Introduction

Combining ferromagnetic and ferroelectric properties has been investigated as multiferroics1, which provides considerable advantages of magnetization control by an electric field (E) as a low-energy power consumption operation without electric current2,3,4,5,6. As spintronics research pursues the suppression of power consumption for device operations, the manipulation of the spins by E is one of the solutions for future device applications7,8,9,10,11,12. Recent developments have focused on the modulation of magnetization M by a low electric field strength using the benchmark of the magnetoelectric (ME) coefficient, \({\alpha }_{E}\), defined as follows: \({\alpha }_{E}={\mu }_{0}\frac{\partial M}{\partial E}\), where \({\mu }_{0}\) is the vacuum permeability13,14,15,16,17,18,19,20. In particular, the interfaces between the ferromagnetic and ferroelectric layers provide various options for both materials, expanding the research field of interfacial multiferroics2,6,10,11,21,22,23,24,25.

To experimentally demonstrate material systems with large \({\alpha }_{E}\) values, magnetostrictive materials such as Fe1−xGax alloys have been utilized as ferromagnetic materials15,16, and piezoelectric Pb(Mg1/3Nb2/3)O3-PbTiO3 (PMN-PT), which can introduce a strain of 0.1% by applying E26,27,28, has been employed as a ferroelectric material. Although a giant \({\alpha }_{E}\) greater than \(1\times {10}^{-5}\) s/m was demonstrated in epitaxial Fe1−xGax/PMN-PT interfacial multiferroics15,16, material systems compatible with spintronic technologies need to be developed29,30. Recently, a study reported that Co2FeSi/PMN-PT interfacial multiferroics exhibited a large \({\alpha }_{E}\) of \(1.8\times {10}^{-5}\) s/m20, where Co2FeSi (CFS) is a Co-based Heusler alloy that was expected to have high spin polarization at room temperature and high Curie temperature31,32,33,34,35. Other studies suggested that the origin of the large \({\alpha }_{E}\) value in CFS-based interfacial multiferroics was associated with strain-mediated magnetic anisotropy modulation36,37. The modulation of the strain-induced magnetocrystalline anisotropy energy (MAE) for CFS has thus far been discussed, whereby a relatively large MAE modulation in CFS was theoretically determined, compared with Fe3Si, via the application of the in-plane lattice strain20. However, at this moment, the microscopic origin of the giant ME effects in the CFS/PMN-PT interfacial multiferroics is still unknown.

X-ray magnetic circular dichroism (XMCD) using core-level excitation enables the detection of element-specific spin magnetic moments (ms) and orbital magnetic moments (morb). This technique with magnetooptical sum rules has been utilized for the characterization of spintronics materials because of its precise determination of finite morb through symmetry breaking at the film interfaces. Because the changes in magnetic anisotropy are related to the anisotropic morb, the relationship between piezo strain and morb need to be explicitly clarified for interfacial multiferroic systems24. Although XMCD under the application of an external E for manipulating interfacial spin has also been investigated38,39,40,41,42,43, minimal research on orbital moment modulation by reversible strain has been performed44,45. To initiate novel research into the physics of the relationship between the lattice distortion and orbital magnetic moments, the anisotropic morb needs to be explored even in interfacial multiferroics. Since operando XMCD can probe element-specific orbital modulation when a reversible strain is applied, it can be used to address a fundamental and unsolved problem in the scientific research field of novel spin and orbital physics. When we consider the relationships among ms, morb, and strain (\(\varepsilon\)), the spin-orbit interaction is related to ms and morb, and the magnetostriction is related to ms and \(\varepsilon\)24\(.\) However, the relationship between morb and the lattice distortion has not been studied, even though they have conjugate relations that govern the controllability in interfacial multiferroic systems through the piezoelectric field. Therefore, the microscopic understanding between morb and \(\varepsilon\) is directly connected to the investigations of the giant ME effect.

