Ab initio study of alloying of MnBi to enhance the energy product

High energy density magnets are preferred over induction magnets for many applications, including electric motors used in flying rovers, electric vehicles, and wind turbines. However, several issues related to cost and supply with state-of-the-art rare-earth-based magnets necessitate development of high-flux magnets containing low-cost earth-abundant materials. Here, by using first-principles density functional theory, we demonstrate the possibility of tuning magnetization and magnetocrystalline anisotropy of one of the candidate materials, MnBi, by alloying it with foreign elements. By using density functional theory in the high-throughput fashion, we consider the possibility of various metal and non-metal elements in the periodic table occupying empty sites of MnBi and found that MnBi-based alloys with Rh, Pd, Li, and O are stable against decomposition to constituent elements and have larger magnetization energy product compared to MnBi. Combined with other favorable properties of MnBi, such as high Curie temperature and earth abundancy of constituent elements, we envision the possibility of MnBi-based high-energy-density magnets.

MnBi, a member of Mn-A alloys, where A can be any element with 3+ oxidation state, such as Al, Ga, In, has potential to be a good magnet. [1][2][3][4] The compound has sizeable magnetocrystalline anisotropy energy ($0.163 meV uc À1 (ref. 5)), coercivity ($7 kOe at room temperature 6 ), and energy product as large as 17 MGOe, 5 which are very attractive for magnet applications. Additionally, the compound has a large Curie temperature of 628 K, 1 and both coercivity and magnetocrystalline anisotropy increase with increasing temperature making it well suited for high-temperature applications. However, even with such favorable properties, the MnBi and MnBi-based compounds suffer from relatively low saturation magnetization. 7 Various ideas, including defect engineering, 2,8-13 exchange coupling with so magnets, 7,10,14-21 and microstructural engineering, [22][23][24][25][26] have been tested to improve the energy products in these compounds. However, they have so far yielded mixed results. While most of the doping with various foreign elements and microstructure renement has shown an increase of coercivity at the expense of magnetization, 2,3,8,27 Sn doping 3 is found to increase magnetization and magnetic anisotropy, leading to an increase in energy product.
In this manuscript, we test an alternative approach of alloying MnBi with foreign elements to enhance magnetization and magnetocrystalline anisotropy, leading to enhanced energy product (BH) max . We use the high throughput density functional theory for screening suitable elements to form an alloy with MnBi. The structure of MnBi offers clues on why alloying may be possible in MnBi. The low-temperature phase of the MnBi crystallizes in hexagonal NiAs type structure (space group no. 194), as shown in Fig. 1. The lattice vectors of such a crystal are Wyckoff's positions with D 3h symmetry (1/3A 1 + 2/3A 2 + 1/4A 3 ) and (2/3A 1 + 1/3A 2 + 3/4A 3 ) while the other high symmetry Wyckoff's 2d positions: (2/3A 1 + 1/3A 2 + 1/4A 3 ) and (1/3A 1 + 2/3A 2 + 3/4A 3 ) are empty. We incorporate foreign elements to these sites and search for the elements that increase magnetization and coercivity. We consider two cases for each elemental (el) insertion. (i) Only one of two vacant sites is lled with foreign elements (half-lled case) such that effective concentration of alloy is 0.2/uc; the other half-lled conguration obtained by lling the rest vacant position is expected to behave similarly because of the similarity of lattice symmetry and the local environment surrounding the elements when placed in those positions, and (ii) both of the vacant sites are lled (full-lled case) such that effective concentration of alloy is 0.33/uc. Our strategy to nd the suitable element for alloying is rst to evaluate the formation energy of each alloy to nd the stable one; second, determine saturation magnetization of stable alloys; and nally, calculate magnetic anisotropy energy of alloys that are stable and have magnetization larger than undoped MnBi.
For the stable alloy, the formation energy, DH, must be negative, which ascertains energy gain by alloying. The formation energy of the MnBi:el is evaluated as DH ¼ E(MnBi:el) À E(MnBi) À E(el). Here E(MnBi) and E(el) are energies of MnBi and energy of element, el. Similarly, E(MnBi:el) is the energy of the conguration of the alloy. We determine the energy of the alloy by comparing the energies of two congurations: rst, ferromagnetic (FM) conguration in which magnetic moment of alloying elements aligns parallel to the magnetic moments of Mn; and second antiferromagnetic (AFM) congurations in which the magnetic moment of alloying element is aligned antiparallel to the magnetic moments of Mn. The calculated formation energy from these calculations is likely to be in the order of $0.1 eV per formula unit, corresponding to $10 kJ mol À1 . 28 Fig . 