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[Yoshio Miura](https://orcid.org/0000-0002-5605-5452), [Keisuke Masuda](https://orcid.org/0000-0002-6884-6390)

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[First-principles calculations on the spin anomalous Hall effect of ferromagnetic alloys](https://mdr.nims.go.jp/datasets/e6c612a5-84d6-49ee-a410-7addfcb579af)

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First-principles calculations on the spin anomalous Hall effect of ferromagnetic alloysPHYSICAL REVIEW MATERIALS 5, L101402 (2021)LetterFirst-principles calculations on the spin anomalous Hall effect of ferromagnetic alloysYoshio Miura 1,2,* and Keisuke Masuda 11Research Center for Magnetic and Spintronic Materials (CMSM), National Institute for Materials Science (NIMS),1-2-1 Sengen, Tsukuba, Ibaraki 305-0047, Japan2Center for Spintronics Research Network (CSRN), Graduate School of Engineering Science, Osaka University,Machikaneyama 1-3, Toyonaka, Osaka 560-8531, Japan(Received 15 June 2021; revised 4 October 2021; accepted 12 October 2021; published 29 October 2021)The spin anomalous Hall effect (SAHE) in ferromagnetic metals, which can generate a spin-orbit torque torotate the magnetization of another ferromagnetic layer through a nonmagnetic spacer in magnetic junctions, hasattracted much attention. We theoretically investigated the spin anomalous Hall conductivity (SAHC) of the L10-type alloys XPt (X = Fe, Co, Ni) on the basis of first-principles density functional theory and linear responsetheory. We found that the SAHC of FePt is much smaller than the anomalous Hall conductivity (AHC), leadingto very small polarization for the anomalous Hall effect ζ = SAHC/AHC of around 0.1. On the other hand, theSAHC increases with an increasing number of valence electrons (Nv), and CoPt and NiPt show relatively largevalues of |ζ |, greater than 1. The negative contribution of the spin-down-down component of AHC is the originof the large SAHC and ζ in CoPt and NiPt, which is due to the antibonding states of Pt around the Fermi levelin the minority-spin states.DOI: 10.1103/PhysRevMaterials.5.L101402Reducing the power consumption for magnetization rever-sal in magnetic tunnel junctions (MTJs) is one of the mostimportant tasks for the realization of magnetic random ac-cess memory (MRAM) [1]. Current-induced switching, suchas spin transfer torque [2–5] (STT)-induced switching, hasemerged as a promising method for magnetization reversalin MTJs. In STT-induced switching, the spin flip of a con-duction electron flowing perpendicular to the plane gives atorque to the local spin moment due to the conservation ofangular momentum, and a continuous spin-flip scattering ofconduction electrons can rotate the magnetization directionof a free layer in MTJs. In order to realize ultrahigh-densityMRAM, we must reduce the critical current density Jc0 in themagnetization reversal by STT up to 105 A/cm2 [1].Spin-orbit torque [6–8] (SOT)-induced switching also hasattracted much attention in recent years because it enableshigh-speed and reliable operation in three-terminal MRAM,where the magnetization of a free layer can be switched by anin-plane current flowing in the nonmagnetic layer attached tothe free layer in the current in-plane geometry. SOT is causedby spin current injection into a ferromagnetic layer from anonmagnetic layer due to the spin Hall effect (SHE) in non-magnetic metals [9,10]. The spin current Js generated by theSHE can be given by Js ∝ αSH[ŝ × Jc], where αSH is the spinHall angle given by the ratio of the spin Hall conductivity σspinxyto the conductivity of the charge current σxx, i.e., σspinxy /σxx, ŝis the direction of the quantization axis of an electron spin,and Jc is the charge current. To obtain an efficient spin-orbittorque in ferromagnetic layers, a large