Strain-induced magnetocrystalline anisotropy in L2₁ Co₂FeSi: first-principles study

Abstract

The Co₂FeSi/Fe/Pb(Mg_1/3Nb_2/3)O₃-PbTiO₃ interface exhibits the strong magnetoelectric coupling arising from the magnetocrystalline anisotropy (MCA) of Co₂FeSi. To enable the design of the MCA based on crystal structures and plane orientations, we examined the relationship between the atomic configurations and the MCA of Co₂FeSi under strain within the (001) and (422) planes. Despite exhibiting similar MCA under strain within the (001) and (422) planes, the MCA origin differs. Under strain within the (422) plane, the MCA arises from the orbital magnetic moments projected onto the 3d orbitals extended in the plane perpendicular to the magnetization direction. In contrast, under strain within the (001) plane, Fe atoms are coordinated by Si atoms within the (100) and (010) planes, which suppresses the orbital magnetic moments of Fe and their contributions to the MCA. Furthermore, under strain within the (422) plane, Co and Fe atoms contribute oppositely to the MCA due to differences in the behavior of the orbital magnetic moments.

Introduction

Electric-field control of magnetization based on the magnetoelectric (ME) coupling observed in multiferroics¹ is expected to become a key component for realization of energy-efficient magnetoresistive random access memory (MRAM). As a promising non-volatile magnetic memory device, MRAM has been extensively investigated for power reduction and miniaturization^{2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22}. The energy required to generate an electric field is lower than that needed for a magnetic field^2,3,4, because electric-field control reduces energy loss due to Joule heating in conventional current-driven methods, enabling power saving^5,6,7. Magnetization can be controlled by methods such as adjusting the perpendicular magnetic anisotropy at the tunnel junction interface^8,9 or generating pure spin currents in magnetic tunnel junctions (MTJs)¹⁰.

A strong converse ME coupling is observed in Co₂FeSi/Fe/Pb(Mg_1/3Nb_2/3)O₃-PbTiO₃ (PMN-PT) heterostructures attributed to the strain-induced magnetocrystalline anisotropy (MCA) of the Co₂FeSi^23,24. The ME coupling and its origin have been investigated in various system beyond this system^{24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71}. In Co₂FeSi/Fe/PMN-PT heterostructures, an electric field applied to the PMN-PT substrate induces strain via the inverse piezoelectric effect. The strain is transferred to the Co₂FeSi layer across the interface, thereby enabling control of the MCA through magnetoelastic coupling^23,24,70,71. The strength of this coupling of Co₂FeSi depends on the plane orientation, exhibiting a high ME coupling coefficient α = 1.8 × 10⁻⁵ s m⁻¹ when oriented along the (422) plane on the substrate²⁴. This value exceeds the practical requirement of α > 10⁻⁵ s m^−172,73,74. Additionally, Co₂FeSi is a ferromagnetic Heusler alloy with high spin polarization and the Curie temperature, making it a promising candidate for spintronics applications^{75,76,77,78,79,80,81}. Heusler alloys have been studied for their magnetic properties due to their potential for spintronics^{82,83,84,85,86,87,88,89}. Spin polarization and the Curie temperature in Heusler alloys depend on structural order. For example, Co₂CrAl shows a rapid decrease in spin polarization due to Co-Cr disorder⁸⁶. In contrast, Co₂FeSi maintains high spin polarization at room temperature and has the high Curie temperature of about 1100 K^75,76.

