First principles study of the structural phase stability and magnetic order in various structural phases of Mn$_2$FeGa

TL;DR

This study uses density functional theory to analyze the structural stability and magnetic order of Mn$_2$FeGa across different phases, revealing phase transformation mechanisms and magnetic interactions.

Contribution

It provides a detailed first-principles analysis of phase stability, transformation pathways, and magnetic structures in Mn$_2$FeGa, highlighting the Jahn-Teller effect and magnetic interactions.

Findings

Hexagonal to tetragonal transformation involves a Heusler-like phase.

Tetragonal phase exhibits collinear ferromagnetic order due to exchange interactions.

Hexagonal phase has in-plane magnetic moments with no frustration.

Abstract

We investigate the structural and magnetic properties of Mn$_{2}$FeGa for different phases(cubic, hexagonal and tetragonal) reported experimentally using density functional theory. The relative structural stabilities, and the possible phase transformation mechanisms are discussed using results for total energy, electronic structure and elastic constants. We find that the phase transformation form hexagonal to ground state tetragonal structure would take place through a Heusler-like phase which has a pronounced electronic instability. The electronic structures, the elastic constants and the supplementary phonon dispersions indicate that the transition from the Heusler-like to the tetragonal phase is of pure Jahn-Teller origin. We also describe the ground state magentic structures in each phase by computations of the exchange interactions. For Heusler-like and tetragonal phases, the ferromagnetic exchange interactions associated with the Fe atoms balance the dominating antiferromagnetic interactions between the Mn atoms leading to collinear magnetic structures. In the hexagonal phase, the direction of atomic moment are completely in the planes with a collinear like structure, in stark contrast to the well known non-collinear magnetic structure in the hexagonal phase of Mn$_{3}$Ga, another material with similar structural properties. The overwhelmingly large exchange interactions of Fe with other magnetic atoms destroy the possibility of magnetic frustration in the hexagonal phase of Mn$_{2}$FeGa. This comprehensive study provides significant insights into the microscopic physics associated with the structural and magnetic orders in this compound.

Tables

Table 1. Table 1: The calculated lattice parameters(in Å), total(M) and atomic magnetic moments(M X )(in μ B subscript 𝜇 𝐵 \mu_{B} per formula unit) of Mn 2 FeGa in Cu 3 Au, X a , L1 0 and DO 19 phases. The results from experiments and other DFT calculations are given in parenthesis

Structure	Lattice parameter $a$	Lattice parameter $c$	M	MMnI	MMnII	MFe	MGa
Cu3Au	3.69(3.7526 GasiAPL13 )	3.69	0.68	-3.08	2.55	1.28	-0.03
Xa	5.78(5.80 LuoJAP08 ,5.79 WollmannPRB14 )	5.78	1.04(1.03 WollmannPRB14 )	-2.07	2.80	0.29	0.01
L10	3.68(3.7915 GasiAPL13 ,3.68 WollmannPRB15 )	7.29(7.1858 GasiAPL13 ,7.29 WollmannPRB15 )	-0.80(0.96 WinterlikAM12 ,-0.78 WollmannPRB15 )	-2.43	2.96	-1.41	0.04
DO19	5.25(5.3151 LiuAPL16 )	4.20(4.3050 LiuAPL16 )	1.22(1.26 LiuAPL16 )	-2.84	2.54	1.55	-0.03

Table 2. Table 2: The possible configurations due to different site occupancy patterns in DO 19 phase of Mn 2 FeGa. E f (eV/f.u.) is the formation enthalpy for a given configuration. M tot ( μ B / f . u . formulae-sequence subscript 𝜇 𝐵 𝑓 𝑢 \mu_{B}/f.u. ) is the calculated total magnetic moment in a given configuration.

Sites and occupancies
No.	T1	T2	T3	T4	B1	B2	B3	B4	Ef	Mtot
1	Mn	Fe	Mn	Ga	Mn	Fe	Mn	Ga	-0.460	3.03
2	Mn	Fe	Mn	Ga	Mn	Mn	Fe	Ga	-0.459	1.63
3	Mn	Fe	Mn	Ga	Fe	Mn	Mn	Ga	-0.504	1.28
4	Mn	Mn	Fe	Ga	Mn	Mn	Fe	Ga	-0.509	1.30
5	Mn	Fe	Fe	Ga	Mn	Mn	Mn	Ga	-0.456	1.46
6	Fe	Mn	Fe	Ga	Mn	Mn	Mn	Ga	-0.456	1.46
7	Fe	Fe	Mn	Ga	Mn	Mn	Mn	Ga	-0.456	1.45

Table 3. Table 3: The calculated elastic constants (in GPa) of Mn 2 FeGa in Cu 3 Au, X a , L1 0 and DO 19 phases.

