MnBi Alloy Anisotropy for Permanent Magnets: DFT Study

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Transformation of in‑plane
to out‑of‑plane anisotropy in MnBi
alloy for permanent magnet
application: a First‑principles study
Khoveto Vero & J. P. Borah
*
The low‑temperature phase (LTP) MnBi exhibits remarkable ferromagnetic properties at room
temperature. However, below its Curie temperature (
TC
), a phase transition occurs around 613 K due
to diusion of Mn into interstitial sites, raising concerns about its structural and magnetic properties.
Furthermore, the presence of in‑plane anisotropy in LTP‑MnBi alloy at low temperatures raises
concerns about its suitability for use in permanent magnet applications, even at higher temperature.
Therefore, this study examines the structural and magnetic properties of pure LTP‑MnBi and its
successive Ni‑doped and Fe‑substituted alloys using rst‑principles study based on density functional
theory (DFT). To prevent Mn diusion into interstitial sites, Ni doping is employed. Additionally, the
incorporation of Ni successfully addresses the in‑plane anisotropy issue in LTP‑MnBi, transforming
it into out‑of‑plane anisotropy. Moreover, we explored the potential advantages of substituting Fe
for one of Mn site. This substitution aims to improve the observed dynamical instability in Ni‑doped
alloy and to further enhanced the magnetocrystalline anisotropy energy (MAE) of the material,
resulting in an MAE of 3.21 MJ/m3, along with a
TC
of 523 K. Therefore, the coexistence of high MAE
and moderate
TC
in the Mn0.5Fe0.5Bi–Ni alloy presents viable option for its application in permanent
magnet technology.
Keywords Permanent magnets, Magnetic anisotropy energy, Curie temperature, In-plane anisotropy,
Maximum energy products
e applications of permanent magnets (PM) play a major role in technological advancements and moderniza-
tion of the world. In the realm of PM, the utilization of rare-earth elements has signicantly transformed the
landscape of high-performance applications. Rare-earth based permanent magnets, including those composed of
Neodymium (Nd), Samarium (Sm), and Dysprosium (Dy), have emerged as the cornerstone of numerous tech-
nological advancements over the past few decades1–6. e unique properties of rare-earth elements, particularly
their strong magnetic moments resulting from the presence of unpaired electrons in their electronic congura-
tion, make them highly desirable for PM applications. Among these, Neodymium-based magnet, Nd2Fe14B is
known for its exceptional strength and magnetic properties, oering the maximum energy product (BH)max of
60 MGOe5, which is higher than any material known. However, despite their exceptional performance, rare-
earth-based magnets face challenges related to the scarcity and geopolitical concentration of rare-earth elements,
particularly Neodymium and Dysprosium2,7–10. e reliance on a limited supply chain has led to concerns about
potential disruptions and uctuations in prices, prompting eorts to explore alternative magnet materials and
recycling strategies to mitigate these risks11,12. In addition, the maximum energy product (BH)max of Nd2Fe14B,
decreased rapidly above 373 K3 which is accompanied by low Tc of 312 K5, which hampers its eectiveness in
high-temperature PM applications. Hence, there is a pursuit for materials that are free of rare-earth elements,
cost-eective, and capable of achieving performance comparable to that of rare-earth based PM. In this context,
Manganese (Mn) has attracted considerable attention due to its high magnetic moment and cost-eectiveness13.
However, in nature, Mn is known to poses anti-ferromagnetic nature. us, alloying Mn with other elements
is essential for altering the magnetic behavior of Mn towards a ferromagnetic phase, overcoming its natural
anti-ferromagnetism. One of the Mn-based alloy which have shown a remarkable magnetic properties is the
low-temperature phase (LTP) MnBi alloy with NiAs-type hexagonal structure14–17. It has maximum energy
OPEN
Department of Physics, National Institute of Technology Nagaland, Chűmoukedima, Nagaland 797103, India.