The MCA arises from spin-orbit coupling (SOC). By affecting electronic structure and magnetism^{25,26,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107,108,109,110,111}, SOC plays a key role in spintronics. In several transition metal-based systems, second-order perturbation with respect to the SOC Hamiltonian¹¹² is valid^{25,26,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104}, since the first-order perturbation term vanishes¹¹³. Based on perturbative approximations, the MCA of Co₂FeSi has often been described by the Bruno and van der Laan terms from each atom. The Bruno term arises from spin-conserving virtual excitations from occupied down-spin to unoccupied down-spin states and is related to the orbital magnetic moments⁹⁰. Alternatively, the van der Laan term arises from the spin-flip virtual excitations from occupied up-spin to unoccupied down-spin states and is related to the quadrupole moments of the spin density^91,92,93. The contributions of these terms were examined for the energy difference for magnetization along the 〈100〉 directions under a highly symmetric (001) in-plane strain. Here, the MCA arises mainly from the van der Laan term from Co, i.e., the quadrupole moments of the spin density at the Co site enhance the MCA. The Bruno term from Co and the contributions from Fe have little effect on or suppress the MCA^25,26. In the system exhibiting a high ME coefficient α = 1.8 × 10⁻⁵ s m⁻¹, Co₂FeSi is epitaxially grown on the PMN-PT(011) substrate following the relation of Co₂FeSi(422)∥PMN-PT(011)²⁴. Therefore, the contributions were also examined for the energy difference between magnetization along the $[01\bar{1}]$ and $[1\bar{1}\bar{1}]$ directions within the (422) plane under in-plane strain. In this case, the orbital magnetic moments at the Fe site are modulated and enhance the MCA⁷⁰. The Bruno and van der Laan terms have not been explicitly evaluated for the (422) plane.

Despite the dependence of the MCA and the ME coupling on the plane orientation, the relationship between the atomic configurations and the MCA remains unclear. In this study, we investigate the MCA and its origin in Co₂FeSi under strain within the (001) and (422) planes based on first-principles calculations. The purpose is to reveal the effects of the atomic configurations on the MCA by comparing these results. This understanding is expected to enable the design of the MCA based on crystal structures and plane orientations. In particular, Co₂MnSi and Co₃Mn exhibit the MCA and the ME coupling similar to Co₂FeSi^{25,68,114,115}, indicating their potential for applications.

Results

Unit cell and total density of states

Co₂FeSi crystallizes in the L2₁-ordered structure with space group $Fm\bar{3}m$, as shown in Fig. 1(a) and (b). The Fe, Si, and Co atoms occupy the 4a, 4b, and 8c Wyckoff sites, respectively. The system is ferromagnetic, as all other antiferromagnetic and ferrimagnetic configuration within the 1 × 1 × 1 unit cell were unstable, which is consistent with results reported in ref. ⁸¹. The systems with in-plane strain applied within the (001) and (422) planes are denoted as CFS(001) and CFS(422), respectively. For CFS(001), the [100], [010], and [001] directions are set as the a-, b-, and c-axes, while for CFS(422), the $[0\bar{1}1]$, $[1\bar{1}\bar{1}]$, and [211] directions are set, respectively. For both systems, strains ε^a and ε^b ranging from −2% to + 2% were independently applied along the a- and b-axes, respectively. Here, ε^a = (a − a₀)/a₀ and ε^b = (b − b₀)/b₀, where a₀ and b₀ are the unstrained lattice constants. Figure 1(c) shows the total density of states (DOS) of unstrained Co₂FeSi, which exhibits half-metallicity.

Fig. 1: Unit cells of L2₁-ordered Co₂FeSi.

a a-, b-, and c-axes are set to [100], [010] and [001], respectively. b a-, b-, and c- axes are set to $[0\bar{1}1],[1\bar{1}\bar{1}],{\rm{and}}[211]$, respectively. These cells (a) and (b) were generated using VESTA¹²⁴. c Total density of states (DOS) per atom of unstrained Co₂FeSi.

