\ndefinition has been widened to incorporate quaternary\nmaterialswithin the familyofHeuslercompounds. Inthe\nrepresentation of SG 216 ( F¯43m), the structure contains\nfour highly symmetric Wyckoff positions: 4 d(3\n4,3\n4,3\n4),\n4c(1\n4,1\n4,1\n4), 4b(1\n2,1\n2,1\n2), and 4a(0,0,0). Depending on\nthe occupation of the crystallographic positions, two or-\ndering possibilities for ternary alloys ( X=X′) are ob-\ntained (Table I). In this study, gallium was chosen as\ntheZelement, whereas one manganese atom, X, occu-\npies position 4 d. The second manganese atom, X′, and\nthe other transition metal, Y, are located at 4 cor 4b,respectively, depending on the formation of the regular-\norinverse-type Heusler material, as shown in Table I.\nOther Heusler-related structures [ 23] are the tetragonal\nderivatives of the cubic parent phases, which have been\nwidely treated in the context of magnetic shape-memory\nalloys. The relationship and the unit cell transformation\nbetweencubic andtetragonalphasesisdepicted in Fig. 1.\nIt is seen that a conventional cubic unit cell can be de-\nscribed in terms of a tetragonal lattice exhibiting a c/a\nratio of√\n2. The cell parameters are interrelated accord-\ning toctet=ccub,atet=acub/√\n2. In this study, a set of\nMn2-based materials, including transition metals of peri-\nods IV, V, and VI, are considered, namely, Mn 2Y(3d)Ga,\nMn2Y(4d)Ga, and Mn 2Y(5d)Ga.\nIII. COMPUTATIONAL DETAILS\nThe numerical work was done within density func-\ntional theory as implemented in the all-electron FP-\nLAPW code, WIEN2k [24], employing the generalized\ngradient approximation (GGA) in the parametrization\nof Perdew, Burke, and Enzerhof as exchange-correlation\nfunctional [ 25]. The angular momentum truncation was\nset tolmax= 9 and the number of plane-waves deter-\nmined by RKmax= 9 to ensure well-converged calcu-\nlations. The energy convergence criterion for the self-\nconsistent field calculations was set to 10−5Ry, whereas\nthe charge convergence was set to 10−3. All calculations\nwhere done on a 20 ×20×20k-mesh. For a set of given\nc/aratios, the volumes were optimized and fitted to the\nBirch–Murnaghan equation of state [ 26,27]. From these,\nthe optimal ratio and volume were obtained and the lat-\ntice parameters evaluated. On the basis of the crystallo-\ngraphic details, complex magnetic properties such as the\nexchange parameters and the corresponding Curie tem-\nperatures( TC)werecomputedbymeansoftheKorringa–\nKohn–Rostoker (KKR) Green’s function method as im-\nplemented by the Munich SPR-KKR package [ 28]. The3\nc/a (cubic reference cell) \n0.8 1.0 1.2 1.4 1.6 1.8(a) (b) \n0.6\n0.4\n0.2\n0\n-0.2\n-0.40.6\n0.4\n0.2\n0\n-0.2\n-0.40.8 1.0 1.2 1.4 1.6 1.8\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4\nc/a (tetragonal reference cell) 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4Mn  CoGaMn  FeGa Mn  Ga\nMn  TiGa Mn  CrGaMn  ScGa Mn  VGa 2 2\n2 23 Mn  Ga\nMn  CuGaMn  NiGa 2 2\n2 23 ∆E     [ eV/ f.u. ] t-c \nFIG. 2. (Color online) Energetic response to volume-conser ving elongations and compressions of the crystal structure along\nthec-axis for the Mn 2Y(3d)Ga-series. Panel (a) includes systems with a valence electr on count NV(Y)≤7 (early transition\nmetals, ETM) and panel (b) shows the remaining combinations of Mn 2Y(3d)Ga with NV(Y)≥7, with so-called late transition\nmetals (LTM).\nangularmomentumexpansionwastruncatedfor lmax= 3\nwhich corresponds to the f-wave symmetry. The energy\nintegration was done on a complex energy mesh with 48\npoints along the integration path, using the Lloyd’s for-\nmula [29] for an improved estimate of the Fermi energy.\nThe computation of the exchange parameters is based\non the classical Heisenberg model, which was evaluated\nby means of the real-space approach [ 30]. This provides\nsite- and distance-dependent exchange between sites via\ninfinitesimal rotation of the magnetic moments at a par-\nticular site in real space. To account for distance depen-\ndence, an appropriate truncation of the cluster radius, r,\naround each atomic site had been chosen. This radius\nwas set to 3 .5alattice spacings to capture even small\ninteractions, as the largest contributions to the effective\nexchange constants are found for radii smaller than 1 .5a\nlattice spacings [ 15,31].\nIV. RESULTS\nA. Lattice Relaxation\nThe total energy E(c/a) as a function of the c/aratio\nwas calculated and the results are shown in Figs. 2–5.\nThe energy zero are defined with respect to the cubic\nparent compound, and consequently the energy differ-\nence for all phases can be compared easily. The case of\nMn3Ga is used as a benchmark and is repeatedly plotted\nin Figs.2and3. TableIIIcontains the numerically opti-\nmized lattice parameters. The study reveals that a large\nnumber of the herein-treated materials are most stable\nin their respective tetragonal structures, with c/a >√\n2.Tetragonally compressed structures are described by\nc/a <√\n2, whereas tetragonally elongated lattices are\ncharacterized by c/a >√\n2, as compared to the cubic\nparent or austenite phase. Elongation occurs with an\nincrease in the length of the c-axis, whereas the ab-\nplane is compressed, leaving the volume approximately\nunchanged. In this study, the volume of the unit cell,\nVcell, was optimized in addition to the c/a-ratio, and it\nwas found that no significant change occurred for most\ncases (Table II).\nFigs.2a and2b show the E(c/a) curves of the\nMaterial Ct/c Material Ct/c\nMn2ScGa 5 .08 Mn2OsGa 2 .32\nMn2CrGa 6 .11 Mn2IrGa 2 .44\nMn3Ga 9 .08 Mn2PtGa −3.13\nMn2FeGa 1 .93 Mn2AuGa −1.49\nMn2CoGa 1 .82 Mn2OsSn\nMn2NiGa −1.09 Mn2IrSn −1.94\nMn2CuGa 2 .37 Mn2PtSn −0.35\nMn2MoGa 3 .46 Mn2OsIn −0.36\nMn2RuGa 1 .63 Mn2IrIn −1.01\nMn2RhGa 1 .48 Mn2PtIn −0.86\nMn2PdGa −2.78 Mn3Ge 3 .98\nMn2FeGe 3 .80\nTABLE II. Relative volume change between the austenite and\nmartensite phase in percent, Ct/c= (Vtet−Vcub)·100/Vcub\nMn2Y(3d)Ga series for early transition metals (ETM,\nY= Sc,Ti,V,Cr) and late transition metals (LTM,4\nc/a (cubic reference cell) \n0.8 1.0 1.2 1.4 1.6 1.8(a) (b) \n0.8 1.0 1.2 1.4 1.6 1.8\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4\nc/a (tetragonal reference cell) 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.40.6\n0.4\n0.2\n0\n-0.2\n-0.40.6\n0.4\n0.2\n0\n-0.2\n-0.4Mn  Ga Mn  NbGa Mn  RuGa\nMn  ZrGa Mn  MoGa 2 22Mn  AgGa Mn  RhGa2 2Mn  PtGa2 2 3\nMn  OsGaMn  Ga\nMn  AuGa Mn  IrGaMn  PtGa\n2 Mn  TaGa2Mn  WGa 2 2\n2 23 ∆E     [ eV/ f.u. ] t-c \nFIG. 3. (Color online) Energetic response to volume-conser ving elongations and compressions of the crystal structure along\nthec-axis for the (a) Mn 2Y(4d)Ga and (b) Mn 2Y(5d)Ga series. Systems involving late transition metals and sys tems exhibiting\nat least a local tetragonal energy minimum are shown.\nY= Mn,Fe,Co,Ni,Cu), respectively. A preference\nfor tetragonal structures is seen for materials including\nLTM for Mn 2Y(3d)Ga, with Ybeing Mn, Fe, or Ni.\nIn the group of ETM systems, Mn 2ScGa was found to\nbe tetragonal, which is an exception. The situation for\nMn2Y(4d)Ga and Mn 2Y(5d)Ga is seen to be similar.\nIn contrast to Mn 2CoGa, for which a cubic structure\n(c/a=√\n2) is preferred, the systems with the same va-\nlence electron count ( NV= 26) involving heavier species\n(ruthenium and osmium) exhibit a global energy mini-\nmum for the tetragonal structure.\nDiscontinuities in the E(c/a) curves are observed for the\ncases of Mn 3Ga and Mn 2FeGa, resembling a first-order\ntransition, whereas E(c/a) for Mn 2NiGa is continuous\n(Fig.2b). Strongly composition-dependent modulated\nmartensitic phases [ 32,33] and premartensitic phases\n[34,35] have been experimentally produced; in combi-\nnation with first-principles calculations [ 36], the onset of\nthe martensitic transition has been thought to be initi-\natedbyadisplacementofatomicplanesorthogonaltothe\ncrystallographic c-axis of the tetragonalcell. Experimen-\ntally [35] and theoretically [ 36], it has been shown that\nphonon softening along [ ζ,ζ,0] exists in shape-memory\nmaterials, and thus the transition has been related to\nthe occurrence of the tetragonal distortion.\nThe approach undertaken in the current study resulted\nin elongated structures in those cases, which were unsta-\nble towards a tetragonal distortion. Additionally, global\nenergyminimafordistortedphaseswerefoundmostlyfor\ntheinversestructures, as depicted in Fig. 4. In contrast,\ncompressed variants possessing a global energy minimum\nwerenot observedin the Mn–GaHeusler family. The cal-\nculatedlatticedataandthefiguresshown(Figs. 2–3)lead\nto the conclusion that stable tetragonal structures mayonly be formed in the series of Mn 2YGa that includes\nLTMs, making them derivatives of inverseHeusler sys-\ntems. The information obtained from the lattice opti-\nmization in terms of the relative positions of the energy\nminima is condensed in Fig. 4, which gives the preferred\ncrystal structure visualized in the manner of the periodic\ntable.\nEach compound in the family of Mn 2YGa materials\nshown in this figure is depicted by one cell that symbol-\nizes a transition metal Yof the 3d,4d, or 5dseries. The\ncorresponding cell is built up by three subcells, which\nrepresent the two variants of chemical coordination (the\nfirst or second subcell) and the existence of a global en-\nergy minimum for c/a/negationslash=√\n2 (third subcell). The color\ncode symbolizes the energy level of a configuration on\nthe energy landscape relative to one another: dark green\n- global minimum, light green - local minimum, red - no\nminimum. In cases in which the investigated materials\ndid not exhibit a cubic minimum, the first two subcells\nare understood to be the type of coordination around\nthe 4dcrystallographic site: the symmetry of the coordi-\nnating shell is either centrosymmetric (in relation to SG\n225) or non-centrosymmetric (in relation to SG 216). As\nshown in a preceding publication on the cubic variants\n[15], systems involving ETMs adopt the L2 1-type struc-\nture, whereas compounds containing LTMs are found to\nhave the inverseHeusler structure (X a-type).\nInspection of Fig. 4in combination with Figs. 2and3\nreveals interesting details, such as the fact that tetrag-\nonal derivative phases of cubic Heusler alloys, implying\nthat a global energy minimum is present, are observed\nonly for a valence electron count of NV≥24. It is also\nclearly seen that the onset of the formation of tetrago-\nnally elongated structures evolves over the periods from5\n225 216 139 225 216 139 225 216 139 225 216 139 225 225 216 216 119 119 225 216 119 225 225 216 216 119 119 225 225 216 216 119 119 225 225 216 216 119 119 \n225 216 139 225 216 139 225 216 139 225 216 139 225 225 216 216 119 119 225 216 119 225 225 216 216 119 119 225 225 216 216 119 119 225 225 216 216 119 119 \n225 216 139 225 216 139 225 216 119 225 216 119 225 216 119 225 216 119 225 216 119 Y Zr Zr Zr Nb Nb Nb Nb Nb Mo Mo Mo Mo Mo Mo Mo Tc Tc Tc Tc Ru Ru Ru Ru Ru Ru Rh Rh Rh Rh Rh Pd Pd Pd Pd Pd Ag Ag Ag Ag Ag Ag Ag Ag \nTa Ta Ta Ta Ta WWWWW Re Re Re Re Re Re Os Os Os Os Os Ir Pt Pt Pt Pt Au 225 216 139 Sc Sc Sc Sc Sc Sc Sc Ti Ti V Cr Cr Cr Cr Cr Cr Mn Co Ni Ni Ni Ni Ni Ni Cu Fe Fe Fe Fe Fe 225 216 139/119 \nL2 \nL2 X Tet Tet Tet Tet Cub a\nXa1\n1Hf Hf Hf Hf \nFIG. 4. (Color online) Schematic overviewof the preferred s ite occupancy and crystal structure of Mn 2YGa Heusler compounds.\nStable, metastable, and instable lattices are marked by dar k-green, light-green, and red subcells, respectively.\nleft to right and from lower to higher NV.\nThis leads to the question of the mechanism behind this\ndistortion, which we are goingto approachin Sec. IVA1.\nThe lattice parametersfor all cubic compounds arefound\nwithin a range of ∆ ac/√\n2= 0.35˚A for the Ga series. The\nvalues increase from the borders of the series towards\nthe middle of the range. The same behavior is found\nfor tetragonal compounds, where the range spans from\natet= 3.68 to 4.11 (∆atet= 0.43˚A) andctet= 6.91 to\n7.48(∆ctet= 0.54˚A), whereasMn 2ScGaisanexception.\nThec/acoordinate exhibits inverted behavior, decreas-\ning from the middle to the left and right borders of the\nseries. The overallsimilarity of the lattice data opens the\npossibility of intermixability with each other and thus\ntunability of the whole class of materials. Therefore, the\nmagnetization may be adjusted over a large range, al-\nlowing for the formation of tetragonal compensated fer-\nrimagnets via adequate substitution.\nHaving discussed the structural trends as a function\nof the valence electron change in the d-electron sys-\ntem, the effects of the variation of the main group el-\nement,Z, from Ga to Ge, In, and Sn in a small sam-\nple ofcompounds[Mn 2(Mn,Fe)Ge, Mn 2(Os,Ir,Pt)In, and\nMn2(Os,Ir,Pt)Sn] are briefly discussed. The correspond-\ning data, including those for Mn 3Ga, are graphed in\nFig.5.\nFig.5revealsthattheplacementofInandSnatthe Zpo-\nsition leads to the emergence of a deep energy minimum\nfor Mn 2IrSn with c/a <√\n2 and even deeper minima\nfor elongated phases with large c/a-ratios of Mn 2OsIn,\nMn2IrIn and Mn 2OsSn whose lattice parameters resem-\nble those of layered structures. Heusler alloys are of-\nten interpreted in terms of a rigid, band model-like ap-\nproach, and an electron-filling scheme is employed for\nthe prediction of magnetic moments in Co 2-based al-\nloys. Interpretation of the tetragonal instabilities us-\ning such an approach leads to the assumption of Mn 3Ge(NV= 25) behaving analogouslyto Mn 2FeGa (NV= 25)\nand of Mn 2FeGe (NV= 26) being similar to Mn 2CoGa\n(NV= 26). A comparison of the corresponding c/a\ncurves demonstrates that this is approximately true and\nthus that isoelectronicityis an appropriateconcept in the\nchemistry and physics of Heusler materials.\nAlthough our calculations agree well with those of previ-\nous works[ 37], comparisonwith the experimental data of\nMn–Ni–Ga systems exhibits an interesting discrepancy.\nThedeviationbetweenexperiment andtheorywastraced\nbackto the deviating structural model in terms ofthe oc-\ncupationoftheinvolvedsites. Neutrondiffractionstudies\non Mn 2NiGa highlighted the fact that the order is differ-\nent from the expected Xa-type in the austenite phase.\nThus, the chemical formula reads as (Mn,Ni) 2MnGa and\nis called L2 1b-type because the point symmetry includes\ninversion symmetry through random occupation of 4 d\nand 4csites with Ni and Mn [ 38]. Similarly in Mn 2FeGa,\na deviation from theory was found in an experimental\nstudy because the site occupation was expected to differ\nfromthe perfect MnFeMnGa ordering[ 16]. Similarissues\nare present for the Mn 2Y(4d)Ga series. From available\ndata, including neutron diffraction studies, the site occu-\npancies have been clarified. Several authors found that\nmembers of the Mn 2RuZseries (Z= Ga,Ge,Si,Sn)\nand Mn 2RhZseries (Z= Ga) [ 17,18,39] not to be\ntetragonal under the respective reaction conditions. In\ncontrast, they have been realized as cubic Heusler alloys\nexhibiting a strong degreeof anti-site disorder, which has\nbeen characterized as an alloying tendency [ 18].\nOrthorhombic deformations of the unit cell have been\nobserved in some systems such as Mn 2NiGa [37]. Thus,\nthe restrictionto tetragonaldistortionsand orderedcom-\npounds in conducted studies leadsto a simplified descrip-\ntion of these materials. Nevertheless, a general under-\nstanding can be obtained in approaching the Mn 2-based\nHeusler systems through this ansatz. In future studies6\nthat aim to predict ground-statestructures and magnetic\nconfigurations, the parameter space for the atomic sites\nand relaxations paths for the electronic and spin degrees\nof freedom has to be enlarged and restrictions that are\nwidely used have to be dropped. As a consequence of\ntheappliedrestrictions,disordereffectswerenotincorpo-\nrated into this study, thus leaving open any explanations\nof the deviations from experimental results.\nc/a (cubic reference cell) \n0.8 1.0 1.2 1.4 1.6 1.8\n1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4\nc/a (tetragonal reference cell) 0.6\n0.4\n0.2\n0\n-0.2\n-0.4Mn  Ga\nMn  FeGe Mn  IrIn Mn  IrSn\nMn  Ge Mn  PtInMn  OsIn Mn  OsSn\n2\n22\n22\n23\n3 ∆E     [ eV/ f.u. ] t-c \nFIG. 5. (Color online) The energy landscapes, E(c/a),\nof Mn 3Ge, Mn 2FeGe, Mn 2(Ir,Pt)Sn, and Mn 2(Ir,Pt)In are\nshown.\n1. Analysis of the Densities of States\nVarious attempts to explain the instability of the cu-\nbic phase have been given in the literature using mod-\nels such as the band Jahn–Teller effect [ 5,16], anoma-\nlous phonon vibrations [ 49,50], and Fermi-surface nest-\ning [51]. These different approaches describe the same\nbehavior, i.e., the instability of the cubic phase, from\ndifferent perspectives and extracting different types of\ninformation. Commonly, the densities of states (DOSs)\nof related austenite and martensite are compared and\ncontrasted marking the starting and the endpoint of the\ntransition. As example for the Mn 2Y(3d)Zseries, the\nDOSs of Mn 3Ga, Mn 2FeGa, and Mn 2NiGa are shown\nin Figs. 6–8. For the cubic variants, the partial DOSs\n(PDOSs) areshownin theircorrespondingprojectionson\nthe sites and in terms of projections on the irreducible\nrepresentations. The peaks in the PDOS of Mn(4 b) are\nwell separated on the energy scale. On the one hand,\nthis separation is due to the strong crystal field split-\nting of the Mn(4 b)d-states, where the occupied egstates\nare located in a range between −4 and−3 eV, whereas\nthet2gPDOS is found between −1 eV and the Fermi\nedge,εF. On the other hand, the separation of occu-\npied and empty states follows from the strong exchange\nsplit of the Mn(4 b)d-states, in contrast to the d-statesofY(4c) and Mn(4 d), which are found to be more widely\ndispersed even though the majority and minority states\nare separated owing to exchange splitting. The PDOSs\nof Mn(4d) andY(4c) are strongly dispersed, with con-\nsiderable overlap of the spectral weight between the t2g\nandegstatesin the majoritychannel, whereasthe minor-\nity channel is gapped, with t2gcharacterizing the lower\nboundary and egcomprising the upper boundary of the\ngap. As the DOS is gapped in the minority spin channel\nthe study of the tetragonal system is strongly facilitated\nas the states at the Fermi edge in the majority channel\nmostly constitute the origin of the tetragonal distortion.\nThe majority PDOSs in the range of −5 eV up to εF\nexhibit a characteristically shaped peak structure. From\nMn3Ga to Mn 2CoGa, the majority spin channel (lower\npanels of Figs. 6a–8c) is continuously filled. The Fermi\nenergy is consequently shifted to higher band energies\nand thus εFsweeps over a range of the majority DOSs,\nwhereastheminorityspinchannelremainsunchanged. It\nisclearlyseenfromFigs. 6and7that thetetragonaltran-\nsition correlates with the peak structure of the majority\nDOSs. For εFbeing centered on a peak of the majority\nDOS, the tetragonal distortion can be triggered. These\nlocal maxima are mainly composed of states of the Y(4c)\n(Mn, Fe, Co, Ni) and Mn (4 d) atoms. These energy lev-\nels are of t2gsymmetry mainly. In simple interpretation,\nthe DOS can be understood in a rigid-band-like fashion.\nThelimitofthis interpretationisreachedwith Mn 2NiGa,\nwhere the d-PDOS is rearranged and the Slater–Pauling\nrule is no longer valid for the cubic phase [ 15]. Compar-\ning this to the PDOS of the tetragonally distorted sys-\ntems, it is observed that the resulting PDOSs are widely\ndispersed and significantly less structured. Mn 2NiGa,\nhowever, behaves differently. Further filling of the ma-\njority states, as intuitively expected, does not occur. In\ncontrast, the gap in the minority channel closes as the\nstates of egsymmetry are pulled towards εF. Thus, the\ntetragonal distortion in Mn 2NiGa is formed by another\nmechanism, which may explain why Mn 2NiGa is found\nto be a magnetic shape memory alloy, whereas Mn 3Ga\nand Mn 2FeGa are found in their respective tetragonal\ncrystal structures, although the total energy differences\nare comparable to that of Mn 2NiGa.\nWe emphasize that these findings differ from other mod-\nels in which the instability is thought to depend solely\non states of Mn(4 b) that is found in a tetrahedral envi-\nronment. The DOS at εFin mainly composed of states\nof the 4cand 4dpositions with minor contributions from\nMn(4b). Here the instability removes the strong peaks of\nMn, Fe, and Co at the 4 cposition, whereas the states of\nMn(4b) are not rearranged.\n2. Spin-Polarization\nHalf-metallicity [ 52,53] (complete or nearly complete\nspin-polarization P(εF)≈100%) is generally observed\nin cubic Co 2- and Mn 2-based Heusler compounds. The7\nTABLE III. Calculated lattice parameters of the cubic paren t phases and corresponding tetragonal structures, in compa rison\nwith existing literature data. The lattice parameters are g iven in˚Angstroms. The saturation magnetization data, MS, are\ngiven in µB/f.u.The listed literature data refer to theoretical ( xt) or experimental ( xe) investigation of the mentioned cubic\n(cy) or tetragonal (t y) phases, respectively.\nNVSGc/a a catMSSGc/a a tctMSSGc/a a tctordering Lit.\nMn2ScGa 20 225√\n2 6.15 4.35 −4.00139 1.94 3.98 7.70 −5.36\nMn2TiGa 21 225√\n2 5.95 4.21 −2.97139 225√\n2 5.95 ct MnMnTiGa [ 40]\nMn2VGa 22 225√\n2 5.82 4.12 −1.98139 225√\n2 5.91 ce MnMnVGa [ 41],[42]\nMn2CrGa 23 225√\n2 5.76 4.07 −1.00139 1.82 3.82 6.95 −2.75225√\n2 5.77 ct MnMnCrGa [ 43]\nMn3Ga 24 225√\n2 5.82 4.12 0.01139 1.82 3.90 7.08 −1.89139 1.77 3.77 7.16 tt MnMnMnGa [ 4]\n139 1.82 3.90 7.09 te MnMnMnGa [ 4]\nMn2FeGa 25 216√\n25.79 4.09 1.03119 1.98 3.68 7.29 −0.78119 1.90 3.79 7.19 te Mn(Fe,Mn) 2Ga [16]\nMn2CoGa 26 216√\n2 5.78 4.09 2.00119 1.93 3.71 7.14 0.17216√\n2 5.86 ce MnCoMnGa [ 44]\nMn2NiGa 27 216√\n2 5.85 4.14 1.18119 1.82 3.79 6.91 1.00216√\n2 5.91 ce [ 45]\n119 1.72 3.91 6.70 te [ 45]\n225√\n2 5.94 ce (Mn,Ni) 2MnGa [ 38]\n139 1.74 3.89 6.77 te (Mn,Ni) 2MnGa [ 38]\n216√\n2 5.84 ct MnNiMnGa [ 37]\nMn2CuGa 28 216√\n2 5.94 4.20 0.33 216√\n2 5.94 ct MnCuMnGa [ 37]\nMn2ZrGa 21 225√\n26.14 4.34 −3.00119\nMn2NbGa 22 225√\n26.00 4.24 −2.00119\nMn2MoGa 23 225√\n2 5.91 4.18 −1.01119 1.81 3.89 7.04 −2.99\nMn2RuGa 25 216√\n2 5.96 4.22 1.03119 1.96 3.80 7.45 −0.24216√\n2 6.00 ce (Mn 2\n3,Ru1\n3)3Ga [17]\nMn2RhGa 26 216√\n2 5.98 4.23 1.64119 1.94 3.82 7.43 0.10225√\n2 6.03 ce (Mn,Rh) 2MnGa [ 18]\nMn2PdGa 27 216√\n2 6.12 4.33 0.55119 1.84 3.93 7.23 0.93\nMn2AgGa 28 216√\n2 6.22 4.40 0.34119\nMn2HfGa 21 225√\n2 6.12 4.33 −2.99119\nMn2TaGa 22 225√\n2 6.00 4.24 −1.99119\nMn2WGa 23 225√\n2 5.92 4.19 −0.94119\nMn2OsGa 25 216√\n2 5.95 4.21 1.02119 1.97 3.80 7.48 −0.28\nMn2IrGa 26 216√\n25.97 4.22 2.00119 1.95 3.83 7.44 0.11\nMn2PtGa 27 216√\n2 6.13 4.33 0.44119 1.87 3.91 7.31 0.75119 1.38 4.37 6.05 te MnPtMnGa [ 19]\nMn2AuGa 28 216√\n2 6.26 4.42 0.19119 1.73 4.11 7.13 0.14\nMn2OsSn 25 216√\n2 6.21 4.39 1.50119 1.95 3.97 7.75 −0.02\nMn2IrSn 26 216√\n2 6.31 4.46 0.41119 1.91 4.01 7.67 0.45119 1.54 4.29 6.59 te MnIrMnSn [ 46]\nMn2PtSn 27 216√\n2 6.39 4.52 0.19119 1.81 4.15 7.52 −0.02119 1.35 4.51 6.08 te MnPtMnSn [ 1]\nMn2OsIn 25 216√\n2 6.26 4.43 0.62119 2.02 3.93 7.93 −0.27\nMn2IrIn 26 216√\n2 6.30 4.45 0.68119 1.98 3.97 7.85 0.07\nMn2PtIn 27 216√\n2 6.37 4.51 0.31119 1.84 4.12 7.57 0.38119 1.57 4.32 6.77 te MnPtMnIn [ 21]\nMn3Ge 25 216√\n2 5.76 4.07 1.01119 1.90 3.74 7.10 −0.98225 1.91 3.81 7.26 te MnMnMnGe [ 47]\n225 1.90 3.75 7.12 tt MnMnMnGe [ 6]\nMn2FeGe 26 216√\n2 5.73 4.05 2.01119 2.05 3.63 7.42 −0.06216√\n2 5.80 ct MnFeMnFe [ 48]\nhighly symmetric structure and the peculiar electronic\nproperties due to covalent bonding lead to the appear-\nance of a gap in the minority density of states (DOS).\nThe emergence of the tetragonal distortion reduces the\nspin-polarizationofthehalf-metalliccubic parentphases.\nThe degeneracy of the t2gandegstates is lifted due to\nthe changein local coordinationcausedby the distortion,\nwhich is seen in Figs. 6–8and Fig. 16, and the emergenceof apseudo-gap in one spin-channel is observed instead.8\nMn(4b) a) \nb) Mn(4d) \n-7 -6 -5 -4 -3 -2 -1 \n ε 1 0  2  3  4  5 \nFtotal \ne\nt2g g 0  4 \n 4 \n 0  4 \n 4 \n2\nxy 2 2x  - y \nz\nxz+yz Mn(2b) 139 225 \nMn(2d) c) \nd)  0  4 \n 4 \n 0  4 \n 4 DOS  [1/eV] \nE     [eV] -\nFIG. 6. (Color online) DOS of cubic (SG 225) and tetragonal\n(SG 139) Mn 3Ga.\nB. Magnetic Ground State\n1. The Slater–Pauling Rule\nReferring to a previous publication [ 15] on the cu-\nbic parent compounds of the investigated materials, the\nresults are presented by means of the Slater–Pauling\ncurves. As can be seen in Fig. 10, the Slater–Pauling\nrule experiences strong changes, so that calling it the\nSlater–Pauling rule is done for reasons of convenience.\nFig.10visualizes the fact that the magnetization of\nall tetragonal alloys of the Mn 2Y(d)Ga family experi-\nences a shift to smaller values. In the case of the\nMn2Y(3d)Ga group, this shift is found to be constant\nthroughout the set of compounds, which results in a\nlinear dependence of the net moment on the valence\nelectron count, thus giving rise to pseudo-Slater–Pauling\nbehavior, even though half-metallicity and thus integer\nnet moments are not observed (Fig. 9). A decrease of\nthe net moment is also found for the Mn 2Y(4d)Ga and\nMn2Y(5d)Ga compounds. Unlike the lighter compounds,\nthe changesare not constant overthe series and thereforeMn(2b) \nFe(2c) \nMn(2d) Mn(4b) \nY(4c) \nMn(4d)  0  4 \n 4 \n 0  4 \n 4 \n 0  4 \n 4 total \ne\nt2g g\n119 216 \n 0  4 \n 4 \n 0  4 \n 4 \n 0  4 \n 4 2\nxy 2 2x  - y \nz\nxz+yz a) \nb) \nc) \nd) \ne) \nf) \n-7 -6 -5 -4 -3 -2 -1  1 0  2  3  4  5 DOS  [1/eV] \n εFE     [eV] -\nFIG. 7. (Color online) DOS of cubic (SG 216) and tetragonal\n(SG 119) Mn 2FeGa.\nresult in nearly vanishing net moments for Mn 2RuGa,\nMn2RhGa, Mn 2PdGa, Mn 2OsGa, Mn 2IrGa, Mn 2PtGa,\nand Mn 2AuGa, as is seen in Fig. 10. Nevertheless, com-\npensation of spin moments may be achieved for an elec-\ntron count close to NV= 25.7, which can be realized by\nintermixtures of stoichiometric phases of Mn 2YGa, such\nasmMn3Ga=−1.89µBand a corresponding proportion\nof Mn 2NiGa with mMn2NiGa= 1.00µBor Mn 2CoGa.9\n-7 -6 -5 -4 -3 -2 -1  1  2  3  4  5  0  4 \n 4 \n 0  4 \n 4 \n 0  4 \n 4 DOS  [1/eV] Mn(2b) a) \nNi(2c) b) \nMn(2d) c) 119 \n2\nxy 2 2x  - y \nz\nxz+yz Mn(4b) a) \nNi(4c) b) \nMn(4d) c) total \ne\nt2g g216 \n 0  4 \n 4 \n 0  4 \n 4 \n 0  4 \n 4 \n0\n εFE     [eV] -\nFIG. 8. (Color online) DOS of cubic (SG 216) and tetragonal\n(SG 119) Mn 2NiGa.\nThus, fractions of 0.435 mMn2NiGaand 0.565 mMn2FeGa\ncould ideally lead to complete compensation of the mag-\nnetization. A similar approachwas undertaken by Nayak\net al.that obtained a compensated ferrimagnet by vary-\ning the Mn/Pt-ratio in Mn 3−xPtxGa leading to com-\nplete compensation of magnetization for x≈0.59 theo-\nretically [ 20]. In Mn 2RuxGa thin films the compensation\nof the spin-moment has been achieved through variation100\n80 \n60 \n40 \n20 \n 0 Spin- Polarization P(ε ) / % F\n22 21 20 23 24 25 26 27 28 5 4 3 6 7 8 9 10 11 \nMn Y     Ga 2(4d )\nMn Y     Ga 2(5d )Mn Y     Ga 2(3d )Group Number Y(  )\nNumber of valence electrons   NV\nFIG. 9. (Color online) The spin-polarization at the Fermi en -\nergy for both cubic and tetragonal phases. For a system with\nNV<27, the spin-polarization is reduced by the tetragonal\ndistortion.\nNb Mo \nZr Ru Rh Pd \nAg Os \nW\nHf Ta Ir Pt Au \nMn Y     Ga 2(5d) Mn Y     Ga 2(4d) 16 20 24 28 16 20 24 28 32 \n4b) c) \na) 0\n-4 \n4812 \n0\n-4 Magnetic moment  m [    ] B\nSc Ti VCr Fe Ni \nCu Co \nMn  Ga3\n16 20 24 28 32 36 40 44 48 Cr Cr \nCo Fe \nNi \nMn Y     Ga 2(3d) µ\nNumber of valence electrons   NV\nFIG. 10. (Color online) Slater–Pauling curves of\n(a) Mn 2Y(3d)Ga, (b) Mn 2Y(4d)Ga compounds, and (c)\nMn2Y(5d)Ga compounds.\nof the ruthenium concentration [ 54].10\nTi (a) (b) (c)\nSc V Cr Mn Fe Co Ni Cu \nTotal magnetic moment m [µ   / f.u.] B4\n02\n-2 4\n02\n-2 \n21 20 22 23 26 25 24 27 28 L2 1 Xa Xa XaMo Tc Ru Rh Pd Ag \n23 26 25 24 27 28 26 25 27 28 Os Ir Pt Au \nL2 1\nmmm\nMn(4d)Y(4c) Mn(4b)\nmmm\nMn(4d) Y(4c) Mn(4b) \nmmm\nMn(4d) Y(4c) Mn(4b) \nm\nmMn(8c)Y(4b) m\nmMn(8c)Y(4b) Mn Y     Ga 2(3d) Mn Y     Ga 2(5d)Mn Y     Ga 2(4d)\nNumber of valence electrons   NV\nFIG. 11. (Color online) Atomic magnetic moments in (a) Mn 2Y(3d)Ga, (b) Mn 2Y(4d)Ga, and (c) Mn 2Y(5d)Ga compounds.\nOpen symbols denote the L2 1-type coordination. Filled symbols denote the X a-type coordination. The squares stand for the\ncubic materials, whereas the tetragonal systems are repres ented by rectangles.\n2. Local Magnetic Moments\nThe change of the total magnetic moment is to be un-\nderstood in terms of the site moments. Inspection of\nFig.11clearly reveals the dependencies. The change in\nthelocalmomentsindicatesthespecificimpactofthedis-\ntortiononthesinglesites. Thelocalmomentsresponding\nstrongest to elongation/compression of the crystal axes\nare members of the magnetic sublattices formed by the\nMn(8c), Mn(4d), andY(4c) sites (the former so-called\ntetrahedral sites as compared to SG 225). Depending\non the characters of the local moments, which are either\nitinerant orlocalizedin nature[ 12,52], we find majordif-\nferences in the influence of the tetragonal distortion on\nthese sites. In Fig. 11, the localized moment of Mn(4 b) is\nfound in the upper part of the plot for a positive value of\nthe magnetic moment of approximately 3 µB. The effect\nof elongation along the c-axis and compression of the ab-\nplane has a stronger influence on moments of itinerant\ncharacter found in the lower part of the plot, referring\nto the 4dand 4csites. These are located in the same\nlattice plane (compare the atoms depicted as red and\nblue spheres in Fig. 1). Manganese on site 4 b[Mn(4b)]\nexhibitsalargelocalizedmomentof3 µB, isthusisgener-\nally much less affected. Apart from the magnitude of the\nlocal moments, Fe(4 c) in Mn 2FeGa exhibits a spin-flip\nfrom parallel to anti-parallel alignment of the Mn(4 b)-\nFe(4c) interaction upon the tetragonal distortion. Apart\nfrom the changes in magnitude of the local atomic mo-\nments, the effective anti-parallel coupling of the nearest\nneighbormanganeseatomsdoesnotsufferfromthestruc-\ntural transformation, as will be quantified in terms of the\nexchange interaction constants in Sec. IVC. Since the\nvolume change Ct/cis in the order of 1-2% for the mostsystems, the nearest-neighbordistance basically does not\nchange, whereas the direction of the nearest-neighbor in-\nteraction does (Fig. 16). In the case of Mn 3Ga,Ct/c\nis approximately 9%, and thus may suppress the shape\nmemory effect in this material.\n3. Magnetocrystalline Anisotropy\nInherentinnoncubiccrystalsisadirectionalpreference\nof magnetization that is absent in cubic materials, which\nis related to the tetragonal modification of the crystal\naxes. The magnetocrystalline anisotropy (MCA) energy\nis here defined as the energy difference between states\nwith magnetization pointing along the z-axis and the\nx- ory-axis, that is, EMCA=E(100)−E(001), whereas\nother crystallographicdirections are not considered. The\nanisotropy energy is phenomenologically thought to de-\npend on the value of the c/aratio, which is more or\nless equal for most of the compounds investigated in the\npresent study. Therefore, the underlying mechanism is\nunderstood as a band-filling effect, which affects the spin\norbit coupling (SOC) symmetry. This interpretation can\nbe directly taken from Figs. 12and13. Increasing the\nSOC strength by varying the Yelement through the 3 d,\n4d, and 5dseries increases the MCA energy by a factor of\napproximately 3 for Y= Fe,Ru,Os. The effect of band\nfilling is deciphered by sweeping the Yelements along a\nseries. Going from left to right in any set of compounds\nMn2Y(xd)Ga, the MCA is altered from preferred out-of-\nplanetoin-plane orientation. The same situation holds\nfor Mn 2Y(5d)Zcompounds, whose preferred orientation\nis graphed in Fig. 13. Two compounds in this figure de-\nservespecialattention, which areMn 2PtIn and Mn 2IrSn.11\nPreviously they were predicted to possess a non-collinear\nmagnetic order just as Mn 2RhSn, which was described\nin great detail in Ref. [ 46]. Although we did not con-\nsider non-collinear order in the present case, it might be\nworthwhile topoint out that it is the Mn-atomon site 4 d\nthat reacts by canting to the incipient spin-reorientation\nthat is seen clearly in Figs. 12. Similar physics is ob-\nserved in the famous Rare-Earth magnets Nd 2Fe14B and\nEr2Fe14B. Manganese thus shares properties with Rare\nEarths [55–57].\n-1 123\n0MCA E         3 4 5 6 7 8 9 10 11 \nMn  CrGaMn  NiGa Mn  ScGa\n2Mn  MoGaMn  RhGa 22 Mn  PdGa\nMn  PtGaMn  AuGa2\n22Mn  RuGa2Mn  IrGa2Mn  OsGa2\n2\nMn  CoGa2Mn  FeGa2\nMn Y     Ga 2(3d) Mn Y     Ga 2(4d) Mn Y     Ga 2(5d) in plane (x,y)out of plane (z)2\nMn  Ga3L2 1 aX\n20 21 22 23 24 25 26 27 28 \nNumber of valence electrons   NVGroup Number Y(  )meV / f.u. [                 ] \nT K[   ] 11.60\n-11.60023.2134.81\nFIG. 12. (Color online) Calculated MCA energy of Mn 2YGa.[                 ] Mn  IrIn2\nMn  IrSn2Mn  RhSn2\nMn  PtIn2Mn  PtSn2Mn  Ge3Mn Y     2(3d) \nMn Y      2(5d) Sn Mn Y      2(5d) In Ge \n22 23 24 25 26 27 28 29 \nNumber of valence electrons   NVMCA E         \nT KmeV / f.u. \n[   ] \n-1 123\n0in plane (x,y)out of plane (z)L2 1 aX\n11.60\n-11.60023.2134.81\nFIG. 13. (Color online) Calculated MCA energy of some cho-\nsenMn 2YZalloys: Mn 3Ge, Mn 2IrIn, Mn 2IrSn,Mn 2PtIn, and\nMn2PtSn.\nC. Exchange Coupling and Curie Temperatures\nThedetails ofthe calculationspertainingtoCurietem-\nperatures ( TC) have been described by us previously [ 15].\nEven though the in-plane next-nearest-neighbor distance\ndecreases due to the tetragonal distortion, the next-\nnearest neighbor Mn(8 c)-Mn(8c) coupling is still positive\nfor the L2 1-derived tetragonal phases; thus, the overallmagnetic order does not change. A decrease or change\nof sign of the coupling constant JMn(4d)−Mn(4d)is typical\nowing to preferential anti-parallel coupling as caused for\nshort Mn–Mn distances. Instead, the cubic to tetragonal\ntransition even results in increased values, thus leading\nto an increase of TC. Prima facie, the overall trend of\nincreased TCs cannot be traced back to a common mech-\nanism. In contrast, there is no exchange interaction that\nexhibits a similar behavior over the series. For instance,\nthe findings in the case of Mn 2FeGa are related to the re-\nduction of magnetic frustration, which is due to the com-\npeting anti-parallelinteractionMn(4 b)–Mn(4d), andpar-\nallel interactions of Fe(4 c) with both Mn(4 d) and Mn(4 b)\nneighbors. Upon the tetragonal distortion a spin-flip of\nthe Fe(4c) local moment is observed, due to the change\nof the sign of Mn(4 b)–Fe(4c) interaction. Upon the dis-\ntortion, the strength of Mn(4 b)–Mn(4d) interaction is al-\ntered by approximately ∆ JMn(4b) −Mn(4b)= 56 meV in\nMn2FeGa.\nA similar, but smaller effect is found in Mn 2CoGa, indi-\ncating the magnetic frustration that had been present in\nthe cubic phase and the weakening of the exchange inter-\nactionY(4c)–Mn(4b) (Y= Fe, Co) due to the tetragonal\ndistortion. This might be one of the contributions pro-\nhibiting the shape memory effect in Mn 2FeGa. The main\ncontribution to TC, the Mn(4 d)–Mn(4b) exchange, does\nnot suffer from the structural transition. Similarly, the\ninfluence of the distortion on the exchange causes the\nsmaller contributions to be altered and some frustration\nto be diminished. For example, the Mn(4 d)–Mn(4d) in-\nteraction vanishes, with preferred anti-parallel alignment\nin the cubic case, whereas the major Mn(4 d)–Mn(4b) in-\nteraction remains unchanged.\nExceptions to the general observation of increased TCin\ntetragonally distorted phases are Mn 2NiGa, Mn 2PdGa,\nand Mn 2PtGa systems, in which the TCs are reduced\nupon the tetragonal distortion. The significant reduc-\ntion (Fig. 14) is caused by a weakened Mn(4 d)–Mn(4b)\ninteraction (Fig. 15) that may indicate an unstable mag-\nnetic groundstate. A relation to the Heusler compound\nMn2RhSn [46] can theoretically be established as these\nmaterials possess the same number of valence electrons,\nNV. Mn2RhSn has been shown to exhibit a non-collinear\nmagneticgroundstateduetocantingofthedifferentman-\nganese moments [ 46]. Non-collinear spin configurations\nhave not been considered in this work, so that canting of\nthe spins or spin spiral groundstates in these materials\nmay be present.\nV. SUMMARY\nUsing total energy calculations within density func-\ntional theory, we investigated in detail the response to\ntetragonaldistortionsforalargesetofcubicHeuslercom-\npounds, Mn 2Y(3d,4d,5d)Ga, and some other chosen mate-\nrials. We were able to single out the systems that remain\ncubic from those that favor a tetragonal structure. The12\nPcPtTC,cTC,t∆TC,t−cEMCA Ku\nMn2ScGa87 35 464 0 .616 1 .62\nMn2TiGa83 557\nMn2VGa94 587\nMn2CrGa97 50 578 970 392 0 .779 2 .46\nMn3Ga96 56 221 610 389 0 .906 2 .7\nMn2FeGa95 56 601 848 247 0 .359 1 .16\nMn2CoGa93 60 928 1124 196 0 .236 0 .77\nMn2NiGa35 42 1005 750 −255 0.193 0 .62\nMn2CuGa53 1491\nMn2ZrGa82 207\nMn2NbGa98 289\nMn2MoGa85 65 140 335 196 0 .636 1 .91\nMn2RuGa95 1 619 1315 696 0 .564 1 .68\nMn2RhGa15 59 576 1351 776 0 .322 0 .95\nMn2PdGa 7 46 809 335 −473 0.040 0 .11\nMn2AgGa24 1240\nMn2HfGa89\nMn2TaGa96\nMn2WGa83\nMn2OsGa96 273 1075 802 3 .270 9 .72\nMn2IrGa74 5 411 1122 711 2 .388 7 .02\nMn2PtGa23 51 799 326 −472−0.293−0.84\nMn2AuGa 8 26 1027 897 −130−0.731−1.94\nTABLE IV. The calculated Curie temperatures in Kelvin of\ntetragonal TC,tand cubic TC,cparent compounds (taken from\nRef.15). The changes due to the tetragonal transformation\nare listed as ∆ TC,t−c.EMCArepresents the magnetocrys-\ntalline anisotropy energies in meV per formula unit, wherea s\nthe anisotropy constant, Ku, is given inMJ\nm3.\n1400\n1200\n1000\n800\n600\n400\n200Curie Temperature  T  [K] c\n21 22 23 24 25 26 27 28 4 5 6 7 8 9 10 11 \nOs Ir \nPt Au \nZr Nb Mo Ru Rh \nPd Ag \nTi V Cr Mn Fe Co Ni Cu \nMn Y     Ga 2(4d)\nMn Y     Ga 2(5d)Mn Y     Ga 2(3d)\nNumber of valence electrons   NVGroup Number Y(  )\nFIG. 14. (Color online) The calculated Curie temperatures\nof the Heusler compounds containing Ga. The results shown\nhere are obtained in the Mean-Field Approximation (MFA)\nand highlight the consequence of the structural relaxation .\nSquares correspond to cubic compounds and rectangles to\ntetragonal compounds.Ti Sc V Cr Mn Fe Co Ni Cu \nL2 1 Xa\n21 20 22 23 24 25 26 27 28 50 \n-500\n-100\n-150\n-200\n-250J    [meV] eff \nMn(4d) - Mn(4d) \nMn(4d) - Mn(4c) \nY       -Y Mn(4d) - Y Mn(4d) - Mn(4d) \nMn(4d) - Y\nMn(4b) - Mn(4b) Mn(4d) - Mn(4b) \n(4b) (4b) (4c)\n (4b)\nNumber of valence electrons   NV\nFIG. 15. (Color online) The evaluated effective exchange in-\nteraction parameters are shown and compared for all cubic\nand tetragonal cases. Thus, the underlying mechanism of\nthe increase of TCdue to the distortion is visualized. Circles\ncorrespond to cubic compounds and rectangles to tetragonal\ncompounds.\nMn \nMn Mn Z\nZ\nFIG. 16. (Color online) A two-dimensional projection of\nthe nearest-neighbor coordination of Mn(4 b) is shown. The\nsymmetry of the coordination changes while undergoing the\nmartensitic transition , whereas the nearest-neighbor distance\nremains unchanged as the volume change is on the order of\nonly 1–9%.\ndetails of the total energy as a function of the distortion\nwerefoundtobe similarformaterialsexhibitingthesame\nnumber of valence electrons. The magnetizations of the\ntetragonal alloys were found to be shifted to smaller val-\nues, which we could attribute to changes of the itinerant\nlocal moments. This led to characteristic modifications\nof the Slater–Pauling curve. By means of partial den-\nsities of states, the changes to the electronic structures\nrevealed the microscopic origin of the observed trends.\nAs compared to the cubic parent phases, a strengthening\noftheexchangeinteractionbetweenneighboringsiteswas\nobserved, which resulted in an increase of the Curie tem-\nperature. 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Wosi ński  1 \n \n1 Institute of Physics, Polish Academy of Sciences, 0 2-668 Warszawa, Poland \n2 MAX-IV Laboratory, Lund University, P.O. Box 118, S E-221 00 Lund, Sweden \n \n \nEffect of misfit strain in the layers of (Ga,Mn)(Bi ,As) quaternary diluted magnetic \nsemiconductor, epitaxially grown on either GaAs sub strate or (In,Ga)As buffer, on their \nmagnetic and magneto-transport properties has been investigated. High-resolution X-ray \ndiffraction, applied to characterize the structural  quality and misfit strain in the layers, proved \nthat the layers were fully strained to the GaAs sub strate or (In,Ga)As buffer under \ncompressive or tensile strain, respectively. Ferrom agnetic Curie temperature and magneto-\ncrystalline anisotropy of the layers have been exam ined by using magneto-optical Kerr effect \nmagnetometry and low-temperature magneto-transport measurements. Post-growth annealing \ntreatment of the layers has been shown to enhance t he hole concentration and Curie \ntemperature in the layers.  \n \nPACS: 75.50.Pp; 61.05.cp; 78.20.Ls; 73.50.Jt; 75.30 .Gw. \n \n*levchenko@ifpan.edu.pl  \n  2 1. Introduction \nHighly mismatched ternary semiconductor compound Ga (Bi,As) has recently emerged \nas promising material for possible applications in a new class of photonic and spintronic \ndevices. The replacement of a small fraction of As atoms by much larger Bi atoms in GaAs \nrequires highly non-equilibrium growth conditions, such as low-temperature molecular-beam \nepitaxy (LT-MBE) at the substrate temperatures of 2 00 −400°C [1, 2], i.e. far below 580°C, \nwhich is optimal for the MBE growth of GaAs. The ba nd-gap energy of Ga(Bi,As) alloys \ndecreases rapidly with increasing Bi content due to  an interaction of Bi 6p bonding orbitals \nwith the GaAs valence band maximum [3, 4]. Moreover , the replacement of As atoms by \nmuch heavier Bi atoms results, owing to a large rel ativistic correction to the GaAs band \nstructure, in a strong enhancement of spin-orbit co upling, accompanied by a giant separation \nof the spin-split-off hole band [5, 6]. The increas ed spin-orbit coupling is especially \nfavourable for spintronic materials where spin prec ession can be electrically tuned via the \nRashba effect [7]. \nOn the other hand, another GaAs-based ternary compo und (Ga,Mn)As, in which a few \npercent of Ga lattice atoms have been substituted b y Mn impurities, has become a prototype \ndiluted ferromagnetic semiconductor, which exhibits  spintronic functionalities associated with \ncollective ferromagnetic spin ordering. Substitutio nal Mn ions in (Ga,Mn)As become \nferromagnetically ordered below the Curie temperatu re owing to interaction with spin-\npolarized holes. The sensitivity of the magnetic pr operties, such as the Curie temperature and \nmagnetic anisotropy, to the hole concentration allo ws for tuning those properties by post-\ngrowth annealing, photo-excitation or electrostatic  gating of the (Ga,Mn)As layers [8, 9]. \nMoreover, appropriate nanostructurization of thin ( Ga,Mn)As layers offers the prospect of \ntaking advantage of magnetic domain walls in novel spintronic devices [10, 11]. \nIn the present paper we report on an effect of misf it strain in thin epitaxial layers of \n(Ga,Mn)(Bi,As) quaternary compound, grown under eit her compressive or tensile strain, on \ntheir magnetic and magneto-transport properties.  \n \n2. Experimental details \nWe have investigated (Ga,Mn)(Bi,As) thin layers of 15 and 50 nm thicknesses, with 6% \nMn and 1% Bi contents, grown by the low-temperature  MBE technique at a temperature of \n230°C on either semi-insulating (001)-oriented GaAs  substrate or the same substrate covered \nwith a 0.63- µm thick In 0.2 Ga 0.8 As buffer layer. After the growth the samples were cleaved into \ntwo parts. One part of each sample was subjected to  the low-temperature annealing treatment  3 performed in air at the temperature of 180°C during  50 h. Annealing at temperatures below \nthe growth temperature can substantially improve ma gnetic and transport properties of thin \n(Ga,Mn)As layers due to outdiffusion of charge- and  moment-compensating Mn interstitials \nfrom the layers [12].  \nBoth the as-grown and the annealed samples, contain ing the (Ga,Mn)(Bi,As) layers of \n50 nm thicknesses, have been subjected to high-reso lution X-ray diffraction (XRD) \ncharacterization at room temperature. Lattice param eters and misfit strain in the layers were \ninvestigated using reciprocal lattice mapping and r ocking curve techniques for both the \nsymmetric 004 and asymmetric 422 Bragg reflections of Cu K α1 radiation. The thinner \n(Ga,Mn)(Bi,As) layers of 15 nm thicknesses, grown u nder the same conditions as the thicker \nones, have been subjected to investigations of thei r magnetic and magneto-transport \nproperties. Magnetic properties of the (Ga,Mn)(Bi,A s) layers were examined using magneto-\noptical Kerr effect (MOKE) magnetometry. The MOKE e xperiments were performed both in \nlongitudinal and polar geometries using He-Ne laser  as a source of linearly polarized light \nwith the laser spot of about 0.5 mm in diameter. Th e angle of incidence of light on the sample \nwas about 30 ° for the longitudinal and 90 ° for the polar geometry. The standard lock-in \ntechnique with photo-elastic modulator operating at  50 kHz and a Si diode detector was used. \nMeasurements of magnetic hysteresis loops were perf ormed in the temperature range \nT = 6−150 K and in external magnetic fields up to 2 kOe a pplied in the plane of the layer and \nperpendicular to the layer for the longitudinal and  polar geometry, respectively. Magneto-\ntransport properties of the layers have been measur ed at liquid helium temperatures in \nsamples of Hall-bar shape supplied with Ohmic conta cts to the (Ga,Mn)(Bi,As) layers using a \nlow-frequency lock-in technique, as described in ou r earlier paper [13].  \n \n3. Experimental results and discussion \nHigh-resolution X-ray diffraction characterization of the investigated layers confirmed \ntheir high structural perfection and showed that th e layers grown on GaAs substrate were \npseudomorphically strained to the substrate under c ompressive misfit strain. The 004 \ndiffraction patterns recorded from the layers displ ayed symmetric (Ga,Mn)(Bi,As) peaks with \nstrong interference fringes, indicating homogeneous  layer compositions and good interface \nquality. An addition of a small amount of Bi to the  (Ga,Mn)As layers resulted in a distinct \nincrease in their lattice parameter perpendicular t o the layer plane and an increase in the in-\nplane compressive strain [14, 15]. On the other han d, the (Ga,Mn)(Bi,As) layers grown on the  4 (In,Ga)As buffer layer, with distinctly larger latt ice parameter, were pseudomorphically \nstrained to the buffer under tensile misfit strain,  as obtained from the 422 reciprocal lattice \nmaps, cf. Fig. 1. The annealing treatment applied t o the (Ga,Mn)(Bi,As) layers resulted in a \ndecrease in their lattice parameters and the strain . Taking into account the recent Rutherford \nbackscattering spectrometry results by Puustinen et  al. [16], which gave no evidence of Bi \ndiffusing out of Ga(Bi,As) layers during annealing at temperatures of up to 600°C, we assume \nthat the observed decrease in lattice parameters of  the (Ga,Mn)(Bi,As) layers under the \nannealing treatment resulted mainly from outdiffusi on of Mn interstitials from the layers. The \nin-plane misfit strain (lattice mismatch) values in  the annealed (Ga,Mn)(Bi,As) layers, \ncalculated from the XRD results, were 4.6 ×10 -3 (compressive strain) and −9.7 ×10 -3 (tensile \nstrain) for the layers grown on the GaAs substrate and on the (In,Ga)As buffer, respectively.  \nThe longitudinal MOKE measurements were performed a s a function of magnetic field \napplied along the main in-plane crystallographic di rections: [100], [110] and ] 10 1 [ . The \nrepresentative MOKE magnetization hysteresis loops of the (Ga,Mn)(Bi,As) layer grown on \nGaAs are presented in Fig. 2. The layers grown on G aAs displayed nearly rectangular \nhysteresis loops in the magnetic reversal measured with longitudinal MOKE and no magnetic \nreversal measured with polar MOKE, evidencing for t he in-plane magnetization. Closer \ninspection of the MOKE hysteresis loops obtained un der a magnetic field along the main in-\nplane crystallographic directions indicates easy ma gnetization axes along the in-plane 〈100 〉 \ncubic directions and hard axes along two magnetical ly nonequivalent in-plane 〈110 〉 \ndirections, with the ] 10 1 [  direction being magnetically easier than the perpendicular [110] \none. Such a rather complicated magneto-crystalline anisotropy is characteristic of (Ga,Mn)As \nlayers grown under compressive misfit strain [11, 1 7]. \nOn the other hand, the (Ga,Mn)(Bi,As) layers grown on the (In,Ga)As buffer displayed \nclear hysteresis loops while measured in polar MOKE  geometry, as shown in Fig. 3, and no \nmagnetic reversal measured in longitudinal MOKE geo metry. These results evidence for the \neasy magnetization axis along the [001] growth dire ction, characteristic of (Ga,Mn)As layers \ngrown under tensile misfit strain [17]. The coerciv e field for the annealed (Ga,Mn)(Bi,As) \nlayer grown on the (In,Ga)As buffer was 670 Oe at T = 6 K (cf. Fig. 3) and it was much larger \nthan the coercive field of 75 Oe at the same temper ature for the annealed (Ga,Mn)(Bi,As) \nlayer grown on GaAs (cf. Fig. 2). From the analysis  of temperature dependences of MOKE \nmagnetization hysteresis loops we have determined t he ferromagnetic Curie temperatures, TC, \nin the investigated layers. The as-grown layers dis played the TC values of about 60 K, which  5 have been enhanced as a result of the annealing tre atment to the values of 100 K and 125 K \nfor the layers grown under compressive and tensile misfit strain, respectively. \nThe electrical-transport properties of ferromagneti c materials are strongly affected by \ntheir magnetic properties resulting in an anisotrop ic magneto-resistance, which reflects the \nmaterials’ magneto-crystalline anisotropy [18]. Mag neto-transport properties of the \n(Ga,Mn)(Bi,As) layers have been measured as a funct ion of magnetic field applied both along \nthe main in-plane crystallographic directions and p erpendicular to the layer. While sweeping \nthe magnetic field up and down in the range of ±1 kOe, the layer resistance varied non-\nmonotonously displaying double hysteresis loops cau sed by a rotation of the magnetization \nvector in the layers between equivalent easy axes o f magnetization [13]. The magneto-\ntransport results fully confirmed the in-plane and out-of-plane easy axes of magnetization in \nthe layers grown under compressive and tensile stra in, respectively. Moreover, those results \nclearly evidenced a significant increase in the hol e concentration as a result of the annealing \ntreatment, which was the main reason of the observe d enhancement of the layer Curie \ntemperature.  \n \n4. Summary and conclusions \nHomogeneous layers of the (Ga,Mn)(Bi,As) quaternary  diluted magnetic semiconductor \nhave been grown by the low-temperature MBE techniqu e on either GaAs substrate or the \nsame substrate covered with a thick (In,Ga)As buffe r layer. High-resolution X-ray diffraction \ncharacterization of the layers showed that in both cases they were grown pseudomorphically, \nunder compressive or tensile misfit strain, respect ively. Magnetic properties of the layers were \nexamined with the MOKE magnetometry and low-tempera ture magneto-transport \nmeasurements. The obtained results revealed the in- plane and out-of-plane easy axis of \nmagnetization in the layers grown under compressive  and tensile misfit strain, respectively. \nPost-growth annealing of the layers, causing outdif fusion of self-compensating Mn \ninterstitials, results in significant increase in t he layer Curie temperature and the hole \nconcentration. Incorporation of a small amount of B i into the (Ga,Mn)As layers, which results \nin distinctly stronger spin-orbit coupling in the l ayers, does not markedly change their \nmagnetic properties.  \nAcknowledgment \nThis work was supported by the Polish National Scie nce Centre under grant No. \n2011/03/B/ST3/02457.  6 References \n[1] F. Bastiman, A. R. B. Mohmad, J. S. Ng, J. P. R . David, S.J. Sweeney, J. Cryst. Growth  \n338 , 57 (2012). \n[2] R. B. Lewis, M. Masnadi-Shiraz, T. Tiedje, Appl. Phys. Lett.  101 , 082112 (2012). \n[3] K. Alberi, J. Wu, W. Walukiewicz, K. M. Yu, O. D. Dubon, S. P. Watkins, C. X. Wang, X. \nLiu, Y.-J. Cho, J. Furdyna, Phys. Rev. B  75 , 045203 (2007). \n[4] Z. Batool, K. Hild, T. J. C. Hosea, X. Lu, T. T iedje, S. J. Sweeney, J. Appl. Phys.  111 , \n113108 (2012). \n[5] B. Fluegel, S. Francoeur, A. Mascarenhas, Phys. Rev. Lett.  97 , 067205 (2006). \n[6] M. Usman, C. A. Broderick, Z. Batool, K. Hild, T. J. C. Hosea, S. J. Sweeney, E. P. \nO’Reilly, Phys. Rev. B  87 , 115104 (2013). \n[7] Yu. A. Bychkov, E. I. Rashba, J. Phys. C: Solid State Phys. 17 , 6039 (1984). \n[8] T. Dietl, H. Ohno, Rev. Mod. Phys.  86 , 187 (2014). \n[9] M. Tanaka, S. Ohya, P. N. Hai, Appl. Phys. Rev.  1, 011102 (2014). \n[10] T. Figielski, T. Wosinski, A. Morawski, A. Mak osa, J. Wrobel, J. Sadowski,  Appl. Phys. \nLett . 90 , 052108 (2007). \n[11] T. Wosinski, T. Andrearczyk, T. Figielski, J. Wrobel, J. Sadowski, Physica E  51 , 128 \n(2013). \n[12] I. Kuryliszyn-Kudelska, J. Z. Domagała, T. Woj towicz, X. Liu, E. Łusakowska, W. \nDobrowolski, J. K. Furdyna, J. Appl. Phys.  95 , 603 (2004). \n[13] K. Levchenko, T. Andrearczyk, J. Z. Domagala, T. Wosinski, T. Figielski, J. Sadowski, \nActa Phys. Polon A  126 , 1121 (2014). \n[14] O. Yastrubchak, J. Sadowski, L. Gluba, J.Z. Do magala, M. Rawski, J. Żuk, M. Kulik, T. \nAndrearczyk, T. Wosinski, Appl. Phys. Lett . 105 , 072402 (2014). \n[15] K. Levchenko, T. Andrearczyk, J. Z. Domagala, T. Wosinski, T. Figielski, J. Sadowski, \nPhys. Status Solidi C  (2015) (in print). \n[16] J. Puustinen, M. Wu, E. Luna, A. Schramm, P. L aukkanen, M. Laitinen, T. Sajavaara, M. \nGuina, J. Appl. Phys.  114 , 243504 (2013). \n[17] F. Matsukura, M. Sawicki, T. Dietl, D. Chiba H . Ohno, Physica E  21, 1032 (2004). \n[18] K. Hong, N. Giordano, Phys. Rev. B  51 , 9855 (1995). \n \n  7 Figure captions \nFig. 1. Reciprocal lattice map of the annealed (Ga, Mn)(Bi,As)/(In,Ga)As/GaAs \nheterostructure for the 422 XRD reflection where the vertical and horizontal a xes are along \nthe out-of-plane [001] and in-plane [110] crystallo graphic directions, respectively, in \nreciprocal lattice units. The solid and dashed line s denote the reciprocal lattice map peak \npositions calculated for pseudomorphic and fully re laxed layers, respectively. \nFig. 2. Magnetization hysteresis loops of the annea led (Ga,Mn)(Bi,As) layer grown on GaAs \nsubstrate recorded at various temperatures (written  in the figure) with the longitudinal MOKE \nmagnetometry under an in-plane magnetic field along  the [100] crystallographic direction. \nThe curves have been vertically offset for clarity.  \nFig. 3. Magnetization hysteresis loops of the annea led (Ga,Mn)(Bi,As) layer grown on the \n(In,Ga)As buffer recorded at various temperatures ( written in the figure) with the polar \nMOKE magnetometry under a magnetic field along the out-of-plane [001] crystallographic \ndirection. The curves have been vertically offset f or clarity. \n \n-3.30 -3.25 -3.20 -3.15 -3.10 -3.05 -3.00 -2.95 -2.90 4.35 4.40 4.45 4.50 \n[-2-2 4] (Ga,Mn)(Bi,As) / GaAs  \n Qy (rlu) \nQx (rlu) \nGaMnBiAs \nInGaAs GaAs \n(Ga,Mn)(Bi,As) / \n(In,Ga)As / GaAs \npseudomorphic line relaxation line \n[-2-2 4] \n \nFig. 1  8  \n-0.4 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 0.4 04812 16 20 Magnetisation (arb. units) \nH (kOe)  142 K \n 85 K \n 65 K \n 45 K \n 25 K \n 6 K \n \nFig. 2 \n \n \n-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 012345678Magnetisation (arb. units) \nH (kOe)  135 K \n 120 K \n 90 K \n 60 K \n 30 K \n 6 K \n \nFig. 3 \n "    },    {        "title": "1507.02711v1.Validity_of_the_Néel_Arrhenius_model_for_highly_anisotropic_Co_xFe__3_x_O_4_nanoparticles.pdf",        "content": "1 \n  \nValidity of the Néel -Arrhenius model for highly anisotrop ic  \nCo xFe 3-xO4 nanoparticles  \n \n \n \nT.E. Torres1,2,3,†, E. Lima Jr .4, A.Mayoral1,3, A.Ibarra1,3,  C. Marquina2,5 M. R. Ibarra1,2,3 and G. F. Goya1,2  \n \n1 Instituto de Nanociencia de Aragón (INA), Universidad de Zaragoza, Zaragoza , Spain.   \n 2 Departamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad de Zaragoza, \nZaragoza , Spain.  \n3 Laboratorio de Microscopias Avanzadas (LMA), Universidad de Zaragoza, Zaragoza , Spain.  \n4 División  Resonancias Magnéticas, Centro Atómico Bariloche/CONICET, S. C. Bariloche 8400, \nArgentina.  \n5 Instituto de Ciencia de Materiales de Aragón (ICMA), CSIC - Universidad de Zaragoza, Zaragoza , \nSpain.  \n \n \n \n \nAbstract.  \nWe report  a systematic study on the structural and magnetic properties of CoxFe3-xO4 \nmagnetic nanoparticles  with sizes between 5 to 25 nm , prepared by thermal \ndecomposition of Fe(acac) 3 and Co(acac) 2. The large magneto -crystalline  anisotropy of \nthe synthesized particles  resulted in high blocking temperatures  (42K< TB< 345K  for 5 \n< d < 13 nm) and l arge coercive fields (HC ≈ 16 00 kA/m for T = 5 K) . The smallest \nparticles ( <d>=5 nm)  revealed the existence of a magnetically hard , spin-disordered  \nsurface . The thermal dependence of static and dynamic magnetic properties of the \nwhole series of samples could be explained within the Neel –Arrhenius relaxation \nframework  by including the thermal dependence of the magnetocrystalline anisotr opy \nconstant K1(T), without the need of ad-hoc corrections . This approach, using the \nempirical Brükhatov -Kirensky relation , provided K1(0) values very similar to the bulk \nmaterial from either static or dynamic magnetic measurements, as well as realistic \nvalues for the response time s (0 ≈ 10-10s). Deviations from the bulk anisotropy values \nfound for the smallest particles could be qualitatively explained based on Zener ’s \nrelation be tween K1(T) and M(T).  \n \n \n \n \n                                                 \n† Corresponding author: teo@unizar.es  \n 2 \n  \nI. INTRODUCTION                 \nFerrites are s pinel oxides with formula MFe2O4 (M = 3d transition metal)  with cubic \ncrystal structure  and a multiplicity of complex  magnetic configurations arising from the \ndivers e interactions between the M and Fe magnetic ions. When  M = Co2+, the resulting \ncobalt ferrite (CoFe 2O4) has distinctive magnetic properties due to its large  first order \nmagnetocrystalline anisotropy constant (K1= 2x105 J/m3), which is about  an order of \nmagnitude greater than any other spinel oxide .1 Together with its chemical stability, th is \npropert y make CoFe 2O4 magnetic nanoparticles (MNPs) a fundamental material for \nmagnetic recording applications and ferrofluids.2 Considerable efforts have been made \nto obtain homogenous and stable water -based nanofluids through different synthesis \nroutes such as hydrothermal, coprecipitation, microemulsion, forced hydrolysis, and \nreduction -oxidation methods .2 In particular, the t hermal  decomposition of \norganometallic precursors in a boiling solution  of organic solvents has been successfully \nused to  produce  MNPs  with narrow size dispersion ,3,4 and thus they are being \nincreasingly exploited in those  applications with critical specifications about size \ndispersion of the MNPs .5 \nThe ferrimagnetic order in CoFe 2O4 result s from the competing super -exchange \ninteraction s between the two magnetic sublattices of tetrahedral (A) and octahedral (B) \nsites in the structure. The  Fe+3 ions within the B sublattice are ferromagnetically \nordered, as well as the Co+2 ions within the A  sublattice. On the other hand, the \ninteractions between A  and B spin sublattices  are antiferromagnetic, resulting in an \nuncompensated net magnetic moment . The exchange energy  in this material has been \nreported to be as large as  JAF = -24 kB.6 It is well known that t he relation between the \nanisotropy and exchange energies  determines the  critical size  (Dcr) for the single domain \nconfiguration.  The existence of a critical  diameter Dcr of a (spherical ) particle implies \nthat below a certain diameter value d such that d < Dcr, the lowest free energy state is \nthat of uniform magnetization, as proposed by Brown .7 This critical value has been \nestimated 8,9 to be 𝐷𝑐𝑟=5.1√𝐴\n𝜇0𝑀𝑆2, where 𝐴 is the exchange stiffness10 and M S is the \nsaturation magnetization of the material. Usin g 𝐴 = 15 x10-12 J/m; MS = 425 A/m (bulk  \nCoFe 2O4)11 and 𝜇0=4𝜋𝑥10−7 H/m,  a critical diameter  𝐷𝑐𝑟=40,7 nm is obtained. \nAccordingly, reported values of the single domain critical size for CoFe 2O4 are between 3 \n 30 and 70 nm.12 As a consequence of  the large magnetic anisotropy , single domain \nparticles of  CoFe 2O4 of a few-nanometer size can retain the blocked regime up to room \ntemperature. This particularity allows observing the thermal evolution of some magnetic \nparameters of MNPs such as saturation magnetization and coercivity  of the blocked \nstate in a wide range of temperatures before the superparamagnetic transition wipes out \nthis information.   \n \nThe energy E of an assembly of uniaxial particles with their easy axes parallel to the z \naxis under an external applied field is usually described (at T=0) by: \n \n                                    𝐸(𝑉)=𝐾𝑒𝑓𝑓𝑉 𝑠𝑖𝑛2 𝜃+𝐻𝑀𝑆𝑉cos𝜃                                    (1) \n \nwhere  is the angle between field H and saturation magnetization MS, V the particle \nvolume and Keff is the effective magnetic anisotropy. Assuming the energy of a single \nparticle given by equation (1), the unblocking process occurs through an energy barrier  \nE given by : \n \n                                            ∆𝐸=𝐾𝑒𝑓𝑓 𝑉(1−𝐻𝑀𝑆\n2𝐾𝑒𝑓𝑓)2\n (2)  \n \nAt a fixed  temperature T the reversal of the magnetic moment occurs through the energy \nbarrier  given  by equation (2) . This  thermally -activated process  is described by the  Néel -\nArrhenius model , which gives a simple expression for the relaxation time  𝜏=\n𝜏0𝑒𝐾𝑒𝑓𝑓𝑉\n𝑘𝑇⁄. Taking  = 102 s for the measuring time window and 0 = 10-9 s we get \nKeffV = 25 k BT the coercive field H C (T) can be expressed as : \n  \n                               𝑯𝑪(𝑻)=𝟐𝑲𝒆𝒇𝒇\n𝑴𝑺[𝟏−(𝟐𝟓𝒌𝑩𝑻\n𝑲𝒆𝒇𝒇𝑽)𝟏/𝟐\n]                                (3) \n \nThis is the well -known H C vs. T1/2 relation often used for fitting the temperature \nevolution of the coercive field in the blocked state, i.e., at low temperatures. It is worth \nto note here that t he thermal dependence of Keff in equation (3) is neglected , although 4 \n previous studies of bulk spinel  oxide s have reported  large variations of the anisotropy \nbelow room temperature .13   \nIn this work, we  report a systematic study on the magnetic properties in a series of Co \nferrite  magnetic nanoparticles within 5 and 25 nm. An exhaustive study by high \nresolution electron transmission microscopy (HRTEM) techniques has been performed \nin order to explore the influence of MNPs size and shape on the observed \nmagnetocrystalline anisotropy ,14 with a precise observation  of the crystallographic \nstructure with atomic resolution. The chemical compositi on at the single -particle level \nwas performed  to assess the levels of stoichiometric homogeneity of samples . \nSystematic measurements of magnetization, coercive field and magnetic anisotropy \nwere performed for increasing particle size to study the temperatu re evolution of the \nmagnetic parameters in the blocked regime. The validity of the Neel -Arrhenius law for \nexplaining the temperature dependence of the relaxation time has been re -gained  by \ntaking into account the variation of the anisotropy constant with the temperature.  \n \nII. EXPERIMENTAL  \n \nCoxFe3-xO4 nanoparticles of different sizes were prepared by thermal decomposition 3 of \niron acetylacetonate Fe(acac) 3 and cobalt acetylacetonate Co(acac) 2 as precursors4. \nDifferent solvents (phenyl ether, benzylether, 1 -octadecene, and trioctylamine) with \nincreasing boiling temperatures were used in order to control the final particle size. For \na standard  preparation, 10.4 mmol of Fe( acac )3 and 5.2 mmol of Co( acac )2 were \ndissolved in 52 mmol of Oleic acid (OA), 65.4 mmol of Oleylamine, 86.5 mmol of 1,2 \nOctanediol  and 150 ml of the chosen solvent. Then , the mixture was heated up to the \nstabilization temperature T St (200 °C in this case) under mechanical stirring under a \nflow of nitrogen gas  for the nucleation step . This temperature was kept constant for 120 \nminutes, and then the solution was heated to the boiling temperature of the solvent  (260-\n330°C), that is the final synthesis temperature, T FSt, in nitrogen atmosphere. After \nwaiting a few mi nutes  (depending on the sample)  at this temperature  the solution was \ncooled down to room temperature. The resulting CoxFe3-xO4 MNPs were washed three \ntimes with ethanol, and then magn etically -assisted precipitated until the supernatant 5 \n solution became clear. Afterwards, the final product , composed by ferrite nanoparticles \ncoated with a layer of oleic acid,  was re -dispersed in hexane.  \nThe samples were labeled as AVXX, where the number XX represents the average \nparticle diameter  (in nanometers)  obtained from the core distributions observed in TEM \nimages  (see below). The resulting samples showed average particle diameters ranging \nfrom 5 to 25 nm. Details of the  ether/alkenes used as solvents in each case , together \nwith the stabilization (TSt) and final synthesis  (TFSt) temperature s used in each synthesis \nare given in Table SI of the supplemental material.  In the case of sample AV11, the only \nsample synthesized in trioctylamine , the temperature was carefully raised for 10 \nminutes, from 320 °C up to TFSt=330 °C. Once TFSt was reached the sample was \nimmediately cooled down . It is also worth to mention  that s amples AV16 and AV18 \nwere obtained from  the same dispersion of nanoparticles by magnetically -assisted \nprecipitation : sample  AV18 was collected as the precipitated MNPs after applying a \nferrite permanent  magnet for 10 s to the as synthesized colloid  and re-dispers ing this \nprecipitate  in hexane . The supernatant  resulting from this separation was precipitated a \nsecond time applying the  magnet for 5 minutes , and re -dispersed in hexane . This latter \nsample was labeled as  AV16.  Sample  AV25 was grown using the heterogeneous \nmethod  3 starting from already existing MNPs (sample  AV13 ) as seeds, and mixing 80 \nmg with the same molar concentration of reactants.  \nThe morphology and stoichiometry of the MNPs were studied by Transmission and \nScanning Electron Microscopies (TEM and SEM, respecti vely). TEM images were \nobtained using a thermo -ionic LaB 6 200 kV Tecnai T20 microscope operating at an \naccelerating voltage of 200 kV. STEM –HAADF (Scanning  Transmission Electron \nMicroscopy  using a High Angle Annular Dark Field  detector ) images were acquired \nusing a XFEG TITAN 60 –300 kV, operated at 300 kV, equipped with  monochromator \nand with a CEOS hexapole aberration corrector for the electron probe . TEM specimens \nwere prepared by placing a drop of a hexane solution containing the MNPs onto a holey \ncarbon coated copper micro -grid. The mean particle size <d> and size di stribution were \nevaluated by measuring about 150 -500 particles found in arbitrarily chosen areas of \nenlarged micrographs of different regions of the micro -grid. SEM measurements were  \ncarried out in a FEI INSPECT F with INCA PentaFETx3  system operating at 20 keV.  \nThe ratio between iron and cobalt content was determined through Energy -Dispersive \nX-ray spectroscopy (EDX) performed on a macroscopic zone of a powder sample 6 \n (about 10 000 m2) in SEM analyses, and on a small area (about 1 000 nm2) containing \nmany  particles as well as on single particles using the  TEM.  \n \nThe total iron concentration was determined from  UV/Vis spectroscopy in a Varian \nCary 50 Spectrophotometer operating at a fix wavelength of 478 nm. For the \nabsorbance measurements , Potassium thiocy anate (KSCN ) was used following the \nstandard protocol described  elsewhere .15,16  \nMagnetization measurements M (T, H) and ac magnetic susceptibility measureme nts \nwere performed on a MPMS -XL SQUID Quantum Design  magnetometer. All \nmeasurements were performed on dried samples, after conditioning the dry powder \ninside plastic capsules. The temperature dependence of the magne tization was measured \nfollowing zero -field-cooling (ZFC) and field cooling (FC) protocols, applying 7.9 \nkA/m, and the data were collected increasing the temperature from 5 to 400 K. The \nmagnetization isotherms  were measured between 5 and 400 K up to a maxi mum \nmagnetic field of 3.96 MA/m. The susceptibility versus temperature was measure d \napplying an excitation ac field of 0.24 kA/m, at frequencies from 0.1 to 103 Hz, under \nzero external dc magnetic field.  \n \nIII. RESULTS AND DISCUSSION  \n \nA. Particle morphology and composition analyses  \n \nThe an alysis of the TEM images (Fig.  1) showed that  for each particular synthesis, the \nMNPs obtained can be considered as  uniform in size.  The statistical analysis of the \nMNPs size distribution done by  fitting the respective size -histograms to a Gaussian \ndistribution yielded mean diameters ranging from d = 5 to 25  nm and standard \ndeviations  size distribution widths  between  0.7 and 3 nm (see Table I).  7 \n  \nFigure 1. TEM images of CoFe 2O4 nanoparticles for the AV XX series. The corresponding size histograms \nare shown below each image, together with the Gaussian fit, (solid lines) and the obtained mean size <d> \nand distribution width ( ). All micrographs were taken at the same magnification.  \n \nAs previously reported for this synthesis route, t he final average p article size  reflected \nthe influence of both the boiling point of the solvent  and boiling time .3, 17 Specifically , a \nsystematic increase of the average particle size <d> for increa sing boiling temperature \nof the solvent was observed . In the case of s ample AV11 the  final size is a combination \nof the higher boiling temperature and a shorter time at T Fst (10 minutes , see Table  SI of \nsupplemental material ).18 Regarding the MNPs morphology, the analysis of HRTEM \nimages showed that for <d>  13 nm  a noticeable population of  rounded -shaped  \nparticles were present , whereas the largest ones  showed  a more faceted structure  (see \n0 10 20 30 40050100\n Counts\nd(nm)    AV05\n<d> = 5 nm\n = 0.7 nm\n0 10 20 30 40050100150200\n Counts\nd(nm)       AV08\n<d> = 8.8 nm\n = 1.3 nm\n0 10 20 30 40050100\n d(nm)\n Counts     AV11\n<d>=11 nm\n = 1.5 nm\n0 10 20 30 40050100150\n Counts\nd(nm)       AV13\n<d>=13.3 nm\n = 1.3 nm\n0 10 20 30 40050100\n  \n d (nm)Counts       AV14\n<d>=14.3 nm\n= 2.6 nm\n0 10 20 30 40050\n Counts\nd(nm)       AV16\n<d> =16.8 nm\n     = 1.7 nm\n10 20 30 40050\n Counts\nd(nm)      AV18\n<d>=18.6 nm\n = 1.7 nm\n10 20 30 40035\n  Counts \nd (nm)     AV25\n<d>= 25 nm\n   = 2.1 nm8 \n Figs 1 and 2  and Fig. S1 in supplementary material) . It has been proposed that the \ndifferent morphologies are related to the rate of the temperature increase from  the \nstabilization temperature (T St) to the final synthesis temperature (T FSt), and to the to tal \nreaction time at T Fst.18 Assuming  the thermal decomposition as an autocatalytic \nreaction4 it is expected that the conce ntration of precursor in the solution , which is \ninversely proportional to the volume of the particle,  has a time dependence described by \nthe logistic equation  19. For the sample s prepared with T FSt = 320 and 330 °C a linear \ndependence of <d> with the time of solution at T FSt has been observed , suggesting  that \nthe reaction is in an intermediat e time regime without the complete consumption of the \nprecursor. In addition, the composition  analyse s presented later on  also show a time \ndependence of the composition on TFSt, for TFSt = 320 and 330 °C, indicating that the \nchemical kinetic s of Co and Fe i ncorporation onto the particle are different . For lower \nTFSt temperatures, we observe smaller values of <d> . \n \nFigure 2. (a) Cs-STEM –HAADF image of sample AV13 and (b) Cs-STEM –BF image of sample AV18 . \nSpherical and faceted  morphologies  are observed . \n \n \nThe C S-corrected STEM -HAADF analys is at atomic resolution reveal ed that all the \nsynthesized nanoparticles  crystallize d in the spinel structure with Fd-3m space group \nand unit cell parameter a = 8.394 Å. The data showed no evidence of distortions , crystal \ndefects or any  preferential orient ation of the nanoparticles. As an example, Fig.  3(a) \nshows a high resolution CS-STEM -HAADF  image of a particle of  sample  AV13 . \n1 0  n m\n(b) \n5  n m\n (a) 9 \n  \nFigure 3. (a) High -resolution Cs STEM –HAADF image of a particle of the AV1 3 sample  with its  \ncorrespond ent FFT inset . (b) Simulated  image (c) a magnified region displaying the atomic distribution \nwith the model superimposed.  \nThe inset corresponds to the Fast Fourier Transform ( FFT)  in the [111] zone axis, \nshowing the spots corresponding t o (02̅2) and (022̅) planes . Fig.  3(b) shows a simulated \nimage using the parameters of the CoFe 2O4 structure from the Crystallography Open \nDatabase  (St. sample card Nº 22 -1086 of the JCPDS -International Centre for \nDiffraction Data® -ICDD®) . Finally, Fig.  3(c) displays  the superposition of the \nsimulated and real crystal structures , showing  the coincidence of both of them.  \nThe relative abundance of cobalt an d iron in the samples was obtained from EDX in \nSEM analyses,  taking spectra in different zones of the sample . In SEM the electron \nbeam spot has a diameter between 10 -100 nm and the emitted X -rays are collected from \nan underly ing sample volume of a bout 1 -3 m deep . Therefore the information of the \natomic composition corresponds to a volume around 0.2 m3 and therefore  these results \nreflect the  ‘macroscopic’ average composition of the sample . As an example , a \nmicrograph  corresponding to sample AV14 ( <d> = 14 .3 nm)  is shown in Fig. S2(a) of \nthe supplementary material , indicating  the squared -defined area  for EDX –SEM \nsampling . The corresponding EDX spectrum from this area  (Fig.  S2(b) of the \nsupplementary material)  show ed the peaks associated with the K and L edges of iron \nand cobalt  atoms . A minimum of five  areas within the sample holder were studied for \neach sample, and in all cases the results were coincident within the experimental error, \nsupporting the macroscopically ho mogeneous nature of the samp les. The  results are  \nsummarized in  Table I. A deviation from the stoichiometry  (i.e., atomic ratio \nFe]/[Co] =2.0) can be noticed , showing an  excess of iron  in all the samples . The \n5  n m\n1  n m\n(a) \n(02̅2) \n[111]  \n(022̅) \n(b) \n(c) 10 \n resulting composition of the CoxFe3-xO4 MNPs extracted for these analysis yielded x \nvalues ranging from 0.90 (sample AV05) to 0. 54 (for sample AV08) . \n \nTable I:  Average particle diameter d with deviation , atomic Fe/Co ratio  ( obtained by \nEDX -SEM and EDX -TEM, and the resulting chemical composition Co xFe3-xO4  \nSample  <d>   Fe]/[Co]Fe]/[Co]CoxFe3-xO4 \n (nm)  (nm)  EDX -SEM  EDX -TEM   \nAV05  5.0 0.8 2.3 2.9 Co0.90Fe2.10O4 \nAV08  8.8 1.3 4.5  Co0.54Fe2.46O4 \nAV11  11.0 1.6 2.9  Co0.77Fe2.23O4 \nAV13  13.3 1.3 3.4  Co0.68Fe2.32O4 \nAV14  14.3 2.6 3.5 4.6 Co0.67Fe2.33O4 \nAV16  16.8 1.7 3.9  Co0.61Fe2.39O4 \nAV18  18.6 1.7 3.5  Co0.66Fe2.34O4 \nAV25  25.0 2.1 3.3 3.6 Co0.70Fe2.30O4 \n      \n \nAnalysis of the chemical composition was also performed through TEM  at the single -\nparticle level,  by acquiring the EDX s pectra of individual particles and small aggregates  \nof MNPs  for samples AV05, AV 14 and AV25 . Typical results obtained for sample \nAV25 (<d>= 25 nm) are displayed in Fig. 4. For all analyzed samples,  the Fe :Co ratios \nderived from individual particles and from particle clusters  coincide , as in the EDX -\nSEM analysis . The close values of both TEM and SEM analysis in each case (see  Table \nI) indicat e that the chemical composition of the MNPs is  homogeneous  throughout the \nsamples and, more importantly, within individual particles . Clearly, this analysis of the \nhomogeneous internal structure of single MNPs is performed  only for a few selected \nMNPs . However,  the consis tency of  these data from several particles  has been  verified \nin all synthesized samp les and therefore, gives support to the statistical confidence of \nthese results.  \n                             11 \n  \nFigure 4. EDX -TEM  carried out on sample AV25 (<d>= 25 nm). Spectra in (c) and (d) correspond to  the \nnanoparticles in the area selected in (a) and to the particle selected in (b).   \n \n \nB. Temperature dependence of the magnetization . \n \nThe main features of magnetization M(T)  curves,  taken in zero-field cooling and field -\ncooling ( ZFC/FC ) mode s for all samples exhibit ed similar trends , as can be seen from \nFig. 5(a). The blocking temperature distribution s were  obtained from the plot of  \n(1\n𝑇)𝑑(𝑀𝑍𝐹𝐶 −𝑀𝐹𝐶)\n𝑑𝑇 vs. T  (Fig. 5b), and the mean blo cking value <T B> was extracted  by \nfitting the experimental data with a  Gaussian  distribution . Large <TB> values were \nobtained  even for the smalle r samples (TB = 42 K for particles with <d>= 5nm ), \nreflecting  the large magnetic anisotropy of CoFe 2O4.17,20,21,22,14 For those particles larger \nthan 14 nm, the blocki ng temperatures were beyond the maximum of our experimental \nsetup . It is interesting to note that  the shift of <TB> to higher temperatures  with \nincreasing particle size  was not linear with particle volume V as expected  from the \nfunctional definition of <T B> in equation (3), i.e., 𝑇𝐵=𝐾𝑒𝑓𝑓𝑉\n25𝑘𝐵⁄ . Instead, a nearly \nlinear dependence on particle diameter  <d> was observed .  \n \n20 nm  \n(b) \n5 0  n m\n  \n(a) \n012345678910(b)CO\nCoKFeK\nCoLFeL\n \n  \nEnergy (keV)Intensity (A.U)\n012345678910C\nO\nCoLFeL\nCoKFeK\n \n  Intensity (A.U)\nEnergy (keV)(d)12 \n  \n0,00,40,8\n05\n48\n0612\n0 80 160 240 320 4001020\n \n  \n<TB> 345 K<TB> 306 K<TB> 259 K<TB> 182 K<TB> 42 K\n  \n   \n  \n0 80 160 240 320 400\n   (b)\n  M(Am2/Kg)\n (a)     AV05\n<d> =5.5 nm\n        AV13\n<d> =13.3 nm     AV08\n<d> =8.8 nm\n \n  \n      AV11\n<d> =11 nm\n   \n \n           AV14\n<d> =14.3 nm\nT(K)  \nT(K)\n \nFigure 5. (a) M (T) data taken in z ero-field-cooled (ZFC) and field -cooled (FC) modes for samples AV05 \nto AV14 (<d> between 5 to 14.3 nm).  (b) Blocking temperature distribution s fitted with a Gaussian \nfunction (solid line) .  \n \nC. Magnetic field dependence of the magnetization . \n \nThe magnetization of all samples was studied at temperatures from 5 K to 400 K , in \napplied field H up to 11.2 MA/m (14 T) . For the M(H) performed at T = 400 K, t he \nobtained coercive field values HC decreas ed with decreasing particle size  (Table II) \nattaining zero for the samples with d  13 nm, in agreement with the  blocking \ntemperatures observed from  ZFC/FC curves.  At T = 5 K, the h ysteresis loops (figure 6 ) \nshowed similar  features  for all samples , i.e., large coercive fields H C and saturation \nmagnetization values M S around 60 Am2/kg. The values of M S at 5 K collected in Table \nII are l ower than the typical ( MS = 80  Am2/kg) found for bulk Co Fe2O4.11 This \nreduction of M S has been previously observed  for small particles (1 -10 nm) and thin \nfilms23, 24 and it could be related to changes in the inversion degree of the spinel \nconfiguration. Indeed, there is no clear consensus about the inversion degree of cobalt \nferrite in bulk and in nanostructured forms , probably because the relative occupancy of \nthe A and B sites by Co and Fe seems to depen d on sample preparation  details . While 13 \n neutron diffraction studies25 have indicated that bulk CoFe 2O4 has an inverted spinel \nconfiguration, latter Mossbauer and X -ray spectroscopy data26, 27 indicated  a partially \ninverted  configuration, consist ent with inversion degree s as high as  i=0.76  in the \nformula [𝐶𝑜1−𝑖𝐹𝑒𝑖]𝐴[𝐶𝑜𝑖𝐹𝑒(2−𝑖)]𝐵𝑂4.28 A second explanation for the observed  \nreduction in M S could be the existence of spin canting at the particle surface  11, 29 \noriginated from competing interactions between A and B sublattices when a symmetr y \nbreak and oxygen vacancies are produced at the particle surface.  Monte Carlo \nsimulations using different models30,31 and approximations have shown that the \nreduction of MS is size dependent, and is related to the canted configuration of the spins \nat the surface . \n \n \nTable II: Blocking temperature <T B>, coercive field H C and saturation m agnetization M S of \nCoxFe3-xO4 samples  with different average particle diameters, <d>.  \n*Values of M S and H C for sample AV05 obtained from the high -field M(H) cycles (up to H = 11.4 \nMA/m) at 5 K and 400 K .  \n \nFor all but AV05 and AV08  samples (i.e., the two smallest particle sizes) , the \nmagnetization was nearly  saturate d at H  = 2x103 kA/m. Samples AV05  showed  a \nmarked decrease in the magnitude of M, and no signs of saturation  up to the highest \nfield. We further investigate this behavior of sample AV05  through measuring the M(H) \ncurves up to H = 11.2 MA/m at 400 K  and at 5 K (see Figure 7). As expected for a \nminor loo p, saturation was not re ached even at this high field  and the cycle remained Sample  <d> <TB > HC (kA/m)  MS (Am2/kg) \n (nm)  (K) 5 K 400 K  5 K 300 K  400 K  \nAV05 5 (0,8 ) 42 390 0 30 24 17 \nAV05*   1060  0 31  61 \nAV08  8,8(1,3 ) 182 1600 0 54 43 40 \nAV11  11 (1,6 ) 259 920 0 61 51 47 \nAV13  13,3(1,3 ) 306 1600 0 86 76 67 \nAV14  14,3(2,6)  345 1600 2 66 57 53 \nAV16  16,8(1,7 ) >400  1400 3 55 47 45 \nAV18  18,6(2,1 ) >400  1500 10 57 47 51 \nAV25  25(4,1)  >400  1030 500 53 48 44 14 \n open showing that the  irreversibility field Hirr, defined as the field where  the two \nbranches of the hysteresis loop merge, was larger  than our attainable maximum field.  \n-80-4004080\n-4000 -2000 0 2000 4000-60-3003060M(Am2/kg)\n AV05\n AV08\n AV11\n AV13\n AV14\n AV16\n AV18\n AV25\nT = 400 KM(Am2/kg)\nH(kA/m)T = 5 K\n \nFigure 6. M (H) curves for all samples measured at (a) T = 400  K and (c ) 5 K \n \n \nThe hypothesis of the surface spin canting that could explain the reduction of \nmagnetization, also would originate the  non-saturating behavior of the M(H) curves \neven at large applied fields , similarly to previous reports on small -sized  ferrite  \nnanoparticles . 21, 28, 32 This is likely  to be the case in our sample s AV05  and AV08, with \na less pronounced effect in AV08 since surface effects are attenuated in particles with \nincreasing volume . \nFor the rest of the samples, however, the decreasing surface/volume ratio would imply \nthat surface spin canting cannot be a major cause for magnetization reduction. \nAdditionally, for these samples the observed reduction i n MS is not accompanied by the \nlinear increase of the magnetization at high field s. On the contrary, the M (H) curves 15 \n showe d that the magnetic saturation is attained at moderate fields (H  2 MA/m), \nconsistent with previous findings  using  polarization -analyzed small -angle neutron \nscattering  experiments on Co -ferrite nanoparticles of 11 nm. 33,34 These results are in \nagreement with our observation  of the concurrent low value of the saturation \nmagnetization and the small fields required to reach M S.33,34 There is e xperiment al \nevidence  that the above mentioned spin canted structure extends over the whole particle \nvolume , instead of  forming  a shell .28 In moderate/ high magnetic fields the meas ured \nmagnetization  is due to the net sum of spin components parallel to the applied field, and  \nthe reduction with respect to the bulk magnetization is due to the cancellation of the \ncomponents perpendicular to the field , as the result of the comp etition be tween Zeeman \nand anisotropy energies. This might be the case of our nanoparticles  with <d>  11 nm , \nbeing the particles with <d> = 13 nm, those in which the canting angle is lower (and \ntherefore the magnetization  is higher). However, local probe and/or neutron scattering \nexperiments  would be necessary to confirm thi s hypothesis.  \n-12 -9 -6 -3 0 3 6 9 12-60-40-200204060\n9 10 1156586062\n M (Am2/kg)\nH (x103 kA/m)  T = 5K\n   T = 400K\nAV05\n  M (Am2/kg)\nH (MA/m)\n \nFigure 7. Magnetization hysteresis curves measured at 400 and 5 K for sample AV05 measurements taken \nuntil 11.2 MA/m (14T) . The inset shows the high -field irreversibility from the T = 5 K data.  \n \nFor AV05 , the drastic reduction of magnetization observed in Figure 6 goes together \nwith a clear non -saturating  behavior up to H = 4 MA/m, also observed (although much \nless pronounced) for sample AV08. Additionally, irreversible behavior up to the largest \nfields (i.e., non -closure of the M(H) loops)  could be observed for AV05 sample.  These \neffects have been observed  in many systems like ZnFe 2O4 29 and CuFe 2O4 35 ferrites and \nwas first explained by J.M.D Coey 36 as originate d from a spin-canted configuration of \nthe surface spins due to broken symmetry at the surface and/ or to oxygen -defici ent \nstoichiometry.  To get further understanding of this process, high -field measurements of 16 \n sample AV05 were performed up to H ≤ 11 MA/m at both T = 5 K and 400 K (see \nFigure 7). The non-saturation observed in AV05  at the highest fields of ∼11 MA/m  \nimplies  anisotropy fields much larger than the expected  from magnetocrystalline or \nshape anisotropy  as sources of magnetic anisotropy, and suggests that spin canting \n(originated in exchange interaction s) must be  operative. In agreement with our results, \nprevious reports by Respaud et al.37 attributed the linear increase of M(H) up to fields of \n28 MA/m observed in ultrasmall coba lt nanoparticles to the major influence of surface \natoms as particle size decrease. Given the small particle size of AV05 samples , the \nincreasing contribution from surface atoms to the overall magnetic moment is the more \nlikely explanation for this M(H) be havior. T he existence of a large number of broken \nexchange bonds at the surface of the particle , associated to the lack of neighboring \natoms has been modeled by a shell of misaligned spins that surrounds  a magnetically \nordered core .38  \nThe values of H C measured at T = 5 K and 400 K ( see Table II) are in agreement  with \npreviously  reported results  in nanosized cobalt ferrite .21, 39-41 The values observed at low \ntemperature are within 1< H C < 1.6 MA/m . As the magnetization isotherm  of sample \nAV05 corresponds  likely to a minor loop , its corresponding small HC value cannot be \ncompared with those of the rest of the series . The value observed for particles with \n<d>= 25 nm is in good agreement with the theoretical calculations performed by \nKachkachi et al.31 that predicted lower coercitivity in faceted nanoparticles as compared \nto spherical ones, due to the higher symmetric coordination of surface atoms and lower \namount of missing coordinating oxygen atoms. However, du e to the synthesis  protocol  \nmentioned in Section II, a mixture of spherical and faceted particles cannot be \ndiscarded. The se synthesis conditions might have also resulted in a distribution of Fe \nand Co atoms among  A and B crystallographic sites different than the rest of the series. \nThe change in site populations  would lead to a different local anisotropy of Co2+ ions, \nwhich could explain the observed lower value of H C.  \n \nD. Temperature  dependence  of the coercive field . \n \nWe have studied the evolution of the coercive field , HC, with the temperature by \nplotting the experimental H C(T,V) data for 5 ≤ T ≤ 400 K . The expected  decrease of  \nHC(T) for increasing temperature  was observed  in all samples , reaching the HC=0 value \nat the  corresponding superparamagnetic transition temperatures . The exact functional 17 \n dependence of H C with temperature for single/domain magnetic nanoparticles in the \nblocked state has been discussed  since decades ago. Within the simple Neel -Arrhenius \nmodel already presented in the introduction  section, a 𝐻𝐶∝ 𝑇1/2 is expected. However, \nequation (3) neglects the particle  size dispers ion existing in any real sample, which is an \noversimplification in most cases .42 Recent works have pointed out the difficulties of \ninclud ing the size distribution  into a realistic model43 because  the measured HC is not a \nsimple superposition of individual particle coercivities. An analytical expression for the \ndependence of HC(T) with T and  particle size has been proposed,44 obtaining a T3/4 for \nthe thermal dependence in a randomly -oriented ensemble of particles. The fact that this \napproximation was unable to fi t our experimental data for any sample, together with the \nquite narrow size distributions observed in our samples (see Figure 1) suggest that \ndeviations from the T1/2 law for H C were not due to size distributions.  \n The departures observed from the H C(T) vs. T1/2 graphs of our samples (see \nFigure S3 in the supplementar y material) were increasingly marked for the larger \nparticles, strongly suggesting that this feature was related with some neglected T-\ndependence of the magnetic parameters involved. As equation (3) assumes  that the \nmagnetocrystalline anisotropy  is a tempe rature -independent parameter , the \ncorresponding H C expression should  be a valid  approximat ion only for a narrow T-\nrange where K 1 is not expected to vary substantially .45, 46 This is the case for particles \nwith low blocking temperatures , since  only in the blocked state H C>0 can be effectively \nmeasured. Indeed, a good T1/2 fits have been reported for small and/or low -anisotropy \nMNPs  (e.g., T< 50 K).47,48, 49 However, this approximation fails completely for particles \nwith large size and/or anisotropies like CoFe 2O4, for which the blocked state may  span a \ntemperature range from 5 to  400 K . In such a wide temperature interval K1(T) can \nchange markedly50 and therefore, the T1/2 dependence of H C is no longer valid.  The \nimportance of the temperature dependence of the anisotropy has been pointed out  in \nprevious works in relation to the thermal dependence of H C of metallic Fe, Co and \nNi8,51,52 nanoparticles, as well as in  Co-containing ferrites.53,54,55,56 However, an explicit \nthermal dependence of the magnetoc rystalline anisotropy has not been so  far included in \nthe expression  of H C(V,T), to the best of our knowledge.  \n \nThe classical theory by Zener  57 on the effect of temperature on the magnetic anisotropy \nprovides a relation between the magnetization M and K 1 of the form58 \n 18 \n 𝑲𝟏(𝑻)\n𝑲𝟏(𝟎)=[𝑴(𝑻)\n𝑴(𝟎)]𝒏\n  (4) \n \nwith n = 10 for full correlation between adjacent spins and n=6 for incomplete  \ncorrelation.59 In cubic ferromagnetic crystals like spinel oxides, this relation is expected \nto hold for temperatures below 0.9T C, being T C the Curie temperature of the material. \nBased on these relationships Shenker60 has demonstrated that for bulk cobalt ferrite \nK1(T) can be expressed by the empirical Brukhatov -Kirensky relation 60 \n \n               𝑲𝟏(𝑻)=𝑲𝟏(𝟎)𝐞𝐱𝐩  (−𝑩𝑻𝟐)                         (5) \n \nvalid  for the 20 K< T < 350  K temperature range , with 𝐾1(0)=1.96𝑥106 𝐽/𝑚3 and \n𝐵=1.9𝑥10−5 𝐾−2. Incorporating this dependence into the H C(T) expression  given by \neq. (3)  and considering  that Keff  as the first magnetocrystalline anisotropy constant K1 \nwe obtain :  \n \n             HC(𝑻)=𝟐𝑲𝟏(𝟎) 𝒆−𝑩𝑻𝟐\n𝝁𝟎𝑴𝑺[𝟏−(𝟐𝟓𝒌𝑩𝑻\nV𝑨𝒆−𝑩𝑻𝟐)𝟏/𝟐\n]             (6) \n \nAs seen in Figure 8, this expression provides an  excellent  fit of the experimental data  \nfor a wide range of particle sizes and temperature , and makes clear that any atte mpt of  \ndescribing the thermal evolution of any magnetic parameter depending on Keff over \nmore than a few -degrees temperature range should consider the impact of K1(T).  19 \n \n0 100 200 300 4000,00,51,01,5\n  AV05\n AV08\n AV11\n AV13\n AV14\n AV16\n AV18\n AV25HC(MA/m)\nT(K) \nFigure 8. Temperature dependence of  the coercive  field HC. The dashed lines are the corresponding  fit \nusing HC(T) given by eq. (6).  \nThe values of K1(0) and B obtained from Figure 8 using eq.(6) are listed in Table III. \nThey are in excellent agreement with previous ex perimental reports53 ,61, 56, 62 and \ntheoretical calculations63,64 for this material . For those samples with <d> between 13 - 25 \nnm the obtained K1(0) values spanned a narrow range 2.8 -5.4x106 J/m3, with a  \nmaximum difference of ≈60% from the bulk value in sample AV16 .  \nThe magnetocrystalline anisotropy of CoFe 2O4 is due to the spin -orbit coupling, mainly \nfrom the contribution of the Co+2 cations at the octahedral B sites . Therefore, changes in \nthe occupancy  factor of A and B sites usually reported in many spinel ferrites65,66 could \nbe expected to yield  changes in K1 values. The fact that the chemical composition of our \nnanoparticles is off -stoichiometric would have led us to expect this departure  in the \ncobalt content to influence the magnetocrystalline anisotropy as well. Our data showed \nno major deviations from nominally stoichiometric bulk samples regarding magnetic \nanisotropy. For the smallest samples AV05 and AV08, a n increase in both K 1 and B \nfitted parameters can be  noticed . As the B parameter is related to the n exponent of \nZener’s relation, it seems plausible  that the non -saturation behavior due to the spin \ncanting will translate in large deviations of the M(T)/M(0) ratio, thus affecting the B \nparameter. Similar arguments could be applied to qualitatively explain the  additional \ncontribution to the anisotropy  observed for K1(0) in AV05 and AV08 samples . \n \nTable III. Parameters K1(0) and B obtained from a) fitting the HC(T) data using eq.(6); and b)  \nNéel –Arrhenius model using the eq. (8). For the latter, the values of 0 are also listed.  20 \n  \n \nE. Temperature and frequency dependence of the AC magnetic susceptibility  \n \nIn order to get a deeper insight into the effective magnetocrystalline anisotropy obtained \nfrom dc data,  the magnetic dynamics o f the se nanoparticles was studied through the \ntemperature dependence of  and at fixed field amplitude and increasing frequency \nfrom 100 mHz to 1 kHz. Typical ly, both  (T) and   (T) components for all samples \nexhibit ed the peak at a temperature T P expected for a single -domain magnetic particle, \nwhich shifted towards higher T values with increasing  frequency . Typical curves are \nshown in  Figure 9 as examples for <d> = 8.8 and 1 1 nm (samples AV08  and AV11 , \nrespectively ). The dynamic response of an ensemble of single -domain magnetic \nnanoparticles can be described by the thermally -assisted magnetic relaxation  of a single -\ndomain magnetic moment over the anisotropy energy  barrier E a.47 The relaxation  time  \nassociated to thi s process is given by  a Neel–Arrhenius law  \n \n                         𝝉=𝝉𝟎𝒆𝒙𝒑 (𝑬𝒂\n𝒌𝑩𝑻)                               (7) \n \nwhere 0 is in the 10-9 – 10-11 s range for SPM systems.  \nIn the absence of an external magnetic field, the energy barrier  Ea can be assumed to \ndepend on the particle volume V and the effective magnetic anisotropy Keff through the \nexpression  𝐸𝑎=𝐾𝑒𝑓𝑓𝑉 𝑠𝑖𝑛2 𝜃, where  represents  the angle between the magnetic \nmoment  of the particle and its easy magnetiz ation axis.  A linear dependence of \nln𝑓 𝑣𝑠.  𝑇𝑃−1 is expected from eq.(7) if Keff is assumed to be temperature -independent.  Diameter  \n(nm)  (a) (b) \n K1(0) \n(x106 J/m3) B \n(x10-5  K-2) K1(0) \n(x106 J/m3) B \n(x10-5  K-2) 0 \n(x10-10 s) \nAV05  5.0 7.3(2)  87(7)  0.41  8.12  16.7 \nAV08  8.8 5.6(1)  8.7(1)  0.59  2.45  8.14 \nAV11  11.0 2.77(4)  4.6(2)  2.12 1.94  5.61 \nAV13  13.3  2.86(2)  2.8(1)  3.61 2.47  3.81  \nAV14  14.3 4.60(4)  2.8(1)     \nAV16  16.8 5.41(5)  2.7(1)     \nAV18  18.6 5.28(5)  2.7(1)     \nAV25  25.0 3.79(9)  1.9(2)     \nBulk *   1.96 1.9    \n* Values from Ref.60 21 \n However, the extrapolation of the linear fit of the experimental data to T-1 = 0 usually \ngives too small, unphysical values of 0, from 10-12 to less than 10-32 s.51 Several \nattempts to fit the frequency dependence of the AC susceptibility maxima included the \nVogel -Fulcher law  67 and critical slowing down  68 approaches. The sophisticated \nDormann -Bessais -Fiorani model69 of interparticle interactions  tried to solve this \ndifficulty through an interaction term in the expression of the anisotropy energy E a. This \nattempt provided a general expression that resulted rather hard to contrast with \nexperimental data, since it includes parameters depending on the relative location of the \nindividual particles.  \n02468\n0481216\n160 240 320 4000246\n160 240 320 4000481216   ´ (m3/kg)x 10-4     AV08\n<d>= 8,8 nm(a)´´ (m3/kg)x10-4 \n  \n      AV11\n<d> = 11 nm(b)\n T(K) \n  \nT(K)\n \nFigure 9. Temperature dependence of the in -phase (real) component of the magnetic susceptibility ´ (T), \nat different excitation frequencies for selected samples . (a) AV08 and (b) AV11. Arrows indicate \nincreasing frequencies. Insets: Temperature dependence of the o ut of phase (imaginary) component , ´´ \n \nFollowing t he same approach discussed above for  the temperature dependence of HC, \nwe propose  to describe the TP(f) experimental data by i ncluding the explicit 𝐾1(𝑇)=\n𝐾1(0) 𝑒−𝐵𝑇2 dependence into eq.(7). By doing this a non -linear expression for ln𝜏 vs. \nT-1 is obtained:  \n  \n                           𝐥𝐧𝝉=𝐥𝐧𝝉𝟎+𝑲𝟏(𝟎)𝑽\n𝒌𝑩𝑻𝒆𝒙𝒑 (−𝑩𝑻𝟐)                         (8) \n \nFigure 10 shows the good agreement between fitted curves using eq.(8) and  \nexperimental data from those  samples measured within  our accessible frequency range , \ndemonstrating the suitability of the Néel–Arrhenius model to describe the magnetic 22 \n relaxation. At low -temperatures, eq.(8) gives the expected linear behavior in the \nln [𝜏(T)] vs. T-1 plot, whereas at high temperatures the exponential term  dominates the \napproach to the independent ln 0 term, yielding realistic values of 0≈10-10 s.  \n0 2 4 6 8 10 12 14 16-25-20-15-10-505\n  \n AV05  \n AV08  \n AV11  \n AV13  Ln()\nT-1 (x103 K-1)\n \nFigure 10. (a) Arrhenius plot of the relaxation time  vs.  𝑻𝑷−𝟏 obtained  from the imaginary component \n´´(T). The lines are the corresponding fit s using Eq. (7).   \n \nThe K1(0) and B parameters obtained from dynamic data were found to be in agreement \nwith the previously discussed values obtained from the fit of H C(T) curves, and \nconsistent  to those reported for bulk CoFe 2O4 (see Table III). These values should be \nconsidered as the actual effective magnetic anisotropy ( Keff), since additional \nshape/stress contributions to the en ergy barrier could not be discarded. However, the \nclose values obtained from both methods to the bulk counterpart indicate that these \neffects, if present, have no major influence over the overall magnetic anisotropy.  Also \nconsistent with the results from H C of the previous section, the two smallest particles \nAV08 and AV05  showed deviations of both K1(0) and B. Nonetheless, as our \nmeasurements of dynamic data was limited those four samples with T B < 400 K, further \nmeasurements at T < 400 K would be needed to draw conclusions for the actual \nbehavior of these parameters.  \nThe effective magnetic anisotropy reported for many small and ultrasmall MNPs has \nbeen found to be largely e nhanced with respect to the corresponding bulk materials. \nFurtherm ore, theoretical  calculations  have also led to expect an increase in Keff as the \nparticle size decreases .70-73 Models for this increased value have been attempted through 23 \n an additional surface contribution to the total anisotropy,74 of the form 𝐾𝑒𝑓𝑓=𝐾𝑉+\n6\n𝑑𝐾𝑆 with K V and K S being volume and surface anisotropies for a particle of diameter 𝑑, \nalthough it is not clear how this approach could be applied to spherical particles, for \nwhich symmetry arguments yield a  zero net cont ribution from the surface term.  In any \ncase, the Néel -Arrhenius or any other simple model would be expected to fail for ultra-\nsmall  particles , composed by  a few number of atomic layers , and a more complete \napproach such as the Landau -Lifschitz -Gilbert equa tion should be employed.75  \n \nV. Conclusions.   \n \nOur systematic exploration of these high -anisotropy particles having  d between 5 -25 \nnm showed a consistent magnetic behavior over a wide range of temperatures . \nInterestingly, some deviations in the stoichiometry of the samples measured in \nmacroscopic sample volumes were found to extend to the single -particle level, opening \nquestions about the actual magnetic structure in cobalt -ferrite nanoparticles. For the \nsmallest sample s (d = 5 and 8 nm), non-saturating  behavior of M(H) was found at \n400 K and  5 K, consistent with the development of a spin -canted surface layer for \ndecreasing particle sizes.  Larger particles of the series  showed some  reduction of MS \nwith respect to the bulk , pointing to the existence of partial inversion degree.  \nFurthermore, our systematic measurements of the static and dynamic magnetic \nproperties in the series of CoxFe3-xO4 nanoparticles provided an experimental framework \nto check the validity of the Néel -Arrhenius model for single -domain nanoparticles. The \nsystemati c analysis of the thermal dependence of coercive field for different particle \nsizes showed that the deviations , usually reported in high -anisotropy MNPs, from the \nNéel -Arrhenius magnetic relaxation model  can be accounted for by considering the \ntemperature dependence of the K1(T) in the fit of the experimental data. The same \nstraightforward approach of including the thermal variation of Keff explained the \nmagnetic dynamics of our nanoparticles as obtained from ac susceptibility \nmeasurements. Indeed, making use of an empirical expression for K1(T) in bulk \nmaterials  we were able not only to fit the frequency  dependence of the ac susceptibility \npeaks but to obtain values of the characteristic response time 0 more realistic than those \nusually reported in the literature.  Our approach demonstrates that it is possible to 24 \n analyze the temperature dependence of the magnetic parameters of high -anisotrop y \nMNPs without the need of artificial corrections to the Neel –Arrhenius relaxation \nframework, which correctly describes the dynamic response of single -domain magnetic \nnanoparticles.  \n \nAcknowledgements  \n \nThis work was supported by the Spanish Ministerio de Economia y Competitividad \n(MINECO, project  MAT2010 -19326 and MAT2013 -42551).  The authors are indebted \nto Dr. A. Goméz Roca and Dr. M.P. Morales for their advice on sample preparation . \nGFG wish  to thank Professor R.F. Jardim for valuable discussions that stimulated this \ninvestigation.  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Wysin, Physical Review B 50, 3077 (1994).  \n \n \n "    },    {        "title": "1507.06799v1.Theory_of_perpendicular_magnetocrystalline_anisotropy_in_Fe_MgO__001_.pdf",        "content": "arXiv:1507.06799v1  [cond-mat.mtrl-sci]  24 Jul 2015Theory ofperpendicular magnetocrystallineanisotropy in Fe/MgO(001)\nDorj Odkhuu1,2, Won Seok Yun1,3, S. H. Rhim1,∗and Soon-Cheol Hong1†\n1Department of Physics and Energy Harvest Storage Research C enter,\nUniversity of Ulsan, Ulsan, 680-749, Republic of Korea\n2Department of Physics, Incheon National University, Inche on, 406-772, Republic of Korea\n3Department of Emerging Materials Science, DGIST, Daegu, 71 1-873, Republic of Korea\n(Dated: September 25, 2021)\nThe origin of large perpendicular magneto-crystalline ani sotropy (PMCA) in Fe/MgO (001) is revealed by\ncomparing Fe layers with and without the MgO. Although Fe-O p-dhybridization is weakly present, it cannot\nbe the main origin of the large PMCA as claimed in previous stu dy. Instead, perfect epitaxy of Fe on the MgO\nis more important to achieve such large PMCA. As an evidence, we show that the surface layer in a clean free-\nstanding Fe (001) dominantly contributes to EMCA, while inthe Fe/MgO, those by the surface and the interface\nFelayerscontributealmostequally. ThepresenceofMgOdoe snotchangepositivecontributionfrom /angbracketleftxz|ℓZ|yz/angbracketright,\nwherease itreduces negative contributionfrom /angbracketleftz2|ℓX|yz/angbracketrightand/angbracketleftxy|ℓX|xz,yz/angbracketright.\nExplorationformagneticmaterialswithfutureapplicatio ns\ndatesbackmorethantwodecades,whichincludesgiantmag-\nnetoresistance (GMR) and many applications in spintronics\nsuch as magneto-resistive random-access-memory(MRAM),\nmagnetic sensors, and novel programmable logic devices[1] .\nAmong those, perpendicular magnetocrystalline anisotrop y\n(PMCA) has attracted greatly as materials with large PMCA\ncan offer more opportunities to realize magnetic devices. I t\ncan provide an ideal tool to realize spin transfer torque ex-\ncludingexternalfields. Alsoitofferslargebitdensityinp rac-\ntical applications. Magnetic tunnel junctions (MTJs) usin g\nMgO as an electrode, in particular, have drawn huge atten-\ntion owing to enormous magnetoresistance reaching as high\nas180%[2]. Furthermore,largePMCAinFe/MgOhasshown\ngreatpromisewithtremendousperformance[3–5].\nTherehavebeennumeroussubsequentworkseversincein-\ncluding Fe/MgO and FeCoB |MgO. In theoretical sides, sev-\neral wayshavebeenproposedto enhancePMCA in Fe/MgO:\nByapplyinganexternalelectricfield[6,7],addingheavytr an-\nsition metal layer on top of Fe[8], and so forth. While it is\nstillon-goingendeavortoenhancePMCA,itlooksquiteindi s-\npensable to identify the physics origin of PMCA in Fe/MgO\nwithout aforementioned effects. Although it has been at-\ntributed to the p-dhybridization between Fe and O atoms at\nthe interface[9, 10] that is responsible for large PMCA, we\nargue that the hybridizationcannotbe the main driving forc e.\nIf the hybridizationis indeedthe main sourceofPMCA, why\nPMCAisabsentinFegrownonAlO x[11,12]despitethepres-\nence of apparent hybridization between Fe and O? Further-\nmore, if the hybridizationis really the main originof PMCA,\nthenitshouldbesmallwhenFe/MgOisunderoxidizeddueto\nthereductionofhybridization[9].\nIn this paper, we show that the most dominant contribu-\ntion to PMCA is not the p-dhybridization despite its weak\npresence in Fe/MgO. Instead, perfect epitaxy of the interfa ce\nwouldbethekey,whichisanindicatorofhighlevelofsample\nfabrication. Tosupportouridea,thecontributionfromthe sur-\nface and the interface Fe in the presence of the MgO is com-\npared, which is almost the same in magnitude. This clearly\ncounter-argues that the hybridization plays the main role o fPMCA, whose detailed analysis of the electronic structure i s\nprovided.\nDensity functional calculations are performed using the\nhighly precise full-potential linearized augmented plane\nwave (FLAPW) method[13] and Vienna Ab-initio Simula-\ntion Package (VASP)[14, 15]. Generalized gradient approx-\nimation (GGA) by Perdew, Burke, and Ernzerhof (PBE)\nparametrization[16] isemployedforthe exchange-correla tion\npotential. In VASP calculations with projector augmented\nbasis sets[17], full atomic relaxations have been carried o ut\nwithinforcecriteria 0.001eV/ ˚A, wherecutoffenergy400eV\nforwavefunctionexpansionand16 ×16×1kmeshintheirre-\nducibleBrillouinzonewedgearechosen. InFLAPWcalcula-\ntions, cutoffs for wave function and potential representat ions\nare 16 and 256 Ry, respectively. Charge densities and poten-\ntialinsidemuffin-tin(MT)sphereswereexpandedwithlatti ce\nharmonics ℓ≤8withMTradiiof2.1,1.4,and1.8a.u. forFe,\nO, and Mg atoms, respectively. For kpoint summation, we\nFe(S) \nFe(S-1) \nFe(C) \nFe(I) \nO(I) \nFe \nMg \nO\n(a) (b) (c) (d) O(I) \nFIG.1. (color online)(a)MgO-free FelayersandFe/MgOwith sub-\nstrate with (b) 1-ML, (c) 3-ML, and (d) 4-ML Fe. Spheres with y el-\nlow, red, and green color denote Fe, Mg, and O atoms, respecti vely.\nS and I denote atoms at the surface and the interface, S-1 that of\nsubsurface.2\nTABLE I. Magnetic moments (in µB) of Fe atoms in the clean Fe and Fe/MgO, where Fe(S), Fe(S-1), Fe(C), and Fe(I) stand for Fe atom at\nsurface, subsurface, center, and interface, respectively . Without MgO, Fe(S-1) and Fe(C) are not well defined for 1-, 2- ML, so is Fe(S-1) for\n3-ML.\n#of Felayers cleanFe Fe/MgO\nFe(S) Fe(S-1) Fe(C) Fe(S) Fe(S-1) Fe(I) O(I)\n1 3.14 – – – – 3.09 0.03\n2 2.82 – – 2.85 – 2.62 0.02\n3 2.96 2.39 – 2.96 2.45 2.83 0.02\n4 2.94 2.45 – 2.92 2.44 2.72 0.02\n5 2.96 2.43 2.62 2.95 2.44 2.80 0.02\n6 2.95 2.44 2.56 2.95 2.44 2.79 0.02\n7 2.95 2.43 2.42 2.95 2.44 2.79 0.02\nused24×24×1meshintheirreducibleBrillouinzonewedge.\nA self-consistent criteria of 1.0 ×10−5e/(a.u.)3was imposed,\nwhere convergence with respect to the numbers of the basis\nfunctions and kpoints was also seriously checked. MCA en-\nergies(EMCA)areobtainedusingFLAPW. Forthecalculation\nofEMCA, torquemethod[18] was employedto reducecompu-\ntationalcosts,whosevalidityandaccuracyhavebeensucce ss-\nfully proved in conventional FM materials and others[8, 19–\n25]\nThe model geometry in our study is depicted in Fig. 1: (a)\nMgO-freeFe-layersand(b-d)Fe-layersontheMgOsubstrate ,\nwherewehaveconsiderednumberofFelayersfrom1to7and\nfivelayersofMgO.TheexperimentallatticeconstantofMgO\n(4.214˚A)wasadoptedforthein-planelattices. Magneticmo-\nmentsofFe atomsare listedin Table I,whereFe atomsat the\nsurface, subsurface, (one layer beneath the topmost surfac e),\nand the interface with MgO, are denoted by Fe(S), Fe(S-1),\nand Fe(I), respectively, as in Fig. 1. The center layer in the\nabsence of MgO is labelled as Fe(C). Since Fe(S) and Fe(I)\nareidentical1-MLFe/MgO,onlyFe(I)isshown. Intheclean\nFe layers, i.e. MgO-free Fe layers, moments of Fe(S) are\nenhanced with respect to the bulk giving maximum value of\n3.14µBfor 1-ML Fe. Moments of Fe(S-1) are comparable\nor smaller than those of Fe(C). Fe(C) moments are slightly\nlargerthanbulkFe,whichisduetoenlargedlatticeconstan ts.\nThe presence of MgO also enhances moments of Fe(S) up to\n2.96µB. In all cases, Fe(S) have the largest moments mostly\naround2.95 µB. Interestingly,Fe(I)momentsaresmallerthan\nbut comparable to those of Fe(S). This feature has been al-\nreadyaddressedin previouswork[26].\nEMCAas function of the number of Fe layers is plotted in\nFig.2withandwithoutMgO.ThepresenceofMgOsystemat-\nically increasestotal EMCAby∼20% with respect to the free-\nstanding Fe-layer except 2-ML Fe. As seen clearly, the satu-\nrationbehaviorisevidentasthenumberofFelayerincrease s.\nThe singular behavior of 2-ML Fe will be discussed later. To\nrevealtheroleofFe3 d-O2phybridization, EMCAisplottedas\nfunctionoftheFe-OdistanceinFig. 2(b)forthecaseof6-ML,\nwherethedottedlinedenotes EMCAof6-MLFewithoutMgO.\nEMCAdecreasesastheFe-Odistanceincreases. Whenthedis-\ntance exceeds 3.0 ˚A, MCA energy becomes equal to that of\nMgO-free Fe layers. Definitely, this implies that hybridiza -tion affects EMCA. However, in forthcoming discussion, we\nwill show that the hybridization, though not completely ig-\nnorable,isnotthemaincontributionto PMCA asclaimed.\nThe atomically decomposed EMCAis presented as function\nofnumberofFelayerinFig. 2(c)and(d)fortheclean-Feand\nFe/MgO, respectively. In the clean-Fe, the surface contrib u-\ntion is largest,while those fromthe subsurfaceandthe cent er\nlayers are much smaller. Contributions from the subsurface\nare evennegative. On the otherhand,with MgO the interface\nlayers in contact with the MgO contribute almost equally as\nthesurfacelayerwhenthenumberofFelayersexceedsthree.\nContributionsfromlayersotherthaninterfaceandsurface are\nnegligiblysmall. [SeeSupplementaryInformation]. Ifthe p-d\nhybridizationisreallythemainsourceofthelargePMCAob-\nserved in the Fe/MgO, then why the surface Fe layer with no\nhybridizationcontributes almost equally as Fe(I)? This qu es-\ntionwill beexploredinforthcomingdiscussions.\nThe spin-channel decomposition is presented in Fig. 3for\nFIG.2. (a) EMCAas function of number of Fe layers with(blue line)\nand without MgO (black line). (b) EMCAas a function of Fe-O dis-\ntance for 6-Fe/MgO. Atomic resolution of EMCAversus number of\nFelayer for the (c) cleanFeand (d) Fe/MgO3\nFIG. 3. (color online) Spin-channel decomposition of EMCAfor (a)\nclean-Fe and (b) Fe/MgO, where red (blue) symbols denote ↑↑(↓↓)\nchannel, andblack ones for ↑↓channel.\n(a) the clean-Fe and (b) the Fe/MgO. Again, it is plotted as\nfunction of number of Fe layers, where the saturation behav-\nior is evident. Remarkably, in both cases −with and without\nMgO,the ↓↓channeldominatesovertheotherchannelssince\nthemajorityspinstatesarealmostfullyoccupiedasinthec ase\nofFe multilayers[19,20].\nTo elucidate the physics origin of PMCA, band structures\nofFeMLwithoutandwithMgOareshowninFig. 4. Wefirst\nanalyze the 1-ML case for simplicity, to clarify the physics\nwithoutlossofgenerality. Theoverallbandslooksosimila rin\ntwocases. ThepresenceofMgOaffects dz2anddxz,yzorbitals.\nSmall shifts of dxz,yzare apparent due to MgO. At X,dxzare\nsplit into two states, one 0.6 eV below EFand the other near\nEF. On the contrary,in-plane orbitalsare less affectedby the\npresenceofMgO.\nFor the MCA analysis, we follow the recipe by Wang\net al.[19], which has proved to be successful in various\nmaterials[8,19–25]. TheincreaseofMCAbythepresenceof\nMgO can be viewed in two ways. First, positive contribution\nby/angbracketleftxz|ℓZ|yz/angbracketrightremainsthesameinspiteoflittereducedcontri-\nbution by /angbracketleftx2−y2|ℓZ|xy/angbracketrightaroundΓ. Second, negative contri-\nbutionsby /angbracketleftz2|ℓX|yz/angbracketrightarereducedastheoccupied dyzstatebe-\ncomesunoccupiedabout1\n2M-Γ. Third,negativecontributions\nsuch as/angbracketleftxy|ℓX|xz,yz/angbracketrightinX-Mare reduced owing to enlarged\nenergy differences. As such, the presence of MgO slightly\nFIG. 4. (color online) Band structure of the free-standing F e (1-ML\nFe)andFe/MgO.Onlytheminorityspinbands are shown. (a)1- ML\nFe,and (b) 1-ML Fe/MgO. The orbital contribution of Fe dstate is\nemphasized in colors: red ( dz2), black (dx2−y2), blue (dxy), orange\n(dxz),and green( dyz),respectively.affects some bands due to weak hybridization. Nonetheless,\nthe overallbandsdonot changemuchwith a little increase in\nMCA.\nForcompleteness,the case of 6-MLFe/MgO areanalyzed.\nContributions from Fe(I) and Fe(S) to bands are shown in\nFig.5(a) and (b), respectively. Despite the complexity of\nbands, it is noticeable that the energy levels of those orbit als\ninvolving matrices /angbracketleftx2−y2|ℓZ|xy/angbracketrightand/angbracketleftxy|ℓX|xz,yz/angbracketrightdo not\ndiffer very much in both Fe(I) and Fe(S). This implies that\nthe hybridization is so weak that MCA from Fe(I) and Fe(S)\nare comparable. The large PMCA in Fe/MgO is a result of\ninterplayof ℓZandℓXmatrices.\nTo get more insights on the role of Fe(S) and Fe(I), DOS\nof the free-standing6-ML Fe and the 6-ML Fe/MgO are pre-\nsented in Fig. 5(c-f), where the left panel is for Fe(S) of the\ncleanFeandonMgO,andtherightpanelforFe(I)andO(I)in\nFe/MgO. DOS of Fe(S) is little affectedby MgO. The major-\nity spin states of Fe(I) and Fe(S) are almost filled with peaks\nfromdx2−y2anddz21∼2 eV above EF. In particular, Fe(I)\npeaksfrom dz2in the majorityspin state are closer to EF. On\nthe other hand, minority spin dxyanddxz,yzstates close to EF\nformpeaks. TheminorityspinstatesofFe(I)retainalmostt he\nsame feature of the free-standing 6-ML Fe. DOS of O(I) is\nalso shown in Fig. 5(f), where pstates are prominent around\n-6∼-4 eV, which is rather far away from Fe dxz,yzanddz2\nstates. As clearly seen from the DOS plots, the hybridizatio n\nisweakinFe/MgO.\nWhile MCA shows convergent behavior with the increase\nof the number of Fe layers, the singular behavior of 2-ML is\npuzzling. To tackle this issue, the uniquestructural featu reof\n2-ML should be emphasized. Without MgO, both Fe layers\nare symmetrically equivalent. On the other hand, with MgO,\nwhile one layer is the surface layer the other is the interfac e\nlayer. Moreover, the 2-ML exhibits rather unique electroni c\nstructure [See Supplementary Information]: dx2−y2state are\nprominent in DOS for both spin states just below EF. These\nstates couple with the minority-spin dxz,yzstates, hence pos-\nitive contribution is compensated by negative one. Also, Fe\n2p1/2core-levels are analyzed. The core-level shifts reflect\neither charge transfer or change of internal electric field[ 27].\nRegardlessofthepresenceofMgO,2 p1/2energyofthecenter\nFelayerislowest,whilethoseoftheinterfaceandthesurfa ce\nare higher. [See Supplementary Information for core-level s.]\nIn other cases, Fe layers are thick enough to screen out in-\nternal field by MgO, which is not the case for 2-ML Fe, as\nmanifestedinFe 2 p1/2corelevels.\nSo far in our calculations, the stoichiometry is assumed to\nbe perfect. On the other hand, in over- and under-oxidized\nFe/MgO[9], the Fe layer in contact with the over- orunder-\noxidized MgO is expected to show ripples Here, we argue\nthat perfectepitaxyat the interfaceis the key to achievela rge\nPMCA in Fe/MgO. Results clearly show that MCA contribu-\ntions from Fe(S) and Fe(I) are similar in magnitudes regard-\nless of the hybridization. Hence, the hybridizationis weak as\nevidencedinDOS,whichcannotbethemaindrivingforcefor\nlargePMCA.althoughitscontributiontoPMCAcannotcom-4\nFIG. 5. (color online) Band structure of (a) Fe (I) and (b) Fe ( S) in\n6-ML Fe/MgO. Color representing dorbitals are the same as Fig. 4.\nDensity of states (PDOS) of Fe and Fe/MgO with 6-ML Fe. The\nsurface Fe (c) without MgO, (d) Fe(I), (e) Fe(S)with MgO, and (f)\nO(I) 2porbitals. The dorbital states are shown in different colors:\nred (dz2), black (dx2−y2), blue (dxy), and green ( dxz,yz), respectively.\nForthe 2porbitals:px,py,andpz, respectively.\npletely neglected. Before we conclude, we emphasize here\nthat the physics we have demonstrated occurs in other lattic e\nconstantsas well. Even if we cannotexclude the strain effec t\nin Fe/MgO, we arguethat the maindrivingforceofPMCA is\nthe perfect epitaxywhich is intrinsicallyachievedin ab initio\ncalculations.\nIn summary, we have shown that the p-dhybridization,\nthough not completely ignorable, cannot be the main driv-\ningforceforlargePMCA, whichis differentfromcommonly\naccepted. The hybridization is overall weak, and indeed in-\ncreases systematically PMCA with MgO. However, MCA\nwith andwithout hybridizationarealmost equal aswell man-\nifested in Fe(S) and Fe(I). From the detailed analysis, cont ri-\nbutions from the surface and the interface Fe are dominant.\nAs in the Fe multilayers, since dstates of the majority spin\nbandsare completelyfilled, it is the ↓↓channelthat playsthe\nmost dominant role in PMCA. 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Vaishnava1∗\n1Department of Physics, Kettering University, Flint, MI 485 04\n2Department of Physics and Astronomy,\nWayne State University, Detroit, MI 48202 and\n3Department of Chemistry and Biochemistry,\nKettering University, Flint, MI 48504\n(Dated: June 19, 2021)\nAbstract\nWe report a novel method of determining the average N´ eel rel axation time and its temperature\ndependenceby calculating derivatives of the measured time dependenceof temperature for a frozen\nferrofluid exposed to an alternating magnetic field. The ferr ofluid, composed of dextran-coated\nFe3O4nanoparticles (diameter 13.7 nm ±4.7 nm), was synthesized via wet chemical precipita-\ntion and characterized by x-ray diffraction and transmission electron microscopy. An alternating\nmagnetic field of constant amplitude ( H0= 20 kA/m) driven at frequencies of 171 kHz, 232 kHz\nand 343 kHz was used to determine the temperature dependent m agnetic energy absorption rate\nin the temperature range from 160 K to 210 K. We found that the s pecific absorption rate of the\nferrofluid decreased monotonically with temperature over t his range at the given frequencies. From\nthese measured data, we determined the temperature depende nce of the N´ eel relaxation time and\nestimate a room-temperature magnetocrystalline anisotro py constant of 40 kJ/m3, in agreement\nwith previously published results.\nPACS numbers: 75.47.Lx, 75.50.Mm, 75.50.Tt\n∗Corresponding author: pvaishna@kettering.edu\n1INTRODUCTION\nColloidal suspensions of superparamagnetic iron-oxide nanopartic les, specifically those of\nmagnetite (Fe 3O4) and maghemite ( γ-Fe2O3), have been extensively investigated for their\npotential applications such as cell separation, use as contrast ag ents for magnetic resonance\nimaging, targeted drug delivery and magnetic fluid hyperthermia (MF H) [1, 2]. Among\nthese applications, MFH has piqued the interest of researchers fr om various disciplines in-\ncluding biophysics, biomedicine and oncology due to its potential applic ations in various\ntechnologies for the treatment of cancer without the side effects inherent to radiation and\nchemotherapy-based methods [3]. MFH involves the excitation of ma gnetic nanoparticles\nsuspended in a fluid medium (a ferrofluid) using an oscillating magnetic fi eld of the form\nH(t) =H0cos(2πft), where H0is the field amplitude (typically between 5 - 30 kA/m) and\nfis the frequency (typically in the range from 150 kHz - 350 kHz). For a single-domain\nnanoparticle, hysteresis losses are absent and energy absorptio n from the field can occur\nvia two excitation mechanisms: N´ eel and Brownian relaxation. This a bsorbed magnetic en-\nergy is eventually transformed to thermal energy and the temper ature of the ferrofluid rises.\nN´ eel relaxation [4] involves the alignment of the nanoparticle momen t with the external field\nwithin a fixed, non-rotating nanoparticle. As the excitation occurs against the anisotropy\nenergy barrier of the particle the relaxation time depends strongly on the nanoparticle’s\nmagnetic volume, Vm, magnetocrystalline anisotropy constant, K, temperature, Tand char-\nacteristic relaxation time, τ0. While there are a few models to describe this relaxation, the\nmost commonly used has the form [4]\nτN=√π\n2τ0exp/parenleftbiggKVm\nkBT/parenrightbigg/radicalBigg\nKVm\nkbT. (1)\nBrownian relaxation [5] involves the alignment of the particle’s moment with the external\nfield via the physical rotation of a fixed-moment nanoparticle within t he carrier fluid. This\nalignment is affected by the hydrodynamic properties of the carrier fluid and nanoparticles,\nit is described by [6]\nτB=3ηVH\nkBT, (2)\nwhereηis the viscosity of the carrier fluid and VHis the hydrodynamic volume of the\ncomposite particle which includes any surfactant layer used to give c olloidal stability to\nthe ferrofluid. When both mechanisms are active these processes occur in parallel and the\n2effective relaxation time which describes the energy transfer rate is given by\nτ=τNτB\nτN+τB. (3)\nFrom this relation, it is evident that the shorter relaxation time domin ates the overall\ndissipative characteristics and thus determines the heating rate o f the sample.\nThe N´ eel mechanism plays a dominant role in the relaxational charac teristics of the\nnanoparticles particularly when they are embedded in a cancerous t issue undergoing MFH\ntreatment, since in the tissue’s environment, due to immobilization of the particles, the\nBrownian mechanism is highly damped. Several studies have shown th at if the magnetic\nnanoparticles are internalized by the cancer cells, they are either lo cked in the cell plasma or\nadhere to the cell walls [7]. Consequently, the role of Brownian relaxa tion is insignificant for\nnanoparticles used in the hyperthermia treatment of cancer when the ferrofluid is applied\ndirectly to the tumor tissues via injection. In such situations, N´ ee l relaxation produces\nthe local heating of the cancerous mass; however, one must be ca reful about comparing\nexperiments done in the laboratory with therapies that will be perfo rmed under real-life\nconditions.\nThe experimental studies of the heating rates of ferrofluids in the presence of alternating\nmagnetic fields can be broadly placed into one of two categories: tho se focusing on varying\nonly the intrinsic physical and magnetic parameters of the nanopar ticles [8, 9] and those\nthat investigate the influence of varying only the parameters of th e applied field (i.e. H0\nandf) [10]. Among these two broad classifications, studies of how the intr insic parameters\nof nanoparticles influence magnetic heating still remain an active are a of research. However,\nas the effects of the intrinsic parameters are reflected through t he relaxation processes (N´ eel\nand Brownian) which act in parallel, it is difficult to understand the cont ribution of each\nprocess on the heating characteristics of the ferrofluid.\nKnowing thesignificance oftheN´ eel mechanism inhyperthermia, ma ny studies have been\nperformed to estimate this parameter using ac and dc magnetic sus ceptibility measurements\n[11, 12]; however, these methods often require large volumes of sa mple, accurate determi-\nnations of sample volume and mass and costly equipment. In this pape r, we present a new\nmethod to determine this parameter using a simple induction heating s ystem. This method\nrequires only measurements of the sample heating rate under the in fluence of an oscillating\nmagnetic field of constant amplitude at two different frequencies. A dditional measurements\n3of sample mass, volume and magnetic anisotropy are not needed. Th e goal of this investiga-\ntion is to determine the temperature dependence of the N´ eel rela xation time ( τN) when the\nBrownian mechanism is completely quenched. This suppression of the Brownian relaxation\nis achieved by performing the experiments in the temperature rang e from 160 K to 210 K\nwhere the carrier fluid (DI water) is completely frozen, thus locking the nanoparticles in\nplace.\nExperimentally, the heating rate in MFH is expressed in terms of the s pecific absorption\nrate (SAR) which is defined to be the power absorbed per unit mass of the nano particles\nin the ferrofluid. This can be calculated using the thermodynamic rela tion for rate of heat\nenergy absorbed by the sample as [13]\nSARheating=Msample\nmFe3O4C(T)∆Theating\n∆t, (4)\nwhereMsampleis the overall mass of the sample, mFe3O4is the mass of the Fe 3O4nanoparti-\ncles,C(T) is the temperature dependent specific heat of the carrier fluid an d ∆Theating/∆t\nis the time rate of change of the sample’s temperature as it absorbs energy from the applied\nmagnetic field. When the sample begins to warm under the influence of the applied mag-\nnetic field, it is important to note that heat exchange with the enviro nment is also occurring\ndue to the temperature differential between the sample and its sur roundings. For accurate\ndetermination of SAR, it is important to estimate this heat exchange between the sample\nand the environment due to convective, conductive and radiative p rocesses. While this pro-\ncess is quite hard to do from a calculational standpoint, we can expe rimentally estimate\nthe collective effects of heat exchange via these processes by mea suring the heating of the\nsample placed into the experimental apparatus while keeping the ext ernal field turned off.\nFrom this data, we determine the specific power gain (SPG) as heat is transferred from the\nenvironment to the sample as [14]\nSPGwarming=Msample\nmFe3O4C(T)∆Twarming\n∆t, (5)\nwhere ∆Twarming/∆tis the warming rate of the sample due only to heat exchange between\nthe sample and the environment. Once this SPG is determined, we can correct the measured\nSARfor the sample as a function of temperature [14]\nSAR=SARheating−SPGwarming (6)\n4Due to the suppression of the Brownian mechanism, the values of SARobtained over this\ntemperature interval are dependent only upon N´ eel relaxation. This relaxation time can\nbe determined using a theoretical model based on a direct relations hip between the power\ndensity dissipated by the nanoparticles and the out-of-phase, dis sipative component of the\nferrofluid’s susceptibility [15]\nP=πµ0H2\n0χ0f2πfτN\n1+(2πfτN)2, (7)\nwhereχ0is the equilibrium susceptibility and µ0is the permeability constant. In this model,\nthe energy dissipation by the nanoparticles, which is related to the o ut-of-phase component\nof the magnetic susceptibility, is expressed in terms of the equilibrium susceptibility and the\nN´ eel relaxation time τN. Measuring the SARat two different frequencies f1andf2while\nkeeping the field amplitude ( H0) fixed, we find τNfrom Equation 7 as\nτN=1\n2πf1f2/radicalBigg\nf2\n1−αf2\n2\nα−1, (8)\nwhere\nα=SARf1\nSARf2=/parenleftBig∆Theating\n∆t−∆Twarming\n∆t/parenrightBig\nf1/parenleftBig∆Theating\n∆t−∆Twarming\n∆t/parenrightBig\nf1. (9)\nIt is interesting to note that τNis determined only from the values of f1,f2and the sam-\nple heating rates (corrected for heat exchange with the environm ent) and do not require\nmeasurement of the sample mass, nanoparticle mass, or specific he at values.\nEXPERIMENTAL DETAILS\nIron oxide nanoparticles were synthesized using a standard co-pr ecipitation technique in\nwhich an aqueous solution of FeCl 3·6H2O and FeCl 2·4H2O were mixed in a 2:1 molar ratio\nand Fe 3O4nanoparticles were precipitated by the drop-wise addition of 1M NH 4OH. During\nprecipitation, N 2gas was bubbled through the solution to protect against oxidation o f the\nFe2+ionsintoFe3+ions. Theprecipitatewasseparatedfromthesolutionbyastrongm agnet,\nwashed with DI water and re-suspended in a metastable 0.5 M NaOH so lution. In order to\nsuspend the precipitated nanoparticles in a carrier solution (DI wat er) they were coated in\ndextran by the drop-by-drop addition of the metastable solution o f Fe3O4to a solution of\n15-20 kDa dextran (MP Biomedicals) in 0.5M NaOH while simultaneously pr obe sonicating.\n5The product was rinsed and resulted in a water-based suspension o f Fe3O4nanoparticles\nwith a concentration of 20 mg of Fe 3O4per mL of solution. A portion of the sample was\nlypholized and characterized via x-ray diffraction (Rigaku MiniFlex 600 ) and transmission\nelectron microscopy (JEOL HR TEM 2010 operating at 200 keV).\nFor calorimetric measurements in the 160 K to 210 K range, the ferr ofluid sample was\nfirst cooled to 77 K via the immersion of seald vials in liquid nitrogen. The s ample was\nthen allowed to warm under ambient conditions to 160 K at which time a 2 0 kA/m (250\nOe) ac magnetic field was applied using an Ambrell EasyHeat 2.4 kW induc tion heating\nsystem with a water-cooled, 8-turn, 2-cm-diameter coil. Tempera ture versus time data\nwere collected in the 160 K to 210 K region using an Optocon FOTEMP1- H fiber optic\ntemperature monitoring system equipped with a TS5 optical temper ature sensor with 0.1 K\naccuracy. The sample vial was thermally insulated using cotton padd ing and foam rubber\nto minimize heat exchange with the environment; however, as therm al interaction with the\nenvironment cannot be completely suppressed, experiments were performed to determine\nthe extent of the heat exchange with the surrounding environmen t and this estimation was\naccounted for in all measurements as discussed above.\nRESULTS AND DISCUSSION\nThe left panel of Figure 1 shows the powder x-ray diffraction (XRD) spectrum taken\nfrom the lypholized dextran coated Fe 3O4nanoparticles. The open symbols represent the\nobserved counts recorded for different d-spacing values between 1.25 and 3.25 ˚A. The solid\nline is a full-profile Le Bail fit to the data, the vertical bars indicated t hed-spacing positions\nof the Bragg reflections and the lower trace is the difference curve between the observed and\ncalculated counts. The fit confirms that the sample consists of a sin gle nanocrystalline phase\nof Fe3O4with cubic Fm¯3msymmetry and lattice constant a= 8.36˚A. Using the full-width\nat half maximum of the (311) reflection in Scherrer’s equation, an av erage nanoparticle\ndiameter of 14 nm was estimated for this sample. The right panel of F igure 1 shows the\nmagneticfielddependenceoftheferrofluid’smagnetizationmeasur edat150Kwhichexhibits\nno hysteresis thus confirming the superparamagnetic nature of t he nanoparticles within the\nexperimental temperature range between 160 K and 210 K.\nTransmission electron microscopy (TEM) was performed to determ ine the mean diameter\n6and size distribution of the nanoparticles in addition to confirming the crystallographic\ninformationdetermined via XRD. Figure 2(a) shows a histogramof pa rticle sizes determined\nfrom the bright-field TEM micrograph shown in Figure 2(b) and similar im ages at the same\nmagnification. The histogram shows a log-normal particle size distrib ution having mean\ndiameter /angbracketleftD/angbracketright=13.4nmandstandarddeviationfromthemeanof σD=4.7nm. Thisnumber\nis in agreement with the 14 nm average diameter determined using Sch errer’s equation in\nconjunction with the collected XRD spectrum. Figure 2(c) shows a h igh-resolution TEM\nmicrograph of a portion of a single nanoparticle. The visible lattice plan es in the image\nshow a spacing of 2.9 ˚A marking them as the (220) set of planes in Fe 3O4. Lastly, Figure\n2(d) shows a selected area electron diffraction (SAED) pattern wh ich was used to to confirm\nthe cubic symmetry determined using XRD.\nThe left panel of Figure 3 shows the temperature versus time data collected while heating\nthe ferrofluid using an alternating magnetic field of amplitude H0= 20 kA/m at 171 kHz\n(open circles), 232 kHz (open triangles) and 343 kHz (open square s). All three curves\nwere fit to a polynomial (solid lines) and differentiated in order to dete rmine the slope,\n∆T/∆t. To account for any heat exchange with the environment, a contr ol experiment was\nperformed in which the ferrofluid was placed inside the field coil and allo wed to warm in the\nabsence of the magnetic field. This data is plotted in the right panel o f Figure 3 and will\nbe referred to as the ambient curve. From this data, we were able t o calculate the specific\nabsorption rate ( SARheating) for each frequency and the specific power gain ( SPGwarming)\nfor the ambient experiment all as functions of temperature using t he temperature-dependent\nspecific heat of ice [16] and Equations 4 and 5, respectively. From th ese calculations, we\nwere able to determine the specific absorption rate ( SAR) due only to energy absorbed from\nthe alternating magnetic field using Equation 6. The results of these calculations are shown\nin Figure 4 and yield average values of SARover the temperature interval of 57 W/g, 88\nW/g and 130 W/g at 171 kHz, 232 kHz and 343 kHz, respectively.