Giant converse magnetoelectric effect in a multiferroic heterostructure with polycrystalline Co2FeSi

To overcome a bottleneck in spintronic applications such as those of ultralow-power magnetoresistive random-access memory devices, the electric-field control of magnetization vectors in ferromagnetic electrodes has shown much promise. Here, we show the giant converse magnetoelectric (CME) effect in a multiferroic heterostructure consisting of the ferromagnetic Heusler alloy Co2FeSi and ferroelectric-oxide Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT) for electric-field control of magnetization vectors. Using an in-plane uniaxial magnetic anisotropy of polycrystalline Co2FeSi film grown on PMN-PT(011), the nonvolatile and repeatable magnetization vector switchings in remanent states are demonstrated. The CME coupling coefficient of the polycrystalline Co2FeSi/PMN-PT(011) is over 1.0 × 10−5 s/m at room temperature, comparable to those of single-crystalline Fe1-xGax/PMN-PT systems. The giant CME effect has been demonstrated by the strain-induced variation in the magnetic anisotropy energy of Co2FeSi with an L21-ordered structure. This approach can lead to a new solution to the reduction in the write power in spintronic memory architectures at room temperature. A material harnessing strong electricity–magnetism interactions, useful for efficient information processing, has been developed by researchers in Japan and UK. Rather than using just the electrical charge of an electron, spintronics takes advantage of a magnetic electron property known as spin. Spintronics has great potential for low power-consumption devices, such as random-access memory. Materials with both strong electric and strong magnetic properties, necessary for electrical control of the spin, are, however, rare. Kohei Hamaya from Osaka University, Japan, and colleagues from Osaka University, Japan, University of York, UK, and Tokyo Institute of Technology, Japan, demonstrated a large magnetoelectric effect in an alloy of cobalt, iron, and silicon. The team layered their alloy with a ferroelectric material to optimize the strain in a way that induced a so-called giant converse magnetoelectric effect. We show giant converse magnetoelectric effect in a multiferroic heterostructure consisting of ferromagnetic Heusler alloy Co2FeSi and ferroelectric-oxide Pb(Mg1/3Nb2/3)O3-PbTiO3 for an electric-field control of magnetization vectors. In this system, the non-volatile and repeatable magnetization vector switchings in remanent states are also demonstrated. This approach can lead to a new solution to the reduction in the write power in spintronic memory architectures at room temperature.


Introduction
Switching the magnetization vectors via spin transfer torque using an electric current has been utilized as a method of writing information for next-generation spintronic nonvolatile memories such as magnetoresistive random-access memory (MRAM) devices [1][2][3][4] . In general, a high electric current density is required to switch the magnetization vectors of ferromagnetic electrodes in magnetic tunnel junctions (MTJs) during the information writing process, and it is a serious bottleneck in the application of spintronic devices. For this problem, several methods for manipulating magnetization vector via an electric field are expected to markedly reduce the power consumption 5 . However, there are limitations on the usage environment of ferromagnetic semiconductors 6,7 and ultrathin ferromagnets 8,9 to control and/or change the magnetization vectors. In practical applications, it is very desirable to switch the magnetization vectors by an electric field without using an assist-magnetic field above room temperature.
With respect to the giant ME coupling coefficients of more than 1.0 × 10 −5 s/m, single-crystalline Fe 1-x Ga x alloys grown on PMN-PT are promising materials 34,35 , as Fe 1-x Ga x alloys are good magnetostrictive materials for strain sensors, actuators, and energy harvesters 36,37 . However, spin polarization at the Fermi level of Fe 1-x Ga x alloys is less than that of spintronic materials 36 . Thus, highperformance spintronic materials, such as FM half-metals grown on PMN-PT, should be explored to realize magnetization switching with ultralow power consumption [23][24][25] . Here, we have chosen Co-based Heusler alloys, since they are expected to be half-metallic materials with high Curie temperatures [38][39][40] . In this work, we focus particularly on Co 2 FeSi as an FM material because the L2 1 -ordered Co 2 FeSi has already shown a half-metallic nature [40][41][42] . We demonstrate giant ME coupling coefficients of more than 1.0 × 10 −5 s/m by utilizing highperformance polycrystalline Co 2 FeSi films with the L2 1 -ordered structure on PMN-PT(011). The findings in the present study have the potential to overcome the current bottleneck in spintronic devices.

