Impact of Coulomb Correlations on Magnetic Anisotropy in Mn3Ga Ferrimagnet

Traditional density functional theory (DFT) miserably fails to reproduce the experimental volume and magnetic anisotropy of D022 Mn3Ga, which has recently become one of the most sought-after materials in order to achieve a stable spin switching at low current density. Despite great progress over the last 10 years, this issue has hitherto remained unsolved. Here, taking into account the effects of strong electronic correlations beyond what is included in standard DFT, we show by comparison with the experiment that the DFT+U method is capable of quantitatively describing the volume and the magnetic anisotropy energy (MAE) in this alloy with physically meaningful choice of onsite Coulomb-U parameter. For the first time using a plane-wave code, we decompose MAE into spin channel-resolved components in order to determine spin-flip and spin-conserving contributions. The Mn atom at the tetrahedral site is identified as the primary source of the high perpendicular MAE with the most dominant spin-orbit coupling (SOC) occurring between its two orbital pairs: ↑↑ coupling and ↓↓ coupling between \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{d}}}_{{{\boldsymbol{x}}}^{{\bf{2}}}-{{\boldsymbol{y}}}^{{\bf{2}}}}$$\end{document}dx2−y2 and d xy, and ↑↓ coupling between d yz and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{d}}}_{{{\boldsymbol{z}}}^{{\bf{2}}}}$$\end{document}dz2. Using the SOC-perturbation theory model, we provide interpretation of our numerical results. These results are important for quantitative microscopic understanding of the large perpendicular MAE observed in this material, and should assist in harnessing its potential for applications in futuristic spintronic devices.