*email: jpborah@redimail.com

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product (BH)max of 17.7 MGOe at 300 K18, along with reasonable Curie temperature (
TC
) of 650 K19 and satura-
tion magnetization of 0.8T at room temperature20. In addition, one of the interesting properties of LTP-MnBi is
that the anisotropy constant increases with temperature unlike most magnets. Below 90K, LTP-MnBi exhibits
an in-plane anisotropy with − 0.2MJ/m3 at 0K and reaches to 2MJ/m3 at room temperature21. is unique mag-
netic property of LPT-MnBi inspired the pursuit of extensive research and further exploration into this alloy for
PM applications. However, at higher temperature around 603K, the LTP-MnBi undergoes rst-order structural
phase transition due to the diusion of Mn into the interstitial sites22. As a result, doping of third element into the
interstitial site has been suggested to prevent diusion of Mn into interstitial site at higher temperature. In this
regard, we doped Ni at the interstitial site to prevent the diusion of Mn into the interstitial sites. Consequently,
the addition of Ni altered the material’s anisotropy from in-plane to out-of-plane. However, this introduction of
Ni signicantly aected its structural properties, resulting in dynamic instability. To address this, we substituted
one Mn atom with Fe in our investigation. A noteworthy nding of our study is the reorientation of the magnetic
moment in LPT-MnBi, resulting in pronounced out-of-plane anisotropy and a substantial increase in uniaxial
magnetic anisotropy energy (Ku).
Computational details
To explore the structural and magnetic properties of the compounds, we performed rst-principles study based
on DFT using the Vienna Abinitio Simulation Package (VASP)23. e generalized gradient approximation
(GGA) with the Perdew–Burke–Ernzerhof (PBE)24 exchange–correlation functional has being employed. To
ensure accuracy and reliability, the optimization of crystal structures were performed with the converging of
forces below 0.01eVÅ−1. We utilize an energy cuto of 600eV to adequately capture electronic behavior within
the system. For Brillouin zone integration, we employ the Gamma-scheme with 13 × 13 × 9k-point mesh for
structure relaxation. For determining the magnetic anisotropy energy (MAE), we consider the discrepancy in
total energy between two magnetic orientations, incorporating spin–orbit coupling (SOC) eects. Specically,
we computed MAE as the energy dierence between the a- and c-axis, denoted as MAE = Ea − Ec, where Ea and
Ec correspond to the total energies along the respective axes. To ensure the accuracy and convergence of MAE
calculations, we employ the tetrahedron method coupled with Blöchl corrections, which is an eective approach
known for its eciency in handling spin–orbit interactions. Additionally, in non-collinear computations, we
implement a denser k-point mesh of 21 × 21 × 17. is enhanced k-mesh density contributes to well-converged
MAE values, providing reliable insights into the magnetic properties of the system. Utilizing the optimized lat-
tice constants obtained from the VASP computation, we calculated the exchange interactions using the Munich
spin-polarized relativistic Korringa–Kohn–Rostoker (SPR-KKR) package25. roughout these calculations, we
adopted the full-potential mode coupled with the spin-polarized scalar-relativistic mode, ensuring a compre-
hensive analysis. e angular momentum cuto is set to lmax = 3 to facilitate the expansion of the Green function
and enhance computational eciency. Pair exchange coupling parameters up to rmax = 5.0a, where “a” is the lat-
tice constant, have been considered for the calculation of exchange interactions, ensuring the convergence of
TC
with respect to the real-space cluster radius. While the electron charge density calculation were done through
WIEN2K packages26. Further, the structural dynamical stability has been tested using PHONOPY code27 by
considering 2 × 2 × 2 supercell.