Polycrystalline Co 2 FeSi/PMN-PT(011) heterostructures
We use an FE rhombohedral PMN-PT pseudocubic substrate with a large piezoelectric constant 43 . Unlike in the previous report on PMN-PT(001) 29 , the PMN-PT(011) single crystal is chosen because there have been many studies on the electric-field control of the magnetization vector [26][27][28] . The detailed growth procedure of Co 2 FeSi films on PMN-PT(011) is described in the Materials and methods.
A schematic of the Co 2 FeSi/PMN-PT(011) heterostructure is shown in Fig. 1a, where a 0.3-nm-thick Fe layer is inserted between Co 2 FeSi (30 nm) and PMN-PT(011). This layer was added to improve the crystallinity of the   Co 2 FeSi films, since films with poor structural quality are unlikely to have a substantial converse ME (CME) effect 29 . In Fig. 1b Fig. 1c. The two different RHEED patterns indicate that the growth mode of the Co 2 FeSi layer on the (011) surface of PMN-PT is anisotropic. Figure 1d is a high-resolution transmission electron microscopy (HRTEM) image from a region at the interface that shows the PMN-PT substrate and the Co 2 FeSi film. The HRTEM clearly shows that between the substrate and film, an amorphous layer forms during growth, since the in situ RHEED from the substrate shows a well-ordered crystalline PMN-PT(011) surface. The EDX spectra from the amorphous interface layers (not shown here) showed that the amorphous layer in Fig. 1d is an oxide with little Co, Fe or Si. The grown film is polycrystalline with a grain width of 13 ± 4 nm. Co 2 FeSi grains are highly textured; hence, the RHEED patterns indicate epitaxial like film growth. The digital diffractograms from the labeled regions of the grains (Fig. 1d) show the grain orientations and the L2 1 structure of the Co 2 FeSi grains by observing the (111) ordering spots. We also performed nanodiffraction (see Figs. S1 and S2 of Supplementary information 44 ) from individual grains that show the preferred grain growth along the [422] and [220] directions, and the results are further confirmed and discussed in the next paragraph.
The structural analysis from the X-ray diffraction (XRD) ω-2θ scan (out-of-plane) for the Co 2 FeSi/PMN-PT(011) heterostructure is shown in Fig. S3 of the Supplementary information 44 . Weak diffraction peaks from the (220) and (422) planes were observed in the out-of-plane XRD measurement (Fig. S3a), indicating that the grown Co 2 FeSi layer was not a highly oriented structure on PMN-PT(011), consistent with the data shown in Fig. 1d. On the other hand, from the pole figure measurement (2θ = 27.55 degrees) shown in Fig. 1e, we clearly observed {111} diffraction peaks, indicating the presence of the L2 1 -ordered structure of Co 2 FeSi. These results are consistent with the results observed in the HRTEM image and diffractogram shown in Fig. 1d and the results presented in Figs. S1 and S2 of the Supplementary information 44 . From these structural characterizations, we conclude that the grown Co 2 FeSi layer is a textured polycrystalline film on PMN-PT(011), in which the polycrystalline Co 2 FeSi film includes the high spin polarization L2 1 -ordered structure [40][41][42] .

Initial in-plane uniaxial anisotropy
Prior to the investigation of magnetic properties of the Co 2 FeSi/PMN-PT(011) heterostructure, we briefly present the well-known unpoled state of the PMN-PT(011) substrate 21 , as schematically shown in Fig. 2a. The spontaneous piezoelectric polarizations of (011) cut PMN-PT lie along the diagonals of the (011) plane and (011) plane, as shown in the top section of the figure 21 . In this situation, the (011) plane of the PMN-PT pseudocubic single crystal has a rectangular shape with a long axis along the [011] direction, as depicted at the bottom of the figure.
We first measure general magnetic properties of the Co 2 FeSi/PMN-PT(011) heterostructure at room temperature. Magnetic-field (H)-dependent magnetization, measured along the PMN-PT[011] and PMN-PT[100] crystallographic directions in the film plane, is presented in Fig. 2b. An anisotropic feature of the magnetization curves is present along the two different crystal axes, where the two different RHEED patterns are observed during the growth in Fig. 1c. Because the value of the saturation magnetization (M S ) (1090 ± 30 kA/m) is nearly the same as that in our previous works on Co 2 FeSi films 29,42,45 , we regard the relatively high M S value as a consequence of the formation of the L2 1 -ordered structure. Figure 2c shows a polar plot of the squareness of the  Fig. 2a), the inplane uniaxial magnetic anisotropy can be understood by the anisotropic lattice distortion induced from the (011) plane of the PMN-PT substrate. This evidence shows that by utilizing our growth method, as illustrated in Fig. 1, moderate in-plane uniaxial magnetic anisotropy can be induced even in polycrystalline Co 2 FeSi/PMN-PT(011) heterostructures. Because the (011) surface of the pseudo-cubic PMN-PT unit cell is distorted with shear strain along the red polarization vectors 46 , we infer that the small deviation of the uniaxial easy axis from PMN-PT[011] is an intrinsic property of this system.