Structure	B	C11	C12	C13	C33	C44	C66	C′
Cu3Au	131.3	141.6	126.1	-	-	33.9	-	7.8
Xa	147.2	166.8	137.4	-	-	127.3	-	14.7
L10	140.2	292.2	0.56	124.2	179.5	100.2	11.6	145.8
DO19	126.3	210.8	40.8	27.2	224.4	55.2	84.9	85

Equations

$$H=-\sum _{\mu ,\nu }\sum _{i,j}{J}_{ij}^{\mu \nu }{\mathbf{e}}_i^{\mu }.{\mathbf{e}}_j^{\nu }$$

Full text

I INTRODUCTION

Mn2YZ compounds in Heusler and Heusler-like structures have been in the attention of spintronics community due to their multiple possible applications in spin-transfer torque random access memory(STT-MRAM) WinterlikAM12 , spin valves SinghPRL13 , magnetic tunnel junction KubotaAPL09 , spin gapless semiconductor OuardiPRL13 and magnetic shape memory effect LiuAPL05 . Such traits of Mn2YZ compounds are artefacts of various possible structures in which the systems can crystallise as well as of different orientations of the Mn spins giving rise to interesting magnetic configurations LiuAPL05 ; BarmanPRB08 ; PaulJAP11 ; MeshcheriakovaPRL14 ; WinterlikPRB11 ; GasiAPL13 ; WollmannPRB14 ; WollmannPRB15 ; NayakPRL13 ; Kundumodulation17 . Among the compounds in this family, Mn3Ga KurtPSS11 ; RodePRB13 ; KharelJPCM14 ; ZhangJPCM13 ; KhmelevskyiPRB16 and Mn2NiGa LiuAPL05 ; Kundumodulation17 have been studied extensively. Mn2NiGa is a recently discovered ferrimagnetic shape memory system which martensitically transforms to a low temperature tetragonal structure through a series of complex modulated phases of orthorhombic and monoclinic symmetries Kundumodulation17 ; the martensitic transformation being driven by phonon softening and Fermi surface nesting PaulJPCM15 . Mn3Ga, on the other hand, is found to crystallise in a cubic Cu3Au-like KharelJPCM14 , a hexagonal DO19 KurtAPL12 and a tetragonal DO22 phase RodePRB13 ; KurtPRB11 , resulting in a high perpendicular magneto-crystalline anisotropy in the DO22 phase and a large Exchange Bias in the DO19 phase, which are useful for STT-MRAM RodePRB13 and magnetic tunnel junction KurtAPL12 applications, respectively. First-principles Density Functional Theory (DFT) calculations predicted a Heusler-like metastable structure in Mn3Ga with a half-metallic gap, which phase transforms to the DO22 structure WollmannPRB14 ; WollmannPRB15 . This phase, however, has not yet been synthesised experimentally. Comprehensive DFT calculations inferred that the phase transition from DO19 to DO22, as observed in the experiments KharelJPCM14 , happens via the Heusler-like phase ZhangJPCM13 . First-principles computations of the magnetic exchange interactions KhmelevskyiPRB16 and the magnetic anisotropy RodePRB13 concluded that the novel magnetic properties of Mn3Ga in DO19 and DO22 structures are due to non-collinear magnetic structures arising out of frustrations due to geometry as well as competing exchange interactions between in-plane and out-of-plane Mn atoms.

Inspite of enough promises towards a variety of magnetism related applications, both Mn3Ga and Mn2NiGa have low saturation magnetisations originating from the predominantly antiferromagnetic interactions between the Mn atoms. This hinders the exploitations of their complete potentials in the respective applications. The low saturation magnetisation in Mn3Ga limits it’s applicability in potential permanent magnet applications, inspite of having strong uniaxial magnetic anisotropy and large Curie temperature. For Mn2NiGa, much lower saturation magnetisation LiuAPL05 ; PaulJAP11 , in comparison to Ni2MnGa, the prototype magnetic shape memory system in the Heusler family, is an obstacle to obtain significant magnetic field induced strain required for actuator applications. Attempts have, therefore, been made to combat the dominant antiferromagnetic interactions in these systems by replacing one of the Mn with ferromagnetic elements like Co and Fe. Complete replacement of one Mn atom by Fe, resulting in the compound Mn2FeGa, was expected to circumvent the problems. Synthesis of Mn2FeGa, however, exhibited several important aspects of structure-property relationship. Various crystallographic phases with possible magnetic structures were observed and predicted experimentally. Gasi et al. GasiAPL13 reported an inverse tetragonal Heusler structure with a low saturation magnetic moment and a high Curie temperature for samples annealed at 4000C. The system behaved like an exchange spring which was attributed to the distribution of Fe atoms among the Mn sites resulting in two different magnetic states of Fe. A Cu3Au-like structure was observed when the system was annealed at 8000C. A giant tuneable Exchange Bias was later obtained in the inverse tetragonal structure with a saturation magnetisation as low as 0.09 $\mu_{B}$/f.u. NayakNM15 suggesting a near compensation of moments from Mn and Fe atoms. DFT calculations, like Mn3Ga, predicted a meta-stable inverse Heusler phase for Mn2FeGa, which transforms to the inverse tetragonal structure observed in the experiments WollmannPRB14 ; WollmannPRB15 . The electronic structure of inverse Heusler Mn2FeGa revealed a pronounced instability associated with the minority spin band WollmannPRB15 . However, unlike Mn2NiGa, the transformation from inverse Heusler to inverse tetragonal phase didn’t turn out to be volume conserving, implying the possibility of absence of shape memory effect in this system. Investigations into sputter deposited thin films of Mn2FeGa NiesenIEEE16 find that the system crystallises either in inverse tetragonal or in the Cu3Au-like cubic phase. Mossbauer spectroscopy on epitaxial thin films suggested that Fe might be statistically distributed among the two Mn sites leading to a low spin moment in the inverse tetragonal phase BettoarXiv17 . Heteroepitaxially grown thin films of Mn-Fe-Ga with composition near that of Mn2FeGa also exhibited strong perpendicular anisotropy and moderate coercivity suggesting that this system can be used in mid-range permanent magnet applications KalachearXiv17 . Like Mn3Ga, this system could also be synthesised in DO19-type hexagonal structure which yielded a giant Exchange Bias field upto 1.32 kOe LiuAPL16 . The large Exchange Bias was attributed to the presence of substantial ferromagnetic matrix due to Fe-Mn pairs in an antiferromagnetic host.