In this study, we assess the microscopic origin of the giant ME effect in CFS/PMN-PT via element-specific precise measurements of the anisotropic morb and the lattice distortion by strain through operando spectroscopic techniques using XMCD with X-ray absorption spectroscopy (XAS) and extended X-ray absorption fine structure (EXAFS) spectroscopy. Notably, we find that the modulation of only Fe sites in Co2FeSi predominantly contributes to the giant ME effect. The density functional theory (DFT) calculations agree with this finding, and the growth of the high-index (422) plane in Co2FeSi causes a giant ME effect. The role of several elements, Fe and Co, can be clarified using operando XMCD and EXAFS; our study is the first report detecting the ‘orbital elastic effect.’ The findings from this study provide material designs for functional ME effects in interfacial multiferroics from the viewpoint of orbital states.

Results

Strain introduced into the samples

First, we describe the crystallographic orientation of the CFS layer on PMN-PT in the initial stage before introducing strain by applying E based on our previous structural characterization of CFS/PMN-PT heterostructures20. Here, we used a concentration of x = 0.3 in (1−x)Pb(Mg1/3Nb2/3)O3-xPbTiO3 because of the appearance of the large piezoelectric effect26,27. The out-of-plane c-axis in PMN-PT is [\(011\)], and the in-plane [100] and [\(01\bar{1}\)] directions are defined as unit cells26,27,28. On this surface, a 10-nm-thick CFS layer with an L21-ordered structure was grown along the out-of-plane [422] direction and in-plane [\(01\bar{1}\)] and [\(1{\bar{1}}{\bar{1}}\)] directions with a 0.3-nm-thick Fe buffer layer to avoid interfacial mixing20,46, as illustrated in Fig. 1a. The detailed structural analyses are described in Supplemental Fig. S1. The surface was capped with a 2-nm-thick amorphous Si layer. The electrodes were mounted on the sample surface and bottom side of the PMN-PT substrate, and the voltage of the sample surface was set to ground. Since the 0.5-mm-thick PMN-PT substrates were used in this study, an applied voltage of 400 V corresponded to an electric field of 0.8 MV/m. Without an applied E, the CFS layer included a tensile strain of 0.8% with lattice matching to PMN-PT through a comparison of the bulk lattice constants, thereby resulting in the formation of an in-plane uniaxial magnetic easy axis along the [\(01\bar{1}\)] direction in the CFS; this is the same as [100] in the PMN-PT orientation.

Fig. 1: Crystalline orientation at the Co2FeSi/PMN-PT interface and magnetic properties under an external electric field.
figure 1

a Crystalline orientations of CFS and PMN-PT in three-dimensional and side views from the [\(01\bar{1}\)] and [\(1{\bar{1}}{\bar{1}}\)] directions. b Magnetization curves estimated from the magneto-optic Kerr effect measurement under an applied electric field (E) when a magnetic field (H) is applied along the in-plane PMN-PT [100] direction. c E dependence of the estimated remanent magnetization (Mrem). d Schematic illustrations of the changes in the magnetic anisotropy by an applied E in the strain directions.

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Next, under an applied E, the piezoelectric stress along the PMN-PT [100] direction could be introduced in the tensile and compressive directions using positive and negative bias voltages, respectively. The tensile [100] and compressive [\(01\bar{1}\)] strains in the PMN-PT were detected20. Thus, the magnetic easy axis direction in the CFS changed from the [\(01\bar{1}\)] to [\(1{\bar{1}}{\bar{1}}\)] direction under a negative applied E; this change was detected by the longitudinal magnetooptical Kerr effect (MOKE), as shown in Fig. 1b. Owing to the abrupt changes in the magnetization curves caused by the magnetic field along the [\(01\bar{1}\)] direction with an electric field, the domain structures in the CFS were almost completely controlled by the piezo strain. The MAE (K) was estimated to be \(2.5\times {10}^{4}\) J/m3 using the area of the easy and hard axes loops. The application of a magnetic field along the orthogonal [\(1{\bar{1}}{\bar{1}}\)] direction is shown in Fig. S2. The opposite behaviors of magnetic anisotropy changes were clearly observed. Furthermore, E applications were reproducibly applied for at least 20 times. This showed that the piezoelectric control along the CFS [\({01{\bar{1}}}\)] direction strongly affected the MAE modulation.

