Anomalous magneto-structural behavior of MnBi explained: a path towards an improved permanent magnet
AAnomalous magneto-structural behavior of MnBi explained:a path towards an improved permanent magnet
N.A. Zarkevich, a) L.-L. Wang, and D.D. Johnson
1, 2 The Ames Laboratory, U.S. Department of Energy, Ames, Iowa 50011 USA Materials Science and Engineering, Iowa State University, Ames, Iowa 50011 USA (Dated: 11 November 2018)
Low-temperature MnBi (hexagonal NiAs phase) exhibits anomalies in the lattice constants ( a , c ) and bulkelastic 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) above180 K. We calculate the total energy and magneto-anisotropy energy (MAE) versus ( a , c ) using DFT+Umethods. We reproduce and explain all the above anomalies. We predict that coercivity and MAE increasedue to increasing a , suggesting means to improve MnBi permanent magnets.PACS numbers: 75.30.Gw, 81.40.Rs, 65.40.De, 02.70.-cMnBi in its low-temperature phase (LTP) has one ofthe most extraordinary magnetic properties among ferro-magnetic materials. Uniquely, its coercivity increaseswith temperature (T), and its value is larger than that ofNd Fe B above 423 K, making it potentially an excellentpermanent magnet for higher-temperature applications.MnBi does not contain critical rare-earth elements and,thus, it has a potential for technological impact. If mag-netic anisotropy energy (MAE) is better controlled andtuned, use of MnBi magnets could be broadened. Belowwe provide theoretical explanation for the long-standingexperimental puzzles in the measured coercivity, spin ori-entation, 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,
MnBi exhibits several puzzlingand unexplained behaviors versus T.
First, the lat-tice constant a exhibits minimal thermal expansion below70 K and then expands rapidly during the spin reori-entation, while c shows a chaotic zigzag behavior below150 K. Second, there is a measured kink in the bulkmodulus (B) near 39 GPa at 100 K. Third, a spin reori-entation is observed at T SR ≈
90 K, when the mag-netization M(T) easy axis changes from in-plane to c -axisabove T SR . Next, coercivity is near zero at T <
180 K, andincreases with T above 180 K. Finally, above 628 K MnBitransforms to a high-T oP10 phase (stable between 613 Kand 719 K) with M=0. We explain all these observations by examining depen-dence of the calculated total energy (E) and MAE on thelattice geometry (Figs. 2, 3 and 4). The total energy isanisotropic versus ( a, c ), like a “flat-bottom canoe,” andits asymmetry causes abnormal thermal expansion. Dueto the nature of the potential energy surface, the secondderivative of the total energy with respect to volume isnot monotonic, producing a kink in B=
V d E/dV near39 GPa, whose origin can be traced to features in elec-tronic density of states (DOS). Spin reorientation arises a) Electronic mail: [email protected] from a change of sign in MAE, which depends on in-creasing a , This suggests simple means to control MAE:by thermal expansion (observed), or by strain or alloying,e.g., coherent interfacing or doping. While temperatureand strain affect mostly ( a, c ), doping can induce compet-ing effects on MAE, some of which can be beneficial. Pre-liminary results suggest that doping with selected metals(Ni, Rh, Pd, Ir) increases MAE and coercivity and sta-bilize the spin orientation along c at all temperatures. Computational method:
We use a DFT+U methodimplemented in the Vienna ab initio simulation pack-age (VASP).