The experimental observations clearly indicate that investigations into the phase stability, sub-lattice occupancy and magnetic properties of Mn2FeGa would be insightful. The following issues, in particular, are worth looking into, in order to understand the physics associated with Fe substitution in either Mn3Ga or in Mn2NiGa, the two compounds having very different functional aspects: (i) the energetics of various phases observed experimentally or are predicted theoretically but not observed experimentally, (ii) the magnetic interactions among three different magnetic atoms and their dependencies on the sub-lattice occupancies of the atoms in different structural phases. This can be particularly relevant in the hexagonal phase where the Fe atom has more than one choice of the crystallographic site it can occupy, and (iii) the effect of anti-site disorder on the magnetic properties. In this paper, we have undertaken a DFT based comprehensive investigation into the structural and magnetic properties of the four crystallographic phases of Mn2FeGa, the Cu3Au-like, the hexagonal DO19-like, the inverse Heusler and the inverse tetragonal. Emphasis has been given on identifying the magnetic structure in the hexagonal phase as it is supposed to be most complex in this phase as was observed in Mn3Ga. We have computed the magnetic exchange interactions in each structural phase in order to understand the magnetic structures. The electronic structures of the structural phases are computed and analysed in order to provide a possible picture of the phase transformations. The elastic constants and the phonon dispersion relations for select phases are computed to supplement the analysis from the energetics and the electronic structures. In this work, we have not incorporated anti-site disorder in order to avoid dealing with prohibitively large number of possible configurations, in particular for the hexagonal structure. In absence of concrete quantitative estimate of the anti-site disorder from the experiments, this is justified. The paper is organised as follows: in the section II, computational details are given. The results and discussion are presented in section III followed by conclusions.

II Calculational details

All calculations were performed with spin-polarised density functional theory (DFT) based projector augmented wave(PAW) method PAW94 as implemented in Vienna Ab-initio Simulation Package (VASP) VASP196 ; VASP299 . Perdew-Burke-Ernzerhof (PBE96) implementation of Generalised Gradient Approximation (GGA) for exchange and correlation PBEGGA96 part in the Hamiltonian was used throughout. An energy cut-off of 500 eV and a Monkhorst-Pack MP89 $25\times 25\times 25$ $k$-mesh for Cu3Au-like and inverse Heusler structures , a $15\times 15\times 13$ $k$-mesh for the tetragonal structure and a $13\times 13\times 11$ $k$-mesh for the hexagonal structure were used for self consistent calculations. Larger $k$-meshes were used for the densities of states calculations of all the structures. For all calculations, the total energy convergence criteria and the force convergence criteria were set to 10*-6* eV and to $10^{-2}$ eV/Å respectively. The elastic constants were calculated from the second derivatives of the total energies with respect to the strain tensors Vitosbook07 .

The magnetic pair exchange parameters were calculated with multiple scattering Green function formalism as implemented in SPRKKR code EbertRPP11 . In here, the spin part of the Hamiltonian is mapped to a Heisenberg model

[TABLE]

$\mu$, $\nu$ represent different sub-lattices, i, j represent atomic positions and $\mathbf{e}^{\mu}_{i}$ denotes the unit vector along the direction of magnetic moments at site i belonging to sub-lattice $\mu$. The $J^{\mu\nu}_{ij}$s are calculated from the energy differences due to infinitesimally small orientations of a pair of spins within the formulation of Liechtenstein et al. LiechtensteinJMMM87 . In order to calculate the energy differences, full potential spin polarised scaler relativistic Hamiltonian with angular momentum cut-off $l_{max}=3$ is used along with a converged $k$-mesh for Brillouin zone integrations. The Green’s functions are calculated for 32 complex energy points distributed on a semi-circular contour. The energy convergence criterion is set to 10*-5* eV for the self-consistency. The equilibrium lattice parameters and optimised atomic positions as obtained from the PAW calculations are used to obtain the self-consistent potentials in the multiple scattering Green’s function method.

III Results and Discussions

III.1 Structural parameters and magnetic structures in various crystallographic phases

Experimentally, Mn2FeGa has been observed to crystallise in three different phases, the Cu3Au-like, the tetragonal DO22 like and the hexagonal DO19 GasiAPL13 ; LiuAPL16 , depending on the annealing temperature. DFT calculations predicted a inverse Heusler phase WollmannPRB14 ; WollmannPRB15 ; LuoJAP08 , yet undetected in experiments. In these investigations, the possible magnetic structures associated with different structural phases have been indicated through indirect evidences. In this sub-section, we present results on the structural properties and possible ground state magnetic structures, along with atomic moments for all four structural phases of Mn2FeGa. Throughout the manuscript, the Cu3Au-like phase, the hexagonal phase, the inverse Heusler phase and the inverse tetragonal phases are referred to as the Cu3Au, the DO19, the Xa and the L10 phases respectively. The results are summarised in Table 1.