Operando XAS and XMCD

Figure 2 shows the Fe and Co L-edge XAS and XMCD spectra obtained in the partial fluorescence yield (PFY) mode under an applied electric field E of \(\pm\)0.8 MV/m. The sample surfaces were connected to the ground, and E was applied to the back side of the PMN-PT substrates. Note that the PFY mode probed depth information approximately 10 nm beneath the surface because of the photon-in and photon-out processes47. In addition, total-electron-yield (TEY) mode, which measures the drain currents and detects signals of 3 nm from the sample surface, was used to analyze the XAS and XMCD line shapes without an applied E. The XAS spectral line shapes shown in Fig. 2a, b originated from metallic Fe and Co features in CFS. In particular, the satellite structures in Co XAS appeared in the cases of highly ordered Co-based Heusler alloys48,49,50,51. The spectral line shapes of XMCD and the slight change in the L3-edge peak were modulated by E only at the Fe L-edge, despite the fixed sample measurement position. A comparison of the results for positive and negative E showed a slight variation in the peak asymmetries between the L3- and L2-edges. Since the convergent values of the integrals of the XMCD are proportional to morb within the framework of the XMCD sum rule52,53,54, these results indicated that the morb were modulated by the applied E, thereby resulting in changes in magnetic anisotropy55. The values of ms and morb in Fe were estimated to be 1.84 and 0.176 μB from the spectra recorded in TEY mode, corresponding to the case of +0.8 MV/m electric field. Details are described in Fig. S3. For an electric field of \(-\)0.8 MV/m, the changes at the L3-edge in the PFY mode contributed to the integral of XMCD. The modulation of morb by 0.01 μB under an applied E was related to the induced lattice distortion from the PMN-PT substrates. Moreover, after releasing E to zero, the spectral line shapes reverted to those of the pristine state. The modulation of ms was less than 0.01 μB despite being one order larger ms than morb. Evidently, ms did not contribute to the MAE in the case of the CFS (422) orientation, and the effect of the magnetic dipole term mT was negligible. In contrast to the case of Fe, no spectral changes were detected for the case of the Co L-edge XMCD. The ms and morb values at the Co sites were estimated to be 0.60 and 0.131 μB, respectively. These results indicated that orbital moment anisotropy in Fe was essential, whereas that in Co remained unchanged. Therefore, these results were caused by the modulation of the orbital moments rather than the spin moments, thus indicating that the inverse magnetostriction effects, which are discussed as macroscopic phenomena, were derived from the changes in morb from the viewpoint of electronic structures.

Fig. 2: XAS and XMCD under an applied electric field in Co2FeSi/PMN-PT.
figure 2

a Fe and b Co L-edges at \(\pm\)0.8 MV/m. Expanded views around L3 edges are shown in the bottom panels. Four measurements are repeatedly displayed by plus and minus electric fields. Insets show the expanded views near the L3-edge XMCD. Magnetic field dependence of L3-edge XMCD under an applied electric field. c Fe and d Co L-edges at \(\pm\)0.8 MV/m. Four measurements are displayed repeatedly by plus and minus electric fields. In (c), positive and negative electric field cases are aligned in the positive magnetic field region to emphasize the difference in the negative magnetic field side.

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The element-specific XMCD hysteresis curves (M-H curves) at the Fe and Co L3-edges during the application of E in the oblique-incidence setup are also shown in Fig. 2c, d. In the case of +8 MV/m, easy axis behavior was clearly observed, whereas the \(-\)0.8 MV/m case indicated the hard axis. The measurements were repeatedly performed four times in the \(+\), \(-\), \(+\), \(-\) sequence under the fixed conditions of incident beam helicity and sample position. This result showed reproducible behavior of the changes in the magnetic easy-axis direction. In the case of Fe shown in Fig. 2c, the vertical axis strength was slightly different between \(\pm\)0.8 MV/m, thus affecting the changes in XMCD in the Fe L3-edge intensity, whereas the M-H curves of Co by \(\pm\)0.8 MV/m overlapped above the saturation magnetic fields (Fig. 2d). Therefore, only the change in morb for Fe was experimentally found. In addition, the changes in the easy axis direction were similar for both Fe and Co because of the exchange coupling between these sites.