Strain-induced converse magnetoelectric effect
To characterize the electric field (E) effect on magnetic properties for the Co 2 FeSi/PMN-PT(011) heterostructures, we perform magneto-optic Kerr ellipticity (η) measurements at room temperature by applying E, where H is applied to the crystallographic direction along PMN-PT[011] or [100] while E is applied to the PMN-PT[011] direction. Here, as a reference, the reported polarization switching process of a PMN-PT(011) single crystal is described in Fig. S4 of the Supplementary information 44 . Figure 3a shows the plots of the Kerr-ellipticity magnitude in the remanent state (η R ) as a function of E at room temperature, in which each point is obtained by measuring H-dependent Kerr-ellipticity curves along the PMN-PT[011] and PMN-PT[100] direction, as shown in Fig. 3b. Because we can clearly observe the saturation behavior of the Kerr-ellipticity magnitude (η S ) in Fig. 3b, we can assume that the value of η S {(611 ± 3) × 10 −6 radian} corresponds to the value of M S (1090 ± 30 kA/m) measured in Fig. 2b. On the basis of the assumption, the M R values in the right axes in Fig. 3a are determined as M R = M S (η R /η S ).
Both η R − E curves shown in Fig. 3a indicate the presence of two magnetization states at E = 0. These features greatly differ from the conventional strain-induced magnetization vector switching processes that can be easily predicted from the polarization switching of PMN-PT(011), as shown in [011] [100] [011]  (Fig. S6c), we should consider the competition between the initial in-plane uniaxial magnetic anisotropy and the strain-induced magnetic anisotropy in the Co 2 FeSi film at the two states. The two different states of the lattice deformation of the (011) plane at E = 0 are schematically shown in Fig. 3c, d. When E is increased from −0.8 MV/m to 0, the tensile strain along PMN-PT[011] is maintained at E = 0 (Fig. 3c). In this situation, the straininduced uniaxial magnetic anisotropy of the Co 2 FeSi film along the PMN-PT[011] direction is added to the initial uniaxial magnetic anisotropy along PMN-PT[011] with a small off-axis orientation. As a result, the magnetization direction of the Co 2 FeSi film is maintained along a magnetic easy axis in the PMN-PT[011] direction. On the other hand, when E is decreased from +0.8 MV/m to 0, compressive strain along PMN-PT[011] is observed (Fig. 3d); this compressive strain decreases the magnitude of the uniaxial magnetic anisotropy of the Co 2 FeSi film along the PMN-PT[011] direction. In this situation, the remanent magnetization direction of the Co 2 FeSi film can switch from near the PMN-PT[011] to the PMN-PT[100] direction. Thus, we can qualitatively explain the origin of the two magnetization states at E = 0, shown in Fig. 3a, b, by considering the presence of the anisotropic piezostrain of the (011) surface of the PMN-PT substrate along the PMN-PT[ 011] direction at E = 0. However, if the initial in-plane uniaxial anisotropy of the Co 2 FeSi film along the PMN-PT[ 011] direction was weaker or stronger than that in the present study, the two magnetization states at E = 0 could have been unstable. To quantitatively understand its origin, we should further investigate the correlation among the magnitude of the initial in-plane uniaxial magnetic anisotropy, local domain structures, and the magnitude of the global piezostrain 46,47 for many Co 2 FeSi/PMN-PT(011) heterostructures at E = 0.
