Anomalous magneto-structural behavior of MnBi explained: a path towards an improved permanent magnet

Low-temperature MnBi (hexagonal NiAs phase) exhibits anomalies in the lattice constants (a, c) and bulk elastic modulus (B) below 100 K, spin reorientation and magnetic susceptibility maximum near 90 K, and, importantly for high-temperature magnetic applications, an increasing coercivity (unique to MnBi) above 180 K. We calculate the total energy and magneto-anisotropy energy (MAE) versus (a, c) using DFT+U methods. We reproduce and explain all the above anomalies. We predict that coercivity and MAE increase due to increasing a, suggesting means to improve MnBi permanent magnets.

MnBi in its low-temperature phase (LTP) has one of the most extraordinary magnetic properties among ferromagnetic materials.  Uniquely, its coercivity increases with temperature (T), and its value is larger than that of Nd 2 Fe 14 B above 423 K, making it potentially an excellent permanent magnet for higher-temperature applications. MnBi does not contain critical rare-earth elements and, thus, it has a potential for technological impact. If magnetic anisotropy energy (MAE) is better controlled and tuned, use of MnBi magnets could be broadened. Below we provide theoretical explanation for the long-standing experimental puzzles in the measured coercivity, spin orientation, lattice constants, and bulk modulus of MnBi. We also suggest a means to further increase the MAE.
Despite its simple NiAs hexagonal structure ( Fig. 1), stable below 628 K, 5,14 MnBi exhibits several puzzling and unexplained behaviors versus T. [11][12][13][14][15] First, the lattice constant a exhibits minimal thermal expansion below 70 K and then expands rapidly during the spin reorientation, while c shows a chaotic zigzag behavior below 150 K. 9,13,15 Second, there is a measured kink in the bulk modulus (B) near 39 GPa at 100 K. 8 Third, a spin reorientation is observed at T SR ≈ 90 K, 8,11,12 when the magnetization M(T) easy axis changes from in-plane to c-axis above T SR . Next, coercivity is near zero at T<180 K, and increases with T above 180 K. Finally, above 628 K MnBi transforms to a high-T oP10 phase (stable between 613 K and 719 K) with M=0. 7 We explain all these observations by examining dependence of the calculated total energy (E) and MAE on the lattice geometry (Figs. 2, 3 and 4). The total energy is anisotropic versus (a, c), like a "flat-bottom canoe," and its asymmetry causes abnormal thermal expansion. Due to the nature of the potential energy surface, the second derivative of the total energy with respect to volume is not monotonic, producing a kink in B= V d 2 E/dV 2 near 39 GPa, whose origin can be traced to features in electronic density of states (DOS). Spin reorientation arises a) Electronic mail: zarkev@ameslab.gov from a change of sign in MAE, which depends on increasing a, This suggests simple means to control MAE: by thermal expansion (observed), or by strain or alloying, e.g., coherent interfacing or doping. While temperature and strain affect mostly (a, c), doping can induce competing effects on MAE, some of which can be beneficial. Preliminary results suggest that doping with selected metals (Ni, Rh, Pd, Ir) increases MAE and coercivity and stabilize the spin orientation along c at all temperatures.
Computational method: We use a DFT+U method implemented in the Vienna ab initio simulation package (VASP). 23 (0001) projection (left), and primitive unit cell (right) with two Mn (red) and two Bi (green) atoms.
are order of meV. We improve description of the electronic structure (and, hence, magnetization and lattice parameters) by combining the spin-polarized, generalized gradient approximation (GGA) 26 with the rotationally invariant DFT+U formalism. 27 GGA includes local value and gradient of the electron density n = n ↑ + n ↓ and spin density n σ (σ=↑, ↓) in the exchange-correlation functional E GGA xc [n ↑ , ∇n ↑ , n ↓ , ∇n ↓ ]. 28 DFT+U corrects the total energy 27 for presence of localized states, i.e., where n m,σ is the occupation number of state m (m = 2 for d-orbital on Mn). See textbook 29 for more details. After testing, we set (U−J)=2 eV for correlated Mn d-electrons to better reproduce the measured (a, c) and M (Table 1). Note, a single U−J parameter cannot be adjusted to reproduce both (a, c) and M perfectly. At 0 K, we find a (4.363Å) and c (6.123Å) in good agreement with those measured 11 at 10 K (Table 1) 11 While the GGA+U better describes stronglycorrelated systems, like MnBi, there still remains a small systematic DFT error in the lattice constants, arising from the approximation in the exchange-correlation functional (which introduces a small shift in pressure, but not in the curvature of the total energy). Notably, the measured lattice constants differ by 1%, e.g., at T=50 K, c=6.05Å 13 and 6.11Å. 15 The MAE (Fig. 4) is small and very sensitive to (a, c). For proper comparison, we plot in Fig. 4 both the measured (a, c) and those shifted by 0.8% to account for a DFT bias in the GGA+U lattice constants for a given alloy.
Comparison to previous DFT calculations: Without the Hubbard U correction, GGA gives M(0) of 3.455 µ B and distorts the cell, underestimating its volume (Table  1). For comparison, previous DFT results are 3.50, 33 3.49, 11 and 3.52 µ B . 34 Fixing a to 4.170Å and c to 5.755Å gives a total moment of 4.01 µ B in the fullpotential LMTO, 35 while fixing a to 4.26Å and c to 6.05Å gives 3.7 µ B in augmented spherical methods (ASM). 34 Magnetization of MnBi increases with volume. The calculated lattice constants, volume, and magnetization increase with the value of (U−J).

