First principles study of the structural phase stability and magnetic order in variousstructural phases of Mn2FeGa

Ashis Kundu∗ and Subhradip Ghosh†

Department of Physics, Indian Institute of Technology Guwahati, Guwahati-781039, Assam, India.(Dated: June 13, 2017)

We investigate the structural and magnetic properties of Mn2FeGa for different phases(cubic,hexagonal and tetragonal) reported experimentally using density functional theory. The relativestructural stabilities, and the possible phase transformation mechanisms are discussed using resultsfor total energy, electronic structure and elastic constants. We find that the phase transforma-tion form hexagonal to ground state tetragonal structure would take place through a Heusler-likephase which has a pronounced electronic instability. The electronic structures, the elastic constantsand the supplementary phonon dispersions indicate that the transition from the Heusler-like to thetetragonal phase is of pure Jahn-Teller origin. We also describe the ground state magentic structuresin 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 anti-ferromagnetic interactions between the Mn atoms leading to collinear magnetic structures. In thehexagonal phase, the direction of atomic moment are completely in the planes with a collinear likestructure, in stark contrast to the well known non-collinear magnetic structure in the hexagonalphase of Mn3Ga, another material with similar structural properties. The overwhelmingly large ex-change interactions of Fe with other magnetic atoms destroy the possibility of magnetic frustrationin the hexagonal phase of Mn2FeGa. This comprehensive study provides significant insights intothe microscopic physics associated with the structural and magnetic orders in this compound.

I. INTRODUCTION

Mn2YZ compounds in Heusler and Heusler-like struc-tures have been in the attention of spintronics commu-nity due to their multiple possible applications in spin-transfer torque random access memory(STT-MRAM) [1],spin valves [2], magnetic tunnel junction [3], spin gap-less semiconductor [4] and magnetic shape memory ef-fect [5]. Such traits of Mn2YZ compounds are arte-facts of various possible structures in which the sys-tems can crystallise as well as of different orientationsof the Mn spins giving rise to interesting magnetic con-figurations [5–14]. Among the compounds in this family,Mn3Ga [15–19] and Mn2NiGa [5, 14] have been studiedextensively. Mn2NiGa is a recently discovered ferrimag-netic shape memory system which martensitically trans-forms to a low temperature tetragonal structure througha series of complex modulated phases of orthorhombicand monoclinic symmetries [14]; the martensitic trans-formation being driven by phonon softening and Fermisurface nesting [20]. Mn3Ga, on the other hand, is foundto crystallise in a cubic Cu3Au-like [17], a hexagonalDO19 [21] and a tetragonal DO22 phase [16, 22], resultingin a high perpendicular magneto-crystalline anisotropyin the DO22 phase and a large Exchange Bias in theDO19 phase, which are useful for STT-MRAM [16] andmagnetic tunnel junction [21] applications, respectively.First-principles Density Functional Theory (DFT) calcu-lations predicted a Heusler-like metastable structure in

∗k.ashis@iitg.ernet.in†subhra@iitg.ernet.in

Mn3Ga with a half-metallic gap, which phase transformsto the DO22 structure [11, 12]. This phase, however, hasnot yet been synthesised experimentally. ComprehensiveDFT calculations inferred that the phase transition fromDO19 to DO22, as observed in the experiments [17], hap-pens via the Heusler-like phase [18]. First-principles com-putations of the magnetic exchange interactions [19] andthe magnetic anisotropy [16] concluded that the novelmagnetic properties of Mn3Ga in DO19 and DO22 struc-tures are due to non-collinear magnetic structures arisingout of frustrations due to geometry as well as competingexchange interactions between in-plane and out-of-planeMn atoms.