As shown in Fig. 3b, the magnetization directions at two different states at E = 0 are nearly switched from the uniaxial hard axis to the uniaxial easy axis or the uniaxial easy axis to the uniaxial hard axis during the E sweeping process. Therefore, nearly 90°magnetization vector switching can occur in the remanent state (H = 0) after the application of positive or negative E values, as schematically shown in Fig. 4a. Figure 4b

Giant CME coupling coefficient
We quantitatively evaluate the CME effect of the Co 2 FeSi/PMN-PT(011) heterostructures. To estimate the CME coupling coefficient (α E ), we define the value of α E as follows: α E = μ 0 dM R dE , where μ 0 is the vacuum permeability. Figure 5a displays the value of α E as a function of E, estimated from the data in Fig. 3a directions. Very interestingly, the value of α E was over 1.0 × 10 −5 s/m when E~−0.25 MV/m at room temperature, as shown in Fig. 5a. In the relevant fields, an α E of more than 1.0 × 10 −5 s/m has thus far been reported in multiferroic heterostructures consisting of PMN-PT substrates and single-crystalline magnetostrictive materials such as FeRh 22 and FeGa alloys 34,35 . On the other hand, we observe the giant α E of more than 1.0 × 10 -5 s/m even in the polycrystalline Co 2 FeSi/PMN-PT(011) heterostructure, where Co 2 FeSi is one of the most representative spintronic Co-based Heusler alloys 38,39 . This evidence indicates that magnetostrictive materials and single-crystalline structures are not strict conditions for obtaining a giant α E of more than 1.0 × 10 −5 s/m.
To verify the importance of Co 2 FeSi, we also investigate the CME effect of the Fe 3 Si/PMN-PT(011) heterostructure, where Fe 3 Si is a binary Heusler alloy 42 , and the growth of the heterostructure, including its characterizations, is also shown in Fig. S7  In Table 1, we summarize the reported values of the giant α E for various multiferroic heterostructures and compared them to those of this study. Apart from the Fe 50 Rh 50 / BaTiO 3 (001) system 22 , a giant CME effect was observed in various FM/PMN-PT systems 28,34,35 . As previously described, because ferroelectric PMN-PT has a piezoelectric constant 43 that is relatively large compared to that of BaTiO 3 , a large piezostrain can be induced from PMN-PT to FM layers via multiferroic heterointerfaces [26][27][28][29]34,35 . For magnetostrictive materials such as Fe 1-x Ga x , one has to achieve a metastable bcc (A2) phase with x~30% as a single crystalline film on PMN-PT to obtain a giant α E of more than 1.0 × 10 −5 s/m 35 . For spintronic materials such as Co 40 Fe 40 B 20 28 and Co 2 FeSi in this study, amorphous and polycrystalline films grown on PMN-PT have also shown giant values of α E . Although these spintronic materials did not have a large magnetostrictive constant, an induced uniaxial magnetic anisotropy (6 kJ/m 3 for Co 40 Fe 40 B 20 in ref. 28 , 5.8 kJ/m 3 for Co 2 FeSi in this work) can be largely and steeply modulated in the film plane by applying E. The above results show that electric-field control of the magnetization vector can be efficiently achieved by utilizing the strain-mediated magnetic anisotropy for high-performance spintronic materials with a giant α E .

Discussion
For the Co 2 FeSi/PMN-PT(011) multiferroic heterostructures, we should consider the strain-mediated variation in the magnetic anisotropy, which is an extrinsic mechanism. To discuss a substantial contributor to the CME effect in the Co 2 FeSi/PMN-PT(011) heterostructure, we focus on the correlation between the magnetocrystalline anisotropy energy for Co 2 FeSi and the extrinsic lattice strain in Co 2 FeSi in a simple model. Here, the extrinsic lattice strain is imposed through the in-plane lattice vectors a and b of the conventional 16-atom unit cell, as shown in the inset of Fig. 6a. Figure 6a shows the magnetocrystalline anisotropy energy (MAE) estimated by first-principles calculations for Co 2 FeSi, together with Fe 3 Si as a reference, as a function of b/a, where b/a is obtained by changing the lattice vector a while optimizing the b vector. The MAE of Co 2 FeSi changes linearly with b/a, where the tetragonal in-plane lattice distortion results in in-plane magnetization pointing to the elongated axis: the positive and negative MAEs for b/a > 1 and b/a < 1, respectively. In contrast, Fe 3 Si shows relatively small changes in the MAE with the opposite sign compared to Co 2 FeSi. This tendency supports that the value of α E for [011] [100] [011] the Fe 3 Si/PMN-PT(011) heterostructure is smaller than that for the Co 2 FeSi/PMN-PT(011) heterostructure. From these results, we interpret that the modulation of the lattice strain induced by the application of E causes changes in the MAE in Co 2 FeSi and Fe 3 Si. Even though the result in Fig. 6a is obtained with the constraint of c = a for simplicity, the behavior of the in-plane anisotropy favoring the magnetization along the elongated axis remains unchanged for the case of full relaxation, as shown in Fig. 6b.