We use 16 × ×
10 Monkhorst-Pack k -point grid with the Γ-point, a 337 . . e V augmentation charge cut-off, for both energy and magnetic anisotropy energy. Amodified Broyden’s method is used for electronic self-consistency. Bulk moduli are found from dependence ofthe total energy E( a, c ) on volume V = ca √ /
2. MAEis the energy difference with moments along (cid:104) (cid:105) andthen (cid:104) (cid:105) , i.e., E[1¯210] − E[0001]. Generally, the MAEcan be the order of µe V to me V; in MnBi for changes in a , pertinent to thermal lattice expansion effects, changesFIG. 1: MnBi hexagonal structure ( hP
4, P63/mmc,No.194), with 0 . e/ ˚A charge density isosurfaces.(0001) projection (left), and primitive unit cell (right)with two Mn (red) and two Bi (green) atoms. a r X i v : . [ c ond - m a t . m t r l - s c i ] F e b are order of me V.We improve description of the electronic structure(and, hence, magnetization and lattice parameters) bycombining the spin-polarized, generalized gradient ap-proximation (GGA) with the rotationally invariantDFT+U formalism. GGA includes local value and gra-dient of the electron density n = n ↑ + n ↓ and spin den-sity n σ ( σ = ↑ , ↓ ) in the exchange-correlation functional E GGAxc [ n ↑ , ∇ n ↑ , n ↓ , ∇ n ↓ ]. DFT+U corrects the totalenergy for presence of localized states, i.e., E DF T + U = E DF T + (U − J) (cid:80) σ ( n m,σ − n m,σ ), where n m,σ is theoccupation number of state m ( m = 2 for d -orbital onMn). See textbook for more details. After testing, weset (U − J)=2 e V for correlated Mn d -electrons to betterreproduce the measured ( a, c ) and M (Table 1). Note,a single U − J parameter cannot be adjusted to repro-duce both ( a, c ) and M perfectly. At 0 K, we find a (4 . c (6 . at 10 K (Table 1), with an overestimate by1.86% and 0.21%, respectively. The calculated M(0) is3 . µ B /MnBi (with site-projected moments of 4.231 and − . µ B on Mn and Bi, respectively); it agrees with theextrapolated to 0 K values of 4.0 and 3 . µ B ; or themeasured values of 3 . ± . µ B at 4 . . µ B at10 K, or 3 . µ B at room T. While the GGA+U better describes strongly-correlated systems, like MnBi, there still remains a smallsystematic DFT error in the lattice constants, arisingfrom the approximation in the exchange-correlationfunctional (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., atT=50 K, c =6 .
05 ˚A and 6 .
11 ˚A. The MAE (Fig. 4)is small and very sensitive to ( a, c ). For proper com-parison, we plot in Fig. 4 both the measured ( a, c ) andthose shifted by 0.8% to account for a DFT bias in theGGA+U lattice constants for a given alloy.
Comparison to previous DFT calculations:
Withoutthe Hubbard U correction, GGA gives M(0) of 3 . µ B TABLE I: a and c , c/a , cell V , and M ( µ B /MnBi) ofLTP-MnBi from experiment and our (or former a ) DFTresults. a (˚A) c (˚A) c/a V (˚A ) µ B Ref.4.2827 6.1103 1.4267 97.0574 4.18 @10 K4.286 6.126 1.4293 97.4567 Note: a and c were fixed in 35, 34, and 33. and distorts the cell, underestimating its volume (Table1). For comparison, previous DFT results are 3.50, and 3 . µ B . Fixing a to 4 .
170 ˚A and c to5 .
755 ˚A gives a total moment of 4 . µ B in the full-potential LMTO, while fixing a to 4 .
26 ˚A and c to6 .
05 ˚A gives 3 . µ B in augmented spherical methods(ASM). Magnetization of MnBi increases with volume.The calculated lattice constants, volume, and magneti-zation increase with the value of (U − J).
Results and Discussion:
Around equilibrium,E(∆ a ,∆ c ) looks like a flat-bottom canoe, cantedfrom a constant volume direction towards c (Fig. 2).Because the energy penalty for changing c by 0.5% isclose to zero, even low-energy defects can alter c , and anyvalue of c within that range is accessible in experimentbelow 100 K. Indeed, this predicted behavior of c withchaotic amplitude within ∼ .
5% is observed.
Below 6 me V (70 K), E(∆ a ,0) in Fig. 3 is symmet-ric with E(+∆ a,
0) = E( − ∆ a, a ) = ¯ mω ( a − a ) , where theunit cell mass is ¯ m = 2( m Mn + m Bi ) = 527 .