2 shows the formation energy of the alloy with the various elements in the periodic table. Out of all those calculations, the stable alloys are the ones with the negative formation energy. In the half-lled case, only F and Pd doped alloys have negative formation energies. These are marked with light color in Fig. 2. Similarly, in the full-lled case, F, Y, and Rh doped alloys also have negative formation energies, indicating the alloys to be stable with respect to decomposition to the elemental phase. Other half-lled alloys with Rh, Pt, O, and Sc have formation energies less than 0.2 eV; these dopants only partially occupy these vacant sites. We calculate site occupancy N occ /N ¼ exp(ÀDH/k B T) at 500 K, where N is the number of available sites, k B is the Boltzmann constant, and T is the temperature. We found that Rh, Pt, O, and Sc occupy 40%, 25%, 24%, and 2% of empty sites, respectively. Similarly, full-lled MnBi alloys with La, Ca, and Pt have slightly positive formation energy (<0.2 eV) and only partially occupy the empty sites: La occupies 6%, Ca occupies 9%, and Pt occupies 23% of the empty sites. When DH > 0.2 eV occupation of interstitial sites of MnBi reduces to less than 1%. While ternary phase diagrams of many of these elements with MnBi are unavailable, experimental studies including Mn-Rh-Bi ternary phase diagram shows the presence of various phases including Mn 1.05 Rh 0.02 -Bi, 29 Mn 0.8 RhBi 0.2 , 30 and Mn 5 Rh 6 Bi 18 and MnRhBi 3 (ref. 31) consistent with our prediction of high solubility of Rh in MnBi.
All the alloys with negative formation energy are found to have a larger volume compared to that of MnBi. The lattice volume increases, but modestly, according to the increase in the atomic radius. 32 The smallest volume increase is observed in F (1%), and the largest volume increase is observed in the case of Y (42%). Volume increases more in the full-lled than half-lled case; however, the increase is not linear; for example, the volume increase of full-lled MnBiRh is $19%, whereas that of half-lled MnBiRh 1/2 is $14%. This nonlinear increase in volume indicates additional hybridization between the alloying elements and MnBi while both empty sites are occupied. Despite the increase in volume, the volume formation, which is dened in a similar way as the formation energy, where n is number of doped elements, Ve l is volume of doped element, and V MnBi is the volume of MnBi, is generally negative (except for MnBi:Pt) as shown in Table 1. The negative DV F also indicates bonding of the alloying elements with Mn and Bi in MnBi.
Having found the stable alloys, we next analyze their magnetic properties. Table 1 shows changes in magnetization with respect to the 3.54m B per Mn of bulk MnBi. Here magnetization per Mn atom is determined by dividing total magnetization density that minimize total energy of system by number of Mn atoms, assuming magnetism is mostly carried by Mn atoms. Out of the alloys with negative formation energy, alloying with F, Sc, Y reduces the net magnetization, where alloying with Li, O, Rh, Pd, and Pt increases net magnetization by up to 14% compared to the undoped case. In the case of F doping, F-p orbital hybridizes strongly with Bi-p orbitals (Fig. 3c), which couples to Mn-d orbitals antiferromagnetically like that in bulk MnBi, thereby reducing the net moment. In the case of Sc and Y, their 3d-states, as shown in Fig. 3d and e, mainly lie in the conduction band; thus, they hybridize with Mn minority d-states, inducing small moments in themselves and reducing moments in Mn, and the moments in Y and Sc couple antiferromagnetically with Mn moments further reducing the overall magnetizations.
Alloying with Rh, Pd, Pt increases the net magnetization of the system. 4d orbitals of the Rh (Fig. 3f) and Pd (Fig. 3g) and 5d orbitals of the Pt (Fig. 3h) lies mainly in the valence band and hybridize strongly with the majority spin-channel of Mn, which induces small magnetic moments on themselves and enhances magnetic moments on Mn. These moments on Rh, Pd, and Pt, however, couple ferromagnetically with Mn moments, which is opposite to interaction between Mn at the regular sites and interstitial sites, which couples antiferromagnetically, 33,34 or interaction between Sc, Y in interstitial sites and Mn at regular sites, as we discussed previously. The difference comes from the large exchange splitting of Mn-d bands leading to almost no interaction between electrons with different spins located at the different sites. However, in Rh, Pd, and Pt, even though intra-atomic exchange interaction leads to d-orbitals splitting according to spin like that in Mn, the splitting is much smaller. As a result, bands corresponding to both spin channels are occupied though there is a slight shi between them, which results in a small magnetic moment. Additionally, they have a signicant overlap with Mn dorbitals occupied by the majority spin channel, resulting in ferromagnetic direct exchange.
