Strain effects on the magnetic anisotropy of Y₂Fe₁₄B examined by first-principles calculations

Strain effects on the magnetic properties of sintered-magnet main phases are important, because sintering and post-sinter thermal processes inevitably induce residual strain. Although a few experimental studies have been made on the dependence of strain for intermetallic compounds consisting of a rare-earth element, iron, and boron (R₂Fe₁₄B),^1–4 the present understanding of this issue is far from complete. In particular, the magnetocrystalline anisotropy, which is a key factor for understanding the high coercivity of Nd-Fe-B sintered permanent magnets having Nd₂Fe₁₄B as the main phase,^5–7 is not well understood. In contrast with localized 4f electrons, which are relatively insensitive to their surrounding environment, 5d electrons and Fe 3d electrons are typically strongly affected by lattice strain due to their itinerancy. In this sense, Y₂Fe₁₄B is suitable for studying the effects of strain on itinerant d electrons, because Y is a prototypical f ⁰ example of rare-earth elements. While the previous calculations for Y₂Fe₁₄B have provided insights into magnetic moments and the magnetic anisotropy of unstrained single crystals,^8–12 strain effects have not been examined from first principles.

In this work, we report a first-principles study of the strain effects for Y₂Fe₁₄B. We show that the magnetic anisotropy of Y₂Fe₁₄B comes mostly from Fe d electrons, while the contribution of Y d electrons is considerably smaller. We find the magnetic anisotropy is significantly enhanced upon compression of the lattice. We also clarify the contribution of each Fe site to the magnetic anisotropy by a quantitative approach using second-order perturbation theory, where the coupling at an Fe j₂ site among occupied and unoccupied states of $3dx2−y2$ and 3d_xy components with the same spin is found to be prominent in enhancing the perpendicular anisotropy, i.e., perpendicular to the (001) plane (c plane). Our method to identify dominant d-orbital couplings should be a practical tool for uncovering insights for the design of microstructures and even compounds for permanent magnets, as well as nanostructures with high magnetocrystalline anisotropy.^13,14

The electronic-structure calculations were made within density functional theory using the OpenMX code,¹⁵ in which pseudoatomic-orbital basis sets and norm-conserving pseudopotentials are used. We employed the Perdew– Burke–Ernzerhof (PBE) exchange-correlation functional in the generalized gradient approximation (GGA). Same as Nd₂Fe₁₄B, Y₂Fe₁₄B has a tetragonal unit cell with P4₂/mnm space-group symmetry, containing 68 atoms with six crystallographically distinct Fe sites, two Y sites, and one B site.^16–18 In our calculations, the atomic positions are fully relaxed with the criteria for convergence of the total energy and the force acting on atoms of 10⁻⁶ Hartree and 10⁻⁵ Hartree/Bohr, respectively. We carefully checked the convergence of the total energy with respect to the number of k points, finding a 9 × 9 × 7 grid to be sufficiently accurate. The magnetic anisotropy energy (MAE) is defined as the change in the total energy by the rotation of the magnetic moment from the c direction to the a direction, i.e., the [100] direction. Unless otherwise stated, the MAE is calculated self-consistently, where the Kohn–Sham Hamiltonian includes the one-body spin–orbit coupling explicitly as a scalar triple product of the electric field, the momentum, and the Pauli matrix. In addition, we employ our approach using perturbation theory, which is compared with self-consistent calculations, to decompose the MAE into contributions from each orbital coupling. In this approach, the following formulae are used to analyze contributions from each atomic site τ as well as couplings among specific atomic orbitals μ and ν: $ESO=∑τESOτ, ESOτ=∑μνESOτμν$⁠, and

where i and j are occupied and unoccupied Kohn–Sham eigenstates. Note that the indices i, j, μ, and ν include the spin index. In this perturbation scheme, the spin–orbit coupling is approximated as $HSO=ξl·s$ considering only on-site contributions of d orbitals. The coupling constants ξ for d electrons are ξ_Fe = 62.69 meV and ξ_Y = 41.11 meV, as obtained by calculations for atoms in Ref. 19. By projecting onto the atomic orbitals μ and ν at a specific atomic site τ, we can use the matrix elements of H_SO for each atomic orbital. The matrix elements $⟨τμ|HSO|τν⟩$ for a single atom have been reported for the purpose of qualitative understanding.^20,21 In contrast, we will use the matrix elements for quantitative identifications of the MAE contributions based on the above formulae, as discussed below. The MAE is calculated by rotating the spins of the Bloch states i and j, whereas the spatial part is kept fixed.