Operando EXAFS

To further explore the local environment around the Fe and Co atoms, XAFS measurements with EXAFS analysis of the Fe and Co K-edges during the application of E were performed to deduce the changes in the nearest-neighbor distance using a Fourier transform. As shown in Fig. 3a, the Fe and Co K-edge absorption spectra with EXAFS oscillatory behaviors are observed in the fluorescence yield mode. Clear XAFS oscillations are detected and plotted at wavenumber k in the inset of Fig. 3b. The XAFS oscillation function \({k}^{2}\chi (k)\) can be fitted by the FEFF8 program56. The Fourier transformed EXAFS profile for the real-space view of the local structure is displayed in Fig. 3b and considers the information up to the second-nearest neighbor sites and using the range of k = 3 − 13 Å−1. Owing to the effect of the phase shift, the peak position in the EXAFS was not directly related to the bond length. Using the fitting procedure shown in the Supplemental Information, the nearest bond lengths in both Fe and Co sites are estimated to be 2.22 and 2.33 Å, respectively, for the +0.8 MV/m case; these values are almost identical to the bulk CFS57,58. The linearly polarized incident beam is used to detect the absorption along the [011] direction in the CFS. Under a negative applied E on the PMN-PT substrate as a compressive strain, the nearest neighbor distances in the Fe and Co sites change \(-\)0.27% and 0.09%, respectively; these potentially originate from the orientation of the CFS layer in the high-index (422) plane. These element-specific distortions could not be detected by X-ray diffraction in the 10-nm-thick CFS films. These findings indicated that the tensile local distortion in Fe triggered a change in morb in the CFS layer.

Fig. 3: XAFS and EXAFS analysis under an applied electric field in Co2FeSi/PMN-PT.
figure 3

a XAFS spectra of Fe and Co K-edges under \(\pm\)0.8 MV/m. b EXAFS analysis through the Fourier transform of the XAFS oscillation shown in the insets.

