Theoretical design of tetragonal rare-earth-free alloys with high magnetisation and high magnetic anisotropy

Tetragonal alloys, such as D022-Mn3Ga, are potential candidates for rare-earth free permanent magnets due to their high Curie temperature and uniaxial magnetic anisotropy. For high-performance permanent magnets, high saturation magnetisation is necessary. However, the saturation magnetisation of D022-Mn3Ga is small due to ferrimagnetic ordering. We investigated the possibility of developing ferromagnetic Heusler alloys with high magnetic anisotropy and saturation magnetisation using the first-principles calculation. We focused on the effects of Fe substitution for Mn in D022-Mn3Ga as well as the consequent volume expansion; the ferromagnetic tetragonal XA phase is stabilized in Fe2MnGa by an 8% volume expansion. This tetragonal XA-Fe2MnGa has desirable properties for a high-performance permanent magnet, such as high magnetisation (1350 emu cc−1), perpendicular magnetic anisotropy (2.12 MJ m−3), and Curie temperature (1047 K). In addition, the substitution of Sn and increasing the Ga composition in the Fe2MnGa alloy results in volume expansion, which stabilizes the ferromagnetic tetragonal XA phase.


Introduction
The high-performance permanent magnet is widely used in various industrial applications, including in motors for electric vehicles, power generators in wind turbines, and hard disk drives. The development of a new rare-earth free permanent magnet is important due to the cost and limited availability of rare-earth elements. Mn-Ga alloys such as L10-MnGa and DO 22 -Mn 3 Ga have been attracting much attention as candidates for spintronic materials and rare-earth-free permanent magnets because of their high uniaxial magnetic anisotropy and Curie temperature exceeding 700 K. [1][2][3] Moreover, a high saturation magnetisation is required for the application of permanent magnets. However, the saturation magnetisation of D0 22 -Mn 3 Ga is much smaller than that of L1 0 -MnGa (845 emu cc −1 ) 4) owing to the ferrimagnetic ordering of the magnetic moments at the two Mn sublattices, 2.8 μ B and 1.6 μ B per Mn atom at the 4d and 2b sites, respectively. 1) If the magnitude of the magnetic moment at each Mn site is assumed to be the same even in the ferromagnetic state, a magnetisation of about 1.7 T can be expected for ferromagnetic Mn 3 Ga.
Furthermore, Mn-Fe-Ga alloys have interesting properties, which can enable their application in spintronics and magnetic shape memory alloys. Mn 2 FeGa, which is a tetragonal Heusler alloy, has also been identified as a suitable candidate for use in spin-transfer torque devices, as the magnetisation is nearly compensated, and the high perpendicular magnetic anisotropy is maintained. 5) The Fe 2 MnGa alloy has a complex phase diagram for temperature and composition. Martensitic transformation [6][7][8][9][10] and anti-ferromagnetic to ferromagnetic phase transformation [11][12][13][14] have been reported for stoichiometric and off-stoichiometric Fe 2 MnGa.
The exchange coupling between the Mn atoms in most alloys and compounds empirically shows a universal dependence on the Mn-Mn distance. 15) In this study, we theoretically investigated the possibility of realizing ferromagnetic ordering by substituting Fe for Mn in Mn 3 Ga and volume expansion.

Material and methods
We investigated the relationship between the ferrimagnetic and ferromagnetic formation energies and volumes for D0 22 -Mn 3 Ga, Mn 2 FeGa, and Fe 2 MnGa, by first-principles density-functional calculations using the Vienna ab-initio simulation package based on the plane-wave basis set and projector augmented wave method. 16) We adopted a generalised density gradient approximation (GGA) parameterised by Perdew, Burke and Ernzerhof (PBE) for the exchangecorrelation potential. 17) The cut-off energy of the planewave basis set was 500 eV. A 12 × 12 × 12 k-point mesh was employed for Brillouin zone integrations. The formation energy of Mn 3-x Fe x Ga alloys is given as where E(Mn 3-x Fe x Ga) is the total energy of the Mn 3-x Fe x Ga alloys per formula unit; E(Mn), E(Fe), and E(Ga) denote the total energies of the bulk α-Mn, bcc-Fe, and α-Ga per atom, respectively. The magnetic anisotropy energy (MAE) values were estimated based on the magnetic force theorem 18) using a 24 × 24 × 24 k-point mesh.
In addition, we evaluated the Heisenberg exchange coupling parameter (J ij ) between the magnetic atoms as a function of volume using the Liechtenstein formula 19) with the spin-polarized relativistic Korringa-Kohn-Rostoker code. 20) The angular momentum cut-off was set at 4 in the multiple scattering expansion. We used a 15 × 15 × 15 kpoint mesh and 50 energy points on the complex energy path for the self-consistent calculation. We adopted a GGA-PBE for the exchange-correlation potential. A fine k-mesh with 34 × 34 × 34 k-points was adopted for the evaluation of the exchange coupling parameters. The Curie temperature was estimated from the exchange coupling parameters in the framework based on the mean-field approximation (MFA) for multi-sublattice systems. 21,22) Furthermore, we also investigated the stable structure and magnetic property of the Ga-rich Fe 2 MnGa compound using a supercell approach. To simulate the off-stoichiometric compound, we constructed a 40 atom "special quasi-random structure" (SQS) 23) to model the off-stoichiometric Fe 2 MnGa compound. The SQSs used in this work were generated using the "alloy theory automated toolkit" package. 24)

