Rare-earth free, ferromagnetic MnBi shows a positive temperature coefficient of coercivity from room temperature to 400 K and energy product (*BH*)_{max} of 17.7 MGOe at 300 K. However, MnBi undergoes a first-order structural phase transformation from a ferromagnetic low-temperature phase (LTP) to a paramagnetic high-temperature phase at 613 K below the Curie temperature (*T*_{c}) of 716 K. The transformation is attributed to Mn diffusion into the interstitial site of LTP MnBi unit cell. Interstitial Mn antiferromagnetically couples with the Mn at lattice 2*a* site, lowering the magnetization. Cu-occupied bipyramidal sites are investigated as a possible means to suppress Mn diffusion into the bipyramidal sites using first-principles calculations based on the density functional theory. Saturation magnetization, magnetocrystalline anisotropy constant (*K*), and *T*_{c} of (Mn_{0.5}Bi_{0.5})_{100−x}Cu_{x} (x = 0–33) are reported. The magnetocrystalline anisotropy changes to the out-of-plane direction (x = 13) from the in-plane direction (x = 0.0). *T*_{c} decreases gradually to 578 K at x = 33 from 716 K at x = 0.0. The calculations show a slightly lower (*BH*)_{max} of 15.6 MGOe while it is expected that Cu-occupied interstitial sites will significantly suppress Mn diffusion and raise the temperature of the phase transformation.

## I. INTRODUCTION

The rare-earth (RE) permanent magnet Nd_{2}Fe_{14}B has the highest maximum energy product (*BH*)_{max} among commercial permanent magnets because it has both high magnetization and high magnetic anisotropy. The high concentration of Fe is the origin of the high magnetization. The high magnetic anisotropy originates from the crystal structure that is related to the CaCu_{5}-type hexagonal crystallographic phase^{1} and strong coupling of the spins to the lattice that is mediated by the high spin–orbit coupling of Nd. The theoretical upper limit of (*BH*)_{max} for Nd–Fe–B is 512 kJ/m^{3} (64.8 MGOe),^{2} and commercial products show about 440 kJ/m^{3} (55.5 MGOe). Practical permanent magnets must possess high saturation magnetization (*M*_{s}) and magnetocrystalline anisotropy (*K*_{u}) with a corresponding high coercivity (*H*_{c}) and Curie temperature (*T _{c}*), and Nd–Fe–B is the best available choice for a wide range of applications.

However, RE material availability may present problems for more widespread use of RE permanent magnets and provides motivation for the discovery of RE-free permanent magnet materials. From an application perspective, Nd–Fe–B does suffer from a relatively low Curie temperature of less than 600 K, so the energy product at moderate operating temperatures is significantly reduced. As an alternative, the RE-free permanent magnet MnBi has a Curie temperature of 650 K and an unusual increase in magnetic coercivity with temperature that offsets the reduction in magnetization with temperature, suggesting that MnBi outperforms Nd–Fe–B at elevated temperatures.

Mn-based magnets offer a higher magnetic moment per Mn atom. According to Hund's rule, body-centered cubic Mn in Fig. 1(a) possesses the highest magnetic moment per atom 3.60 *μ*_{B}/atom among all the 3d transition metals. However, Mn metal is antiferromagnetic since the exchange integral changes sign at short distances. When Mn is doped with B in Fig. 1(b), the distance between Mn atoms increases; therefore, the magnetic phase transition from antiferromagnetic to ferromagnetic appears.^{3} As a result, the magnetization of MnB increases to 851 emu/cm^{3} (1.08 T) at 300 K and the Curie temperature is about 578 K. However, MnB is magnetically soft with a low coercivity of ∼16 Oe.^{4}