\nAdditionally, using the temperature dependent slopes of the data p resented in Figure 3\nfound through differentiation of the polynomial fits, we are able to u se Equations 8 and 9\nto calculate the temperature dependence of the N´ eel relaxation time over the temperature\nregime in question. The results of this calculation are shown in Figure 5 and show a value\nnear 5×10−7s for all possible permutations of f1andf2. Using Equation 1 with this\ndetermined value of τN, the experimentally determined particle diameter of 13.7 nm, and\n7assuming a characteristic relaxation time of τ0= 10−9s we find the room temperature\nmagnetocrystalline anisotropy constant, K, for this sample to be 40 kJ/m3, which agrees\nwith the range of values previously reported in the literature [17, 18 ].\nIt is important to emphasize that for quantitative determination of τNusing Equation 8,\none needs the time derivatives of the temperature at two different frequencies and for the\nambient (no applied field) case. Using these values in Equation 9, we de termine the temper-\nature dependence of the N´ eel relaxation time using Equation 8. In our case, by taking data\nat three different frequencies, we were able to provide three differ ent estimations of the tem-\nperature dependence of τNand found, as expected, that the value was roughly the same for\neach pair of frequencies. We see in Figure 5 a convergence of the va lues ofτNas we approach\n210 K. In addition, it is important to note that the magnitude of the N ´ eel relaxation time\nis on the order of 10−7s as expected for a system of non-interacting magnetic nanopart icles\nhaving characteristic time constant τ0∼10−9−10−13s [4, 19]. The non-interacting nature\nof this ensemble was confirmed through calculation of the relative va riation of the blocking\ntemperature per frequency decade ( φ= ∆T/Tlog10f) in which we found a value of φ= 0.18,\nanorderofmagnitudelargerthanthatfoundininteracting samples exhibiting spin-glass-like\ntransitions [20–23]. The details of this investigation will be published in a subsequent paper\n[24]. At the macroscopic level, τN, as described by Equation 1, references the relaxation of\nan individual nanoparticle of volume Vmhaving magnetocrystalline anisotropy constant K\nand characteristic relaxation time τ0=/radicalBig\nπKVm/4kBT; however, in a polydisperse system\nthese parameters may vary from particle to particle and conseque ntly their N´ eel relaxation\ntimes do vary. In SARmeasurements, one measures the temperature rise of the ferro fluid\ndue to the average thermal energy of its constituents (i.e. nanop articles, surfactant and\ncarrier fluid) in contrast to the relaxation time described by Equatio n 1 which refers to a\nsingle particle. Thus, our measured τNrepresents an average N´ eel relaxation time of the\nensemble of particles–this is what is responsible for the measured va lues ofSAR. However,\nfor a monodisperse system, Equations 1 and 8 may be applied as what is true for any one\nparticle is true for any other in the system. The use of Equation 1, h owever, requires an\naccurate determination of the particle volume, Vm, and the magnetocrystalline anisotropy\nconstant, K. On the other hand, the method outlined in this paper requires only t he value\nof the applied field frequencies and the derivatives of the temperat ure versus time curves\nwhich can be accurately determined using simple to understand meth odology.\n8CONCLUSION\nWe studied the temperature dependence of the specific absorptio n rate (SAR) of a frozen\nferrofluid at different frequencies at a constant field amplitude. Fr om the determined SAR\nvalues we were able to calculate the temperature dependence of th e N´ eel relaxation time\nusing a novel and simple approach. This method is useful for cases in which there are\nuncertainties related to the magnetocrystalline anisotropy const ant,K, and nanoparticle\nsize. In our experimental temperature window between 160 K and 2 10 K, our measured\nSARdecreases with temperature; however, this trend may not contin ue as the experimental\nwindow is widened to include temperatures above 210 K where other e ffects may begin\nto play a role in the relaxation characteristics of the nanoparticles. We believe that this\ninvestigation offers a new and simpler method of determining an avera ge N´ eel relaxation\ntime in a colloidal suspension of magnetic nanoparticles.\nACKNOWLEDGEMENTS\nThis work was supported by the NSF though DMR-1337615 which pro vided funding for\nthe Miniflex 600 powder x-ray diffractometer. RJT, REK, CR and PPV would like to thank\nDr. JamesZhangforhissupportthroughtheFacultyResearchFe llowship. Inaddition, PPV\nwould like to thank Dr. Zhang for his support through the Rodes Pro fessorship. Lastly,\nREK would like to thank the Society of Physics Students for the supp ort.\n9[1] Q.A. Pankhurst, J. Connolly, S.K. Jones and J. Dobson, J. 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Tr iboulet,Solid State Comm. 49,\n10417 (1984).\n[22] M.S. Seehra, V. Singh, P. Dutta, S. Neeleshwar, Y.Y. Che n, C.L. Chen and C.C. Chen, J.\nPhys. D: Appl. Phys. 43, 1450032 (2010).\n[23] K.L. Pisane, E.C. Despeaux and M.S. Seehra, J. Magn. Magn. Mater. 384, 148-154 (2015).\n[24] R.J. Tackett, J. Thankur, N. Mosher, E. Abdelhamid and P .P. Vaishnava J. Appl. Phys. (to\nbe submitted).\n11FIG. 1: (Left Panel) A powder x-ray diffraction spectrum of a ly pholized portion of the sample.\nThe open symbols correspond to the oberserved data and the so lid line is a full-profile Le Bail\nfit to the data indicated the Fm¯3mcubic symmetry and 8.36 ˚A lattice constant characteristic\nof Fe3O4. The Miller indices of the more intense reflections are indic ated and the bottom trace\nis the difference curve between the observed data and the fit. (R ight Panel) The magnetic field\ndependence of the sample’s magnetization at 150 K showing th at the nanoparticles are in the\nsuperparamagnetic state in the experimental temperature r egime between 160 K and 210 K.\n12FIG. 2: (a) A histogram of the particle size distribution det ermined using the micrograph shown\nin (b) and other similar micrographs taken at the same magnifi cation. (c) A high-resolution TEM\nmicrograph of a portion of a single nanoparticle with an outl ine of the particle provided for clarity.\nThe spacing of the observed lattice planes was found to be 2.9 ˚A indicative of the (220) set of\nplanes. (d) A selected area electron diffraction (SAED) patte rn which was used to confirm the\ncubic nature of the sample.\n13/s48 /s53/s48 /s49/s48/s48 /s49/s53/s48 /s50/s48/s48/s49/s55/s48/s49/s56/s48/s49/s57/s48/s50/s48/s48/s50/s49/s48\n/s32/s65/s109/s98/s105/s101/s110/s116/s84/s32 /s40/s75/s41\n/s116/s32 /s40/s115/s41/s48 /s49/s48 /s50/s48 /s51/s48 /s52/s48 /s53/s48/s49/s55/s48/s49/s56/s48/s49/s57/s48/s50/s48/s48/s50/s49/s48\n/s67\n/s66\n/s65/s32/s45/s32 /s49/s55/s49/s32/s107/s72/s122\n/s66/s32/s45/s32 /s50/s51/s50/s32/s107/s72/s122\n/s67/s32/s45/s32 /s51/s52/s51/s32/s107/s72/s122/s84/s32 /s40/s75/s41\n/s116/s32 /s40/s115/s41/s65\nFIG. 3: (Left Panel) Temperature versus time measurements m ade with the ferrofluid in an alter-\nnating magnetic field of amplitude H0= 20 kA/m at frequencies of 171 kHz (A), 232 kHz (B) and\n343 kHz (C). (Right Panel) Temperature versus time measurem ents made with the ferrofluid in\nthe field coil but in the absence of an applied field.\n14/s49/s55/s48 /s49/s56/s48 /s49/s57/s48 /s50/s48/s48 /s50/s49/s48/s54/s48/s56/s48/s49/s48/s48/s49/s50/s48/s49/s52/s48\n/s32/s49/s55/s49/s32/s107/s72/s122\n/s32/s50/s51/s50/s32/s107/s72/s122\n/s32/s51/s52/s51/s32/s107/s72/s122/s83/s65/s82/s32 /s40/s87/s47/s103/s41\n/s84/s32 /s40/s75/s41/s67\n/s105/s99/s101\n/s49/s50/s51/s52/s53\n/s67\n/s105/s99/s101/s32/s40/s74/s47/s103 /s75/s41\nFIG. 4: The temperature dependence of the specific absorptio n rate (SAR) of the ferrofluid mea-\nsuredat frequenciesof171kHz(opencircles), 232kHz(open triangles)and343kHz(opensquares).\nThe temperature dependence of the specific heat of ice used to calculate the specific absorption\nrate is plotted as a solid line\n15/s49/s55/s48 /s49/s56/s48 /s49/s57/s48 /s50/s48/s48 /s50/s49/s48/s50/s52/s54/s56/s49/s48\n/s32/s102\n/s49/s32/s61/s32/s49/s55/s49/s32/s107/s72/s122/s32/s32/s32/s32/s32 /s102\n/s50/s32/s61/s32/s50/s51/s50/s32/s107/s72/s122\n/s102\n/s49/s32/s61/s32/s49/s55/s49/s32/s107/s72/s122/s32/s32/s32/s32/s32 /s102\n/s50/s32/s61/s32/s51/s52/s51/s32/s107/s72/s122\n/s32/s102\n/s49/s32/s61/s32/s50/s51/s50/s32/s107/s72/s122/s32/s32/s32/s32/s32 /s102\n/s50/s32/s61/s32/s51/s52/s51/s32 /s107/s72/s122/s78/s32/s40/s49/s48/s45/s55\n/s32/s115/s41\n/s84/s32 /s40/s75/s41\nFIG. 5: The temperature dependence of the N´ eel relaxation t ime calculated using equations 8 and\n9 for frequencies f1= 171 kHz and f2= 232 kHz (open circles), f1= 171 kHz and f2= 343 kHz\n(open triangles) and f1= 232 kHz and f2= 343 kHz (open stars).\n16"    },    {        "title": "1509.01106v1.Investigation_of_ferromagnetic_domain_behavior_and_phase_transition_at_nanoscale_in_bilayer_manganites.pdf",        "content": "Investigation of ferromagnetic domain behavior and phase\ntransition at nanoscale in bilayer manganites\nC. Phatak,1,\u0003A. K. Petford-Long,1, 2H. Zheng,1J.\nF. Mitchell,1S. Rosenkranz,1and M. R. Norman1\n1Materials Science Division, Argonne National Laboratory,\n9700 S. Cass Avenue, Argonne, IL 60439, USA\n2Dept. of Materials Science and Engineering, Northwestern University,\n2220 Campus Drive, Evanston, IL 60208, USA\nAbstract\nUnderstanding the underlying mechanism and phenomenology of colossal magnetoresistance in\nmanganites has largely focused on atomic and nanoscale physics such as double exchange, phase\nseparation, and charge order. Here we consider a more macroscopic view of manganite materials\nphysics, reporting on the ferromagnetic domain behavior in a bilayer manganite sample with a\nnominal composition of La 2\u00002xSr1+2xMn2O7with x= 0:38, studied using in-situ Lorentz trans-\nmission electron microscopy. The role of magnetocrystalline anisotropy on the structure of domain\nwalls was elucidated. On cooling, magnetic domain contrast was seen to appear \frst at the Curie\ntemperature within the a\u0000bplane. With further reduction in temperature, the change in area\nfraction of magnetic domains was used to estimate the critical exponent describing the ferromagntic\nphase transition. The ferromagnetic phase transition was accompanied by a distinctive nanoscale\ngranular contrast close to the Curie temperature, which we infer to be related to the presence of\nferromagnetic nanoclusters in a paramagnetic matrix, which has not yet been reported in bilayer\nmanganites.\nPACS numbers: 75.47.Gk, 75.70.Kw, 75.47.Lx\n1arXiv:1509.01106v1  [cond-mat.str-el]  3 Sep 2015I. INTRODUCTION\nBilayer manganites such as La 2\u00002xSr1+2xMn2O7exhibit a rich phase diagram based on\ntheir doping level, which includes ferromagnetic (FM), antiferromagnetic (AF), and charge-\nordered phases1,2. Due to the complex crystal structure, there are several di\u000berent exchange\ninteractions within these materials that contribute to their behavior. For example the inter-\nbilayer exchange along the c\u0000axis is weaker than the intra-bilayer exchange within the a\u0000b\nplane, as a result of the intrinsic two-dimensional layered structure3,4. These anisotropic\nexchange interactions along with the competition among orbital, charge, and spin order, as\nwell as lattice distortions, lead to interesting and complex magnetic and transport prop-\nerties. The double exchange interaction between the Mn3+and Mn4+ions results in the\nmaterial undergoing a phase transition from a paramagnetic (PM) insulator to a ferromag-\nnetic (FM) metallic state below the Curie temperature5. As a result of this dramatic change\nin conductivity, the layered manganites exhibit a colossal magnetoresistance (CMR) e\u000bect,\nwhich has garnered much attention in the past two decades from both a fundamental as well\nas an applications context.\nOne of the proposed mechanisms for the CMR e\u000bect is that small FM regions, which are\nconnected in a percolative manner6, form as the material is cooled through the transition\ntemperature. Magnetic interactions and domains in these manganites and related materials\nhave previously been studied using various techniques such as neutron scattering7,8and mag-\nnetic force microscopy9, as well as Kerr microscopy10. The majority of the research e\u000borts\ntowards understanding the formation of FM domains has been done using reciprocal space\nand scattering methods. Only recently there have been some e\u000borts towards direct real space\nvisualization of the way in which the small FM regions form and become connected, leading\nto the formation of FM domains within the material below Tc. Lorentz transmission electron\nmicroscopy (LTEM) has also been used to study cubic manganites since it o\u000bers high spatial\nresolution and a direct visualization of the magnetic domains11{13. Furthermore with current\nadvanced in-situ capabilities, LTEM o\u000bers unique possibilities to study the magnetic phase\ntransitions as a function of temperature, while simultaneously obtaining information about\nstructural and charge ordering using electron di\u000braction. Phase-reconstruction methods\nenable quantitative magnetic induction maps to be obtained that can provide information\nabout the nature of the magnetic domain walls as well as physical parameters such as the\n2exchange sti\u000bness of the sample.\nIn this work, we have explored the behavior of La 2\u00002xSr1+2xMn2O7withx= 0:38, which\nhas been reported to show a ferromagnetic transition with a Curie temperature of Tc=\n125 K14. At this doping level, the magnetic moments in the unit cell are oriented such\nthat the crystal has a strong easy plane ( a\u0000bplane) anisotropy with Ku\u0019\u00002:5\u0002105\nJ/m3.15The behavior of the magnetic domains and the relationship between the crystal\nstructure and domain structure is discussed in detail together with a derivation of magnetic\nparameters obtained directly from the nanoscale imaging. Furthermore, we also describe\nthe ferromagnetic phase transition and the observation of a granular nanoscale contrast that\nprovides direct evidence of the coexistence of FM and PM phases in a bilayer manganite.\nII. EXPERIMENTAL METHODS\nSingle crystals of La 2\u00002xSr1+2xMn2O7withx= 0:38 i.e., La 1:24Sr1:76Mn2O7were synthe-\nsized using the \roating zone method16. TEM samples were prepared from these crystals\nusing focused ion-beam milling method as well as conventional polishing, followed by gentle\nmilling by low energy Ar+ions to improve electron transparency. In order to fully under-\nstand the magnetic domain behavior and elucidate the role of magnetocrystalline anisotropy\non the formation of domain walls, two samples were fabricated with di\u000bering geometry; (1)\nS1 - with the hard axis ( h001i) in the plane of the TEM sample, (2) S2 - with the hard axis\n(h001i) perpendicular to the plane of the TEM sample. The magnetic domain behavior in\nthe samples was then analyzed in the Lorentz TEM mode using a Tecnai F20 transmission\nelectron microscope. Through-focus series of images were acquired with a nominal defo-\ncus ranging between \u0001 f= 500\u00001000\u0016m. It should be noted that Lorentz microscopy\nis only sensitive to magnetization components that are perpendicular to the direction of\nthe electron beam. The local magnetization was analyzed using the gradient of the phase\nshift of electrons passing through the sample. This phase shift was recovered using the\ntransport-of-intensity equation method17.In situ experiments were performed using a liquid\nN2stage that is capable of cooling the sample to 90 K, in order to observe the magnetic\ndomain behavior during the magnetic phase transition from the paramagnetic state to the\nferromagnetic state.\n3III. RESULTS\nA. Magnetic domain walls\nFigure 1(a) shows an under-focused Lorentz TEM image from sample S1 with the hard\nmagnetic axis,h001i, in the plane of the TEM sample. The inset (top right) shows the\ndi\u000braction pattern viewed along the h100izone axis and the orientation of the crystallo-\ngraphic axes in the sample plane is indicated. The sample was cooled to 95 K, which is well\nbelow the Curie temperature. As expected, 180\u000edomain walls are present, seen as bright\nand dark sharp lines, running vertically in the image. The magnetization map within this\nregion was reconstructed from the phase shift of the electrons and is shown as a color map\noverlaid on the bottom left of the image. The color indicates the direction of magnetization\nas given by the color wheel. The additional curved lines seen running horizontally in the\nimage are bend contours, which are related to strong electron di\u000braction e\u000bects. This com-\nposition of La 2\u00002xSr1+2xMn2O7is expected to have an easy plane anisotropy, which means\nthat the magnetization prefers to lie in the a\u0000bplane. Due to the speci\fc geometry of\nthis TEM sample and its crystallographic orientation, we are observing these a\u0000bplanes\nedge on, thereby e\u000bectively creating a strong uniaxial anisotropy in the TEM sample, with\ndomain walls separating domains running perpendicular to the h001idirection. This also\nmanifests itself via the formation of needle-like domains seen in the magnetization color map\nnear the bottom of the image, which is also the edge of the sample. This type of domain is\nformed in order to minimize the stray \feld energy. The widths of the domains near the edge\nand inside the sample are determined by a balance between the domain wall energy and\nthe closure (stray) \feld energy. The domain pattern observed here is an example of two-\nphase branching, which re\fnes the domain pattern near an edge18. This e\u000bect is observed\nin sample S1 because it has a strong e\u000bective uniaxial anisotropy along the h010idirection\nresulting in the magnetization lying along only two easy magnetization directions: [010] and\n[0\u001610].\nSince the hard axis for magnetization is in the plane of the sample, the domain walls\ncan be expected to be of Bloch type where the magnetization rotates out-of-plane across the\nwall. The width of the domain wall can be related to physical constants such as the exchange\nsti\u000bness and magnetocrystalline anisotropy using the relation: \u000e\u0018\u0019q\nA=jKuj18. Using the\n4classical approximation of spin rotation across a 180\u000edomain wall, the distribution of the\nin-plane component of the magnetic induction can be approximated using the relation:\nBy=a+btanhf\u0019(x\u0000c)=\u000eg; (1)\nwherea;b;c are constants and \u000eis the domain wall width. Figure 1(b) shows a plot of the in-\nplane component of the projected magnetic induction across the domain wall (black squares).\nThe values were averaged over the region showed by dashed lines in Figure 1(a). A non-linear\nleast-squares \ft to the measured data was performed (shown in red) using equation 1, from\nwhich, the domain wall width was determined to be 77 nm. Furthermore, using the value\nofKu= 2:5\u0002105J/m3from the literature15, a value for the exchange sti\u000bness constant for\nLa1:24Sr1:76Mn2O7was determined to be A= 1:45\u000210\u000010J/m. This demonstrates that we\ncan determine the magnetic parameters of a material directly using nanoscale imaging. The\nexchange sti\u000bness constant can be related to the exchange interactions and is dependent on\nthe crystal structure of the material. The relationship is well established for cubic materials\nbut not for bilayer manganites.\nThe magnetic domain structure in sample S2, which has the hard axis of magnetization\nperpendicular to the plane of the sample, is shown in Figure 2. Figure 2(a) shows an\nunder-focused LTEM image from this sample. The top-right inset shows the di\u000braction\npattern along the h001izone axis and the in-plane crystallographic directions are indicated.\nThe domain walls are not seen as sharp lines as they were for sample S1, but now show\na broad band-like contrast as highlighted by the white lines. The width of the band-like\ncontrast varies from narrow at the edge of the sample to broad inside the sample. In this\norientation, the easy plane ( a\u0000b) of magnetization is in the plane of the TEM sample, and\nthe surface termination and sample edges lead to formation of a closure domain con\fguration\nto minimize the stray \felds. This is clearly seen from the colored magnetization map shown\nin Figure 2(c). The magnetization direction within each region is close to a h110itype\ndirection. It has previously been estimated from bulk magnetic measurements that although\nthere is an easy plane anisotropy in La 1:24Sr1:76Mn2O7, there is a small uniaxial anisotropy\nof about 7\u0002103J/m3along theh110idirection15. It is interesting to note that at the\nlocation where two of the domain walls intersect, a bright white line contrast is observed\n(indicated by the arrow). The broadening of the domain walls can be attributed to either\na large domain wall width or the presence of inclined domain walls. As we have already\n5estimated that the domain wall width for this material is 77 nm from the images of Sample\nS1, this cannot explain the broad contrast extending over a range of 300 nm. Hence we\ncan infer that the domain walls must be inclined with respect to the viewing direction.\nThis was further investigated by tilting the sample to observe the e\u000bect on the domain wall\ncontrast. Figure 2(b) shows an under-focused Lorentz TEM image of the same region after\ntilting by 22\u000eabout the axis shown in (b). Figure 2(d) shows the corresponding colored\nmagnetization map. The e\u000bective broadening of the domain wall contrast has decreased\nalong with a decrease in the length of the bright white line contrast.\nIn order to con\frm the origin of the contrast, we performed image simulations as shown in\nFigure 3. Figure 3(a) shows a simulated underfocus image and (b) shows the corresponding\ncolored magnetization map. The magnetic con\fguration with inclined domain walls (gray)\nused for these image simulations is shown schematically in Figure. 3(c). There is excellent\nagreement between the simulated images and the experimental ones, corroborating our view\nthat the domain walls observed for this sample are indeed inclined with respect to the\nviewing direction ( \u0018h001i). By comparison with the model, we can interpret the features\nindicated by the solid and dashed line in Fig. 2(a) as the intersection of the domain wall\nwith the top surface and bottom surface of the TEM sample, respectively.\nB. Ferromagnetic transition\nNext we explored the magnetic domain behavior as a function of temperature across the\nphase transition from paramagnetic to ferromagnetic state for both the sample geometries.\nFigure 4 shows the phase transition for TEM sample S1 (hard axis in the plane of the\nsample). As the temperature decreases from 120 K, the 180\u000edomain walls are seen to\nnucleate at the edge of the sample (bottom of the images) and then grow across the TEM\nsample. The \frst appearance of magnetic domain wall contrast was observed at T= 118\nK. This temperature is about 7 K lower than the Curie temperature of the same sample as\nmeasured from magnetometry to be Tc= 125 K. This di\u000berence can attributed to the fact\nthat at temperatures very close to Tc, the ferromagnetic domain signal is too weak to be\ndetected using Lorentz TEM. Similar di\u000berences between temperature at which magnetic\ncontrast is observed and the Curie temperature have previously been reported13?. The area\nfraction of the sample that was ferromagnetic was calculated as a function of temperature\n6from this series of images. The area fraction roughly corresponds to the total magnetization\nof the sample under the assumption that it is uniform through the thickness of the sample.\nA power-law \ft to the area fraction (which is representative of the magnetization) and\nthe reduced temperature, t= (1\u0000T=Tc), using the relation A/t\fyields the exponent,\n\f= 0:36. Figure. 4(b) shows the plot of the measured area fraction as a function of\ntemperature (symbols) together with the power-law best \ft to the data (red line). This\nvalue of\fis close to the literature reported value of \f= 0:32 for a three-dimensional Ising\nmodel?. Previous reports have determined the value of \f= 0:13 which indicate that the\nphase transition below Tcis still explained by the 2D Ising model7, however a crossover to\nthree-dimensional scaling close to Tchas been also been suggested8.\nAs for the domain wall structure, a distinctive di\u000berence was observed during the phase\ntransition for sample S2 compared with that for S1. Figure 5 shows a series of under-\nfocused Lorentz TEM images during cooling to below Tc. As the sample is cooled, there is\nno immediate formation of magnetic domain walls, but rather the formation of a nanoscale\ngranular contrast starting from T= 118 K, which increases in density as the temperature\ndecreases. The granular nanoscale contrast was only observed in the out-of-focus images\nand not in the in-focus image, indicating that it is magnetic in origin. Eventually these\nnanoscale magnetic clusters merge together to form magnetic domains separated by domain\nwalls, leading to a decrease in the total number of clusters. Finally at 100 K, most of\nthe nanoscale clusters disappear leaving behind domain walls that form a closure domain\ncon\fguration to minimize the stray \feld energy. A movie showing the in situ cooling of\nthe sample from two di\u000berent regions is included in the supplementary information19. It\nshould also be noted that at T= 108 K, there is a region in the center of the sample\nmarked by red arrow that does not show any black and white granular contrast related to\nthe nanoscale clusters, although it is surrounded by this contrast. Eventually at T= 103 K,\nthe granular contrast is seen inside the region, which slowly disappears by T= 100 K. This\nsuggests that there are local inhomogenieties (for example due to strain in the sample) that\ncan result in a di\u000berence in Curie temperature. The e\u000bect of such local inhomogenieties is\noften missed in bulk measurements as they are averaged over the entire sample. However\nusing LTEM, we are able to observe the coexistence of sub-micron size regions that are non-\nferromagnetic in the surrounding ferromagnetic region. Similar coexistence of charge-ordered\n(insulating) and charge-disordered (metallic FM) domains has been previously observed in\n7La5=8\u0000yPryCa3=8MnO 36. Additionally as the sample was cooled, the bend contour contrast\nin the TEM sample was seen to change sharply over a narrow temperature range just above\nthe Curie temperature, indicating a change in the strain state of the sample. This can be\ndirectly related with the magnetostriction of the sample as it undergoes the phase transition\nfrom the PM to FM phase. Note that the bend contour contrast stays stable over the\nrest of the temperature range analyzed. The abrupt change in volume and the resulting\nmagnetostriction e\u000bect at Tchas previously been reported in bilayer manganites and is\nassociated with the insulator-to metal transition in these materials20.\nFigure 6(a), (b) and (c) show the under-focus, over-focus, and in-focus Lorentz TEM\nimages respectively of the granular contrast for T= 108 K. The granular contrast arising\nfrom the nanoscale clusters (highlighted by the red circle) shows a distinctive white and\nblack intensity on either side of each cluster. The inset at the top right of Figure 6(a) and\n(b) shows the magni\fed view of the region circled in red. This black and white intensity\nreverses between the under-focus and over-focus images as shown by the plot of normalized\nintensity in Figure 6(d), and disappears for the in-focus images. This type of contrast is\nobserved for a spatial distribution of \fnite objects in the sample that lead to a phase shift of\nthe electron wave passing through it, resulting in the observation of the contrast only in out-\nof-focus images. Thus the contrast could be related to a distribution of magnetic objects\nor to e\u000bects such as strain related to the phase transition. If the origin of the contrast\nwas crystallographic, i.e. strain, then changes in the bend contour contrast would also be\nexpected. However this was only observed prior to the appearance of the granular contrast\nas mentioned earlier. We therefore infer that the origin is magnetic and is evidence for the\nformation of a random distribution of ferromagnetic clusters in a non-magnetic matrix. Since\nthe spins of individual atoms within these ferromagnetic clusters are aligned, each cluster\ncan be described as a nanoscale single domain magnetic object. The expected contrast\nin the out-of-focus images that is associated with such a single domain magnetic object is\nschematically shown in the bottom-inset of Figure 6(a) and (b). Further evidence for this\ninterpretation comes from the fact that the nanoclusters eventually merge to form domains.\nAn example of a wall segment that has formed is indicated by the red arrow in Figure 6(a)\nand (b).\nA similar granular contrast of nanoclusters has previously been observed, although only\nin cubic manganites such as Nd 0:5Sr0:5MnO 311and La 0:55Ca0:45MnO 313. In both cases, the\n8granular contrast was associated with the presence of ferromagnetic nanoclusters. However,\nin the case of Nd 0:5Sr0:5MnO 3, the granular constrast was observed only during the phase\ntransition from the AF phase to FM phase. In the case of La 0:55Ca0:45MnO 3, the ferromag-\nnetic nanoclusters with an ordered superstructure were seen to form within a matrix that\nwas already ferromagnetic with sub-micron size magnetic domains. Here we have observed\nthe formation of these nanoclusters in bilayer manganites during both cooling through the\nCurie temperature as well as heating through it, without the presence of a charge-ordered\nphase or any other form of superstructure. The lack of any structural or long-range charge\nordering was con\frmed using electron di\u000braction during the heating and cooling. From the\nplot of the intensity (Figure 6(d)), the size of these nanoclusters can be measured as roughly\n40 nm (peak to peak distance). However, it must be noted that the high defocus value\nused in these images results in additional magni\fcation. Therefore the true size of these\nnanoclusters is expected to be smaller than 40 nm. This size is still signi\fcantly larger than\nthe lattice spacing in the a\u0000bplane of\u00180:4 nm or the inter-bilayer distance of \u00182 nm.\nThis suggests that we are only able to image the clusters once they reach a size that their\nnet magnetic moment is detectable using Lorentz TEM.\nIV. SUMMARY\nIn summary, we have studied the magnetic domain wall structure in La 1:24Sr1:76Mn2O7\nin the ferromagnetic regime and its relation to the crystallography of the sample. Using the\nfreedom to prepare the TEM sample along di\u000berent crystallographic orientations, we inves-\ntigated the detailed structure of the domain walls and were able to conclude that fabrication\nof the TEM sample does not signi\fcantly alter the domain wall behavior as compared to\nthe bulk. When the hard axis of magnetization was in the plane of the TEM sample, 180\u000e\nBloch walls are observed. By measuring the domain wall width from the nanoscale imaging,\nwe determined the exchange sti\u000bness of the material. In the sample with the hard axis of\nmagnetization perpendicular to the sample plane, we observed broad band-like contrast for\nthe domain walls. By comparing the experimental images with simulated ones, we were\nable to conclude that the domain walls are inclined which results in the broadening of the\ncontrast. By analyzing the in-situ growth of magnetic domains as a function of temperature\nduring cooling, we were able to determine the nature of the ferromagnetic transition by\n9\ftting a power law to the magnetization versus temperature data and estimating the critical\nexponent\fto be 0:36. We infer that this corresponds to a crossover to three-dimensional\nscaling close to Tc. We were also able to visualize the formation of nanoclusters during the\nphase transition close to T=Tcwhich showed a direct evidence of co-existence of mag-\nnetic and non-magnetic phases in bilayer manganites. Additionally, we also observed that\nthere are local sub-micron scale regions which become ferromagnetic at slightly di\u000berent\ntemperatures as compared to their surroundings. Both the formation of nanoclusters and\nsub-micron scale regions suggest that this phase transition is percolative in nature. Further\ndetailed image analysis of the nanoclusters to determine their relative size, and density as a\nfunction of temperature could yield more insights into the details of the phase transition.\nNote added in proof. Recent work by Bryant et. al.21, reported on imaging the magnetic\ndomain walls as a function of temperature in La 1:2Sr1:8Mn2O7(x= 0:40) using low temper-\nature MFM. They measured Tcclose to 118 K, however, they observed that the magnetic\ndomain walls disappear at about 20 K below Tc. This observation could be related to sup-\npression of the magnetization at the surface which has been previously reported22. However,\nthey are only able to observe surface e\u000bects and do not report on the formation of magnetic\ndomain walls or the formation of nanoclusters close to Tc.\nACKNOWLEDGMENTS\nThis work was supported by the U.S. Department of Energy, O\u000ece of Science, Materials\nSciences and Engineering Division. Use of Center for Nanoscale Materials was supported\nby the U.S. Department of Energy, O\u000ece of Science, O\u000ece of Basic Energy Sciences, under\ncontract no. DE-AC02-06CH11357.\n\u0003cd@anl.gov\n1D. Argyriou, J. Mitchell, and P. Radaelli, Physical Review B 59, 8695 (1999).\n2C. D. Ling, J. E. Millburn, J. F. Mitchell, D. N. Argyriou, J. Linton, and H. N. Bordallo,\nPhysical Review B 62, 15096 (2000).\n3Y. Moritomo, A. Asamitsu, H. Kuwahara, and Y. Tokura, Nature , Nature 380, 141 (1996).\n104T. G. Perring, D. T. Adroja, G. Chaboussant, G. Aeppli, T. Kimura, and Y. Tokura, Physical\nReview Letters 87, 217201 (2001), 0105230.\n5Y. Konishi, T. Kimura, M. Izumi, M. Kawasaki, and Y. Tokura, Applied Physics Letters 73,\n3004 (1998).\n6M. Uehara, S. Mori, C. Chen, and S. Cheong, Nature 399, 560 (1999).\n7R. Osborn, S. Rosenkranz, D. N. Argyriou, L. Vasiliu-Doloc, J. W. Lynn, S. K. Sinha, J. F.\nMitchell, K. E. Gray, and S. D. Bader, Physical Review Letters 81, 3694 (1998).\n8S. Rosenkranz, R. Osborn, J. F. Mitchell, L. Vasiliu-Doloc, J. W. Lynn, and S. K. Sinha,\nJournal of Applied Physics 87, 5816 (2000).\n9Q. Lu, C.-C. Chen, and A. de Lozanne, Science 276, 2006 (1997).\n10A. Gupta, G. Gong, G. Xiao, P. Duncombe, P. Lecoeur, P. Trouilloud, Y. Wang, V. Dravid,\nand J. Sun, Physical Review B 54, R15629 (1996).\n11T. Asaka, Y. Anan, T. Nagai, S. Tsutsumi, H. Kuwahara, K. Kimoto, Y. Tokura, and Y. Matsui,\nPhysical Review Letters 89, 207203 (2002).\n12J. Loudon, N. Mathur, and P. Midgley, Nature 420, 797 (2002).\n13J. Tao, D. Niebieskikwiat, Q. Jie, M. A. Scho\feld, L. Wu, Q. Li, and Y. Zhu, Proceedings of\nthe National Academy of Sciences 108, 20941 (2011).\n14M. Medarde, J. Mitchell, J. Millburn, S. Short, and J. Jorgensen, Physical Review Letters 83,\n1223 (1999).\n15U. Welp, A. Berger, V. K. Vlasko-Vlasov, H. You, K. E. Gray, and J. F. Mitchell, Journal of\nApplied Physics 89, 6621 (2001).\n16J. F. Mitchell, D. N. Argyriou, J. D. Jorgensen, D. G. Hinks, C. D. Potter, and S. D. Bader,\nPhysical Review B 55, 63 (1997).\n17V. V. Volkov, Y. Zhu, and M. De Graef, Micron 33, 411 (2002).\n18A. Hubert and R. Sch afer, Magnetic domains: the analysis of magnetic microstructures\n(Springer, Berlin, 1998).\n19Supplementary information showing the in-situ movies of the ferromagnetic phase transition is\navailable online.,.\n20M. Matsukawa, H. Ogasawara, T. Sasaki, M. Yoshizawa, M. Apostu, R. Suryanarayanan,\nA. Revcolevschi, K. Itoh, and N. Kobayashi, Journal of the Physical Society of Japan 71,\n1475 (2002).\n1121B. Bryant, Y. Moritomo, Y. Tokura, and G. Aeppli, Physical Review B 91, 134408 (2015).\n22J. W. Freeland, J. J. Kavich, K. E. Gray, L. Ozyuzer, H. Zheng, J. F. Mitchell, M. P. Warusaw-\nithana, P. Ryan, X. Zhai, R. H. Kodama, and J. N. Eckstein, Journal of physics. Condensed\nmatter 19, 315210 (2007).\n12FIGURES\n500 nm [010] \n[001] \n[100] \n(a) (b) \n-100 0 -200 100 200 -40 -20 \n-60 020 40 \nDistance (nm) Projected Mag. Induction  (T.nm) \nFIG. 1. (Color online)((a) shows the under-focused LTEM image of sample S1 at 95 K. The\ntop right inset shows the di\u000braction pattern along the h100izone axis and the schematic shows\nthe orientation of the crystallographic axes. The bottom left inset shows the magnetization color\nmap overlaid on the image showing the presence of 180\u000edomain walls. (b) shows the plot of the\nprojected magnetic induction (black squares) across the domain wall computed by averaging the\nvalues shown in the dashed region in (a) and a \ft obtained to determine the domain wall width\n(red).\n13500 nm \n(a) (b) \n(c) (d) β = 22° \nFIG. 2. (Color online) (a) shows the under-focused LTEM image from sample S2 at 95 K. The\ninset shows the di\u000braction pattern obtained along the h001izone axis and the orientation of the\ncrystallographic axes is shown schematically. (b) shows the underfocus LTEM image of the same\nregion after tilting the sample by 22\u000eabout the axis shown in the \fgure. (c) and (d) show the\nreconstructed magnetization color map for (a) and (b) respectively. The colorwheel indicates the\ndirection of magnetization.\n14(a) (b) \n(c) \nxyzFIG. 3. (Color online)(a) shows the simulated under-focus LTEM image and (b) shows the\ncorresponding magnetization color map for a model with inclined domain walls forming a closure\ndomain con\fguration as shown schematically in (c).\n151 μm 120 K 118 K \n116 K 110 K 0.4 \n0.2 \n0.0 1.0 \n0.8 \n0.6 \n (1-T/T c)Area fraction (a) (b) \n0.04 0.08 0.12 0.16 Tc = 125 K FIG. 4. (a) shows a series of under-focus LTEM images acquired during the cooling of sample S1\nfrom 120 K to 110 K. (b) shows a plot of the area fraction of magnetic domains as a function of\nreduced temperature (1-T/T c) as calculated from the in-situ cooling image series (diamonds) and\na power-law \ft to the data (red line).\n16500 nm 118K 108K \n103K 100K FIG. 5. shows a series of under-focus LTEM images from sample S2 during in-situ cooling from 120\nK to 100 K. The nanoscale granular contrast starts to appear at 118 K and eventually disappears,\nleaving magnetic domain walls at 100 K.\n17100 nm \n0 20 40 60 80 \nDistance (nm) Normalized Intensity overfocus underfocus (a) (b) \n(c) (d) FIG. 6. (a)-(c) show the under-focus, over-focus and in-focus LTEM images respectively of an\narea showing the nanoscale granular contrast at 109 K from sample S2. A magni\fed view of the\nregion in red circle is shown in the top left inset showing the black and white contrast associated\nwith the nanoclusters. The schematic in the bottom right inset shows the relation between the\nblack and white contrast and the magnetization of the local cluster. (d) shows the plot of the\nnormalized intensity across the dashed red line in (a) and (b).\n18"    },    {        "title": "1509.04126v1.From_soft_to_hard_magnetic_Fe_Co_B_by_spontaneous_strain__A_combined_first_principle_and_thin_film_study.pdf",        "content": "arXiv:1509.04126v1  [cond-mat.mtrl-sci]  14 Sep 2015From soft to hard magnetic Fe-Co-B by spontaneous strain: A\ncombined first principle and thin film study\nL. Reichel∗and L. Schultz\nIFW Dresden, P.O. Box 270116, 01171 Dresden, Germany and\nTU Dresden, Faculty of Mechanical Engineering,\nInstitute of Materials Science, 01062 Dresden, Germany\nD. Pohl, S. Oswald, and S. F¨ ahler\nIFW Dresden, P.O. Box 270116, 01171 Dresden, Germany\nM. Werwi´ nski\nDivision of Materials Theory, Department of Physics and Ast ronomy,\nUppsala University, Box 516, SE-751 20, Uppsala, Sweden and\nInstitute of Molecular Physics, Polish Academy of Sciences , 60-179 Pozna´ n, Poland\nA. Edstr¨ om, E. K. Delczeg-Czirjak, and J. Rusz\nDivision of Materials Theory, Department of Physics and Ast ronomy,\nUppsala University, Box 516, SE-751 20, Uppsala, Sweden\nAbstract\nIn order to convert the well-known Fe-Co-B alloy from a soft t o a hard magnet, we propose\ntetragonal strain by interstitial boron. Density function al theory reveals that when Batoms occupy\noctahedral interstitial sites, the bcc Fe-Co lattice is str ained spontaneously. Such highly distorted\nFe-Co is predicted to reach a strong magnetocrystalline ani sotropy which may compete with shape\nanisotropy. Probing this theoretical suggestion experime ntally, epitaxial films are examined. A\nspontaneous strain up to 5% lattice distortion is obtained f or B contents up to 4at%, which leads\nto uniaxial anisotropy constants exceeding 0 .5MJ/m3. However, a further addition of B results in\na partial amorphization, which degrades both anisotropy an d magnetization.\nPACS numbers: 75.30.Gw, 71.15.Mb, 71.15.Nc, 81.15.Fg\nKeywords: Fe-Co, rare earth free permanent magnet, magneto crystalline anisotropy, tetragonal strain, DFT\n1I. INTRODUCTION\nFe-Co exhibits one of the highest magnetizations among all magnetic materials1,2and\nis thus very attractive for applications3. Fe-Co-B alloys are well known as soft magnetic\nmaterials due to the glass forming ability of boron4–7. In such amorphous materials the\nmagnetocrystalline anisotropy is suppressed. In this study, we de monstrate that Fe-Co-B\nis also a promising hard magnetic material, when the amorphization is av oided and the B\natoms induce a spontaneous strain in the crystalline lattice.\nWithin this introduction, we will first summarize criteria for soft and h ard magnetic\nproperties. We then use Density Functional Theory (DFT) to show that introducing B on\ninterstitials sites in the Fe-Co lattice results in spontaneously strain ed phases with substan-\ntial magnetocrystalline anisotropy (MCA). In order to probe the lim its of this approach\nexperimentally, we study a series of epitaxial thin films. Results are d iscussed with respect\nto the concepts of epitaxial and spontaneous strain in Fe-Co for o btaining rare-earth free\npermanent magnets. In particular, we relate Fe-Co-B presented here to recently published\nFe-Co-C films8.\nSince both, soft and hard magnets require a high magnetization, Fe -Co1,2is in focus of\nresearch. For soft magnetic materials, the magnetic anisotropy s hould be as low as possi-\nble, which is commonly achieved by reducing the grain size towards nan ocrystalline or even\namorphous materials5,7,9. Aiming at hard magnetic properties, a crystalline structure is es-\nsential as a magnetocrystalline anisotropy can favor particular dir ections of magnetization.\nThe bcc crystal structure of binary Fe-Co, however, exhibits on ly a low cubic anisotropy10.\nReducing the crystal symmetry by uniaxial strain is a route to achie ve high magnetocrys-\ntalline anisotropy in Fe-Co as proposed in different theoretical stud ies11–14. According to\nthese studies, tetragonally strained Fe-Co is considered a possible rare-earth free permanent\nmagnet material. Due to its high magnetostriction15,16, Fe-Co indeed appears to be more\nsusceptible to strain compared to e.g.Fe-Ni, which makes Fe-Co les s favorable regarding soft\nmagnet applications. Though inverse magnetostriction only describ es the influence of low\nstrains, it allows a speculation that large strains are beneficial for a high magnetocrystalline\nanisotropy.\nThetetragonalstrainiscommonly described by thelatticeparamet ers ofthestrained axis\ncand the compressed axes a, perpendicular to c. There are two main routes to strain the Fe-\n2Co lattice experimentally. First, in coherent epitaxial growth, Fe-C o thin films adapt the in-\nplanelatticeparameter afromthesubstrate, whichresultsinaninducedstrain( c/a >1)due\ntocompressive stresswithinthefilm’splane. However, thedrivingfo rcesforstrainrelaxation\naretoo highto maintain tetragonal distortion infilms thicker than15 monolayers17–21, which\nis equivalent to 2nm. In situ studies8revealed that binary Fe-Co films of 4nm thickness\nare again cubic with c/a= 1. In order to stabilize the strain, calculations based on Density\nFunctional Theory (DFT)22motivated an introduction of C atoms on interstitial sites in\nFe-Co. Delczeg et al.22proposed that about 6at% of these small atoms establish a c/aratio\nof 1.12. In experiments, however, the solubility on interstitial sites is much more limited.\nUsing non-equilibrium preparation methods like Pulsed Laser Depositio n (PLD), the limit\nof solubility, which is only about 0 .1at% for C or B in bcc Fe-Co in thermal equilibrium24,\ncan be shifted to significantly higher values. PLD prepared Fe-Co film s containing 2at%\nC exhibit such spontaneous strain with c/a= 1.03 up to at least 100nm thickness as\nshown in our previous study8. In contrast to the induced tetragonal strain at a coherent\nepitaxial interface, interstitials can result in a minimum of total ener gy at a certain lattice\ndistortion. This second approach is thus not limited to ultrathin films a nd accordingly,\nthe strain is also present in thicker films. In analogy to martensitic tr ansformations, we\nassign this distortion as spontaneous strain, though a direct exam ination of an associated\ntransformation is difficult, as common DFT only gives the ground stat e and in thin films, a\ntransformationisoftenhinderedbytherigidsubstrate23. Nevertheless, theenergydifferences\ncalculated between cubic and tetragonal state are within a factor of two comparable to\nthermal energies at room temperature and thus transformation s are expected to occur in\nbulk materials.\nComputational modeling can act as a powerful guide to experimenta l studies. Hence,\nwe begin our work by utilizing a combination of the most relevant and hig hly accurate,\nspin-polarized DFT methods. In the first step, structural prope rties are modeled with low\ndopant concentrations being reproduced by using large supercells and alloying treated in\nthe coherent potential approximation (CPA). The result of these structural studies are then\nused as input for the next step, where magnetic properties, includ ing magnetic moments\nof both Fe and Co atoms on each inequivalent site, as well as MAE and o rbital moments,\nare evaluated in fully relativistic calculations including the effect of the crucial spin-orbit\ncoupling.\n3In PLD, ions with high kinetic energy around 100eV are deposited25. Thus, besides\nregular deposition, an implantation of the material within the topmos t film monolayer takes\nplace26and allows for a supersaturation of Fe-Co films with C or B. However, this high\nenergy impact also supports the formation of lattice defects. Sup ersaturated PLD prepared\nFe-Co films with 2at% C retain c/aratios of 1.03 up to high film thicknesses8. In such\nspontaneously strained Fe-Co-C, magnetocrystalline anisotropy energies (MAE) of about\n0.4MJ/m3were observed independently from film thickness. Fe-Co with spont aneous strain\nshould not have limitations in film thickness and is thus considered more promising than\nFe-Co films with induced strain.\nAn open question, though, is how the spontaneous strain and the M CA may be further\nincreased. Since the solubility of C in Fe-Co is limited also in PLD prepared films8, we\nfocus on boron as atom of comparable size. The location of B in bcc Fe (and similar in bcc\nFe-Co) is still under debate. Boron can substitute Fe in the ideal bc c positions27–29, can\noccupy interstitial positions30–33, and B solution in Fe can be stabilized by Ni, N and C34,35\nboth, interstitially andsubstitutionally36. In a more recent theoretical investigation37, it was\nshown that B prefers the substitutional chemical disorder in bulk F e-B. However, this work\ndid not take a local relaxation of the strain around interstitial B ato ms into account. The\nauthors further found that surfaces stabilize the occupation of octahedral interstitials. Near\nfree surfaces, the geometrical pressure on B interstitials relaxe s. Occupation of octahedral\ninterstitials is followed by an increase of the B-Fe distances and thus a distortion of the\nbcc lattice. Baik et al.37thus give the motivation to further study the straining effect of B\ninterstitials in Fe(-Co) thin films which are also dominated by free surf aces.\nConsidering the different interstitial sites in a bcclattice, a tetrago nal strainis exclusively\nexpected, when the B atoms occupy octahedral interstitials. Sub stitutional arrangements\nor occupation of tetrahedral interstitials would not change c/a, but only affect the unit cell\nvolume. Octahedral interstitials as defined in Ref.38exist along all three spatial directions\n(see supplementary39). There are six octahedral interstitial sites in a bcc lattice, indicat ing\na high theoretical solubility, when the atomic radius of the interstitia l atom is sufficiently\nlow and chemical conditions are beneficial. If such an interstitial site is occupied by a small\natomlike B,the atomsoftheapexes ofthe octahedrondisplace alon gitsaxis, i.e. uniaxially.\nA lattice strain by means of c/a >1 is thus only possible, when the octahedral interstitials\nalong the caxis as e.g. (0;0;1/2) within the bcc unit cell are preferentially occup ied. The\n4question, howanadditionaloccupationoftheotheroctahedralin stitialsaffectsthestructural\nproperties and the total energy will be discussed based on DFT calc ulations.\nBesides the discussed lattice preferences of boron, amorphizatio n of the Fe-Co lattice is\nexpected to begin at a certain B content. Depending on the film prep aration conditions,\namorphous Fe-Co-B phases were reported at B concentrations o f e.g.7.56, 157or 22at%5.\nIn order to establish a high magnetocrystalline anisotropy in Fe-Co- B, amorphization has to\nbe avoided. Our study thus presents theoretical calculations of t he preference of B atoms to\noccupy the same type of octahedral interstitials due to lower ener gy of such configuration.\nThen it focuses on the MCA evaluation of ideal Fe-Co-B crystals ass uming all B impurities\noccupying the octahedral interstitials perpendicular to the subst rate. We then introduce the\nproperties of PLD prepared epitaxial Fe-Co-B films, taking the diffe rent possibilities how B\nmay affect the Fe-Co lattice into consideration.\nII. METHODS\nA. Computational methods\nFirst principles electronic structure calculations within the generaliz ed gradient approxi-\nmation (GGA)40were used to identify stable or metastable body centered tetrago nal (bct)\nstructures for B doped Fe-Co alloys.\nIn the first step, different B concentrations were modeled by thre e FeyB supercells, Fe 8B,\nFe16B and Fe 24B with 1 ×2×2, 2×2×2 and 2×2×3 supercells, respectively. Fe atoms\noccupy the ideal bcc positions, while B atoms are placed in the octahe dral positions, which\nare the most stable interstitial positions37. B dopants in the octahedral positions reproduce\nthe upper limit for tetragonal distortion. Substitutional B dopant s were not considered as\nthey do not contribute to the formation of distortion. The conjug ate-gradient algorithm\nas implemented in the Vienna Ab Initio Simulation Package (VASP)41–44was used to fully\nrelax these structures. The k-point mesh was set to 16 ×16×8, 8×8×8, 8×8×6 for\nFe8B, Fe16B and Fe 24B, respectively, within the Monkhorst-Pack scheme45. The plane-wave\ncut off and the energy convergence criterion of the scalar relativis tic calculations were set\nto 500eV and 10−7eV, respectively.\nIn the second step, the alloying effect on the equilibrium parameters (c/a)eq-ratio and\n5volumeVeqof these relaxed structures was studied using the Exact Muffin-Tin Orbitals -\nFull Charge Density (EMTO-FCD) method46–53. The accuracy of the EMTO method is\ncontrolled by the optimized overlapping muffin-tin potential49,51,54. The muffin-tin potential\noptimization procedure is described in details in Ref.55. The potential parameter ηwas cho-\nsen to match the equilibrium parameters obtained by EMTO to those o btained by the VASP\ncalculations. The potential parameter ηis 0.75, 0.85 and 0.9 for Fe 8B, Fe16B and Fe 24B,\nrespectively. The chemical disorder for (Fe 1−xCox)yB alloys was treated via the Coherent\nPotential Approximation (CPA)56,57. The one-electron equations were solved within the\nscalar-relativistic and soft-core approximations. The Green’s fun ctions were calculated for\n18 complex energy points distributed exponentially on a semi-circular contour with radius\nof 1.2Ry. The 3 dand 4sstates of Fe, Co, and the 2 sand 2pstates of B were treated as\nvalence electrons. s,p,d,forbitals were included in the muffin-tin basis set ( lmax= 3).\nThe one center expansion of the full-charge density was truncate d atlh\nmax= 8. Between 40\nand 500 uniformly distributed k-points were used in the irreducible wedge of the Brillouin\nzone. The electrostatic correction to the single-site CPA was desc ribed using the screened\nimpurity model58,59with a screening parameter of 0.6. Other screening parameters60have\nbeen tested for the smallest system and it turned out that their eff ect on the equilibrium\nparameters is less than 1%.\nThe magnetic properties, including magnetic moments and the magne tocrystalline\nanisotropy energy (MAE), of the various (Fe 1−xCox)yB systems were evaluated using the\nspin-polarized relativistic KKR (SPR-KKR)61,62method in the atomic spheres approxi-\nmation (ASA), in a similar manner as was done in Ref.22. Calculations were performed\nusing GGA40for the exchange-correlation potential and at least several tho usandk-points\n(depending on system size) were used for numerical integration ov er the Brillouin zone in\norder to obtain well converged values of MAE. Alloying was treated w ith the CPA56and\nthe MAE was obtained by total energy difference for two different m agnetization directions,\ni.e. MAE = Etotal(ˆm/bardbl100)−Etotal(ˆm/bardbl001).\nAll structures from subsection IIIB were fully relaxed, c/aratios and volumes were\noptimizedtogetherwithatomicrelaxationforevery c/aratioandvolume. Thosecalculations\nwereperformedwiththefullpotentiallinearizedaugmentedplanew avemethod(FP-LAPW)\nimplemented intheWIEN2k code63. Fortheexchange-correlation potential the GGAform40\nwas used. Calculations were performed with a plane wave cut-off par ameterRKmax= 6.5,\n6total energy convergence criterion 10−6Ry, with 4 ×4×4k-points and with radii of the\natomic spheres 1.96 a0for Fe/FeCo and 1.45 a0for B, where a0is the Bohr radius. The\nVirtualCrystalApproximation(VCA)wasusedtostudydifferentF e0.4Co0.6-Bcompositions,\nfollowing the same procedure as described in Ref.22. All parameters were carefully tested to\nprovide well converged values.\nB. Experimental\nThe Fe-Co-B samples were prepared as thin films performing Pulsed L aser Deposition\n(PLD) in ultra-high vacuum (5 ×10−9mbar) at room temperature. The used KrF excimer\nlaser (Coherent LPXpro 305) operates with a wave length of 248nm and a pulse length of\n25ns. On MgO(100) single crystal substrates, 3nm Cr seed layers and 30nm Au 0.55Cu0.45\nbuffer layers were deposited prior to the deposition of the 20nm thic k Fe-Co-B films. The\nAu-Cu and the Fe-Co-B layers were prepared in pseudo-co-depos ition, i.e. by repetitive\nchanging of the targets during PLD to achieve the aimed composition s. Elemental (Au,\nCu, Fe and Co) and a FeB composite target were used therefore. T he deposition rates were\nmeasured prior the preparation with a quartz crystal rate monito r.\nEnergy dispersive x-ray spectroscopy (EDX) measurements on a Bruker EDX in a JEOL\nJSM6510-NX electron microscope confirmed the intended film compo sitions. The boron\ncontent in the Fe-Co-B films was measured exploiting Auger Electron Spectroscopy (AES)\non a JEOL JAMP-9500F Field Emission Auger Microprobe device. The at omic concentra-\ntions64were calculated using standard single element sensitivity factors fr om mean values\nof sputter depth profiles measurements carried out with 1keV Ar+ions. Atomic force mi-\ncroscopy (AFM) measurements on an Asylum Research Cypher AFM were performed to\nstudy the surface morphologies and to confirm a high flatness of th e film surfaces. A Bruker\nD8 Advance diffractometer operating with CoK αradiation was used for X-ray diffraction\n(XRD) in Bragg-Brentano geometry. The Fe-Co-B lattice strains w ere derived from pole fig-\nure measurements, which had been carried out on an X’pert four cir cle goniometer (CuK α\nradiation). Transmission electron microscopy (TEM) studies were p erformed on a Titan3\n80-300 microscope, which was equipped with a C Scorrector and a Schottky field emission\nelectron source. The lamella was cut using focused ion beam on a FEI m icroscope (Helios\nNanoLab 600i). For magnetic hysteresis measurements at 300K, a vibrational sample mag-\n7netometer (VSM) mounted in a Quantum Design Physical Property M easurement System\nwas used.\nIII. RESULTS AND DISCUSSION OF DFT STUDIES\nA. Structure optimization\nIn order to prepare crystal structures of chemically disordered (FexCo1−x)yB, the simpler\nFeyB systems have to be considered as an initial step. Fe yB structures are modeled with\nB atoms placed into octahedral interstitial positions along the caxis. This configuration\ndetermines the upper limit of the tetragonal distortion. Internal relaxation of Fe atoms\naround the B interstitials leads to the elongation of cand shrinking of alattice parameter.\nIn other words, the c/aratio is now higher than 1, where 1 corresponds to a cubic lattice.\nThe total energies Eas a function of c/aratio and volume Vfor three Fe yB systems are\ncomputed by VASP and their plots are presented in Figure 1.\nFIG. 1. (Color online) Contour plots of total energy (meV/Fe atom) as a function of c/a-ratio and\nunit volume V, for Fe 8B, Fe16B and Fe 24B, respectively. The zero reference values for the energies\nareE(c/a=1,Vc/a=1\neq). The zero energy isolines are denoted by dashed lines. Tota l energy and\nvolume are normalized per Fe atom.\nAteachc/aratio,theequilibriumvolume( Veq)isobtainedbyusingaMorse-typeequation\nof state as minimum of E(V) dependency. Next, the equilibrium ratio ( c/a)eqis attained\nby fitting E(Veq) with a second order polynomial. Results of ( c/a)eqandVeqare listed\nin Table I. It is observed that both the c/aratio and volume increase with B content, as\nexpected.\nThree types of (Fe xCo1−x)yB supercells corresponding to different B concentrations have\nbeen considered, i.e. (Fe xCo1−x)8B, (Fe xCo1−x)16B and (Fe xCo1−x)24B, see Figure 2. Ini-\ntially, (Fe xCo1−x)8B and(Fe xCo1−x)16B were taken into account as reasonable crossing point\nbetween computational effort and expected experimental B solub ility. Preliminary results\nsuggested that higher B concentration gives higher c/aratio and MAE. Unfortunately, nei-\nther 11.1at% B ((Fe xCo1−x)8B) nor 5.9at% B ((Fe xCo1−x)16B) could be obtained experi-\n8TABLE I. Equilibrium ( c/a)eqratios and volumes ( Veq) of Fe yB systems obtained by EMTO and\nVASP.\nAlloyat% B (c/a)eq Veq(˚A3/Fe atom)\nVASP EMTO ∆ VASP EMTO\nFe8B11.11.126 1.132 -0.006 12.73 12.88\nFe16B5.91.089 1.084 0.005 12.21 12.25\nFe24B4.01.050 1.044 0.006 11.93 11.95\nFIG. 2. (Color online) Crystallographic models of (Fe xCo1−x)8B, (Fe xCo1−x)16B and\n(FexCo1−x)24B, respectively. The bigger red spheres represent Fe/Co ato ms while smaller black\nspheres denote B atoms.\nmentally, and what was shown instead, maximum B solubility producing m aximum tetrag-\nonal strain is about 4at%. At that point of this study the (Fe xCo1−x)24B structures (4at%\nB) were introduced and calculated.\nThe effect of substituting Fe by Co on the structural parameters of FeyB has been taken\ninto account by using the CPA approximation implemented in the EMTO m ethod. The\nequilibrium ratios ( c/a)eqand the lattice parameter aeqhave been evaluated for the rigid\nsupercells taken from the preceding VASP calculations. Thus, it is imp ortant to assess the\naccuracy of the EMTO method compared to VASP for the pure Fe yB. For each c/aratio,\na fully relaxed structure is generated by VASP at Veqand used as input for the EMTO.\n9Total energies are then calculated for each c/aratio and five different V′swhile keeping the\ninternal parameters fixed. Results obtained by EMTO, using the ab ove mentioned fitting\nprocedure for VASP, are listed in Table I. The error bar of the ( c/a)eqratio is defined by ∆\n= (c/a)VASP\neq- (c/a)EMTO\neq, withc/aratios obtained by VASP and EMTO. Good agreement\nbetween these two theoretical methods, regarding both ( c/a)eqandVeq, is observed.\nAfter validation of the EMTO method, the desired CPA structures o f (FexCo1−x)yB have\nbeen evaluated. Results are presented in Table II. It is observed t hat an increase of Co\nconcentration in (Fe xCo1−x)yB enhances the tetragonal distortion ( c/a)eq. At the same time\ntheaeqandVeqdecrease. Decrease in Veqis ascribed to the lower atomic volume of Co\ncompared to Fe.\nNote, that the tetragonal distortion predicted for B-doped Fe- Co alloys is larger than for\nC-doped alloys22. This can be understood by the fact that the B atomic radius is bigge r\nthan the C atomic radius. By adding 4at% of B a distortion around 1.05 –1.07 should be\nachieved, while for C-doping it is merely about 1.03–1.04. This comparis on suggests that\nB-doping could be a better way to achieve tetragonally distorted Fe -Co alloys.\nNote further, that with B-doping, the tetragonal distortion can be obtained for a broader\nrange of Co concentrations, compared to C-doped counterpart s. This suggests a possibility\nto prepare tetragonal Fe-Co alloys with lower Co contents. In ord er to find out whether this\nbrings any advantages, we discuss their magnetic characteristics in the following section.\nB. Preferential orientation of octahedral interstitials\nFor a permanent magnet it is decisive that the easy axis of all unit cells is aligned along\none particular direction, in which the magnet will be magnetized. The d ifferent orientations\nof octahedral interstitials38, which could be occupied by the B atoms, thus have to be consid-\nered. Consequences of a random distribution have to be discussed as well. The occupation\nof tetrahedral sites, which just result in a volume change, are not considered in the following\nsince they are energetically unfavorable. Baik et al.37computed that the energy difference\nbetween octahedral andtetrahedral sites amountsto0 .70eVforacubic Fe-Bsupercell, with-\nout taking into account an optimization of lattice parameters. The c orresponding energy\ndifference calculated by us is 0 .77eV/Batom, based on bcc Fe 2x2x2 supercells (Fe 16B) with\ngeometry optimization. This means, that B atoms favor the octahe dral sites even stronger\n10TABLE II. Summary of calculated structural (EMTO) and magne tic (SPR-KKR) characteristics\nof (FexCo1−x)yB systems.\nComposition ( c/a)eqaeq Veq mSmLµ0MsMAE MAE\n(˚A) (˚A3\nFe atom) (µB\natom) (µB\natom) (T) (µeV\natom) (MJ\nm3)\n(Fe0.5Co0.5)8B 1.247 2.716 12.49 1.71 0.066 1.87 118 1.51\n(Fe0.50Co0.50)16B 1.086 2.809 12.04 1.97 0.068 2.09 46 0.62\n(Fe0.45Co0.55)16B 1.091 2.802 12.00 1.91 0.068 2.04 51 0.69\n(Fe0.40Co0.60)16B 1.103 2.788 11.95 1.86 0.069 2.00 52 0.69\n(Fe0.35Co0.65)16B 1.116 2.774 11.91 1.82 0.069 1.96 43 0.58\n(Fe0.65Co0.35)24B 1.049 2.835 11.95 2.16 0.064 2.26 23 0.31\n(Fe0.60Co0.40)24B 1.051 2.832 11.94 2.12 0.066 2.22 30 0.40\n(Fe0.55Co0.45)24B 1.053 2.827 11.90 2.08 0.067 2.19 30 0.40\n(Fe0.50Co0.50)24B 1.056 2.822 11.87 2.04 0.068 2.15 28 0.38\n(Fe0.45Co0.55)24B 1.059 2.816 11.82 1.99 0.069 2.12 27 0.37\n(Fe0.40Co0.60)24B 1.063 2.810 11.79 1.95 0.070 2.08 29 0.40\n(Fe0.35Co0.65)24B 1.068 2.803 11.76 1.91 0.070 2.04 35 0.48\nthan Baik et al. showed. A similar set of calculations for the alloy (Fe 0.4Co0.6)16B, treated\nwithin the VCA, gives the energy difference 0 .39eV/Batom, which also suggests to consider\nonly the octahedral interstitials in the subsequent models.\nAs our calculations reveal a tetragonal distortion with uniaxial anis otropy along the c\naxis, independent of the B content, we can take any tetragonal b uilding block for the follow-\ning considerations, which are borrowed from martensite theory as this can be extrapolated\ntowards the unit cell level65,66. We consider martensite theory as reasonable since the diffu-\nsion of B as interstitial33is much faster incontrast to Fe(or Co) atoms68. In this picture, the\norientation of the tetragonal distortion of the Fe-Co ”martensit e” changes, when B moves\nfrom one octahedral site to another.\nThe connection of different tetragonal building blocks is possible by t win boundaries, but\nthey require additional twin boundary energy. A twin boundary is co nnected with a 90◦do-\nmain wall in these uniaxial ferromagnets, which further requires ma gnetic exchange energy.\nIn this simplified picture, both, elastic and magnetic contributions ma ke a different orienta-\n11tion of neighboring tetragonal building blocks energetically unfavor able. Twin boundaries\nmust be introduced, however, to minimize elastic stress energy and magnetostatic energy,\nand in bulk samples all three orientations are thus equivalent and sho uld occur together.\na) Fe54B2orthogonal b) Fe 54B2uniaxial\nFIG. 3. 3x3x3 bcc Fe supercells with two B atoms in octahedral interstitial positions (Fe 54B2) as\na) orthogonal and b) uniaxial configuration of interstitial axes.\nIn order to support this picture on the atomic scale, we performed DFT calculations of\na configuration with orthogonal B interstitials and compare it with a u niaxial orientation of\nB interstitials. For this comparison, one has to consider at least two B atoms per unit cell,\nwhereby the Fe/FeCo supercell has to be enlarged in order to keep low B concentration. In\nthe first step, two Fe based models are considered, both with two B atoms in octahedral\ninterstitial positions in bcc Fe 3x3x3 supercells (Fe 54B2), see Fig. 3. In the orthogonal\nmodel (Fig.3(a)), the positive strains are generated along two or thogonal axes, which leads\nto ac/aratio below 1. In the second model (Fig.3(b)), the octahedral int erstitial axes\nare oriented parallel, which forms strain exclusively along the caxis and leads to c/aratio\nabove 1. Calculations show that the uniaxial configuration is favore d with a difference\nof total energy equal to 129meV/Batom. In order to relate the t heoretical result to the\nexperimental Fe-Co-B system one has to go beyond the pure Fe mo del and, as we did in the\nprevious section, consider the influence of alloying with Co. For this r eason, in the second\n12step, the two Fe 54B2models described above have been recalculated for a selected Fe 0.4Co0.6\nconcentration within the VCA. Results show once again that the unia xial configuration is\nmorestable, withslightlylowertotalenergydifferenceof126meV/B atom. CPAcalculations\nare prohibitively computationally expensive for the EMTO method due to a large number of\ninequivalent atoms in the unit cells. Some crystallographic data of the optimized structures\nare collected in Table III. An interesting result is that the supercell volumes, which for\northogonal and uniaxial configurations were optimized independen tly, have the same values\nup to 5 significant digits, which indicates the high accuracy of these r esults.\nTABLE III. Crystallographic data of optimized Fe 54B2and (Fe 0.4Co0.6)54B2structures and differ-\nences of total energies between orthogonal and uniaxial con figurations of B interstitials.\northogonal uniaxial\nV a c/a V a c/a ∆E\n[˚A3] [˚A] [˚A3] [˚A]meV\nB atom\nFe54B2635.74 8.652 0.982 635.74 8.492 1.038 129\n(Fe0.4Co0.6)54B2625.31 8.636 0.971 625.31 8.454 1.035 126\nWith three B atoms per 3x3x3 bcc Fe supercell it is possible to constr uct a system with\noctahedral interstitials in each of the three spatial directions, re sulting in a cubic structure.\nSuch a Fe 54B3structure with a simulated random occupation of the octahedral in terstitials,\nwhile keeping their mutual distances as large as possible within the sup ercell, has been\ncarefully optimized together with its uniaxial Fe 54B3counterpart. The total energy differ-\nence between these structures is 49meV/Batom and the preferr ed configuration of boron\nis again the uniaxial one. After dividing the latter value by Boltzmann c onstant one ob-\ntains 570K/Batom, the same order of magnitude as room temperat ure. This implies, that\na small fraction of B atoms will occupy positions not yielding a tetrago nal strain. Never-\ntheless, calculations of the presented ideal cases, which may not c ompare the energies of\nall theoretical possibilities, but compare the most extreme scenar ios, suggest that an occu-\npation of neighboring orthogonal octahedral sites with varying or ientation is energetically\nunfavorable.\nFrom an experimental point of view, the preferential site occupat ion is tedious to be\nmeasured directly due to the much lower atomic number of B compare d to Fe-Co. However,\n13the reduced symmetry of thin films compared to bulk may result in a fa vorable orientation,\nwhich could be induced by coherent epitaxial growth, followed by a re laxation towards the\nspontaneously strained state as shown for Fe-Co-C films8. A preferential lattice orientation\nallows using integral methods like texture measurements to probe t he present tetragonal\ndistortion andmagnetization measurements to determine the magn etocrystalline anisotropy.\nAn agreement of these global measurements with the local DFT calc ulations would also\nconfirm the unique alignment of all easy axes, which is beneficial for p ermanent magnet\napplications.\nIn other words, when comparing DFT calculations with experiments, one has to consider\nthe different length scales. In DFT calculations, the preferential caxis is the result of\nthe spontaneous strain at the atomic scale, while in thin film experimen ts, the out-of-plane\norientationisgivenandmayactonthewholesample. Theexpectedpr eferential alignmentof\nthe strained caxis along the out-of-plane orientation thus has to be probed by th e following\nexperiments.\nC. Saturation magnetization\nFor the considered (Fe xCo1−x)yB systems, the influence of B on magnetic moments is re-\ngarded first. The average magnetic moments per atom, obtained b y SPR-KKR calculations,\nare listed in Table II. It is observed that the average magnetic mome nt decreases by increas-\ning the B content. This can besimply explained by addition of a non-mag netic B component\nitself, and furthermore by the reduction of the magnetic moments on Fe/Co atoms around\nthe B interstitial. In order to look closer at the latter effect an insigh t into atom and site\nspecific magnetic moments is necessary. Internal relaxation chan ges the distances between\nFe/Co atoms, especially around B interstitials. These reconfigurat ions affect the exchange\ncoupling between Fe/Co atoms and thus affects magnetic moments.\nInTableIVthelocalmagnetic moments forselected (Fe 0.5Co0.5)yBsystems arepresented.\nThe Fe/Co atoms that are the closest neighbors to the B atoms hav e the magnetic moments\nmost reduced compared to the moments on Fe/Co atoms far from B . The latter values alter\nrarely, indicating that the Fe/Co atoms far from B have bulk-like cha racteristics already for\n(Fe0.5Co0.5)8B. This suggests that the influence of a B impurity is relatively short- ranged.\nAn increase of Co content leads to a decrease of the total magnet ic moment for the\n14TABLE IV. Magnetic moments of (Fe 0.5Co0.5)yB systems ( µB/atom) obtained by SPR-KKR. Mo-\nments on Fe and Co atoms in the nearest neighborhood of B octah edral interstitial and as far as\npossible from B.\nAlloy close to B far from B\nat% BmFemComFemCo\n(Fe0.5Co0.5)8B11.11.74 0.98 2.89 1.95\n(Fe0.5Co0.5)16B5.91.52 0.82 2.85 1.93\n(Fe0.5Co0.5)24B4.01.52 0.85 2.76 1.91\nconsidered Co concentrations in the (Fe xCo1−x)24B system (see Table II). This negative\ndependence is not necessarily true beyond the studied range, sinc e for bct Fe/Co alloys a\nmagnetization maximum is observed for lower Co concentrations1,2. In our case the drop of\nmagnetic moment is accompanied by an increase of ( c/a)eq.\nD. Magnetocrystalline anisotropy energy\nTable II summarizes the MAEs of the (Fe xCo1−x)yB systems, as obtained by SPR-KKR\ncalculations. Trends are in many aspects similar as those obtained fo r interstitial carbon22.\nIn particular, the tetragonal distortion of the crystal structu re leads to significant values\nof the MAE under particular c/aand alloy concentrations. Here, however, the reduction\nin MAE is rather small when going from y= 16 to y= 24, most likely – as pointed out\nabove – because the c/avalues are larger for y= 24 with B than with C, which leads to\nlarger MAEs compared to similar dopant concentrations of C. Consid ering the MAE as a\nfunction of Fe concentration xfor the case of y= 24, there is a rather flat behavior, but\na maximum value of 0 .48 MJ/m3is obtained for x= 0.35. Fory= 24 and x= 0.40 we\nobtain MAE = 0 .40 MJ/m3, which can be compared to MAE = 0 .19 MJ/m3obtained with\nC-doping instead of B. This means that a rather small amount of inte rstitial B dopants\nappears to be enough to yield a significant MAE which is promising in a per manent magnet\ncontext.\nFor systems with y= 24, we have performed test calculations (not shown) utilizing a\nsimpler approach to the problem of alloying, the virtual crystal app roximation (VCA), fol-\n15lowing the same procedure as described in Ref.22. Although these calculations overestimate\nthe MAE by a factor between 2 and 4 compared to CPA calculations, n evertheless they\nresult in a similar trend and thus provide a further support to the fin ding that a substantial\nMAE can be obtained also for lower Co contents. In contrast to CPA results, VCA ones\nwere obtained within a full potential method.\nIV. EXPERIMENTAL RESULTS AND DISCUSSION\nA. Structural properties of Fe-Co-B films\nUsing PLD, we prepared a boron composition series and studied the s tructural and\nmagnetic properties of Fe-Co-B films with 20nm thickness. All films we re deposited on\nAu0.55Cu0.45buffer layers, which provide a reduced in-plane lattice parameter co mpared to\nFe-Co and thus induce a caxis oriented film growth67,69. Based on the theoretical results,\nwhere the Fe to Co ratio was varied, a composition close to Fe 0.4Co0.6was chosen as base for\nour experimental investigations. For the films of this (Fe xCo1−x)-B series, EDX measure-\nments gave x= 0.38(2). Corresponding XRD patterns are shown in Figure 4(a). The (002)\nreflection is the only intensity originating from the Fe-Co-B films and in dicates an epitaxial\nfilm growth of Fe-Co-B on Au-Cu. No additional phases as e.g. borid es, are detected. Pole\nfigure measurements (see supplementary39) confirm the epitaxial growth without formation\nof twins. Compared to the position of the Fe 0.38Co0.62(002) reflection given in literature70,\nwe observe a shift to lower 2 θangles for all samples. This indicates already an expanded c\naxis of the Fe-Co-B lattice, which will be discussed later in detail.\nThe AES depth profiles, which were carried out to determine the B co ntent of the Fe-Co\nfilms, did not reveal a significant variation of the B content in depend ence on the sputter\ndepth. ThisallowedustodetermineerrorbarsoftheBcontentsfo reachconsideredfilmfrom\nits statistical variation. With increasing B content, both a reduced scattering intensity and\na peak broadening of the Fe-Co-B(002) reflection is detected. Ap plying Scherrer’s formula,\nwe estimate the x-ray coherence length of the Fe-Co-B films, which can be taken as measure\nfor the Fe-Co-B crystal size. The results are plotted in Figure 4(b ). An increase in B content\nleads to a substantial decrease of coherence length from almost 1 0nm in binary Fe 0.38Co0.62\nto about 3nm for the film with 9 .6at% B.\n16/s53/s48 /s54/s48 /s55/s48 /s56/s48\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s50/s52/s54/s56/s49/s48/s40/s98/s41\n/s70/s101/s45/s67/s111/s40/s48/s48/s50/s41/s77/s103/s79/s40/s48/s48/s50/s41\n/s65/s117\n/s48/s46/s53/s53/s67/s117\n/s48/s46/s52/s53/s40/s48/s48/s50/s41\n/s50/s48/s32/s110/s109/s32/s70/s101\n/s48/s46/s51/s56/s67/s111\n/s48/s46/s54/s50/s32/s43\n/s32/s110/s111/s32/s66\n/s32/s50/s46/s53/s32/s97/s116/s37/s32/s66\n/s32/s52/s46/s50/s32/s97/s116/s37/s32/s66\n/s32/s57/s46/s54/s32/s97/s116/s37/s32/s66/s108/s111/s103/s46/s32/s105/s110/s116/s101/s110/s115/s105/s116/s121\n/s50 /s32/s47/s32/s176\n/s98/s117/s108/s107/s32/s70/s101\n/s48/s46/s51/s56/s67/s111\n/s48/s46/s54/s50/s40/s48/s48/s50/s41/s40/s97/s41\n/s32/s88/s82/s68/s32/s99/s111/s104/s101/s114/s101/s110/s99/s101/s32/s108/s101/s110/s103/s116/s104 /s32/s47/s32/s110/s109\n/s66/s32/s99/s111/s110/s116/s101/s110/s116 /s32/s47/s32/s97/s116/s37\nFIG. 4. (Color online) (a) XRD patterns of 20nm Fe 0.38Co0.62-B films with varying B content on\nAu0.55Cu0.45buffers. Relevant reflections are marked. The (002) equilibri um 2θangle of binary\nFe0.38Co0.62is given as broken line. The XRD pattern of a Fe 0.38Co0.62film without B addition is\nadded for comparison. The intensity maxima near 60◦and 77◦are attributed to PLD droplets with\nthe composition of the used targets. (b) shows the x-ray cohe rence length according to Scherrer’s\nformula against the AES determined B content.\nA TEM study of a Fe-Co-B film with 4 .2at% B confirms the crystallinity and continuity\nof the film (Figure 5). The Fourier Transform (FT) of the Fe-Co-B fi lm (inset) gives a very\nregular pattern as expected for an epitaxially grown crystal. Minor contrasts within the\nFe-Co-B film may indicate the formation of separated grains with a gr ain size of approx.\n17FIG. 5. (Color online) TEM image of a 20nm Fe 0.38Co0.62film with 4 .2at% B deposited on\nAu0.55Cu0.45. The zone axis is [010]. The inset shows a FT image of the Fe-Co -B layer. The\nmarked reflections were used for determination of the lattic e parameters canda.\n5nm. This would be in agreement with the observed x-ray coherence length (Figure 4(b)),\nbut may also be linked to thickness variations of the lamella.\nThe strong reduction of crystal size at higher B contents as indica ted by the XRD mea-\nsurements (Fig. 4(b)) has been confirmed by an additional TEM inve stigation of the film\nwith the highest B content (see supplementary39), which revealed oriented nanocrystals of\nthe same diameter surrounded by nanocrystalline and amorphous p hases. In literature,\nboth, a formation of Fe-Co-B nanocrystals9or amorphous Fe-Co-B7is reported for films of\nsimilar B contents. Asai et al. also observed an amorphous structu re already at B contents\nabove 5at% in addition to crystalline Fe-Co-B in Fe 0.7Co0.3-B films7. Our measurements\nthus confirm their findings of a decreasing size and fraction of Fe-C o-B crystals with in-\ncreasing B content in the films. However, all of the presented films c ontain a significant\ncrystalline fraction which is grown epitaxially. These crystals compris e a certain amount of\nthe alloyed B atoms as is clearly indicated by the change of their lattice parameter c, when\ncompared to the binary Fe 0.38Co0.62film, which is given as reference in Figure 4(a).\nIn order to characterize the tetragonal strain in the epitaxially gr own Fe-Co-B films,\n{011}pole figures were measured for all films. As described in Ref.8, thec/aratios were\n18/s49/s46/s48/s48/s49/s46/s48/s50/s49/s46/s48/s52/s49/s46/s48/s54\n/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s40/s98/s41\n/s32/s32\n/s32/s88/s32/s61/s32/s66\n/s32/s88/s32/s61/s32/s67/s99/s47/s97/s40/s97/s41\n/s32/s32/s75\n/s85/s32/s47/s32/s77/s74/s47/s109/s179\n/s66/s32/s99/s111/s110/s116/s101/s110/s116/s32/s47/s32/s97/s116/s37\nFIG. 6. (Color online) (a) Tetragonal distortion c/afor 20nm thick Fe 0.38Co0.62-B films in depen-\ndence of the B content (full squares). Results of Fe-Co-C film s8(open diamonds) have been added\nfor comparison. Note that the film with the highest C content i s of Fe-C type and did not contain\nCo. (b) Corresponding uniaxial magnetic anisotropy consta ntsKUof the films. The results from\nthe DFT calculations Fe 0.4Co0.6-B (Table II) have been added as open stars in both graphs. Cub ic\nFe-Co (c/a= 1) with no uniaxial MCA ( KU= 0) has been added, although it was not treated in\nour calculations. Broken lines are guides to the eye.\ndetermined from the tilt angles of these planes. In all films, one pred ominant variant with\nlongercthan in-plane axes was found. No indications for variants with deviat ingcaxis\norientation were found. The c/aratios are plotted in Figure 6(a) for the different boron\ncontents. An increase of tetragonal distortion is observed with in creasing B content up to\n4.2at%. The maximum observed c/aratio is 1.045 and is achieved for this composition. We\ncan compare this finding to Fe 0.4Co0.6-C films (open diamonds), which exhibit a maximum\ntetragonal strain of c/a= 1.03 and a lower dependence on the C content as discussed in our\nprevious study8. The observation of an increased strain in the Fe-Co-B films qualitat ively\nconfirms our theoretical calculations, which predicted anenhance dc/aratioby about0.03in\nFe-Co-B when compared to Fe-Co-C. The maximum observed strain withc/aof nearly 1.05\n19is also reasonably close to the predicted c/a= 1.063 for Fe 0.4Co0.6with 4at% B (Table II).\nHowever, a further increase of B content is not followed by a furth er increased tetragonal\nstrain, which was predicted by DFT. There are different possible rea sons for this difference:\n(a)ThelimitedsolubilityofboroninFe-Cowhichisnearlyzerointhermal equilibrium24, but\nis obviously increased substantially by PLD, an effect known for many systems71. A limited\nsolubility favors nanocrystalline or amorphous B-rich phases which r educe the effective B\ncontent in the Fe-Co matrix and thus also decreases the tetragon al strain. (b) The possible\nsubstitutional alloying of B asdiscussed within theintroduction, whic h would benefit acubic\nlattice. And/or (c) the possibility that not all B atoms occupy octah edral interstitials along\nthe axis perpendicular to the film surface, but also interstitials along the in-plane axes are\noccupied, as e.g.(1/2;0;0) within the bcc unit cell, which benefit a cer tain in-plane strain,\ni.e.a reduction of the measured c/a.\nAll reasons have in common that the amount of B atoms, which contr ibute in a lattice\nstrain along the caxis, is effectively reduced. Our DFT calculations were based on the\nassumption that all B atoms occupy octahedral positions along the caxis. As shown in the\ncalculationsinsectionIIIB,theBatomsindeedpreferentiallyalignino ctahedralinterstitials\nalong one particular axis, although all three crystal axes are equiv alent in a Fe-Co lattice.\nThe experimental results thus confirm these calculations. Howeve r, because of the named\nreasons, not all B atoms contribute to the caxis oriented strain above a B content of 4 .2at%.\nA likely explanation, why DFT calculations are not able to confirm the ex periments exactly\nis, that they do not consider kinetic effects. Films are prepared with in a finite time, which\nmay lead to local variations of B density and thus result in an occupat ion of different lattice\nsites than expected from DFT, which describes the ground state.\nThe comparison of the B and C doped Fe-Co films8implies that a higher amount of B\nthan C atoms can be solved in the PLD prepared Fe-Co films and prefe rentially occupies\ninterstitial sites along the caxis, which favors a higher strain. An additional contribution\nmay come from the size of the B atoms, which are bigger than C atoms . However, when a\ncertain limit is reached, further added B (or C) atoms lead to a decre ase ofc/a. This limit is\nreached at around 4at% B (or C). Since the caxis length is not altered when adding more B\nor C (see Figure 4), we conclude that most of the additional atoms o ccupy sites along the a\naxes or are dissolved from the Fe-Co crystals. The latter explains t he decreasing crystal size\nwith increasing B (or C) content as discussed with Figure 4(b) and in R ef.8, respectively.\n20For the Fe 0.38Co0.62film with 4 .2at% B, the lattice distortion was determined from the\nFT of the TEM images (Figure 5). Compared to the XRD result, where c/awas 1.045\n(Figure 6(a)), we observe a reduced strain of 1.02 ±0.01. Such strain reduction after TEM\nlamella preparation was already reported for Fe-Co-C films and was a ttributed to a lattice\nrelaxation due to a reduced constraint of a thin TEM lamella compared to a continuous\nfilm8.\nFrom growth studies of Fe-Co-C films, we concluded that the misfit d islocations which\nform during film growth affect the whole underlying film69. Most of the crystal volume is\nthus expected to exhibit the same strain state. Only a small fractio n of the film close to\nthe buffer interface had an increased strain8, expressed by a c/aratio increased by 0.02.\nThis also holds for the Fe-Co-B films examined here. The tetragonal strain determined\nclose to the Au-Cu buffer from TEM images of the 4 .2at% B containing Fe 0.38Co0.62film\nis 1.05±0.04. This is about 0.03 higher than the value for the film’s volume as det ermined\nin TEM, which may be explained by the lower in-plane lattice parameter o f Au0.55Cu0.45\ncompared to Fe-Co69. The TEM studies thus prove that only a very small fraction of the\nfilms is influenced by this induced strain at the buffer-film interface an d most of the film\nexhibits a spontaneous strain.\nThe structural characterisation of the films clearly supports our picture of tetragonally\nstrained Fe-Co-B lattices, where the axis perpendicular to the film s urface is preferentially\nstrained and the in-plane axes are compressed. The decisive quest ion, why a predominant\namount of B atoms preferentially occupies the octahedral interst itials along this axis, which\nthus is the strained caxis of the tetragonal lattice (Fig.2), may be answered as follows. DFT\ncalculations show that there is a strong energetical preference f or two nearby B atoms to oc-\ncupy the same type of octahedral interstitial position. The subst rate may act as a seed layer\nto establish that the first B atoms occupy octahedral interstitial in direction perpendicular\nto the substrate. The DFT results then suggest that such domain will grow preferentially,\ncompared to a random interstitial occupation or formation of multip le domains, both hav-\ning higher energy. Multiple variants with differently aligned caxes are unfavorable due to\nthe required interface, i.e. twin boundary, energy. Thus, Fe-Co -B films exhibit a quasi\nsingle crystalline state with one preferentially oriented strained caxis. This axis aligns\nperpendicularly to the film surface due to the square surface symm etry of the Au-Cu buffer.\nAs second main result, the structural measurements revealed sp ontaneously strained\n21Fe0.38Co0.62-B phases similar to those in Fe-Co-C8. The 20nm thick films are much thicker\nthan coherently strained films. The tetragonal strain is not depen dent on film thickness:\n100nm thick films with the same compositions (see supplementary39) exhibit the same c/a\nratios as the films presented here. The studied spontaneously str ained Fe-Co-B films con-\nfirm our DFT calculations, which suggested minima of total energy de pending on the B\ncontent (Table II). However, and this is an important difference to the Fe-Co-C films, the\nspontaneous strain indeed depends on the particular B content – a t least for low B contents\nup to about 4at%.\nB. Magnetic properties of the Fe 0.38Co0.62-B films\nIn order to compare the magnetic properties of spontaneously st rained Fe-Co-B films, the\nanisotropy constant KUof the strain related uniaxial MCA was determined from hysteresis\nmeasurements as described in Ref.8. The results are summarized in Figure 6(b). Similar\ntoc/a,KUalso has a maximum in dependence of the B content. The highest MCA w ith\nKU= 0.53MJ/m3isreachedinthe2 .5at%BcontainingFe 0.38Co0.62film. Filmswithsmaller\nB content have a lower MCA because of their reduced tetragonal s train. B contents above\n2.5at% in Fe 0.38Co0.62do not increase the MCA, although the c/aratio’s maximum is at a\nB content of 4 .2at% (Figure 6(a)). The observed decrease of MCA with further in creased B\ncontent occurssimultaneously withasignificant reduction ofsatur ationmagnetizationinthe\nfilms (Figure 7). While our theoretical results described a reduction ofMSby about 1 .5%\nwith each percent B (open stars), the experiments (squares) re veal a stronger dependence\nbetween B content and magnetic saturation: The decrease is abou t 2.5% with each 1at%\nB. A possible reason might be a decreased Curie temperature due to the addition of B.\nHowever, this is not expected72for B additions in Fe. We rather attribute the higher\ndependence of MSon the B content to a stronger perturbation of the Fe-Co lattice b y the\nB atoms in the PLD prepared films than it is described by the CPA appro ach, where the B\natoms occupy regularly distributed interstitials. The calculations alr eady revealed that Fe\nand Co atoms have a strongly reduced magnetic moment, when B is th eir nearest neighbor.\nIn comparison to our experimental study, magnetron sputtered Fe-Co-B films5, where the\nmaterial was deposited with substantially lower kinetic energy, exhib it a lower decrease of\nMSwith increasing B content, namely 1 .5% per added 1at% B, which is consistent with\n22/s48 /s50 /s52 /s54 /s56 /s49/s48 /s49/s50/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53\n/s32/s70/s101\n/s48/s46/s51/s56/s67/s111\n/s48/s46/s54/s50/s45/s66/s32/s102/s105/s108/s109/s115\n/s32/s70/s101\n/s48/s46/s52/s67/s111\n/s48/s46/s54/s45/s66/s32/s40/s68/s70/s84/s41\n/s32/s32/s181\n/s48/s77\n/s83/s32/s47/s32/s84\n/s66/s32/s99/s111/s110/s116/s101/s110/s116/s32/s47/s32/s97/s116/s37\nFIG. 7. (Color online) Magnetic saturation µ0MSin dependence of the B content for the\nFe0.38Co0.62-B films (squares) and the corresponding calculated structu res (stars) according to\nTable II. Linear fits have been added as broken lines.\nour theoretical results (Table II).\nWe argue that both, the strong reduction of MSand the decrease of MCA observed in our\nsamples originate from an interplay of the proceeding amorphization due to the increasing\nB content and the high energy impact of the ions during PLD25which may disturb the\nlocal environment of the Fe and Co atoms drastically. However, whe n we compare the\ntheoretical and the experimental results of KU(Figure 6(b)), we find increased values for\nthe Fe 0.38Co0.62-B thin films at low B contents. We attribute this discrepancy to a lack of full\npotential effects inDFT. At Bcontents above 4at%, this underest imation is overcome by the\ndiscussed structural effects. We thus observe a very good agre ement of the experimentally\nmeasured magnetic anisotropy with the theoretically determined va lue for (Fe 0.4Co0.6)24B,\nbut no longer for (Fe 0.4Co0.6)16B, i.e. 6at% B (open circles in Figure 6(b)).\nConcluding the magnetic measurements of the Fe 0.38Co0.62-B films, we observe a direct\ndependence of the magnetocrystalline anisotropy on the spontan eous strain for low B con-\ntents. However, an increased tetragonallity in the Fe-Co-B lattice did not necessarily imply\nanincreased MCAforthecompletefilms duetotheformationofnon- crystalline ordisturbed\nregions.\n23/s48/s46/s52 /s48/s46/s53 /s48/s46/s54/s49/s46/s48/s48/s49/s46/s48/s50/s49/s46/s48/s52/s49/s46/s48/s54/s99/s47/s97\n/s120/s32/s105/s110/s32/s70/s101\n/s120/s67/s111\n/s49/s45/s120/s48/s46/s48/s48/s46/s53/s49/s46/s48/s32\n/s75\n/s85/s32/s47/s32/s77/s74/s47/s109/s179\nFIG. 8. (Color online) Measured tetragonal lattice strain ( full squares) and uniaxial magnetocrys-\ntalline anisotropy constants KU(full circles) in dependence on Fe-Co ratio in 20nm films with\n4at% B on Au 0.55Cu0.45. DFT results for c/aandKUare added (open stars).\nC. Structural and magnetic properties in dependence on the Fe/Co ratio\nThe study of various Fe/Co ratios by DFT calculations (Table II) mot ivated for a com-\nparison of films with unaltered B content. We chose Fe xCo1−xfilms with 4at% B due to\nthe observed strain maximum (Figure 6(a)). The measured tetrag onal strain and the ob-\nserved perpendicular anisotropy constants KUin these 20nm thick films are summarized in\nFigure 8. With regard to the tetragonal strain, we observe a slight decrease with increasing\nFe content. Although there is only little variation between the films, t his finding is qualita-\ntively consistent with the theoretical predictions for (Fe xCo1−x)24B in Table II, which also\ngive higher c/aratios for higher Co contents. Due to already described reasons, we do not\nreach as highstrains ascalculated by DFT. The highest experimenta lly observed c/ais 1.045\nforx= 0.36.\nWhen comparing the magnetocrystalline anisotropy energies KU, we do not observe a\nstrong dependence on the Fe/Co ratio, which in principle fits to the w eak dependency\nproposed by DFT (see Figure 8). KUis about 0 .5MJ/m3on average, but slightly lower\non the Co rich side and slightly higher in the films with more Fe than Co. Th e maximum\nobserved KUis 0.54MJ/m3forx= 0.6. The DFT calculations, however, predicted the\nhighest MCA for the lowest Fe content.\nFromthe Fe xCo1−xfilms with 4at% B we thus conclude that higher Fe contents ( x= 0.6)\n24not only lead to a higher magnetic saturation MS, but are also beneficial for a high MCA\nat low tetragonal strain. Such Fe rich films or additional concepts t o stabilize higher strains\nbeyondc/a= 1.05 obtained here are considered promising for further research. As one way\nto fulfill the latter task, we suggest the choice of a preparation me thod with less energetic\nimpact, where the effect of lattice perturbation due to the B atoms might be smaller.\nV. CONCLUSION\nIn this combined theory-experimental study, Fe-Co-B is introduc ed as an alloy with spon-\ntaneousstrainandhardmagneticproperties. Structurecalculat ions basedonEMTO predict\nstrong tetragonal lattice distortions up to c/a= 1.25 in dependence on the B content, if all\nB atoms occupy octahedral interstitials along the caxis. This arrangement is preferential,\nwhen compared to an occupation of other possible interstitial sites . SPR-KKR results for\nthe distorted Fe-Co-B lattices give strong magnetocrystalline anis otropies when the former\ncubic symmetry is broken. For 11at% B, KUshould be comparable to shape anisotropy.\nExperiments, however, show that the amount of B atoms which str ains the Fe-Co lattice is\nlimited. At a B content higher than 4at%, the strain decreases. A su persaturation of the\noctahedral lattice sites with B atoms is considered to be responsible for this deviation from\nthe ideal behavior of B interstitials as treated in DFT. For low B conte nts, our theoretical\nresults are well confirmed by experiments, in particular when regar dingKU. The highest\nmagnetocrystalline anisotropies are observed for Fe 0.38Co0.62with 2.5at% B or in Fe richer\nFe0.6Co0.4with 4at% B. KUof these alloys is above 0 .5MJ/m3. 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Here, we use scanning tunnel-\ning spectroscopy to identify the di\u000berent quantum ground states of Manganese phthalocyanine on\nPb(111). We observe Shiba states, which are split into triplets by magnetocrystalline anisotropy.\nTheir characteristic spectral weight yields an unambiguous proof of the nature of the quantum\nground state.\nMagnetic adsorbates on a superconductor create a\nmagnetic scattering potential for the quasi-particles of\nthe superconductor. A single spin gives rise to so-called\nYu-Shiba-Rusinov (Shiba) states [1{3]. Recently, it was\nargued that hybridization of the Shiba states can lead to\nShiba bands with nontrivial topological character [4{6].\nThis is essential for the formation of Majorana modes,\nwhich have been detected in ferromagnetic chains of Fe\natoms on a Pb(110) surface [5]. If not only one but sev-\neral Shiba states are present, the hybridization will lead\nto a more complex band structure. Di\u000berent origins of\nmultiple Shiba states are discussed theoretically. These\ninclude di\u000berent angular momentum scattering contribu-\ntions, individual dorbitals acting as separate scatter-\ning potentials, or low-energy excitations due to magnetic\nanisotropy or vibrations [7{12]. However, experimentally,\nthe origin of multiple Shiba states is often di\u000ecult to de-\ntermine [13, 14].\nConcomitantly with the formation of a Shiba state,\nthe exchange coupling between the magnetic adsorbate\nand the substrate also drives the formation of a singlet\nKondo state [10, 15{17]. If the Kondo energy scale\nkBTKis much larger than the superconducting pairing\nenergy \u0001, the \"Kondo screened\" state is the ground\nstate. On the other hand, if kBTK\u001c\u0001, an unscreened\n\"free spin\" state with S > 0 is the ground state. A\nShiba resonance as \fngerprint of this magnetic interac-\ntion can be found in the quasiparticle excitation spectra\nwithin the superconducting energy gap if kBTK\u0018\u0001 [18].\nThis resonance corresponds to a transient state, in which\nan electron is added/removed to/from the ground state.\nThus, the electron occupation changes and the spin is\naltered by \u0001 S=\u00061=2. In the \"Kondo screened\" case,\nthe Shiba resonance is found with a binding energy Eb\nbelow the Fermi level EF. The weaker the exchange\ncoupling and Kondo screening, the closer is the Shiba\nstate to the Fermi level. The crossing of the Shiba state\nthroughEFmarks the quantum phase transition from\nthe \"Kondo screened\" S= 0 to the \"free spin\" state\non the superconductor, i.e.,S > 0. This transition oc-curs atkBTK\u00180:3\u0001 and has been described theoreti-\ncally [15, 17, 19] and experimentally [16, 20, 21].\nSpin-1=2 systems feature one pair of Shiba resonances\nat\u0006Eb[18]. If the adsorbate carries a higher spin, mul-\ntiple Shiba states may appear inside the gap as discussed\ntheoretically by \u0014Zitko and co-workers [10]. They argue\nthat such systems may involve multiple Kondo screen-\ning channels with di\u000berent coupling strengths Jkand,\nhence, multiple Shiba states. Furthermore, a mutual cou-\npling of the spins may lead to a splitting of the peaks\nin the excitation spectra in the presence of magnetic\nanisotropy. Here, we resolve triplets of Shiba states on\nsingle paramagnetic molecules. A multitude of di\u000ber-\nent adsorption sites provides access to a large range of\nmagnetic coupling strengths with the substrate. We de-\ntect the splitting of the Shiba states even throughout\nthe quantum phase transition from a \"Kondo screened\"\nto a \"free spin\" ground state. The intensities of the\nShiba resonances yield an unambiguous proof of the na-\nture of the spin states and their splitting by magnetic\nanisotropy. The basic understanding of the in\ruence of\nmagnetic anisotropy on many-body interactions is crucial\nfor the design of quantum states with controlled proper-\nties. Furthermore, its knowledge may provide interesting\napproaches for creating and addressing Majorana states\nin proximity coupled magnetic nanostructures [12].\nRESULTS\nDetection of Shiba states\nThe e\u000bects of magnetic anisotropy on Shiba states can\nbe explored experimentally by bringing a metal-organic\nmolecule into contact with a superconductor. The or-\nganic ligand is responsible for the splitting of the spin\nstates with S\u00151 of the transition metal core [22, 23].\nManganese phthalocyanine (MnPc) has a spin S= 3=2 in\ngas phase and retains a magnetic moment on metal sur-\nfaces [24{26]. In particular, its magnetic moment inter-arXiv:1509.09108v1  [cond-mat.mes-hall]  30 Sep 20152\nMnPcPb(111)2∆tip+2∆sample\n(a) (b)\n(c)\n-3.0 -2.5 -2.001\n dI/dV (μS)\nSample bias (mV) Sample bias (mV)2 0 -2dI/dV (μS)2\n01\nFIG. 1. Manganese-phthalocyanine (MnPc) on Pb(111): (a)\nSTM topography of a MnPc monolayer island on Pb(111)\n(V= 50 mV, I= 200 pA), scale bar is 2 nm. (b) dI=dV spec-\ntra on the pristine Pb(111) surface ( top, o\u000bset for clarity) and\non the center of a MnPc molecule ( bottom ) inside a molecular\nisland (opening feedback loop at: V= 5 mV, I= 200 pA).\n(c) Zoom on the subgap excitations below EFin the MnPc\nspectrum in (b). The inset shows a MnPc structure model.\nacts with the superconductor Pb and shows single Shiba\nstates when measured at 4.5 K [16].\nDeposition of MnPc molecules at room temperature re-\nsults in self-assembled, densely-packed monolayer islands\n[see Fig. 1(a)]. The molecules appear clover-shaped with\nfour lobes around a central protrusion of the Mn ion.\nThe nearly square lattice of the molecular adlayer accom-\nmodates many di\u000berent adsorption sites of the Mn core\non the hexagonal Pb lattice. Consequently, this Moir\u0013 e-\nlike pattern involves variations in the electronic and mag-\nnetic coupling strength between adsorbate and substrate\n[16, 25]. This rich system allows us to identify di\u000berent\nquantum ground states with distinct \fngerprints of their\nmagnetic excitations.\nWe use tunneling spectroscopy with a superconducting\nPb tip at 1.2 K to detect \fngerprints of magnetic inter-\naction of MnPc with the superconducting substrate. As\na reference, we plot the di\u000berential conductance ( dI=dV )\nspectrum of the bare Pb surface in Fig. 1(b). A re-\ngion of zero conductance around the Fermi energy ( EF),\ni.e., the superconducting gap, is framed by quasiparticle\nresonances at eV=\u0006(\u0001sample + \u0001 tip) =\u00062.63 meV.\nThe doubling of the size of the SC gap is due to\nthe superconductor-superconductor tunneling geometry.\nThe observed presence of the two quasiparticle reso-\nnances at each side of the gap is explained by the two-\nband superconductivity of the Pb single crystal as de-\nscribed recently [28].\nInterestingly, the spectra on the MnPc molecules show\ntwo triplet sets of peaks inside the superconducting en-\nergy gap, which are symmetric in energy around EF, butasymmetric in intensity [Fig. 1(b)]. In the limit of small\ntunneling rates { as in our experiment { the asymmetric\nintensity is an expression of the di\u000berent hole and elec-\ntron components of the Shiba wavefunctions [14]. The\ndi\u000berent weights arise from the particle-hole asymmetry\nin the normal state and an on-site Coulomb potential at\nthe scattering site [9, 17, 27].\nThe triplets consist of very sharp peaks (50 to 100 \u0016eV\nfull width at half maximum), which are separated by up\nto 400\u0016eV [e.g., Fig. 1(c)]. To observe such narrow peaks\nat 1:2 K, a superconducting tip is required, because then\nthe resolution is not limited by the Fermi-Dirac distribu-\ntion anymore [28, 29].\nThe varying coupling strength within the Moir\u0013 e-like\nstructure leads to di\u000berent bound state energies [16].\nWe use this property to further investigate the origin of\nthe splitting of the Shiba resonances and perform tun-\nneling spectroscopy on more than 130 molecules. All\nspectra exhibit two triplets of peaks, one in the bias\nvoltage window\u00002\u0001=e < V bias<\u0000\u0001=eand one in\n\u0001=e < V bias<2\u0001=e. The additional resonances in\nthe energy interval [ \u0000\u0001tip;\u0001tip] are due to tunneling\ninto/out of thermally excited Shiba states (Supplemen-\ntary Fig. 1, Supplementary Note 1). In Fig. 2(a), we\nordered the spectra according to the energy of the most\nintense Shiba resonance. The false color plot shows a\ncollective \"shift\" of the Shiba triplets through the su-\nperconducting gap. The spectra can be categorized into\nthree di\u000berent regimes. For each of these, we plot a spec-\ntrum in Fig. 2(b). In spectrum I, the intensity of the\ntriplet is larger for tunneling out of the occupied states;\nin spectrum II, the peaks are close to EF, which hinders\na clear distinction of the triplet; spectrum III exhibits\nlarger intensity of the triplet when tunneling into unoc-\ncupied states. The energetic position of the higher in-\ntensity subgap peaks corresponds to the binding energy\nEbof the Shiba states [15{17]. The order of the spectra\nfrom top to bottom thus represents a decreasing coupling\nstrengthJwith the substrate, which comes along with\ndi\u000berent adsorption sites. The three spectra are represen-\ntative for the \"Kondo screened\" case [ Eb<0, spectrum\nI in Fig. 2(b)], the \"free spin\" case ( Eb>0, spectrum\nIII), and a case close to the quantum phase transition\n(Eb\u00190, spectrum II).\nPossible origins of a Shiba state splitting\nThe collective shift of the Shiba states through a wide\nrange of the gap (and even the quantum phase transi-\ntion) suggests a correlated origin of the peaks within the\ntriplet. In principle, three di\u000berent scenarios may ac-\ncount for the occurrence of multiple Shiba bound states\nin a type I superconductor: (i) di\u000berent angular momen-\ntum scattering channels [7{9, 12, 13], (ii) independent\nscattering at spins in di\u000berent dorbitals [10], and (iii)3\n(b)\nIIIIII\n02\n13\n-2 0 2dI/dV (arb. units)\nSample bias (mV)3\n12\n0Spectral intensity (arb. units)\nEnergy (meV)0 -1 1IIIIII(c) Eb2\nSample bias (mV)-3 3 0IIIIIIdI/dV low high (a)\nEb3Eb1 2∆tip\n2∆tip\nFIG. 2. Di\u000berential conductance spectra of MnPc in Moir\u0013 e-like pattern: (a) False color plot of dI=dV spectra of 137 MnPc\nmolecules ordered by the energy of the most intense Shiba resonance (feedback: V= 5 mV, I= 200 pA). (b) Spectra of three\nMnPc molecules with bound state energies in three di\u000berent coupling regimes. (c) Spectral intensity obtained by deconvolution\nof the spectra shown in (b).\nbound state excitations coupled to other low-energy ex-\ncitations, such as spin excitations or vibrations [10, 11].\nBound states, which originate from higher angular mo-\nmentum scattering channels ( l= 1;2;etc:), always reside\nclose to the gap edge, while the l= 0 channel may shift\nthrough the superconducting energy gap depending on\nthe coupling strength J[9, 12]. This is in contrast to our\nobservation, and, hence, we can rule out (i) as possible\norigin for the split bound states. In the case of inde-\npendent scattering of spins in di\u000berent dorbitals (ii), a\nsimilar shift of all Ebiis unexpected, because dorbitals\nexhibit di\u000berent symmetries and interactions with the\nsurface.\nFor the coupling to other degrees of freedom at a sim-\nilar energy scale (iii), a collective shift is expected. As\nwe will show in the following, a detailed analysis of the\nintensities of the Shiba states allows us to unambiguously\nidentify a magnetocrystalline origin of the splitting of the\nShiba states as predicted by \u0014Zitko and co-workers [10] for\nS\u00151 systems.\nShiba intensities as \fngerprint of a quantum phase\ntransition\nThe evaluation of the Shiba intensities also sustains the\nassigned regimes of \"Kondo screened\" and \"free spin\"\nground states. Such an analysis requires the spectral\ndensity of the molecule-substrate system, which can be\ndirectly related to the relative weight in the tunneling\nprocesses. For this, we remove the e\u000bect of the supercon-\nducting density of states of the tip by numerical decon-\nvolution of the spectra as described in the Supplemen-\ntary Note 2 (see Supplementary Fig. 2 for \ft quality)\n[Fig. 2(c)].\nWe observe a distinct change in the relative peak ar-\neas within the triplets when crossing the quantum phase\ntransition. In the \"Kondo screened\" regime, i.e., forShiba states with negative binding energies, the individ-\nual peaks within a triplet exhibit equal areas [Fig. 3(a)].\nThus, their relative areas Ai=\u00063\nj=1Ajare close to 1 =3\n[left part in Fig. 3(c)]. In contrast, in the \"free spin\"\nregime, i.e., positive Shiba binding energies, the areas\nare considerably di\u000berent [Fig. 3(b) and (c)]. Here, the\nrelative area of the subgap excitations decreases from the\noutermost, Eb3, to the innermost, Eb1[30]. The ratio of\npeak areas decreases with increasing energy separation\n[Fig. 3(d)]. Such a behavior is reminiscent of a Boltz-\nmann distribution, indicating a thermal occupation of a\nsplit many-body ground state.\nThe characteristic change in the relative peak areas at\nthe point of the quantum phase transition is a direct \fn-\ngerprint of the origin of the splitting. Independent bound\nstates of di\u000berent dorbitals (ii) would not change their\nrelative weight at the phase transition. Furthermore, the\nphase transition should not occur simultaneously for all\nscattering channels. Additional vibronic resonances (iii)\nwould appear as satellite peaks at higher absolute en-\nergy for both ground states. Their spectral weight should\nscale with the electron-phonon coupling strength, albeit\nbeing substantially lower than the weight of the main\nresonance [11]. This is clearly not the case in our data.\nHence, both scenarios can safely be ruled out.\nAssignment of scattering channels and anisotropy\nsplitting of Shiba states\nTo conclude on the correct model for the description of\nthe multiple subgap resonances, we summarize the essen-\ntial properties of the Shiba states: On the one hand, the\nequal peak areas for the \"Kondo screened\" ground state\nre\rect that our system is characterized by a single level\nwith three possible excitation levels of equal probability.\nOn the other hand, the Boltzmann-like distribution of\nthe areas in the \"free spin\" case indicate a triplet-split4\nMean Shiba energy (meV)Ai/(A1+A2+A3)\n0 -10.20.40.6\n0(a) (c)\n(b)\nEnergy (meV)0 0.4 0.8Spectral intensity (arb. u.)1\n02-0.6 -1 -1.40\nEnergy (meV)123Spectral intensity (arb. u.)\nEnergy separation (meV)0.1 0.201peak area ratio(d)  A1/A3 A2/A3 pBoltzmann at 1.2K A1 A2 A3\n-0.5 0.5\nFIG. 3. Shiba state analysis: (a) Zoom on the framed part of\nthe deconvoluted spectrum I in Fig. 2(c). We model the spec-\ntrum with three Lorentzian peaks of di\u000berent width (shaded\nin blue, red, and gray) and a broadened step function at the\ngap edge (see Supplementary Material for details). (b) As (a),\nbut on spectrum III in Fig. 2(c). (c) Peak areas relative to\nthe sum of all three subgap peaks ( Ai=P3\ni=1Ai) plotted vs.\nthe mean energy ( Ebi=P3\ni=1Ebi) of the respective set of sub-\ngap peaks. (d) Peak area ratios plotted vs. their respective\nenergy splitting from all spectra in the \"free spin\" state. A\nBoltzmann distribution for T= 1:2 K is sketched as dashed\nline.\nground state with one excited state. We can correlate\nthese levels to the magnetic interaction channels of the\nMnPc molecule with the substrate. The MnPc molecule\ncarries a spin S= 3=2 in gas phase [31]. Theory predicts\nthat on Pb(111), the spin in the dxz;yzforms a singlet\nwith the organic ligand states, which reduces the e\u000bective\nspin seen by the substrate's quasiparticles to S= 1 [32].\nThe unpaired spin in the dz2orbital is subject to strong\ncoupling with the electronic states of the substrate lead-\ning to sizable Kondo screening, whereas the spin in the\ndxyorbital is not expected to show a signi\fcant coupling\nwith the substrate [24, 32, 33]. The occurrence of the\nShiba states in tunneling spectra is thus linked to the\ninteraction of the dz2orbital with the substrate. We la-\nbel this scattering channel as k= 1 (sketches in Fig. 4).\nThe unscreened spin in the dxyorbital (which we label\nask= 2) does not give rise to an observable Shiba state\nin agreement with the theoretical predictions [32]. How-\never, it couples to the spin in the dz2orbital and leads to\nan anisotropy splitting of the Shiba state with k= 1 as\ndiscussed below.\nWe can describe the whole set of spectra on the di\u000ber-\n(a) (b)\nk = 1 k = 2  \nk = 2 k = 1S*=1\nS=1/2|0〉\n|−〉|+〉D+E\nD-E\nS=1S*=1/2\n|0〉\n|−〉|+〉FIG. 4. Schematic representation of the many-body ground\nand excited states and the corresponding energy level dia-\ngrams. (a) In the \"Kondo screened\" ground state the spin\n(white ) in scattering channel k= 1 is screened and the tunnel-\ning electron can enter with its spin ( blue) parallel to the spin\nink= 2, increasing the excited state's total spin to S\u0003= 1.\nThe excitation scheme including the anisotropy-split excited\nstate is shown on the right. (b) In the \"free spin\" ground\nstate, the spin in k= 1 ( red) is only partially screened. The\ntunneling electron has to enter this state in an anti-parallel\nalignment, obeying the Pauli exclusion principle and reducing\nthe spin to S\u0003= 1=2. Red lines in the excitation scheme sym-\nbolize the thermal occupation of the anisotropy-split ground\nstate.\nent molecules by these spin states and their interactions:\nThe coupling strength J1of channelk= 1 depends on the\nadsorption site of MnPc on the Pb(111) surface. In the\ncase of strong coupling, the spin is totally Kondo screened\n[white arrow, Eb<0, Fig. 4(a)], but k= 2 remains un-\nscreened (red arrow). Hence, the e\u000bective total spin is\nreduced to S= 1=2 by Kondo screening. Tunneling into\nthe Shiba state re\rects the excitation to S\u0003= 1. In the\ncase of weak coupling in k= 1 [left red arrow, Eb>0,\nFig. 4(b)], the total spin in the ground state multiplet\nisS= 1. The excited state probed by the Shiba reso-\nnance is aS\u0003= 1=2 state, because the electron attached\ntok= 1 must obey Pauli's exclusion principle and align\nanti-parallel.\nBoth spin-1 states, i.e.,S\u0003= 1 andS= 1, can be\nsplit by magnetic anisotropy. A breaking of spherical\nsymmetry of the Mn orbitals by the organic ligand and\nthe adsorption on a substrate leads to a splitting of these\nspin states [22, 23]. The corresponding Spin Hamilto-\nnianHS=DS2\nz+E\u0000\nS2\nx\u0000S2\ny\u0001\n, where the Siare the\nspin operators in Cartesian coordinates, accounts for the\naxial anisotropy and additional rhombicity with the pa-\nrametersDandE, respectively. This yields a new set\nof spin eigenstates j0i,j+i, andj\u0000i, with the latter two\nbeing linear combinations of the former eigenstates with\nms= 1 andms=\u00001 [23]. Schemes of the excitations ob-\nserved in the tunneling spectra are shown in Fig. 4(a,b).\nIn the strongly coupled regime (\"Kondo screened\"), the\nexcitation of the system yields three excited states with\nthe energy splittings being related to the anisotropy pa-\nrametersDandE. It should be noted, that their values5\nare not a direct measure of the anisotropy energies of the\nmolecule on the surface, but rather represent a renor-\nmalized value due to the many-body interactions with\nthe substrate [10]. Since the energy separation between\nEb1andEb2is smaller than between Eb2andEb3, the\nanisotropy parameter Dis negative, which means easy-\naxis anisotropy. For the weakly coupled system, i.e., the\n\"free spin\" case, the ground state is split by anisotropy.\nThe excitation spectra re\rect transitions from the states\nj\u0000i,j+iandj0iinto the excited S\u0003= 1=2 state.\nThe characteristic variations of the peak areas are\nalso well captured in this scenario of anisotropy-split\nShiba states: All three spin excitations in the \"Kondo\nscreened\" regime are equally probable, therefore leading\nto the same relative peak area [Fig. 3(c)]. On the other\nhand, a splitting of the ground state, as it is found in\nthe \"free spin\" regime, leads to a Boltzmann occupa-\ntion of the levels. In the zero temperature limit, which\nis discussed in Ref. [10], only one excitation would be\ndetected in the \"free spin\" regime. At \fnite temper-\nature, the levels are occupied according to Boltzmann\nstatistics. The excitation probabilities are proportional\nto the state occupation and should thus directly re\rect\nthis distribution. Our data in Fig. 3(d) is in agreement\nwith a Boltzmann distribution at 1.2 K or slightly higher,\nwhich re\rects a temperature-induced population of the\nsplit ground state.\nDISCUSSION\nOur study shows the importance of anisotropy e\u000bects\non the subgap excitations which determine the elec-\ntron transport properties. They provide an unambigu-\nous \fngerprint of the nature of the ground and excited\nstate throughout the whole range of magnetic interaction\nstrengths, which drive the phase transition between the\ntwo quantum ground states. Although the anisotropy\nenergies are renormalized by the coupling to the sub-\nstrates quasiparticles, this method can be used to extract\nknowledge about magnetic anisotropy and, hence, about\nspin-orbit coupling of magnetic adsorbates on supercon-\nductors.\nPeculiar consequences of the split Shiba states may\noccur for coupled magnetic impurities, which lead to the\nformation of extended Shiba bands. If the exchange cou-\npling strength is similar to or smaller than the anisotropy\nenergy, the Shiba bands are expected to re\rect the split-\nting of the individual Shiba states. Recently, subgap\nbands have gained particular importance in the search of\ntopological phases and Majorana states in ferromagnetic\nchains coupled to an s-wave superconductor [5]. If an odd\nnumber of spin-polarized bands crosses the Fermi level,\nMajorana end states can form in the presence of an in-\nduced (non-trivial) topological gap. Considering that the\nenergy level splitting amounts to about one third of thesuperconducting gap, one may expect that a split Shiba\nband structure a\u000bects the number of crossing bands and\nthe topological gap width, which is also in the order of\n100\u0016eV [34].\nMETHODS\nThe experiments were carried out in a commercial\nSpecs JT-STM operating at a base temperature of 1.2 K\nand a base pressure below 10\u000010mbar. The Pb(111) sin-\ngle crystal surface was cleaned by repeated cycles of Ne+\nsputtering and annealing to 430 K until a clean, atomi-\ncally \rat, and SC surface was obtained. From a Knudsen\ncell held at 673 K, MnPc was thermally evaporated onto\nthe clean surface kept at room temperature, which then\nwas directly transferred into the precooled STM.\nTo gain energy resolution beyond the Fermi-Dirac limit\nof a normal metal tip, we indented the chemically-etched\nW tip into the clean SC surface applying 100 V tip bias\nuntil a Pb covered, superconducting tip was obtained\n(energy resolution better than 45 \u0016eV) [28]. The result-\ning spectrum as acquired on the clean Pb(111) surface is\nshown in Fig. 1(b) top. 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Kondo E\u000bect in Self-Assembled Man-\nganese Phthalocyanine Monolayer on Pb Islands. Chin.\nPhys. Lett. 27,087202 (2010).\n[26] Str\u0013 o_ zecka, A., Soriano, M., Pascual, J. I., Palacios, J.\nJ. Reversible Change of the Spin State in a Manganese\nPhthalocyanine by Coordination of CO Molecule. Phys.\nRev. Lett. 109, 147202 (2012).\n[27] Salkola, M. I., Balatsky, A. V., Schrie\u000ber, J. R. Spec-\ntral properties of quasiparticle excitations induced by\nmagnetic moments in superconductors Phys. Rev. B 55,\n12648{12661 (1997).\n[28] Ruby, M., Heinrich, B. W., Pascual, J. I., Franke, K. J.\nExperimental Demonstration of a Two-Band Supercon-\nducting State for Lead Using Scanning Tunneling Spec-\ntroscopy. Phys. Rev. Lett. 114, 157001 (2015).\n[29] Yet, a \fnite temperature leads to \fnite lifetime e\u000bects,\nwhich broaden the superconducting coherence peaks and\nsubgap states. Hence, already at 4.5 K, the triplet peaks\noverlap and give rise to a single peak [16].\n[30] Close to the quantum phase transition, i.e., close to EF,\nthe Shiba resonances overlap with their thermal excita-\ntions. This prohibits a clear distinction of the peak areas\nand they are therefore omitted in the data plot.\n[31] Liao, M.-S., Watts, J.D., Huang, M.-J. DFT Study of\nUnligated and Ligated ManganeseIIPorphyrins and Ph-\nthalocyanines. Inorg. Chem. 44,1941{1949 (2005).\n[32] Jacob, D., Soriano, M., Palacios, J. J. Kondo e\u000bect\nand spin quenching in high-spin molecules on metal sub-\nstrates. Phys. Rev. B 88,134417 (2013).\n[33] K ugel, J., Karolak, M., Kr onlein, A., Senkpiel, J., Hsu,\nP.-J., Sangiovanni, G. Bode, M. State identi\fcation and\ntunable Kondo e\u000bect of MnPc on Ag(001). Phys. Rev. B\n91,235130 (2015).\n[34] Ruby, M., Pientka, F., Peng, Y., von Oppen, F., Hein-\nrich, B. W., Franke, K. J., End states and subgap struc-\nture in proximity-coupled chains of magnetic adatoms.\narXiv:1507.03104 (2015)."    },    {        "title": "1510.05325v1.Minimum_Anisotropy_of_a_Magnetic_Nanoparticle_out_of_Equilibrium.pdf",        "content": "arXiv:1510.05325v1  [cond-mat.mes-hall]  19 Oct 2015Minimum Anisotropy of a Magnetic Nanoparticle out of\nEquilibrium\nW. Jiang,1P. Gartland,1,∗and D. Davidovi´ c1,†\n1School of Physics, Georgia Institute of Technology,\n837 State Street, Atlanta, Georgia 30332, USA\n(Dated: April 19, 2021)\nAbstract\nIn this article we study magnetotransport in single nanopar ticles of Ni, Py=Ni 0.8Fe0.2, Co,\nand Fe, with volumes 15 ±6nm3, using sequential electron tunneling at 4.2K temperature. We\nmeasure current versus magnetic field in the ensembles of nom inally the same samples, and obtain\nthe abundances of magnetic hysteresis. The hysteresis abun dance varies among the metals as\nNi:Py:Co:Fe=4:50:100:100(%), in good correlation with th emagnetostatic andmagnetocrystalline\nanisotropy. The abrupt change in the hysteresis abundance a mong these metals suggests a concept\nof minimum magnetic anisotropy required for magnetic hyste resis, which is found to be ≈13meV.\nThe minimum anisotropy is explained in terms of the residual magnetization noise arising from the\nspin-orbit torques generated by sequential electron tunne ling. The magnetic hysteresis abundances\nare weakly dependent on the tunneling current through the na noparticle, which we attribute to\ncurrent dependent damping.\n1Magnetic anisotropy in ferromagnets is vital in magneto-electronic applications, such as\ngiant magnetoresitance1,2and spin-transfer torque.3–5For example, in some applications,\na strong spin-orbit anisotropy is desired in order to establish a hard or fixed reference\nmagnetic layer, while in other applications, it is beneficial to use a weak er anisotropy in\norder to fabricate an easily manipulated, soft or free magnetic laye r. The ability to tune\nthe degree of anisotropy for various applications is therefore of u tmost importance. In\nthermal equilibrium, the minimum anisotropy necessary for magnetic hysteresis is temper-\nature dependent6–8. In this article, we address the minimum anisotropy in the case of a\nvoltage-biased metallic ferromagnetic nanoparticle First studies of discrete levels and mag-\nnetic hysteresis in metallic ferromagnetic nanoparticles have been d one on Co nanoparti-\ncles9–11. Here we discuss the magnetic hysteresis abundances in single elect ron tunneling\ndevices containing similarly sized single nanoparticles of Ni, Py=Ni 0.8Fe0.2, Co, and Fe. At\n4.2K temperature. the probability that a given nanoparticle sample w ill display magnetic\nhysteresis in current versus magnetic field, at any bias voltage, wa s found to vary as follows:\nNi:Py:Co:Fe=0 .04:0.5:1:1. The very small (high) probability of magnetic hysteresis in the\nNi (Fe and Co) nanoparticles suggests a concept of minimum magnet ic anisotropy necessary\nfor magnetic hysteresis, comparable to the average magnetic anis otropy of Py nanoparticle.\nThe minimum magnetic anisotropy is explained here in terms of the fluct uating spin-orbit\ntorques exerted on the magnetization by sequential electron tun neling. These torques lead\nto the saturation of the effective magnetic temperature at low tem peratures. In order for\nthe nanoparticle to exhibit magnetic hysteresis at 4.2K, the blocking temperature must be\nlarger than the residual temperature. The magnetic hysteresis a bundances are found to be\nindependent of the tunneling current through the sample, which su ggests that the damping\nis proportional to the tunneling current.\nI. EXPERIMENT\nOur samples consist of similarly-sized ferromagnetic nanoparticles t unnel-coupled to two\nAl leads via amorphous aluminum oxide barriers. First, a polymethilmet achryllate bridge\nis defined by electron-beam lithography on a SiO 2substrate using a technique developed\npreviously, as sketched in Fig. 1A. Next, we deposit 10nm of Al along direction 1. Then, we\nswitch the deposition to direction 2, and deposit 1 .5nm of Al 2O3, 0.5-1.2nm of ferromagnetic\n2FIG. 1: A: Sketch showing the sample fabrication process. B: Scanning electron micrograph of a\ntypical sample.\nmaterial, 1.5nm of Al 2O3, topped off by 10nm of Al, followed by liftoff in acetone. The\ntunnel junction is formed by the small overlap between the two lead s as shown in the circled\npart in Fig. 1B. The nanoparticles are embedded in the matrix of Al 2O3in the overlap.\nThe nominal thickness of deposited Co, Ni, and Py is 0 .5-0.6nm. At that thickness, the\ndeposited metals form isolated nanoparticles approximately 1 −5nm in diameter. We find\nthat if we deposit Fe at the nominally thickness 0 .6nm, then the resulting samples are\ngenerally insulating. Thus, the deposited Fe thickness is increased t o 1−1.2nm, which\nyields samples in the same resistance range as in samples of Co, Ni, and Py. We suppose\nthat because Fe can be easily oxidized, Fe nanoparticles are surrou nded by iron oxide shells.\nThus, our sample characterization suggests that it is appropriate to attribute the difference\namong Co, Ni, and Py samples to intrinsic material effects rather tha n size discrepancies,\nwhile, in Fe nanoparticles, the comparison is complicated by the uncer tainty in the size of\nthe metallic core. Still, we find the comparison with Fe to be fair, becau se of the wide range\nof nanoparticle diameters involved.\nWe obtain the transmission electron microscope (TEM) image of the d eposited pure\naluminum oxide surface, and the aluminum oxide surface topped with n ominally 1.2nm of\nFe, 0.55nm of Ni, and 0 .6nm of Co, as shown in Fig. 2. The deposition is done immediately\nprior to loading the sample in the TEM. Pure aluminum oxide surface app ears completely\namorphous, with no visible signs of crystalline structure. In compar ison, single crystal\nstructure can be identified in the TEM images for Fe, Ni, and Co. From the TEM image,\nthe areal coverage of Ni nanoparticles is 44% and the nanoparticle density is 3 .6·104µm−2.\n3FIG. 2: EDS spectra and TEM images of aluminum oxide surface t opped with 0.6nm of Co (top\nrow), 0.6nm of Ni (second row), 1.2nm of Fe (third row), and co ntrol Al 2O3on TEM grid (bottom\nrow).\n4Assuming that the nanoparticles have pancake shape, the averag e area and the height of\nthe particles are 0 .44/3.6·104µm−2≈12nm2and 0.55nm/0.44 = 1.25nm, respectively. The\nstandard deviation of the nanoparticles area is 40% of the average area, which is estimated\nby the shape analysis in 120 Ni neighboring nanoparticles. Thus, the volume of the Ni\nnanoparticles is 15 ±6nm3, where 6nm3is the standard deviation. The volume distribution\ninCo nanoparticles is similar to that inNi. Wetake theenergy dispersive x-rayspectroscopy\n(EDS) for the samples we imaged, which confirms the materials depos ited on the aluminum\noxide surface.\nII. MEASUREMENTS\nFIG. 3: Current versus voltage in four representative sampl es at 4.2K temperature.A.Co, B.Ni,\nC.Fe nanoparticles, and D.Leaky pure Al 2O3tunneling junctions.\n5The IV curves are measured using an Ithaco model 1211 current p reamplifier and are\nreproducible with voltage sweeps. Figures 3 A, B, and C display the IV curves of three rep-\nresentative tunneling junctions with embedded nanoparticles of Co , Fe, and Ni, respectively,\nin samples immersed in liquid Helium at 4.2K. The IV curves display Coulomb b lockade\n(CB) which confirms electron tunneling via metallic nanoparticles, but no discrete levels are\nresolved. We also measure tunneling junctions containing only the alu minum oxide, without\nembedded metallic nanoparticles. Those junctions are generally insu lating but some are\nnot, e.g., there may be leakage. The IV curves of those leaky junct ions are linear, as shown\nin Fig. 3D, as expected for simple tunnel junctions. We show the IV c urve in a pure alu-\nminum oxide junction, to demonstrate that the CB in the samples with embedded metallic\nnanoparticles originate from tunneling via those nanoparticles. The issue here is that, as\nwill be shown immediately below, some of the samples with embedded nan oparticles do not\ndisplay any hysteresis in current versus magnetic field at 4.2K. Since those samples also\ndisplay Coulomb blockade in the I-V curve, we can conclude that the a bsence of hysteresis\nis intrinsic to the nanoparticles, rather than an artifact from tunn eling through a possibly\nleaky aluminum oxide.\nCurrent versus applied magnetic field (parallel to the film plane) is obt ained by measur-\ning the current at a fixed bias voltage while sweeping the magnetic field slowly. Fig. 4A-D\nshow the representative magnetic hysteresis loops for Co, Fe, Ni, and Py samples at 4.2K,\nrespectively. As seen in Fig. 4C and D, the Ni-nanoparticle and one Py nanoparticle have no\nmagnetic hysteresis. All the samples that lack hysteresis do so at t he lowest resolved tunnel-\ningcurrent. Thestatistics ofthepresence ofhysteresis fordiffe rent materialsaredisplayed in\nTable. 1. All the Co (over 50) and Fe (6), one half of the Py (out of 1 0), and only 2 of the 46\nmeasured Ni samples, with the switching field 0 .05T and 0 .12T, display magnetic hysteresis.\nThe abrupt change in the hysteresis abundance between Co, Fe, P y, and Ni suggests a con-\ncept of minimum magnetic anisotropy necessary for magnetic hyste resis, akin to the concept\nof Mott’s maximum metallic resistivity. Note that the Ni samples and th e non-hysteretic\nPy samples still display significant magneto-resistance. Among the h ysteretic samples, the\naverage magnetic switching field varies as Ni:Py:Co:Fe=0 .085:0.114:0.233:0.257 (Tesla).\nThe magnetic energy EM(/vector m) of the Fe/Ni/Py and Co nanoparticles can be written as\nS0[K1(m2\nxm2\ny+m2\nym2\nz+m2\nzm2\nx)+K2m2\nxm2\nym2\nz+Ks/vector mˆN/vector m] andS0[K1m2\nz+K2m4\nz+Ks/vector mˆN/vector m],\nrespectively, where S0¯his the total spin in the nanoparticle, /vector mis the magnetization unit\n6FIG. 4: Examples of tunneling current versus magnetic field a t 4.2K, for samples with A.Co, B.Fe,\nC.Ni, and D.Py nanoparticles. In D, two pairs of curves corre spond to two different samples.\nRed(black) lines correspond to decreasing(increasing) ma gnetic field.\nvector, and ˆNis the demagnetization tensor which is set to have 3 eigenvalues equa l to 0.2,\n0.3, and 0.5. In the calculations, Euler angles between the principal a xes of shape anisotropy\nand magnetocrystalline anisotropy axes are all equal to π/5. The results of the calculation\nof the energy barrier EBare shown in table 1, assuming the nanoparticle average volume\nobtained in Sec. 2. The error bar reflects the standard deviation in the nanoparticle volume.\nThe abrupt change in magnetic hysteresis abundance is monotonic w ith the calculated EB.\nSince 50% of Py nanoparticles display hysteresis, it follows that the m inimum anisotropy\nenergy barrier required for magnetic hysteresis in our samples at 4 .2K is≈13meV. This\nenergy barrier is too large to explain our findings in terms of the redu ction in the blocking\ntemperature among these metals. Experimentally, the magnetome try on similarly-sized\n7TABLE I: Hysteresis percentage versus magnetic anisotropy in different materials12,13\nMaterial Hysteresis(%) Ks(µeV/spin) K1(µeV/spin) K2(µeV/spin) EB(meV)\nCo14100 105 64.1 8.3 97±39\nFe15100 128 4.0 1.3 51±20\nPy1650 67.1 0 0 13±5\nNi174 37.9 -24.4 4.1 7.7±3\nnanoparticles show that the blocking temperature varies between 13−30K for Co and\n6−20K for Ni.18–22Though Co nanoparticles appear to have higher blocking temperatu re\nthan Ni nanoparticles, the difference in blocking temperature is not sufficient to explain\nthe vast contrast in the hysteresis abundance in our nanoparticle s under electron transport.\nTheoretically, the Arrhenius flipping rate can be estimated as νatexp(−EB/kBT). Assuming\nνat= 1010Hz, we obtain the magnetization flipping time of 100 hours, much longe r than the\ntime it takes to measure a magnetic hysteresis loop. We conclude tha t the breakdown in\nmagnetic hysteresis we observe reflects the effect of sequential electron tunneling through\nthe nanoparticle on magnetization. This effect will be discussed in Sec . 4.\nIII. NUMERICAL SIMULATIONS\nThe magnetic Hamiltonian for Ni and Fe nanoparticles can be written a s\nH(n,/vectorSn) =S0[K1,n\n2(α2β2+β2α2+α2γ2+γ2α2+β2γ2+γ2β2)+K2,n\n6(α2β2γ2\n+α2γ2β2+β2α2γ2+β2γ2α2+γ2β2α2+γ2α2β2)]−Ks/vectorSnˆN/vectorSn\nCo has uniaxial magnetocrystalline anisotropy, thus the magnetic H amiltonian is\nH(n,/vectorSn) =−S0[K1,nγ2+K2,nγ4]−Ks/vectorSnˆN/vectorSn\nHere,nis the number of electrons in the nanoparticle while S0¯his the total spin. K1,n,\nK2,n, andKsrepresent magnetocrystalline anisotropy constants and the sha pe anisotropy\nconstant per spin, respectively. α,β,γ=Sx,Sy,Sz/S0.ˆNis the demagnetization tensor. The\nmagnetocrystalline anisotropy constants per unit volume K1,VandK2,Vare obtained from\nRefs. 14–17 . We obtain 2 S0/Nafrom Ref.12, whereNais the total number of atoms in\n8the nanoparticle. Then, S0/V=ρS0/NaMA, whereρis the mass density and MAis the\natomic mass. The average value of the magnetocrystalline anisotro py constants per spin\n(K1,K2) are (K1,V,K2,V)V/S0, respectively, while, Ks=µ0M2\nsV/2S0, whereMsis the satu-\nration magnetization obtained in Ref. 13,16. Because of spin-orbit a nisotropy fluctuations,\n(K1,n,K2,n) fluctuate around ( K1,K2), respectively, according to the number of electrons\nn. We expect that the average magnetic anisotropy has strong mat erial dependence, while\nthe mesoscopic fluctuations in the total magnetic anisotropy ener gy due to single electron\ntunneling on/off, are independent of the material.23–25In Co nanoparticles, the change in\ntotal magnetic energy after electron tunneling-on is S0∆K∼0.4meV.24In Ni nanoparticles\nofthe samesize at 4.2Kor below, S0andKareboth ≈1/3ofthevalues inCo nanoparticles.\nSo, the relative fluctuations in magnetic anisotropy ∆ K/Kin Ni nanoparticles are enhanced\n9−10 times compared to Co nanoparticles with the same size. At the sam e time, the mag-\nnetic energy barrier in Ni nanoparticles is suppressed because of t he cubic symmetry and\nweak shape anisotropy. As a result, the magnetization in Ni nanopa rticles is significantly\nmore susceptible to perturbation by electron transport compare d to Co or Fe nanoparticles.