As reported later in detail, we find the magnetic anisotropy of Y₂Fe₁₄B is enhanced upon compression of the lattice. Here, we first describe our most significant findings: By examining all the Fe and Y sites as well as all the possible d-orbital couplings by our perturbation scheme, we have identified that the $dx2−y2$ and d_xy orbitals at the Fe j₂ site play an essential role. Their local density of states (LDOS) and the positions of nearest-neighbor (NN) atoms of the Fe j₂ site are shown in Fig. 1. The j₂ site in Y₂Fe₁₄B are sandwiched by double kagome layers¹⁶ of Fe having large in-plane interaction, while Y atoms are located nearly along the c axis as seen in Fig. 1(b). This local environment of the j₂ site makes the band widths of $dx2−y2$ and d_xy more sensitive to the compression than the other orbitals. As a result, both peaks of the $dx2−y2↓$ and $dxy↓$ states just above the Fermi energy shift downwards by compression, which results in a significant increase in the LDOS near the Fermi energy.

Since the LDOS is proportional to the absolute square of the coefficients $⟨i|τμ⟩$ and $⟨τν|j⟩$ for the Bloch states i and j in Eq. (1), the high LDOS near the Fermi energy contributes markedly to the MAE. On the other hand, in considering the matrix element $⟨τμ|HSO|τν⟩$⁠, the coupling between the $dx2−y2$ and d_xy orbitals, which correspond to angular momenta of $±2ℏ$ in the z direction, with parallel spins makes the largest contribution to the anisotropy in the z direction, as identified by Kyuno et al.²⁰ Consequently, combining these two factors, couplings between the occupied (unoccupied) $dx2−y2↓$ component and the unoccupied (occupied) $dxy↓$ component, respectively, close to the Fermi energy are explained as the most significant origin of the MAE enhancement upon compression.

Coming back to the unstrained Y₂Fe₁₄B single crystal, the calculated value of the magnetic moment is 31.3 μ_B/f.u., in excellent agreement with the experimental value of 31.4 μ_B/f.u. measured at 4.2 K.¹ The electronic states in the vicinity of the Fermi energy are mainly associated with the Fe 3d and the Y 4d orbitals. The spin polarization of Fe 3d bands is much larger than that of Y 4d bands. As for the MAE, the magnetic easy axis of Y₂Fe₁₄B is the c axis and this result qualitatively agrees with experiments, even though the calculated value, 0.45 MJ/m³, underestimates the experimental one (0.77 MJ/m³ at 4.2 K (Ref. 6)). This discrepancy in the MAE has been understood as being due to a limitation of the present first-principles methods. It should be noted that the antiferromagnetic coupling among Fe 3d and Nd 5d orbitals plays a crucial role in the appearance of strong magnetic anisotropy in Nd₂Fe₁₄B (4.3 MJ/m³ at room temperature⁶). The strong anisotropy of the Nd 4f electrons cannot couple with the Fe 3d electrons directly, where most of the magnetic moment of the compound comes from.

We now proceed to the lattice-constant dependence of the magnetic moments and the magnetic anisotropy for Y₂Fe₁₄B, where the lattice constants, i.e., a and c, are changed by a 3% compression and a 3% expansion independently. Figure 2 shows the behavior of the MAE and magnetic moments with respect to the variation in the lattice constants. The MAE is significantly enhanced upon compression of the lattice, while the magnetic moment decreases. By the compression of both a and c by 3%, the MAE increases from 0.45 MJ/m³ (2.6 meV/u.c.) to 2.16 MJ/m³ (11.2 meV/u.c.), while the magnetic moment decreases from 31.3 μ_B to 25.7 μ_B. The changes in the MAE and the magnetic moment are much smaller for the expansion. As seen in Fig. 1(b), the number of NN Fe atoms for the j₂ site is 12. The average of these NN distances changes from 2.67 Å to 2.59 Å upon compression, which is comparable with that for the ferromagnetic fcc Fe theoretically determined as 2.57 Å and is much larger than that for bcc Fe of 2.45 Å. This is consistent with the fact that R₂Fe₁₄B compounds have much lower Curie temperatures compared with bcc Fe.

The DOS of the compressed and expanded Y₂Fe₁₄B are compared with the case for the equilibrium lattice constants in Fig. 3(a). The figure shows that the majority-spin (up-spin) d bands become partially occupied upon compression of the lattice, where the DOS attributable to d electrons becomes broader indicating the enhancement of the electron itinerancy. In contrast, the DOS of the expanded system exhibits slight narrowing of the d-band width attributable to the localization of d electrons. To identify the origin of the strong enhancement of the MAE and the decrease in the magnetic moment, we decompose the total MAE and the magnetic moment into contributions from each atomic site. We decompose the magnetic moment by the LDOS projected onto Fe atomic orbitals [Fig. 3(b)]. Without strain, the j₂ site has the largest magnetic moment of 2.74 μ_B, while the e site exhibits the smallest magnetic moment of 2.13 μ_B, as seen in Fig. 3(c). Under the compression, they decrease to 2.54 μ_B and 1.73 μ_B, respectively. As Fig. 3(c) shows, the Fe j₁ site has the largest changes under the compression and the expansion. Upon compression, the magnetic moment at the Fe j₁ site decreases considerably from 2.24 μ_B to 1.40 μ_B. This drastic decrease in the magnetic moment can be understood from the fact that the up-spin d bands at the j₁ site, which are essentially fully occupied for the case without strain, become partially occupied by the compression-induced modification to the orbital hybridization.²² 