Replacing the low atomic number B with high atomic number Bi results in a dramatic increase of the magnetocrystalline anisotropy constant (*K*_{u}) due to the strong spin–orbit coupling of Bi and the formation of a NiAs-type hexagonal crystal structure in Fig. 1(c). This low-temperature phase (LTP) of MnBi exhibits a theoretical upper limit of (*BH*)_{max} of 141 kJ/m^{3} (17.7 MGOe) at 300 K. A remarkable feature of MnBi is that it exhibits a positive temperature coefficient of coercivity (*β *= Δ*H*_{c}/ΔT) up to 400 K^{5} because of the magnetocrystalline anisotropy constant (*K*_{u}) increases with temperature and (*BH*)_{max} is moderately reduced to 13 MGOe at 400 K.^{6} This value is in good agreement with the experimental result.^{7}

Recently, (*BH*)_{max} of 13 MGOe was experimentally achieved at 300 K.^{8} Cui *et al*. have also reported that the theoretical (*BH*)_{max} of MnBi is estimated to be 20 MGOe at 300 K and 17 MGOe at 400 K using the formula (*BH*)_{max }= ¼*M*_{s}^{2}, where *M*_{s} is the saturation magnetization and 9.1 kG at 300 K and 8.4 kG at 400 K.^{9} These two (*BH*)_{max} are higher than the results in the study by Park *et al.*^{6} In 2015, 16.3 MGOe of Mn_{50}Bi_{50} was reported,^{10} which is very close to the theoretical upper limit of 17.7 MGOe in Ref. 6.

The spin orientation of LTP MnBi can be controlled by partially replacing Bi of MnBi with the third element or temperature. Sakuma *et al*. have performed first-principles calculations on Sn-doped Mn(Bi_{1−x}Sn_{x}) and found that *K*_{u} hugely increases to about 3 MJ/m^{3} at x = 0.1 from −0.5 MJ/m^{3} at x = 0.0 and then remains unchanged up to x = 0.3.^{11} The magnetic spin direction changes from the in-plane (*K*_{u} < 0) to the out-of-plane (*K*_{u }> 0) direction at x = 0.02. Bi at 2*c* sites in MnBi is partially replaced by Sn to push the (*BH*)_{max} of LTP MnBi to its theoretical upper limit. As a result, 114 kJ/m^{3} (14 MGOe) and 86 kJ/m^{3} (10.8 MGOe) were experimentally achieved at 300 and 473 K (200 °C), respectively.^{12} The magnetocrystalline anisotropy constant of MnBi_{1−x} Sb_{x} increases as x increases, but *K*_{u}'s sign does not change at x = 0.125. Thus, no spin reorientation occurs until x = 0.125.^{13} Furthermore, the easy axis of magnetization of LTP MnBi magnetocrystalline anisotropy changes from in the hexagonal plane and perpendicular to the *c* axis (*K*_{u }< 0) to out of the hexagonal plane along the *c* axis (*K*_{u }> 0) at about 90 K.^{14}

From the application of point of view, for example, a permanent magnet should be sustainable at elevated temperatures for applications of permanent magnet motors for electric vehicles. To evaluate the thermal stability of permanent magnets above 473 K (200 °C), Park *et al*. have theoretically predicted (*BH*)_{max} of LTP MnBi as a function of temperature compared to Nd–Fe–B. The (*BH*)_{max} of Nd–Fe–B is smaller than that of the LTP MnBi above 450 K.^{6} This is ascribed to a rapid reduction of magnetization of Nd–Fe–B associated with its lower Curie temperature of 585 K. LTP MnBi has a much higher Curie temperature of 720 K.^{15} Nd–Fe–B also has a large negative temperature coefficient of coercivity while the temperature coefficient of coercivity for LTP Mn Bi is still increasing at 450 K. In addition, frequent temperature excursions approaching the Curie point will result in demagnetization and poor performance. Thus, Nd–Fe–B is not suitable for high-temperature applications. Recent attempts at improving Nd–Fe–B involve addition of Dy to improve thermal stability; however, this is not practical due to the rarity and associated high cost of Dy.