Inspite of enough promises towards a variety of mag-netism related applications, both Mn3Ga and Mn2NiGahave low saturation magnetisations originating from thepredominantly antiferromagnetic interactions betweenthe Mn atoms. This hinders the exploitations of theircomplete potentials in the respective applications. Thelow saturation magnetisation in Mn3Ga limits it’s appli-cability in potential permanent magnet applications, in-spite of having strong uniaxial magnetic anisotropy andlarge Curie temperature. For Mn2NiGa, much lower sat-uration magnetisation [5, 7], in comparison to Ni2MnGa,the prototype magnetic shape memory system in theHeusler family, is an obstacle to obtain significant mag-netic field induced strain required for actuator applica-tions. Attempts have, therefore, been made to combatthe dominant antiferromagnetic interactions in these sys-tems by replacing one of the Mn with ferromagnetic el-ements like Co and Fe. Complete replacement of oneMn atom by Fe, resulting in the compound Mn2FeGa,was expected to circumvent the problems. Synthesisof Mn2FeGa, however, exhibited several important as-

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pects of structure-property relationship. Various crystal-lographic phases with possible magnetic structures wereobserved and predicted experimentally. Gasi et al. [10]reported an inverse tetragonal Heusler structure with alow saturation magnetic moment and a high Curie tem-perature for samples annealed at 4000C. The system be-haved like an exchange spring which was attributed tothe distribution of Fe atoms among the Mn sites result-ing in two different magnetic states of Fe. A Cu3Au-likestructure was observed when the system was annealedat 8000C. A giant tuneable Exchange Bias was later ob-tained in the inverse tetragonal structure with a satura-tion magnetisation as low as 0.09 µB/f.u. [23] suggestinga near compensation of moments from Mn and Fe atoms.DFT calculations, like Mn3Ga, predicted a meta-stableinverse Heusler phase for Mn2FeGa, which transformsto the inverse tetragonal structure observed in the ex-periments [11, 12]. The electronic structure of inverseHeusler Mn2FeGa revealed a pronounced instability as-sociated with the minority spin band [12]. However, un-like Mn2NiGa, the transformation from inverse Heuslerto inverse tetragonal phase didn’t turn out to be volumeconserving, implying the possibility of absence of shapememory effect in this system. Investigations into sput-ter deposited thin films of Mn2FeGa [24] find that thesystem crystallises either in inverse tetragonal or in theCu3Au-like cubic phase. Mossbauer spectroscopy on epi-taxial thin films suggested that Fe might be statisticallydistributed among the two Mn sites leading to a low spinmoment in the inverse tetragonal phase [25]. Heteroepi-taxially grown thin films of Mn-Fe-Ga with compositionnear that of Mn2FeGa also exhibited strong perpendic-ular anisotropy and moderate coercivity suggesting thatthis system can be used in mid-range permanent magnetapplications [26]. Like Mn3Ga, this system could alsobe synthesised in DO19-type hexagonal structure whichyielded a giant Exchange Bias field upto 1.32 kOe [27].The large Exchange Bias was attributed to the presenceof substantial ferromagnetic matrix due to Fe-Mn pairsin an antiferromagnetic host.

The experimental observations clearly indicate thatinvestigations into the phase stability, sub-lattice oc-cupancy and magnetic properties of Mn2FeGa wouldbe insightful. The following issues, in particular, areworth looking into, in order to understand the physicsassociated with Fe substitution in either Mn3Ga or inMn2NiGa, the two compounds having very different func-tional aspects: (i) the energetics of various phases ob-served experimentally or are predicted theoretically butnot observed experimentally, (ii) the magnetic interac-tions among three different magnetic atoms and theirdependencies on the sub-lattice occupancies of the atomsin different structural phases. This can be particularlyrelevant in the hexagonal phase where the Fe atom hasmore than one choice of the crystallographic site it canoccupy, and (iii) the effect of anti-site disorder on themagnetic properties. In this paper, we have under-taken a DFT based comprehensive investigation into the

structural and magnetic properties of the four crystallo-graphic phases of Mn2FeGa, the Cu3Au-like, the hexago-nal DO19-like, the inverse Heusler and the inverse tetrag-onal. Emphasis has been given on identifying the mag-netic structure in the hexagonal phase as it is supposedto be most complex in this phase as was observed inMn3Ga. We have computed the magnetic exchange in-teractions in each structural phase in order to understandthe magnetic structures. The electronic structures of thestructural phases are computed and analysed in order toprovide a possible picture of the phase transformations.The elastic constants and the phonon dispersion relationsfor select phases are computed to supplement the analy-sis from the energetics and the electronic structures. Inthis work, we have not incorporated anti-site disorder inorder to avoid dealing with prohibitively large number ofpossible configurations, in particular for the hexagonalstructure. In absence of concrete quantitative estimateof the anti-site disorder from the experiments, this is jus-tified. The paper is organised as follows: in the section II,computational details are given. The results and discus-sion are presented in section III followed by conclusions.