Results
First, we evaluated the formation energies (E f ) of the ferromagnetic and ferrimagnetic states in the tetragonal Heusler structure. A tetragonal X 2 YZ Heusler compound can have a regular (L2 1 ) or inverse (XA) Heusler structure, as shown in Fig , where E f (ferro) and E f (ferri) denote the formation energies in the ferromagnetic and ferrimagnetic states, respectively, and the value is 0.39 eV f.u. −1 at the equilibrium volume of each magnetic structure. This value is relatively larger than the elastic energy, and therefore, it is difficult to stabilize the ferromagnetic state of the D0 22 structure by volume modulation alone [see Fig. 2(a)]. In Mn 2 FeGa, the ΔE f values are −0.02 and 0.33 eV f.u. −1 for the L2 1 and XA structures, respectively.
In the L2 1 structure, the ferromagnetic state becomes stable, but the L2 1 structure is energetically unfavourable as compared with the XA structure for all the volumes studied here. A discontinuous point is seen in the energy-volume curve of the ferromagnetic L2 1 and XA phases, as shown in Figs. 2(b), 2(c), which is attributed to the transition from the low-spin to high-spin ferromagnetic phase. The magnetisation is enhanced from 6.16 (5.83) to 7.37 (8.49) μ B f.u. −1 for the L21 (XA) phase at these points. For Fe 2 MnGa, the ΔE f values are 0.19 and −0.14 eV f.u. −1 for the L2 1 and XA structures, respectively, and the ferromagnetic XA phase has formation energy of 0.01 eV f.u. −1 , and it is more stable than the ferrimagnetic L2 1 phase. In the ferromagnetic XA phase, the magnetic moments of the A site Fe, C site Mn, and B site Fe are 2.19, 2,80, and 2.30 μ B , respectively, and the magnetisation reaches 1350 emu cc −1 .
The tetragonal L2 1 -Mn 2 FeGa and XA-Fe 2 MnGa, wherein the Fe atom occupies the B site, favour the ferromagnetic state over the ferrimagnetic state. Table I shows the J ij between the nearest-neighbour magnetic atoms at the A(C) site and the B site. When Mn occupies the B site, the J ij values are negative, and the ferrimagnetic spin alignment becomes stable. Conversely, the J ij values become positive when Mn is replaced with Fe at the B site, and the ferromagnetic spin alignment becomes energetically favourable.
Next, we compared the E f and magnetisation of several crystal structures of Mn 2 FeGa and Fe 2 MnGa (see Table II). For Mn 2 FeGa, the ferrimagnetic state in the tetragonal XA structure is the most energetically favourable, as mentioned in a previous theoretical report. 25) For Fe 2 MnGa, the ferrimagnetic state in the cubic L2 1 structure becomes the most energetically favourable. In the previous theoretical report, the ferromagnetic L1 2 structure is more stable than the L2 1 structure. 26) However, the E f difference between the tetragonal XA structure with ferromagnetic ordering and the cubic L2 1 structure is very small, 0.016 eV f.u. −1 . Figure 3 shows the volume dependence of E f for Fe 2 MnGa with several crystal structures. The equilibrium volumes are 46.09, 48.05, and 49.76 Å 3 f.u. −1 for the cubic L2 1 , tetragonal L2 1 , and tetragonal XA structures, respectively, which are different. The ferromagnetic XA phase is stabilized by the 8% volume expansion from the equilibrium volume of the cubic L2 1 phase. The data in Table II reveal that the E f of the ferromagnetic D0 19 phase is also close to that of the cubic L2 1 . Experimental evidence for bcc to the ferromagnetic hexagonal phase transformation upon thermal annealing has been reported. 27) Afterward, we evaluate the Curie temperature and MAE of the tetragonal XA-Fe 2 MnGa compound. The Curie temperature obtained for the XA-Fe 2 MnGa by MFA is 1047 K, which  is much higher than that for D0 22 -Mn 3 Ga (730 K). This high Curie temperature is predominantly attributed to the strong ferromagnetic exchange coupling (29.5 meV) between the Fe atoms at the A and B sites (Table I). A high perpendicular MAE of 2.12 MJ m −3 is also obtained for XA-Fe 2 MnGa; however, this value is slightly smaller than that for the D0 22 -Mn 3 Ga (2.80 MJ m −3 ). Additionally, we evaluate the contribution from each constituent atom to the MAE. In