Alloying group II elements like Li and O also increases overall magnetization. The s-states of Li and p-states of O lie on the opposite side of the Fermi level: The O-p states lie primarily in the valence band as shown in Fig. 3j, while Li-s states lie in the conduction band as shown in Fig. 3i. As a result, the hybridization between Li-s states and Mn-d states, and O-p and Mn-d states is spin-dependent; the oxygen states hybridize mostly with the Mn-d majority states, while Li-s states hybridize mostly with Mn-d minority states leading to the enhancement of Mn moments and development of small moments in Li and O themselves. The magnetic moments of Li and O also couple ferromagnetically because of direct exchange with Mn moments and help increasing overall magnetization. We verify this hypothesis by explicitly comparing the energy of the system with the moments of Mn and moments of Li and O coupling ferromagnetically and antiferromagnetically and found that the ferromagnetically coupled system has lower energy. Using the energy difference DE ¼ E FM À E AFM we estimate the Curie temperature in the mean-eld approximations by using T c ¼ DE/3. 34 For the bulk MnBi, we calculate DE ¼ 150 meV per Mn and T c z 580 K, which is reasonably close to extrapolated experimental value of 775 K; 35 the structural transition at 628 K hinder measuring real T c of MnBi. Using the same approach, we evaluate the T c of stable MnBi alloys with enhanced magnetizations. The second to last column of the Table 1 shows the calculated results, which shows that T c reduces in general upon doping. This is consistent with the fact that upon doping antiferromagnetic magnetic exchange J 1 between nearest neighbor Mn-atoms in c-directions 34 increases because of enhancement of volume, reduction of c/a ratio and increase in orbital overlap. The other exchange parameters related with second (J 2 ), third (J 3 ), fourth (J 4 ), h (J 5 ) and sixth (J 6 ) nearest neighbor Mn atoms are ferromagnetic and smaller than J 1 . 5 With the increase of J 1 the net exchange J ¼ 2J 1 + 6J 2 + 12J 3 + 2J 4 + 2J 5 + 2J 6 can be expected to decrease and along with it the T c , as while the other J s are also expected to be affected upon alloying, their large coefficients will remain same so that J remains negative and alloy still is ferromagnetic. Next, we calculate change in magnetic anisotropy energy MAE, DMAE ¼ MAE(MnBi:el/el 1/2 ) À MAE(MnBi), in stable Table 1 Change in saturation magnetization (DM), anisotropy (DAE), volume formation (DV F ), c/a, energy difference (DE ¼ E FM À E AFM ) between antiferromagnetic (AFM) and ferromagnetic (FM) configurations, estimated Curie temperature (T c ) and change in energy product, D(BH) max (%) as a function of alloying elements (el) in half-filled MnBi:el 1/2 and full-filled MnBi:el alloys  , and projected to various alloying elements (el) (c) to (j) resolved according to spin: majority in positive axis and minority in negative axis and orbitals: color red, blue, and green corresponds to d, p and s orbitals respectively. In the cases (F, Rh and Pd), where both half-filled and full-filled alloys are stable, the DOS of element in full-filled alloy is shown with dotted line in the background of solid line that corresponds to DOS of the element in half-filled case. el Â x, where x is 5 or 10 in (c-f) and (g-j) corresponds to the enlargement factor for DOS for better visualization.
where the remnant eld B r ¼ 4pM r ¼ 4pM s . 37 The formula allows us to estimate the energy-product using only the intrinsic properties such as actual magnetization (emu cm À3 ) and homogeneous sample without using coercivity that relies on the extrinsic factors such as shape and size of the sample. The last column of Table 1 shows the calculated percentage change in energy product to calculated energy product of 20 MGOe, which is similar to the value of 17 MGOe 38,39 measured experimentally. The energy product enhances when MnBi is alloyed with Pd, Pt, Rh, Li, and O per the change in magnetizations.
In summary, we used the rst-principles density functional theory to nd an alloy with the increased magnetization and anisotropy. We predict MnBi alloy with Pd, Pt, Rh, Li, and O are stable against decomposition to constituent elements and have larger magnetization and anisotropy compared to MnBi and have a high energy product. Magnetic easy axis of MnBi alloy with Pd 1/2 and Rh remains the same as bulk MnBi and lies outof-plane direction, while the magnetic-easy-axis of MnBi alloy with Li, O, Pd and Pt lie in-plane. Pd can rotate the magneticeasy-axis 40 from in-plane to out-of-plane depending upon the percentage of Pd that would be incorporated in MnBi. We anticipate this comprehensive study of MnBi alloy spurs more theoretical and experimental study. We note that the alloying of these elements with MnBi may require non-equilibrium growth methods as their formation energies are relatively small.

Computational methods
We employ the projected augmented wave (PAW) 41 method for the electron-ion potential and the gradient density approximation (GGA) for exchange-correlation potential, as implemented in the Vienna ab initio simulation package (VASP) 42,43 with the recommended set of pseudopotentials. 44 The calculations are carried out using the kinetic energy cutoff of 340 eV and 6 Â 6 Â 1 k-point mesh for Brillouin zone integration of pseudocubic unit cells. The k-points are scaled according to size. We fully relax ionic coordinates with the force convergence limit of 0.001 eV per atom. For the anisotropy calculations, we include an additional onsite Coulomb interaction Hubbard (U-J) parameter of 3 eV to Mn-3d states so that the sign of magnetocrystalline of bulk MnBi is consistent with the experiments and includes spin-orbit coupling terms explicitly.

Data availability
The data that support the ndings of this are available from the corresponding author upon reasonable request.

Conflicts of interest
There are no conicts to declare.