II. CALCULATIONAL DETAILS

All calculations were performed with spin-polariseddensity functional theory (DFT) based projector aug-mented wave(PAW) method [28] as implemented in Vi-enna Ab-initio Simulation Package (VASP) [29, 30].Perdew-Burke-Ernzerhof (PBE96) implementation ofGeneralised Gradient Approximation (GGA) for ex-change and correlation [31] part in the Hamiltonian wasused throughout. An energy cut-off of 500 eV and aMonkhorst-Pack [32] 25× 25× 25 k-mesh for Cu3Au-likeand inverse Heusler structures , a 15 × 15 × 13 k-meshfor the tetragonal structure and a 13 × 13 × 11 k-meshfor the hexagonal structure were used for self consistentcalculations. Larger k-meshes were used for the densitiesof states calculations of all the structures. For all cal-culations, the total energy convergence criteria and theforce convergence criteria were set to 10−6 eV and to10−2 eV/A respectively. The elastic constants were cal-culated from the second derivatives of the total energieswith respect to the strain tensors [33].

The magnetic pair exchange parameters were calcu-lated with multiple scattering Green function formalismas implemented in SPRKKR code [34]. In here, the spinpart of the Hamiltonian is mapped to a Heisenberg model

H = −∑µ,ν

∑i,j

Jµνij eµi .eνj (1)

µ, ν represent different sub-lattices, i, j represent atomicpositions and eµi denotes the unit vector along the di-rection of magnetic moments at site i belonging to sub-lattice µ. The Jµνij s are calculated from the energy differ-ences due to infinitesimally small orientations of a pair of

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FIG. 1: (a) Possible magnetic configurations in Cu3Au phaseof Mn2FeGa. (b) Total energy as a function of lattice con-stant for different magnetic configurations in Cu3Au phase ofMn2FeGa.

spins within the formulation of Liechtenstein et al. [35].In order to calculate the energy differences, full potentialspin polarised scaler relativistic Hamiltonian with angu-lar momentum cut-off lmax = 3 is used along with aconverged k-mesh for Brillouin zone integrations. TheGreen’s functions are calculated for 32 complex energypoints distributed on a semi-circular contour. The en-ergy convergence criterion is set to 10−5 eV for the self-consistency. The equilibrium lattice parameters and op-timised atomic positions as obtained from the PAW cal-culations are used to obtain the self-consistent potentialsin the multiple scattering Green’s function method.

III. RESULTS AND DISCUSSIONS

A. Structural parameters and magnetic structuresin various crystallographic phases

Experimentally, Mn2FeGa has been observed to crys-tallise in three different phases, the Cu3Au-like, thetetragonal DO22 like and the hexagonal DO19 [10, 27],depending on the annealing temperature. DFT calcu-lations predicted a inverse Heusler phase [11, 12, 36],yet undetected in experiments. In these investigations,the possible magnetic structures associated with differentstructural phases have been indicated through indirectevidences. In this sub-section, we present results on thestructural properties and possible ground state magneticstructures, along with atomic moments for all four struc-tural phases of Mn2FeGa. Throughout the manuscript,the Cu3Au-like phase, the hexagonal phase, the inverseHeusler phase and the inverse tetragonal phases are re-

ferred to as the Cu3Au, the DO19, the Xa and the L10phases respectively. The results are summarised in TableI.