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DFT calculations

The spin- and orbital-resolved electronic structures were examined using first-principles DFT calculations and depended on the film orientation and strain. Since the CFS layers were deposited along the CFS [422] c-axis direction on PMN-PT (011), the same unit cell structures as in the experimental situations were employed. The in-plane b-axis and other orthogonal axes were defined as the [\(01\bar{1}\)] and [\(1{\bar{1}}{\bar{1}}\)] directions of the CFS layer, respectively. In a previous report, the strain dependence of the band structures along the [001] direction was also investigated, along with the highly symmetric [100] and [010] directions20. Here, the strain dependence of the MAE on the b-axis and Δb/b0 in terms of the equilibrium cubic structure are displayed in Fig. 4a. The CFS layer grown on PMN-PT was strained by +0.8% along the [\(01\bar{1}\)] b-axis direction at the initial stage. The lattice parameters along the other directions were relaxed to minimize structural formation energies. MAE is defined as the in-plane total energy difference (E[\(01\bar{1}\)]\(-\)E[\(1{\bar{1}}{\bar{1}}\)]). The changes in the easy axis were observed by the b-axis strain. However, the modulation of the easy-hard axis that occurred with the strain was different from the experimental results because the effective experimental strain was difficult to estimate. For the MAE in the highly symmetric case between the [100] and [010] directions, the strain was relaxed completely in the bulk form. To elucidate the changes in magnetic anisotropy, the values of morb, depending on the strain, were estimated, as shown in Fig. 4b. Evidently, a large modulation was observed in the case of morb(\(\varepsilon\)) along the Fe [\(01\bar{1}\)] direction; this result was comparable to the XMCD results. The slope of morb(\(\varepsilon\)) depending on the strain differed between the Fe [\(01\bar{1}\)] and [\(1{\bar{1}}{\bar{1}}\)] directions, and the strong dependence of Fe [\(01\bar{1}\)] morb led to orbital control by the Fe sites with the application of b-axis strain, indicating a linear relationship between the difference in morb(\(\varepsilon\)) and MAE. In contrast, in the case of the Co sites, similar tendencies were displayed for all directions, thus indicating a cubic local environment. Furthermore, its strain dependence of morb in the cases of [100] and [010] was also plotted, and the tendency was low compared with that of the Fe [\(01\bar{1}\)] case (see Supplemental Information). Therefore, the giant ME effects in CFS/PMN-PT(011) relied on high-index (422) growth and the modulation of morb along [\(01\bar{1}\)] by strain. To further analyze the orbital states, the dependence of the element-specific density of states (DOS) on the orientation and strain was examined, as shown in Fig. 4c. In the cases of \(\pm\)2% strain, the projected DOS exhibited half-metallic properties, with the energy gap in the minority spin states using the intra-Coulomb repulsion energy using the DFT + U method. Since morb originated from the spin-conserved band transition in the scheme of second-order perturbation for spin-orbit interaction, the changes in DOS by strain slightly affected morb(\(\varepsilon\)). The unoccupied DOS in the Fe 3d state was modulated by the strain, which primarily corresponded to the modulation of morb[011].

Fig. 4: Strain-induced magnetic anisotropy energy, orbital magnetic moments, and density of states by DFT calculation in Co2FeSi.
figure 4

a Total energy differences along the in-plane across two axial directions in the (422) and (001) orientations depending on the b-axis strain (Δb/b0). b Element-specific orbital magnetic moments along two axis directions in the (422) orientation. c Element-specific density of states depending on the strain for the (422) orientation.

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Discussion

First, we discuss the physical meaning of morb(\(\varepsilon\)) obtained from XMCD and DFT calculations. The changes in magnetic anisotropy originated from the anisotropic morb and are calculated by Δmorb = morb([\(01\bar{1}\)]; 0 MV/m)\(-\)morb([\(01\bar{1}\)]; \(-\)0.8 MV/m); this corresponds to the orbital elastic anisotropy. The DFT calculations found four cases of morb ([\(01\bar{1}\)], [\(1{\bar{1}}{\bar{1}}\)]; equilibrium) and morb ([\(01\bar{1}\)], [\(1{\bar{1}}{\bar{1}}\)]; strained). The differences by strain are comparable to the E-induced XMCD and are defined as Δmorb(\(\varepsilon\)) (the difference between morb([\(01\bar{1}\)]; equilibrium) and morb([\(01\bar{1}\)]; strained)). Notably, according to anisotropic orbital moment theory, the magnetic anisotropy energy K = 1/4\(\xi \alpha\)Δmorb(\(\varepsilon\)) can be applied using the perturbation of the spin-orbit interaction \(\xi\)Fe = 50 meV. Furthermore, \(\alpha\) of 0.2 is a constant that depends on the band structure, and this is applicable for metallic Fe59,60. Using these parameters and the unit cell volume of CFS, the value of K in the Fe sites is estimated to be \(1.1\times {10}^{5}\) J/m3 by strain; this value is comparable to that of K = \(2.5\times {10}^{4}\) J/m3 from the magnetization measurements. The values of ms, morb, and MAE are listed in Table 1. In the case of the Co sites, since Δmorb(\(\varepsilon\)) = 0, no contributions to the MAE were observed. Compared with the DFT calculation, the difference originated from the disordering in the CFS layer because the calculation was performed in a perfectly ordered L21 structure. In the case of the CFS layer grown at relatively low temperatures, D03-type disordering was introduced61, which could suppress the magnetic moment values. However, because the DFT calculation did not include Hund’s second rule, that is, a relativistic effect is not explicitly included, the values of morb were underestimated. The value of ms in Fe estimated by the calculations was larger than that in bulk Fe (2.3 μB), thus indicating electron localization in the Fe 3d states. Note that the MAE values estimated from the orbital moment anisotropy are generally one order of magnitude larger than those obtained from the magnetization measurements. Therefore, the changes in anisotropy due to E can be explained by the modulation of morb. These phenomena indicate a novel orbital control through the elastic effect, where the macroscopic magnetoelastic effect can be interpreted from the viewpoint of electronic structures.