As described above, a ferromagnetic MnBi crystallizes with a hexagonal NiAs-type structure at 300 K. When the MnBi is heated above 603 K, MnBi undergoes a first-order structural phase transition at 603 K. This transition occurs due to the diffusion of Mn into interstitial sites of the NiAs structure^{16} and Mn at 2*a* site couples antiferromagnetically with Mn in the interstitial site.^{17} Therefore, the ferromagnetism of LTP MnBi disappears well below the Curie temperature. At higher temperatures, MnBi decomposes to Mn and liquid Bi at 718 K.^{18} During cooling, the ferromagnetic state is restored below 613 K.^{19}

To enhance the structural stability, interstitial doping of Co^{20} and Co/Fe^{21} was studied, and it was found that doping with Co and Fe significantly degrades the Curie temperature but increases the magnetization and magnetocrystalline anisotropy. For comparison, interstitial doping of Al in MnBi lowers the spin–orbit coupling interaction due to the hybridization of Al and Bi states and increases Mn's moment.^{22} Suppressing Mn diffusion into the interstitial sites may shift the phase transformation a temperature near or higher than the Curie temperature.

In this paper, Cu atoms are intentionally inserted into the bipyramidal (interstitial) sites of MnBi unit cells to suppress Mn's diffusion into the interstitial sites. Thus, the phase transition temperature of LTP MnBi is expected to be higher than 613 K. Cu-occupied interstitial sites serve to weaken antiferromagnetic coupling between Mn at 2*a* site and Mn in the interstitial site. We perform first-principles calculations on (Mn_{0.5}Bi_{0.5})_{100−x}Cu_{x} to calculate electronic structure and estimate the temperature dependence of saturation magnetization and composition dependence of magnetocrystalline anisotropy constant. We also report magnetic spin reorientation, which is tuned by interstitial Cu.

## II. METHOD OF CALUCLATIONS AND CRYSTAL STRUCTURE

### A. Method of calculations

After having relaxed (Mn_{0.5}Bi_{0.5})_{100−x}Cu_{x} (x = 0–33), the Kohn–Sham equation, $H^KS\Psi (r)=\epsilon i\Psi (r),$ is solved with the relaxed lattice constants. The Hamiltonian $(H^KS)$ for the Kohn–Sham equation consists of kinetic energy, nuclei energy (lattice potential), and a sum of the electron exchange and correlation energies (interaction),

The corresponding wave function is *Ψ*(r) = ∑C_{kn}φ_{kn}, where the wave (φ_{kn}) consists of a partial atomic wave in an atomic sphere (*as*) and a plane wave of the interstitial region (*ir*) of MnBi unit cell. The total electron density is as follows:

The partial atomic wave is $\varphi kn=[AlmKul(r,\epsilon )+BlmKu\u02d9l\u2061(r,\epsilon )]Ylm(r),$ and the plane wave is $\varphi kn=ei(k+Kn)r$. $AlmK$ and $BlmK$ are the coefficients for matching the plane wave.$ul$ is the numerical solution of the radial Schrödinger equation in a given spherical potential. $u\u02d9l$ is the energy derivative of $u\u02d9lYlmisthespherical$ harmonic of angular momentum *l* and quantum number *m*. After summing *ρ*_{as} and *ρ*_{ir}, we obtain the total electron density and construct the electron density map.

The WIEN2k package, based on the density functional theory (DFT) within the local-spin-density approximation (LSDA) and using the full-potential linearized augmented plane wave (FPLAPW) method, is used to conduct the first-principles calculations.^{23} All calculations use 19 × 19 × 27 k-point mesh, generating 1400 k-points in the irreducible part of the Brillouin zone.