Results and Discussion:
Around equilibrium, E(∆a,∆c) looks like a flat-bottom canoe, canted from a constant volume direction towards c (Fig. 2). Because the energy penalty for changing c by 0.5% is close to zero, even low-energy defects can alter c, and any value of c within that range is accessible in experiment below 100 K. Indeed, this predicted behavior of c with chaotic amplitude within ∼0.5% is observed. 13,15 Below 6 meV (70 K), E(∆a,0) in Fig. 3 is symmetric with E(+∆a, 0) = E(−∆a, 0), and can be well described by a parabola E(a) = 1 2m ω 2 (a − a 0 ) 2 , where the unit cell mass ism = 2(m M n + m Bi ) = 527.836 a.m.u., and ω = 1.2 · 10 13 s −1 is harmonic frequency for vibrations along a. Quantization of this potential results in a descrete spectrum with the equidistant levels E n = ω(n + 1 2 ), with ω = 7.7 meV (90 K). Due to the symmetric potential and absence of vibrational excitations, there is no thermal expansion along a at T < 70 K.
Above 9 meV (100 K), E(∆a,0) is asymmetric with E(+∆a, 0) < E(−∆a, 0). It can be approximated by a cubic polynomial E(a) = E 0 + 2 ∆a 2 + 3 ∆a 3 , with 2 = 3.8 eV/Å 2 and 3 = −2.1 eV/Å 3 . This fit has χ 2 = 6 · 10 −7 , RMS relative error of 7.9 · 10 −6 , and Theil U coefficent of 7.8 · 10 −6 . For N = 4 ions per unit cell, our theoretical estimate of the linear thermal expansion ) is 1.153 · 10 −5 K −1 , in agreement with experiment, 15 i.e., 1.168 · 10 −5 K −1 . Hence, the potential energy surface in Fig. 2 predicts no thermal expansion along a at low T < 70 K, and a positive expansion at higher T above 100 K, as observed. 15 The spin reorientation in MnBi near 90 K was not fully understood in experiments. 8,11,12 Moreover, previous DFT calculations of MAE found the easy axis to be always in-plane ( Table 3 in Ref. 35). We calculate dependence of the MAE on (a, c), and find that it is strongly affected by a and very weakly by c, see Fig. 4. Thus, thermal expansion of a causes the MAE to change from negative (in-plane oriented moments) to positive (moments oriented along the c-axis). This sign change causes a spin reorientation, experimentally observed around 90 K. 8,11,12 Magnetic susceptibility has maximum at MAE=0, 8,10,11 when spins easily reorient along the external applied magnetic field. Coercivity is zero if |MAE| < kT, but increases with MAE at T > 180 K. 12 Thus, dependence of MAE on (a, c) causes spin reorientation and explains the thermal behavior of magnetic susceptibility and coercivity.
Another consequence of the anomalous potential energy surface E(a, c) is the observed kink in B near 39  GPa at 100 K, a long-standing puzzle. 8 We calculate B= V d 2 E/dV 2 from dependence of E(a, c) on V = ca 2 √ 3/2 at isotropic expansion (∆a = ∆c in Fig. 2). We find that B versus a (Fig. 5) is not monotonic near B= 39 GPa, as observed. 8 This kink originates from a change in DOS at the Fermi level ( Fig. 5; see also Fig. 4 in Ref. 34). The Fermi level (E F ) is in a pseudo-gap, and the minimum in the minority-spin DOS passes through E F with thermal expansion of a; the DOS minimum corresponds to the a at B=39 GPa (inset, Fig. 5).
Summary: We calculated dependence of the total energy and magneto-anisotropy energy on the lattice geometry for MnBi low-T phase. Our results explain the unusual structural and magnetic properties, heretofore unexplained. From the potential energy surface, we reproduced and explained the observed anomalous behavior of (i) the lattice constants and (ii) bulk modulus. The is shifted by 0.83% (filled circles), see text. calculated MAE changes sign with a small increase in a, which causes spin reorientation during thermal expansion. (iii) The magnetic susceptibility has a maximum at MAE=0 (at spin reorientation). (iv) Further increase of MAE with thermally expanding a increases coercivity at T > 180 K, where |MAE| > kT. Due to its sensitivity on a, the MAE can be altered by temperature, pressure, doping, or interfacial strain. [36][37][38][39] To test whether doping can achieve a positive MAE at all temperatures, we performed preliminary, small-cell calculations that find that doping with selected (Ni, Rh, Pd, Ir) metals increases coercivity and stabilizes the spin orientation along c. More extensive calculations for <3% cationic or anionic doped (substitutions and interstitials) cases are planned to establish the effects on lattice, magnetism, and stability. Our understanding of the anomalous magneto-structural behavior offers an opportunity to develop improved MnBi-based permanent magnets.