Result and discussion
Structural properties
As shown in Fig.1, the LTP-MnBi with NiAs-type hexagonal structure has a space group of P63/mmc (#194),
with two Mn atoms in 2a sites (0, 0, 0) and (0, 0, 1/2) and two Bi atoms at the 2c sites of (1/3, 2/3, 1/4) and (2/3,
1/3, 3/4) respectively. e calculated lattice parameters for the three dierent crystal structures are provided in
Table1. e lattice parameters for the pure LTP-MnBi are obtained as a = b = 4.317Å and c = 5.741Å, where “a”
overestimates and “c” underestimates the experimental value [a = b = 4.285, c = 6.113]28. However, it is in agree-
ment with the earlier reported theoretical value29. We introduced Ni at the interstitial bipyramidal 2d sites of
(2/3, 1/3, 1/4) and (1/3, 2/3, 3/4) as shown in Fig.2a, as a result we observed an increased in lattice parameters
along “a” and “b” with 5.6% while “c” decreased by 0.6% for MnBi–Ni alloy. Further, with Fe substitute on one
of the Mn atom at 2a site as shown in Fig.2b, the lattice parameter for “a” reduced to 4% while c increased by
Figure1. Unit cell of LTP-MnBi crystal structure.

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0.36% comparing to pure LTP-MnBi. Moreover, the ferromagnetic (FM) and anti-ferromagnetic (AFM) structural
phases stability has been tested for Ni-doped and Fe-substituted alloys by using the formula, ΔE = EAFM − EFM. e
positive value of ΔE represents the ferromagnetic phase while negative indicates the anti-ferromagnetic phase.
It is found that for both MnBi–Ni and Mn0.5Fe0.5Bi–Ni alloys, it gives a positive ΔE value of 0.15eV and 0.14eV
respectively, indicating the ferromagnetic phase stability of the alloys.
Electron charge density and dynamical stability analysis
To ensure doping at interstitial sites without distorting the NiAs-type hexagonal crystal structure, it is crucial
to examine the charge distribution around the targeted doping region. e (11
2
0) plane of LTP-MnBi and
Mn0.5Fe0.5Bi–Ni has been illustrated in Fig.3a and b. us we investigated the electron charge density as shown
in Fig.3c and d along the (11
2
0) plane. As can be seen from the Fig.3c, there is two interstitial sites at (2/3,
1/3, 1/4) and (1/3, 2/3, 3/4) suggesting the possibility to dope foreign atom. e electron density map illustrates
the variances in valence electronic charge density within the plane containing Bi and vacant interstitial sites.
Following the doping of Ni, as depicted in the Fig.3d, there is no overlapping of charge distribution between
the adjacent atoms, conrming the viability of substituting Ni at these specic sites. Next, we investigated the
dynamical stability of the LTP-MnBi, MnBi–Ni, and Mn0.5Fe0.5Bi–Ni alloys by analyzing the phonon frequencies
along high symmetry points using the PHONOPY code. e phonon band structure were calculated using the
force constant obtained by density functional perturbation theory (DFPT) approach. As shown in Fig.4a, we
observed that for pure LPT-MnBi there is no imaginary frequencies across the high symmetry path, indicating
that the structure is dynamically stable. However, the MnBi–Ni exhibits dynamical instability as indicated by
the presence of imaginary frequencies as shown in Fig.4b. is critical aw renders the structure unsuitable for
practical applications, highlighting the necessity to address its instability. As a result, we substituted Fe on one of
the Mn atom at 2a sites to improve the structure dynamical instability of the alloy. In the Mn0.5Fe0.5Bi–Ni alloy, the
calculated phonon band structure demonstrates dynamical stability, exhibiting no imaginary frequencies across
all high symmetry points, as illustrated in Fig.4c. Consequently, the substitution of Fe for Mn eectively resolves
the structural dynamical instability observed in the MnBi–Ni alloy, rendering it viable for practical applications.