836 a.m.u.,and ω = 1 . · s − is harmonic frequency for vi-brations along a . Quantization of this potential re-sults in a descrete spectrum with the equidistant levels E n = (cid:126) ω ( n + ), with (cid:126) ω = 7 . m eV (90 K). Due to thesymmetric potential and absence of vibrational excita-tions, there is no thermal expansion along a at T <
70 K.Above 9 me V (100 K), E(∆ a ,0) is asymmetric withE(+∆ a, < E( − ∆ a, a ) = E + (cid:15) ∆ a + (cid:15) ∆ a , with (cid:15) = 3 . and (cid:15) = − . . This fit has χ = 6 · − , RMS relative error of 7 . · − , and TheilU coefficent of 7 . · − . For N = 4 ions per unit cell,our theoretical estimate of the linear thermal expansioncoefficient ( α a = a dadT ≈ − a N k (cid:15) (cid:15) ) is 1 . · − K − , inagreement with experiment, i.e., 1 . · − K − .Hence, the potential energy surface in Fig. 2 predictsno thermal expansion along a at low T <
70 K, and a pos-itive expansion at higher T above 100 K, as observed. The spin reorientation in MnBi near 90 K was notfully understood in experiments.
Moreover, previ-ous DFT calculations of MAE found the easy axis tobe always in-plane (Table 3 in Ref. 35). We calcu-late dependence of the MAE on ( a, c ), and find thatit is strongly affected by a and very weakly by c , seeFig. 4. Thus, thermal expansion of a causes the MAEto change from negative (in-plane oriented moments)to positive (moments oriented along the c -axis). Thissign change causes a spin reorientation, experimentallyobserved around 90 K. Magnetic susceptibility hasmaximum at MAE=0, when spins easily reorientalong the external applied magnetic field. Coercivityis zero if | MAE | < k T, but increases with MAE atT >
180 K. Thus, dependence of MAE on ( a , c ) causesspin reorientation and explains the thermal behavior ofmagnetic susceptibility and coercivity.Another consequence of the anomalous potential en-ergy surface E( a, c ) is the observed kink in B near 39FIG. 2: Change in the total energy E at 0 K vs.(∆ a, ∆ c ), with 3 me V/cell (or 34 . ca √ /
2) is the line through (0,0). -6 -4 -2 0 2 4 6050100150200250300350 E ne r g y ( m e V ) DFTCubic fitQuadratic ∆ E -6 -4 -2 0 2 4 6 ∆ a (%) -60-40-200204060 ∆ E ( m e V ) -1 0 1 ∆ a (%) E ( m e V ) FIG. 3: E(∆ a, ∆ c = 0). GGA+U results (circles) withcubic (line) and quadratic (dashed) fits, and theirdifference ∆E (black line, right scale). (cid:126) ω = 7 . m eV(90 K) is the horizontal dotted line in the inset.GPa at 100 K, a long-standing puzzle. We calculate B=
V d E /dV from dependence of E( a, c ) on V = ca √ / a = ∆ c in Fig. 2). We find thatB versus a (Fig. 5) is not monotonic near B= 39 GPa, asobserved. This kink originates from a change in DOS atthe Fermi level (Fig. 5; see also Fig. 4 in Ref. 34). TheFermi level (E F ) is in a pseudo-gap, and the minimum inthe minority-spin DOS passes through E F with thermalexpansion of a ; the DOS minimum corresponds to the a at B=39 GPa (inset, Fig. 5). Summary:
We calculated dependence of the total en-ergy and magneto-anisotropy energy on the lattice ge-ometry for MnBi low-T phase. Our results explain theunusual structural and magnetic properties, heretoforeunexplained. From the potential energy surface, we re-produced and explained the observed anomalous behav-ior of (i) the lattice constants and (ii) bulk modulus. The FIG. 4: MAE vs. ( a, c ) with 0 . me V/cell steps incontours from zero (grey line). Assessed data (circles)is shifted by 0.83% (filled circles), see text. -1 0 1E − E F (eV)-3-2-101 D O S a/a B ( G P a ) FIG. 5: B vs. a at isotropic expansion ( a/a = c/c )relative to a = 4 . c = 6 . (Inset)Spin DOS (states/[cell · eV]) for 3 values of a/a .calculated MAE changes sign with a small increase in a ,which causes spin reorientation during thermal expan-sion. (iii) The magnetic susceptibility has a maximum atMAE=0 (at spin reorientation). (iv) Further increase ofMAE with thermally expanding a increases coercivity atT >
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