Magnetocrystalline anisotropy energy

Figure 2 shows the diagram of the magnetization easy axes and the MCA energy of the Co₂FeSi alloy as a function of the in-plane strains, ε^a and ε^b. The magnetization easy axis in unstrained Co₂FeSi is 〈110〉 direction. Except for ε^a ≈ ε^b in CFS(001), either the a- or b-axis is the magnetization easy axis under strain. Therefore, the MCA energy E_MCA is defined as the difference between ${E}_{{\rm{SOC}}}^{a}$ and ${E}_{{\rm{SOC}}}^{b}$, i.e., ${E}_{{\rm{MCA}}}={E}_{{\rm{SOC}}}^{b}-{E}_{{\rm{SOC}}}^{a}$. Here, ${E}_{{\rm{SOC}}}^{a}$ and ${E}_{{\rm{SOC}}}^{b}$ denote the energy arising from the SOC with magnetization aligned along a- and b-axes, respectively. For CFS(422), the a- and b-axes are inequivalent; therefore, E_MCA ≠ 0 holds even for the unstrained CFS(422). We evaluated the partial derivative of the MCA energy with respect to strain, based on an ordinary least squares (OLS) method. For CFS(001), ∂E_MCA/∂ε^[100] = − ∂E_MCA/∂ε^[010] = 0.7732 meV atom⁻¹. In the case of CFS(422), $\partial {E}_{{\rm{MCA}}}/\partial {\varepsilon }^{[0\bar{1}1]}=0.8634\,{\rm{meV}}\,{{\rm{atom}}}^{-1}$ and $\partial {E}_{{\rm{MCA}}}/\partial {\varepsilon }^{[1\bar{1}\bar{1}]}=-0.8644\,{\rm{meV}}\,{{\rm{atom}}}^{-1}$. These results show that the tensile strain in the a-axis decreases E_MCA and the tensile strain in the b-axis increases E_MCA, in both CFS(001) and CFS(422). By the definition ${E}_{{\rm{MCA}}}={E}_{{\rm{SOC}}}^{b}-{E}_{{\rm{SOC}}}^{a}$, the decrease in MCA energy makes the state with magnetization along the a-axis lower in energy than that along the b-axis. Conversely, the increase in MCA energy makes the state with magnetization along the b-axis lower in energy than that along the a-axis. Those indicate that tensile strain makes the strained axis the magnetization easy axis. Considering compressive strain as well, compressive strain makes the strained axis the magnetization hard axis. Furthermore, the strain within the (422) plane induces a larger change in the MCA energy than the strain applied within the (001) plane. This trend is consistent with previous studies under different strains or interface configurations^24,25,26.

Fig. 2: The magnetization easy axes and the MCA energy under strain.

The magnetization easy axes depending on the strains within the (a) (001) and (b) (422) planes. The MCA energy E_MCA per atom as a function of strains within the (c) (001) and (d) (422) planes.

Contributions to the MCA can be decomposed into four terms, based on the spin orientations of unperturbed occupied and virtual states in the second-order perturbation theory^{94,98,100,103}:

$${E}_{{\rm{MCA}}}={E}_{{\rm{MCA}}}^{\uparrow \uparrow }+{E}_{{\rm{MCA}}}^{\downarrow \uparrow }+{E}_{{\rm{MCA}}}^{\uparrow \downarrow }+{E}_{{\rm{MCA}}}^{\downarrow \downarrow }.$$

(1)

Here, ${E}_{{\rm{MCA}}}^{{\sigma }_{1}{\sigma }_{2}}$ denotes the MCA energy due to the virtual excitation from the occupied spin σ₁ state to the unoccupied spin σ₂ state. When the up-spin states are essentially fully occupied, the ${E}_{{\rm{MCA}}}^{\uparrow \uparrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \uparrow }$ can be considered negligible. Applying additional approximations, ${E}_{{\rm{MCA}}}^{\uparrow \downarrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ are related to the van der Laan and Bruno terms, respectively^90,91,92,93. In a previous study of CFS(001)^25,26, the ${E}_{{\rm{MCA}}}^{\uparrow \uparrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \uparrow }$ have small values because there are few unoccupied up-spin bands. The ${E}_{{\rm{MCA}}}^{\uparrow \downarrow }$ from Co 3d states mainly enhance the MCA, whereas the ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ terms have little effect on or suppress the MCA. In contrast, The ${E}_{{\rm{MCA}}}^{\uparrow \downarrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ from Fe 3d states have little effect on the MCA^25,26.