Spin and orbital magnetic moments
e calculated spin and orbital magnetic moments are listed in Table2. In pristine LPT-MnBi, the spin magnetic
moments of Mn and Bi are recorded as 3.540
µB
and − 0.159
µB
along with an orbital moments of 0.077
µB
and
− 0.024
µB
, which align well with the reported theoretical values19. e polarization of Bi magnetic moment in
the opposite direction of Mn can be attributed to the hybridization of Bi p-orbitals with Mn d-orbitals, leading
to a reduction in the overall magnetization of the compound. In MnBi–Ni, the Ni doping inuence the spin
magnetic moment of Mn leading to an increased spin magnetic moment with 3.585
µB
while the interstitial dop-
ing element Ni contributes a relatively low magnetic moment of 0.1578 µB. In Mn0.5Fe0.5Bi-Ni, the spin magnetic
moment of Mn at (0, 0, 0) site and Ni at interstitial sites further increased to 3.648
µB
and 0.203
µB
while the
spin magnetic moment of Fe is obtained as 2.548
µB
. e Ni doping and Fe substitution has a major inuence on
Table 1. Optimized lattice parameters and volume.
System a c c/a V3(Å)
MnBi 4.317 5.741 1.32 92.68
MnBi–Ni 4.576 5.705 1.24 103.54
Mn0.5Fe0.5Bi–Ni 4.501 5.762 1.28 101.14
Figure2. Crystal structure of (a) MnBi–Ni and (b) Mn0.5Fe0.5Bi–Ni alloy.

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the spin magnetic moment of Bi, leading to increase from − 0.159
µB
in case of pure LPT-MnBi to − 0.075
µB
in
Mn0.5Fe0.5Bi–Ni. In transition metals, the orbital magnetic moment is oen suppressed due to the inuence of
the crystal eld, leading to a relatively small orbital moment. With the introduction of Fe through substitution,
it has a pronounced eect on the overall orbital magnetic moments. A substantial portion, exceeding half of the
total orbital moment, arises from the presence of Fe, contributing 0.163
µB
. However, we observed a smaller
saturation magnetization in Mn0.5Fe0.5Bi–Ni alloy compared to pure LPT-MnBi due to Fe substitution on Mn,
which contributes a lower magnetic moment than that of the Mn atom.
Figure3. Lattice plane along (11
2
0) plane of (a) LTP-MnBi (b) Mn0.5Fe0.5Bi–Ni and charge density plot of (c)
LTP-MnBi (d) Mn0.5Fe0.5Bi-Ni.
Figure4. Phonon band structure of (a) LTP-MnBi, (b) MnBi–Ni and (c) Mn0.5Fe0.5Bi–Ni alloy.
Table 2. e calculated spin ms (
µB
) and orbital magnetic moments ml (
µB
) for total and each element of the
three dierent crystal structures.
System msMn msBi msFe msNi mstotal mlMn mlBi mlFe mlNi mltotal
MnBi 3.540 − 0.159 – – 6.763 0.077 − 0.024 – – 0.105
MnBi–Ni 3.587 − 0.093 – 0.157 7.31 0.042 − 0.007 – 0.022 0.112
Mn0.5Fe0.5Bi–Ni 3.648 − 0.075 2.548 0.203 6.45 0.040 − 0.009 0.163 0.025 0.235

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Density of states
e density of states (DOS) of LPT-MnBi, MnBi–Ni and Mn0.5Fe0.5Bi–Ni has been plotted in the Fig.5a–c. In
LPT-MnBi, it has been observed that the major contribution to the DOS is mainly from the Mn 3d-orbitals. e
Mn, with an [Ar] 3d54s2 electronic conguration, has ve unpaired electrons in its 3d-orbitals. In ferromagnetic
materials like MnBi, the exchange interaction causes neighboring spins to align parallel to each other. is align-
ment minimizes the system’s energy by reducing the Coulomb repulsion among electrons with parallel spins.