Cu3Au phase: In the ordered Cu3Au structure of Mn2FeGa, the Ga atom occupies the corners of the cubic unit cell while the two Mn, MnI and MnII, along with the Fe, occupy the face centres (Fig. 1(a)). Depending upon the relative orientations of the moments of the three magnetic atoms, three inequivalent collinear spin configurations are possible in this structure as shown in Fig.1(a). Our total energy calculations show that the energetically lowest configuration is T-I where the two Mn atoms orient anti-parallel (Fig. 1(b)). The calculated lattice parameter agrees well with experimental reported one (Table I). The total magnetic moment is only 0.68 $\mu_{B}/f.u.$, with MnI having a larger moment among the two Mn atoms. The Fe moment is only 1.28 $\mu_{B}$, much less than that in it’s elemental solid phase. Mossbauer spectra of Mn2FeGa, annealed at 800*∘C GasiAPL13 , confirmed the Cu3Au-like structure but the measurements of hyperfine field distributions suggested that the average moment at the Fe site is only $0.5\mu_{B}$ as Fe may occupy Mn sites as well. Since we have not included the anti-site disorder, that is mixing of Fe and Mn atoms at all crystallographic sites, the discrepancy with the experimental results is natural. Since the experimental measurements did not indicate any possibility of non-collinearity in the magnetic structure, our consideration of the collinear magnetic structure is justified, and thus, our results provide the possible ground state magnetic configuration if Mn2FeGa crystallises in a perfectly ordered Cu3Au phase. A comparison with Mn3Ga in the ordered Cu3Au phase KharelJPCM14 shows that replacing one Mn atom by Fe has led to a reduction of the total moment of the system. In Mn3Ga, the MnI and MnII moments were -2.93 $\mu_{B}$ and 2.24 $\mu_{B}$ respectively, leading to a total moment of 1.55 $\mu_{B}$ per formula unit. The atomic moments in Mn2FeGa, as obtained here, suggest that the Mn moments have not changed significantly, but the reduction in the total moment, in comparison to Mn3*Ga, happens due to weaker exchange splitting associated with the Fe atoms.

Xa phase: DFT calculations predict a Hg2CuTi structure (space group no. 216; $F\bar{4}3m$) for Mn2FeGa with four inequivalent Wyckoff positions(4a, 4b, 4c, 4d) in the unit cell LuoJAP08 ; WollmannPRB14 . In this structure, also known as Xa structure, the MnI and MnII atoms occupy 4a(0,0,0) and 4c(0.25, 0.25, 0.25) positions respectively. The 4b(0.5, 0.5, 0.5) and 4d(0.75, 0.75, 0.75) sites are occupied by Fe and Ga atoms respectively. The Xa structure of Mn2FeGa is shown in Fig. 2(a). The equilibrium lattice constant (Table 1) obtained in this work is consistent with the reported results LuoJAP08 ; WollmannPRB14 . The calculated magnetic moment is close to 1.0 $\mu_{B}/f.u$, in very good agreement with the existing DFT results. The atomic moments (Table 1) show that the Fe moment is quenched substantially, and that the low total moment arises due to anti-parallel alignment of the two Mn atoms. The ground state magnetic configuration in this structure is consistent with that of Mn2NiGa PaulJAP11 , in which Fe substitution at the Ni site gives rise to the present compound under investigation. Interestingly, the total and the atomic moments of Mn2FeGa in this structural phase are very similar to Mn2NiGa. In contrast, there is substantial differences between the total and the atomic moments of Mn2FeGa and Mn3Ga in this phase. In Mn3Ga, the total moment is nearly zero in this phase WollmannPRB14 . This is due to the fact that unlike Mn2FeGa, Mn3Ga has a Heusler like coordination (space group 225) and thus the Mn moments at 4a and 4b sites compensate the moment at 4c sites associated with the other Mn aligning anti-parallel. The alteration in the coordination in Mn2FeGa, thus, introduces magnetic interactions, very different from those in Mn3Ga.