For CFS(422), we analyze the site-decomposed E_MCA given by Eq. (1), as shown in Fig. 3. Similar to CFS(001)^25,26, the ${E}_{{\rm{MCA}}}^{\uparrow \uparrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \uparrow }$ contribute little to the MCA because there are few unoccupied up-spin bands as shown in Fig. 1(c). For Co, the ${E}_{{\rm{MCA}}}^{\uparrow \downarrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ change significantly under strains. However, their opposite dependence on the strain causes partial cancellation, resulting in a small net contribution. In contrast, for Fe, the ${E}_{{\rm{MCA}}}^{\uparrow \downarrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ change significantly and same dependence on the strain. Unlike CFS (001), Fe plays a significant role in the enhancement of the MCA.

Fig. 3: Strain-induced changes in the site-decomposed MCA energy.

The partial derivatives of the site-decomposed MCA energy per atom ${E}_{{\rm{MCA}}}^{\uparrow \uparrow },{E}_{{\rm{MCA}}}^{\downarrow \uparrow },{E}_{{\rm{MCA}}}^{\uparrow \downarrow },{\rm{and}}\,{E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ with respect to (a) a-axis strain ${\varepsilon }^{[0\bar{1}1]}$ and (b) b-axis strain ${\varepsilon }^{[1\bar{1}\bar{1}]}$.

Orbital magnetic moment

We focus on the spin-conserving term ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }={E}_{{\rm{SOC}}}^{\downarrow \downarrow }(b)-{E}_{{\rm{SOC}}}^{\downarrow \downarrow }(a)$. The ${E}_{{\rm{SOC}}}^{\downarrow \downarrow }(\zeta )$ is related to the Bruno terms^90,94,103:

$${E}_{{\rm{SOC}}}^{\downarrow \downarrow }(\zeta )\approx -\frac{1}{4}\sum \xi \langle {\hat{L}}_{\zeta }\rangle =-\frac{1}{4{\mu }_{{\rm{B}}}}\sum \xi {m}_{{\rm{orb}}}(\zeta ).$$

(2)

Here, ζ is the magnetization direction, μ_B is the Bohr magneton, and ξ is the SOC constant. Equation (2) indicates that ${E}_{{\rm{SOC}}}^{\downarrow \downarrow }\propto {m}_{{\rm{orb}}}$ and ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }\propto \Delta {m}_{{\rm{orb}}}={m}_{{\rm{orb}}}(b)-{m}_{{\rm{orb}}}(a)$, where Δm_orb denotes the anisotropy of the orbital magnetic moment. Furthermore, E_SOC(E_MCA) decreases with increasing m_orb(Δm_orb).

Because ${E}_{{\rm{MCA}}}^{{\sigma }_{1}\downarrow }$ arises from the virtual excitation from the occupied spin σ₁ state to the unoccupied down-spin state, the modulation of unoccupied down-spin density of states (DOS) contributes to ${E}_{{\rm{MCA}}}^{{\sigma }_{1}\downarrow }$. This modulation also contributes to m_orb since m_orb arises from the virtual excitation from the occupied down-spin state to the unoccupied down-spin state, as indicated in Eq. (2). In a previous study of CFS(422)⁷⁰, for Fe, the local DOS of the unoccupied down-spin 3d states is modulated by strain, which primarily corresponds to the modulation of Δm_orb⁷⁰. This strain-induced modulation of DOS also corresponds to ${E}_{{\rm{MCA}}}^{\uparrow \downarrow }$ and ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ as shown in Fig. 3. For Co, the strain-induced modulation of $\Delta {m}_{{\rm{orb}}}={m}_{{\rm{orb}}}^{b}-{m}_{{\rm{orb}}}^{a}$ is small because ${m}_{{\rm{orb}}}^{a}$ and ${m}_{{\rm{orb}}}^{b}$ vary similarly with strain⁷⁰.