\nThe numerical simulation of the magnetization dynamics is based on th e master equation\nmodified from that in Ref.26. In the sequential tunneling regime, the number of electrons\nin the nanoparticle hops between nandn+1. Assuming the electron tunnels through only\none minority single electron state j, which reduces the spin of the nanoparticle by 1 /2 after\nthe electron tunneling-on event (including more electron states do es not affect the result in\na major way), the master equation can be written as\n∂Pα\n∂t=/summationdisplay\nα′/summationdisplay\nl=L,R/summationdisplay\nσ=↑,↓Γlσ/braceleftBig\n|< α′|cjσ|α >|2[−(1−fl(∆EM))Pα+fl(∆EM)Pα′]+\n|< α′|c†\njσ|α >|2[−fl(−∆EM)Pα+(1−fl(−∆EM))Pα′]/bracerightBig\n+\nΓB/summationdisplay\nα′,ǫ=±ǫ|< α′|Sx−iǫSy|α >|2[ρB(ǫ∆EM)nB(∆EM)Pα′+ρB(ǫ∆EM)nB(−∆EM)Pα]\n(1)\nThe first part of the master equation describes the magnetic tunn eling transitions. Here,\n|α >and|α′>represent magnetic eigenstates of the nanoparticle , i.e., the eigen state of\nH(n,/vectorSn) withnorn+1 electrons. |α >and|α′>can be obtained as superpositions of the\neigenstates of ˆS2andˆSz, which arethepure spin-states |S0,m >.cjσ(c†\njσ) istheannihilation\n9FIG. 5: Simulations of the time evolution of the magnetizati on vector in a Ni nanoparticle. The\nblue-yellow-red scale indicates the magnetic anisotropy e nergy landscape from low to high versus\nthe polar angle Θ and the azimuthal angle Φ in a Ni nanoparticl e. The value of t is the past time\ncounted from the beginning of electron tunneling. The white dots signify possible angle positions\nat the specified time for each simulation. As time progresses , the likely magnetization position is\nspread out over phase space and approaches a random isotropi c distribution.\n(creation) operator for an electron with spin σon levelj. Γlσdenotes the tunneling rate to\nleveljthrough the leads l=L,Rfor electron with spin σandflis the Fermi distribution\nin the leads. ∆ EM=Eα−Eα′.< α′|c†\njσ|α >is the tunneling transition matrix element\nfor transition between the initial state |α >and the final state |α′>. The second part of\nthe master equation describes the magnetic damping due the couplin g to the bosonic bath.\nΓBis the rate related to the damping rate 1 /T1.ρB(∆EM) is the spectral density of the\nboson which is set to be constant because it varies very slowly26.nBis the Bose-distribution\nfunction.\nFig. 5 shows the motion of the magnetization statistical distribution versus time. The\nmagnetic anisotropy energy vs (Θ ,Φ) represents the relation between magnetic energy and\nmagnetization direction, where Θ ,Φ are the two polar angles. Each magnetic state corre-\nsponds to a contour on the magnetic anisotropy energy landscape . Classically, one would\nconsider the magnetic state as the magnetization precessing on th e contour corresponding\nto that state. A random dot is selected on a contour to indicate the presence of the mag-\n10netic state related to that contour. The number of dots on each c ontour is decided by the\nprobability of the corresponding magnetic state. The plotted dots together represent the\ndistribution of the magnetization over Θ ,Φ-space at that time.\nIn the simulation for Fig. 5, S0= 100. Γ L,Rσ= 6×107which corresponds to a current\nabout 5pA. Γ B= 2×103leads to a relaxation time of ∼2.5µs. (K1,n,K2,n) = 1.25(K1,K2)\nand (K1,n+1,K2,n+1) = 0.75(K1,K2).Ksis taken from Table. 1. ˆNis set to have 3 eigenval-\nues equal to 0.2, 0.3, and 0.5. Varying ˆNand the Euler angles, which were earlier defined,\ndoes not affect the qualitative result. The magnetic field is set to 0.00 1T to eliminate\nKramers degeneracy. We set the nanoparticle to be initially at the gr ound state with n\nelectrons. Then we iterate the master equation for 40 µs. The magnetic state distribution\nof the nanoparticle gradually spreads from the ground state to ot her states and eventually\nbecomes isotropic as shown in Fig. 5.\nIV. DISCUSSION AND CONCLUSIONS\nThe transfer of a single electron into the magnetic nanoparticle cre ates a fluctuation in\nthe spin-orbit energy of the nanoparticle.23–25,27Such a fluctuation in turn creates a spin-\norbit torque that is exerted on the magnetization. In the previous section, we show how a\nfluctuating spin-orbit torque can lead to isotropic distribution of th e magnetization. The\nfluctuating spin-orbit torques are mesoscopic effects and do not d epend significantly on the\nmaterial of the nanoparticle.23,25But, the nonfluctuating magnetic anisotropy, such as mag-\nnetocrystalline and magnetostatic shape anisotropy, depends st rongly on the material. As\nthe magnetic anisotropy of the nanoparticle is reduced, the stren gth of the fluctuating spin-\norbit torques relative to the deterministic torques will increase, cr eating a noise floor which\nsets the limit on magnetic anisotropy below which magnetic hysteresis cannot be observed\nin sequential electron tunneling. Such a noise floor is reflected by th e abrupt change in\nmagnetic hysteresis abundance in similarly sized Ni, Py, Fe, and Co nan oparticles. By con-\ntrast, in thermal equilibrium, magnetometery of the ensembles of s imilarly sized Co and Ni\nnanoparticles show much less dramatic change in the blocking temper ature.18–22The mini-\nmummagneticanisotropyalsoexplainspriormeasurements ofvoltag ebiasedsinglemagnetic\nmolecules, in a double tunneling barrier, which showed no signs of magn etic hysteresis, even\nat temperatures much lower than the blocking temperature,28,29notwithstanding that the\n11magnetometryofensembles ofsuch molecules showed magnetichys teresis belowtheblocking\ntemperature.30,31\nIt may be surprising to the reader that the minimum magnetic anisotr opy is found to be\nindependent of the size of the tunneling current through the nano particle. The tunneling\ncurrent we use in the measurements of current versus magnetic fi eld, varies between 1pA\nand 100pA. A Co nanoparticle at 100pA is likely to exhibit magnetic hyst eresis, while a Ni\nnanoparticle at 1pA is highly unlikely to do so. The ratio of the applied tu nneling currents\nis one order of magnitude larger than the ratio of the energy barrie rs between the average Co\nand Ni nanoparticle. Since the data presented in this paper were ga thered, we have studded\nsingle Ni nanoparticles at mK-temperature, and discovered that 2 out of the 5 measured\nNi nanoparticles display magnetic hysteresis at the onset voltage f or sequential electron\ntunneling.32The hysteresis in current versus magnetic field was abruptly suppr essed in the\nbias voltage range starting just above the lowest discrete energy level for single-electron\ntunneling. The abrupt suppression of magnetic hysteresis versus bias voltage was explained\nin terms of the magnetization blockade, which was caused by the bias voltage dependent\ndamping rate.32,33In the magnetization blockade regime, the ordinary spin-transfer33or the\nspin-orbit torques32are damped because of the small bias energy available for single-elec tron\ntunneling. In the voltage region where the magnetization is blocked, the spin-transfer and\ndamping rates are both proportional to the electron tunneling rat e, regardless of which spin-\ntransfer mechanism is at play (e.g. the ordinary spin-transfer or t he spin-orbit torques). A\nchange in the bias-voltage will change the damping rate,32but the effective magnetic tem-\nperature, controlled by the ratio of the damping rate and the spin- transfer rate,33will be\nindependent of the electron tunneling rate. It is reasonable to ass ume that the magnetic\nnanoparticle will exhibit magnetic hysteresis in our experimental time s scales, if the flipping\ntime given by the Arrhenius law, based on the attempt frequency an d the ratio of the energy\nbarrier and the effective magnetic temperature, is longer than the hysteresis measurement\ntime. Since neither the energy barrier nor the effective magnetic te mperature depend on the\ntunneling current, this explains, at least in principle, why magnetic hy steresis abundances in\nour samples are so weakly dependent on the tunneling current. The characteristic tempera-\nture above which the two hysteretic Ni nanoparticles stop displayin g magnetic hysteresis is\n2−3K.32That characteristic temperature corresponds to the magnetiza tionblockade energy,\nwhich is comparable to the single-electron anisotropy. We can conclu de that the minimum\n12magnetic anisotropy is the limiting anisotropy of the nanoparticle belo w which magnetic\nhysteresis cannot be guaranteed. If the magnetic hysteresis do es occur in the nanoparticle\nwith magnetic anisotropy smaller thanthe minimum magnetic anisotrop y, it will do so below\n2-3K temperature, the characteristic temperature of the magn etization blockade.\nIn summary, we have performed magnetoresistance measuremen ts on a variety of ferro-\nmagnetic materials 1-5 nm in diameter at 4.2K, and found an abrupt ch ange in magnetic\nhysteresis abundancebetween Ni, Py, Fe, andConanoparticles. T hisabruptness leadstothe\nconclusion that there is a minimum magnetic anisotropy energy in a met allic ferromagnetic\nnanoparticle or a magnetic molecule out of equilibrium, required to gua rantee magnetic hys-\nteresis at low temperatures. The size of the tunneling current doe s not affect the minimum\nmagnetic anisotropy. Our finding has an implication for the miniaturiza tion of magnetic\nrandom access memory. 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Lett. 91, 247201 (2003).\n15"    },    {        "title": "1511.02097v2.Structural__magnetic_and_electrical_properties_of_sputter_deposited_Mn_Fe_Ga_thin_films.pdf",        "content": "1\nStructural and magnetic properties of sputter\ndeposited Mn-Fe-Ga thin films\nAlessia Niesen1, Christian Sterwerf1, Manuel Glas1, Jan-Michael Schmalhorst1and G ¨unter Reiss1\n1Center for Spinelectronic Materials and Devices, Physics Department, Bielefeld University, Germany\nAbstract —We investigated structural and magnetic properties\nof sputter deposited Mn-Fe-Ga compounds. The crystallinity of\nthe Mn-Fe-Ga thin films was confirmed using x-ray diffraction.\nX-ray reflection and atomic force microscopy measurements\nwere utilized to investigate the surface properties, roughness,\nthickness and density of the deposited Mn-Fe-Ga. Depending on\nthe stoichiometry, as well as the used substrates (SrTiO3(001)\nand MgO (001)) or buffer layer (TiN) the Mn-Fe-Ga crystallized\nin the cubic or the tetragonally distorted phase. Anomalous\nHall effect and alternating gradient magnetometry measurements\nconfirmed strong perpendicular magnetocrystalline anisotropy.\nHard magnetic behavior was reached by tuning the composition.\nTiN buffered Mn2.7Fe0.3Ga revealed sharper switching of the\nmagnetization compared to the unbuffered layers.\nIndex Terms —Heusler compounds, perpendicular magne-\ntocrystalline anisotropy, material science, spintronics\nI. I NTRODUCTION\nFerromagnetic, fully spin polarized materials (half-metals)\nfound a lot of interest in the recent years due to their possible\napplication in spintronic devices as nonvolatile memories [1]\nand field programmable logic devices. [2], [3] To maintain\nthe thermal stability at shrinking device sizes, an out-of-plane\n(oop) oriented magnetization of the material is advantageous.\nTherefore investigation of perpendicularly magnetized Heusler\ncompounds has found attraction. High spin polarization and\nout-of-plane magnetization direction was predicted for the\nMn3-xGa (0:15\u0014x\u00142) compound. [4]–[6] The transition\nfrom the cubic D03into the tetragonal D022phase takes\nplace at temperatures above 500\u000eC. [7] Hence a variation of\nthe deposition temperature is an important criterium in terms\nof the crystallographic properties. The tetragonally distortion\ncould also be obtained by shifting the Fermi energy at the\nVan Hove singularity in one of the spin-channels, which is\nreached by tuning the material with an additional element.\n[8] Mn-Fe-Ga is one possible material, which is calculated\nto be 95% spin polarized at the Fermi level for the cubic\nphase (Mn2Fe1Ga). [9] The predicted low total magnetization\nM= 1:03\u0016B[9] and high Curie temperature TC= 550 K\n(lowest measured value for Mn1.4Fe1.6Ga) makes this material\ninteresting to serve as an electrode in magnetic tunnel junc-\ntions (MTJ’s). [10] The replacement of Mn atoms by Fe leads\nto an enhancement of the magnetic moment. The measured\nmagnetic moment of pure Mn-Ga (prepared by arc melting) is\n1\u0016B. The Fe - rich Fe2Mn1Ga showed the highest magnetic\nmoment of 3:5\u0016B. [10]\nThe tunable magnetic behavior makes this material in-\nteresting for investigations and a promissing candidate forapplications. In this work, we focused on the preparation and\ninvestigation of the crystallographic, structural and magnetic\nproperties of the ternary Mn-Fe-Ga compound thin films. The\ninfluence of different stoichiometries, substrates and deposi-\ntion temperatures on the material properties was analyzed.\nWith regard to the preparation of magnetic tunnel junctions\nand the aim to increase their applicability, additionally a TiN\nseed-layer was used. Sputter deposited TiN is a material with\nlow electrical resistivity ( 16µ\ncm) and a surface roughness\nbelow 1nm. [11], [12] Thus it provides a good electrical\nconnection to the MTJ. High thermal stability (melting point\n2950\u000eC [13]) is another advantage, which prevents chemical\nreactions of TiN with the on top deposited material. The lattice\nconstant of TiN (fcc structure) is 4:24˚A and therefore suitable\nfor various Heusler compounds. It was already shown, that TiN\nis a suitable seed-layer for Mn3-xGa and Co2FeAl. [14]\nII. E XPERIMENTAL\nMn-Fe-Ga thin films ( 40nm thickness) were prepared in\nan ultra-high-vacuum (UHV) sputtering system with a base\npressure below 5\u000210\u000010mbar. DC magnetron co-sputtering\nfrom a pure Mn, Fe and a Mn45Ga55composite target was\nused to prepare the samples. The Ar pressure was set to\n1:7\u000210\u00003mbar. The amount of Mn, in the MnyFexGa com-\npound, was varied in the range of 1:5\u0014y\u00143and the\namount of Fe in the range of 0:3\u0014x\u00141. Deposition\ntemperatures from 190\u000eC to 595\u000eC were chosen in order\nto achieve crystalline growth and the tetragonally distorted\nphase of Mn-Fe-Ga. MgO (100) ( aMgO= 4:21˚A) and SrTiO3\n(STO) (100) ( aSTO = 3:91˚A) single crystalline substrates\nwere utilized. Additionally TiN buffered Mn-Fe-Ga thin films\non MgO and STO substrates were prepared. The TiN layers\n(30nm) were deposited using reactive sputtering in an Ar and\nN atmosphere which results in stoichiometric Ti1N1thin films.\nDuring the sputtering process a N flow of 2sccm and an Ar\nflow of 20sccm was used, leading to a deposition pressure of\n1:6\u000210\u00003mbar. The stoichiometry of TiN was verified via\ndensity, resistivity and x-ray absorption spectroscopy measure-\nments. Additionally the superconductance of TiN was proved.\nFurther information on the TiN seed-layer is given in [14].\nOn top of the Heusler compound a 2nm thick MgO layer was\ndeposited to prevent the surface from contaminations.\nIII. C HARACTERIZATION OF THE MN-FE-GA COMPOUND\nA. Crystal structure\nCrystallographic and structural properties of the Mn-Fe-Ga\nthin films were determined via x-ray diffraction (XRD) andarXiv:1511.02097v2  [cond-mat.mtrl-sci]  1 Aug 20162\n101 103 105 107 log intensity (arb. units)\n120 10080 60 40\n2θ/s32/s40/s161/s41(004) D022(002) D022\n(008) D022MgOMgO\n Mn2.7Fe0.3Ga\n TiN/ Mn2.7Fe0.3Ga\n \n Mn2.7Fe0.3Ga\n TiN/ Mn2.7Fe0.3Ga\n Tdep=365 °C\nTdep=280 °C(004) D03(002) D03b)101 103 105 107 log intensity (arb.units) MgO\n STO(004) D022(002) D022\n(008) D022STO\nSTOSTOSTO\nMgOMgO\n(006) D022\nMn2Fe1GaMn2.5Fe0.5GaMn2.7Fe0.3GaMn-Fe-Ga: D022 (c = 7.15 Å)\n a)\nFig. 1. a) XRD patterns of the Mn3-xFexGa compound deposited at 450\u000eC\non MgO (blue curves) and SrTiO3(dashed red curves) substrates. b) XRD\npatterns of Mn2.7Fe0.3Ga deposited at 365\u000eC and 280\u000eC with and without\na TiN buffer layer.\n2.5\n2.0\n1.5\n1.0\n0.5\n0.0Fe amount\n3.02.52.01.51.0\nMn amount D03\n D022a)\n8\n6\n4\n2\n0c lattice constant (Å)400300200\nTdep (°C) D03\n D022\n no crystalline growthb)\nFig. 2. a) Crystal structure of the Mn-Fe-Ga compounds deposited on MgO\nsubstrates in dependence on the Mn and Fe amount. b) Dependence of the\ncrystal structure of Mn2.7Fe0.3Ga on the deposition temperature.\nx-ray reflection (XRR) measurements using a XRD Philips\nX’Pert Pro diffractometer (Cu anode). The lowest roughness\nvalues ( \u00141nm) combined with the highest crystallinity were\nmeasured on samples deposited at temperatures of 407\u000eC and\n450\u000eC. All samples with the Mn3-xFexGa (0:3\u0014x\u00141)\nstoichiometry crystallize in the D0 22tetragonally distorted\nphase (Fig. 1 and Fig. 2) which is in good agreement with\nthe previously reported data. [10] The determined in-plane and\noop lattice constants are a=3:9\u00060:01˚A andc=7:15\u00060:04˚A\nfor Mn-Fe-Ga deposited on both substrate types, leading\nto ac=a ratio of 1:8. The substrate type obviously does\nnot influence the crystallographic phase of the Mn3-xFexGa.\nTherefore this stoichiometry is stabilized in the tetragonally\ndistorted phase. Increasing the Fe and decreasing the Mn\namount leads to a formation of the cubic phase (D0 3) (c=\n6˚A) on MgO substrates (Fig. 2). On STO each composition\nresults in a formation of the tetragonally distorted phase,\ndue to the low lattice mismatch of 0:5%with the in-plane\nlattice constant. The dependence of the crystal structure on the\n32.521.510.5032.521.510.50X[µm]Y[µm]\n 5.00 nm\n 0.00 nm\n32.521.510.5032.521.510.50X[µm]Y[µm]\n 50.00 nm\n 0.00 nmMgO/ Fe2.6Mn1.2Ga MgO/ Mn2.7Fe0.3Ga 50.00 (nm) \n0.00 (nm) 0.00 (nm) 5.00 (nm) \nX (µm) X (µm)  Y (µm) a) b) Fig. 3. AFM images of a) Mn2.7Fe0.3Ga (tetragonally distorted phase) and\nb) Fe2.6Mn1.2Ga (cubic phase) thin films deposited on MgO (100) substrates\nat450\u000eC.\n32.521.510.5032.521.510.50X[µm]Y[µm]\n 5.00 nm\n 0.00 nm\n32.521.510.5032.521.510.50X[µm]Y[µm]\n 100.00 nm\n 0.00 nmMgO/ Mn1.5Fe0.9Ga STO/ Mn1.5Fe0.9Gab) a)  Y (µm) \nX (µm) X (µm) 100.00 (nm) \n0.00 (nm) 0.00 (nm) 5.00 (nm) \nFig. 4. AFM images of Mn1.6Fe0.9Ga (tetragonally distorted) deposited on\na) MgO (100) and b) STO (100) substrates at 550\u000eC.\n32.521.510.5032.521.510.50X[µm]Y[µm]\n50.00 nm\n0.00 nm\n32.521.510.5032.521.510.50X[µm]Y[µm]\n 100.00 nm\n 0.00 nmMgO/ TiN/ Mn2.7Fe0.3Ga MgO/ TiN/ Mn2Fe1Ga a) b) \nX (µm) X (µm)  Y (µm) \n0.00 (nm) 100.00 (nm) \n0.00 (nm) 50.00 (nm) \nFig. 5. AFM images of TiN buffered a) Mn2Fe1Ga and b) Mn2.7Fe0.3Ga\ndeposited on MgO (100) at 450\u000eC.\ndeposition temperature was investigated for the Mn2.7Fe0.3Ga\ncompound, which showes the tetragonally distortion on each\nsubstrate type, as well as on the TiN buffer layer. The transition\nfrom the cubic D0 3(a=c= 6˚A) into the tetragonally\ndistorted D0 22phase takes place at a deposition temperature of\n320\u000eC (Fig. 2). The density of the samples was determined\nvia XRR measurements and also revealed a dependence on\nthe crystallographic phase. The density of the tetragonally\ndistorted Mn-Fe-Ga is 7:1\u00060:5g/cm3and6\u00060:5g/cm3for the\ncubic phase.\nDeposition on a TiN buffer layer ( aTiN= 4:24˚A) leads to a\nmixture of the cubic D0 3and the D0 22phase, depending on\nthe Mn-Fe-Ga composition and the deposition temperature.\nThe tetragonally distorted phase for the Mn2.7Fe0.3Ga, on a\nTiN seed-layer, appears already at a deposition temperature\nof280\u000eC (Fig. 1), which is a lower temperature compared\nto the unbuffered sample. The seed-layer obviously influences3\n2.0\n1.5\n1.0\n0.5\n0.0Hc ·104 (Oe)\n2.5 2.0 1.5 1.0 0.5\nx  Mn3-xFexGa on STO\n Mn3-xFexGa\n MnyFexGa\ny=1.2y=1.3y=1.5y=1.6y=3\nFig. 6. Coercivity dependence on the stoichiometry of Mn-Fe-Ga deposited\non MgO and STO substrates. The blue squaress and black triangles give the\ncoercivity values for Mn3-xFexGa. The red circles point out the values for the\nMnyFexGa stoichiometries (y gives the values for the amount of Mn).\nthe crystalline growth of this compound leading to a lower\ndeposition temperature at which the tetragonally distorted\nphase is formed. We already observed this behavior for the\nMn-Ga compound. [14] TiN buffered Mn2.5Fe0.5Ga forms a\nmixture of the cubic and the tetragonally distorted phase. In\ncase of TiN buffered Mn2Fe1Ga, only the cubic phase was\nformed.\nB. Surface properties\nAtomic force microscopy (AFM) was carried out to inves-\ntigate the surface topography of the samples. Fig. 3 shows a\ncomparison of (a) the tetragonally distorted Mn2.7Fe0.3Ga and\n(b) the cubic Fe2.6Mn1.2Ga layer deposited on MgO at 450\u000eC.\nThe Mn2.7Fe0.3Ga forms 200nm - 400nm broad grains with\nsteep grain boundaries. The measured rms roughness value is\n3:4\u00060:05nm. The cubic Fe2.6Mn1.2Ga compound forms small\ngrains and a smooth surface (roughness = 0:8\u00060:05nm). Mn-\nFe-Ga with lower Mn ratio shows island growth (Fig. 4 a). The\nobtained roughness value is 17\u00060:5nm. The equivalent film\n(temperature and stoichiometry) on STO revealed no island\ngrowth and low roughness of 0:43\u00060:05nm (Fig. 4 b), which\ncan be attributed to the low lattice mismatch with the in-plane\nlattice constant of the tetragonally distorted Mn-Fe-Ga.\nWe indicate that the applied roughness analysis is leading\nto an overestimation of the roughness for samples, which\nconsist of big grains with steep grain boundaries. The structure\nof Mn-Fe-Ga deposited on a TiN seed-layer also showed\na strong dependence on the stoichiometry. As previously\nmentioned, TiN buffered Mn2Fe1Ga crystallizes in the cubic\nD03phase. Due to the high lattice mismatch of the cubic phase\nwith the lattice constant of TiN ( 8:3%) island growth and\nhigh roughness ( 14:25\u00060:05nm) appears (Fig. 5 a). The TiN\nbuffered Mn2.7Fe0.3Ga, crystallized in the D0 22phase, shows\na similar morphology and roughness value as the unbuffered\nsample (Fig. 3 a) and (Fig. 5 b). Without a TiN buffer layer\nwe determined a roughness value of 3:4\u00060:05nm (see above)\ncompared to 3:54\u00060:05nm with a TiN buffer.\nC. Magnetic properties\nThe magnetic properties were investigated via AGM (al-\nternating gradient magnetometry) and AHE (anomalous Hall\n\t\r sample Hc (Oe) SR MS (kA/m) Mn1.2Fe2.6Ga  0.1 × 10!   0.5 144±14  Mn1.5Fe0.9Ga 0.3 × 10!  0.95 350±10 Mn1.6Fe0.3Ga 0.9 × 10!  0.97 317±13 TABLE I\nCOERCIVITY Hc,SQUARENESS RATIO SRAND SATURATION\nMAGNETIZATION MSVALUES OF MN-FE-GA DEPOSITED ON STO\nDETERMINED VIA AHE AND AGM MEASUREMENTS .\n-1.00.01.0\nfield·104 (Oe)Mn1.5Fe0.9Ga\n M \n ρyxb)\n-2-1012\nfield ·104 (Oe)Mn1.6Fe0.3Ga\n M\n ρyxc)\n-1.0-0.50.00.51.0normalized M or ρyx\n-1.00.01.0\nfield·104 (Oe)Mn1.2Fe2.6Ga\n M\n ρyxa)\nFig. 7. oop measured AHE (red) and AGM (dashed blue) hysteresis curves for\na) Mn1.2Fe2.6Ga, b) Mn1.5Fe0.9Ga and c) Mn1.6Fe0.3Ga thin films deposited\non STO. The measured magnetization Mand Hall voltage values UHare\nnormalized for a better comparison.\neffect) measurements. The coercivity of the Mn-Fe-Ga thin\nfilms was determined via AHE measurements in a 4-terminal\narrangement, carried out in a closed cycled He-cryostat. High\ncoercivity fields ( Hc\u00142\u0002104Oe) in the oop direction can\nbe reached by tuning the composition. The coercivity de-\ncreases with increasing Fe and decreasing Mn amount for both\nsubstrate types. The highest coercivity of 1:8\u0002104Oe was\nmeasured for Mn3Fe0.4Ga. The lowest value of 0:2\u0002104Oe\nshowed the cubic Mn1.3Fe1.4Ga (Fig. 6). Each sample revealed\na hard magnetic axis in the in-plane direction. Since the\nsamples could not be saturated in the in-plane direction even\nat an applied field of 4\u0002104Oe, a strong perpendicular mag-\nnetocrystalline anisotropy (PMA) could be experimentally ver-\nified. Fig. 7 shows a comparison of normalized magnetization\nMand AHE curves ( UH) for the tetragonally distorted Mn-\nFe-Ga deposited on STO (smooth films with roughness values\nbelow 1nm). In case of the Mn1.2Fe2.6Ga and Mn1.5Fe0.9Ga\ndeposited on STO substrates the hysteresis curves are in\ngood agreement. The coercivity, the squareness ratio and the\nsaturation magnetization values are given in Tab. I. We defined\nthe squareness as SR=Mr=MsorSR=\u001ayxr=\u001ayxs(ther\norsindex denotes the remanence or the saturation value of\nthe magnetization Mor resistivity \u001a). Increasing the Mn and\nlowering the amount of Fe obviously also leads to an increase\nof the squareness ratio. The magnetization values, determined\nvia AGM measurements, are in the range of 140kA/m and\n350kA/m, which is comparable to the values of the Mn-Ga\ncompound. [15] Mn1.6Fe0.3Ga shows a feature in the AGM\ncurves around 0Oe field, which could not be observed via\nAHE measurements. This was attributed to a second phase4\n \t\r Tdep=365 °C Hc (Oe) SR w/o TiN 1.4 × 10!  0.53 w TiN 1.2 × 10!  0.93 Tdep=450 °C   w/o TiN 1.5 × 10!  0.74 w TiN 1.2 × 10!  0.98 \nTABLE II\nCOMPARISON OF THE COERCIVITY AND SQUARENESS RATIO VALUES OF\nTINBUFFERED (WTIN)AND UNBUFFERED (W/OTIN) M N2.7FE0.3GA\nDEPOSITED ON MGOSUBSTRATES .\n-4-2024\nfield·104 (Oe) TiN/ Mn2.7Fe0.3Ga\n Mn2.7Fe0.3GaTdep=365 °C\n-1.0-0.50.00.51.0normalized ρyx\n-4-2024\nfield·104 (Oe) TiN/ Mn2.7Fe0.3Ga\n Mn2.7Fe0.3GaTdep=450 °C\nFig. 8. AHE hysteresis curves of TiN buffered (red) and unbuffered (blue)\nMn-Fe-Ga measured at room temperature. Both samples were deposited on\nMgO substrates. The TiN seed-layer ( 30nm thickness) was deposited at\n405\u000eC. The Mn2.7Fe0.3Ga thin films were deposited at 450\u000eC and 365\u000eC,\nrespectively.\n(soft magnetic) inside the Mn-Fe-Ga thin film, which has a\ndifferent coercive field. Such behavior was already observed\nfor Mn2Fe1Ga samples. [16] The AHE measurements do not\nshow such a feature, which indicates that the second phase\ncould have a high resistance and therefore does not contribute\nto the AHE. A second phase was not detected via XRD\nmeasurements, leading to the assumption that this phase is\nan amourphous part of the material, which might be located\nat the grain boundaries. To increase the applicability of Mn-\nFe-Ga as an electrode in MTJ’s a TiN seed-layer ( 30nm\nthickness) was used. Fig. 8 shows a comparison of the AHE\nmesurements for TiN buffered Mn2.7Fe0.3Ga deposited at two\ndifferent temperatures. In both cases the TiN buffered samples\nrevealed sharper switching of the magnetization compared to\nthe unbuffered layers. The squareness ratio increases and the\ncoercive field decreases for the buffered samples (Tab. II).\nRegarding the similar morphology of the samples, the reason\nfor the changed switching behavior is still unclear and we\nattempt further investigations on this topic.\nIV. C ONCLUSION\nStructural and magnetic properties of sputter deposited\nMn-Fe-Ga compounds were investigated. Depending on the\nstoichiometry, the deposition temperature, as well as the used\nsubstrate (STO (001) and MgO (001)) or buffer layer (TiN)the Mn-Fe-Ga crystallizes in the cubic D03(c=6˚A) or the\ntetragonally distorted phase D022(c=7:15˚A). The main\ndrawback for applications is the island growth, which was\nconfirmed via AFM measurements. Low roughness ( \u00141nm)\nand theD022phase was observed for each used deposi-\ntion temperature and composition of the Mn-Fe-Ga on STO,\nwhereas on MgO substrates the roughness showed strong\ndependence on the deposition temperature and the Mn-Fe-\nGa composition. Strong PMA was confirmed via AHE and\nAGM measurements. High coercivity fields (up to Hc\u00142\n\u0001104Oe) in the out-of-plane direction were reached by tuning\nthe composition. TiN buffered Mn2.7Fe0.3Ga revealed sharper\nswitching of the magnetization compared to the unbuffered\nlayers. Similar results were achieved for TiN buffered Mn-Ga\nand Co2FeAl thin films. [14]\nACKNOWLEDGMENTS\nThe authors gratefully acknowledge financial support by\nthe Deutsche Forschungsgemeinschaft (DFG, Contract No. RE\n1052/32-1).\nAPPENDIX A\nThe roughness values (root mean square) were calculated\nusing a standard deviation of surface heights:\nRMS =r\n1\nN\u0006N\nx=1(z(x;y)\u0000z(N;M ))2 (1)\nwithz(N;M )the arithmetic average height. The surface is de-\nscribed by a matrix with N lines and M columns corresponding\nto the points (x,y) of the height z(x,y).\nREFERENCES\n[1] S. A. Wolf, Science , vol. 294, no. 5546, p. 1488, 2001.\n[2] G. Reiss and D. Meyners, Journal of Physics: Condensed Matter ,\nvol. 19, no. 16, p. 165220, 2007.\n[3] A. Thomas, D. Meyners, D. Ebke, N.-N. Liu, M. D. Sacher, J. Schmal-\nhorst, G. Reiss, H. Ebert, and A. H ¨utten, Applied Physics Letters , vol. 89,\nno. 1, p. 012502, 2006.\n[4] B. Balke, G. H. Fecher, J. Winterlik, and C. Felser, Applied Physics\nLetters , vol. 90, no. 15, p. 152504, 2007.\n[5] J. Winterlik, B. Balke, G. Fecher, C. Felser, M. Alves, F. Bernardi, and\nJ. Morais, Physical Review B , vol. 77, no. 5, p. 12, 2008.\n[6] S. Wurmehl, H. C. Kandpal, G. H. Fecher, and C. Felser, Journal of\nPhysics: Condensed Matter , vol. 18, no. 27, p. 6171, 2006.\n[7] M. Glas, C. Sterwerf, J.-M. Schmalhorst, D. Ebke, C. Jenkins, E. Aren-\nholz, and G. Reiss, Journal of Applied Physics , vol. 114, no. 18, p.\n183910, 2013.\n[8] J. Winterlik, S. Chadov, A. Gupta, V . Alijani, T. Gasi, K. Filsinger,\nB. Balke, G. H. Fecher, C. A. Jenkins, F. Casper, J. K ¨ubler, G.-D. Liu,\nL. Gao, S. S. P. Parkin, and C. Felser, vol. 24, no. 47, p. 6283, 2012.\n[9] L. Wollmann, S. Chadov, J. K ¨ubler, and C. Felser, Physical Review B ,\nvol. 90, no. 21, p. 214420, 2014.\n[10] C. Felser, V . Alijani, J. Winterlik, S. Chadov, and A. K. Nayak, IEEE\nTransactions on Magnetics , vol. 49, no. 2, p. 682, 2013.\n[11] F. Magnus, A. S. Ingason, S. Olafsson, and J. T. Gudmundsson, Thin\nSolid Films , 2011.\n[12] Y . Krockenberger, S.-i. Karimoto, H. Yamamoto, and K. Semba, Journal\nof Applied Physics , vol. 112, no. 8, p. 083920, 2012.\n[13] M. Pritschow, “Titannitrid- und titan-schichten f ¨ur die nano-\nelektromechanik,” Ph.D. dissertation, Institut f ¨ur Mikroelektronik\nStuttgart, Mechanical Engineering Department, 2007.\n[14] A. Niesen, M. Glas, J. Ludwig, J.-M. Schmalhorst, R. Sahoo, D. Ebke,\nE. Arenholz, and G. Reiss, Journal of Applied Physics , vol. 118, no. 24,\np. 243904, 2015.5\n[15] M. Glas, D. Ebke, I. M. Imort, P. Thomas, and G. Reiss, Journal of\nMagnetism and Magnetic Materials 333 , p. 134, 2013.\n[16] T. Gasi, A. K. Nayak, J. Winterlik, V . Ksenofontov, P. Adler, M. Nicklas,\nand C. Felser, Applied Physics Letters , vol. 102, no. 202402, 2013."    },    {        "title": "1511.07892v1.Ferromagnetic_Resonance_of_a_YIG_film_in_the_Low_Frequency_Regime.pdf",        "content": "Ferromagnetic Resonance of a YIG film in the  \nLow Frequency Regi me \n \nSeongjae Lee,1 Scott Grudichak,2 Joseph Sklenar,2 C. C. Tsai,3 Moongyu Jang,4 Qinghui \nYang ,5 Huaiwu Zhang ,5 and John B. Ketterson2 \n \n1Department of Physics, Research Institute for Natural S ciences, Hanyang University, Seoul, \n133-791 South Kore a \n2Department of Physics and Astronomy, Northwestern University, Evanston IL, 60208 USA  \n3Department of Engineering & Management of Advanced Technology, Chang Jung Christian \nUniversity, Tainan 71101, Tai wan \n4Department of Materials Science and Engineering, Hallym University, Chuncheon, 200 -702 \nSouth Korea  \n5State Key Laboratory of Electronic Films and Integrated Devices, University of Electronic \nScience and Technology, Chengdu, Sichuan, 610054, China  \n \nAbst ract \nAn improved method for characterizing the magnetic anisotropy  of films with cubic \nsymmetry is described  and is applied to an yttrium iron garnet (111) film .  Analysis of the \nFMR spectra performed both in -plane and out -of-plane from 0.7 to 8 GHz yielde d the \nmagnetic anisotropy constants as well as the saturation magnetization. The field at which \nFMR is observed turns out to be quite sensitive  to anisotropy constant s (by more than a factor \nten) in the low frequency (< 2 GHz) regime and when the orientati on of the magnetic field is \nnearly normal to the sample plane ; the restoring force on the magnetization arising from  the \nmagnetocrystalline anisotropy fields is then comparable to that from the  external field, \nthereby allowing  the anisotropy constants to be determined with greater  accuracy. In this \nregion, unusual dynamic al behaviors are observed such as multiple resonances and a \nswitching of FMR resonance with only a 1 degree change in field orientation at 0.7 GHz.   \nIntroduction  \nYttrium iron garnet (YIG , Y3Fe5O12) is a well -known ferromagnetic  insulator with \nextremely low magnetic damping that has been widely used in microwave devices , and  its \nspin wave properties have been extensively studied for decades.[1,2] Recently there have \nbeen important discoverie s of spin-related phenomena such as spin pumping,[3,4,5] spin \nSeebeck effect,[6,7] and spin Hall magnetoresistance [8,9] in which YIG films played a \ncentral role . As an example, several authors have reported that the injection efficiency of a \nspin current f rom YIG to Pt is strongly enhanced at low resonant frequencies (\n 3 GHz) or \nlow fields (\n 2 kG). [4,10 -13] Thus it is important  to have an accurate characterization of the \nmagnetic response, in particular as it affects magnetization direction  in YIG thin films ; this is \nespecially important  at low frequencies  and near -normal field orientations . \nFMR is a powerful tool to probe the internal magnetic field of magnetic materials \n[14,15] and has become a standard te chnique to provide information  abou t  \nmagnetocrystalline anisotropy, as well as the uniaxial anisotropy that typically emerges in \nthin films.[16] Several authors have reported high -quality YIG thin films prepared by pulsed \nlaser deposition, [17,18,19] sput ter deposition,  [20] or chemical vapor deposition [21] along \nwith FMR characterization. They employed a conventional FMR setup involving a narrow \nband cavity resonator where one sweep s the magnetic field  through the resonance  at different \norientation s but at a fixed frequency (typically  9 GHz ); the anisotropy constants are \nobtained by  fitting the data to a theoretical ly predicted form . [18,19,21 ] The quantity  \n4πM  \nis usually  measured independently  when obtaining  the magnetic anisotrop y constants. Since \nthe required external field for the resonance is much higher than the anisotropy fields of YIG, \nthe accuracy with which the anisotropy constants can be determined  is limited.  \nIn this paper, we present a new method to obtain the anisotro py constants of a YIG film \nwith better accuracy by using a broadband FMR spectrometer in which resonant fields are \nmeasured not only as a function of the field orientation , as specified by both the polar  angle, \n\n, but also  over frequenc ies that range from 0.7 GHz to 8 GHz. Our method ha s three \nadvantages  over those of angle -resolved FMR at fixed frequency[22,23] : 1) the magnetization \nvalue can be directly extracted from the frequency -dependent FMR data without resorting to \nindepende nt magnetometry measurements;  2) the azimuthal angle of the field orientation with respect to the crystal axis is determined  as a part of the fitting process , thereby eliminating  \nany error associated with  aligning sample ; 3) the resonant field change resul ting from a  shift \nin the  anisotropy constant turns out to  be strongly  amplified  at low frequenc ies (\n2 GHz) , by \nmore than 10 times compared to that at 8 GHz. Furthermore, in this region we observe \nmultiple resonances that can furth er enhance  the accuracy. Finally, a t a low frequency of 0.7 \nGHz it is observed that a 1 degree change in field orientation induces an abrupt switching \nphenomenon  in the FMR resonance . All these behaviors  follow from  the Landau -Liftshitz \ntheory  as augmented  by the inclusion of appropriate anisotropy energies . [24]  \n \nTheoretical Background  \n Figure 1 shows  a schematic illustration of the coordinate system used for  our (111) \noriented epitaxial YIG film , where the film normal is taken  as the z-axis and the polar angles \nand azimuthal angles for the effective magnetization \nM  and an external  magnetic field \nH  \nare denoted as \n , and ,    , respectively. The free energy density  describing  the \nmagnetization o f a ferromagnetic film having a cubic structure is given by [24]  \n \n  FHM2MNMK1Mx2My2My2Mz2Mz2Mx2 K2Mx2My2Mz2   (1) \nHere the first term is the Zeeman energy, the second is the demagnetization energy  involving  \na demagnetization tensor \nN , the third a nd fourth terms represent the cubic first and second \norder magnetic anisotropy energies (MAE) characterized by constants  \n K1  and \n K2  and \nMx,MyandMz\n are the vector components of \nM  along the crystal coordinates: x, y, z.  \nAssuming the film as an infinite plane sheet where \nx y zN N 0, N 1    and \ntaking the z-axis and y -axis parallel to the \n[111]  and \n[011]  direction s in th e crystal \ncoordinates, respectively, Eq. (1) becomes   \n  \n \n\n 2 2 2\nH M u\n4 2 3 1M\n642 2\n3 2 6\nMMF HM cos cos sin sin cos 2 M cos K cos\nK7sin 8sin 4 4 2 sin cos cos312\nK24sin 45sin 24sin 4108\n2 2 sin cos 5sin 2 cos3 sin cos6 .          \n      \n    \n          (2) \nHere we included the out -of-plane uniaxial anisotropy energy in the third term  as \ncharact erized by a  constant \n Ku . For conveniences in the analys is, we introduce \ndimensionless variables: \nu 12u 1 2 2 2 2K KK Hh , k , k and k4M 2 M 2 M 2 M        and then the \nabove equation becomes  \n \n  \n \n\n 2\nH M u 2\n4 2 3 1M\n642 2\n3 2 6\nMMF2h cos cos sin sin cos 1 k cos\n2M\nk7sin 8sin 4 4 2 sin cos cos312\nk24sin 45sin 24sin 4108\n2 2 sin cos 5sin 2 cos3 sin cos6         \n\n      \n    \n              (3) \nNote  that the cubic anisotropy  energy terms each result in  a three -fold symmetry in the \nazimuthal angle \n M for a (111) oriented film, leading to different resonant fields when  \n\n as will be discussed  below . We also note that the anisotropy terms contain higher \norder terms in \n  and \n Mthus producing higher order contributions in their second \nderivatives when \n0\n . Since the FMR frequency depends on the second derivatives, this \nterm will play an important role in the angular dependence of the FMR  response. For small , \nthe second derivative of Zeeman term with respect to \n M  is comparable with those  of MAE \nterm and cancel s out, leading to a soft mode , even when the external field H is much larger \nthan internal fields arising from  MAE ; i.e., \n HA12K1/M4Mk1    and \n HA22K2/M4Mk2  \n. For bulk single crystal YIG, the anisotropy field values are \nHA187.2 Oe \nand \nHA23.71Oe  while \n4M  1760 gauss. [25]  Given the set of par ameters \nH h, and that define  the external field , the \nequilibrium orientation of the magnetization , specified by  \nM and  , is determined by \nminimizing the free energy:  \n F\n0,F\nM0,\n        (4) \nwith the additional conditions of positive curvatures \n 2F\n20  and \n 2F\nM20 . A small \norientational perturbation of  the magnetization from this equilibrium position is counteracted  \nby the restoring forces that depend on curvatures of the free energy , which dictate the \ndynamics of the magnetization. The general expression for the  FMR frequency  \n fres  was \nderived by Suhl, Smit and Beljers in 1955: [26,  27] \n fres\n2Msin2F\n2\n\n2F\nM2\n\n2F\nM\n\n2 \n\n\n\n\n\n1/2\n    (5) \nwhere \n  is the gyromagnetic ratio.  \n  There are no general analytical solutions of Eq. 5 for the FMR frequency . A special \ncase occurs when the field is applied normal to the film ; i.e., where  \n  = 0 . When  \n h1ku+2k1\n3+2k2\n9\n the mag netization align s with H  \n 0) and the FMR frequency  \nbehaves linearly with the field strength:  \n fres\n2M=h  1  ku+2k1\n3+2k2\n9\n\n\n   \n  = 0)    (6) \nUsing Eq. ’s (3) through ( 5) we numerically solved  for the resonant frequency , \n fresh, \nusing  \n ku,k1,k2,H  as parameters that were then varied to obtain a least squares fit when \ncompared with  the experimental FMR data  (this was accomplished using the commercially \navailable MATLAB program); t he quantit y \n4πM  is obtained by comparing the data set for  fresH,0with Eq. 6. Note we take the origin of the experimental azimuthal angle \n H \nas a fitting parameter , so as to minimize its uncertainty re lative to the [110] axis. [ 18] \n \nExperimental  Methods  \nThe magnetic systems studied here are 6 -m thick yttrium iron garnet (YIG) single \ncrystalline films of composition Y 3Fe5O12, grown by a liquid phase epitaxial technique on the \n(111) oriented Gd 3Ga5O12 (GGG) garnet substrate. [28, 29]  Data were acquired with  an FMR \nspectrometer capable of varying the frequency (f), DC magnetic field (h), and orientation ( ). \nIt operate s in a transmission  mode  with a sample cell placed in the rotatable Varian electro -\nmagnet  with maximum  fields of 6 kOe. The cell  consist s of a copper housing with  input and \noutput microwave coax connectors and an internal  chamber that clamped  our YIG film \nadjacent to a meander line  formed from copper  wire which generates the RF magnetic  fields ; \ndetails concerning the construction of the cell  are presented elsewhere .[30] The microwave  \nsource  is a HP model 83623A synthesizer with a frequency  range of 10 MHz –20 GHz . The \ninput RF signal was modulated in amplitude at a frequency of 4000 Hz and fed i nto the \nmeanderline in contact with the YIG film. The output RF signal  from the meanderline was \nrectified by a micro wave diode and the resulting signal was sent to a lock -in amplifier  while \nsweeping the DC magnetic field , which directly yields the FMR abso rption line.  For each , \nwe carried out sweeps with increasing and decreasing magnetic field  as well as positive and \nnegative fields.  The field was swept by a bipolar Kepco operational amplifier power supply \ndriven from the analog output of a National Instruments D/A board inse rted in a computer \noperating under software created in Labview; the same board digitized the lock -in output. \nWhen necessary, the signal to noise could be enhanced by collecting data in a signal \naveraging mode with a programmable number of sweeps.  \n \nResults and Discussions  \nThe r esonant modes of the YIG film were measured by sweeping the magnetic field  \nat seven different frequencies 0.7, 1, 1.6, 2.5, 4, 6, and 8 GHz and fixed  polar angle s from  \n90 deg to +90 deg. The frequency vs. field behavior for the resona nces observed with the field normal to the film  (\n 0 ) is plotted in Fig. 2 , and shows  a linear behavior that is fitted \nas \n fres=2.818 H1.612 , as expected from theory (Eq. 6), where the units of \nfres  and H \nare GHz and kOe.  From the slope of this relation  we obtain  \nγ/2π2.818 GHz/kOe , which \nis close to the free electron ’s value 2.803 GHz/kOe. From the intercept, we have an important \nrelation between  the saturation magnetization in kG and the terms  of ku, k1, and k2 :   \nu 1 21.6124πM1 k +2k /3+2k /9\n.            (7) \nNote  the quantity \n4πM  of our sample is determined as soon as \n ku,k1,andk2  are fixed. \nAs previously mentioned, it is important to obtain the magnetization  value without resorting \nto independent measurement s using a magnetometer,  which can result in  as much as a 5~10% \nerror in extracting  the anisotropy constants. [19]  \nFig. 3  plots the resonant field (H) vs. angle ( ) for seven frequencies  with various \nsymbols. At lower frequencies  and with small  we observed multiple resonances in field \nsweep s where we denote  the highest resonant field as the primary resonance and refer to  the \nothers  as secondary. For simpl icity we display only the primary resonance field data in this \nfigure. However, we took  the data for all of the (multiple ) resonance into account in our fit . \nThe resulting  parameters are: \n ku=0.01143 , \n k1 0.05040 , \n k20.01276 , and \n H= 4.599\n. Theoretical curves using  these values are displayed  as the  solid lines in the \nfigure. The fit between theory and experiment is consistent for all frequencies. Substituting  \nthese  three k -parameters into Eq. 7 yiel ds \n4πM 1.644 kG , which is comparable to the bulk \nvalue 1.76 kG.[23] The corresponding conv entional  magnetic anisotropy constants (fields) are \n33\n1K 5.421 10 erg/cm \n HA182.87Oe \n, \n K21.372103erg/cm3\n HA220.98Oe , \nand \n Ku1.230103erg/cm3 \n Hu18.80Oe. The first order anisotropy field \nA1H 82.87 Oe\n is in good agreement with previous work on bulk YIG, \n87.2 Oe [25] or \non thin films, which are centered on  \n80 Oe . [17, 18, 19, 21] However  the second order \nanisotropy field \n HA22K2/M  is about an order of magnitude higher than the bulk value  \nof \n3.71 Oe [25]. Judged from the two constants \nK1  and  \nK2 , we see that the easy axis for \nYIG film is in the (111)  direction.  One important feature that follows from  Fig. 3.  is the asymmetry when\n  due \nto the three -fold symmetry of the free energy in  \nM . The asymmetric behavior of resonant \nfields is more pronounced as we lower frequency. In the inset, we show the data at some \nlower frequencies 0.7, 1, 1.6 GHz at near-normal angle s to emphasize this asymmetric nature \nof the FMR spectra. This asymmetry in  improv es the  accuracy in fitting the data. In \nparticular at 1 GHz, a sudden drop of resonant field from 1.878 kOe to 0.562 kOe is observed \nas  changes from \n2  to \n4  that is contrasted from the smooth behav ior at the \ncorresponding positive angles. Note  that this behavior follows from  the theory , as seen by the \nsolid lines. We emphasize again that this abruptly changing behavior is only apparent in a low \nfrequency regime , \n 1 GHz. For the case of 0.7 GHz, data are absent except for several \nsmall angles near zero because resonance fields are so low that domain dynamics prevail over \nuniform magnetization precession.  \nIn order to investigate the sensitivity of the data  to the  fitted anisot ropy constants , we \ncomputed the fractional change in the resonant field when k 1 is shifted by  5% from the best -\nfit value, k 1(0) = -0.05040 , \n \n(0) (0) (0) (0)\nres res res 1 1 1 1H (k 0.05k ) H (k ) H (k ),   \nfor various frequencies and angles ; the results are shown in Fig. 4. At the highest freq uency 8 \nGHz, the most sensitive orientation is \n 45 with the sensitivity ~ 0.2 %, but as \nfrequency is lower ed, the polar angle of the most sensitive orientation increases gradually. \nFor 1.6 GHz, the sensitivity r ises up to 3 % at \n ;6.8 , which is more than ten  times \nlarger than  for 8 GHz , corresponding to an increase in resulting accuracy. Th is increase in the \nsensitivity at lower frequenc ies for near-normal orientation s of external field is due to the fact \nthat the second derivative of magnetocrystalline energy term with respect to \nM  is \ncomparable to that of Zeeman energy. In addition, there is a sign change when passing \nthrough \n0 ; this is related to the asymmetry  mentioned above , which enhanc es the \naccuracy of the fitting. The sensitivity of the resonant field to a 5% deviation of \n H  is also \ncomputed in the same manner and is shown in the inset where the maximum of an ~ 2 % \nchange occurs near \n 5.5  at 0.7 GHz. This degree of sensitivity allows for a  pinpoint \ndetermination of the azimuthal angle of the field; precise alignment of experimental apparatus  \nwith crystal axis is not required, thereby  avoiding  the error associated w ith it.  The asymmetric feature due to MAE stands out best in the frequency vs field plot of \nFMR data together with theoretical curves for fixed angles , as shown in Fig. 5 in which we \nrestrict the data to the primary resonance and the ang ular range  \n20 20    for clear \nviewing. Here the FMR data and theoretical curves for positive angles are denoted by filled \nsymbols and solid lines respectively and those for negative angles by open symbols with \ndotted lines. We can see the soft mode behavior s in the low frequency range (<1.2 GHz) with \nnear normal orientation  (\n4   ). It is attributed to the fact that the restoring forces on the \nmagnetization in azimuthal direction cancels out for small ; that is, the second derivatives o f \nZeeman term with respect to \nM  is comparable to that of MAE term in the free energy. \nFrom this figure, we can clearly note the difference between branches with positive and \nnegative polar angles. For a given angle, the positive br anch bends downward more rapidly \nthan the negative branch near \nH4πM\n . At 1 GHz, for example, the resonant field is 1.729 \nkOe for \n4  which is to be contrasted with 0.562 kOe for \n4  . We can also easily \nunderstand the rapid drop in \n Hres  in the inset of Fig.  3 wh ere we see the local minimum \npoint is lifted up above 1 GHz as \n  decreases from \n2  to \n4 . It is noteworthy that for \nsmall , the soft mode feature ha s a local minimum  which leads to multiple resonant fields  \nfor frequenc ies less than ~1.2 GHz. In this region, the FMR spectrum is very sensitive to \neither the frequency or the angle , an example  being  the curve for \n2  denoted by a \ndotted blue line in Fig. 5. At 1 GHz, we have multiple resonances. If we lower the frequency \nto 0.7 GHz, the resonance at high field disappears and only one resonance , at a low field \n0.285 kOe , survives since the minimum point near  1.7 kOe is lifted up above 0.7 GHz (a \nhorizontal solid line). If we change the angle to \n1   (red dotted line) while we fix the \nfrequency 0.7 GHz, a pair of resonances reemerges at high fields.  \nIn order to examine  the multiple resona nces at 0.7 GHz more close ly, we show  the \nmicrowave absorption spectra for \n2, 1, 0, 1, 2 and 4   in Fig. 6 ; here  the curves are \noffset horizontally and vertically for clear er viewing along with a symbol indicating the \nposition predicted by the theoreti cal calculation s. Here we use filled circles, open squares, \nand open triangles as markers for the primary, secondary and tertiary resonances respectively. \nWe can check the evolution of the spectrum from \n2  to \n4   by following the \ncorresponding f(H) curves in Fig. 5 as previously noted. The clear triple resonance at 2  is to be contrasted to the very weak single resonance at \n2  , clearly \ndemonstrating  the asymmetry.  We can also identify three resonances for \n1, 0,1 , as \npredicted by theory. A double resonance feature was reported earlier for a  (001) YIG film by \nManuilov et. al.[31], however  a triple resonance has never been observed. Unlike the prim ary \nresonance peaks, however, secondary and tertiary resonances a re broad and hence less \ndistinct. It is interesting to observe the huge change in the FMR spectrum induced by only a \none degree change from \n1   to \n2  .  \n \nConclusion  \nFrequency -dependent and a ngle-resolved FMR spectra of a (111) YIG film were \nmeasured with a field sweep method using a broadband FMR spectrometer  for frequenc ies in \nthe range  0.7 ~ 8 GHz . By comparing to a theoretical model  based on the Liftshitz -Landau \nformalism that incorporates  uniaxial and magnetocrystalline anisotropy through second -order, \nthree anisotropy constants  and the saturation magnetization  were determined:  \n \n 33\n1 A1K 5.421 10 erg/cm H 82.87 Oe  , \n 33\n2 A2K 1.372 10 erg/cm H 20.98 Oe  \n \n 33\nuuK 1.230 10 erg/cm H 18.80 Oe  \n \n4πM 1.644 kG\n.  \nAn improvement in accuracy of the se constants was accomplished by 1) an enhanced \nsensitivity of the resonant field to the anisotropy in the low frequency (< 2 GHz ) for a near-\nnormal orientation of the field, where the second derivative of the MAE term with respect to \nthe azimuthal angle is comparable to the Zeeman term ; and 2) excluding the error associated \nwith determining  the a zimuthal orientation of the field relative to the in -plane [110] axis of \nthe epitaxia l YIG film,  which is obtain ed through the fitting process itself . In addition, \nmultiple resonances observed at low frequencies  (0.7 GHz and 1.0 GHz ) boost the accuracy \nof data fitting process. In particular  we observed an abrupt  “switch ing-on” of a sharp \nresonance induced by a  less than 1 -degree change of the field angle , which can potentially have application s. \n \nAcknowledgement  \nThis work was supported by the U. S. Department of Energy under grant DE-SC0014424 and \nthe National Science Foundation under grant  DMR -1507058 . The film growth was supported \nby the National Natural Science Foundation of China (NSFC) under Grants 51272036 and \n51472046.  This work was also supported by the National Research Foundation of \nKorea(NRF) grant funded by the Korean government( MSIP) (NRF -2015R1A4A1041631).  \n  \nReferences  \n1. V . Cherepanov, I. Kolokolov, and V . Lvov, The saga of YIG: spectra, thermodynamics, \ninteraction and relaxation of magnons in a complex magnet, Phys. Rep. 229 (1993 ) \n81-144.  \n2. A. A. Serga, A. V . Chumak, and B. Hill ebrands, YIG magnonics, J. Phys. D: Appl. \nPhys. 43 (2010)  264002  (16pp).   \n3. Y . Kajiwara, K. Harii, S. Takahashi, J. Ohe, K. Uchida, M. Mizuguchi, H.  Umezawa, \nH. Kawai, K. Ando, K. Takanashi, S. Maekawa, and E. Saitoh,  Transmission of \nelectrical signals by sp in-wave  interconversion in a magnetic insulator , Nature 464 \n(2010) 262-267. \n4. H. Kurebayashi, O. Dzyapko, V . E. Demidov, D. Fang, A. J. Ferguson, and S. O. \nDemokritov, Controlled enhancement of spin -current emission  by three -magnon \nsplitting , Nat. Mater. 10 (2011) 660-664. \n5. H. L. Wang, C. H. Du, Y . Pu, R. Adur, P. C. Hammel,  and F. Y . Yang , Large spin \npumping from epitaxial Y 3Fe5O12 thin films to Pt and W  layers , Phys. Rev. B 88 \n(2013) 100406 -1-5. \n6. K. Uchida, J. Xiao, H. Adachi, J. Ohe, S. Takahashi, J.  Ieda, T . Ota, Y . Kajiwara, H. \nUmezawa, H. Kawai,  G. E.W. Bauer, S. Maekawa, and E. Saitoh, Spin Seebeck \ninsulator, Nat. Mater.  9 (2010)  894-897. \n7. E. Padron -Hern andez, A . Azevedo, and S. M. Rezende, Amplification of spin waves \nby thermal spin-transfer torque , Phys.  Rev. Lett. 107 (2011) 197203 -1-5. \n8. C. Hahn, G. de Loubens, O. Klein, M. Viret, V . V . Naletov, and J. Ben Youssef, \nComparative measurements of inverse spin Hall effects and magnetoresistance  in \nYIG/Pt and YIG/Ta , Phys. Rev. B  87 (2013)  174417 -1-8. \n9. H. Nakaya ma, M. Althammer, Y . -T. Chen, K. Uchida, Y . Kajiwara, D. Kikuchi, T.  \nOhtani, S. Geprags, M. Opel, S. Takahashi, R. Gross, G. E. W. Bauer, S. T. B. \nGoennenwein, and E. Saitoh, Spin Hall Magnetoresistance Induced by a Nonequilibrium Proximity Effect , Phys. R ev. Lett. 110 (2013) 206601 -1-5. \n10.  R. Iguchi, K. Ando, R. Takahashi, T. An, E. Saitoh, and T. Sato, Spin pumping \nwithout three -magnon splitting in polycrystalline Bi 1Y2Fe5O12/Pt bilayer structure , \nJap. J. Appl. Phys. 51 (2012)  103004 -1-5. \n11. O. Dzyapko, H. Kur ebayashi, V . E. Demidov, M. Evelt, A. J. Ferguson, and S. O. \nDemokritov , Effect of the magnetic film thickness on the enhancement of the spin \ncurrent by multimagnon  processes , Appl. Phys. Lett.  102 (2013) 25240 9-1-4.  \n12. V . Castel, N. Vlietstra, B. J. van Wee s, and J. B. Youssef, Frequency and power \ndependence of spin -current emission by spin pumping in a  thin-film YIG/Pt system , \nPhys. Rev. B 86 (2012) 134419 -1-6. \n13.  V . Castel, N. Vlietstra, B. J. van Wees, and J. B. Youssef, Yttrium iron garnet \nthickness and fr equency dependence of the  spin-charge current conversion in YIG/Pt \nsystems , Phys. Rev. B 90 (2014) 214434 -1-5. \n14. J. H. E. Griffiths , Anomalous high-frequency resistance of ferromagnetic metals, \nNature , 158 (1946) 670-671. \n15. C. Kittel , Interpretation of anomalo us Larmor frequencies in ferromagnetic resonance \nexperimen t, Phys. Rev.  71 (1947) 270-271.  \n16. M. Farle, Ferromagnetic resonance of ultrathin metallic layers , Rep. Prog. Phys. 61 \n(1998) 755-826. \n17. P. C. Dorsey, S. E. Bushnell, R. G. Seed, and C. Vittoria, Epita xial yttrium iron garnet \nfilms grown by pulsed laser deposition , J. Appl. Phys. 74 (1993)  1242 -1246 . \n18. A. A. Jalali -Roudsar, V . P. Denysenkov, and S. I. Khartsev, Determination of \nmagnetic anisotropy constants for magnetic  garnet epitaxial films using \nferrom agnetic resonance , J. Magn. Magn. Mater. 288 (2005)  15-21. \n19. S. A. Manuilov, S. I. Khartsev, and A. M. Grishin,  Pulsed laser deposited Y3Fe5O12 \nfilms: Nature of magnetic anisotropy I , J. Appl. Phys. 106 (2009)  123917 -1-11. \n20. Tao Liu, Houchen Chang, Vincent Vla minck, Yiyan Sun, Michael Kabatek, Axel \nHoffmann, Longjiang Deng, and Mingzhong Wu, Ferromagnetic resonance of sputtered yttrium iron garnet nanometer films , J. Appl. Phys. 115 (2014)  17A501 -1-3. \n21. C. Vittoria, H. Lesshoff, and N. D. Wilsey, Induced in-plane magnetic anisotropy  in \nYIG films, IEEE Trans. Mag. 8 (1972)  273-275. \n22. H. Makino and Y . Hidata, Determination of magnetic anisotropy constants for bubble \ngarnet epitaxial films using field orientation dependence in ferromagnetic resonances,  \nMater. Res. Bull. 16 (1981)  957-966. \n23. A. M. Zyuzin, V . V . Radaikin, and A. G. Bazhanov, Determination of the magnetic \ncubic anisotropy field in (111) oriented films by FMR , Tech. Phys. 42 (1997) 155 -159. \n24. L. D. Landau and E. M. Lifshitz , Electrodynamics of C ontinuous Media , Pergamon, \n1984.  \n25. D. Stancil, A. Prabhakar, Spin Waves, Theory and Applications , Springer, New York, \n2009  \n26. J. Smit and H. G. Beljers, Ferromagnetic resonance absorption in BaFe 12O19, a highly \nanisotropic crystal , Philips Res. Rep. 10 (1955) 113-130. \n27. H. Suhl, Ferromagnetic resonance in Nickel ferrite  between one and two \nkilomegacycles , Phys. Rev. 97(1955) 555-557.  \n28. H. J. Levinstein, S. Licht, R. W. Landorf, and S. L. Blank, Growth of high‐quality \ngarnet thin films from supercooled melts, Appl. Phys. Lett. 19 (1971)  486-488. \n29. S. L. Blank and J. W. Nielsen, The growth of magnetic garnets by liquid phase \nepitaxy , Cryst. Growth 17 (1972)  302-311. \n30. C. C. Tsai, J. Choi, S. Cho, S. J. Lee, B. K. Sarma, C. Thompson, O.  \nChernyashevskyy, I. Nevirkovets, and  J. B. Ketterson, Microwave absorption \nmeasurements using a broad -band meanderline  approach , Rev. Sci. Instrum.  80 \n(2009) 023904 -1-8. \n31. S. A. Manuilov and A. M. Grishin, Pulsed laser deposited Y 3Fe5O12 films: Nature of \nmagnetic anisotropy II , J. Appl. Phys. 108 (2010)  013902 -1-9. Figure Captions  \nFig. 1  Spherical coordinate system for the FMR study of an epitaxial YIG film. The film  \nnormal is taken as z -axis, which is also the \n111  crystal axis; x- and y -axis are parallel to \n[211]\n and \n[011]  in crystal axis.  \n \nFig. 2 FMR frequency data as a function of magnetic field normal to the film plane. Data are \nrepresented as filled circles w hile the line shows least squares fit to a straight line.    \n \nFig. 3 Resonant field versus polar field angle at frequencies  of 0.7, 1, 1.6, 2.5, 4, 6, and 8 \nGHz. The experimental d ata are shown as the  various symbols in the legend and the solid \nlines show a best fit. . Inset shows  a magnified view for 0.7, 1.0, and 1.6 GHz  near =0. \n \nFig. 4 Fractional changes in resonant field as a function of  at different frequencies resulting \nfrom a  5% change of k 1 relative to  the best -fit value. Inset shows the same with \n H  \n \nFig. 5 FMR frequency vs. field for various  values of . For positive  (negative) angles, the \ndata are represented with filled  (open) symbols and the theoretical lines are represented by \nsolid  (dotted) l ines. A horizontal line is drawn to indicate points for 0.7 GHz.  \n \nFig. 6 FMR absorption spectra a t 0.7 GHz for \n2, 1, 0,1, 2 and 4   . For clear viewing, \ncurves are offset both horizontally and vertically. The positions of the primary resonances \npredicted by the model  are denoted  by filled circles ; open squares and open triangles denote  \nthe second ary and tertiary resonances.  \n Figures  \nFig. 1  \n \n \n  \nFig. 2  \n  \nFig. 3  \n  \nFig. 4  \n \n  \nFig. 5  \n \n  \nFig. 6  \n \n "    },    {        "title": "1512.03470v2.Piezomagnetic_effect_as_a_counterpart_of_negative_thermal_expansion_in_magnetically_frustrated_Mn_based_antiperovskite_nitrides.pdf",        "content": "Piezomagnetic e\u000bect as a counterpart of negative thermal expansion in magnetically\nfrustrated Mn-based antiperovskite nitrides\nJ. Zemen,1;2Z. Gercsi,1;3and K.G. Sandeman1;4;5\n1Department of Physics, Blackett Laboratory, Imperial College London, London SW7 2AZ, United Kingdom\n2School of Chemistry, University of Nottingham,\nUniversity Park, Nottingham NG7 2RD, United Kingdom\n3CRANN and School of Physics, Trinity College Dublin, Dublin 2, Ireland\n4Department of Physics, Brooklyn College, CUNY,\n2900 Bedford Ave., Brooklyn, NY 11210, USA and\n5The Graduate Center, CUNY, 365 Fifth Avenue, New York, New York 10016, USA\n(Dated: June 9, 2021)\nElectric-\feld control of magnetization promises to substantially enhance the energy e\u000eciency of\ndevice applications ranging from data storage to solid-state cooling. However, the intrinsic linear\nmagnetoelectric e\u000bect is typically small in bulk materials. In thin \flms electric-\feld tuning of spin-\norbit interaction phenomena (e.g., magnetocrystalline anisotropy) has been reported to achieve a\npartial control of the magnetic state. Here we explore the piezomagnetic e\u000bect (PME), driven by\nfrustrated exchange interactions, which can induce a net magnetization in an antiferromagnet and\nreverse its direction via elastic strain generated piezoelectrically. Our ab initio study of PME in\nMn-antiperovskite nitrides identi\fed an extraordinarily large PME in Mn 3SnN available at room\ntemperature. We explain the magnitude of PME based on features of the electronic structure and\nshow an inverse-proportionality between the simulated zero-temperature PME and the negative ther-\nmal expansion at the magnetic (N\u0013 eel) transition measured by Takenaka et al. in 9 antiferromagnetic\nMn3AN systems.\nI. INTRODUCTION\nEmerging non-volatile magnetic random access mem-\nory (MRAM) devices represent bits of information as a\nmagnetization direction which needs to be stabilised by\nmagnetic anisotropy. A spin-transfer torque (STT) is\ntypically used to overcome the energy barrier between\ntwo stable directions. STT is induced by passing spin-\npolarized current which leads to Joule heating and sets\nlimits on the storage density. Much research is focused on\nalternative switching mechanisms based on direct or in-\ndirect electric-\feld control of magnetic anisotropy which\ncan reduce the dissipated energy by a factor of 100.1An-\nother recent alternative to STT-RAM devices replaces\nthe ferromagnetic components with a single active an-\ntiferromagnetic (AFM) layer with a bistable alignment\nof the staggered moments. The switching then utilizes\na spin-orbit torque (SOT) induced by an unpolarised\nelectric current.2,3There is no dipolar coupling between\nneighbouring elements and they are insensitive to exter-\nnal magnetic \felds. Again this alternative promises a\nhigher storage density and energy e\u000eciency. Note that\nboth aforementioned alternatives to STT use the rela-\ntivistic spin-orbit interaction (SOI) to achieve the ther-\nmal stability and the switching between distinct magnetic\nstates.\nHere we explore an ambitious approach combining the\nelectric-\feld control with the noncollinear antiferromag-\nnetic structure of Mn-antiperovskite nitrides. The re-\nquired coupling between the spin and orbital degrees of\nfreedom is not due to the relativistic SOI but due to\ngeometrically frustrated exchange interactions. The in-\ndirect magnetoelectric e\u000bect (ME) is hosted by a piezo-magnetic Mn-antiperovskite layer elastically coupled to\na piezoelectric substrate. We focus on the piezomagnetic\ne\u000bect (PME) which is characterised by a net magnetiza-\ntion directly proportional to the applied lattice strain.4,5\nFully compensated AFM states are hard to track and uti-\nlize in general but the PME o\u000bers a valuable technique\nto probe and control the AFM ordering via the strain-\ninduced magnetic moment.\nIn order to substantiate the future use of the PME\nin magnetoelectric composites, we perform a systematic\nabinitio study of PME in 9 cubic antiperovskites Mn 3AN\n(A = Rh, Pd, Ag, Co, Ni, Zn, Ga, In, Sn). We explain the\nvariation of the magnitude of PME across this range of\nbased on features of the electronic structure. The PME\nin Mn 3SnN predicted here is an order of magnitude larger\nthan PME modelled so far in Mn 3GaN.5Moreover, the\nsimulated PME is shown to be inversely proportional to\nthe measured magnetovolume e\u000bect (MVE) at a mag-\nnetic (N\u0013 eel) transition temperature6across the full set\nof 9 studied systems. This agreement with experimental\ndata is remarkable as both the PME and MVE originate\nin the frustrated AFM structure but we simulate PME\nat zero temperature whereas MVE was measured at the\nmagnetic (N\u0013 eel) transition temperature. MVE has not\nbeen modelled for this set of systems before. In addition\nto applications in spintronics our results can be used as\na tool in search for materials with large negative thermal\nexpansion (NTE) and barocaloric e\u000bect (BCE) which are\nboth directly related to MVE.\nMn-based antiperovskite nitrides were \frst examined\nin 1970s.7,8More recent experimental work on these\nmetallic compounds includes a demonstration of large\nNTE in Mn 3AN (A = Ga, Zn, Cu, Ni)9{12at thearXiv:1512.03470v2  [cond-mat.mtrl-sci]  26 Sep 20162\n\frst order phase transition to a PM state. A large\nbarocaloric e\u000bect was measured in Mn 3GaN at T N=\n288 K13and the Mn-antiperovskites were consequently\nproposed as a new class of mechanocaloric materials.\nMore importantly for spintronic applications, the baro-\nmagnetic e\u000bect (BME) closely related to the PME was\nreported in Mn 3G0:95N0:94very recently,14the exchange\nbias e\u000bect was observed in Mn 3GaN/Co 3FeN bilayers,15\nperpendicular magnetic anisotropy was demonstrated in\nMn67Ga24N on MgO substrate, and the magnetocapaci-\ntance e\u000bect was measured in Mn 3GaN/SrTiO 3bilayers.16\nTheoretical work on Mn-antiperovskites includes an\nearly tight binding study17suggesting that the proximity\nof the Fermi energy to a sharp singularity (narrow N p-\nMndband) in the electronic density of states has a large\nin\ruence on the stability of the structural and magnetic\nphases. However, this model considers only the nearest\nneighbour Mn-N hopping and neglects any hybridization\nwith atom A. Phenomenological studies analysed phase\ntransitions,18magnetoelastic, and piezomagnetic4prop-\nerties with respect to the symmetry of the crystal and\nmagnetic structure. More recently ab initio modelling\nof the noncollinear magnetic structure has been carried\nout. The NTE and MVE are attributed to the frustrated\nexchange coupling between the three Mn atoms.11,19,20\nThe local spin density has been simulated for Mn 3GaN\nand Mn 3ZnN revealing its distinctly nonuniform distri-\nbution and localized character of the 3d Mn moment.21\nThe piezomagnetic5and \rexomagnetic e\u000bect22were sim-\nulated in Mn 3GaN by the same group. The strain-\ninduced net magnetic moment predicted for Mn 3GaN is\nan order of magnitude lower than that of Mn 3SnN pre-\ndicted in this work.\nThe PME is de\fned by a linear dependence of the\nnet magnetization on elastic stress tensor components,\nin contrast to the magnetoelastic e\u000bect where the de-\npendence on stress is quadratic. Both e\u000bects can be de-\nscribed phenomenologically by adding appropriate stress-\ndependent terms to the thermodynamic potential, i.e.,\nthe free energy:\nF(T;H;\u001b) =F0(T;H)\u0000\u0015i;jkHi\u001bjk\u0000\u0016i;jkHi\u001b2\njk;(1)\nwhere\u0015i;jkis an axial time-antisymmetric tensor repre-\nsenting the PME, Hiare components of magnetic \feld,\n\u001bjkis the elastic stress tensor, and \u0016i;jkis the magnetoe-\nlastic tensor. Non-vanishing elements of \u0015i;jkcorrespond\nto terms of eq. (1) which are invariant under operations\nfrom the magnetic symmetry group.23These elements\nthen contribute to the magnetization:\nMi=\u0000@F\n@Hi=\u0000@F0\n@Hi+\u0015i;jk\u001bjk+\u0016i;jk\u001b2\njk:(2)\nThe PME was \frst proposed by Voigt24in 1928. The\nlinear character limits its existence to systems without\ntime inversion symmetry or with magnetic group that\ncontains time inversion only in combination with other\nelements of symmetry.25Hence, the PME is forbidden in\nFIG. 1: (Color online) Mn-aniperovskite magnetic unit cell,\ncubic and strained lattice assuming Poisson's ratio of 0.5, the\ncanting and changes of size are not to scale; (a) unstrained\nstructure of Mn 3GaN with local moments on Mn sites accord-\ning to \u00005grepresentation; (b) tensile strained magnetic order\nin (111) plane, Mnetindicates the direction of the induced\nnet moment; (c) compressively strained unit cell; (d) tensile\nstrained unit cell.\nall paramagnetic and diamagnetic materials. The most\nstriking manifestation of PME is in antiferromagnets\nwhere the zero spontaneous magnetization acquires a \f-\nnite value upon application of strain. The \frst AFM sys-\ntems where PME was proposed26,27and later observed28\nwere transition-metal di\ruorides. In Mn-anitiperovskite\nnitrides PME was predicted quantitatively in 20085and\nit has not been observed experimentally so far.\nThe noncollinear magnetic structure of Mn 3AN which\nhosts the PME and NTE considered in this work is\nshown in Fig. 1. (The direction of canting of the Mn\nlocal moments is speci\fc for Mn 3GaN.) The ground\nstate presented in Fig. 1(a) is the fully compensated\nAFM structure with symmetry corresponding to \u00005g\nrepresentation.29(The magnetic unit cell belongs to the\ntrigonal space group P31mand has the same size as the\ncubic paramagnetic unit cell belonging to space group\nPm3m.) The exchange coupling between the neighbour-\ning Mn atoms is antiferromagnetic which leads to the\nfrustration in the triangular lattice in (111) plane (high-\nlighted as orange online). The three Mn local magnetic\nmoments (LMM) are of the same size and have an angle of\n2\u0019=3 between their directions. A simultaneous rotation\nof all three LMMs by \u0019=2 within the (111) plane results\nin another fully compensated AFM structure correspond-\ning to \u00004grepresentation where the LMMs all point inside\n(outside) the triangle in a given (adjacent) plane.7The\nenergy di\u000berence between \u00004gand \u00005gordering is purely\ndue to the spin-orbit coupling whereas the noncollinearity\nand magneto-structural coupling is due to the exchange3\ninteraction. It should be noted that the origin of PME in\nexchange interaction distinguishes it from magnetostric-\ntion which is due to spin-orbit coupling5(PME can be\ndescribed as linear exchange-striction).\nII. RESULTS\nWe calculate the total energy, magnetic moments, and\nprojected density of states (DOS) for the noncollinear\nmagnetic structure of biaxially strained Mn 3AN (A =\nRh, Pd, Ag, Co, Ni, Zn, Ga, In, Sn) from \frst principles.\nOur computational procedure is the following:\n(1) We \fnd the equilibrium lattice parameter a0, bulk\nmodulusK, and the Poisson's ratio \u0017for each material\nwith \fxed AFM order by \ftting the total energies ob-\ntained for a range of lattice parameters ( a;c=a ) to Birch-\nMurnaghan equation of state.30We also allowed for relax-\nation of individual atomic positions but we found no bond\nbuckling in agreement with an earlier ab initio study.5\nThe results are summarized in Table I.\n(2) We relax the magnetic moments with a \fxed lat-\ntice for a range of biaxial strains to evaluate the PME.\nWe perform two independent sets of calculations with\nthe vertical lattice parameter cset: (a) to conserve the\nunstrained unit cell volume - data labelled as \"V\"; (b) ac-\ncording to the calculated Poisson's ratio - data labelled\nas \"P\"; The initial AFM local moment directions and\nsizes are either relaxed by the VASP code31in a self-\nconsistent loop or explicitly by searching for minima in\na total energy pro\fle E tot(\u000f;\u00121) as shown in Fig. 2. The\nquantitative agreement of these two methods gives us\ncon\fdence that we found the physically relevant energy\nminimum. All calculations include the spin-orbit cou-\npling and con\frm that its impact on PME is negligible\nin case of period 4 and 5 elements.\n(3) Finally, we increase the density of k-points and\ncalculate the projected DOS for the converged strained\nand unstrained noncollinear structures in order to iden-\ntify features in the electronic structure that would ex-\nplain the variation of PME across the material range.\nOur results do not con\frm a proximity of the Fermi en-\nergy to a sharp peak in DOS as suggested by an earlier\ntight-binding study.17\nFig. 1(c) and (d) represent a qualitative overview of the\nsimulated response of the magnetic structure to the ten-\nsile and compressive strain, respectively. A comparison\nwith the ground state in Fig. 1(a) shows that Mn mag-\nnetic moments cant and change size which are two inde-\npendent contributions to PME. This behaviour is due to\nthe strain induced reduction of symmetry from P31mto\nPm0m0morthorhombic magnetic space group and from\nPm3mtoP4=mmm tetragonal space group in the para-\nmagnetic case (the system is no longer invariant under\nthe third order rotation about the (111) axis).\nFor more clarity, Fig. 1(b) shows the tensile strained\n(\u000f= \u0001a=a0>0) magnetic order in the (111) plane. The\ncanted angles \u0012iwithin the (111) plane and LMM mag-\nFIG. 2: (Color online) Total energy as a function of biaxial\nstrain and canted angle for Mn 3GaN. No interpolation is used\nin the surface plot. The equilibrium angle depends linearly\non the strain. The reference energy corresponds to E(\u00121= 0)\nfor each strain.\nnitudes M ion the three Mn sites are introduced. The\nmoments in the (100) and (010) planes cant in opposite\ndirections, \u00121=\u0000\u00122, to become more parallel (antipar-\nallel) in case of positive (negative) \u00121. The moment in\nthe (001) plane does not change direction.\nThe change of moment size \u0001 Mi=Mi\u0000M0is strongly\ndependent on the c=aratio of the tetragonal lattice. ( M0\nis the LMM size common to all Mn sites in the un-\nstrained system.) The changes plotted in Fig. 1(b) cor-\nrespond to unit cell volume conservation when \u0001 M1=\n\u0001M2\u0019\u0000\u0001M3=2 for all studied systems. M3universally\nincreases (decreases) with compressive (tensile) strain.\nWith realistic Poisson's ratios all three Mn moments\nincrease (decrease) for tensile (compressive) strain fol-\nlowing the volume change of the unit cell (Mn 3RhN is\nthe only exception where M3is almost independent on\nstrain). Atom A develops a moment two orders of mag-\nnitude lower than the Mn local moment for small applied\nstrain,j\u000fj<1%, so its role in PME is negligible.\nThe unstrained ground state (plotted \u00005g) has no spon-\ntaneous magnetization but a net moment Mnetaligned\nwith M 3develops upon straining. Our calculations con-\n\frm that the canted angle \u0012i, the change of moment size\n\u0001Mi, and consequently Mnet= 2M1cos(2\u0019=3+\u00121)+M3\ndepend linearly on applied strain as required by Eq. (2).\nThe dependence departs slightly from linearity for larger\nstrainj\u000fj>1%, our study is limited to the interval\n\u000f2h\u0000 2:5;2:5i%. A striking feature of PME is the change\nof orientation of Mnetwhen switching between tensile\nand compressive biaxial strain. Note that such control\nof net moment orientation cannot be achieved by mag-\nnetostriction. (The same description holds also for \u00004g\norder butMnetkM3is then rotated by \u0019=2 in (111)\nplane.)\nTable I list all relevant measured properties and re-\nsults calculated in this work. Our Mn magnetic moment4\nATN[K]a0[\u0017A]at\n0[\u0017A]!s[10\u00003]\u0017tKt[GPa]Mt\n0[\u0016B]\nRh 226 3.918 3.88 2.07 0.19 148.4 2.84\nPd 316 3.982 3.94 3.60 0.20 140.7 3.15\nAg 276 4.013 3.98 5.79 0.20 118.9 3.08\nCo 252 3.867 3.80 5.64 0.13 149.5 2.48\nNi 256 3.886 3.84 8.18 0.15 136.5 2.83\nZn 170 3.890 3.87 20.44 0.13 126.0 2.64\nGa 288 3.898 3.86 19.10 0.13 129.4 2.43\nIn 366 4.000 3.99 9.24 0.18 115.0 2.70\nSn 475 4.060 3.97 0.0 0.18 102.0 2.52\nTABLE I: Physical properties of Mn 3AN: N\u0013 eel temperature,\nlattice parameter at 10 K, calculated lattice parameter, spon-\ntaneous volume change, Poisson's ratio, bulk modulus, size\nof Mn local moment in unstrained system; all measured data\nare taken from Ref. [6] except a0andTNfor Mn 3SnN which\nare from Ref. [32]. Calculated data are markedt.\nfor Mn 3GaN is in good agreement with a previous the-\noretical study.5Our Poisson's ratios do not vary much\nacross the range of compounds and are slightly smaller\nthan\u0017=0.25-0.3 predicted by an ab initio study of elas-\ntic properties in Mn 3(Cu,Ge)N.33All calculated lattice\nparameters are 1-2% smaller than the values measured\nat low temperatures.\nFig. 3 presents our results on PME and the related\nfeatures of electronic band structure. The net moment\nMnetplotted for the nine Mn-antiperovskite systems sub-\nject to tensile strain \u000f= 1% is a natural measure of PME.\nPositive (negative) value of Mnetcorresponds to net mo-\nment induced parallel (antiparallel) to M3irrespective of\nbelonging to the \u00004gor \u00005grepresentation.\nFig. 3(a) compares the PME obtained assuming unit\ncell volume conservation (Poisson's ratio \u0017= 0:5) and\nusing our calculated Poisson's ratios, \u0017, listed in Ta-\nble I which correspond to smaller vertical distortion for\na given strain. The latter is our lower estimate of the\nexperimentally accessible PME as our calculated values\nof\u0017are lower than expected for metallic materials. The\nformer version of PME neglects the elastic properties of\nthe lattice and represents the response of the frustrated\nmagnetic system to a lattice symmetry breaking (nor-\nmalized tetragonal distortion). As a result, the predicted\nMnet(V) should be regarded as an upper estimate of the\nexperimentally accessible PME. In both cases Mn 3SnN is\npredicted to have Mnetan order of magnitude larger than\nMn3GaN, the only PME value available in literature.5\nA. Fitting PME by Heisenberg model\nIn order to interpret the calculated PME in terms of\nthe AFM pairwise exchange interactions Jij(\u000f) between\nthe three Mn atoms in the (111) plane we resort to the\nclassical Heisenberg model:\nE(\u00121;\u000f) =\u0000J12M1M2cos(2\u0019=3\u00002\u00121)\u00002J13M1M3cos(2\u0019=3 +\u00121); (3)\nwhere the values of the exchange parameters J13=J236=\nJ12and the local moments M1=M26=M3introduced\nin Fig. 1(b) are restricted by the tetragonal symmetry.\nWe \fnd the canted angle minimizing the exchange energy\n(@E=@\u0012 1= 0) and insert it into the expression for the net\nmomentMnet= 2M1cos(2\u0019=3 +\u00121) +M3. We obtain a\nrelationship between PME and changes of the exchange\ninteraction due to strain:\nMnet\nM3= 1\u0000J13\nJ12(4)\n\u0019J0\u0000\u0001J\u0000(J0+ \u0001J)\nJ0\u0000\u0001J\u0019\u00002\u0001J\nJ0\nMJ\nnet\u0011 \u00002M3\nJ0\u0001J=2M3\nJ0@J12\n@\u000f\u0001\u000f; (5)\nwhereJ0<0 is the exchange parameter in the unstrained\nlattice and \u0001 Jis the induced change of J12andJ13.\nWe \ftted our ab initio total energy as a function of the\ncanted angle to the Heisenberg model of eq. (3) to extract\nJ12andJ13for each value of strain. In all compounds we\nobserved:J12\u0019J0\u0000\u0001JandJ13\u0019J0+\u0001Jwhich allows\nus to de\fne MJ\nnetin eq. (5) that is directly proportional\nto the derivative of the exchange parameters Jijwith\nrespect to the biaxial strain \u000f.\nFig. 3(a) shows that MJ\nnetis in good agreement with\nMnet(V) extracted directly from our calculated LMMs\n(without any \ftting). The small di\u000berences are due to\ndeviations of the magnetic system from the Heisenberg\nbehaviour (e.g., LMMs change size as they cant even in\nan unstrained lattice) and deviations from linearity as-\nsumed in eq. (4). The key conclusion based on Fig. 3(a)\nin combinations with eq. (4) is that a large PME cor-\nresponds to a large di\u000berence between J12(bond in the\nplane of the biaxial strain) and J13(bond with a compo-\nnent perpendicular to this plane.)\nB. Linking PME to band structure\nFig. 3(b) relates the total induced moment Mnet(V) to\nthe mean band energy of the valence pord-states of atom\nA. This quantity is often called the band center and we\nextract it from our projected DOS, \u001aAp;d(E), as follows:\n\u0016Ap;d= 1=\nR\nE\u001aAp;d(E)dE, where \n =R\n\u001aAp;d(E)dEis\na normalization. We consider only the d-band (\u001aAd(E))\nwhen atom A is a transition metal and only the p-band\n(\u001aAp(E)) for the rest. The wide s-band does not seem to\nplay an important role in PME. The right vertical axis\nof Fig. 3(b) measures the inverse of \u0016Ap;dwith respect to\nthe Fermi energy ( EF), marked as \f1\u00111=j\u0016Ap;dj, and\nthe same quantity with respect to the Mn d-band center,\n\u0016Mnd, weighted by the relative occupation of pord-band\nof atom A, marked as \f2\u0011nAp;d=j\u0016Ap;d\u0000\u0016Mndj, where\nnAp;d= 1=NREF\u001aAp;d(E)dEandNis the occupation of\na fully \flled pord-band.5\nFIG. 3: (Color online) Comparison of the net moment Mnet\ninduced by 1% of tensile strain: (a) Mnetassuming unit cell\nvolume conservation (V) and Poisson's rations of Table I (P)\nandMJ\nnet\ftted according to eq. (5); (b) comparison of PME,\nmeasured by Mnet(V), to the inverse of the energy separation\nbetweenpord-states of atom A and d-states of Mn (weighted\nby the relative band \flling), marked as \f.\nBased on the remarkable match between jMnetjand\nboth variants of \f1;2we conclude that piezomagnetism\nin Mn-antiperovskite nitrides is governed by the mutual\ncon\fguration of Mn d-states and pord-states of atom\nA. More speci\fcally, a greater proximity (a potential for\nhybridization) of the valence band of atom A to the spin-\npolarizedd-band of Mn increases the di\u000berence between\nJ12andJ13per unit strain which manifests itself as a\nlarger induced net moment. On the other hand, when\nthe triangular magnetic order of Mn moments is undis-\nturbed by hybridization with pord-states of atom A then\nJ12\u0019J13and only a small net moment is induced. The\nbest example is Mn 3ZnN where the narrow fully \flled\nd-band is about 7 eV below the Fermi energy and the in-\nduced net moment is negligible. This trend is analogous\nto a scaling of the N\u0013 eel temperature with the number of\nvalence electrons of atom A in the same class of materials\ndetected in 1977.8\nIt should be noted that Mn 3AgN and Mn 3RhN do not\nshare the triangular AFM order according to earlier neu-\ntron di\u000braction studies,32whereas the magnetic structure\nof Mn 3CoN and Mn 3PdN is yet to be con\frmed experi-\nmentally. We include these four compounds in our study\nas their composition, AFM order, and experimentally re-\nsolved MVE6makes them potential candidates for piezo-\nmagnetic behaviour.In more general terms, we perform a computational ex-\nperiment when the magnetic system is initialized in the\ntriangular state (\u00004gor \u00005g) even if it was only a local\nenergy minimum for Mn 3AN (A = Ag, Co, Pd, Rh) and\nthe response (induced Mnet) to a tetragonal distortion\nis detected. The consistency of the piezomagnetic re-\nsponse across the whole set of materials motivates us to\nuse this procedure as a probe of the level of frustration of\nthe exchange interaction even if the real systems did not\nhost piezomagnetism. In the following paragraphs, we\ncompare our simulated PME to the spontaneous magne-\ntovolume e\u000bect which is a measure of the magnetic frus-\ntration and experimental data is available for all nine\nMn3AN compounds.6\nC. Comparing PME to MVE\nTo draw an analogy between the strain and an external\n\feldHthat can induce magnetization, we introduce a\npiezomagnetic susceptibility:\nMJ\nnet\nM3=2\nJ0@J12\n@\u000f\u0001\u000f\u0011\u001fP(\u0016Ap;d)\u0001\u000f; (6)\nwhere the change of applied train \u0001 \u000freplacesHand\nMJ\nnet(\u0001\u000f) was introduced in eq. (5). Based on Fig. 3(b)\nwe can say that the susceptibility \u001fP(\u0016Ap;d) is inversely\nproportional to the mean valence band energy of atom A\nin the unstrained system.\nFig. 4 compares the measured magnetovolume e\u000bect6\nto our calculated piezomagnetic susceptibility \u001fP. The\nMVE is a spontaneous change of volume due to a change\nof magnetic ordering (typically the size of magnetic mo-\nment). It was \frst observed in Ni-Fe Invar below its TC.34\nTakenaka et al. measure a spontaneous volume increase\nupon the transition from PM to AFM state and sub-\ntract the phononic contribution so their MVE data are\npurely of magnetic origin.6They investigate a wide range\nof Mn-antiperovskite nitrides and conclude that MVE is\na property of the frustrated triangular AFM state which\nis strongly dependent on the number of valence electrons.\nMVE is the largest when there are two s-electrons and\none or nop-electrons (A = Zn, Ga). When the number of\nvalencesandp-electrons changes then the systems trans-\nforms to a di\u000berent crystal/magnetic structure with no\nMVE (A = Cu, Ge, As, Sn, Sb).\nIn addition, Takenaka et al. have observed an increase\nin MVE as the d-band of atom A moves away from EF.\nThis general trend reminds us of the scaling of suscep-\ntibility\u001fPwith the mean band energy of atom A \u0016Ap;d\ndescribed above. We include Fig. 4(a) to check if the de-\npendence on \u0016Ap;dfurnishes a clear link between PME\nand MVE. The \fgure shows that our piezomagnetic sus-\nceptibility\u001fPis inversely proportional to the measured\nvolume change as expected. In other words, a large MVE\nimplies a small PME and vice versa. Atoms A belonging\nto periods 4 and 5 of the periodic table have di\u000berent\ncoe\u000ecients of proportionality. This implies that not only6\nFIG. 4: (Color online) Calcualted PME characterised by j\u001f\u00001\nP j\nas a function of the measured MVE weighted by bulk modu-\nlus, triangles indicate systems with con\frmed triangular mag-\nnetic ground state, red and green symbols indicate a positive\nand negative canted angle at tensile strain, respectively; the\nblue lines are least square linear \fts; (a) two di\u000berent trends\nfor atom A from period 4 and 5; (b) j\u001f\u00001\nP jweighted by a strain\ninduced shift of mean band energy of two Mn atoms - one\ntrend for all systems with con\frmed triangular magnetism.\nthe position of A-band with respect to EFbut also the\nsize of atom A plays a role in weakening the triangular\nAFM structure. Such di\u000berence between period 4 and 5\nwas \frst seen also in case of the scaling of TNwith the\nnumber of valence electrons of atom A in 1977.8\nThe agreement of a calculated zero temperature sus-\nceptibility ( \u001f\u00001\nP) with a spontaneous volume change !s\nat the PM-AFM phase transition (weighted by K) is re-\nmarkable and requires further analysis. Magnetovolume\ne\u000bects in itinerant electron magnets were \frst analysed\nby the Stoner-Edwards-Wohlfarth theory.35The free en-\nergy can be approximated by F(T;M;! ) =F0(T;!) +\n1\n2KV!2+1\n2a(T;!)M2and minimized with respect to\nthe volume strain != \u0001V=V to obtain KV! =\ncmvM2wherecmv=\u00001\n2@a(T;!)=@! is the magneto-\nvolume coe\u000ecient, Mis the spontaneous magnetiza-\ntion,Kis the bulk modulus, and Vis the reference\nvolume. After considering the spin \ructuations at the\n\frst-order phase transition the above formula becomes:\nKV! =cmv(M2\u0000\u00182) where\u0018is the amplitude of spin\n\ructuations.36?\nIn the case of Mn-antiperovskites the local moments\nare relatively well localized21so we can approximate the\nmagnetic energy of the triangular AFM system on a cubic\nlattice by eq. (3) with zero canted angle: E(\u00121= 0) =\n3\n2J0M2\n0. The balance of elastic and magnetic energy thenleads to an expression for the spontaneous volume strain\n(\u0001V=V):\n!sK=\u00003M2\n0\n2V@J0\n@!\u0018@J0\n@!\u0011tv; (7)\nwhere we neglect the change of local moments M0with\nchanging volume, (\u0001 M0)2, as a higher order contribu-\ntion. The magnetic stress per Mn-Mn bond tvis intro-\nduced following the work of Filippetti and Hill.37The\nmagnetic stress at the phase transition can be then ex-\npressed as: Tv=@E\u00005g=@! =3\n2tvM2\n0, whereE\u00005gis\nagain the magnetic energy E(\u00121= 0).\nAfter establishing the link between MVE and the mag-\nnetic volume stress Tv, we attempt the same for PME\nand the magnetic biaxial stress: Tb=@E(\u00121)=@\u000f\u0018tb,\nwhereE(\u00121) is a magnetic energy of the canted AFM\nstructures and the magnetic stress per Mn-Mn bond tbis\nproportional to the susceptibility \u001fPof eq. (6):\n\u001fP=2\nJ0@J12\n@\u000f\u0018@J12\n@\u000f\u0011tb: (8)\nFinally, based on the comparison of eqs. (7) and (8)\nwe can conclude that both !sKand\u001fPare proportional\nto derivatives of the exchange parameters with respect to\nstrain and thereby to the magnetic stress of the triangular\nAFM system. Hence the linear relationship of Fig. 4(a)\nindicates a trade-o\u000b between two complementary stress\nrelief mechanisms.\nIII. DISCUSSION\nIn principle, the stress arising at the onset of AFM or-\ndering atTNcan be relieved by a volume change or a\nlattice distortion. However, our calculations and subse-\nquent \ftting to Heisenberg model \fnd that the magnetic\nenergy saved by a tetragonal distortion (linear in \u000f) be-\ncomes smaller than the elastic energy cost (quadratic in\n\u000faround unstrained lattice) for negligibly small distor-\ntions. This is con\frmed by x-ray di\u000braction6which has\nnot indicated a tetragonal distortion in any compound\nstudied in this work. Nevertheless, \u001fPre\rects how much\nmagnetic stress could be relieved by a tetragonal distor-\ntion and this quantity is inversely proportional to !sKas\nshown by Fig. 4. We plot \u001fPvs!sKrather than !sto\ncompare only quantities related to magnetism and factor\nout the system dependent elastic properties.\nIt should be noted that the sign of \u001fPindicates\nwhich type of tetragonal distortion is energetically more\nfavourable. A brief demonstration of this neglects the de-\npendence of Miand\u0012ion strain in eq. (3) - then we can\n\fnd a spontaneous biaxial strain \u000fs(analogous to vol-\nume strain !s) from the balance of elastic and magnetic\nenergy:\u000fs=@J12=@\u000fM2\n0=C=\u00001\n2\u001fPjJ0jM2\n0=C, where\nC > 0 is an e\u000bective elastic modulus. Immediately, we\ncan see that all systems in this study with \u001fP>0 tend\nto a distortion with \u000fs<0 (c=a> 1) and vice versa.7\nWe conclude that a system with robust triangular mag-\nnetic order undisturbed by the proximity of electronic\nstates of atom A (large \u0016Ap;d) tends to relieve its mag-\nnetic stress via a volume change, whereas a system more\nin\ruenced by atom A but with persisting triangular or-\nder (small\u0016Ap;d) prefers to relieve its magnetic stress via\na tetragonal distortion should the elastic energy cost al-\nlow it. (If the tetragonal distortion is enforced externally,\nthen the system develops a large net magnetization.)\nThe slight deviations of j\u001f\u00001\nPjfrom!sKseen in Fig. 4\nmay originate in: (a) spin \ructuations which we ne-\nglected in eq. (7), the small size of the deviations sug-\ngests that the spin \ructuation contribution to MVE\n(KV! =cmv(M2\u0000\u00182)) is signi\fcantly suppressed by the\nstrong frustration; (b) limited numerical accuracy, e.g.,\nMn3ZnN is most a\u000bected as it has almost trivial \u001fPand\nits large relative error is ampli\fed by the inversion; (c)\nNitrogen de\fciency (8-16%) varying across the range of\nsamples where MVE was measured,6e.g., magnetic order\nin Mn 3SnN is known to be sensitive to N concentration;32\n(d) a material-speci\fc elastic property that was not fac-\ntored out of the plotted quantities, e.g., the use of bulk\nmodulusK= 130 GPa for all compounds when subtract-\ning the phononic contribution to MVE6(consequently, in\nthe plot we use K= 130 GPa instead of our calculated\nKof Table I).\nTo further explore the inverse proportionality between\nPME and MVE with respect to features of the electronic\nstructure we analyse the strain dependence of mean band\nenergy of Mn-states. We extract the mutual shift of mean\nband energy of Mn1\nd-states (site in (100) plane of the unit\ncell) and Mn3\nd(site in (001) plane) from the projected\nDOS\u001aMn1\nd(E;\u000f;\u0012 1) and\u001aMn3\nd(E;\u000f;\u0012 1) of the strained\nsystem before canting( \u000f=1%,\u00121=0) in analogy to evalu-\nation of\u0016Ap;dshown in Fig. 3(b). The obtained quantity\nj\u00161\u0000\u00163jdirectly measures the response of the spin po-\nlarized electronic structure to the tetragonal distortion.\nSuch information is missing in \u0016Ap;dof the unstrained\nstructure.\nFig. 4(b) showsj\u001f\u00001\nPjweighted by the mutual band\nshiftj\u00161\u0000\u00163jas a function of !sK. Compounds with\natom A from period 4 and 5 now follow the same linear\ntrend with the exception of A = Ag, Co, Rh. Our hy-\npothesis based on Fig. 4 is that the factor j\u00161\u0000\u00163jincor-\nporates the dependence of PME on the size of atom A for\nsystems with stable triangular AFM ordering. Mn 3AgN\nand Mn 3RhN do not have triangular AFM ground state\nwhich has explanation in their band structure proper-\nties and become apparent in Fig. 4(b). Extending the\nsame argument to the unknown magnetic order, we ex-\npect Mn 3PdN (Mn 3CoN) to have a triangular (other)\nAFM ground state.\nThe linear scaling of the spontaneous MVE with j\u001f\u00001\nPj\nimplies a signi\fcant suppression of spin \ructuations by\nthe strong frustration in these systems. At the sametime it can be used as a tool in theory led design of\nnon-stoichiometric materials with large MVE and con-\nsequently BCE where the entropy change is propor-\ntional to the spontaneous volume change according to\nthe Clausius-Clapeyron relation:\nS(Tt;p)\u0000S(Tt;0) =V!s\u0012dTt\ndp\u0013\u00001\n: (9)\nModelling the pressure dependence of the transition tem-\nperaturedTt=dpgoes beyond the capability of density\nfunctional theory at zero temperature and is the subject\nof our ongoing work.38\nWe hope that the successful comparison of our pre-\ndicted PME to the measured MVE and the coherent in-\nterpretation of the PME based on features of the elec-\ntronic structure will provide guidance for further investi-\ngations of the unique physical properties of the frustrated\nAFM structure of Mn-antiperovskites and enable devel-\nopment of applications including data storage, memory,\nand solid-state cooling.\nIV. METHODS\nAll our calculations employ the projector augmented-\nwave (PAW) method implemented in VASP code31\nwithin the Perdew- Burke-Ernzerhof (PBE) generalized\ngradient approximation.39This approach allows for re-\nlaxation of fully unconstrained noncollinear magnetic\nstructures.40We use a 12x12x12 k-point sampling in the\nself-consistent cycle and 17x17x17 k-point sampling to\nobtain the site and orbital resolved DOS. The cuto\u000b en-\nergy is 400 eV. The local magnetic moments are eval-\nuated in atomic spheres with the default Wigner Seitz\nradius as they are not very sensitive to the projection\nsphere radius.5\nWe constrain the Mn local moment directions using an\nadditional penalty energy as implemented in the VASP\ncode in order to obtain the projected DOS \u001aMn3\nd(E;\u000f;\u0012 1)\nof the strained system. 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Marrows,3and\nHans Fangohr1\n1)Faculty of Engineering and the Environment, University of Southampton, Southampton, SO16 7QF, United\nKingdom.\n2)Department of Physics, Ningbo University, Ningbo 315211, China\n3)School of Physics & Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom.\n(Dated: 25 October 2018)\nWe demonstrate that chiral skyrmionic magnetization con\fgurations can be found as the minimum energy\nstate in B20 thin \flm materials with easy-plane magnetocrystalline anisotropy with an applied magnetic\n\feld perpendicular to the \flm plane. Our observations contradict results from prior analytical work, but are\ncompatible with recent experimental investigations. The size of the observed skyrmions increases with the\neasy-plane magnetocrystalline anisotropy. We use a full micromagnetic model including demagnetization and\na three-dimensional geometry to \fnd local energy minimum (metastable) magnetization con\fgurations using\nnumerical damped time integration. We explore the phase space of the system and start simulations from a\nvariety of initial magnetization con\fgurations to present a systematic overview of anisotropy and magnetic\n\feld parameters for which skyrmions are metastable and global energy minimum (stable) states.\nSkyrmions are topological defects1that can be ob-\nserved in the magnetization con\fguration of materials\nthat lack inversion symmetry,2either due to a non-\ncentrosymmetric crystal lattice,3,4or at interfaces be-\ntween di\u000berent materials.5This lack of inversion sym-\nmetry results in a chiral interaction known as the\nDzyaloshinskii-Moriya (DM) interaction.3,4The DM in-\nteraction results in a rich variety of chiral magnetization\ncon\fgurations, including helical, conical, and skyrmionic\nmagnetization con\fgurations. Skyrmionic con\fgura-\ntions were predicted6and later observed in helimag-\nnetic materials,7{10and materials with an interfacial DM\ninteraction.11{15\nSkyrmions demonstrate potential for applications in\ndata storage and processing devices. Skyrmions have\nbeen observed with diameters of the order of atom spac-\nings in mono-atomic Fe layers,16which is signi\fcantly\nsmaller than the magnetic domains proposed for the\nracetrack memory design.17This results in a greater\nstorage density. The movement of skyrmions has also\nbeen demonstrated18,19using spin-polarized current den-\nsities of the order 106Am\u00002, which is orders of magni-\ntude less than what is required to move magnetic do-\nmain walls.17,20These observations demonstrate poten-\ntial for skyrmion-based racetrack memory technology21\nand other data storage and processing devices.22\nCertain material restrictions need to be overcome be-\nfore skyrmions can be used in such technologies. While\nskyrmions can be stabilized, they are only stable in a\nlimited region of the parameter space de\fned by an ap-\nplied magnetic \feld and the temperature. This region is\nnarrow in bulk materials,7larger in thin \flm materials,9\nand further stabilized in laterally con\fned geometries23\nand materials with pinning defects.24Analytical analy-\nsis of helimagnetic thin \flm material models \fnd that\nskyrmion lattice states are ground states in helimagnetic\nthin \flms with an applied magnetic \feld only in sys-\ntems with easy-axis magnetocrystalline anisotropy,2,25where the easy axis and the applied \feld are perpen-\ndicular to the plane of the \flm. However, simulated an-\nnealing methods \fnd that skyrmions can be the ground\nstate in two-dimensional helimagnetic thin \flms with\neasy-plane anisotropy.26Skyrmions have also been iden-\nti\fed in two-dimensional surface-inversion breaking sys-\ntems with easy-plane anisotropy and Rashba spin-orbit\ncoupling.27Furthermore, experimental studies of easy-\nplane helimagnetic thin \flms identify an additional con-\ntribution to the Hall resistivity beyond the ordinary and\nanomalous contributions.28{33This may be interpreted\nas the topological Hall e\u000bect, which arises through real\nspace Berry phase e\u000bects,34,35and is an indication of po-\ntential skyrmion presence. Skyrmions have been directly\nobserved with Lorentz transmission electron microscopy\n(LTEM) in easy-plane MnSi with the \feld applied along\nall principal crystallographic directions.36\nIn this letter, prior analyses are extended by consider-\ning a three-dimensional thin \flm with demagnetization\nto determine whether or not skyrmions are stable in thin\n\flms with easy-plane anisotropy. For a magnetization\ncon\fguration to be stable, it must be a con\fguration with\nthe lowest possible energy. Variational techniques have\nbeen used to minimize the energy in simpli\fed model\nsystems analytically.37Here we use numerical simulation\nmethods to solve a more complete model system.\nWe consider a cuboidal simulation cell representing a\nthin \flm of Fe 0:7Co0:3Si. The cell has lateral dimensions\nLxandLy, and \fnite thickness Lz\u001cLx;Ly, wherex,\ny, andzare Cartesian axes with origin at the center\nof the geometry. Lxis equal to the helical period, and\nLy=Lxp\n3 to support hexagonal skyrmion lattice mag-\nnetization con\fgurations in the simulation cell. Periodic\nboundary conditions are imposed on the Heisenberg and\nDM exchange interactions in the lateral directions. The\nmacrogeometry approach38is used to model periodicity\nof the demagnetizing \feld, with a disc macrogeometry\nof radius equal to 26 times the helical period and thick-arXiv:1602.02064v2  [cond-mat.mtrl-sci]  22 Apr 20162\nnessLz= 5 nm. The boundary conditions pose a math-\nematically di\u000berent problem from analytical work con-\nducted previously, which considers rotationally symmet-\nric magnetization.2Our method allows arbitrary magne-\ntization con\fgurations that do not satisfy this symmetry,\nsuch as helical states.\nThe standard numerical micromagnetics approach of\nsolving the Landau-Lifshitz-Gilbert (LLG) equation as\nan initial value problem is employed here. Unlike pre-\nvious work, demagnetization e\u000bects are incorporated in\nthis energy model since demagnetization is known to\na\u000bect the stability of skyrmions.23A three-dimensional\nsimulation domain with \fnite thickness is used because\nmagnetization variation in the thickness direction can\nstabilize skyrmionic con\fgurations.39These extensions\ndi\u000berentiate this study from previous works.2,25,26\nThe micromagnetic representation of the system en-\nergy is modelled here as\nW(m) =Z\nV(we+wdmi+wz+wa+wd) dV; (1)\nwhereVis a cuboid region of volume Lx\u0002Ly\u0002Lz,\nandm(x;y;z ) = M(x;y;z )=MSis the magnetization\nvector \feld normalized by saturation magnetization MS\nsuch thatjmj= 1. The terms we=A(rm)2and\nwdmi=Dm\u0001(r\u0002m) are the energy density contri-\nbutions from Heisenberg and DM exchange interactions\nrespectively. The term wz=\u0000\u00160MSH\u0001mis the Zeeman\nenergy density contribution from the applied magnetic\n\feld. The term wa=K1(1\u0000\u0000\nm\u0001^z)2\u0001\nis the energy den-\nsity contribution from the magnetocrystalline anisotropy.\nThis anisotropy is easy-axis when K1>0 and is easy-\nplane when K1<0. The term wd=\u0000\u00160MS(Hd\u0001m)=2\nis the energy density contribution from demagnetization,\nwhere the demagnetizing \feld Hdis calculated using\nthe Fredkin-Koehler \fnite element method-boundary el-\nement method (FEMBEM).40\nThe energy model uses material parameters from\nexperiments on Fe 0:7Co0:3Si30,41as follows: the sym-\nmetric exchange coe\u000ecient A= 4:0\u000210\u000013Jm\u00001,\nthe DM exchange coe\u000ecient D= 2:7\u000210\u00004Jm\u00002,\nthe magnetocrystalline anisotropy coe\u000ecient\nK1=\u00003:0\u0002104Jm\u00003, and the saturation magne-\ntizationMS= 9:5\u0002104Am\u00001. To obtain a systematic\ndata set and understanding, the magnetocrystalline\nanisotropy K1is varied in the range [ \u00000:5K0;1:25K0],\nwhereK0=D2=A= 1:8\u0002105Jm\u00003. This range con-\ntains the anisotropy value K1=\u00000:16K0calculated for\nFe0:7Co0:3Si. The applied \feld parallel to the zdirection\njHj, is varied in the range [0 ;15:4MS].\nMultiple initial magnetization con\fgurations are used\nto determine the minimum energy state of the micromag-\nnetic system for each combination of jHjandK1values.\nThese initial states are shown in Fig. 1. A \fnite ele-\nment method-based simulator has been used to solve the\ndamped LLG equation as an initial value problem for\nthe aforementioned system. The simulator is similar to\nthe software Nmag42and uses the FEniCS43\fnite ele-\n−1.0 0 .0 1 .0\nm·ˆzFIG. 1. Initial magnetization con\fgurations to be relaxed us-\ning damped Landau-Lifshitz-Gilbert (LLG) dynamics. From\nthe top, left to right, these are hexagonal and rectangular\nskyrmion lattice states, canted uniform states parallel to the\nx,y, and zdirections, and a variety of helical states. The\nperiod of the helices is varied to support metastable states\nwith di\u000berent helical periods.\nment framework. The edge lengths of the tetrahedral\nelements of the \fnite element mesh do not exceed one\nnanometer, which is chosen to be smaller than both the\nBloch parameterp\nA=jK1j>1:3 nm and the exchange\nlengthp\n2A=\u0016 0M2\nS= 8:4 nm to correctly resolve micro-\nmagnetic behavior. The same \fnite element mesh is used\nto discretize the magnetization domain for all results re-\nported here. When the magnetization precesses more\nslowly than 1\u000ens\u00001everywhere, it is considered relaxed.\nAll initial con\fgurations shown in Fig. 1 are relaxed in-\ndependently for each pair of jHjandK1values. The\nenergies of all relaxed con\fgurations at the same jHjand\nK1are compared, and the con\fguration with the lowest\nenergy out of all these states is classi\fed as the ground\nstate. A selection of these states are shown in Fig. 2 (a)\nto (f). Table I shows the normalized jHjandK1values\nfor each of the six con\fgurations (a) to (f).\nFig. 2 (top) shows a phase diagram which groups re-\nlaxed con\fgurations into uniform, skyrmionic, helical,\nand unclassi\fed con\fgurations. This phase diagram\nshows the obtained ground state con\fguration for each\npoint in the applied \feld and anisotropy parameter space.\nThe locations of the ground states shown in Fig. 2 (a) to\n(f) are indicated in the phase diagram.\nStates are considered as uniform if m\u0001\u0016m>0:85 ev-\nerywhere, where \u0016mis the spatially-averaged magnetiza-\ntion direction. States are considered skyrmionic if the\nskyrmion number S(m)>0:5, where\nS(m) =1\n4\u0019Z\nTm\u0001\u0012@m\n@x\u0002@m\n@y\u0013\ndxdy; (2)\nandTis the surface at z= 0 contained by V. States\nare considered helical if the magnetization con\fguration\ncontains a full rotation along a single direction in the\nthin \flm. Magnetization con\fgurations that do not sat-3\n-0.50 -0.25 0.00 0.25 0.50 0.75 1.00 1.25\nK1/K00.01.42.84.25.67.08.49.811.212.614.015.4|H|/MSEasy-plane Easy-axis\nS≈0S≈0\nS≈0S≈1\nS≈2\na bcdef\nUniform\nSkyrmionic\nHelical\nUnclassified(a) (b) (c) (d) (e) (f)\nFIG. 2. Top: Phase diagram showing ground state variation\nwith normalized anisotropy K1=K0and applied \feld mag-\nnitudejHj=MS. Skyrmions are ground states in both easy-\nplane and easy-axis anisotropy regions. Parameters below the\nsolid line in the uniform region support metastable skyrmionic\nstates. Bottom: Selected magnetization con\fgurations.\nisfy any of these conditions are unclassi\fed, but are still\nconsidered when energies of relaxed states are compared.\nThe helical con\fguration is found for anisotropy val-\nuesK1in the range [\u00000:2K0;0:85K0], and weak ap-\nplied \felds, in agreement with previous predictions44\nand observations.9The uniform state is observed when\nthe applied \feld and easy-axis anisotropy dominate, and\nthese states have magnetization aligned with the out-of-\nplane direction ( z). This is expected, since the energy\ncontributions from the applied \feld and the easy-axis\nmagnetocrystalline anisotropy are minimized in this case.\nThe magnetization con\fgurations in the unclassi\fed re-\ngion, exempli\fed by Fig. 2 (c), are driven by the easy-\nplane anisotropy contribution, which causes the magneti-\nzation to orient mostly within the plane of the thin-\flm.\nSkyrmion lattice states (Fig. 2 (d) to (f)) are minimum\nenergy states for both positive (easy-axis) and negative\n(easy-plane) values of the magnetocrystalline anisotropy\ncoe\u000ecient. Analytical work suggests that skyrmion lat-\ntice states are minimum energy states only for positive\nanisotropy values K1in the range [0 ;0:48K0],25which\nTABLE I. Normalized anisotropy and applied \feld values for\nthe selected magnetization con\fgurations in Fig. 2 (a) to (f).\n(a) (b) (c) (d) (e) (f)\nK1=K00:85\u00000:20\u00000:35 0 :55 0 :00\u00000:35\njHj=MS 0:0 0 :0 0 :7 2 :1 5 :6 9 :8\nm ·ˆz\n0 (Ly/2)−ry\ny-Axis Displacement from Core-1.00.01.0|H|/MS=10.5\nK1/K0\n0.5 (g)\n0.0 (h)\n-0.5 (i)\n0 |r2−r1|\nDisplacement from Core 1 to 2-1.00.01.0K1/K0=0\n|H|/MS\n15.4 (j)\n7.0 (k)\n0.0 (l)\n(g) (h) (i) (j) (k) (l)\nFIG. 3. Top: Metastable skyrmion pro\fles varying with\nanisotropy in the rectangular lattice (left), and applied \feld\nmagnitude in the hexagonal lattice (right), where ris the co-\nordinate of the skyrmion core. Bottom: Magnetization con-\n\fgurations of the pro\fles shown above.\nis in contrast to the wider [ \u00000:35K0;0:55K0] range ob-\nserved in this work, which includes both positive and\nnegative anisotropy values. There are di\u000berences be-\ntween our work and the model used in Ref. 25 that we\nbelieve causes this discrepancy. Firstly, unlike the ana-\nlytical work, our model includes demagnetization energy,\nwhich is known to change skyrmion energetics.23,39Sec-\nondly, their work identi\fes skyrmion lattice states as lo-\ncal energy minima and the cone state as the ground state\nin easy-plane anisotropy systems. In our work, the thin-\nner \flms suppress conical states,9resulting in skyrmion\nlattice states having the lowest energy. Finally, the chi-\nral twist in the magnetization at the top and bottom of\nthe \flm, that are known to contribute to the skyrmion\nstability,23,45are accounted for in our model, unlike the\ntwo-dimensional analytical model where the skyrmion\nmagnetization does not vary in the thickness direction.\nThe parameter sets within the solid line in the\nskyrmion region of Fig. 2 exhibit a hexagonal skyrmion\nlattice as the minimum energy con\fguration, as shown in\nFig. 2 (e). The magnetization in this simulation cell has\na skyrmion number S\u00192. In the remaining skyrmionic\nregion, one skyrmion per simulation cell is found (as in\nFig. 2 (d) and (f)), which corresponds to a rectangular\nskyrmion lattice ( S\u00191). The rectangular skyrmion lat-\ntice has a lower energy than the hexagonal lattice over\nmost of the parameter space. This occurs because the\nskyrmion spacing that minimizes the energy changes with\nthe anisotropy, meaning that the hexagonal skyrmions\nare frustrated in the simulation cell. These two skyrmion\nlattice con\fgurations compete.26\nParameter sets below the solid line in the uniform re-\ngion of Fig. 2 support local energy minimum (metastable)4\nskyrmion states, which encompasses most of the param-\neter space. Fig. 3 shows how metastable skyrmion size\nchanges with anisotropy (snapshots (g), (h) and (i)) and\napplied \feld magnitude (snapshots (j), (k) and (l)) at\nz= 0, as parameterized by Table II. Skyrmion size in-\ncreases with decreasing magnetic \feld magnitude and in-\ncreasing easy-plane anisotropy because the magnetiza-\ntion component in the plane is more favorable energeti-\ncally, which increases the length over which the skyrmion\ntwists. Fig. 3 (i) shows that skyrmions expand to \fll the\ngeometry that contains them when easy-plane anisotropy\nis strong enough. This causes the skyrmion to stretch\ninto two separate objects.26\nTo summarize, skyrmions are minimum energy states\nin magnetic systems with easy-plane magnetocrystalline\nanisotropy and demagnetizing e\u000bects under micromag-\nnetic simulation. This conclusion is in counterpoint to\n\fndings with analytical models,2,25but agrees with nu-\nmerical work that does not consider demagnetization\nor thickness e\u000bects.26Systems with weaker easy-plane\nanisotropy result in smaller skyrmions. Skyrmion lattice\nstates are also metastable in a wide range of anisotropy\nvalues. 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Commun. 6, 7638 (2015)."    },    {        "title": "1602.03141v1.Magnetocrystalline_anisotropy_of_Fe_and_Co_slabs_and_clusters_on_SrTiO___3__by_first_principles.pdf",        "content": "Magnetocrystalline anisotropy of Fe and Co slabs and clusters on SrTiO 3by first-principles\nDongzhe Li,1Cyrille Barreteau,2, 3and Alexander Smogunov2,\u0003\n1SPEC, CEA, CNRS, Universit ´e Paris-Saclay, CEA Saclay, F-91191 Gif-sur-Yvette Cedex, France\n2Service de Physique de l’Etat Condens ´e, Centre National de la Recherche Scientifique,\nUnit´es Mixtes de Recherche 3680, IRAMIS/SPEC, CEA Saclay,\nUniversit ´e Paris-Saclay, F-91191 Gif-sur-Yvette Cedex, France\n3DTU NANOTECH, Technical University of Denmark, Ørsteds Plads 344, DK-2800 Kgs. Lyngby, Denmark\n(Dated: March 17, 2022)\nIn this work, we present a detailed theoretical investigation of the electronic and magnetic properties of\nferromagnetic slabs and clusters deposited on SrTiO 3via first-principles, with a particular emphasis on the\nmagneto-crystalline anisotropy (MCA). We found that in the case of Fe films deposited on SrTiO 3the effect\nof the interface is to quench the MCA whereas for Cobalt we observe a change of sign of the MCA from in-\nplane to out-of-plane as compared to the free surface. We also find a strong enhancement of MCA for small\nclusters upon deposition on a SrTiO 3substrate. The hybridization between the substrate and the d-orbitals of\nthe cluster extending in-plane for Fe and out-of-plane for Co is at the origin of this enhancement of MCA. As a\nconsequence, we predict that the Fe nanocrystals (even rather small) should be magnetically stable and are thus\ngood potential candidates for magnetic storage devices.\nPACS numbers: 75.30.Gw, 75.50.Ss, 75.70.Ak, 71.15.-m\nI. Introduction\nThe fine-tuning of the interfacial magnetocrystalline\nanisotropy (MCA) in ferromagnet-oxide insulator systems\nrepresents a key issue for several technological applica-\ntions such as perpendicular magnetic tunnel junctions (p-\nMTJs)1–3and tunneling anisotropic magnetoresistive (TAMR)\nsystems4,5. It is well known that the physical origin of the\nMCA is the spin-orbit coupling (SOC). For the 3 dtransition-\nmetals the SOC being of the order of a few tens of meV , the\nMCA per atom is extremely small (10\u00003meV) in the bulk\nphase of cubic materials but can get larger ( \u001810\u00001meV)\nat surfaces/interfaces due to reduced symmetry. In order to\nobtain even larger MCA, traditionally, the MCA of nanos-\ntructures of 3 delements is enhanced by introducing 4 dor 5d\nheavy elements with large SOC as a substrate such as Co/Pt6\nor Co/Pd7multilayers as well as in small 3 dclusters on heavy\nelements substrate8. However, despite the weak SOC at the\ninterface, a strong MCA has been observed in Co and Fe thin\nfilms on metallic oxides such as AlO xand MgO9,10. The ori-\ngin of this large MCA is attributed to electronic hybridization\nbetween the metal 3 dand O- 2porbitals11. More recently, Ran\netal. have shown that it was possible to reach the magnetic\nanisotropy limit (\u001860 meV) of 3 dmetal atom by coordinating\na single Co atom to the O site of an MgO surface12. Enhanc-\ning MCA of nanostructures provides a route towards future\nminiaturization of data storage at ultimate length scales13,14\nIn our previous work, we demonstrated that for both Fe and\nCo nanocrystals, the MCA of free nanocrystals is mainly dom-\ninated by the (001) facets resulting in an opposite behavior:\nout-of-plane and in-plane magnetization direction favored in\nFe and Co nanocrystals, respectively15,16. Therefore, the study\nof magnetic properties of nanocrystals deposited on a SrTiO 3\nas experimentallly obtainable16,17is essential, since depend-\ning on the bonding between the substrate and (001) facets this\ncan influence greatly the overall behaviour of the nanocrystal.\nIn this paper, we report first-principles investigations of theMCA of bcc-Fe(001) and fcc-Co(001) deposited on a SrTiO 3\nsubstrate, namely Fe(Co) jSrTiO 3interface. Next, we also in-\nvestigated the MCA of very small (five atoms) Fe and Co clus-\nters on SrTiO 3.\nII. Calculation method\nWe carried out the first-principles calculations by using\nthe plane wave electronic structure package QUANTUM\nESPRESSO (QE)18. Generalized gradient approximation in\nPerdew, Burke and Ernzerhof parametrization19was used for\nelectronic exchange-correlation functionals and a plane wave\nbasis set with the cutoffs of 30 Ry and 300 Ry were employed\nfor the wavefunctions and for the charge density, respectively.\nThe MCA was calculated from the band energy difference\nbetween two magnetic orientation ^m1and ^m2using force\ntheorem16, as we implemented recently in QE package:\nMCA =X\n\u000bocc\u000f\u000b(^m1)\u0000X\n\u000bocc\u000fi(^m2): (1)\nWhere\u000f\u000b(^m)are the eigenvalues obtained after a single di-\nagonalization of the Hamiltonian including SOC, but starting\nfrom an initial charge/spin density of a self-consistent scalar-\nrelativistic calcutation that has been rotated to the appropriate\nspin orientation axis as explained in Ref. 16.\nThe Fe(Co)jSrTiO 3interface was simulated by 10 layers\nof bcc-Fe(001)[fcc-Co(001)] slab deposited on a SrTiO 3(001)\nwith 5 layers. In the ionic relaxation, the Brillouin-zone has\nbeen discretized by using 10 \u000210 in-plane k-points mesh\nand a smearing parameter of 10 mRy. Two bottom layers of\nSrTiO 3were fixed while other three layers of substrate and\nferromagnetic slabs were relaxed until the atomic forces are\nless than 1 meV/ ˚A. To obtain reliable values of MCA, the\nconvergence of calculations has been carefully checked. A\nmesh of 20\u000220 in-plane k-points has been used for SCFarXiv:1602.03141v1  [cond-mat.mtrl-sci]  9 Feb 20162\ncalculation with scalar-relativistic PPs with a smaller smear-\ning parameter of 5 mRy. In non-SCF calculation with full-\nrelativistic PPs including SOC the mesh was increased to 60\n\u000260 and smearing parameter was reduced down to 1 mRy\nwhich provides an accuracy of MCA below 10\u00002meV .\nFor small Fe and Co clusters on SrTiO 3, the interface was\nsimulated by a (4\u00024)in-plane TiO 2-terminated SrTiO 3(001)\nsubstrate with 5 atomic layers containing one Fe(Co) cluster\nmade of 5 atoms. Two bottom layers were fixed while other\nthree layers of substrate and Fe(Co) cluster were relaxed until\natomic forces are less than 1 meV/ ˚A. For both scalar and full\nrelativistic calculations, a (8\u00028\u00021)k-points mesh and a\nsmearing parameter of 1 mRy was used. In addition, the effect\nof unphysical interaction in the direction zwas minimized by\ntaking a vacuum space of about 15 ˚A.\nIII. Results and discussions\nA. Fe(Co)jSrTiO 3interfaces\nThe SrO and TiO 2planes in the perovskite cubic SrTiO 3\nalternate in the (001) direction, here SrTiO 3(001) surface was\nchosen to be TiO 2-terminated since it is energetically more\nfavorable than SrO-terminated one20. The lattice constants of\nbulk bcc-Fe, fcc-Co and SrTiO 3are 2.85, 3.53 and 3.93 ˚A, as\ncompared to the experimental values of 2.87, 3.54 and 3.91\n˚A. When deposited on SrTiO 3the in-plane lattice parameter\nof Fe(Co) slab is imposed by the one of bulk SrTiO 3since it\nhas been shown that the Co layer can nicely be grown on this\nsubstrate21,22. In order to obtain a better match, the Fe and\nCo slabs are rotated by 45\u000ewith respect to the substrate, and\neach layer of the ferromagnetic slab is made of 2 atoms per\nsupercell. The TiO 2layer at the interface in Fe(Co) jSrTiO 3is\ndenoted as S(see Fig. 1). Layers toward the SrTiO 3bulk are\nlabeled as S\u00001,S\u00002, etc., while Fe(Co) layers towards the\nsurface are labeled as S+1,S+2,S+3, etc.\nWe found that the most stable configuration is, in all cases,\nwhere the Fe(Co) sites in layer S+1 are on top of the O sites in\nlayerSwith the distance of 1.961(1.968) ˚A. This is in agree-\nment with previous study in Ref. 23. We used 12 ˚A of vac-\nuum space in the zdirection in order to avoid the unphysical\ninteractions between two adjacent elementary unit cells. The\nmismach with SrTiO 3was found to be about \u00002.5 and 10.1\n% for Fe and Co, respectively. The Fe and Co slabs have\nbeen strained and relaxed to accomodate the lattice structure\nof the SrTiO 3substrate, respectively. As a result, one finds\nthat the distances beween SandS+1 of about 1.501 ˚A and\n1.378 ˚A which should be compared with the bulk values of\n1.425 ˚A and 1.765 ˚A for Fe and Co, respectively.\n1. Magnetic spin moment\nWe plot in Fig. 2 the local spin moments of a free Fe(Co)\nslab (blue circles) but for which the ionic positions are the\none obtained after relaxation in presence of the SrTiO 3(001)\nsubstrate. In this way we can evaluate the role of the relaxation\nFIG. 1: Atomic structure of bcc-Fe(001) and fcc-Co(001) slabs on\ntop of TiO 2-terminated (001) surface of SrTiO 3. The ferromagnetic\nslab is rotated by 45\u000ewith respect to substrate in order to better\nmatch with the SrTiO 3lattice. Note that each layer of ferromagnetic\nslabs is made of 2 atoms per supercell. Layers S+3, ...,S\u00002 are\nshown and the distances in the zdirection between different layers\nare also indicated.\non the free surface as compared to the interface. The local spin\nmoments of the full system Fe(Co) jSrTiO 3(001) are shown in\nred squares. For free slabs, the magnetic moment of S+1\nlayer are enhanced up to 3.07 and 1.97 \u0016Bwith respect to\ntheir bulk values of 2.15 and 1.79 \u0016BinS+5 layer for Fe and\nCo, respectively. However, in the case of Fe(Co) jSrTiO 3, the\nsurface spin moment is reduced to 2.61 and 1.74 \u0016B(it is even\nsmaller than its bulk value) due to bonding and charge transfer\nat the interface. In addition, the hybridization between Fe 3 d\nand states of TiO 2at the interface induces spin moments on\nTi and O atoms. It has been found that the induced magnetic\nmoment of the interface O atom in Slayer is\u00180.05 (0.06)\n\u0016Band is parallel to the magnetic moment of Fe(Co). A much\nlarger induced but opposite spin moment in Slayer has been\nfound on Ti atoms : \u0018\u00000.27 (\u00000.29)\u0016B.\n0 10 20 30-101234\n0 10 20 30-101234(a) (b)\nFe Co\nAtomic site Atomic siteSpin moment ( µΒ)\nSrTiO3SrTiO3\nTi Ti\nFIG. 2: Layer-resolved magnetic spin moment (in \u0016B) at\nFejSrTiO 3(001) (a) and CojSrTiO 3(001) (b) interfaces. Blue circles\nand red squares correspond to free slab and slab on SrTiO 3substrate,\nrespectively.3\n2. Electronic properties\nIn order to explain the origin of the induced magnetic mo-\nments at the interface, we investigated the electronic structure\n(PDOS) of the free Fe(Co) slab as well as the Fe(Co) jSrTiO 3\ninterface compared to the corresponding PDOS in bulk phase\nof bcc-Fe (fcc-Co) and SrTiO 3.\nAs shown in Fig. 3 (a), the DOS of the interfacial Fe(Co) 3 d\n(S+1) (red line) for free slab differs from the DOS of the bulk\nFe(Co) 3d(S+5) (black line) as a result of the reduced coor-\ndination. A significant minority spin states at \u00180.1 and 0.7\neV (\u00000.4 and 0.2 eV) with respect to the Fermi level has been\nfound for Fe(Co) at the interface.These states are the origin of\nthe increase of spin moment for the surface atom as shown in\nFig. 2.\nPDOS (a.u.) PDOS (a.u.)Co 3 d (S+1)\nCo|SrTiO3up\ndown\nTi 3d (S)\n<dzx, dzy>Free Co slab\nCo 3 d (S+1)\nCo|SrTiO3\n-6 -4 -2 0 2 4PDOS (a.u.)\nE - EF (eV)O 2p (S)Co|SrTiO3pz\n-6 -4 -2 0 2 4PDOS (a.u.)\nE - EF (eV)O 2p (S)(d)Fe|SrTiO3pzPDOS (a.u.) PDOS (a.u.) PDOS (a.u.)Fe 3d (S+1)\nFe|SrTiO3up\ndown(a)\n(b)\n(c)Fe|SrTiO3\n<dzx, dzy>Free Fe slab\nFe 3d (S+1)\nTi 3d (S)Fe Co\nFIG. 3: (a) Free Fe slab: Scalar-relativistic projected density of\nstates (PDOS) of the surface Fe 3 dorbitals in layer S+1; b)\nFejSrTiO 3(001): PDOS of Fe 3 dorbitals in layer S+1, (c) Ti 3d\nand (d) O 2porbitals in layer S. The DOS of atoms in the central\nmonolayer of Fe slab (a, b) or (c, d) TiO 2in layer S\u00002 are plot-\nted as black lines. Positive and negative PDOS are for spin up and\nspin down channels, respectively. The vertical dashed lines indicate\nthe Fermi level ( EF). It is the same for Co as presented in the right\npanels.\nFig. 3 (b) - (d) show the PDOS of Fe(Co) 3 d(S+1), Ti\n3d(S) and O 2p(S) orbitals at Fe(Co) jSrTiO 3interface, indi-\ncating the presence of hybridizations between the orbitals. It\nis well known that the degree of hybridization at the interface\ndepends on the strength of the orbital overlap and inversely onthe energy seperation between them. Although there is a direct\natomic bonding between the interfacial Fe(Co) and O atoms,\nthe induced magnetic moment on the O atom was found to be\nrelatively small (\u00180.05\u0016B). This is due to the fact that O\n2p(S) orbitals lie well below the Fermi level and, therefore,\nhave a small overlap with the Fe(Co) 3 dstates. However, the\nTi 3dorbitals that are centered at about 2 eV above the Fermi\nlevel [the black lines in Fig. 3 (c)] have a strong hybridization\nwith the minority-spin Fe(Co) 3 dorbitals which have a signif-\nicant weight at these energies [the black lines in Fig. 3 (b)].\nThe most important consequence of this hybridization is the\nformation of the hybridized states in the interval of energies\n[\u00000.5,+0.5 ] eV and [\u00001,+1 ] eV for Fe and Co, respec-\ntively. As shown in Fig. 3 (c), the DOS of the Ti 3 dSlayer\nat the Fe(Co)jSrTiO 3interface, the minority-spin states which\noriginates from the dzxanddzyorbitals at\u0018\u00000.5 eV (the two\npeaks at\u00001 eV and\u00000.5 eV) are occupied, whereas the cor-\nresponding majority-spin states are found at \u0018+1.5 eV (the\ntwo peaks at +0.5 eV and +1 eV) are unoccupied. This leads\nto an induced magnetic moment of \u00000.27 and\u00000.29\u0016Bon\nthe Ti ( S) for Fe and Co based interfaces, respectively.\n3. Local analysis of MCA\nWe now investigate the MCA of the Fe(Co) jSrTiO 3inter-\nface. The MCA is calculated as band energy difference be-\ntween the spin quantization axes perpendicular and parallel\nto the slab surface, explicitely, MCA =Eband\n?\u0000Eband\nk, and\nfor the sake of simplicity we have chosen the most symmetric\nin plane orientation. By definition a positive (negative) sign\nin MCA means in-plane (out-of-plane) magnetization axis. It\nshould be noted that, the full relativistic Hamiltonian includ-\ning spin-orbit coupling is given in a basis of total angular mo-\nmentum eigenstates jj;mj>withj=l\u00061\n2. Although the\n(l; ml; ms) is not a well defined quantum number for the\nfull relativistic calculations, the MCA can still be projected\ninto different orbital and spin by using local density of states.\nSince the spin-orbit coupling in 3 d-electron systems is rel-\natively small, this approximate decomposition introduces a\nnegligible numerical inaccuracy.\nAs shown in Fig. 4 (a) and (b), we have calculated the atom-\nresolved MCA of the Fe(Co) jSrTiO 3system (red squares) and\ncompared it with the free Fe(Co) slab (blue circles) containing\n10 atomic layers (but relaxed in presence of the substrate).\nFor free Fe(Co) slab, the total MCA reaches \u0018\u00000.49 (1.60)\nmeV per unit-cell favouring an out-of-plane (in-plane) axis of\nmagnetization. If the Fe(Co) slab is in contact with SrTiO 3\nsubstrate, the axis of magnetization is preserved but the total\nMCA is reduced to \u0018\u00000.38 (1.02) meV .\nFrom the atom-resolved MCA, one finds that the MCA\ncurves for free slabs are not symmetrical, particularly\npronouced for Co, due to (asymmetrical) relaxation effect.\nThe main contribution to MCA is located in the vicinity of the\ninterface, from Slayer to S+3 layer, marked as vertical dotted\nline in Fig. 4 (a) and (b), and it converges to the expected bulk\nvalue in the center of the slab ( S+5 layer). Interestingly, at\nthe interface, in comparison with free Fe(Co) slab it appears4\nthat the contact with SrTiO 3strongly favors in-plane and out-\nof-plane for Fe and Co, respectively.\nFor Fe( S+1), upon adsorption on SrTiO 3, the MCA de-\ncreases from\u0018\u0000 0.15 to\u0018\u0000 0.06 meV/atom and the out-\nof-plane magnetization remains. However, in the case of\nCo(S+1), the MCA abruply changes from \u00180.22 to\u0018\u00000.25\nmeV/atom exhibiting magnetization reversal from in-plane to\nout-of-plane at the same time. For S+2 layer, we find a sign\nchange of MCA between free slab and slab on SrTiO 3for\nboth elements, with the MCA difference of \u00180.04 meV/atom\nand\u00180.15 meV/atom for Fe and Co, respectively. For S+3\nlayer, the MCA enhances slightly ( \u00180.05 meV/atom) in-plane\nMCA when depositing slabs on SrTiO 3for both elements.\nFurthermore, the Ti atom in Slayer [indicated by arrows in\nFig. 4 (a) and (b)] presents a rather large in-plane MCA of \u0018\n0.1 meV/atom and a much smaller in-plane MCA of \u00180.03\nmeV/atom for Fe and Co-based interfaces, respectively. As a\nresult, for free slabs, the MCA values from S+1 layer to S+3\nlayer sum up to the total value of \u0018\u00000.22 meV (out-of-plane)\nand 0.45 meV (in-plane) for Fe and Co. However, when the\nslabs are supported on SrTiO 3, the overall out-of-plane MCA\nin the vicinity of the surface (here, the Slayer is also taken\ninto account) is almost quenched for Fe by \u00180 meV , and in\nthe case of Co, a spin transition from in-plane to out-of-plane\nmagnetization has been found with a MCA value of \u0018\u00000.10\nmeV .\nIn order to understand the origin of this difference in MCA\nbetween free Fe(Co) slab and Fe(Co) jSrTiO 3system, we in-\nvestigated the d-orbitals-resolved MCA of the Fe(Co) atom as\nshown in Fig. 4 (c) and (d). Here, due to symmetry, the con-\ntributions to MCA from ( dzx;dzy) and (dx2\u0000y2;dxy) pairs are\nalmost equal, therefore, their averaged values are presented\nfor simplicity.\nIn the case of Fe, we notice that going from the free Fe slab\nto the FejSrTiO 3system, the MCA of the dz2(in-plane mag-\nnetization) and ( dx2\u0000y2;dxy) (out-of-plane magnetization) or-\nbitals decreases in magnitude, while the MCA of ( dzx;dzy)\norbitals is almost not affected. In addition, quantitatively, the\nreduction of MCA is larger for ( dx2\u0000y2;dxy) than fordz2due\nto stronger hybridization between (Fe- dx2\u0000y2;xy, Ti-dzx;zy )\norbitals than between (Fe- dz2, O-pz) orbitals. This is at-\ntributed to the fact that, shown in Fig. 3, close to the Fermi\nlevel, the shape of the electron density for O and Ti suggest\nthat this density has a pzcharacter and dzx(dzy) character, re-\nspectively. Moreover, the strong in-plane MCA in Ti ( S) layer\noriginates from the Ti- dzx;zy orbitals since there is a signifi-\ncant weight close to Fermi level of minority-spin (Ti- dzx;zy )\norbitals [see Fig. 3 (c) left panel]. As a result, the MCA at\nthe interface appears to almost quench the out-of-plane mag-\nnetization when the Fe slab is deposited on SrTiO 3. More-\nover, if we sum over the contribution of the first three layers\nof Fe slab at the interface, we found that dzx;dzyorbitals tend\nto maintain the out-of-plane MCA while dx2\u0000y2;xyorbitals\ntend to favor the in-plane MCA. A similar result has also been\nreported in Ref.24in FejMgO magnetic tunnel junctions.\nIn the case of Co, we find that the hybridization between\npzorbitals of O and dz2(and, to a slightly lesser extent with\ndzx;zy ) of Co plays a crucial role to decrease in-plane MCAof the free Co slab. On the other hand, the MCA from in-\nplane (dx2\u0000y2;xy) orbitals of Co is less affected due to rather\nsmall minority-spin states of (Ti- dzx;zy ) close to the Fermi\nlevel [see Fig. 3 (c) right panel]. This leads to induce an in-\nverse spin orientation transition from in-plane to out-of-plane\nin CojSrTiO 3system. A similar result has also been reported\nin Ref.25at C 60jCo interface.\nB. Fe and Co clusters on SrTiO 3\nWe now investigate the electronic and magnetic properties\nof Fe and Co clusters deposited on SrTiO 3surface. As shown\nin Fig. 5, two geometries are examined, namely top (a) and\nhollow (b) adsorption sites. The base atoms of Fe(Co) clusters\nare always on top of O atom for both geometries however the\napex atom is either on top of a Ti atom (top geometry) or of\nan underneath Sr atom (hollow geometry). We found that a\nhollow adsorption site is more energetically stable for both el-\nements, with an energy difference of \u00180.65 eV and\u00180.88 eV\nfor Fe and Co, respectively. In the following, we concentrate\non the lowest energy configuration.\nThe strength of the cluster-SrTiO 3interaction can be quan-\ntified by calculating the binding energy via the energy differ-\nence:\nEb=E[cluster ] +E[SrTiO 3]\u0000E[clusterjSrTiO 3](2)\nwhere E[cluster], E[SrTiO 3] and E[clusterjSrTiO 3] are the to-\ntal energy of the free cluster, the free SrTiO 3substrate and the\ncluster-SrTiO 3system, respectively. The calculated binding\nenergy was found to be \u00184.23 (4.58) eV for Fe(Co) cluster\non SrTiO 3substrate, showing strong chemisorption mecha-\nnism (see Tab. I).\nFe Co\nFree cluster Cluster on SrTiO 3Free cluster Cluster on SrTiO 3\nEb(eV) — 4.23 — 4.58\nd1(˚A) 2.31 2.55 2.17 2.20\nd2(˚A) 1.73 1.45 1.80 1.74\nMtot\ns(\u0016B) 18.00 16.63 13.00 7.67\njMtot\nsj(\u0016B) 18.34 17.96 13.41 11.06\nMbase\ns(\u0016B) 3.62 3.33 2.54 1.75\nMtop\ns(\u0016B) 3.58 3.32 2.84 1.57\nTABLE I: Binding energies ( Eb), atomic bonds, total/total absolute\nspin moments ( Mtot\ns/jMtot\nsj), spin moment of base ( Mbase\ns) and top\n(Mtop\ns) atoms of the free clusters and clusters deposited on SrTiO 3\nfor the lowest energy configuration.\nCompared to free Fe cluster, the Fe-Fe distance in basal\nplane (d1) is elongated from 2.31 ˚A to 2.55 ˚A while the Fe-Fe\ndistance in vertical distance from apex to basal plane ( d2) is\ncompressed from 1.73 ˚A to 1.45 ˚A (see Tab. I). However,\nin the case of Co, the geometry optimization of Co 5jSrTiO 3\nresults in a rather small (negligible) distortion compared to its\nfree Co 5cluster. In addition, the bond length between Fe(Co)\nand O is\u00182˚A.5\n5 10 15 20 25 30-0.3-0.2-0.100.10.20.30.4\n5 10 15 20 25 30-0.3-0.2-0.100.10.20.30.4\nAtomic site Atomic siteE⊥ − Ε||(meV)(a) (b)\nin-plane\nout-of-planeTiTi\n5 10 15 20 25 30-0.1-0.0500.050.1\n5 10 15 20 25 30-0.2-0.100.10.20.30.4\nAtomic site Atomic siteE⊥ − Ε||(meV)(c) (d)\ndz2\n<dzx, dzy>\n<dx2-y2, dxy>Fe Co\nFIG. 4: Atom-resolved MCA at Fe jSrTiO 3(a) and CojSrTiO 3(b) interfaces, blue circles and red squares correspond to free slab and slab on a\nSrTiO 3substrate, respectively. d-orbitals-resolved MCA for Fe (c) and Co (d) slabs on SrTiO 3, we plot only the part of ferromagnetic slabs.\nDue to symmetry, contributions from different orbitals in ( dzx;dzy) and (dx2\u0000y2;dxy) pairs are very similar so that their averaged values are\npresented for simplicity. Note that positive and negative MCA represent in-plane and out-of-plane magnetization, respectively.