L10 phase: The tetragonal L10 phase of Mn2FeGa (Fig. 2(b)) has been synthesised GasiAPL13 ; BettoarXiv17 and been the subject of intense investigations due to the possible PMA and high coercivity in this structure. Investigations into the magnetic structure concluded that the system has a ferrimagnetic order, and unlike Mn3Ga RodePRB13 , the spins are collinear. The magnetisation measurement on polycrystalline samples at 10 T field yielded a non-saturating magnetisation of about 1.5 $\mu_{B}$ GasiAPL13 . Using results from magnetometry, Betto et. al. BettoarXiv17 estimated the atomic magnetic moments and the spin structure (Fig. 2(b)). Their estimation of atomic moments, based upon magneto-optical sum rules resulted in equal moments on MnI and Fe sites with MnII sites having a 30$\%$ less moment. Incorporation of Fe anti-sites in their calculations increased the moment at MnII sites resulting in nearly equal moments at all three sites, which would result in a very small net moment in the system. This was in contrast to Mn3Ga, where the estimated moments associated with the MnI and MnII were 2.67 $\mu_{B}$ and 4.74 $\mu_{B}$ respectively, producing a net moment of about 0.6 $\mu_{B}$ per formula unit RodePRB13 . DFT calculations by Wollmann et al. WollmannPRB15 , on the other hand, obtained unequal magnetic moments at MnI and MnII sites, along with a moment at the Fe site which is nearly 50$\%$ of the moment which is larger among the two Mn moments, producing a net moment of about 0.8 $\mu_{B}$ per formula unit. Our results in Table 1 are in excellent agreement with those of Wollmann et al. Particularly interesting is the gain in Fe moment in comparison to the Xa phase, and the change in it’s orientation with respect to the Mn sub-lattices. Such behaviour of the atomic moments, in particular of Fe in Mn2FeGa, is in stark contrast to that of Ni in Mn2NiGa. In prototype shape memory alloys like Ni2MnGa and in Mn2NiGa, the systems undergo martensitic transformations from L21 Heusler and Xa structures, respectively to L10 structures, conserving their volumes with little changes in their atomic and total magnetic moments. The very different behaviour of the atomic magnetic moments in Xa and L10 phases of Mn2FeGa indicate that the physics associated with the system in these phases would be different from those of shape memory Ni2MnGa and Mn2NiGa.

In order to elucidate this point, we have optimised the structural parameters in the L10 phase of Mn2FeGa by computing the total energy as a function of the tetragonal deformation, represented by the $c/a$ ratio, keeping the volume fixed at that of the Xa phase. The results are presented in Fig. 3. The energetics of Ni2MnGa and Mn2NiGa are presented for comparison. We find a local energy minima at $c/a=1.0$ and a global energy minima at $c/a=1.40$. For Mn2FeGa, the energy surface looks very different compared to those of Mn2NiGa and Ni2MnGa. We further computed the total energies for the L10 structure by varying the volume. The results are shown in the inset of Fig. 3. We clearly see that the optimised volume of the L10 phase is higher by 2.54% than the volume of Xa structure which indicates that the Xa to L10 transformation in this system is not a martensitic one and the system would not exhibit shape-memory effect. The optimised structural parameters, presented in Table 1 are in good agreement with the results obtained from experiments and previous DFT calculations.

DO19 phase: The hexagonal DO19 phase of Mn2FeGa has been synthesised only recently LiuAPL16 after annealing the polycrystalline sample to 6000C. In the hexagonal(DO19) structure(space group no. 194; $P6_{3}/mmm$) there are two formula units of atoms(8 atoms) arranged in two different planes of the unit cell. (T1, T2, T3, T4) and (B1, B2, B3, B4) are the lattice sites in top and bottom planes respectively, as shown in Fig. 4(a). The atomic positions at top plane are: T1(1/6, 1/3, 3/4), T2(2/3, 5/6, 3/4), T3(1/6, 5/6, 3/4), T4(2/3, 1/3, 3/4), and at bottom plane are: B1(1/3, 1/6, 1/4), B2(5/6, 2/3, 1/4), B3(5/6, 1/6, 1/4), B4(1/3, 2/3, 1/4). In hexagonal Mn3Ga, the Ga atoms occupy T4 and B4 sites in two different layers of the unit cell. The Mn atoms occupy other six positions in the unit cell. Such a geometry inherently produces a frustrated magnetic structure in Mn3Ga. Accordingly, different non-collinear spin configurations were investigated to obtain the ground state magnetic configuration in the system by DFT calculations ZhangJPCM13 . The configuration with the lowest energy was found to be a triangular structure with a 1200 mutual orientations of the neighbouring Mn atoms in the same plane as well as of the neighbours in the adjacent planes. This was in confirmation with the results of Neutron diffraction krenSSC70 . Subsequently, the magnetic moment came out to be nearly zero with the moment at Mn site to be $\sim$ 2.5 $\mu_{B}$ krenSSC70 ; ZhangJPCM13 . The magnetisation measurements on DO19 Mn2FeGa obtained a saturation moment of 1.26 $\mu_{B}$ per formula unit LiuAPL16 . The authors proposed that this is an effect of increasing Fe-Mn local ferromagnetic matrix in otherwise antiferromagnetic host as Fe can occupy Mn sites either in a single layer or across the layers of the hexagonal planes. If hexagonal Mn2FeGa is considered to be formed out of hexagonal Mn3Ga by replacing one Mn atom with Fe, in the DO19 unit cell, two Mn atoms out of six are to be replaced by Fe atoms. Since these Fe atoms can occupy any two positions out of the six inequivalent sites among the two planes (keeping the position of Ga atoms fixed), many inequivalent combinations of site occupancy are possible in this phase. Consideration of symmetry, however, reduces the possible number of configurations to only seven which are given in Table 2. Due to the geometrical frustration arising out of triangular networks in the DO19 structure, consideration of non-collinear spin structures as the starting point of the calculations are necessary. In absence of any experimental information regarding the spin structure of this system in DO19 phase, we started our calculations with the reported spin configuration of Mn3Ga in the literature ZhangJPCM13 . We optimised the magnetic structures and calculated the formation enthalpies of all seven configurations. The results are presented in Table 2. The results suggest that the configuration with configuration number 4 in Table 2 has the lowest formation enthalpy. The corresponding magnetic structure is shown in Fig. 4(a). The calculated total magnetic moment for this configuration is 1.30 $\mu_{B}/f.u.$ which agrees very well with the experimental magnetisation value of 1.26 $\mu_{B}/f.u.$ LiuAPL16 .