In Table I, the total MAE calculated by the perturbation scheme is compared with that by the self-consistent calculations that explicitly include the spin–orbit coupling in the Kohn–Sham Hamiltonian. It is clearly seen that the change in the total MAE by the compression obtained by the self-consistent calculations is reproduced by the perturbation scheme, even though the energy scale is much smaller than the chemical accuracy. The error of the perturbation scheme compared with the self-consistent one is mainly attributed to the on-site approximation with the form of $HSO=ξl·s$⁠. In addition, we also compare the perturbation scheme with self-consistent calculations for another intermetallic compound, YFe₃, in which the MAE obtained by the perturbation scheme (−0.65 MJ/m³) differs only 0.15 MJ/m³ from the self-consistent MAE, −0.80 MJ/m³.

Next, we discuss the MAE of inequivalent Fe sites as well as the f and g sites of Y listed in Table I. We observe that the symmetry of the MAE is lower than that of the atomic structure for 16k₁, 16k₂, and 4c sites; e.g., among 16 k₁ sites, eight k₁ sites exhibit one value for the site-decomposed MAE, while the other eight k₁ sites exhibit another. Note that the symmetry of the atomic structure remains unchanged with strain. For each case, $ESOτ(x)$ becomes inequivalent, providing two different values, while the original site symmetry is retained in $ESOτ(z)$⁠. For E_SO (x), the spins of the Bloch states point in the x direction. The symmetry created by a set of vector quantities can be lower than that created by scalar quantities, i.e., the atomic positions in this case. It is clearly seen that the MAE of Y₂Fe₁₄B essentially comes from the Fe sublattice, where the contribution of Y atoms is negligibly small. For the case of the equilibrium lattice constants, the contribution from the j₁ site is most significant for the perpendicular anisotropy. This fact is consistent with the fact that the LDOS at the Fermi energy for the up spin is among the highest for the j₁ site, as seen in Fig. 3(b). It is also evident that the enhancement of the MAE by the compression comes from that at the Fe j₂ site, where the site-decomposed MAE increases from 0.06 to 0.58 meV. This enhancement of the MAE at the j₂ site is closely related to the increase in the LDOS in the vicinity of the Fermi energy for the down spin shown in Fig. 3(b).

We also analyze the couplings among specific atomic orbitals on the basis of $ESOτμν$⁠. Table II shows the MAE contribution from couplings among the 3d orbitals at the j₂ site. We can clearly see that the enhancement of the MAE is related to an increase in the coupling among $dx2−y2$ and d_xy orbitals from −0.55 to 0.23 meV: The contribution even changes from in-plane anisotropy to perpendicular anisotropy under compression. In more detail, the contribution of $dx2−y2−dxy$ coupling at the j₂ site is most significant for the down spins. Under compression, the coupling among $dx2−y2↓$ and $dxy↓$ increases from 0.79 meV to 1.43 meV. The other MAE contributions for the compressed system from these d orbitals at the j₂ site are −1.36 meV for $dx2−y2↓−dxy↑$⁠, −0.11 meV for $dx2−y2↑−dxy↓$⁠, and 0.27 meV for $dx2−y2↑−dxy↑$⁠. We also found that couplings among $dx2−y2$ and d_xy orbitals at the e site have an increase in the MAE upon compression from −0.06 to 0.45 meV, where the coupling among down spins contributes to a MAE change from 0.94 to 1.37 meV.

In summary, we investigated the dependence of the MAE of Y₂Fe₁₄B on the lattice constants by using first-principles calculations. We have found that the magnetic anisotropy of Y₂Fe₁₄B comes from the 3d electrons of Fe atoms. The MAE is enhanced by lattice compression. We also proposed a convenient and computationally inexpensive method, based on second-order perturbation theory, to quantify the contribution of each atomic orbital to the MAE. This method quantitatively considers both the effect of the LDOS through matrix elements such as $⟨i|τμ⟩$ and the effect of orbital couplings $⟨τμ|HSO|τν⟩$⁠. With this method, we clarified that the Fe j₂ site makes the highest contribution to the enhanced MAE in the compressed system, where coupling among the $3dx2−y2↓$ and $3dxy↓$ orbitals of the j₂ site plays the most significant role. With predictive descriptions of couplings among atomic orbitals for the magnetic anisotropy, our method should be useful for computational designs of permanent-magnet materials utilizing the anisotropy of itinerant d electrons, even without the use of 4f electrons of rare-earth elements. On the other hand, it is also interesting to clarify the anisotropy couplings among the Fe sublattice and the 4f electrons.

Part of this work has been supported by the Elements Strategy Initiative Center for Magnetic Materials under the outsourcing project of MEXT as well as by Grants-in-Aid for Scientific Research through Contract Nos. 25104705 and 22104006 from MEXT, Japan. The calculations were partly performed on the K-computer (Grant No. hp120086) and supercomputers at ISSP and ITC, The University of Tokyo.