spin-orbit interactionis necessary in the nonmagnetic layer. Thus, heavy metals*miura.yoshio@nims.go.jpsuch as Pt and Ta have been used as nonmagnetic under-layers. While SOT-induced switching leads to more efficientmagnetization switching in MRAM, the direction of the spinquantum axis of injected electrons related to the direction ofthe torque is limited by the geometry of the device, such asthe direction of the current flow and the magnetic anisotropyof the ferromagnetic layers.Recently, Taniguchi et al. proposed a new type of SOT-induced switching [11], in which the nonmagnetic layer isreplaced by a ferromagnetic layer, and spin current due tothe spin anomalous Hall effect (SAHE) in the ferromagneticlayer can provide torque to rotate the magnetization of theferromagnetic upper layer through the nonmagnetic spacer.Figures 1(a) and 1(b) show schematic viewgraphs of SAHEand SHE in ferromagnetic materials (see Supplemental Mate-rial for a detailed explanation of the definition of SAHE [12]).In this case, the spin current Js generated by the SAHE can begiven by Js ∝ (ζ − β )αAH[m̂ × Jc], where m̂ is the directionof magnetization of the bottom ferromagnetic layer, αAH isthe anomalous Hall angle σxy/σxx, β is the spin polarizationof the bottom ferromagnetic layer, and ζ is the ratio of thespin current to the transverse charge current by the AHE,i.e., σspinxy /σxy. Thus, ferromagnetic materials showing largeζ are promising as high-efficiency spin current sources bythe SAHE. So far, several experiments have been carried outto evaluate the efficiency of the spin-orbit torque originatingfrom the spin current in ferromagnetic metals for varioussystems [13–17]. Theoretical studies on the SAHE of typicalferromagnetic metals such as Fe, Ni, and Co have been per-formed using first-principles calculations [18]. Furthermore,the correlation between AHE and SHE of CoPt was theo-retically investigated [19]. However, the material dependenceof SAHE in ferromagnetic alloys is still unclear. Here, weinvestigate and discuss the efficiency of SAHE on the basis2475-9953/2021/5(10)/L101402(5) L101402-1 ©2021 American Physical Societyhttps://orcid.org/0000-0002-5605-5452https://orcid.org/0000-0002-6884-6390http://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevMaterials.5.L101402&domain=pdf&date_stamp=2021-10-29https://doi.org/10.1103/PhysRevMaterials.5.L101402YOSHIO MIURA AND KEISUKE MASUDA PHYSICAL REVIEW MATERIALS 5, L101402 (2021)xyzXXXXXXXXXXXXPtPtPtPtacX=Fe, Co or Ni(c)xyzSHE(b)xyzSAHE+SHE(a)FIG. 1. Schematic viewgraph of (a) spin anomalous Hall effect(SAHE: magnetization m̂ is perpendicular to electric field Ê ) and(b) magnetization-independent spin Hall effect (SHE: m̂ is parallelto Ê ) in ferromagnets. (c) Crystal structure and coordinate system ofL10-XPt (X = Fe, Co, Ni).of first-principles calculations and linear response theory inorder to clarify the possible origin of the SAHE. To this end,we focus on the Pt-based L10 binary alloys, FePt, CoPt, andNiPt, as sources of SOT in ferromagnetic materials.We performed density functional theory (DFT) calcula-tions on L10-type ferromagnetic alloys using the PHASE/0code [20] with the projector augmented-wave (PAW) potential[21] including the spin-orbit interaction. The spin-polarizedgeneralized gradient approximation was adopted for the ex-change and correlation energy [22]. The cutoff energies ofthe plane wave and the charge density were set to 70 and700 Ry, respectively. The spin-orbit interaction was includedby adding the spin-orbit Hamiltonian to the nonlocal PAWpotential part in the DFT calculations with plane-wave basissets [23,24]. We used the tetragonal unit cell for L10-XPt(X = Fe, Co, and Ni) as shown in Fig. 1(c), which includesone X atom and one Pt in the unit cell. The lattice parametersof c along the ẑ direction and a along the x̂ or ŷ direction weredetermined by structure optimization calculations. We usedthe lattice constants a = 2.7437 Å and c = 3.7641 Å for FePt,a = 2.6925 Å and c = 3.7286 Å for CoPt, and a = 2.7407 Åand c = 3.5945 Å for NiPt.To calculate the intrinsic transverse Hall conductivity, weassume that the electric field is applied along the [100] di-rection (x̂), the Hall current flows in the [010] direction (ŷ),and the magnetization is directed along [001] (ẑ) when theanomalous Hall effect (AHE) is considered [see Figs. 1(a)and 1(b)]. The anomalous Hall conductivity (AHC) and thespin Hall conductivity (SHC) can be obtained by using linearresponse theory for electronic conductivity [25,26] as follows,σαxy = BαV∑k�αxy(k), (1)�αxy(k) is the so-called Berry curvature of the charge or spin[27,28], which is given by�αxy(k) = 2h̄2m2e∑n′>n( fkn − fkn′ )Im〈kn|pαy |kn′〉〈kn′|px|kn〉(εkn′ − εkn)2,(2)where n and n′ are the band indices of occupied and unoccu-pied states. V , me, εkn, and fkn are the cell volume, electronmass, band energy, and occupation function for each k pointk, band n. pαy and px are the momentum operators for the yand x directions. α denotes “charge” or “spin” current, whereTABLE I. Number of valence electrons Nv, anomalous Hall con-ductivity σ AHCxy , magnetization m̂-dependent spin anomalous Hallconductivity (SAHC) σ SAHC+SHCxy , spin Hall conductivity (SHC)σ SHCxy , SAHC σ SAHCxy , and Hall conductivity polarization ζ =eh̄ σ SAHCxy /σ AHCxy , which are calculated according to Eqs. (1)–(3).The units of AHC and (S)AHC are (� cm)−1 and ( h̄e )(� cm)−1,respectively.Nv σ AHCxy σ SAHC+SHCxy σ SHCxy σ SAHCxy ζFePt 18 1031 445 163 282 0.273CoPt 19 481 563 115 448 0.931NiPt 20 −826 2371 378 1993 −2.41pchargey = px and pspiny = (pysz + sz py)/2, respectively. sz is thePauli spin matrix. Our definition of the momentum operatorof spin current is consistent with Eq. (11) in Ref. [29] andEq. (23) of Ref. [18]. Bα = e2/h̄ for α = charge and Bα = efor α = spin. pβ = (me/h̄)∂H/∂kβ (β = x, y), where H isthe Hamiltonian of the present system and kβ is the wavevector of the β direction. σchargexy indicates the AHC. σspinxyindicates the spin anomalous Hall conductivity (SAHC) ina ferromagnet depending on the magnetization direction m̂,i.e., σ SAHC+SHCxy (m̂), which includes m̂-dependent SAHC andm̂-independent SHC as shown in Fig. 1(a). Thus, the SAHCcan be obtained byσ SAHCxy = σ SAHC+SHCxy (m̂ ‖ ẑ) − σ SHCxy (m̂ ‖ x̂), (3)where m̂ ‖ ẑ and m̂ ‖ x̂ indicate perpendicular [001] and in-plane [100] magnetization directions, respectively [18]. Theeigenstate |kn〉 and the band energy εkn can be obtained fromthe DFT calculations, including the spin-orbit interaction. Theconvergence of σαxy as a function of the k points was carefullychecked for each alloy, and 93 × 93 × 65 k points in the firstBrillouin zone were used for the electronic structure calcula-tions and linear response calculations of L10-FePt, CoPt, andNiPt.In Table I, we show the calculation results for AHC σ AHCxy ,SHC σ SHCxy , and SAHC σ SAHCxy for L10-type FePt, CoPt, andNiPt alloys. AHC and SAHC show a clear chemical trend forL10-XPt (X = Fe, Co, Ni). Larger AHC and smaller SAHCwere obtained for FePt, while smaller AHC and larger SAHCwere found for CoPt and NiPt. As a result, |ζ | of FePt isvery small and less than 1, while the values of |ζ | for CoPtand NiPt are larger than 1. Thus, we can say that CoPt andNiPt are more favorable for obtaining large SOT than FePt.The small value of ζ in FePt is not consistent with the recentexperimental result by Seki et al. [17], where a large ζ around6 was estimated from the experiment.Furthermore, a longitudinal resistivity of FePt is estimatedaround 93 μ� cm in the experiment, which correspondsto σxx ≈ 1.1 × 104 (� cm)−1. Thus, an efficiency of SAHEcan be given by αSAHE = [( eh̄σSAHC − βσAHC)]σxx = [(282 −0.4 × 