Consequently, the majority spin states are lower in energy compared to the minority spin states. In the MnBi alloy,
this results in the majority spin states being predominantly occupied, as the contribution to the DOS in MnBi
mainly arises from the Mn 3d-orbitals, as illustrated in Fig.5a. Moreover, since Mn has a half-lled 3d-orbital,
the DOS contribution from Mn atom split almost equally in both occupied and unoccupied states with major-
ity state being occupied and minority state unoccupied. e DOS around the Fermi level (EF) of LPT-MnBi is
degenerate with the majority-spin state lies just below the EF, while the minority-spin state lies slightly above the
EF. e doping of Ni with electronic conguration [Ar] 3d84s2, which is almost fully lled d-orbital, we observed
that the contribution from Ni is mostly from the occupied states dominated by the minority-spin states. With Ni
substitution, there is a higher degeneracy energy states around the EF, which explained the increased in magnetic
moment of both Mn and Bi in MnBi–Ni alloy. Furthermore, in the Mn0.5Fe0.5Bi–Ni alloy, where Fe substitutes one
of the Mn sites, additional highly degenerate energy states emerge, predominantly associated with the presence
of Fe atom around the EF in the minority states. e heightened DOS at the EF in the minority states, resulting
from Fe substitution, leads to an increased number of states contributing to the anisotropy in the Mn0.5Fe0.5Bi–Ni
alloy. is eect elucidates the observed enhancement in the MAE in the Fe-modied compound, providing
further insight into its magnetic properties.
Magnetic anisotropy energy analysis
e magnetocrystalline anisotropy energy (MAE) is an inherent magnetic property that plays a primary role in
determining the coercivity of a magnetic material. It represents the energy needed to align the magnetization
away from the easy axis relative to the crystallographic axis when subjected to an applied magnetic eld. In the
case of a hexagonal material featuring a single easy axis perpendicular to the hard axis, the formulation of the
uniaxial anisotropy energy density is expressed as
E
V
= K1sin2θ + K2sin4θ + K3sin6θ + …, where, K1, K2, and K3
represent the magnetic anisotropy constants, while θ denotes the polar angle, indicating the angle between the
easy axis and the magnetization vector30. Comparatively, the higher-order constant K3 and above are typically
much smaller in magnitude than K1 and K2. Hence, when θ = π/2, Ku approaches approximately K1 + K2. us
the Ku can be formulated by considering the total energy density dierence along two distinct direction θ = π/2
and θ = 0 i.e. Ku = {E(θ = π/2) − E(θ = 0)}/V. A positive (negative) value of Ku signies uniaxial (planar) anisot-
ropy. erefore, within DFT calculations, we can determine Ku by conducting non-self-consistent computations
along two distinct directions: the a-axis and the c-axis, while considering the eects of spin–orbit coupling as,
Ku = (Ea − Ec)/V, where Ea and Ec correspond to the total energies along the a-axis and c-axis, while V is the volume
of the unit cell of the crystal structure. e calculated Ku for the three dierent crystal structures are listed in
Table3. e Ku value for LPT-MnBi is determined to be -0.82MJ/m3, exhibiting a larger magnitude compared to
the experimentally reported Ku of -0.21MJ/m3. However, it is consistent with the anticipated in-plane anisotropy
in LPT-MnBi and aligns more closely with experimental ndings than the previously reported theoretical pre-
diction of -1.97MJ/m322. Introducing Ni into the interstitial sites alters the sign of Ku from negative to positive,
signifying a transition in anisotropy from in-plane to out-of-plane in the material of MnBi–Ni as illustrated in
Figure5. e total density of states (TDOS) and atom projected density of state (PDOS) of (a) LTP-MnBi, (b)
MnBi–Ni, (c) Mn0.5Fe0.5Bi-Ni.
Table 3. e calculated saturation magnetization (µoMs), uniaxial magnetic anisotropy constant (Ku),
maximum energy product ((BH)max), and curie temperature (TC) of the dierent crystal structures.
System µoMs(T) Ku (MJ/m3) (BH)max(MGOe)
TC
MnBi 0.82 − 0.82 17.05 712
MnBi–Ni 0.82 0.88 16.95 482
Mn0.5Fe0.5Bi–Ni 0.74 3.68 13.80 523