To examine these contributions, we analyzed the orbital magnetic moments projected onto the d orbitals of Co and Fe in Co₂FeSi. The orbital angular momentum eigenstate $\left\vert \zeta ,l,m\right\rangle$ is defined with ζ, l, and m denoting the quantization axis, azimuthal quantum number, and magnetic quantum number, respectively. It can be expressed as a linear combination of the d-orbital basis states $| \mu \rangle =| {{\rm{d}}}_{{x}^{2}-{y}^{2}}\rangle ,| {{\rm{d}}}_{{z}^{2}}\rangle ,| {{\rm{d}}}_{yz}\rangle ,| {{\rm{d}}}_{zx}\rangle ,| {{\rm{d}}}_{xy}\rangle$ defined with the quantization axis along the z-axis:

$$\left\vert x,2,0\right\rangle =-\frac{1}{2}\left\vert {{\rm{d}}}_{{z}^{2}}\right\rangle +\frac{\sqrt{3}}{2}\left\vert {{\rm{d}}}_{{x}^{2}-{y}^{2}}\right\rangle .$$

(3)

$$\left\vert x,2,\pm 1\right\rangle =-\frac{1}{\sqrt{2}}\left\vert {{\rm{d}}}_{xy}\right\rangle \mp \frac{{\rm{i}}}{\sqrt{2}}\left\vert {{\rm{d}}}_{zx}\right\rangle .$$

(4)

$$\left\vert x,2,\pm 2\right\rangle =-\frac{1}{2}\sqrt{\frac{3}{2}}\left\vert {{\rm{d}}}_{{z}^{2}}\right\rangle -\frac{1}{2\sqrt{2}}\left\vert {{\rm{d}}}_{{x}^{2}-{y}^{2}}\right\rangle \pm \frac{{\rm{i}}}{\sqrt{2}}\left\vert {{\rm{d}}}_{yz}\right\rangle .$$

(5)

$$\left\vert y,2,0\right\rangle =-\frac{1}{2}\left\vert {{\rm{d}}}_{{z}^{2}}\right\rangle -\frac{\sqrt{3}}{2}\left\vert {{\rm{d}}}_{{x}^{2}-{y}^{2}}\right\rangle .$$

(6)

$$\left\vert y,2,\pm 1\right\rangle =-\frac{1}{\sqrt{2}}\left\vert {{\rm{d}}}_{yz}\right\rangle \mp \frac{{\rm{i}}}{\sqrt{2}}\left\vert {{\rm{d}}}_{xy}\right\rangle .$$

(7)

$$\left\vert y,2,\pm 2\right\rangle =\frac{1}{2}\sqrt{\frac{3}{2}}\left\vert {{\rm{d}}}_{{z}^{2}}\right\rangle -\frac{1}{2\sqrt{2}}\left\vert {{\rm{d}}}_{{x}^{2}-{y}^{2}}\right\rangle \pm \frac{{\rm{i}}}{\sqrt{2}}\left\vert {{\rm{d}}}_{zx}\right\rangle .$$

(8)

The d orbitals corresponding to the $\left\vert \zeta ,2,\pm 2\right\rangle$ states have the maximum magnitude of the azimuthal orbital angular momentum, i.e., ∣m∣ = 2, extending in the ζ-plane perpendicular to the magnetization direction ζ. Based on the relationship the d orbitals and the states given by Eqs. (3–8), the d orbitals corresponding to the $\left\vert x,2,\pm 2\right\rangle$ states are ${{\rm{d}}}_{{z}^{2}}$, ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$, and d_yz, whereas the d orbitals corresponding to the $\left\vert y,2,\pm 2\right\rangle$ states are ${{\rm{d}}}_{{z}^{2}}$, ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$, and d_zx. Therefore, significant contributions to the MCA are expected from the ${{\rm{d}}}_{{z}^{2}}$, ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$, and d_yz(d_zx) orbitals when magnetization is aligned along the a(b)-axis.