1031)](1.1 × 104) ≈ −0.012. Here, we assume that aspin polarization of the longitudinal conductivity is around0.4, which comes from the spin polarization of total density ofstates of FePt. Thus, the theoretically estimated αSAHE is oneorder magnitude smaller than that of the experiments (≈0.25).L101402-2FIRST-PRINCIPLES CALCULATIONS ON THE SPIN … PHYSICAL REVIEW MATERIALS 5, L101402 (2021)FIG. 2. (a) Anomalous Hall conductivity (AHC) (� cm)−1 and(b) spin anomalous Hall conductivity (SAHC) ( h̄e )(� cm)−1 of L10-FePt, CoPt, and NiPt as a function of the number of valence electronsNv. The AHC and SAHC listed in Table I correspond to the val-ues of Nv = 18 for FePt, Nv = 19 for CoPt, and Nv = 20 for NiPt,respectively.Since the present calculation does not include the extrinsicpart of the transverse Hall conductivity, such as the skewscattering and the side jump effect due to impurities, the largevalue of ζ found in the experiment might be attributed to theextrinsic part of σαxy.To clarify the differences in AHC and SAHC for L10-XPt(X = Fe, Co, Ni), we show AHC and SAHC as a function ofthe number of valence electrons (Nv) for each alloy in Fig. 2.To obtain the Nv-dependent AHC and SAHC, we evaluated theoccupation function fkn in Eq. (2) with changing Nv. Note thatAHC and SAHC shown in Table I correspond to the values ofNv = 18 for FePt, Nv = 19 for CoPt, and Nv = 20 for NiPt. Ascan be seen in Fig. 2, the AHCs of FePt, CoPt, and NiPt show asimilar valence dependence. The same is true for SAHC. TheAHC roughly decreases with increasing Nv, and the sign ofAHC changes from positive to negative around Nv = 19–20,while the SAHC increases with increasing Nv and reachesa maximum around Nv = 20–21. These results indicate thatthe differences in AHC and SAHC for FePt, CoPt, and NiPtcan be explained by the difference in the number of valenceelectrons within the rigid-band model.To obtain further understanding of this point, we dividedthe anomalous Hall conductivity for each spin componentfor perpendicular magnetization m̂ ‖ ẑ. Each spin componentσ ↑↑xy , σ ↓↓xy , σ ↑↓xy , and σ ↓↑xy indicates the spin component ofoccupied and unoccupied states in Eq. (2), which can beobtained by calculating the eigenvalues of the sz operatorfor each eigenstate |kn〉 along the spin quantum axis. Thesedecompositions to up-spin and down-spin states for eacheigenstate are approximate because the spin-orbit interactionmixes the up-spin and down-spin states in the DFT calcula-tions. Nonetheless, it would be very useful to understand AHCand SAHC in terms of electronic structures. For m̂ ‖ ẑ, theσ AHCxy and σ SHCxy (m̂) can be given byσ AHCxy = σ ↑↑xy + σ ↓↓xy + σ ↑↓xy + σ ↓↑xy , (4)eh̄σ SAHC+SHCxy (m̂ ‖ ẑ) = σ ↑↑xy − σ ↓↓xy . (5)Equation (5) is consistent with Eq. (23) in Ref. [30]. It isnatural that the AHC can be given by the sum of all spincomponents in matrix elements of momentum operator.TABLE II. Spin-decomposed anomalous Hall conductivity σ ↑↑xy ,σ ↓↓xy , σ ↑↓xy and σ ↓↑xy (� cm)−1 of FePt, CoPt, and NiPt, where theanomalous Hall conductivity can be given by σ AHCxy = σ ↑↑xy + σ ↓↓xy +σ ↑↓xy + σ ↓↑xy .(� cm)−1 σ ↑↑xy σ ↓↓xy σ ↑↓xy σ ↓↑xyFePt (Nv = 18) 545 100 160 226CoPt (Nv = 19) 472 −91 85 15NiPt (Nv = 20) 583 −1788 289 90Table II shows the spin-decomposed AHC at the Fermilevel for FePt, CoPt, and NiPt. In the case of FePt, all spincomponents have positive values, which provides a largerAHC according to Eq. (4). It was found that the spin-up-upcomponent is dominant, and the spin-down-down componentis relatively small in FePt. Furthermore, σ ↑↓xy and σ ↓↑xy showrelatively large positive values, but these spin-flip terms donot contribute to the spin current shown by Eq. (5). Thismeans that the spin current of FePt is dominated only byσ ↑↑xy according to Eq. (5), which is much smaller than σ AHCxy .Thus, we obtained