Contrary to the expectations, our calculations show the minimum energy configuration of the spins to be one where they are confined to the planes and pointing either along the $a$-direction or against it, a collinear arrangement when observed in the $a-b$ plane. Since the magnetic arrangement is like a collinear one, we calculated the total energy of this configuration by aligning the spins along the $z$-direction. The differences in energies of the two configurations turned out to be less than 1 meV per formula unit. The magnetic moment of the configuration with spins directed either along or against $z$-direction is 1.22 $\mu_{B}$ per formula unit (Table 1), very close to that obtained for the minimum energy configuration with spins in the $a-b$ plane (Table 2). This justifies using the collinear configuration with $z$-axis the easy axis for further calculations. The lattice parameters obtained by optimising this configuration, presented in Table 1, agree quite well with the experiments. Thus, the presence of Fe in the system appears to have taken care of the frustration in the system and produces a collinear kind of magnetic structure which explains the significant non-zero magnetisation of Mn2FeGa in this phase. It is worth mentioning that such collinear like magnetic structures are also obtained for some of the other configurations considered here, but they are energetically higher, yet possible to form as is obvious from the negative values of the formation enthalpies (Table 2). The optimised magnetic structure corresponding to each of the configurations considered here are presented in the supplementary information.

III.2 Stabilities of various phases: analysis from energetics, electronic structure and elastic constants

Fig. 5 shows the variations in the total energies with volume for the four phases of Mn2FeGa. We relaxed all the structural parameters to obtain the most stable structure in a given phases. The tetragonal phase comes out to be energetically the most stable phase while the Cu3Au phase is having the highest total energy. The hexagonal, and the Xa phases lie in between with the Xa phase having a total energy in between the hexagonal and the tetragonal phase. The relative energies of these phases obtained this way agrees with the experiments in a way as the tetragonal phase is observed to be the low temperature phase GasiAPL13 while annealing to a high temperature stabilises the Cu3Au phase; the hexagonal phase being obtained only when the sample is annealed at temperatures in between the annealing temperatures to obtain the L10 and the Cu3Au phases. The other interesting feature of the relative energetics is that the DO19 and the Xa phases are energetically very close near the equilibrium volume of the DO19 phase. Since there is no possibility of a direct DO19 to L10 transformation due to symmetry constraints ZhangJPCM13 , and that a transformation between the Xa and DO19 is possible, as illustrated in case of Mn3Ga ZhangJPCM13 , our results imply that the phase transformation from DO19 to L10 can take place via the Xa phase. Thus, although the Xa phase has not yet been obtained experimentally, possibly due to requirement of growth by non-equilibrium methods as it appears to be metastable (Fig. 3), it’s presence is very important to understand the stabilities of the DO19 and the L10 phases and the possible path of transformation between the two phases.

In order to gain further insight into the stabilities of these phases in Mn2FeGa, we take a look at their electronic structures. In Fig. 6, we show the total and atom-projected densities of states of Mn2FeGa in various structural phases. These also help us understanding the behaviour of atomic magnetic moments in each phase. Across all the structures, we find that the minority(majority) band of MnI is nearly full(nearly empty) while the opposite is true for MnII bands. This explains the anti-parallel orientation of the moments associated with the two Mn atoms. For Cu3Au and DO19 phases, there is strong hybridisation and delocalisation of the MnII and Fe bands in the majority spin channel, while they are more localised in the Xa and L10 phases. In the minority spin channel, the hybridisations between MnI and Fe bands are minimal for Cu3Au, DO19 and Xa phases while it is substantial in the L10 phase. In spite of less hybridisation, both MnI and Fe minority states in the Cu3Au and DO19 phases are delocalised while they are more localised in the Xa phase. Thus, the densities of states in the Xa phase stand out in contrast to the other three phases; the highlights being more localised atomic densities of states and thus more structured total densities of states in comparison to relatively structureless total densities of states in other phases. The other major difference observed in the densities of states in the Xa phase is a sharp peak at the Fermi level in the majority band and a half-metallic gap in the minority band. The peak at the fermi level is due to hybridisation of MnI and Fe $t_{2g}$ states while the gap in the minority band is flanked by the well separated $t_{2g}$ and the $e_{g}$ states of mainly Fe atom. The peak in the majority band densities of states implies instability in this structure which triggers the tetragonal distortion and subsequent stabilisation of the L10 phase. The $t_{2g}$ states at the Fermi level in the Xa phase now splits into three non-degenerate states due to the lowering of the symmetry, thus washing away the peak. The symmetry lowering gets rid of the half-metallic gap too as the system stabilises by filling in the gap, that is by re-distributing the states into the lower energy states. The delocalisation and strong hybridisation of the MnI and Fe minority states in L10 structure is the outcome of this. The quenching of the Fe moment in the Xa structure can also be understood from the electronic structure. The nearly full Fe states in both spin channels reduces the exchange splitting associated with Fe. When the system tetragonally distorts towards the L10 structure, the Fe states fill out significant part of the occupied region in the minority band, while the spectral weight associated with the Fe states in the majority band shifts towards the unoccupied region, thus recovering the exchange splitting observed in the Cu3Au and DO19 phases. The MnII states in the majority bands associated with the Cu3Au and DO19 structures are distributed in a nearly identical way, providing an explanation to the almost equal MnII moments in these two phases. The proximity of the MnI moments in these two phases also emerge due to the near identical minority spin densities of states of MnI which implies nearly same exchange splitting associated with the MnI atoms. The Fe moments are also nearly equal in these phases which can be explained in a similar way as Fe hybridises with both Mn in relevant spin channels. The moments associated with MnII in these two phases are less than that of MnI. The relatively less exchange splitting associated with the MnII atoms can be understood as the effect of stronger hybridisation with the Fe states. In the Xa and L10 states, the exchange splitting of MnII is more than that of MnI. This happens due to appearance of more localised MnII and Fe states in the majority spin band, thus reducing hybridisations.