\n1. Magnetic spin moment\nWe next investigated the local magnetic spin moment. In\nTab. I, the local spin moments for both free clusters and the\nclusters on SrTiO 3are given. The binding between Fe(Co)\nand O atoms reduces the total spin moment from 18.00 \u0016B\n(free Fe 5) to 16.63\u0016Band from 13.00 \u0016B(free Co 5) to 7.67\n\u0016Bfor the deposited clusters. We also calculated the abso-\nlute total spin moment jMtot\nsjand compared to corresponding\ntotal spin moment Mtot\ns. Interestingly, a substantial differ-\nence of\u00183.4\u0016Bhas been found between jMtot\nsjandMtot\nsfor\nCo5jSrTiO 3. In order to understand the origin of this differ-\nence, we plot in Fig. 6 the real-space distribution of magnetic\nspin moment of Co cluster on SrTiO 3. Note that the red (blue)\ncorresponds to positive (negative) spin moment. We can see\nclearly the negative magnetic moment is mainly localized on\nTi atoms at the interface and around the Co top atom of clus-\nter. However, for Fe cluster, the positive spin moment is very\nlocalized on the Fe atoms and the negative part is negligible.2. Electronic structure properties\nTo gain more insight into the electronic structure of\nFe5jSrTiO 3and Co 5jSrTiO 3, we plot the scalar-relativistic\nprojected density of states (PDOS) on d-orbitals of Fe(Co)\nbase atom and top atom of the cluster in Fig. 7 (a) and (b).\nFor the base atom of both clusters, the density of majority\nstates is almost completely occupied (situated below \u00000:6eV)\nand negligibly small around the Fermi level, while the den-\nsity of minority states is partially occupied. Around the Fermi\nlevel, there is a higher density of ( dx2\u0000y2;dxy,dzy) states for\nFe while the most dominant states are the out-of-plane dor-\nbitals for Co, namely ( dz2;dzx;dzy) orbitals. For top atom,\nin the interval of energies [ \u00000.25, +0.25] eV , the density of\nstates for both majority and minority spins is negligibly small\nfor both clusters.\n3. Local analysis of MCA\nThe MCA is calculated by the formula MCA =Eband\nz\u0000\nEband\nx0using as usual the magnetic force theorem. The MCA\nin thexyplane is found to be extremely small. we have chosen6\nFIG. 5: Top (upper panels) and side (lower panels) views of the opti-\nmized geometries of Fe and Co cluster absorbed on TiO 2-terminated\nSrTiO 3(001). Two different adsorption configurations are presented\nin (a) and (b), the latter one is the most stable configuration for both\nFe and Co clusters. The bond length d1between base atoms and the\nvertical distance d2between base and top atoms are indicated.\nFIG. 6: Real-space distribution of magnetic spin moment of Fe (left)\nand Co (right) cluster on SrTiO 3. Note that red (blue) corresponds to\npositive (negative) spin moment. The nonnegligible negative part of\nspin moment has been found around the Ti atoms at the interface and\nthe Co top atom of cluster.\nthe most symmetric in-plane direction x0(see Fig. 8) which\nhas an azimuthal angle of \u001e= 45\u000ewith respect to x. Due\nto symmetry, this definition gives us almost similar contribu-\ntion for each pair of ( dzx;dzy) and of (dx2\u0000y2;dxy) Fe(Co)\norbitals, therefore, their averaged values are presented for the\nsake of simplicity.\nIn Fig.8 (a) and (b) the local decomposition of MCA with\ndifferent atomic sites as well as with different d-orbitals is\npresented for Fe 5jSrTiO 3and Co 5jSrTiO 3, respectively. Note\nthat only the contributions of clusters is shown. Interestingly,\nwe find the opposite behavior of MCA for Fe and Co clus-\nters deposited on SrTiO 3. The easy axis of magnetization is\ndirected along out-of-plane for Fe cluster with a total MCA\nof\u0018\u0000 5.08 meV , on the contrary it is in-plane for Co with\na total MCA of\u00184.72 meV . For both elements, the atom-\nically resolved MCA (black lines) reveals that the MCA is\nmainly dominated by the base atoms (numbered as 1 \u00184)\n-0.6 -0.4 -0.2 0 0.2 0.4 0.6PDOS (a.u.) PDOS (a.u.)base atom\ntop atomup\ndown\nE - EF (eV)-0.6 -0.4 -0.2 0 0.2 0.4 0.6base atom\ntop atomup\ndown\nE - EF (eV)Fe Co\ndz2\ndzx\ndzy\ndx2 - y2\ndxyFIG. 7: Scalar-relativistic d-orbitals projected density of states\n(PDOS) for Fe(Co) base atom (a) and top atom (b) of the cluster\nabsorbed on SrTiO 3. Positive and negative PDOS are for spin up and\nspin down channels, respectively. The vertical dashed lines mark the\nFermi level ( EF)\nand a relatively much smaller contribution from the top atom\n(numbered as 5). The value of MCA per atom is as large as\n\u0018\u00001.22 (1.08) meV/atom for base atom and \u0018\u00000.18 (0.38)\nmeV/atom for the top atom of Fe(Co) cluster.\nIt is also interesting to note that the MCA mainly origi-\nnates from the d-orbitals of the cluster extending in-plane for\nFe, namely ( dx2\u0000y2;dxy) orbitals, and out-of-plane for Co,\nnamely, (dz2;dzx;dzy).\nFinally in Fig. 8 (c) and (d), we present the real-space\ndistribution of MCA for Fe 5jSrTiO 3and Co 5jSrTiO 3. The\nred colors represent in-plane magnetization direction, whereas\nthe blue colors are out-of-plane easy axis. We can clearly se\nthat the MCA mainly originates from the base atoms for both\nclusters, and for Fe(Co) the MCA originates from d-orbitals\nof the cluster extending in-plane (out-of-plane). In addition,\ndue to hybridization between the states of TiO 2surface and\nd-orbitals of the cluster, the Ti and O atoms close to the clus-\nter gives a rather small contribution to MCA. For Fe, Ti atom\nslightly favors the in-plane easy axis and the easy axis of O\natom is out-of-plane. In the case of Co, both Ti and O atoms\naround the cluster favor to in-plane magnetization direction.\nAs a consequence, we predict that the Fe 5nanocrystals\nshould be magnetically stable and are thus good potential can-\ndidates for magnetic storage devices.7\nTotal\ndz2\n<dzx, dzy>\n<dx2-y2, dxy>\n5 1 2 3 4 5-0.500.511.5\nAtomic siteCo\n1 2 3 4 5-1.5-1-0.500.5\n-0.5\nAtomic siteEz - Ex’ (meV)Fe\n24\n1 35\n(a) (b)\n(c) (d)\nFIG. 8: Atom/ d-orbitals-resolved MCA of Fe (a) and Co (b) clusters deposited on SrTiO 3. Due to symmetry, contributions from different\norbitals in (dzx;dzy) and (dx2\u0000y2;dxy) pairs are very similar so that their averaged values are presented for simplicity. Clear out-of-plane and\nin-plane MCA have been found for Fe and Co clusters, respectively. Real-space distribution of MCA for Fe (c) and Co (d) clusters is given.\nNote that red (blue) colors represent the regions favoring in-plane (out-of-plane) magnetization orientation. The MCA mainly from the base\natoms for both clusters, and for Fe (Co) the MCA originates from d-orbitals of the cluster extending in-plane (out-of-plane).\n4. MCA analysis from perturbation theory\nLet us consider the perturbation of the total energy due to\nthe spin-orbit coupling Hamiltonian HSO26–30. Since the first-\norder term vanishes the second order perturbation term \u0001E(2)\nof the total energy has to be evaluated:\n\u0001E(2)=\u0000X\nn\u001bocc\nn0\u001b0unoccjhn\u001bjHSOjn0\u001b0ij2\nEn0\u001b0\u0000En\u001b(3)\nwherejn\u001bi(jn0\u001b0i) is an uperturbed occupied (unoccupied)\nstate of energy En\u001b(En0\u001b0) ,ndenotes the index of the state\nand\u001bits spin (which is still a good quantum number for the\nunperturbed state). Writing the eigenstates in an orthogonal\nbasis of real atomic spin orbitals \u0015\u001bcentered at each atomic\nsitei, one can derive a rather cumbersome equation written\nexplicitely in the Appendix of Ref. 28 (Eq. C.8). However it\nis possible to drastically simplifiy Eq. C.8 by retaining onlythe diagonal terms of the density matrix which leads to the\nfollowing expression:\n\u0001E(2)=A\u0000\u00182X\n\u0015\u0016jh\u0015\"jHSOj\u0016\"ij2X\ni\u001b\u001b0\u001b\u001b0Ii(\u0015;\u0016;\u001b;\u001b0)\n(4)\nwhereAis a constant isotropic term and\nIi(\u0015;\u0016;\u001b;\u001b0) =ZEF\n\u00001dEZ1\nEFdE0ni\u0015\u001b(E)ni\u0016\u001b0(E0)\nE0\u0000E(5)\nni\u0015\u001b(E)(ni\u0016\u001b0(E0)) being the projected density of states\nof occupied (unoccupied) states. The dominant terms\nIi(\u0015;\u0016;\u001b;\u001b0)are the ones corresponding to a transition be-\ntween an occupied and an unoccupied state presenting a high\ndensity of states below and above the Fermi level respectively.\nThe MCA defined as the difference of energy between the\ndirectionzandxcan be decomposed in local atomic contri-8\nbutions MCA i:\nMCAi=\u00182X\n\u0015\u001b\n\u0016\u001b 0\u001b\u001b0T\u0015;\u0016Ii(\u0015;\u0016;\u001b;\u001b0) (6)\nT\u0015;\u0016is the difference of the square of the spin-orbit matrix\nelements between two orientations of the magnetization M:\nT\u0015;\u0016=jh\u0015\"jL:Sj\u0016\"ij2\nMkx\u0000jh\u0015\"jL:Sj\u0016\"ij2\nMkz(7)\nSinceIi(\u0015;\u0016;\u001b;\u001b0)is always positive, the sign of the ma-\ntrix elements \u001b\u001b0T\u0015;\u0016for a given transition between an occu-\npied state\u0015\u001band an unocuppied state \u0016\u001b0will define the sign\nof the corresponding anisotropy. In practice there are a limited\nnumber of transitions and in addition spin-flip transitions are\noften negligible, therefore in most case \u001b\u001b0= 1.\nLet us now apply this perturbation expansion to the case of\nIron and Cobalt clusters on SrTiO 3. First, it is clear from the\nPDOS analysis that the top atom will contribute negligibly\nto the total MCA. In contrast for both atoms the PDOS of\nthe base atom shows that there are dominantly four occupied-\nunoccupied transitions that will dominate the MCA. Namely\nthe transition dx2\u0000y2!dzx,dx2\u0000y2!dxy,dzy!dzx, and\ndzy!dxyfor Fe anddzx!dzx,dzx!dzy,dz2!dzx,\nanddz2!dzyfor Co. From Eq. A3 it comes out that for\nFe two transitions are large with a negative sign ( dx2\u0000y2!\ndxy/ \u00004,dzy!dzx/ \u00001) and two are small with a\npositive sign ( dx2\u0000y2!dzx/1=2,dzy!dxy/1=2).\nFor Co we find the opposite trend: two transition have a large\nand positive sign ( dz2!dzx/3=2,dz2!dzy/3=2),\none has a negative sign ( dzx!dzy/\u00001) and the last one\nis diagonal and do not contribute ( dzx!dzx= 0). Overall\nthis shows that Fe pyramid favors out-of-plane magnetization\nwhile Co favors in plane magnetization. The main orbitals\ninvolved are (dxy;dx2\u0000y2)for Fe and (dzx;dzy;dz2)for Co\nin agreement with the results presented in Sec. III B 3\nThis type of analysis remains qualitative and applies pref-\nerentially to low dimensional systems presenting sharp fea-\ntures in their PDOS. Nevertheless, the arguments put forward\nare rather general and could be be very useful in the design\nof atomic-scale devices with optimized magnetic anisotropy.\nNote however, that if the nonsphericity of the Coulomb and\nexchange interaction31starts to play a dominant role in the\nelectronic structure of the system, then orbital polarization ef-\nfects arise32,33and our analysis of the MCA based on a per-\ntubation treatment of the SOC only non longer applies, and\nmore complex scenarii can occur as in the case of the giant\nmagnetic anisotropy of single adatoms on MgO34.\nIV . Conclusion\nWe investigated the electronic properties and MCA of Fe\nand Co slabs and nanoclusters interfaced with SrTiO 3under-\nlayer. Interestingly, a comparative study of Fe and Co free-\nstanding slabs with their interface with SrTiO 3, revealed atremendous impact of the latter on the MCA. Namely, the\nMCA contribution from the interfacial Fe layer in Fe jSrTiO 3\nis quenched resulting in the loss of the perpendicular mag-\nnetic anisotropy (PMA) while for Co jSrTiO 3, the anisotropy\nis changed from in-plane to the out-of-plane. This is explained\nby the orbital resolved analysis of hybridizations of Fe and Co\nd-orbitals of with those of Ti and pzorbital of O.\nWe also find a strong enhancement of out-of-plane and in-\nplane MCA for small Fe and Co clusters (containing only\nseveral atoms) upon deposition on a SrTiO 3substrate. The\nhybridization between the substrate and the d-orbitals of the\ncluster extending in-plane for Fe and out-of-plane for Co is at\nthe origin of this enhancement of MCA. As a consequence,\nwe predict that the Fe nanocrystals (even rather small) should\nbe magnetically stable and are thus good potential candidates\nfor magnetic storage applications.\nV . Acknowledgement\nThe research leading to these results has received funding\nfrom the European Research Council under the Euro-\npean Unions Seventh Framework Programme (FP7/2007-\n2013)/ERC grant agreement No. 259297. This work was\nperformed using HPC computation resources from GENCI-\n[TGCC] (Grant No. 2015097416).\nA. Expression of the Tmatrix\nThe matrix elements of the spin-orbit coupling Hamiltonian\nin thedorbital basis (ordered as dxy;dzy;dzx;dx2\u0000y2;dz2)\nare written explictely in Apppendix A of Ref. 28 for an arbi-\ntrary orientation of the magnetization defined by the altitude\nangle and the azimuth angle (\u0012;\u001e). If we define the MCA as\nthe total energy difference between a magnetization along z\n(\u0012=\u001e= 0) and a magnetization along an arbitrary direction\nn(\u0012;\u001e)the corresponding Tmatrix reads:\n1\n42\n666640 sin2\u0012sin2\u001esin2\u0012cos2\u001e\u00004 sin2\u0012 0\nsin2\u0012sin2\u001e 0\u0000sin2\u0012 sin2\u0012cos2\u001e3 sin2\u0012cos2\u001e\nsin2\u0012cos2\u001e\u0000sin2\u0012 0 sin2\u0012sin2\u001e3 sin2\u0012sin2\u001e\n\u00004 sin2\u0012 sin2\u0012cos2\u001esin2\u0012sin2\u001e 0 0\n0 3 sin2\u0012cos2\u001e3 sin2\u0012sin2\u001e 0 03\n77775\n(A1)\nIfnis alongx(\u0012=\u0019=2;\u001e= 0),Ttakes the form:\n1\n42\n66640 0 1\u00004 0\n0 0\u00001 1 3\n1\u00001 0 0 0\n\u00004 1 0 0 0\n0 3 0 0 03\n7775(A2)\nWhile if instead of xwe take the more symmetric in plane9\nx0direction ( (\u0012=\u0019=2;\u001e=\u0019=4) we find for T:\n1\n42\n66640 1=2 1=2\u00004 0\n1=2 0\u00001 1=2 3=2\n1=2\u00001 0 1=2 3=2\n\u00004 1=2 1=2 0 0\n0 3=2 3=2 0 03\n7775(A3)A positive sign means an easy axis along nand a negative\nsign an easy axis along z.\n\u0003Electronic address: alexander.smogunov@cea.fr\n1L. 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Wu, Physical Review Let-\nters115, 257201 (2015)."    },    {        "title": "1602.08377v1.Tuning_the_magnetocrystalline_anisotropy_in__R_CoPO_by_means_of__R__substitution__a_ferromagnetic_resonance_study.pdf",        "content": "Tuning the magnetocrystalline anisotropy in RCoPO by means of Rsubstitution: a\nferromagnetic resonance study.\nG. Prando,1, 2,\u0003A. Alfonsov,2A. Pal,3, 4V. P. S. Awana,3B. B uchner,1, 2and V. Kataev2\n1Center for Transport and Devices of Emergent Materials,\nTechnische Universit at Dresden, D-01062 Dresden, Germany\n2Leibniz-Institut f ur Festk orper- und Werksto\u000bforschung (IFW) Dresden, D-01171 Dresden, Germany\n3National Physical Laboratory (CSIR), New Delhi 110012, India\n4Department of Physics, Indian Institute of Science, Bangalore 560012, India\n(Dated: June 19, 2018)\nWe report on broad-band electron spin resonance measurements performed within the itinerant\nferromagnetic phase of RCoPO (R= La, Pr, Nd and Sm). We reveal that the Rsubstitution is\nhighly e\u000bective in gradually introducing a sizeable easy-plane magnetocrystalline anisotropy within\nthe Co sublattice. We explain our results in terms of a subtle interplay of structural e\u000bects and of\nindirect interactions between the fanddorbitals from Rand Co, respectively.\nPACS numbers: 75.50.Cc, 76.30.-v, 76.50.+g\nI. INTRODUCTION\nThe interest for RMX O oxides (R,MandXbeing\nrare-earth, transition metal and pnictide ions, respec-\ntively) has arisen dramatically after the recent discov-\nery of high- TcSC (superconductivity) in this class of\nlayered materials.1{3The prototype RFeAsO 1\u0000xFxsys-\ntems reach remarkable Tcvalues higher than 50 K,3{5\nwhile lower Tc's are typically achieved for the di\u000berent\ncompositions RFe1\u0000xCoxAsO.5{12Both the O 1\u0000xFxand\nthe Fe 1\u0000xCoxdilutions nominally introduce one extra-\nelectron per substituted ion (however, see Refs. 13 and\n14) leading to SC for values x\u00180:05\u00000:1.5Still, it is\nvery interesting to consider how the electronic ground\nstate evolves in the opposite limit x!1, where SC\nis completely suppressed. An itinerant FM (ferromag-\nnetic) phase is achieved in RCoAsO and RCoPO, with\nan ordered magnetic moment per Co ion in saturation\nstrongly suppressed if compared to its value in the para-\nmagnetic regime.15While itinerant ferromagnetism can\nbe predicted for these materials by means of ab-initio\ncomputations,15,16a detailed investigation of their prop-\nerties can possibly lead to interesting insights also in the\nsuperconducting state in view of the closeness of these\ntwo ground states in the phase diagram.\nIn particular, this is the case for the impact of di\u000berent\nRions on the whole electronic properties of the systems.\nPreviously,17,18we showed by means of \u0016+SR (muon spin\nspectroscopy) that the felectronic degrees of freedom as-\nsociated with Rions do not play an active role in RCoPO\nbut, on the contrary, Rions should be thought as \\pas-\nsive\" sources of chemical pressure which ultimately tune\nTC, i. e., the FM transition temperature.17,18As further\ncon\frmation of the crucial importance of structural ef-\nfects, we also demonstrated the full equivalence of chemi-\ncal and external pressures on a quantitative level as long\nasTCis considered.17,18These results can be interest-\ning in view of the analogy with superconducting samples\nand, in particular, with the strong dependence of the Tcvalue on the actual Rion at optimal doping.4,19It should\nbe stressed that we could not demonstrate a full analogy\nbetween chemical and external pressures for supercon-\nducting samples, as here quenched disorder contributes\nin a complicated and non-negligible way as well.20\nIn this paper we report on ESR (electron spin res-\nonance) measurements performed in the FM phase of\nRCoPO (R= La, Pr, Nd and Sm). We analyse the ESR\nsignal in a wide range of temperature ( T), magnetic \feld\n(H) and frequency of the employed microwave electro-\nmagnetic radiation ( \u0017). We observe for all the samples a\nclear crossover from a high- Tparamagnetic region, where\nthe ESR line shows a Dysonian distortion, to a low- Tre-\ngion, where the ESR line arises instead from the macro-\nscopic magnetization of the whole sample (FMR, ferro-\nmagnetic resonance). Remarkably, within the FM phase,\nwe unambiguously detect the gradual development of a\nsizeable easy-plane magnetocrystalline anisotropy upon\nincreasing chemical pressure. We discuss our experimen-\ntal results in the light both of the distortion of the local\ntetrahedral crystalline surroundings of Co ions and of the\nanisotropic properties introduced by the strong indirect\ninteraction between fanddelectronic degrees of freedom\nfromRand Co orbitals, respectively.\nII. EXPERIMENTAL DETAILS\nA. Samples' characterization\nWe reported details about the synthesis of polycrys-\ntallineRCoPO (R= La, Pr, Nd, Sm) in our previous\npublications, together with thorough investigations of the\nconsidered samples by means of dc magnetometry and ZF\n(zero-\feld) \u0016+SR under pressure.17,18In this paper we\ndiscuss ESR measurements performed on the same sam-\nples already studied by means of the other techniques\nmentioned above. In the whole text, we refer to ground\npowders composed to a \frst approximation of sphericalarXiv:1602.08377v1  [cond-mat.str-el]  26 Feb 20162\ngrains with similar dimensions for all the samples. The\npowders were embedded in Double Bubble 2-part epoxy\n(Loctite) for the aim of avoiding sample movement and\ngrain re-orientation triggered by H, i. e., preserving con-\nstant powder-average properties of the ESR signal for all\nthe accessed experimental conditions.\nWe measured M(macroscopic magnetization) for the\nfour samples as a function of Tat \fxed sample-dependent\nvalues ofHby means of a Magnetic Property Measure-\nment System based on a superconducting quantum inter-\nference device (by Quantum Design).\nB. Electron spin resonance\nWe performed continuous wave ESR measurements at\n\fxed\u0017while sweeping H. For this aim, we employed two\ndi\u000berent experimental con\fgurations.\n1. X-band regime\nWe accessed the low-frequency regime ( \u0017'9:56 GHz)\nby means of a commercial Bruker EMX X-band spec-\ntrometer equipped with an Oxford Instruments ESR900\ncontinuous4He \row cryostat ( T= 4:2\u0000300 K). Mea-\nsurements were always performed upon gradually warm-\ning the sample from the lowest accessed Tvalue after\na zero-\feld cooling protocol. Standing electromagnetic\nmicrowaves were induced in a rectangular cavity (Bruker\nX-band resonator ER4104OR, TE 102mode). The sample\nwas placed in the cavity's centre where the Hcomponent\nis maximum and we measured the P(power) resonantly\nFIG. 1: (Color online) Representative result for the \frst-\nderivative X-band ESR measurements (squares) together with\nthe \ftting curve (black continuous line) according to Eqs. (1),\n(2) and (3). First-derivative data are numerically-integrated\nto obtain the actual P(H) behaviour (red continuous line),\nwhich is enlarged in the inset showing the de\fned resonance\n\feldHrand the two half-height \feld values HlandHh.absorbed by it as a function of H(t) =H0+Ha(t). Here,\nthe quasi-static component H0was in the range 0 \u00009\nkOe and it was swept with a typical rate \u001850 Oe/s. Si-\nmultaneously, the t(time) dependent \feld Ha(jHaj\u001420\nOe) was sinusoidally-modulated with frequency 100 kHz\nand superimposed to H0. By means of a lock-in detec-\ntion at the modulation frequency, we directly measured\nthe \frst derivative d P/dHrather than P(see Fig. 1 for\na representative experimental curve).\nWe \ftted the d P/dHdata by means of the expression\nf(H) =pdfAbs\nL(H)\ndH+ (1\u0000p)dfDisp\nL(H)\ndH+rH+q;\n(1)\nwhere the coe\u000ecients randqallow for a small linear\nbackground while\nfAbs\nL(H) =AL\n\u0019\"\n\u0000L\n\u00002\nL+ (H\u0000HrL)2\n+\u0000L\n\u00002\nL+ (H+HrL)2#\n(2)\nand\nfDisp\nL(H) =AL\n\u0019\"\n(H\u0000HrL)\n\u00002\nL+ (H\u0000HrL)2\n+(H+HrL)\n\u00002\nL+ (H+HrL)2#\n(3)\nare the absorptive (Abs) and dispersive (Disp) compo-\nnents of the employed Lorentzian model (hence the sub-\nscript L) weighted by the parameter 0 \u0014p\u00141.21{24\nHere,ALis the signal amplitude and HrLthe resonance\n\feld, while for the linewidth the relation \u0000 L= \u0001H=2\nholds with \u0001 Hrepresenting the FWHM (full width at\nhalf maximum). Eqs. (2) and (3) already incorporate the\ncontribution from negative magnetic \felds arising from\nthe linear polarization of the electromagnetic radiation\nin the cavity.23,25This correction is mostly relevant for\nbroad ESR lines, namely whenever \u0000 L&HrL.\nThe choice of Eqs. (1), (2) and (3) gives excellent \ft-\nting results in LaCoPO at all Tvalues, except for a\nnarrow region around the onset of the long-range or-\ndered FM phase where the signal is slightly distorted.\nSimilar distortion e\u000bects around the ordering tempera-\ntures of magnetic phases have been reported before for\nother materials.26For highTvalues, \fts by Eqs. (1), (2)\nand (3) still yield to excellent results also in the case of\nPrCoPO, NdCoPO and SmCoPO (see in Sect. III). How-\never the situation for these materials is di\u000berent in the\nwhole low-TFM regime, where the signal is always so dis-\ntorted that it can not be \ftted properly. For this reason,\nwe took an alternative empirical approach to data analy-\nsis. In particular, we numerically-integrated the d P/dH\ndata to give the actual P(H) behaviour, from which we3\nextracted important quantities such as\nI(T) =Z+1\n0PT(H)dH; (4)\nnamely the integrated intensity of the ESR signal at \fxed\nT, and the characteristic \feld values Hr(resonance \feld),\nHlandHh(half-height \felds) de\fned as shown in the\ninset of Fig. 1. Accordingly, we de\fned the FWHM as\n\u0001H\u0011Hh\u0000Hland introduced the empirical parameter\n\u0011\u0011Hh\u0000Hr\nHr\u0000Hl(5)\nto quantify the half-width asymmetry of the ESR line. In\nparticular, \u0011= 1 corresponds to a symmetric line with\nrespect toHr, while\u0011 > 1 is found when experimental\nlines are broadened on the high-\felds side.\nAs is well-known, an asymmetry ( \u0011 > 1) of the ESR\nline may have di\u000berent physical origins. One possibility\nis the so-called Dysonian distortion typical of metallic\nsamples.27Here, the impinging electromagnetic radiation\nis mostly screened and it penetrates the material over\nthe skin-depth \u000es.28Accordingly, the resonance process\nonly takes place in the non-screened fraction of the sam-\nple, namely within \u000es. The resonance signal may arise\nboth from localized magnetic moments interspersed in\nthe metallic background and from conduction electrons\nthemselves. In the latter case, two main characteristic\ntimes govern the resonance process, i. e., the intrinsic\ntransverse relaxation time of electrons Tesand the so-\ncalled di\u000busion time TD=\u000e2\ns=D.27Here,\u000esis the length-\nscale of interest for the electron di\u000busion while Drepre-\nsents a constant characteristic of the process. When the\nelectron di\u000busion can be neglected, i. e., when\n1\nTes\u001dD\u000e\u00002\ns; (6)\nthe absorbed Pcan be conventionally expressed in terms\nof the sample impedance. In the ideal case of metallic\nspherical grains with diameter d, the condition \u000es\u001dd\nimpliesP\u0018\u001f00(bulk-impedance limit) while P\u0018\n(m\u001f0+n\u001f00) holds with m=nin the opposite limit\n\u000es\u001cd(surface-impedance limit), with \u001f0(\u001f00) the real\n(imaginary) component of the magnetic susceptibility.27\nWhen a conventional Lorentzian relaxation process is be-\ning considered, the former condition implies A=jBj'1\nfor the ratio of the two quantities de\fned in the main\npanel of Fig. 1, while the latter condition typically re-\nsults inA=jBj'2:55 and, accordingly, in a broadening\nofP(H) on the high-\felds side.27,29\nOn the other hand, anisotropic magnetic properties\ngenerally cause an inhomogeneous broadening of mag-\nnetic resonance lines for randomly-oriented powders as\nwell.30It should be recalled that only the anisotropy-\nbased distortion would still be detected in case the exper-\nimental apparatus allowed one to independently measure\n\u001f0and\u001f00. While this is not feasible with our X-band in-\nstrumentation, we could successfully disentangle the two\nsignals by means of a di\u000berent setup, as discussed below.2. High-frequency/high-\feld regime\nWe performed ESR measurements at higher \u0017andH0\nat selected Tvalues by means of a home-made spec-\ntrometer based on a PNA network analyser N5227A\n(Keysight Technologies), generating and detecting mi-\ncrowaves with broad-band tunable frequency \u0017= 10 MHz\n\u000067 GHz. We extended the upper \u0017limit to 330 GHz by\nmeans of complementary millimiter-wave modules (Vir-\nginia Diodes, Inc.). We also accessed the 20 GHz \u0000\n30 GHz regime by means of a home-made spectrometer\nbased on a MVNA vector network analyser (AB Millime-\ntre). We performed measurements at selected Tvalues in\na transmission-con\fguration31by exploiting gold-plated\ncopper mirrors, German silver waveguides and brass con-\ncentrators to properly focus the radiation on the sample.\nWe could generate quasi-static H0values up to 160 kOe\n(with a typical ramping rate \u0018150 Oe/s) by means of a\nsuperconducting solenoid (Oxford Instruments) equipped\nwith a4He variable temperature insert.\nThis experimental setup allowed us to directly measure\nthe complex impedance of the whole system (sample and\nwaveguides) and to associate anomalies induced by H0\nto the resonant Pabsorption in the sample. Di\u000berently\nfrom the X-band setup, the network analyser allowed us\nto disentangle real and imaginary components of the sig-\nnal, i. e., dispersive and absorptive components of the\nsample's uniform magnetic susceptibility \u001f(see Fig. 2\nfor a representative experimental curve). After a proper\nbackground-subtraction and phase-correction, we de\fned\nHr,HlandHhfrom the absorptive component, analo-\ngously to the case of X-band (see the inset of Fig. 1).\nFIG. 2: (Color online) Representative results for high-\n\feld ESR, obtained after proper background-subtraction and\nphase-correction. The continuous lines are relative to a si-\nmultaneous best-\ft to the dispersive (Disp.) and absorptive\n(Abs.) components according to a Lorentzian model.4\nIII. RESULTS\nA. Summary of the main magnetic properties of\nRCoPO\nRCoPO materials display a metallic behaviour for the\nTdependence of the electrical resistivity (typical values\n\u00181\u000210\u00001m\n cm) with negligible qualitative and quan-\ntitative dependences on the actual Rion.15,32They ex-\nhibit interesting magnetic properties with the appearance\nof an itinerant FM phase below a characteristic critical\ntemperature TC. This FM state is understood in terms\nof a conventional Stoner criterion after computing D(EF)\n(density of states at the Fermi energy) which, as a result,\nis mainly of dcharacter and arising from Co orbitals.15,18\nIn a simple covalent picture, the valency of Co ions is 2+\nand the measured value for the ordered magnetic moment\nper Co ion is\u00180:3\u0016Bfor LaCoPO. This value slightly\ndecreases with decreasing rI(ionic radius of the Rion)\nor, equivalently, the equilibrium unit cell volume V{ see\nlater on in Tab. I. While density functional theory cal-\nculations are able to reproduce this trend, the absolute\nvalues typically overestimate the experimental ones by\na factor\u00181:7.15,18This is highly reminiscent of what is\nobserved for the isostructural oxides based on Fe, as asso-\nciated to the di\u000eculties in describing these materials only\nfrom a fully-itinerant or a fully-localized perspective.33{36\nWe observed a linear relation for the TCvs.Vtrend\nand, as already discussed based on ZF- \u0016+SR measure-\nments, we quantitatively veri\fed this dependence also\nFIG. 3: (Color online) Phase diagram of RCoPO after \u0016+SR\nin zero magnetic \feld.17,18Values ofTC(squares) and TN\n(circles) are reported as a function of the equilibrium unit cell\nvolume. The upper x-axis indicates external pressure and it\nis referred to TCvalues only. The continuous red curve is\na best-\ftting linear function to TCdata. The dashed blue\ncurve is a guide to the eye. The black hatched area denotes\nuncertainty about the emergence of the AF phase.with further decreasing Vby means of hydrostatic pres-\nsure, pointing to a one-to-one correspondence between\nchemical and external pressures in these materials (see\nFig. 3).17,18According to this picture, the active role of\nRions is limited to the generation of chemical pressure as\nlong as the itinerant FM phase is concerned. Otherwise\nsaid, thefelectronic degrees of freedom localized on the\nRions do not in\ruence TCsigni\fcantly.\nUpon gradually increasing the chemical pressure, a sec-\nond magnetic phase appears at lower Tvalues, below the\ncritical temperature TN. Here, the Co sublattice enters\nan AF (antiferromagnetic) phase, as marked by the sud-\nden vanishing of the macroscopic magnetization and by\nclear modi\fcations in the Mvs.Hhysteresis curves.18,32\nThe AF phase is observed in NdCoPO and SmCoPO but\nnot in LaCoPO and PrCoPO. Since the localized mag-\nnetic moments on Pr3+and Nd3+ions are comparable,37\nthis observation provides further evidence for the ine\u000bec-\ntiveness offelectronic degrees of freedom in driving the\noverall magnetic properties of RCoPO. Once in the AF\nstate, we observed a gradual orientation of the Nd3+and\nSm3+magnetic moments, giving rise to a peculiar Tde-\npendence of the internal magnetic \feld at the \u0016+site.18\nB. ESR. Low-frequency regime (X-Band)\nWith decreasing Tand for all the investigated com-\npounds, the onset for the detection of a well-de\fned ESR\nsignal isT\u001890\u0000120 K, i. e., well above the TCvalues\nestimated in zero magnetic \feld by means of \u0016+SR.\n1. Signal intensity\nWe show the behaviour of the integrated intensity I(T)\nfor the four samples in comparison to Min Fig. 4. We\nmeasured the latter quantity at sample-dependent Hval-\nues comparable to those of the central resonance \feld Hr\n(see later on). The good agreement between I(T) and\nMis an indication that the ESR signal is indeed intrin-\nsic for every sample and not associated to, e. g., extrinsic\nmagnetic impurities. In particular, we notice that I(T) is\nmonotonously increasing with decreasing Tin LaCoPO\nand PrCoPO. On the other hand, both MandI(T) go\nthrough a maximum at around \u001815 K and\u001840 K\nfor NdCoPO and SmCoPO (respectively), i. e., in cor-\nrespondence to the TNvalues detected by ZF- \u0016+SR. The\nfast suppression of I(T) in the AF phase for NdCoPO\nand, in particular, for SmCoPO is a clear indication that\nESR is actually probing the signal associated to the FM\nphase. For this reason, we will refer to FMR38,39rather\nthan ESR from now on. Moreover, in view of the gen-\neral arguments discussed above about RCoPO and after\nconsidering the fact that Co2+is the only source of mag-\nnetism in LaCoPO, we are con\fdent to assign the ob-\nserved signal to Co electrons for all samples. We notice\nthat a quantum mechanical treatment of the Co2+ion5\nFIG. 4: (Color online) A comparison of the ESR line intensity (squares) and dc magnetization (diamonds) is presented for each\nsample. The estimates of ZF- \u0016+SR data for TCandTNare indicated by the vertical dashed lines.17,18\nin a tetrahedral crystalline environment (strong-ligand-\n\feld approach) would lead to an orbital singlet ( S= 3=2)\nassociated with the upper t2gtriplet.40\nWe need to discuss SmCoPO data further. As men-\ntioned in Sect. II, P(H) curves are distorted for T <T C,\nmaking \ftting attempts according to Eqs. (1), (2) and\n(3) unsuccessful in this whole Tregion. Moreover, for\nT <T N, not only is I(T) sharply suppressed but the sig-\nnal is also sizeably shifted to low Hvalues. This means\nthat a non-negligible amount of the signal is located in\nnegative \felds for T <T Nand, as such, it is simply not\nmeasured (see Fig. 5). This e\u000bect implies that a numeri-\ncal integration of d P/dHdata is also no longer possible\nin this limit, as a wide section of the resulting P(H) curve\nwould take unphysical negative values for H > 2 kOe ir-\nrespective of any attempted background subtraction. We\nalso notice that the FMR signal for T <T Nlikely arises\nfrom an inhomogeneous state, where FM islands inter-\nspersed in an AF background still give a non-zero contri-\nbution to the overall FMR signal. This may introduce an\nadditional not-controllable signal distortion. For all the\nreasons mentioned above, our data analysis for SmCoPO\nis limited to 35 K as lowest temperature.\nFinally, we stress that in SmCoPO a second signal de-velops forT <T Nat highHvalues, as is shown in Fig. 5.\nThis is strongly shifting to even higher Hvalues with de-\nFIG. 5: (Color online) d P/dHcurves for SmCoPO for T <\nTN, displaying the appearance of a second signal possibly be-\nlonging to an AF resonance branch. Data in the inset are\nvertically shifted for the aim of better visualization.6\nFIG. 6: (Color online) Colour plots presenting the Tdependence of the P(H) curves normalized with respect to the maximum\nvalue observed at Hrfor the four investigated samples. The black dots denote the actual position of Hr.\ncreasingTand it is completely lost for T < 25 K. We\ninterpret this as a signature of the AF phase, possibly\nbelonging to an AF resonance branch. This signal will\nnot be discussed in the rest of the manuscript.\n2. Asymmetry of the FMR line\nAfter estimating I(T), we normalized the P(H) curves\nwith respect to the relative maximum value observed at\nHr. The results are shown for the four samples in Fig. 6,\nglobally displaying all the main features of the FMR line\nto be discussed in detail below. We \frst focus on the\nline asymmetry quanti\fed by the parameter \u0011, whose be-\nhaviour for the four samples is displayed in Fig. 7. In the\nhighTparamagnetic limit, i. e., for T\u001dTC,\u0011takes\nvalues\u00181:2\u00001:3 independently on the actual sample.\nAccordingly, the origin of the slight asymmetry in this\nlimit must be a common mechanism for all RCoPO ma-\nterials which, we argue, is associated with an incomplete\nDyson-like distortion [Eqs. (1), (2) and (3) lead indeed\nto excellent \ftting results for all the samples in this T\nregion]. On more physical grounds, as main argument\nin favour of the given interpretation, we notice that the\npowders' morphology is identical for the four samples\nand, moreover, no substantial di\u000berences are noticed for\ntheTdependence of the electrical resistivity.32As a con-\nsequence,\u000esanddhave similar values for all the samples.\nA quanti\fcation of the A=jBjratio (see Sect. II) leads to1:25\u00001:4 as typical values, which are quite far from 2 :55\nas expected in the case of full Dyson-like distortion with\na negligible e\u000bect of electrons' di\u000busion [or, equivalently,\nassuming that Eq. (6) is satis\fed]. We conclude that our\nsamples are in an intermediate regime \u000es.d. By con-\nsidering\u001a'0:2 m\n cm as typical value for the electrical\nFIG. 7: (Color online) Tdependence of the asymmetry pa-\nrameter.\u0011= 1 corresponds to a symmetric line with respect\nto its central resonance \feld Hr, while values \u0011 >1 is corre-\nsponding to lines broadened on the high-\felds side.7\nFIG. 8: (Color online) d P/dHnormalized data for the four\nsamples at the common value T= 45 K, safely above the AF\nphase for both NdCoPO and SmCoPO. In spite of the qualita-\ntive resemblance to Dysonian lines, none of the experimental\ncurves for PrCoPO, NdCoPO and SmCoPO can be precisely\n\ftted by Eqs. (1), (2) and (3).\nresistivity at 100 K,32we deduced&\u000es\u001810\u0016m, which\nshould be considered as a reasonable order of magnitude\nford.\nTheTdependence of \u0011for LaCoPO evidences a sharp\ncrossover at around T'55 K. In particular, below this\ntemperature, the FMR line gets perfectly symmetric with\n\u0011'1 (as also observed by eye in Fig. 1). We argue that\nthe signal for T&55 K arises from a set of moments with\nFM correlations, hence the applicability of the Dyson's\ntheory and the resulting distortion, though partial. On\nthe other hand, for T.50 K, the resonance signal is\nof collective nature, associated with an isotropic macro-\nscopic magnetization of the sample which is not subject\nto the microscopic origin of the Dysonian distortion. This\nis in agreement with previous reports on itinerant ferro-\nmagnets with sizeable magnetization values.29We also\nnotice that the onset of the FM state is not straightfor-\nward to locate precisely from these data alone but would\nbe for sure higher than TC= 33:2 K estimated by means\nof ZF-\u0016+SR. Considering that the current measurements\nare performed with a non-zero Hvalue, this result is con-\nsistent with the development of a FM phase.\nFor PrCoPO, NdCoPO and SmCoPO, decreasing T\nalso results in a clear departure from the \u0011\u00181:2\u00001:3\ncondition. However, for these materials, the behaviour\nis opposite if compared to LaCoPO, i. e., the FMR line\nasymmetry strongly increases. This is further displayed\nin Fig. 8 where we report d P/dHcurves for all the sam-\nples at the common value T= 45 K, i. e., safely above the\nAF phase for both NdCoPO and SmCoPO. As already\nmentioned above, there is no sign in the Tdependence of\nthe electrical resistivity for the four samples which could\nexplain the origin of this strong asymmetry in terms of\nDyson-like distortions. Moreover, despite the qualitative\nFIG. 9: (Color online) Temperature dependence of the Hr\n(empty symbols) and HrL(full symbols) values for the reso-\nnance \feld of the four samples.\nresemblance to Dysonian lines, none of the experimental\ncurves for PrCoPO, NdCoPO and SmCoPO can be pre-\ncisely \ftted by Eqs. (1), (2) and (3). We rather argue that\nthis e\u000bect should be associated to an intrinsic magne-\ntocrystalline anisotropy gradually developing within the\nFM phase of PrCoPO, NdCoPO and SmCoPO, leading\nto an inhomogeneous broadening of the FMR line.\n3. Resonance \feld, e\u000bective gfactor and linewidth\nAs is evident after inspecting Fig. 6, the central reso-\nnance \feld Hris not shifting for LaCoPO in the whole\naccessed experimental range but its Tdependence be-\ncomes gradually more and more marked when consider-\ning PrCoPO, then NdCoPO and \fnally SmCoPO. These\narguments are made clearer in Fig. 9. The origin of the T\ndependence of Hrcannot be ascribed to the development\nof an internal magnetic \feld within the FM phase, as this\ncannot explain the almost complete lack of any shift for\nLaCoPO. In this respect, it should be further stressed\nthat in LaCoPO the ordered magnetic moment per Co\nion even takes its strongest value within the considered\nsamples' series (see later on in Tab. I). Overall, this is\nanother indication that an increasing magnetocrystalline\nanisotropy is developing in RCoPO compounds while de-\ncreasing the volume of the unit cell V.\nA more detailed data analysis is needed in the param-\nagnetic regime where the signal distortion is arising af-\nter a Dyson-like mechanism. As is well known, a simple\nestimate of characteristic \felds as done in the inset of\nFig. 1 is indeed not accurate in the presence of a Dyso-\nnian distortion. In particular, this analysis introduces\nsystematic shifts in Hrwhich should be merely consid-\nered as artefacts.29,41A proper way of accounting for\nthese e\u000bects is to perform a conventional \ftting proce-\ndure of the Dysonian line in the high- Tregion by means8\nFIG. 10: (Color online) Temperature dependence of the \u0001 H\n(empty symbols) and 2\u0000 L(full symbols) values for the FWHM\nof the four samples. Vertical arrows de\fne the Tminvalues\ndiscussed in the text.\nof Eqs. (1), (2) and (3). Accordingly, with decreasing T,\nwe followed this strategy down to the point where the\ncontribution of the magnetocrystalline anisotropy starts\nto introduce a severe distortion in the FMR line. With\nfurther decreasing T, the line \ftting is no longer possible\nand we then refer to the more empirical data analysis\ndescribed in the inset of Fig. 1. As already mentioned\nin Sect. II, in LaCoPO a sizeable line distortion is only\nobserved around the onset of the FM phase, i. e., in the\nnarrow range 35 K .T.45 K and, accordingly, the two\n\ftting approaches are equivalent for T < 35 K. Still, we\nconsider the empirical approach of Fig. 1 (inset) in this\nTregion for consistency with the other samples.\nResults of both the analyses are presented in Fig. 9. A\ndiscrepancy between HrandHrLdata is con\frmed in the\nparamagnetic regime. Here, while Hrshows a marked de-\npendence on T,HrLtakes indeed a constant value for La-\nCoPO, PrCoPO and NdCoPO. The latter result re\rects\nthe intrinsic physical behaviour and it allows us to derive\nthe e\u000bective ge\u000bfactor values 2 :08\u00060:005, 2:05\u00060:005\nand 1:995\u00060:005 for Co2+in LaCoPO, PrCoPO and Nd-\nCoPO, respectively. All these values are far from the re-\nportedg0= 2:25\u00002:30 for Co2+in tetrahedral crystalline\nenvironments.40This estimate cannot be performed for\nSmCoPO in the accessed Trange, where HrLis still show-\ning a strong Tdependence in the paramagnetic regime,\nsuggesting a \rattening only at higher T.\nTheTdependence of the FWHM is displayed in Fig. 10\nfor the four samples. Similarly to Fig. 9, we report data\nfrom both the analysis procedures described above with\nthe same meaning of symbols (however, only 2\u0000 Ldata\nare reported in the paramagnetic regime for the aim of\nclarity). In LaCoPO, a fast decrease is observed with\ndecreasing Tin the paramagnetic regime until a mini-\nmum value \u0001 Hminis reached at T=Tmin. With further\ndecreasingT, the linewidth increases again with a much\nlower rate than in the paramagnetic regime. The ob-\nFIG. 11: (Color online) The \u0001 Hand 2\u0000 Ldata (already pre-\nsented in Fig. 10) are here reported after normalizations by\nTminand \u0001Hminvalues on the x-axis and on the y-axis, re-\nspectively. Inset: dependence of Tminfor the four samples as\na function of the TCvalues estimated by means of ZF- \u0016+SR.\nThe dashed line is a linear guide to the eye.\nserved result is in qualitative agreement with previous\nobservations in itinerant compounds with diluted mag-\nnetic moments even if, in these systems, the observed\nrates are opposite (i. e., slow decrease and fast increase\nabove and below Tmin, respectively).27,29,41A qualita-\ntively similar Tdependence of the FWHM is observed\nalso for PrCoPO, while a new feature emerges for Nd-\nCoPO. Here, below T'55 K, \u0001His further suppressed\nupon decreasing Tgiving rise to a local maximum. We\nargue that this additional feature is associated to the\nincreased magnetocrystalline anisotropy and, possibly,\nalso to an additional dynamical contribution associated\nwith the onset of antiferromagnetic correlations prelud-\ning to the AF phase. Finally, we stress that a similar\ne\u000bect is observed for SmCoPO as well. However, in this\ncompound, the strong e\u000bects of the magnetocrystalline\nanisotropy (and, possibly, of additional dynamical con-\ntributions) set in at much higher Tvalues, making the\noverall \u0001Hvs.Tbehaviour qualitatively di\u000berent from\nthe ones discussed above. Still, an in\rection point can\nbe distinguished at Tmin\u001890 K.\nIn Fig. 11, we report the data already presented in\nFig. 10 after normalization by Tminand \u0001Hminvalues\non thex-axis and on the y-axis, respectively (the mean-\ning of the used symbols is preserved). Remarkably, the\nnormalized experimental points collapse onto one single\nwell-de\fned trend for T=T min&1. At the same time,\nas shown in the inset, we notice that Tminlinearly cor-\nrelates with the TCvalues estimated by means of ZF-\n\u0016+SR. Accordingly, we deduce that the FWHM is inti-\nmately governed by the growing ferromagnetic correla-\ntions within the Co sublattice for T&Tminand that\nthese latter show similar properties for all the samples.\nAs already commented above, the deviations observed9\nFIG. 12: (Color online) Experimental FMR lines at compara-\nble frequencies for the four investigated samples at di\u000berent\nTvalues safely within the FM phase. The vertical dashed line\ndenotes the position of Hrfor LaCoPO. Curves are vertically\nshifted for the aim of clarity.\nforT.Tminshould be ascribed to di\u000berent contribu-\ntions from the magnetocrystalline anisotropy and, possi-\nbly, from dynamical e\u000bects preluding to the AF phase.\nC. ESR. High-frequency regime\nWe performed measurements in the high-frequency\nregime atTvalues selected in such a way that all the\nfour samples are properly tuned within the FM phase\n(see Fig. 3). A comparison of the observed FMR lines\nat comparable \u0017values is presented in Fig. 12. Here, we\nclearly observe that the asymmetric line broadening is\nsizeably increasing when substituting the R3+ion from\nLa3+to Pr3+, Nd3+and \fnally Sm3+. Accordingly, we\nmainly recognize a further indication of what we have\nalready argued above, namely that the Rsubstitution\ninRCoPO gradually induces an increasing magnetocrys-\ntalline anisotropy and, accordingly, an inhomogeneous\nbroadening of the powder-averaged FMR line. The line-\nshapes presented in Fig. 12 are highly reminiscent of\nhard-axis anisotropy limit,42as discussed in more detail\nin the next section.\nIV. DISCUSSION\nIn Fig. 13 we report Hrdata extracted from Fig. 12\nand from similar measurements performed at \fxed T\nand at several \u0017values. As enlightened in the inset of\nFig. 13, we observe a non-linear behaviour in the \u0017(Hr)\ntrends of PrCoPO, NdCoPO and SmCoPO for small \u0017\nvalues. We also recognize that the non-linearity of the \u0017\nvs.Hrdatasets is progressively increasing for PrCoPO,\nthen NdCoPO and \fnally SmCoPO. On the other hand,\nFIG. 13: (Color online) Hrdata for the four samples extracted\nfrom Fig. 12 and from similar measurements performed at\n\fxedTand at several \u0017values. Continuous lines are best \fts\nto experimental data according to Eq. (7). The inset shows\nan enlargement of data in the low \u0017regime.\nLaCoPO displays a linear behaviour over the whole ac-\ncessed experimental window. By referring to the theory\nof FMR,38,39this property of LaCoPO can be considered\nas an a posteriori con\frmation of our original assump-\ntion about the sample morphology, namely, the powder\nis composed of approximately spherical grains. Accord-\ningly, we can neglect the e\u000bect of demagnetization factors\n(shape anisotropy) on the actual \u0017(Hr) trend, assuming\nthat the same holds for the other compounds as well.\nIn the light of the observed phenomenology, we analyze\n\u0017(Hr) data by referring to a basic model for magnets with\nuniaxial symmetry.43Data in Fig. 13 are indeed highly\nreminiscent of the hard-axis (easy-plane) limit for the\nmagnetocrystalline anisotropy43\n\u0017?=\r\n2\u0019p\nH(H+jHAnj) (7)\nwhereHAnis an e\u000bective magnetic \feld quantifying the\nmagnetocrystalline anisotropy within the FM phase and\n\ris the gyromagnetic ratio. The expression\nHAn=2K\nM(8)\nrelates this latter parameter to the usual magne-\ntocrystalline anisotropy constant K < 0 (easy-plane\nanisotropy) via the sample magnetization.43,44Results\nof the \ftting procedure to experimental data are shown\nin Fig. 13, denoting an excellent agreement with the ex-\nperimental data upon properly setting \randHAnvalues.\nIt should be remarked that Eq. (7) is relative to one\nspeci\fc branch of the K < 0 limit and, in particular, to\nthe case of the external magnetic \feld lying within the\neasy plane ( H?z, wherezdenotes the hard axis). In10\nFIG. 14: (Color online) Qualitative illustration of the inho-\nmogeneous broadening of the FMR line of SmCoPO arising\nfrom grains with di\u000berent orientations with respect to the\nexternal magnetic \feld. The continuous black lines repre-\nsent the two branches described by Eqs. (7) and (9), with\n\r=2\u0019= 2:91\u000210\u00003GHz/Oe and HAn= 9 kOe obtained for\nSmCoPO from the \ftting procedure in Fig. 13.\nthe opposite case ( Hkz), one expects43\nH <jHAnj:\u0017k= 0 (9)\nH >jHAnj:\u0017k=\r\n2\u0019(H\u0000jHAnj):\nThe exemplary trends for both \u0017?and\u0017kare visualized\nin Fig. 14 as continuous lines, after selecting \r=2\u0019=\n2:91\u000210\u00003GHz/Oe and HAn= 9 kOe, i. e., the values\npreviously obtained from a \ftting procedure to SmCoPO\ndata. Fig. 14 also reports some selected experimental\ncurves for SmCoPO at di\u000berent \u0017values. By consid-\nering each curve, it is clear that the overall \u001f00vs.H\nbehaviour is the result of the contribution of grains with\ndi\u000berent orientations with respect to H, giving rise to\na powder-averaged, inhomogeneously-broadened absorp-\ntion line. We further stress that the well-de\fned maxi-\nmum observed in the experimental curves at Hrshould\nTABLE I: Summarizing results from previous studies of dc\nmagnetization on the currently investigated samples.17,32For\neach sample, we report the ordered value for the magnetic mo-\nment per Co ions, \u0016, and the corresponding saturation mag-\nnetizationMs. Estimates were performed at temperatures\ncomparable to the conditions of the FMR measurements (see\nFig. 12).\nCompound \u0016(\u0016B/Co)Ms(erg/Oe cm3)\nLaCoPO 0.295 \u00060.01 40.5 \u00061.5\nPrCoPO 0.27 \u00060.01 37.1 \u00061.5\nNdCoPO 0.24 \u00060.01 32.8 \u00061.5\nSmCoPO 0.225 \u00060.01 31 \u00061.5\nFIG. 15: (Color online) Summarizing results for the estimated\nanisotropy parameters Kaccording to the model described in\nthe text. The dashed line is a guide to the eye. The inset\nshows the\r=2\u0019values estimated from the \ftting procedure\nof Eq. (7) to data in Fig. 13.\nbe associated to grains where the H?zcondition holds.\nThe stronger intensity compared to the Hkzbranch is\neasily explained from geometrical considerations of the\npowder average. Remarkably, the low- \u0017line is less asym-\nmetric than the other ones. We attribute this e\u000bect to\nthe low typical values of Hin this limit, which may result\nin an undetectable signal from the \u0017kbranch.\nEstimates of HAnvalues from Fig. 13 enable us to di-\nrectly estimate the Kanisotropy constants for the four\nsamples via Eq. (8). We used magnetization values de-\nrived from independent measurements17,32and, in par-\nticular, we employed the saturation values Msestimated\nat temperatures close to the conditions of FMR mea-\nsurements (see Fig. 12). The results are displayed in\nFig. 15, showing a well de\fned trend for Kas a function\nof the unit cell volume V. These results suggest that\ndecreasing the Vvalue is the physical origin for trigger-\ning anisotropic magnetic properties for RCoPO. Since\npreviously reported data18evidence that smaller Rions\ninduce a reduction in aandcaxes such that the c=aratio\nis approximately constant (i. e., the lattice is contracting\nisotropically), one reasonable conclusion is that the pnic-\ntogen height hPis reducing faster, making the local tetra-\nhedral environment progressively more distorted. How-\never, we also expect a sizeable interaction between fand\ndelectronic degrees of freedom from rare-earth and pnic-\ntogen ions, respectively, which is typically measured by\nmeans of local-probe techniques for RMX O oxides.45{47\nWhile this e\u000bect may be well enhanced by the increas-\ning chemical pressure, we argue that it may introduce\nan indirect transfer of anisotropic properties from the f\norbitals of Rions to the dbands arising from Co or-\nbitals and ultimately in\ruencing the magnetocrystalline\nanisotropy. In this respect, extending our measurements\nto a more complete set of RCoPO samples with di\u000berent\nprolaticity properties for the Rorbitals would lead to a11\nimportant check on which of the two proposed mecha-\nnisms is indeed the dominant one.\nAnother scenario can be considered in order to under-\nstand the origin of the observed behaviour. One robust\noutput of our investigation is that LaCoPO shows almost\nfully-isotropic magnetic properties within the FM phase,\na fact which is surprising in the light of the typically\nanisotropic properties of uniaxial magnets.44These fea-\ntures may be only apparently isotropic if one assumes\nthat a strong magnetocrystalline anisotropy could be ef-\nfectively compensated by shape anisotropy e\u000bects, under\nthe hypothesis that the spherical grains composing the\ninvestigated samples are coupled among them. While it\nseems quite unlikely that this compensation e\u000bect leads\nto completely symmetric lineshapes as the experimental\nones measured for LaCoPO, the main conclusions out-\nlined above (i. e., the magnetocrystalline anisotropy is\nenhanced by Rsubstitution) are robustly preserved also\nwithin this scenario.\nV. CONCLUSIONS\nWe reported on ferromagnetic resonance measure-\nments inRCoPO for di\u000berent Rions. 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B 83, 094526 (2011)."    },    {        "title": "1603.00253v1.Magnetic_properties_of_Fe__5_SiB__2__and_its_alloys_with_P__S__and_Co.pdf",        "content": "arXiv:1603.00253v1  [cond-mat.mtrl-sci]  1 Mar 2016Magnetic properties of Fe 5SiB 2and its alloys with P, S, and Co\nMiros/suppress law Werwi´ nski∗\nDepartment of Physics and Astronomy, Uppsala University, B ox 516, SE-751 20 Uppsala, Sweden and\nInstitute of Molecular Physics Polish Academy of Sciences,\nul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland\nSofia Kontos, Klas Gunnarsson, and Peter Svedlindh\nDepartment of Engineering Sciences, Uppsala University, B ox 534, SE-751 21 Uppsala, Sweden\nJohan Cedervall, Viktor H¨ oglin, and Martin Sahlberg\nDepartment of Chemistry, Uppsala University, Box 538, SE-7 51 21 Uppsala, Sweden\nAlexander Edstr¨ om, Olle Eriksson, and J´ an Rusz\nDepartment of Physics and Astronomy, Uppsala University, B ox 516, SE-751 20 Uppsala, Sweden\n(Dated: September 25, 2018)\nFe5SiB2has been synthesized and magnetic measurements have been ca rried out, revealing that\nMsat= 0.92 MA/m at T= 300 K. The MvsTcurve shows a broad peak around T= 160 K. The\nanisotropy constant, K1, estimated at T= 300 K, is 0.25 MJ/m3. Theoretical analysis of Fe 5SiB2\nsystem has been carried out and extended to the full range of F e5Si1−xPxB2, Fe 5P1−xSxB2, and\n(Fe1−xCox)5SiB2compositions. The electronic band structures have been cal culated using the\nFull-Potential Local-Orbital Minimum-Basis Scheme (FPLO -14). The calculated total magnetic mo-\nments are 9.20, 9.15, 9.59 and 2.42 µBper formula units of Fe 5SiB2, Fe5PB2, Fe5SB2, and Co 5SiB2,\nrespectively. In agreement with experiment, magnetocryst alline anisotropy energies (MAE’s) cal-\nculated for T = 0 K changes from a negative (easy-plane) aniso tropy −0.28 MJ/m3for Fe 5SiB2\nto the positive (easy-axis) anisotropy 0.35 MJ/m3for Fe 5PB2. Further increase of the number of\np-electrons in Fe 5P1−xSxB2leads to an increase of MAE up to 0 .77 MJ/m3for the hypothetical\nFe5P0.4S0.6B2composition. Volume variation and fixed spin moment calcula tions (FSM) performed\nfor Fe 5SiB2show an inverse relation between MAE and magnetic moment in t he region down to\nabout 15% reduction of the spin moment. The alloying of Fe 5SiB2with Co is proposed as a prac-\ntical realization of magnetic moment reduction, which ough t to increase MAE. MAE calculated in\nvirtual crystal approximation (VCA) for a full range of (Fe 1−xCox)5SiB2compositions reaches the\nmaximum value of 1.16 MJ/m3at Co concentration x= 0.3, with the magnetic moment 7.75 µBper\nformula unit. Thus, (Fe 0.7Co0.3)5SiB2is suggested as a candidate for a rare-earth free permanent\nmagnet. For the stoichiometric Co 5SiB2there is an easy-plane magnetization, with the value of\nMAE = −0.15 MJ/m3.\nPACS numbers: 71.20.Be, 75.30.Gw, 75.50.Bb, 75.50.Cc, 75. 50.Ww\nI. INTRODUCTION\nVolatile prices of the rare-earth elements observed in\nthe recent years have motivated wide research initia-\ntives in an effort to find replacement materials for the\npermanent magnets industry, which would use little or\nnone of the rare-earth elements. Late 3 dtransition el-\nements, such as Mn, Fe or Co are the primary candi-\ndates for such materials, thanks to their large and stable\nmagnetic moments and high magnetic transition tem-\nperatures found in many of the compounds they form.\nHowever, weak spin-orbital interaction strength typicall y\ncauses low magnetocrystalline anisotropy energy (MAE),\nwhich in turn means that they often lack appreciable co-\nercivity, forming soft magnetic materials. Under certain\ncircumstances, however, the MAE of 3 dtransition metal\n∗Corresponding author: werwinski@ifmpan.poznan.plcompounds can be substantial, for example the tetrago-\nnally distorted Fe/Co alloys1–5, L1 0phases such as FeNi,\nCoNi, MnAl6,7, Fe 16N28, (Fe/Co) 2B9–11or Fe 2P-based\nalloys12,13. A general overview on rare-earth-free perma-\nnent magnets is available in Ref.14. All of the above men-\ntioned systems are hexagonal or tetragonal, i.e., uniaxial .\nFor this reason, in this work we have focused on an iron-\nbased system Fe 5SiB2and its alloys (Fe 1−xCox)5SiB2\nand Fe 5Si1−xPxB2- they are tetragonal and rich on mag-\nnetic elements.\nFe5SiB215and Fe 5PB216are isomorphous with Cr 5B3.\nThey were initially reported by Aronsson and En-\ngstr¨ om15and Rundqvist16in 1960 and 1962. Several\nother borides crystallize within the same structure type.\nIn the more general formula X5SiB2for the silico-borides,\nXmay stand for (Mn, Fe, Mo)15, V17, Nb18or W19. The\nrecently discovered Nb 5SiB2and W 5SiB2have been rec-\nognized as superconductors. Isomorphous phases with\nphosphorus, X5PB2, were identified with X= (Mn, Fe,2\nCo)16, Cr20and Mo21.\nFIG. 1. (a) The crystallographic structure of Fe 5SiB2; (b)\nFe2-B layer; (c) Fe 1layer; (d) Si layer; (e) Fe 2-B layer limited\nto the unit cell boundaries.\nThe crystal structure of Fe 5PB2was determined by\nRundqvist16and Fe 5SiB2by Aronsson and Engstr¨ om15,\nsee Fig. 1. Rundqvist stated that the distribution of P\nand B atoms in Fe 5PB2was ordered and had a narrow ho-\nmogeneity range16. Both Fe 5PB2and Fe 5SiB2crystallize\nin the Cr 5B3-type structure16with body-centered tetrag-\nonal (bct) unit cell, space group I4/mcm, see Fig. 1(a).\nSingle unit cell of Fe 5XB2contains 32 atoms divided into\n4 formula units. Fe 1atoms are distributed on the 16-fold\nposition, B on the 8-fold, Fe 2and Si/P on the 4-fold\nposition, respectively.\nThe Fe 5SiB2and Fe 5PB2structures have a princi-\npally ordered arrangement of Si/P and B on their respec-\ntive crystallographic sites, but some intermixing occurs\nas well as interstitial accommodations of B, Si and P.\nFurthermore, octahedral holes of sufficient size for small\nnon-metal atoms exist in the system. Aronsson and En-\ngstr¨ om15found it likely that some of the boron atoms\nmay be interstitially accommodated in these holes. More-\nover, Rundqvist16described two types of holes in the\nCr5B3-type lattice, trigonal prismatic and anti-prismatic\nones. In the Fe 5PB2system the radii of the trigonal pris-matic and anti-prismatic holes are 0.93 ˚A and 1.08 ˚A, re-\nspectively, while the B radius is 0.88 ˚A. A homogeneity\nrange of Fe 5SiB2with the substitutions of B for Si has\nbeen observed previously. Fe 5SiB2measured by Arons-\nson and Engstr¨ om15have B/Si ratio slightly larger than\n2.00. The authors even proposed a new general formula,\nFe5Si1−xB2+x(0< x < 0.1). Another feature of Fe 5XB2\nstructure is its layered character. The structure is built\nup of three types of layers, which alternate along the c-\naxis. A single unit cell consists of 8 layers: 2 Fe-B, 4 Fe,\nand 2 Si layers, see Fig. 1.\nAt room temperature Fe 5SiB2and Fe 5PB2are fer-\nromagnets with an orientation of spins parallel to the\ntetragonal axis22. The Curie temperature ( TC) is\nslightly varying with the deviation from the stoichiom-\netry (Fe 5X1−xB2+x),TC= 628 ±11 K for Fe 5PB223and\nTC∼784 K for Fe 5SiB222. For Fe 5PB2the tetrago-\nnal easy axis of the magnetization direction is valid from\n80 K up to TC22. The low temperature M¨ ossbauer study\nof Fe 5SiB2revealed a significant change of the spectral\nshape in the temperature range of 130 to 150 K22. Eric-\nsson et al.22concluded that below 140 K the spins were\noriented in the ab-plane or close to that. There were\nalso indications that below 140 K the pure ferromagnetic\ncoupling is partly destroyed22. In the temperature re-\ngion from 140 K up to TCthe Fe 5SiB2was confirmed to\nbe a ferromagnet with the spins parallel to the c-axis22.\nH¨ aggstr¨ om et al.23referred that an average magnetic mo-\nment of Fe 5PB2was reported to be 1 .73µBper Fe atom.\nThe individual moments calculated by H¨ aggstr¨ om et al.\nfrom the values of the magnetic hyperfine fields are 1 .6µB\nfor Fe 1and 2 .2µBfor Fe 2.\nTo the best of our knowledge there is no literature data\nconcerning Co 5SiB2, but only its homologue Co 5PB216.\nVery recently, Co- and Mn-doped Fe 5SiB2and Fe 5PB2\nwere synthesized and characterized24. Another recent\nmanuscript reports a study of single crystal of Fe 5PB225.\nBoth these studies also report basic characterization\nof the electronic structure of the parent compounds,\nFe5SiB2and Fe 5PB2, respectively.\nII. EXPERIMENTAL AND COMPUTATIONAL\nDETAILS\nA. Experimental details\nFe5SiB2samples were prepared with a conventional arc\nfurnace in an argon atmosphere. Stoichiometric amounts\nof iron (Leico Industries, purity 99.995 %, surface ox-\nides were reduced in H 2-gas), silicone (Highways Interna-\ntional, purity 99.999 %) and zone melted boron (Wacher)\nwere used as raw materials. The samples were crushed,\npressed into pellets and sealed in silica tubes, annealed\nat 1273 K for 19 days and then quenched in cold water.\nX-ray powder diffraction intensities (XRD) were\nrecorded using Bruker D8 diffractometer with a V˚ antec\nposition sensitive detector with a 4° opening using CuK α13\nradiation, λ= 1.540598 ˚A. The measurements were taken\nin a 2 θ-range of 20-90°. Phase analysis and crystal\nstructures refinements were performed using the Rietveld\nmethod26implemented in the software FullProf27and\ncell parameters were refined using the software UNIT-\nCELL28.\nMagnetization measurements on the powder samples\nwere performed using a Quantum Design PPMS 6000\nVibrating Sample Magnetometer. The low field magne-\ntization temperature dependence was recorded between\nT= 300 K and T= 10 K at a constant magnetic field of\n8 kA/m. The magnetization, M, vs field, H, at constant\nT= 300 K was measured in the range from −5.6 MA/m\nto +5 .6 MA/m.\nB. Computational details\nFully relativistic electronic band structure calculation s\nwere carried out by using the Full-Potential Local-Orbital\nMinimum-Basis Scheme (FPLO-14)29within the gener-\nalized gradient approximation (GGA) for the exchange-\ncorrelation potential in the Perdew, Burke, Ernzerhof\nform (PBE)30. Calculations were performed with 16 ×\n16×16k-mesh and convergence criterion 10−8Ha. An\ninitial spin splitting is applied to the 3 d-atoms, assuming\nferromagnetic structures, as observed experimentally.\nTABLE I. The calculated optimized crystallographic parame -\nters for Fe 5SiB2, Co 5SiB2and Fe 5PB2. Space group I4/mcm,\nno. 140.\nsystem a[˚A]c[˚A]xFe1/Co 1zFe1/Co 1xB\nFe5SiB25.546 10.341 0.169 0.136 0.382\nCo5SiB25.511 9.953 0.169 0.134 0.376\nFe5PB25.503 10.347 0.172 0.138 0.383\nFe5SB25.466 10.594 0.172 0.138 0.383\nFor Fe 5SiB2, Co 5SiB2, Fe 5PB2, and Fe 5SB2the lat-\ntice parameters and the internal atomic positions were\noptimized, see Table I. The optimized structural param-\neters of Fe 5SiB2and Fe 5PB2are in an excellent agree-\nment with the corresponding experimental values16,22.\nWe note that the computationally relaxed structural pa-\nrameters reported here are for the stoichiometric phases\nof Fe 5SiB2and Fe 5PB2, while the previously measured\nvalues are estimated from non-fully stoichiometric sam-\nples.\nVirtual crystal approximation (VCA) is used to study\nthree ranges of compositions, the first one between\nFe5SiB2and Co 5SiB2, the second one between Fe 5SiB2\nand Fe 5PB2, and the last one between Fe 5PB2and\nFe5SB2. The general idea of the VCA is to imitate a\nhomogeneous random on-site occupation of two types of\natoms by using a virtual atom with a corresponding aver-\naged value of the nuclear charge and number of electrons.\nApplication of VCA for Fe/Co alloying is motivated by\nthe fact that Co atomic number is one above Fe and fur-thermore that similar VCA calculations have already pro-\nduced valuable results for other Fe/Co alloys3,5,11. Since\nP and Si are also neighbors in the periodic table, and as\nthep-elements are basically supplying itinerant electrons\nto the crystal, the VCA is expected to be a good approxi-\nmation of the substitutional disorder also for this class of\nsystems. The same argument holds for P/S alloying. It\nis important to comment that although VCA is expected\nto give the right qualitative trends of MAE changes with\nFe/Co concentration, the exact values of MAE are usu-\nally overestimated by a factor between 2 and 4 in compar-\nison to experiment. More accurate results of MAE can\nbe calculated within the coherent potential approxima-\ntion (CPA)11,31–33. Unfortunately, the utilized version of\nthe FPLO code, which allows to perform the spin-orbit\ncoupling (SOC) calculations together with fixed spin mo-\nment (FSM), does not allow to combine SOC with CPA.\nCalculations for Fe 5Si1−xPxB2compositions start from\nthe Fe 5SiB2structure with a full Si site occupation. The\nSi atom is then replaced by a virtual atom to mimic\nthe specific composition. The atomic number of virtual\natom is ZVCA = (1−x)·ZSi+x·ZP. The lattice param-\neters of the intermediate compositions are interpolated\nlinearly between the lattice parameters of the extreme\nsystems, aVCA = (1−x)·aFe5SiB 2+x·aFe5PB2. For ev-\nery composition the internal atomic positions are opti-\nmized. The VCA calculations for (Fe 1−xCox)5SiB2se-\nries are performed with lattice parameters and internal\natomic positions interpolated as the linear combinations\nof crystal parameters of boundary compositions Fe 5SiB2\nand Co 5SiB2.\nVESTA code34was used for visualization of crystal\nstructures.\nIII. EXPERIMENTAL RESULTS\nA. Phase analysis and crystal structure\nThe XRD intensities, presented in Fig. 2, confirm that\nFe5SiB2crystallizes in the tetragonal Cr 5B3-type struc-\nture with the space group I4/mcm15. Synthesized sample\ncontains less than 2 at.% of an additional Fe 3Si impurity\nphase.\nTABLE II. Atomic coordinates and site occupancies for\nFe5SiB2derived from crystal structure refinement of XRD\ndata.\nAtom Site x y z occupancy\nFe116(l) 0.1683(1)1\n2+x0.1384(1) 0.500\nFe24(c) 0 0 0 0.125\nSi 4(a) 0 01\n40.125\nB 8(h) 0.378(1)1\n2+x 0 0.250\nTable IIpresents the atomic coordinates of Fe 5SiB2\nstructure derived from crystal structure refinement of\nXRD data. Determined lattice parameters are a=4\n5.5561(3) ˚A and c= 10.3476(10) ˚A, estimated standard\ndeviations are in the parenthesis. Previously reported\nFe5SiB2lattice parameters obtained by Ericsson22are\na= 5.43˚A and c= 10.33˚A, in close agreement with our\nresults. The Fe 5PB2lattice parameters are a= 5.482˚A\nand c= 10.332˚A16. It indicates that there are only\nfew promiles differences between the calculated and mea-\nsured lattice parameters for Fe 5SiB2and Fe 5PB2, com-\npare with Table I.\nFIG. 2. X-ray powder diffraction pattern of Fe 5SiB2refined\nwith the Rietveld method. The black dots and the red line\ncorrespond to the observed and calculated patterns, respec -\ntively, while the blue line shows the difference between ob-\nserved and calculated data. Bars show the theoretical Bragg\npeaks of the Fe 5SiB2phase (upper) and the impurity phase,\nFe3Si (lower).\nB. Measured magnetic properties\nThe value of Msat(MatH= 5.6 MA/m) at T= 300 K\nis 0.92 MA/m (6 .3µB/f.u.) which is considerably smaller\nthan expected from M¨ ossbauer measurements, 1.2 MA/m\n(8.2µB/f.u.). However, the M¨ ossbauer data was reported\nforT= 80 K. Further, due to surface effects, Msatfor\nsmall particles should be lower than the bulk value. Also,\nthere are non-magnetic impurities in the sample, e.g.\nFe3Si, decreasing the value of Msat. Assuming the sam-\nple to consist of crystallites with uniaxial anisotropy, th e\nanisotropy constant, K1, was estimated using the 1 /H2-\nterm in the law of approach to saturation. This term\nis assumed to originate from the magnetization rotating\nagainst the magnetocrystalline anisotropy:\nM(H)\nMsat=/parenleftBig\n1−a2\nH2/parenrightBig\n, (1)\nwhere\na2=4\n15/parenleftbiggK1\nMsat/parenrightbigg2\n(2)Using the MvsHmeasurement at T= 300 K, the\nresult for Fe 5SiB2isK1= 0.25 MJ/m3. Since Eq. 1con-\ntains only the 1 /H2dependence of M(H), the selection\nof field interval for the K 1estimation is crucial. Here,\nthe field interval 0.6–1.0 MA/m was used, corresponding\ntoM(H)/Msat= 0.95–0.98. However, the need of a more\naccurate method, e.g. involving a single crystal sample, is\nemphasized and will be the aim for further experiments.\nConcerning the proposed reorientation of spins at T=\n140 K reported from M¨ ossbauer experiments, the follow-\ning observation was made from magnetization measure-\nments, see Fig. 3: The MvsTcurve shows a broad\npeak around T= 160 K and then Mdecreases continu-\nously with decreasing temperature to an approximately\nconstant level at T= 10 K. The decrease in Mfrom\nT= 160 K to T= 10 K is 30%.\n0 100 200 30056789\nT (K)M (emu/g)\nFIG. 3. The temperature dependency of magnetization as\nmeasured for Fe 5SiB2.\nIV. DENSITY FUNCTIONAL THEORY\nCALCULATIONS\nA. Fe 5SiB 2Electronic Structure\nThe basic electronic configurations of the elements\nforming Fe 5SiB2are Fe: [Ar] 4 s23d6, Si: [Ne] 3 s23p2\nand B: [He] 2 s22p1. This is also reflected in the densities\nof electronic states (DOS), see Fig. 4. The total DOS is\na sum of the individual atomic contributions, from which\nthe dominant role is played by Fe 3 dbands. Fe 3 d, Si 3 s,\nSi 3p, B 2 s, and B 2 pbands are presented on the spin\nprojected DOS plots. The Fe bands Fe 13dand Fe 23d,\nfor two inequivalent Fe sites, are both strongly spin po-\nlarized. The corresponding spin magnetic moments are\n1.87 and 2.24 µB/atom, respectively. The spin, orbital\nand total magnetic moments are summarized in Table III.\nThe Si and B spin magnetic moments are both equal to\n−0.25µB.\nFrom −2.5 eV up to the Fermi level the character of\nDOS originates almost entirely from Fe 13dand Fe 23d5\n0510\n-10-50\n012\n-2-10\n012\n-2-10\n00.5\n-0.50\n00.5\n-14 -12 -10 -8 -6 -4 -2 0 2-0.50\nE - EF (eV)total Fe5SiB2\nFe1 3d \nFe2 3d DOS (states / (eV · spin · (formula unit or atom)))spin up\nspin dn\nSi 3s Si 3p\nB 3sB 2p\nFIG. 4. The spin projected partial and total densities of sta tes\nDOS for Fe 5SiB2. Calculations done within GGA(PBE)+so\napproximation.\nbands. It means that the main contribution to the DOS\nat the Fermi level is provided by the Fe 3 dstates, what\nmoreover leads to high spin polarization on the Fermi\nlevel, of about 66 %. Furthermore, calculations show that\nthe orbital magnetic moments are present only on the Fe\natoms (see Table III). Their values 0 .039µBand 0 .048µB\nare close to the value 0 .043µBfor bcc iron, calculated\nwithin the same frame. Furthermore, the all theoreti-\ncal values are reduced in comparison to the experimental\nvalue 0 .086µBfor the bcc iron35. The total magnetic\nmoment per formula unit is equal to 9 .20µB, which is\nin good agreement with the recently determined exper-\nimental value 9 .35µB/f.u. (5 ∗1.87µB/Fe) from M(H)\nmeasurements at 10 K36.\nTABLE III. The spin, orbital and total magnetic moments\n[µB/(atom or formula unit)] for Fe 5SiB2, Co 5SiB2, Fe 5PB2,\nand Fe 5SB2.\nFe5SiB2 Co5SiB2 Fe5PB2 Fe5SB2\natom µs µlµs µlµs µlµs µl\nFe1/Co 11.87 0.039 0.40 0.016 1.83 0.034 1.91 0.031\nFe2/Co 22.24 0.048 0.90 0.025 2.22 0.052 2.26 0.048\nSi/P/S -0.25 0.001 -0.05 0.000 -0.14 0.003 -0.03 0.002\nB -0.25 0.001 -0.06 0.001 -0.22 0.001 -0.23 0.002\nµtot 9.20 2.42 9.15 9.59B. MAE of Fe 5Si1−xPxB2and Fe 5P1−xSxB2\nThe M¨ ossbauer study revealed the ”in-plane” spin ori-\nentation for Fe 5SiB2below 140 K22, in contrast to the\n“easy-axis” spin orientation for Fe 5PB2down to 80 K.\nAs we are interested in materials with high magne-\ntocrystalline anisotropy, we extend theoretical inquires\nof Fe 5SiB2by the promising uniaxial case of Fe 5PB2. In\nthis paper MAE is determined as the difference between\ntotal energies obtained for orthogonal spin quantization\naxes, MAE = E100−E001. Positive MAE value there-\nfore denotes the uniaxial magnetocrystalline anisotropy\nalong with easy axis along the [001] direction (tetrago-\nnal axis), whereas for negative MAE there is an easy-\nplane magnetization ( ab-plane). Calculated MAE of\nFe5SiB2and Fe 5PB2are−0.28 and 0 .35 MJ/m3(−140\nand 170 µeV/f.u.), respectively. The negative and pos-\nitive signs agree well with the experimentally observed\nmagnetic structures22. For magnetic moment, going from\nFe5SiB2to Fe 5PB2, the most noticeable change is that of\nthe induced spin magnetic moment on 3 p-element, from\n−0.25 to −0.14µB(see Table III). The calculated total\nmagnetic moments on Fe 1and Fe 2sites of Fe 5PB2are\n1.86 and 2.27 µB/atom, respectively. They differ from\nexperimental values 1 .6µBfor Fe 1and 2 .2µBfor Fe 2, de-\nrived from the magnetic hyperfine fields23, however the\nmentioned experimental values are only indirect approx-\nimate estimations.\n0.2 0.4 0.6 0.8-0.4-0.200.20.40.60.8\n0.2 0.4 0.6 0.8Si P S\nP concentration x S concentration xMAE (MJ/m3)(a) (b)\nFIG. 5. The concentration dependency of the MAE for a)\nFe5Si1−xPxB2and b) Fe 5P1−xSxB2.\nA priori it is not obvious, what will be the evolution\nof MAE as a function of concentration x. As an exam-\nple of non-monotonic behavior we note the Fe/Co1or\n(Fe/Co) 2B alloys9. Therefore we have performed VCA\nstudies of the Fe 5Si1−xPxB2for the whole range of com-\npositions, 0 ≤x≤1. The resulting VCA calculations of\nMAE vs x are presented in Fig. 5a). The x-dependence\nof MAE is monotonic except for the smallest values of P\nconcentration. The MAE values for both stoichiometric6\nconcentrations are near extremal. With an increase of the\nP concentration the system is heading towards the uniax-\nial anisotropy, with a cross-over at x>∼0.7. The transition\nfrom Si 3 p2to P 3 p3can be understood as an increase in\nthe number of electrons in Fe 5Si1−xPxB2system. Fur-\nther increase in the number of electrons can be realized\nby transition from P 3 p3to S 3 p4. As shown in Fig. 5b)\nfor the hypothetical alloys with sulfur Fe 5P1−xSxB2sig-\nnificant values of MAE ∼0.7 MJ/m3are obtained for a\nbroad range of sulfur concentration 0 .4≤x≤1.\nIn order to get deeper insight into the origin of MAE,\nthe energy scale of the band structure has to be changed\nfrom tens of eV into tenths of eV, since the SOC con-\nstant of 3 d-metals is in the order of 0.05 eV, which is also\nwhy the spin-orbit splitting does not exceed this value,\nsee Fig. 6. Fully relativistic calculations result in an ad-\nditional splitting of the electronic bands, due to spin-\norbit coupling. The figure presents the Fe 5SiB2bands in\nthe vicinity of the Fermi level, together with the MAE\ncontributions per k-point as obtained by the magnetic\nforce theorem37,38. Spin-orbit splitting results in differ-\nent band structures for quantization axes 100 and 001\nindicated by red and blue lines.\nThe calculated total MAE of Fe 5SiB2is−140µeV/f.u.\n(−0.28 MJ/m3). The MAE value ∼10−4eV/f.u. indi-\ncates a fine compensation of the much bigger negative\nand positive contributions to MAE of about 10−3eV per\nk-point. Moreover, it is easy to notice the step changes\nof MAE( k) whenever the band crosses the Fermi level.\nThus, when so many bands cross it, a very accurate\nmodel of the electronic band structure and fine k-mesh\nare crucial to calculate MAE.\nC. Fe 5SiB 2Fixed Spin Moment and Volume\nDependencies\nFor Fe 5SiB2the M¨ ossbauer study revealed the ”in-\nplane” spin orientation below 140 K and spins parallel to\nc-axis above 140 K22. A similar transition from negative\nto positive magnetocrystalline anisotropy constant was\nobserved for Fe 2B with a transition temperature about\n520 K9. Since the goal of this study is to explore for\npotential candidates for permanent magnets, the experi-\nmentally revealed increase of MAE with temperature for\nFe5SiB2has motivated theoretical studies which should\nallow for better understanding of this phenomena. To\nanswer the question, whether the increase of MAE with\nTis related to the volume expansion, we have calcu-\nlated volume-dependency of MAE. Experimentally ob-\nserved increase of volume between 16 and 165 K is only\nabout 0.01%, but for this small volume expansion no sig-\nnificant change of MAE is observed, see Fig 7. The per-\nformed calculations cover a much bigger range of volume\nchange, namely from −10% to +10% and, what is strik-\ning, for a very large volume reduction the negative MAE\nchange its sign, and for V/V 0= 0.9 reaches significant\nvalue 0.52 MJ/m3. By a rough estimation, such volumereduction would correspond to about 20 GPa pressure,\nreadily available in high pressure experiments39.\nFig.8shows non-linear volume dependency of the to-\ntal magnetic moment for Fe 5SiB2. From figures 7and\n8one could contemplate that MAE might increase when\nmagnetic moment decreases. To address the question,\nwhether the MAE changes in the same manner with\nonly magnetic moment variation and without change\nof volume, a set of fully relativistic fixed spin moment\n(FSM) calculations was performed. Fig. 9presents the\ntotal magnetic moment ( MS+ML) dependency of MAE.\nMS+MLis evaluated for every fixed spin magnetic mo-\nment MS.\nFSM calculations show for Fe 5SiB2that MAE starts\nto increase with the decrease of total magnetic moment,\nand reaches the maximal value of MAE = 1.78 MJ/m3\nforMS+ML= 6.15 µB/f.u., while equilibrium MS+\nML= 9.20 µB/f.u. We suggest that significant increase\nof MAE might occur if we would find a way to reduce\nthe magnetic moment of Fe 5SiB2by 1/3. A possible\nvenue to achieve this could be alloying of magnetic ele-\nment Fe with an element carrying a lower magnetic mo-\nment, Co. Following this idea the VCA calculations of\n(Fe1−xCox)5SiB2are carried out, expecting that alloying\nwith Co will reduce magnetic moment, and hopefully in\nconsequence increase the MAE. It is important to com-\nment that reducing the magnetic moment by FSM or\nby alloying with another element will necessarily result\nin different changes to electronic structure, therefore we\ncan’t expect this analogy to hold, at least not up to large\nmoment reduction. Nevertheless it is of interest to find\nout to what extent is this parallel realistic. Moreover,\nalloying of iron-based magnetic materials with Co has\nalready been shown to lead to an increased MAE in sev-\neral cases1,3,4,11.\nD. (Fe 1−xCox)5SiB 2\nVCA calculations of the whole range of\n(Fe1−xCox)5SiB2concentrations start from optimizing\nthe extreme structures of Fe 5SiB2and Co 5SiB2(see\nSect. IIand Table I) and interpolating crystallographic\nparameters for intermediate compositions. The spin\nmagnetic moments on Co 1and Co 2atoms in Co 5SiB2\nare 0.41 and 0.90 µB, respectively. They induce the small\nopposite spin moments on Si and B, equal to −0.05 and\n−0.06µB, respectively. The total magnetic moment per\nCo5SiB2formula unit is equal to 2.42 µB(see Table III).\nWe expect that Fe/Co alloying will reduce the mag-\nnetic moment, and thus may increase MAE of Fe 5SiB2.\nThe results of MAE(x) presented in Fig. 10con-\nfirms our predictions partially. The maximum of\nMAE = 1.16 MJ/m3is obtained for x= 0.3 together with\nMS+MLreduced from 9.20 to 7.75 µB/f.u., and not for\n6.15µB/f.u., as predicted for Fe 5SiB2from VCA. Fig. 11\nshows explicitly the dependence of MAE on MS+MLfor\n(Fe1−xCox)5SiB2.7\n-0.1-0.05 0 0.05 0.1\nΓ X P Γ3 Z Γ Γ1 Γ2E - EF (eV)\n-0.1-0.05 0 0.05 0.1\n 0\n-0.02-0.01 0 0.01 0.02K-point resolved MAE (eV)\nFIG. 6. Band structure of Fe 5SiB2calculated in fully relativistic approach for quantizatio n axes 100 (red lines) and 001 (blue\nlines), together with the MAE contribution of each k-point (green lines) as obtained by the magnetic force theor em.\n0.9 0.92 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 1.1-0.4-0.200.20.4\nV/V0MAE (MJ/m3)\nFIG. 7. The volume dependency of the MAE for Fe 5SiB2.\n0.95 1 1.050.850.900.951.001.051.101.15\nV/V0M/M0\nFIG. 8. The volume dependency of the magnetic moment for\nFe5SiB2- black circles. For a better perception of its deviation\nfrom linearity the arbitrary (dashed) straight line is draw n.\nOur studies of MAE( MS+ML) are then summarized\nin Fig. 12, where the total magnetic moment MS+ML\nwas varied by changing volume, fixed spin moment\n(FSM) or Fe/Co concentration. The MAE( MS+ML)\nvariations induced by FSM or volume scaling are very\nsimilar. The behavior of MAE( MS+ML) related to\nalloying Fe 5SiB2with Co exhibit significant differences\nfrom the other two, which most probably roots in the0 1 2 3 4 5 6 7 8 9 1000.511.52MAE (MJ/m3)\nMS+ML (µB/formula)\nFIG. 9. The total magnetic moment dependency of the MAE\nfor Fe 5SiB2. Based on fully relativistic fixed spin moment\n(FSM) calculations.\n0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-2-1.5-1-0.500.51MAE (MJ/m3)\nCo concentration x\nFIG. 10. The Co concentration dependency of the MAE for\n(Fe1−xCox)5SiB2.\nchange of the number of electrons in the system, accom-\npanied with the Co alloying.\nThese differences can be examined based on the de-\ntailed band structure analysis as presented in Fig. 13.\nThe energy window from −0.5 to 0 .5 eV allows to keep\ntrack on the variation of Fermi level with fixing spin mo-8\n6 7 8 900.51MAE (MJ/m3)\nMS+ML (µB/formula)x = 0.0x = 0.1x = 0.2x = 0.3\nx = 0.4\nx = 0.5\nFIG. 11. The total magnetic moment dependency of the MAE\nfor (Fe 1−xCox)5SiB2.\n8 8.5 9 9.5 10-0.500.511.5\n V/V0\n FSM\n (Fe1-xCox)5SiB2MAE (MJ/m3)\nMS+ML (µB/formula)x = 0.1x = 0.2x = 0.3\nV/V0 = 0.92\nV/V0 = 1.08\nFIG. 12. The total magnetic moment dependencies of\nthe MAE for Fe 5SiB2, mediated by volume variation\n(V/V 0), fixed spin moment (FSM) and alloying with Co\n((Fe 1−xCox)5SiB2). Dashed line denotes equilibrium state\nwith x= 0, V/V 0= 1, and M/M 0= 1.\nment FSM and doping with Co. Together with electronic\nbands the MAE contributions of each k-point are pre-\nsented as obtained by the magnetic force theorem37,38.\nThe direction between high symmetry points XandP\nhas been chosen because of the exceptionally simple form\nof bands in comparison to the other directions in Fig. 6.\nForX-Pdirection it is easy to notice how the band is\nfilled with an increase of Co concentration x(panels d),\ne), f), and g)) or a corresponding behavior induced by\nreduction of spin moment (panels d), c), b), and a)).\nThe filling of the bands entails the change in MAE. In\nthe energy window from −0.5 to 0 .5 eV the differences\nin spin-orbit splitting for two quantization axes 100 and\n001 might be difficult to notice. Nonetheless, the band\nstructures calculated for two perpendicular quantization\naxes are marked in red and blue, respectively. Differences\nin spin-orbit splitting can be observed on several bands\ncrossings. For Co concentration x= 0 (Fig. 13d)) it is\nparticularly easy to notice that the extra positive contri-bution to MAE is related to the band crossing Fermi\nlevel. For this band the 001 solution (blue line) lies\nslightly below the corresponding 100 band (red line), thus\nfor the whole filled region, up to Fermi level, the positive\ncontributions to MAE is observed.\nThe substantial differences in MAE(M S+M L) medi-\nated by FSM or Co alloying raises the question whether\nthis two mechanism work together. In Fig. 14the FSM\nand VCA dependencies of MAE are presented in a form of\ntwo-dimensional color map in superposition with the the-\noretical equilibrium total magnetic moments (M S+M L).\nThe MAE landscape reveals that by going from equi-\nlibrium magnetic moment of pure Fe 5SiB2towards the\nmuch lower equilibrium magnetic moment of Co 5SiB2the\nequilibrium MAE path (black dots) starts from negative\nsign, goes across the range of positive values and follows\nthrough a steep hollow of negative values. This equi-\nlibrium path crossing the MAE landscape correspond to\nthe MAE(x) function presented in Fig. 10. Focusing on\nthe uniaxial anisotropy, the highest MAE value along the\nequilibrium path is reached for the (Fe 0.7Co0.3)5SiB2al-\nloy and is equal to 1.16 MJ/m3. We note that on the\ncalculated MAE landscape, there is a region with about\ntwice higher values of MAE for x= 0.1 and M S+M L\nabout 5.8 µB/f.u. To approach this region, starting, e.g.,\nfrom the (Fe 0.7Co0.3)5SiB2composition, both the mag-\nnetic moment and the d-electrons number (proportional\ntox) should be reduced. The desired magnetic moment\nreduction, from 7.75 to around 5.8 µB/f.u., is around\n25%. This can be obtained by alloying (Fe 0.7Co0.3)5SiB2\nwith 25% of a suitable non-magnetic element. The non-\nmagnetic alloying should at the same time decreases x\n(which may be understood as the number of delectrons\nbeyond those of Fe) by about 0.2. Such decrease can be\nobtained by alloying (Fe 0.7Co0.3) with elements having\nless of delectrons, i.e., the elements from columns of the\nperiodic table preceding the column with Fe. Among 3 d-\nelements which fulfill the latter condition are Cr or Mn,\nbut these often carry rather large magnetic moments,\ntherefore they would not fulfill the first condition – re-\nduction of average moment per atom. From 4 d- and 5 d-\nelements possible candidates are Mo and Tc, or W and\nRe, respectively.\nSimilar analysis of MAE landscape as presented above\nhas been previously performed for (FeCo) 2B11and led\nto very similar results. The additional calculations for\n(FeCo) 2B with W and Re confirmed that these doped\nsystems should exhibit significant increase of MAE. Sub-\nsequently conducted synthesis succeeded in producing\n(Fe0.675Co0.3Re0.025)2B sample for which the theoreti-\ncally predicted increase of MAE has been observed11.\nThus we propose that also for (Fe 0.7Co0.3)5SiB2, alloying\nit with with W and Re might help to improve MAE. This\nis, however, outside the scope of the present manuscript\nand is left for future investigations.\nThe VCA yields similar results of MAE as a func-\ntion of xin (Fe 1−xCox)5SiB2when comparing with\n(Fe1−xCox)2B11. Nevertheless, the VCA may quanti-9\n-0.4-0.3-0.2-0.1 0 0.1 0.2 0.3 0.4\nX Pa) 7.7µBE - EF (eV)\nX Pb) 8.2µB\nX Pc) 8.7µB\nX Pd) 9.2µB;  x = 0\nX Pe) x = 0.1\nX Pf) x = 0.2\nX Pg) x = 0.3\n-4-3-2-101234K-point resolved MAE (meV)\nFIG. 13. Band structures calculated in fully relativistic a pproach for quantization axes 100 (red lines) and 001 (blue l ines),\ntogether with the MAE contributions of each k-point (green line) as obtained by the magnetic force theore m. Panels d), c),\nb), and a) depict results for Fe 5SiB2with an increasing fixed spin moment (FSM), in representatio n of total moment M S+M L\n[µB] per formula unit. Panels d), e), f), g) present increase of C o concentration xin (Fe 1−xCox)5SiB2alloys.\ntatively overestimate the MAE11,32,33. Hence, calcula-\ntions have also been attempted using the SPRKKR40,41\nmethod in the atomic sphere approximation (ASA) with\nthe coherent potential approximation (CPA) to treat the\nalloying. However, it turns out that the SPRKKR-ASA\ncalculations yield magnetic moments in poor agreement\nwith the full potential calculations performed with the\nFPLO, as illustrated in Table IV. We suspect that the\nprimary reason for this discrepancy is the lack of full-\npotential effects, since our SPRKKR calculations were\ndone with the same exchange-correlation functional30\nand the ASA might not give an accurate description of\nmagnetic properties in non-close packed systems11. To\nconfirm this suspicion, full potential calculations were\nalso performed for Fe 5SiB2in SPRKKR and as seen in\nTable IV, the agreement with the FPLO is much better.\nThe failure of the ASA is likely to be related to the empty\nspaces observed in the structure of this system15, since\nit should expected to be a good approximation for close\npacked structures. The magnetocrystalline anisotropies\ncalculated with the CPA in SPRKKR-ASA (not shown)\nqualitatively disagree with the FPLO VCA results but\nas the MAE is a delicate magnetic property depending\nsensitively on the electronic structure, we do not con-\nsider MAE values obtained in SPRKKR-ASA to be reli-\nable when the magnetic moments are not accurately de-\nscribed. Full potential MAE calculations using CPA are\nmore challenging and beyond the scope of the current\nwork. We point out that VCA prediction agrees with theTABLE IV. Spin magnetic moments, in µB, in Fe 5SiB2as\ncalculated in the SPRKKR-ASA, SPRKKR-FP, and FPLO.\nFe1Fe2Si B formula\nSPRKKR-ASA 1.73 2.63 -0.12 -0.15 9.22\nSPRKKR-FP 1.86 2.22 -0.15 -0.16 9.20\nFPLO 1.87 2.24 -0.25 -0.25 8.97\nrecent experimental results24of MAE increase for alloy-\ning Fe 5SiB2with Co. McGuire and Parker24have ob-\nserved for (Fe 0.8Co0.2)5SiB2(Fe4CoSiB 2) the increase of\nanisotropy field accompanied by 20% reduction of mag-\nnetic moment, in comparison to the pure Fe 5SiB2. Their\nmeasurements done on Fe 5SiB2are also in good agree-\nment with our experimental results. For Fe 5PB2Lamich-\nhane et al.25report experimental values of MAE = 0.5\nMJ/m3, and total magnetic moment equal to 8.6 µB/f.u.\nMcGuire and Parker24calculated in plane anisotropy\nwith K 1equal to -0.42 (-0.28) MJ/m3with total magnetic\nmoment 9.15 (9.20) µB/f.u. for Fe 5SiB2, and uniaxial\nanisotropy with K 1equal to 0.46 (0.35) MJ/m3with to-\ntal magnetic moment 8.95 (9.15) µB/f.u. for Fe 5PB2, for\ncomparison the results of our calculations in parenthesis.\nThe differences between MAE calculated by us and by\nMcGuire and Parker most likely come from the fact that\nwe have used the optimized lattice parameters instead of\nthe experimental ones. The fact that we employed the\nFPLO and McGuire and Parker used the WIEN2k should\nnot lead to substantial differences11.10\n 0  0.2  0.4  0.6  0.8  1 0 2 4 6 8 10\n-2-1 0 1 2MAE (MJ/m3) \n22\n1.51.51.51.511\n11\n0.50.5\n0000\n00\n-0.5-0.5\n-0.5-0.5-0.5-0.5\n-1-1-1-1-1.5-1.5\n 0  0.2  0.4  0.6  0.8  1\nCo concentration x 0 2 4 6 8 10MS+ML (µB/formula unit)\n 0  0.2  0.4  0.6  0.8  1 0 2 4 6 8 10\nFIG. 14. MAE as a function of xand total magnetic moment\n(MS+M L) for (Fe 1−xCox)5SiB2. Disorder was treated by the\nVCA. M S+M Lwas stabilized with fixed spin moment (FSM)\napproach. Equilibrium M S+M Lare denoted by the black\ndots.V. SUMMARY AND CONCLUSIONS\nWe have presented an experimental study of struc-\ntural and magnetic properties of Fe 5SiB2polycrystalline\nmaterials. This study was computationally extended\nacross the whole range of Fe 5Si1−xPxB2, Fe 5P1−xSxB2,\nand (Fe 1−xCox)5SiB2alloys, with an effort to evaluate\nthe magnetocrystalline anisotropy energies. Theoreti-\ncal study of volume variation and fully relativistic fixed\nspin moment calculations of stoichiometric Fe 5SiB2ev-\nidences a strong inverse dependency between MAE and\ntotal magnetic moment, leading to a maximal value of\nMAE = 1.78 MJ/m3forMS+ML= 6.15 µB/f.u., while\nequilibrium magnetic moment is 9.20 µB/f.u. Alloying of\nFe5SiB2with Co is suggested to reproduce reduction of\nmagnetic moment. The whole range of (Fe 1−xCox)5SiB2\nconcentrations is calculated and evidence of a MAE max-\nimum is obtained for Co concentration x= 0.3, with\nthe value of MAE = 1.16 MJ/m3and with the mag-\nnetic moment 7.75 µB/f.u. Thus, (Fe 0.7Co0.3)5SiB2ap-\npears to be a promising candidate for a rare-earth free\npermanent magnet. Also further increase of MAE of\n(Fe0.7Co0.3)5SiB2is expected by doping with some of 4 d-\nor 5d-elements. For Fe 5Si1−xPxB2a monotonic trend in\nMAE has been obtained, suggesting that at low tem-\nperatures and below x≈0.8 the alloys should have in-\nplane magnetization, while above x≈0.8 the magnetiza-\ntion should point along the c-axis. The overall MAE val-\nues are rather low, suggesting that Fe 5Si1−xPxB2class\nof materials can only form soft magnets. On the other\nhand the hypothetical Fe 5P1−xSxB2compositions with\nsulfur exhibit increased MAE with the highest value of\n0.77 MJ/m3for Fe 5P0.4S0.6B2and MAE = 0.67 MJ/m3\nfor Fe 5SB2.\nVI. ACKNOWLEDGEMENTS\nWe gratefully acknowledge the financial support of\nG¨ oran Gustafsson’s Foundation, Swedish Research Coun-\ncil and an EU-FP7 project REFREEPERMAG.\n1T. Burkert, L. Nordstr¨ om, O. Eriksson, and O. Heinonen,\nPhys. Rev. Lett. 93(2004) 027203.\n2G. Andersson, T. Burkert, P. Warnicke, M. Bj¨ orck, B.\nSanyal, C. Chacon, C. Zlotea, L. Nordstr¨ om, P. Nordblad,\nand O. Eriksson, Phys. Rev. Lett. 96(2006) 037205.\n3E. K. 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Phys. 4(2011) 096501.\n41H. Ebert, The munich spr-kkr package, version 7.2,\nhttp://ebert.cup.uni-muenchen.de/SPRKKR"    },    {        "title": "1603.03794v2.Atomic_scale_control_of_magnetic_anisotropy_via_novel_spin_orbit_coupling_effect_in_La2_3Sr1_3MnO3_SrIrO3_superlattices.pdf",        "content": "Atomic -scale control of magnetic anisotropy via novel spin -orbit coupling \neffect in La 2/3Sr1/3MnO 3/SrIrO 3 superlattices  \n \nDi Yi‡*1, Jian Liu‡*2,3,4, Shang -Lin Hsu1,5, Lipeng Zhang7, Yongseong Choi6, Jong -Woo Kim6, Zuhuang \nChen1, James D. Clarkson1, Claudy  R. Serrao1, , Elke Arenholz8, Philip J. Ryan6, Haixuan Xu7, Robert \nJ. Birgeneau1,3,4 and Ramamoorthy Ramesh1,3,4 \n \n \n1. Department of Materials Science and Engineering, University of California, Berkeley, California 94720, \nUSA \n2. Department of Physics and Astronomy, University of Tennessee, Knoxville, Tennessee 37996, USA  \n3. Department of Physics, University of California, Berkeley, California 94720, USA  \n4. Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA  \n5. National  Center for Electron Microscopy, Lawrence Berkeley National Laboratory, Berkeley, California \n94720, USA  \n6. Advanced Photon Source, Argonne National Laboratory, Argonne, Illinois 60439, USA  \n7. Department of Materials Science and Engineering, University of Tennessee, Knoxville, Tennessee \n37996, USA  \n8. Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA  \n \n* Email: yid@berkeley.edu , jianli u@utk.edu  \n‡ These authors contributed equally to this work  \n \n \n \n \n \n \n Abstract  \n \nMagnetic anisotropy (MA) is one of the most important material properties for modern  spintronic \ndevices . Conventional manipulation of the intrinsic MA, i.e. magnetocrystalline  anisotropy (MCA), \ntypically depends upon  crystal symmetry. Extrinsic  control over the MA is usually  achieved by \nintroducing shape anisotropy  or exchange bias from another magnetically ordered material.  Here \nwe demonstrate a pathway to manipulate MA  of 3d transition metal oxides (TMOs) by digitally \ninserting non -magnetic 5 d TMOs with pronounced spin-orbit coupling ( SOC ). High quality \nsuperlattices comprised of ferromagnet ic La2/3Sr1/3MnO 3 (LSMO)  and paramagnet ic SrIrO 3 (SIO) \nare synthesized with the precise control of thickness at atomic scale. Magnetic easy axis \nreorientation is observed by controlling the dimensionality of SIO, mediated through the emergence \nof a novel spin -orbit state within the nominally paramagnetic  SIO.  \n \n \n \n \n \n \n \n \n \n \n \n \n Magnetic anisotropy (MA) is one of the fundamental properties of magnetic materials. The widespread \nscientific interest in MA originates from its decisive role in determining a rich spectrum of physical \nresponses, such as the Kondo effe ct (1), the magneto -caloric effect  (2), magnetic skyrmions  (3), etc. From \na technological viewpoint, it is an important and promising approach to control MA by external stimuli, \nsuch as electric field  (4). In general, there are two approaches to design MA of a ferromagnet. In the first \napproach one manipulates the intrinsic magnetocrystalline anisotropy (MCA), deriving from the local \ncrystal symmetry and spin -orbit coupling (SOC) of the magnetic ion  (5-7). Alternatively, one can tune the \nMA though extrinsic contributions to the anisotropy such as shape  (8) or exchange coupling to a strong \nantiferromagnet  (9). \nOne focus of magnetism research is 3 d transition metal oxides (TMOs), a class of materials that exhibit \nvarious functionalities including ferromagnetism due to the strong electron -electron correlation. However \nSOC is usually weak or negligible in 3 d TMOs. On the other hand, the pronounced SOC of heavy elements \nhas drawn attention in recent years due to the emergence of new topological states of matter  (10-12) and \nspintronics  (13, 14). In contrast to 3 d TMOs, the correlation strength is often too small in 5 d TMOs to \nhost magnetism. Therefore it is an interesting approach to design systems that combine the merits of these \ntwo fundamental interactions. Similar ideal has been studied in metal multilayers  (15, 16). However it still \nremains an important challenge to explore the ideal in complex oxides, where a variety of emergent \nphenomena have been discover ed due to the power of atomic -scale confinement and interfacial coupling  \n(17-21). \nHere we present an approach towards accomplishing this goal by atomic -scale synthesis. By fabricating \nhigh quality superlattices comprised of 3 d and 5d TMOs, we address two o pen questions: the effect of \nSOC on the functionality of 3 d TMOs and the possible emergent magnetic state of 5 d TMOs. So far this \napproach has been limited and overlooked. To the best of our knowledge, SrTiO 3/SrIrO 3 is the only 3 d/5d superlattice that has been experimentally studied  (22), which reveals the effect of dimensional \nconfinement. However the 3 d state is rather inactive in that system. Here we study a model system \ncomprised of ferromagnetic La 2/3Sr1/3MnO 3 (LSMO) and paramagnetic SrIrO 3 (SIO). We h ave discovered \nthat the magnetic easy axis of LSMO rotates between two crystallographic directions, i.e. 〈100 〉 and 〈110〉 \n(pseudo -cubic) by digitally reducing the SIO thickness down to one monolayer. Remarkably, the \nreorientation of MA is accompanied by the emergence of a large spontaneous, orbital -dominated magnetic \nmoment of the 5 d electrons, revealing a heretofore -unrepor ted spin-orbit coupled state.   \nResults and Discussion  \nThe colossal magnetoresistive system LSMO is a 3 d ferromagnet with a high Curie temperature  (23). Due \nto the potential for applications in all -oxide spintronics , the MA of LSMO thin films has been investigated \nextensively. Previous studies have established the magnetic easy axis of an epitaxial LSMO thin film on \n(001) oriented SrTiO3 (STO) substrate to be in-plane  along the crystallographic 〈110 〉 (7, 24) as a \nconsequence of the strain superimposed on the intrinsic rhombohedral symmetry. SIO is an end member \nof the Ruddlesden -Popper  (RP) series Sr n+1IrnO3n+1(25). It has been identified to be a spin -orbit -coupled \nparamagnet without any signature of long -range magnetic o rdering  (26, 27), likely due to the topological \nnature of its metallicity  (28). Owing to the structural compatibility, it has been theoretically proposed as \na key building block for stabilizing topological phases at interfaces and in superlattices  (12, 29) . To \ninvestigate the impact of artificial confinement and interfacial coupling, we intentionally insert m unit \ncells (uc) of SIO every 3uc of LSMO with m being varied from 10 to 1 in order to scale down the SIO \nlayer from 4nm to 0.4nm (labeled as SL3m). Al l the superlattices are deposited on SrTiO3 (STO) (001) \nsubstrates. The precise control of thickness is achieved by monitoring the intensity oscillations of the \nreflection high -energy electron diffraction (RHEED) pattern during the growth ( SI Appendix,  Fig. S1), \nrevealing the layer -by-layer growth mode of both LSMO and SIO. This growth mode is critical for synthesizing high quality superstructures. Details of the synthesis protocols used in our study are provided \nin the Methods section.  \nThe high quality of the superlattice s characterized by several techniques  demonstrates the precise control \nof thickness  at the atomic -scale. Fig. 1A  shows a scanning transmission electron microscopy (STEM) \nimage of a superlattice with differing periodicitie s in repeated patterns. The sharp Z -contrast of B -site \nspecies across the interface, supplemented by the line profile of the electron energy loss spectroscopy \n(EELS) of A -site La atoms ( SI Appendix,  Fig. S2), indicates minimal inter -diffus ion at the interf ace. Fig. \n1B shows the high -resolution θ -2θ x -ray diffraction of the SL31 and SL35. The satellite peaks \ncorresponding to the superlattice structure and the finite size oscillations arising from the thickness are \npronounced, suggesting the high degree of in terface abruptness and agreement with the  intended \nperiodicity. Fig. 1C  shows the reciprocal spacing mapping (RSM) of sample SL35, revealing that the \nsuperlattice is coherently strained by the STO substrate. Further structural characterization data are sho wn \nin SI Appendix,  Fig. S1.  \nThe temperature -dependence of the magnetization of the superlattices ( SI Appendix,  Fig. S4) is similar to \nthat of pure LSMO thin film (albeit with a decrease of T c as m increases), indicating that the overall \nmagnetization is do minated by the ferromagnetic LSMO component. In order to study the MA, \nmagnetization loops are measured along different cry stallographic axes of SL3 m (Materials and Methods \nsection). First, the magnetic easy axis is revealed to be in the film plane by comp aring the in -plane and \nout-of-plane magnetic loops ( SI Appendix,  Fig. S4), consistent with our expectations (due to the strain \neffect and shape anisotropy). Additionally, magnetization loops along symmetry -equivalent in -plane \ndirections, e.g. [100] and [01 0] (Fig. 2B ), demonstrate that the MA is biaxial (four -fold rotational \nsymmetry with π⁄2 periodicity), which is indicative of MCA and thus rule s out the influence of shape \nanisotropy  (8). The impact of SIO is demonstrated by the systematic influence of the SIO layer thickness on the MA (Fig. \n2A), which is represented by the normalized difference between the remnant magnetization along 〈100〉 \nand 〈110〉 directions. The positive sign corresponds to the 〈100〉 easy axis while the negative sign \nindicates a π⁄4 shift to 〈110〉. As can be seen, the superlattices with long periodicity ( m>5, i.e. 2nm) exhibit \n〈110 〉 easy axis, identical to that of pure LSMO thin film (purple dot, Fig. 2A ). Intriguingly, as m reduces \n(m<5), a reorientation of the easy axis to 〈100 〉 is observed . The magnitude of the normalized difference \nsystematically increases as m decreases, revealing a tunability of ~40% (theoretical limit ~58% of the \nbiaxial MA, magnetic moments aligning along one direction have a π⁄4 projection on the other direction). \nWe also carried out anisotropic magnetoresistance (AMR) measurements to validate the observed MA. \nThe longitudinal resistance is measured along 〈100 〉 direction and the magnetic field is ro tated in -plane \nwith respe ct to the current direction ( Materials and  Methods section). F ig. 2C  shows the polar plots of \nAMR of SL33 and SL310. The four -fold rotation of AMR reflects the same symmetry  of the MCA. The \nπ/4 phase shift of AMR between SL33 and S L310 is coincident with the MA evolution shown in Fig. 2A . \nFurther analysis of the AMR of SL3 m is shown in SI Appendix  Fig. S5 and confirms the change of MA \nas the SIO thickness reduces. The temperature dependence of the novel MA (〈100〉 easy axis) is acqu ired \nby measuring magnetic hysteresis loops in different orientations at multipl e temperatures. Fig. 3A  shows \nthat the MA persists to the Curie temperature (~270K) of the superlattice.  \nA close examination of the results discussed above reveals the unique n ature of the MA tailoring. First, as \npointed out, the MA with a π/4 phase shift of easy axis is not due to the shape anisotropy  (8). Since the \nLSMO dominates the aggregate magnetization , one must consider the potential contribution from LSMO \ncrystal sy mmetry change. Previous studies have revealed a possible mechanism that could lead to the \nreorientation of in -plane easy axis of LSMO thin films. It has been demonstrated that a moderate biaxial \ncompressive strain on LSMO could lead to the orthorhombic structure and 〈100 〉 easy axis due to  the asymmetry of octahedral rotation patterns  (5). RSM measurements ( represented by SL35 in Fig. 1C ) \nreveal that our superlattices are coherently constrained by the substrate, confirming that LSMO is under \nbiaxial tensile  strain. Another possible contribution is the interfacial octahedral coupling  (30), considering \nthe difference between LSMO and SIO  (23, 26). In this scenario, one expects the rotational pattern of the \nLSMO to be unaltered by the thinner SIO  (30), therefor e the short period superlattice (SL31) would have \na reduced tendency than the long period superlattice (SL310) to show the reorientation of magnetic easy \naxis compared to pure LSMO film. This is however opposite to the observed thickness evolution in Fig. \n2A. In fact, x -ray diffraction measurements of several half ordering reflections  of SL31 rule out the \nalteration of octahedral rotation pattern as the origin ( SI Appendix section  b and Fig. S3). The results thus \nimply a distinct role of the strong SOC in S IO to engineer the MA.  \nTo gain more insight in the spin -orbit interaction, we investigated the valence and magnetic state of both \nLSMO and SIO by carrying out element  selective X -ray absorption spectroscopy (XAS) and X -ray \nmagnetic circular dichroism (XM CD) measurements  (31, 32). Fig. 3C and D  show the XAS spectra of \nMn (red curve) and Ir (blue curve) of the SL31 taken at the resonant L 2,3 edges. As a comparison, XAS \nspectra of reference samples o f LSMO (purple curve, Fig. 3C ) and SIO (purple curve, Fig. 3D) thin films \nwere taken simultaneously. The absence of peak position shift and the identical multiplet features suggest \na minimal effect of charge transfer between  Mn and Ir cations. Figure 3E  shows the Mn and Ir XMCD \nspectra. The large dichroism at Mn e dge is expected for the highly spin -polarized ferromagnetic LSMO \nand consistent with magnetometry. However the presence of a sizable XMCD at the Ir edge reveals the \nonset of a net magnetization, unexpected for the paramagnetic SIO  (26). To validate this ob servation, the \nXMCD spectra were taken by multiple measurements with alternated x -ray h elicity and magnetic field \n(Materials and Methods section). The opposite sign of dichroism of the two cations indicates that the Mn \nand Ir net moments are antiparallel t o each other. This nontrivial coupling is further demonstrated by the coincident reversal of LSMO magne tization and Ir -XMCD (Fig. 3B ). Furthermore, the temperature \ndepend ence of Ir edge XMCD (Fig. 3A ) reveals a relatively high onset temperature (near room \ntemperature), which is closely related to the Curi e temperature of LSMO. The combination of  these results \nsuggest s the emergence of magnetic ordering in the nominally paramagnetic SIO in the ultrathin limit.  \nTo understand the origins of the Ir  moments, sum -rules analy sis of XMCD spectra in Fig. 3E  was applied \nto differentiate the spin component from the orbital counterpart  (33), which yields an unexpected result. \nA relatively large orbital moment 𝒎𝒍=(𝟎.𝟎𝟑𝟔 ±𝟎.𝟎𝟎𝟑 ) 𝝁𝒃/𝑰𝒓 is obtained  for SL31 compared with \nthe effective spin component 𝒎𝒔𝒆=(𝟎.𝟎𝟎𝟐 ±𝟎.𝟎𝟎𝟑 ) 𝝁𝒃/𝑰𝒓 (SI Appendix section e  and Fig. S6 ). Such \na large ratio of 𝒎𝒍𝒎𝒔𝒆⁄  is to date unreported even in the 5d TMOs. As a comparison, sum rules analysis \nwas also applied to the Mn L edge, which yields a 𝒎𝒍𝒎𝒔𝒆⁄  ratio less than 0.01 and  is consistent with the \ndominant role of the spin moment for 3d TMOs ( SI Appendix section e  and Fig. S7 ). \nIn order to further understand the magnetic behavior of SIO within the confines of the superstructure \nenvironment, we also performed first -principles density functional calculations with generalized gradient \napproximation (GGA) + Hubbard U + spin -orbit coupling (SOC) (SI Appendix section f). We compared \nthe energies of configurations where the Mn moments ali gn in the 〈100〉 and 〈110 〉 directions in SL31. \nSince correlated oxides are notoriously challenging for GGA, we explored a variety of U parameter \ncombinations. While the magnitude of the energy difference depends on the details of the parameters, the \n〈100〉 direction is energetically more favorable than the 〈110 〉 direction in SL31, consistent with the \nexperiments. Moreover, the monolayer of SIO in SL31 develops a canted in -plane antiferromagnetic \nordering (weak ferromagnetism), which is similar to the mag netic ordering of Sr 2IrO 4 (34). However , \nwhile the moments of Sr 2IrO 4 are known to align along 〈110 〉 direction (35, 36) , the moments of SIO in \nSL31 prefer 〈100〉, highlighting a key distinction in terms of MA ( Fig. S8 ). In summary, XMCD and first -\nprinciples calculations both reveal the emergence of the weak fe rromagnetism in the ultrathin SIO, which shows an orbital -dominated moment and a different magnetic anisotropy compared to the Ruddlesden -\nPopper series iridates (discussed in SI Appendix sectio n g). \nThis distinctive character of the Ir moment in the superlattices presents an unconventional spin -orbit state \nin the iridate family. Due to the strong spin -orbit coupling, the low -spin d5 configuration of Ir4+ valence \nstate within the octahedral crystal field fills the d-shell up to ha lf of a spin -orbit coupled doublet  (Jeff=1/2 \nstate), which has been theoretically established  (37) and experimentally observed (34) for several iridates, \nfor example, Sr 2IrO 4. This Jeff=1/2 state, regarded as the main driver of Ir related physics, is characterized \nby the distinct orbital character and orbital mixing of t2g bands, leading to the ratio of 𝒎𝒍𝒎𝒔𝒆⁄  ~0.5 (37). \nExperimentally it is characterized by the absence of the L2 edge magnetic dichroism, due to dipole \nselection rules (34, 38). Our XMCD results of SL31 unambiguously reveal the breakdown of the Jeff=1/2 \npicture in the superlattices by considering how the sign and amplitude of L 2 and L 3 edge XMCD signatures \nare equivalent and comparable respectively, thereby yielding a large 𝒎𝒍𝒎𝒔𝒆⁄  ratio as discussed above. In \norder to enhance the orbital component relative to the spin component, the new spin -orbit state is  likely \nto be formed by mix ing the Jeff=1/2 state with the  Jeff=3/2 state, where the two components are antiparallel \n(discussed in SI Appendix  section e). Moreover this spin -orbit state was not reported before in the \nSTO/SIO example (22) that is dominated by dimensional confinement,  which also clearly implies the \ndecisive role of interfacial coupling, beyond  the dimensionality effects.  \nThe emergent weak ferromagnetism in SIO exhibits a close correlation to the control of MA. As the \nthickness m reduces, the stability of  〈100〉 easy axis (Fig. 2A ), the emergent  weak ferromagnetism (Fig. \n3E) and the m𝑙/m𝑠𝑒 ratio (SI Appendix section e, Fig S6 ) become more significant . In addition, the \nemergent weak ferromagnetism shows a similar  temperature dependence to  that of the MA with < 100> \neasy axis (Fig. 3A ). Thus the results suggest a crucial  contribution  to the overall MA from the interfacial \nmagnetic coupling between the Mn spin moments and the emergent Ir orbital moments. The effective in -plane biaxial anisotropic  energy is commonly defined as:  𝑬=𝑲𝐞𝐟𝐟𝐌𝟒𝐬𝐢𝐧𝟐𝟐𝜽 (39), where  M is the in -\nplane magnetization and θ is the angle to 〈100〉 (Fig. 4A ). Taking  into a ccount the superlattice geometry, \nthe effective anisotropy  𝑲𝐞𝐟𝐟 is determined by the competition between MCA of LSMO ( 𝑲𝐜) and SIO-\ninduced anisotropy ( 𝑲𝐢𝐧) (Fig. 4D )). The sign of 𝑲𝐜 remains negative due to the absence of structure \nchange discussed before, which favors the 〈𝟏𝟏𝟎 〉 easy axis (Fig. 4A ). The sign of  𝑲𝐢𝐧 is positive which \nfavors the 〈𝟏𝟎𝟎〉 easy axis (Fig. 4B ). Therefore in the short -period superlattices where the emergent weak \nferromagnetism is more significant, 𝑲𝐢𝐧 overcomes 𝑲𝐜 and becomes dominant in 𝑲𝐞𝐟𝐟 (Fig. 4C ). As m \nincreases digitally,  𝑲𝐞𝐟𝐟 evolves from positive ( 𝑲𝐢𝐧 dominated) to negative ( 𝑲𝐜 dominated), manifesting \nitself as a system atic evolution of MA (Fig. 4E ). The sign of 𝑲𝐢𝐧 reflects the magnetic an isotropy of the \nemergent Ir moment that the Mn couples to. As discussed above, in contrast to the Jeff=1/2 RP phases \nSr2IrO 4 (34) and Sr 3Ir2O7 (40), the new spin -orbit state  in the superlattices features as a mixture of Jeff=1/2 \nand Jeff=3/2. Unlike Jeff=1/2 which is actua lly an atomic J=5/2 spin -orbit  state, Jeff=3/2 is not an eigenstate \nof the SOC operator albeit being an eigenstate of the octahedral crystal field operator. As a result, the \nJeff=3/2 wavefunctions are further  hybridized with eg orbitals and their energies acquire corrections \nproportional to the ratio of SOC and crystal field. The refore the crystal field tends to lock the total angular \nmoment along its principle axis, e.g. 〈100〉, leading  to a large single -ion anisotropy which is absent in \nJeff=1/2 ( discussed in SI Appendix  section g). Thus the mixture of Jeff=1/2 and Jeff=3/2, which can be \nengineered in the superlattices, controls  the magnetic anisotropy  of the emergent Ir moment . This result  \nproffers a new control paradigm in correlated electron behavior.  \nIn conclusion, we present the ability to engineer the MA of ferromagnet ic LSMO by inserting the strong \nSOC paramagnet SIO with atomically controlled thickness. The origin is attributed to a novel spin -orbit \ncoupled state with a relatively large orbital -dominated moment that develops in the typically paramagnetic \nSIO. Our resu lts demonstrate the potential of combined artificial confinement and interfacial coupling  to discover new phases as well as to control the  functionalities . This study particularly expands the current \nresearch interest of the atomic -scale engineering towards the strong SOC 5d TMOs,  which also paves the \nway towards all -oxide  spintronics.  \nMethods  \nSynthesis  (LSMO) 3(SIO) m superlattices with different m were grown  by RHEED -assisted Pulsed Laser \nDisposition on low -miscut STO  substrates. Before the growth, the substrates were wet -etched by buffered \nHF acid, followed by a  thermal annealing process at 1000 ºC for 3 hours in oxygen atmosphere. Both \nLSMO and SIO sublayer s were deposited at 700 ºC and 150 mTorr oxygen partial pressure from the \nchemical stoichiometric ceramic target by using the KrF excimer laser (248nm) at the energy density of \n1.5J/cm2. The repetition rate was 1Hz and 10Hz for each sublayers. During the g rowth, in -situ RHEED \nintensity oscillations were monitored to control the growth at the atomic scale. After growth the samples \ncooled down at the rate of 5 ºC/min in pure oxygen atmosphere.  \n Magnetic and transport measurement  Magnetic measurements were pe rformed on the Quantum \nDesign SQUID magnetometry with an RSO option, which provides a sensitivity of 10-7 emu. In order to \nstudy the MA, magnetization loops were measured along different crystallographic directions of SL3 m: \nin-plane [100], in -plane [110] and out -of-plane [001]. Also magnetization loops were measured along \nsymmetry -equivalent in -plane directions, e.g. [100] and [010], to check the angle dependence of MA. \nTransport measurements were performed using the Quantu m Design Physical Property Measurement \nSystem  (PPMS, 14T). The longitudinal resistance is measured by the four -point probes method with the \nexcitation current (10 μA) flowing in the film plane (crystallographic [100], shown in Fig. 2 (c )). The \nrelative ang le between the magnetic field and current was controlled by rotating the sample holder. The \nmagnetic hysteresis loops and AMR curves reported here have been reproduced on multiple samples.  XAS,  XMCD  and XRD measurement  The XAS and XMCD characterization at  the Mn edge were \ncarried out at beamline 4.0.2 at the Advanced Light Source, Lawrence Berkeley National Lab. The \nmeasurements were performed using the total -electron -yield (TEY) mode and the angle of incident beam \nis 30º to the sample surface. The XAS and  XMCD characterizations at Ir edge were carried at beamline 4 -\nID-D at APS in Argonne National Lab. The results were taken by collecting the fluorescence yield signal \nand the incident beam is 3º to the sample surface. All of the XMCD spectra were measured b oth in \nremanence and in saturation field . Experimental artifacts were ruled out by changing both the photon \nhelicity and the magnetic field direction. Since the XMCD spectra of the Ir -edge is relatively weak, \nmultiple measurements were repeated to increase  the signal -to-noise ratio of the spectra (5 times for each \nspectrum at each field). Also we measured the spectra at different times and on different samples.  The \nhysteresis loop of the Ir -XMCD was measured with energy fixed at 12.828 keV (maximum of L 2 XMCD) \nby altering the photon helicity at each magnetic field. Synchrotron XRD measurements were carried out \nat sector 33BM and 6 -ID-B at APS in Argonne National Lab.  \nAcknowledgement  \nWe acknowledge Dr. Daniel Haskel , E. Karapetrova , S. Cheema , and Dr. J.H. Chu and for experiment \nassistance and Professor L.W. Martin , Professor M . van Veenendaal  and Professor E. Dagotto  for critical \ndiscussions. This work was funded by the Director, Office of Science, Office of Basic Energy Sciences, \nMaterials Science  and Engineering Department of the U.S. Department of Energy (DOE) in the Quantum \nMaterials Program (KC2202) under Contract No. DE -AC02 -05CH11231.  D.Y. is supported by the \nNational Science Foundation (NSF) through Materials Research Science and En gineering  Centers \n(MRSEC) Grant  No. DMR 1420620. We acknowledge additional support for the research at UC Berkeley \nthrough the U.S. Department of Defense (DOD) Army Research Office (ARO) Multidisciplinary \nUniversity Research Initiatives (MURI) Program, Defense Adva nced Research Projects Agency (DARPA) and Center for Energy Effi cient Electronics Science (E3S) . Research at  the University of Tennessee ( J.L., \nL. Z. and H.X) is sponsored by the  Science Alliance Joint  Directed Research and Development Program. \nThis resear ch used resource of  The National Institute for Computational Sciences (NICS) at the University \nof Tennessee under contract UT -TENN0112.  This research used resources of the Advanced Photon Source, \na U.S. Department of Energy (DOE) Office of Science User Fac ility operated for the DOE Office of \nScience by Argonne National Laboratory under Contract No. DE -AC02 -06CH11357. Use of t he \nAdvanced Light Source is supported by the Director, Office of Science, Office of Basic Energy Sciences, \nof the U.S. Department of E nergy under Contract No. DE -AC02 -05CH11231. 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(A) High -angle annular dark -field \n(HAADF) STEM images of LSMO/SIO superlattice  with designed periodicities in one sample. The four \nhigh magnification images correspond to the regions of (LSMO) 1/(SIO) 1, (LSMO) 2/(SIO) 2, \n(LSMO) 3/(SIO) 3 and (LSMO) 5/(SIO) 5 from top to bottom (the number refers to the thickness in unit cell). \nThe atoms are marked by different colors: Ir (brightest contrast) in orange , Mn (darkest contrast ) in green , \nand A -site atoms in blue. (B) θ-2θ X -ray diffractograms of a SL31  (top)  and a  SL35 (bottom) superlattice. \nBoth the superlattice peaks and the thick ness fringes reveal the high degree of interface abruptness. (C) \nX-ray reciprocal spacing mapping of a SL35 superlattice around (103) peak, confirming coherent growth \nof the superlattice.  \nFig.2 Magnetic and transport characterization of the LSMO/SIO super lattice.  (A) Dependence of \nmagnetic anisotropy (MA) on SIO thickness ( m) in the superlattice series SL3m. MA is defined as the \ndifference of remnant moment MR between two crystallographic directions normalized by the saturation \nmoment MS ( (𝑀𝑅[100]−𝑀𝑅[110])𝑀𝑆⁄). The positive sign corresponds to the 〈100〉 easy axis while the \nnegative sign indicates a 𝜋4⁄ shift to 〈110〉. The purple dot shows the anisotropy of LSMO thin film. \nMagnetic easy axis is shown by the arrow in the oxygen octahedral for the series of superlattices. Error \nbars are derived from measurements on multiple samples. (B) Magnetic hysteresis loops of a SL31 \nsuperlattice with magnetic field H in [100] (blue), [110] (red) and [010] (black) crystallographic \norientation. The inset is the plot of ma gnetic hysteresis loops of LSMO (20nm) thin film on STO as a \ncomparison. Magnetization is averaged by LSMO thickness in this study. (C) Schematic and polar plots \nof in -plane anisotropic magnetoresistance (AMR). The current is along [100] direction and the magnetic field (1T) is rotated within the film plane. The polar plots show a phase shift of π/4 between SL33 and \nSL310, consistent with the thickness evolution of MA in (A).  \nFig.3 XAS and XMCD spectra of the LSMO/SIO superlattice.   (A) Temperature depende nce of the \nmagnetization, MA and the XMCD (Ir edge) of SL31 . (B) Field dependence of magnetization measured \nby magnetometer and sign of Ir -XMCD. The opposite sign corresponds to the antiparallel configur ation \nof Mn and Ir moments in (E ). (C), (D) Core -level XAS spectra of Mn and Ir of the superlattice SL31  along \nwith the spectra of SIO and LSMO thin film (purple curve). Peak positions of the XAS spectra of the \nsuperlattice are same as the pure thin films and the multiplet  features are id entical in both Ir and Mn edge \nwithin the experimental limit, suggesting the minimal effect of charge transfer in the superlattice. (E) \nXMCD spectra of the multiple superlattices measured at 10K with 1T applied along [100] direction with \nthe same photon he licity and field direction . The magnitude of XMCD is normalized by the magnitude of \nL3 XAS for Mn and Ir respectively.  For comparison, the magnitude of XMCD of Ir is multiplied by a \nfactor of 25.  \nFig.4 Schematic diagram of the origin of the MA engineering . (A) Magnetic anisotropy energy of \nLSMO. The anisotropy energy is defined by the formula: 𝐸=𝐾∙M4sin22𝜃, whereas M is the in -plane \nmagnetization and 𝜃 is the angle to [100] (red arrows). The MCA of LSMO ( Kc <0) favors 〈110〉 easy \naxis as shown by the black solid lines (energy minimum). (B) SIO -induced magnetic anisotropy energy \n(Kin >0), which favors the 〈100〉  easy axis. (C) Effective magnetic anisotropy energy of SL31, in which \nthe Kin overcomes Kc. Therefore SL31 shows π/4 s hift of easy axis compared to LSMO.  (D) Schematic \ndiagram of the anisotropy contributions . The SIO -induced anisotropy is determined by the interfacial \nexchange coupling to the emergent weak ferromagnetism in SIO (green arrows), which effectively shifts \nthe magnetic easy axis of LSMO by π/4. (E) Thickness evolution of the MA of the superlattice and the \nemergent magnetism in the strong SOC SIO (FM: ferromagnetic, PM: paramagnetic).   \n \n \n \n \nFigure 1  \n \n \n \n \n \n \n \n \n \nFigure 2 \n \n \n \n \n \n \n \nFigure 3  \n \n \n \nFigure 4  \n \n \n \n \nSupplementary information:  \na. Sample preparation and structural characterization (Fig. S1 and S2)  \nFig. S1 (a) shows the RHEED patterns at different growth stages. All of the sublayers exhibit 2D RHEED \npatterns. Fig. S1 (b) shows the RHEED oscillations of SL33, a typical example for the series SL3 m. The \nRHEED oscillations indicate the layer -by-layer growth mode of both LSMO and SIO, the key factor to \nfabricate high -quality superlattice. The total thickness of the superlattices is kept around 20 -25nm (e.g. \n[SL31] 13, [SL32] 10, [SL33] 10, [SL35] 8, [SL37] 6, [SL310] 6, the number refers to the repetitions). Fig. S1 \n(c) is the atomic force microscopy (AFM) image of SL33, which clearly shows the unit -cell step terraces \nwith the width of 600 -800 nm, identical to that of the low -miscut TiO 2-terminated  STO substrate. All the \nsuperlattices in the SL3 m series show the similar topology as Fig. S1 (c). The width of the steps and the \nconsistent orientation of the steps of the substrates used for the SL3m also indicate that th e new ly observed  \nanisotropy change described  in the main text doesn’t originate from step -terrace structure (8). Furthermore, \nthe shape anisotropy i nduced  from the step -terrace structure is uniaxial while the observed anisotropy is \nbiaxial. Fig. S1 (d) sho ws the θ-2θ X-ray diffraction measurements of the superlattice series SL3 m. The \nsatellite peaks correspond to the periodicity and the thickness fringes reveal the high -quality of the samples.  \nFig. S2 shows the STEM image and  electron energy loss spectroscopy (EELS) of La M-edge across the \ninterface in the area (LSMO) 5(SIO) 5. The STEM images shown in this study were filtered by Average \nBackground Subtraction filter (ABSF) to reduce noise. The Z -contrast (Fig. S2 (a)) is dominated by the \nlarge difference of atomic numb er (Z) between B -site atoms Mn and Ir. Therefore the EELS line profile \nof La across the interface is used to check the A -site interfusion. Fig. S2 (b) shows the EELS line profile \nof the (LSMO) 5(SIO) 5 region. The blue and red background correspond to the LS MO and SIO sublayers \nidentified by the Z -contrast, which match with the EELS line profile. The results suggest the A -site inter -\ndiffusion is within half unit cell at the interface.  b. Octahedral rotation pattern of superlattices (Fig. S3)  \nPrevious study has r evealed crystal symmetry  change may lead to a reorientation of magnetic easy axis of \nLSMO from 〈110 〉 to 〈100 〉. As demonstrated by Boschker and coauthors  (5), LSMO thin films under \nmoderate compressive strain have the orthorhombic structure with [001 ]o in the film plane (larger \ncompressive strain would change the easy axis to the out -of-plane direction). The phenomenon  is \nsuggested to arise from the asymmetry of octahedral rotation pattern along two in -plane directions of the \northorhombic distorted LSMO, which leads to a different degree of overlap of d orbitals. The superlattices \nin this study are coherently strained, confirming that LSMO is under tensile strain, same as LSMO thin \nfilm on STO substrate. On the other hand,  SIO thin film is orthorhombic with [001 ]o in the film plane. It \nhas been demonstrated that octahedral rotation pattern could be altered due to interfacial octahedral \nmismatch  (30). Therefore we examined the possibility  whether the  LSMO layer  in the superlattice has the \northorhombic distortion w ith [001] o in the film plane due to the SIO adjacent layer. Here we concentrate \non SL31 (the one shows the stro ngest MA with <100>  magnetic easy axis) and SL310.  \nIn perovskite oxides, octahedral rotations offset the oxygen atoms from the face -centered positions \neffectively doubling the pseudocubic unit cell, producing a distinctive set of half -order B ragg peaks \ndepending on the octahedral rotations  pattern  (41). To describe the phase of the octahedral rotations along \neach axis, we employ the Glazer nota tion, in which  a superscript is appended to each axis to indicate \nwhether neighboring octahedra l rotate  is in-phase (+), out -of-phase ( -) or no rotation (0).  For example, \nrhombohedral unit cell  shows the 𝑎−𝑎−𝑎− rotation pattern and orthorhombic  unit c ell shows the 𝑎+𝑏−𝑐− \nwith [001] o along a axis. The method to determine the rotation pattern is illustrated in details in the \nreference  (41), which is based on presence or absence of specific half -ordering peaks. The rotation patterns \ncan generally be iden tified by measuring a series of [H/2, K/2, L/2] reflections. Therefore there are three \ntypes of half ordering peaks: peaks with one half -integer and two integers, peaks with two half -integers and one integer and peaks with three half -integers. Peaks with t hree half -integers correspond to the \nrhombohedral  structure ( 𝑎−𝑎−𝑎−, out-of-phase rotation along all directions). Peaks with one half -integer \nand two half -integers both correspond to the orthorhombic  structure ( 𝑎+𝑏−𝑐−, in-phase rotation along a \nand o ut-of-phase along b, c).  \nTo examine possible orthorhombic distortion of LSMO as the origin of MA engineering, we need to check \nthe two classes of peaks corresponding to different rotation patterns. Each half ordering reflection  has a \ndifferent structure factor and thus a different intensity. Zhai and coauthors have carefully measured 50 \ndifferent half -ordering peaks on manganite system, which provides a solid reference to differentiate the \nrotation patterns  (42). Here we cho se the peaks with high intensity in our study. Specifically we measure d \nthe (0.5, 0.5, 1.5), (1.5, 2.5, 2.5) and (0, 0.5, 3), (0.5, 0, 3). If LSMO develops the orthorhombic structure \ndue to the SIO, (0, 0.5, 3) and (0.5, 0, 3) peaks should be observed. Fig. S3  shows the X -ray diffraction \nmeasurements of the half-ordering peak s of SL31  and SL310 taken at beamline 6 -ID-B and 33BM , APS \nin Argonne National. As for SL31 (blue), the three half -integer peaks are observed (Fig. S 3 (a) and (b)) \nand the one half -integer peaks ar e absent ( Fig. S 3 (c) and (d)), which can be explained by the octahedral \nrotation without any in -phase rotations. Thus it should be 𝑎−𝑎−𝑐− considering the biaxial strain. This \nresults therefore rule out the mechanism suggested by Boschker and coauthors  (5) as the origin of the MA \nengineering observed here . For SL310 (red), the one half -integer peaks are clearly observed ( Fig. S 3 (c) \nand (d)), which is consistent with the orthorhombic structure of SIO ( 𝑎+𝑏−𝑐−). In conclusion, the \nmeasurements qualitatively rule out th e change of structure of LSMO as the origin of MA engineering by \nconsidering the absence of peaks corresponding to orthorhombic distortion.  \nc. Magnetization measurements (Fig. S4)  \nFig. S4 (a) shows the temperature dependence of magnetization of the superlatti ce series SL3 m. The curve \nwas measured during warming with 200 Oe field applied along the [100] after 1T field cooling. The superlattices show a decrease of T c as m increases, similar to what has been observed in the LSMO/ STO \nsuperlattices  (43, 44) . The effect is probably related to the interlayer coupling of LSMO  (43) or the \nemergent ferromagnetism in the spacer layer  (44), both of which depend on the spacer layer thickness.   \nFig. S4 (b) shows the different crystallographic directions along which the magnetic loops were measured \n(defined in the pseudo -cubic  notion). Fig. S4 (c) shows the magnetization loops of SL31 along in -plane \n[100] and out -of-plane [001] directions, revealing that the easy axis of SL31 is in the film plane. Fig. S4 \n(d)-(h) show  the magnetic loops along two in -plane axis (black for [100] and red for [110]) for SL3 m \n(m=2,3,5,7 and 10), from which the thicknes s dependence of MA in Fig. 2A  is extracted. The π/4 shift of \nmagnetic easy axis is clearly demonstrated. All the hysteresis loops were taken at 10K and the \nmagnetization were averaged by the LSMO thickness.   \nd. Transport measurements (Fig. S5)  \nFig. S5 (a) shows the temperature dependence of resistance (normalized by the resistance at 300K) for the \nsuperlattice series SL3 m. As m increases from 1 to 10, the temperature dependence behavior changes from \nLSMO -like to SIO -like. Fig. S5 (b) shows the AMR of 4uc SIO thin films as a reference sample. The \nblack curve is the AMR measured with 1T magnetic field, which shows the magnitude of  0.01% (close to \nthe resolution of the set -up). The amplitude of AMR of the 4uc SIO increases with the magnitude of \nmagnetic field. AMR at 14T shows 4 -fold rotation dependence with the amplitude of 0.1% and minimum \nat 45 degree ([110]) instead of 90 degree  [100] (the definition of angle is the same as main text). The \namplitude of AMR for the superlattices measured with 1T is in the order of 1%, which is one order higher \nthan that of SIO shown in Fig. S5 (b). Therefore the results suggest that the AMR signal  is mainly \noriginated from  the colossal magnetoresistive LSMO, which is strongly coupled with SIO.  \nFig. S5 (c) shows the AMR measured at 10K with 1T field rotated in -plane for SL3 m. The measurement \ngeometry is the same as main text (Fig. 2C). For SL31, the  AMR is dominated by the uniaxial component (2-fold rotation dependence), which is commonly observed in ferromagnetic metals and independent on \nthe lattice symmetry . As the thickness m increases, the biaxial anisotropy (four -fold) component gradually \ndomin ates the AMR. Despite ongoing debate on the physical origin , the 2 -fold and 4 -fold AMR has been \nobserved and discussed in manganite systems before (45), establish ing several  experimental facts. First, \nthe four -fold AMR reflects the magnetocrystalline aniso tropy of the manganite system (45). Second, there \nis a four -fold to two -fold AMR transition as the applied magnetic field increases. The transition due to the \nmagnetic field was also observed in the superlattice SL33 as shown in Fig. S5 (d), which again co nfirms \nthe dominant contribution from LSMO. The phase of the AMR (the angle of the maximum/minimum) has \nan interesting thickness dependence, which shows the π/4 shift by comparing m=2,3,5 with m=10. The \nresults are consistent with the magnetic easy axis ev olution shown in Fig. 2A and therefore confirm the \nMA evolution. Fig. S5 (e) shows the temperature dependence of AMR for SL33. The π/4 p hase shift of \nAMR (maximum at 45º instead of 90º) persists to high temperature around Curie temperature, which is \nconsistent with the magnetization measurement in Fig. 3A.  \ne. XMCD characterization and analysis (Fig S6, S7)  \nThe sum -rules is applied as following  (46) to both the Mn and Ir L edge to analyze the XMCD spectra:  \n𝑚𝑜𝑟𝑏=−4∫ (𝜇+−𝜇−)𝑑𝐸(𝐿2+𝐿3)\n3∫ (𝜇++𝜇−)𝑑𝐸(𝐿2+𝐿3)(10−𝑛3𝑑) \n𝑚𝑠𝑒=−6∫ (𝜇+−𝜇−)𝑑𝐸−4∫ (𝜇+−𝜇−)𝑑𝐸(𝐿3+𝐿2) (𝐿3)\n∫ (𝜇++𝜇−)𝑑𝐸(𝐿2+𝐿3)(10−𝑛3𝑑) \n \nFig. S6 (a) shows the schematic on the application of the sum -rules for XMCD at Ir -edge. For the hard X -\nray XMCD, the normalization is conventionally  performed by setting th e edge jump of L 3 as 1 (38). For the soft X -ray XMCD, the normalization is conventionally  performed by setting the maximum of XAS as \n1. However no matter which normalization method is applied, the results yielded by sum -rules are \nnormalization -independent  because the XAS and XMCD are multiplied by the same factor during the \nnormalization process. In this study, the Ir -XMCD here is normalized by the maximum of L 3 XAS as \nshown in Fig. S6 (a). In the normalization process, the ratio of L 3 edge jump to L 2 edge jump is kept as 2. \nIn Fig. S6 (a), the black dash line is the background absorption spectra expressed by the two step functions.  \nThe blue shadow areas correspond to the integration of L 2 and L 3 XAS: ( ∫ (𝜇++𝜇−)2⁄𝑑𝐸(𝐿2) and \n∫ (𝜇++𝜇−)2⁄𝑑𝐸(𝐿3)). The red shado w areas correspond to the integration of L 2 and L 3 XMCD \n(∫ (𝜇+−𝜇−)𝑑𝐸(𝐿2) and ∫ (𝜇+−𝜇−)𝑑𝐸(𝐿3)).  \nFig. S6  (b)-(g) shows the integration (“Sum” in each figure) of XAS and XMCD for Ir L 3 and L 2 edges of \nboth the SL31 and SL33. By using the values of integration (Sum) in the formula of  sum-rules, the orbital \nangular momentum and the effective spin momentum can be obtained. For SL31, the analysis yields 𝑚𝑙=\n(0.036 ±0.003 ) 𝜇𝑏/𝐼𝑟 and 𝑚𝑠𝑒=(0.002 ±0.003 ) 𝜇𝑏/𝐼𝑟. (Error bar is estimated by performing the \nsum-rule analysis on multiple measurements) The isotropic  spin component cannot be extracted from the \neffective spin momentum  without a priori  knowledge of  the value of the 〈𝑇𝑧〉 (𝑚𝑠𝑒=𝑚𝑠+7〈𝑇𝑧〉, 〈𝑇𝑧〉 is \nthe ma gnetic dipole operator), which requires further investigations. Besides, i t is not always accurate to \ncorrelate the value of m l and m s from the sum -rule analysis to that of the real system, which  is primarily \nbecause the absolute number of holes within the  valence shell is difficult to extract from x -ray absorption . \nNevertheless, one can still compare reliably the obtained ratio between the orbital and effective spin \nmomenta with that of an ideal Jeff=1/2 state  (38).  \nMoreover, the Ir  edge XMCD spectra show an interesting thickness  dependence. As shown in Fig. 3 E, the \nXMCD signal reduces as m increases. A careful analysis of the results suggests that the reduction cannot be simply attributed to the decreased interface density within th e superlattices . First of all, no XMCD \nsignal was observed on SL310 as shown in Fig. 3E, which would otherwise show a small but observable \nsignal. Second, the comparison between SL31 and SL33 XMCD further supports the conclusion. As \nshown in Fig. S6 (d) an d (g), the XMCD signal of SL33 is roughly 1/4 of that of SL31 on L 3 edge and 1/5 \non L 2 edge, neither one of which  scales with the total thickness. More interestingly, L 2/L3 ratio reduces as \nm increases, which would lead to a change of  𝑚𝑙/𝑚𝑠𝑒 by applying the sum -rules. All these results show  \nthe SIO thickness dependence of  the emergent magnetism and new spin-orbit state. The effect of thickness \nconfinement on SIO has also been shown in other systems  (22, 25) , albeit without the novel spin -orbit \nstate observed here. The possible origin of the effect could be the change of nearest -neighboring coupling  \nor the change of octahedral rotation of SIO  (27) as the m increases.  \nThe sum -rule analysis also suggests that the novel spin -orbit state is likely  to be a mixture of Jeff=1/2 and \nJeff=3/2. Jeff=1/2 originates from the atomic Jeff=5/2 with same sign of orbital and spin moment. The \nincrease of m l/ms, observed by XMCD, suggests that Jeff=1/2 is likely to mix with another state that has \nthe opposite sig n of orbital and spin moment, i.e. the Jeff=3/2 ( Jeff=3/2 originates from the atomic Jeff=3/2 \nwith opposite sign of orbital and spin moment). The mixture is possible if the degeneracy of t2g orbitals is \nlifted, for example, by the interfacial orbital reconstruction (in the ideal Jeff=1/2, the ratio of xy:yz:xz is \n1:1:1) . Alternately, the mixture could also arise from the interface magnetic coupling, which will be \ndiscussed in SI Appendix sect ion g.   \nFig. S7 show the integration (“Sum” in each figure) of XAS and XMCD of Mn edge for the SL31. The \nsum-rules tends to underestimate the magnetic moment when spin -orbit splitting of the transition -metal \n2p core levels is not large enough compared to the 2 p-3d exchange splitting interaction, leading to \nuncertainties in the estimation process for the low 3 d TMOs  (47). However by considering the geometry \nfactor of the measurement (incident angle) and the ‘correction factor’ given in the reference  (42),  the effective  spin component of the Mn moment here is estimated to be around 2.7 𝜇𝑏/𝐼𝑟 and the orbital \ncomponent is estimated to be less than 0.02  𝜇𝑏/𝐼𝑟. Unlike SIO, the 𝑚𝑙/𝑚𝑠𝑒 is drastically  smaller, \nconsistent with the dominant role of the  spin moment for 3d TMOs. The calculated value is consistent \nwith our macroscopic SQUID measurement.  \nf. First  principle calculations (Fig. S8  and Table S1)  \nDensity functional theory calculations were carried out using  the Vienna Ab -Initio Simulation Package  \n(48) (VASP ) with the projector augmented  wave (PAW) method . The corresponding electronic \nconfigurations for each element are Sr: 4s4p5s; Ir: 5d6s; O: 2s2p ; La:  5s5p6s5d; Mn: 3d4s . The cutoff \nenergy is chosen to be 550 eV  based on the convergence tests. The revised version Perdew -Burke -\nErnzerhof functional for solids (PBEsol) under the generalized gradient approximation (GGA)  was chosen \nas the exchange -correlation function. S pin-orbital coupling  is included in the simu lations . For the spin -\norbit coupling  (SOC)  calculation, unconstrained noncollinear magnetism  (49) setting are employed.  The \nquantization axis for spin was set as [0 0 1]. The  local magnetic moment of Ir and Mn atoms in x, y, and \nz directions was initialize d and then  subsequently relaxed.  In additi on, the rotationally invariant +U \nmethod  (50) introduced by Liechtensitein et al. were applied. The chosen effective coulomb -U parameters \nfor each element are discussed in the following part. We used 4× 4 × 2 K-points following the Monkhorst -\nPack scheme  in our systems. The convergence criterion for the electronic relaxation is 10-6 eV.  \nTo simulate the SL31 superlattice , we worked  with the basic  cell depicted  in Fig. S8 (a), which comprises \n4 perovskite layers in the (001) orientation with totally 20 atom s. In periodic calculations it is not possi ble \nto properly mimic the ionic disorder  La/Sr. In this calculation, we treated it in an ordered way by keeping \nthe ra tio as 2/1. When relaxing the cell parameters and atomic positions, the in -plane lattice constant  (a) \nis constrained to that of STO  and we make sure that  c/a fits the experiments .     We used different combinations of U values for Mn, Ir and La in our U -correc ted DFT approach to the \nLSMO/SI O superlattice  as shown in Table S1. The Coulomb interaction parameter  U for Ir 5 d orbital was \nset to be 2 eV, which was successfully used in calculation of perovskite Iridates  (51). The U of Mn 3 d \norbital was chosen to be 3 ev or 4eV, commonly used for the manganites  (18). For La,  we considered La \nwithout U and with U=3 eV  on 5d orbital. All of the parameters yield simil ar structure as shown in Fig. \nS8(a) . The rotation pattern is out -of-phase along the c direction (c-), consistent with the experimental \nresults shown in Fig. S3. In order to determine the magnetic easy axis, the magnetic moment of Mn was \nset to be along the pseudo -cubic [100] or [110] and the magnetic moment of Ir was free to relax in each \ncase to minimi ze the energy. Then the obtained energy minimum of the two configurations were compared \nto determine the magnetic easy axis (lower energy). All of the parameters in Table S1 yield equivalent \nresult with 〈100〉 magnetic easy axis despite of slight numerical di fferences in the energy. Therefore the \ncalculations indicate that  the shift of magnetic easy axis to 〈100 〉 in SL31  is a robust feature . \nFig. S8 (b) shows the relaxed magnetic structure of the low -energy state  (〈100 〉 easy axis)  obtain ed from \nthe calculation. The moment of each Mn cation aligns along [ 1̅00] direction (~3.4 u b/Mn with dominant \nspin component). In the SIO monolayer, an in -pane canted antiferromagnetic ordering (weak \nferromagnetism) emerges with the moment of each Ir  cation along [010] direction (perpendicular to the \nmoment of Mn cation). The canting of Ir moments yields a net moment (~0.03 u b/Ir) that is antiparallel to \nmoment of Mn cation, consistent with the experiment result. This magnetic structure is consistent with \nthe “spin -flop” mechanism observed at many ferromagnet/antiferromagnet interfaces.     \nTo further validate the calculation, we also carried out the same calculation on LSMO without the SIO \nmonolayer by using the same parameters. The spin direction of M n is set to be along [100] and [110] \ninitially. After the spin -orbit calculation, the energy of each relaxed magnetic state is obtained as shown \nin table S1.  The result suggests the easy axis along [110], consistent with experiment, although the energy difference is much smaller compared with SL31. The significant reduction of energy difference is due to \nthe weak SOC of Mn (SOC is proportional  to Z4, Z is the atomic number) , which is the origin of the \nmagnetocrystalline anisotropy. In general, it is a diffi cult task to calculate the magnetic anisotropy energy \nof 3d TMOs by first -principle since the energy difference in many cases are extremely small  (52). The \nreferences on this topic are very limited. To our best knowledge, we cannot find other reference that \ncalculates the magnetic anisotropy energy of manganite. However calculation results on other 3d TMOs \nyield the energy difference in the similar order of magnitude  (53). Therefore our first -principle calculation \nresults qualitatively support the trend we observed experimentally.  \ng. Further discussions of the magnetic anisotropy (Fig. S9)  \nInsights into the interface -controlled magnetic anisotropy observed in the superlattices can be gained \nthrough comparison with the  long-range magnetic ordering s in Sr 2IrO 4 and Sr 3Ir2O7. Sr 2IrO 4 shows the \nin-plane canted antiferromagnetic ordering with moment axis 〈110 〉 (35, 36).  Sr3Ir2O7 shows the c -axis \ncollinear antiferromagnetic ordering w here moment axis [001]  without canting (40). Our experimental \nresults suggest that , while the dimensional magnetic evolution is different from that of the RP series,  the \nemergent magnetic ordering is likely to be the canted in -plane antiferromagnetism (like Sr 2IrO 4). However \nthe magnetic anisotropy of the superlattice , combined with ab -initio calculation, suggests a 〈100〉 easy \naxis for  Ir moments , different from 〈110 〉 in Sr 2IrO 4. The mechanism of the 〈110 〉 easy axis of Sr 2IrO 4 \nhas be en discussed by considering the competition between quantum fluctuations and interlayer coupling  \n(35, 36) . This model suggests that , although  moments of a 2D IrO 2 plane prefers to align along 〈100〉 \ndirection s, the interlayer coupling lowers the energy when moments align along 〈110 〉 due to the layer -\nstacking sequence  of Sr2IrO 4(214) phase (Ir  cations shift by [0.5, 0.5] between neighboring IrO 2 planes).  \nIn contrast, in the superlattice, the interlayer coupling between IrO 2 planes is weak since the two adjacent \nlayers are separated by 3uc LSMO. More importantly, the stacking with a perovskite -type superlattice renders a  crystal symmetry completely different f rom 214 phase. Therefore , applying  the same theoretical \nmodel would conclude  that the moments of Ir in the superlattice geometry should align along 〈100〉 \ninstead of 〈110〉. However, a shortcoming  of this model is that it is only applicable to the single IrO 2 plane \nlimit and the energy scale of this anisotropy would be rather weak due to limitation of the quantum zero -\npoint energy.  \nNotice that all the models applied to  the RP phases are based on  the validity of Jeff=1/2 state  as a good \napproximation. As we discussed above, the new spin -orbit state in the superlattice is significantly deviated \nand likely to be a mixture of Jeff=1/2 and Jeff=3/2. Therefore one must consider the magnetic anisotropy \ncontributed by the mixed Jeff=3/2 component , which, unlike Jeff=1/2, has a relatively large single -ion \nanisotropy as discussed in the following.  \nTo understand the origin of the anisotropy, one can  write down the Hamiltonian matrix that  includes the \nspin-orbit coupling energy (λ), cubic crystal field energy (10Dq) and magnetic energy (h, an effective \nmagnetic field coupled to the onsite moment  along the quantization axis which is set to be along the Z \naxis) in the cubic limit. The stability of a certain magnetic moment direction, i.e. single -ion anisotropy, is \nultimately determined  by the size of the Zeeman splitting. The matrix can be reorganized into four \nindependent blocks with the corresponding ten states in d-shell as shown  in Fig. S9(a). There are clearly \noff-diagonal matrix elements  due to SOC and Zeeman splitting that mix different state s. The presence of \nthese matrix elements is due to the fact that these “ideal ” states are not eigenstates of the SOC operator or \nthe angular momentum operators.  \nThe first block reveals a hybridization between Jeff=3/2( -3/2), Jeff=1/2( -1/2) and eg(x2-y2, up) orbitals by \nthe off -diagonal matrix elements. Based on the perturbation theory, the second order  correction to \neigenenergy of Jeff=1/2 is proportional to h2/λ (hybridization with J eff=3/2) and h2/10Dq (hybridization \nwith x2-y2). Both are purely due to Zeeman splitting.  On the other hand, the change of eigenenergy of Jeff=3/2 is proportional to ( h+λ)2/10Dq (with x2-y2). For iridates, the crystal field 10Dq is round 3  eV (54) \nand SOC energy λ is about 0.4  eV (37). It is not easy to evaluate h . However, even if we  treat it as a large \neffective exchange field (e.g. because of the magnetic coupling of Ir -Mn), a reasonable estimation could \nonly be as large as the order of 10 meV, which leads to 10Dq, λ > > h. Thus the correction is small for \nJeff=1/2. In other words, the Jeff=1/2 state ha s no significant single -ion magn etic anisotropy since the \nenergy is not sensitive to the orientation of moment with respect to the c rystal field. On the other hand for \nJeff=3/2 state, because h* λ/10Dq >> h2/10Dq , the contribution from any effective exchange  field would \nbe effectively amplified by a factor of  ~ λ /h through the hybridization with the  eg (x2-y2) state. Meanwhile, \nsince this effect relies on the mixing with the  eg (x2-y2) state on the order of λ2/10Dq , it is maximized \nwhen the Zeeman quantization axis is aligned with a crystal field principle axis.  In other words, the Jeff=3/2 \nshould have a large single -ion anisotropy since the energy is sensitive to the orientation of moment with \nrespect to the crystal field.  \nTo demonstrate this effect , we also simulated the angle dependence (in the XZ plane) of Zeeman splitting \nenergy (“ up”-state minus “down ”-state) by replacing Zeeman term (Lz+2S z)h with \n[cosθ(L z+2S z)+sinθ(L x+2S x)]h in the Hamiltonian. In the simulation, we set 10Dq=3, λ=0.4 and h=0.02  \nin the unit of eV . The simulation results are shown in Fig. S9 (b). The Zeeman splitting energy of Jeff=3/2 \nstate shows a significant angle dependence, reflecting a large single -ion anisotropy with easy axis along \nthe crystal bond direction s (θ=nπ/2, 〈100 〉). On the other hand,  Zeeman splitting energy of Jeff=1/2 shows \na very weak dependence, consistent with the small single -ion anisotropy. To further demonstrate the origin \nof the anisotropy  of the Jeff=3/2 state, we also examined situations where  the SOC energy  (λ) and crystal \nfield (10Dq) are zero respectively , which reveal their essential role in determining the anisotropy. In total, \nthe simulations suggest that the Jeff=3/2 state has a large single -ion anisotropy with easy axis 〈100 〉, which \ncontributes the anisotropy of the superlattices.   In sum, the emergent magnetic state of SIO in the superlattice shows both similarities and distinctions \ncompared to the RP -phase iridates. The difference in magnetic anisotropy is likely to arise f rom the novel \nspin-orbit state of SIO in the superlattice, which is a mixture of Jeff=1/2 and Jeff=3/2. Moreover, the \ndiscussions above also reveal that magnetic coupling between LSMO and SIO at the interface could \nactually contribute to the mixing of Jeff=1/2 and Jeff=3/2 (see SI Appendix section e) . \n \n \nSI Figure Legends  \nFig. S1  (a) RHEED patterns of STO substrate, LSMO sublayer and SIO sublayer . All of the sublayers \nshow the 2D character. (b) The typical RHEED intensity oscillations of the LSMO (bl ue) and SIO (red) \nsublayers for  SL33. (c) AFM image of SL33.The topography shows  that the surface pr eserves the step -\nterrace structure  of the substrate.   (d) θ-2θ X -ray diffractograms of the series SL3 m. \nFig. S2 (a) High -angle annular dark -field (HAADF) STEM image  of LSMO/SIO superlattice in \n(LSMO) 5(SIO) 5 periodic region. (b) EELS line profile of the La M-edge  of the same region. The scan line \nis shown by the red arrow.   \nFig. S3  L-scan of several half -ordering peaks of the superlattice  SL31 (Blue) and SL310 (red): (a) (0.5, \n0.5, 1.5), (b) (1.5, 2.5, 2.5), (c) (0, 0.5, 3), (d) (0.5, 0, 3). Inset in (c) and (d) show the results of SL31.   \nFig. S4  (a) Temperature dependence of magnetization of SL3 m. The dependence was measured during \nwarmi ng with 200 Oe applied in [100] after 1T field cooling. (b) Schematic of the different \ncrystallographic directions along which the magnetic loops were measured. (c) Magnetic hysteresis loops \nof SL31 with field applied in -plane ([100], black) and out -of-plane ([001], green). (d) -(h) Magnetic \nhysteresis loops of SL3 m (m=2, 3, 5, 7, 10) along two in -plane directions: [100] (black) and [110] (red).  Fig. S5  (a) Temperature dependence of normalized resistivity of SL3 m. (b) AMR of 4uc SIO thin film on \nSTO at 10K ( measurement geometry is the same as Fig. 2C)). (c) Thickness dependence of the AMR for \nSL3m at 10K with 1T field applied in -plane. (d) Field dependence of AMR for SL33 at 10K. (e) \nTemperature dependence of AMR for SL33 with 1T field.  \nFig. S6 (a) Schematic diagram of the application of sum -rules to the Ir -edge XAS and XMCD. (b), (c), (d) \nshow the results on Ir-L3 edge: integration (“Sum” in each figure) of (b) XAS, (c) XMCD for SL31 and \n(d) XMCD for SL33. (e), (f), (g) show the results on Ir-L2 edge: integration of (e) XAS, (f) XMCD for \nSL31 and (g) XMCD for SL33.  \nFig. S 7 Application of sum -rules to the Mn L -edge of SL31.  \nFig. S 8 (a) The basic cell used for calculation (black line) and the relaxed crystal structure of the SL31. \n(b) Relaxed magne tic structure of the SL31.  \nFig. S9  (a) The Hamiltonian Matrix that includes the spin -orbit coupling (λ), crystal field (10Dq) and \nmagnetic  energy (h). (b) Numeric simulations of angle dependence of Zeeman splitting energy (ZS , “up”-\nstate minus “down ”-state) for J eff=3/2 and J eff=1/2 (λ=0.4, 10Dq=3 and h=0.02 or otherwise stated). θ is \nthe angle of moment with respect to the quantization axis Z in XZ plane (θ=nπ/2 represents the moment \nalong the bond direction 〈100〉). To compare different plots, Zeeman s plitting energy (ZS) is presented as \nZS(θ) -(ZS(max)+ZS(min))/2. The maximum of ZS corresponds to the direction of easy axis.  (Zeeman \nsplitting energy of J eff=3/2 represents the difference of eigenenergy between J eff=3/2(3/2) and J eff=3/2( -\n3/2)).  \nTable S 1 Energy minimum calculated for different combinations of U parameters with magnetic moments \nof Mn along two crystallographic directions for SL31 and LSMO single layer  \n S1 \n \n \n \n \n \nS2 \n \n \n \n \n \n \n \n \nS3 \n \n \n \n \n \n \n \n \nS4 \n \n \nS5 \n \n \nS6 \n \n \n \nS7 \n \n \nS8 \n \n \n \n \nS9 \n \n \nTable S1  \n \n \n Energy Minimum (eV) \n Parameters (eV)  M // [100]  M // [110]  \n \nSL31  U(Mn)=3, U(Ir)=2, U(La)=0  -314.9843  \n -314.9244  \n U(Mn)=3, U(Ir)=2, U(La)=3  -311.9687  \n -311.9137  \n \nU(Mn)=4, U(Ir)=2, U(La)=3  -308.6973  \n -308.5817  \n \n \nLSMO  U(Mn)=4, U(La)=0  -246.041255  -246.041262  \nU(Mn)=4, U(La)=3  -242.999605  -242.999608  \n \n"    },    {        "title": "1603.05053v2.Illustrative_view_on_the_magnetocrystalline_anisotropy_of_adatoms_and_monolayers.pdf",        "content": "arXiv:1603.05053v2  [cond-mat.mtrl-sci]  20 Apr 2016Illustrative view on the magnetocrystalline anisotropy of adatoms and monolayers\nO.ˇSipr,1,∗S. Mankovsky,2S. Polesya,2S. Bornemann,2J. Min´ ar,2,3and H. Ebert2\n1Institute of Physics ASCR v. v. i., Cukrovarnick´ a 10, CZ-162 53 Prague, Czech Republic\n2Universit¨ at M¨ unchen, Department Chemie, Butenandtstr. 5-13, D-81377 M¨ unchen, Germany\n3New Technologies Research Centre, University of West Bohem ia, Pilsen, Czech Republic\n(Dated: August 9, 2021)\nEven though it has been known for decades that the magnetocry stalline anisotropy is linked to\nthe spin-orbit coupling (SOC), the mechanism how it arises f or specific systems is still subject of\ndebate. We focused on finding markers of SOC in the density of s tates (DOS) and on employing\nthem for understanding the source of magnetocrystalline an isotropy for the case of adatoms and\nmonolayers. Fully relativistic ab-initio KKR-Green function calculations were performed for Fe,\nCo, and Ni adatoms and monolayers on Au(111) to investigate c hanges in the orbital-resolved DOS\ndue to a rotation of magnetization. In this way one can see tha t a significant contribution to\nthe magnetocrystalline anisotropy for adatoms comes from p ushing of the SOC-split states above\nor below the Fermi level. As a result of this, the magnetocrys talline anisotropy energy crucially\ndepends on the position of the energy bands of the adatom with respect to the Fermi level of the\nsubstrate. This view is supported by model crystal field Hami ltonian calculations.\nPACS numbers: 75.30.Gw,75.70.Tj\nKeywords: magnetic anisotropy,adatom,monolayer,spin-o rbit coupling\nI. INTRODUCTION\nMagneticanisotropy,i.e., thepreferenceofasystemfor\nbeing magnetized in a certain direction, is one of the key\nproperties underlying practical use of magnetic materi-\nals. One contribution to the magnetic anisotropy comes\nfrom the classical interaction of magnetic dipoles. This\nmechanism stands behind the so-called shape anisotropy\nand can be described using classical physics — although\na quantum-mechanical description has been developed as\nwell.1,2Another contribution, which becomes important\nin particular for small systems such as atomic clusters\nor nanostructures, comes from the spin-orbit coupling\n(SOC). This magnetocrystalline anisotropy can only be\ndescribed within a relativistic quantum-mechanical for-\nmalism. We will deal exclusively with this SOC-induced\ncontribution in this work.\nA quantitative measure of the magnetocrystalline\nanisotropy is the magnetocrystalline anisotropy energy\n(MAE), i.e., the difference between total energies of the\nsystem for two orientations of the magnetization M.\nEvaluating the MAE is often numerically very difficult\nbecause one has to subtract two large numbers to get a\nsmall difference between them. To get accurate results,\none has to tune severaltechnical parameterssuch as inte-\ngration mesh setup in k-space3,4or the adequacy of the\nbasisset. Forsupportednanostructures,thetreatmentof\nthe substrate is also very important.5,6A lot of attention\nwas devoted to these issues recently.\nNevertheless, there is also another line of research on\nthe magnetocrystalline anisotropy, namely, the effort to\nunderstand its mechanism intuitively and, in particular,\nto see which electronic structure features participate in\nthe phenomenon. One possibility is to use perturbation\ntheory and to describe spin-orbit interaction approxi-\nmately within the two-component formalism by the SOCtermHSOC=ξL·S, whereLandSare the orbital\nand spin angular momentum operators and ξis the SOC\nstrength. For systems studied here the lowest-order non-\nvanishing contribution to the total energy is the second-\norder term,\n∆E(2)=−/summationdisplay\ni∈occ\nj∈unocc|/angbracketleftψi|HSOC|ψj/angbracketright|2\nEj−Ei.(1)\nRelying on second order perturbation theory has led to\nconcepts such as scaling of the MAE with the square of\nthe SOC strength or the frequently used Bruno and van\nder Laan formulae relating the MAE to the anisotropy of\nthe orbital magnetic moment.7–11On the other hand, as\nthe sum in Eq. (1) involves a large number of summands\nwhich may be of comparable magnitude, it may be very\ndifficult to identify just a few terms as the dominant ones\nand in this way to link MAE of a particular system to\nspecific features in the electronic structure. Getting a\nsimple intuitive understanding of the MAE by looking\non the interaction between individual states thus may\nbe very hard to achieve — despite the effort and inter-\nesting results obtained.12–14Approaches that focus on\nintegral quantities such as a corresponding susceptibil-\nity (still within second order perturbation theory) could\nhave a more general use.11\nHowever, other mechanisms of generating the magne-\ntocrystalline anisotropy, not accounted for by second or-\nder perturbation theory, are also possible and were dis-\ncussed in the past. In particular, Eq. (1) cannot be\nused if degenerate levels are coupled. For that situation\nanother mechanism contributing to the MAE was sug-\ngested, namely, a SOC-induced splitting of states that\nwould be degenerate otherwise.3,11,15–17If some of these\nstates are pushed above or below the Fermi level EF,\na large change of the total energy occurs. For layered2\nand bulk systems this effect may not be dominant be-\ncause relevant states occupy only a restricted region in\nk-space.13,17,18However, the situation could be different\nfor adatoms and clusters, where there is no dispersion in\nk-space.\nThe question then remains whether there exist in\nreality systems where the origin of magnetocrystalline\nanisotropy can be traced to a SOC-induced splitting of\notherwise degenerate states at EFand where this mech-\nanism can be effectively visualized in terms of integral\nquantities such as the density of states (DOS). Such a\nmechanism could give rise to a large MAE. In fact lately\nthere have been intensive efforts to understand how the\nMAE could be made as large as possible.19–23A better\nintuitive insight into the magnetocrystalline anisotropy\nbeyond the perturbation theory might be useful in this\ncontext. From a more general point of view, it is desir-\nable to have a framework that would enable to visualize\nthe emergence of the magnetocrystalline anisotropy by\nmeans of simple concepts.\nWe decided therefore to perform a detailed ab ini-\ntio, i.e., material-specific study of magnetocrystalline\nanisotropy for Fe, Co, and Ni adatoms and monolayers\non Au(111). The motivation for this choice is that only\nlittle hybridization between 3 dstates and Au states is\nexpected.24For adatoms, the 3 dstates could thus have\nan atomic-like character where the effect of SOC-induced\nsplitting of states should be larger than for delocalized\nstates. Comparison between adatoms and monolayers\ncould further elucidate the role of different factors. We\nemploy a fully relativistic framework (solving the four\ncomponent Dirac equation) to treat the SOC as accu-\nrately as possible. The application of Green function\nformalism allows a proper treatment of adatoms, avoid-\ning possible artefacts that might arise from a supercell\napproach.\nOur paper is organized as follows. First we introduce\nour computational method and the investigated systems.\nThen we present numerical values of MAE and magnetic\nmoments. The main emphasis is put on showing how the\nSOC affects the DOS resolved into components accord-\ning to the magnetic quantum numbers. We demonstrate\nthat the effect ofSOC is much largerif the magnetization\nis perpendicular to the plane than if it is in-plane. This\neffect is reproduced using a simple crystal-field Hamil-\ntonian. Some technical details related to projecting the\nDOS onto magnetic quantum number components for a\nmagnetic system are described in the appendix.\nII. METHODS\nA. Computational scheme\nThe electronic structure is calculated within the ab\ninitiospin density functional framework, relying on\nthe local spin density approximation with the Vosko,\nWilk and Nusair parametrization for the exchange andcorrelation potential.25The electronic structure is de-\nscribed, including all relativistic effects, by the Dirac\nequation, which is solved using the spin-polarized rel-\nativistic multiple-scattering or Korringa-Kohn-Rostoker\n(KKR) Green function formalism26as implemented in\nthesprkkr code.27The potential was treated within the\natomic sphere approximation (ASA). For the multipole\nexpansion of the Green function, the angular momen-\ntum cutoff ℓmax=3 was used. The energy integrals were\nevaluated by contour integration on a semicircular path\nwithin the complex energy plane using a Gaussian mesh\nof 32 points. The integration over the kpoints was done\non a regular mesh, using 10000 points in the full surface\nBrillouin zone.\nThis work deals with adatoms and monolayers on a\nsubstrate. The Green function formalism allows to treat\nthe substrate as truly semi-infinite: the electronic struc-\nture is relaxed within the topmost seven substrate lay-\ners while at the bottom of this relaxation zone the elec-\ntronic structure is matched to the bulk via the decima-\ntion technique. Monolayers are dealt with in the same\nmanner as the clean substrate, just adding a layer of 3 d\natoms on top. The vacuum is represented by four lay-\ners of empty spheres. Adatoms are treated as embedded\nimpurities: first one calculates the electronic structure of\na semi-infinite host and then solves the Dyson equation\nfor an embedded impurity cluster.28The impurity clus-\nters we used contain 62 sites in total; this includes one\n3dadatom, 25 substrate atoms and the rest are empty\nspheres.\nThe MAE is calculated as a difference of total energies\nforM/bardblˆxandM/bardblˆz,\nEMCA=E(x)−E(z). (2)\nAccordingly, a positive MAE means that the easy axis of\nmagnetization is out-of-plane.\nIf the Dirac equation is used, the influence of SOC can-\nnot be isolatedin a straightforwardway. One can achieve\nit, nevertheless, using an approximate two-component\nscheme29where the SOC-related term is identified via\nrelying on a set of approximate radial Dirac equations.\nThis scheme was used in the past to investigate the influ-\nence of SOC on various properties including the MAE.6\nIn this work we use this scheme to suppress the SOC\nwhen investigating the DOS in Sec. IIIB. If SOC is to\nbe included, the DOS can be calculated either using the\nfull Dirac equation or using the approximative scheme;29\nthe corresponding curves in the graphs agree within the\nthickness of the line, demonstrating that both schemes\nare equivalent as concerns the DOS. On the other hand,\nthere are small yet identifiable differences between both\nschemes concerning the MAE (about 10 % in case of\nadatoms and about 20 % in case of monolayers). The\nresults presented in Sec. IIIA were obtained using the\nfully relativistic scheme.3\nFIG. 1. (Color online) Structure diagrams for an adatom\nand a monolayer on an Au(111) surface. The 3 datoms are\nrepresented by blue (dark) circles, various shades of orang e\n(grey) represent Au atoms in different layers.\nTABLE I. Vertical distances z3d-Auin˚A between the layer\ncontaining the 3 datoms and the nearest layer containing Au\natoms.\nz3d-Au z3d-Au\n3d adatom monolayer\nFe 1 .889 2 .088\nCo 1 .856 2 .035\nNi 1 .820 2 .016\nB. Investigated systems\nWe investigated Fe, Co, and Ni adatoms and mono-\nlayers on the fcc Au(111) surface. The corresponding\nstructures are shown in Fig. 1. To get proper interatomic\ndistances, we performed geometry optimization using the\nvaspcode.30,31These calculations were done for slabs of\nthree layers of substrate atoms covered either by a com-\nplete layer of 3 datoms (for monolayers) or by a 3 ×3 sur-\nface supercell of 3 datoms (for adatoms). The positions\nof the substrate atoms in the two lowermost layers were\nfixed while the positions of topmost substrate atoms and\n3datoms were relaxed. This led to a mild buckling of the\ntopmost Au layer for the adatoms (about 0.02 ˚A), which\nwe ignored in subsequent KKR-Green function calcula-\ntions. Using a three layers thick slab instead of a\nsemi-infinite substrate is justified if one is interested in\nrelaxing the positions of the 3 datoms above the host.\nHowever, using it for evaluating the MAE would be in-\nappropriate — for that, either a much thicker slab (as in\nRef. 5) or a proper semi-infinite crystal (as in this work)\nshould be employed.\nThe optimized structural parameters as we took them\nfromvaspcalculations and used in the sprkkr calcu-\nlations are summarized in Tab. I. As concerns the dis-\ntances between Au substrate layers, we used the bulk\ninteratomic distance 2.396 ˚A everywhere except for the\ndistance for the topmost Au layer, which we took 2.431 ˚A\nfor adatoms and 2.427 ˚A for monolayers (as obtained via\nthevaspcalculations).TABLE II. Magnetic properties of 3 dadatoms and monolay-\ners on Au(111). The first two columns identify the system,\nthe third column shows the MAE obtained as a difference\nof total energies (in meV per 3 datom), the fourth column\nshows spin magnetic moments for M/bardblˆz, and the fifth and\nsixth columns show orbital magnetic moments for M/bardblˆzand\nM/bardblˆx, respectivelly. Magnetic moments are in units of µB.\nEMCA µ(z)\nspin µ(z)\norb µ(x)\norb\nFe adatom 4 .07 3 .40 0 .536 0 .062\nmonolayer 0 .97 3 .08 0 .127 0 .073\nCo adatom 4 .42 2 .13 0 .218 0 .206\nmonolayer 0 .42 2 .01 0 .156 0 .168\nNi adatom −1.63 0 .67 0 .063 0 .158\nmonolayer −1.97 0 .73 0 .118 0 .191\nIII. RESULTS\nA. MAE and magnetic moments\nThe results obtained for the MAE and magnetic mo-\nments are presented in Tab. II. One can see that the\neasy axis of the magnetization is perpendicular to the\nsurface for Fe and Co adatoms and monolayers and par-\nallel to the surface for Ni adatom and monolayer. The\nmagnetic moments were evaluated within atomic spheres\naround the 3 datoms. Magnetic moments for Au atoms\nare small. In case of adatoms, the total magnetic mo-\nment induced in the Au substrate amounts to about 5 %\nof the 3dadatom moment and is oriented parallel to the\nmoment of the adatom. In the case of monolayers, the\ntotal magnetic moment induced in the Au substrate per\na 3datom is about 2 % of the 3 datom moment and is\noriented antiparallel to the moments of the 3 datoms.\nThe spin moments µspinpractically do not depend on\nthe magnetization direction, while the orbital moments\nµorbstrongly depend on it. For Fe and Ni atoms, µorb\nis significantly larger if Mis parallel to the easy axis of\nthe magnetization than if Mis parallel to the hard axis\n— in agreement with the expectations based on second\norder perturbation theory.7–9Surprisingly, this is not the\ncase for Co, where for the adatom the value of µorbonly\nslightly depends on the Mdirection and for the mono-\nlayer the trend is even reversed.\nB. Density of states\nWe first look at the spin-projected density of states\nin a range covering the whole valence region. This is\npresented in Fig. 2. The data correspond to M/bardblˆzbut\nthe plot would look practically the same also for M/bardblˆx\nat this scale. There is a considerable overlap between\n3dmajority-spin states and Au states, implying that\nmajority-spin states are affected by hybridization while\nminority-spin states are more atomic-like.\nOne can see that the majority-spin states are nearly4\n024n↑[sts/eV]3d\nAu next\nto 3d\nAubulk\n024n↓[sts/eV]\n-4 -2 0 2\nenergy [eV]Fe\nadatom\n0123n↑[sts/eV]3d\nAu next\nto 3d\nAubulk\n0123n↓[sts/eV]\n-4 -2 0 2\nenergy [eV]Fe\nmonolayer024n↑[sts/eV]\n024n↓[sts/eV]\n-4 -2 0\nenergy [eV]Co\nadatom\n0123n↑[sts/eV]\n0123n↓[sts/eV]\n-4 -2 0\nenergy [eV]Co\nmonolayer024n↑[sts/eV]\n024n↓[sts/eV]\n-2 0\nenergy [eV]Ni\nadatom\n0123n↑[sts/eV]\n0123n↓[sts/eV]\n-2 0\nenergy [eV]Ni\nmono-\nlayer\nFIG. 2. (Color online) Spin-projected DOS for 3 dadatoms and monolayers on Au(111) (in states per eV) for M/bardblˆz. Full lines\nrepresent the DOS related to 3 datoms, dashed lines represent the DOS related to those Au ato ms which are nearest neighbors\nto 3datoms, dotted lines represent the DOS for bulk Au.\nfully occupied. The Fermi level EFis around the middle\nof the minority-spin band. Thus if we are interested in\npossibleeffects ofshiftingthe statesacross EF, weshould\nfocus on the influence of the SOC on the minority-spin\nstates. Restricting ourselves to the minority-spin states\nwill greatly simplify further analysis without missing the\nimportant aspects.\nStudying how m-resolved DOS varies upon the rota-\ntion of the magnetization requires some clarifications.\nThe projection of the DOS according to the quantum\nnumbermhas to be done always in the same reference\nframe, disregarding the orientation of M. We call this\nframethe “globalreferenceframe”—itisfixedtotheun-\nderlying crystal lattice. If the m-projections are done in\ndifferent reference frames for different magnetization di-\nrections, the definitions of the m-components themselves\nalso vary, because they are linked to the spherical har-\nmonicsYℓmwhich are defined with respect to the x,y,\nzaxes. On the other hand, if one wants to retain and\nemphasize the difference between spin-up and spin-down\ncontributionsto the DOS, onehas tomakethe projection\nin a “local reference frame”, rotated so that the zaxis\ncoincides with the magnetization direction. The need for\nthis can be easily seen from the Stern-Gerlach term inthe Pauli equation, σ·B, which is diagonal only if the\neffective magnetic field Bis parallel to the zaxis. If the\nspin quantization axis is not parallel to the magnetiza-\ntion direction, the chosen representation strongly mixes\nspin-up and spin-down components.\nThese two circumstances suggest that if one wants to\nstudy the DOS for different directions of the magne-\ntization M, one has to renounce either having a uni-\nversal definition of the m-projections or retaining well-\nseparated spin-resolved DOS components. This is not an\nissue if the SOC is ignored because then the direction of\nthe magnetization has no effect on the electronic struc-\nture anyway. However, if the SOC is accounted for and\nthe dependence of the DOS on the direction of Mis in\nfocus, this is a serious obstacle.\nFortunately, this restriction can be by-passed in our\ncase. It is possible to get well-defined spin-minority DOS\nm-decomposed in a global reference frame even if M\nis is not parallel to the zaxis, relying on an approxi-\nmate procedure which is described in appendix A. The\nprocedure combines results for an m-decomposition in\nglobal and local reference frames. Employing this tech-\nnique, we obtained the density of minority-spin d-states\nresolved according to the magnetic quantum number m5\n0.00.51.0n↓[sts/eV]\n-1 0 1no\nSOC\n0.00.51.0n↓[sts/eV]\n-1 0 1with\nSOC=0\n0.00.51.0n↓[sts/eV]\n-1 0 1\nenergy [eV]with\nSOC=90012EMAE[eV]\n-1 0 1\nEband-EF[eV]adatom\nFe\n0.00.5n↓[sts/eV]\n-1 0 1 2noSOC\n0.00.5n↓[sts/eV]\n-1 0 1 2with SOC =0\nm = -2\nm = +2\nm= 0\nm= -1\nm= +1\n0.00.5n↓[sts/eV]\n-1 0 1 2\nenergy [eV]with SOC =90-2-101EMAE[eV]\n-1 0 1 2\nEband- EF[eV]monolayer\nFe\nFIG. 3. (Color online) The d-component of the minority-spin\nDOS for a Fe adatom (left) and a Fe monolayer (right) on\nAu(111), resolvedaccordingtothemagnetic quantumnumber\nm. ThecasewhentheSOCissuppressedispresentedtogether\nwith the case when the SOC is included. The magnetization\nis either perpendicular to the surface ( θ=0◦) or parallel to the\nsurface ( θ=90◦). The dependence of the MAE on the position\nof the top of the valence band is shown in the top graphs.\nas shown in Figs. 3–5. The magnetization is either out-\nof-plane ( M/bardblˆz,θ=0◦) or in-plane ( M/bardblˆx,θ=90◦) and\nthem-projections are defined in the same (global) refer-\nence frame in both cases. To highlight the effect of the\nSOC, we present results obtained with SOC suppressed\nand with SOC accounted for.\nIt can be seen readily from the plots in Figs. 3–5that if\nthe SOC is suppressed, the DOS does not depend on the\nsign ofm. Components for + |m|and−|m|are the same\nin this case, the only splitting comes from the crystal\nfield. If the SOC is taken into account, then the DOS\nfurther depends on whether mis positive or negative.\nThere is a significant difference in how the ±|m|states\nare split for out-of-plane magnetization and for in-plane\nmagnetization (especially for the m=±2 case).0.00.51.0n↓[sts/eV]\n-1 0 1no\nSOC\n0.00.51.0n↓[sts/eV]\n-1 0 1with\nSOC=0\n0.00.51.0n↓[sts/eV]\n-1 0 1\nenergy [eV]with\nSOC=90-20246EMAE[eV]\n-1 0 1\nEband-EF[eV]adatom\nCo\n0.00.5n↓[sts/eV]\n-1 0 1noSOC\n0.00.5n↓[sts/eV]\n-1 0 1with SOC =0\nm = -2\nm = +2\nm= 0\nm= -1\nm= +1\n0.00.5n↓[sts/eV]\n-1 0 1\nenergy [eV]with SOC =9001020EMAE[eV]\n-1 0 1\nEband- EF[eV]monolayer\nCo\nFIG. 4. (Color online) Density of the minority-spin dstates\nresolved according to the magnetic quantum number mand\nthe dependence of the MAE on the position of the top of\nthe valence band for a Co adatom (left graphs) and a Co\nmonolayer (right graphs) on Au(111). Otherwise, this figure\nis analogous to Fig. 3.\nThe procedure outlined in appendix A can be applied\nonly if the SOC-induced splitting of the majority-spin\nstates is negligible in the energy region in which we are\ninterested in, i.e., around EF. This assumption is well\njustified for Fe and Co. However, it is not so good for Ni,\nwhere the exchange-splitting is quite small (cf. Fig. 2)\nand the influence of the SOC on the majority-spin DOS\nis significant up to about 0.5 eV below EF. Therefore,\nfor Ni we present the data only for for E >−0.6 eV and\neven there they are less reliable than analogous data in\nFigs. 3–4. The full energy range for M/bardblˆxis covered\nby Figs. 9–11 in appendix B, where we present the m-\nresolvedDOS projected in a localreferenceframe rotated\nso that the z(loc)axis is parallel to M. (For M/bardblˆz,\nthe global reference frame and the local reference frame\ncoincide, because z(loc)is then identical with z.)\nThe definitions of individual m-components employed6\n0.00.51.01.5n↓[sts/eV]\n-1 0no\nSOC\n0.00.51.01.5n↓[sts/eV]\n-1 0with\nSOC=0\n0.00.51.01.5n↓[sts/eV]\n-1 0\nenergy [eV]with\nSOC=90-20020EMAE[eV]\n-1 0\nEband-EF[eV]adatom\nNi\n0.00.51.0n↓[sts/eV]\n-1 0no SOC\n0.00.51.0n↓[sts/eV]\n-1 0with SOC =0\nm = -2\nm = +2\nm= 0\nm= -1\nm= +1\n0.00.51.0n↓[sts/eV]\n-1 0\nenergy [eV]with SOC =90-200-1000EMAE[eV]\n-1 0\nEband-EF[eV]monolayer\nNi\nFIG. 5. (Color online) Density of the minority-spin dstates\nresolved according to the magnetic quantum number mand\nthe dependence of the MAE on the position of the top of the\nvalencebandforaNiadatom (leftgraphs) andaNimonolayer\n(right graphs) on Au(111). Otherwise, this figure is analogo us\nto Fig. 3.\nin appendix B and employed in this section obviously\ndiffer. One cannot, therefore, directly compare the\nplots where the DOS was resolved into m-components in\nthe global reference frame (Figs. 3–5) with plots where\nthe DOS was resolved in the local reference frame (ap-\npendix B). What is common in both reference frames is\nthat the SOC-induced splitting of the ±|m|components\nit significantly smaller for M/bardblˆxthan for M/bardblˆz.\nLet us summarize the picture obtained by inspecting\nthe DOS. First, note that the minority-spin DOS for the\nadatomshasquiteanatomiccharacter: iftheSOCissup-\npressed, it can be seen as representing just three broad-\nened atomic levels, depending on |m|. For monolayers,\nthe hybridization between states from different 3 datoms\nisconsiderable,sotheDOSdoesnothaveanatomicchar-\nacter any more. The second point to emphasize is that\nthe influence of the SOC is significantly larger for θ=0◦TABLE III. Energy levels of delectrons for an axial-crystal-\nfieldHamiltonianifthereisnoexchangesplittingandnoSOC .\nenergy Yℓm\nE1=P m=±2\nE2=−2(P+Q) m=0\nE3=Q m=±1\nthan forθ=90◦. More specifically, for θ=0◦the SOC\nsplits them=±2 peak into two and shifts their positions\nin different directions while for θ=90◦, the peak positions\nremain the same (only their intensities change).\nThe splitting of m-resolved DOS peaks by the SOC\nsuggests that the MAE could be very sensitive to the po-\nsition ofEFwith respect to these peaks. Therefore we\ncalculated the MAE while varying the position of the top\nof the valence band Eband, i.e., the band filling. The re-\nsults are shown in top panels of Figs. 3–5. One can see\nthat for the adatoms there is indeed a sharp peak in the\nMAE just at the energy where there is a peak for the\n|m|=2 component in the case of no SOC. This is espe-\ncially visible for Co and Ni adatoms. For the Fe adatom\nthis aspect is overshadowed by another strong feature\nstemming from the fact that, in this case, also the |m|=1\nstates are affected by SOC. The situation for monolayers\nis more complicated because the m-components are not\natomic-like any more. Nevertheless, even here a strong\npeakinthecurvefor EMAEasafunctionof Ebandappears\nat the energy where the DOS components for |m|=2have\ntheir maximum. We would like to note in this context\nthatthe densityofthe Ebandmeshusedin thecalculation\nisthesameforadatomsandmonolayers. Thismeansthat\nthe observation that the EMAE(Eband) oscillations are\nmuch wilder for monolayers than for adatoms describes\na real effect. Probably this is connected with hybridiza-\ntion between 3 datoms which is present for monolayers\nbut absent for adatoms.\nC. Effect of SOC on the energy levels via model\nHamiltonian\nWecouldseeinSec.IIIBhowtheSOCsplitselectronic\nstates for different orientationsof the magnetization. Let\nus check to what extent this can be described within a\nsimple model with only the crystal-fieldeffects taken into\naccount. This corresponds to a situation where the elec-\ntron feels only the Coulombic field generated by charges\nlocated at the positions of the nuclei.\nTohighlighttheessentialfeatures, werestrictourselves\ntodelectrons in an axial field (correspondingto D4d, i.e.,\nantiprism symmetry). If there is no magnetic order or\nSOC, the crystal-field Hamiltonian is given as described\nfor example in the book of Bersuker32[Eqs. (4.9)–(4.10)\nand Tab. 4.1]. The Hamiltonian is determined by two\nparameters (if the constant energy shift is omitted), re-\nsulting in three spin-degenerate energy levels as given in7\nTab. III. The order of levels E1,E2, andE3depends on\nthe values of parameters PandQ. LevelsE1andE3are\ndouble degenerate. Non-zero terms of the crystal-field\nHamiltonian are\nH(cry)\nms,m′s′=\n\nP m =±2, s=s′,\nQ m =±1, s=s′,\n−2P−2Q m = 0, s=s′.(3)\nThe subscript mscombines the magnetic quantum num-\nbermand the spin quantum number s, meaning that our\nHamiltonian is represented by a 10 ×10 matrix.\nThe magnetization of the system is reflected by the ex-\nchange field Hamiltonian H(ex). To distinguish between\ntwo orientations of the magnetization, we keep the spin\nquantizationaxisfixed(parallelto z)andvarytheHamil-\ntonianH(ex). The non-zero terms of H(ex)forM/bardblˆxare\nH(ex)\nms,m′s′=B m =m′, s/negationslash=s′(4)\nand forM/bardblˆzthey are\nH(ex)\nms,m′s′=B m=m′, s=s′=−1/2,(5a)\nH(ex)\nms,m′s′=−B m=m′, s=s′= +1/2.(5b)\nThe third contribution to the model Hamiltonian\ncomes from the SOC. The spin quantization axis is kept\nparallel toz, so the Hamiltonian H(SOC)=ξL·Scan be\nrepresented as (cf. St¨ ohr)33\nH(SOC)=\n−ξ0 0 0 0 0 ξ0 0 0\n0−ξ\n20 0 0 0 0ξ√\n6\n20 0\n0 0 0 0 0 0 0 0ξ√\n6\n20\n0 0 0ξ\n20 0 0 0 0 ξ\n0 0 0 0 ξ0 0 0 0 0\n0 0 0 0 0 ξ0 0 0 0\nξ0 0 0 0 0ξ\n20 0 0\n0ξ√\n6\n20 0 0 0 0 0 0 0\n0 0ξ√\n6\n20 0 0 0 0 −ξ\n20\n0 0 0 ξ0 0 0 0 0 −ξ\n.\n(6)\nThe total Hamiltonian we have to diagonalize is\nH=H(cry)+H(ex)+H(SOC). (7)\nWe want to apply this model for an adatom, where\nthe hybridization is small and the crystal-field effects will\nbe important. Looking at the adatom-related panels of\nFigs. 3–5, we can see that in the absence of SOC the\nminority-spinDOSindeedresemblesthreeatomic-likeen-\nergy levels, as in Tab. III. It is convenient to introduce\nlevel spacings\n∆1≡E2−E1, (8)\n∆2≡E3−E2, (9)TABLE IV. Parameters (in eV) for the model Hamiltonian\nsimulating 3 dadatoms on Au(111) by means of an axial crys-\ntal field model.\nFe Co Ni\n∆1 0.24 0.21 0.06\n∆2 0.05 0.06 0.20\nE↓−E↑ 2.81 1.96 0.57\nξ 0.065 0.085 0.108\nthrough which we can express the model Hamiltonian\nparameters as\nP=−3∆1+2∆2\n5, (10)\nQ=2∆1+3∆2\n5. (11)\nTo simulate 3 dadatoms on Au(111), one should read the\n∆1, ∆2splittings from Figs. 3–5 to get the values for the\nparameters P,Qand the exchange splitting from Fig. 2\nto get the parameter BusingE↓−E↑= 2B. The SOC\nparameters ξcanbe obtainedvia ab-initio calculations.34\nThe appropriate values are given in Tab. IV.\nA general idea how the SOC affects the energy levels\ncan be obtained by diagonalizing the Hamiltonian (7) for\ndifferent orientations of Mwhile the SOC strength ξis\ngradually increased from zero to a realistic value. The\ncorresponding results are presented in Fig. 6 where we\nshow energy levels E↑(ξ) andE↓(ξ) for parameters given\nin Tab. IV. The “proper” value of ξfor each element is\nmarked by a thin dashed line. To avoid confusion, we\nshould note that our categorizing of levels as E↑orE↓is\ndone just for convenience, by comparing their positions\nto the spin-projected DOS shown in Fig. 2. We care only\nabout the energy levels in this context and not about the\nwave functions, so the issue of mixed spin character for\nθ=90◦, discussed in Sec. IIIB and in appendix A, does\nnot interfere with our analysis.\nA prominent feature of Fig. 6 is that the effect of ξ\nis much less for in-plane magnetization ( θ=90◦) than for\nperpendicular magnetization ( θ=0◦). This is especially\ntrue for the lowest energy which corresponds to m=±2.\nBy comparing this observation to Figs. 3–5, we see that\nthe simple crystal-field model indeed accounts for the\ntrends in the m-resolved DOS for the 3 dadatoms. It\nis worth noting that if the exchange-field parameter B\ndecreases (i.e., going from Fe to Co to Ni), the m=±2\nenergy levels split also for the θ=90◦case (in-plane mag-\nnetization). A similar trend can be seen also in the DOS\nin Sec. IIIB: the difference between m=±2 curves in the\nlowermost left panels in Figs. 3–5 increases when going\nfrom Fe to Co to Ni.\nIV. DISCUSSION\nOur aim was to investigate whether markers of MAE\ncan be seen in intuitive quantities such as the m-resolved8\n0.0 0.05 0.1\n[eV]1.21.41.6E↓[eV]\n-1.6-1.4-1.2E↑[eV]=0Fe\n0.0 0.05 0.1\n[eV]=90\n0.0 0.1\n[eV]0.81.01.2E↓[eV]\n-1.2-1.0-0.8E↑[eV]=0Co\n0.0 0.1\n[eV]=90\n-0.6-0.4-0.20.00.20.40.6E↑[eV] E↓[eV]\n0.0 0.1 0.2\n[eV]=0Ni\n0.0 0.1 0.2\n[eV]=90\nFIG. 6. (Color online) Dependence of eigen-energies of the m odel Hamiltonian given by Eq. (7) on the SOC strength ξfor two\norientations of M. The model parameters for Fe, Co, and Ni adatoms are given in T ab. IV. Thin dashed lines mark ξvalues\nappropriate for each element. The zero of energy correspond s to the case with no magnetization and no crystal field.\nDOS. Figs. 3–5 (in conjunction with Figs. 9–11) show\nhowSOCaffectstheDOSdependingontheorientationof\nthe magnetization M. The corresponding changes in the\nDOS can be linked to the magnetocrystalline anisotropy\nof adatoms. Particularly for the Fe and Co adatoms one\ncan see that for θ=0◦the SOC splits the |m|=2 compo-\nnent of the DOS in such a way that one of the peaks\nis pushed above EF(or at least an essential part of it).\nThe band-energy contribution to the total energy is thus\nsubstantially reduced. As this effect does not occur for\nθ=90◦, the out-of-plane orientation of Mis energetically\nmore favored and the corresponding MAE is positive, in\nagreement with Tab. II\nThe SOC-induced splitting of the |m|=2 peak occurs\nforθ=0◦also for the Ni adatom. However, in that case\nboth peaks remainbelow EFandthe changein the band-\nenergy is therefore much smaller. The influence of SOC\nfor theθ=90◦case is best seen if the m-projectionis done\nin a rotated local reference frame, as in Fig. 11. This is\nbecause the isolation of the minority-spin DOS in the\nglobal reference frame cannot be done properly due to\nthe small energy separation between the majority-spin\nand minority-spin states of Ni. The lowermost graphs in\nFig. 5 have to be seen as primarily illustrative in this re-\nspect because they are affected by the fact that majority-\nspin states are still influenced by the SOC in this region.\nFocusing on the unambiguous data in Fig. 11 one can\nsee that for θ=90◦the states with |m(loc)|=2 are split\nin such a way that part of one of the SOC-split peaks is\npushed above EF. This effect overrunsthe corresponding\neffect on the |m|=2 states for θ=0◦(Fig. 5) and, accord-ingly, the easy axis of magnetization is in-plane for the\nNi adatom.\nEffects of this kind can hardly be identified for mono-\nlayers. In this case the hybridization between the 3 d\nstates distorts the atomic-like character of the states and\none would have to consider a lot of contributions, simi-\nlarly as if the E(k) band-structure of layered systems is\nanalyzed.12,14,16\nThe simple crystal field model accounts qualitatively\nfor many aspects of the magnetocrystalline anisotropy\nof adatoms, indicating that this anisotropy can be un-\nderstood intuitively as an interplay between the axial\ncrystal field, the exchange field and the spin-orbit cou-\npling. However, there are also differences between the\npictures offered by the model Hamiltonian and by the\nDOS obtained from ab initio calculations. For exam-\nple, the model Hamiltonian suggests that for an in-plane\nmagnetization ( θ=90◦), the splitting between the m=±1\nlevels is larger than the splitting between the m=±2 lev-\nels (Fig. 6); however, we do not observe this feature in\nFigs. 3–5. This means that effects not included in the\nsimple model of Sec. IIIC, such as hybridization, are im-\nportant as well.\nIt should be noted that by monitoring SOC-induced\nchanges in the DOS one accounts only for the band-\nenergy contribution to the total energy, omitting thus\nthe terms that explicitly depend on the change of the\npotential upon rotation of M(see, e.g., chapter 6 of the\nmonograph of Weinberger35for more details). This is\nequivalent to relying on the so called force theorem. If\nthe MAE is evaluated accounting for the band energy9\ncontribution only (by means of the torque method),36,37\nwe obtainEMCA=5.7 meV for the Fe adatom, 1.9 meV\nfor the Co adatom, and −0.8 meV for the Ni adatom.\nComparison with Tab. II that gives EMCAas a differ-\nence of total energies shows that the change in the band\nenergy does not fully account for the magnetocrystalline\nanisotropy but nevertheless constitutes a significant part\nofit. Oneshouldalsokeepinmind thattheSOC-induced\nsplitting of the DOS is not the only way the band en-\nergy is changed upon rotation of M. For example, all\neffects contained in Eq. (1) contribute as well. Accord-\ningly, what has been done here is identifying and visual-\nizing one important mechanism contributing to the mag-\nnetocrystalline anisotropy. We suggest (following earlier\nhints)3,11,15–17that this mechanism may be the domi-\nnant one for some adatoms and small clusters on sur-\nfaces — including those that attracted a lot of attention\nrecently.21,23,38,39\nAnother interesting system to be mentioned in this\ncontext is lithium nitridoferrate Li 2[(Li1−xFex)N] which\nattracted a lot of attention due to its very high mag-\nnetocrystalline anisotropy.11,40–42This system can be\nviewed as a collection of semi-isolated Fe impurities. A\nsimilar effect as the one investigated here could therefore\nbe important for Li 2[(Li1−xFex)N] and attention was in-\ndeed paid to it in this respect.11,42Generally, the mecha-\nnism we explored here should be important whenever the\nwidth of the electronic bands becomes comparable to the\nSOC-induced changes in the orbital-resolved DOS upon\nthe rotation of the magnetization.\nIf the magnetocrystalline anisotropy is generated via\npushing some SOC-split levels across the Fermi level, it\nmust crucially depend on their mutual positions. Specif-\nically in our case, it must depend on the position of the\nenergy bands of the adatom with respect to the Fermi\nlevel of the substrate (cf. also the top graphs of Figs. 3–\n5). Therefore, one might be able to manipulate the MAE\nby changing the substrate EF, e.g., via doping.\nEven though the aim of this study is not to reproduce\nexperimental MAE for specific systems, it is useful to\ncompare our values of MAE with available experiments.\nTherearenodatafor adatomson Au(111)but there have\nbeen several experimental studies of Fe and Co layers on\nAu(111). Beforecomparisonwithexperimentisdone,the\ndipole or shape anisotropy energy for monolayers must\nbe given. It is −0.18 meV, −0.08 meV, and −0.01 meV\nfor Fe, Co, and Ni monolayer, respectively. These val-\nues are smaller than the magnetocrystalline anisotropy\nenergy given in Tab. II. So we predict that Fe and Co\nmonolayers on Au(111) have out-of-plane easy axes of\nmagnetization and a Ni monolayer (for which there are\nno experiments available) has an in-plane easy axis of\nmagnetization. Earlier calculations for a Co monolayer\non Au(111) predicted an in-plane easy axis of magneti-\nzation for this system;6,43the reason for the difference\nis almost certainly the structural relaxation which was\naccounted for here but not in the two earlier works.\nDespite several experimental studies of magnetocrys-talline anisotropy of Fe and Co layers on Au(111) done\nin the past, drawing conclusions from them is not easy\nor unambiguous because the growth conditions vary and\ntypically do not favor formation of a single monolayer.\nA critical analysis of experiments is beyond our scope.\nFor a Fe monolayer it is probably safe to say that exper-\niment suggests an out-of-plane easy axis,44–47as our cal-\nculations do. For a Co monolayer, the situation is more\ncomplicated. For bilayer islands on Au(111) one gets an\nout-of-planeeasy axis.48–50Again growthconditions may\nbe crucial.51No data seem to exist for a single monolayer\non Au(111). As a whole, even though we cannot verify\nour results by a comprehensive comparison with experi-\nment, agreement with available data as well as the fact\nthat our values of MAE are in the same range of values\nas those obtained for similar systems indicate that our\nresults are reliable and can be used as a basis for the\nanalysis we performed in Secs. IIIB and IIIC.\nV. CONCLUSIONS\nThe effect of spin-orbit coupling on adatoms that only\nweakly hybridize with a substrate consists in splitting\natomic-likelevelsthatwouldbedegenerateinitsabsence.\nThe splitting is much larger if the magnetization is ori-\nented perpendicular to the surface than if it is oriented\nparallel to the surface and can be viewed as a combined\nresult of crystal field, exchange splitting and spin-orbit\ncoupling. If the originally degenerate level is close to the\nFermi level, one of the peaks can be pushed above it,\ndecreasing thereby the energy of the system. This effect\nrepresents a significant contribution to the magnetocrys-\ntalline anisotropy of adatoms. If hybridization smears\nout the atomic-like character of energy levels, as it is the\ncase for monolayers, this effect is not so important.\nACKNOWLEDGMENTS\nThis work was supported by the Grant Agency of\nthe Czech Republic within the project 108/11/0853, by\nthe Deutsche Forschungsgemeinschaft within the project\nSFB 689 “Spinph¨ anomene in reduzierten Dimensionen”\nand by Ministry of Education, Youth and Sports within\nthe project CENTEM PLUS (LO1402).\nAppendix A: Spin-resolved and m-resolved DOS for\nM∦ˆz\nIt was argued in Sec. IIIB that if one wants to see\nhow individual m-components of the DOS are affected\nby the rotation of the magnetization M, one should per-\nformthem-projectionsalwaysin aglobalreferenceframe\nso that the definitions of the m-components remain the\nsame. However, in case that M∦ˆz, projecting the DOS\nin a global reference frame mixes the spin components10\n024n↑, n↓[sts/eV]\n-4 -2 0\nenergy [eV]adatom\nn↑n↓\n0123n↑, n↓[sts/eV]\n-4 -2 0\nenergy [eV]monolayer\nn↑n↓\nFIG. 7. (Color online) Spin-projected DOS for a Co adatom\nand a Co monolayers on Au(111) for M/bardblˆz(solid blue lines)\nand forM/bardblˆx(red cross markers).\nbecause the spin quantization axis is no longer parallel\ntoM. In this appendix we present a method to restore\nthe separation of spin components in the DOS even in\nsuch a case. Our goal is achieved by a detour, combining\nresults of projections in the global and local reference\nframes. Effectively, it could be seen as a way to make\nthe spin-projection and the m-projection in different ref-\nerence frames.\nLet us recall that inside an atomic sphere the DOS for\na spin channel can be represented by means of the Green\nfunctionG(E) as\nn(E) =−1\nπℑ/integraldisplay\nd3r/angbracketleftr|G(E)|r/angbracketright.(A1)\nWe omit the spin labels here for brevity. Angular-\nmomentumprojectionsof n(E)canbeobtainedbymeans\nof the spherical harmonics. These spherical harmonics\nYℓmcan be defined in a global reference frame (fixed to\nthe crystal lattice) or in a local reference frame chosen so\nthat thez(loc)axis is parallel to M. The way the DOS\ncomponents nLaredefined thus depends on the reference\nframe. We can write schematically (again, for each spin\nchannel)\nn(glo)\nL(E) =−1\nπℑ /angbracketleftY(glo)\nL|G(E)|Y(glo)\nL/angbracketright,(A2)\nn(loc)\nL(E) =−1\nπℑ /angbracketleftY(loc)\nL|G(E)|Y(loc)\nL/angbracketright.(A3)\nIntegration over the radial coordinate is implicitly as-\nsumed.\nWe start by projecting the DOS in the local reference\nframe, where M/bardblˆz(loc). In this way we perform the sep-\naration of the spin components. We assume that this\nseparation was performed “once for all times”, i.e., it will\nbe preservedduring the whole subsequent procedure. All\nthe manipulations will be applied to minority-spin DOS\nand to majority-spin DOS separately.\nThis requires a further comment. By doing the\nspin-projection in the rotated local reference frame, we\nget spin-up and spin-down states assuming that the\nspin quantization axis is in a general direction whilewhen dealing with the global reference frame, the spin-\nquantization axis is fixed and parallel to z. However, this\ndifference can be neglected in our case: we checked that\nthe spin-projected DOS (without any m-decomposion)\nlooks practically the same no matter whether the mag-\nnetization is in-plane or out-of-plane. As an illustration,\nspin-projected DOS for a Co adatom and a Co mono-\nlayer is shown in Fig. 7 for two magnetization directions.\nThese spin projections were obtained in local reference\nframes defined so that the z(loc)axis is always parallel\ntoM. One can see that there is hardly any difference\nbetween the DOS for M/bardblˆxandM/bardblˆz. Another hint that\nthe spin projections can be maintained upon rotating M\ncomes from the fact that if the magnetization is rotated,\nthe spin magnetic moments almost do not change. There\nis a common experience that this is the case for all mag-\nnetic systems. By assuming that the spin-projected DOS\ndoes not depend on the direction of the magnetization,\nwe make an effective decoupling of spin and orbital de-\ngrees of freedom. This enables us to focus on changes\nin them-resolved components. A similar decoupling is\nused, e.g., when deriving useful relations for the angular-\ndependence of the magnetic dipole term Tzfor analyzing\nthe x-ray magnetic circular dichroism spectra.52,53\nSo far we have obtained minority-spin DOS and\nmajority-spin DOS, m-resolved in the local frame. Now\nwe need to transform the spin-polarized DOS from the\nbasis spanned by Y(loc)\nLto the basis spanned by Y(glo)\nL.\nA straightforward transformation between the n(loc)\nLand\nthen(glo)\nLcomponents is generally not possible — one al-\nways has to start with the Green function Gto getnL\nin a new basis. However, the transformation can be done\nprovided that Gis diagonal in the basis in which nLhas\nbeen initially known. Indeed, if we assume that\n/angbracketleftY(glo)\nL|G(E)|Y(glo)\nL′/angbracketright=δLL′/angbracketleftY(glo)\nL|G(E)|Y(glo)\nL/angbracketright,\nwe obtain\nn(loc)\nL(E) =−1\nπℑ /angbracketleftY(loc)\nL|G(E)|Y(loc)\nL/angbracketright,\n=−1\nπℑ/summationdisplay\nL′L′′/angbracketleftY(loc)\nL|Y(glo)\nL′/angbracketright /angbracketleftY(glo)\nL′|G(E)|Y(glo)\nL′′/angbracketright\n× /angbracketleftY(glo)\nL′′|Y(loc)\nL/angbracketright,\n=/summationdisplay\nL′/vextendsingle/vextendsingle/vextendsingle/angbracketleftY(loc)\nL|Y(glo)\nL′/angbracketright/vextendsingle/vextendsingle/vextendsingle2\n×/parenleftbigg\n−1\nπ/parenrightbigg\nℑ /angbracketleftY(glo)\nL′|G(E)|Y(glo)\nL′/angbracketright,\n=/summationdisplay\nL′ULL′n(glo)\nL′(E). (A4)\nSpecifically in our case we need to describe the situation\nfor in-plane magnetization, i.e., M/bardblˆx. The local ref-\nerence frame is then defined by the rotation y→y(loc),\nz→x(loc),x→ −z(loc). Considering the explicit forms11\nofY(glo)\nℓmandY(loc)\nℓmforℓ=ℓ′=2, one gets for the dstates\nUmm′≡/vextendsingle/vextendsingle/vextendsingle/angbracketleftY(loc)\n2m|Y(glo)\n2m′/angbracketright/vextendsingle/vextendsingle/vextendsingle2\n=\n1\n161\n43\n81\n41\n16\n1\n41\n401\n41\n4\n3\n801\n403\n8\n1\n41\n401\n41\n4\n1\n161\n43\n81\n41\n16\n.\n(A5)\nMore generally, the transformation between the bases is\ngiven by Wigner matrices.54\nStrictly speaking, Eq. (A4) with matrix Udefined\nin (A5) can be used only if the Green function Gis diag-\nonal in the Lindices. This is generally not the case (de-\npending on the symmetry of the system). Fortunately,\nnon-diagonal elements of /angbracketleftY(glo)\nL|G|Y(glo)\nL′/angbracketrightare small and\ncan be neglected for the systems we are dealing with. We\nverified this explicitly: If n(loc)\nLis obtained from n(glo)\nL\nby the transformation (A4), the m-resolved DOS curves\nobtained thereby agree within the thickness of the line\nwith curves obtained directly from the Green function\nvia Eq. (A3). It should be noted that this verification\nought to be applied to a sum of the spin components, be-\ncause for M∦ˆzthe spin components in n(glo)\nLare mixed\nif they are evaluated directly. Additionally, the SOC has\nto be suppressed to get exact equalities.\nSofar wefound atransformationfromthe globalframe\nto the local frame. However, we started our procedure by\nfinding spin-projected DOS in the local reference frame,\nso we need an opposite transformation, from the local\nframe to the global frame. A procedure analogous to\nthat we used to derive Eqs. (A4) and (A5) cannot be\nused, because if the Green function is evaluated in the\nrotated local reference frame, its non-diagonal elements\n/angbracketleftY(loc)\nL|G|Y(loc)\nL′/angbracketrightcannot be neglected (the z(loc)axis of\nthe rotated frame is chosen in an “inconvenient” way\n— parallel to the surface). That means we have only\nEq. (A4) at our disposal and the transformation from\nn(loc)\nLton(glo)\nLhas to be accomplished by inverting it.\nThe inversion of the transformation matrix Udefined\nby Eq. (A5) cannot be done straightforwardly because\nthis matrix is singular. However, the singular 5 ×5 ma-\ntrixUof Eq. (A5) can be folded down to a regular 3 ×3\nmatrixU(fold)if we assume that the m-components do\nnot depend on the sign of m, i.e., if\nn(glo)\n|m|=n(glo)\n−|m|. (A6)\nIn such a case one does not have to deal with all five\nindependent components n(glo)\n−2, ...,n(glo)\n2. It is enough\nto keep three of them, n(glo)\n−2,n(glo)\n−1,n(glo)\n0and the re-\nmaining two can be recoveredby taking n(glo)\n2=n(glo)\n−2and\nn(glo)\n1=n(glo)\n−1. This means that the original Eq. (A4),which we write here in a more explicit form as\nn(loc)\nm(E) =2/summationdisplay\nm′=−2Umm′n(glo)\nm′(E), m=−2,...,2,\nis reduced to\nn(loc)\nm(E) =0/summationdisplay\nm′=−2U(fold)\nmm′n(glo)\nm′(E), m=−2,−1,0\n(A7)\nwith\nU(fold)\nmm′=\n1\n81\n23\n8\n1\n21\n20\n3\n401\n4\n. (A8)\nThe matrix U(fold)is regular and can be inverted. Its\ninversion yields a matrix\nV(fold)\nmm′=\n−2\n32\n31\n2\n34\n3−1\n2−2 1\n(A9)\nwhich transforms the m-resolved DOS from the local ref-\nerence frame to the global reference frame:\nn(glo)\nm(E) =0/summationdisplay\nm′=−2V(fold)\nmm′n(loc)\nm′(E), m=−2,−1,0.\nIf we assume that the m-resolved DOS is independent on\nthe sign ofmnot only in the global frame but also in the\nlocal frame,\nn(loc)\n|m|=n(loc)\n−|m|\n[consistently with the fact that the matrix Uin Eq. (A5)\nis symmetric], we can unfold the 3 ×3 matrixV(fold)to a\nfull 5×5 matrixV,\nVmm′=\n−1\n31\n311\n3−1\n3\n1\n32\n3−12\n31\n3\n1−1 1−1 1\n1\n32\n3−12\n31\n3\n−1\n31\n311\n3−1\n3\n,(A10)\nto get a complete transformation of the m-resolved DOS\nfrom the local reference frame to the global reference\nframe:\nn(glo)\nm(E) =2/summationdisplay\nm′=−2Vmm′n(loc)\nm′(E), m=−2,...,2.\n(A11)\nFrom the way the transformation Eq. (A11) was de-\nrived it follows that it can be used only if the m-resolved12\n0.00.51.0n↑, n↓[sts/eV]\n-1 0 1\nenergy [eV]n↑n↓adatom\n0.00.5n↑, n↓[sts/eV]\n-1 0 1\nenergy [eV]n↑n↓monolayerm= -2\nm= +2\nFIG. 8. (Color online) The m=-2and m=2components of the\nmajority- and minority-spin DOS for a Co adatom and a Co\nmonolayer on Au(111). Magnetization is out-of-plane. Data\nfor majority-spin DOS are labeled by n↑, data for minority-\nspin DOS by n↓.\nDOS for the + |m|states is the same as the DOS for the\n“−|m|states”. As a whole, this is not the case because\nthe SOC splits the “ ±|m|states”. Therefore, one cannot\nsimply apply the transformation (A11) to the minority-\nspin DOSm-resolved in the local reference frame to get\nthe minority-spin DOS m-resolved in the global refer-\nence frame: the minority-spin ±|m|-states are split by\nthe SOC, therefore the folding of Eq. (A4) to Eq. (A7)\ncannot be done and the unfolding of the matrix (A9) to\na full matrix (A10) cannot be done either. However, in\nthe energy region we are interested in, i.e., in the region\nwhere the minority-spin states dominate, there is only\nlittle SOC-induced splitting of the ±|m|-states for the\nmajority-spin . This can be checked explicitly by look-\ning on the m-resolved majority-spin DOS curves in the\nenergy region around Fermi level. As an example, we\nshow here the m=-2 andm=+2 DOS components for\na Co adatom and a Co monolayer (Fig. 8). We select\nthe case for M/bardblˆz, where the SOC-induced splitting is\nthe largest. One can see that indeed the splitting of the\nmajority-spin DOS (labeled by n↑) aroundEFis much\nless than the splitting of the minority-spin DOS (labeled\nbyn↓). So even though the transformation (A11) cannot\nbe applied to the minority-spin DOS, it can be applied\nto the majority-spin DOS.\nThem-resolved minority-spin DOS in the global refer-\nence frame can thus be recovered in the following way:\nFirst, let us evaluate the m-resolved DOS directly in the\nglobal reference frame, as indicated in Eq. (A2). Both\nglobal spin channels are strongly mixed for in-plane mag-\nnetization, so there is only a very small difference be-\ntween “spin-up” and “spin-down” m-resolved DOS com-\nponents; if there is no SOC, even this difference disap-\npears. By adding contributions from both spin chan-\nnels, we get a “total” m-resolved DOS in a global frame,\nwith spin components unresolved. In a second step, we\ntake them-resolved DOS in the local (rotated) reference\nframe, keep only its majority-spin component and trans-\nform it to the global reference frame via Eq. (A11). This\nprovides us with a well-defined majority-spin m-resolved\nDOS in the global reference frame. Finally, we subtract0.00.51.0n↓[sts/eV]\n-1 0 1no\nSOC=90\nFe\nadatom\n0.00.51.0n↓[sts/eV]\n-1 0 1\nenergy [eV]with\nSOC=900.00.5n↓[sts/eV]\n-1 0 1 2no SOC =90\nFe\nmonolayer\n0.00.5n↓[sts/eV]\n-1 0 1 2\nenergy [eV]withSOC =90\nm(loc)=-2\nm(loc)=+2\nm(loc)= 0\nm(loc)= -1\nm(loc)= +1\nFIG. 9. (Color online) The d-component of the minority-\nspin DOS for a Fe adatom (left) and a Fe monolayer (right)\non Au(111) for magnetization parallel to the surface, with\nSOC either ignored (top) or included (bottom). The DOS is\nresolved according to the magnetic quantum number min a\nlocal reference frame where the z(loc)axis is parallel to the\nAu(111) surface.\nthismajority-spinDOS m-resolvedinaglobalframefrom\nthe totalm-resolvedDOS obtained in the first step. This\nleaves us with minority-spin m-resolved DOS in a ba-\nsis defined in the global reference frame. This detour\n(getting minority-spin DOS by subtracting majority-spin\nDOS from the total DOS) provides more accurate values\nthan what would be obtained if the transformation(A11)\nwas applied to the minority-spin DOS, because the con-\ndition (A6) is satisfied much better for the majority-spin\nstates than for the minority-spin states in the energy re-\ngion of interest.\nThe procedure described in this appendix should be\nused only for systems where there is a substantial ex-\nchangesplitting between the majority-spinand minority-\nspin states. Only then one can neglect the SOC-induced\n±|m|splitting of the majority-spin states with respect\nto the splitting of the minority-spin states (for ener-\ngies where the minority-spin DOS is much larger than\nthe majority-spin DOS). As an indicative parameter\nwhether the procedure can be applied or not we suggest\nthe ratio between the exchange splitting E↓-E↑and the\nSOC constant ξ. Using the parameters given in Tab. IV,\nonegetsthefollowingvaluesforthe( E↓-E↑)/ξratio: 43.2\nfor the Fe adatom, 23.1 for the Co adatom, and 5.3 for\nthe Ni adatom. This illustrates why our procedure\nworks nicely for Fe and Co but not so well for Ni, as\nacknowledged in Sec. IIIB.13\n0.00.51.01.5n↓[sts/eV]\n-1 0 1no\nSOC=90\nCo\nadatom\n0.00.51.01.5n↓[sts/eV]\n-1 0 1\nenergy [eV]with\nSOC=900.00.5n↓[sts/eV]\n-1 0 1no SOC =90\nCo\nmonolayer\n0.00.5n↓[sts/eV]\n-1 0 1\nenergy [eV]withSOC =90\nm(loc)=-2\nm(loc)=+2\nm(loc)= 0\nm(loc)= -1\nm(loc)= +1\nFIG. 10. (Color online) The d-component of the minority-\nspin DOS for a Co adatom (left) and a Co monolayer (right)\non Au(111) for magnetization parallel to the surface. This\nfigure is analogous to Fig. 9.\nAppendix B: DOS for M/bardblˆxresolved in a local\nreference frame\nThe DOS presented in Sec. IIIB was resolved accord-\ning to the magnetic quantum number min a global refer-\nence frame, with the zaxis perpendicular to the surface.\nThis ensured the same meaning of the m-components no\nmatter how the magnetization is oriented. However, one\nhad to apply an additional procedure described in ap-\npendix A to resolve the spin components. As this proce-\ndure assumes that SOC does not split the majority-spin\nDOS which is not quite the case here (especially for sys-\ntems with low exchange splitting such as Ni adatom and\nmonolayer), one might wonder whether the conclusions\nbased on Figs. 3–5 can be trusted.\nTherefore, we present in this appendix the m-resolved\nDOS for M/bardblˆxwhere the magnetic quantum number m\nrefers to a local reference frame, with the spin quantiza-\ntion axisz(loc)parallel to M. The outcome is presented\nin Figs. 9–11. Analogous plots for M/bardblˆzwould be\nthe same as respective panels in Figs. 3–5, because in\nsuch a case the local and global reference frames coin-\ncide. Note that the individual m-components pre-sented in Figs. 9–11 cannot be directly compared to\nanalogous components in Figs. 3–5 because their defini-\ntions differ. This can be clearly seen when comparing\nthe DOS for systems without SOC, when there can be\nin principle no dependence on the magnetization direc-\ntion. The graphs in the second from the top panels\nof Figs. 3–5 and in the top panels of Figs. 9–11 describe\nthe same physical situation and yet the individual curves\ndiffer — because the magnetic quantum numbers are\ndefined with respect to different reference frames.\nEventhough the m-components aredefined differently,\n0.00.51.01.5n↓[sts/eV]\n-1 0no\nSOC=90\nNi\nadatom\n0.00.51.01.5n↓[sts/eV]\n-1 0\nenergy [eV]with\nSOC=900.00.51.0n↓[sts/eV]\n-1 0no SOC =90\nNi\nmonolayer\n0.00.51.0n↓[sts/eV]\n-1 0\nenergy [eV]with SOC =90\nm(loc)=-2\nm(loc)=+2\nm(loc)=0\nm(loc)=-1\nm(loc)=+1\nFIG. 11. (Color online) The d-component of the minority-\nspin DOS for a Ni adatom (left) and a Ni monolayer (right)\non Au(111) for magnetization parallel to the surface. This\nfigure is analogous to Fig. 9.\none can still qualitatively compare how SOC affects the\nDOS forθ=0◦and forθ=90◦. 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Rose, Elementary Theory of Angular Momentum\n(Wiley, New York, 1957)."    },    {        "title": "1604.00176v3.Magnetocrystalline_anisotropy_of_FePt__a_detailed_view.pdf",        "content": "arXiv:1604.00176v3  [cond-mat.mtrl-sci]  14 Oct 2016Magnetocrystalline anisotropy of FePt: a detailed view\nSaleem Ayaz Khan,1Peter Blaha,2Hubert Ebert,3Jan Min´ ar,1,3and Ondˇ rej ˇSipr1,4\n1New Technologies Research Centre, University of West Bohem ia,\nUniverzitn´ ı 2732, 306 14 Pilsen, Czech Republic\n2Institute of Materials Chemistry, TU Vienna, Getreidemark t 9, A-1060 Vienna, Austria\n3Universit¨ at M¨ unchen, Department Chemie, Butenandtstr. 5-13, D-81377 M¨ unchen, Germany\n4Institute of Physics ASCR v. v. i., Cukrovarnick´ a 10, CZ-162 53 Prague, Czech Republic\n(Dated: November 6, 2018)\nTo get a reliable ab-initio value for the magneto-crystalli ne anisotropy (MCA) energy of FePt, we\nemploy the full-potential linearized augmented plane wave (FLAPW) method and the full-potential\nKorringa-Kohn-Rostoker (KKR) Green function method. The M CA energies calculated by both\nmethods are in a good agreement with each other. As the calcul ated MCA energy significantly\ndiffers from experiment, it is clear that many-body effects be yond the local density approximation\nare essential. It is not really important whether relativis tic effects for FePt are accounted for by\nsolving the full Dirac equation or whether the spin-orbit co upling (SOC) is treated as a correction\nto the scalar-relativistic Hamiltonian. From the analysis of the dependence of the MCA energy on\nthe magnetization angle and on the SOC strength it follows th at the main mechanism of MCA in\nFePt can be described within second order perturbation theo ry. However, a distinct contribution\nnot accountable for by second order perturbation theory is p resent as well.\nKeywords: magnetism; anisotropy; relativity\nI. INTRODUCTION\nThe variousab-initio electronicstructurecodesuse dif-\nferent approaches to solve the Schr¨ odinger equation for\na solid. Usually different codes and/or methods yield\nresults that are similar but show sometimes important\ndifferences in the details. These details start to matter\nif one aims at high-precision calculations with predic-\ntive power. Therefore an effort has lately intensified to\nstandardize ab-initio calculations and to find the condi-\ntions that have to be met so that reliable “true” quanti-\ntative values are obtained. So far the attention has been\npaid mostly to total energies, equilibrium lattice param-\neters and bulk moduli [1–4]. We want to extend this\neffort to another numerically sensitive area, namely, to\nthe magneto-crystalline anisotropy (MCA).\nThe MCA is manifested by the fact that the energy\nof a magnetically ordered material depends on the direc-tion of the magnetization Mwith respect to the crystal\nlattice. It is an interesting phenomenon both for funda-\nmental and technological reasons, as the MCA is impor-\ntant among others for the design of magnetic recording\nmedia. Theoretical research on MCA proceeds in two\ndirections. First, one tries to understand the mechanism\nbehind the MCA in simple intuitive terms, so that one\nwould have guidance in search for materials with a high\nMCA energy [5–7]. Second, one tries to find which com-\nputational procedures have to be employed so that one\ncan make quantitative predictions on the MCA energy\n[8–10].\nGetting an accurate value of the MCA energy EMCA\nis quite difficult as one has to, at least in principle, sub-\ntract two very large numbers (total energies for two ori-\nentations of magnetization) to get a very small number,\nnamely, EMCA. Several conditions for getting accurate\nwell-converged results were explored in the past. In par-2\nticular, the importance of a sufficiently dense mesh in\nthe Brillouin zone (BZ) for the k-space integration was\nrecognized [11–13]. When dealing with supported sys-\ntems such as adatoms or monolayers, the semi-infinite\nsubstrate has to be properly accounted for [14, 15]. De-\nspite all the efforts, getting accurate and reliable predic-\ntions of the MCA energy is still a problem. Numerical\nuncertainties severely restrict the practical usefulness of\ncalculations of the MCA. They hinder our understanding\nof the underlying physics as well, because lack of reliable\nnumerical values means that it is not really possible to\ndetermine which physicalapproximationsand modelsare\nacceptable and which are not.\nIn this workwe focus on MCA ofbulk FePt. This com-\npound has the largest MCA energy of all bulk materials\nformed by transition metals and its crystal structure is\nquite simple, so it is a good candidate for a reliablecalcu-\nlation. At the same time, the presence of Pt — a heavy\nelement — suggests that relativistic effects should be sig-\nnificant, offering thus an interesting possibility to check\nhow different methods of dealing with relativistic effects,\nin particular with the spin-orbit coupling, influence the\nresults. Besides, a deeper understanding of the MCA of\nFePt is important regarding current search for suitable\nrare-earth-freemagnetic materials. Transitionmetals are\nnatural candidates in this respect and attracted a lot of\nattention recently [16, 17].\nPrevioustheoretical studies on FePt based on the local\ndensity approximation (LDA) give a large spread of the\nresults — from 1.8 meV to 4.3 meV [9, 12, 18–24]. If one\nrestrictstofullpotentialmethodsonly,onestillgetsarel-\natively large difference between various studies: EMCAof\nFePt was determined as 2.7 meV by FP-LMTO calcula-\ntion of Ravindran et al.[12] and FLAPW calculation of\nShick and Mryasov [23], 3.1 meV by plane-waves calcu-\nlation of Kosugi et al.[25], and 3.9 meV by FP-LMTO\ncalculation of Galanakis et al.[22]. The differences be-\ntween various LDA calculations are comparable to the\ndifferences between LDA results and the experimentalvalue of 1.3 meV [26]. Even though part of the spread of\nthe LDA results can be attributed to the use of different\nLDAexchange-correlationfunctionals, thedifferencesare\nstill too large to be acceptable. Besides, they occur also\nforstudies which usethe sameexchange-correlationfunc-\ntional (e.g., both Ravindran et al.[12] and Galanakis et\nal.[22] use von Barth and Hedin functional [27]). This\nsuggests that the accuracy of ab-initio MCA energy cal-\nculations may not even be sufficient to answer the\nfundamental question whether the LDA itself is able to\nreproduce the experimental MCA energy of FePt or not.\nDeciding which method gives better MCA results than\nthe other is quite difficult, among others because differ-\nent computational approachesused by different codes are\nintertwinedwith differentwaysofimplementingrelativis-\ntic effects. Recall that as the MCA is intimately related\nto the spin orbit coupling (SOC), the way the relativity\nis included can be an important factor. To verify that\na calculated MCA energy really represents the true LDA\nvalue, one has to use two different methods and make\nsure that the calculations are properly converged.\nThe aim ofour workis to perform arobust and reliable\nLDA calculation of the MCA energy of FePt to find out\nwhether treating relativistic effects via an explicit SOC\nHamiltonian is sufficient for MCA calculations, whether\nthe MCA of FePt can be described within the LDA, and\nwhat is the accuracy of current MCA energy calculations\nin general.\nThe first computational method we employ is the well-\nestablished and recognized full potential linearized aug-\nmented plane wave (FLAPW) method as implemented\nin thewien2k code [28]. This method was used as a\nreference in the recent study of the accuracy of total en-\nergies and related quantities [2–4]. Relativistic effects\nare treated approximately in wien2k, by introducing a\nseparate SOC-related term to the Hamiltonian. As the\nsecondmethodweoptedforafullyrelativistic fullpoten-\ntial multiple scattering KKR (Korringa-Kohn-Rostoker)\nGreen function formalism as implemented in the sprkkr3\ncode [29, 30].\nMany aspects of the MCA of FePt were theoretically\ninvestigated in the past already. Daalderop et al.[18]\nand Ravindran et al.[12] studied the influence of the\nband-filling on EMCAof FePt. Many groups studied the\ninfluence of the temperature on the MCA of FePt [31,\n32]. The dependence of the Curie temperature on the\nFePt grain size was investigated via model Hamiltonian\ncalculations [33]. Burkert et al.[9], Lukashev et al.[34]\nand Kosugi et al.[25] studied how EMCAdepends on the\nstrain (i.e., the c/aratio). Cuadrado et al.[35] gradually\nsubstituted the Fe atom by Cr, Mn, Co, Ni, or Cu to find\nthat the MCA energy of Fe 1−yXyPt alloys can be tuned\nby adjusting the content of the substituting element.\nTo facilitate the understanding of the MCA of FePt\nand related systems further, we focus on some aspects\nthat have not been paid attention so far. We show that\nifwien2k andsprkkr calculations are converged, they\nyield comparable values for the MCA energy. Dealing\nwith relativity by introducing an additional SOC-related\nterm to the Hamiltonian is thereby justified. The the-\noretical MCA energy of FePt (3.0 meV) is significantly\nlarger than the experimental value (1.3 meV), implying\nconclusively that the LDA cannot properly describe the\nMCA of FePt. We also analyze how the total energy\nvaries with the magnetization angle and how MCA en-\nergy scales with the SOC strength. Based on this we\nconclude that even though the MCA of FePt is domi-\nnated by a second order perturbation theory mechanism,\nthere isa smallbut distinct contribution originatingfrom\nthe Pt sites which is not accountable for by second order\nperturbation theory.\nII. COMPUTATIONAL DETAILS\nA. Technical details\nWe studied bulk FePt with the L1 0layered structure.\nThe lattice parameters of a tP2 unit cell are a=2.722˚A\nFIG. 1. Crystal structure of bulk L1 0FePt\nandc=3.714˚A. Fe atoms and Pt atoms are at the\n(0.0,0.0,0.0) and (0.5,0.5,0.5) crystallographic positions,\nrespectively, resulting in a compound with alternating Fe\nand Pt atomic layersstacked alongthe caxis (see Fig. 1).\nWe usedtwodifferentcomputationalmethods, namely,\nthe FLAPW method as implemented in the wien2k code\n[28] and the multiple scattering KKR Green function\nmethodasimplementedinthe sprkkrcode[29,30]. Our\ncalculations are based on the LDA. The values presented\nin the section Results (Sec. III) were obtained using\nthe Vosko, Wilk and Nusair(VWN) exchange-correlation\nfunctional [36]. Use of different LDA functionals leads to\nsmall but identifiable changes in EMCA, as explored in\nSec. IIIE.\nThe KKRGreenfunction calculationsweredone in the\nfull-potential mode. Only when studying the scaling of\nEMCAwith the SOC strength (Sec. IIID), we rely on the\natomic spheres approximation (ASA), because in that\ncase many evaluations of EMCAhave to be done and the\nfocusin that partis ontrendsand notsomuch onnumer-\nical values. The energy integrals were evaluated by con-\ntour integration on a semicircular path within the com-\nplex energy plane, using a Gaussian mesh of 40 points.\nAn important convergence parameter is the maximum\nangular momentum ℓ(KKR)\nmaxused for the multipole ex-\npansion of the Green function (see Appendix1). To get\nas accurate results as possible, we mostly use ℓ(KKR)\nmax=7.\nHowever, if a lot of calculations with different settings4\nhas to be done (Secs. IIIC and IIID), we use ℓ(KKR)\nmax=3,\nwhich issufficient ifthe focusis on how EMCAvarieswith\nthe magnetization angle or with the SOC strength and\nnot on particular values.\nTheconvergenceofFLAPWcalculationsisdetermined\nby the size of the basis. We treated Fe 3 p, 3d, 4sand Pt\n5p, 5d, 6sstates as valence states and Fe 1 s, 2s, 2p, 3s\nand Pt 1 s, 2s, 2p, 3s, 3p, 3d, 4s, 4p, 4d, 4f, 5sstates as\ncore states. The expansion of the wave functions into\nplane waves is controlled by the plane wave cutoff in\nthe interstitial region. This cutoff is specified via the\nproduct RMTKmax, whereRMTis the smallest muffin-\ntin (“atomic”) sphere radius and Kmaxis the magnitude\nof the largest wave vector. We use RMTKmax=8 in this\nstudy. The convergence of EMCAwithRMTKmaxis in-\nvestigated in the Appendix2. The expansion of the wave\nfunctions into atomic-like functions inside the spheres is\ncontrolled by the angular-momentum cutoff ℓ(APW)\nmax. We\nuseℓ(APW)\nmax=10throughoutthispaper. Note thatthecut-\noff’sℓ(APW)\nmaxandℓ(KKR)\nmaxhave different roles in FLAPW\nandKKR-Greenfunctionmethods, sotheirvaluescannot\nbe directly compared.\nAs concerns the muffin-tin radii in wien2k calcu-\nlations, the atomic spheres are chosen so that they\nare smaller than the touching spheres for the MCA\nenergy calculations ( R(Fe)\nMT=2.2 a.u., R(Pt)\nMT=2.3 a.u.,\nR(touch)\nMT=2.527 a.u.) because in this way the basis avoids\nthe linearization error. On the other hand, for analyzing\nsite-related magnetic moments we use touching muffin-\ntin spheresbecausein this waywe minimize the moments\nassociated to the interstitial region. In this way we are in\na better position to compare the wien2k results with the\nsprkkr data, where the site-related magnetic moments\nare determined as moments within Voronoi polyhedra.\nThe stability of EMCAwith respect to RMT’s variation is\ndemonstrated in the Appendix3.\nOnce the Greenfunction componentsor the wavefunc-\ntions have been determined, the charge density is ob-\ntained via the k-space integration over the BZ. Whenusing the wien2k code, the BZ integration was carried\nout using the modified tetrahedron method [37]. When\nusing the sprkkr code, the BZ integration was carried\nout via sampling on a regular k-mesh , making use of\nthe symmetry [38]. The convergence of EMCAwith re-\nspect to the the number of k-points is explored in the\nAppendix4. Based on it, we used 800000 k-points in the\nfull BZ for wien2k calculations and 100000 k-points in\nthe full BZ for sprkkr calculations.\nConsidering the convergence tests as a whole, we ar-\ngue that that the numerical accuracy of our EMCAval-\nues is about 0.1 meV for wien2k calculations and about\n0.2 meV for sprkkr calculations.\nB. Treatment of relativistic effects\nThesprkkrcodeworksfullyrelativistically,itsolvesa\nfour-componentDirac equationby default. SOC is there-\nfore implicitly fully included for all states. Nevertheless,\nthebareeffectoftheSOCcanbeinvestigatedvia sprkkr\nif one employes an approximate two-component scheme\n[39] where the SOC-related term is identified by rely-\ning on a set of approximate radial Dirac equations. This\nschemewasusedrecentlytoinvestigatehowtheMCAen-\nergy of adatoms and monolayers on noble metals varies\nif SOC is selectively switched on only at some sites [15].\nWe employed it here for the same purpose.\nAs concerns the wien2k code, SOC is included differ-\nently for core and valence electrons. The core electrons\naretreatedfully relativisticallybysolvingthe atomic-like\nDirac equation. For the valence electrons the SOC is in-\ncluded in atomic spheres via an approximative scheme\nthat introduces an additional term\nHSOC=ξ(r)L·S (1)\nto the spin-polarized Schr¨ odinger-like scalar relativistic\nequation. Technically, the influence of the term (1) is in-\ncluded by starting with a scalar-relativistic FLAPW cal-\nculation without SOC. The eigenfunctions thus obtained5\nare then used as a basis in which another diagonalization\nis done and this time also the SOC term Eq. (1) is taken\ninto account. This procedure is often called second vari-\national step [40]. Usually this second variational step is\napplied only to a subset of FLAPW eigenstates to gain a\nsubstantial speed-up. This subset is defined so that it in-\ncludesallscalar-relativisticeigenstatesuptoenergy Emax\nabove the Fermi level. The Emaxparameter thus plays\nan analogous role as RMTKmax. Moreover, relativistic\nlocal orbitals ( p1/2wavefunctions) were added to the ba-\nsis [41]. To achieve the highest accuracy, we set Emax\nas large as needed to include all FLAPW eigenfunctions\nin the second step (this can achieved by setting Emax\nof 100 Ry or higher). More details can be found in the\nAppendix. As concerns the interstitial region, valence\nelectrons are treated in a non-relativistic way. In the\nrest of this paper “fully relativistic calculation” implies\nuse of the Dirac equation for sprkkr and Schr¨ odinger\nequation plus separate SOC term (1) in the Hamiltonian\nforwien2k, unless it is explicitly said otherwise.\nThesprkkr andwien2k codes allow for non-\nrelativistic and scalar-relativistic calculations as well. In\nthe first case, both valence and core electrons are treated\nnon-relativistically. In the second case, the valence elec-\ntrons are treated using the scalar-relativistic approach\nwhile for the core electrons atomistic Dirac equation is\nsolved (this applies to both codes).\nC. Scaling of the spin-orbit coupling\nFora deeperunderstandingwe wantto investigatehow\nEMCAdepends on the SOC. More specifically, we are in-\nterestedin how EMCAvariesifthe SOCstrengthisvaried\nat the Fe and Pt sites separately, i.e., we assume that the\nHamiltonian Eq. (1) can be symbolically rewritten as\nHSOC=/summationdisplay\niλiξi(r)Li·Si, (2)\nwhereλiis the scaling factor for site i. Such calculations\nwere done via the sprkkr code, using the approximateTABLE I. Spin magnetic moments (in µB) related either to\na FePt unit cell or just to the Fe site, for different ways of\nincluding the relativistic effects.\nsprkkr wien2 k\nµ(cell)\nspin µ(Fe)\nspin µ(cell)\nspin µ(Fe)\nspin\nnon relativistic 3 .17 2 .86 3 .15 2 .86\nscalar relativistic 3 .21 2 .86 3 .21 2 .87\nfully relativistic 3 .17 2 .83 3 .17 2 .84\nTABLE II. Orbital magnetic moments (in µB) related to the\nFe and Pt atoms in FePt for magnetization either parallel to\nthezaxis (µ(M/bardblz)\norb ) or perpendicular to the zaxis (µ(M/bardblx)\norb ).\nFe Pt\nsprkkr wien 2ksprkkr wien 2k\nµ(M/bardblz)\norb 0.065 0 .065 0 .044 0 .042\nµ(M/bardblx)\norb 0.062 0 .062 0 .060 0 .054\nscheme [39] mentioned in the beginning of Sec IIB.\nIII. RESULTS\nA. Magnetic moments\nThe presence of Pt in FePt suggests that the way rela-\ntivistic effects are treated could be important. There-\nfore, we calculated magnetic moments in FePt using\na non-relativistic Schr¨ odinger equation, using a scalar-\nrelativistic approach, and using a relativistic scheme.\nSpin magnetic moments related either to the unit cell\nor only to the Fe site are shown in Tab. I. We can see\nthatrelativityhasonlyamarginaleffectonthe spinmag-\nnetic moments in FePt. Orbital magnetic moments are\nmore interesting in this respect — they would be zero\nin the absence of SOC. Our results in Table II give the\norbital magnetic moment at the Fe and Pt sites for two\norientations of the magnetization.\nOne can see that both codes lead to very similar val-\nues forµspinandµorb. In particular, the anisotropy of\nµorbat Fe and at Pt sites is nearly the same. Small6\ndifferences between the codes in the local magnetic mo-\nments may be due to the fact that the moments are de-\nfined in different regions: Wigner-Seitz cells (or more\nprecisely Voronoi polyhedra) in sprkkr and touching\nmuffin-tin spheres in wien2k. The difference would be\nlarger if we used “standard” setting of muffin-tin radii\ninwien2k (R(Fe)\nMT=2.2 a.u. and R(Pt)\nMT=2.3 a.u. instead of\nR(Fe)\nMT=R(Pt)\nMT=2.527a.u.): in that case, the local spin mo-\nments obtained via wien2k would be smaller by about\n3 % and orbital moments by about 10 %.\nB. Magneto-crystalline anisotropy energy\nCalculating the MCA energy by subtracting total en-\nergies for two orientations of the magnetization as\nEMCA≡E(M/bardblx)−E(M/bardblz)(3)\nis very challenging, because the total energies and the\nMCA energy differ by about eight or nine orders of mag-\nnitude. We paid a lot of attention to the issues of conver-\ngence to get accurate numbers. The details can be found\nin the Appendix. Here we only mention two issues which\nhave to be given special attention.\nFor full-potential sprkkr calculations, attention has\nto be paid to the multipole expansion of the Green func-\ntion governed by the cutoff ℓ(KKR)\nmax. KKR calculations\nhave known behavior concerning the ℓ(KKR)\nmaxconvergence\nwhich play role if one aims at high-accuracy total en-\nergy calculations [42, 43]. Part of the problem are nu-\nmerical difficulties connected with the evaluation of the\nMadelungcontributiontothefullpotentialforhighangu-\nlarmomenta[44,45]. NotethattoobtaintheGreenfunc-\ntion components up to ℓ(KKR)\nmax, one needs potential com-\nponents up to 2 ℓ(KKR)\nmaxand shape functions components\nup to 4ℓ(KKR)\nmax. Another difficulty is an efficient treat-\nment of the so-called near-field corrections [44, 46]. Var-\nious ways to deal with these issues have been suggested\n[43, 46–48]. We performed a test of the ℓ(KKR)\nmaxconver-\ngence (Appendix1) which indicate if that the ℓ(KKR)\nmax=7TABLE III. MCA energy Emaxof FePt (in meV) calculated\nby two approaches.\nsprkkr wien 2k\nsubtracting total energies 3 .04 2 .99\nmagnetic force theorem 3 .12 2 .85\ncutoff is used, that the numerical accuracy of the MCA\nenergy is about 0.2 meV.\nFor accurate MCA energy calculations using the\nwien2k code, one has to pay special attention so that\ntheenergyparameters Eℓusedforcalculatingradialwave\nfunctions uℓ(r,Eℓ)aredeterminedverypreciselyandcon-\nsistently. Thisapplies, inparticular,alsofortherelativis-\ntic local orbitals. In wien2k this is done by searching\nfor the energies where uℓ(RMT,E) changes the sign to\ndetermine Etop, and where it has zero slope to deter-\nmineEbottom. The arithmetic mean of these two ener-\ngies gives Eℓ. For the calculations presented here these\nenergies had to be determined with an accuracy better\nthan 0.1 mRy. A parameter specific for relativistic cal-\nculations via wien2k isEmax, which controls how many\nscalar-relativistic eigen-states are considered when SOC\nis included (Appendix5). We used Emax=100 Ry, mean-\ning that all eigen-states were included.\nThe MCA energy obtained by subtracting the total\nenergies is shown in the first line of Tab. III. Values\nobtained via sprkkr andwien2k show good agreement.\nConsidering the convergence analysis we performed, this\nallows us to state that the magnetic easy axis of FePt\nis out-of-plane and the MCA energy is 3.0 meV within\nthe LDA framework (for the VWN exchange-correlation\nfunctional).\nObtaining the MCA energy by subtracting the total\nenergiesiscomputationallyverycostly. Theneed forself-\nconsistent calculations for two magnetization directions\ncan be avoided if one relies on the magnetic force theo-\nrem. In this approach the MCA energy is calculated us-\ning a frozen spin-dependent potential [49, 50]. The MCA7\nenergy is then obtained either by subtracting the band-\nenergies or by evaluating the torque at magnetization tilt\nangle of 45◦[15, 51]. As the magnetic force theorem is\nfrequently employed, we applied it here as well. The re-\nsults are shown in the second line of Tab. III. We can\nsee that the magnetic force theorem yields very similar\nvalues as if total energies are subtracted.\nRelation between EMCAand anisotropy of µorb\nFor the sake of completeness we checked also the\nBruno formula [52], which links the MCA energy to the\nanisotropy of orbital magnetic moment. The Bruno for-\nmula [52] (as well as the slightly more sophisticated van\nder Laan formula [5]) can be derived from second order\nperturbation theory if some additional assumptions are\nmade. It is often employed in the context of x-ray mag-\nnetic circular dichroism experiments that give access to\nthe anisotropy of orbital magnetic moment via the so-\ncalled sum rules.\nEventhoughthe formulawasoriginallyderivedforsys-\ntems with only one atomic type, the relation between the\nMCA energy and the anisotropy of orbital magnetic mo-\nments has been frequently applied also for multicompo-\nnent systems [12, 53–57]. In such a case an estimate of\nEMCAcan be made by evaluating (cf. Ravindran et al.\n[12] and Andersson et al.[58])\nEMCA=/summationdisplay\niξi\n4/parenleftBig\nµ(i,M/bardblz)\norb−µ(i,M/bardblx)\norb/parenrightBig\n,(4)\nwhereilabels the constituting atoms. This equation is\nvalid only if off-site spin-flip terms are neglected [12, 58,\n59].\nWe evaluated Eq. (4) using SOC parameters\nξ(Fe)=65 meV and ξ(Pt)=712 meV, as obtained from\nab-initio calculations for FePt relying on the method\ndescribed by Davenport et al.[60]. We obtained\nEMCA=−2.62 meV using sprkkr results and EMCA=\n−2.09 meV using wien2k results. The sign of EMCA\nevaluated from Eq. (4) is wrong, indicating that this for-mula does not provide a suitable framework for studying\nthe MCA of FePt. Technically, the reversal of the sign\nofEMCAobtained via Eq. (4) is due to µorbat Pt (see\nTab. II): we have µ(M/bardblz)\norb> µ(M/bardblx)\norbat the Fe site and\nµ(M/bardblx)\norb> µ(M/bardblz)\norbat the Pt site. As ξ(Pt)is much larger\nthanξ(Fe), the Pt-related term dominates in Eq. (4).\nThe failure of the Bruno formula (4) does not auto-\nmatically imply that second order perturbation theory\ncannot be used for describing the MCA of FePt. Namely,\nit is likely that additional assumptions employed in the\nderivation of Eq. (4) are not fulfilled; in particular, for\nPt atoms, the exchange splitting and SOC will be of the\nsame order of magnitude. Two more indicative tests\nwhether second order perturbation theory itself pro-\nvides a good framework for understanding the MCA of\nFePt are presented below.\nC. Dependence of the total energy on the\norientation of the magnetization axis\nAccurate calculations can provide information on the\nfull form of the dependence of the total energy on the\nangleθbetween the magnetization direction and the z\naxis. For tetragonal systems the first two terms in the\ndirectional cosines expansion of the total energy are\nE(θ)−E0=K1sin2θ+K2sin4θ .(5)\nHere we omit the azimuthal dependence, keeping φ=0◦.\nIf the influence of SOC is included via the explicit term\nEq. (1), then application of second order perturbation\ntheory leads to a simple dependence of the total energy\non the angle θas\nE(θ)−E0=K1sin2θ ,\nmeaning that only the first term survives in Eq. (5)\n[52, 61]. Inspecting the full E(θ) dependence as obtained\nvia fully-relativistic ab-initio calculations thus provides\nthe possibility to estimate to what degree a treatment of\nMCA based on second order perturbation theory is ade-8\n0123E()-E0(meV)\n0 30 60 90\nmagnetizationangleab-initioK1sin2+K2sin4\nK1sin2\n2.72.82.93.03.1E() -E0(meV)\n70 80 90\nmag.angle\nFIG. 2. Dependence of the total energy on the magnetization\nangleθ(circles) and its fit either as K1sin2θ(dashed line) or\nasK1sin2θ+K2sin4θ(dash-dotted line). An overall view is\nin the left panel, a detailed view on the region close to θ=90◦\nis in the right panel.\nquate: large K2coefficient implies large deviations from\nsecond order perturbation theory.\nWe performed aseriesofcalculationsfor differentmag-\nnetizationtiltangle θ, usingthe sprkkrcode. TheMCA\nenergy was evaluated as a difference of total energies.\nThe results are shown via circles in Fig. 2. Because we\nwanted to have a fine θ-mesh, we had to perform a lot of\ncalculations; therefore, we used ℓ(KKR)\nmax=3 in this section.\nThe numerical value for θ=90◦thus differs a bit from\nTab. III, where the ℓ(KKR)\nmax=7 cutoff was used.\nThe ab-initio data were fitted via Eq. (5). If only\ntheK1sin2θterm is employed (taking K2=0), we ob-\ntainK1=3.085 meV. If both terms in Eq. (5) are em-\nployed, we obtain K1=3.008 meV and K2=0.092 meV.\nEven though both fits look nearly the same in the over-\nall view, a detailed analysis shows that the fit with both\nterms is significantly better (cf. the right panel in Fig. 2).\nUsing even higher order terms in the fit did not lead to\na significant improvement.\nTo summarize, our calculations show that the depen-\ndence of the total energy on the magnetization angle is\nfully described by Eq. (5). The ratio of the coefficients0510EMCA(meV)\n0.0 0.5 1.0 1.5\nSOCfactor =Fe=Ptscalingboth\nFeandPt\nFIG. 3. Dependence of EMCA on the SOC scaling factor λ.\nThe markers denote calculated values of EMCA, the line rep-\nresents a fit to these data within the λ∈[0; 0.4] interval.\nK2/K1is 0.03, thus we deduce that the MCA of FePt is\ndominated by the second order perturbation theory but\nthere is also a small but identifiable contribution which\ncannot be described by it.\nD. Dependence of the MCA energy on spin orbit\ncoupling\nIf the magnetocrystalline anisotropy is described\nwithin second order perturbation theory, it scales with\nthe square of the SOC-scaling parameter λ,EMCA∼λ2\n[5, 52, 61]. Inspecting the EMCA(λ) dependence thus\nprovides another criterion to what degree second order\nperturbationtheoryissufficient todescribemagnetocrys-\ntalline anisotropy of FePt. To get type-specific informa-\ntion, one should scale λFeandλPtseparately. In that\ncase, however, the scaling of EMCAwith SOC takes a\nsomewhat more complicated form [58]\nEMCA(λFe,λPt) =Aλ2\nFe+BλFeλPt+Cλ2\nPt.(6)\nThe scaling of EMCAwith SOC will thus retain a\nquadratic form only if the scaling is uniform ( λFe=λPt)\nor if SOC for one of the atomic types is zero (recovering\nthus the case of a single-component system [5, 52, 61]).\nWe start by calculating EMCAfor a uniform SOC scal-\ning, i.e., λFe=λPt. We vary λfrom 0 to 1.5 to cover the\nnon-relativistic as well as the relativistic regime: if λis9\nzero, there is no spin orbit coupling, if λis 1, we recover\nthestandardrelativisticcase. Thecalculationsweredone\nwith the sprkkr code, employing the scheme described\nin Sec. IIC and evaluating EMCAby subtracting total\nenergies. To reduce the computer requirements, we per-\nformed all the calculations in this section with ℓ(KKR)\nmax=3\nin the ASA mode; this enables us to use a fine λmesh so\nthat the curve fitting is reliable. The results are shown\nby points in Fig. 3. Employment of the ASA obviously\nleads to less acurate results than for full-potential cal-\nculations: EMCAobtained within the ASA is by about\n1 eV larger than EMCAobtained for full potential. How-\never, this does not affect our conclusions concerning the\nscaling of EMCAwith strength of the SOC.\nTo verify the predictions of the perturbation theory,\nwe fit calculated EMCA(λ) with the quadratic function,\nEMCA(λ) =aλ2. (7)\nPerturbation theory should work well for small values of\nλwhile it can be less appropriatefor large values of λ. So\nthe fit to the function (7) is performed in such a way that\ntheacoefficient is sought only for λin the range between\nzero and 0.4 (the upper value was arbitrarily chosen just\nfor convenience). One can see from Fig. 3 that while\nthe fit describes the ab-initio data very well within the\nλ∈[0;0.4] range, there are small but clear deviations for\nlargerλ. This suggests that while second order pertur-\nbation theory accounts for the dominant mechanism of\nmagnetocrystalline anisotropy of FePt, some effects be-\nyond it are also present.\nTo learn more about atom-specific contributions to\nMCA, let us scale the SOC at the Fe and Pt sites sep-\narately. When varying λFeorλPtwe further distinguish\ntwo cases — either the SOC at the remaining species is\ntotally suppressed ( λ=0) or it is kept at its “normal”\nvalue (λ=1). Results for scaling SOC at the Fe sites are\nshown in Fig. 4, results for scaling SOC at the Pt sites\nare shown in Fig. 5. Fits to the quadratic dependence\nofEMCAonλFeor onλPtwere done only in case that0.00.51.0EMCA(meV)\n0.0 0.5 1.0 1.5\nSOCfactor Fescaling Fe\nPt=0\n46EMCA(meV)\n0.0 0.5 1.0 1.5\nSOCfactor Fescaling Fe\nPt=1\nFIG. 4. Dependence of EMCA on the SOC scaling factor at\nthe Fe sites λFe. The markers denote calculated values of\nEMCA, the line in the left panel represents a fit to these data\nwithin the λFe∈[0; 0.4] interval.\nSOC at the other site is suppressed. Namely, if λat\nthe other atomic type is non-zero, the functional depen-\ndence is more complicated — see Eq. (6) — and fitting\nEMCA(λ) with the simple Eq. (7) would not make sense.\nSimilarly as in the case of the uniform scaling, the fits\nwere attempted for λin the [0;0.4] interval.\nConcerningthecasewhenSOCisvariedattheFesites,\none can see that if λPt=0, the dependence of EMCAon\nλFeis perfectly accounted for by second order perturba-\ntion theory: the quadratic fit describes the EMCA(λFe)\ndependence very well also outside the [0;0.4] interval in\nwhich the acoefficient was sought (left graph in Fig. 4).\nThis suggests that it must be the strong SOC at Pt sites\nwhich makesthe EMCA(λ) curvein Fig. 3 to deviate from\na perfect parabola. Indeed, if SOC at Pt sites is switched\non (right graph in Fig. 4), the EMCA(λ) functional de-\npendence changes completely.\nLet us turn now to the case of varying λPt. If there\nis no SOC at the Fe sites, the EMCA(λPt) dependence is\ndescribed by the fitted parabola only for low values of\nλPt(left graph in Fig. 5). If λPtincreases beyond the fit-\nting interval of [0;0.4], deviations of ab-initio data points\nfrom the fit by Eq. (7) are similar as for uniform SOC fit\npresented in Fig. 3. So it follows from our analysis that\nthe effect of SOC at the Fe sites can be accounted for by\nsecond order perturbation theory while the effect of SOC10\n051015EMCA(meV)\n0.0 0.5 1.0 1.5\nSOCfactor Ptscaling Pt\nFe=0\n051015EMCA(meV)\n0.0 0.5 1.0 1.5\nSOCfactor Ptscaling Pt\nFe=1\nFIG. 5. Dependence of EMCA on the SOC scaling factor at\nthe Pt sites λPt. The markers denote calculated values of\nEMCA, the line in the left panel represents a fit to these data\nwithin the λPt∈[0; 0.4] interval.\nat the Pt sites goes beyond it.\nE. Dependence of the MCA energy on the LDA\nexchange-correlation functional\nUsually the calculated properties of solids do not cru-\ncially depend on which form of the LDA exchange-\ncorrelation functional is used. However, as the MCA\nenergy is a very sensitive quantity, it is useful to inves-\ntigate how the EMCAvaries if different LDA exchange-\ncorrelation functionals are used. Apart from the VWN\nexchange-correlation functional used throughout this\nwork we include in the comparison the Perdew and\nWang exchange-correlation functional [62] (the default\nforwien2k) and functionals suggested by von Barth and\nHedin [27] and by Moruzzi, Janak and Williams [63].\nWe evaluated EMCAby subtracting total energies for\nthis test. The results are summarized in Tab. IV. One\ncan see that different LDA functionals lead to MCA en-\nergies that differ from each other by 0.1–0.2 meV.\nF. Relativistic effects in the density of states\nFig. 6 depicts the influence of relativity on the density\nof states (DOS) resolved in angular momentum compo-\nnents respective to Fe and Pt sites. The data presentedTABLE IV. The MCA energy of FePt (in meV) calculated by\nsubtracting total energies for different exchange and corre la-\ntion functionals.\nsprkkr wien 2k\nVosko and Wilk and Nusair [36] 3 .04 2 .99\nPerdew and Wang [62] — 3 .02\nvon Barth and Hedin [27] 3 .29 3 .18\nMoruzzi, Janak and Williams [63] 2 .97 —\n/s45/s57 /s45/s54 /s45/s51 /s48 /s51/s45/s48/s46/s49/s54/s45/s48/s46/s48/s56/s48/s46/s48/s48/s48/s46/s48/s56/s48/s46/s49/s54\n/s70/s101/s32/s115/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s110/s111/s110/s32/s114/s101/s108/s97/s116/s105/s118/s105/s115/s116/s105/s99\n/s32/s115/s99/s97/s108/s97/s114/s32/s114/s101/s108/s97/s116/s105/s118/s105/s115/s116/s105/s99\n/s32/s102/s117/s108/s108/s121/s32/s114/s101/s108/s97/s116/s105/s118/s105/s115/s116/s105/s99\n/s45/s57 /s45/s54 /s45/s51 /s48 /s51/s45/s48/s46/s49/s50/s45/s48/s46/s48/s54/s48/s46/s48/s48/s48/s46/s48/s54/s48/s46/s49/s50\n/s32/s110/s111/s110/s32/s114/s101/s108/s97/s116/s105/s118/s105/s115/s116/s105/s99\n/s32/s115/s99/s97/s108/s97/s114/s32/s114/s101/s108/s97/s116/s105/s118/s105/s115/s116/s105/s99\n/s32/s102/s117/s108/s108/s121/s32/s114/s101/s108/s97/s116/s105/s118/s105/s115/s116/s105/s99/s32/s80/s116/s32/s115/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41\n/s45/s57 /s45/s54 /s45/s51 /s48 /s51/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48\n/s70/s101/s32/s112/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s45/s57 /s45/s54 /s45/s51 /s48 /s51/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s48/s46/s48/s53/s48/s46/s49/s48/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s32/s32/s80/s116/s32/s112\n/s32\n/s45/s57 /s45/s54 /s45/s51 /s48 /s51/s45/s49/s46/s54/s45/s48/s46/s56/s48/s46/s48/s48/s46/s56/s49/s46/s54/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s70/s101/s32/s100\n/s45/s57 /s45/s54 /s45/s51 /s48 /s51/s45/s49/s46/s48/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s68/s79/s83/s32/s40/s115/s116/s97/s116/s101/s115/s47/s101/s86/s41\n/s101/s110/s101/s114/s103/s121/s32/s40/s101/s86/s41/s80/s116/s32/s100\nFIG. 6. Partial spin-resolved angular-momentum-projecte d\ndensity of states for Fe and Pt sites calculated within a non-\nrelativistic, a scalar-relativistic and a fully-relativi stic frame-\nwork.\nherewereobtainedusingthe sprkkrcode; dataobtained\nusing the wien2k code look practically the same.\nGenerally, there is a significant change in the DOS\nwhen going from non-relativistic to scalar-relativistic\ncase and only a minor change when going from scalar-11\nrelativistic to the fully relativistic case. The largest dif-\nferencebetweennon-relativisticandrelativisticcaseisfor\nthesstates. This may be due to the fact that selectrons\nhave a large probability density near the nucleus where\nrelativistic effects (mass-velocity and Darwin term) are\nstronger than at larger distances. Largest difference be-\ntween scalar relativistic and fully relativistic calculations\nare for the Pt dstates, where also the SOC is expected\nto be stronger than for the other cases.\nFor Ptsanddstates one can make an interesting com-\nparison with atomic results for Au [64] which are often\nquoted when relativistic effects in solids are discussed. It\nfollowsfromFig.6thatrelativisticeffectsshiftvalencePt\n6sstates to lower energies due to the orthogonality con-\nstrains to the more localized 1 sstate and Pt 5 dstates\nto higher energies due to a better screening of the nu-\ncleus by innermost electrons. The same happens for 6 s\nand 5datomic states of Au, respectively. So we can in-\nfer that the mechanism through which relativity affects\nPt states is essentially atomic-like and common to all 5 d\nnoble metals.\nIV. DISCUSSION\nOur aim was to get realiable quantitative information\non the MCA of FePt, which we take as an archetypal\nlayered system of magnetic and non-magnetic transition\nmetals. We employed two quite different computational\nprocedures. Both of them yield similar values for the\nMCA energy. Numerical stability ofresults is well docu-\nmented by convergence tests presented in the Appendix.\nTherefore the results can be trusted to represent the true\nLDA value of the MCA energy. Our data can be used as\na benchmark for LDA calculations.\nRelativistic effects are implemented in the wien2k\ncode in an approximative way, accounting for the SOC\nby a separate term (see Eq. (1)) which is added to the\nscalar-relativistic Hamiltonian. Most codes rely on this\napproachwhentheydealwithSOC.The sprkkrscheme,ontheotherhand,solvestheDiracequationsoitdoesnot\nuse approximationswhen dealing with relativistic effects.\nGood agreement between MCA energies obtained via the\nwien2k code and via the sprkkr code shows that deal-\ning with relativity by invoking the separate term Eq. (1)\nis justified in our case. As we are studying FePt, i.e., a\ncompound containing an element with a strong SOC, it\nis likely that the approximative scheme associated with\nEq. (1) is sufficiently accurate for most common situa-\ntions and/or systems. One should only make sure that\na sufficiently large basis for the second variation step is\ntaken (see Appendix).\nWe calculated EMCAboth via subtracting total en-\nergies and via the magnetic force theorem. Using the\nmagnetic force theorem is technically much more con-\nvenient than subtracting total energies. Knowing limits\nof its reliability it thus vital. For pure Fe monolayers\nthe magnetic force theorem was shown to be valid to a\nhigh accuracy [65, 66]. However, there are indications\nthat this may no longer be true for systems with nor-\nmally non-magnetic atoms with large induced moments\nand strong SOC [10, 67]. For such atoms one would ex-\npect rather large changes of the spin-polarized electron\ndensity upon rotation of the magnetization. This applies\nalso for the Pt atoms in FePt. Our results indicate, nev-\nertheless, that the magnetic force theorem yields quite\naccurate values for EMCAfor FePt (Tab. III). One can\nconjecture that this would be the case for similar layered\nsystems as well.\nWhen comparing our EMCAwith experiment (1.3–\n1.4 meV) [26], it is evident that the LDA result does\nnot quite agree with it. Clearly one has to go beyond\nLDA for a quantitative description of MCA of FePt. It\ndoes not matter in this respect which specific form of the\nLDA functional is used. Nevertheless, as different LDA\nfunctionalslead to similarbut still visibly different values\nofEMCA(cf. Tab. IV), each calculation of the MCA en-\nergyshouldbe alwaysaccompaniedbyinformationwhich\nparametrization of the LDA functional was employed.12\nEmploying the generalized gradient approximation\n(GGA) does not lead to substantial improvement with\nrespect to the LDA. We obtained EMCA= 2.73 meV\nfor the frequently used PBE-GGA form [68] (using the\nwien2k code and evaluating the MCA energy as a differ-\nence of total energies). It is worth to note in this respect\nthat Shick and Mryasovwere able to obtain the MCA en-\nergy of FePt as 1.3 meV by using the LDA+ Uapproach\nand searching for suitable site-related values of the Upa-\nrameter [23]. Interestingly, if many-body effects are de-\nscribed via the orbital polarization term of Brooks [69],\ncalculated EMCAis not significantly improved in com-\nparison with the LDA [12, 20, 21, 23] — despite the fact\nthat this approach proved to be useful when calculating\norbital magnetic moments of transition metals [70, 71].\nThe Bruno formula, derived originally for single-\ncomponentsystemsonly, hasrecentlybeenemployedalso\nfor systems where there is more than one magnetic ele-\nment [12, 53, 54, 56]. In our case the Bruno formula\nsuggests a wrong magnetic easy axis, hence it not a suit-\nable tool for understanding the MCA of FePt. Similar\nobservations were made earlier for other compounds con-\ntaining3 dand5delements[58,59, 72], sowesuggestthat\nintuition based onanalysisoforbitalmoments shouldnot\nbe used for these systems — despite its appeal and suc-\ncess in monoelemental systems.\nConcerning a more detailed view on the mechanism of\nMCA, we found that even though MCA of FePt is domi-\nnated by a second order perturbation theory mechanism\n(as found earlier by Kosugi et al.[25] by analyzing the\ndependence of EMCAof FePt on c/a), effects beyond it\nare clearly present as well. These effects could be iden-\ntified (i) by analyzing the full angular dependence of the\ntotal energy and (ii) by inspecting how the MCA energy\ndepends on the SOC strength. Separate scaling of SOC\nat Fe and Pt sites allows us to deduce that the deviations\nfrom a pure secondorderperturbation theorymechanism\nhave their origin at the Pt sites. One possible mecha-\nnism that is beyond the standard second order perturba-tion theory is reoccupation of states close to the Fermi\nlevel [7, 18].\nAnother implication comming from our analysis of the\nfull angular dependence of the total energy is that one\ncan indeed use the torque implemenetation of the mag-\nnetic force theorem: replacing the difference of energies\nE(90◦)−E(0◦) by the torque at 45◦can be done only if\nEq. (5) is valid [15, 51]. It follows from the results shown\nin Fig. 2 that this indeed is the case.\nV. CONCLUSIONS\nIfelectronicstructurecalculationsperformedbymeans\nof FLAPW and KKR methods are properly converged,\nthey yield the same results even for such sensitive quan-\ntities as the magnetocrystalline anisotropy energy. The\nproper LDA value of the MCA energy for FePt ( 3.0 meV\nfor the VWN exchange-correlation functional) is signifi-\ncantlylargerthaninexperiment(1.3meV),meaningthat\nthe MCAofFePtcan bedescribed properlyonlyifmany-\nbody effects beyond the LDA are included. As our value\nofEMCAwas obtained by two different methods and the\nconvergenceof both of them was carefully checked, it can\nbe used as a benchmark in future calculations.\nItisnotreallyimportantwhetherrelativisticeffectsfor\nFePt are accounted for by solving the full Dirac equation\nor whether the spin-orbit coupling is treated as a cor-\nrection to the scalar-relativistic Hamiltonian. The main\nmechanism of MCA in FePt can be described within the\nframeworkof second order perturbation theory but a sig-\nnificant contribution not accountable for by the second\norder perturbation theory is present as well.\nACKNOWLEDGMENTS\nWe would like to acknowledge CENTEM project\n(CZ.1.05/2.1.00/03.0088), CENTEM PLUS (LO1402)\nand COST CZ LD15147 of The Ministry of Education,\nYouthandSports(CzechRepublic). Computationaltime13\nhas been provided with the MetaCentrum (LM205) and\nCERIT-SC (CZ.1.05/3.2.00/08.0144) infrastructures. In\naddition we would like to thank for travel support from\nEU-COST action MP1306 (EUSpec).\nAppendix: Convergence tests\nThe total energies and the MCA energy can differ by\nabout eight or nine orders of magnitude. Very well con-\nverged calculations are thus required for precise values of\nthe MCA energy. We checked the influence of different\ntechnical parameters on the MCA energy if the wien2k\nandsprkkr codes are used. Some results which may\nbe interesting for those practicing such calculations are\npresented in this appendix.\nTheEMCAvalues presented in this appendix some-\ntimes differ from the values presented in the Results sec-\ntion of the main paper. This is because in order to save\ncomputer resources, when studying the dependence of\nEMCAon a particular convergence parameter, the other\nparametersweresometimes set to lowervalues than what\nwould lead to the most accurate results. These circum-\nstances do not influence the outcome of the convergence\ntests.\nUnless explicitly stated otherwise, the setting of\ntechnical parameters in this appendix is the following\n(cf. Sec. IIA): ℓ(KKR)\nmax=3 (forsprkkr),R(Fe)\nMT=2.2 a.u.,\nR(Pt)\nMT=2.3 a.u., RMTKmax=8,ℓ(APW)\nmax=10,Emax=100 Ry\n(forwien2k). Reciprocal space integrals were evaluated\nusing a mesh of 100000 k-points in the full BZ (both\ncodes).\nBased on the convergence tests presetend here, we ar-\nguethatthatthenumericalaccuracyof EMCAvaluespre-\nsented in the main paper is about 0.1 meV for wien2k\ncalculations and about 0.2 meV for sprkkr calculations.TABLE V. Convergence of EMCA obtained via the sprkkr\ncode with the angular momentum cutoff ℓ(KKR)\nmax .EMCA was\nevaluated by subtracting total energies.\nℓ(KKR)\nmax EMCA (meV)\n2 1.289\n3 3.101\n4 3.437\n5 3.407\n6 3.217\n7 3.039\n1. Convergence of sprkkr calculations with ℓ(KKR)\nmax\nKKRcalculationsoftotalenergiesarequitesensitiveto\ntheℓ(KKR)\nmaxcutoff. Therefore, we explore the dependence\nofourresultson this parameter. Theresultsareshownin\nTab. V. It followsfrom the table that cutting the angular\nmomentum expansion at ℓ(KKR)\nmax=3 (as it is commonly\ndone for transition metals) yields qualitatively correct\nvalue for the MCA energy.\nOne can see from Tab. V that even for ℓ(KKR)\nmax= 7,\na full convergence still has not been reached. However,\nincreasing ℓ(KKR)\nmaxfurther would be computationally very\ndemanding and, moreover, the issue of ℓ(KKR)\nmaxconver-\ngence would get intertwinned with numerical problems\nin evaluating the Madelung potential and near-field cor-\nrections, so the real benefit of it would be dubious. We\nconcludethat this limits the numericalaccuracyof EMCA\ncalculations to about 0.2 meV.\n2. Convergence of wien2k calculations with\nRMTKmax\nAn important parameter for the FLAPW calculations\nis the size of the basis set. It can be controlled by the\nRMTKmaxproduct. The value RMTKmax= 7.0 is set by\ndefault in wien2k. We increased the product RMTKmax\nstep by step from 6.0 up to 11.0 and calculated the MCA\nenergy. The results are shown in Tab. VI. It is clear14\nfrom this that reliable values for the MCA energy can be\nobtainedforabasissetdeterminedbythe RMTKmax=8.0\ncondition.\nTABLE VI. Convergence of EMCA obtained via the wien 2k\ncode with RMTKmax.EMCA was evaluated via subtracting\ntotal energies (second column) and via the magnetic force\ntheorem (third column).\nRMTKmax EMCA (meV) EMCA (meV)\nviaEtot via force th.\n6.0 2.851 2.772\n7.0 3.046 2.954\n8.0 3.051 2.967\n9.0 3.081 2.900\n10.0 2.993 2.908\n11.0 3.013 2.917\n3. Stability of wien2k calculations with respect to\nRMTvariations\nRecently the stability of the results with respect to\nvarying the muffin-tin radii was adopted as an informa-\ntivetestwhethertheFLAPWbasissetissufficientornot.\nNamely, in this way one changes the regions where the\nwave functions are expanded in terms of plane waves and\nwhere they are expanded in terms of atomic-like func-\ntions. Only if both expansions are appropriate the result\nwill be stable againstthis variation. We adopted this test\nin our study, the results are summarized in Tab. VII. We\ncan see from a good agreement between the MCA ener-\ngies obtained for different muffin-tin radii settings that\nthe basis we used for our wien2k calculations is appro-\npriate.\n4. Convergence of sprkkr andwien2k calculations\nwith the number of k-points\nA very important parameter is the number of k-points\nused in evaluating the integrals in the reciprocal space.TABLE VII. Dependence of EMCA obtained via the wien 2k\ncode on muffin-tin radii RMT.EMCA was evaluated via sub-\ntracting total energies (third column) and via the magnetic\nforce theorem (fourth column).\nR(Fe)\nMT(a.u.) R(Pt)\nMT(a.u.) EMCA (meV) EMCA (meV)\nviaEtot via force th.\n2.100 2.200 3.012 2.910\n2.180 2.280 3.083 2.973\n2.200 2.300 3.051 2.967\n2.220 2.320 3.004 2.848\n2.300 2.400 3.021 2.944\nTABLE VIII. Convergence of EMCA calculated by the sprkkr\ncode (second column) and the wien 2k code (third and fourth\ncolumns) with the number of k-points in the full BZ. EMCA (in\nmeV) was evaluated via subtracting total energies (second a nd\nthird columns) and via the magnetic force theorem (fourth\ncolumn).\nEMCA[sprkkr ]EMCA[wien 2k]\nno. ofk-points via Etot viaEtot via force th.\n1000 2.894 2.996 2.897\n10000 3.174 3.052 2.967\n60000 3.129 3.009 2.896\n100000 3.101 3.051 2.967\n140000 3.091 3.024 2.966\n180000 3.092 2.944 3.008\n220000 3.099 3.090 2.848\n260000 3.103 3.001 2.897\n500000 3.099 2.997 2.894\n800000 3.096 2.989 2.848\nWe performed corresponding tests for both codes. The\ndependenceof EMCAonthenumberof k-pointsinthefull\nBZ is shown in Tab. VIII. One can see that using about\n100000k-points in the full Brillouin zone is sufficient to\nget stable and reliable results.15\n5. Convergence of wien2k calculations with Emax\nWhen including the SOC within the second variation\nstep, the size of the new basis set is determined by the\nEmaxparameter (Sec. IIB). If Emaxis sufficiently large,\nall scalar-relativistic eigenstates are involved. 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B78, 144403 (2008)."    },    {        "title": "1604.07137v1.Room_temperature_tetragonal_noncollinear_antiferromagnet_Pt__2_MnGa.pdf",        "content": "arXiv:1604.07137v1  [cond-mat.mtrl-sci]  25 Apr 2016Room temperature tetragonal noncollinear antiferromagne t Pt2MnGa\nS. Singh,1S. W. D’Souza,1E. Suard,2L. Chapon,2A. Senyshyn,3\nV. Petricek,4Y. Skourski,5M. Nicklas,1C. Felser,1and S. Chadov1\n1Max Planck Institute for Chemical Physics of Solids, N¨ othn itzer Str. 40, D-01187 Dresden, Germany\n2Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, Fra nce\n3Forschungsneutronenquelle Heinz Maier-Leibnitz FRM-II,\nTechnische Universit¨ at M¨ unchen, Lichtenbergstrasse 1, 85747 Garching, Germany\n4Institute of Physics ASCR, Department of Structure Analysi s, Na Slovance 2, 18221 Praha, Czech Republic\n5Dresden High Magnetic Field Laboratory (HLD-EMFL),\nHelmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, G ermany\nHere we present the tetragonal stoichiometric Heusler comp ound Pt 2MnGa with the noncollinear\nAFM order stable up to350 K. Itis resolved bythe neutrondiffr action as a helical spiral propagating\nalong the tetragonal axis. Ab-initio calculations suggest a pure exchange origin of the spiral an d\nexplain its helical character being stabilized by a large ba sal plane magnetocrystalline anisotropy\n(MCA). Together with the inversion-symmetric crystal stru cture, this provides a bi-stability of a\nspiral with respect to the right- and left-handed magnetic h elices. Despite the large MCA, the long\nperiod of a helix might greatly facilitate the switch of the h elicity by the precessional reorientation,\nsuggesting Pt 2MnGa as a potential candidate for the vector-helicity based non-volatile magnetic\nmemory.\nAntiferromagnets(AFMs) gain an increasing attention\nin the state-of-the-art applied and academic research.\nTheir auxiliary role of a static support, enhancing the\nhardness of ferromagnetic electrodes through the ex-\nchange bias effect in the conventional microelectronics,\nhas been broadly extended by the new prospectives in\nspintronics applications. For instance, by studying the\nmagnetoresistance effects typically exploited in spintron-\nics [1], it has been demonstrated that Ir-Mn AFM, uti-\nlized as an active medium in a tunneling magnetoresis-\ntance device, exhibits a160% tunneling anisotropicmag-\nnetoresistance at 4K in weak magnetic fields of 50mT or\nless. AFMs also facilitate the current-induced switch-\ning of their order parameter [2–4] due to the absence\nof the shape anisotropy and the action of spin torques\nthrough the entire volume. For instance, a relatively low\ncritical current density of 4.6MA/cm2was reported for\nthe collinear AFM CuMnAs [5]. Additional nontrivial\nspintronic effects originating from a non-vanishing Berry\nphase might occur in the noncollinear AFMs [6]. For in-\nstance, the noncollinear planar AFMs with the absence\nof mirror symmetry, such as Mn 3Ir, where predicted to\nexhibit the anomalous Hall [7, 8], Kerr and other effects\ncharacterized by the same spatial tensor shape [9], which\nwhere not encountered in AFM systems so far.\nAnother set of specific properties, alternative to\nthe above mentioned systems, are provided by one-\ndimensional long-range AFM modulations, such as\ncycloidal - /vector q⊥(/vector ei×/vector ej) and screw (or helical) -\n/vector q/bardbl(/vector ei×/vector ej), with/vector ei,jbeing the spin directions on iand\njneighboring atomic sites sitting along the spiral prop-\nagation vector /vector q. These systems possess a specific or-\nder parameter /vector κij=/vector ei×/vector ej, denoted as “chirality” or\n“helicity”. E.g., in cycloidal AFM insulators /vector κis cou-\npled to the polarization vector /vectorP∼/vector q×/vector κ, by leading tothe first-order ferroelectic effect [10–13]. For the screw-\nspiral order ( /vector q×/vector κ= 0) it becomes possible only upon\nthe additional specific condition, namely, when the crys-\ntal structure remains invariant under inversion and ro-\ntations around /vector κ, but non-invariant under 180◦rotation\nof the/vector κ-axis [14, 15]. The information transfer in cy-\ncloidal spirals along the one-dimensional atomic chains\nwith the fixed /vector κ, stabilized by the surface Dzyaloshinskii-\nMoriya mechanism, was demonstrated by switching their\nphase with external magnetic field [16]. Such scheme is\ninapplicable to the screw spirals due to their energy de-\ngeneracy with respect to the /vector κreversal, even if they are\ndeposited on a surface [17]. Similar to the situation with\nthe ferroelectric effect, to fix a helicity of the screw spi-\nral would require additional symmetry constrains on the\ncrystal structure [18]. Despite that the cycloidal order\nseems to be more ubiquitous for the applications, the\naforementioned degeneracy between the left- and right-\nhanded magnetic screws in crystals with inversion sym-\nmetry might be considered as an alternative advantage.