To elucidate the origin of strain-induced MAE modulation, we evaluated the orbital-resolved MAE for Co 2 FeSi on the basis of perturbation theory (see Materials and methods). As we decompose the total MAE of Co 2 FeSi, it is found that the dominant contribution to the MAE comes from Co atoms relative to those from Fe and Si atoms.   Figure 6c displays the projected density of states (DOS) of Co 3d orbitals in Co 2 FeSi. Here, the e g (t 2g ) states consist of degenerate d x 2 Ày 2 and d 3z 2 Àr 2 (d xy , d xz , d yz ) states because the octahedral ligand field is valid due to the cubic structure. Even though the degeneracy is lifted by the interfacial strain, we also use this classification for the strain-perturbed states. With lateral strain, we find that the MAE is dominated by up spins in the occupied states and down spins in the unoccupied states, in particular, ht 2g ; " jH SO je g ; #i couplings, where the bra (ket) in the matrix element corresponds to occupied (unoccupied) states. These dominant orbitals are shown in Fig. 6d for Δa/a 0 = 3.55% and Δb/b 0 = − 3.55%. Subtle changes are seen from the DOS that the unoccupied e g orbital with the down spins shifts toward the Fermi level (ε F ) by the in-plane deformation. The change in the electronic structures contributes to the negative MAE favoring the magnetization in the a direction. It should be noted that changes in the occupied down spins in Fig. 6d are irrelevant because the matrix element is small due to the very low density of states in unoccupied up-spin states. Therefore, the origin of the strain-induced MAE modulation in the Co 2 FeSi/PMN-PT(011) heterostructures is related to the modulation of the Co-3d orbitals occupied by up-spin states in Co 2 FeSi. In the field of next-generation spintronic nonvolatile memories such as MRAMs 1-4 and spintronic logic devices 48,49 , effective switching of the magnetization vector via spin transfer torque by using an electric current is one of the bottlenecks for low energy power consumption because of the heat dissipation process. In contrast, the CME effect in multiferroic heterostructures can provide a solution to overcome the heat dissipation of the magnetization switching at room temperature [23][24][25] . In this study, we have presented a giant CME effect in Co 2 FeSi/PMN-PT(011) multiferroic heterostructures with α E of more than 1.0 × 10 −5 s/m at room temperature. The value of giant α E is the largest of the high-performance Heusler-based spintronic materials. We infer that the giant α E in the Co 2 FeSi/PMN-PT(011) heterostructures is strongly related to the straininduced in-plane magnetic anisotropy derived from the Co 3d orbitals in Co 2 FeSi, in addition to a relatively large M S . Additionally, repeatable and nonvolatile magnetization vector switching without applying H has been demonstrated at room temperature. Thus, the present study provides a new solution for achieving magnetization switching with ultralow power consumption with hundreds of orders of magnitude of attoJoules in heat dissipation [23][24][25] .
Although the present study was performed by utilizing PMN-PT substrates, some technologies for PMN-PT films without the influence of substrate clamping, such as piezoelectric layers, have been demonstrated 24,50 . Furthermore, the giant CME effect based on the Co-based Heusler alloys/ PMN-PT multiferroic heterostructures can be utilized for MTJs 38,51 and current-perpendicular-to-plane giant magnetoresistance (CPP-GMR) devices with polycrystalline Cobased Heusler alloy electrodes 52 and a new spintronic logic architecture, such as magnetoelectric spin-orbit devices 23,53 .