Figure 4 shows the partial derivatives of the anisotropy in the orbital magnetic moments projected onto the d orbitals with respect to strains. For Co, the anisotropy of the orbital magnetic moments projected onto the d orbitals with m = ± 2 generally increases (decreases) under tensile strain along the a(b)-axis in both CFS(001) and CFS(422). This likely suppresses the MCA shown in Fig. 3. In contrast, for Fe, the change in the anisotropy appears to be opposite to that of Co, indicating its enhancement of the MCA. As for CFS(001), this change is relatively minor and makes only a small contribution to the MCA.

Fig. 4: Strain-induced changes in the anisotropy of the orbital magnetic moment.

The partial derivatives of anisotropy in the orbital magnetic moment projected onto the d orbitals of Co and Fe with respect to the (a) a-axis strain ε^a and (b) b-axis strain ε^b. The anisotropy in the orbital magnetic moment Δm_orb(μ^a, μ^b) is defined as a difference between ${m}_{{\rm{orb}}}^{a}({\mu }^{a})$ and ${m}_{{\rm{orb}}}^{b}({\mu }^{b})$. Here, ${\mu }_{{\rm{orb}}}^{\zeta }({\mu }^{\zeta })$ is the orbital magnetic moments projected onto orbital μ with magnetization aligned along ζ-axis.

D0₃-ordered Co₃Mn exhibits a similar atomic configuration and the MCA as Co₂FeSi. By identifying Co atoms at the 8c sites (Co^I) and 4b sites (Co^II), the structure can be considered as the L2₁ structure. Similar to Co₂FeSi, the axis under tensile strain becomes the magnetization easy axis⁶⁸. However, when strain is applied within the (001) plane, both the Co^I and the Mn atoms enhance the MCA, while the Co^II atoms suppress it⁶⁸. In contrast, in Co₂FeSi, only the Co atoms at the 8c site enhance the MCA under the same conditions.

Figure 5 (a) shows the orbital magnetic moments projected onto the d orbitals of Co and Fe in unstrained Co₂FeSi. Figure 5(b) and (c) show the partial derivatives of the orbital magnetic moments with respect to strains. Except for Fe in CFS(001), the orbital magnetic moments projected onto the d orbitals with m = ± 2 of unstrained Co₂FeSi are large. As a result, the orbital magnetic moments change is also enhanced, indicating a large change in this anisotropy. In contrast, for Fe in CFS (001), the orbital magnetic moments with m = ± 1 is relatively large, whereas that with m = ± 2 is small. This can be attributed to Fe atoms coordinated by covalently bonded Si atoms in the (100) or (010) planes. This difference between CFS(001) and CFS(422) affects the partial DOS of Fe ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$ and ${{\rm{d}}}_{{z}^{2}}$ states in unstraind Co₂FeSi as shown in Fig. 5(d). The partial DOS of occupied down-spin ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$ and ${{\rm{d}}}_{{z}^{2}}$ bands in CFS(422) are larger than those in CFS(001), which increases the contributions from the couplings $\langle {{\rm{d}}}_{{x}^{2}-{y}^{2}}^{\downarrow }| {\hat{L}}_{x}| {{\rm{d}}}_{yz}^{\downarrow }\rangle$, $\langle {{\rm{d}}}_{{z}^{2}}^{\downarrow }| {\hat{L}}_{x}| {{\rm{d}}}_{yz}^{\downarrow }\rangle$, $\langle {{\rm{d}}}_{{x}^{2}-{y}^{2}}^{\downarrow }| {\hat{L}}_{y}| {{\rm{d}}}_{zx}^{\downarrow }\rangle$, and $\langle {{\rm{d}}}_{{z}^{2}}^{\downarrow }| {\hat{L}}_{y}| {{\rm{d}}}_{zx}^{\downarrow }\rangle$ in CFS(422). As a result of these contributions, the orbital magnetic moments ${m}_{{\rm{orb}}}^{a}$ of ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$, ${{\rm{d}}}_{{z}^{2}}$, d_yz in CFS(422) are larger than that of CFS(001), whereas the orbital magnetic moments ${m}_{{\rm{orb}}}^{b}$ of ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$, ${{\rm{d}}}_{{z}^{2}}$, d_zx in CFS(422) are larger than that of CFS(001).