Cu3Au phase: In the ordered Cu3Au structure ofMn2FeGa, the Ga atom occupies the corners of the cubicunit cell while the two Mn, MnI and MnII, along withthe Fe, occupy the face centres (Fig. 1(a)). Depend-ing upon the relative orientations of the moments of thethree magnetic atoms, three inequivalent collinear spinconfigurations are possible in this structure as shown inFig.1(a). Our total energy calculations show that theenergetically lowest configuration is T-I where the twoMn atoms orient anti-parallel (Fig. 1(b)). The calcu-lated lattice parameter agrees well with experimental re-ported one (Table I). The total magnetic moment is only0.68 µB/f.u., with MnI having a larger moment amongthe two Mn atoms. The Fe moment is only 1.28 µB ,much less than that in it’s elemental solid phase. Moss-bauer spectra of Mn2FeGa, annealed at 800◦C [10], con-firmed the Cu3Au-like structure but the measurementsof hyperfine field distributions suggested that the aver-age moment at the Fe site is only 0.5µB as Fe may oc-cupy Mn sites as well. Since we have not included theanti-site disorder, that is mixing of Fe and Mn atoms atall crystallographic sites, the discrepancy with the ex-perimental results is natural. Since the experimentalmeasurements did not indicate any possibility of non-collinearity in the magnetic structure, our considerationof the collinear magnetic structure is justified, and thus,our results provide the possible ground state magneticconfiguration if Mn2FeGa crystallises in a perfectly or-dered Cu3Au phase. A comparison with Mn3Ga in theordered Cu3Au phase [17] shows that replacing one Mnatom by Fe has led to a reduction of the total moment ofthe system. In Mn3Ga, the MnI and MnII moments were-2.93 µB and 2.24 µB respectively, leading to a total mo-ment of 1.55 µB per formula unit. The atomic momentsin Mn2FeGa, as obtained here, suggest that the Mn mo-ments have not changed significantly, but the reductionin the total moment, in comparison to Mn3Ga, happensdue to weaker exchange splitting associated with the Featoms.

Xa phase: DFT calculations predict a Hg2CuTi struc-ture (space group no. 216; F 43m) for Mn2FeGa withfour inequivalent Wyckoff positions(4a, 4b, 4c, 4d) in theunit cell [11, 36]. In this structure, also known as Xa

structure, the MnI and MnII atoms occupy 4a(0,0,0) and4c(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 byFe and Ga atoms respectively. The Xa structure ofMn2FeGa is shown in Fig. 2(a). The equilibrium lat-tice constant (Table I) obtained in this work is consis-tent with the reported results [11, 36]. The calculatedmagnetic moment is close to 1.0 µB/f.u, in very goodagreement with the existing DFT results. The atomicmoments (Table I) show that the Fe moment is quenchedsubstantially, and that the low total moment arises due toanti-parallel alignment of the two Mn atoms. The ground

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TABLE I: The calculated lattice parameters(in A), total(M) and atomic magnetic moments(MX)(in µB per formula unit)of Mn2FeGa in Cu3Au, Xa, L10 and DO19 phases. The results from experiments and other DFT calculations are given inparenthesis

Structure Lattice parameter a Lattice parameter c M MMnI MMnII MFe MGa

Cu3Au 3.69(3.7526 [10]) 3.69 0.68 -3.08 2.55 1.28 -0.03Xa 5.78(5.80 [36],5.79 [11]) 5.78 1.04(1.03 [11]) -2.07 2.80 0.29 0.01L10 3.68(3.7915 [10],3.68 [12]) 7.29(7.1858 [10],7.29 [12]) -0.80(0.96 [1],-0.78 [12]) -2.43 2.96 -1.41 0.04DO19 5.25(5.3151 [27]) 4.20(4.3050 [27]) 1.22(1.26 [27]) -2.84 2.54 1.55 -0.03

MnI MnII Fe Ga(b)(a)

FIG. 2: (a) The Xa structure and (b) the L10 structure ofMn2FeGa. The spin configurations shown are the groundstate configurations obtained from our calculations.

state magnetic configuration in this structure is consis-tent with that of Mn2NiGa [7], in which Fe substitutionat the Ni site gives rise to the present compound un-der investigation. Interestingly, the total and the atomicmoments of Mn2FeGa in this structural phase are verysimilar to Mn2NiGa. In contrast, there is substantial dif-ferences between the total and the atomic moments ofMn2FeGa and Mn3Ga in this phase. In Mn3Ga, the to-tal moment is nearly zero in this phase [11]. This is dueto the fact that unlike Mn2FeGa, Mn3Ga has a Heuslerlike coordination (space group 225) and thus the Mn mo-ments at 4a and 4b sites compensate the moment at 4csites associated with the other Mn aligning anti-parallel.The alteration in the coordination in Mn2FeGa, thus, in-troduces magnetic interactions, very different from thosein Mn3Ga.