very small Hall conductivity polarizationζ = eh̄σ SAHCxy /σ AHCxy for FePt, less than 1. In the case of CoPtand NiPt, the spin-up-up components are similar to that ofFePt, while the spin-down-down components show negativevalues. In particular, NiPt shows large negative σ ↓↓xy , corre-sponding to −1700 (� cm)−1. The opposite signs of σ ↑↑xyand σ ↓↓xy in the spin-conserving AHC cancel out the chargecurrent in Eq. (4), but enhance the spin current in Eq. (5).This provides a small AHC and large SAHC, leading to alarger Hall conductivity polarization |ζ | of more than 2.4 forNiPt. In Fig. 3, we show the spin-decomposed AHC as afunction of the number of valence electrons. It is apparent thatthe spin-up-up component slightly decreases, while the spin-down-down component significantly decreases and changessign from positive to negative with increasing number of va-lence electrons around Nv = 18–20. The change of the sign inthe σ ↓↓xy in Fig. 3(d) around Nv = 18–20 can be attributed toFIG. 3. Spin-decomposed AHC (a) σ ↑↑xy , (b) σ ↑↓xy , (c) σ ↓↑xy , and(d) σ ↓↓xy (� cm)−1 for FePt, CoPt, and NiPt as a function of thenumber of valence electrons Nv.L101402-3YOSHIO MIURA AND KEISUKE MASUDA PHYSICAL REVIEW MATERIALS 5, L101402 (2021)FIG. 4. Color map of Berry curvatures �chargexy (k) in Eq. (2) for(a) FePt, (b) CoPt, and (c) NiPt and the nodal line N (k) in Eq. (6) for(d) FePt, (e) CoPt, and (f) NiPt plotted on the Fermi surface in thethree-dimensional Brillouin zone of the tetragonal unit cell, whichare visualized with FERMISURFER [31].appearance of d (yz, zx) and d (xy) states in the minority-spinstates at the Fermi level, which will be explained later. Thesetrends also suggest that the behaviors of the AHC and SAHCfor FePt, CoPt, and NiPt could originate in the difference inthe number of valence electrons within the rigid-band model.Next, we consider the Berry curvature to gain insight intothe large positive σ ↑↑xy and negative σ ↓↓xy spin, especially inCoPt and NiPt. In Fig. 4, we show the Berry curvature ofcharge �chargexy (k) in Eq. (2) and N (k) in Eq. (6) correspond-ing to the denominator of Eq. (2) with occupation functionsmapped onto the Fermi surface of FePt, CoPt, and NiPt in thethree-dimensional Brillouin zone,N (k) =∑n =n′( fkn − fkn′ )1(εkn′ − εkn)2. (6)The N (k) will diverge if two eigenvalues with occupied andunoccupied states are close to each other at the same k point.This situation will occur when the nodal lines of band disper-sions (band crossing points) are located around the Fermi leveland the spin-orbit interaction opens a gap in the degeneratestates. Thus, N (k) will reflect the nodal line of band structureswithout the spin-orbit interaction in the Brillouin zone. First,we can find that the distribution of Berry curvature �chargexy (k)is not the same as the distribution of the nodal line N (k).This means that there are many k points in which the matrixelements of the momentum operator are very small for kpoints on the nodal line, due to the forbidden transition of themomentum operator. The matrix of the momentum operatorcorresponding to the dipole moment operator has nonzeroelements only for transitions between states of different parity.Thus, the value of matrix elements with the same parity iszero, where the parity corresponds to the eigenvalue of thespace inversion operator for eigenstates |kn〉.In Fig. 4(a), the positive peaks of the Berry curvatures forFePt are expressed in yellow or red, especially on the  -Z lineand the A-M line. Positive peaks also appear along the nodalline on the Fermi surface around the  -Z line see Figs. S1–S3for detailed plots of kx-ky, ky-kz, and kz-kx planes for the Berrycurvatures and the nodal lines in the three-dimensional (3D)Brillouin zone [12]. We confirmed that these positive peaksof Berry curvature come from the