Discussion
Volume expansion is necessary for the stabilization of the tetragonal XA-Fe 2 MnGa compound. We focus on the effect of substituting the Ga atom with other typical elements having a large atomic radius. Figure 4 shows the volume dependence of E f for Fe 2 MnGa 0.75 Sn 0.25 , where 25% of the Ga atoms are replaced by Sn atoms. The equilibrium volume of the cubic L2 1 and tetragonal XA phases expands by 4.1% and 3.6%, respectively, due to the Sn substitution. Due to the lattice expansion, the E f of the ferromagnetic state in the tetragonal XA becomes lower than that of the cubic L2 1 . The magnetisation, Curie temperature, and MAE of the tetragonal XA-Fe 2 MnGa 0.75 Sn 0.25 are determined to be 1314 emu cc −1 , 1049 K, and 2.23 MJ m −3 , respectively. Thus, Fe 2 MnGa 0.75 Sn 0.25 is a good candidate for permanent magnet applications. However, the formation energy difference between the cubic L2 1 and tetragonal XA phases is not enough to stabilize the XA phase around the Curie temperature.
Next, we evaluated the effect of volume expansion by increasing the Ga composition, which has a larger atomic radius than those of Fe and Mn. First, we determine the site preference of the excess Ga atoms in the Ga-rich L2 1 and XA-Fe 2 MnGa alloys. We calculated the formation energies of the Fe-deficient alloy with a composition of Fe 1.9 Mn 1.0 Ga 1.1 and the Mn-deficient alloy with a composition of Fe 2.0 Mn 0..9 Ga 1.1 . In the present work, we consider 2, 2, 4 and 3 site-occupation configurations for the     In the composition region of the blue circles, the tetragonal XA structure is more stable than the cubic L2 1 structure, as shown in Fig. 5(b). The difference between the formation energies of the cubic L2 1 and tetragonal XA structures is 0.10 eV f −1 .u. with a composition of Fe 1.5 Mn 1.0 Ga 1.5 in the composition range investigated, and this is enough to stabilize the XA phase around the Curie temperature. Next, we determine the formation energy difference between the disordered B2 phase and the ordered XA phase for the Fe 1.5 Mn 1.0 Ga 1.5 alloy, and the estimated value is 0.11 eV f.u. −1 . Further, we estimate the composition dependence of the magnetisations, MAEs and Curie temperatures of the Fe 2-x Mn 1-y Ga 1+x+y alloys with the tetragonal XA structure, as shown Fig. 6. The magnetisation decreases upon replacing the magnetic atoms of Fe and Mn with non-magnetic Ga atoms. Contrarily, the K u is maintained at a high value with replacing Fe with Ga, since the excess Ga atoms replace the B site Fe atoms that do not contribute to the MAE; in addition, tetragonal distortion (c/a) is enhanced with increasing Ga composition. The substitution of Ga with Fe is efficient for stabilizing the tetragonal XA phase and obtaining a high K u . For the Fe 1.5 Mn 1.0 Ga 1.5 alloy, B s = 1.3 T, K u = 2.3 MJ m −3 , and T c = 850 K, and large magnetisation, magnetic anisotropy and Curie temperature are expected.

Conclusions
We theoretically investigated the stabilization of the ferromagnetic state in tetragonal Heusler alloys. We focused on the effects of Fe atom substitution for Mn and the lattice expansion in the Mn 3 Ga alloy. When the volume of Fe 2 MnGa is expanded by about 8%, the ferromagnetic tetragonal XA phase becomes stable. The tetragonal XA-Fe 2 MnGa alloy can be applied in permanent magnets because of its high saturation magnetisation, perpendicular magnetic anisotropy, and Curie temperature. Sn substitution at the Ga sites results in volume expansion so that the ferromagnetic tetragonal XA phase is stabilized. Finally, we examined the stability of the XA phase and magnetic properties of Fe 2 MnGa in relation to the Ga composition and found that Fe 1.5 Mn 1.0 Ga 1.5 is a good material that can be utilized as a permanent magnet.