A tetragonal (D0 22 ) Heusler alloy Mn 3 Ga has recently created increasing interest among researchers because of its excellent combination of properties, such as low Gilbert damping constant (α < 0.008) 1 , small saturation magnetization (M s ~ 250 emu/cm 3 ) 2 , high Curie temperature (T C > 770 K) 3 , large spin polarization close to that of a half-metal (P ~ 88%) 4 , and strong perpendicular magnetocrystalline anisotropy (K u > 10 Merg/cm 3 ) 2 . Low Gilbert damping and saturation magnetization but high Curie temperature and spin polarization are necessary preconditions for advanced spintronic applications in order to realize low switching currents and high efficiency of spin injection 5 . High values of perpendicular magnetocrystalline anisotropy (PMA) are preferred to stabilize the perpendicular magnetization against thermal fluctuations, ensuring non-volatility of the stored information particularly when scaling down materials for high density magnetic data storage.
Mn 3 Ga bulk in its tetragonal D0 22 structure (I4/mmm space group, number 139) has experimental lattice parameters of a = 3.90 Å and c = 7.12 Å 3 . Its structure optimization using density functional theory (DFT) with the Perdew-Burke-Ernzerhof (PBE) functional gives a = 3.78 Å and c = 7.10 Å. These values lead to about 6% smaller lattice volume compared to the experimental one, standing in stark contrast to the well-known trend that the PBE calculations 2 systematically overestimate the experimental lattice volume. This contrast indicates that a strong Coulomb correlation beyond the traditional DFT is likely in operation. Furthermore, experimental measurements are known to yield the perpendicular (⊥) magnetocrystalline anisotropy energy (MAE) of 14 × 10 6 erg/ cm 3 (1 meV) 6 . The value theoretically obtained using first-principles DFT overestimates the experimental value by a factor of about 2. This inspires us to investigate the effect of intra-site Coulomb correlation on the magnetic anisotropy of Mn 3 Ga using DFT+U approach (see Method section).
In D0 22 Mn 3 Ga crystal, Mn atoms occupy two different positions [see Fig. 1(a)]. The first position (Mn I ), with multiplicity 1, is located at the Wyckoff position 2b (0, 0, 0.5) [octahedral site] and the second position (Mn II ), with multiplicity 2, is at 4d (0, 0.5, 0.25) [tetrahedral site] 7 , indicating that the effective U can be potentially different for these two different sites. A proper choice of the effective U parameter in PBE+U formalism is crucial in understanding and interpreting the results of first-principles calculations. Therefore, we scan the U I , U II parameter space [U I = U(Mn I ), U II = U(Mn II )] and obtain the best set of U I and U II values (U I = 2.6 eV and U II = 0 eV), yielding in-plane lattice constant, out-of-plane lattice constant, magnetic moment of Mn I and magnetic moment of Mn II close to the respective experimental ones simultaneously [see Fig. 1(b-e), the light-grey color corresponds to the experimental value in these subfigures].
With U I = 2.6 eV (U I = 0 eV), the optimized in-plane lattice parameter is determined to be 3.9054 Å (3.78 Å), while the out-of-plane lattice parameter is 7.0518 Å (7.10 Å). Both lattice parameters with U I = 2.6 eV are in close agreement with existing experimental lattice parameters of 3.90 Å and 7.12 Å 3 . The optimized structure possesses a ferrimagnetic ordering, with the Mn I atoms aligned antiparallel to the Mn II atoms. For U I = 2.6 eV (U I = 0 eV), the Mn II atoms are separated from their nearest Mn II neighbors by 2.76 Å (2.67 Å) and from the nearest Ga atoms   of orbital magnetic moment. This anisotropy is smaller for the PBE+U calculation in agreement with the finding that the PBE+U rather than PBE gives a smaller MAE for Mn 3 Ga.
Within the framework of second-order perturbation theory 8 , ε ε ′ σ σ respectively stand for eigenstates and eigenvalues of unoccupied (occupied) states in spin state σ(σ′), ξ is the SOC coefficient, and L z and L x are the angular momentum operators. Relative contributions of the nonzero matrix elements with the d-states are as follows: d L d 1 . For these nonvanishing matrix elements (two for L z and three for L x operators), the most dominant contribution to the MAE comes from the states near the Fermi level and its behavior is essentially determined by the denominator of Eq. (1). The SOC interaction between states with the same (different by 1) magnetic quantum number(s), m, is through the L z (L x ) operator. For parallel mutual spin orientations (σσ′ = ↑↑ or ↓↓), positive (negative) contribution comes from the L z (L x ) coupling; whereas for antiparallel mutual spin orientations (σσ′ = ↑↓ or ↓↑), Eq. (1) has opposite sign and thus positive (negative) contribution comes from the L x (L z ) coupling.  Fig. 3(a) shows that large negative (in-plane) contributions to the MAE come from σσ′ = ↑↓ coupling and ↓↑ coupling between the − d x y 2 2 and d xy orbitals, while relatively small positive (perpendicular or out-of-plane) contributions from ↑↑ coupling and ↓↓ coupling also occur between these orbitals. Additional positive contributions come from ↓↑ coupling and ↑↓ coupling between d x y 2 2 − and d yz orbitals, d z 2 and d yz orbitals, and d xz and d xy orbitals, while ↑↓ coupling and ↓↑ coupling between the d yz and d xz orbitals gives a small negative contribution. The most notable difference between the orbital-resolved MAE of Mn II and that of Mn I is that the d xy and − d x y 2 2 orbitals of Mn II atoms contribute to larger PMA due to stronger ↓↓ coupling and ↑↑ coupling, while these orbitals of Mn I atoms give larger in-plane contribution due to stronger ↓↑ coupling and ↑↓ coupling. Like Mn I atoms, in the case of Mn II atoms, the d yz and d z 2 orbitals also make a significant ⊥ contribution through ↑↓ coupling and ↓↑ coupling; while small in-plane contributions come from the ↑↑ coupling and ↓↓ coupling between the d xz and d xy orbitals, and from ↑↓ coupling and ↓↑ coupling between the d xz and d yz orbitals. With applying the Hubbard U correction of 2.6 eV at Mn I site, similar trends are found as − bands, mostly near the Γ point on the Γ-X line. The ↑↓ coupling also occurs between the unoccupied d yz ↑ and occupied d xz ↓ states mostly around the X point, leading however to a small in-plane contribution.
In the presence of on-site U correction of 2.6 eV at Mn I , one feature common to both Mn I and Mn II atoms is that the correction results in a large shift of spectral weight away from the Fermi level. Pushing away the bands near the Fermi level by on-site U causes significant reduction of electron states at and near E F which eventually leads to a reduction of about 25% of total MAE in Mn 3 Ga.
In conclusion, taking into account the effects of strong electronic correlations, we show by comparison with the experiment that the DFT+U method is capable of quantitatively describing the volume and the MAE in D0 22 Mn 3 Ga ferrimagnet. For the first time using a plane-wave code, we decompose MAE into spin channel-resolved components to determine spin-flip and spin-conserving contributions. The Mn atom at the tetrahedral site is identified as the main source of the high ⊥ MAE with the most dominant spin-orbit coupling (SOC) occurring between its two orbital pairs: ↑↑ coupling and ↓↓ coupling between − d x y 2 2 and d xy , and ↑↓ coupling between d yz and d z 2. Using the SOC-perturbation theory model, we provide interpretation of our numerical results. These results are important for quantitative microscopic understanding of the large PMA in this material, and should assist in the development of the futuristic spintronic devices.

Method
Our calculations are performed using the VASP 9 implementation of DFT, with the Perdew-Burke-Ernzerhof exchange-correlation functional and projector augmented wave (PAW) potentials [10][11][12][13][14] . Kohn-Sham wave functions are represented using a plane-wave basis truncated at an energy cutoff of 40 Ry. Brillouin zone integrations are done on a uniform Monkhorst-Pack 15 k grid of 19 × 19 × 11. The effect of Coulomb correlation is incorporated using DFT+U approach of Dudarev, in which an effective, rotationally-invariant, screened, onsite Coulomb U (U d − J) is added to the DFT functional [16][17][18] . Atomic positions are fully relaxed using the conjugate gradient algorithm until all inter-atomic forces are smaller than 0.1 meV/Å. The MAE is determined by applying spin-orbit coupling and comparing the total energy values for in-plane and out-of-plane magnetization orientations, according to the following equation: MAE = E 100 − E 001 , where (100) and (001) representing the in-plane and out-of-plane orientations, respectively.