Fig. 5: The orbital magnetic moments and the partial DOS.

a The orbital magnetic moments in unstrained Co₂FeSi. The partial derivatives of the orbital magnetic moments with respect to (b) a-axis strain ε^a and (c) b-axis strain ε^b. Each orbital magnetic moment is projected onto the d orbitals of Co and Fe. d The partial DOS of Fe ${{\rm{d}}}_{{x}^{2}-{y}^{2}}$ and ${{\rm{d}}}_{{z}^{2}}$ states in unstraind CFS(001) and CFS(422).

Furthermore, for Co, strain causes a larger change in the orbital magnetic moments with the magnetization perpendicular to the strain than the parallel case. In contrast, for Fe, the orbital magnetic moments change more when magnetization is parallel to the strain. This difference affects both anisotropy of the orbital magnetic moments and its role in contributing to the MCA.

Figure 6 shows partial derivatives of site-decomposed spin-conserving term ${E}_{{\rm{SOC}}}^{\downarrow \downarrow }$ with respect to strain applied within the (422) plane. As indicated by Eq. (2), the changes in the ${E}_{{\rm{SOC}}}^{\downarrow \downarrow }$ correspond to the changes in the orbital magnetic moment. Specifically, for Co, the energy changes more when magnetization aligns perpendicular to the strain, whereas for Fe, it changes more when magnetization aligns parallel. These trends correspond to the changes observed in the orbital magnetic moment.

Fig. 6: Strain-induced changes in site-decomposed SOC energy.

The partial derivatives of site-decomposed SOC energy ${E}_{{\rm{SOC}}}^{\downarrow \downarrow }$ from Co and Fe with respect to (a) a-axis strain ε^a and (b) b-axis strain ε^b.

Discussion

Co₂FeSi exhibits the MCA such that tensile strain makes the strained axis the magnetization easy axis, whereas compressive strain makes the strained axis the magnetization hard axis. However, while only the spin-flip terms ${E}_{{\rm{MCA}}}^{\uparrow \downarrow }$ from Co enhance the MCA under strain within the (001) plane, Fe atoms also enhance under strain within the (422) plane. To reveal the correlation between atomic configurations of metallic and non-metallic elements and the MCA, we investigated the orbital magnetic moments projected onto structurally anisotropic d orbitals and the spin-conserving terms. The d orbitals extended in the plane perpendicular to the magnetization direction play a significant role in the MCA. Except for Fe in CFS(001), the orbital magnetic moments projected onto these orbitals are large. As a result, they change significantly under strain and contribute to the MCA. In contrast, Fe in CFS(001) is coordinated by Si atoms within the (100) and (010) planes, which suppresses these orbital magnetic moments and the contributions to the MCA. This correlation between atomic configurations and the MCA holds promise as a new design principle based on crystal structures and plane orientations. Furthermore, for CFS(422), the orbital magnetic moment and the ${E}_{{\rm{MCA}}}^{\downarrow \downarrow }$ from Co changes more significantly when magnetization aligns perpendicular to the strain. In contrast, those from Fe change more when magnetization are parallel.