L10 phase: The tetragonal L10 phase of Mn2FeGa(Fig. 2(b)) has been synthesised [10, 25] and beenthe subject of intense investigations due to the possiblePMA and high coercivity in this structure. Investiga-tions into the magnetic structure concluded that the sys-tem has a ferrimagnetic order, and unlike Mn3Ga [16],the spins are collinear. The magnetisation measurementon polycrystalline samples at 10 T field yielded a non-saturating magnetisation of about 1.5 µB [10]. Using re-sults from magnetometry, Betto et. al. [25] estimated theatomic magnetic moments and the spin structure (Fig.2(b)). Their estimation of atomic moments, based uponmagneto-optical sum rules resulted in equal moments on

MnI and Fe sites with MnII sites having a 30% less mo-ment. Incorporation of Fe anti-sites in their calculationsincreased the moment at MnII sites resulting in nearlyequal moments at all three sites, which would result ina very small net moment in the system. This was incontrast to Mn3Ga, where the estimated moments asso-ciated with the MnI and MnII were 2.67 µB and 4.74 µBrespectively, producing a net moment of about 0.6 µBper formula unit [16]. DFT calculations by Wollmann etal. [12], on the other hand, obtained unequal magneticmoments at MnI and MnII sites, along with a momentat the Fe site which is nearly 50% of the moment whichis larger among the two Mn moments, producing a netmoment of about 0.8 µB per formula unit. Our resultsin Table I are in excellent agreement with those of Woll-mann et al. Particularly interesting is the gain in Fe mo-ment in comparison to the Xa phase, and the change init’s orientation with respect to the Mn sub-lattices. Suchbehaviour of the atomic moments, in particular of Fe inMn2FeGa, is in stark contrast to that of Ni in Mn2NiGa.In prototype shape memory alloys like Ni2MnGa and inMn2NiGa, the systems undergo martensitic transforma-tions from L21 Heusler and Xa structures, respectivelyto L10 structures, conserving their volumes with littlechanges in their atomic and total magnetic moments.The very different behaviour of the atomic magnetic mo-ments in Xa and L10 phases of Mn2FeGa indicate that thephysics associated with the system in these phases wouldbe different from those of shape memory Ni2MnGa andMn2NiGa.

In order to elucidate this point, we have optimised thestructural parameters in the L10 phase of Mn2FeGa bycomputing the total energy as a function of the tetrag-onal deformation, represented by the c/a ratio, keepingthe volume fixed at that of the Xa phase. The resultsare presented in Fig. 3. The energetics of Ni2MnGa andMn2NiGa are presented for comparison. We find a localenergy minima at c/a = 1.0 and a global energy min-ima at c/a = 1.40. For Mn2FeGa, the energy surfacelooks very different compared to those of Mn2NiGa andNi2MnGa. We further computed the total energies forthe L10 structure by varying the volume. The results areshown in the inset of Fig. 3. We clearly see that the op-timised volume of the L10 phase is higher by 2.54% thanthe volume of Xa structure which indicates that the Xa

to L10 transformation in this system is not a martensiticone and the system would not exhibit shape-memory ef-

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FIG. 3: Total energy as a function of tetragonality(c/a) forMn2FeGa in the L10 phase. The total energy curves ofNi2MnGa and Mn2NiGa are presented for comparison. Thezero energy is taken to be the energy corresponding to the Xa

phase(c/a=1). Total energy of Mn2FeGa as a function of c/afor different volumes in L10 phase are shown in inset. Eachcurve in the inset represents the percentage change in volumewith respect to the equilibrium volume in the Xa phase.

MnI MnII Fe Ga(a)

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FIG. 4: (a) Left: Hexagonal(DO19) structure of Mn3Ga. Theatomic sites marked T and B correspond to sites in the topand the bottom planes respectively. (a) Right: The DFT cal-culated magnetic structure of Mn2FeGa. (b) The magneticstructure of Mn2FeGa in DO19 phase modelled after the ac-tual one shown in (a), with the magnetisation axis along z-direction.

fect. The optimised structural parameters, presented inTable I are in good agreement with the results obtainedfrom experiments and previous DFT calculations.