\nIn particular, it allows to directly associate a bit of infor-\nmation with the helicity. The switch of /vector κcan be realized\ne.g., by an external magnetic pulse /vectorHext⊥/vector κwhich re-\norients the spins precessionally (see Fig. 1). In this case,\nboth stable ±/vector κstates would be connected over the en-\nergy barrier representing the cycloidal-ordered state.\nHere we report on a similar AFM spiral magnetic or-\nder in the tetragonal Pt 2MnGa Heusler system, revealed\nby the neutron diffraction. The present first-principles\nanalysis justifies the non-relativistic exchange origin of a\nspiral, confirms its experimentally deduced wave-vector\n/vector q≈(0,0,1/5) inunitsof2 π/candsuggeststhescrew-type\norder caused by a moderate hard axis (tetragonal c-axis)\nmagnetocrystalline anisotropy (MCA).\nTo our knowledge, there are no reliable experimental2\nHext\nq qq\nκκκEnergy\nFIG. 1: Application of the external magnetic pulse /vectorHext\nperpendiculartothespiral wavevector /vector qcauses theprecession\nof local moments (along the dashed green circles). In case\nof the easy-plane MCA, the long period of a spiral greatly\nfacilitates reorientation of helicity from /vector κ↑↓/vector q(blue) to /vector κ↑↑/vector q\n(red) or vice versa, since the top of the energy barrier betwe en\ntwo stable screws is a cycloid /vector κ⊥/vector q(magenta), in which only\nfew atomic planes with magnetic moments orthogonal to /vectorHext\nacquire the high energy.\nresults on this material in the literature. The single old\nreportonPt 2MnGa[19] brieflyrefersitas L21AFM with\nTN= 75K, but no further details are given. Later on,\nPt2MnGa has been studied by ab-initio assuming ferro-\nmagneticorderingand revealedthat the tetragonalphase\nis more stable [20, 21]. In [20] it was only mentioned that\nin PtxNi2−xMnGa alloy series “the AFM correlationsbe-\ncome stronger by increasing x”. In more recent ab-initio\nstudy [22] the nearest neighbor AFM order was found\nto be noticeably higher in energy compared to the fer-\nromagnetic. To clear the actual crystal and magnetic\nstructure we prepared a polycrystalline Pt 2MnGa sam-\nple (details are given in the Supplementary). The room-\ntemperature crystal structure (Fig. 2) was deduced from\nthe Rietveld refinement of the x-ray diffraction data. All\nBraggreflectionscanbeindexedbyassumingthetetrago-\nnalspacegroup I4/mmm. Therefinedlatticeparameters\narea=b= 4.02˚A,c= 7.24˚A; Ptoccupies4 d(0,1/2,1/4),\nwhile Mn and Ga - 2 a(0,0,0) and 2 b(0,0,1/2) Wyckoff\nsites, respectively (see the inset in Fig. 2).\nThe low-field M(T) curves measured within the zero-\nfield-cooled(ZFC)andfield-cooled(FC) cyclesareshown\nin Fig. 3a. The ZFC M(T) shows a maximum at\nT≈65K, absent in the FC regime. This observation\ntypical for many Heusler alloys results from the poly-\nFIG.2: Crystal structureofPt 2MnGa. Rietveldrefinementof\nthe room-temperature XRD pattern assuming the tetragonal\nunit cell with I4/mmmsymmetry. Observed and calculated\npatterns, as well as their difference are shown by black open\ncircles, red and green solid lines, respectively. Blue vert ical\nticks indicate the Bragg peak positions. The sketch of the\nunit cell is shown in the inset: red, yellow and gray spheres\nindicate Mn, Ga and Pt atoms in 2 a, 2band 4dWyckoff\npositions, respectively.\n0.0016\n0.0014\n0.0012M (µΒ/f.u.)\n300200100\nT (K)  \n \n  0.1 TeslaFC\nZFC \n  \n0.11\n0.10\n0.09\n0.08\n300200100\nT (K)  7 Tesla\n \n \n \n \n  \n FC\nZFC \n0.10\n0.05\n0.00\n-0.05\n-0.10M (µΒ/f.u.)  \n-404\nH (T)  300 K\n 2 K\n-0.40.00.4 \n-0.40.00.4 0.8\n0.6\n0.4\n0.2\n0.0\n40200\nH (T)   257 K\n 1.5 Kab\nc d\nFIG. 3: Magnetization of Pt 2MnGa as a function of tempera-\ntureTand magnetic field H:M(T) at (a) 0.01T and (b) 7T;\n(c) at 300K and 2K up to 7 T; M(H) in the magnetic pulse\nof 60T at 257K and 1.5K (d).\ncrystalline configuration of the anisotropic crystallites.\nThe high-temperature behavior is similar in both ZFC\nand FC regimes and indicates the change of the mag-\nnetic ordering at 350K. Overall, the amplitude of M(T)\nis very small in both ZFC and FC regimes, even at low\ntemperatures (2K) and higher fields (7T), (Fig. 3b).\nBoth isothermal M(H) curvesat 300K (Fig. 3c) and 2K3\n(Fig. 3d) exhibit a non-saturating(straight line) increase\nup to 7T, similar to the antiferromagnetic or paramag-\nnetic materials. Only a narrow field hysteresis (Fig. 3d,\ninset) indicates a very weak ferromagnetism at low tem-\nperature (2K). To probe the behavior of the system at\nvery high magnetic fields, we performed the 60T pulse\nmeasurements. Corresponding M(H) curves measured\nat 257 and 1.5K (Fig. 3e, f) also do not exhibit any sat-\nuration, by scaling almost linearly with the field. Only at\n1.5K a small hysteresis within 0 < H <35T is observed\nindicating the metamagnetic transition around 14T in-\nduced by the non-equilibrium magnetic pulse. All this\nclearly suggests that Pt 2MnGa is not a ferromagnet.\nTo determine the actual magnetic order, the pow-\nder neutron diffraction measurements were performed at\n500K (above the magnetic ordering), 300K and 3K. The\ncrystal structure refinement of Pt 2MnGa of powder neu-\ntron diffraction data at 500K gives similar results as\nof the room-temperature XRD, however, the substantial\ndifference in the nuclear scattering amplitude (of 0.96,\n−3.73,and 7.29fm forPt, Mn, andGa, respectively)sug-\ngests a certain degree of disorder to be present in the ac-\ntual atomic occupancies. Fig. 4a shows a comparison of\nthe Rietveld refinement with the high-temperature spec-\ntrum for (002) and (110) Bragg peaks, assuming several\ndifferent configurations. The chemically ordered model\ngives a clear mismatch in the fit (upper panel of Fig. 4a),\nthe presence of random Mn(2 a)-Pt(4d) disorder does not\nimproveit either. The most reasonableagreementgivesa\nmodel with 33% of Mn(2 a)-Ga(2b) disorder (lower panel\nin Fig. 4a). The refinement details for the full range\ndiffraction pattern at 500K is given in the Supplemen-\ntary.\nBy going to lower temperatures, we assumed the mag-\nneticunitcellwithtwoMntypesin2 aand2bsiteshaving\ncorresponding occupancies. The Rietveld refinement of\nthe magnetic phases observed at 300 and 3K was per-\nformed by accounting for the magnetic and atomic struc-\ntures simultaneously. The comparison within a narrow\nangular range 20◦<2θ <35◦between 500, 300 and 3K\nneutron diffraction patterns is given in Fig. 4b. At 300\nand3K,thelong-rangemagneticorderingisevidencedby\nthe presenceoftwoadditionalBraggpeaks, at2 θ≈24.1◦\nand 29.3◦(indicated by the red arrows at 300 and 2K in\nFig. 4b), which closely correspondsto the commensurate\nreciprocal vector /vector q= (0,0,1/5) in units of 2 π/c. Precise\nanalysis reveals slightly incommensurate temperature-\ndependent variation: /vector q= (0,0,0.2066(1)) at 300K and\n(0,0,0.19(5)) at 3K.\nTo specify more details of the magnetic ground state,\nwe focus on the 3K data exhibiting the highest intensity\nof the magnetic Bragg peaks. Its analysis suggests that\nthe spiral order of the magnetic moments on Mn atoms\n(in 2aand 2b) would be more favorable, as it delivers\nthe magnetic moments amplitudes close to a reasonable\nMn value of 4 µB. In contrast, the collinear spin-wavemodel leads to the values substantially exceeding 5 µB.\nIn the next step, we tried to distinguish which type of\na spin spiral is more preferable. By assuming the spi-\nral magnetic structure rotating in the bc-plane (cycloidal\nspiral) we obtained the moments of 4.33(13) µBfor both\nMn in 2aand 2bsites. Although these values are ac-\nceptable, they slightly exceed those reported in the liter-\nature for Mn(2 a) in Mn-based Heusler alloys. Finally, by\nassuming the spiral rotating within the ab-plane (screw\nspiral) leads to 3.93(11) µB, a value which is somehow\ncloser to those reported in the literature. Since the rigor-\nous experimental answer requires the polarized neutron\nspectroscopy data, in the following we will focus on the\nfirst-principle analysis.\nTo complete the experimental information we com-\nputed the /vector q-dependent total energies using the ab-initio\nLMTO method [23] adopting the local spin-density ap-\nproximation to the exchange-correlation [24]. Since the\nmethod assumes perfectly orderedsystems, we do not ac-\ncount for Mn/Ga chemical disorder indicated by neutron\nscattering. The unit cell parameters were taken from the\npresent experimental refinement.\nFirst, we determined the preferential /vector qvector. Since\nthe parameters of the present magnetic modulation are\ndefined mostly by the interplay of the isotropic exchange\ninteractions which have a largest energy scale (note, that\nthe anisotropic Dzyaloshinskii-Moriya interactions must\nbe largely canceled by crystal symmetry), it is practi-\ncal to perform the non-relativistic calculations first. The\nabsence of the spin-orbit coupling allows to apply the\ngeneralized Bloch theorem and to study the spin spirals\nwithin the chemical unit cell without going to the large\nsupercells. The energy dispersion computed along sev-\neral symmetric directions is shown in Fig. 5a. In order\nto plot several curves along the same coordinate axis, we\ngivetheqlengthinthe2 π/dunits, where disthedistance\nbetweenthenearestMn-containingatomicplanesorthog-\nonal to/vector q. In this notation, q= 0.5 always corresponds\nto the antiparallel orientation of the spin moments in\nthe nearest planes. As we see from Fig. 5a, the energy\ndispersion in the ab-plane ([100] and [110] directions) is\nmonotonous, being characterized by a single minimum\natq= 0 (ferromagnet) and a single maximum at q= 0.5\n(shortest AFM order). For the out-of- ab-plane direc-\ntions one observesa formation of a local minimum within\n0< q <0.5. Whereas along [111] direction it forms at\nrather high energy, along [001] ( c-axis) it turns to global,\nsupporting the experimental conclusions. Energy min-\nimum vector is /vector q≈(0,0,0.11) = (0,0,0.22·(2π/c)) (see\nalso a more detailed plot in Fig. 5b), which is very close\nto the experimental one /vector q≈(0,0,1/5·(2π/c)). Addi-\ntional Monte-Carlosimulation of the classical Heisenberg\nmodel parametrized by the ab-initio exchange coupling\nconstants(seetheSupplementary)reasonablyreproduces\nthe magnetic ordering temperature ( TN≈350K) and re-\nveal that the AFM order is set by the interplay be-4\n \n \nNo  disorder  500 K\n \n \nMn-Pt  disorder  500 K\n322824\n2θ  (degree)         \n \nMn-Ga  disorder  500 K\n322824\n2θ  (degree) \n500 K\n 300 K\n 3 K\n14012010080604020\n2θ  (degree)                3 K\n \n Observed\n Calculated\n Difference\n Bragg positionsa b c\nFIG. 4: Powder neutron diffraction studies on Pt 2MnGa. (a) Rietveld refinements of 500K neutron diffraction pa ttern where\nthe (002) and (110) Bragg peaks (black circles) have been fitt ed (red solid lines) by assuming (i) no disorder, (ii) Mn(2 a)/Pt(4d)\ndisorder, and (iii) Mn(2 a)/Ga(2b) disorder. The green curve shows the difference between obse rved and calculated patterns.\nVertical ticks are nuclear Bragg peak positions. (b) Compar ison of neutron diffraction patterns at 500, 300 and 2K. Magne tic\npeaks are indicated by red arrows at 300 and 2K. (c) The observ ed (black circle) and modeled (red solid line) neutron diffra ction\npattern for Pt 2MnGa at 3K. The vertical arrows indicate the magnetic peaks. Upper vertical ticks are nuclear Bragg peak\npositions; lower vertical - magnetic.\n∆E [eV/atom]\n∆E [meV/atom]\nq [2π/d] q [2π/d](a) (b)\n0 0.1 0.2 0.3 0.4 0.500.010.020.03[100]\n[110]\n[111]\n[001]\n00.05 0.1 0.15 0.2-1-0.500.51\n[001] non-rel.\n[001] screw\n[001] cycloid\n[001] cycloid\nFIG. 5: Total energies ∆ Ecalculated as functions of /vector q(in units of 2 π/d, where dis the distance between the nearest Mn-\ncontaining planes orthogonal to /vector q). (a) Non-relativistic regime: black solid and dashed line s refer to [100] ( d=a/2) and [110]\n(d=a/√\n2)directions; reddashedandsolid lines-to[111] ( d= (c/2)//radicalbig\n2(c/a)2+1)and[001] ( d=c/2)directions, respectively.\nThe energy zero is taken at q= 0 (ferromagnet). (b) Detailed comparison of different regi mes along [001]: red, blue and green\nlines/points refer to the non-relativistic (same as [001] i n (a)), screw- and cycloidal-type spirals, respectively. C ycloidal order\nhas two variants: the first one (dashed green line, open circl es) which represents at q= 0 a hard- c-axis oriented ferromagnetic\nstate (θ0= 0); the second one (solid green line, filled circles) repres ents atq= 0 an easy- ab-plane oriented ferromagnetic state\n(θ0= 90◦). The energy zero corresponds to the easy- ab-plane ferromagnetic order. All lines are given for an eye gu ide. For\nseveral specific configurations the Mn(2 a) magnetic sublattice is shown explicitly; spin moments are colored from red to blue\naccording to their phase.5\ntween the strong short-range parallel and the weaker\nlong-range (7-th shell) antiparallel interactions along the\nc-axis within the 2 asublattice.\nNext, we determine the type of the spin spiral (screw-\nor cycloidal), which results from the relativistic effects.\nIn this case the Bloch theorem does not hold and the\nmagnetic order can be studied only in supercells. Since\nin this case we can account only for the commensurate\nmodulations, the supercells must be sufficiently large to\nprovide the energies at long wavelengths: a minimal su-\npercell hosting a (0 ,0,1/5·(2π/c)) modulation contains\nat least five standard units. As we see from Fig. 5b, the\nrelativistic effects substantially deepen the spiral energy\nminimum and almost do not change the corresponding\n/vector qvector (q≈0.11). The ab-plane appears to be an easy\nplane,sincethescrew-typespiralismorepreferableinthe\nwhole range of the wavelengths. The energy difference\nbetweenthe cycloidalandthe screwspiralsis contributed\nby the MCA energy which has a rather large magnitude\nfor the Heusler class, being close to 0.65meV/atom (or\n2.6meV/f.u.) at q= 0. At the global minimum this en-\nergy difference is reduced more than twice. Due to the\nhard-c-axis MCA, the cycloidal spiral can still be opti-\nmized in terms of a homogeneity (which must be dis-\ntorted by the MCA), however it will have a higher en-\nergy compared to the screw order anyway. At the same\ntime, as it is follows from the growing energy difference\nbetween θ0= 0 and 90◦cases, the phase optimization\nof the cycloidal spiral makes sense only for the small\nq <0.06, far from the global minimum. The amplitude\nof the Mn magnetic moment in the screw spiral has a\ntendency to grow by going from q= 0 (ferromagnetic)\ntowards q= 0.5 (shortest AFM order), though its abso-\nlute increase is relatively small: from about 3.7 to 3.8 µB,\nwhich agrees with neutron data refinement.\nTo conclude, we present a newly synthesized stoichio-\nmetric tetragonal ( I4/mmm) Pt2MnGa Heusler system\nexhibiting the room-temperature AFM spiral order with\nthe wave vector /vector q= (0,0,1/5·2π/c), as it follows from\nthe neutron-diffraction refinement analysis. Certain de-\ngree (∼30%) of Mn-Ga chemical disorder was indicated.\nAb-initio calculations(assumingthe orderedsystem) rea-\nsonably reproduce the experimental /vector qvector indicating\nthe exchange origin of the spiral. Monte-Carlo simula-\ntionsofthe classicalHeisenbergmodelparametrizedwith\ntheab-initio exchange coupling constants reasonably re-\nproducethe Neeltemperatureandsuggestthe long-range\nantiparallel Mn-Mn exchange (beyond the 6-th Mn shell)\nas a driving mechanism for the AFM order. Relativistic\ncalculationsindicate aneasy- ab-planeMCA, which stabi-\nlizes the screw (proper screw, or fully helical) spiral type.\nDue to inversion symmetry, the left- and right-handed\nspirals are stable and degenerate in energy. In spite of\na large MCA, the energy barrier between them can be\nefficiently overcome via the precessional magnetization\nreorientation induced by the magnetic pulse perpendic-ular to the spiral axis. In this case, the barrier reduces\nto the energy difference between the screw and cycloidal\nspiral orders. In particular, this suggests Pt 2MnGa as a\nconvenient candidate for the non-volatile magnetic mem-\nory based on the helicity vector as a bit of information.\nS.S. thanks to Alexander von Humboldt foundation\nfor fellowship. S.C. thanks to A.N. Yaresko (MPI-\nFKF Stuttgart) for providing his program code and dis-\ncussions. The work was financially supported by the\nERC AG 291472 “IDEA Heusler!”\n[1] B. G. Park, J. Wunderlich, X. Mart´ ı, V. Hol´ y,\nY. Kurosaki, M. Yamada, H. Yamamoto, A. Nishide,\nJ. Hayakawa, H. Takahashi, et al., Nature Mat. 10, 347\n(2011).\n[2] A. S. N´ u˜ nez, R. A. Duine, P. Haney, and A. H. MacDon-\nald, Phys. Rev. B 73, 214426 (2006).\n[3] H. V. Gomonay and V. M. Loktev, Phys. Rev. B 81,\n144427 (2010).\n[4] E. V. Gomonay and V. M. Loktev, J. Low Temp. Phys.\n40, 17 (2014).\n[5] P. Wadley, B. Howells, J. ˇZelezn´ y, C. Andrews, V. Hills,\nR. P. Campion, V. Nov´ ak, K. Olejn´ ık, F. Maccherozzi,\nS. S. Dhesi, et al., Science (2016).\n[6] O. Gomonay, Phys. 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Nusair, Can. J. Phys. 58,\n1200 (1980).arXiv:1604.07137v1  [cond-mat.mtrl-sci]  25 Apr 2016Supplementary information\nRoom temperature tetragonal noncollinear antiferromagne t Pt2MnGa\nS. Singh,1S. W. D’Souza,1E. Suard,2L. Chapon,2A. Senyshyn,3\nV. Petricek,4Y. Skourski,5M. Nicklas,1C. Felser,1and S. Chadov1\n1Max Planck Institute for Chemical Physics of Solids, N¨ othn itzer Str. 40, D-01187 Dresden, Germany\n2Institut Laue-Langevin, BP 156, 38042 Grenoble Cedex 9, Fra nce\n3Forschungsneutronenquelle Heinz Maier-Leibnitz FRM-II,\nTechnische Universit¨ at M¨ unchen, Lichtenbergstrasse 1, 85747 Garching, Germany\n4Institute of Physics ASCR, Department of Structure Analysi s, Na Slovance 2, 18221 Praha, Czech Republic\n5Dresden High Magnetic Field Laboratory (HLD-EMFL),\nHelmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, G ermany\nEXPERIMENTAL DETAILS\nPolycrystalline ingot of Pt 2MnGa was prepared by\nmelting appropriate quantities of Pt, Mn and Ga of\n99.99% purity in an arc furnace. Ingots were then an-\nnealed at 1273K for 5 days to obtain homogeneity and\nsubsequently quenched into ice water. Powder x-ray\ndiffraction (XRD) at room temperature (RT)was done\nto investigate the sample quality and homogeneity using\nCuKαradiation. Thecompositionofthesamplewascon-\nfirmed using energydispersive analysisof x-rays(EDAX)\nanalysis which gave a composition Pt 50.98Mn25.05Ga23.98\n(Pt2.04Mn1.0Ga0.96), which we refer to as Pt 2MnGa\nhenceforth in the manuscript. The temeprature de-\npendent ( M(T)) and field dependent ( M(H)) magneti-\nzation measurementsweredone using SQUID-VSM mag-\nnetometer. The low-field M(T) curves measured within\nthe zero-field-cooled (ZFC) and field-cooled (FC) cycles\nFor the ZFC-measurement, the sample wascooled in zero\nfield down to 2K and after 0.1T field was applied and\nthen the data were recorded in heating cycle up to 400K.\nSubsequently, the data were recorded in the same field\n(0.1T) by cooling from 400 down to 2K (FC).The mag-\nnetic isotherms, M(H), at 1.2K and 257 K have been\ndone in a pulsed fields up to 60T. The pulsed magnetic\nfield experiments were performed at the Dresden High\nMagnetic Field Laboratory. Neutron diffraction mea-\nsurements were done at the D2B high-resolution neu-\ntron powder diffractometer (ILL, Grenoble). A vana-\ndium cylinder was used as sample holder. The data were\ncollected at 500K, 300K and 3K using a neutron wave-\nlengthof1.59 ˚A inthehigh-intensitymode. Theanalysis\nof diffraction patterns were done with Fullprof software\npackage[1].\nNuclear structure from neutron diffraction\nFig. 1 shows the observed and calculated neutron\ndiffraction pattern of at 500K (paramagnetic phase)\nin the 2 θrange of 20-100◦. All the neutron diffrac-\ntion peaks can be indexed well by the tetragonal unit\nIntensity (arb. unit)\n14012010080604020\n2θ  (degree)         \n \nMn-Ga  disorder  500 K  Observed \n  Calculated\n  Difference\n Bragg positions\nFIG. 1: Rietveld refinement of powder neutron diffraction\npattern of 500K in the 2 θrange of 20-100◦. The observed,\ncalculated and difference patterns are shown by black,red an d\ngreen solid lines, respectively. The vertical ticks indica te the\nnuclear Bragg peak positions.\ncell with refined lattice parameters are a=b= 4.03˚A,\nc= 7.23˚A. The Rietveld refinement was performed us-\ning space group I4/mmmas in case of x-ray diffraction.\nIn order case the Pt occupy the 4 d(0,0.5,0.25) position,\nwhile Mn and Ga occupied at 2 a(0,0,0) and 2 b(0,0,0.5)\nWyckoff positions, respectively. However, the Rietveld\nanalsyis shows that a substantial ( 33%) Mn(2 a)-Ga(2b)\nantisite disorder exist in the sample. Therefore,33% Mn\noccupied at Ga (2 b) site and similarly 30% Ga occupied\nat Mn 2asite.\nTHEORETICAL DETAILS\nIt is instructive to understand which mechanisms are\nresponsible for setting such ground-state modulation.\nSincetherelativisticeffectsdonotaffecttheground-state\n/vector qvector substantially, in the following we will calculate\nthe isotropic exchange coupling constants Jusing the2\ndistance,  R/a\nT [K]M / Ms exchange coupling,   Jij  [meV]  Mn(2a)-Mn(2a)\nMn(2a)-Mn(2a)\nMn(2b)-Mn(2b)\nMn(2a)-Mn(2b) (× 0.1)\nT [K]{{ordered\nR/a = 2.0\nR/a = 2.06R/a = 2.0\nR/a = 2.06(a)\n(b) (c)ordered 30% Mn(2 a)/Ga(2b)30% Mn(2 a)/Ga(2b) \n disorder  \n1 1.5 2 2.5 3 3.5 4-2024\n1/√2\n0 100 200 300 400 50000.20.40.60.81\nT [K] T [K]\nspecific heat  \nC\nV [kB / atom] \n0 100 200 300 400 500R/a = 2.87 R/a = 2.69AFMspecific heat  \nC\nV [kB / atom] \n0 100 200 300 400 50000.20.40.60.81\n0 100 200 300 400 50000.20.40.60.81\nFIG. 2: (a) Isotropic exchange coupling constants Jijcal-\nculated as functions of distance R(in the units of a) be-\ntweeniandjMn sites for the fully ordered and partially\ndisordered cases. (b) and (c) represent the Monte-Carlo sim -\nulatedM/MStemperature dependencies of the Heisenberg\nmodel parametrized by the computed Jij, in the fully ordered\nand partially disordered cases, respectively. Solid line c orre-\nsponds to the minimal cluster size needed to set the AFM\norder, dashed - by one shall smaller cluster. The insets show\nthe temperature dependency of the specific heat CV(com-\nputed for the largest cluster size) which indicates the posi tion\nofTNby the local maximum.\nreal-space approach [2] implemented in the SPR-KKR\nGreen’s function method [3]. In Fig. 2a they are plotted\nas functions of distance between the interacting sites i\nandj. Here, we drop all interactions involving Pt and\nGa atoms as insignificant, by leaving only those between\nMn atoms. In the fully ordered case, all nearest Mn(2 a)-\nMn(2a) interactions are parallel ( J >0), whereas the an-\ntiparallel ones ( J <0) are encountered by starting from\nR/a= 2 (6-th shell within (2 a)-sublattice). As it is\nshown by M(T)/MScurves (Fig. 2b) obtained by the\nMonte-Carlo simulation (ALPS package [4]) of the classi-\ncal Heisenberg model ( H=−/summationtext\ni>jJij/vector ei·/vector ej, where/vector ei,j\nare the unity vectors along the local magnetization direc-\ntionsoniandjsites), the AFMordersetsinbyincluding\nall interactions at least up to R/a≈2.06 (7-th shell); ac-\ncounting of the higher shells does not affect the M(T)behavior anymore. Such a superposition of the strong\nnearest parallel and the weaker long-range antiparallel\nexchange interactions typically allows for the long-range\nspin-spiralorder. Itsdirection( /vector q/bardbl[001])followsfromthe\nsymmetry reasons: the 7-th shell, critical for setting up\nthe AFM order, contains 8 atoms at /vectorR= (±a,0,±c) and\n(0,±a,±c), situated above and below the ab-plane of the\ncentral atom. The corresponding Neel temperature can\nbe estimated from the peak of the magnetic specific heat\nCV(T) computed for the largest cluster size ( TN≈350K\natR/a≈2.87, see the inset in Fig. 2b). This reasonably\nagrees with experimental M(T) slope change, well seen\nat about the same temperature.\nThe influence of chemical disorder can be estimated by\nmeans of the CPA alloy theory [5, 6], implemented in the\nSPR-KKRmethod. Byassuming30%ofMn(2 a)/Ga(2b)\ndisorder, as it was identified in experiment, modifies the\nexchange picture (Fig. 2a). The exchange interactions\nwithin Mn(2 a)sublattice, especiallythe nearest-neighbor\nones, become slightly weaker compared to the ordered\ncase. Inaddition,itsstatisticalfactorof(1 −0.3)2= 0.49\nmight noticeably reduce the Neel temperature. Alto-\ngether, this coupling alonewouldleadto the similarmag-\nnetic order as in the fully ordered case. Similar depen-\ndencyisexhibitedwithin the extrasublatticeMn(2 b), ex-\ncept for the 2-nd shell, which shows a noticeable antipar-\nallel coupling to 8 atoms at ( ±a/2,±a/2,±c/2), which\nalso leads to /vector q/bardbl[001], though favoring a shorter wave-\nlength. However, since these interactions enter with the\nlow weight of 0 .32= 0.09, they play a minor role for\na final picture. The most important here is Mn(2 a)-\nMn(2b) coupling, which has a moderate statistical weight\nof0.3·(1−0.3) = 0.21,butalsoahugebareamplitudeof\nthe near-neighborinteractions (note that, JMn(2a)−Mn(2b)\nvalues shown in Fig. 2a are downscaled by a factor of\n10). In particular, within the ab-plane there is a huge\nantiparallel interaction of −28 meV with the 4 first-shell\nneighbors at ( ±a/2,±a/2,0), but also strong +18 meV\nparallel out-of-plane interaction with two second-shell\nneighbors at (0 ,0,±c/2). Alone, this combination would\nlead to the vertical parallel coupling of the neighboring\ncollinear-AFM ordered ab-planes, however the last im-\nportant∼ −9meV antiparallel coupling to 8 neighbors\nin a 3-rdshell at ( ±a,0,±c/2), (0,±a,±c/2)distorts this\npicture. Thus, to get a rough idea about the behavior of\nsuch complicated magnetic system, we simplified the cor-\nresponding Heisenberg model, by parametrizing it with\neffective interactions (i.e., by multiplying the bare val-\nues with corresponding statistical weights). As it follows\nfrom Fig. 2c, the minimal cluster size which sets the\nAFM order remains the same, however, the TNdrops\ndown to about 220K (see the inset). Since this drives us\nawayfromthe experimentalresult, it mostprobablyindi-\ncatesthat the effective parametrizationofthe Heisenberg\nmodel is inapplicable (despite that the bare exchange\ninteractions are correct) and the adequate description3\nmight be achieved by treating the local effects explicitly,\nwhich requires substantially larger cluster sizes.\n[1] J. R. Carvajal, FULLPROF, a Rietveld refinement and\npattern matching analysis program (Laboratoire LeonBril-\nlouin, CEACNRS, France, 2000).\n[2] A. I. Liechtenstein, M. I. Katsnelson, V. P. Antropov, an dV. A. Gubanov, J. Magn. Magn. Materials 67, 65 (1987).\n[3] H.Ebert, D. K¨ odderitzsch, andJ. Min´ ar, Rep.Prog. Phy s.\n74, 096501 (2011).\n[4] B. Bauer, L. D. Carr, H. G. Evertz, A. Feiguin, J. Freire,\nS. Fuchs, L. Gamper, J. Gukelberger, E. Gull, S. Guertler,\net al., J. Stat. Mech.: Theory and Experiment 2011,\nP05001 (2011).\n[5] P. Soven, Phys. Rev. 156, 809 (1967).\n[6] D. W. Taylor, Phys. Rev. 156, 1017 (1967)."    },    {        "title": "1605.02381v1.Exchange_bias_like_effect_in_TbFeAl_intermetallic_induced_by_atomic_disorder.pdf",        "content": "Exchange bias-like effect in TbFeAl intermetallic induced by atomic disorder\nHarikrishnan S. Nair1,\u0003and Andr ´e M. Strydom1, 2\n1Highly Correlated Matter Research Group, Physics Department,\nP . O. Box 524, University of Johannesburg, Auckland Park 2006, South Africa\n2Max Planck Institute for Chemical Physics of Solids (MPICPfS), N ¨othnitzerstraße 40, 01187 Dresden, Germany\nExchange bias-like effect observed in the intermetallic compound TbFeAl, which displays a magnetic phase\ntransition at Th\nc\u0019198 K and a second one at Tl\nc\u0019154 K, is reported. Jump -like features are observed in the\nisothermal magnetization, M(H), at 2 K which disappear above 8 K. The field-cooled magnetization isotherms\nbelow 10 K show loop-shifts that are reminiscent of exchange bias, also supported by training effect . Signifi-\ncant coercive field, Hc\u00191.5 T at 2 K is observed in TbFeAl which, after an initial increase, shows subsequent\ndecrease with temperature. The exchange bias field, Heb, shows a slight increase and subsequent leveling off\nwith temperature. It is argued that the inherent crystallographic disorder among Fe and Al and the high magne-\ntocrystalline anisotropy related to Tb3+lead to the exchange bias effect. TbFeAl is recently reported to show\nmagnetocaloric effect and the present discovery of exchange bias makes this compound a multifunctional one.\nThe result obtained on TbFeAl generalizes the observation of exchange bias in crystallographically disordered\nmaterials and gives impetus for the search for materials with exchange bias induced by atomic disorder.\nPACS numbers:\nReviews on exchange bias[1–4] in materials point\ntoward the importance of this effect in read-heads in\nmagnetic recording,[5] giant magnetoresistive random\naccess memory devices[6] and in permanent magnets.[7]\nInterpreted as phenomena occurring at the interface be-\ntween magnetically ordered microscopic regions, exchange\nbias interfaces can be ferromagnetic/antiferromagnetic or\nferromagnetic/antiferromagnetic/spin-glass. Mixed magnetic\ninteractions are deemed to be an important ingredient for this\neffect to occur. Recent work[8] promotes the importance of\nferromagnetic spin structure in exchange bias and explains\nsome of the anomalous features of exchange bias field. In\nthis Letter we report on the observation of exchange bias-like\neffect in TbFeAl. RFeAl (R= rare earth) were first inves-\ntigated by Oesterreicher.[9–11] Only the heavier rare earths\nwere observed to form the stable MgZn 2-type structure.\nTbFeAl is recently identified as a magnetocaloric[12] (albeit,\na weak effect) which becomes a possibel ”multifunctional”\ncompound with the observation of exchange bias.\nAs noted above, TbFeAl crystallizes in hexagonal MgZn 2\ntype crystal structure with P63=mmc space group. Tb\noccupies 4fWyckoff position while Fe and Al are situated\nat2aand6hsites respectively.[10, 11] Mixed occupation\nof Fe and Al is possible in this structure. Magnetically,\nTbFeAl is a ferrimagnet with a transition temperature of\n195 K.[12] A saturation-like effect of magnetization which\ndisplayed an S-curve was observed[9] which was described\nas a consequence of partial chemical disorder of Fe and Al.\nSignificantly high magnetocrystalline anisotropy leading\nto the formation of thin domain walls which are pinned to\ndefects influences the magnetization, and to some extent, is\nthe reason for the magnetocaloric effect.[12]\nPartial or total crystallographic disorder is a favourable\ningredient for the observation of exchange bias. Granular\n\u0003Electronic address: h.nair.kris@gmail.com, hsnair@uj.ac.za\nFIG. 1: Top: The powder x ray diffraction pattern of TbFeAl along\nwith Rietveld refinement. The black circles are the experimentally\nobserved intensity, the red solid line is the calculated intensity as-\nsumingP63=mmc space group. The difference curve is shown as\ngreen solid line and the allowed Bragg peaks as vertical bars. Bot-\ntom: The hexagonal framework of TbFeAl shown as a projection on\nto theab-plane.\nnanoparticles in a structurally and magnetically disordered\nmatrix[13] or interacting magnetic defects embedded in an\nantiferromagnetic matrix with high degree of disorder[14] are\nexample systems where disorder brings about exchange bias\neffect. In the oxide Y 2CoMnO 6, below 8 K prominent stepsarXiv:1605.02381v1  [cond-mat.str-el]  8 May 20162\nin magnetization and significant coercive field of \u00192 T were\nobserved.[15] Martensitic-like growth of ferromagnetic do-\nmains, formed as a result of antisite disorder, was postulated\nas the reason for exchange bias in Y 2CoMnO 6. Motivated\nby the prospect of obtaining a general feature of exchange\nbias induced by disorder , we have extended our research to a\ndisordered intermetallic – TbFeAl.\nThe polycrystalline sample used in this study was prepared\nusing arc melting method. The elements Tb, Fe and Al\n(all4Npurity) were melted in the water-cooled Cu hearth\nof an Edmund Buehler furnace in Argon atmosphere. The\nonce-melted buttons were remelted four or five times to\nensure homogeneity. Post melting, powder x ray diffrac-\ntograms were recorded on pulverized samples in a Rigaku\nSmartLab x ray diffractometer which used Cu K- \u000bradiation.\nMagnetic properties were recorded using a Magnetic Property\nMeasurement System from Quantum Design Inc., San Diego.\nMagnetization as a function of temperature in the range 2 -\n350 K in both zero field-cooled (ZFC) and field-cooled (FC)\nprotocols as well as magnetic field in the range 0 - 7 T and ac\nsusceptibility measurements were performed.\nThe experimentally obtained powder x ray diffractogram is\npresented in Fig 1 as black circles. The observed peaks could\nbe indexed in hexagonal space group P63=mmc (MgZn 2\ntype). FullProf suite of programs[16] was used to perform\nRietveld analysis[17] of the x ray data which yielded a(˚A) =\n5.3975(4) and c(˚A) = 8.7526(3). The results of the Rietveld\nrefinement are presented in Fig 1. In the bottom panel of\nFig 1, the hexagonal structure of TbFeAl is shown as a\nprojection on to the abplane.\nThe magnetic phase transitions reported in the literature\nfor TbFeAl[12, 18, 19] are reproduced in the magnetization\ncurve,M(T), obtained at 200 Oe which is presented in Fig 2.\nA bifurcation between the ZFC and FC arms is observed at\n\u0019275 K (see inset (b)) followed by two humps at 186 K\nand at 145 K. The transition temperatures are determined\nby taking the derivative of M(T)and is plotted as dM=dT\nin the inset (a). From this, Th\nc\u0019198 K andTl\nc\u0019154 K\nare determined. In the literature, the second transition is\nattributed to the existence of two crystallographic regions\nin the sample with different occupation of Fe and Al on the\nsites 2aand6h. The FC arm of M(T)suggests that upon\napplication of magnetic field, ferromagnetic like enhance-\nment of magnetization results. In the inset (b) of Fig 2, the\nhigh temperature region of M(T)forT\u0015Th\ncis shown\nmagnified to highlight a ”loop”-like structure. It is to be\nnoted that no significant linear region is observed up to 350 K\nand hence a description of magnetic susceptibility following\nideal Curie-Weiss formalism does not hold. The ”loop”-like\nstructure in M(T)at high temperature might suggest the\npresence of magnetic correlations extending above Th\nc.\nThe ferromagnetic feature of TbFeAl is evident from the\nisothermal magnetization curve, M(H), at 5 K and 10 K\npresented in Fig 3 (a). Significant coercive field ( Hc\u00191.5 T)\nand magnetic saturation-like effects are observed at 2 K. The\nM(H)curves in Fig 3 (a) are comparable to the S-curve\nreported in TbFeAl.[9, 11, 12] Interestingly, sharp jumps of\nmagnetization are observed in the zero field-cooled magneti-\n/s32/s70/s67\n/s50/s48/s48/s32/s79/s101\n/s126/s32/s50/s55/s53/s32/s75/s48 /s49/s48/s48 /s50/s48/s48 /s51/s48/s48/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49\n/s32/s32/s100/s77/s47/s100/s84\n/s40/s101/s109/s117/s47/s103/s46/s32/s75/s41\n/s84 /s32/s40/s75/s41/s84/s104\n/s99/s32/s61/s32\n/s49/s57/s56/s32/s75/s84/s108\n/s99/s61/s49/s53/s52/s32/s75\n/s40/s98/s41/s32\n/s32/s40/s97/s41FIG. 2: Magnetization curves (ZFC and FC) of TbFeAl obtained at\n200 Oe. Two magnetic transitions at Th\nc\u0019198K andTl\nc\u0019154K\nare observed. Inset (a) shows the derivative plot, dM=dT , used to\ndetermine the transition temperatures. Inset (b) magnifies the region\nclose to 250 K ( >Th\nc) where a ”loop”-like feature is seen.\nzation isotherm at 5 K, see Fig 3 (a). As the temperature is\nreduced, the sharp jumps inM(H)are enhanced. The M(H)\ncurves in ZFC mode at 2 K presented in Fig 3 (b) display the\njumps . In (b), the magnetization isotherms at 2 K obtained in\n/s32/s53/s32/s75/s77 /s32/s40\n/s66/s47/s102/s46/s117/s46/s41\n/s72 /s32/s40/s84/s41/s90/s70/s67\n/s40/s97/s41\nFIG. 3: (a) Isothermal magnetization curves, M(H), in zero field-\ncooled mode for TbFeAl measured at 5 K and 10 K. At 10 K, no\njumps in magnetization are visible while at 5 K they are present.\nSignificant coercive field is observed in both the cases and the sat-\nuration magnetization attains \u00195.8\u0016B=f.u. (b) The magnetization\nisotherms at 2 K obtained under different values of field cooling. The\nZFC curve is also plotted for a comparison. (c) The magnetization\nisotherms at 4, 6, 8 and 10 K after field cooling in 1000 Oe from\n300 K. The field-cooled curves display clear evidence for loop-shifts\nreminiscent of exchange bias effect.3\nfield cooled mode under different applied fields are presented.\nWith the application of magnetic field, the hysteresis loops\nshift from the ZFC location though the displacement is\nnot highly systematic. This observation is reminiscent of\nexchange bias. Similar features are observed in (c) where\nisotherms at different temperatures are presented, all obtained\nafter field-cooling in 1000 Oe from 300 K to the temperature\nof measurement. As the temperature is increased, the width\nof the hysteresis loop decreases and the jumps disappear. At\n2 K the maximum magnetization attained by the application\nof 5 T is \u00195.8\u0016B=f.u., which is significantly reduced from\nthe free-ion moment of Tb3+which is 9\u0016B.[9, 11]\nThe exchange bias field Heb=(H++H\u0000)=2(whereH\u0006\nare the positive and negative intercepts of the magnetization\ncurve with the field axis) and the coercive field, Hc, were\nestimated from the data in Fig 3 (b,c). Figure 4 (a,b) present\nthe evolution of HcandHebas a function of temperature.\nThough TbFeAl displays significant Hc(about 1.5 T at 6 K),\nit is lower than that of Y 2CoMnO 6which has similar domain-\nrelated structure.[15] Generally, a monotonous decrease\nofHcwith increasing temperature is favoured. However,\ndisordered granular systems are reported to show a variation\nofHcsimilar to what has been observed for TbFeAl.[13] The\nanomalous temperature dependence of Hebcould be related\nto the recent work on the role of ferromagnetic layers or do-\nmains in exchange bias[8] where, contrary to the conventional\ncase, an increase of Hebwith temperature is explained. The\natomic disorder in TbFeAl could be held responsible for such\na behaviour. Interestingly, though the Hcof TbFeAl shows\nan initial increase and subsequent decrease with temperature,\nHebincreases first and attains a near-constant value up to\n15 K.\nExchange bias effect results from interfaces between\nferromagnetic, antiferromagnetic or spin-glass regions. In\norder to probe the presence of spin glass in TbFeAl, ac\nsusceptibility measurements were carried out at frequencies\nranging from 0.1 Hz to 999 Hz. The results are presented in\nFig 4 (c). The susceptibility peaks at Tl\ncandTh\ncare observed\nto show no frequency dependence other than weak damping.\nThe peak positions were determined by taking the derivative\nd\u001f0(f;T)=dT. It is then clear that the disorder in this material\nonly pertains to structural aspects. It was unable to quantify\nthe degree of disorder related to the mixed-occupancy\nbetween Fe and Al from the x ray data. However, with the\nintroduction of mixed-occupancy, an improvement in the\ngoodness-of-fit of refinement was observed. Presence of nano\nmagnetic domains in RFeAl were experimentally observed in\nthe case of TmFeAl[20] where the crystallographic disorder\nand the high magnetocrystalline anisotropy of Tm were the\nreason. By comparison, the experimental data presented\nhere for TbFeAl suggests a similar scenario. Finally, the\nloop-shifts observed in training effect experiment presented\nin Fig 5 confirms the exchange bias in TbFeAl. Hysteresis\nloops were measured at 2 K for 4 continuous loops after\nfield cooling the sample using 1000 Oe. It can be seen\nthat the hysteresis curves begin to shift with increasing\nnumber of loops. For clarity, only a part of the hysteresis\nis shown in (a). A fully magnified view of the loop-shift is\n/s32\n/s32/s32/s109/s39 /s32/s40/s49/s48/s45/s52\n/s101/s109/s117/s41\n/s40/s100/s41/s40/s99/s41\n/s84 /s32/s40/s75/s41/s40/s98/s41 /s40/s97/s41\n/s32/s32/s72\n/s99/s32/s40/s84/s41\n/s84 /s32/s40/s75/s41\n/s72\n/s101/s98/s32/s40/s84/s41/s32/s32\n/s84 /s32/s40/s75/s41\n/s32/s57/s57/s57/s32/s72/s122 /s32/s51/s51/s32/s72/s122\n/s32/s51/s51/s51/s32/s72/s122 /s32/s49/s49/s32/s72/s122\n/s32/s49/s49/s49/s32/s72/s122 /s32/s49/s32/s72/s122\n/s32/s48/s46/s49/s32/s72/s122/s32/s57/s57/s57/s32/s72/s122 /s32/s51/s51/s32/s72/s122\n/s32/s51/s51/s51/s32/s72/s122 /s32/s49/s49/s32/s72/s122\n/s32/s49/s49/s49/s32/s72/s122 /s32/s49/s32/s72/s122\n/s32/s48/s46/s49/s32/s72/s122\n/s32/s32/s109/s34 /s32/s40/s49/s48/s45/s53\n/s32/s101/s109/s117/s41\n/s84 /s32/s40/s75/s41FIG. 4: The temperature-dependence of coercive field Hcand ex-\nchange bias field Hebare shown in (a) and (b) respectively (the error-\nbars were comparable to the size of data points). The real part of the\nac susceptibility, \u001f(T), is presented for various frequencies 0.1 Hz to\n999 Hz in (c) and the imaginary part in (d). No significant frequency\ndispersion is discernible, ruling out the possibility of the presence of\ncanonical spin glass phase.\nprovided in (b). We now employ a similar analysis as was\nemployed in the case of Y 2CoMnO 6which also exhibited\ndomain-related effects.[15] The scenario of pinning of\ndomain walls with associated lattice strain has been modeled\nin phase-separated manganites.[21] Ferromagnetic clusters\nin structurally disordered materials present strain fields that\ncan pin the domain walls. The formalism of pinning of\nelastic objects described above is similar to the pinning of\nvortices in high- Tcsuperconductors.[22] According to the\nmodel developed by Larkin, the surface tension of a typical\ndomain wall is approximated as \u000ft\u00184q\nJS2K\nawhereJis\nthe exchange constant between nearest neighbour spins of\nmagnitudeS,Kis the magnetocrystalline anisotropy energy\nper unit volume and ais the distance between nearest neigh-\nbour spins. The Jvalue for TbFeAl can be approximated by\nthe valueJTm\u0000Tm\u00191 K for TmFeAl.[23] The spin value\nfor Tb3+isS= 3 and a value of K\u0019103J/m3is adopted\nfrom the typical values for TmFeAl.[24] From the structural\nrefinement using x ray data, the nearest neighbour distance\nafor TbFeAl (Tb-Tb distance) is obtained as 0.343 nm.4\n/s45/s49/s46/s54 /s45/s48/s46/s56 /s48/s46/s48 /s48/s46/s56 /s49/s46/s54/s45/s49/s48/s48/s45/s53/s48/s48/s53/s48/s49/s48/s48\n/s49/s46/s48/s56 /s49/s46/s49/s48 /s49/s46/s49/s50/s45/s52/s48/s45/s51/s48\n/s32/s108/s111/s111/s112/s32/s49\n/s32/s108/s111/s111/s112/s32/s50\n/s32/s108/s111/s111/s112/s32/s51\n/s32/s108/s111/s111/s112/s32/s52/s40/s98/s41\n/s32/s32/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s72 /s32/s40/s84/s41/s40/s97/s41\n/s50/s32/s75\n/s32/s32/s32\n/s77 /s32/s40/s101/s109/s117/s47/s103/s41\n/s72 /s32/s40/s84/s41/s108/s111/s111/s112/s32/s49/s108/s111/s111/s112/s32/s52\nFIG. 5: The result of training effect experiment at 2 K is shown\nin (a) for 4 loops of hysteresis. A shift of the hysteresis curve with\nincreasing number of loops is visible. In (b), the loops are shown\nmagnified, which confirms the loop-shifts and hence, exchange bias.\nWith these values, an \u000ft\u00182.1\u000210\u00004J/m2. The amplitude\nof surface excitations follows the relation \u0018\u0018q\nU\n2\u000ft. The\nac susceptibility data of TbFeAl above the Th\nccould be\nroughly described by a thermally activated behaviour, \u001f(T)\n=\u001f0+\u001f1exp(\u0000U=kBT)yielding a value, U\u001883 meV .\nSubstituting these values, \u0018\u00195.5 nm.\nAmongRFeAl (R= rare earth) compounds, TmFeAl is\nreported to display unique magnetic properties.[20, 23, 24]\nThe formation of nano-magnetic domains in TmFeAl was dis-\ncovered through M ¨oßbauer spectroscopy.[24] The magnetic\nstructure of TmFeAl describes ferromagnetic sublattice of\nTm moments aligned along the c-axis of the hexagonal cell.Fe and Al are arranged disordered in the unit cell and because\nof the low moment of Fe ( \u00180.52\u0016B), it is not easily detected\nin neutron diffraction studies. However, at low temperature\na development of ferromagnetic short-range order of the\nsize of 1 nm is inferred. Interestingly, this size corresponds\nto the size of nano magnetic domains observed through\nneutron depolarization investigations.[20] The disorder in the\nFe-Al sublattice combined with the high magnetocrystalline\nanisotropy of Tm moments were attributed as the reason for\nthe development of nano domains. The anisotropy field of Tm\nis about 100 T[23] and hence, can prevent the nano domains\nfrom ordering under applied fields.\nExchange bias in ferromagnetic TbFeAl with significant\ncoercive field is experimentally demonstrated. Hc\u00191.5 T\nis observed at 2 K. Magnetic saturation is observed in\nisothermal magnetization curves below 5 K however, not\nreaching the full ferromagnetic moment of Tb3+. The crys-\ntallographic disorder among Fe-Al sublattice and the strong\nmagnetocrystalline anisotropy of Tb3+are argued to be the\nreason for observed domain effects that lead to exchange bias.\nThe results presented in this work attains a general feature\nfollowing the extension from exchange bias effect observed\nin double perovskite Y 2CoMnO 6which was driven by the\ntheme exchange bias induced by atomic disorder.\nHSN acknowledges FRC/URC for the grant of a postdoc-\ntoral fellowship and AMS thanks the SA NRF (93549) and\nUJ URC/FRC for financial assistance.\n#Present address: Department of Physics, Colorado\nState University, Fort Collins, CO 80523\n[1] S. Giri, M. Patra, and S. Majumdar, J. Phys.: Condens. Matter\n23, 073201 (2011).\n[2] I. K. Schuller and J. Nogu ´es, J. Magn. Magn. Mater 192, 203\n(1999).\n[3] M. Kiwi, J. Magn. Magn. Mater. 234, 584 (2001).\n[4] A. E. Berkowitz and K. Takano, J. Magn. Magn. Mater. 200,\n552 (1999).\n[5] Z. Zhang, Y . C. Feng, T. Clinton, G. 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Matter 19, 225007 (2007).\n[14] M. Gruyters, Phys. Rev. B 79, 134415 (2009).\n[15] H. S. Nair, T. Chatterji, and A. M. Strydom, Appl. Phys. Lett.\n106, 022407 (2015).\n[16] J. Rodriguez-Carvajal, LLB, CEA-CNRS, France (2010).\n[17] H. M. Rietveld, J. Appl. Crystall. 2, 65 (1969).\n[18] M. Klimczak and E. Talik, in J. Phys.: Conf. Ser. (IOP Publish-\ning, 2010), vol. 200, p. 092009.\n[19] L. Li, D. Huo, Z. Qian, and K. Nishimura, Intermetallics 46,\n231 (2014).\n[20] A. M. Mulders, W. H. Kraan, P. C. M. Gubbens, K. H. J.\nBuschow, N. St ¨ußer, and M. Hofmann, J. Alloys and Comp.\n299, 88 (2000).\n[21] D. Niebieskikwiat and R. D. S ´anchez, J. Phys.: Condens. Mat-\nter24, 436001 (2012).\n[22] A. I. Larkin, Sov. Phys. JETP 31, 784 (1970).\n[23] A. M. Mulders, Ph.D. thesis, TU Delft, Delft University of\nTechnology (1998).\n[24] A. M. Mulders, W. H. Kraan, A. R. Ball, E. Br ¨uck, K. H. J.\nBuschow, and P. C. M. Gubbens, Hyperfine inter. 97, 109\n(1996)."    },    {        "title": "1605.05739v1.Interfacial_magnetic_anisotropy_from_a_3_dimensional_Rashba_substrate.pdf",        "content": "arXiv:1605.05739v1  [cond-mat.mes-hall]  18 May 2016Interfacial magnetic anisotropy from a 3-dimensional Rash ba substrate\nJunwen Li1,2, Paul M. Haney1\n1. Center for Nanoscale Science and Technology,\nNational Institute of Standards and Technology,\nGaithersburg, MD 20899\n2. Maryland NanoCenter, University of Maryland,\nCollege Park, MD 20742, USA\nWe study the magnetic anisotropy which arises at the interfa ce between a thin film ferromagnet\nand a 3-d Rashba material. The 3-d Rashba material is charact erized by the spin-orbit strength α\nand the direction of broken bulk inversion symmetry ˆ n. We find an in-plane uniaxial anisotropy\nin the ˆz׈ndirection, where ˆ zis the interface normal. For realistic values of α, the uniaxial\nanisotropy is of a similar order of magnitude as the bulk magn etocrystalline anisotropy. Evaluating\nthe uniaxial anisotropy for a simplified model in 1-d shows th at for small band filling, the in-plane\neasy axis anisotropy scales as α4and results from a twisted exchange interaction between the spins\nin the 3-d Rashba material and the ferromagnet. For a ferroel ectric 3-d Rashba material, ˆ ncan\nbe controlled with an electric field, and we propose that the i nterfacial magnetic anisotropy could\nprovide a mechanism for electrical control of the magnetic o rientation.\nI. INTRODUCTION\nInterfacial magnetic anisotropy plays a key role in\nthin film ferromagnetism. For ultra thin magnetic lay-\ners (less than 1 nm thickness), the reduced symmetry at\nthe interface and orbital hybridization between the fer-\nromagnet and substrate can lead to perpendicular mag-\nnetic anisotropy [1–3]. Perpendicular magnetization in\nmagnetic multilayers can enable current-induced mag-\nnetic switching at lower current densities [4, 5]. Inter-\nfacial magnetic anisotropy is also at the heart of several\nschemes of electric-field based magnetic switching. In\nthis case an externally applied field can modify the elec-\ntronic properties of the interface, changing the magnetic\nanisotropy and leading to efficient switching of the mag-\nnetic layer [6–9]. The combination of symmetry break-\ning at the interface and the materials’ spin-orbit cou-\npling generally leads to an effective Rashba-like interac-\ntion acting on the orbitals at the interface [10, 11]. The\ninterfacial magnetic anisotropy can be studied in terms\nof a minimal model containing both ferromagnetism and\nRashbaspin-orbitcoupling[12]. Theinterfacialmagnetic\nanisotropy direction is a structural property of the sam-\nple leading to easy- or hard-axis out-of-plane anisotropy,\nand isotropic in-plane anisotropy.\nThere has been recent interest in materials with strong\nspin-orbit coupling which lack structure inversion sym-\nmetry in the bulk. These are known as 3-d Rashba ma-\nterials, and examples include BiTeI [13] and GeTe [14].\nIn BiTeI, the structure inversion asymmetry results from\nthe asymmetric stacking of Bi, Te, and I layers, and pho-\ntoemission studies reveal an exceptionally large Rashba\nparameter α[13]. In GeTe, a polar distortion of the\nrhombohedral unit cell leads to inversion asymmetry and\nferroelectricity [15–17]. Both materials are semiconduc-\ntors in which the valence and conduction bands are de-\nscribed by an effective Rashba model with symmetry-\nbreaking direction ˆ n, which is determined by the crystalstructure. There is interest in finding other ferroelectric\nmaterials with strong spin-orbit coupling, motivated by\nthe desire to control the direction of ˆ nwith an applied\nelectric field [17–22].\nIn this work, we study the influence of a 3-d (non-\nmagnetic) Rashba material on the magnetic anisotropy\nof an adjacent ferromagnetic layer. The interface be-\ntween these materials breaks the symmetry along the ˆ z-\ndirection,andtheadditionofanothersymmetrybreaking\ndirection ˆ nenrichesthemagneticanisotropyenergyland-\nscape. For a general direction of ˆ n, we find a complex de-\npendence of the system energy on magnetic orientation.\nIn our model system, we find the out-of-plane anisotropy\nis much smaller than the demagnetization energy. How-\never an in-plane component of ˆ nleads to a uniaxial in-\nplane magnetic anisotropy which can be on the order\nof (or larger than) the magnetocrystalline anisotropy of\nbulk ferromagnetic materials. Control of ˆ n(for exam-\nple via an electric field in a 3-d Rashba ferroelectric) can\ntherefore modify the magnetization orientation, opening\nup new routes to magnetic control.\nFIG. 1: (a) shows the system geometry of a thin film fer-\nromagnet adjacent to a nonmagnetic material described by\na 3-d Rashba model. (b) shows the unit cell of the model\nsystem.2\nII. NUMERICAL EVALUATION OF 2-D MODEL\nWe first consider a bilayer system as shown in Fig.\n1(a), with unit cell as shown in Fig. 1(b). We take 4\nlayers of the Rashba material and 2 layers of the ferro-\nmagnetic material with stacking along the z-direction,\nand assume a periodic square lattice in the x−yplane\nwith lattice constant a. The Hamiltonian of the system\nis given by H=HTB+HF+HR+HF−R, where:\nHTB=t/summationdisplay\n/angbracketleftij/angbracketrightc†\nicj (1)\nHF=∆\n2/summationdisplay\niγνc†\niγciν/parenleftBig\nˆM·/vector σγν/parenrightBig\n+HTB, (2)\nHR=iα\n2a/summationdisplay\n/angbracketleftij/angbracketrightγνc†\niγcjν[(ˆrij×/vector σγν)·ˆn]+HTB,(3)\nHF−R=HTB+α\n2a/parenleftBig\nic†\nFγcRνσx\nγν+h.c./parenrightBig\n.(4)\nHTBdescribes nearest-neighbor hopping with amplitude\nt.HFis the on-site spin-dependent exchange interaction\nin the ferromagnet. Its magnitude is ∆ and is directed\nalongthemagnetizationorientation ˆM.HRdescribesthe\nRashbalayer: spin-orbitcouplingandthebrokensymme-\ntry direction ˆ nlead to spin-dependent hopping between\nsitesiandjwhich is aligned along the ˆ rij׈ndirection,\nwhere ˆrijis the direction of the vector connecting sites i\nandj.αis the Rashba parameter (with units of energy ×\nlength). HF−Ris the coupling between ferromagnetic\nand Rashba layers - it includes both spin-independent\nhopping and spin-dependent hopping. In Eq. 4, the F\n(R) subscript in the creation and annihilation operators\nlabels the interfacial ferromagnet (Rashba) layer. We\nfind the model results are similar if HF−Rincludes only\nspin-independent hopping.\nThe Fermi energy EFis determined by the electron\ndensityρand temperature Taccording to:\nρ=1\n(2π)2/integraldisplay\ndkf/parenleftbiggEk−EF\nkBT/parenrightbigg\n(5)\nwherekBis the Boltzmann constant, and f(x) is the\nFermi distribution function: f(x) = (1+ ex)−1. The\nintegral is taken over the two-dimensional Brillouin zone.\nFor a given electron density ρ, Eq. 5 determines the\nFermi energy (which generallydepends on ˆM). The total\nelectronic energy is then given by:\nE(ˆM) =1\n(2π)2/integraldisplay\ndkEkf/parenleftbiggEk−EF\nkBT/parenrightbigg\n(6)\nThe default parameterswe use are∆ = t, α= 0.4×ta.\nFort= 1 eV and a= 0.4 nm, this corresponds to a\nRashba parameter of 0 .16 eV·nm (compared to a value\nα= 0.38 eV·nm for BiTeI [13]). We let T= 0.1 K and\nuseaminimumof12002k-pointstoevaluatetheintegrals\nin Eqs. 5-6.Fig. 2(a) shows the total energy versus magnetic ori-\nentation for ˆ n= ˆy. We observe an out-of-plane mag-\nnetic anisotropy, however its magnitude is much smaller\nthan the demagnetization energy, which is typically on\nthe order of 1000 µJ/m2. In the rest of the paper,\nwe assume that the demagnetization energy leads to\neasy-plane anisotropy of the ferromagnet, so that the\nmost relevant features of the Rashba substrate-induced\nanisotropy energy are confined to the x−yplane. The\nenergy versus easy-plane magnetic orientation (parame-\nterized by the azimuthal angle φ) is shown in Fig. 2(b).\nThe anisotropy is uniaxial and favors orientation in the\n±ˆy-directions. This is in contrast to the bulk magne-\ntocrystalline anisotropy of cubic transition metal ferro-\nmagnets, which has in-plane biaxial anisotropy.\nAs a point of reference for the magnitude of the cal-\nculated substrate-induced uniaxial anisotropy, we com-\npare it to the magnetocrystalline anisotropy EMCfor\n2-monolayer thick film of Fe, Ni, and Co, for which\nEMC=(34,3.5,318)µJ/m2, respectively [23]. Fig. 2(b)\nshows that for the material parameters in our model,\ntheRashba-induceduniaxialanisotropyislargerthanthe\nmagnetocrystalline anisotropy of Ni. We also note that\npermalloy, commonly used as a thin film ferromagnet,\nhas a vanishing magnetocrystalline anisotropy [24].\nFIG. 2: (a) The system energy as a function of the magnetic\norientation for ˆ n= ˆy, (c) shows the same for ˆ n= (ˆy+ ˆz)/√\n2.\n(b) shows the system energy versus magnetic orientation\nwithin the x−yplane, parameterized by the azimuthal angle\nφfor ˆn= ˆy(d) shows the same for ˆ n= (ˆy+ ˆz)/√\n2.\nNext we consider Rashba layer with both in-plane and\nout-of-plane components of ˆ n: ˆn= (ˆy+ ˆz)/√\n2. This\nis motivated in part by the fact that the symmetry is\nstrongly broken along the ˆ z-direction by the interface.\nOur interest is in the influence of an out-of-plane com-\nponent ˆnwhen there is also an in-plane component of ˆ n.\nThe resulting E(ˆM) shown in Fig. 2(c). As before, there3\nis an in-plane uniaxial anisotropy, as shown in Fig. 2(d),\nwith a larger energy barrier as the previous ˆ n= ˆycase.\nNote that the ˆ ydirection is now a hard axis. In general\nwe find that for an in-plane component of ˆ n, the ˆn׈z\ndirection can be either a hard or an easy axis, depending\non details of the electronic structure.\nWe define the uniaxial in-plane anisotropy energy EA\nas the difference in energy for ˆM= ˆyandˆM= ˆx. Fig.\n3(a) shows EAas the electron density ρis varied, and\nindicates that sign of the anisotropy can change depend-\ning on the value of ρ. Fig. 3(b) shows the dependence of\nEAonαfor two values of the electron density ρ. We find\nthat the uniaxial anisotropyenergyvariesas a powerof α\nwhich depends on ρ(or equivalently on the band filling).\nThe origin of this dependence is discussed in more detail\nin the analytic model we develop next.\nFIG. 3: (a) EAversus dimensionless electron density ρa2. (b)\nEAversusαfor two values of ρ(ρ1a2= 0.02,ρ2a2= 2.1),\nalong with fitting to powers of α. For small ρ, theα4depen-\ndence can be understood in terms of perturbation theory.\nIII. ANALYTICAL TREATMENT OF 1-D\nMODEL\nTo gainsomeinsight intothe physicaloriginofthe uni-\naxial in-plane anisotropy, we consider a simplified system\nof a 1-d chain of atoms extending in the x-direction with\n2 sites in the unit cell, as shown in Fig. 4(a), and take\nˆn= ˆy. The Hamiltonian is the same as Eqs. 1-4. We\ncompute E(kx) in a perturbation expansion of the spin-\norbit parameter α, then determine the total energy as a\nfunction of αandˆM.\nThe lowest order term in αis of the form E(kx)∝\nαkxMz. This can be understood as a simple magnetic\nexchange interaction between the F and R sites: spin-\norbit coupling leads to effective magnetic field on R along\nthez-direction. This effective magnetic field is propor-\ntional to αkxand is exchange coupled to the effective\nmagnetic field on the F site (see Fig. 4(a)). The linear-\nin-kxterm inE(kx) shifts the energy bands downward\nby an amount proportional to the square of the coeffi-\ncient multiplying kx. The total energy decreases by the\nsame factor. This finally results in a magnetic anisotropy\nenergywhich is proportionalto α2M2\nz, describing an out-\nof-plane anisotropy.As discussed earlier, we assume the ferromagnet is\neasy-plane and are therefore interested in the magnetic\nanisotropy within the plane. In-plane anisotropy in E(k)\nappears only at 2nd order in α. Taking Mz= 0, we find\nE(k) takes a simple form in the limits that α≪∆≪2t\nandka≪1. We present the result for the lowest energy\nband:\nE(kx) =Ak2\nx+Bα2kxMy+C (7)\nWhereA, B, C depend on ∆ and t, whose precise form\nis not essential for this discussion [25]. Fig. 4(b) shows\nthe numerically computed dispersion for two orientations\nof the ferromagnet. For ˆM= ˆx, the energy bands are\npurely quadratic in kx, while for ˆM= ˆy, the energy\nbands acquire a linear-in- kxcomponent, which is of op-\nposite sign for the two lowest energy bands. In the case\nwhere only the lowest band is occupied, the total energy\nagain depends on the square of the linear-in- kxcoeffi-\ncient, resulting in an in-plane anisotropy energy which is\nproportional to α4M2\ny.\nFIG. 4: (a) cartoon of the unit cell of a 1-d model which\nextends along the x-direction. The top site is ferromag-\nnetic and the bottom site includes Rashba spin-orbit cou-\npling. The arrow along the +ˆ z-direction on the R site in-\ndicates the direction of the Rashba effective magnetic field\n(we’ve assumed kx>0 in the figure). The double arrow\nalong the ±x-direction depicts the spin direction of the spin-\ndependent hopping between F and R sites. (b) The energy\ndispersion for the lower two bands for two magnetic orienta-\ntions (∆ = 1 , α= 0.5 for this plot).\nWe can understand the physical origin of the depen-\ndence of the energy on My: the spin-dependent hopping\nbetween F and R sites induces a twisted exchange in-\nteraction between the spins on these sites, which favors\na noncollinear configuration in which both spins lie in\nthey−zplane [26]. The effective magnetic field on the\nR-site is always along the z-direction, so the exchange\nenergy therefore differs when the ferromagnet is aligned\ninthexversusydirection. Thetwisted exchangeinterac-\ntion energy contains a factor proportionalto the effective\nmagnetic field on R (which varies as αkx), and a factor\nproportionaltothe spin-dependent hopping (which islin-\near inα) - so the energy varies as finally as α2kxMy.\nWe find that the lowest energy band of the 2-d system\nof the previous section also contains a linear-in- kxterm4\nwhich varies as α2kxMy, indicating that the physical pic-\nture developed for the 1-d system also applies for the\n2-d system. In the case where multiple bands are oc-\ncupied, we’re unable to find a closed form solution for\nthe anisotropy energy, and find that it can vary with a\npower of αthat depends on the electron density (and\ncorresponding Fermi level). This is shown in Fig. 3(b)\nwhere the uniaxial anisotropy varies as α3for multiply\nfilled bands (we’ve observed several different power-law\nscalings with αfor different system parameters). Never-\ntheless the case of a singly occupied band is sufficient to\nillustratethephysicalmechanismunderlyingthe in-plane\nmagnetic anisotropy.\nIV. CONCLUSION\nWe’ve examined the influence of a 3-d Rashba mate-\nrial on the magnetic properties of an adjacent ferromag-\nnetic layer. A uniaxial magnetic anisotropy is developed\nwithin the plane of the ferromagnetic layer with easy-\naxis direction determined by ˆ n. Depending on mate-\nrial parameters, the easy-axis can be parallel or perpen-\ndicular to ˆ n. For large but realistic values of the bulk\nRashba parameter of the substrate, the magnitude of the\nanisotropy indicates that the effect should be observable.\nFor materials in which the direction of the bulk sym-\nmetry breaking is tunable - for example in a ferroelectric3-d Rashbamaterial - the interfacialmagnetic anisotropy\noffers a novel route to controlling the magnetic orienta-\ntion. The magnitudeoftheRashba-inducedanisotropyis\nmuch less than the demagnetization energy, so its influ-\nence is confined to fixing the in-plane component of ˆM.\nThis control would nevertheless be useful in a bilayer\ngeometry in which electrical current flows in plane. In\nthis case the anisotropic magnetoresistance effect yields\na resistance which varies as ( ˆJ·ˆM)2[27], which can be\nutilized to read out the orientation of ˆM. We also note\nrecent works have utilized the reduced crystal symmetry\nof the substrate to achieve novel directions of current-\ninduced spin-orbit torques [28]. This indicates that the\nsymmetry of the substrate can influence the nonequilib-\nrium properties of the magnetization dynamics, in addi-\ntion to modifying the equilibrium magnetic properties,\nas studied in this work.\nAcknowledgment\nJ. Li acknowledges support under the Cooperative Re-\nsearch Agreement between the University of Maryland\nand the National Institute of Standards and Technol-\nogyCenter forNanoscaleScience and Technology, Award\n70NANB10H193, through the University of Maryland.\n[1] P. F. Carcia, A. D. Meinhaldt, and A. Suna, App. Phys.\nLett.47, 178 (1985).\n[2] H. Yang, M. Chshiev, B. Dieny, J. Lee, A. Manchon, and\nK. Shin, Phys. Rev. B 84, 054401 (2011).\n[3] G. H. O. Daalderop, P. J. Kelly, and M. F. H. Schuur-\nmans, Phys. Rev. B 50, 9989 (1994).\n[4] S. Ikeda, K. Miura, H. Yamamoto, K. Mizunuma, H. D.\nGan, M. Endo, S. Kanai, J. Hayakawa, F. Matsukura,\nand H. Ohno, Nat. Mat. 9, 721 (2010).\n[5] S. Mangin, D. Ravelosona, J. A. Katine, M. J. Carey,\nB.D.Terris, andE. E.Fullerton, Nat.Mat. 5, 210(2006).\n[6] W.-G. Wang, M. Li, S. Hageman, and C. Chien, Nat.\nMat.11, 64 (2012).\n[7] X. Li, G. Yu, H. Wu, P. Ong, K. Wong, Q. Hu,\nF. Ebrahimi, P.Upadhyaya, M. Akyol, N. Kioussis, et al.,\nApp. Phys. Lett. 107, 142403 (2015).\n[8] T. Maruyama, Y. Shiota, T. Nozaki, K. Ohta, N. Toda,\nM. Mizuguchi, A. Tulapurkar, T. Shinjo, M. Shiraishi,\nS. Mizukami, et al., Nat. Nanotech. 4, 158 (2009).\n[9] Y. Shiota, T. Nozaki, F. Bonell, S. Murakami, T. Shinjo,\nand Y. Suzuki, Nat. Mat. 11, 39 (2012).\n[10] J.-H. Park, C. H. Kim, H.-W. Lee, and J. H. Han, Phys.\nRev. B87, 041301 (2013).\n[11] P. M. Haney, H.-W. Lee, K.-J. Lee, A. Manchon, and\nM. D. Stiles, Phys. Rev. B 88, 214417 (2013).\n[12] S. E. Barnes, J. Ieda, and S. Maekawa, Scient. Rep. 4\n(2014).\n[13] K. Ishizaka, M. Bahramy, H. Murakawa, M. Sakano,\nT. Shimojima, T. Sonobe, K. Koizumi, S. Shin, H. Miya-hara, A. Kimura, et al., Nat. Mat. 10, 521 (2011).\n[14] G.Pawley, W.Cochran, R.Cowley, andG.Dolling, Phys.\nRev. Lett. 17, 753 (1966).\n[15] M. Liebmann, C. Rinaldi, D. Di Sante, J. Kellner,\nC. Pauly, R. N. Wang, J. E. Boschker, A. Giussani,\nS. Bertoli, M. Cantoni, et al., Adv. Mat. 28, 560 (2016).\n[16] J. Krempask` y, H. Volfov´ a, S. Muff, N. Pilet, G. Landolt ,\nM. Radovi´ c, M. Shi, D. Kriegner, V. Hol` y, J. Braun,\net al., arXiv preprint arXiv:1503.05004 (2015).\n[17] A. V. Kolobov, D. J. Kim, A. Giussani, P. Fons, J. Tom-\ninaga, R. Calarco, and A. Gruverman, APL Mat. 2,\n066101 (2014).\n[18] D. Di Sante, P. Barone, R. Bertacco, andS. Picozzi, Adv.\nMat.25, 509 (2013).\n[19] A. Narayan, Phys. Rev. B 92, 220101 (2015).\n[20] A. Stroppa, D. Di Sante, P. Barone, M. Bokdam,\nG. Kresse, C. Franchini, M.-H. Whangbo, and S. Picozzi,\nNat. Comm. 5(2014).\n[21] M. Kepenekian, R. Robles, C. Katan, D. Sapori,\nL. Pedesseau, and J. Even, ACS Nano 9, 11557 (2015).\n[22] M. Kim, J. Im, A. J. Freeman, J. Ihm, and H. Jin, Proc.\nNat. Ac. Sc. 111, 6900 (2014).\n[23] B. D. Cullity and C. D. Graham, Introduction to mag-\nnetic materials (John Wiley & Sons, 2011).\n[24] L. Yin, D.Wei, N.Lei, L.Zhou, C.Tian, G. Dong, X.Jin,\nL. Guo, Q. Jia, and R. Wu, Phys. Rev. Lett. 97, 067203\n(2006).\n[25] For completeness we give the forms for the\nlowest two energy bands here, A1,2= 1−5\nα2/64/parenleftbig\n±16+8∆ ∓16∆2+7∆3/parenrightbig\n,B1,2= (∓1+∆/2),\nandC1,2= 1/16/parenleftbig\n−8α2+∆2α2−16∓8∆−∆2/parenrightbig\n. The\nα-dependent terms in these factors lead to higher order\ncorrections to the total energy.\n[26] H. Imamura, P. Bruno, and Y. Utsumi, Phys. Rev. B 69,\n121303 (2004).[27] T. McGuire and R. Potter, IEEE Trans. Mag. 11, 1018\n(1975).\n[28] D. MacNeill, G. M. Stiehl, M. H. D. Guimaraes, R. A.\nBuhrman, J. Park, and D. C. Ralph, arXiv:1605.02712\n(2016)."    },    {        "title": "1605.07141v1.Large_magnetic_anisotropy_in_Fe__0_25_TaS_2.pdf",        "content": "Large magnetic anisotropy in Fe0.25TaS2\nVaideesh Loganathan,1Jian-Xin Zhu,2, 3and Andriy H. Nevidomskyy1\n1Department of Physics and Astronomy, Rice University, Houston, Texas 77005, USA\n2Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n3Center for Integrated Nanotechnologies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA\n(Dated: June 18, 2021)\nWe present a first-principles study of the large magneto-crystalline anisotropy in the intercalated di-\nchalcogenide material Fe 0.25TaS 2, investigated with the DFT+U approach. We verify a uniaxial magnetocrys-\ntalline anisotropy energy(MAE) of 15meV/Fe. in the material. We further analyze the dependence of MAE on\nthe constituent elements and the effect of spin-orbit coupling. Contrary to conventional intuition, we find a small\ncontribution to MAE due to strong spin-orbit coupling in the heavier element, Ta. We show that the electronic\nconfiguration, crystal field environment and correlational effects of the magnetic ion are more important.\nI. INTRODUCTION\nA figure of merit for hard ferromagnets is proportional\nto the magneto-crystalline anisotropy energy (MAE), which\nmeasures the energy cost of deviations from easy-axis mag-\nnetization. Typical strong magnets consist of rare-earth and\ntransition-metal intermetallic compounds. The combination\nof strong spin-orbit coupling and large ordered moment gives\nrise to strong anisotropy. Due to the scarce availability of\nrare-earths, hard magnets without rare-earths are desirable. A\ncandidate material consisting of 3 dand 5dtransition metals\noffers a platform for further exploration.1–3\nTransition metal dichalcogenides (MX2, with M = tran-\nsition metal, X = S,Se,Te) form layered structures. They\nshow interesting properties such as charge density waves\nand superconductivity4–7. These properties can be tuned and\nenhanced by intercalating them with metal ions, resulting\nin changes in superconducting transition temperatures, and\nanisotropic magneto-transport8–15. FexTaS2is an example\nwhere Fe ions are intercalated between the hexagonal layers\nof 2 H\u0000TaS2. The magnetic properties can be varied with\nthe concentration of Fe: it is found that for x < 0:4, the\nmagnetic ordering is ferromagnetic, and switches to being\nantiferromagnetic for higher concentrations of Fe16. The\nCurie temperature, TCis highest for x= 0:25with a value of\n160K.17\nFe0.25TaS2has been observed to display large magneto-\ncrystalline anisotropy, sharp switching of magnetization,\nand anomalous magnetoresistance17. Fe crystallizes in a\n2\u00022superlattice within the TaS2layers, with Fe ions\nforming a hexagonal crystal.18,19. The arrangement is crucial\nas it allows RKKY interactions to maximize TCand also\nresults in a large uniaxial magnetic anisotropy. For easy\naxis magnetization along the hexagonal c-axis, the hysteresis\nloop is almost square,17reminiscent of strong permanent\nmagnets. On the other hand, measurements in an in-plane\nmagnetic field barely show discernible magnetization. An\neasy axis moment of \u00184\u0016Bis found per Fe ion, of which\nan unusually large 1\u0016Bwas found to arise from the orbital\ncomponent by previous X-ray magnetic circular dichoism\n(XMCD) measurements19. The large easy-axis orbitalmoment accounts for a calculated MAE of 15meV/Fe19, a\nvalue comparable to rare-earth magnets.23Fe0.25TaS2is thus\na candidate hard magnet without rare-earth elements.\nFirst-principles electronic structure calculations based\non density functional theory (DFT) have long been used to\nestimate MAE values20–24. Typically, separate calculations\nare done with magnetic moment along and away from the\npreferred easy-axis direction. The energies can be compared\nby the force theorem25and total-energy difference methods22.\nA less computationally intensive method was found with\nthe so-called torque method, which involves restricting the\nmagnetization to the 45\u000eangle to the easy axis and evaluating\nthe angular derivative of the energy (torque)26,27. Previous\nfirst-principles calculations have used the orbital-polarization\nscheme to account for many-body correlational effects.28–30\nIt is essential to incorporate the many-body effects of elec-\ntron correlations into the DFT calculations to reproduce the\nobserved orbital moments, which are critical to the anisotropy\nenergy. Here, we employ the DFT+U method31, which pro-\nvides a better description of the electron correlations. We\nstudy the magnetization and anisotropy energy in Fe0.25TaS2.\nWe break down the dependence of MAE into its constituent\nelements, disentangling the effects of the crystal field envi-\nronment and of spin-orbit coupling. In contrast to earlier first\nprinciples calculations19, we find that a modest value of the\nHubbard on-site interaction U&2:5eV is sufficient to re-\nproduce the measured magnetic moment and the results of the\nX-ray absorption spectroscopy (XAS)19. To further elucidate\nthe origin of the large MAE in Fe0.25TaS2, we study the sub-\nstitutions of Fe and Ta by other 3dand4delements. Contrary\nto the intuitive expectation, we find that the presence of 5d\nelectrons of Ta does not provide a significant source of MAE.\nRather, it is the d6configuration of Fe2+ions that results in\na large orbital moment and is thus responsible for the large\nobserved MAE in Fe0.25TaS2. Our findings bear important\nramifications for the search for strong permanent magnets, in\nparticular without rare-earth elements32.arXiv:1605.07141v1  [cond-mat.str-el]  23 May 20162\nII. METHODS\nWe performed first-principles calculations within the\nDFT+U scheme in the generalized gradient approximation\n(GGA-PBE) for exchange-correlation33. The full-potential,\nlinearized augmented plane wave (FP-LAPW) method was\nused as implemented in the Wien2K code34. A13\u000213\u00026\nMonkhorst-Pack k-point grid was used for BZ integration\nwith the tetrahedron method. We performed a range of cal-\nculations by varying the Hubbard on-site energy parameter, U\non the Fe site. The Hund exchange parameter, J was fixed to\n0.7eV .\nTo choose the optimal value for Hubbard U, the calculated\nmoments were compared with the experimental values. To\nexamine the anisotropy, calculations were done by restricting\nthe magnetization along the easy axis (001) and along a hard\ndirection in the basal plane (010). Noting that spin-orbit cou-\npling(SOC) is the main contributor, MAE was calculated in\ntwo ways: (a) as a difference in values of SOC energy between\nthe two directions: EMAE = \u0001h\u0010L\u0001Si(\u0010being the SOC con-\nstant); and (b) an approximation involving the orbital moment\nanisotropy as: EMAE =1\n4\u0010h\u0001Li\u0001hSi, whereh\u0001Liis the\nchange in orbital moment due to crystalline anisotropy35. We\nfound both methods to give consistent results and to be much\nmore accurate than the brute-force comparison of total ener-\ngies of the two spin configurations. The error in the total en-\nergy difference is an artifact of the Hubbard interaction term\nin DFT+U. This term is linear in U, and involves pair-wise\nproducts of orbital occupations.31As will be explained later,\nthe orbital occupations are significantly different between the\ntwo magnetic directions. This gives rise to a misleading U-\ndependent correction when MAE is calculated as the total en-\nergy difference, one that is an order of magnitude larger than\nthe SOC effect.\nIII. RESULTS\nA. Selection of U\nBare DFT calculations (without Hubbard U interac-\ntion) yielded a ferromagnetic ordering with spin moment,\nms= 3:2\u0016B/Fe and orbital moment, mo= 0:3\u0016B/Fe.\nThese values can be compared with the moments concluded\nfrom XAS measurements19. Whilemsagrees with the\nexperimental value of 3\u0016B,mois largely underestimated\nfrom the experiment value of 1\u0016B. In order to cure this\ndeficiency of the DFT, we have performed a series of DFT+U\ncalculations. As the Hubbard interaction Uon Fe site was\nprogressively increased, morose to 0:7\u0016Band plateaued\nbeyondU\u00182:5eV. 4 We plot the saturated moment\n(msat=ms+mo) in Fig. 1 (green, empty circles). On com-\nparing the calculated saturated moment of the system with\nthe experimental value( msat= 4\u0016B)17, we findU\u00182:5eV\nto be sufficient.\nWe note that the previous first-principles study19was per-\nformed using a higher value of U= 4:5eV , which is ratherlarger than the typically expected value in other correlated Fe\ncompounds.36,37We find that the justification by the authors of\nRef.19 for using such a large value of U= 4:5eV was likely\na result of attempting to match the experimentally observed\nmagnetic moment with the calculated moment exclusively on\nthe Fe-site (blue, filled circles in Fig. 1). However, it is often\nthe case that the interstitial regions outside of the muffin-tin\nsphere provide a non-negligible contribution to the magnetic\nmoment, which is indeed the case here (green, empty circles\nin Fig. 1). We conclude that it is therefore important to the\ninclude the magnetic moments in interstitials into the calcula-\ntion. We note that this contribution is largest to the spin mo-\nment (interstitials contribute \u00187%of the totalmsvalue), and\nis negligible for the orbital moments (less than 1:5%ofmo).\nWe conclude that the Hubbard U&2:5eV is sufficient to\nreproduce the experimentally measured saturated moment17.\n1 2 3 4 53.43.63.84\nU (eV)msat (B/Fe)\n  \nFe\ntotal\nexpt/uni03BC\nFIG. 1: Comparison of saturated moment from entire cell\n(green, empty circles) with experimental value shows\nU&2:5eVto be sufficient. Restricting to only moments\nwithin the Fe sphere (blue, filled circles) would require larger\nU to match experiment.\nA further check on the value of Hubbard interaction Ucan\nbe performed by comparing the calculated easy-axis orbital\noccupations and splittings with the XAS measurements19.\nWhile the majority spin orbitals at the Fe site are fully\noccupied, XAS measurements show that the minority spin\nis occupied mainly by orbitals dmwith orbital momentum\nprojectionm=\u00001andm= 2. This anisotropic distribution\naccounts for the large orbital moment on Fe site. To compare\nwith the XAS results, we plot in Fig. 2 the electron density of\nstates (DOS) projected onto Fe dm-orbitals. Figure 2a) shows\nthe density of states calculated in the absence of SOC and for\nsmall Hubbard U. The orbital splitting corresponds to the\nD3dtrigonal symmetry, caused by the distorted octahedron\ncoordination of S atoms around the Fe site. The symmetry\nbeing lower than the cubic case leads the Fe 3d-orbitals to\nsplit intoe\u0019\ng,a1gande\u001b\ngsub-bands. In this case, the e\u0019\ng\norbitals (consisting of m=\u00061;\u00062) dominate the DOS at the\nFermi level. This however does not match the XAS results\nwhich find the e\u0019\ngmultiplet split.\nIt turns out that in order to reproduce the XAS findings,3\n0123DOS (/eV/cell)eg\na1g\neg\n-2 -1 0 1 2012\nU (eV)DOS (/eV/cell)d-1,2d-2,1d0π\nσ(a)\n(b)\nM || (001)\nFIG. 2: DOS projected onto Fe d-orbitals. (a) Insufficient\nHubbard interaction U ( U\u00142eV), and lack of SOC; (b)\nSufficient U (2.5 eV) with SOC (moment quantization along\nthe easy axis - (001)). Majority spin (bottom panels) and\nminority spin (top panels) contributions are shown. The\ncolors represent the different dmorbitals, the relevant orbitals\nare labeled.\nInclusion of Hubbard U and spin-orbit coupling(SOC) is\nnecessary to reproduce experimental occupation\nboth the spin-orbit coupling and a sufficiently large Hubbard\nU (U\u00152:5eV) must be included in the calculations. In this\ncase, we find that the e\u0019\ngstates at the Fermi level split into\nlower-energy m=\u00001;2states (as in XAS experiment19) and\nhigher-energy unoccupied m=\u00002;1states, as illustrated in\nFig. 2b. As the value of Hubbard Uis increased beyond 2.5eV ,\nthesed\u00001=d2orbitals lower further. If the energy separation\nbetween the occupied and unoccupied parts of the e\u0019\ngorbitals\nwere known experimentally, it would would help ascertain\nthe value of Hubbard U more precisely. For the purpose of\nthis work, in what follows we shall focus on U= 2:5eV,\nwhich also reproduces the experimentally measured satu-\nrated magnetic moment in Fig. 1, as already mentioned above.\nB. Anisotropy energy\nIn our DFT+U calculations with U= 2:5eV , we obtain the\nspin and orbital moment ms= 3:3\u0016Bandmo= 0:7\u0016B, re-\nspectively, resulting in the total moment on Fe site mtot=\n4\u0016B. Although the total moment matches the experimen-\ntal value17, our value of mois less than the previous calcu-lated and measured value19of1\u0016B. This difference could\nbe attributed to the technical differences in the implementa-\ntion of the DFT+U method.38The occupation analysis of d-\nlevels shows that Fe ions have a +2 oxidation state, resulting\nin a high-spin d6configuration. We next performed similar\ncalculations with the moment along the magnetic hard direc-\ntion in the basal plane. Such moment arrangement lowers the\nhexagonal symmetry, resulting in a different splitting of Fe e\u0019\ng-\norbitals, with d\u00061becoming lower in energy (fig 3). This in\nturn leads to a lower orbital moment, mo= 0:1\u0016B, while the\nspin moment is same as earlier. The resulting large anisotropy\nin the magnitude of the orbital moment, \u0001L= 0:6\u0016B, is\ncrucial for the large magnetocrystalline anisotropy. MAE\ncan be estimated from the orbital anisotropy as EMAE =\n1\n4\u0010h\u0001Li\u0001hSi(here\u0010is the spin-orbit coupling constant). The\nresulting MAE dependence on the strength of Hubbard repul-\nsionUis plotted in Fig. 4(a) (blue, filled circles).\n-2 -1 0 1 2012\nU (eV)DOS (/eV/cell)d-1,1d-2,0,2M || (010)\nFIG. 3: DOS projected onto Fe d-orbitals for hard-axis\nmagnetization(010). Majority spin (bottom panels) and\nminority spin (top panels) contributions are shown. The\ncolors represent the different dmorbitals, the relevant orbitals\nare labeled.\nOccupations differ from the easy-axis case, leading to orbital\nanisotropy.\nTo verify the validity of these results, we have also com-\nputed MAE from the variation of the SOC energy, \u0001ESOC\nbetween the easy-axis and hard magnetic direction35(filled\ncircles in Fig. 4). Both sets of data are in reasonable agree-\nment with one another (within 15%) and both show the satu-\nration of MAE as Uexceeds\u00183eV . The Hubbard interaction,\nalong with SOC split the orbitals as explained in the previous\nsection. The resulting variation in orbital moment, \u0001Lis re-\nsponsible for the anisotropy. The two panels in Fig. 4 show\nthe correspondence between MAE and the (easy-axis) orbital\nmoment,mo. We estimate the value of MAE at U= 2:5eV\nas the average of the two methods, yielding EMAE\u001912meV\nper Fe. This large value of MAE is comparable to those\nfound in rare earth magnets23. The corresponding anisotropy\nfield, required to rotate the magnetization from the easy to\nhard direction, can be found from EMAE =\u0016\u0001Bani(where\n\u0016=\u0016B(L+ 2S)is the saturated magnetic moment), yield-\ningBani\u001962T, consistent with the experimental estimate of4\n60 T19.\n51015MAE (meV/Fe)\n  \n ESOC\norb. anisotropy\n1 2 3 40.40.50.60.7\nU (eV)mo (B/Fe)/uni0394/uni03BC(a)\n(b)\nFIG. 4: (a) MAE calculations from \u0001ESOC (green, empty\ncircles), and orbital anisotropy (blue, filled circles). (b)\nEasy-axis moment shows an analogous trend\nC. Origin of MAE\nNaively, one would expect the 3d-electon metal to provide\nmagnetic moment, and its interaction with a 5d-electron metal\nto provide anisotropy. We put this hypothesis to test by study-\ning the dependence of the MAE on the constituent elements,\nFe and Ta. Below, we report on two different approaches:\nwe first study the substitution of the individual elements in\nthe material and its effect on MAE. Next, we artificially tune\nthe strength of spin-orbit coupling on Fe and Ta sites inde-\npendently of one another, in order to disentangle the relative\ncontribution of these sites to the MAE.\n1. Elemental substitution\nThe importance of each of the constituent elements can\nbe examined by substituting them with another carefully se-\nlected element. To test the strong SOC coupling effect due\nto Ta, it can be replaced by one with lower SOC strength.\nNb, the element immediately above Ta in the periodic table\npreserves the crystal structure and the electronic configura-\ntion in Fe0.25NbS2. Nb, being a 4d3element is expected to\nhave lower SOC strength than the 5d3element, Ta. Within\nthe DFT+U implementation, we calculate the SOC constantfor Nb(\u0010Nb= 107 meV) to be smaller than Ta by a factor of\nfour(\u0010Ta= 417 meV). Next, to test the effect of the electronic\nconfiguration of Fe, we replaced it with the neighboring el-\nement, Mn to study Mn0.25TaS2. The minority spin electron\nin3d6Fe2+ion, which was responsible for orbital moment is\nabsent in 3d5Mn2+. (We also attempted to substitute Fe with\nother 3d elements - Cr, Co and Ni. However, unlike Mn, these\nelements did not preserve ferromagnetic ordering.) The MAE\ncalculations were performed on Fe0.25NbS2and Mn0.25TaS2.\nMaterial MAE splitting (meV) mo(\u0016B)ms(\u0016B)\nFe0.25TaS 2Fe : 14 Ta : -1.0 0.7 3.3\nFe0.25NbS 2Fe : 12 Nb : -0.1 0.6 3.2\nMn 0.25TaS 2Mn : 0 Ta : -0.1 0.01 4.1\nTABLE I: Comparison of MAE and the orbital ( m0) and spin\nmoments (ms) on the 3delement site for different\ncompounds. (Note that there are 4 times as many Ta/Nb\natoms as Fe/Mn, so one would expect a larger effect on MAE\nby substituting on Ta site.)\nTable I shows the contribution to MAE from the constituent\nelements. In Fe0.25TaS2, a major part of the MAE value arises\nfrom Fe. The four Ta atoms together account for a small neg-\native value. (This means the easy and hard directions are\nswitched for Ta moments.) On substitution of Ta with Nb,\nthe MAE and moon Fe decrease slightly, while Nb accounts\nfor an even smaller contribution to MAE than Ta. A four fac-\ntor reduction in SOC strength from Ta to Nb, only contributes\na15% decrease in MAE. On the other hand, substitution of\nFe with Mn results in negligible MAE and orbital moment in\nMn0.25TaS2. We conclude that the d6electron configuration\nof Fe was crucial for the resultant orbital moment and MAE.\nThese results show that MAE mainly arises from the Fe site.\nA partial contribution to MAE is caused by the influence of\nTa on the Fe moments.\n2. Scaling of the SOC constants\nThe breakdown of MAE can further be explored by artifi-\ncially scaling down the SOC strengths on Fe and Ta atoms.\nSOC gives rise to the orbital moment, and is responsible\nfor MAE. We were able to scale down the SOC strengths\non Fe and Ta independently. This was done by artificially\ndecreasing the SOC constant, \u0010at each site. The effect was\nexamined by inspecting changes to MAE and mo. Fig 5 (a)\nshows the changes to MAE as the SOC constant is gradually\nreduced on each of the atoms selectively. MAE is seen to\nvanish as the SOC strength on Fe is diminished (green, empty\ncircles), similar to Mn-substitution from the previous section.\nOn the other hand, the change in MAE due to the SOC\nstrength on Ta is an order of magnitude smaller (blue, filled\ncircles), comparable to Nb-substitution from the previous\nsection.\nWe also plot the orbital moment, movariations in Fig 5\n(b).modecreases linearly on reducing the SOC strength5\n0510MAE (meV/Fe)\n  \nTa\nFe\n0 0.25 0.5 0.75 10.550.60.65\nSOC scaling factormo (B/Fe)\n  \nTa\nFe/uni03BC(a)\n(b)\nFIG. 5: Dependence of (a) orbital moment, and (b) MAE on\nthe SOC strengths on different atoms. Blue, filled (green,\nempty) circles indicate varying \u0010on Ta (Fe) only.\nat either of the sites. The orbital moment is affected twice\nas much due to SOC strength at the Fe site (green, empty\ncircles), compared with Ta site (blue, filled circles). As\nMAE depends on both orbital moment and SOC strength,\nMAE drops more rapidly. This set of calculations further\nconfirm what we found in the previous section. As was shown\nearlier, the combination of Hubbard interaction and spin-orbit\ncoupling results in the orbital anisotropy. In this section, we\nfix Hubbard Uto a sufficiently large value ( U\u00182:5eV),\nand only alter the SOC strength. Starting from a state with\nlarge orbital moment, like in Fig.2 (b), the decreasing SOC\nstrength measures deviations to orbital moment. In the limit\nof vanishing SOC strength, mois non-zero as the orbital\nsplitting in Fig.2 (b) is preserved. The same test performed\nwith a small U(= 1 eV) showsmoto vanish as SOC strength\nis decreased to zero(not shown here). In this case, the lack\nof splitting of states at the Fermi level as seen in Fig.2 (a),\nproduces a state with lower orbital moment. As SOC strengthis decreased, orbital moment gets quenched.\nThe moments in Fe0.25TaS2lie mainly at the Fe site. The\ncrystal field splitting and coupling with the lattice result in\na highly anisotropic orbital moment on Fe. This is the main\ncontributor to total MAE. Although the 5d electrons in Ta have\na large SOC strength, the lack of magnetic moment at the Ta\nsite results in a much smaller contribution to the total MAE.\nTheir interactions with the Fe electrons only partly enhances\nthe MAE.\nIV . CONCLUSION\nFe0.25TaS2is an interesting magnetic material composed\nentirely of transition metals. It shows magnetic properties\ncomparable to rare-earth based magnets. We have investi-\ngated its magnetocrystalline anisotropy from DFT+U based\nfirst-principles calculations. We observe a large uniaxial\nMAE that arises due to the orbital anisotropy at the Fe site.\nThe Coulomb interaction strength and crystal field splitting\nof the Fe ion give rise to the orbital moment anisotropy. We\nwere able to test the effect of spin-orbit coupling strength\nof the heavier element in the dichalcogenide layer on the\nmaterial’s MAE. Subsituting Ta with a lighter element\nsuch as Nb greatly reduces the spin-orbit strength. We\nalso artificially vary the spin-orbit constant of the heavier\nelement, and note the changes in orbital moment. In both\ncases, we find the changes to MAE and orbital moment to be\nsmaller than previously expected. The dichalcogenide layer\nlacks magnetic moment, and has no direct overlap with the\nmagnetic intercalant ion. So, its spin-orbit strength weakly\naffects the anisotropy produced by the intercalant. Instead,\nthe dichalcogenide layer plays an important role in forming\na favorable crystal field environment for the intercalant ion.\nOur analysis helps in the search of strong permanent magnets\nwith transition metals. 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Blaha et al., WIEN2K, An Augmented Plane Wave +Local Or-\nbitals Program for Calculating Crystal Properties (Vienna Uni-\nversity of Technology, Vienna, Austria, 2001).\n35G. van der Laan, J. Phys. Condens. Matter 10, 3239 (1998)\n36J. Ferber, Y . -Z. Zhang, H. O. Jeschke, and R. Valent ´ı, Phys. Rev.\nB82, 165102 (2010)\n37G. Rollmann, A. Rohrbach, P. Entel, and J. Hafner, Phys. Rev. B\n69, 165107 (2004)\n38Various implementations of the DFT+U method differ in the way\nthey account for the double-counting correction. While we used\nthe Self-interaction correction form of DFT+U, we speculate the\nprevious authors may have used the Around Mean-field form. Our\nchecks with the second method resulted in a higher movalue of\n0:8\u0016B."    },    {        "title": "1606.03231v1.Spin_orbit_coupling_control_of_anisotropy__ground_state_and_frustration_in_5d2_Sr2MgOsO6.pdf",        "content": "Spin -orbit coupling control of anisotropy, ground state and \nfrustration in 5d2 Sr2MgOsO 6 \n \nRyan Morrow,1* Alice E. Taylor ,2* D. J. Singh,3 Jie Xiong,1 Steven Rodan4, A. U. B . Wolter,4 \nSabine Wurmehl ,4,5 Bernd Büchner ,4,5 M. B. Stone,2 A. I. Kolesnikov,6 Adam  A. Aczel,2 A. D. \nChristianson ,2,7 Patrick M. Woodward1 \n \n1 Department of Chemistry and Biochemistry, The Ohio State University, Columbus, Ohio 43210 -1185, USA  \n2 Quantum Condensed Matter Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  \n3Department of Physics and Astronomy, University of Missouri, Columbia, MO 65211 -7010 , USA  \n4Leibniz Institute for Solid State and Materials Research Dresden IFW, D -01171 Dresden, Germany  \n5Institute for Solid State Physics , Technische Universität Dr esden, D -01062 Dresden, Germany  \n6Chemical and Engineering Materials Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  \n7 Department of Physics and Astronomy , The University of Tennessee, Knoxville, TN 37996, USA  \n* Authors have made equal contributions  \n \nAbstract  \nThe influence of spin -orbit coupling (SOC) on the physical properties of the 5 d2 system \nSr2MgOsO 6 is probed via a combination of magnetometry, specific heat measurements, elastic \nand inelastic neutron scattering , and density functio nal theory calculations . Although  a \nsignificant  degree of  frustration is expected, we find that Sr2MgOsO 6 orders in a type I \nantiferromagnetic structure  at the remarkably high temperature of 108 K . The measurements \npresented allow for the first accurate quantification of the size of the magnetic moment in a 5d2 system  of 0.60(2) μB – a significantly reduced moment from the expected value for  such a \nsystem . Furthermore, s ignificant anisotropy is ident ified via a spin excitation gap, and we \nconfirm by first principles calculations tha t SOC not only provides  the magneto crystalline \nanisotropy , but also plays a crucial role in determining both the ground state magnetic order and \nthe size of the loca l moment in this compound.  Through comparison to Sr 2ScOsO 6, it is \ndemonstrated that  SOC -induced anisotropy  has the ability  to relieve frustration in 5 d2 systems \nrelative to their 5 d3 counterparts , providing an explanation of the high TN found in Sr2MgOsO 6. \nIntroduction  \nThere is a great deal of interest in materials with inherent ly frustrated magnetic exchange \ninteractions  and the result ing unusual  magnetic ground states [1, 2].  One class of materials \ncurrently under investigation in this  context is the dou ble perovskite s, formula A2BB′O6, where A \nand B are non -magnetic cations, and B′ is a magnetic 4 d or 5d transition metal cation  [3]. The \nresultin g network of B′ cations forms a quasi -face-centered cubic  (fcc) type lattice  that is highly \nfrustrated as each B′ cation has twelve nearest neighbor B′ cations  [3]. Particularly complex \nmagnetic states can arise in frustrated systems due to the presence of significant spin-orbit \ncoupling (SOC)  that typically accompani es cations of the  4d and 5 d transition metals . It’s \nimperative to understand  the role SOC plays in such materials  due to its role in lifting  the orbital \ndegeneracy and enhancing  multipolar interactions , leading to rich  phase diagrams [ 4-7]. \nIn the case of double perovskites containing  a single magnetic cation with a d3 electronic \nconfiguration, the relatively large S = 3/2 spins are expected to have  a quenched orbital \ncontribution  resulting in classical behavior [ 6]. While experimental results indicate that SOC  is \nallowed in the d3 configuration for 4 d or 5d cations, the reduction in  the magnetic  moment due to SOC is small relative to the effects of covalency, which is large for cations with high oxidation \nstates [ 8, 9]. Double perovskites with 4 d3 or 5d3 ions tend to exhibit frustration indexes (|Θ|/ TN) \nranging from 4 to 14 and adopt a  type I antiferromagnetic structure upon ordering [8, 10-13], \nalthough incommensurate behavior has also  been reported  [14, 15]. \nIn the case of the much smaller S = 1/2 d1 configuration, SOC is expected to play a significant \nrole and antiferromagnetic, ferromagnetic , and quadrupolar order have  all been  predicted as a \nresult [ 5, 16]. These predictions  appear to be in line with measurements  of antiferromagnetic \norder in Ba 2LiOsO 6 and ferromagnetic order in Ba 2NaOsO 6 [17, 18]. However, th ere are \nadditional examples such as spin glass Sr 2MgReO 6 [19] and spin singlet Ba 2YMoO 6 [20, 21] \nwhich do not easily fit into this framework.  Investigation of magnetism due to the d4 \nconfiguration  has also begun , such as  in A2BIrO 6 (A = Sr, Ba; B = Sc, In, Y) , where questions \nhave arisen concerning the strength of SOC and the magnetism of the resulting ground state [ 22 - \n25]. \nIn this work, we investigate the influence of SOC on the magnetic state for the intermediate 5d2 \nS = 1 configuration , by study ing the double perovskite Sr 2MgOsO 6 [26-28]. Theoretical work [ 6] \nhas predicted a rich phase diagram with seven different phases/regions  for the present 5 d2 S = 1  \nscenario , however there has been difficulty in sorting known materials in this context.   \nBa2YReO 6, Sr 2YReO 6, and Ca 2MgOsO 6 appear to be spin glasses, which have not been predicted \n[28-30], La 2LiReO 6 and Sr 2InReO 6 host non-predicted spin singlet state s [29, 30], while  μSR \nexperiments  indicate that  cubic Ba 2CaOsO 6 order s antiferromagnetically , though with moments \ntoo small for detection in the reported neutron scattering experiment [ 31]. By combining \nmagnetization, specific heat  and neutron scattering measure ments  with first principles calculations  for Sr2MgOsO 6, we are ab le to provide insight into the nature of frustration and \ninfluence of SOC in this material . \nSr2MgOsO 6 orders  at 108 K  with a type I antiferromagnetic structure , shown in Figure 1 , a \nresults that was previously predicted in a model considering the influence of SOC [6] . The Os6+ \nmoments of 0.6 0(2) μB are significantly reduced  from the 2 μB spin only value expected for an S \n= 1 ion. Density functional theory ( DFT ) confirms that this substantial reduction in moment \noccurs through a combination of  both covalency and  SOC , and furthermore predicts that SOC -\ninduced anisotropy is essential in the selection of  the magnetic ground state. The presence of this \nanisotropy is experimentally confirmed by t he observation  of a spin gap in the magnetic \nexcitation spectrum via inelastic neutron scattering . Sr2MgOsO 6 is therefore a rare example of a \ncompound where the N éel order, rather than just the anisotropy, is set by  the spin -orbit \ninteraction.  Furthermore, we find that SOC -induced anisotropy  is responsible for the reduced \nmagnetic frustration in Sr 2MgOsO 6 relative to d3 double perovskites , therefore explaining the \nenhanced TN in Sr2MgOsO 6. \nExperimental  \nPowder samples  of Sr 2MgOsO 6 were synthesized by grinding stoichiometric amounts of SrO 2, \nMgO, Os , and OsO 2 together using a mortar and pestle according to the followin g chemical \nequation:    \n2 SrO 2 + MgO  + 1/2 Os + 1/2 OsO 2 → Sr 2MgOsO 6 \nGround mixtures of up to 3  g were contained in high -density alumina tubes and sealed in \nevacuated silica ampoules (approximate volume 40 mL with 3 mm thick walls)  for heatings  of \n48 hours at 1000  °C in a box fur nace located within a fumehood.  This was followed by regrinding and identical reheating  for an additional t wo cycles . Larger sample sizes were \nproduced by synthesizing multiple aliquots which would be ground togethe r in the intermittent \ngrindings and redistributed for the  subsequent  heatings.  Powdered  Sr2MgWO 6 sample s were \nsynthesized in air following the procedure outlined in the literature  [32]. \nThe temperature dependence of the magnetization of Sr 2MgOsO 6 powders w as measured using a \nQuantum Design MPMS SQUID magnetometer.  Data were collected over the temperature range \n2.5 to 400 K under zero -field-cooled (ZFC) and field -cooled conditions (FC) in an applied field \nof 10 kOe.  Powders were contained in gel capsules and mounted in straws for insertion i nto the \ndevice for measurement.  An analogous data set was collected using an empty sample  mount and \nsubtracted from the temperature depend ent magnetization data of Sr 2MgOsO 6 in order to remove \nthe backgr ound response.  \nPowders of Sr 2MgOsO 6 and Sr 2MgWO 6 were cold pressed and sintered at their synthesis \ntemperatures overnight (in an evacuated ampoule for Sr 2MgOsO 6) to p repare polycrystalline \npellets.  The specific heat measurements were conducted on the pelle ts mounted with Apiezon \ngrease using  a Quantum Design PPMS instrument  using a relaxation technique.  \nLaboratory x -ray powder diffraction measurements were conducted at room temperature on a \nBruker D8 Advance equipped with a Ge (111) monochrom ator and a Cu radiation source.  Time \nof flight neutron powder diffraction (NPD) measurements were conducted on Sr 2MgOsO 6 at Oak \nRidge National Laboratory’s (ORNL) Spallation N eutron Source (SNS) on the POWGEN \nbeamline [ 33] using a sample size of 1 .359 g.  Data were collected at 10, 50, and 300 K using the \nPOWGEN Automatic Changer (PAC) environment . Separate data sets with  the bank 2 and bank \n7 chopper settings corresponding  to respective d -spacing ranges of 0.2760 –3.0906 Å and 2.2076 –10.3019 Å  were collect ed at each temperature . Data were analyzed using the Rietveld \nmethod as implemented in the GSAS EXPGUI software package [ 34, 35]. Additional NPD data \nwere collected at High Flux Isotope Reactor (HFIR) facility  at ORNL  on the triple -axis \nspectrometer HB -1A using a sample size  of 11 g. The sample was  sealed under a He atmosphere  \ninto a cylindrical can made of aluminum  with an inner diameter 0.6 cm.  Data were collected at  a \nconstant wavelength of λ=2.37Å using collimation of 40′ -40′-40′-80′. The data were anal yzed \nusing the Rietveld refinement suite FULLPROF [ 36], and the magnetic form factor for Os6+ from \nRef. [ 37] was assumed.  \nInelastic neutron scattering experiments were performed on an 11 g sample of Sr 2MgOsO 6, and \non a 16.5  g sample  of Sr 2ScOsO 6 that was previously examined in Ref.  [38]. Measurements were \nperformed on the SEQUOIA chopper spectrometer at the Spallation Neutron Source (SNS) at \nOak Ridge National Laboratory (ORNL). The samples were sealed in aluminum  cans, and an \nidentical empty Al can wa s measured as a background. A closed -cycle refrigerator was used to \nreach temperatures between 6 K and 125 K, and measurements were performed using  an incident \nneutron energy 20 meV. Empty -can measurements were subtracted from the data sets, which \nwere the n normalized by a factor mf.u./ms, where mf.u. is the formula unit mass and ms is the \nsample mass for each of Sr 2MgOsO 6 and Sr 2ScOsO 6. The presented magnetic scattering intensity \nis therefore per Os ion.  \n \nFirst principles calculations were performed using the generalized gradient approximation \n(GGA) of Perdew, Burke and Ernzerhof (PBE) [ 39] with the general potential linearized \naugmented planewave  (LAPW)  method [ 40] as implemented in the WIEN2k code [ 41]. We used  \nthe experimental  10 K  crystal structure and  highly converged basis sets, including local orbitals with LAPW sphere radii of 2.0 bohr for the metal atoms and 1.55 bohr for O.  We did \ncalculations both with the PBE -GGA itself and with an additional Coulomb repulsion parameter \nU=3 eV in the PBE+U appro ach with the fully localized limit double counting. The key \ndifference between these treatments is that metallic behavior is predicted with PBE while an \ninsulating gap is obtained with U=3 eV, consistent with experimental reports for the resistivity of \npolycrystalline samples [ 28]. We focus on results that do not depend on U.  \n \nResults  \nSr2MgOsO 6 crystallizes in the tetragonal I4/m space group as previously reported [ 27, 28] and \nshown in Figure 1 , which is common to a number of other Sr 2BOsO 6 compositions [ 42-46]. The \nI4/m space group is associated with the a0a0c− Glazer tilt system, where out of phase tilting \noccurs about the c-axis [47]. Rietveld refinement of the x -ray powder diffraction data did not \nindicate any disorder between Mg and Os cations, a typic al result considering the charge \ndifference of 4+ between the anticipated oxidation states of Mg2+ and Os6+ [3]. Refinements also \ndid not indicate any loss of Os  during synthesis  within experimental error . \nThe results of the refinement of neutron powder di ffraction data on Sr2MgOsO 6 from POWGEN \nare given in Table 1, and the refined pattern  at 10 K is given in Figure 2. The average Os −O \nbond length is typical of recent results for octahedrally coordinated Os6+ in the double perovskite \nstructure [ 42, 46, 48]. Both the MgO 6 and OsO 6 octahedra are slightly e longated, with two \nslightly longer and f our slightly shorter M−O bonds.  For the d2 cation Os6+, this is the ex pected \nJahn-Teller distortion. The unit cell is tetragonally  distorted as compared to the cubic cell, with a \nc-axis which is greater than √2 of the a-axis. The tetragonal distortion is enhanced at lower \ntemperatures, with a c/√2a ratio of 1.00 68 at 300 K and 1.0206 at 10 K.  The distortion  manifest s as a combination  of enhanced  tetragonal elongation of the octahedra and octahedral tilting  as \nevidenced by a reduced Mg−O−Os bond angle.  No structural phase transition or change in \nsymmetry occurs within the temperature range studied.  The short Os −O bond lengths compared  \nto the d3 osmates reflect the contraction and increased covalency that can be anticipated as the \noxidation state is increased.  \nThe temperature dependence of the magnetization of Sr 2MgOsO 6 is given in Figure 3a. A clear \ncusp corresponding to an antiferromagnetic transition occurs with a maximum at 108 K in both \nthe FC and ZFC data sets, in approximate agreement with previous reports [ 27, 28], while the \nFisher heat capacity, d( χT)/dT, shown in Figure 3b, indicates  a maximum value at 102 K  –  a \nlower TN from this approach is typical of double perovskites [31]. A divergence of the FC and \nZFC data  at 15 K  is noted  in Figure 3a  which is absent in previous reports [2 7, 28] despite being \npresent in numerous independent samples and measurements by the present authors.  A Curie -\nWeiss fit was conducted in the temperature range 250 to 400 K , which resulted in an effective \nmoment of 1. 88 μB, in close agreement with reported values,  and a Weiss constant of Θ = −269 \nK, in appro ximate agreement with reported values [2 7, 28]. The effective paramagnetic moment \nis substantially reduced from the theoretical spin -only result of 2.83  μB for S = 1, indicating  that \nthe influence of spin -orbit coupling is significant for Os6+. The ratio between the  Weiss constant  \nand ordering temperature TN yields a  relatively low  frustration index, (|Θ|/TN), of 2.5. \nThe specific heat of Sr 2MgOsO 6 is shown in Figure 4, with a clear second -order type anomaly \npositioned at 108 K indicating that the transition is likely due to long -range antiferro magnetic \norder.  An isostructural  nonmagnetic material with similar mass, Sr 2MgWO 6, was measured to \napproximate the nonmagnetic lattice contr ibutions to the specific heat. The solid line, also shown \nin Fig. 4(a), represents the specific heat data of Sr 2MgWO 6 scaled by mass . The difference of these two data sets, plotted as Cmag/T and shown in Figure 4b, corresponds to the magnetic \ncomponent of the specific heat in Sr 2MgOsO 6. Clearly, there is a large peak at the \nantiferromagnetic transition, but there is also a significant tail up to temperatures much higher \nthan TN, indicating persistent  magnetic fluctuations. The magnetic specific  heat Cmag/T is \nintegrated to obtain  the magnetic entropy, plotted against the right axis in Figure 4b. Analysis of \nthe magnetic entropy over the entire temperature range results in Smag ~ 11.2 J/mol K,  which is in  \nbetween the theoretical  values for a simp le L-S scheme for a d2 cation with an expected total spin \nJ = 2 (Smag = 13.38 J/mol K ) and a spin -only scenario with S = 1 (S mag = 9.134 J/mol K). T hese \nresults clearly contrast from a similar analysis recently conducted on  3d8 S = 1 Sr 2NiWO 6 [49], \nwhere a  spin-only analysis  in a nominally orbitally quenched material  resulted in good \nagreement with the experimentally det ermined magnetic entropy.  Therefore, we conclude that \nthe magnetic entropy of Sr 2MgOsO 6 is significantly impacted by the orbital contributi on to the \nmagnetic moment in this compound.  \nIn order to obtain a microscopic insight into the ordered magnetic structure, we examined the \nlow Q region of the neutron powder diffraction data collected on the POWGEN instrument, \nshown as the inset of Figure 2 . However , no apparent magnetic reflections were observed arising \nbelow the ordering temperature. The specific location of the anticipated peaks associated with \nthe common type I antiferromagne tic order are highlighted in the  inset . This is similar to the case \nof 5d2 Ba2CaOsO 6, where NPD data  from the C2 diffractometer at the Canadian Neutron Beam \nCentre at Chalk River National Laboratories  did not yield any observable magnetic  reflections \ndespite evidence of long range magnetic order from muon spin relaxation experiments  [31]. In \nthat study, it was determined that the ordered moments must be less than an estimated detection \nlimit  of 0.7 μ B per Os6+.  In order to search for the presence of weak magnetic reflection s, additional neutron powder \ndiffracti on data was collected on the HB -1A beamline  using a n 11 g sample, nearly 10 times the \nmass measured on POWGEN , and w ith the sample contained in an a luminum  can to minimize \nincoherent scattering.  HB-1A was utilized  for this particular investigation becau se of its \nexcellent signal -to-noise ratio, arising from the combined use of a double -bounce \nmonochromator and an analyzer.  Below TN, two magnetic reflection s previously anticipated due \nto type I antiferromagnetic order were observed , see Figure 5. The diffraction pattern was \nanalyzed using constant structural values as determined from POWGEN , but varying the \nbackgro und and instrument -dependent  parameters, resulting in a  good fit to the data, Figure 5a. \nExtra peaks ar e visible which are due to the a luminum  sample can scattering  and a small, \nunidentified impurity phase is v isible in this sample,  as indicated in  Figure 5, which is present at \nall temperatures . The magnetic structure refinement  yielded Os6+ moments of 0.60(2) μ B, which \nare aligned with in the a-b plane . The resulting value is just below the proposed detection limit \nfrom the case of Ba 2CaOsO 6 [31]. The temperature dependence of the Q(001) = 0.78 Å−1 peak is \nshown in Fig. 5c. A power -law curve was fit to the data to extract TN, and confirms the TN = \n108(2) K transition temperature associated with this magnetic peak.  This is a remarkably high \nNéel temperature for a double perovskite with B = Mg, since the Os6+ ions are on a quasi -fcc \nlattice and are separated by more than 5.5 Å.  \nFor our DFT calculations, we considered different magnetic orders including ferromagnetic,  the \nobserved type I order, and checkerboard antiferromagnetic order in the basal a-b plane of the \ntetragonal cell. We find that  PBE-GGA calculations without SOC  predict an incorrect magnetic \norder, specifically a ferromagnetic ground state.  Only when spin -orbit coupling is included do \nwe obtain the correct type I order as the lowest energy  state. This conclusion is robust, as when spin orbit coupling is included we obtain type I order independent of the moment direction and \nfor both PBE and PBE+U calculatio ns. It follows that Sr2MgOsO 6 is a rare example of a material  \nwhere the  Néel order  itself, rat her than just the anisotropy  is set by the spin -orbit interaction – a \nreflection of the strong SOC.  \nAlso for both PBE and PBE+U calculations with SOC  we find the lowest energy spin direction \nto be along the tetragonal <1 00> direction in the tetragonal I4/m cell. The anisotropy is sizable \nand increases with U.  For the PBE  calculations we find that the <11 0> direction is disfavored by  \n1 meV/Os, while the <001 > direction is disfavored by 5  meV/Os.  The easy axis and pla ne do not \ndepend on U . This  anisotropy and  the symmetry breaking due to the tetragonal lattice no doubt \npartly explain the relatively high ordering temperature  on a dilute fcc-like lattice.  The other \nneeded ingredient in obtaining the ordering is the intersite exchange interaction. The value of th is \ncoupling depends on the choice of U, and as such we cannot directly predict the precise \nmagnitude of this coupling. However, as seen from the energy differences, regardless of the \nchoice of U the correct ground state is predicted, i.e. there is sufficie nt intersite exchange \ncoupling.  We find that the ferromagnetic ordered state is 31 meV above the ground state in the \nPBE+U calculation and 168 meV above the ground state without U.  \nTurning to the moment size, based on integration within the LAPW spheres (2 .0 bohr for Os) we \nobtain moments that are strongly reduced from the nominal values due both to covalency and \nSOC . The total Os moment in the PBE calculation is 0.48 µ B consisting of a spin moment of 0.77 \nµB and an orbital moment of −0.29 µ B. In the proba bly more realistic PBE+U calculation we \nobtain 0.57 µ B, from a spin moment of 1.07 µ B and an orbital moment of −0.50 µ B, in close \nagreement with the experimental result of 0.60(2) μ B from NPD . We also find sizable moments \non the O ions, which is a result of strong covalency.  This can be seen in the density of states, shown for the ground state with PBE+U in Fig. 6. The O 2p bands extend from −7.4 eV to −1.1 \neV relative to the valence band maximum. The region from the bottom to −5.1 eV comprise O 2p \n– Os eg σ bonding states, and one can see very strong Os character in this region reflecting the \ncovalency. The Os t2g states, which are the active orbitals here, extend from −0.5 eV to +1.8 eV \nand are split by U to give a gap of 0.22 eV.  The t2g band width i s similar in the PBE calculation. \nThe Os eg states extend from 4.3 eV to 6.3 eV. The large  crystal field splitting is another \nreflection  of very strong covalency.  \nA consequence of this covalency is the presence of sizable moments on the O sites. In the dou ble \nperovskite structure each O has only one Os nearest neighbor. The O moments are parallel to \nthose of the neighboring Os regardless of the treatment (PBE or PBE+U), the magnetic order or \nthe inclusion of SOC.  While Os6+ d orbitals can be regarded as lar gely inside a 2 bohr LAPW \nsphere, this is not the case for O2− p orbitals with a 1.55 bohr sphere. In order to estimate the O \ncontribution, we turn to the ferromagnetic case. In the PBE+U calculation the total spin moment \nin the unit cell is 1.93 µ B per formula unit, while the Os spin moment is only 1.14 µ B. Thus \n~40% of the spin moment is distributed over the six neighboring O ions. This moment will be \nactive in susceptibility fits, but not in refinements of neutron diffraction data  as the moment is \nspread across neighboring oxide ions and bonds . The O moments provide an explanation for the \nsizable intersite exchange. Although  the double perovskite lattice can be regarded from a Zintl \nperspective as touching (OsO 6)6− anions held tog ether by interstitial Mg2+ and Sr2+ cations , the \nfact that the O atoms on the exterior of these contacting polyanions carry sizable moments \nprovides a mechanism for the intersite exchange.  \nHaving shown that SOC has a significant influence on the moment siz e obs erved by neutron \nscattering, we anticipate that SOC may have a major effect on the magnetic dynamics in Sr2MgOsO 6. Confirmation of this can be found by examining  the inelastic neutron scattering \nspectra of both Sr 2MgOsO 6 and Sr 2ScOsO 6 show n in Fig. 7.  We compare Sr2MgOsO 6 to \nSr2ScOsO 6 because Sr2ScOsO 6 also shows high -TN type I AFM order with TN = 92 K, but has \nOs5+ 5d3 ions which are expected to show significantly reduced SOC due to a S=3/2 state. In \nboth materials we observe scattering emanating from the type I antiferromagnetic wavevector at \nQ ≈ 0.8 Å−1, Fig. 7. We identify the development of a spin gap in both materials at low \ntemperatures —compare Fig.  7(a) to Fig.  7(c) for Sr 2MgOsO 6 and Fig.  7(b) to Fig.  7(d) for \nSr2ScOsO 6. The Sr 2MgOsO 6 spectrum at 6 K is remarkably similar to that of Sr 2ScOsO 6,in \nwhich the gap has been extensively characterized [ 38], see Figs. 7(a) and (b), respectively. This \nsuggests that the physical mechanisms controlling each system are more similar than previo usly \npredicted [6 ], with SOC having influence in both materials .  \nThe similar size of the gaps in Sr2MgOsO 6 and Sr2ScOsO 6 does, however, support  a picture of \nSOC having stronger influence in Sr2MgOsO 6. The microscopic mechanism by which SOC \ntypically produces the spin gap is associated with either exchange anisotropy or single -ion \nanisotropy (or a combination) [ 10, 15, 38, 50]. For either mechanism, for a fixed strength of SOC \nthe magnitude of the gap ob served by neutron scattering scales with the magnetic moment  size. \nTherefore , as Sr2MgOsO 6 has a smaller magnetic moment, the similarity of observed gap to that \nin Sr2ScOsO 6 demonstrates  that SOC is stronger in Sr2MgOsO 6 resulting in a comparable gap . \nDespite the similarity in the spectra at 6 K, above TN the intensity of the observed scattering is \nvery different between  the two  compounds, see Figs. 7(c) and (d). To examine the temperature \ndependence further, we present in Fig 8(a) the integrated intens ity of  the scattering for the range \n0.7 < Q < 1 Å−1 and 3 < E < 12 meV. For both materials the integrated intensity in this region \nincreases with temperature, because  of both the modification of the scattering due to  the closing of the gap and the Bose th ermal population factor . The integrated intensity per Os ion for each of \nthe samples is similar at low temperatures, with a slightly higher value for Sr 2ScOsO 6, as \nexpected for the larger spin system. However, as the magnetic transition temperatures are \napproached the integrated intensities diverge dramatically. This difference in fluctuation \nintensity above TN is indicative of the level of frustration in each system - a strong signal implies \nstrong correlations despite the absence of long range magnetic or der. \nIn Fig.  8(b) we present the temperature dependence of the scattering at very low energies, i.e. \nwithin the gap at low temperatures. The data is converted to χ′′(T) for  the fixed range 0.7 < Q < 1 \nÅ−1 and 3 < E < 5 meV  following the method described in Ref. [ 10], in which the lowest \ntemperature data  set has been subtracted as a background and a Bose factor correction  has been  \napplied. This confirms that the reduction in the scattering at low energies is far beyond what \nwould be expected due to thermal population, thereby confirming the opening of a gap at low \ntemperature.  \nDiscussion  \nA phase diagram depicting seven potential magnetic ground  states , including three potential \nantiferromagnetic configurations,  has been prop osed for double perovskite s with a single \nmagnetic 5 d2 cation [ 6]. Through a combination of reduced paramagnetic effective moment, an \nenhanced magnetic entropy from a spin -only scenario , a substantially reduced moment refined \nfrom neutron diffraction , and a significant spin -excitation gap observed in neutron spectroscopy , \nwe have  unequivocally  shown that SOC plays a  major role in the magnetic behavior of the Os6+ \n5d2 cation in Sr 2MgOsO 6. Despite this, we have shown that the ground state remains in the \n“AFM100” region of the phase diagram predicted in Ref. [6], similar to many 4 d3 and 5 d3 double perovskites [8, 10 -13]. The predicted influence of SOC on the d2 state would inherently infer  \nanisotropic interactions on the Os6+ ions, with  the strong Os −O hy bridization in Sr 2MgOsO 6 \nensuring  that the anisotropy  has significant influence on the collective properties.  We have \nconfirmed  anisotropy  is present  in Sr2MgOsO 6 via observation of the spin gap in the magnetic \nexcitation spectrum, Fig. 7a .  \nThe moment  observed  by neutron diffraction  of 0.60(2) μ B is considerably reduced  (70%)  from \nthe expected high -field spin -only value of 2 μ B. Covalency has been shown to play a significant \nrole in the reduction of  the moment in the case of 4 d3 and 5 d3 transition meta l oxides , resulting in \n37-47% reductions of the 5 d3 Os5+ 3 μ B moment  to 1.6 to 1.9 μ B [8-13], and we expect a similar \neffect in the Os6+ 5d2 case. Here , however,  the magnetization and  specifi c heat analyses  strongly \nsuggests that an orbital contribution is  also important , consistent with the recent x -ray magnetic \ncircular dichroism (XMCD) study of Os6+ in the related material  Ca2CoOsO 6 [48]. Our DFT \nresults confirm that both  SOC and covalency  together  cause the major reduction in the observed \nspin-moment , predicting a 4 7% reduction from covalency and a further 2 5% reduction from \nSOC, consistent with the total 70% reduction we observed experimentally . The result is a \nmoment which is challenging to observe with standard neutron diffraction instrumentation , but \nvia a high-flux, low -background  experiment  we were able to determin e both the ground state and  \nthe moment size - it would be interesting to revisit Ba 2CaOsO 6 on a similar  instrument in order \nto conclusively place the magnetic ground state among those known and predicted.   \n \nFor 5 d2 double perovskites the type I AFM structure was anticipated via  theory including strong \nSOC in Ref. [6]. The predicted influence of SOC on the d2 state would inherently infer  \nanisotrop y, with  the strong Os −O hybridization in Sr 2MgOsO 6 ensuring  that the anisotropy  has significant influence on the collective properties.  Via inelastic neutron scattering w e have  indeed  \nobserved anisotropy  in Sr2MgOsO 6 via the spin gap .   \nIt is interesting to investigate the comparison between 5 d2 and 5 d3 systems. Sr 2MgOsO 6 orders at \na higher temperature than other double perovskites with a single magnetic ion  [28]. While 4 d3 \nand 5 d3 double perovskites like Sr 2ScOsO 6 have larger magnetic moments, which should yield \nstronger interactions and higher ordering temperatures, they also have significantly larger \nfrustration  indices ranging from 4 to 14 [8, 10 -13] in comparison to 2.5 in Sr 2MgOsO 6. The \nsimilari ty of the  QE-space dependence and the intensity of the  excitation spectra of Sr2MgOsO 6 \nand Sr 2ScOsO 6 at 6 K, shown in Fig. 7, indicates that similar interaction mechanisms are \nresponsible for the collective properties in each . However, Fig. 8a shows that the intensity of \nfluctuations in Sr2ScOsO 6 above TN is far greater than the intensity in Sr 2MgOsO 6 above TN. \nThis implies strong correlations persist in Sr2ScOsO 6 in the absence of long range magnetic order \n– a hallmark of fru stration . While the tetragonal symmetry of Sr 2MgOsO 6 may play some role in \nreducing the geometric frustration, it is less distorted than monoclinic Sr 2ScOsO 6, which has \nangles that deviate more substantially  from 180 ° [8].  \nThe question,  therefore , is why is the frustration relieved in Sr 2MgOsO 6 compared to \nSr2ScOsO 6? The strength of hybridization  plays a major role in determining the strength of the \ninteractions, but does not directly explain relief of frustration.  Similarly , the unit cell volume a nd \nB/B′ site disorder  affect interaction strengths, and all these mechanisms likely contribute to \ndifferences in TNs amongst the AFM d2 DPs Sr 2MgOsO 6 (|Θ|/TN = 2.5) [this work] , Ba 2CaOsO 6 \n(|Θ|/TN = 3.1) [51] and Ca 2CaOsO 6 (|Θ|/TN = 3.0) [5 2], but in all of these materials the frustration \nindex is low compared to d3 DP Sr2ScOsO 6. Our DFT results reveal that the SOC induced \nanisotropy leads to selection of the magnetic ground state , as also  found experimentally  for Sr2ScOsO 6 [38], therefore strong er anisotropy promotes a more robust ground state. Given the \nobservation of the large spin gap in Sr 2MgOsO 6 and the multiple sources of evidence we have \npresented for  stronger SOC  in this d2 material, we conclude that SOC induced anisotropy is the \ndominant f actor in  the relieved frustration in Sr 2MgOsO 6.  \nThis conclusion is supported by an earlier theoretical prediction that TN should vary with the \nstrengt h of anisotropy in such systems, due to reduced competition between possible ground \nstates  [53]. The anis otropy term is proportional to the gap size (probed by INS) divided by the \nmoment size (probed by neutron diffraction) . Given that the gap is of the same magnitude in \nSr2MgOsO 6 and Sr 2ScOsO 6, Fig. 7, but the moment is less than half the size in  Sr2MgOsO 6, we \nconclude that the anisotropy is indeed much larger in Sr 2MgOsO 6. Therefore, t he results  \npresented demonstrate the relief of frustration via SOC induced anisotropy, and represent  the \nexperimental demonstration of the evolution of TN with SOC.  \nConclusion  \nThe d ouble perovskite osmate Sr 2MgOsO 6 has been synthesized and characterized by \nmagnetometry, specific heat measurements,  and elastic and inelastic  neutron scattering . The \ncombined results demonstrate  that spin -orbit coupling is essential to describ e the magnetic \nproperties of the system . The Os6+ moments order antiferromagnetically at 108 K in a type I \nconfiguration on the Os fcc sublattice as theoretically predicted [6].  For the first time in such a \n5d2 material, r efinements of neutron powder diffraction data yields a moment of 0.60(2) μ B per \nOs cation which is significantly reduced due to the combined effects of SOC and covalency.   \nThrough comparison of inelastic ne utron spectra, it is shown that  SOC  induced anisot ropy has the ability to relieve frustration in d2 systems , relative to analogous d3 materials which have \nsystematically higher frustration indices , promoting high magnetic transition temperatures . \n \nAuthor Information  \nCorresponding Author  \nr.c.morrow@ifw -dresden.de  \nNotes  \nThe authors declare no competing financial interest.  \nAuthor Contributions  \nRM synthesized  all samples as well as  conducted and analyzed  XRD and  POWGEN NPD data .  \nRM and JX collected and analyzed magnetometry data.  SR , AUBW, SW, and BB were \nresponsible for the specific heat data collection and analysis.  AET and AAA conducted the HB -\n1A NPD experiment.  AET, MBS, and AIK measured the SEQUOIA INS data .  AET analyzed \nthe results from the HB -1A and SEQUOIA experiments.   ADC supe rvised neutron scattering \nactivities, and PMW supervised synthesis and property characterization activities.  RM and AET \nled the manuscript preparation, and all coauthors contributed.  \n \nAcknowledgements  \nSupport for this research was provided by the Center for Emergent Materials an NSF Materials \nResearch Science and Engineering Center (DMR -1420451 ), and in the framework of  the \nmaterials world network (Deutsche Forschungsgemein schaft DFG  project no. WU595/5 -1 and \nNational Science Foundation  (DMR -1107637 )). S. Wurmehl gratefully acknowledges funding by \nDFG  in project WU 595/3 -3 (Emmy Noether program)  and by DFG in SFB 1143. A portion of this research was carried out at Oak Ridge National Laboratory's Spallation Neutron Source  and \nHigh Flux Isotope Reactor , which is sponsored by the U.S. Department of Energy, Office of \nBasic Energy Sciences.  Work at the University of Missouri (DJS) was funded through the \nDepartment of Energy S3TEC Energy Frontier  Research Center, award  DE-SC0001299/DE -\nFG02 -09ER46577.  The authors would like to acknowledge S. Calder and M. D. Lumsden for \nhelpful discussions, and t he authors  also thankfully acknowledge Ashfia Huq for experimental \nassistance with POWGEN data collection.   \nThis manuscript has been authored by UT -Battelle, LLC under Contract No. DE -AC05 -\n00OR22725 with the U.S. Department of Energy. 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Solid State Chem.  179, 605 -\n612 (2006).  \n52) Feng,  H. L. et al. High -pressure crystal growth and electromagnetic properties of 5d double -\nperovskite Ca 3OsO 6. J. Solid State Chem.  201, 186 -190 (2013).  \n53) Kuz’min,  E. V.,  Ovchinnikov,  S. G. &  Singh,  D. J. Effect of frustrations on magnetism in the \nRu double perovskite Sr 2YRuO 6. Phys. Rev. B  68, 024409 (2003).  \n   \n \n \nFIG. 1 (color online).  The crystal and magnetic structure of Sr 2MgOsO 6 with Mg and Os shown \nas grey and red spheres located within octahedra of the same color with O ions positioned at the \ncorners.  Sr cations are omitted for clarity.  The magnetic moment on Os6+ is shown as a black \narrow, and while shown along the b axis, is  known only to be in the a-b plane.  \n \n \n Temperature (K)  10 50 300 \nSpace Group   I4/m  I4/m  I4/m \na (Å) 5.52776(8)  5.52889(7)  5.56345(6)  \nc (Å) 7.9781(1)  7.9755(1)  7.9217(1)  \nV (Å)3 243.781(9)  243.798(8)  245.193(7)  \nRwp 3.25%  3.43%  3.47%  \nRp 2.00%  2.30%  2.22%  \n        \nMg−O1 (×4, Å)  2.037(1)  2.039(1)  2.041(1)  \nMg−O2 (×2, Å)  2.060(2)  2.059(2)  2.047(2)  \nOs−O1 (× 4, Å) 1.904(1)  1.903(1)  1.911(1)  \nOs−O2 (×2, Å)  1.929(2)  1.928(2)  1.914(2)  \n∠Mg−O1−Os (°)  165.32(4)  165.33(4)  169.01(6)  \n        \nO1 x 0.2264(2)  0.2266(2)  0.2343(2)  \nO1 y 0.2907(2)  0.2909(2)  0.2824(2)  \nO2 z 0.2582(2)  0.2582(2)  0.2584(2)  \n        \nSr U iso 0.0023(1)  0.0025(1)  0.0079(1)  \nMg = Os U iso 0.0013(1)  0.0014(1)  0.0036(1)  \nO1 U eq 0.0043(1)  0.0044(1)  0.0098(1)  \nO2 U eq 0.0043(1)  0.0046(2)  0.0115(2)  \nTABLE 1: Neutron powder diffraction parameters obtained from Rietveld refinement for \nSr2MgOsO 6 at 10, 50 and 300 K.  Mg and Os U iso values were constrained to be equal.  U eq is \ncalculated as a third of the trace of the tensor for the anisotropically refined oxygen ion U’s.  \n \n \nFIG. 2 (color online).  Refined neutron powder diffraction pattern of Sr 2MgOsO 6 at 10 K.  Black \nsymbols, red curve s, and blue curve s correspond to  the observed data, the calculated pattern s, \nand the difference  curve , respectively .  The black hashes correspond to the nuclear peak \npositions of the compound.   The inset shows low Q neutron powder diffraction data on \nSr2MgOsO 6 from POWGEN at 300, 50, and 10 K.  Arrows indicate the position of the potential  \nmagnetic reflections associated with Type I AFM order.  \n \n  \nFIG. 3 (color online).  a) The temperature dependence the field -cooled (red  filled circles ) and \nzero-field-cooled (blue  open circles ) magnetization  under an applied field  of Sr 2MgOsO 6 plotted \nagainst the left axis.  The inverse data (black  circles ) is plotted against the right axis with a red \nline indicating the higher temperature Curie -Weiss fitting.   b) The Fisher heat capacity of \nSr2MgOsO 6 derived from the data in panel a).  \n \n \n \n \nFIG. 4 (color online).  (a) T he temperature dependence of the  specific  heat of Sr 2MgOsO 6 (blue) \nand diamagnetic Sr 2MgWO 6 (red) which is used to approximate the lattice specific heat of \nS2MgOsO 6.  (b)  The magnetic specific heat (black) of Sr 2MgOsO 6 taken by subtracting the \nscaled specific heat of Sr 2MgWO 6 and magnetic entropy (red) obtained by integration . \n \n \nFIG. 5 (color online).  a) Neutron powder diffraction data collected on Sr 2MgOsO 6 using the HB -\n1A beamline at 125 K. b) A close view of the low Q data at 125 K (red open circles) and 6 K \n(blue filled circles)  with a Rietveld fitting to the 6 K data (green line) as described in the text. c) \nThe temperature dependence of the Q(001) = 0.78 Å-1 magnetic reflection with a power -law \nfitting to the data as described in the text . \n \n \nFIG. 6 (color online) PBE+U (U=3 eV) density of states and projection onto Os for the ground \nstate antiferromagnetic order, including SOC . \n \n \n \nFIG. 7 (color online).  Neutron scattering intensity maps, measured with Ei = 20 meV, showing \nthe gap below TN for (a) Sr 2MgOsO 6 and (b) Sr 2ScOsO 6 which closes above TN in (c) and (d) \nrespectively.  \n \n \nFIG. 8 (color online).  (a) Comparison of the evolution of normalized per Os integrated neutron \nscattering intensity of Sr 2MgOsO 6 and Sr 2ScOsO 6 and (b) the temperature dependence of the \nscattering within the gap of Sr 2MgOsO 6, converted to χ′′ as in Ref. [10 ]. \n \n \n"    },    {        "title": "1607.01919v2.Giant_perpendicular_magnetic_anisotropy_energies_in_CoPt_thin_films__Impact_of_reduced_dimensionality_and_imperfections.pdf",        "content": "Giant perpendicular magnetic anisotropy energies in CoPt thin \flms: Impact of reduced\ndimensionality and imperfections\nSamy Brahimi1,\u0003Hamid Bouzar1, and Samir Lounis2\n1Laboratoire de Physique et Chimie Quantique, Universit\u0013 e Mouloud Mammeri, Tizi-Ouzou, 15000 Tizi-Ouzou, Algeria and\n2Peter Gr unberg Institut and Institute for Advance Simulation,\nForschungszentrum J ulich, 52425 J ulich & JARA, Germany\nThe impact of reduced dimensionality on the magnetic properties of the tetragonal L10CoPt\nalloy is investigated from ab-initio considering several kinds of surface defects. By exploring the\ndependence of the magnetocrystalline anisotropy energy (MAE) on the thickness of CoPt thin \flms,\nwe demonstrate the crucial role of the chemical nature of the surface. For instance, Pt-terminated\nthin \flms exhibit huge MAEs which can be 1000% larger than those of Co-terminated \flms. Besides\nthe perfect thin \flms, we scrutinize the e\u000bect of defective surfaces such as stacking faults or anti-sites\non the surface layers. Both types of defects reduce considerably the MAE with respect to the one\nobtained for Pt-terminated thin \flms. A detailed analysis of the electronic structure of the thin\n\flms is provided with a careful comparison to the CoPt bulk case. The behavior of the MAEs is\nthen related to the location of the di\u000berent virtual bound states utilizing second order perturbation\ntheory.\nI. INTRODUCTION\nThe magnetocrystalline anisotropy energy (MAE) is\nat the heart of magnetic properties of materials. It is\nof crucial importance from the fundamental or techno-\nlogical point of views since it provides an energy scale\nfor the stability of magnetic domains where for example\nmagnetic information is stored. When the MAE is large\nand favors an out-of-plane orientation of the magnetic\nmoments, perpendicular magnetic recording or magneto-\noptical recording is possible (see e.g. Refs1,2). CoPt bi-\nnary bulk alloy in the L10structure (see Fig.1) is by now\na classical example of a material exhibiting a large per-\npendicular MAE, around 1 meV4{6,8,9. There has been\na tremendous number of studies related to the magnetic\nproperties of this alloy in its bulk phase, as nanoparti-\ncles or in nanostructures combining Co and Pt (see e.g.\nRefs.1,10{22).\nFIG. 1. The conventional cell of the CoPt L10alloy. The\nprimitive cell is also sketched using dashed lines.\nA large amount of work has been devoted to unveil the\norigin of the large perpendicular MAE in binary bulkalloys, see e.g. Refs23{30. The interplay of the tetrago-\nnality of the alloy, band \flling, hybridization between the\nconstituents a\u000bect certainly the magnitude of the MAE.\nFor instance, tetragonality leads to the lifting of the de-\ngeneracy of the electrons by the tetragonal crystal \feld\nand produces thereby an additional contribution to the\nMAE. Thus, and as expected from perturbation theory,\nthe MAE becomes proportional to \u00182instead of\u00184as\nfound for cubic symmetry, where \u0018is the spin-orbit cou-\npling constant. Indeed, in cubic bulk systems, the high\nsymmetry allows only for a fourth-order anisotropy con-\nstant, and thus they are characterized by a small MAE.