The densities of states imply that Cu3Au, DO19 and L10 phases are stable as was observed experimentally. The strong hybridisations between Mn and Fe states in either spin channel are found to be responsible behind it. The reduced hybridisations can be correlated with the instability in the Xa phase as implied by the large densities of states at the Fermi level. In order to understand the origin of this instability and to predict whether this phase can ever be synthesised, we first check the mechanical stability by computing the elastic constants. The elastic constants for all four phases are given in Table 3.

The calculated elastic constants satisfy all the stability criteria for Cu3Au, Xa, DO19 and L10 phases Vitosbook07 ; OzdemirJAC10 . Among all, the shear modulus(C${}^{\prime}=(C_{11}-C_{12})/2$) gives an insight to the stability of all the structures with respect to the shear(for hexagonal, $C_{66}=C^{\prime}$). For all the structures, $C^{\prime}$ is positive and indicate the relative mechanical stability of the structure against shear. The $C^{\prime}$ value of tetragonal structure is higher than other structures which might indicate that it is the most stable structure against shear instability. For the Xa structure, however, the value of C*′* is sufficiently small, in comparison to C44. The elastic anisotropy ratio $A\left(=C^{44}/C^{\prime}\right)$ is a measure of the stability of a crystal with cubic symmetry against stress along $(110)$ planes ZenerPRB47 . In the Xa phase of Mn2FeGa, this ratio is 8.66, indicating that the system can be unstable against anisotropic stress. In order to gain further insight into the possible consequences of it, and thus understand the mechanism of the Xa to L10 transformation in this system, we have computed the phonon dispersion relations in both Xa and L10 phases (Figs. 8 and 9, supplementary material). The dispersion relations indicate dynamical stability in both phases. None of the acoustic modes in the Xa phase were imaginary, or produced any pronounced softening. These indicate that the transformation from Xa to L10 phase is not phonon driven as was the case for Mn2NiGa PaulJPCM15 . Thus, the phase transition from Xa to L10 can be considered due to the well known Jahn-Teller effect.

III.3 Magnetic Exchange interactions

In order to understand the origin of the magnetic structures in various phases as depicted in sub-section III-A, we now look at the results of various inter-atomic and intra-atomic exchange interactions for the Xa, L10 and DO19 phases. Since the L10 and DO19 phases are the most promising ones from applications point of view and the Xa phase has strong connection with the L10 phase as has been established in the previous sub-sections, we have investigated the exchange interactions for these three phases. The results for the Xa and the L10 phases are presented in Fig. 7(a) and (b), respectively. For both phases, the nearest neighbour strong antiferromagnetic and ferromagnetic interactions between two different pairs of atoms dominate the scene. The higher neighbour interactions are too small to influence the magnetic structure, which would be collinear. In case of the tetragonal DO22 phase of Mn3Ga, neutron diffraction experiments RodePRB13 found a small canting of Mn spins occupying the 2b sites (the sites occupied by Fe in Fig. 2 (b)). The calculated exchange parameters established frustration associated with that particular Mn site, which together with the in-plane magnetic anisotropy at that site, was used to explain the spin canting at that site. In the present case, we do not see any such possibility arising out of the qualitative and quantitative nature of the three major inter-atomic parameters associated with MnI-MnII, Fe-MnI and Fe-MnII pairs. For all three of them, at least the first two nearest neighbour exchange parameters are either ferromagnetic or antiferromagnetic, thus ruling out possibility of magnetic frustration. In Both Xa and L10 structures, the overwhelmingly dominating interaction is the antiferromagnetic MnI-MnII. Same was the case for DO22 Mn3Ga. The qualitative difference between Mn2FeGa and Mn3Ga magnetic interactions is observed in cases of the exchange parameters involving Fe. In the Xa phase, the first neighbour antiferromagnetic Fe-MnI exchange parameter is of the same magnitude as that of the first neighbour ferromagnetic Fe-MnII one although the former pair is at a larger distance. In the L10 phase, we observe a role reversal of these two interactions: the Fe-MnI is now strongly ferromagnetic in the first neighbour, the interaction being ferromagnetic even for higher neighbours, while the Fe-MnII exchange is now weakly antiferromagnetic. This role reversal can be explained from the structural deformation of Xa that leads to the L10. The contraction in the basal plane now reduces the Fe-MnI distances along with a subsequent increase in the Fe-MnII distance due to elongation in the [001] direction. This increase in the Fe-MnII distances weaken the exchange interactions between this pair and makes it weakly antiferromagnetic, while the opposite happens for the Fe-MnI pairs which are now closer. The presence of strong ferromagnetic and antiferromagnetic components in the first neighbour exchange interactions of Mn2FeGa in the Xa and L10 phases along with weak higher neighbour interactions are the reasons behind the collinear ferrimagnetic structure in these phases. Thus, the presence of Fe in the system inflicts a significant ferromagnetic component in the magnetic interactions, in comparison to Mn3Ga.