DO19 phase: The hexagonal DO19 phase ofMn2FeGa has been synthesised only recently [27] af-ter annealing the polycrystalline sample to 6000C. Inthe hexagonal(DO19) structure(space group no. 194;P63/mmm) there are two formula units of atoms(8atoms) arranged in two different planes of the unit cell.

(T1, T2, T3, T4) and (B1, B2, B3, B4) are the latticesites in top and bottom planes respectively, as shownin 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 occupyT4 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 mag-netic structure in Mn3Ga. Accordingly, different non-collinear spin configurations were investigated to obtainthe ground state magnetic configuration in the system byDFT calculations [18]. The configuration with the low-est energy was found to be a triangular structure with a1200 mutual orientations of the neighbouring Mn atomsin the same plane as well as of the neighbours in the ad-jacent planes. This was in confirmation with the resultsof Neutron diffraction [37]. Subsequently, the magneticmoment came out to be nearly zero with the momentat Mn site to be ∼ 2.5 µB [18, 37]. The magnetisationmeasurements on DO19 Mn2FeGa obtained a saturationmoment of 1.26 µB per formula unit [27]. The authorsproposed that this is an effect of increasing Fe-Mn lo-cal ferromagnetic matrix in otherwise antiferromagnetichost as Fe can occupy Mn sites either in a single layer oracross the layers of the hexagonal planes. If hexagonalMn2FeGa is considered to be formed out of hexagonalMn3Ga by replacing one Mn atom with Fe, in the DO19

unit cell, two Mn atoms out of six are to be replacedby Fe atoms. Since these Fe atoms can occupy any twopositions out of the six inequivalent sites among the twoplanes (keeping the position of Ga atoms fixed), manyinequivalent combinations of site occupancy are possiblein this phase. Consideration of symmetry, however, re-duces the possible number of configurations to only sevenwhich are given in Table II. Due to the geometrical frus-tration arising out of triangular networks in the DO19

structure, consideration of non-collinear spin structuresas the starting point of the calculations are necessary. Inabsence of any experimental information regarding thespin structure of this system in DO19 phase, we startedour calculations with the reported spin configuration ofMn3Ga in the literature [18]. We optimised the mag-netic structures and calculated the formation enthalpiesof all seven configurations. The results are presented inTable II. The results suggest that the configuration withconfiguration number 4 in Table II has the lowest forma-tion enthalpy. The corresponding magnetic structure isshown in Fig. 4(a). The calculated total magnetic mo-ment for this configuration is 1.30 µB/f.u. which agreesvery well with the experimental magnetisation value of1.26 µB/f.u. [27].

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TABLE II: The possible configurations due to differ-ent site occupancy patterns in DO19 phase of Mn2FeGa.Ef (eV/f.u.) is the formation enthalpy for a given configura-tion. Mtot(µB/f.u.) is the calculated total magnetic momentin a given configuration.

Sites and occupanciesNo. T1 T2 T3 T4 B1 B2 B3 B4 Ef Mtot

1 Mn Fe Mn Ga Mn Fe Mn Ga -0.460 3.032 Mn Fe Mn Ga Mn Mn Fe Ga -0.459 1.633 Mn Fe Mn Ga Fe Mn Mn Ga -0.504 1.284 Mn Mn Fe Ga Mn Mn Fe Ga -0.509 1.305 Mn Fe Fe Ga Mn Mn Mn Ga -0.456 1.466 Fe Mn Fe Ga Mn Mn Mn Ga -0.456 1.467 Fe Fe Mn Ga Mn Mn Mn Ga -0.456 1.45