\nRazee et al.27argued however that the tetragonal dis-\ntortion of CoPt, given by the axial ration c/a = 0.98,\ncontributes by only 15% of the MAE. It was then con-\ncluded that the compositional order of the alloy is an\nimportant ingredient for a large MAE.\nSakuma24shows that by changing the axial ratio (c/a)\nde\fning the tetragonality of CoPt and FePt alloys, the\nMAE \frst smoothly decreases by increasing c/a till reach-\ning a minimum at \u00180.8 before a smooth increase in mag-\nnitude. Interestingly, except a small window of axial ra-\ntios (0:7< c=a < 0:9), the MAE favors an out-of-plane\norientation of the magnetic moments. The tetragonaliza-\ntion is then thought to provide an e\u000bect similar to the\nband \flling23,24.\nIn the context of thin \flms, Zhang et al.41demon-\nstrated with ab-initio simulations that for CoPt \flms ter-\nminated by Co layers, a thickness of at least 9 monolayers\nexhibit a rather converged MAE, with a bulk contribu-\ntion of 1.36 meV favoring a perpendicular orientation of\nthe magnetic moments and a counter-acting surface con-\ntribution of -0.76 meV favoring, interestingly, an in-plane\norientation of the moments. Their interest in CoPt was\nmotivated by the experimental demonstration of coerciv-\nity manipulation of L10FePt and FePd thin \flms42by\nexternal electric \feld. Their ab-initio simulations pre-\ndicted a higher sensitivity of CoPt to electric \feld than\nthat of FePt \flms. Pustogowa et al.43investigated fromarXiv:1607.01919v2  [cond-mat.mtrl-sci]  12 Oct 20162\n\frst-principles several components made of Co and Pt\ndeposited on Pt(100) and Pt(111) surfaces. They found\nthat ordered superstructures of (CoPt) ndeposited on\nboth mentioned substrates are characterized by a per-\npendicular MAE, which is heavily a\u000bected by chemical\ndisorder in line with the analysis of Razee et al.27.\nThe goal of this manuscript is to present a systematic\nab-initio investigation on the e\u000bect of reduced dimen-\nsionality on the magnetic properties of CoPt(001) \flms\nwith a focus on their MAE and by addressing the impact\nof the termination type of the \flms. Contrary to previ-\nous investigations41, we consider not only Co-terminated\n\flms but also Pt-terminated \flms and several types of\nsurface defects (see Fig.2). For instance, we found that\ndecreasing the thickness of the \flms leads to a sign change\nof the surface MAE. Pt covered thin \flms can boost\nthe total perpendicular MAE by a large amount stabi-\nlizing, thereby, more strongly the out-of-plane orienta-\ntion of the moments. Molecular dynamics simulations\ndemonstrated the likeliness of having Pt on the surface\nof CoPt alloy31,32and thus the pertinence of our predic-\ntions. After a careful study of di\u000berent defective termi-\nnations types (stacking faults, anti-site defects), we pro-\nvide the ingredient to increase the MAE of the thin \flms.\nIf we label the Co and Pt layers by respectively A and\nB, the perfect stacking along the [001] direction is given\nfor example by ABABAB for 6 layers. Possible stacking\nfaults, which are planar defects, could be the sequence\nABABAA (see Figs.2(c-d)). Anti-site defects on the sur-\nface means that instead of having at the surface a pure\nlayer A, or layer B, we have an alloy, for example, made\nof A and B. In our work, we considered an alloy of the\ntype A 0:25B0:75in the surface layer instead of the perfect\nB layer of our example (see Figs.2(e-f)).\nII. METHOD\nWe simulate the thin \flms by adopting the slab ap-\nproach with periodic boundary conditions in two direc-\ntions while the periodic images in the third direction\nare separated by a su\u000ecient amount of vacuum (15 \u0017A )\nto avoid interaction between neighboring supercells. We\nhave chosen to use symmetrical calculation cells with an\nodd number of planes to avoid the pulay stress. Some\nrepresentative slabs are shown in Fig.2. Here it can be\nobserved that for equiatomic L10type of alloys two dif-\nferent surfaces exist when the slabs are stacked along\nthe [001] direction. In the perfect cases, the surface ter-\nmination can be made of either purely Co atoms or Pt\natoms. The self-consistent calculations are carried out\nwith the Vienna ab initio simulation package (VASP) us-\ning a plane wave basis and the projector augmented wave\n(PAW) approach33,34. The exchange-correlation poten-\ntial is used in the functional from of Perdew, Burke and\nErnzerhof (PBE)35,36. The cut-o\u000b energies for the plane\nwaves is 478 eV. The integration over the Brillouin zone\nwas based on \fnite temperature smearing (Methfessel-Paxton method) for the thin \flms while for the bulk case\nthe tetrahedron method with Bl ochl corrections has been\nused. The k-points grids are 14 \u000214\u000211 for the bulk\ncalculation, and 14 \u000214\u00021 for the (001) surface calcu-\nlations. The energy convergence criterion is set to 10\u00008\neV while the geometrical atomic relaxations for the sur-\nfaces calculations were stopped when the forces were less\nthan 0.01 eV/ \u0017A. The MAE is extracted from the di\u000ber-\nence between the total energies of the two con\fgurations:\nout-of-plane versus in-plane orientations of the magnetic\nmoments. A positive value indicates a preference for the\nout-of-plane orientation of the magnetic moments.\nIII. RESULTS\nA. Bulk CoPt\nTo start our investigations, we revisited the bulk alloy\nphase by \fnding the tetragonal lattice structure mini-\nmizing its energy. The optimal value of c/a ratio is equal\nto 0.984 with a lattice parameter value a of 3.80 \u0017A, in\ngood agreement with values available in the literature\n(see e.g.19,28,29,37). The calculated magnetic moments\n(MCo= 1.91\u0016B,MPt= 0.40\u0016B) are also in line with\nprevious works28,29,38{40. We also note the well-known\nemergence of an induced moment in Pt, which is due\nto the hybridization of its 5d orbitals with the exchange\nsplitted 3d orbitals of Co. In order to calculate the MAE,\nwe considered two possible directions for the in-plane mo-\nments orientations, [110] and [100], and we found a negli-\ngible di\u000berence in the obtained perpendicular MAE. For\nthe optimized structure, the MAE reaches a value of 0.91\nmeV when the in-plane orientation of moments is along\n[100] and 0.97 meV for [110] as an in-plane orientation of\nthe moments. Both values are close to the experimental\nvalues that are given around 1 meV3,7.\nIn Fig.3, we plot the bulk MAE as function of the\nratio c/a under the constant unit cell volume in a simi-\nlar fashion then that of Sakuma24. The obtained curve\nagrees well with the one published in the latter article.\nFor ratios between 0.6 and 1.2, the MAE experiences one\nminimum and two maxima. The largest in-plane MAE is\nfound for a ratio of 0.8. As expected, if c/a = 1 =p\n2, i.e.\nc/at= 1, the MAE drops to zero since this corresponds\nto a primitive cell of the cubic B2 structure. As discussed\nby Sakuma, upon tetragonalization, the electronic states\nof the atoms are shifted and the band \flling changes,\nwhich a\u000bect the magnitude and sign of the MAE.\nB. Perfect surfaces\nAs mentioned earlier, we considered di\u000berent \flm\nthicknesses. We start analyzing the results obtained for\nperfect Co-terminated thin \flms. A representative of one\nof the simulated \flms is shown in Fig. 2(a). We plot the\nMAE versus the \flm thickness in a fashion similar to3\nFIG. 2. Supercells used for the simulation of the (001) CoPt thin \flms where the blue and magenta spheres correspond\nrespectively to the Co and Pt atoms: (a) Pure Co surface, (b) pure Pt surface, (c) Co stacking fault, (d) Pt stacking fault, (e)\nPt anti-site and (f) Co anti-site. In the latter two cases, numbers 1 and 2 refer to atoms with di\u000berent magnetic moments. For\neach case, the number of (CoPt) sequences, X, is given.\nLayer Atom M (\u0016B)\nS Co 2.00\nS-1 Pt 0.40\nS-2 Co 1.90\nS-3 Pt 0.40\nCenter Co 1.90\u000ed/d0(%)\n-4.30\n1.40\n-0.80\n-0.05Layer Atom M (\u0016B)\nS Pt 0.42\nS-1 Co 1.99\nS-2 Pt 0.40\nS-3 Co 1.90\nCenter Pt 0.40\u000ed/d0(%)\n-5.60\n1.60\n-0.50\n-0.05\nTABLE I. Magnetic pro\fle and geometrical relaxations of 9 layers think thin \flm of CoPt terminated by Co (a) and by Pt (b).\nS labels the outermost surface layer and \u000ed=d-d0describes the changes of the interlayer distance dwith respect to that of\nthe bulkd0that is equal to 1.871 \u0017A.\nthat of Zhang et al.41, i.e. considering along the x-axis\nthe number of (CoPt) sequences, X(see Fig. 4(a)). A\nsingle CoPt sequence is shown in Fig. 2. Thus, for the\nCo-terminated 9 layers-thick thin \flm, X= 4. We notice\nthat the MAEs of the Co-terminated thin \flms are char-\nacterized by an oscillating behavior, which is induced by\ncon\fnement e\u000bects. Indeed and as indicated by Zhang\net al.41, quantum well states can occur because of con-\n\fnement, which can impact on the electronic structure ofthe thin \flms and thus on the related MAE. For X= 1,\nthe moments prefer interestingly an in-plane orientation\ncontrary to thicker \flms.\nAs done for the bulk, two possible in-plane orientations\nof the magnetic moments are assumed: [100] and [110].\nThey are quasi-equivalent with a slight preference for the\n[100] direction. Thus, in the rest of our analysis we focus\non the latter direction.\nThe MAE can be decomposed, as usually done, into a4\nFIG. 3. MAE of the bulk L10CoPt alloy as function of the\naxial ratio c/a under constant volume. Two possible in-plane\norientation of the magnetic moments are considered, [100] and\n[110], but the obtained MAE are very similar. The closed\ncircle represents the experimental value3, which is well repro-\nduced by our simulations. Other experimental values can be\n50% larger, see e.g. Ref.7.\nbulk contribution, Kb, and a surface contribution, Ks:\nMAE =X\u0002Kb+ 2\u0002Ks: (1)\nWhile this decomposition is reasonable for thick \flms,\nit is questionable for the very thin \flms considered in\nour work. The value of Ksextracted from the previous\nformula is then meant to indicate the impact of low di-\nmensionality on the total MAE. One sees in Fig. 4(b),\nthat forX < 4 the surface interestingly contributes with\na negative value to the total MAE and counteracts the\n\\bulk\" contribution.\nIn order to provide a common reference to compare all\nkind of considered \flm terminations, we extend the con-\ncept used for the Co-terminated \flms and plot the MAEs\nas function of the (CoPt) sequences. A 9-layers thick Pt-\nterminated \flm is then characterized by X= 3. In Fig.4,\nit would then be compared to the Co-terminated \flm con-\ntaining 7 atomic layers. This comparison indicates there-\nfore the impact of Pt coverage of the Co-terminated \flm.\nA general observation deduced from Fig. 4 is the large\nvalue of the MAEs after Pt coverage, in some cases 1000%\nlarger than the one obtained for the Co-terminated \flms.\nInterestingly, the surface contribution to the total MAE\nfavors a perpendicular orientation of the moments in the\ncase of Pt-covered \flms contrary to the Co-terminated\n\flms with thicknesses below X= 4.\nWe believe that a further thickness increase of the \flms\nwould not modify the surface contributions to the total\nMAE. This was already seen by Zhang et al.41for the\nCo-terminated case, and we expect the same to occur for\nthe Pt-terminated case (see Fig. 4). It is thus interest-ing to analyze the electronic and magnetic structure of\nthe 9-layers thin \flm, which would be representative of a\nsurface of CoPt alloy. In Table I, we provide for respec-\ntively Co- and Pt-terminated \flm, the atomic magnetic\nmoments and the ratio of the change in the interlayer\ndistance \u0001dwith respect to the unrelaxed interlayer dis-\ntanced0. As expected, the surface magnetic moments are\nslightly enhanced in comparison to those characterizing\nthe bulk phase. Moreover, for the considered thickness,\nthe bulk moments are recovered in the middle of the \flms.\nIn Fig.5, we show the atom-projected electronic densities\nof states (DOS) for the L10bulk phase and for the sur-\nfaces with di\u000berent types of terminations. As expected,\nthe Pt band is in general larger than the Co band. In\nFig.5(b), corresponding to Co-terminated thin \flm with\nX= 4, we observe a shrinking of the electronic band\nwidth of the outermost Co layer, labeled Co (S), because\nof the hybridization lowering due to the reduced coordi-\nnation, which explains the increase of the corresponding\nmagnetic moment. However, for the other planes, we\nrecover basically the bulk electronic properties. A simi-\nlar behavior is obtained for Pt-termination as shown in\nFig.5(c). On the Pt surface layer, labeled Pt (S), the mo-\nment increased negligibly with respect to the bulk value\nbut the corresponding DOS around the Fermi energy is\nrather di\u000berent from the bulk counter-part.\nBefore analyzing the mechanisms and origins behind\nthe increase of the MAE after Pt-coverage, we address in\nthe next subsections two possible surface imperfections:\nstacking fault defects and alloying at the surface.\nC. Stacking faults\nIn this section, we consider 9-layers thick thin \flms\nwith a stacking fault by substituting the last atomic layer\nof the perfect surfaces by the other type of atomic layer.\nThus, the two possible types of stacking faults would lead\nto a termination made of Pt (Pt-stacking fault) or Co\n(Co-stacking fault) for the last two surface layers (see\nFigs.2(c-d)). After geometrical relaxation, the interlayer\ndistance at the surface of the Pt-stacking fault increases\nby +12.5 %, i.e. outward relaxation, when compared\nto the bulk value, unlike the Co-stacking fault, where\nthe relaxation of the surface layer is strongly inward and\nreaches - 25% (see Table II). This is undoubtedly due to\nthe fact that Pt atoms are heavier and larger compared\nto Co atoms.\nIn order to compare the MAE obtained in this case\nto those extracted in the perfect thin \flms, we consider\nagain the same common reference, i.e. the number of\nbulk (CoPt) sequences. The studied Co-stacking fault\nis then characterized by X= 3 (black square in Fig.4)\nwhile for the Pt-stacking fault X= 2 (magenta square\nin Figs.4). One sees that Co-stacking fault leads to a\ntotal MAE and a surface contribution, Ks, rather similar\nto the one found for a perfect thin \flm terminated by\nCo. This suggests once more the importance of having5\nFIG. 4. MAE of CoPt thin \flms as function of X, the number of (CoPt) sequences. In contrast to (a), where the MAE of total\nthin \flms is plotted, in (b) the surface contribution is depicted. Several cases are considered: Co-terminated thin \flms (red\ntriangles), Pt-terminated thin \flms (green triangles), stacking faults defects (Co with a black square and Pt with a magenta\nsquare), anti-sites (Co with a blue circle and Pt with a green circle). The diamonds represent the data of Zhang et al.41obtained\nfor Co-terminated thin \flms considering the MAE with respect to the direction [110]. For completeness, we consider both type\nof possible in-plane orientation of the moments, along the [110] shown with open symbols and along the [100] direction with\n\flled symbols.\nLayer Atom M (\u0016B)\nS Co 1.88\nS-1 Co 1.78\nS-2 Pt 0.35\nS-3 Co 1.92\nCenter Pt 0.40\u000ed/d0(%)\n-24.80\n2.90\n-2.00\n-0.05Layer Atom M (\u0016B)\nS Pt 0.12\nS-1 Pt 0.33\nS-2 Co 1.96\nS-3 Pt 0.39\nCenter Co 1.93\u000ed/d0(%)\n12.50\n-3.50\n0.70\n-0.5\nTABLE II. Magnetic pro\fle and geometrical relaxations of 9 layers thick thin \flms of CoPt are shown in the left table for the\nCo-stacking fault and in the right table for the Pt-stacking fault.\nPt on the surface in order to enhance the MAE. The\nMAE related to the Pt-stacking fault is larger than the\none of a pure 5-layers \flm terminated by Co but smaller\nthan the one of Pt-terminated \flm. This indicates that\nincreasing the thickness of Pt covering the CoPt \flms\nis not necessarily increasing the total MAE. Here the\nsurface contribution to the MAE is positive and rather\nlarge contrary to the case of a Co-stacking fault.\nAs shown in Table II (left table), the surface magnetic\nmoments in case of Co stacking fault did not increase\non the surface unlike the perfect thin \flms. Here, the\ne\u000bect of coordination lowering that increases the magni-\ntude of the moment is compensated by the e\u000bect of the\nlarge inward relaxations that favors hybridization of the\nelectronic states and therefore decrease the magnitude of\nthe moment. For the case of Pt stacking fault, the results\nare shown in TableII (right table). Here, the magnetic\nmoment of Pt at the outermost surface layer decreases\neven compared to the bulk one. This is the signature of\nthe induced nature of the Pt moment. The closest Co\nlayer to the Pt surface layer is two interlayer distancesaway and certainly the outward surface relaxation is not\na helping factor.\nIn Figs. 5(d-e), we show the DOS for these surfaces.\nIn Fig. 5 (d) corresponding to a Co stacking fault, we\nnote that for the Co layer underneath the surface, la-\nbeled Co (S-1), the width of the density of states curve\nis slightly larger than that of the bulk. Also, one notices\nthat the majority-spin DOS is less occupied for Co (S-1)\nthan those of the rest of Co layers. All of that leads to a\ndecrease in the magnetic moment of the S-1 layer. The\nmoment at the surface is also not that large compared\nto the bulk value since the interlayer hybridization of the\nelectronic states is rather strong in this particular case.\nIndeed, the surface layer experiences a large inwards re-\nlaxation induced by the fact that the bulk lattice param-\neter of Co is much smaller than that of CoPt alloy. We\nalso note that the electronic states of the Pt layers do\nnot change dramatically from those of the bulk phase.\nThis is certainly not the case in the thin \flm with Pt\nstacking fault. The Pt surface atom is characterized by a\nnarrower d-band with a larger DOS than that of the bulk6\nFIG. 5. Spin-dependent atom-projected electronic densities of states of CoPt L10in the bulk phase (a), for Co (b) and Pt (c)\nterminated thin \flms, and \flms with Co (d) and Pt (e) stacking faults. S labels the top surface layer of the thin \flms.\nas shown in Fig. 5 5(e). This is induced by the reduced\ncoordination at the surface. However, the exchange split-\nting between the majority{ and minority{spin bands is\nsmaller than that of the bulk. This indicates once more\nthe induced nature of the Pt magnetic moment since the\nclosest neighboring Co layer to the Pt surface layer is a\nsecond nearest neighbor.\nD. Anti-site defects\nTo realize anti-site defects on the surface of the 9-layers\nthick CoPt thin \flm, the perfect surface layer was re-\nplaced by an alloy of Co xPt1\u0000x, withx= 1=4 if the\noriginally perfect \flm is terminated by Pt or x= 3=4 for\nthe perfect Co-terminated \flm. In this case, every layer\ncontains 4 atoms in the unitcell as depicted in Figs.2(e-f)\nand the \flms are fully relaxed. Similarly to the previ-\nously studied \flms, here we encounter two possibilities:\neither the sub-surface layer (S-1) is of Pt type or of Cotype. The former corresponds then to a Pt anti-site de-\nfect while the latter is a Co anti-site defect.\nThe anti-site defects have a dramatic impact on the\nMAE of the thin \flms as depicted in Fig.4. Obviously\nthe surface contribution favors strongly the in-plane ori-\nentation of the magnetic moments. This is the largest\nin-plane contribution to the MAE found for all investi-\ngated \flms.\nIn the case of Pt anti-site, the Pt defect is repelled from\nthe ideal surface position (+6.69 %)) in line with the re-\nsult found for Pt stacking fault. Pt is a large atom and\nrequires more space, which explains this type of geomet-\nrical relaxation. Note that the Co atoms surrounding the\nPt anti-site relax towards the surface (-4.01 %, -1.66 %),\nwhich leads to a sort of surface roughness with large devi-\nations in the surface atom positions. Unlike Pt anti-site,\nCo anti-site literally sinks (-14.33 %) similarly to the sur-\nrounding neighboring Pt atoms (-11.44 %, -8.13 %). The\nrelaxations trends are rather similar to those obtained for\nthe stacking faults. Table III collects the calculated mag-7\nLayer Atom M (\u0016B)\nS Pt, Co1, Co2 0.33, 2.02, 2.04\nS-1 Pt 0.39\nS-2 Co 1.93\nS-3 Pt 0.40\nCenter Co 1.93Layer Atom M (\u0016B)\nS Co, Pt1, Pt2 2.00, 0.38, 0.42\nS-1 Co 1.95\nS-2 Pt 0.39\nS-3 Co 1.92\nCenter Pt 0.40\nTABLE III. Magnetic pro\fle of thin \flms considering anti-sites at the surface layer. The results related to the case of Pt\nanti-site, wherein 1/4 of Co surface atoms is replaced by Pt, is shown in the left table and the case of Co anti-site is shown in\nthe right table.\nnetic moments. Interestingly, the in-plane reconstruction\nwas e\u000bective for only the subsurface layers in order to ac-\ncommodate the large surface relaxations. In the case of\nthe thin \flm with Pt anti-site, a slight dilatation of 0.74\n% is found in contrast to the thin \flm with Co anti-site\nwhere a contraction is noticed (-4.43 %).\nE. Discussion\nHere we discuss the general trend of the MAE for the\ndi\u000berent investigated systems. The details of the elec-\ntronic structure de\fnitely impact on the SOC related\nproperties. Our aim is to simplify this complicated pic-\nture and grasp the main ingredients needed to a\u000bect\nthe MAE. First we recall that within the framework of\nsecond-order perturbation theory44:\nMAE =\u00182X\no;u;\u001b;\u001b0(2\u000e\u001b\u001b0\u00001)jho\u001bjlzju\u001b0ij2\u0000jho\u001bjlxju\u001b0ij2\n\u000f\u001bu\u0000\u000f\u001b0\no;\n(2)\nwhereu\u001b(o\u001b0) and\u000f\u001b\nu(\u000f\u001b0\no) represent eigenstates and\neigenvalues of unoccupied (occupied) states in spin state\n\u001b(\u001b0);\u0018is the SOC constant. lzandlxare the angu-\nlar momentum operators. In an alloy like CoPt, one\nhas a contribution from each element, Co and Pt, to\nthe MAE given by the previous equation. One has\nto keep in mind that the SOC constant \u0018Ptis larger\nby one order of magnitude than the one of Co, \u0018Co.\nThe positive and negative contributions to the MAE are\ncharacterized by lzandlxoperators, respectively. The\npossible nonzero matrix elements with the d-states are\nhxzjlzjyzi= 1,hxyjlzjx2\u0000y2i= 2,\nz2\f\flxjxz;yzi=p\n3,\nhxyjlxjxz;yzi= 1, and\nx2\u0000y2\f\flxjxz;yzi= 1. Con-\nsidering that all majority-spin states are occupied, Eq.2\nis left with two terms only, the one involving the cou-\npling between the unoccupied and occupied minority-spin\nstates, (\u001b\u001b0) = (##), and the one involving the coupling\nbetween the occupied majority-spin states to the unoccu-\npied minority-spin states, ( \u001b\u001b0) = (\"#). Interestingly, for\nthe##(\"#){term the lz(lx) matrix elements favor an out-\nof-plane easy axis and compete against the lx(lz) matrix\nelements. An interesting analysis related to our work is\ngiven in the context of FeRh \flms for example.45\nIn our discussion we focus on the Co electronic states\nalthough as it will be discussed later on Pt has also atremendous impact on the \fnal MAE. We start by ana-\nlyzing the orbital-resolved DOS for the d-states of the Co\natoms in CoPt bulk as plotted in Fig. 6(a). One notices\nthat around the Fermi energy, there are minority-spin\nvirtual bound states (VBSs) of large amplitude: z2- and\nxy-VBSs as well as unoccupied xy,x2\u0000y2andyz. With\nthis con\fguration the matrix elements active and proba-\nbly important in Eq.2 for the ##{term are\njhxyjlzjx2\u0000y2ij2\n\u000fx2\u0000y2\u0000\u000fxy\u0000j\nz2\f\flxjyzij2\n\u000fyz\u0000\u000fz2\u0000jhxyjlxjyzij2\n\u000fyz\u0000\u000fxy:(3)\nThese three terms counteract each other and therefore\nthe##{contribution to the MAE is expected to be negli-\ngible. Having a rather localized occupied z2-VBS favors\nan in-plane orientation of the moment, while the occupied\nxy-VBS pushes for an out-of-plane easy axis. Consider-\ning the\"#{term, we expect contributions coming from\nthe majority-spin VBSs: xy,yzandxz. The active ma-\ntrix element would then be:\njhyzjlxjxyij2\n\u000fxy\u0000\u000fyz+jhxzjlxjxyij2\n\u000fxy\u0000\u000fxz; (4)\nwhere we considered the coupling to the closest unoc-\ncupied minority-spin VBS, xy, the other possible states\nwould lead to rather large denominators. These terms\nfavor the out-of-plane orientation of the magnetic mo-\nments, which explains the behavior of bulk CoPt.\nFor the thin \flm discussion, we proceed to compar-\nisons involving the same thickness reference, X= 3. The\norbital-resolved DOS for the d-states of the Co-atom at\nthe surface of the Co-terminated thin \flm is shown in\nFig.6(b). By this analysis, we try to explain why the\nsurface contribution to the total MAE is negative and\nfavors an in-plane easy axis. The increase of the mag-\nnetic moment on the surface compared to the bulk value\ncan be grasped from the larger exchange splitting be-\ntween the bands. Interestingly, important changes oc-\ncur in both spin channels. In the majority-spin channel,\nthe bulk-VBS close to the Fermi energy and favoring the\nout-of-plane easy axis disappear. Thus, we are left with\nprocesses contributing to the ##{term of the MAE. The\nsurface allows to better localize the VBSs pointing out-\nof-plane, i.e. z2-,xz- andyz-VBSs, which are then less\nsubject to hybridization. In the minority-spin channel,\nthez2-VBS becomes prominent. Obviously, following the\nupper discussion for the bulk case, the lxcontribution be-\ncomes important favoring then an in-plane orientation of8\nFIG. 6. Spin-dependent and orbital resolved density of states\nof Co in bulk CoPt (a), in the outermost perfect surface of\nCoPt thin \flm ( X= 3) shown in (b) and the layer underneath\nthe surface layer of the Pt-terminated thin \flm ( X= 3) shown\nin (c).\nthe magnetic moment. Moreover the xy-VBS decreases\nin intensity, which is not helping the out-of-plane orien-\ntation of the magnetic moments. However, a new term\ninvolvinglz, i.e. favoring the out-of-plane orientation of\nthe moment, shows up and involves the coupling between\nthe occupied xz-VBS and and the unoccupied yz-VBS.\nOverall, the lxcontribution is certainly more important\nthan the one from lzand thus leading to a negative sur-\nface contribution of -0.37 meV to the MAE.\nIf a Pt-layer is deposited on top of the previous \flm,we noticed a dramatic increase in the MAE with a switch\nof the sign of the surface contribution. This can be un-\nderstood from the orbital resolved DOS of the Co-atom\nunderneath the Pt surface layer. In this con\fguration,\nthe DOS shown in Fig. 6(c) resembles more the one ob-\ntained in bulk CoPt but with slight di\u000berences. The z2-\nVBS decreased in intensity, when compared to the bulk\ncounterpart, thereby the lx-contribution in the ##{term\nfavoring an in-plane orientation of the moment decreases.\nAs shown in Table I, there is a large inward-relaxation\n(-5.6%) of the Pt surface layer while the Co-layer un-\nderneath relaxes upward (+1.6%). The interlayer dis-\ntance between Co and Pt at the surface is therefore much\nsmaller than the bulk interlayer distance, which a\u000bects\nthe intensity of the out-of-plane VBSs. Moreover, the\nVBSs seen in the bulk majority-spin channel are recov-\nered upon deposition of the Pt surface layer. That helps\nto increase the positive contribution of the \"#{term to\nthe MAE.\nAnother path for the analysis of the calculated MAEs\nis to use of the celebrated Bruno's formula46, which trans-\nlates to the neglect of spin-\rip contributions to the MAE\nas given in Eq.2:\nMAE =X\ni\u0018i\n4(Li\n[001]\u0000Li\n[100]); (5)\nwhereilabels the di\u000berent atoms, and Lbeing the orbital\nmagnetic moment calculated when the spin magnetic mo-\nment points along the [100] or the [001] directions. The\nessence of Bruno's formula is to relate the MAE to the\norbital moment anisotropy (OMA), i.e. L[001]\u0000L[100],\nand leads to the conclusion that the orientation of the\nmagnetic moments is favored when the orbital magnetic\nmoment is maximized. This formula is known to work\nreasonably well when the majority-spin states are occu-\npied. Thus, its validity is probably limited to some of\nthe Co atoms discussed in this manuscript but certainly\nnot for Pt. It is however instructive to analyze the results\nobtained with this well known formulation since it should\ncorrelate with the previous discussion. In Fig.7(a), the\nOMAs in the bulk of CoPt is plotted for Co and Pt as\nfunction of the c/a ratio in fashion similar to that used in\nFig.3. One notices that the Pt contribution counteracts\nthe one of Co. While the anisotropy of the Co orbital\nmoment increases with c/a, favoring thereby an out-of-\nplane orientation of the magnetic moment, the anisotropy\nof the Pt orbital moment has an opposite slope and favors\nan in-plane orientation of the magnetic moment. When\nsumming up the two curves, considering the spin-orbit\ncoupling constant, \u0018, to be the same for Co and Pt, which\nis of course is not true since \u0018Ptis one order of magni-\ntude larger than \u0018Co, one recovers the shape of the curve\nobtained in Fig,3, i.e. having a minimum of the curve at\nc/a = 0.8.\nSimilar to the bulk, the behavior of the Co and Pt\nOMA in CoPt thin \flms counteract each other. In gen-\neral, the Co OMA favors an out-of-plane orientation of\nthe moment contrary to the Pt OMA. In Fig.7(b), we9\nFIG. 7. (a) Anisotropy of the orbital magnetic moment, \u0001 L=L[001]\u0000L[100], for Co and Pt calculated in the CoPt bulk case.\n(b) Besides the average Co and Pt OMAs, the surface MAE of the CoPt thin \flms is plotted as function of Pt concentration\nin the layer covering the Co-terminated thin \flm with X= 3.\nplot the surface MAE of the thin \flms characterized by\nX= 3 as function of the Pt concentration on the layer\ndeposited on the Co-terminated thin \flm. Thus, in the\ncase of one perfect Pt overlayer the Pt concentration is\n100%, while the investigated anti-site corresponds to a Pt\nconcentration of 75%. For the speci\fc case of Co stacking\nfault, the Pt concentration is -100%. The surface MAE\nseems to increase with the Pt concentration but not in\na regular manner. We plot on the same \fgure the aver-\nage Co OMA per Co atom and \fnd that this quantity\nincreases smoothly in magnitude with Pt concentration.\nIn addition the contribution of the average Pt OMA per\nPt atom is shown in Fig.7(b). The Pt OMA seems to\ncorrelate the irregular behavior of the surface MAE. In-\nterestingly, we \fnd that thin \flms with large Pt OMA\nper Pt atom compared to the Co OMA leads to an in-\nplane surface MAE. Only the Pt-terminated thin \flm,\nwith a large perpendicular surface MAE, is characterized\nby a large Co OMA.\nIV. CONCLUSION\nWe investigated from ab-initio the magnetic behavior\nof CoPt thin \flms as function of thickness considering\ndi\u000berent types of terminations: perfect Co or Pt layers\nor di\u000berent types of defects: anti-site or stacking faults.\nAfter this systematic study, we found that the MAE is thelargest when the thin \flms are terminated by a perfect Pt\noverlayer. Surprisingly in the latter case, the MAE can\nbe 1000% times larger than the one of Co-terminated thin\n\flms. We also \fnd that all types of investigated defects\nreduce dramatically the MAE. The surface MAE expe-\nriences a sign change when increasing the thickness of\nseveral investigated \flms. Except for the Pt-terminated\n\flms, the surface MAE favors an in-plane orientation of\nthe moments when the thickness Xis smaller then four.\nWe proceeded to an analysis of the electronic structure of\nthe thin \flms with a careful comparison to the CoPt bulk\ncase and related the behavior of the MAE to the location\nof the di\u000berent virtual bound states utilizing second or-\nder perturbation theory. Finally, the correlation between\nthe MAE and the OMA is studied.\nACKNOWLEDGMENTS\nWe are grateful to Claude Demangeat, Vasile Caciuc,\nJulen Ibanez-Azpiroz and Manuel dos Santos Dias for\nhelpful discussions. Also we thank Hongbin Zhang for\ndiscussion and for providing us his data. 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Sabirianov9, Hao Zeng4, Peter Ercius3 & Jianwei Miao1† \n1Department of Physics & Astronomy  and California NanoSystems Institute, University \nof Califor nia, Los Angeles, CA 90095, USA.  2Department of Physics, National Sun Yat -\nsen Univ ersity,  Kaohsiung 80424, Taiwan. 3National Center for Electron Microscopy, \nMolecular Foundry, Lawrence Berkeley National Laboratory, Berkeley, CA 94720, \nUSA. 4Department of Physics, University at Buffalo, the State University of New York, \nBuffalo, NY 14260, USA . 5Nanoscale Physics Research Laboratory, Schoo l of Physics \nand Astronomy, University of Birmingham, Ed gbaston, Birmingham B15 2TT, UK. \n6National Center for Computational Sciences , Oak Ridge National Laboratory, Oak \nRidge , TN 37831, USA.  7Computer Science an d Mathematics Division, Oak Ridge \nNational Laboratory, Oak Ridge , TN 37831, USA. 8Center for Nanophase Materials \nSciences , Oak Ridge National Laboratory, Oak Ridge , TN 37831, USA. 9Department of \nPhysics, University of Nebraska at Omaha, Omaha, NE 68182, US A.  \n *These authors contributed equally to this work . †Email:  miao@physics.ucla.edu   \nCorrelating 3D arrangements of atoms and defects with material properties and \nfunctionality forms the core of several scientific disciplines . Here, we determined \nthe 3D co ordinates of 6,569 iron and 16,627 platinum atoms in a model iron -\nplatinum nanoparticle system to correlate 3D atomic arrangements and chemical \norder/disorder with material properties at the single -atom level. We identified rich \nstructural variety  and chemical order/disorder including 3D atomic composition, 2 \ngrain  boundaries,  anti-phase boundaries, anti-site point defects and  swap defects. \nWe show for the  first time that experimentally measured 3D atomic coordinates  \nand chemical species with 22  pm pr ecision can be used as direct input for first-\nprinciples  calculations of material properties  such as atomic magnetic moments \nand local magnetocrystalline anisotropy . This work not only opens the door to \ndetermining 3D atomic arrangements and chemical order /disorder of  a wide range \nof nanostructured materials with high precision, but also will transform our \nunderstanding of structure -property  relationships at the most fundamental level .  \n Perfect crystals are rare in nature. Real materials are often composed  of crystal \ndefects and chemical order/disorder such as grain boundaries, dislocations,  interfaces, \nsurface reconstructions and point defects that disrupt the periodicity of the atomic \narrangement and determine their properties and performance1-5. One prominent example  \nis intermetallic compounds involving  two or more atomic species , in which chemical \norder/disorder  determine s their mechanical , catalytic , optical , electronic  and magnetic \nproperties6-9. For instance, as-synthesized at room temperature,  FePt nanoparticles  and \nthin films with a near-1:1 composition  have  a chemically disordered face-centered cubic  \n(fcc) structure ( A1 phase)7,10,11. When  annealed  at high temperatures , they undergo a \ntransformation from an A1 phase to an ordered face -cente red tetragonal ( L10) \nphase7,10,11. Due to the chemical ordering and strong spin-orbit coupling12, L10 FePt \nexhibits extremely large magnetocrystalline anisotropy  energy (MAE)  and is among the \nmost promising candidates for next -generation magnetic storage  media11-13 and \npermanent magnet applications14. Recently, it has also been reported that L10 ordered \nFePt nanoparticles show enhanced catalytic  properties for oxygen evolution reaction15. \nHowever, although this material  system has attracted considerable attention, a 3 \nfundamental understanding of 3D chemical order/disorder, crystal defects  and its \nmagnetic properties  at the individual atomic  level remains el usive .  \n On a parallel front, quantum mechanics calculations such as density functional \ntheory (DFT) have made significant progress from modelling ideal bulk systems to \n“real” materials with dopants, dislocations, grain boundaries and interfaces16,17. \nPresently, these calculations re ly heavily on average  atomic  models extracted fro m \ncrystallography.  However , for complex materials such as non -stoichiometric \ncompounds, materials deviating from thermodynamic equilibrium, or materials subject \nto complex processing treatments , it is either impossible or hugely computationally \nexpensive to perform  structural optimization using ab initio  methods16-19. To improve \nthe predictive power of DFT  calculations, there is a pressing need to provide  atomic  \ncoordinates  of real systems beyond average  crystallographic measurements  as input for \ncomputatio n. One powerful method to address this challenge is atomic electron \ntomography  (AET)20-24, in which the 3D positions  of individual atoms with defects  can \nbe measured  without assuming crystallinity or using averaging25-28.  \n Here, we report the precise  determination of  the 3D coordinates  and chemical \nspecies of 23,196 atoms in a single  8-nm Fe 0.28Pt0.72 nanoparticle  using  a generalized \nFourier iterative reconstruction (GENFIRE) algorithm . From these atomic coordinates \nand species, we revealed chemical or der/disorder and crystal defects such as  grain \nboundaries , anti -phase  boundaries , anti -site and swap  defects  in 3D . Furthermore, the \nmeasured atomic coordinates were  used as direct input for  DFT calculations  to correlate \n3D chemical ordering and defects with the local MAE . This work marries  the forefront \nof 3D atomic structure determination  of crystal defects and chemical order/disorder  with 4 \nDFT calculations , which we expect to find broad applications in physics, chemistry, \nmaterials science, nanoscience and nanotechnology.     \n3D determination  of atomic positions and species  \nFePt nanoparticles with a size  of ~8 nm were synthesized by an organic solution phase \nsynthesis technique29 and annealed at 600°C for 25 minutes to induce partial chemical \nordering  (Methods) . Using an aberration -corrected scanning transmission electron \nmicroscope (STEM) operated in annular dark -field (ADF)  mode30-32, we acquired high-\nresolution tomographic tilt series from several FePt nanoparticles. A representative tilt \nseries of 6 8 images with a tilt range from -65.6° to +64.0 ° was chosen for the detailed \nanalysis  due to its structural complexity  (Extended Data Fig. 1) . After being  denoised33 \n(Methods, Extended Data Fig. 2 ) and aligned21, the tilt series was used to compute a  3D \nreconstruction with a newly developed GENFIRE  algorithm (Methods) . GENFIRE \nstarted with  a 3D regular grid in Fourier space and assigned some grid points with \nmagnitude s and phases  derived from the experimental  tilt series through \noversampling34,35  and the discrete Fourier transform. The assembled 3D grid consists of \na fraction of grid points assigned with measured data, while the remaining grid points \nwere  set as undefined. The algorithm  then iterated  between real and reciprocal space  \nusing the fast Fou rier transform  and its inverse . A support and positivity were used as \nreal space constraints , and the measured grid points were enforced in reciprocal space , \nwhile the undefined points were iteratively updated by the algorithm . After 500 \niterations, a converged 3D reconstruction was obtained . Next, we refined each tilt angle \nby projecting the 3D reconstruction  to calculate  a 2D image  (Methods) . An error metric \nwas computed between the calculated and measured images. A more accurate tilt angle \nwas obtain ed by optimizing  the orientation of the 3D reconstruction to produce  a 5 \ncalculated  projection image and minimize the error metric . After refining all the tilt \nangles , we repeated  the GENFIRE reconstruction and produced a final 3D \nreconstruction. Compared to  equal slope tomography36 that has been applied to \nAET21,23 ,25, GENFIRE  not only produces more accurate 3D reconstruction s through the \nrefinement of the tilt angles, but also represents  a more general method applicable to \nany tilt geometry  and capable of solving larger systems in a faster manner .  \n From the final 3D reconstruction, we identified  all local intensity maxima taking \ninto account a minimum distance of 2Å between two neighbouring atoms, which is \njustified as the covalent diameter  of an Fe atom is 2.52Å.  Extended D ata Fig. 3a  shows \nthe histogram of the local intensity maxima , indicating each local maximum belongs to \none of the  three categories : i) potential Pt atoms, ii) potential Fe atoms and iii) potential \nnon-atoms (intensity to o weak to be an atom).  By developing  an unbiased classification \nmethod  (Methods, Extended Data Fig. 3), we classified each atom into one of these \nthree categories . However, e xperimental data  could not be measured beyond -65.6° and \n+64.0° ( known as the missing wedge problem ), resulting in lower average atom \nintensity in the missing wedge region . To mitigate this effect,  we re -classified the local \nintensity  maxima in this region . This atom tracing and classification process yielded  \n17,087 Pt and 6,717 Fe atom candidates . To validate the robustness of our  method with \nregard s to the choice of the minimum distance , we changed the minimum distance to \n1.6Å and repeated  the whole process,  producing  16,551  Pt and 6,639  Fe atom \ncandidates . The majority of the atom  candidates between the two independent  sets are \nconsistent and  identified as real atoms. A  very small fraction of inconsistent atom \ncandidates were  validated  one-by-one using  the 68 experimental images to produce  a \n3D atomic model  (Methods) . The model was further refined using the experimental 6 \nimages to  improve the precision of individual atom coordinates25 (Extended Data Fig. \n4), resulting in  a final 3D model  of 23,196 atoms, consisting of 16,627 Pt and 6,569 Fe \natoms .  \nTo verify the final a tomic model, we applied  multislice STEM simulations37 to \ncalculate 68 images from the model using  the same experimental parameters . Extended \nData Figs. 5a -c show  good agreement between a measured and multislice image. Using  \nthe same reconstruction, atom tracing and classification procedures, we obtained a new \n3D model consisting of 16,579 Pt and 6,7 91 Fe atoms. Compared to the experimental \nfinal model , 99.0% of all atoms are correctly  identified in the new model and a root-\nmean -square deviation  of the common atom positions  is 22  pm (Extended D ata Fig. 5d ). \nTo further confirm the precision of our atomic position measurements, w e performed a \nlattice analysis of the 3D atomic model (Extended Data Figs. 6)  and determined the 3D \natomic displacements of the  nanoparticle (Extended Data Figs. 7) . Based on the atomic \ndisplacements relative to  an ideal fcc lattice , we estimated  an average 3D precision of \n21.6 pm (Extended D ata Fig. 8 a), which agrees well  with the multislice result.  \n3D identification of chemical  order/disorder  \nFrom the 3D positions  of individual  atomic species (Fig. 1a), we classifi ed the 3D \nchemical order/ disorder of the FePt nanoparticle. This was achieved  by determining  the \nshort -range order parameter  (SROP)  of all phases present  in the 3D structure (Methods). \nThe nan oparticle consists of two large  chemically ordered fcc ( L12) FePt 3 grains  that \nform concave shapes and are nearly connected (Fig. 1b). Seven smaller grains are \nlocated at the boundary between the two large L1 2 grains or near the centre of the \nnanoparticle, including three L1 2 FePt 3 grains, three L1 0 FePt grains and a Pt -rich A1 \ngrain (Fig. 1b  and Supplementary Video 1 ). This level of complexity in the 3D structure 7 \nand chemical order/disorder can only be fully revealed by AET. To illustrate this point, \nwe used multislice STEM simulations to calculate 2D projection images from the 3D \natomic model along the [100], [010] and [001] directions (Fig. 1c). Although 2D \nanalysis of the images indicates that the nanoparticle is primarily composed of L1 2 \nFePt 3 grains, several of the ‘L10 grain ’ signatures appearing in the projection images \n(magenta  in Fig. 1c ) are deceptive  structural information , as these derive from the \noverlapping projections of the two  L12 grains, rather than from actual L1 0 grains .  \nNext,  we analysed  the chemical order /disorder  associated with  the 3D grains. \nFigure 2a shows the atomic positions and chemical species of the nanoparticle with \ngrain boundaries marked as black lines. The gr ains identified are more ordered in their \ncores and less ordered closer to the grain surface. Figures 2b -e show four representative \ncut-outs of the atomic model where the SROP  of the corresponding  phase is averaged \nalong the [010] direction and displayed  as the background  colour. T he most h ighly \nchemically ordered region  of the nano particle is at the core of  a large L1 2 grain with the \nSROP  close to 1 (Fig. 2b). Figure 2c shows a grain boundary between two large L1 2 \ngrains with  a varied grain boundary width . Anti-phase boundaries between the two L1 2 \ngrains are also observed (Extended D ata Fig. 8b). The largest L1 0 grain is shown in Fig. \n1b (the 3rd grain from the left) and Fig. 2d . The  L10 ordering can be  identified by the \nhigh concentration of Fe atoms on alternating planes along the vertical direction. This \nL10 grain sits between the two large L1 2 FePt 3 grains with each o f its two Fe sublattices \nmatching  the Fe sublattice of one of the neighbouring L1 2 grains (Extended Data Fig. \n6), suggesting the shared  Fe planes w ith its neighbouring grains may have facilitate d the \nnucleation of the L1 0 phase . The central region  of the nanoparticle has the highest 8 \ndegree of chemical disorder , including a Pt -rich A1 -phase grain (Fig. 2e),  with much \nlower SROP  values than those in the two large L1 2 grains.               \nTo probe the 3D chemical order/disorder  at the single -atom level, we analyse d \nindividual anti -site point defects in the 3D reconstruction of the nanopartic le. Figure s \n3a, b and Extended D ata Fig. 8b show 3D atomic positions overlaid on the reconstructed \nintensity of several  representative anti -site point defect s (arrows)  in the L12 FePt 3 \ngrains , where an Fe atom occupies  a Pt atom site  or vice versa . A perfect L1 2 FePt 3 \nphase is illustrated in Fig. 3d  for reference . As our 3D reconstruction does not use an a \npriori  assumption of crystallinity, the  identification of individual atomic species is \nbased on  the intensity distribution around local maxima of t he 3D reconstruction \n(Extended Data Fig. 4) . The anti-site point defect s in these figures  are clearly visible by \ncomparing their local peak intensity with that of the surrounding Pt and Fe atoms. \nFurthermore, swap  defects are also observed  (Fig. 3c), where a pair of nearest -\nneighbour Fe and Pt atoms are swapped . Overall, the FePt nanoparticle contains a \nsubstantial number of anti -site defects and chemical disorder. Figures 3e and g  show the \nanti-site defect density of the two large L1 2 grains (inset) as a function of the distance \nfrom the grain  surface.  Far outside of each grain, the anti -site defect density approaches \n~50%. This is because two of the four sublattices in the two large L1 2 grains share the \nsame composition (pure Pt), while th e other two sublattices swap Fe for Pt and vice \nversa  (Extended Data Fig. 6) . The anti-site defect density drops to below 40%  at the \nsurface of the two grains and reduces to  ~3% for sites deep inside each grain.  Figures 3f  \nand h show  the SROP of the two la rge L1 2 grains as a function of the distance from the \ngrain surface. The negative SROP  values outside each  grain indicate  the local structure  \nis anti -correlated with that of the other L12 grain .  9 \nThe striking similar ities between the two large L1 2 grains , i.e. each having  a \nconcave shape  with a highly -ordered core,  a similar  chemically disordered boundary \nand a consistent  distribution  of the anti -site defect density  (Figs. 3e -h), suggest  a \npotential pathway  in the nucleation and growth process . We note that  as-synthesized \nFePt nanoparticles show large chemical dis order with a Pt -rich core38. Such a 3D Pt -rich \ncore is observed  in our measurement  (Fig. 2e). During the annealing process , Pt atoms \ndiffuse d out from the core38 and the nucleation of the L1 2 phase likely  occurred \nsimultaneously at multiple sites in the nano particle. The nuclei then gre w and merge d \ninto larger grains  by the Ostwald ripening  process39. This process would  continue until \nthe nano particle became  a single crystal if sufficiently high t emperature/long time \nannealing wa s app lied. However, if the annealing process was stopped  at some \nintermediate stage, two or more  larger grains with similar sizes could coexist since it \nwas difficult for either to annihilate the others . The chemical ordering at the grain \nboundaries would then be frustrated by competition be tween neighbouring grains . \nHowever, determining the particle ’s chemical structure growth pathway with certainty \nwill require adding the dimension of  time to the AET measurements. In pri nciple, i t is \npossible to measure a series of tomographic  data sets  of the same particle  at different \ntimes during heat treatment. This  would u nlock the atomic -scale mechanism  underlying \nthe nucleation and growth process .        \nCorrelating chemical order/ disorder  to magnetic properties  \nTo relate  experimentally determined atomic coordinates and chemical order/disorder to \nmagnetic properties, we performed DFT calculations of the atomic magnetic moments \nand MAE s. We focused on one of the grain boundaries between two large L1 2 grains, \nwhere the largest L1 0 grain is located , and calculated the MAE using two independent  10 \napproaches  (Methods) . First, we cut out a 1, 470-atom supercell from this region, \ncontaining the largest L1 0 grain. B y sliding a 32 -atom -cubic volume in the supercell  \nwith a half -unit-cell per step along each direction , we obtained 1,452 32-atom volumes  \nand calculated t heir local MAEs with DFT . Second , we cropped six nested supercells  of \nthe same region with a size ranging from 32 to 1,372  atoms  and calculated the atomic \nmagnetic moments (Extended Data Fig. 9) and MAEs . Figure 4a and Extended Data  \nFig. 10a show a good agreement of the MAEs between the supercell and sliding volume \ncalculations , which validates the sliding volume approach  for determining the local \nMAEs . The correlation between the MAE and supercell size was also reproduced by \nmodelling a spherical -shaped L1 0 grain, enclosed by cubic L1 2 grains with different  \nsizes. In this model, the MAE of the L1 0 grain was obtained from  the 32-atom supercell \ncalculation , and the radius of the L1 0 sphere and the MAE of the L1 2 grains were fitted \nto the  MAEs  obtained from six nested supercells (Figure 4a and Extended Data Fig. \n10a). To study the influence of measured atomic positions,  we self -consistently relaxed \nfour 32 -atom , one 256-atom  and one 500 -atom volumes  using DFT. The root-mean -\nsquare deviation  between  the measured and relaxed atomic positions is 24.7 pm, which \nagrees  with our  precision estimation. The average MAE difference  is 0.064  meV/atom , \nindicating that our measured atomic coordinates are sufficiently accurate for DFT \ncalculations  and the atomic c omposition and chemical order/disorder  are the main \ndeterminant s of the local MAE . \n Figure 4b and Extended Data Fig. 10b show a strong correlation between the \nlocal MAEs of all sliding 32 -atom volumes and the L10 order parameter difference , \nwhere the L1 0 order parameters were computed from the same 32 -atom volumes . The \n3D distribution of the local MAE s matches well with that of the L10 order parameter 11 \ndifference  inside the 1,470 -atom supercell (Fig. 4c and Extended Data Fig. 10c) . The \nMAEs obtained by our sliding volume approach can be used to provide the local  \nuniaxial anisotropy constants for micromagnetic simulations40,41, which are presently \ntaken from  either bulk or modelled values without considering atomic details. Because \nthere is no perfect L1 0 phase in the nanoparticle, t he largest local MAE in the region \n(0.945 meV/atom ) is smaller  than that of an ideal  L10 phase (1.40 meV/ato m) obtained \nfrom  our DFT calculation . The smallest MAE s exist  in the L1 2 grain and some sharp \ntransitions  from  the large  and small  MAE s are also observed . Fig. 4d and Extended  Data \nFig. 10d show the l ocal MAE distribution at an L1 0 and L1 2 grain boundary , overlaid  \nwith measured atomic positions  and species . The sharp grain boundary is responsible \nfor a sudden  transition  of the local MAE . Although here we used an FePt model sy stem \nto correlate 3D grain boundaries and chemical order/disorder  with magnetic pro perties \nat the single -atom level, this method  can be readily  applied to many other material \nsystems.      \nConclusions   \nWe have developed a new tomographic reconstruction algorithm, termed GENFIRE, to \ndetermine  the 3D coordinates and chemical species of 23,196 atoms in a FePt \nnanoparticle with 22 pm precision . We have identified atomic composition, chemical \norder/ disorder, grain boundaries, anti -phase boundaries , anti -site point defects and swap  \ndefects with unprecedent ed 3D detail . Based on  a statistical  analysis of the chemical \norder/disorder  and anti -site defect densit ies of the two large L1 2 grains, we suggested  a \npotential pathway  in the nucleation and growth process of the chemically ordered grains  \ninside  the nanop article . We also , for the first time , used the experimentally measured 3D \natomic coordinates and species with defects as direct input for DFT calculations to 12 \ncorrelate 3D chemical order/disorder  and grain boundaries with magnetic properties at \nthe single -atom level . This work makes significant advances in characterization \ncapabilities and expands our fundamental understanding of structure -property  \nrelationship s. As a general and powerful 3D reconstruction algorithm, GENFIRE can be \nbroadly applied to  other systems and tomographic fields.  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This work was primarily supported by the Office of Basic Energy Sciences of the US DOE  \n(DE-SC0010378).  This work was partially supported by NSF ( DMR -1437263 ) and ONR MURI \n(N00014 -14-1-0675) . The chemical ordering analysis and  ADF -STEM imaging  with TEAM I were \nperformed at  the Molecular Foundry , which is supported by the Office of Science, Office of Basic Energy 16 \nSciences of the U.S. DOE  under Contract No. DE -AC02 —05CH11231.  M.E. (DFT calculations) was \nsupported by the U.S. DOE , Office of Science, Basic Energ y Sciences, Material Sciences and \nEngineering Division. DFT calculations by P.K. were conducted at the Center for Nanophase Materials \nSciences, which is a DOE Office of Science User Facility. This research used resources of the Oak Ridge \nLeadership Computi ng Facility, which is supported by the Office of Science of the U.S. DOE  under \ncontract DE -AC05 -00OR22725.   \nFigure legends  \nFigure 1 . 3D determination of atomic coordinates, chemical species and \ngrain structure of an FePt nanoparticle . a, Overview of the 3D positions of \nindividual atomic species with Fe atoms in red and Pt atoms in blue. b, The \nnanoparticle consists of two large L1 2 grains, three small L1 2 grains, three small \nL10 grains and a Pt -rich A1 grain. c, Multislice projection ima ges obtained from \nthe experimental  3D atomic model along the [100], [010] and [001] directions, \nwhere several ‘L10 grains’  (magenta) appearing in the 2D images are deceptive \nstructural information, as these derive from the overlapping projections of the \ntwo L1 2 grains . Scale bar, 2 nm.   \nFigure 2 . 3D identification of grain boundaries and chemical \norder/ disorder . a, Atomic coordinates and species of the FePt nanoparticle \ndivided into one-fcc-unit-cell thick slices . The grain boundaries are marked with \nblack lines. b-e,  Four representative cut-outs of the experimental atomic model, \nshowing the most highly chemically ordered L1 2 region of the particle ( b), a \ngrain boundary between the two large L1 2 grains ( c), the largest L1 0 grain ( d), \nand the most chemically disordered region of the particle centered on a Pt -rich \nA1 grain ( e). The location s of the cut-outs are labelled  in (a), and the SROP of \neach cut-out is averaged along the [010] viewing direction and displayed as the \nbackground colour.         17 \nFigure 3 . Observation of anti-site point defects and swap  defect s and \nstatistical analysis  of the chemical order /disorder  and anti-site density . 3D \natomic positions overlaid on the 3D reconstructed intensity illustrating anti -site \npoint defects: a Pt atom occupying a n Fe atom site  (a), an Fe atom occupying a \nPt atom site  (b), a pair of nearest -neighbouring Fe and Pt atoms are swapped \n(swap defect)  (c). d, 3D atomic structure of a n ideal  L12 FePt 3 phase  for \nreference . e, f, The anti-site defect density and SROP for a large  L12 grain , \ninset in ( e), as a function of the distance from the grain surface  (unit cell size = \n3.875 Å). g, h, The anti-site defect density and SROP for the other large L12 \ngrain , inset in ( g), as a function of  the distance from the grain surface. Smooth \nred trendlines are overlaid on the defect density distribution as a guide for the \neye. \nFigure 4.  Local MAE s between the [100]  and [001] directions  determined \nby using measured atomic coordinates  and species  as direct input to DFT . \na, Black dots represent the MAE s calculated from six nested cubic volumes  of \n32, 108, 256 , 500 , 864  and 1,372 atoms . Blue curve shows the results of fitting \na L1 0 sphere inside cubic L1 2 grains with different sizes. Red dots are th e local \nMAE s averaged  by sliding  a 32 -atom volume inside the  corresponding six  \nsupercell s. b, MAE s of all sliding 32-atom volumes as a function of the L10 \norder parameter difference  of the same volume between the [100] and [001] \ndirections . Negative MAE values indicate that their local magnetic easy axis is \nalong the [100] instead of [001] direction.  c, 3D iso -surface rendering of the \nlocal MAE (top) and L10 order parameter difference s (bottom) inside the 1,470 -\natom supercell.  d, Local MAE distribution  at an L10 and L1 2 grain boundary , \ninterpolated from the sliding local volume calculations and overlaid  with \nmeasured atomic positions.  18 \n \n \nFigure 1  \n19 \n \n \nFigure 2  \n20 \n \nFigure 3  \n21 \n \nFigure  4 \n \n"    },    {        "title": "1607.06470v2.Perpendicular_magnetic_anisotropy_of_two_dimensional_Rashba_ferromagnets.pdf",        "content": "arXiv:1607.06470v2  [cond-mat.mes-hall]  3 Nov 2016Perpendicular magnetic anisotropy oftwo-dimensionalRas hba ferromagnets\nKyoung-Whan Kim,1,2,3,4Kyung-Jin Lee,5,6Hyun-Woo Lee,4,∗and M. D. Stiles1,†\n1Center for Nanoscale Science and Technology, National Inst itute of Standards and Technology, Gaithersburg, Maryland 20899, USA\n2Maryland NanoCenter, University of Maryland, College Park , Maryland 20742, USA\n3Basic Science Research Institute, Pohang University of Sci ence and Technology, Pohang 37673, Korea\n4PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang 37673, Korea\n5Department of Materials Science and Engineering, Korea Uni versity, Seoul 02841, Korea\n6KU-KIST Graduate School of Converging Science and Technolo gy, Korea University, Seoul 02841, Korea\n(Dated: February 19, 2018)\nWe compute the magnetocrystalline anisotropy energy withi n two-dimensional Rashba models. For a fer-\nromagnetic free-electron Rashba model, the magnetic aniso tropy is exactly zero regardless of the strength of\nthe Rashba coupling, unless only the lowest band is occupied . For this latter case, the model predicts in-plane\nanisotropy. For a more realistic Rashba model with finite ban d width, the magnetic anisotropy evolves from\nin-plane to perpendicular and back to in-plane as bands are p rogressively filled. This evolution agrees with\nfirst-principles calculations on the interfacial anisotro py, suggesting that the Rashba model captures energetics\nleadingtoanisotropy originatingfromtheinterface provi dedthatthemodel takes account ofthefiniteBrillouin\nzone. The results show that the electron density modulation by doping or an external voltage ismore important\nforvoltage-controlled magnetic anisotropy than the modul ation of the Rashba parameter.\nPACS numbers:\nI. INTRODUCTION\nRecent developments in the design of spintronic devices\nfavor perpendicular magnetization, increasing the intere st in\nmaterials with perpendicularmagnetic anisotropy[1–4]. O ne\nadvantage is that devices with the same thermal stability ca n\nbe switched more easily if the magnetization is perpendicu-\nlar than if it is in plane [4–9]. Since magnetostatic interac -\ntions favor in-plane magnetization for a thin film geometry,\nperpendicular magnetic anisotropy requires materials and in-\nterfaces that have strong magnetocrystalline anisotropy. Nu-\nmerous computational studies [10–17] show the importance\nof interfaces on magnetocrystalline anisotropy. The theor y\ndevelopedbyBruno[18,19],whichprovidesaninsightfulex -\nplanationofthesurfacemagnetocrystallineanisotropyan dits\ncorrelation with orbital moment [20], has been confirmed by\nexperiments[21, 22]. ThecasesforwhichtheBruno’stheory\ndoesnotapply[23]requireacasebycasestudythroughfirst-\nprinciplescalculations,makingitdi fficulttogetmuchinsight.\nSome insight into perpendicular magnetic anisotropy can\nbe gained by studying it within a simple model. One such\nmodel is the two-dimensional Rashba model [24]. A two-\ndimensional Rashba model includes only minimal terms im-\nposed by symmetry breaking. As extensive theoretical stud-\nies have shown, a two-dimensional Rashba model can cap-\nturemostofthequalitativephysicsofspin-orbitcoupling with\nbroken inversion symmetry, such as the intrinsic spin Hall\neffect [25, 26], the intrinsic anomalous Hall e ffect [27], the\nfieldlikespin-orbittorque[28,29],thedampinglikespin- orbit\ntorque [30–33], the Dzyaloshinskii-Moriya interaction [3 4–\n37], chiral spin motive forces [38, 39], and correctionsto t he\n∗Electronic address: hwl@postech.ac.kr\n†Electronic address: mark.stiles@nist.govmagnetic damping [38], each of which has received atten-\ntion because of its relevance for e fficient device applications.\nDespite the extensive studies, exploring magnetocrystall ine\nanisotropy within the simple model is still limited. Mag-\nnetocrystalline anisotropy derived from a two-dimensiona l\nRashbamodelmayclarifythecorrelationsbetweenitandvar -\niousphysicalquantitieslistedabove.\nTherearerecenttheoreticalandexperimentalstudiesonth e\npossible correlationbetween the magnetic anisotropyand t he\nRashba spin-orbit coupling strength. The theories [40, 41]\nreport a simple proportionality relation between perpendi cu-\nlar magnetic anisotropy and square of the Rashba spin-orbit\ncoupling strength and argued its connection to the voltage-\ncontrolled magnetic anisotropy [16, 42–46]. However, thes e\nexperiments require further interpretation. Nistor et al.[47]\nreport the positive correlation between the Rashba spin-or bit\ncoupling strength and the perpendicular magnetic anisotro py\nwhile Kim et al.[48] report an enhanced perpendicularmag-\nnetic anisotropy accompanied by a reduced Dzyaloshinskii-\nMoriya interaction in case of Ir /Co. Considering that the\nDzyaloshinskii-Moriya interaction and the Rashba spin-or bit\ncouplingarecorrelatedaccordingto Ref. [37],the perpend ic-\nular magnetic anisotropy and the Rashba spin-orbit couplin g\nvary opposite ways in the latter experiment. These inconsis -\ntentobservationsimplythatthecorrelationis,evenifite xists,\nnot a simple proportionality. In such conceptually confusi ng\nsituations, simple models, like that in this work, may provi de\ninsightintosuchcomplicatedbehavior.\nIn this paper, we compute the magnetocrystalline\nanisotropy within a two-dimensional Rashba model in or-\nderto explorethecorrelationbetweenthe magnetocryatall ine\nanisotropyand the Rashba spin-orbitcoupling. We start fro m\nRashbamodelsaddedtodi fferentkineticdispersions(Sec.II)\nand demonstrate the following core results. First, a two-\ndimensional ferromagnetic Rashba model with a free elec-\ntron dispersion results in exactlyzero anisotropy once the2\nFermi level is above a certain threshold value (Sec. IIIA).\nThis behavior suggests that the simple model is not suitable\nfor studying the magnetic anisotropic energy in that regime .\nSecond, simple modifications of the model do give a finite\nmagnetocrystalline anisotropy proportional to the square of\nthe Rashba parameter (Sec. IIIB). We illustrate with tight-\nbinding Hamiltonians that a Rashba system acquires perpen-\ndicular magnetic anisotropy for wide parameter ranges once\nthe Brillouin zone and energy band width being finite in size\nistakenintoaccountin themodel. Thisdemonstratesthat th e\nabsenceofmagneticanisotropyisapeculiarfeatureofthef or-\nmerfree-electronRashbamodelandwediscussthesimilarit y\nof this behavior to the intrinsic spin Hall conductivity [26 ].\nThird, we show that the magnetocrystallineanisotropy of th e\nmodifiedRashba modelsstronglydependsonthe bandfilling\n(Sec. IIIB). The system has in-planemagnetic anisotropy fo r\nlow band filling. As the electronic states are occupied, the\nanisotropy evolves from in-plane to perpendicular and back\nto in-plane for high electron density. This suggests that it\nmay be possible to see such behavior in systems in which\nthe interfacial charge density can be modified, for example\nthrough a gate voltage. This also provides a way to recon-\ncilemutuallycontradictoryexperimentalresults[47,48] since\ndifferent band filling can result in opposite behaviors of the\nmagnetocrystalline anisotropy. We make further remarks in\nSec.IIICandsummarizethepaperinSec.IV.Wepresentthe\nanalyticdetailsinAppendix.\nII. MODELANDFORMALISM\nWe first present the model and formalism for a quadratic\ndispersion and then generalize the model to a tight-binding\ndispersion. In this paper, we call a Rashba model with a\nquadraticdispersiona“free-electronRashbamodel”andca lla\nRashbamodelwithatight-bindingdispersiona“tight-bind ing\nRashbamodel”. Allthemodelsincludeferromagnetisminthe\nsamemanner.\nA ferromagnetic free-electron Rashba model is described\nbythefollowingHamiltonian.\nH=p2\n2me+Jσ·m+αR\n/planckover2pi1(σ×p)·ˆz, (1)\nwherepis the momentum operator of itinerant electrons, me\nis the effective electron mass, J>0 is the exchange energy\nbetweenconductionelectronsandthe magnetization, σis the\nvectorof the Pauli spin matrices, αRis the Rashba parameter,\nˆzis the interface normal direction perpendicular to the two-\ndimensional space, and mis a unit vector along the direction\nof magnetization. The terms in Eq. (1) reflect the quadratic\nkineticenergy,theexchangeinteraction,andtheRashbasp in-\norbit coupling, respectively. The second and third term ori g-\ninate respectively from the time-reversal symmetry breaki ng\n(magnetism) and the space-inversionsymmetry breaking (in -\nterface). Thus, the Rashba model is a minimal model tak-\ning account of the symmetry breaking features of the sys-\ntem. There are various types of Rashba models dependingky\nkxE-(kx,k y)= E\nE+(kx,k y)= E(2 π)2N+(E)\nFIG.1: Geometricalmeaningof N+(E),thenumberofminorityelec-\ntrons per unit area that satisfies E+(kx,ky)≤E.N+(E) is given by\nthe area enclosed by the constant energy contour of E+(kx,ky)=E.\nN−(E), the number of majority electrons per unit area that satisfi es\nE−(kx,ky)≤E, has the similarmeaning (not showninthe figure).\non the momentum dependence of spin-orbit coupling Hamil-\ntonian [49]. We confine the scope of the paper to the linear\nRashba model that is linear in p[the last term in Eq. (1)]\nand is the most widely used form. We emphasize that the\nRashbamodelismainlyusefulforitspedagogicalvaluerath er\nthan its ability to make quantitativepredictionsfor real m ate-\nrials [50, 51]. In Ref. [51], the authors find that while it is\npossible to extract an e ffective Rashba parameter for realis-\ntic interfaces, it was not possible to connect this paramete r\nto the calculated magnetocrystalline anisotropy. Still, e ven\nthough the simple Rashba model may have only limited di-\nrect connection to the electronic structure of most interfa ces\nofinterest,itdoesprovideaqualitativeunderstandingof their\nphysicalproperties.\nDiagonalization of Eq. (1) gives the single particle energy\nspectrum of the free-electron Rashba model. For a homoge-\nneousmagnetic texture, Hcommuteswith p, thusk=p//planckover2pi1is\nagoodquantumnumber. Intermsof k,diagonalizationofthe\n2×2 Hamiltonian gives the energy eigenvalues E±(kx,ky) of\nHfor spin majority and minority bands, where +and−refer\ntominorityandmajoritybandsrespectively.\nE±(kx,ky)=/planckover2pi12k2\n2me±/radicalig\nJ2+2JαR(kymx−kxmy)+α2\nRk2,(2)\nwherek=|k|. Since the system has rotational symmetry\naroundˆzaxis[52],we assume my=0fromnowon.\nThe total electron energy is given by summing up single\nparticleenergiesatall electronicstatesbelowtheFermil evel.\nTodothis, wedefine N±(E),the numberofminority /majority\nelectrons per unit area that satisfies E±(kx,ky)≤E. The geo-\nmetricalmeaningof N±(E)istheareaenclosedbytheconstant\nenergy contour E±(kx,ky)=E(Fig. 1). With this definition,3\nthedensityofstatesforeachbandisgivenby dN±/dE. There-\nfore, the expression of the total energy per unit area is give n\nby\nEtot(EF)=/integraldisplayEF\nE−\nminEdN−\ndEdE+η/integraldisplayEF\nE+\nminEdN+\ndEdE,(3)\nwhereE±\nminis the band bottom energy of each band, below\nwhichN±(E)=0.η=0ifEF<E+\nminsothatthereisnooccu-\npied minority state, and η=1 otherwise. Such a factor is ab-\nsentforthefirsttermbecauseweonlyconsidertheFermileve l\nEFaboveE−\nmin. Otherwise,the magnetocrystallineanisotropy\nis trivially zero since there is no occupied state. The total\nenergy density depends on the direction of magnetization in\ngeneral. We then compute the magnetocrystalline anisotrop y\nbythe differenceof thetotal energydensityforperpendicular\nandin-planemagnetizations; ∆E=Etot|m=ˆx−Etot|m=ˆz.\nTo compute∆Efrom Eq. (3), the Fermi levels for m=ˆx\nandm=ˆzneed to be specified. Since the energy dispersion\n[Eq. (2)] is in general dependent on m, the Fermi level also\nchangesasa functionof m, becausethe total electrondensity\ndoes not change for an isolated magnetic system. Thus, we\nfix the total electron density as a constraint. To fix the total\nelectron density as a constraint, we define the total electro n\ndensitybelowenergy E.\nNe(E)=N−(E)+ηN+(E). (4)\nThe domainof EisE≥E−\nminso thatNe(E)≥0. SinceNe(E)\nis a strictly increasing function of Ein the domain, it has an\ninverse function in Ne≥0. We denote the inverse function\nbyεF(Ne).εFhas the same physical meaning as the Fermi\nlevelEFfora givenelectrondensity Ne. However,weusethe\ndifferent symbols to emphasize that εFis given by a function\noftheelectrondensitywhile EFisjustagivenconstant . With\nthisdefinitions,themagnetocrystallineanisotropyisgiv enby\n∆E(Ne)=Etot(εF(Ne))|m=ˆx−Etot(εF(Ne))|m=ˆz.(5)\nThis is the central equation of the formalism to compute the\nmagnetocrystallineanisotropy.\nWe now compute the magnetic anisotropy for a tight-\nbinding Rashba model. To construct a tight-binding Hamil-\ntonian, we discretize Eq. (1) [53, 54]. In the main text,\nwe use a tight-binding Hamiltonian for a two-dimensional\nsquare lattice as an example. The construction and the re-\nsults of a tight-binding Hamiltonian for a two-dimensional\nhexagonal lattice (equivalently a triangular lattice) are pre-\nsentedinAppendixA.Forsimplicity,weuseatwo-bandtight -\nbinding Hamiltonian includingspin degreesof freedomonly ,\nbut ignoringall orbitaldegreesoffreedom. The tight-bind ing\nHamiltonianweconstructhereisgivenby\nH=HK+HJ+HR, (6a)\nwhereHK,HJ, andHRare the discretized versions of the ki-\nnetic energy, the exchange energy, and the Rashba Hamilto-\nnian, respectively. HKis constructedby the hoppingterms to\nthenearestneightborsites.\nHK=−/planckover2pi12\n2mea2/summationdisplay\npqσ(C†\np+1,q,σCp,q,σ+C†\np,q+1,σCp,q,σ)+h.c.,(6b)whereais the lattice constant, pandqare the site indicies,\nandCp,q,σis the electron annihilation operator at site ( x,y)=\n(pa,qa)with spinσ. h.c. refers to hermitian conjugate of all\nthetermsinfrontofit. Eachterminthesummandcorresponds\nto hopping to xandydirections respectively. The hopping\nparameter−(/planckover2pi12/2mea2)isdeterminedbymatchingtheenergy\ndispersion with the free electron dispersion /planckover2pi12k2/2mein the\ncontinuum limit a→0.HJis constructed by on-site energy\nthatmixesthespindegreeoffreedom.\nHJ=J/summationdisplay\npqσσ′/bracketleftig\nC†\np,q,σ(σ)σ,σ′Cp,q,σ′/bracketrightig\n·m,(6c)\nwhere (σ)σ,σ′is the matrix element of the Pauli matrices. HR\nis constructed as following. We impose a hopping term from\nasitetoaneighboringsite,alongadirection ˆu. Sinceˆucorre-\nspondstotheelectronmomentumdirection,thetermacquire s\naspinPaulimatrix( σ׈u)·ˆz. Then,ahoppingtermalongthe\nydirection acquiring σxis given by itC†\np,q+1,σ′(σx)σ,σ′Cp,q,σ,\nwheretis a real hoppingparameter. After consideringall the\nneighboring hopping terms satisfying the hermiticity cond i-\ntion, we determine the hopping parameter by taking contin-\nuumlimituptoO(a2)andmatchingtheenergydispersionwith\nEq.(2). Inthisway,we endupwith\nHR=iαR\n2a/summationdisplay\npqσσ′/bracketleftig\nC†\np,q+1,σ(σx)σ,σ′Cp,q,σ\n−C†\np+1,q,σ(σy)σ,σ′Cp,q,σ/bracketrightig\n+h.c..(6d)\nFor more details of determining the hoppingparameters, see\nthe example in AppendixA for a two-dimensionalhexagonal\nlattice.\nNow we use the same formalism[Eq. (5)]. We use the dis-\ncrete translational symmetry of the lattice to use the Bloch\ntheoremandcomputetheenergydispersionrelationasafunc -\ntionofthecrystal momentum. Onedi fferenceisthat theBril-\nlouinzoneandthebandwidthforatight-bindingHamiltonia n\narefinite (Fig.2),while these are infinitefor thefree elect ron\nmodel Eq. (1). Therefore, the domain of the integration in\nEq.(3) is not onlylimited bythe Fermi contour,but also lim-\nited by the Brillouin zone boundary. We show in Sec. IIIB\nthatthefinitebandwidthisacrucialfeatureforemergenceo f\nperpendicularmagneticanisotropyforwide rangesof param -\neters.\nIII. MAGNETOCRYSTALLINEANISOTROPY\nA. Free-electronRashbamodel\nAlthoughthefreeelectronmodelwepresentabove[Eq.(1)]\nhas a simple form, it still requires complicated mathematic s\nto assess the magnetocrystalline anisotropy predicted by t he\nmodel since a constant energy contour given by Eq. (2) is a\nquarticcurve. In thissection, we first discussresultsof a p er-\nturbative analysis, which assumes αRto be small and keeps\ntermsonlyuptoO(α2\nR). Inthisregime,aconstantenergycon-\ntouris a quadraticcurvewhich allowsthe magnetocrystalli ne4\nΓky\nkxM\nπ/aπ/a\n-π/a\n-π/a\nFIG. 2: Brillouinzone of Eq. (6). We denote the Γand M points for\nlaterpurpose.\nanisotropy to be calculated analytically. The analytic res ults\nshall give useful insight into the model. We then go beyond\nthe perturbative regime and discuss exact results in the non -\nperturbativeregimewith arbitrary αR. In particular,we check\nif the conclusionsfromthe perturbativeanalysis remainva lid\ninthe nonperturbativeregime.\n1. Perturbation theory: Insights intothe model\nAs we show in Appendix. B1, expanding Eq. (2) up to\nO(α2\nR),we obtaina quadraticequationwithrespectto ( kx,ky).\nThe contour E±(kx,ky)=Eforms an ellipse, by which the\narea enclosed is exactly computable. Since N±(E) is exactly\ngiven in a simple way, calculating Eq. (3) is straightforwar d.\nInthisperturbativeregime,therelationbetweenelectron num-\nber density and the Fermi level Eq. (4) is linearly given so\ninverting Ne(E) is also straightforward. Then, the magnetic\nanisotropy∆E(Ne)[Eq.(5)]isevaluatedaftersimple algebra.\nTherearetwodifferentregimes; EF<E+\nminandEF≥E+\nmin.\nFor the first case, there are no minority electrons. For this\ncase,η=0 in Eq. (3). In the second case, the minority band\nis also occupied. For this case, η=1 in Eq. (3). We examine\nthecasesonebyone.\nWhen only majority band is occupied ( η=0), the magne-\ntocrystallineanisotropy[Eq.(5)]is\n∆E(Ne)=−Nemeα2\nR\n2/planckover2pi12/parenleftigg\n1−Ne\nN−(E+\nmin)/parenrightigg\n(majorityonly).(7)\nHereN−(E+\nmin) is the electron density when the Fermi level\ntouches the bottom of the minority band. The result shows\nthat the magnetocrystalline anisotropy is at least quadrat ic in\nαR. Below weshowthisisaresult ofsymmetrythat themag-\nnetocrystalline anisotropy should be an even function of αR.Equation (7) is valid only when there is no minority elec-\ntrons 0<Ne<N−(E+\nmin). We show in Appendix B1 that\nN−(E+\nmin)=Jme/π/planckover2pi12+O(α2\nR),whichisindependentof m[59].\nSinceNe<N−(E+\nmin), Eq. (7) predictsthe magnetocryatalline\nanisotropy to be negative. The sign corresponds to in-plane\nmagnetic anisotropy, which is counter to the na¨ ıve expecta -\ntionthattheRashbaspin-orbitcouplinggeneratestheperp en-\ndicular magnetic anisotropy. However, this observation do es\nnot contradict experimental observations showing perpend ic-\nular magnetic anisotropy since experimental results are us u-\nallyobtainedwhenbothbandsareoccupied.\nNext we examine the second regime where both bands are\noccupied(η=1). Strikingly,thesameformalismleadsus\n∆E(Ne)=0 (bothbandsoccupied) , (8)\nregardless of Ne. There is no magnetocrystalline anisotropy\nfor this case. An intuitive way to understandthis striking b e-\nhavior is observing the absence of angular dependence of Ne\nas a function of the Fermi level. In Appendix B1, we show\nthat,oncebothbandsareoccupied,\n(2π)2Ne(EF)=4πme(meα2\nR+EF/planckover2pi12)\n/planckover2pi14, (9)\nwhich has no mdependence. Therefore, when we increase\nthe number of electrons slightly by dNe, the contribution\nto the additional magnetocrystalline anisotropy is EFdNe=\n[(π/planckover2pi12Ne/me)−(meα2\nR//planckover2pi12)]dNe. Sincethisisindependentofthe\ndirection of magnetization, adding electrons does not chan ge\nthemagnetocrystallineanisotropyatall. BynotingthatEq .(7)\nvanishesNe=N−(E+\nmin), weendupwithEq.(8).\nThere is a recent theory [40] which predicts perpendicular\nmagnetic anisotropy with the free-electron Rashba model. I n\nthatwork,themagnetocrystallineanisotropyisexpressed bya\ncharacteristicenergydenotedby T. Hereweshowthat Ttakes\na value within that model such that the anisotropy is strictl y\nzero.\nTo summarize this section, by using a perturbative ap-\nproach, we make the following observations. First, the free -\nelectron Rashba model model gives the magnetocrystalline\nanisotropy that is at least quadratic in αR. Second, the model\ndoes not give perpendicular magnetic anisotropy. Third, th e\nmagnetocrystalline anisotropy vanishes unless only a sing le\nbandisoccupied. We summarizetheresultinFig. 3.\n2. Beyond perturbation: Extension of validity\nSo far, we examined the properties of the free-electron\nRashba model in the perturbative regime. The perturbative\napproach allows gaining insight into the model easily but it\nworks only for small αR. In this section, we go beyond the\nperturbative regime to see if the conclusions we made in the\nprevioussection changewhen αRis not small. We prove that\nthe qualitative results from the perturbative analysis rem ain\nvalidforlargeαRaswell.\nFirst we prove that the magnetocrystalline anisotropy is at\nleast quadratic inαR. For this, we consider the sign reversal5\nE(kx,ky) EF\nkyMinority band\nNo MCA\nNo MCAMajority band\nIMA´αR2\nFIG.3: Summaryof theresults ofthe magnetocrystalline ani sotropy\n(MCA) from the ferromagnetic free-electron Rashba model. T he\nlower band and the upper band are respectively majority and m i-\nnority band. The horizonal and vertical displacements of ea ch\nband are respectively due to Rashba spin-orbit coupling and the ex-\nchange splitting. The diagram shows behaviors of the magnet ocrys-\ntallineanisotropyforeachregion. Thecalculatedmagneto crystalline\nanisotropy shows in-plane magnetic anisotropy (IMA) when e lec-\ntrons in the ground state occupy only the majority band, and t he\nmagnetic anisotropy energy is at least quadratic in the Rash ba pa-\nrameterαR. On the other hand, the magnetocrystalline anisotropy\nvanishes once bothbands are partiallyoccupied inthe groun d state.\nofαR. Thisdoesnotaffect the energyeigenvaluespectrumof\ntheHamiltonianatallsincetheenergyeigenvaluesatisfies the\nproperty, E(kx,ky;αR)=E(−kx,−ky;−αR)[seeEq.(2)]. Since\nthe total energy density cannot change by a rotational trans -\nformation, it should be invariant under αR→−αR. There-\nfore,themagnetocrystallineanisotropymaybeexpandedas a\npowerseriesofα2\nRwiththeleadingordertermproportionalto\nα2\nR[60]. WhenαRbecomes larger, higher order terms in α2\nR\ncan contribute. In Fig. 4, we numerically compute the mag-\nnetocrystallineanisotropydividedby α2\nR. We see that thefirst\nthree curves almost overlap with each other. However, when\nαRbecomes larger so αRkFis comparable to J, the magne-\ntocryatalline anisotropy divided by α2\nRvaries asαRchanges,\nimplyingthebreakdownofthe perturbativeresult[Eq.(7)] .\nAlthough the perturbation theory breaks down quantita-\ntively, qualitative features remain the same for a wide rang e\nofαR. In particular, Fig. 4 shows that the magnetocrystalline\nanisotropy predicted by the free-electron Rashba model is\nnegative(in-planemagneticanisotropy)forlow electrond en-\nsity and vanishes (within the numerical error of our calcu-\nlation) once the total electron density goes above threshol d\nvalue. Perpendicularmagneticanisotropyisnevergenerat ed.\nIt turns out that Eq. (8) can be rigourously proven for ar-\nbitraryαR. Due to its complexity, here we sketch the proof\nonlybriefly. ThedetailedproofispresentedinAppendixB2.\nThe proof proceeds as follows. First, we consider a situatio n\nwhere both bands are occupied for both m=ˆxandm=ˆz,\nwhich occurs if and only if EF≥J[61]. We then use the\nCauchy integral formalism for complex contour integrals to\nshow that Eq. (9) holds beyond the perturbative regime. As\ndiscussedintheprevioussection,Eq.(9)impliesthatthem ag-30 10 20 40 \n-7 -5 -3 -1 Total Electron Density (nm -2 )MCA / αR (eV -1 nm -4 )2αR (eV·nm)\n0.01\n0.05\n0.10\n0.50\n1.00\nFIG. 4: (color online) Numerical computation of the magneto cry-\natalline anisotropy (MCA) divided by α2\nRwithin the free-electron\nRashba model. The results show in-plane magnetic anisotrop y for\nwide range of the Rashba parameter and zero anistropy after a cer-\ntainthresholdwithinthenumericalerror. Notethatthemag netocrys-\ntallineanisotropy isproportional to α2\nRfor small Rashba parameters,\nconfirmingtheresultof our perturbationapproach. Weuse J=1eV\nfor the simulation.\nnetocrystalline anisotropy is independent of the Fermi lev el\nwhenEF≥J. Next, we show that ∆Evanishes in the large\nEFlimit. When combined together these features prove that\n∆Eshould be exactly zero for EF≥J, which is nothing but\nEq.(8).\nHere we emphasize that although Eq. (8) holds for arbi-\ntraryαR,itisveryunstablewithrespecttothemodelvariation\nsinceit iscruciallydependenton Nebeingindependentofthe\nmagnetization m[Eq.(9)],whichholdsonlyfortheidealized\nfree-electronRashba model [Eq. (1)]. Varioustypes of modi -\nficationoftheRashbamodelwhichmakeitmorerealisticcan\nbreakthisindependenceand result in the violationof Eq. (8 ).\nPossible deformations include the change of dispersion fro m\nstrictly quadratic and truncation of the infinite band width to\nfinite width. In the next section, we consider a tight-bindin g\nRashbamodel,whichismorerealisticthantheidealizedfre e-\nelectron Rashba model in the sense that the former has finite\nband width whereas the latter has infinite band width. This\nmodel shows that Eq. (8) is indeed violated and perpendicu-\nlarmagneticanisotropyemerges. Inpassing,we notethat no t\nonly the magnetocrystallineanisotropy but also other prop er-\nties of the idealized free-electron Rashba model are peculi ar.\nA well known example is the intrinsic spin Hall conductiv-\nity [25, 26]. For the idealized free-electron Rashba model, it\nvanishes identically when both bands are partially filled bu t\nforslightlymodifiedRashbamodels[55,56],it isfinite.\nB. Tight-bindingRashbamodel\nWe consider the tight-binding Rashba model for a square\nlattice. From Eq. (6), we use the discrete crystal sym-\nmetry and the Bloch theorem. We define Ck,σ=\n(1/√\nN)/summationtext\np,qexp(ikxpa+ikyqa)Cp,q,σ, whereNis the total\nnumberofsitesand k=(kx,ky)isthecrystalmomentumwithin6\n(b)(a)\n(c)\nMCA (mJ/m 2)MCA (mJ/m 2)9\n6\n3\n0\n-3 \n-6 \n-9 IMA IMA PMA αR (eV·nm) \nαR (eV·Å) 0.02\n0.05\n0.08\nMajority band \nMinority band \n0.0\n0.0 0.5 1.0 1.5 2.00.2 0.4 0.6 0.8 1.0\nNormalized Electron Density \n40 \n30 \n20 \n10 \n0Tight-binding result \nQuadratic fit (MCA~ αR)2(s\u0012- \nN\\D\u0012˭\u0014\u0011\u0013 \u0013\u0011\u0018 \u0010\u0013\u0011\u0018 \u0017\n\u0016\n\u0015\n\u0014\n\u0010\u0014\u0011\u0013 \nFIG. 5: (color online) (a) Energy dispersion of the tight-bi nding\nRashba model [Eq. (10)]. Energies as a function of ky, forkx=0,\naregivenbytheredandbluecurves formajorityandminority bands\nrespectively. The asymmetry between −kyandkyoriginates from\nthe Rashba interaction which is taken to be αR=0.02 eV·nm in\nthis figure. Near the band minimum, the energy dispersion can be\napproximated by a quadratic dispersion [Eq. (2)]. However, near\nthe Brillouin zone boundary, it di ffers significantly. The magnetiza-\ntion direction is taken to be m=(1/2,0,√\n3/2). Depending on the\nRashba parameter and the direction of magnetization, the di spersion\ncan be even more complicated particularly when the two bands ap-\nproach each other. (b) The magnetocrystalline anisotropy ( MCA) as\na function of the total electron density divided by the elect ron den-\nsity for completely filled bands Nmax=2.2×1019m−2. We use the\nRashba parameters αR=0.02 eV·nm, 0.05 eV·nm, and 0.08 eV·nm.\nThe results show in-plane magnetic anisotropy (IMA) for ver y low\nand very high electron occupation. For a wide range of the int erme-\ndiate electron density, it shows perpendicular magneitc an isotropy\n(PMA). (c) The peak values of the magnetocrystalline anisot ropy as\na function of the Rashba parameter. The blue circles are the s imula-\ntion results and the solid line is a quadratic fitting result. J=1 eV,\nme=9.1×10−31kg,anda=0.3nm. Theareaofthetwo-dimensional\nsystem in this simulation is L×LwhereL=60 nm is the length of\neach direction.the Brillouin zone in Fig. 2, which diagonalizes the Hamil-\ntonian. We define the reduced 2 ×2 Hamiltonian h(k) by\nH=/summationtext\nk,σ,σ′C†\nk,σ′[h(k)]σ′,σCk,σ, where [ h(k)]σ′,σis the ma-\ntrix element of h(k) in the 2×2 spin space. Since h(k) is a\n2×2matrix,wecomputetheeigenvaluesexactly.\nE±(k)=−/planckover2pi12\nmea2(coskxa+coskya)±/bracketleftbigg\nJ2m2\nz\n+/parenleftbigg\nJmx+αR\nasinkya/parenrightbigg2\n+/parenleftbigg\nJmy−αR\nasinkxa/parenrightbigg2/bracketrightigg1/2\n.\n(10)\nWe plot Eq. (10) as a function of kyforkx=0 in Fig. 5(a).\nThe formalism given in Eq. (5) provides a way to compute\nthe magnetocrystallineanisotropy. In this section, we pre sent\nthe results for a two-dimensional square lattice only. The r e-\nsult for a two-dimensional hexagonal lattice is presented i n\nAppendixA.\nFigure 5(b) shows the relation between the magnetocrys-\ntalline anisotropyandthe electrondensity(normalizedto one\nwhenbothmajorityandminoritybandsarecompletelyfilled) .\nFor low electron density ( Ne/lessorsimilar0.25Nmax), the system ac-\nquires in-plane magnetic anisotropy. This is understandab le\nin that a parabolic approximation of the dispersion relatio n\n[Eq. (10)] is equivalent to that of the free-electron Rashba\nmodel [Eq. (2)]. However, as the electron density increases ,\nthe parabolic approximation breaks down, thus the system\ncanacquireperpendicularmagneticanisotropyfromthepoi nt\nwhere the effective mass becomes negative ( Ne≈0.25Nmax).\nAfter this point, the perpendicular magnetic anisotropy pe r-\nsist widely, until Ne≈0.75Nmax, covering the whole regime\nwhere the two spin bands overlap, which is in distinct con-\ntrast to the prediction [Eq. (8)] of the idealized free-elec tron\nRashbamodel.\nOur computation shows a similar behavior to a first-\nprinciples calculation [10] on the band filling dependence o f\nthemagnetocrystallineanisotropy. AlthoughasimpleRash ba\nmodel cannot be exact, it providesmuch insight into the sys-\ntem. Changing the electron density by means of substituting\natoms or an external voltage can change not only the magni-\ntudeofthe magnetocrystallineanisotropybutalsoits sign .\nThere are two key di fferences between the tight-binding\nRashba model and the free-electron Rashba model that give\nrise to finite perpendicularmagnetic anisotropy. The first d if-\nference is the deviation of the dispersion from a quadratic.\nIt allows a nonzero magnetocrystalline anisotropy for a wid e\nrange of band filling, due to breakdown of Eq. (9). Once\nthe relation between NeandEFhas a magnetization depen-\ndence,afinitemagnetocrystallineanisotropycanariseeve nif\nboth bands are occupied. The second di fference is finiteness\nof band width (or Brillouin zone). It plays a crucial role for\nthe sign of the magnetocrystallineanisotropy. Since the ba nd\nwidth is finite, there must be both maximum (band top) and\nminimum(bandbottom)energies. Near the band bottom(the\nΓpointinFig.2),thedispersioniselectron-likewithaposi tive\neffective mass. Thus, the theory in Sec. IIIA is relevant, and\nthe sign of the magnetocrystalline anisotropy corresponds to\nin-planemagneticanisotropyforlowelectrondensity. On t he7MCA (mJ/m 2)3\n2\n1\n0\n-1 \n-2 \n0.0 0.5 1.0 J (eV) \n0.3 \n0.6 \n0.9 \nNormalized Electron Density \nFIG.6: (coloronline) Themagnetocrystalline anisotropy( MCA)for\nvarious theexchange energies J=0.3eV,0.6eV,and0.9eVwitha\nfixed Rashba parameter αR=0.05 eV·nm.me=9.1×10−31kg, and\na=0.3nmareused. Thestrongertheexchange energyis,thehigher\nthe magnetocrystalline anisotropy is. On the other hand, th e weaker\nthe exchange energy is, the wider the range of the electron de nsity\nthat acquires perpendicular magnetic anisotropy.\nother hand, near the band top (the M point in Fig. 2), the dis-\npersion is holelike with a negative e ffective mass. Since the\nbehavior is opposite to the electron-like part, the sign of t he\nmagnetocrystallineanisotropy can correspondto perpendi cu-\nlar magnetic anisotropy. As a result, the magnetocrystalli ne\nanisotropynearthebandtopofthemajoritybandcorrespond s\nto perpendicular magnetic anisotropy [Fig. 5(b)]. We remar k\nthat our analysis is similar to that in Ref. [11], which impli es\nthat most important qualitative features of the anisotropy en-\nergy can be understood by analyzing high symmetry points,\nwherebandmaximumandminimumarelocated.\nFigure5(c)indicatesthatthemagnetocrystallineanisotr opy\nisproportionaltoα2\nRinareasonablerangeof αR. Wearguean-\nalytically in Sec. IIIA that the magnetocryatallineanisot ropy\ncan be expanded in terms of α2\nR. The same argument ap-\nplies to this tight-binding Rashba model. We discuss below\nin Sec. IIIC the implication of the sign independence on ex-\nperimental observation of the correlation between the mag-\nnetocrystalline anisotropy and other spin-orbit coupling phe-\nnomena.\nWe nowfix theRashba parameterandcomputethemagne-\ntocrystalline anisotropy for various exchange strengths. Fig-\nure6showstheresult. Thegeneralbehaviorsdiscussedabov e\nremain the same. The weaker Jis, the wider the range of\nthe emergence of perpendicularmagnetic anisotropy is. Thi s\nis because the band overlap between the majority and minor-\nity bands increases as Jdecreases. On the other hand, the\nstrongerJis, the stronger the magnetocrystalline anisotropy\nis. Therefore,weconcludethatmaterialswithstrong Jaread-\nvantageousto achievea strongmagnetocrystallineanisotr opy\nwith high controllability under an external voltage. On theother hand, materials with weak Jare advantageous for per-\npendicularmagneticanisotropiesthatstablyexistsovera wide\nrangeoftheelectrondensity.\nThe mirror symmetryof the magnetocrystallineanisotropy\nin Fig. 5(b) originates from the symmetry between electrons\nat theΓpoint and holes at the M point. From Eq. (10), the\ntotal energy density for completely filled bands is Efilled=\n(2π)−2/integraltextπ/a\n−π/a/integraltextπ/a\n−π/a[E+(kx,ky)+E−(kx,ky)]d2k=0, thus the\nmagnetocrystalline anisotropy at high electron density ca n\nbe computed by hole contributions near the M point. In\nother words,∆E(Nmax−Ne) is the same as the contribu-\ntion from Nenumber of holes. Equation (10) shows the\nsymmetry between the electron-like Γpoint and holelike M\npoint,E±(kx,ky)=−E∓(π/a−kx,π/a−ky), which implies\n∆E(Nmax−Ne)=∆E(Ne). This is a model-specific property.\nForinstance,inAppendixA,westartfromatwo-dimensional\nhexagonallattice for whichthe dispersiondoesnot havesuc h\nsymmetry [Eq. (A3)] and shows that this mirror symmetry\naroundNe=0.5Nmaxisnotgeneral.\nTherearefourkinksinthemagnetocrystallineanisotropyi n\nFig.5(b). Weobservethatthetwokinksaround Ne≈0.3Nmax\nandNe≈0.7Nmaxcorrespond to the bottom of the minority\nband and the top of the majority band, respectively. Since\nthe minority band starts to be occupied from Ne≈0.3Nmax,\nthebehaviorsofthe magnetocrystallineanisotropybelowa nd\nabove this value are di fferent. Similarly, the majority band\nis no longer occupied above Ne≈0.7Nmax. There are two\nmore kinks near Ne≈0.25NmaxandNe≈0.75Nmax. We\nsee that these occur near the point where each band are half\nfilled. Near these points, electrons at the Fermi level is nea r\ninflectionpointsoftheenergydispersionsothee ffectivemass\nchanges its sign. The existence of kinks is quite general as\npresentedinFig.6 andAppendixA.\nTo summarize this section, we perform tight-binding cal-\nculations for the magnetocrystalline anisotropy within a d is-\ncretized Rashba model. The deviation from a quadratic dis-\npersion allows a nonzero magnetocrystalline anisotropy ev en\nwhen both bands are occupied. The finite band width allows\nemergenceof perpendicularmagnetic anisotropy over a wide\nrangeofthetotalelectrondensity. Theresultingmagnetoc rys-\ntalline anisotropyis proportionalto α2\nRfora reasonablerange\nofαR. Even thoughαRbecomes larger than that, the magne-\ntocrystallineanisotropyisindependentofthesignof αRdueto\nsymmetry,andisconstrainedbysymmetryto beevenpowers\nofαR. Theimplicationsofthesignindependenceandcompar-\nison with experiments are discussed in the next section. We\nperformsimilarcalculationsforatwo-dimensionalhexago nal\nlattice as well as a square lattice discussed here. The resul ts\narepresentin AppendixA.\nC. Remarks\nThedependenceofthemagnetocrystallineanisotropyon αR\ndiffersfromthecorrespondingdependenceofmanyotherphe-\nnomenaofspin-orbitcouplingorigin. Intheprevioussecti ons,\nwe show bysymmetrythat the magnetocrystallineanisotropy\nis independent of the sign of αR. As a result, it is quadratic8\ninαRfor a reasonable range of αR. On the other hand, other\nphenomena of spin-orbit origin such as spin-orbit torque an d\nDzyaloshinskii-Moriya interaction have a linear contribu tion\ninαR.\nThis feature has clear experimental implications. When\na magnetic layer has two interfaces with opposite Rashba\nparameters, the total spin-orbit torques and the total\nDzyaloshinskii-Moriyainteractionarisingfromthebothi nter-\nfaces are zero since they are odd in αRand the contributions\nfrom the two interfaces mutually cancel each other. How-\never, such cancellation does not occur for the magnetocrys-\ntalline anisotropy and the contributions from the two inter -\nfaces add up since the anisotropy is even in αR. A similar\nphenomenonpersists evenwhen only oneinterface of a mag-\nnetic layer is subject to strong inversion asymmetry, if the re\nare multiple energybands. It is demonstrated[57] that mult i-\nple bandsfora giveninterfacemayexperiencedi fferentsigns\nof the Rashba spin-orbit coupling. In such a situation, it is\npossiblethatcontributionsofthosebandstothemagnetocr ys-\ntalline anisotropy can add up whereas their contributions t o\nthe linearspin-orbitphenomenasuch asspin-orbittorquea nd\nDzyaloshinskii-Moriya interaction tend to cancel out. Thi s\nobservation indicates that simple proportionality analys is in\nexperiments may fail to capture the correlation between the\nmagnetocrystalline anisotropy and other phenomena of spin -\norbitcouplingorigin.\nIn this sense, our observation can be consistent with a\nrecent experiment [48] reporting the opposite behaviors of\nthe Dzyaloshinskii-Moriya interaction and the perpendicu lar\nmagnetic anisotropy in Ir /Co/AlOxmultilayers for various\nthickness of Co. According to the work, the Dzyaloshinskii-\nMoriya interaction reduces as the thickness of Co increases ,\nwhile the perpendicular magnetic anisotropy increases. Th is\ndifference may originate from multiple origins of the spin-\norbit coupling phenomena, such as multiple interfaces and\nmultiple orbital bands. As the thickness of Co increases, th e\ncontributions to the Dzyaloshinskii-Moriya interaction m ay\ncancel out while those to the magnetocrystalline anisotrop y\nshould add up. One remark is in order. Although our the-\norydemonstratethatthepositivecorrelationbetweenthem ag-\nnetocrystalline anisotropy and other spin-orbit coupling phe-\nnomena may breakdown,it is not necessarily the explanation\nofthebreakdownobservedinRef.[48]becausethereareothe r\nsourcesofmagnetocrystallineanisotropy.\nWeobservethatthemagnetocrystallineanisotropydepends\non the total electron density [Fig. 5(b)] and it can even\nchangeitssign. Thestrongdependenceofmagnetocrystalli ne\nanisotropyon the total electrondensityis anotherfeature that\nrequires a well-controlled experiment to observe the corre la-\ntion. When one varies the experimental conditions to obtain\nsystemswithvariousspin-orbitcouplingparameters,thet otal\nelectron density at the interface may change, which disturb s\nclear interpretation of the dependence of the magnetocrys-\ntallineanisotropyonthespin-orbitcouplingparameter.\nThe density-dependent magnetocrystalline anisotropy\nopens another route of the voltage-controlled magnetic\nanisotropy [16, 42–46]. The voltage-controlled magnetic\nanisotropy received considerable attention due to its sign ifi-cant potential to enhance the e fficiency of spintronic devices.\nThere are previoustheories[40, 41] suggestingthat modula t-\ningtheRashbaparameterbyapplyinganexternalvoltageisa\npossible route of the voltage-controlled magnetic anisotr opy.\nHowever, it is unlikely to be a main mechanism in metallic\nferromagnetic films in which a nominal potential gradient is\nnot a main mechanism generating Rashba parameters [57].\nAn external electric field is shielded by electron screening in\nthe metal, thus it is di fficult to change the Rashba parameter\nsignificantly. On the other hand,densityvariationsbydopi ng\nor an external voltage can change the electron density at\nthe interface, changing the interfacial contributions to t he\nmagnetocrystalline anisotropy significantly. The conclus ion\nfrom the simple model is consistent with first-principle\nstudies[43,58].\nIV. CONCLUSION\nIn conclusion, we compute the magnetoctrystalline\nanisotropy for simple ferromagnetic Rashba models. The\nproperties dramatically change depending on the dispersio n\nrelations. For a free electron (quadratic) dispersion, the sys-\ntem does not acquire perpendicular magnetic anisotropy at\nall. More interestingly, we analytically show that the mag-\nnetocrystalline anisotropy is exactlyzero regardless of the\nRashbacouplingstrengthifbothmajorityandminorityband s\nare partially occupied in the ground state. This result is no t\nconsistent with experimental observations, implying that a\nfreeelectrondispersionisnotsuitableforstudyingperpe ndic-\nularmagneticanisotropyarisingfromtheRashbainteracti on.\nWe thus generalize the model to have a finite band width,\nwhich necessarily generates deviation from the free electr on\ndispersion. We start from tight-binding Hamiltonians and\nconclude that the system acquires perpendicular magnetic\nanisotropyoverwiderangeofparameters,consistentwithe x-\nperimental observations. A finite band width is a crucial fea -\nture of the tight-binding Hamiltonians that gives rise to pe r-\npendicular magnetic anisotropy. We also observe that the\nmagnetocrystallineanisotropydependsonthebandfillinga nd\nit can even change its sign. We argue that the interface elec-\ntrondensity modulationby voltage is a moreimportantcause\nof voltage-controlled magnetic anisotropy than the voltag e-\ncontrolledmodulationoftheRashbaparameteris.\nOur results show the possibility of breakdown of positive\ncorrelation between perpendicular magnetic anisotropy an d\nother spin-orbit coupling phenomena. In particular, if the re\nare multiple sources of spin-orbit coupling phenomena, suc h\nasmultipleinterfacesandmultipleorbitalbands,experim ental\nobservationofthecorrelationrequirescarefulanalysis.\nAcknowledgments\nThe authors acknowledge R. D. McMichael, C.-Y. You,\nP. M. Haney, and J. McClelland for critical reading of the\nmanuscript. K.W.K. acknowledges support under the Coop-\nerativeResearchAgreementbetweentheUniversityofMary-9\nland and the National Institute of Standards and Technol-\nogy, Center for Nanoscale Science and Technology, Grant\nNo. 70NANB10H193, through the University of Mary-\nland. K.W.K. was also supported by Center for Nanoscale\nScience and Technology, National Institute of Standards\nand Technology, based on a Collaborative Research Agree-\nment with Basic Science Research Institute, Pohang Uni-\nversity of Science and Technology. K.J.L was supported\nby the National Research Foundation of Korea (Grant No.\n2011-028163,2015M3D1A1070465). H.W.L. was supported\nby the National Research Foundation of Korea (Grant No.\n2013R1A2A2A05006237)and the Ministry of Trade, Indus-\ntryandEnergyofKorea(GrantNo. 10044723).\nAppendixA: Tight-bindingRashbamodel for a\ntwo-dimensionalhexagonal lattice\nThe two-dimensional hexagonal lattice we use here is pre-\nsentedinFig.7. We constructatight-bindingHamiltonianb y\nthe same way illustrated in Sec. II. First we define the elec-\ntron annihilation operator Cp,q,σat the site ( p,q) and spinσ.\nTheindicesofasitearedefinedbyassigningitspositiontob e\npaˆu1+qaˆu2whereˆu=ˆxandˆu2=(1/2)ˆx+(√\n3/2)ˆy. Then,\ntheHamiltonianis\nH=HK+HJ+HR, (A1a)\nwhere\nHK=−tk/summationdisplay\npqσ/parenleftig\nC†\np+1,q,σCp,q,σ+C†\np,q+1,σCp,q,σ\n+C†\np+1,q−1,σCp,q,σ/parenrightig\n+h.c., (A1b)\nHJ=J/summationdisplay\npqσσ′/bracketleftig\nC†\np,q,σ(σ)σ,σ′Cp,q,σ′/bracketrightig\n·m, (A1c)\nHR=itR/summationdisplay\npqσσ′/bracketleftig\nC†\np+1,q,σ[(ˆu1׈z)·σ]Cp,q,σ\n+C†\np,q+1,σ[(ˆu2׈z)·σ]Cp,q,σ\n+C†\np+1,q−1,σ[((ˆu1−ˆu2)׈z)·σ]Cp,q,σ/bracketrightig\n+h.c.,\n(A1d)\nwheretkandtRare hoppingparametersto bedetermined.\nByusingBlochtheorem,theHamiltoniancanbewrittenby\n2×2matrix,ofwhichthe eigenvaluesareexactlygiven.\nE±(k)=−2tkcoskxa+2coskxa\n2cos√\n3kya\n2±˜J,(A2)\n˜J2=J2m2\nz+Jmx+2√\n3tRcoskxa\n2sin√\n3kya\n22\n+Jmy−2tRsinkxa\n2cos√\n3kya\n2−2tRsinkxa2\n.\n(A3)(p,q)\nau2\nu1 (p+1,q)(p,q+1)\n(p+1,q-1)\nxy\nFIG. 7: The model for a two-dimensional hexagonal lattice. H ere\nais the lattice constant, pandqare the position indices. There are\ntwoprincipal directions ˆu1andˆu2. Then, the positionvector of each\nsite ispaˆu1+qaˆu2. The labeled sites by ( p,q+1), (p+1,q), and\n(p+1,q−1)arethe neighboring hoppingsites. Theotherthreesites\nare captured byadding hermitianconjugates of these.\nMCA (mJ/m 2)15 \n10 \n5\n0\n-5 \n-10 \n0.0 0.5 1.0\nNormalized Electron Density αR (eV·nm) \n0.02 \n0.05 \n0.08 \nFIG. 8: (color online) The magnetocrystalline anisotropy c omputed\nwithin a tight-binding Rashba model for a two-dimensional h exago-\nnal lattice result (Fig.5). The overall features are the sam e as a two-\ndimensional square lattice,except the absence of the mirro r symme-\ntryaround the normalizedelectron density Ne=0.5Nmax.\nThe next step is determining tkandtR. For a continuum\nlimituptoO(a2),\nE±=−6tk+3\n2tka2k2±/radicalig\nJ2+6JtRa(mxky−mykx)+9t2\nRa2k2.\n(A4)\nThis should coincide with the continuum dispersion Eq. (2)\n(up to a constant energy shift). Therefore, we obtain tk=\n/planckover2pi12/3mea2andtR=αR/3a.\nWe now compute the magnetocrystalline anisotropy by\nthe same way. The result is shown in Fig. 8. The fea-\ntures discussed in the main text are valid, except the model-10\nspecific property of a square lattice that the magnetocrys-\ntallineanisotropyismirrorsymmetricaround Ne=0.5Nmax.\nAppendixB: Detailsof theanalytic theories\n1. Perturbationtheoryfor afree-electron Rashbamodel\nThe aim of this section is to present the mathematical\nderivationsofEqs.(7)–(9)fromEq.(2)upto O(α2\nR). Through-\noutthissection,we discardall termsbeyond O(α2\nR).\nFirst we compute N±(E) from the energy dispersion. Ex-\npandingEq.(2)upto O(α2\nR),thedispersionrelationisapprox-\nimatedbythefollowingquadraticfunction.\nE±(kx,ky)=/planckover2pi12k2\nx\n2˜mx\n±+/planckover2pi12(ky±k0\ny)2\n2˜my\n±±J−meα2\nRsin2θ\n2/planckover2pi12,(B1)\nwhereθis defined by m=(sinθ,0cosθ), the spin-dependent\nband shift is given by k0\ny=meαRsinθ//planckover2pi12, and the renormal-\nizedmassesare\n1\n˜mx\n±=1\nme±α2\nR\nJ/planckover2pi12,1\n˜my\n±=1\nme±α2\nRcos2θ\nJ/planckover2pi12.(B2)\nThen, (2π)2N±(E) is given by the area of the contour of E=\nE±(kx,ky) inkspace (See Fig.1). Since E=E±(kx,ky) forms\nanellipse,the areaisanalyticallycomputable.\n(2π)2N±(E)=2π/radicalbig\n˜mx\n±˜my\n±\n/planckover2pi12(E−E±\nmin).(B3)\nFrom Eq. (B1), we obtain the band bottom energies Emin\n±by\nsubstituting k=(0,∓k0\ny).\nE±\nmin=±J−meα2\nRsin2θ\n2/planckover2pi12. (B4)\nWe arenowreadyto computeEq.(3).\nEquation (7) is derived by putting EF<E+\nminandη=0.\nThenNe(E)=N−(E). Inverting the function, we obtain the\nFermilevelasafunctionofthetotalelectrondensity εF(Ne)=\n(2π/planckover2pi12//radicalbig\n˜mx\n−˜my\n−)Ne+E−\nmin. Equation (3) (as a function of Ne)\nis\nEtot(εF(Ne))=/integraldisplayεF(Ne)\nE−\nminEdNe\ndEdE=/integraldisplayNe\n0εF(Ne)dNe\n=π/planckover2pi12\n/radicalbig\n˜mx\n−˜my\n−N2\ne+E−\nminNe. (B5)\nHere, at the first line, we change the variable from EtoNe\nby Eq. (B3). Keeping in mind that the renormalized masses\nandE−\nminhave angular dependence, we end up with the mag-\nnetocrystallineanisotropy\n∆E(Ne)=Etot(εF(Ne))|θ=π/2−Etot(εF(Ne))|θ=0\n=−Nemeα2\nR\n2/planckover2pi12/parenleftigg\n1−Ne\nN−(E+\nmin)/parenrightigg\n, (B6)whereN−(E+\nmin)=(Jme/π/planckover2pi12)+O(α2\nR) from Eqs. (B3) and\n(B4). ThisisEq.(7).\nTo derive Eq. (8), we start from taking derivative of\nEq.(B3)with respectto E,\ndN±\ndE=/radicalbig\n˜mx\n±˜my\n±\n2π/planckover2pi12. (B7)\nTherefore,Eq.(5) for η=1 isgivenby,aftersomealgebra,\nEtot(EF)=2πme(E2\nF−J2)\n/planckover2pi12, (B8)\nwhich is independent of m. We then combine Eq. (B3) with\nEq.(B4)to endupwith\n(2π)2[N+(EF)+N−(EF)]=4πme(meα2\nR+EF/planckover2pi12)\n/planckover2pi14,(B9)\nwhichis nothingbutEq.(9). Invertingthefunction,\nεF(Ne)=π/planckover2pi12\n2meNe−meα2\nR\n/planckover2pi12. (B10)\nCombining Eqs. (B8) and (B10), Etot(ε(Ne)) has no angular\ndependence. ThusEq.(5)is\n∆E(Ne)=Etot(εF(Ne))|θ=π/2−Etot(εF(Ne))|θ=0=0,(B11)\nwhenbothbandsareoccupied. ThisisEq.(8).\n2. Exact theoryfor afree-electronRashbamodel\nThe purpose of this section is to show that Eq. (8) holds\nregardlessofhowlarge αRis. Theflowoftheproofissketched\nin Sec. IIIA. We first show that i) Eq. (9) is exact above the\ntotal electron density at which both bandsare occupied. Thi s\nimpliesthat the magnetocrystallineanisotropyis indepen dent\nofEFin this density range, which amounts to EF≥J. Then\nwe show that ii) lim EF→∞∆E=0. We prove this by showing\nthat∆EgoesO(E−1\nF)at mostforlarge EFlimit. Combiningi)\nand ii), we end up with the result that the magnetocrystallin e\nanisotropyisexactlyzero[Eq.(8)].\na. Proof of Eq.(9) for large αR\nWe prove Eq. (9) by using the contour integral technique,\nmainly, the Cauchy integral theorem. We do not assume that\nαRis small.\nWe assumethatbothbandsareoccupiedforall m. We first\nprove that this is equivalent to EF≥J. To show the forward\npart of this equivalence, we take m=ˆz. Then,E+\nmin=J,\nthusEFshould be greater than or equal to Jfor the minority\nbandto be occupied. To provethe backwardpart, we assume\nEF≥J. Fork=0,E±(kx,ky)=±J≤J≤EF. Therefore,\nk=0stateisoccupiedforbothbands. Onecorollaryfromthe\nproofisthat k=0isalwaysoccupiedwhen EF≥J.11\nWe start from Eq. (2) with m=(sinθ,0,cosθ) for 0≤\nθ≤π/2. We changethe variables( kx,ky) to a single complex\nvariablez=i(kx+iky). Intermsof z,\nE±(z)=/planckover2pi12z∗z\n2me±/radicalig\nJ2cos2θ+α2\nR(z−w)(z∗−w),(B12)\nwherew=Jsinθ/αR>0.\nFor a given EF,N±(EF) is given by the area enclosed by\nE±(z)=EF(Fig.1). ByGreen’stheorem,theareaisgivenby\n(2π)2N±=/integraldisplay\nD±dkxdky=/integraldisplay\nC±kxdky−kydkx\n2=1\n2i/integraldisplay\nC±z∗dz,\n(B13)\nwhereD±={z|E±(z)≤EF}is the set of occupied states and\nC±={z|E±(z)=EF}is the boundary of D±, that is, the con-\ntouroftheFermilevel. Toperformtheintegration,weexpre ss\nz∗asa functionof z. By equating E±(z)=EFandsolving z∗,\nz∗\n±or∓=2me\nz2/planckover2pi14/bracketleftig\nmeα2\nR(z−w)+EF/planckover2pi12z±/radicalbig\nR(z)/bracketrightig\n,(B14)\nR(z)=[meα2\nR(z−w)+EFz/planckover2pi12]2−z2/planckover2pi14(E2\nF−J2+α2\nRwz).\n(B15)\nHerez∗\n±are functionsof zwhichsatisfy z∗=z∗\n±(z) onC±. We\ndenote the subscript by ±or∓since it is ambiguous which\none corresponds to the majority band and the minority band.\nHowever, it does not a ffect the final result. The total electron\ndensityisthengivenby\n(2π)2Ne(EF)=1\n2i/integraldisplay\nC+z∗\n+dz+1\n2i/integraldisplay\nC−z∗\n−dz.(B16)\nTheCauchyintegraltheoremimpliesthatthecomplexcon-\ntour integrals in Eq. (B16) is equivalent to those around non -\nanalyticalpointsonly. FromEq.(B14),therearetwotypeso f\nnonanalytic points of z∗\n±. The first one is the pole at z=0.\nWe call this the trivial pole. We show at the beginningof this\nsection that ( kx,ky)=0 is occupied for both bands. That is,\nthetrivialpole z=0isalwaysin D±(SeeFig.9). Thesecond\ntype comes from the square root function. Since the square\nroot function is multivalued in the complex plane, there are\nbranch cuts which connect the branch points that are defined\nby the zeros of R(z). The whole branch cuts are nonanalytic\npoints. Thus, it is important to see the behavior of the zeros\nofR(z). SinceR(z)isacubicpolynomial,therearethreezeros\nofR(z). Below we present three properties of the three zeros\nwithoutproofs. TheproofsarepresentedinAppendixC.\nThe first property is that i) all three zeros of R (z)are real\nand nonnegative if E F≥J. We call the zeros r1,r2, andr3,\nsatisfying r1≤r2≤r3. Another important result is that ii)\nri∈D−is equivalent to r i∈D+. Intuitively, we may say\nthat, ifriis inside the contour C−, it is also inside the contour\nC+[62]. Since D+⊂D−,onedirectionoftheproofisobvious,\nbut the other direction is not. The last property is that iii) no\nor two zeros of R (z)are in D±(orinsideC±). As a result, the\nsituation is summarized in Fig. 9. We observe that D−−D+\nisanalytic. Therefore,whenweshrinktheintegralcontour by\nusing the Cauchy integral theorem, we can end up with the\nsamecontour C0+Crforbothtermsin Eq.(B16).By using the Cauchy integral theorem, both terms in\nEq.(B16)sharethesameintegralcontour.\n(2π)2Ne(EF)=1\n2i/integraldisplay\nC0+Cr(z∗\n++z∗\n−)dz.(B17)\nIf no zeros of R(z) is inD±,C0is the only relevant contour.\nHowever,we below show that contributionsfrom Crare can-\ncelled outwhen we addup z∗\n+andz∗\n−. One remarkis in order.\nThe situation becomes complicated if any of riis exactly on\nC±. For this case, defining C±bypassing riwith an infinites-\nimally small radius does not change the result. Another res-\nolution is using continuity of Ne(EF). Since one of rican be\nexactlyon C±onlyatparticularvaluesof EF,wemayexclude\nthe particularpointsin the proofanduse the continuityto g et\nNe(EF) forthewholedomain.\nTheresult greatlysimplifiesthesituation. Thecomplicate d√R(z) terms in z∗\n+andz∗\n−are cancelled out when they are\naddedup.\n(2π)2Ne(EF)=2me\ni/planckover2pi14/integraldisplay\nC0+Crα2\nRme(z−w)+EF/planckover2pi12z\nz2dz\n=4πme\n/planckover2pi12Res\nz=z0α2\nRme(z−w)+EF/planckover2pi12z\nz2\n=4πme(α2\nRme+EF/planckover2pi12)\n/planckover2pi14, (B18)\nwhich is exactly Eq. (9). At the second line, we use the\nCauchy’sresiduetheorem.\nTheimportanceoftheassumptionthatbothbandsareoccu-\npiedinthisproofis twofold. First, theconditionisequiva lent\ntoEF≥Jsothatthezerosof R(z)satisfythepropertiesproven\nin Appendix C. The properties guarantee that the integrands\nin Eq. (B16) are analytic in D−−D+so that we can shrink\ntheintegralcontoursforbothbandstothesamecontour. Sec -\nondandmoreimportantly,thecomplicatedcontributionsfr om\n±√R(z) are cancelled out when we add up the contributions\nfrombothbands. Therefore,we can use theCauchy’sresidue\ntheoremforthetrivialpole z=0 only.\nb. Proofof limEF→∞∆E=0\nFor extremely large EF, the contour of the Fermi level is\nsimple. Therefore, we can define Fermi momenta for each\nband as a function of the azimuthal angle of the momentum.\nWe write k=(kcosφ,ksinφ). Then, the Fermi momentum\nkF,±is defined by E±(kFcosφ,kFsinφ)=EF. For simplicity\nof equations, we assume αR>0, but the flow of the proof is\nthe same for general αR. From Eq. (2) and by putting m=\n(sinθ,0,cosθ),\nkF,±=/radicalbigg\n2meEF\n/planckover2pi12∓meαR\n/planckover2pi12+/radicalbiggme\n8EFmeα2\nR∓2J/planckover2pi12sinθsinφ\n/planckover2pi13\n∓J2\n4αREF(1−sin2θsin2φ)+O(E−3/2\nF). (B19)12\nC+D+\nCr C0C-D-Im[z]=k x\nRe[z]=-ky\ntrivial pole\nNo branch cut in D--D +)(No branch point in D--D +Analytic in D--D +branch cutsr3  r1  r2  \nFIG. 9: (color online) Complex contour integral for Ne. HereC±are the integral contours for N±, andD±are the enclosed region (blue and\nred for+an−) respectively. The white X is the trivial pole and the magent a Xs are the branch points of the integrand. The trivial pole i s at\nz=0 and the branch points are on the real axis and denoted by r1,r2, andr3. In Appendix C, we show that ri∈D+for no or two of riand\nri/nelementD−for the others. It is also shown that 0 ∈D+ifEF≥J. Thus, we define the branch cuts (magenta lines) by connectin gr1andr2, and\nconnecting r3and a complex infinity. Therefore, the integrands in Eq. (B16 ) are analytic in D−−D+. We now can shrink the integral contour\nC±toC0+Cr(yellow)bytheCauchyintegraltheorem, where C0isacontour surrounding thetrivialpole,and Crisa contour surrounding the\nbranch cut definedby r1andr2. Ifevenr1andr2arenot in D+,C0is the onlyrelevant contour and Crisoutside D±. Bothcases give the same\nmathematical results.\nBy using the polar coordinate, the total energy density belo w\ntheFermisea is\nEtot(EF)=1\n(2π)2/integraldisplay2π\n0dφ/parenleftigg/integraldisplaykF,+\n0kE+dk+/integraldisplaykF,−\n0kE−dk/parenrightigg\n(B20)\nwecanexpandtheintegrandwithrespectto1 /kandintegrate\ntermbytermsince kF,±isO(E−1/2\nF). Aftertediousalgebra,we\nendupwith\nEtot(EF)=(θ-independentterms) +O(E−1\nF).(B21)\nTherefore,∆E=O(E−1\nF) at most, which proves that\nlimEF→∞∆E=0.\nAppendixC: Propertiesof zeros of R(z)\nInthissection,weprovesomeimportantpropertiesofzeros\nofR(z)definedbyEq.(B15). Since R(z)isacubicpolynomial,\nit has three zeros. We call these rifori=1,2,3. We below\nshowthatall of riarereal. Therefore,we candenote ribythe\norder of its magnitude r1≤r2≤r3. This section consists of\nthreesubsectionseach ofwhichcorrespondsto eachpropert y\nthatwe mentioninthemaintext.1. Allof riare real andnonnegative if EF≥J\nWe write down R(z)=az3+bz2+cz+d. Then,the coeffi-\ncientsare\na=−α2\nR/planckover2pi14w<0, (C1a)\nb=m2\neα4\nR+2mEα2\nREF/planckover2pi12+J2/planckover2pi14>0,(C1b)\nc=−2meα2\nRw(meα2\nR+EF/planckover2pi12)<0, (C1c)\nd=m2\neα2\nRw2>0. (C1d)\nZeros of a cubic polynomial az3+bz2+cz+dare all real\nif and only if∆=18abcd−4b3d+b2c2−4ac3−27a2d2is\nnonnegative. Aftersomealgebra,\n˜∆=4(E2\nF−J2)(α2+2αEF+J2)2−27α2J4t2\n+4αJ2(α+EF)[(α+EF)2−9(E2\nF−J2)]t,(C2)\nwhere˜∆=∆/J2/planckover2pi112m2\neα2\nRt,α=meα2\nR//planckover2pi12andt=cos2θ. We\ntreat˜∆asafunctionof t.˜∆(t)isquadraticandthedomainof t\nis0≤t≤1. Aftersomealgebra,\n˜∆(0)=4(E2\nF−J2)(α2+2EFα+J2)2≥0, (C3)\n˜∆(1)=(J2−2αEF)2[(α+2EF)2−4J2]≥0,(C4)\n˜∆ext=4\n27[3E2\nF−3J2+(α+EF)2]3>0, (C5)13\nifEF≥J. Here˜∆extis the extremumvalue of ˜∆(t) evaluated\nat the value tsatisfying ˜∆′(t)=0. Since the boundary val-\nuesandthe extremumvalueare all nonnegative, ˜∆(thus∆) is\nnonnegativeon0≤t≤1,provingall of riarereal.\nTo showri≥0 foralli, we see the signsof the coe fficients\nin Eq. (C1). It is easy to see that R(−z)>0 for any real and\npositivez. Therefore, R(z)hasnonegativerealzero.\n2.ri∈D+isequivalentto ri∈D−\nThis statement is equivalent to that any branch point of z∗\n±\ncannotbein D−−D+. Itisoneofthemostimportantproperties\nthat allows us to draw Fig. 9. Since D+⊂D−,r∈D+⇒r∈\nD−isstraightforward,buttheotherdirectionisnot.\nTo prove this, we use the definition of D±thatri∈D±is\nequivalent to E±(ri)−EF≤0. We start from the following\nidentity.\n[E+(z)−EF][E−(z)−EF]=meα2\nR\n/planckover2pi12z−w\nz+EF−/planckover2pi12z∗z\n2me2\n−R(z)\nz2/planckover2pi14. (C6)\nSinceR(ri)=0,thesecondterminthe right-handside iszero\nwhenz=ri. Inaddition,we showthat rishouldbe realin the\nprevious section. Therefore, the first term in the right-han d\nside isnonnegativewhen z=ri.\n[E+(ri)−EF][E−(ri)−EF]≥0. (C7)\nIn the main text, we exclude the case where any riis exactly\nonC±. Thus, we may assume E±(ri)−EF/nequal0. Under this\nassumption, Eq. (C7) implies that E+(ri)<EFis equivalent\ntoE−(ri)<EF. In other words, ri∈D+is equivalent to\nri∈D−foranyrisatisfying R(ri)=0.\n3. Onlyeven numberof riare inD±\nIn the previous section, we show that the branch points of\ntheintegralEq.(B16)( ri)arenotin D−−D+. Whatisimpor-\ntantisnotonlythebranchpointsbutalsothebranchcuts. Th e\nbranchcuts are defined by connectinga pair of branchpoints\n(includingthecomplexinfinityifthenumberofbranchpoint s\nare odd). To show that any branchcut doesnot have an inter-\nsectionwith D−−D+,onlyevennumberof rishouldbein D±\n(SeeFig. 9).\nThe following lemma is useful for the proof: ri∈D±is\nequivalent to E F≥/planckover2pi12r2\ni/2me. This lemma is a corollary of\nthe previous section. With this lemma, we do not need to\ncomputeE±(ri) andcompareto EFin orderto check ri∈D±.\nInstead,we only compare rito√2meEF//planckover2pi1[63]. Therefore,it\nprovidesa usefulcriteriontocheck ri∈D±.\nWe first prove ri∈D±⇒EF≥/planckover2pi12r2\ni/2me. Sinceri∈D±,\nEF≥E+(r)≥/planckover2pi12r2\ni/2me, which is the desired result. We next\nproveEF≥/planckover2pi12r2\ni/2me⇒ri∈D±. Notethat EF≥/planckover2pi12r2/2me>\nE−(ri), thusri∈D−. In the previous section, we show that(a)\nNone of ri in D±R(z)\n(b)+∞ \n-∞ kF,0 r1r2r3z\nr1 and r2 in D±R(z)\n+∞ \n-∞ r1r2r3zkF,0 \nFIG. 10: (color online) Two possibilities of the number of riinD±\nsatisfying Eq. (C9). Since R(kF,0)>0, the only possible domains in\nwhichkF,0can be present are (a) kF,0<r1and (b)r2<kF,0<r3.\nThe number of riless thankF,0is the number of riinD±. Therefore,\neither noor twoof riare inD±,leading us toFig.9.\nri∈D−is equivalent to ri∈D+. Therefore, ri∈D±, which\ncompletestheproof.\nAsaresult,thestatementthatwewanttoshowisequivalent\nto the statement that “only even number of risatisfyri≤kF,0\nwherekF,0=√2meEF//planckover2pi1.”Aftersomealgebra,\nR(kF,0)=2meEFJ2/planckover2pi12cos2θ+meα2\nR(kf,0−w)2(meα2\nR+2EF/planckover2pi12).\n(C8)\nTherefore, R(kF,0) is positive unless θ=π/2 andEF=\n/planckover2pi12w2/2me. The latter case is not our interest because of the\nfollowing argument. Note that R(w)=/planckover2pi14w2J2cos2θ, thus,\nwis a zero of R(z) whenθ=π/2. Since EF=/planckover2pi12w2/2me,\nwis exactly at the Fermi level (on C±). In the main text, we\nexcludethissituation. Asaresult,we nowhave\nlim\nz→∓∞R(z)=±∞,R(kF,0)>0. (C9)\nSinceR(z)hasthreerealandnonnegativezeros,thereareonly\ntwo possibilitiesaspresentedin Figs. 10(a)and10(b)resp ec-\ntively. 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Sinova, N. A. Sinitsyn, and A. H. MacDonald ,\nDependence of the intrinsicspin-Hall e ffecton spin-orbit inter-\naction character, Phys.Rev. B 72, 165316 (2005).\n[57] J.-H.Park,C.H.Kim,H.-W.Lee,andJ.H.Han,Orbitalch iral-\nity and Rashba interaction in magnetic bands, Phys. Rev. B 87,\n041301(R) (2013).\n[58] K. Kyuno, J.-G. Ha, R. Yamamoto and S. Asano, First-\nPrinciples Calculation of the Magnetic Anisotropy Energie s of\nAg/Fe(001) and Au/Fe(001) Multilayers, J. Phys. Soc. Jpn. 65,\n1334 (1996).\n[59] Wecanignore the O(α2\nR)termhere since Eq.(7)isalreadypro-\nportional toα2\nR.\n[60] By the same transformation, we also end up with that the t otal\nenergy density isexpanded by(1 −m2\nz).\n[61] Itisstillpossiblefortheminoritybandtobeoccupied form=ˆx\nbut not for m=ˆz. This case is not eligible for the proof. The\nassumption requires that the minority band should be occupi ed\nregardless ofthedirectionofmagnetization. Formore info rma-\ntion, see Appendix B2.\n[62] Thisstatementprovidesanintuitiveunderstandingof theresult,\nbut itis technicallysubtle ifthe contour C±is not simple.\n[63] The nonnegativity of riplays a crucial rolefor deducing this."    },    {        "title": "1608.07429v1.Optimization_of_nanocomposite_materials_for_permanent_magnets_by_micromagnetic_simulations__effect_of_the_intergrain_exchange_and_the_hard_grains_shape.pdf",        "content": "arXiv:1608.07429v1  [cond-mat.mtrl-sci]  26 Aug 2016Optimization of nanocomposite materials for permanent mag nets by\nmicromagnetic simulations: effect of the intergrain exchan ge and the hard\ngrains shape\nSergey Erokhin and Dmitry Berkov\nGeneral Numerics Research Lab, Moritz-von-Rohr-Strasse 1 A, D-07749 Jena, Germany\nIn this paper we perform the detailed numerical analysis of r emagnetization processes in nanocom-\nposite magnetic materials consisting of magnetically hard grains (i.e. grains made of a material with\na high magnetocrystalline anisotropy) embedded into a magn etically soft phase. Such materials are\nwidely used for the production of permanent magnets, becaus e they combine the high remanence\nwith the large coercivity. We perform simulations of nanoco mposites with Sr-ferrite as the hard\nphase and Fe or Ni as the soft phase, concentrating our efforts on analyzing the effects of ( i) the\nimperfect intergrain exchange and ( ii) the non-spherical shape of hard grains. We demonstrate tha t\n- in contrast to the common belief - the maximal energy produc t is achieved not for systems with\nthe perfect intergrain exchange, but for materials where th is exchange is substantially weakened.\nWe also show that the main parameters of the hysteresis loop - remanence, coercivity and the en-\nergy product - exhibit non-trivial dependencies on the shap e of hard grains, and provide detailed\nexplanations for our results. Simulation predictions obta ined in this work open new ways for the\noptimization of materials for permanent magnets.\nPACS numbers: 75.40.Mg, 75.50.Ww, 75.50.Tt, 75.60.-d\nI. INTRODUCTION\nMagnetic materials which can be used for the\nmanufacturing of permanent magnets belong to\nthe key materials in many high-technology applica-\ntions today1. Modern electromotors, actuators and\nmagnetic-field-basedsensors(to name only a few ap-\nplications) require high-performance magnets. The\neffectivenessofthesemagnetsisusuallyestimatedby\nthe value of the maximal energy product ( BH)max\nachieved in the second quadrant of their B−Hhys-\nteresis loop. One of the most promising ways to\nobtain large values of this parameter is the usage\nof magnetic nanocomposites, i.e.materials which\ncombine a high coercivity of a magnetically hard\nphase (the phase made of a material possessing a\nlarge magnetocrystalline anisotropy) with the high\nsaturationmagnetizationoftheanother(soft)phase.\nAt present, best performance magnets are pro-\nduced basing on rare-earth metals (NdFeB, SmCo)\nand employing the precise fabrication control (see,\ne.g.2). Unfortunately, these magnets are relatively\nexpensive and subject to the availability and price\nfluctuations due to the high volatility of the rare\nearth elements market.\nAnother very important class of materials for per-\nmanent magnets is represented by nanocomposites\ncontaining ferrites as the hard phase. The energy\nproduct of Co-, Ba- and Sr-ferrite-based magnets is\nsufficient for many applications of permanent mag-\nnets, e.g. in microwave devices, telecommunication,recording media, and electronic industry. Another\nimportant advantage of these materials have a much\nbetter temperature and corrosion resistance than\nNdFeB-based magnets (see, e.g. Ch. 12.2 in3). In\naddition, situation with the production of ferrite-\nbased magnets is more stable, and costs are much\nlowerdue to the wideravailabilityofthe correspond-\ning raw materials.\nThe specified performance of a nanocomposite\nmaterial can in principle be achieved by tailoring\nvarious parameters of a nanocomposite, such as rel-\native fractions of the soft and hard phases, size\nand shape of hard grains, mutual arrangement of\ngrains belonging to different phases, and the qual-\nity of the intergrain boundaries. The development\nof new magnets of any type requires the thorough\nunderstanding of the relationship between their mi-\ncrostructure and magnetic properties4. Advanced\nstructural experimental techniques can provide a\nvery important information; well known examples\nare, e.g. the electron Bragg scattering diffraction\nstudies of the grain alignment in sintered NdFeB5or\nthe X-ray diffraction along with the high-resolution\ntransmission electron microscopy applied for the\nmeasurements of the grain size distribution in soft\nmagnetic alloys6.\nRecent publications of different scientific collabo-\nrations have shown that in order to study the ques-\ntion ofthe microstructure-magnetismrelationshipin\ndetails, it is absolutely necessary to employ numeri-\ncal modeling - in particular, micromagnetic simula-tions, combining it with other experimental meth-\nods like 3D atome probe7, energy-dispersive X-ray\nspectroscopy8or magnetic neutron scattering9,10.\nThe usage of micromagnetic modeling in the de-\nvelopment process allows the a priori performance\noptimization of permanent magnets, by predicting\nmagnetic characteristics of a nanocomposite mate-\nrial before its actual manufacturing.\nIn general, micromagnetic simulations are per-\nfectly suited for modeling of magnetic compos-\nites, because the typical micromagnetic character-\nistic length - a few nanometers - allows to resolve\nvery well the magnetization distribution inside the\nnanocomposite grains with sizes of several tens of\nnanometers. However,correspondingsimulationsre-\nquire an enormous computational effort due to two\nmain reasons. First, a very fine mesh of finite el-\nements is required for the adequate approximation\nof each single grain as a geometrical object with a\ncomplicated shape. Second, a large number of soft\nand (especially) hard grains should be present in the\nsimulated volume, in order to study the magneti-\nzation reversal as a collective phenomenon and to\nobtain a sufficiently accurate statistics for these dis-\nordered systems (see a detailed discussion of these\nissues in11–13).\nIn recent years, major simulation effort was de-\nvoted to the understanding of rare-earth-based ma-\nterials, with the strategic goal ”to push the border”\nof the conventional (non-superconducting) mag-\nnets. In the first line, a large amount of nu-\nmerical research was carried out for composites\nbased on NdFeB7,8,14–16and PrFeB17–19; some re-\nsults have been published also for SmCo and similar\ncompounds20.\nFor NdFeB-based materials, the question of the\ninternal magnetization structure (mainly vortex) in\na single grain have been studied14. The influence of\nthe soft phase concentration (Fe or FeB), the hard\ngrain size15and (very recently) magnetic behaviour\nof hard grains for several different shapes16was in-\nvestigated. Special attention was paid to Nd-rich\ncompositions, where it has been shown that the en-\nrichment with Nd leads to the coercivity enhance-\nment due to the concentration of the additional Nd\non intergrain boundaries7,8.\nStudies of composites based on PrFeB and con-\ntaining Fe as the soft phase have been devoted to\nthe increasingroleofthe magnetodipolarinteraction\nwith the growingsoft phase fraction17, correlationof\nthe magnetization reversal of soft and hard grains18\nand the effect of the hard grains alignment19. For\nSmCo-like materials different mechanisms of the\nmagnetization reversal when changing the angle be-tween the anisotropy axis and the applied field were\nidentified20.\nSomewhat apart from the main road lie the sim-\nulations of a highly interesting yet not really ap-\nplicable class of MnBi-based materials. Here the\ninfluence of the soft phase concentration (Co and\nFeCo) and the orientation degree of the hard grain\nanisotropy axes was studied numerically in21.\nIn contrast to the rare-earth-based composites,\nmaterials based on various ferrites as the hard phase\nhavenot been studied - up to ourknowledge - by mi-\ncromagnetic simulations, although this class of ma-\nterials becomes increasingly important for reasons\nlisted above. In the present paper we address this\nchallenge, starting with detailed studies of the effect\noftheintergrainexchangecouplingandthe influence\nof the hard grain shape on the material properties in\nnanocomposites SrFe 12O19/Fe and SrFe 12O19/Ni.\nOur polyhedron-based micromagnetic algorithm\nprovides a high statistical accuracy of simulated re-\nsults, because we are able to handle systems con-\ntaining a few thousand grains, including the ability\nto resolve a possibly non-trivial magnetization dis-\ntribution in every grain. Due to the flexibility of our\nmesh generation method, a nearly arbitrary grain\nshape can be adequately approximated, so that a\nsimulated sample may include the grains of different\nshapes and sizes.\nThe paper is organized as follows: in Sec. II we\nexplain the mesh generation method which we use\nto create a polyhedron mesh for a system containing\nnon-spherical hard grains embedded into a magnet-\nically soft matrix. Evaluation of micromagnetic en-\nergy contributions in our methodology is also briefly\npresented. Sec. III contains simulation results.\nHere, subsection IIIA is devoted to the analysis of\nthe influence of the intergrain exchange weakening\non the magnetization reversal, whereas subsection\nIIIB deals with the effect of non-spherical shapes of\nthe hard grains. Both subsections contain a detailed\nphysical discussion of results obtained. We conclude\nwith the summary of our findings and possible per-\nspectives for the improvement of ferrite-based com-\nposites in Sec. IV.\nII. OUR MICROMAGNETIC\nMETHODOLOGY FOR SIMULATION OF\nNANOCOMPOSITES\nTo overcome the difficulties in modeling mag-\nnetic nanocomposites using standard micromagnetic\nmethods (finite difference and tetrahedral finite ele-\nments), we have developed13a novel micromagnetic\n2methodology based on a discretization of magnetic\nmaterials using polyhedrons of a special type. This\nmethodology combines the flexibility of general fi-\nnite element schemes for the geometrical description\nof a nanocomposite structure with the possibility to\nuse Fast Fourier Transformation for the calculation\nofthe most time consumingcontribution tothe total\nenergy - magnetodipolar interaction.\nA. Mesh generation for grains of a general\nshape\nOne of the main questions discussed in this pa-\nper is the influence of the non-spherical (spheroidal)\nshape of hard grains on the magnetic behavior\nof nanocomposites. For such a system, we had\nto introduce two additional steps into the mesh\ngeneration procedure described in our previous\npublications11–13. Namely, in the present work the\nmesh for hard grains is generated separately from\nthe soft phase mesh (Fig.1) (additional step 1), and\nthen both systems are merged (Fig. 2) (additional\nstep 2).\nFIG. 1. (color online) Examples of the spatial distribu-\ntion of hard crystallites (soft crystallites not shown) in\nsimulated samples for different aspect ratio a/bof corre-\nsponding ellipsoids of revolution (see text for details).\nAsinourstandardmethodology,westartfromthe\ngeneration of mesh consisting of small nearly spheri-\ncal polyhedrons with the sizes less the characteristic\nmicromagnetic length. These polyhedrons will be\nused in micromagnetic simulations as corresponding\nfinite elements (see11–13for details).\nNext, we generate a system of non-overlapping el-\nlipsoidal particles (additional step 1) with sizes and\nshapes corresponding to the hard grains of our com-\nposite. We point out that the generation of an en-\nsemble of non-overlappingellipsoids is computation-\nally challenging: the evaluation of an overlapping\nFIG. 2. (color online) Example ofa microstructure (hard\nandsoftphases)usedinourmodelingofnanocomposites.\nof two ellipsoids and the introduction of the suitable\noverlappingcriterionrequire a development ofa spe-\ncial numerical scheme. We use a method described\nby Donev et al.22, which is based on the Perram-\nWertheim overlap potential and provides a suitable\nparameter describing the overlapping degree of two\nellipsoids. This parameter is then used in the model\nof interacting particles with a short-range repulsive\npotential, where initially ellipsoids are placed ran-\ndomly, but due to the nature of this potential the\nnumber of overlaps is continuously decreasing.\nAt the second additional step, these ellipsoids are\nmapped on the system of (much smaller) polyhe-\ndrons used as mesh elements in our micromagnetic\nsimulation. By this mapping all mesh elements with\ncenters inside ellipsoids, are assigned to the hard\nphase and the rest of elements - to the soft phase.