Table 1 Element-specific ms and morb and anisotropic morb by strain (Δmorb) estimated from XMCD and DFT calculations.
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Next, the reasons for the morb changes reliance only on the Fe sites are discussed. In Co-based Heusler alloy compounds, the DOS of Co is generally dominant at EF for half-metallic conductivity, as shown in Fig. 4c; however, the DOS of Fe is located deep in the valence band and thus contributes to the local magnetic moments. Herein, an analogous property of B-site localization in Heusler alloy A2BX is also discussed in the Mn sites of Co2MnSi, considering U for the B-sites62. The Fe sites are relatively localized and contribute to the orbital polarization and formation of morb, and the Co sites are dominant for the highly spin-polarized transport properties. The nonmagnetic element X can be valid for structural stability and tuning of the energy band gap and the position of the Fermi level. The role of each element is relevant in the CFS layer and can be tuned by strain. Therefore, the control of the morb only in Fe with an applied strain is plausible for the MAE control in Co-based Heusler alloy compounds. These element-specific analyses can be achieved only by operando XMCD.

Third, although magnetostriction or magnetoelastic effects have been recognized as strain effects in magnetism as phenomenologically macroscopic understanding for some materials, a microscopic understanding of the electronic structures and orbital states is lacking. Recent studies on ultrathin ferromagnetic films need to include a more detailed analysis of the restrictive effects at the interfaces. A novel concept of orbital strictive or orbital elastic effects is a potential candidate. The strain-dependent orbital magnetic moments are formulated as a linear relationship, as follows: morb(\(\varepsilon\))= morb0 (1+\({\rm{\lambda }}\varepsilon\)), where morb0 is the equilibrium constant and \({\rm{\lambda }}\) is the coefficient. The linear relationship is obtained from the enhancement of morb in the spin-conserved electron motion in the in-plane direction at the interfaces. The magnetostriction constant for CFS is estimated to be \(12.22\times {10}^{-6}\)63\(,\) and this value is comparable with our definition of the orbital-striction coefficient \({\rm{\lambda }}\). Since the direction of the applied strain and the enhancement of morb were the same along the b-axis, the strain along the other directions was low. These phenomena were significant for the high-index (422) plane in the CFS (422)/PMN-PT (011) system.

Finally, we discuss the enhancement of the ME effect in the CFS/PMN-PT interface. Regarding the crystalline orientation, the growth of the CFS along the [422] direction can be energetically stabilized in the PMN-PT (011) case compared with the PMN-PT (001) orientation. The strain in the CFS (422) plane triggers the modulation of the lattice constants in the highly symmetric [100], [010], and [001] directions, thereby resulting in the modulation of morb in Fe. Therefore, the elemental selectivity and crystalline orientation are indispensable for designing materials with large ME effects.

In conclusion, we developed E-induced XMCD to detect the changes in morb by the piezoelectric strain and clarified the relationship between the strain and morb in an interfacial multiferroic system with a giant ME effect; the system consisted of the Heusler alloy CFS on a ferroelectric PMN-PT substrate. Element-specific investigations of the orbital states by operando XMCD and the local environment by EXAFS analysis showed that the modulation of only the Fe sites in CFS contributed to the giant ME effect. The DFT calculations agreed this finding, and the growth in the high-index (422) plane in the CFS resulted in a giant ME effect. Our findings resulted from the element-specific orbital control by reversible strain recognized as the ‘orbital elastic effect’ and could provide guidelines for material designs with a giant ME effect using orbital-elastic highly spin-polarized Heusler alloys to advance interfacial multiferroics.