spin-up-up, spin-up-down,and spin-down-up components, providing the large AHC ofFePt. Then, in Fig. 4(b), hot-spot-like positive and negativeBerry curvatures can be found on the nodal line of CoPt. Wefound that the positive peaks are from the spin-up-up compo-nent, while the negative peaks are due to the spin-down-downcomponent in the Berry curvature of CoPt. Since the negativespin-down-down component decreases the AHC and increasesthe spin current, the SAHC of CoPt exceeds the AHC, leadingto the Hall polarization being larger than 1. This trend is moreremarkable in NiPt. In Fig. 4(c), the negative Berry curvaturesexpressed in blue can be found on the A-M line and alongthe nodal line [indicated by the black and white arrows inFigs. 4(c) and 4(f)]. Some positive peaks also appear at the Xpoint and on the  -Z line. The negative Berry curvatures aredue to the spin-down-down components, resulting in the hugenegative SAHC of NiPt. We also confirmed that the spin-up-down and spin-down-up components in the Berry curvaturesof NiPt are much smaller than those of FePt and CoPt. Thespin-down-up component contributes to AHC, but not to thespin current. Therefore, the SAHCs are not so large for FePtand CoPt. These trends are consistent with the dependence ofAHC and SAHC on the number of valence electrons in Figs. 2and 3.According to Table II and Eqs. (4) and (5), a large negativeσ ↓↓xy is essential to obtain a large SAHC and Hall polarizationζ . The spin-orbit interaction of Pt is one order of magnitudelarger than that of Fe, Co, and Ni, indicating that the role ofPt atoms will be more significant than those of Fe, Co, andNi. Thus, to clarify the origin of the negative σ ↓↓xy in CoPt andNiPt from the viewpoint of electronic structures, we show inFig. 5 the projected density of states (PDOS) on each atomicorbital of Pt atoms in FePt, CoPt, and NiPt. Since the Berrycurvature and the nodal line are broadly distributed over thewhole range of the Brillouin zone as shown in Fig. 4, thePDOS will be better to present the orbital properties in theBerry curvature as compared with the band dispersion along ahigh-symmetry line. We found a systematic shift in the PDOSof FePt, CoPt, and NiPt and a relatively large PDOS in theminority-spin states around the Fermi level for Pt d (yz, zx)and d (xy) of NiPt as compared to those of CoPt and FePt.These minority-spin states of Pt in NiPt around the Fermilevel are the antibonding states with Ni, providing negativeBerry curvatures along the nodal line on the Fermi surface,as shown in Fig. 4(c). Note that antibonding properties areL101402-4FIRST-PRINCIPLES CALCULATIONS ON THE SPIN … PHYSICAL REVIEW MATERIALS 5, L101402 (2021)FIG. 5. Projected density of states (DOS) onto each atomic or-bital (a) d (yz, zx), (b) d (3z2 − r2), (c) d (xy), and (d) d (x2 − y2) forFePt, CoPt, and NiPt as a function of energy relative to the Fermienergy.not directly related to the negative contribution to the Berrycurvature. However, the appearance of the antibonding statesin the minority-spin states around the Fermi level results inthe negative spin down-down term for the Berry curvature ofNiPt.In summary, we have investigated the SAHC of theferromagnetic alloys L10-FePt, CoPt, and NiPt. The Hall con-ductivity polarization ζ = eh̄σ SAHCxy /σ AHCxy of FePt is around0.1. This is due to the contributions of each spin componentof AHC in FePt being positive. On the other hand, the SAHCincreases and AHC decreases with an increasing number ofvalence electrons (Nv), leading to large Hall conductivity po-larizations |ζ | of greater than 1.0 for CoPt and NiPt. Sincethese alloys show a similar Nv dependence, the differenceof FePt, CoPt, and NiPt in AHC and SHC can be under-stood from the dependence in the number of valence electronswithin the rigid-band model. 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