Regarding magnetocrystalline anisotropy calculation, after adding spin–orbit coupling Hamiltonian $(Hso),HSO=\u210f4me2c2\sigma [\u2207V(r)\xd7p]=14me2c21rdVdr(l\u22c5s)$, where *c* is the speed of light, *V* is the potential energy of the electron, *σ* is the spin, *l* is the angular momentum, and *s* is the spin angular momentum, to the Hamiltonian (*H*_{ks}), the Kohn–Shame equation is solved for the magnetocrystalline anisotropy energy (MAE) $(\Delta EMAE)$,

where $n^1andn^2are$easy and hard spin directions, respectively. Magnetocrystalline anisotropy energy (MAE) was calculated using the total energy difference between ⟨001⟩ and ⟨100⟩ spin configurations (Δ*E* = *E*_{⟨001⟩}–*E*_{⟨100⟩}).

To estimate the temperature dependence of saturation magnetization, *M*_{s}(T), one needs the Curie temperature (*T*_{c}). For the *T*_{c} calculation, the exchange integrals (*J*_{0j}) are calculated by the energy difference between the ground and excited states. The exchange integral can be expressed by$Jij=(\Delta ij\u2212\Delta i\u2212\Delta j)/(4SiSjnizij)$, where *S _{i}* is the quantum spin of the

*i*th Mn atom, Δ

*is the exchange energy difference between the ground and excited states when*

_{i}*i*th Mn atom is reversed,

*n*is the number of the

_{i}*i*th atom, and

*z*is the number of neighboring the

_{ij}*j*th atom to the

*i*th atom.

^{4}The exchange integrals (

*J*) consider interactions over all neighboring spins, then $J0=\u2211jJ0j$.

_{0j}^{5}The

*T*

_{c}is then calculated with

*J*

_{0}using the following mean-field-approximation (MFA),

^{24}

where *J*_{0} is the molecular field parameter calculated by the summation of the exchange integrals *J*_{0j}, and *k*_{B} is the Boltzmann constant (1.38 × 10^{−23} J/K). The factor *γ* equals *S*(*S* + 1)/*S*^{2}, where *S* is the spin angular momentum.

Now, the temperature dependence of saturation magnetization *M*_{s} (T) is described by Eq. (2), i.e., the Brillouin function [*B*(*J*, *a*′)].^{28} Thus, one incorporates both *M*_{s} (0) and *T*_{c} into Eq. (2) to calculate *M*_{s} (T),^{25}

$wherea\u2032=M/M0T/Tc(3JJ+1)$ and *J* is the total angular momentum quantum number.

### B. Crystal structure

Figure 2 shows the hexagonal crystal structures of LTP MnBi and Cu-doped MnBi used in this study. Cu atoms occupy the bipyramidal sites of (2/3, 1/3, 1/4) and (1/3, 2/3, 3/4). The LTP MnBi unit cell has two Mn atoms at the octahedral 2*a* sites of (0, 0, 0) and (0, 0, 1/2) and two Bi atoms at the 2*c* sites of (1/3, 2/3, 1/4) and (2/3, 1/3, 3/4).^{26} Lattice constants are obtained after having relaxed (Mn_{0.5}Bi_{0.5})_{100x}Cu_{x} (x = 0, 11, 20, 33) unit cell and summarized in Table I. Both *c*/*a* ratio and volume decrease at x = 11 and increase with increasing Cu concentration from x = 11 to 33. However, the *c*/*a* for x > 0 is smaller than that at x = 0. The relaxed lattice constants for x = 0 are in good agreement with the results in Refs. 18 and 27–32.

x . | Lattice constant (Å) . | Volume (Å ^{3})
. | c/a
. | |
---|---|---|---|---|

a . | c . | |||

0 | 4.287 | 6.118 | 97.375 | 1.427 |

11 | 4.460 | 5.308 | 91.439 | 1.190 |

20 | 4.550 | 5.505 | 98.699 | 1.210 |

33 | 4.606 | 5.671 | 104.193 | 1.231 |

x . | Lattice constant (Å) . | Volume (Å ^{3})
. | c/a
. | |
---|---|---|---|---|

a . | c . | |||

0 | 4.287 | 6.118 | 97.375 | 1.427 |

11 | 4.460 | 5.308 | 91.439 | 1.190 |

20 | 4.550 | 5.505 | 98.699 | 1.210 |

33 | 4.606 | 5.671 | 104.193 | 1.231 |

## III. MAGNETIC PROPERTIES

### A. Electron density map

Figure 3 shows (a) $(112\xaf0)$ plane and (b) electron density map of LTP MnBi. The total valence electronic charge density differences in the plane containing Bi and vacant interstitial sites are shown. The red color represents the least electron charge density. MnBi exhibits low electron charge density at two interstitial sites of the unit cell; therefore, large trigonal-bipyramidal interstitial sites are available for the dopant or Mn atom. This is in good agreement with the results of Antropov *et al.*^{33} The electron density map confirms that interstitial sites are available for Cu dopants. Thus, our assumption for Cu-occupied site is valid.

### B. Saturation magnetization

Figure 4 shows the density of states for (Mn_{0.5}Bi_{0.5})_{100−x}Cu_{x} (x = 0, 11, 20, 33). Our fully relativistic LSDA density of states calculation (x = 0) produces a spin-polarized magnetic moment of 3.504 *μ _{B}*/Mn at the 2

*a*site and −0.104

*μ*

_{B}/Bi at the 2

*c*site in the MnBi unit cell, that is antiparallel to the Mn at 2

*a*site. The orbital magnetic moment is 0.122

*μ*

_{B}/Mn and −0.022

*μ*

_{B}/Bi. An additional empty sphere also carries a small spin-polarized moment of 0.112

*μ*and an orbital magnetic moment of 0.001

_{B}*μ*

_{B}. The net magnetic moment per formula unit is 3.613

*μ*

_{B}. This is in good agreement with the results in Refs. 6 and 27.

We are interested in interstitial doping of the third element in MnBi to suppress Mn's diffusion into interstitial sites of hexagonal MnBi. It is reported that after doping Al atoms into interstitial sites of MnBi remains, the NiAs hexagonal type structure and crystal parameters are unchanged. Therefore, the Al atoms may preferentially enter into the interstitial sites.^{22,34} Jaswal *et al*. reported that interstitial doping (33.3 at. %) of Al in MnBi slightly increases the Mn moment, but the magnetic moment on Bi decreases. This is due to the hybridization between the *s*-*p* states of Bi and Al.^{22}

When Cu is interstitially doped, the Fermi energy level shifts toward the conduction band. Beyond 11 at. %, the magnetic moment gradually decreases to 3.54 *μ _{B}* as the Cu concentration increases to 33 at. %. The calculated magnetic moment and saturation magnetization are summarized in Table II. The corresponding saturation magnetization in the unit of T monotonically decreases to 0.792 from 0.865 T at 0 K as the Cu concentration increases to 33 at. %. It is noted that magnetization is not much degraded by interstitial doping of Cu.

x . | Formula unit (f.u.)
. | Magnetic moment ( μ/_{B}f.u.)
. | M
. _{s} | K_{u}(MJ/m ^{3})
. | |
---|---|---|---|---|---|

emu/cm^{3} (emu/g)
. | (T) . | ||||

0 | Mn_{1}Bi_{1} | 3.612 | 688 (76.4) | 0.865 | −0.202 |

11 | Mn_{4}Bi_{4}Cu_{1} | 13.219 | 670 (65.9) | 0.842 | −0.047 |

20 | Mn_{2}Bi_{2}Cu_{1} | 6.910 | 650 (65.2) | 0.817 | 0.075 |

33 | Mn_{1}Bi_{1}Cu_{1} | 3.540 | 630 (60.4) | 0.792 | 0.224 |

x . | Formula unit (f.u.)
. | Magnetic moment ( μ/_{B}f.u.)
. | M
. _{s} | K_{u}(MJ/m ^{3})
. | |
---|---|---|---|---|---|