Methods

First-principles calculations were performed based on density functional theory (DFT)^116,117 by using pseudopotentials and pseudoatomic orbitals, as implemented in the OpenMX code¹¹⁸. We used the generalized gradient approximation for the exchange-correlation functional^119,120. The basis sets of the s3p2d2 configuration were adopted for Co and Fe, while the s2p2d1 configuration was adopted for Si. To describe the structural and electronic properties, we employ the DFT + U method with effective Hubbard repulsion U_eff = U − J as 2.5 and 2.6 eV for Fe and Co 3d states, respectively^{25,26,76,121,122}. The k-point grids were set to 15 × 15 × 15 and 21 × 9 × 12 for CFS(001) and CFS(422), respectively. As for convergence criteria, the maximum forces on the unit cell were 10⁻⁴ Hartree Bohr⁻¹,while the total-energy variation was set as within 10⁻⁸ Hartree in general. For ε^[100] = ε^[010] in CFS(001), where ${E}_{{\rm{SOC}}}^{[100]}={E}_{{\rm{SOC}}}^{[010]}\approx {E}_{{\rm{SOC}}}^{[110]}$, the total-energy variation was set as within 10⁻¹¹ Hatree in order to accurately evaluate the energy difference.

Figure 1 shows the unit cells of unstrained L2₁-ordered Co₂FeSi. We chose the 1 × 1 × 1 and $\sqrt{2}/2\times \sqrt{3}\times \sqrt{6}/2$ unit cells, referred to as CFS(001) and CFS(422), respectively. For CFS(001), the [100], [010], and [001] directions are set as the a-, b-, and c-axes, while for CFS(422), the $[0\bar{1}1]$, $[1\bar{1}\bar{1}]$, and [211] directions are set. For both systems, strains ε^a and ε^b ranging from −2% to + 2% were independently applied along the a- and b-axes, respectively. We apply not only uniaxial but also biaxial strain. Here, ε^a = (a − a₀)/a₀ and ε^b = (b − b₀)/b₀, where a₀ and b₀ are the unstrained lattice constants. Under these constraints, the length of the c-axis was optimized to minimize the total energy.

We evaluated the energy arising from the SOC E_SOC, the MCA energy E_MCA, the orbital magnetic moments m_orb, and the anisotropy of the orbital magnetic moments Δm_orb by including the SOC explicitly in the Kohn-Sham Hamiltonian of DFT. The MCA energy is defined as a difference between ${E}_{{\rm{SOC}}}^{a}$ and ${E}_{{\rm{SOC}}}^{b}$. Here, ${E}_{{\rm{SOC}}}^{a}$ and ${E}_{{\rm{SOC}}}^{b}$ denote the energy arising from the SOC per atom with magnetization aligned along a- and b-axes, respectively. Δm_orb is defined as ${m}_{{\rm{orb}}}^{b}-{m}_{{\rm{orb}}}^{a}$, where the orbital magnetic moment ${m}_{{\rm{orb}}}^{a}$ and ${m}_{{\rm{orb}}}^{b}$ are defined in the same fashion. From the energy with magnetization aligned along the 〈100〉, 〈110〉, and 〈111〉 directions within the (001) and (422) planes, we determined the magnetization easy axis. We evaluated the energy ${E}_{{\rm{SOC}}}^{{\sigma }_{1}{\sigma }_{2}}$ and its anisotropy ${E}_{{\rm{MCA}}}^{{\sigma }_{1}{\sigma }_{2}}$ using second-order perturbation theory with respect to the SOC Hamiltonian, on the basis of DFT calculations without including SOC explicitly in the Kohn-Sham Hamiltonian^{25,26,94,98,100,103}. The SOC constants were assigned as 4.6079 Ry for Fe and 5.8063 Ry for Co, respectively¹²³. To evaluate the effect of strain on the MCA-related quantities $y={E}_{{\rm{MCA}}},{E}_{{\rm{MCA}}}^{{\sigma }_{1}{\sigma }_{2}},{E}_{{\rm{SOC}}}^{{\sigma }_{1}{\sigma }_{2}},{m}_{{\rm{orb}}},\Delta {m}_{{\rm{orb}}}$, an ordinary least squares (OLS) regression was performed based on the following linear model: y = y₀ + (∂y/∂ε^a)ε^a + (∂y/∂ε^b)ε^b. Here, y₀ denotes the y of unstrained Co₂FeSi. We quantitatively extracted the strain derivatives ∂y/∂ε^a and ∂y/∂ε^b via OLS regression.