Method

Sample growth

The samples were grown using ultrahigh-vacuum molecular beam epitaxy on [011]-oriented 0.5-mm-thick PMN-PT single-crystal substrates. Therefore, a bias voltage of \(\pm\)400 V applied between the top and bottom electrodes resulted in an electric field of \(\pm\)0.8 MV/m. Before the deposition of a 10-nm-thick CFS layer, a 0.3-nm-thick Fe layer was deposited onto the substrate at 300 °C. The flat surfaces of the PMN-PT substrates were prepared by annealing at 400 °C for 20 min. The CFS layer was grown at 300 °C and covered with 2-nm-thick Si to prevent oxidization at room temperature. The details of the surface and interface conditions and fabrication procedures are reported in ref. 20. The magnetic properties were characterized using longitudinal MOKE measurements under an applied E and magnetometry using a vibrating-sample magnetometer (VSM). The magnetization values were estimated from the VSM. The changes in M-H curves under an applied E were monitored by MOKE.

XMCD and EXAFS

XAS and XMCD measurements of the Fe and Co L-edges were performed using the High-Energy Accelerator Research Organization, Photon Factory (KEK-PF) BL-7A beamline, Japan, at room temperature. A magnetic field of \(\pm\)0.2 T was applied along the incident polarized soft X-rays to sufficiently saturate the magnetization along the normal direction of the surface of the sample. TEY mode was used to detect the drain currents from the samples for measurements recorded without the application of a magnetic field. The electrodes were mounted on the surface of the sample and the rear of the substrate to perform the E-induced XMCD measurements. E-induced XMCD measurements were performed using the PFY mode to probe signals 10 nm below the surfaces of the samples using a bipolar electric power source (Keithley 2410) to apply E to PMN-PT. The fluorescence signals were detected using a silicon drift detector (Princeton Gamma-Tech. Instrument Inc., SD10129) and mounted 90° to the incident beam. The XAS and XMCD measurements were performed in an oblique incidence setup of 60° from the normal of the sample surface from the incident beam and magnetic field, and the signals of the in-plane components in the films were detected. We changed the magnetic field directions to obtain right- and left-hand polarized X-rays while fixing the polarization direction of the incident X-rays. To avoid the saturation effects in PFY mode, the intensities of the XAS measurements were carefully examined by their comparison with those obtained in TEY mode.

The EXAFS measurements at the Fe and Co K edge under an applied E were performed at BL-12C at KEK-PF using the fluorescence yield mode with a 19-element solid-state detector at room temperature. A linearly polarized incident beam arrived at the sample from the 45° direction and enabled the detection of the signals along the [\(01\bar{1}\)] direction. The system that applied the E was the same as that employed in E-induced XMCD.

Computational details in DFT calculations

First-principle calculations were performed based on DFT using the OpenMX code64. The generalized gradient approximation was used with the PBE exchange-correlation functional65. In addition, the DFT + U method with effective Hubbard repulsion Ueff = 2.6 eV and Ueff = 2.5 eV was employed for the Co 3d and Fe 3d orbitals of Co2FeSi, respectively. The k-point grids were set to 15 \(\times\) 15 \(\times\) 15 and 11 \(\times\) 13 \(\times\)15 for CFS (001) and CFS (422), respectively. The in-plane strain was generated by changing the in-plane lattice parameters a and b in the CFS (422) layer. Here, the strain was approximated by Δa/a0 and Δb/b0, where Δa = a \(-\) a0, Δb = b \(-\) b0. For CFS (001), a0 (= b0) was the equilibrium lattice constant without strain. For the CFS (422), b0 was set to the equilibrium CFS lattice parameters. The MAE was evaluated as the total energy difference obtained from the calculations, including the spin-orbit coupling for the magnetization fully self-consistently along the [100] and [010] directions.