emu/cm^{3} (emu/g)
. | (T) . | ||||

0 | Mn_{1}Bi_{1} | 3.612 | 688 (76.4) | 0.865 | −0.202 |

11 | Mn_{4}Bi_{4}Cu_{1} | 13.219 | 670 (65.9) | 0.842 | −0.047 |

20 | Mn_{2}Bi_{2}Cu_{1} | 6.910 | 650 (65.2) | 0.817 | 0.075 |

33 | Mn_{1}Bi_{1}Cu_{1} | 3.540 | 630 (60.4) | 0.792 | 0.224 |

### C. Magnetocrystalline anisotropy energy and Curie temperature

The calculated magnetocrystalline anisotropy constant (*K _{u}*) is −0.202 MJ/m

^{3}for undoped MnBi, which is in good agreement with the measured value −0.2 MJ/m

^{3}, and the easy magnetic plane (the in-plane direction) is consistent with the experimental results at low temperature.

^{11,35}The

*K*increases to −0.047 at x = 11 of (Mn

_{0.5}Bi

_{0.5})

_{100−x}Cu

_{x}and changes its sign to positive 0.075 at x = 20, indicating the change of magnetocrystalline anisotropy to the out-of-plane direction (i.e.,

*c*axis). Therefore, the direction of magnetocrystalline anisotropy can be controlled by chemical composition. It was reported that the change of lattice parameters is not sufficient to change the sign of MAE of the MnBi system,

^{36}but the temperature changes the sign at 90 K from negative to positive.

^{14}Therefore, the direction of magnetocrystalline anisotropy for MnBi can be tuned by either chemical composition or sample temperature. The calculated

*K*with Cu concentration is summarized in Table II. According to the calculated

_{u}*K*, we have built the magnetic spin configuration for MnBi and MnBi–Cu (x = 33) in Fig. 5. Figure 6 shows spin reorientation at about x = 15 from the in-plane to the out-of-plane direction, i.e., the

_{u}*c*axis.

The Curie temperature (*T*_{c}) for (Mn_{0.5}Bi_{0.5})_{100−x}Cu_{x} was calculated using Eq. (1) to use it to estimate the magnetization at the elevated temperature. The supercell structure was built using more than two NiAs-type hexagonal MnBi unit cells to include the significant exchange interactions over all the neighboring spins. The exchange interaction between spins is inversely or exponentially proportional to the square of their distance. Table III summarizes *z*_{ij} and *r*_{ij} along with the calculated *J*_{0} and *T*_{c}. The calculated *J*_{0} decreases from 69.523 to 56. 678 meV as the concentration increases to 33 at. % Cu. *J*_{0} for undoped MnBi is in good agreement with the results in Ref. 11 As a result, the Cu-doped MnBi shows a reduction in calculated *T*_{c}. The calculated molecular field parameters (*J*_{0}) of Mn_{50−x}Bi_{50−x}Cu_{2x} were 69.523, 65.382, 58.853, and 56.014 meV and their corresponding *T*_{c} is 716, 674, 607, and 578 K, respectively. The decrease in *T*_{c} with respect to the Cu concentration is attributed to the decrease in *M*_{s} and exchange integral. These calculated Curie temperatures are used to calculate the temperature dependence of saturation magnetization in Sec. III D.

x . | No. of nearest neighbors . | Distance (Å) . | Exchange integral, J_{0} (meV)
. | Curie temperature, T (K)
. _{c} | ||||
---|---|---|---|---|---|---|---|---|

z_{01}
. | z_{02}
. | z_{03}
. | r_{01}
. | r_{02}
. | r_{03}
. | |||

0 | 2 | 6 | 2 | 3.056 | 4.287 | 6.118 | 69.523 | 716 |

11 | 2 | 6 | 2 | 2.654 | 4.460 | 5.308 | 65.382 | 674 |

20 | 2 | 6 | 2 | 2.753 | 4.550 | 5.505 | 58.853 | 607 |

33 | 2 | 6 | 2 | 2.836 | 4.606 | 5.671 | 50.678 | 578 |

x . | No. of nearest neighbors . | Distance (Å) . | Exchange integral, J_{0} (meV)
. | Curie temperature, T (K)
. _{c} | ||||
---|---|---|---|---|---|---|---|---|