\nAll mesh elements belonging to the same hard grain\nhave the same direction of the anisotropy axes; this\ndirection coincides with the rotational symmetry\naxis of the ellipsoid. Note, that the discretization\nof crystallites of the soft and hard phases is based\non mesh elements of the same size.\nB. Energy contributions and minimization\nprocedure\nIn our simulations we take into account all four\nstandard contributions to the total magnetic free\nenergy: energy in the external field, magnetocrys-\ntalline anisotropy energy, exchange stiffness and\nmagnetodipolar interaction energies11–13.\nThe local energy parts - energy in the external\nfield and the magnetocrystalline anisotropy energy -\narecomputedinastandardway,multiplyingthecor-\nresponding energy densities by the volume of finite\n3elements (polyhedrons in our case) and summing\nover all these elements. The magnetodipolar field\nand energy are computed using the optimized ver-\nsion of the lattice Ewald method for disordered sys-\ntems. In this algorithm, the mapping of the initial\n(disordered) system of mesh elements on the trans-\nlationally invariant regular lattice allows to keep the\nhigh speed of the lattice method (fast Fourier trans-\nformation), at the same time making the mapping\nerrors negligibly small.\nIn this work, we concentrate ourselves in particu-\nlar on the effect of the intergrain exchange. Hence\nwe remind that the exchange energy in our method-\nology is computed in the nearest neighbouring ap-\nproximation as\nEexch=−1\n2N/summationdisplay\ni=1/summationdisplay\nj⊂n.n.(i)2AijVij\n∆r2\nij(mimj),(1)\nwhereVij= (Vi+Vj)/2, ∆rijis the distance be-\ntween the centers of i-th and j-th finite elements\nwith volumes ViandVj. The exchange constant\nAijfor the homogeneous bulk material is equal\nto the corresponding exchange stiffness constant\nA, but is obviously site-dependent in composite\nmaterials12,13.\nFor the minimization of the total magnetic en-\nergy, obtained as the sum of all four contributions\ndescribed above, we use the simplified version of a\ngradient method employing the dissipation part of\nthe Landau-Lifshitz equationof motion for magnetic\nmoments23,24. The minimization is considered as\nconverged, when the condition for the local torque\nmax{i}|[mi×heff\ni]|< εis fulfilled (here miis a nor-\nmalized magnetic moment of the i-th mesh element\nandheff\niisthecorrespondingeffectivefield; thevalue\nε= 10−3was found to be small enough for our qua-\nsistatic minimization procedure).\nFurther details of our method can be found in13.\nIII. RESULTS AND DISCUSSION\nDue to the high performance of our methodology,\nwe are able to simulate bulk nanocomposites with\nhard grains of any prescribed shape, whereby the\nsimulated system may contain up to 600 hard grains\nwith an average discretization of 300 mesh elements\npro grain. Employing this algorithm, we could ob-\ntain systematic results with the high statistical ac-\ncuracy for two model systems: SrFe 12O19/Fe and\nSrFe12O19/Ni.For simulations of these nanocomposites we have\nused standard magnetic parameters of correspond-\ning materials (see, e.g.3, Chap. 5.3 and 11.6) which\nare summarized in the table below:\nSrFe12O19Fe Ni\nMs(G) 400 1700 490\nAnis. kind uniaxial cubic cubic\nK(erg/cm3)4.0·1065.0·105−4.5·104\nA(erg/cm) 0.6·10−62.0·10−60.8·10−6\nThe average grain volume for all phases was cho-\nsen to be equal to the volume of a spherical grain\nwith the diameter D= 25nm. The volume concen-\ntrationofthe hardphasein allpresentedsimulations\nwaschard= 40%.\nA. Effect of the exchange weakening in\nSrFe12O19/Fe\nOne of the central questions for permanent mag-\nnets made of nanocomposite materials is the depen-\ndence of magnetic properties on the exchange weak-\nening between different grains. This weakening is\nunavoidable in real systems, because it is nearly\nimpossible to obtain perfect intergrain boundaries.\nThe quality of these boundaries strongly depends\non the concrete method used for the manufactur-\ning of a nanocomposite and substantial efforts has\nbeen devoted to obtaining materials with more per-\nfect intergrain boundaries (and especially bound-\naries between grains belonging to different phases)\nin order to achieve better exchange coupling. How-\never, recently25is was demonstrated both experi-\nmentally and theoretically that the perfect inter-\ngrain exchange may strongly decrease the perfor-\nmance of a magnetic nanocomposite material, so\nthat this question requires a detailed theoretical\nstudy.\nThe exchange weakening in our methodology is\ndefined bymultiplying the exchangeenergyofneigh-\nboring mesh elements belonging to different crystal-\nlitesbyafactor0 ≤κ≤1. Thisexchangeweakening\ncoefficient is introduced into the expression (1) for\nthe exchange energy as\nEexch=−1\n2N/summationdisplay\ni=1/summationdisplay\nj⊂n.n.(i)κ2AijVij\n∆r2\nij(mimj),(2)\nif neighboring magnetic moments iandjare located\nin different grains (crystallites). From (2) it can be\n4seen that κ= 1 correspondsto the perfect intergrain\nexchange (equal to the exchange within a bulk ma-\nterial) and κ= 0 means no exchange interaction at\nall between different grains.\nDependence of magnetic properties on this ex-\nchange weakening was studied for the composite\nSrFe12O19/Fe with approximately spherical hard\ngrains (obtained from the random placement of\nspheres with D= 25nm, see Sec. IIA).\nFIG. 3. (color online). Simulated hysteresis curves of the\nnanocomposite SrFe 12O19/Fe with spherical hard grains\nfor different exchange weakening constant.\nThe overall trend is shown in Fig. 3, where the\nevolution of hysteresis curves by increasing the ex-\nchange coupling ( κ= 0.0→0.5) between grains is\ndemonstrated. Systems without ( κ= 0.0) or with a\nstrongly reduced ( κ= 0.05) exchange coupling be-\ntween grains exhibit the two-steps magnetization re-\nversal. The first step - large jump on the hystere-\nsis loop in small negative fields (see the panel for\nκ= 0.05 in Fig. 3) - represents the magnetization\nreversal of the soft phase, which volume fraction is\nrelatively high. The second step - reversal of hard\ngrains in much higher fields - leads to the closure\nof the loop. Reversal of the hard phase occurs in\nfields∼Hhard\nK, where the anisotropy field is defined\nasHK= 2K/Ms=βMs(β= 2K/M2\nsdenotes the\nreduced anisotropy constant). The large magnitude\nof the magnetization jump during the first reversal\nstep is due to the dominating contribution of thesoft phase to the system magnetization: msoft=\ncsoft·Msoft/(chard·Mhard+csoft·Msoft)≈0.86.\nFor the detailed analysis of the magnetization re-\nversal in a magnetic composite it is very useful to\nplot hysteresis loops for the soft and hard phase sep-\narately. Such loops can be easily obtained from sim-\nulated magnetization configurations by summing up\ncontributions from finite elements belonging to ei-\nther soft or hard phase and calculating correspond-\ning total magnetizations of these phases.\nWe begin our consideration from systems with\nan absent or very low intergrain exchange coupling,\nwhere the dominant interaction is the magnetodipo-\nlar one. To clarify the effect of this interaction, the\nabove mentioned magnetization reversal curves of\nhard and soft phases are plotted in Fig. 4 for the\ncomposite without any intergrain exchange coupling\n(κ= 0).\nIn order to understand the hysteretic behavior of\nboth phases, it is useful to calculate the reduced\nanisotropy constant β= 2K/M2\ns, which magnitude\ngives (roughly speaking) the relation of the magne-\ntocrystalline anisotropy field to the magnetodipo-\nlar field from the nearest neighbor in the system\nof spherical particles. Substitution of magnetic pa-\nrameters of our materials (see the table above) re-\nsults in the values βh= 50(≫1) for the hard phase\n(SrFe12O19) andβs= 0.34(∼1) for the soft phase\n(Fe). We note that the much higher value of βfor\nthe hard phase is due not only to its largeanisotropy\nconstant K(which is ’only’ 8 times larger than by\nFe), but mainly due to the much higher value of the\nsoftphasemagnetization MFe/MSrFeO= 4.25,which\ngives an additional factor of ≈18.\nConsidering the magnetization reversal of the soft\nphase first, we note that this phase would exhibit\nin the absence of the magnetodipolar interaction\nthe ’ideal’ hysteresis loop for a system of non-\ninteracting particles with the cubic anisotropy con-\nstantKcub>0 (as it is the case for Fe). Such a loop\nhas the remanence j(0)\nR≈0.83 and the coercivity\nH(0)\nc≈0.33HK= 0.33βMs≈195Oe (see, e.g.26).\nThe relatively low value of the reduced anisotropy\nforoursoftphase βs(Fe) = 0.34meansthat themag-\nnetodipolar interaction can considerably modify the\ncorresponding’ideal’ hysteresis. This influence man-\nifests itself primarily in the smoothing of the ’ideal’\nloop26, as it can be seen in Fig. 4, where the loop\nfor the soft phase of our system is shown in red. The\nremanence jR≈0.836is nearly the same and the co-\nercivityHc≈260Oe increased by ≈30% compared\nto the non-interacting case.\nUnfortunately, we are not awareof any systematic\n5theoreticalstudiesofthe magnetodipolarinteraction\neffects in systems of ’cubic’ particles, except for the\npaper27, whereonlysimulationresultsfortheHenkel\nplots are shown; any quantitative comparison with a\ndetailed study of these effects for ’uniaxial’ particles\npresented in28is meaningless due to very different\nenergy landscapes for these two anisotropy types.\nFor this reason, we can only suggest that the nearly\nunchanged remanence (compared to the ’ideal’ sys-\ntem) is due to the interplay of the magnetodipolar\ninteractions within the soft phase and between the\nsoft and hard phases. The increase of Hcis most\nprobably due to the ’supporting’ action of the mag-\nnetodipolar field from the hard phase onto the soft\ngrains. Magnetization of the hard phase in our sys-\ntem is rather low, so that the corresponding effect is\nrelatively small.\nFIG. 4. (color online). Simulated hysteresis loops for\nSrFe12O19/Fe (with spherical hard grains) without the\nintergrain exchange ( κ= 0) presented for hard (solid\nblue line) and soft (solid red line) phases separately.\nDashed line represents the unsheared loop of the SW\nmodel with particle parameters as for SrFe 12O19, solid\ngreen line - the SW loop sheared according the aver-\naged internal field (see text for details). External field\nis normalized by the anisotropy field of the hard phase\nHK=βhMh= 20kOe.\nThe non-interacting hardphase consisting of\ngrains with the uniaxial anisotropy (as for\nSrFe12O19) would reverse according to the ideal\nStoner-Wohlfarth (SW) loop29withjR= 0.5 and\nHc≈0.48HK≈10kOe shown in Fig. 4 with the\nthin dashed green line. The very large value of the\nreduced single-grain anisotropy βh(SrFeO) = 50 for\nthis phase means that intergraincorrelationsof hard\nphase magnetic moments are negligible. However,\nin our composite material hard grains are ’embed-ded’ into the soft phase. Hence, in order to prop-\nerly compare (at least in the mean-field approxi-\nmation) the simulated hard phase loop - blue solid\nline in Fig. 4 - with the SW model, we have to\ntake into account the average magnetodipolar field\n/angbracketleftHmd,z/angbracketright= (4π/3)/angbracketleftMsoft\nz/angbracketrightacting on a spherical par-\nticle inside a continuous medium with the average\nmagnetization of the soft phase /angbracketleftMsoft\nz/angbracketright.\nCorrection of the SW loop using this internal field\n(which depends on the external field via the corre-\nsponding dependence /angbracketleftMz(Hz)/angbracketright) leads to the loop\nshown with the thick solid green line in Fig. 4.\nIt can be seen that this corrected SW loop is in\na good agreement with the simulated hard phase\nloop. Remainingdiscrepanciesaredue to localinter-\nnal field fluctuations (always present in disordered\nmagnetic systems) which are especially pronounced\nin our composite due to the high difference between\nthe magnetizations of soft and hard phases.\nThis analysis reveals that the first jump on the\nhard phase loop in small negative fields is due to\nthe abrupt change in the internal averaged dipolar\nfield due to the magnetization reversal of the soft\nphase. The second jump - for Hz/Hk≈ −0.3 -\nis the manifestation of the singular behavior of the\nSW loop of the hard phase itself, which occurs for\nthe unsheared loop at Hcr=−Hk/2 (near this field\nMz∼/radicalbig\n−(Hz−Hcr) forHz< Hcr30).\nIn summary, despite a relatively high saturation\nmagnetization Ms= 1180G, the corresponding\ncomposite without any intergrain exchange coupling\nwould have only a relatively small maximal energy\nproduct of ≈15kJ/m3(see Fig. 5b). The reason is\nits very small coercivity Hc≈250Oe, which is de-\ntermined entirely by the magnetization reversal of\nthe soft phase in small negative fields.\nBefore we proceed with the analysis of the effect\nof the intergrainexchange coupling on the hysteretic\nproperties of a nanocomposite, an important me-\nthodical issue should be clarified. Namely, we have\ntodetermine the maximalvalue ofthe exchangecou-\npling(maximalvalueof κ), forwhichoursimulations\ncan produce meaningful results.\nThe problem is that with increasing the cou-\npling strength, the interaction between the grains\nincreases, so that grains are starting to form clus-\nters, inside which magnetic moments of constituting\ngrains reverse nearly coherently. The average size of\nsuch a cluster /angbracketleftdcl/angbracketrightobviously growths with increas-\ningκ. In order to obtain statistically significant re-\nsults, we have to assure that /angbracketleftdcl/angbracketrightis significantly less\n(ideally much less) than the maximal system size ac-\ncessible for simulations. Otherwise we might end up\n6with the case where we are simulating the magne-\ntization reversal of a system consisting of a single\n(or very few) cluster(s), so that corresponding re-\nsults will be non-representative for the analysis of\nreal experiments.\nThe best quantitative method to determine /angbracketleftdcl/angbracketright\nis the calculation of the spatial correlation function\nof magnetization components perpendicular to the\napplied field (in our case MxandMy): the average\nvalue of these components should be zero, and the\ndecay length of their correlation functions Cx(r) =\n/angbracketleftMx(0)Mx(r)/angbracketright(the same for My) would provide a\nmost reliable estimation of /angbracketleftdcl/angbracketright.\nHowever, taking into account a complex 3D char-\nacter of Cx,y(r), we have adopted another crite-\nrion to determine the approximate number of in-\ndependent clusters contained in our simulated sys-\ntem. Namely, as the figure of merit we have em-\nployed the maximal value of the perpendicular com-\nponent of the total system magnetization m⊥=/radicalBig\nM2x+M2y/Msduring the magnetization reversal.\nIf the system contains only one (or very few) clus-\nter(s), than for some field during the reversal pro-\ncess this component should be large (close to 1), be-\ncause one cluster reverses nearly in the same fashion\nas a single particle, i.e., its magnetization rotates\nas a whole without significantly changing its magni-\ntude. Hence at some reversal stage m⊥would un-\navoidably become relatively large. In the opposite\ncase, where a system contains many nearly indepen-\ndent clusters ( Ncl≫1), their components Mx,iand\nMy,i(i= 1,...,N), being independent variableswith\nzero mean, would averagethemselves out, leading to\nsmall values of m⊥.\nA simple statistical analysis based on the assump-\ntion of the independence of different clusters shows\nthat the number of such clusters can be estimated\nasNcl≥1/m2\n⊥. This means that up to m⊥≈0.3\nwe produce statistically significant results, because\nin this case Ncl≥10. Corresponding analysis shows\nthat for our systems (containing about ∼5·105fi-\nnite elements) this is the case up to κ≈0.5, so\nbelow we show results only in this range of exchange\ncouplings.\nSimulation results showing basic characteristics\nof the hysteresis loop - remanence jR, coercivity\nHcand energy product Emax= (BH)max- for\nthe SrFe 12O19/Fe composite as functions of the ex-\nchange weakening κare presented in Fig. 5. We\nremind that for these simulations approximately\nspherical hard grains were used.\nFrom Fig. 5 it can be clearly seen that the rema-\nnencejRof this material depends on the intergrain(a)\n(b)\n(c)\nFIG. 5. (color online). (a)Remanence, (b)coerciv-\nity and(c)energy product of simulated nanocomposite\nSrFe12O19/Fe with spherical hard grains as a functions\nof exchange weakening on the grain boundaries. Inset\nin (a) represents the maximal value of perpendicular (to\nthe directions of applied field) component of magnetiza-\ntion during the remagnetization process. Dashed lines\nare paths for the eye.\nexchange coupling relatively weak. The reason is\nthatjRis very high already for the fully exchange\ndecoupled composite ( jR(κ= 0)≈0.8). Such a high\nvalue, in turn, is due to the fact that the remanence\nis governed by the soft phase consisting of cubical\ngrains. The remanenceofthe non-interacting(ideal)\nensemble of such grains is j(0)\nR≈0.83. This high re-\nmanence can not be significantly increased by the\nexchange interaction within the soft phase (as it is\nthecaseforthesystemof uniaxialparticleswithran-\ndomly distributed anisotropy axes, where j(0)\nR= 0.5;\nsee also31for the analysis of a corresponding 2D\n7system). Neither can this remanence be substan-\ntially decreased by the exchange coupling with hard\ngrains, because their magnetizationat Hz= 0is still\nnearly aligned along the initial field direction due\nthe strong magnetizing field from the Fe soft phase,\n(with its high magnetization MFe= 1700 G).\nIn contrastto jR, the coercivity Hcexhibits a pro-\nnounced maximum as the function of the exchange\ncoupling κ, resulting in the corresponding maximum\nof theκ-dependence of the maximal energy product\n(BH)max(κ). We will explain the reasons for the\nappearance of this maximum below, analyzing the\nhysteretic behavior of our nanocomposite for vari-\nousκ.\nFor the smallest non-zero κstudied here the mag-\nnetization reversal process is visualized in Fig. 6,\nwhere hysteresis loops for soft and hard phases are\nshown separately and the magnetization configura-\ntion is displayed for several characteristic external\nfields. First, it can be clearly seen that the mag-\nnetizations of soft and hard phases reverse sepa-\nrately. The inspection of magnetization configura-\ntions shows that the reversal of magnetic moments\nstarts within the soft phase (see panel (a)) around\nthe hard grains which anisotropy axis are directed\n’favorably’(i.e. deviatestronglyfromtheinitialfield\ndirection). Then the reversed area expands, occupy-\ning even larger regions of the soft phase (panel (b))\nuntil nearly the entire soft phase is reversed (panel\n(c)). Note that in the negative field corresponding\nto this nearly complete reversal of the soft phase,\nthe majority of the hard phase is still magnetized\napproximately along the initial direction. Only in\nmuchlargernegativefields(rightdrawingofhystere-\nsis loops) the hard phase magnetization also starts\nto reverse (see panel (d)).\nWe emphasize here two important circumstances:\nalthoughtheexchangecouplingbetweenthesoftand\nhard phases is very weak ( κ= 0.05) and the concen-\ntration of the hard phase is moderate (40%), the\n’supporting’ action of the hard phase is enough to\nnearly double the coercivity of the soft phase and\nhence - of the whole system, when compared to the\ncase ofκ= 0 - see Fig. 5. At the same time, due\nto this low exchange coupling, hard grains reverse\nseparately from the soft phase and nearly separately\nfrom each other (see panel (c)), leading to a high\ncoercivity of the hard phase (right drawing of hys-\nteresis loops).\nFor the larger exchange coupling κ= 0.1 (see Fig.\n7) the ’supporting’ effect of the hard phase increases\nthe coercivity of the soft phase even further (com-\npared to κ= 0.05). At the same time, this larger\ncoupling also leads to the much earlier reversal of\nFIG. 6. (color online). Magnetization reversal process\nfor the composite with exchange weakening κ= 0.05.\nFrom top to bottom: microstructure of the system\n(warm colors - soft, cold colors - hard grains); hystere-\nsis shown as separate curves for the soft (red) and hard\n(blue line) phases (note different scales of the H-axis);\nmagnetization configurations shown as mz-maps for field\nvalues indicated on the hysteresis plots shown above.\nthe hard phase, significantly decreasing its coerciv-\nity - see hysteresis plots in Fig. 7. Magnetization\nreversal for this coupling starts in those system re-\n8FIG. 7. (color online). Magnetization reversal for the\ncomposite with the exchange weakening κ= 0.10 pre-\nsented in the same manner as in Fig. 6.\ngions where the hard phase is nearly absent (due to\nlocal structural fluctuations) - see panel (b) in Fig.\n7 - and is much more cooperative compared to the\ncase ofκ= 0.05.\nThe resulting coercivity of the entire system is at\nits maximum, because the interphase coupling is,\non the one hand, large enough to prevent the soft\nphase from the reversal in small fields, but on an-other hand, small enough to enable to the reverse of\nthe hard phase in much higher negative fields than\nthe soft phase.\nFIG. 8. (color online). Magnetization reversal for the\ncomposite with the exchange weakening κ= 0.20 pre-\nsented in the same way as in Fig. 6. Simultaneous re-\nversal of the hard and the soft phases is clearly visible.\nWhen the intergrain exchange coupling is in-\ncreased further, magnetization reversal of the sys-\ntem becomes fully cooperative, so that the soft and\nhardphasesreversesimultaneously(inthesameneg-\native fields) - see hysteresis loops shown in Fig. 8\nforκ= 0.2. Spatial correlations between the mi-\ncrostructure and the nucleation regions for the mag-\nnetization reversal become weak, as it can be seen\nfrom microstructural and magnetic maps presented\nin this figure. It is also apparent that the correlation\ndistance of the magnetization configuration strongly\nincreases, as it was noted in the discussion above.\nThe overallresult is the decreaseof the system co-\n9ercivity, because the soft phase causes the much ear-\nlierreversalofthehardphase,sothatthesupporting\neffect of the high anisotropy of the hard phase be-\ncomes smaller. However, for this relatively low value\nofκ= 0.2 this ’supporting’ effect is still present:\nHc(κ= 0.2) is nearly twice as large as Hc(κ= 0).\nWhen the exchange coupling increases even fur-\nther, the magnetizationreversalbecomes completely\ndominated by the soft phase due to its larger mag-\nnetization and volume fraction. In particular, for\nκ= 0.5 both the coercivity and the energy product\nare nearly the same as for κ= 0. We note that hys-\nteresis loops for these two cases ( κ= 0 and κ= 0.5)\nlook qualitatively different, but this physically im-\nportant difference (two-step vs one-step magnetiza-\ntion reversal) does not matter for the performance\nof the nanocomposite from the point of view of a\nmaterial for permanent magnets.\nThe non-monotonous dependence of the max-\nimal energy product on the exchange coupling\n(BH)max(κ) can be easily deduced from the depen-\ndenciesjR(κ) andHc(κ). When κincreases from 0\nto≈0.1, bothremanenceandcoercivityincrease,re-\nsultingintherapidgrowthof( BH)max. Forκ >0.1,\nthe small increase of the remanence (up to κ≈0.2)\ncan not compensate the large drop of coercivity, re-\nsulting in the overall decrease of the energy prod-\nuct. We point out here that such a behavior occurs\nonly when the dependence of the coercivity on the\ncorresponding parameter (in our case the exchange\nweakening κ) is really strong. The case when the co-\nercivity depends relatively weak on the parameter of\ninterest, is analyzed in detail in the next subsection.\nSummarizing this part, we have shown that, in\ncontrast to the common belief, there exist an op-\ntimalvalue of the interphase exchange coupling in\na soft-hard nanocomposite which provides the max-\nimal energy product. This optimal value obviously\ndepends on the fractionsof the soft and hard phases,\nbut it is very likely that the optimal coupling should\nbesignificantlylessthantheperfectcoupling( κ= 1)\nfor all reasonable compositions in this class of mate-\nrials.\nThis important insight opens a new route for the\noptimization of the permanent magnet materials.\nB. Effect of the grain shape of the hard phase\ninSrFe12O19/FeandSrFe12O19/Nicomposites\nOne of the intensively discussed questions when\noptimizing the nanocomposite materials for perma-\nnent magnets is whether the materials containing\nthe hard grains with the non-spherical shape couldprovide an improvement of the energy product for\ncorresponding composites (see corresponding refer-\nences in the Introduction).\nThe standard argument in favor of the possi-\nble improvement of Emaxis the additional shape\nanisotropy of non-spherical particles. For an elon-\ngated(prolate)ellipsoidofrevolutionthisanisotropy\ncould increase the already present magnetocrys-\ntalline anisotropy (mc-anisotropy), thus enhancing\nthe coercivity of the hard phase and hence - the en-\nergy product. Below we will demonstrate that this\nline of arguments is not really conclusive and that\nthe grainshape effect may be even the opposite - the\nenergy product can be larger for a material contain-\ningoblatehard grains.\nBefore proceeding with the analysis of our results,\nwe emphasize, that the relativecontribution of the\nshape anisotropy can be approximately the same for\nrare-earth and ferrite-based materials. The former\nmaterials have a much larger mc-anisotropy Kcr, so\nthat on the first glance shape effects for rare-earth\n’hard’ grains should be much smaller. But the the\nrelation between the shape anisotropy and the mc-\nanisotropy contributions is determined not only by\nthe value of Kcr, but by the reduced anisotropy con-\nstantβ= 2Kcr/M2\ns, which gives, roughly speaking,\nthe relation between the mc-anisotropy energy and\nthe self-demagnetizing energy of a particle.\nThe presence of the material magnetization in the\ndenominator of the expression for βmakes this con-\nstants for both material classes very similar. For ex-\nample, the mc-anisotropy Kcr≈4.6×107erg/cm3\nfor Nd 2Fe14B is more than one order of magnitude\nlarger than its counterpart Kcr≈4×106erg/cm3\nfor SrFe 12O19. However, the much lower magne-\ntization Ms≈400G of SrFe 12O19compared to\nMs≈1300G of Nd 2Fe14B makes the difference be-\ntween reduced anisotropies of these materials quite\nsmall:βNdFeB≈60, whereas βSrFeO≈50.\nIn the language of the anisotropy field we have to\ncompare the values of the mc-anisotropy field HK=\nβMs= 2Kcr/Mswith the values of the magnetiz-\ningmagnetodipolar field, which attains its maximal\nvalueHmax\ndip= 2πMsfor a needle-like particle. Cor-\nresponding relation Hmax\ndip/HK=πM2\ns/Kcr= 2π/β\nis≈10.5 for Nd 2Fe14B and≈12.5 for SrFe 12O19.\nThis means that in the best case the effect of\nthe shape anisotropy for both material classes can\nachieve≈20%, what would be a non-negligible im-\nprovement on a highly competing market of modern\npermanent magnet materials.\nUnfortunately, severalcircumstancesare expected\nto strongly diminish the shape anisotropy contribu-\n10tion. First, the estimate above holds for a strongly\nelongated particle; for ellipsoidal particles with a re-\nalistic aspect ratio a/b∼2−3 (ais the length of\nthe axis of revolution) the shape anisotropy field is\nonly about half its maximal value. Second, this es-\ntimation holds for a single-domain particle, whereas\nstrongly elongated or nearly flat particles acquire a\nmulti-domain state much easier than the spherical\nones, because the domain wall energy for strongly\nnon-spherical particles is much smaller, than for a\nsphere. Finally, the relation derived above is true\nonly for an isolated particle, and hard grains in\nnanocomposites are always embedded into a soft\nphase or are in a close contact with another hard\ngrains.\nFor these reasons we have performed a detailed\nnumerical study of the dependence of hysteresis\nproperties on the hard grain shape for nanocom-\nposite SrFe 12O19/Fe and - for comparison - for\nSrFe12O19/Ni . For this purpose we have simulated\nmagnetization reversal in these composites with the\nhard grains having the shape of ellipsoids of rev-\nolution (spheroids) with the aspect ratio a/b=\n0.33,0.5,1.0,2.0,3.0; aspect ratios a/b >1 corre-\nspond, as usual, to prolate spheroids. For all aspects\nratios the volume of a single hard grain was kept\nthe same (and equal to the volume of the approxi-\nmately spherical grains with D= 25 nm). Volume\nconcentration of the hard phase chard= 40% was\nthe same, as for simulations reported in the previ-\nous Sec. IIIA. The exchange weakening parameter\nκ= 0.1 was chosen close to the optimal value for\nspherical hard grains obtained above.\n1. Grain shape effect for SrFe12O19/Fe\nFirst we discuss simulation results obtained for\nthe composite SrFe 12O19/Fe - see Figs. 9, 10 and\n11. In Fig.9, magnetization reversal curves for dif-\nferentaspectratios a/bareshown; both theloopsfor\nthe entire system and for the soft and hard phases\nseparately are presented. The most interesting ob-\nservation here is the pronounced difference between\nthe reversal curves of ’soft’ and ’hard’ phases for\na/b= 1 and nearly synchronous magnetization re-\nversalofboth phasesforotheraspectratiosshownin\nthe figure. This is a key feature for the understand-\ning of the system behavior and will be discussed in\ndetail below.\nOverall dependencies of basic hysteresis parame-\ntersjR,Hcand (BH)maxon the aspect ratio a/bis\npresented in Fig. 5. Both main parameters of the\nhysteresis - remanence jRand coercivity Hcexhibita/b = 0.33 a/b = 1.0\na/b = 2.0 a/b = 3.0\nH (kOe) H (kOe)Mz/Ms Mz/Ms\nFIG. 9. (color online). Simulatedhysteresis curvesof the\nnanocomposite SrFe 12O19/Fe for the exchange weaken-\ningκ= 0.1 and differentaspect ratios of hardcrystallites\nas indicated on the panels. Black loops -hysteresis of the\ntotal system, blue curves - upper part of the hysteresis\nloop for the hard phase, red curve - the same for the soft\nphase.\na highly non-trivial dependence on this aspect ratio,\nwhich should be carefully analyzed.\nThe dependence jR(a/b) shown in Fig. 10\nis clearly counter-intuitive, because normally one\nwould expect a higherremanence for a system con-\ntaining elongated particles - in our case for a/b >1\n- due to the positive shape anisotropy constant for\nsuch particles. The simulated dependence shows\nthe opposite trend - the remanence increases with\ndecreasing the aspect ration a/b, i.e.,jRbecomes\nlarger for a composite with oblatehard grains.\nThis behavior can be explained taking into ac-\ncount that hard ellipsoidal grains are mostly embed-\ndedintothesoftmagneticmatrix(softphase), which\nmagnetization is larger than that of the hard phase:\nMFe> MSrFe12O19. This means that hard grains\nrepresent magnetic ’holes’ inside a soft matrix, what\nmeans, in turn, that the total magnetodipolar field\nacting on the magnetization of the hard grain, is\ndirected (on average) towards the initially applied\nfield. With another words, this field acts as a mag-\nnetizing field, i.e. it increases the remanence of the\nhard phase.\nThe magnitude ofthis magnetizingfield is propor-\ntional to the difference between magnetizations of\n11(a)\n(b)\n(c)\noblateprolate\nFIG. 10. (color online). (a)Simulated reduced re-\nmanence, (b)coercivity and (c)energy product of\nnanocomposites SrFe 12O19/Fe with different aspect ra-\ntios (a/b) of hard grains. Inset in (a) shows the demag-\nnetizing factor in dependence on a/b. Dashed lines are\nguides for an eye.\nthe soft and hard phases and is of the order Hmag\ndip∼\nNdem·(MFe−MSrFe12O19) =Ndem·∆M. Foroursys-\ntem parameters ∆ M= 1300G, so that, taking into\naccount that Ndem∼π, we obtain Hmag\ndip∼4kOe.\nThis value is comparable to the mc-anisotropy field\nof the hard grain itself ( HK(SrFe12O19) = 20kOe),\nso the effect of this magnetodipolar field can be sig-\nnificant.\nTo explain the trend jR(a/b) seen in Fig. 11, it\nremains only to note that this magnetizing field is\nlarger for oblatespheroids, for which it can achieve\nthe magnitude of 4 π∆Ms- the limiting case for a\nthin disk with the revolution axes along the mag-\nnetizing direction of the system. In contrast, for\nthe prolate spheroid Hmag\ndipbecomes weaker when\na/bincreases (spheroid becomes more prolate), be-cause the main contribution to this field comes from\nthe soft phase regions near the ends of this prolate\nspheroid.\nThe result of this complicated interplay is the\nbetter alignment of magnetic moments of the hard\nphase consisting of oblate particles. This leads to\nthe higher remanence of the whole system for two\nreasons: ( i) the remanence of the hard phase itself\nis larger and ( ii) the ’supporting’ action of the hard\nphase on the soft phase - due to the interphase ex-\nchange coupling - is more significant.\nThe explanation of the non-trivial dependence of\nthe coercivity on the aspect ration Hc(a/b) - with\nthe maximum between a/b= 0.5 anda/b= 1.0 -\nrequires a detailed understanding of the magnetiza-\ntion reversal mechanism in composites with partial\ninterphase exchange coupling.\nNamely, magnetizationreversalofthese nanocom-\nposites always occurs according to the following sce-\nnario: the soft phase switches first, and then exhibit\na torque on the hard grains due to the interphase\nexchange interaction. For non-negligible interphase\nexchange this torque is the main interaction mech-\nanism between the phases and leads (together with\nthe applied field) to the magnetization reversal of\nthe hard phase in larger negative external fields.\nIn order to understand, why the coercivity has\nits maximum for particles with a weak shape\nanisotropy, we have to recall that the interphase ex-\nchange interaction is a surface effect and as such is\nproportional to the interphase surface area. In our\ncase this is the surface area of hard grains, which are\nmostly surrounded by the soft phase. This means,\nthat exchange torque which the soft phase exhibits\non the hard grains, is proportional to the surface\narea of these grains. Hence, this torque should be\nminimalforthehardgrainswiththesphericalshape,\nbecause the surface area of an ellipsoid of revolu-\ntion with the given volume is minimal for a/b= 1\n(sphere).\nFor this reason hard phase with grains having the\nshape close to spherical will have the maximal coer-\ncivity, i.e. reverse in the largest negative field. Such\ngrains will be also able to ’support’ soft phase up to\nnegative fields larger than non-spherical hard grains\nwould do, leading to the largest coercivity of the\nwhole sample.\nTo provide further proof of this hypothesis, we\nhave plotted in Fig. 11 the coercivities of the hard\nand soft phases separately (see curves for Hhard\ncand\nHsoft\ncon the panel (a)) and the difference between\nthem∆Hconthepanel(b)asfunctionsoftheaspect\nratioa/b. The excellent qualitative agreement be-\ntween ∆Hc(a/b) and the inverse of the surface area\n12hard phasehard phase\nsoft phase(a)\n(b)\nFIG. 11. (color online) (a)Coercivities of the hard (blue\ncircles)Hhard\ncand soft (red circles) Hsoft\ncphases and (b)\ndifference ∆ Hcbetween these coercivities as functions of\nthe aspect ratio a/b; inset in (b) - inverse of the surface\narea of an ellipsoid of revolution in dependence on a/b\n. Dashed lines are guides for an eye. See text for the\ndetailed explanation.\nof an ellipsoid of revolution 1 /Sell(a/b) (see inset to\nthis panel) as the functions of a/bclearly shows that\nthe observed effect is due to the surface-mediated\ninteraction, what in our case clearly means the in-\nterphase exchange interaction.\nWe finish this subsection with the explanation\nwhy the dependence of the maximal energy prod-\nuct on the aspect ratio Emax(a/b) (panel (c) in\nFig.10)foroursystemcloselyfollowsthecorrespond-\ning trend of the remanence jR(a/b) (see panel (a)),\nbut is not influenced by the dependence Hc(a/b)\n(panel (b)).\nTo understand this phenomenon, we recall that\ntheenergyproductisdefinedasthemaximalvalueof\nthe product ( BH) within the second quadrant ofthe\nhysteresis loop, i.e. for external fields −Hc< H <0\n(here and below we omit for simplicity the index z\nbyH,BandM):\nEmax= max\n−Hc<H<0[B(H)·H] = max[( H+4πM(H))·H],\n(3)\nwhere it is important, that the energy product de-\npends on Hboth explicitly and implicitly - via the\ndependence M(H).Let us now assume that for some reference pa-\nrameter value (e.g. in our case for a/b= 1) with\nthe magnetization vs. field dependence given by the\nfunction Mref(H) the product (3) reaches its maxi-\nmumE(0)\nmaxfor the field value H0. This means that\nthe corresponding derivative of the energy product\ndE/dHvanishes at this point, leading to the condi-\ntion\nd\ndH[(H+4πMref(H))·H]|H=H0=\n=H0+2π/bracketleftBigg\nMref(H0)+H0dMref\ndH/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=H0/bracketrightBigg\n= 0(4)\nIf the parameter in question changes (i.e. we take\nanother value of a/b), then the hysteresis loop also\nchanges, becoming Mnew(H) =Mref(H)+∆Mand\nthe maximum of the energy product is achieved at\nanother field Hnew=H0+∆H. The new maximal\nenergy product then is\nEnew\nmax= (Hnew+4πMnew(Hnew))·Hnew].(5)\nAssuming that ∆ Mand ∆Hare small, we can\nexpand the functions Mref(H) and ∆M(H) in the\nvicinity of the point H0. Starting from Eq.5 and re-\ntaining only terms linear in ∆ Mand ∆H, we obtain\nforEnew\nmaxthe expression\nEnew\nmax=E(0)\nmax+4πH0∆M(H0)+\n+2∆H/braceleftBigg\nH0+2π/bracketleftBigg\nMref(H0)+H0dMref\ndH/vextendsingle/vextendsingle/vextendsingle/vextendsingle\nH=H0/bracketrightBigg/bracerightBigg\n(6)\nThe coefficient in braces after ∆ His exactly the\nexpression (4) at the point where the reference en-\nergy product reaches its maximum and as such is\nequal to zero. Hence we are left with the expression\nof the new maximal energy product as\nEnew\nmax=E(0)\nmax+4πH0∆M(H0) (7)\nThis expression shows that the change of the en-\nergy product due to the variation of some exter-\nnal system parameter is controlled mainly by the\nchange of the magnetization curve (upper part of\ntheM−Hhysteresis loop) at the external field\nH0where the reference energy product reached its\nmaximum. Obviously, this magnetization change\nis roughly proportional to the remanence change,\n13which explains the semiquantitative correspondence\nbetween the dependencies jR(a/b) andEmax(a/b).\nWe point out once more that this statement is true\nonlyifthe coercivityshift due tothe variationofthis\nparameter is relatively small. If this is not the case,\nthen the position of the vertical side of the rectangle\n(in the ( B−H) plane) used for the determination\nof (BH)maxcan strongly depend on the coercivity\nvalue, making the derivation of Eq. (6) invalid.\n2. Grain shape effect for SrFe12O19/Ni\nThe second composite which we have used to\nstudy the grain shape effect - SrFe 12O19/Ni - is\nqualitatively different from the previous material\n(SrFe12O19/Fe) due to the much lower magnetiza-\ntion of the soft phase: Ms(Ni)≈490G. The idea\nbehind the usage of a soft phase with such a low\nmagnetization is that the coercivity of the resulting\nmaterial should be much higher due to the weaker\nresponse of the soft phase with a smaller magneti-\nzation to the external field. This higher coercivity\nmight compensate the decrease of the net magneti-\nzation, resulting in a competitive energy product.\na/b = 0.33 a/b = 1.0\na/b = 3.0 a/b = 2.0\nFIG. 12. (color online).Simulated hysteresis curves of\nthe composite SrFe 12O19/Ni forκ= 0.1 and different as-\npect ratios of hardcrystallites as indicated on thepanels.\nBlack loops - total hysteresis, blue curves - upper part\nof the hysteresis loop for the hard phase, red curve - for\nthe soft phase.(a)\n(b)\n(c)\nFIG. 13. (color online). (a)Remanence, (b)coer-\ncivity and (c)energy product of the nanocomposite\nSrFe12O19/Ni in dependence on the aspect ratio a/bof\nhard grains. Dashed lines are guides for an eye.\nSimulation results for SrFe 12O19/Ni with various\ngrain shapes are presented in Fig. 12 (hysteresis\nloops) and Fig. 13 (basic characteristics of the hys-\nteresis). As it can be clearly seen, the Ni-containing\ncomposite behaves itself qualitatively different com-\npared to Fe-containing material.\nThe main new feature is - as expected - the higher\ncoercivity of both soft and hard phases. The consid-\nerably larger coercivity of the Ni phase ( Hc(Ni)≈\n1000Oe - see Fig. 12, red loops) compared to the\ncoercivity of the Fe phase in the previously stud-\nied composite ( Hc(Fe)≈600Oe - see Fig. 11a) is\nmainlyduetothelowermagnetizationofNi, asmen-\ntioned above.\nThe much larger coercivity of the hard phase (we\nremind that this phase is the same for both stud-\nied materials) for SrFe 12O19/Ni can be most prob-\nably explained by three reasons. First, due to the\n14lowerMs(Ni) the reversal of the soft phase starts\nfor the Ni-containing composite in higher negative\nfields, what should by itself lead to larger Hhard\nc\nfor the hard phase also. However, this reason alone\ncould not be responsible for the more than fourfold\nincrease of Hhard\nc(Hhard\nc(SrFe12O19/Ni)≈4000Oe\nvsHhard\nc(SrFe12O19/Fe)<1000Oe).\nThesecondreasonisthatthemagnetodipolarfield\nof the soft phase acting on hard grains is much\nsmaller for Ni- than for Fe-containing composites\n(due to the same much lower magnetization of Ni).\nAfter the reversal of the soft phase this magne-\ntodipolar field is directed opposite to the initial ma-\nterial saturationand thus assiststhe reversalof hard\ngrains. Much smaller magnitude of this field thus\nleads to much higher external field required to re-\nverse the hard phase.\nFinally, the considerably smaller exchange con-\nstant of Ni compared to Fe (see Table 1) results in\nthe lower exchange torque acting on hard grains af-\nter the reversal of the soft phase, also decreasing the\ntotal torque acting on the hard phase and increasing\nits coercivity.\nThis qualitatively new situation - the non-\ncorrelated magnetization reversals of the soft and\nhard phases - leads to the another type of the co-\nercivity dependence on the aspect ratio a/bof hard\ngrains: coercivity Hc(a/b) decreases when this ratio\nincreases (see Fig. 13b), behaving itself in a very\nsimilar way as the remanence jR(a/b) (Fig. 13a).\nThe most likely explanation of this behavior is\nthe following: the degree of the magnetization align-\nment ofhardgrainsfortheir differentaspectratiosis\nnearly the same for various values of a/b. Hence the\nmain effect on the soft phase reversal is due to the\ndifference in the magnetodipolar field distributions\ncaused by hard grains having various shapes. Pro-\nlate ellipsoids of revolution produce a non-uniform\nmagnetodipolar field which maximal value is higher\nthan for oblate ellipsoids. However, the field of\nprolate particles is strongly concentrated near their\n’sharp’ ends, whereas the dipolar field of oblate el-\nlipsoids (magnetized on average along their axes of\nrevolution) occupies a much larger region near their\n’flat’ surfaces. For this reason the dipolar field of\noblate hardgrainssupportsthe magnetizationofthe\nsoft phasein largerspaceregions,thus leadingto the\nincrease of the soft phase (and total) coercivity with\ndecreasing a/b, i.e. when hard grains become more\noblate.\nAs the result of the decrease of both jRandHc,\nthe energy product also decreases with increasing\na/b, as shown in Fig. 13c. We point out that al-\nthough the resulting behavior is qualitatively some-what similar to the case of the Fe-containing ma-\nterial (compare to Fig. 10) - the energy product\ndecreases with increasing aspect ratio - the physical\nreasonsfor this behavior are fundamentally different\nfor these two composites.\nIt also interesting to note that despite the larger\ncoercivity of Ni-containing composite, its energy\nproduct remains smaller than for the Fe-containing\nmaterial due to its much lower net magnetization.\nThis relation can change for materials with different\nfractions of soft and hard phases.\nIV. CONCLUSION\nIn this paper we have presented a detailed numer-\nical study for the dependence of magnetic proper-\nties of Sr-ferrite-based nanocomposites on two very\nimportant material parameters: ( i) exchange cou-\npling between various crystallites (what includes the\ncoupling between soft and hard grains) and ( ii) the\nshape of the hard grains.\nFirst, we have demonstrated - in contrast to the\ncommon paradigm - that the maximal energy prod-\nuctEmax= (BH)maxis a non-monotonous func-\ntion of the intergrain exchange coupling κand that\nthe optimal κ-value (for which the energy product\nreachesitshighestvalue)isfarbelowtheperfectcou-\npling. This non-monotonous character of the func-\ntionEmax(κ)isduetothecorrespondingdependence\nof the coercivity on the exchange coupling Hc(κ).\nSecond, we have studied the dependence of the\nhysteresis properties and the maximal energy prod-\nuct on the shape ofhard grainsfor twovery different\nnanocomposite materials - SrFe 12O19/Fe (Ms(Fe)≈\n1700G) and SrFe 12O19/Ni (Ms(Ni)≈490G). Hard\ngrains have been assumed to have (approximately)\na shape of ellipsoids of revolution, which aspect ra-\ntio was varied in the range 1 /3≤a/b≤3 (a/b >1\ncorresponds to a prolate ellipsoid). We have shown,\nthat for both materials the aspect ratio dependence\nofthemaximalenergyproduct Emax(a/b)essentially\nfollows the corresponding dependence of the hys-\nteresis loop remanence jR(a/b) and have supported\nthis observation by analytical considerations. For\nboth materials, the maximal value of jR(a/b) - and\nhenceofEmax(a/b)-wasobtainedforthe oblatehard\ngrains with the smallest aspect ration a/b= 1/3\n(also in contrast with common expectations). Phys-\nical reasons for this behavior are revealed.\nFinally, we have also analyzed the dependence\nof the coercivity on the shape of hard grains\nHc(a/b) and have shown that this dependence for\n15the two composites under study is qualitatively dif-\nferent. For SrFe 12O19/Fe the function Hc(a/b) has\na pronounced maximum for approximately spher-\nical grains, whereas for SrFe 12O19/Ni coercivity\nmonotonously decreases with increasing a/b. This\ndifference is explained by analyzing the dominating\ninteraction mechanisms between the hard and soft\nphases in these materials.V. ACKNOWLEDGMENT\nFinancialsupport ofthe DFG projectBE2464/10-\n3 and the EU-FP7project ”NANOPYME”(310516)\n[www.nanopyme-project.eu] is greatly acknowl-\nedged. 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Gorn,\nPhysical Review B 57, 14332 (1998).\n16"    },    {        "title": "1609.00612v2.Potential_Energy_Driven_Spin_Manipulation_via_a_Controllable_Hydrogen_Ligand.pdf",        "content": "Potential Energy Driven Spin Manipulation via a Controllable Hydrogen Ligand\nPeter Jacobson,1,\u0003Matthias Muenks,1Gennadii Laskin,1Oleg O. Brovko,2\nValeri S. Stepanyuk,3Markus Ternes,1and Klaus Kern1, 4\n1Max Planck Institute for Solid State Research, Stuttgart, Germany\n2The Abdus Salam International Centre for Theoretical Physics (ICTP), Trieste, Italy\n3Max Planck Institute of Microstructure Physics, Halle, Germany\n4Institute de Physique, \u0013Ecole Polytechnique F\u0013 ed\u0013 erale de Lausanne, Lausanne, Switzerland\nSpin-bearing molecules can be stabilized on surfaces and in junctions with desirable properties\nsuch as a net spin that can be adjusted by external stimuli. Using scanning probes, initial and \fnal\nspin states can be deduced from topographic or spectroscopic data, but how the system transitioned\nbetween these states is largely unknown. Here we address this question by manipulating the total\nspin of magnetic cobalt hydride complexes on a corrugated boron nitride surface with a hydrogen-\nfunctionalized scanning probe tip by simultaneously tracking force and conductance. When the\nadditional hydrogen ligand is brought close to the cobalt monohydride, switching between a corre-\nlatedS=1=2Kondo state, where host electrons screen the magnetic moment, and a S= 1 state with\nmagnetocrystalline anisotropy is observed. We show that the total spin changes when the system\nis transferred onto a new potential energy surface de\fned by the position of the hydrogen in the\njunction. These results show how and why chemically functionalized tips are an e\u000bective tool to\nmanipulate adatoms and molecules, and a promising new method to selectively tune spin systems.\nINTRODUCTION\nThe magnetic behavior of adatoms and single molecular\nmagnets on surfaces is usually de\fned by static param-\neters such as the local symmetry, the spin-orbit interac-\ntion, or the exchange coupling with the electron bath of\nthe host. [1{5] However, there is widespread interest in\nactively controlling molecular and adatom spin states for\nswitching applications. [6, 7] Beyond imaging and spec-\ntroscopy, scanning probes are atomically precise manipu-\nlation tools. [8, 9] When manipulation and spectroscopy\noperate in tandem, it is possible to observe the formation\nof chemical bonds and continuously tune the exchange in-\nteraction between magnetic impurities. [10{13] Tip func-\ntionalization, now routinely used to create chemically\nprecise contacts where a molecule acts as the transducer,\nis one promising method for controlling spins. [14{18]\nThis strategy has its roots in small molecule adsorption\non metal bearing porphyrins and phthalocyanines [19, 20]\nand capitalizes on two strengths of local probes: the abil-\nity to address speci\fc atomic sites and the variable width\nof the tunnel junction. With magnetic adatoms gaining\nprominence as model quantum systems, it is highly desir-\nable to understand how chemically reactive probes couple\nto and in\ruence the measurement process and eventually\ncontrol the resulting magnetic state.\nHere, we reversibly control the total spin of cobalt hy-\ndride (CoH) spin centers adsorbed on the h{BN=Rh(111)\nmoir\u0013 e by manipulating a single hydrogen atom with the\ntip of a combined scanning tunneling (STM) and non-\ncontact atomic force microscope (AFM). As the dis-\ntancezbetween the probing tip and the CoH com-\nplex decreases, hydrogen initially adsorbed on the tip\napex weakly bonds to the CoH complex, inducing rapid\ntransitions between a correlated S=1=2Kondo state\nand an anisotropic S= 1 state. Local spectroscopyidenti\fes a stable total spin at high and low values of\nthe conductance, while at intermediate conductance dy-\nnamic switching is observed. Combining conductance-\ndistance,G(z), and force-distance, F(z), measurements\ntogether with density functional theory (DFT) calcula-\ntions, we unravel the microscopic potential energy land-\nscapes present within the tunnel junction. We demon-\nstrate that by coupling a functionalized tip to an under-\ncoordinated adatom, the reactivity of the adatom can be\nharnessed to drive transitions between di\u000berent total spin\nstates. The spin within the tunnel junction can therefore\nbe actively monitored and reversibly controlled with sin-\ngle atom precision.\nRESULTS\nFigure 1A shows a constant current image of CoH com-\nplexes on h{BN=Rh(111). The lattice mismatch between\nthe Rh(111) substrate and the single monolayer of h{BN\nresults in a strongly corrugated surface with 3 :2 nm pe-\nriodicity on which the CoH complexes appear as bright\nprotrusions. A clear indication of hydrogen adsorption\non the tip apex are the sharp changes in tip height, re-\nduced by 20 pm (Figure 1A, red dashes), while imaging\ntheh{BN=Rh(111) surface in constant current mode. [21]\nFigure 1B shows an individual CoH complex located near\nthe rim-valley boundary of h{BN=Rh(111) imaged with\na hydrogen-functionalized tip. At low junction conduc-\ntance (G=IS=VS= 1:61\u000210\u00004G0;G0= 77:48\u0016S,\nthe quantum of conductance), corresponding to relatively\nlarge tip-sample separations z, the increased contrast due\nto hydrogen in the junction partially overlaps a CoH (Fig-\nure 1B, bottom panel). As Gis increased and zdecreases,\nthis boundary region transitions to a noise speckled cir-\ncle with a brighter appearance, i.e. largerz-height, to\ncompensate for an overall increase in the conductance.arXiv:1609.00612v2  [cond-mat.mes-hall]  15 Dec 20162\n1-10 -5 5 10dI/dV (arb. units)\nbias voltage V (mV)012.9(10‒4 G0)6.45B\n01\nconductance\nA C\nFigure 1: In\ruence of Hydrogen Functionalized Tips on Imaging and Spectroscopy. (A) Constant current STM\nimage (approximately 5 \u00025 nm2;V=\u000015 mV,I= 20 pA;G= 1:72\u000210\u00005G0) of CoH complexes on the h{BN=Rh(111)\nmoir\u0013 e obtained with a hydrogen functionalized tip. Areas with enhanced contrast due to hydrogen in the junction are outlined\nin red. (B) Constant current STM images (1 :2\u00021:2 nm2, top to bottom: V=\u00000:3,\u00000:7,\u00001:0,\u00001:3,\u00001:6 mV,I= 20 pA,\ncorresponding to G= 8:60, 3:69, 2:58, 1:99, 1:61\u000210\u00004G0) of a CoH complex highlighting the strong conductance (tip-sample\ndistance) dependence of imaging with a hydrogen-functionalized tip. (C) Local spectroscopy obtained on the CoH complex in\npanel (B), the tip was centered on the dark lobe ( G= 1:61\u000210\u00004G0). AtG= 6:45\u000210\u00004G0(blue), a set of double steps is\nobserved, indicative of a spin 1 complex with magnetic anisotropy. Increasing the conductance in steps of \u0001 G= 0:16\u000210\u00004G0\nleads to the unstable spectroscopy until a spin1=2Kondo peak emerges at high conductance (red, G= 12:9\u000210\u00004G0). All\nspectra are normalized to the di\u000berential conductance at \u000010 mV, normalized spectra are o\u000bset by 0 :5.\nGiven the strong Gdependence within such a narrow\nrange, these results hint that the observed contrast is not\nsolely due to the local topography, but also due to me-\nchanical and electronic changes in the junction. Indeed,\nthese images are qualitatively similar to measurements\nof undercoordinated metal adatoms in the presence of\nadsorbed hydrogen. [22, 23] As the hydrogen content of\nthe CoH xcomplex governs the spin state, [5] ( dI/dV)\nspectroscopy was performed while varying the setpoint\nconductance Gwith the tip positioned over the central\nregion. At the lowest conductance, G= 6:45\u000210\u00004G0\n(Figure 1C, bottom curve), the spectra show two sym-\nmetric steps around zero bias with increasing di\u000berential\nconductance. These steps originate from the inelastic\nspin excitations of a CoH complex with total spin S= 1\nwhere magnetocrystalline anisotropy has removed the 3 d\nlevel degeneracy. Increasing Gresults in progressively\nunstable spectra until the emergence of a stable zero bias\npeak atG= 12:9\u000210\u00004G0, identi\fed as a S=1=2Kondo\nresonance of CoH 2. [5] This transition is fully reversible\nand the initial S= 1 total spin state restored when the\njunction conductance is reduced (see Fig. S1). We ob-\nserve a metastable state, when Gis between 8 \u000210\u00004G0\nand 11 \u000210\u00004G0, where the hydride complex randomly\ntransitions between the S= 1 andS=1=2states on a\ntimescale of 100 ms. The change in tip-sample separa-\ntion for this conductance range corresponds to a \u0001 zof\nless than 25 pm. Note, that this metastable behaviordoes not depend on the bias voltage during the spectro-\nscopic measurement. Di\u000berential conductance ( dI=dV )\nspectroscopy not only identi\fes the spin state, but it\nalso aids in the interpretation of the STM images in Fig-\nure 1B. The constant current images in Figure 1B were\nobtained over a bias range (0 :3\u00001:6 mV) where the to-\npographic appearance is closely linked to features in the\ndI=dV measurements and therefore, at small bias volt-\nages, is dominated by the Kondo resonance.\nTo investigate the switching behavior in detail, G(z)\nmeasurements were performed over CoH xcomplexes and\nbare h{BN. Approaching h{BN as well as CoH xcom-\nplexes with a bare tip reveals a strictly exponential in-\ncrease in conductance, G(z) =G0exp(\u00002\u0014G(z0+z))\nwith\u0014Gthe decay rate, and z0the tip-height at the initial\nsetpoint conductance G(Figure 2A). Functionalizing the\ntip apex with hydrogen alters the junction conductance\ncharacteristics, with G(z) showing a less than exponen-\ntial increase and a reduced \u0014Gcompared to data obtained\nwith a bare tip (Figure 2B). This characteristic behavior\nis similar to the observations of Weiss et al. on a complex\norganic molecule with a hydrogen-functionalized tip. [18]\nHowever, approaching a CoH complex with a hydrogen-\nfunctionalized tip, G(z) closely follows the h{BN trace\nuntil the conductance rapidly decreases by a factor of 2 :5,\nindicating the S= 1 toS=1=2transition (Figure 2B,\nred) observed by local spectroscopy in Figure 1C. There-\nfore, the drop in conductance stems from both the direct3\nconductance G (10‒4G0)100\n10\n1\nrelative height z (pm)0 100 200 0 100 200-3-2-10\ni ii iiiA B C\nCo partial DOSE‒EF (eV)\nFigure 2: Conductance { distance spectroscopy. (A) Conductance-distance, G(z), curves obtained with a bare Pt tip on\nh{BN (black), CoH (dashed blue), and CoH 2(dotted yellow) at a tip-sample bias V=\u000010 mV. (B) Using a functionalized tip,\nCoH+H tip(red), a conductance discontinuity, corresponding to the S= 1 toS=1=2total spin change, is observed at a relative\nheightzof 70 pm. The functionalized tip approaching the substrate, h{BN + H tip(dashed green), shows no discontinuity\nand a non-exponential character. For direct comparison, the CoH G(z) measurement from panel (A) is plotted again (dashed\nblue). Inverse decay constants, \u0014G[nm\u00001]: (A) h{BN (black): 8 :7\u00060:1; CoH (dashed blue): 9 :9\u00060:1; CoH 2(dotted yellow):\n9:8\u00060:1; (B) h{BN + H tip(dashed green): 6 :6\u00060:3 (0<z< 70 pm), 7:7\u00060:4 (70<z< 200 pm); CoH + H tip(red): 6:9\u00060:4\n(0<z< 70 pm), 8:5\u00060:5 (70<z< 200 pm). The color- and linetype-coded insets schematically depict the junction geometry.\n(C) Plots of the majority (left) and minority (right) spin projected density of states (blue: dorbitals, red: sporbitals) of a\nS= 1 CoH complex, magnetic moment of 2 :0\u0016B, without tip (i), one with a slight magnetic moment reduction (1 :6\u0016B) due\nto the presence of a hydrogen functionalized tip (ii), transition from S= 1 toS=1=2(1:2\u0016B) at close tip distances (iii). The\nchange in Stoner splitting between majority and minority bands is schematically depicted with vertical grey arrows.\ntransfer of hydrogen within the junction, altering the ge-\nometry and modifying the tunnel barrier, and the relative\nchange in total conductance between the CoH and CoH 2\ncomplexes.\nTo understand the electronic structure modi\fcation in\nthe spin switching process, we performed density func-\ntional theory calculations for various representative junc-\ntion geometries (Figure 2C). A CoH complex on h{BN\nexhibits a nearly free-atom-like electronic structure (Fig-\nure 2C-i) with Stoner split 3 dlevels giving it a magnetic\nmoment of 2 :0\u0016B(Bohr magneton). [5] Approaching\nthe complex with a Pt tip which has a hydrogen atom\nbound to its apex (Figure 2C-ii) gradually introduces in-\ndirect bonding via the H of the CoH complex, resulting in\nthe reduction of the Co magnetic moment (Figure 2C-ii).\nThe magnetic moment of Co remains close to 2 \u0016B, an ef-\nfectiveS= 1 state, until the approaching functionalized\ntip brings the H sporbitals into direct overlap with the\nCodorbitals (Figure 2C-iii). Bonding of two H atoms to\nCo, as shown in our previous work, is strong enough to\npartially quench the magnetic moment of Co, reducing it\nto 1:2\u0016B, an e\u000bective S=1=2state.\nTo reveal the microscopic forces at work in the spin\ntransition, we track the frequency shift, \u0001 f, of the os-\ncillating tuning fork from its non-interacting resonance\nfrequencyf0= 29;077 Hz. We measure \u0001 f(z) curves\nover switching complexes and the bare h{BN. To re-\nmove the long-range forces between the extended tip andthe sample we subtract the background from the data,\ni.e.\u0001f= \u0001fCoH\u0000\u0001fh{BN(see Methods and Fig. S2).\nThe \u0001fis small and negative before rapidly decreasing\non approach (Figure 3A, black). This sharp drop in \u0001 f\ncoincides with a change in the G(z) measurement simi-\nlar to Figure 2B, however here this feature is broadened\ndue to averaging over the 100 pm oscillation amplitude\n(Figure 3A, gray). Short range forces, F(z0), were quan-\nti\fed by converting \u0001 f(z) using the method of Sader\nand Jarvis. [24] Prior to the S= 1 toS=1=2transition,\nthe force between the tip and sample is weakly attractive\nand grows exponentially upon approach. As the hydro-\ngen on the tip apex couples to the CoH-complex, the\nattractive force grows steeply over a transition region of\n35 pm before leveling o\u000b (Figure 3B). The instantaneous\njunction conductance, G(z0), deconvoluted to remove the\nin\ruence of an oscillating tip, [25] reveals that the force\nand conductance transition regions coincide.\nFrom theF(z0) measurement, we reconstruct the 1D\npotential energy landscape, U(z0), across the spin tran-\nsition by integrating F(z0) (Figure 3C). The U(z0) curve\nshows a steep change in slope as the tip brings the hydro-\ngen closer to the CoH complex, suggestive of a transition\nbetween potential energy surfaces. This interpretation is\nbroadly in line with a framework recently developed by\nHapala et al. to describe high-resolution AFM imaging\nwith functionalized tips. [26] One key component of this\nmodel is that the probe particle, hydrogen in our case,4\nA\n0 100 2002\n0\n-2\n-4\n-6\n-8\n-10 0.00.20.40.60.81.01.2conductance G (10‒3 G0)frequency shift Δf (Hz)\nrelative height z (pm)\n0 100 200102\n1010.1− force F (pN)\nrelative height z' (pm)B\n1deconv. cond. G' (10‒3 G0)C\n0 200 4000\n-50\n-100\n-150potential energy U (meV)\nrelative height z' (pm)total energy E tot (meV)\ndCo-Pt (pm)D\n0400\n200\n-200600\n300 500 700 400 600-400\n-600\nFigure 3: Force Measurements on a Switching Complex. (A) Simultaneous frequency shift-distance, \u0001 f(z) (black),\nand conductance-distance, G(z) (gray), measurements on a CoH S= 1 complex with hydrogen-functionalized tip. The spin\ntransition, occurring at a relative zheight of 50 pm, is evident in both force and conductance channels. (B) Frequency shift\nwas converted to short-range forces (black) and the conductance was deconvoluted to remove averaging over the oscillation\namplitude (gray). On either side of the transition region the deconvoluted conductance and force increase exponentially and\ncan be described by the expressions, G(z) =G0exp(\u00002\u0014G(z0+z)) andF(z0) =F0exp(\u00002\u0014F(z0+z0)), respectively. Inverse\ndecay constants, \u0014G[nm\u00001]: 13:0\u00060:5 (0<z0<30 pm), 9:5\u00060:1 (70<z0<200 pm);\u0014F[nm\u00001]: 10:0\u00060:5 (0<z0<30 pm),\n4:2\u00060:3 (70< z0<200 pm). (C) Interaction potential energy surface during the S= 1 toS=1=2transition (black)\ndetermined by integrating the experimental F(z0) data. Dashed lines highlight the change in slope and indicate the point\nwhere a lower potential energy surface becomes accessible. Vertical dashed lines in (B) and (C) indicate the transition regime.\nFor all curves, zero distance corresponds to the point of closest approach. (D) Simulated diabatic potential energy curves for\na CoH=h{BN=Rh(111) complex approached by a H-functionalized Pt tip (dashed blue) and a CoH 2approached with a bare\ntip (dash-dotted red). The approximate adiabatic curve is shown as grey dotted line. The reaction coordinate dCo\u0000Ptis the\ndistance between the Co and the apex Pt atoms.\nnot only follows the lowest potential energy surface but\nalso undergoes relaxation within the junction. In our\nU(z0) measurement, the kink corresponds to the relative\nz0distance where the hydrogen on the tip apex can relax\nonto a lower potential energy surface. Chemically, the\nCoH is transformed into CoH 2, a magnetically distinct\ncomplex with a di\u000berent potential energy surface. DFT\ncalculations show that the transition between the two\ndistinct chemical con\fgurations proceeds via a contin-\nuum of intermediate transition states. This intermediate\nregime is characterized by hydrogen on the tip sharing its\nbond with the tip apex and the CoH while simultaneously\nundergoing reorientation within the junction. To tackle\nthis complexity, it is instructive to examine the diabatic\npotential energy surfaces in the limiting cases, that is,\nwhen CoH interacts with a hydrogen-functionalized tipand CoH 2interacting with a clean metallic tip. In Fig-\nure 3D we plot the dependence of the total energies of the\ntwo above con\fgurations (dash-dotted red and dashed\nblue curves, respectively) on the tip-sample distance d\n(see methods section and Figure S6), the chosen reac-\ntion coordinate for our system. The intersection of the\ntwo potential energy curves with varying junction width\ncon\frms that hydrogen transfer drives the spin transi-\ntion and is consistent with experimental observations. As\nmentioned above, the system does not follow the diabatic\ncurves, rather, it undergoes a barrierless transition along\na reaction path di\u000ecult to precisely identify by ab ini-\ntiomethods due to the shallowness and unevenness of\nthe hydrogen potential energy landscape in the junction.\nAn approximate shape of this pathway is shown in Fig-\nure 3D with a grey dashed line linking the two asymptotic5\ncases. The presence of a transition region (grey regions\nin Figure 3B,C) further indicates that the spin switching\nis driven by the approaching hydrogen and that the resi-\ndence time is shorter than the measurement time, leading\nto an averaging and smearing-out when probed with the\noscillating tip. Furthermore, the increased mechanical\ndissipation during the transition ( \u001855 meV=cycle, see\nFigure S3) is in good agreement with the potential en-\nergy di\u000berence across the transition region of the U(z0)\nmeasurement ( \u001835 meV) and points to the existence\nof hysteresis on the timescale of the oscillation period\n(2f0)\u00001\u001815\u0016s.\nDISCUSSION\nWe have reversibly switched the spin state of cobalt hy-\ndride complexes in a tunnel junction by using the hydro-\ngen on the tip apex as a tunable ligand. During the total\nspin change, a transition region exists where the hydro-\ngen can occupy two nearly equivalent sites separated only\nby a negligible barrier. As the junction conductance is\naltered, site equivalence is removed and the tip displace-\nment shifts the system into a preferred site. Beyond total\nspin, interactions with the tip determine how the cobalt\nspin couples with the host electron bath and whether a\ncorrelated Kondo state emerges. The relative stability of\nthe double potential well and the possible occurrence of\ncorrelations is expected to be dependent on the materials\nused and the reactivity of the adsorbed molecule on the\ntip and the undercoordinated adatom. Reconstructing\nthe 1D potential energy landscape shows that chemical\ntransformation within the junction is responsible for the\nchange of the total spin. Our measurements highlight\nhow tip functionalization can in\ruence the spin under\ninvestigation and suggests that they may be used to tai-\nlor molecular spins in ways di\u000ecult to achieve through\ntraditional chemical synthesis.\nMATERIALS AND METHODS\nThe clean Rh(111) surface was prepared by multiple cy-\ncles of argon ion sputtering and annealing to 1100 K\nin an ultrahigh vacuum chamber. During the \fnal an-\nnealing cycle borazine (B 3N3H6) gas was exposed to the\nhot sample at a pressure of 1 :2\u000210\u00006mbar for about\n\u00182 minutes, resulting in a monoatomic layer of h{BN.\nSubsequently, the sample was cooled down ( \u001820K) and\ncobalt was deposited via an electron beam evaporator.\nAs hydrogen is the predominant component of the resid-\nual gas background it is responsible for the formation\nof the cobalt hydride complexes. STM/AFM experi-\nments were performed on a homebuilt instrument operat-\ning in ultra-high vacuum at a base temperature of 1 :1 K.\nAll spectroscopic ( dI=dV ) measurements were obtained\nby adding a small sinusoidal voltage ( Vmod= 0:2 mV,\nfmod= 600\u0000800 Hz) to the bias voltage Vand using an\nexternal lock-in ampli\fer. We used V=\u000010 mV for ad-justing the conductance Gprior recording dI=dV curves\nby varyingVat constant z, as well as for recording G(z)\nand \u0001f(z) curves at constant V. Hydrogen pickup oc-\ncurs fortuitously while performing our experiments, for\nexample, during the approach of an initially CoH x(x\n=1{2) complex with a non-functionalized tip. Hydrogen-\nterminated tips were initially identi\fed by features such\nas enhanced contrast in constant current imaging and\nthen veri\fed by the characteristic non-exponential be-\nhavior inG(z) measurements (Figure 2B, green).\nThe quartz tuning fork has a resonance frequency of\nf0= 29;077 Hz and a quality factor of approximately\n10,000. Oscillation amplitudes of 100 pm were used. For\nthe frequency shift curves, \u0001 f(z), the tip was positioned\nabove the CoH complex with the oscillating tuning fork\natG= 1:72\u000210\u00005G0. The feedback loop was disabled\nand the oscillating tip then approached 200 \u0000300 pm\ntowards the sample and back. Immediately after com-\npletion, a retract curve was obtained by moving 2000 pm\naway from the surface (see Figure S2). When this se-\nquence was completed, the tip was moved at constant\nheight to the bare h{BN and \u0001 fh{BNwas obtained. The\nfrequency shift due to short-range forces are obtained by\ntaking the di\u000berence \u0001 f= \u0001fCoH\u0000\u0001fh{BN. We distin-\nguish between time averaged signals at tip heights zand\ndeconvoluted (instantaneous) signals at tip heights z0.\nFirst principles calculations have been carried out in\na manner consistent with the calculations employed in\nour previous work. [5] In brief, a density functional the-\nory approach, based on the projector-augmented-wave\nmethod [27] and a plane-wave basis set, [28] was used as\nimplemented in the Vienna Ab-initio Simulation Package\n(VASP). [28, 29] Exchange and correlation were treated\nwith the gradient-corrected functional as formalized by\nPerdew, Burke and Ernzerhof. [30] On-site Coulomb in-\nteraction corrections were accounted for in the framework\nof the LSDA+U formalism as introduced by Dudarev et\nal.[31] The considered geometry of the system was identi-\ncal to the one introduced in Ref. 5, i.e.a Rh(111) surface\nwas simulated by 5 Rh layers on top of which an h{BN\nsheet was deposited (considered to be commensurate for\ncomputational feasibility reasons). A CoH complex was\nresiding on top of an N atom of the h{BN sheet. The\ntip was simulated by a Pt pyramid of 4 atoms attached\nto a Rh surface. Prior to the tip approach simulation,\nthe tip and the sample were allowed to relax fully. After\nplacing the tip in the vicinity of the CoH xcomplex, the\nCoH xcomplex and the apex atom of the tip (and the at-\ntached hydrogen atom, where appropriate) were allowed\nto relax again, assuming an equilibrium static con\fgu-\nration. A full relaxation of the tip and the h{BNatoms\naround the CoH xcomplex was tested and not found to\nintroduce any change to the results above the calculation\nerror level. In the experimentally relevant case of a CoH\ncomplex being approached with a singly hydrogenated\ntip we are thus left with four atoms allowed to relax in 36\ndimensions amounting to a 12-dimensional con\fguration\nspace. In e\u000bect, however, both the tip apex Pt and the Co\natoms were found to experience only vertical relaxations\nwhich, according to our tests, result in minor quantita-\ntive change of the total energies and magnetic moments.\nThis complies fully with our understanding of the mech-\nanism of CoH xcomplex spin switching, the latter being\ngoverned fully by the Co \u0000H bonding, which in turn is al-\ntered through the rearrangement of the hydrogen atoms.\nTo underline this fact, the calculation results presented\nin Figures 3D and S4 were obtained in a diabatic fash-\nion, meaning that the initial relative orientation of the\nCoH xand the TipH 2\u0000xatoms were kept frozen and only\nthe distance between the tip and the sample (character-\nized by the distance dbetween Co and apex Pt atoms\n- see Figure S3) was varied. For hydrogen adsorption\nand transfer energies as well as the adiabatic transition\ncalculation the junction was allowed to relax fully.\nACKNOWLEDGEMENTS\nThe authors thank U. Schlickum for fruitful discussions,\nF. Giessibl for providing quartz tuning forks, and M.\nMausser for focused ion beam cutting of the Pt tips.\nFunding: P.J. acknowledges support from the Alexan-\nder von Humboldt Foundation. M.M. and M.T. acknowl-\nedge support from the SFB 767. O.B. and V.S. acknowl-\nedge support from the SFB 762. O.B. acknowledges\nsupport from the ERC Grant No. 320796, MODPHYS-\nFRICT.\nAuthor contributions: M.T. and K.K. conceived\nthe project. P.J., M.M., and G.L. performed the mea-\nsurements. P.J., M.M., and M.T. analyzed the data.\nO.B. and V.S. performed the density functional theory\ncalculations. 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B 54, 11169\n(1996), ISSN 0163-1829, URL http://link.aps.org/\ndoi/10.1103/PhysRevB.54.11169 .\n[29] G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993),\nISSN 0163-1829, URL http://link.aps.org/doi/10.\n1103/PhysRevB.47.558 .\n[30] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.\nLett. 77, 3865 (1996), ISSN 1079-7114, URL http://\nwww.ncbi.nlm.nih.gov/pubmed/10062328 .\n[31] S. L. Dudarev, S. Y. Savrasov, C. J. Humphreys, and\nA. P. Sutton, Physical Review B 57, 1505 (1998),\nISSN 1098-0121, URL http://link.aps.org/doi/10.\n1103/PhysRevB.57.1505 .8\nSUPPLEMENTAL INFORMATION\n0 100 200 3001\n0conductance G (10−3 G0)\nrelative height z (pm)A3\n2\n0 -5 -10 5 10\nbias voltage V (mV)1\n0dI/dV (arb. units)B\nABCD\nA,CB,D\nFigure S1: Reversibility of the Switching Process. (A) Forward (black) and backward (red) sweeps of a conductance-\ndistance measurement over the CoH complex shown in Figure 3 of the main text without tuning fork oscillation with V=\n\u000010 mV. Approach and retract curves were taken at 5 \u000010 pm=s. The switch between both spin states occurs at approximately\nthe samez-height (\u0001z<25 pm), thus we do not observe a signi\fcant hysteresis. (B) Bias spectroscopy over the same switching\ncomplex at high ( G\u001813\u000210\u00004G0) and low conductance ( G\u00186:5\u000210\u00004G0).\n100 200\nrelative height z (pm)0-30-25\n500 1000 1500 50 150-20-15-10-50frequency shift Δf (Hz)switching complex\nbare h-BN\ndifference\nFigure S2: Long-Range Background Subtraction. Raw Data of the original frequency shift curves that were taken over the\nswitching complex in Figure 3 (black) and a reference h{BN background (red). Both curves were then substracted to account\nfor only the short range interactions (blue).\ndissipation D (meV / cycle)\nrelative height z (pm)30405060708090switching complex\nbare h-BN\n100 200 0 500 1000 1500 50 150100\nFigure S3: Measured Dissipation Across the Spin Transition. Dissipation loss from the recorded excitation voltage of\nthe frequency shift and current curves in Figure 3 for the S= 1 toS=1=2switch. To \fnd the power dissipation (Figure S2)\nwe calculate the energy stored in the mechanical motion of the tuning fork ( E=1\n2k0A2,k0is the spring constant, A{ the\noscillation amplitude) and \fnd the intrinsic energy loss per oscillation cycle D= (\u0019k0A2)=Q, whereQis the Q-factor of the\ntuning fork. For the dataset in Figure 3 the parameters: Q\u001910000,k0= 1800 N=m andA= 100 pm, lead to a power\ndissipation per cycle of around D\u001935 meV. This value increases by \u001955 meV when the spin transition occurs at z\u001950 pm.9\n1.21.61.8\n1.4\ndCo-Pt (pm)mCo (μB)\n3004005006007002.02.2\n1.0\nFigure S4: Calculated Magnetic Moment as a Function of Co{Pt Separation. Evolution of the local magnetic moment\nof Co along the diabatic tip approach curves in Figure 3D. In the case of a hydrogen-functionalized tip approaching a Co H\ncomplex (blue line and circles), the magnetic moment is reduced as the hydrogen on the tip approaches Co by forming a\n(partial) chemical bond. If a bare tip approaches a CoH 2complex (red line, rectangles), at small junction widths the tip forms\na chemical bond with both hydrogen atoms partially claiming their valency and reducing the bonding to Co, resulting in a\npartial restoration of the magnetic moment.\n1C\nB0\n-4\n1nm\n-10 -5 5 10dI/dV (arb. units)\nbias voltage V (mV)001 -2-1\n-5-31234\n-18-4\n-6\n-8\n-10\n-12\n-14\n-16-2current (nA)frequency shift Δf (Hz)\nrelative height z (pm)300 100 0 200A\nF\nA\nB\nCEF\nDA\nB\nC\nD\nE\nF-8.0\n-4.0\n-0.1\n0.1\n4.0\n8.0bias (mV)AA\nFigure S5: Bias and Polarity Dependent \u0001f(z)Curves. (A) Constant current STM image ( V=\u000015 mV,I= 20 pA,\nG= 1:72\u000210\u00005G0) of a CoH complex on the h{BN=Rh(111) moir\u0013 e obtained with a non-functionalized tip. (B) Local\nspectroscopy obtained on the CoH with the tip positioned at the center of the CoH complex (Figure S5A, red dot) using a\nconductance setpoint G= 4:30\u000210\u00004G0. (C) Simultaneous frequency shift-distance, \u0001 f(z) (A{F, black{light gray), and\nconductance{distance, G(z) (red), measurements at di\u000berent bias voltages. The tip was stabilized at V=\u000015 mV,I= 20 pA\nbefore turning o\u000b the feedback loop and changing the bias for the di\u000berent approach curves. The change in bias and polarity\nstrongly in\ruences the tunneling current (red curves), but has almost no e\u000bect on the frequency shift signal.10\nd\nRh(111)h-BNCoHxPt tipRh(111)\nH\n420\nmeV\n-490\nmeV720\nmeV\nCoHH\nB N Co H Pt RhA B\nC\nD\nFigure S6: Schematic Drawing of the Simulated Junction Geometry. (A) Schematic representation of the junction\ngeometry used in \frst principles calculations: A 4-atomic functionalized Pt tip connected to a Rh lead suspended over a CoH\ncomplex residing on a h{BN=Rh(111) surface. For more details see the methods section. (B-D) Selected adsorption and transfer\nenergies with schematic representations of the relevant processes. The adsorption energy of a single H on Co =h{BN=Rh(111)\nfrom molecular hydrogen gas phase, calculated as E(H) =E(CoH) \u0000E(Co)\u0000E(H2)=2, is found to be 720 meV in our\ncalculations. The transfer of an H atom from the CoH complex to the Pt tip raises the total energy of the system by 420 meV,\nwhile the a\u000enity of the second hydrogen in a CoH 2complex is found to be 490 meV lower than its a\u000enity to the tip, which\nmakes the spin-manipulation of a CoH complex with a functionalized STM tip a reversible process."    },    {        "title": "1609.00841v1.Domain_structure_of_ultrathin_ferromagnetic_elements_in_the_presence_of_Dzyaloshinskii_Moriya_interaction.pdf",        "content": "Domain structure of ultrathin ferromagnetic elements in the\npresence of Dzyaloshinskii-Moriya interaction\nCyrill B. Muratov\u0003Valeriy V. Slastikovy\nMarch 31, 2022\nAbstract\nRecent advances in nanofabrication make it possible to produce multilayer nanos-\ntructures composed of ultrathin \flm materials with thickness down to a few monolayers\nof atoms and lateral extent of several tens of nanometers. At these scales, ferromagnetic\nmaterials begin to exhibit unusual properties, such as perpendicular magnetocrystalline\nanisotropy and antisymmetric exchange, also referred to as Dzyaloshinskii-Moriya inter-\naction (DMI), because of the increased importance of interfacial e\u000bects. The presence\nof surface DMI has been demonstrated to fundamentally alter the structure of domain\nwalls. Here we use the micromagnetic modeling framework to analyse the existence\nand structure of chiral domain walls, viewed as minimizers of a suitable micromagnetic\nenergy functional. We explicitly construct the minimizers in the one-dimensional set-\nting, both for the interior and edge walls, for a broad range of parameters. We then use\nthe methods of \u0000-convergence to analyze the asymptotics of the two-dimensional mag-\nnetization patterns in samples of large spatial extent in the presence of weak applied\nmagnetic \felds.\n1 Introduction\nThe exploding amount of today's digital data calls for revolutionary new high density,\nfast and long-term information storage solutions. Spintronics is one among the emerging\n\felds of nanotechnology o\u000bering a great promise for information technologies, whereby\ninformation is carried and processed, using the electron spin rather than its electric charge\n[1,3,44,54]. It brings about many opportunities for creating the next generation of devices\ncombining spin-dependent e\u000bects with conventional charge-based electronics. Despite being\na relatively new \feld of applied physics, it has already \frmly established its presence in\neveryday life through the development of new magnetic storage devices. The discovery\nof giant magnetoresistance (GMR), for which A. Fert and P. Gr unberg were awarded the\n\u0003Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ 07102, USA\nySchool of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom\n1arXiv:1609.00841v1  [cond-mat.mes-hall]  3 Sep 20162007 Nobel Prize in Physics, allowed an ability to \\read\" the magnetization states of\na ferromagnet through electric resistance measurements. This e\u000bect has been used in\nGMR-based spin valves, which transformed magnetic hard-disk drive technology, leading\nto increases in storage density by several orders of magnitude. Yet, the GMR magnetic\nstorage technology has already been superseded by novel spin-dependent devices based on\nthe e\u000bect of tunneling magnetoresistance (TMR), another exciting development in the \feld\nof spintronics [3].\nRecent discoveries of new physical phenomena that become prominent at nanoscale\nopen up a possibility of unprecedented data storage densities and read/write speeds. These\ninclude spin transfer torque (STT), chiral domain walls and magnetic skyrmions, spin\nHall e\u000bect, spin Seebeck e\u000bect, electric \feld control of the magnetic properties, etc. (see,\ne.g., [3, 8, 11, 18, 35, 41, 51]). The ability to manipulate the magnetization, using electric\ncurrents suggests novel designs for magnetic memory. One popular concept is the so-called\nracetrack memory [3, 43], which uses a two-dimensional array of parallel nanowires where\nmagnetic domains { \\bits\" { may be read, moved, and written through an application of\na spin current. Another promising type of memory and logic devices is based on storing\nand manipulating the data bits, using magnetic skyrmions, rather than magnetic domain\nwalls. The existence of magnetic skyrmions was predicted theoretically more than twenty-\n\fve years ago [5,6], but their experimental observations are much more recent [24,41,52].\nThe topological stability, small size and extremely low currents and \felds required to move\nmagnetic skyrmions make them natural candidates for the use in spintronic memory and\nlogic devices [18,52,53].\nA successful design of novel spintronic devices that make use of magnetic domain walls\nor skyrmions is strongly dependent on a deep theoretical understanding of static and dy-\nnamic behaviors of the magnetization in magnetic nanostructures. The manipulation and\ncontrol of magnetic domain walls and topologically protected states (e.g., magnetic vortices\nand skyrmions) in ferromagnetic nanostructures has been the subject of extensive experi-\nmental and theoretical research (see, e.g., [7,9,11,22,45,47,50]; this list is certainly far from\ncomplete). Recent advances in nanofabrication techniques [48] have lead to the production\nof ultrathin \flms with thickness down to several atomic layers and a lateral extent down\nto tens of nanometers. These ultrathin magnetic \flms and multilayer structures often\nexhibit unusual magnetic properties, attributed to an increased importance of interfacial\ne\u000bects. The most important features of these ultrathin magnetic structures include the\nappearance of perpendicular magnetic anisotropy [23, 27] and the Dzyaloshinskii-Moriya\ninteraction (DMI) [15,39]. The latter is closely related to re\rection symmetry breaking in\nsuch \flms and leads to emergence of magnetization chirality [17,25,50].\nThe experimental discovery of the symmetry breaking Dzyaloshinskii-Moriya interac-\ntion in ferromagnetic multilayers has generated a lot of interest in the physics commu-\nnity [4, 24, 46]. There has been a lot of work focusing on the in\ruence of DMI on mag-\nnetization con\fgurations within a ferromagnetic sample [4, 45, 50]. One of the interest-\ning features of DMI is its in\ruence on the pro\fle and the dynamic properties of domain\n2tailored composition or structure. Following this idea, we insertedepitaxially grown Ir layers between Pt(111) substrates and[Co/Ni]nstacks. By tuning the thicknessdIrof Ir layers in the[Co/Ni]n/Ir/Pt(111) system, we observed that DW structureindeed transitions as a function ofdIr. Here we find three verydifferent types of DW structures: right-handed chiral Ne´el wallsfordIr¼0.6 atomic monolayer (ML; Fig. 3a), achiral Bloch wallsfordIr¼2.5 ML (Fig. 3b) and left-handed chiral Ne´el walls fordIr¼3 ML (Fig. 3c). To track this spin texture transition in moredetail, we plot in Fig. 3d the histograms of the angleaas afunction ofdIr, where we find a single peak near 180/C176(right-handed chiral Ne´el wall) fordIrin the range of 0–1 ML, followedby a transition region betweendIr¼1.5 and 2.5 ML, where twopeaks in the histograms of the angleaare located at/C090/C176and\nca2 ML1 ML2 ML1 ML2 ML0.4 ML\n–90Number (normalized)BlochLeftNéelRightNéelBlochBlochmn\n/afii9825 (degree)/afii9825dbe\n09 0180 270NiCoNiCoNiCo\nPt\nFigure 1 | Real-space observation of chiral Ne´el walls in [Co/Ni]nmultilayer.(a) Sketch of [Co/Ni]3grown on Pt(111) substrate. (b) Compound SPLEEMimages highlighting the DW. The colour wheel represents the direction of in-plane magnetization in each image pixel. White arrows show the in-planespin orientations in the DWs. Scale bar, 2mm. (c) Sketch of right-handed Ne´el wall. (d) Histogram of angleain DW boundary counted pixel-by-pixel inbshows a Gaussian distribution peaked at 180/C176, which corresponds to right-handed Ne´el wall. Inset shows the definition of the anglea, wherenisa unit vector perpendicular to the domain boundary and always points from black domain to grey domain,mis the in-plane direction of the magnetizationwithin the DW. (e) Monte Carlo simulation result of perpendicular magnetized film with the DMI shows right-handed chiral Ne´el walls with similarspin structures as seen experimentally inb.\nac d\nbIrNiCoNiCo2 ML1 ML2 ML1 MLn=4IrNiCoNiCoNi2 ML1 ML2 ML1 ML2 ML\nPtNiCoNiCo2 ML1 ML2 ML1 MLn=9\nFigure 2 | The dependence of the DW chirality on substrates and thickness of films.(a) Compound SPLEEM images of Ni/[Co/Ni]2/Ir(111). Scale bar,2mm. (b) Sketch of left-handed Ne´el wall. (c) Compound SPLEEM images of [Co/Ni]9/Pt(111). Scale bar, 1mm. (d) Compound SPLEEM images of[Co/Ni]4/Ir(111). Scale bar, 1mm. Sketches on left sides in panelsa,canddshow multilayer structures of the samples. The colour wheel represents thedirection of the in-plane magnetization. White arrows in the SPLEEM images additionally highlight the in-plane spin orientations inside the DWs.NATURE COMMUNICATIONS | DOI: 10.1038/ncomms3671ARTICLE\nNATURE COMMUNICATIONS| 4:2671 | DOI: 10.1038/ncomms3671 | www.nature.com/naturecommunications3&2013Macmillan Publishers Limited. All rights reserved.cd\n2𝜇m Figure 1: Experimental and numerical observations of chiral domain walls in ultrathin\nferromagnetic \flms in the presence of DMI. (a) The schematics of the multilayer structure.\n(b) A colormap of the magnetization exhibiting chiral domain walls. (c) A histogram of\nthe in-plane magnetization orientation angle relative to the in-plane normal to the domain\nwall showing a preferred rotation direction. (d) A comparison to the result of a Monte-\nCarlo simulation of a discrete spin model. In (b), gray indicates the domains with the\nmagnetization up, black indicates the domains with the magnetization down, and the rest\nof the colors correspond to the directions of the in-plane component, as shown in the\ncolor-wheel. Adapted from Ref. [11], with permission; see that reference for further details.\nwalls [11,16,45,50]. In addition, it is well-known that DMI may be responsible for forma-\ntion of magnetic skyrmions { topologically protected states with a quantized topological\ndegree observed in ultrathin \flms [36, 41]. DMI also plays a crucial role in de\fning the\norientation of the domain walls and chiral behavior of the magnetization inside the wall,\nleading to the formation of a new type of chiral domain walls , also referred to as the\nDzyaloshinskii walls [50], having rather di\u000berent properties than the conventional Bloch\nand Neel walls [26]. For an illustration of chiral domain walls observed experimentally and\nnumerically, see Fig. 1. In a recent theoretical work [45], it was reported that the inter-\nplay between DMI and the boundary of an ultrathin ferromagnetic sample is responsible\nfor creating another type of domain walls { chiral edge domain walls . These walls play a\ncrucial role in producing new types of magnetization patterns inside a ferromagnet. For\ninstance, in the presence of a transverse applied \feld, chiral edge domain walls provide a\nmechanism for tilting of an interior domain wall in a ferromagnetic strip [7,40]. Moreover,\nthey also signi\fcantly modify the dynamic behavior of the interior domain wall under the\naction of current and an applied \feld [50].\nIn this paper, we study chiral domain walls in ultrathin ferromagnetic \flms, using rig-\norous analytical methods within the variational framework of micromagnetics. Our goal is\nto understand the formation of chiral interior domain walls and chiral edge domain walls,\nviewed as local or global energy minimizing con\fgurations of the magnetization, in sam-\n3ples with perpendicular magnetocrystalline anisotropy in the presence of surface DMI and\nweak applied magnetic \felds. The multi-scale nature of the micromagnetic energy allows\nfor a variety of distinct regimes characterized by di\u000berent relations between the material\nand geometric parameters, and makes its investigation a very challenging mathematical\nproblem. Many of these regimes have been investigated analytically, using modern tech-\nniques of calculus of variations in the context of various ferromagnetics nanostructures (see,\ne.g., [14]).\nOur starting point is a reduced two-dimensional micromagnetic energy, in which the\nstray \feld contributes only a local shape anisotropy term to the leading order (see (2.2)\nbelow). This energy gives rise to a non-convex vectorial variational problem, with a non-\ntrivial interplay between the boundary and the interior of the domain due to the DMI term.\nWe seek to understand the formation and structure of the domain walls { transition layers\nbetween constant magnetization states { that correspond to minimizers of the micromag-\nnetic energy. The framework for this analysis is provided by the variational methods of the\ngradient theory of phase transitions [37]. These types of problems have been extensively\nstudied in the mathematical community in both scalar [31,37,38,42] and vectorial [20,49]\nsettings. The nontrivial in\ruence of the boundary within the gradient theory of phase\ntransitions was investigated in [38,42].\nWe begin by investigating the one-dimensional problems on the in\fnite and semi-in\fnite\ndomains. Here we provide a complete analytical solution for the global energy minimizers\nof these one-dimensional problems, see Theorem 1 and Theorem 4, respectively. Our main\ntool is a careful analysis of the case of equality in the vectorial Modica-Mortola type lower\nbound for the energy of one-dimensional magnetization con\fgurations. Our analysis yields\nexplicit pro\fles for one-dimensional chiral interior and edge domain walls. These optimal\npro\fles are used later on in the constructions for the full two-dimensional problem. Our\none-dimensional results con\frm the physical intuition of [45] for a slightly reduced range\nof the DMI constants.\nWe then investigate the full two-dimensional energy in the regime of large domains and\nsmall applied \felds, using methods of \u0000-convergence. After a rescaling, this amounts to a\nstudy of the asymptotic behavior of the energy E\"(m) in (4.2) as \"!0. We note that\nour original problem is vectorial, constrained ( jm(x)j= 1), and the energy contains linear\ngradient terms in the interior, as well as boundary terms (after integration by parts),\nboth coming from DMI. Even though the original problem is vectorial { and these are\nnotoriously di\u000ecult phase transition problems { we show that one can reduce our problem\nto a scalar setting by decoupling the behavior of the normal magnetization component mk,\npreferring to be equal to \u00061, and the in-plane component m?, preferring to be 0, outside\nthe transition layer and proving that the optimal con\fguration of m?is a function of mk\nand the layer orientation. This nontrivial observation signi\fcantly simpli\fes the analysis\nof the problem and allows us to use the methods developed in [38, 42] to obtain the \u0000-\nlimit of the family of micromagnetic energies. The rest of the proof follows the pattern of\nthe gradient theory of phase transitions [37], with some modi\fcations to account for the\n4vectorial and constrained nature of the problem.\nWith the above tools, we obtain the \u0000-limit, given by (4.3), of the family of energies\nin (4.2) with respect to the L1convergence of m\"\nk. The limit energy is geometric, and\nits minimizers determine the locations of the chiral domain walls, which are now curves\nseparating the regions in which m0\nkchanges sign. As a consequence, we also obtain an\nasymptotic characterization of the energy minimizers of E\"as\"!0. Our main result,\nstated in Theorem 6, indicates that the presence of DMI signi\fcantly modi\fes the mag-\nnetization behavior in ultrathin magnetic \flms by creating both interior and edge chiral\ndomain walls.\nThe paper is organized as follows. In section 3, we present the solution of the one-\ndimensional global energy minimization problem for both the interior and boundary chiral\ndomain walls. Then, in section 4 we investigate the full two-dimensional energy (2.2) in\nthe regime of large domains and small applied \felds and study the behavior of the family\nof micromagnetic energies in (4.2) in the limit as \"!0. Finally, in section 5 we summarize\nour \fndings and discuss several additional modeling aspects of our problem, together with\nsome possible extensions of our analysis.\n2 Model\nWe start by considering a ferromagnetic \flm of thickness doccupying the spatial domain\n\n\u0002(0;d)\u001aR3, where \n\u0012R2is a two-dimensional domain specifying the shape of the\nferromagnetic element. Within the micromagnetic framework [26], the magnetization in the\nsample is described by the vector M=M(x;y;z ) of constant length jMj=Ms, whereMs\nis referred to as the saturation magnetization. The micromagnetic energy in the presence\nof an out-of-plane uniaxial anisotropy and an interfacial Dzyaloshinskii-Moriya interaction\n(DMI) may be written in the SI units in the form [5,6,50]\nE(M) =A\nM2sZ\n\n\u0002(0;d)jrMj2d3r+K\nM2sZ\n\n\u0002(0;d)jM?j2d3r\u0000\u00160Z\n\n\u0002(0;d)M\u0001Hd3r\n+\u00160Z\nR3Z\nR3r\u0001M(r)r\u0001M(r0)\n8\u0019jr\u0000r0jd3rd3r0+Dd\nM2sZ\n\n\u0010\nMkr\u0001M?\u0000M?\u0001rMk\u0011\nd2r:\n(2.1)\nHere we wrote M= (M?;Mk), where we de\fned M?2R2andMk2Rto be the\ncomponents of the magnetization vector Mthat are perpendicular and parallel to the\nmaterial easy axis (the z-axis), respectively, and introduced Mwhich is the trace of M\non \n\u0002f0g. In (2.1),Ais the exchange sti\u000bness, Kis the magnetocrystalline anisotropy\nconstant, Mhas been extended by zero outside the sample and r\u0001Mis understood\ndistributionally in R3,\u00160is the permeability of vacuum, H=H(x;y;z ) is the applied\nmagnetic \feld, and Dis the Dzyaloshinskii-Moriya interaction constant, following the\nstandard convention to write Din the units of energy per unit area. In writing the DMI\n5term in this speci\fc form, we took into account that it arises as a contribution from the\ninterface between the magnetic layer and a non-magnetic material and should, therefore,\nenter as a boundary term in the full three-dimensional theory.\nIn the above framework, the equilibrium magnetization con\fgurations in the ferromag-\nnetic sample correspond to either global or local minimizers of a non-local, non-convex\nenergy functional in (2.1). This energy includes several terms, in order of appearance:\nthe exchange term, which prefers constant magnetization con\fgurations; the magnetocrys-\ntalline anisotropy, which favors out-of-plane magnetization con\fgurations; the Zeeman, or\napplied \feld term, which prefers magnetizations aligned with the external \feld; the mag-\nnetostatic term, which prefers divergence-free con\fgurations; and the surface DMI term,\nwhich favors chiral symmetry breaking. The origin of the latter is the antisymmetric ex-\nchange mediated by the spin-orbit coupling in the conduction band of a heavy metal at\nthe ferromagnet-metal interface [13,17,19].\nThe variational problem associated with (2.1) poses a signi\fcant challenge for analysis.\nTherefore, in the following we introduce a simpli\fed version of the energy in (2.1) that\nis suitable for ultrathin ferromagnetic \flms of thickness d.`ex=p\n2A=(\u00160M2s), where\n`exis the material exchange length. In this case a two-dimensional model is appropriate\nin which the stray \feld energy can be modeled by a local shape anisotropy term (see,\ne.g., [21]; for a more thorough mathematical discussion of the stray \feld e\u000bect in ultrathin\n\flms with perpendicular anisotropy, see [28]). Measuring the lengths in the units of `exand\nthe energy in the units of Ad, we can rewrite the energy associated with the magnetization\ncon\fguration M(x;y;z ) =Msm(x;y), where m: \n!S2, as\nE(m) =Z\n\nn\njrmj2+ (Q\u00001)jm?j2\u00002hkmk\u00002h?\u0001m?\n+\u0014\u0000\nmkr\u0001m?\u0000m?\u0001rmk\u0001o\nd2r; (2.2)\nwhere we de\fned m?2R2andmk2Rto be the respective components of the unit magne-\ntization vector mand introduced the dimensionless quality factor Qand the dimensionless\nDMI strength \u0014:\nQ=2K\n\u00160M2s; \u0014 =Ds\n2\n\u00160M2sA; (2.3)\nwhereDis the DMI constant [50]. In (2.2), we also introduced a dimensionless applied\nmagnetic \feld h= (h?;hk) =H=Ms, with h?2R2andhk2R.\nWe are interested in the regime in which the \flm favors magnetizations that are normal\nto the \flm plane, i.e., when Q> 1. Also, since the energy is invariant with respect to the\ntransformation\n\u0014!\u0000\u0014; m?!\u0000m?; h?!\u0000h?; (2.4)\nwithout loss of generality we can assume \u0014to be positive.\n6(a)(b)\nxx\nθθ-10-505100.00.51.01.52.02.53.0\n02468100.00.51.01.52.02.53.0Figure 2: Two types of one-dimensional domain walls due to DMI: (a) interior wall; (b)\nedge wall. In the upper panels, \u0012stands for the angle between mand thez-axis. The\nvector mrotates in the xz-plane (lower panels).\n3 The problem in one dimension\nWe begin by considering an idealized situation in which the ferromagnetic \flm occupies\neither the whole plane or a half-plane, which leads to two basic types of domain walls\nconsidered below (see Fig. 2). These are the magnetization con\fgurations that vary in one\ndirection only. In the case of the half-plane, the magnetization is also assumed to vary in\nthe direction normal to the \flm edge. Throughout this section, we set the applied magnetic\n\feldhto zero.\n3.1 Interior wall\nConsider \frst the whole space situation, in which case we may assume that\n\n =f(x;y)2R2:x2R;0<y< 1g; (3.1)\nwith periodic boundary conditions at y= 0 andy= 1. We then take mto be a one-\ndimensional pro\fle, i.e., m=m(x). Then we may write the energy associated with min\nthe form\nE(m) =Z1\n\u00001n\njm0j2+ (Q\u00001)jm?j2+\u0014\u0010\nmk(^x\u0001m?)0\u0000(^x\u0001m?)m0\nk\u0011o\ndx; (3.2)\nwhere primes denote the derivative with respect to the xvariable and ^ xis the unit vector\nin the direction of the x-axis. We are interested in the global energy minimizers of the\nenergy in (3.2) that obey the following conditions at in\fnity:\nlim\nx!\u00061mk(x) =\u00061; lim\nx!\u00061m?(x) = 0: (3.3)\n7On heuristic grounds, one expects that the optimal domain wall pro\fle has the form of\ntheDzyaloshinskii wall [50]. Namely, one expects that in the domain wall the magnetization\nrotates around the direction of the y-axis. Hence, introducing an ansatz\nm= (sin\u0012;0;cos\u0012); (3.4)\none can rewrite the energy in (3.2) as [45]\nE(m) =Z1\n\u00001n\nj\u00120j2+ (Q\u00001) sin2\u0012+\u0014\u00120o\ndx: (3.5)\nObserve, however, that a priori the energy in (3.5) is not well de\fned in the natural class\nof\u00122H1\nloc(R), since the last term in the energy is not sign de\fnite and does not necessarily\nmake sense as the Lebesgue integral on the whole real line. This fact is closely related to\nthe chiral nature of DMI, favoring oscillations of the magnetization vector. A simple\ncounterexample, in which the \frst two terms of the energy in (3.5) are well de\fned, while\nthe last one is not, is given by the function \u0012(x) =\u0019\n2\u0000Si(x), where Si( x) =Rx\n0t\u00001sintdt\nis the sine integral function. It is also worth noting that if one were to de\fne the energy\nin (3.5) as the limit of the energies on large \fnite domains, then its minimum value would\nbe strictly greater than that obtained from the integral on the whole real line due to the\npresence of edge domain walls [45] (see also Sec. 3.2 for further details).\nTo \fx the issue above, one needs to assume that \u001202L1(R), which introduces a bound\non the total variation of \u0012onR. This, in turn, implies that the limit of \u0012(x) asx!\u00061\nexists, and the last term in (3.5) becomes a boundary term. Furthermore, in order for the\nenergy to be bounded the limits of \u0012(x) at in\fnity must be integer multiples of \u0019, and\nwithout loss of generality we may assume\nlim\nx!\u00001\u0012(x) =\u0019n; lim\nx!+1\u0012(x) = 0; n2Z: (3.6)\nThe energy then becomes\nE(m) =Z1\n\u00001n\nj\u00120j2+ (Q\u00001) sin2\u0012o\ndx\u0000\u0014\u0019n; (3.7)\nfor\u00122H1\nloc(R) with\u001202L1(R) and\u0012obeying (3.6), with n6= 0 to exclude the trivial case.\nIt is easy to see that the energy in (3.7) is uniquely minimized in the above class if and\nonly ifn= 1 and\u0014<\u0014c, where\n\u0014c=4pQ\u00001\n\u0019: (3.8)\nIn this case the optimal pro\fle is, up to translations, given by [45]\n\u0012(x) = 2 arctan e\u0000xpQ\u00001; (3.9)\n8and the wall energy is given by\n\u001bwall= 4p\nQ\u00001\u0000\u0019\u0014> 0: (3.10)\nIndeed, minimizers of (3.7) with n=\u00061 among all admissible \u0012are well known to exist\ndue to the good coercivity and lower semicontinuity properties of those terms (for technical\ndetails in a related problem, see [12]). The pro\fle in (3.9) is then the unique solution, up\nto translations and sign, of the Euler-Lagrange equation associated with (3.7) satisfying\n(3.6). At the same time, for jnj\u00152 the energy is easily seen to satisfy E(\u0012)\u0015jnj\u001bwall.\nHence, by inspection the minimizer with n= +1 corresponds to the global minimizer for\nalln6= 0, with the sign of ncorresponding to the wall chirality imparted by DMI.\nWe remark that, in contrast to the above situation, the problem associated with (3.2)\ndoes not admit minimizers for \u0014>\u0014c, since in this case the energy is not bounded below\nand favors helical structures [45].\nThe following theorem establishes existence and uniqueness of the minimizers of the\none-dimensional domain wall energy in (3.2) among all pro\fles satisfying (3.3) without\nassuming the ansatz in (3.4). In view of the discussion above, an appropriate admissible\nclass for the energy is given by\nA=\b\nm2H1\nloc(R;S2) :m02L1(R;R3)\t\n: (3.11)\nThe theorem below con\frms the expectation that the domain wall pro\fle is given by (3.4)\nand (3.9) for all \u0014below a critical value, although the latter turns out to be slightly lower\nthan the expected threshold value of \u0014=\u0014cgiven by (3.8).\nTheorem 1. Let0< \u0014 <pQ\u00001. Then there exists a unique, up to translations, min-\nimizer m2A of(3.2) satisfying (3.3) . The minimizer mhas the form in (3.4) with\u0012\ngiven by (3.9) , and the minimal energy is given by \u001bwallfrom (3.10) .\nProof. The proof proceeds by showing directly that the pro\fle given by (3.4) and (3.9) is\nthe unique minimizer via establishing a sharp lower bound for the energy. Assume without\nloss of generality that E(m)<+1. Then by dominated convergence theorem we have\nE(m) =Z1\n\u00001\u0010\njm0j2+ (Q\u00001)jm?j2\u0011\ndx+\u0014lim\nR!1ZR\n\u0000R\u0010\nmk(^x\u0001m?)0\u0000(^x\u0001m?)m0\nk\u0011\ndx;\n(3.12)\nandjm?(x)j!0 asx!\u00061 [10, Corollary 8.9]. Using integration by parts [10, Corollary\n8.10], the last integral may be rewritten as\nZR\n\u0000R\u0010\nmk(^x\u0001m?)0\u0000(^x\u0001m?)m0\nk\u0011\ndx= (^x\u0001m?(x))mk(x)\f\f\f\fR\n\u0000R\u00002ZR\n\u0000R(^x\u0001m?)m0\nkdx:\n(3.13)\n9Therefore, passing to the limit we obtain that\nE(m) =Z1\n\u00001\u0010\njm0j2+ (Q\u00001)jm?j2\u00002\u0014(^x\u0001m?)m0\nk\u0011\ndx: (3.14)\nWe now trivially estimate the DMI term from below to obtain\nE(m)\u0015Z1\n\u00001\u0010\njm0j2+ (Q\u00001)jm?j2\u00002\u0014jm?jjm0\nkj\u0011\ndx: (3.15)\nNext, we use the standard trick [29] to estimate the exchange energy by the term involving\nonlyjm0\nkj. In the following, we spell out the details of the argument, paying special attention\nto the optimality of the obtained estimates. We start by applying the weak chain rule [10,\nProposition 9.5] to the identity jm?j2+m2\nk= 1. This yields:\nm2\nkjm0\nkj2=jm?\u0001m0\n?j2\u0014jm?j2jm0\n?j2for a.e.x2R: (3.16)\nTherefore, for a.e. x2Rsuch thatjmkj<1 we can write\nm2\nkjm0\nkj2\n1\u0000m2\nk\u0014jm0\n?j2: (3.17)\nThus\nZ1\n\u00001jm0j2dx=Z1\n\u00001\u0010\njm0\n?j2+jm0\nkj2\u0011\ndx\u0015Z\nfjmkj<1gjm0\nkj2\n1\u0000m2\nkdx: (3.18)\nWriting the lower bound for the energy in terms of mk, with the help of (3.15) and\n(3.18) we obtain\nE(m)\u0015Z\nfjmkj<1g jm0\nkj2\n1\u0000m2\nk+ (Q\u00001)(1\u0000m2\nk)!\ndx\u00002\u0014Z1\n\u00001q\n1\u0000m2\nkjm0\nkjdx: (3.19)\nThis inequality may be rewritten in the following Modica-Mortola type form\nE(m)\u00152Z1\n\u00001\u0010p\nQ\u00001\u0000\u0014q\n1\u0000m2\nk\u0011\njm0\nkjdx\n+Z\nfjmkj<1g0\n@jm0\nkj\nq\n1\u0000m2\nk\u0000q\n(Q\u00001)(1\u0000m2\nk)1\nA2\ndx; (3.20)\nwhere we extended the domain of integration in the \frst term to the whole real line in view\nof the fact that by (3.16) we have m0\nk= 0 wheneverjmkj= 1.\n10We now turn to showing that the energy is minimized by the pro\fle given by (3.4) with\n\u0012given by (3.9). Indeed, from (3.20) we have for any R> 0:\nE(m)\u00152ZR\n\u0000R\u0010p\nQ\u00001\u0000\u0014q\n1\u0000m2\nk\u0011\njm0\nkjdx\n\u00152ZR\n\u0000R\u0010p\nQ\u00001\u0000\u0014q\n1\u0000m2\nk\u0011\nm0\nkdx\n=n\n2mk(x)p\nQ\u00001\u0000\u0014\u0010\nmk(x)q\n1\u0000m2\nk(x) + arcsin(mk(x))\u0011o\f\f\f\fR\n\u0000R; (3.21)\nwhere we used the assumption that \u0014 <pQ\u00001 to go from the \frst to the second line.\nFinally, passing to the limit as R!1 and using (3.3), we obtain\nE(m)\u0015\u001bwall; (3.22)\nwhere\u001bwallis de\fned in (3.10). At the same time, by the computation at the beginning of\nthis section the inequality above is an equality when mis given by (3.4) with \u0012from (3.9).\nIt remains to prove that the pro\fle given by (3.4) with \u0012from (3.9) is the unique, up\nto translations, minimizer of the energy that satis\fes (3.3). Without loss of generality,\nwe may assume that mk(0) = 0, in view of the continuity of mk(x) and (3.3). Since the\nminimal value of the energy is attained by dropping the last term in (3.20) and replacing\njm0\nkjwithm0\nk, we havem0\nk(x)\u00150 for a.e.x2R, andmksatis\fes\nm0\nk=p\nQ\u00001(1\u0000m2\nk) for a.e. x2I; (3.23)\nwhereI= (a;b) with\u00001\u0014a < 0< b\u00141 . Since the right-hand side of (3.23) is\ncontinuos,mkis the unique classical solution of (3.23) that satis\fes mk(0) = 0, which is\nexplicitlymk(x) = tanh(xpQ\u00001 ). Lastly, the inequality in (3.16) becomes equality when\nm0\n?is parallel to m?and, hence, m?=gbfor some constant vector b2R2and a scalar\nfunctiong:R![\u00001;1]. In turn, to make an inequality in (3.15) an equality, one needs\nto choose b= ^xandg\u00150. In view of the unit length constraint for jmj, this translates\nintom?= ^xsech2(xpQ\u00001). The obtained pro\fle m= (m?;mk) is then precisely the\none given by (3.4) with \u0012from (3.9).\nWe note that the arguments in the proof of Theorem 1 do not carry over to the rangepQ\u00001<\u0014\u0014\u0014c, since in this range we can no longer reduce the energy by passing to the\ncon\fgurations in the form given by (3.4). Nevertheless, an inspection of the proof shows\nthat the statement of Theorem 1 remains true for all m= (m?;mk) such that mk(x) is a\nnon-decreasing function of x. Hence, we have the following result.\nTheorem 2. For any\u0014>0, there exists a unique, up to translations, minimizer of (3.2)\namong all m= (m?;mk)2A satisfying (3.3) andm0\nk\u00150. The minimizer mhas the\nform in (3.4) with\u0012given by (3.9) , and the minimal energy is given by \u001bwallfrom (3.10) .\n11Remark 3. We point out that due to the presence of the edge domain walls (see the\nfollowing subsection) the minimizers of the energy in (2.2) in the form of a Dzyaloshinskii\nwall on a strip \n =R\u0002(0;L)are not one-dimensional for any L > 0. Nevertheless, if\none assumes periodic boundary conditions instead of the natural boundary conditions at the\nedges of the strip, an examination of the proof of Theorem 1 shows that the global minimizer\nis still given by (3.4) and(3.9) in this case.\n3.2 Edge wall\nConsider now the half-plane situation, in which case we may assume that\n\n =f(x;y)2R2:x>0;0<y< 1g; (3.24)\nwith periodic boundary conditions at y= 0 andy= 1. Taking mto be a one-dimensional\npro\fle, i.e., m=m(x), we write\nE(m) =Z1\n0n\njm0j2+ (Q\u00001)jm?j2+\u0014\u0010\nmk(^x\u0001m?)0\u0000(^x\u0001m?)m0\nk\u0011o\ndx; (3.25)\nwhere, as before, ^ xis the unit vector in the direction of the x-axis. Once again, in order\nfor this energy to be bounded, we must have jm?(x)j!0 asx!1 . Hence, in view of\nthe symmetry\nm?!\u0000m?; mk!\u0000mk; (3.26)\nwithout loss of generality we may assume that\nlim\nx!1mk(x) = 1: (3.27)\nNote, however, that the value of m(0) is not \fxed and needs to be determined for the\noptimal domain wall pro\fle at the material edge. Such edge domains walls were \frst\ndiscussed in [45].\nSince for\u0014 > \u0014c, where\u0014cis given by (3.8), the energy favors helical structures [45]\nand, hence, is not bounded below on the semi-in\fnite interval as well as on the whole line,\nthroughout the rest of this section we assume that \u0014<\u0014c. Assuming also the ansatz from\n(3.4) and arguing as in the previous subsection, for \u00122H1(R+) with\u001202L1(R+) we may\nwrite the energy in (3.25) as\nE(m) =Z1\n0n\nj\u00120j2+ (Q\u00001) sin2\u0012o\ndx\u0000\u0014\u0012(0); (3.28)\nwhich is easily seen to be minimized at \fxed \u0012(0) =\u001202(0;\u0019) by\n\u0012(x) = 2 arctan e(x0\u0000x)pQ\u00001; x 0=ln tan\u0010\n\u00120\n2\u0011\npQ\u00001: (3.29)\n12Indeed, using the Modica-Mortola trick [37], we rewrite the energy in (3.28) as\nE(m) = 2p\nQ\u00001Z1\n0jsin\u0012jj\u00120jdx+Z1\n0\u0010\nj\u00120j\u0000p\nQ\u00001jsin\u0012j\u00112\ndx\u0000\u0014\u00120\n\u0015\u0000Z1\n0\u0010\n2p\nQ\u00001jsin\u0012j\u0000\u0014\u0011\n\u00120dx=Z\u00120\n0\u0010\n2p\nQ\u00001jsin\u0012j\u0000\u0014\u0011\nd\u0012: (3.30)\nIn particular, the inequality above becomes an equality when \u0012is given by (3.29).\nWe now show that there exists a unique value of \u00120=\u0012\u0003\n02(0;\u0019) for which the function\nfrom (3.29) yields the absolute minimum of the energy in (3.28) for \u0014<\u0014c. Denoting the\nright-hand side in (3.30) by F(\u00120), we observe that F(0) = 0,F0(0)<0, andF(\u00120) =\nF(\u00120\u0000\u0019) +\u001bwall, where\u001bwall>0 is given by (3.10), for all \u00120\u0015\u0019. Therefore, for \u00120\u00150\nit is enough to consider the values of \u001202(0;\u0019), for which we have explicitly\nF(\u00120) = 2p\nQ\u00001 (1\u0000cos\u00120)\u0000\u0014\u00120: (3.31)\nA simple computation then shows that for \u00120\u00150 the function F(\u00120) is uniquely minimized\nby\n\u0012\u0003\n0= arcsin\u0012\u0014\n2pQ\u00001\u0013\n; (3.32)\nand the minimal value of F(\u00120) is given by\n\u001bedge= 2p\nQ\u00001 \n1\u0000s\n1\u0000\u00142\n4(Q\u00001)!\n\u0000\u0014arcsin\u0012\u0014\n2pQ\u00001\u0013\n<0: (3.33)\nIn fact, this is also an absolute lower bound for E(m) in (3.28), since for \u00120<0 the energy\nremains positive. Furthermore, since \u0012\u0003\n02(0;\u0019), this minimum value is attained by the\npro\fle in (3.29) with \u00120=\u0012\u0003\n0. Interestingly, we \fnd that \u0012\u0003\n02(0;arcsin2\n\u0019), spanning the\nrange from 0\u000eat\u0014= 0 to about 39 :5\u000efor\u0014=\u0014c. Thus, the global minimizer of the energy\nin (3.25) among all pro\fles satisfying (3.4) has the form of an edge domain wall whose\npro\fle is given by (3.29), up to a sign, with an optimal value of \u0012at the edge.\nWe now prove, once again, that this picture remains true without the ansatz in (3.4)\nfor a slightly smaller range of the values of \u0014 < \u0014c. The appropriate admissible class for\nthe energy in (3.25) is now\nA+=\b\nm2H1\nloc(R+;S2) :m02L1(R+;R3)\t\n: (3.34)\nTheorem 4. Let0< \u0014 <pQ\u00001. Then there exists a unique minimizer m2A+of\n(3.25) satisfying (3.27) . The minimizer mhas the form in (3.4) with\u0012given by (3.29) and\n\u00120=\u0012\u0003\n0from (3.32) , and the minimal energy is given by \u001bedge from (3.33) .\n13Proof. The proof proceeds exactly as in the case of Theorem 1, except that there is now an\nextra contribution from the boundary of the domain at x= 0. Namely, instead of (3.14)\nwe obtain\nE(m) =Z1\n0\u0010\njm0j2+ (Q\u00001)jm?j2\u00002\u0014(^x\u0001m?)m0\nk\u0011\ndx\u0000\u0014mk(0)(^x\u0001m?(0)):(3.35)\nEstimating both terms coming from DMI from below as\nE(m)\u0015Z1\n0\u0010\njm0j2+ (Q\u00001)jm?j2\u00002\u0014jm?jjm0\nkj\u0011\ndx\u0000\u0014jmk(0)jjm?(0)j;(3.36)\nand retracing the steps in the proof of Theorem 1, we obtain\nE(m)\u00152Z1\n0\u0010p\nQ\u00001\u0000\u0014q\n1\u0000m2\nk\u0011\njm0\nkjdx\u0000\u0014jmk(0)jq\n1\u0000m2\nk(0)\n+Z\nfjmkj<1g0\n@jm0\nkj\nq\n1\u0000m2\nk\u0000q\n(Q\u00001)(1\u0000m2\nk)1\nA2\ndx: (3.37)\nWith the help of the identity jm0\nkj=jjmkj0j[32, Theorem 6.17] and our assumption on \u0014,\nwe can further estimate the right-hand side in (3.37) from below as\nE(m)\u00152ZR\n0\u0010p\nQ\u00001\u0000\u0014q\n1\u0000m2\nk\u0011\njm0\nkjdx\u0000\u0014jmk(0)jq\n1\u0000m2\nk(0)\n\u00152ZR\n0\u0010p\nQ\u00001\u0000\u0014q\n1\u0000m2\nk\u0011\njmkj0dx\u0000\u0014jmk(0)jq\n1\u0000m2\nk(0)\n=n\n2jmk(x)jp\nQ\u00001\u0000\u0014\u0010\njmk(x)jq\n1\u0000m2\nk(x) + arcsin(jmk(x)j)\u0011o\f\f\f\fR\n0\n\u0000\u0014jmk(0)jq\n1\u0000m2\nk(0): (3.38)\nSimplifying the expression above and passing to the limit, we arrive at\nE(m)\u00152p\nQ\u00001 (1\u0000jmk(0)j)\u0000\u0014arccosjmk(0)j: (3.39)\nHowever, the right-hand side of (3.39) is nothing but F(arccosjmk(0)j), whereFis given\nby (3.31). Thus, E(m)\u0015\u001bedge, and equality holds for the pro\fle given by (3.4) and (3.29).\nFurthermore, as in the case of Theorem 1, the inequality above is strict for any other wall\npro\fle. This concludes the proof.\nRemark 5. According to Theorem 4, the magnetization vector in the edge wall that asymp-\ntotes tomk= +1 in the sample interior acquires a component that points along the inner\nnormal at the sample edge. At the same time, by (3.26) the magnetization vector in the\nedge wall that asymptotes to mk=\u00001in the sample interior acquires a component that\npoints along the outer normal at the sample edge.\n14•×Figure 3: Schematics of a magnetization con\fguration containing edge walls and a\nDzyaloshinskii wall. The arrows show the in-plane components of the magnetization vec-\ntor, the colors correspond to the out-of-plane component (\\red\" is up, \\violet\" is down,\nalso indicated by up/down symbols).\n4 The problem in two dimensions\nWe now go back to the original two-dimensional problem and consider the regime in which\nthe Dzyaloshinskii domain walls are present (for an illustration, see Fig. 3). The appear-\nance of these domain walls requires that the lateral extent of the ferromagnetic sample\nbe su\u000eciently large. Therefore, we introduce the domain \n \"=\"\u00001\n, where\"\u001c1, and\nrede\fne the energy in (2.2) on \n \":\nE(m) =Z\n\n\"n\njrmj2+ (Q\u00001)jm?j2\u00002h\"\u0001m+\u0014\u0000\nmkr\u0001m?\u0000m?\u0001rmk\u0001o\nd2r;\n(4.1)\nwhere we also de\fned a rescaled applied \feld h\"= (h\"\n?;h\"\nk) =\"(h0\n?;h0\nk) =\"h0, chosen\nto have an appropriate balance between the Zeeman and the domain wall energies (see\nbelow). We then rescale the domain back to \n and the energy by a factor of \", which leads\nto the following family of energies:\nE\"(m) =Z\n\nn\n\"jrmj2+\"\u00001(Q\u00001)jm?j2\u00002h0\nkmk\u00002h0\n?\u0001m?\n+\u0014\u0000\nmkr\u0001m?\u0000m?\u0001rmk\u0001o\nd2r: (4.2)\nThe purpose of this section is to understand the behavior of global energy minimizers of\nE\"as\"!0, which corresponds to the regime of interest. Throughout the rest of this\n15paper, \n\u001aR2is assumed to be a bounded domain with boundary of class C2. This is\ndone merely to reduce the technicalities of the proofs and focus on the vectorial aspects of\nthe problem involving DMI. With slight modi\fcations, the proof should apply to the case\nwhen@\n is a union of \fnitely many curve segments of class C1(see also [38, Remark 1.3]).\nOur main tool for the analysis of the variational problem associated with (4.2) will be\nthe following \u0000-convergence result.\nTheorem 6. Leth0= (h0\n?;h0\nk)2L1(\n;R3),Q > 1and0< \u0014 <pQ\u00001. Then, as\n\"!0, we haveE\"\u0000\u0000!E0with respect to the L1convergence, where\nE0(mk) =\u001bedgeH1(@\n) +\u001bwallH1(@\u0003\n+)\u00002Z\n\nh0\nkmkd2r; (4.3)\nin whichmk2BV(\n;f\u00001;1g)and@\u0003\n+is the reduced boundary of the set \n+, where\n\n\u0006=fx2\n :mk(x) =\u00061g: (4.4)\nMore precisely:\ni) For any sequence of m\"= (m\"\n?;m\"\nk)2H1(\n;S2)such that lim sup\"!0E\"(m\")<+1\nthere is a subsequence (not relabeled) and a function m0\nk2BV(\n;f\u00001;1g)such that\nm\"\nk!m0\nkandjm\"\n?j!0inL1(\n)as\"!0, and\nlim inf\n\"!0E\"(m\")\u0015E0(m0\nk): (4.5)\nii) For any m0\nk2BV(\n;f\u00001;1g)there is a sequence of m\"= (m\"\n?;m\"\nk)2H1(\n;S2)such\nthatm\"\nk!m0\nkandjm\"\n?j!0inL1(\n)as\"!0, and\nlim sup\n\"!0E\"(m\")\u0014E0(m0\nk): (4.6)\nProof. The proof follows the classical argument of Modica [38] adapted to the vectorial\nmicromagnetic setting and taking into account the boundary contributions to the energy.\nThe latter arise after integration by parts:\nE\"(m) =Z\n\n\u0010\n\"jrmj2+\"\u00001(Q\u00001)jm?j2\u00002h0\nkmk\u00002h0\n?\u0001m0\n?\u00002\u0014m?\u0001rmk\u0011\nd2r\n+\u0014Z\n@\nemk(em?\u0001\u0017)dH1(r);\n(4.7)\nwhere\u0017is the outward unit normal to @\n and (em?;emk) is the trace of ( m?;mk) on@\n.\nThe proof proceeds in three steps.\n16Step 1: Compactness. Given an admissible sequence of m\"= (m\"\n?;m\"\nk) satisfying E\"(m\")\u0014\nCas\"!0 for some C > 0 independent of \", with the help of (4.7) and an elementary\nbound on the DMI term we can write\nZ\n\n\u0010\n\"jrm\"j2+\"\u00001(Q\u00001)jm\"\n?j2\u00002\u0014jm\"\n?jjrm\"\nkj\u0011\nd2r\n\u0014C+ 2kjh0jkL1(\n)j\nj+\u0014H1(@\n): (4.8)\nTherefore, from (3.17) we obtain\nZ\n\n\\fjm\"\nkj<1g \"jrm\"\nkj2\n1\u0000jm\"\nkj2+\"\u00001(Q\u00001)(1\u0000jm\"\nkj2)!\nd2r\n\u00002\u0014Z\n\nq\n1\u0000jm\"\nkj2jrm\"\nkjd2r\u0014C0; (4.9)\nfor some constant C0>0 independent of \". Applying the Modica-Mortola trick to the \frst\nline in (4.9) and using the fact that by (3.16) we have jrm\"\nkj= 0 wheneverjm\"\nkj= 1, we\nobtain\n2Z\n\n\u0010p\nQ\u00001\u0000\u0014q\n1\u0000jm\"\nkj2\u0011\njrm\"\nkjd2r\u0014C0: (4.10)\nThis is equivalent toR\n\njr\b(m\"\nk)jd2r\u0014C0, where\n\b(s) = 2Zs\n0\u0010p\nQ\u00001\u0000\u0014p\n1\u0000t2\u0011\ndt= 2sp\nQ\u00001\u0000\u0014sp\n1\u0000s2\u0000\u0014arcsins (4.11)\nis a continuously di\u000berentiable, strictly increasing odd function of s2[\u00001;1]. Furthermore,\nby our assumption on \u0014we have 0<2(pQ\u00001\u0000\u0014)\u0014\b0(s)\u00142pQ\u00001. Therefore, by\nweak chain rule [10, Proposition 9.5] we have\nkm\"\nkkW1;1(\n)\u0014C00; (4.12)\nfor someC00>0 independent of \". In turn, by compactness in BV(\n) and the compact\nembedding of BV(\n) intoL1(\n) [2], this yields, upon extraction of a subsequence, that\nm\"\nk!m0\nkinL1(\n) for some m0\nk2BV(\n).\nTo prove thatjm0\nkj= 1 and, as a consequence, that jm\"\n?j!0 inL1(\n), we combine\n(4.9) and (4.12) to get\n\"\u00001(Q\u00001)Z\n\n\u0010\n1\u0000jm\"\nkj2\u0011\nd2r\u0014C0+ 2\u0014C00: (4.13)\nTherefore, the integral in the left-hand side of (4.13) converges to zero as \"!0 and,\nhence,m\"\nk(x)!\u0006 1 for a.e.x2\n. This concludes the proof of the compactness part of\nour \u0000-convergence result.\n17Step 2: Lower bound. We now proceed to establish (4.5). By the Modica-Mortola type\narguments in Step 1, we can estimate the energy from below as\nE\"(m\")\u0015Z\n\n\u0010\njr\b(m\"\nk)j\u00002h0\nkm\"\nk\u00002h0\n?\u0001m\"\n?\u0011\nd2r\u0000\u0014Z\n@\njem\"\nkjq\n1\u0000jem\"\nkj2dH1(r):\n(4.14)\nLetu\"= \b(m\"\nk). Then the lower bound in (4.14) may be rewritten as\nE\"(m\")\u0015Z\n\n\u0010\njru\"j\u00002h0\nkm\"\nk\u00002h0\n?\u0001m\"\n?\u0011\nd2r+Z\n@\n\u001b(eu\")dH1(r); (4.15)\nwhere\u001b(u) =\u0000\u0014j\b\u00001(u)jp\n1\u0000j\b\u00001(u)j2andeu\"is the trace of u\"on@\n, noting that\nu= \b(s) de\fnes a continuously di\u000berentiable one-to-one map from [ \u00001;1] to\nI=\u0002\n\u00002pQ\u00001 +1\n2\u0019\u0014;2pQ\u00001\u00001\n2\u0019\u0014\u0003\n. We next de\fne\ne\u001b(u) =juj+ min\nt2I(\u001b(t)\u0000jtj)u2I: (4.16)\nA straightforward calculation shows that we have explicitly\ne\u001b(u) =juj\u0000p\n4(Q\u00001)\u0000\u00142+\u0014arcsins\n1\u0000\u00142\n4(Q\u00001): (4.17)\nIn particular, e\u001b(u) is a 1-Lipschitz function of u, and by de\fnition e\u001b(u)\u0014\u001b(u). Therefore,\nby [38, Proposition 1.2] and the fact that jm\"\n?j!0 inL1(\n), proved in Step 1, we have\nlim inf\n\"!0E\"(m\")\u0015lim inf\n\"!0\u0012Z\n\njru\"jd2r+Z\n@\ne\u001b(eu\")dH1(r)\u0013\n\u00002Z\n\nh0\nkm0\nkd2r\n\u0015Z\n\njru0jd2r+Z\n@\ne\u001b(eu0)dH1(r)\u00002Z\n\nh0\nkm0\nkd2r; (4.18)\nwhereu02BV(\n;f\u00002pQ\u00001 +1\n2\u0019\u0014;2pQ\u00001\u00001\n2\u0019\u0014g) andu\"!u0inL1(\n). In (4.18),\nthe \frst integral in the last line denotes the total variation of u0, and the second term is\nunderstood as an integral of the trace of a BV function [2]. Notice that by (4.17) we have\ne\u001b(eu0) =\u001bedgeandjru0j=1\n2\u001bwalljrm0\nkj, after straightforward algebra. Therefore, the last\ninequality is equivalent to\nlim inf\n\"!0E\"(m\")\u0015\u001bwall\n2Z\n\njrm0\nkjd2r+\u001bedgeH1(@\n)\u00002Z\n\nh0\nkm0\nkd2r; (4.19)\nwhich coincides with (4.5) [2].\nStep 3: Upper bound. Without loss of generality, we may assume hk= 0 and h?= 0. Since\nwe have to preserve the constraint jmj= 1, we will construct an upper bound, using the\n18angle variables \u0012and\u001e. Namely, we de\fne m= (sin\u0012cos\u001e;sin\u0012sin\u001e;cos\u0012) and rewrite\nthe energy in (4.2) in terms of \u0012and\u001e(assumed to be su\u000eciently smooth) as follows:\nE(m) =Z\n\n\u0010\n\"jr\u0012j2+\"sin2\u0012jr\u001ej2+\"\u00001(Q\u00001) sin2\u0012\u0011\nd2r\n+\u0014Z\n\n(sin\u0012cos\u0012\u0000\u0012)r\u0001v(\u001e)d2r+\u0014Z\n@\n\u0012v(\u001e)\u0001\u0017dH1(r); (4.20)\nwhere v(\u001e) = (cos\u001e;sin\u001e), and we used integration by parts.\nLet \n\u0006be de\fned as in (4.4) with mk=m0\nk. Without loss of generality, we assume\nthat@\u0003\n+hasC2regularity, and that @\u0003\n+intersects@\n transversally, if at all. We de\fne\n\u0012\u0003(x) =(\n0x2\n+\n\u0019 x2\n\u0000; \u0012b(x) =(\n\u0012\u0003\n0x2@\nn@\n\u0000\n\u0019\u0000\u0012\u0003\n0x2@\nn@\n+; (4.21)\nwhere\u0012\u0003\n0is de\fned in (3.32), and take a sequence of \u0012\"2C1(\n) such that\n0\u0014\u0012\"\u0014\u0019; \u0012\"!\u0012\u0003inL1(\n); \u0012\"!\u0012binL1(@\n): (4.22)\nNotice that we also have \u0012\"!\u0012\u0003inLq(\n) for every q>1.\nNow, for a \fxed 1 <p< 2 we take two functions \u001e\u0006\n\u00032W1;p(\n\u0006) with values in [0 ;2\u0019)\nsuch that\nv(~\u001e\u0006\n\u0003(x)) =\u0007\u0017\n\u0006(x) for a.e. x2@\n\u0006; (4.23)\nwhere\u0017\n\u0006is the outward normal to \n\u0006and~\u001e\u0006\n\u0003are the traces of \u001e\u0006\n\u0003on@\n\u0006, respectively.\nSuch functions exists, for example, by [34, Theorem 2], since ~\u001e\u0006\n\u0003areC1functions of the\narclength, except at a \fnite number of isolated points where they have jump discontinuities,\nand, hence, belong to the appropriate Besov spaces in the assumptions of [34]. Next, we\nde\fne\u001e\u00032W1;p(\n) as\n\u001e\u0003(x) =(\n\u001e\u0000\n\u0003(x)x2\n\u0000\n\u001e+\n\u0003(x)x2\n+; (4.24)\nand observe that by construction we have\nv(~\u001e\u0003) =8\n><\n>:\u0017\n on@\n\u0000\\@\n\u0000\u0017\non@\n+\\@\n\u0017\u0003 on@\u0003\n+; (4.25)\nwhere\u0017's are the corresponding outward normals to the respective boundaries and ~\u001e\u0003is\nthe trace of \u001e\u0003on those boundaries. We can then construct, using a regularization and a\ndiagonal argument, a sequence of \u001e\"2C1(\n) such that\n\u001e\"!\u001e\u0003inW1;p(\n) and \"jr\u001e\"j2!0 inL1(\n): (4.26)\n19It is then clear that, as \"!0, we have\nZ\n\n\u0012\"r\u0001v(\u001e\")d2r!\u0019Z\n\n\u0000r\u0001v(\u001e\u0003)d2r=\u0019H1(@\u0003\n+) +\u0019H1(@\n\u0000\\@\n); (4.27)\nZ\n\nsin\u0012\"cos\u0012\"r\u0001v(\u001e\")d2r!0; (4.28)\nZ\n@\n\u0012\"v(\u001e\")\u0001\u0017dH1(r)!\u0000\u0012\u0003\n0H1(@\n+\\@\n) + (\u0019\u0000\u0012\u0003\n0)H1(@\n\u0000\\@\n): (4.29)\nPassing to the limit as \"!0 in the energy (4.20) and combining the terms, we obtain\nlim sup\n\"!0E(m\") = lim sup\n\"!0Z\n\n\u0010\n\"jr\u0012\"j2+\"\u00001(Q\u00001) sin2\u0012\"\u0011\nd2r\n\u0000\u0019\u0014H1(@\u0003\n+)\u0000\u0014\u0012\u0003\n0H1(@\n): (4.30)\nIn order to conclude, we need to construct a sequence of \u0012\"2C1(\n) satisfying (4.22)\nsuch that\nlim sup\n\"!0Z\n\n\u0010\n\"jr\u0012\"j2+\"\u00001(Q\u00001) sin2\u0012\"\u0011\nd2r\n=E0(mk) +\u0019\u0014H1(@\u0003\n+) +\u0014\u0012\u0003\n0H1(@\n): (4.31)\nThis construction was done in a more general setting in [42, Lemma 2]) and, therefore,\nusing this result we conclude that lim sup\"!0E(m\") =E0(mk), where\nm\"= (sin\u0012\"cos\u001e\";sin\u0012\"sin\u001e\";cos\u0012\") and (\u0012\";\u001e\") are as above.\nAs an immediate consequence of \u0000-convergence, we have the following asymptotic char-\nacterization of minimizers of the energy E\"in terms of the minimizers of E0.\nCorollary 7. Under the assumptions of Theorem 6, let m\"= (m\"\n?;m\"\nk)2H1(\n;S2)be\na sequence of minimizers of E\". Then, after extracting a subsequence, we have m\"\nk!m0\nk\nandjm\"\n?j!0inL1(\n), wherem0\nk2BV(\n;f\u00001;1g)is a minimizer of E0.\nWe note that by classical results for problems with prescribed mean curvature (see,\ne.g., [33] and references therein), the minimizers of E0are functions, whose jump set \u0000 \u001a\nis a union of \fnitely many C1;1curve segments satisfying weakly the equation\n\u001bwallK(x) = 4h0\nk(x); x2\u0000\\\n; \u00000(x)?@\n; x2\u0000\\@\n; (4.32)\nwhereKis the curvature of \u0000, positive if the set \n+is convex, and the prime denotes\narclength derivative. Physically, these are interpreted as the Dzyaloshinskii domain walls\nseparating the domains of opposite out-of-plane magnetization under the external applied\n\feld. We also note that the limit energy E0contains a contribution from the edge domain\nwalls, which, however, is independent of the magnetization orientation near the edge and\nthus only adds a constant term to the energy.\n20Remark 8. We note that by the results of [31], we can also say that if m0\nkis an isolated\nlocal minimizer of E0, then there exists a sequence of local minimizers m\"ofE\"such that\nm\"\nk!m0\nkandm\"\n?!0inL1(\n).\nBefore concluding this section, let us comment on some topological issues related to\nthe result in Theorem 6. We note that our upper construction in Theorem 6 uses the\nmagnetization con\fgurations that have topological degree zero. This has to do with the\nrepresentation of the test con\fgurations m\"adopted in the proof in terms of the angle\nvariables (\u0012\";\u001e\"), which are assumed to be of class C1up to the boundary. Therefore, the\nproof does not immediately extend to the admissible classes with prescribed topological\ndegree distinct from zero. This is not a problem, however, in view of the fact that away\nfrom the domain walls one could insert skyrmion pro\fles [36], suitably localized, into our\ntest functions to prescribe a \fxed topological degree for \"su\u000eciently small. Our result\nwould then not be altered, in view of the fact that in the considered scaling the energy of\na skyrmion is a lower order perturbation to that of chiral walls. In other words, under the\nconsidered scaling assumptions our energy does not see magnetic skyrmions.\n5 Discussion\nTo summarize, we have analyzed the basic domain wall pro\fles in the local version of\nthe micromagnetic modeling framework containing DMI, which is governed by the energy\nin (2.2). Speci\fcally, we performed an analysis of the one-dimensional energy minimizing\ncon\fgurations on the whole line and on half-line and showed that the magnetization pro\fles\nexpected from the physical considerations based on speci\fc ans atze are indeed the unique\nglobal energy minimizers for j\u0014j<pQ\u00001. This is slightly below (about 30%) the threshold\nvalue ofj\u0014j=\u0014c=4\n\u0019pQ\u00001, beyond which helical structures emerge. Our methods rely\non a sharp Modica-Mortola type inequality and do not extend to the narrow range ofpQ\u00001\u0014j\u0014j<4\n\u0019pQ\u00001. It is natural to expect that our result persists all the way to\nj\u0014j=\u0014c, but to justify this statement one would need to develop new analysis tools for the\nvectorial variational problem associated with the domain walls.\nOur one-dimensional analysis in section 3 identi\fed two basic types of chiral domain\nwalls: the interior and the edge domain walls. These one-dimensional domain wall solutions\nare the building blocks of the more complicated two-dimensional magnetization con\fgu-\nrations in ultrathin \flms subjected to su\u000eciently small applied magnetic \felds. This can\nbe seen from the analysis of \u0000-convergence of the energy in (4.2) performed in section 4.\nEither global or local energy minimizers for \"\u001c1 may then be approximated by those of\nthe energy in (4.3), which determines the geometry of the magnetic domains in the sample.\nOur \fndings indicate that in the considered limit the magnetization con\fgurations solve\nthe prescribed mean curvature problem in (4.32), again, for j\u0014j<pQ\u00001. We note that\nour variational setting could similarly be used to study the gradient \row dynamics gov-\nerned by (4.2) (for a related study, see [42]). Other physical e\u000bects, however, need to be\n21incorporated to account for some unusual properties of chiral domain walls such as their\ntilt in su\u000eciently strong external \felds [7,40].\nFinally, we would like to comment on the assumptions that lead to the model in (4.2),\nand on its possible generalizations. As was already mentioned, this energy functional is\nlocal, with the e\u000bect of the stray \feld surviving in the renormalized magnetocrystalline\nanisotropy term only. This is justi\fed in the limit of arbitrarily thin ferromagnetic \flms\n[21]. In practice, this contribution is only the leading order term in the expansion of the\nenergy in the \flm thickness for \flms whose thickness is less than the exchange length `ex\nof the material. Going to higher order, two types of contributions appear. The \frst is the\none coming from the sample boundary. In the limit of the dimensionless \flm thickness\n\u000e=d=`exgoing to zero, this contribution becomes local and adds an extra penalty term\nfor the in-plane component of the magnetization at the edge [30]:\nEedge\n\"(m) =\u000ejln\u000ej\n2\u0019Z\n@\n(\u0017\u0001m?)2dH1(r); (5.1)\nwhere\u0017is the outward unit normal to @\n. Here we took into account that in a perpendicu-\nlar material the magnetic \\charge\" at the sample boundary would be smeared on the scale\nof`ex. In the interior, the leading order contribution from the stray \feld energy beyond\nthe shape anisotropy can be shown to be [28]:\nEbulk\n\"(m) =\u0000\u000e\n8\u0019Z\n\nZ\n\n(mk(r)\u0000mk(r0))2\njr\u0000r0j3d2rd2r0+\u000e\n4\u0019Z\n\nZ\n\nr\u0001m?(r)r\u0001m?(r0)\njr\u0000r0jd2rd2r0:\n(5.2)\nFurthermore, for \u000e=\u0015jln\"j\u00001it was shown in the case \u0014= 0 and periodic boundary\nconditions in the plane that as \"!0 the e\u000bect of the stray \feld energy is to renormalize\nthe one-dimensional wall energy to a lower value, as long as \u0015 < \u0015c= 2\u0019pQ\u00001 [28]. It\nis natural to expect from the results of [28] that, as \"!0, the wall energy for \u0014>0 will\nbecome\n\u001bwall= 4p\nQ\u00001\u0000\u0019\u0014\u00002\u0015\n\u0019: (5.3)\nSimilarly, one would expect that in this regime the edge wall energy \u001bedgewould also be\nrenormalized to minimize the sum of the exchange, anisotropy, DMI energies (all contained\nin (4.2)) and the stray \feld energy contributions from (5.1) and (5.2). This study is\ncurrently underway. At the same time, for \u0015 > \u0015cone expects spontaneous onset of\nmilti-domain magnetization patterns and qualitatively new system behavior (for a recent\nexperimental illustration, see [52]).\nAcknowledgements. The work of CBM was supported, in part, by NSF via grants\nDMS-1313687 and DMS-1614948. VS would like to acknowledge support from EPSRC\ngrant EP/K02390X/1 and Leverhulme grant RPG-2014-226.\n22References\n[1] D. A. Allwood, G. Xiong, C. C. Faulkner, D. Atkinson, D. Petit, and R. P. 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