The collinear-like magnetic configuration obtained in the DO19 phase of Mn2FeGa, as opposed to the non-collinear magnetic structure in Mn3Ga where the neighbouring atoms(both in-plane and out-of plane) align themselves in a triangular structure with an angle of 1200 between them indicates that Fe plays a vital role. The substantial magnetisation obtained in the measurements LiuAPL16 on the DO19 phase of Mn2FeGa was connected with the possibility of creation of Fe-Mn ferromagnetic matrix in the system. The magnetic exchange interactions calculated for this phase of Mn2FeGa and presented in Fig. 9 helps understanding the origin of the collinear like magnetic structure obtained from the total energy calculations. The network of the nearest neighbour magnetic atoms in the two neighbouring planes in the DO19 structure is shown in Fig. 8. It is clear from the structure that there are two types of triangular networks in the basal planes-one with the shared Ga neighbours and one without. Consequently, the strengths of the interactions between the same pair of specie at the same distance would be different in two networks. This was found out to be the case for Mn3Ga where the in-plane interaction between Mn atoms with shared Ga were comparable to the interaction without shared Ga, which ensured the 1200 orientation between the Mn neighbours belonging to different networks KhmelevskyiPRB16 . In the present case of Mn2FeGa, we find that the in plane exchange parameters for the three pairs of magnetic atoms are heterogeneous. While the MnI-MnII and MnI-Fe first neighbour interactions without shared Ga are strongly antiferromagnetic, the MnII-Fe interactions are strongly ferromagnetic with magnitudes of the three interactions comparable. The same qualitative feature are observed for the interactions with networks having Ga atoms in between, although the strength of the interactions are reduced, yet non-negligible. These ensure that the frustration in the system is destroyed and we obtain a collinear like magnetic structure. The out-of-planes interactions too fail to induce any frustration as the dominating second neighbour out-of-plane ferromagnetic interactions are about 3 times smaller than the first neighbour out of plane antiferromagnetic interaction. Thus, the presence of Fe in the system indeed introduces substantial Fe-Mn ferromagnetic interactions in the system getting rid of the possible frustration as was conjectured LiuAPL16 .

IV Conclusions

The presented investigation of the structural and magnetic properties of four different structural phases of Mn2FeGa in the framework of same first-principles method enables us to understand the microscopic physics associated with different physical aspects as well as the similarities and differences with the prototype Mn3Ga from which the system under present investigation can be derived, and which is having similar promising applications in the area of spintronics. We find striking similarity between the two systems with regard to relative stabilities of the structural phases. Particularly important was to investigate the importance of the Xa phase which has not been synthesised experimentally. We find that the Xa phase provides an important connection between the hexagonal DO19 and the tetragonal L10 phase. However, we find that the phase is mechanically unstable to anisotropic stress. The origin of the electronic instability associated with this phase appears to be Jahn-Teller effect. The analysis of the electronic structures reveal that the significant hybridisations between Mn and Fe atoms are responsible for the stabilities and atomic magnetism of the other three phases investigated. While Mn2FeGa has substantial similarities with Mn3Ga in these aspects, there is significant qualitative differences in the magnetic structures. Unlike the tetragonal DO22 phase of Mn3Ga, where a canted spin structure associated with one of the Mn sites was observed, the tetragonal L10 phase of Mn2FeGa has a collinear magnetic structure. The results of magnetic exchange interactions clearly showed that the equally strong ferromagnetic Fe-Mn interactions and antiferromagnetic Mn-Mn interactions are responsible for the collinear ferrimagnetic structure in this phase as well as in the Xa phase. The magnetic structure in the hexagonal DO19 phase is where we observe substantial impact of the presence of Fe in the system. The non-collinear magnetic structure in Mn3Ga, which was an artefact of the geometric frustrations and magnetic frustrations due to Mn-Mn exchange interactions now gives way to a collinear like magnetic structure with spins confined to be $a-b$ plane, which is energetically almost degenerate to a collinear configuration with $z$-axis as the easy axis. The origin of this is found to be strong ferromagnetic Fe-Mn components in an antiferromagnetic host as was suggested in the experiment LiuAPL16 .

ACKNOWLEDGMENT

Computational support from C-DAC, Pune, India, PARAM-ISHAN Supercomputer, IIT Guwahati and Department of Physics, IIT Guwahati, India through DST-FIST programme are gratefully acknowledged.