z_{01}
. | z_{02}
. | z_{03}
. | r_{01}
. | r_{02}
. | r_{03}
. | |||

0 | 2 | 6 | 2 | 3.056 | 4.287 | 6.118 | 69.523 | 716 |

11 | 2 | 6 | 2 | 2.654 | 4.460 | 5.308 | 65.382 | 674 |

20 | 2 | 6 | 2 | 2.753 | 4.550 | 5.505 | 58.853 | 607 |

33 | 2 | 6 | 2 | 2.836 | 4.606 | 5.671 | 50.678 | 578 |

### D. temperature dependence of saturation magnetization

Since the operating temperature of most applications for the hard magnet is above room temperature, the temperature-dependent *M*_{s}(T) was estimated using Eq. (2) with *M*_{s}(0) and *T*_{c}. In this calculation, *J *= 2 was used. Figure 7 shows *M _{s}*(T) and

*M*(x), where x is the concentration (at. %) of Cu.

*M*(0.0) is in good agreement with the experimental results in Ref. 37 as shown in Fig. 7(a).

*M*of Mn

_{s}_{50−x}Bi

_{50−x}Cu

_{2x}decreases to 0.764 from 0.814 T as x increases from 0.0 to x = 11 at 300 K. Figure 7(b) shows that the

*M*gradually decreases as the Cu concentration increases at both 0 and 300 K.

_{s}*M*(11.0) are 0.842 and 0.764 T at 0 and 300 K, respectively. The gap between 0 and 300 K gets wider as x increases. This is attributed to a decrease in

_{s}*T*

_{c}with x. As seen in Fig. 6, the magnetocrystalline anisotropy constant (

*K*) changes to the out-of-plane (

_{u}*c*axis) direction from the in-plane direction at about 11 at. % of Cu, where the magnetization is 0.842 T. The Curie temperature is 674 K. This Curie temperature is much higher than the phase transition temperature of 613 K. Therefore, about 11 at. % of Cu is appropriate to maintain reasonable magnetic properties while suppressing diffusion of Cu into interstitial sites of MnBi unit cell. The interstitial doping of Cu in MnBi gives advantages over undoped MnBi and Nd–Fe–B in high-temperature applications. Experimental verification of phase transition temperature of Cu-doped MnBi is needed. Successful fabrication of Cu-doped MnBi at interstitial sites will shed light on high-temperature application permanent magnet without rare-earth. It is foreseeable to observe a higher phase transformation temperature than 613 K of undoped MnBi.

## IV. CONCLUSION

Interstitially doped Cu in MnBi unit cell exhibits decreased magnetization and Curie temperature. The in-plane anisotropy to the out-of-plane anisotropy, i.e., the *c* axis, occurs at about 13 at. % Cu. Furthermore, the out-of-plane magnetocrystalline anisotropy constant increased to 0.224 MJ/m^{3} at 33 at. % Cu. Magnetization of (Mn_{0.5}Bi_{0.5})_{100−x}Cu_{x} (x = 13) is 0.75 T at 300 K and the Curie temperature is about 650 K. These calculated magnetic parameters lead to 14 MGOe. Even though magnetic properties are sacrificed a bit, the interstitial doping is beneficial in high-temperature applications for suppressing Mn diffusion, therefore weakening antiferromagnetic coupling between Mn at 2*a* site and Mn at the interstitial site. Phase transformation temperature is expected to be higher than the temperature of undoped MnBi.

## ACKNOWLEDGMENTS

This work was supported in part by the National Science Foundation (NSF) CMMI and IUCRC under Grant Nos. 1463301, 1463078, and 1650564.

## DATA AVAILABILITY

Computational simulation code from this study is available from Yang-Ki Hong (email: ykhong@eng.ua.edu) upon reasonable request.