Growth and characterization of Co 2 FeSi on PMN-PT(011)
The Co 2 FeSi/PMN-PT(011) multiferroic heterostructure was grown by molecular beam epitaxy (MBE). Prior to the growth of the Co 2 FeSi film, heat treatment was performed at 450°C for 20 min to obtain a flat surface of the singlecrystal PMN-PT(011) substrates with a size of 5.0 × 5.0 × 0.5 mm 3 . After cooling to a growth temperature of 300°C, a 0.3-nm-thick Fe layer was grown on top of the cleaned PMN-PT(011) surface. When we did not use the 0.3-nmthick Fe layer, the grown polycrystalline Co 2 FeSi film became completely nonoriented. To improve the crystallinity of the Co 2 FeSi film with an L2 1 -ordered structure, the insertion of the 0.3-nm-thick Fe layer is essential. Thus, 30nm-thick Co 2 FeSi film was grown by co-evaporation using Knudsen cells, where we set the supplied atomic composition ratio of Co:Fe:Si to 2.0:1.0:1.0 during the growth 29,45 . Here, the heat treatment temperature of 450°C and the growth temperature of 300°C were lower than those described in the literature, showing a very high α E of 1.5 − 4.5 × 10 −5 s/m 34 . After growth, we characterized the Co 2 FeSi/PMN-PT(011) multiferroic heterostructure. First, the multiferroic heterostructure was evaluated by X-ray diffraction (XRD) (Rigaku SmartLab) for out-of-plane and in-plane analyses. High-resolution transmission electron microscopy (HRTEM) and bright-field transmission electron microscopy (BF-TEM) were performed using an aberration-corrected JEOL 2200FS TEM, and nanobeam diffraction was performed on a JEOL 2100+TEM. The specimen preparation for TEM analysis was performed using focused ion beam techniques.
To measure the conventional magnetic properties of the grown multiferroic heterostructures, we used a vibrating sample magnetometer (VSM) at room temperature, while magneto-optic Kerr ellipticity measurements using the LED with a wavelength of 670 nm were performed to examine the CME effect at room temperature. To apply an E to the PMN-PT substrate along the [011] direction, a Au(100 nm)/Ti(3 nm) electrode was deposited on the backside of the PMN-PT substrate, where the Co 2 FeSi film was also utilized as a top electrode. Prior to the evaluation of the CME effect, we first applied an E of −0.8 MV/m. Then, the amplitude of E was gradually changed from −0.8 MV/m to + 0.8 MV/m, then back to E = −0.8 MV/m. At each step, the Kerr-ellipticity magnitude was obtained by measuring the hysteresis loops as a function of H along the PMN-PT[011] or [100] direction.

Computational details for first-principles calculations
We performed first-principles calculations on the basis of density functional theory by using the VASP code 54 , where the inner-core electrons were treated by the projector augmented-wave method 55,56 . The generalized gradient approximation parameterized by the Perdew-Burke-Ernzerhof functional was used for the exchangecorrelation functional 57 . In addition, the DFT + U method with the effective Hubbard repulsion U eff = 2.6 eV and U eff = 2.5 eV was employed for the Co 3d and Fe 3d orbitals of Co 2 FeSi, respectively 58 . The cutoff kinetic energy for the plane-wave basis set was set to 520 eV, and k-point grids were set to 10 × 10× 10 and 15 × 15× 15 for ionic relaxation and static calculations, respectively. The in-plane strain was generated by changing the in-plane lattice parameters a and b. Here, the strain was approximated with Δa/a 0 and Δb/b 0 , where Δa = a − a 0 , Δb = b − b 0 , and a 0 (=b 0 ) is the equilibrium lattice constant without strain.
The MAE was evaluated as the total-energy difference obtained from calculations including the spin-orbit coupling for magnetization along the [100] and [010] directions while fixing the electron density that was obtained by the self-consistent collinear calculation. For the orbitaldecomposed MAE, contributions from each atomic site τ and couplings among atomic orbitals μ were derived from second-order perturbation theory 59,60 : E τ 1 τ 2 μ 1 μ 2 SO % À X occ: i X unocc: j X τ 3 τ 4 X μ 3 μ 4 hijτ 1 μ 1 ihτ 1 μ 1 jH SO jτ 2 μ 2 ihτ 2 μ 2 jjihjjτ 3 μ 3 ihτ 3 μ 3 jH SO jτ 4 μ 4 ihτ 4 μ 4 jii ε j À ε i ð2Þ where i and j are indices of occupied and unoccupied eigenstates with eigenenergies ε i and ε j , respectively. Note that the indices i, j, and μ include the spin index. The spin-orbit coupling is approximated as H SO = ξl · s, where ξ is the coupling constant, l is the orbital-angular momentum quantum number, and s is the spin angular momentum quantum number. The values of ξ were compiled in a prior study 61 . In analyzing the MAE decomposition, electronic states obtained by the OpenMX code were used 62 . The x, y, and z directions in the MAE decomposition were defined as along the a, b, and c axes, respectively.