Contrary to the expectations, our calculations showthe minimum energy configuration of the spins to be onewhere they are confined to the planes and pointing ei-ther along the a-direction or against it, a collinear ar-rangement when observed in the a − b plane. Since themagnetic arrangement is like a collinear one, we calcu-lated the total energy of this configuration by aligningthe spins along the z-direction. The differences in ener-gies of the two configurations turned out to be less than1 meV per formula unit. The magnetic moment of theconfiguration with spins directed either along or againstz-direction is 1.22 µB per formula unit (Table I), veryclose to that obtained for the minimum energy config-uration with spins in the a − b plane (Table II). Thisjustifies using the collinear configuration with z-axis theeasy axis for further calculations. The lattice parametersobtained by optimising this configuration, presented inTable I, agree quite well with the experiments. Thus, thepresence of Fe in the system appears to have taken careof the frustration in the system and produces a collinearkind of magnetic structure which explains the significantnon-zero magnetisation of Mn2FeGa in this phase. It isworth mentioning that such collinear like magnetic struc-tures are also obtained for some of the other configura-tions considered here, but they are energetically higher,yet possible to form as is obvious from the negative val-ues of the formation enthalpies (Table II). The optimisedmagnetic structure corresponding to each of the configu-rations considered here are presented in the supplemen-tary information.

B. Stabilities of various phases: analysis fromenergetics, electronic structure and elastic constants

Fig. 5 shows the variations in the total energies withvolume for the four phases of Mn2FeGa. We relaxed allthe structural parameters to obtain the most stable struc-ture in a given phases. The tetragonal phase comes out tobe energetically the most stable phase while the Cu3Auphase is having the highest total energy. The hexago-nal, and the Xa phases lie in between with the Xa phasehaving a total energy in between the hexagonal and the

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tetragonal phase. The relative energies of these phasesobtained this way agrees with the experiments in a wayas the tetragonal phase is observed to be the low temper-ature phase [10] while annealing to a high temperaturestabilises the Cu3Au phase; the hexagonal phase beingobtained only when the sample is annealed at temper-atures in between the annealing temperatures to obtainthe L10 and the Cu3Au phases. The other interestingfeature of the relative energetics is that the DO19 andthe Xa phases are energetically very close near the equi-librium volume of the DO19 phase. Since there is nopossibility of a direct DO19 to L10 transformation due tosymmetry constraints [18], and that a transformation be-tween the Xa and DO19 is possible, as illustrated in caseof Mn3Ga [18], our results imply that the phase trans-formation from DO19 to L10 can take place via the Xa

phase. Thus, although the Xa phase has not yet beenobtained experimentally, possibly due to requirement ofgrowth by non-equilibrium methods as it appears to bemetastable (Fig. 3), it’s presence is very important to un-derstand the stabilities of the DO19 and the L10 phasesand the possible path of transformation between the twophases.

In order to gain further insight into the stabilities ofthese phases in Mn2FeGa, we take a look at their elec-tronic structures. In Fig. 6, we show the total andatom-projected densities of states of Mn2FeGa in vari-ous structural phases. These also help us understand-ing the behaviour of atomic magnetic moments in eachphase. Across all the structures, we find that the minor-ity(majority) band of MnI is nearly full(nearly empty)while the opposite is true for MnII bands. This ex-plains the anti-parallel orientation of the moments as-sociated with the two Mn atoms. For Cu3Au and DO19

phases, there is strong hybridisation and delocalisationof 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 betweenMnI and Fe bands are minimal for Cu3Au, DO19 and Xa

phases while it is substantial in the L10 phase. In spite

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L10

FIG. 6: Spin polarised total and atom-projected densities ofstates of Mn2FeGa in various structural phases.

of less hybridisation, both MnI and Fe minority states inthe Cu3Au and DO19 phases are delocalised while theyare more localised in the Xa phase. Thus, the densitiesof states in the Xa phase stand out in contrast to theother three phases; the highlights being more localisedatomic densities of states and thus more structured totaldensities of states in comparison to relatively structure-less total densities of states in other phases. The othermajor difference observed in the densities of states in theXa phase is a sharp peak at the Fermi level in the ma-jority band and a half-metallic gap in the minority band.The peak at the fermi level is due to hybridisation ofMnI and Fe t2g states while the gap in the minority bandis flanked by the well separated t2g and the eg states ofmainly Fe atom. The peak in the majority band densi-ties of states implies instability in this structure whichtriggers the tetragonal distortion and subsequent stabil-isation of the L10 phase. The t2g states at the Fermilevel in the Xa phase now splits into three non-degeneratestates due to the lowering of the symmetry, thus wash-ing away the peak. The symmetry lowering gets rid ofthe half-metallic gap too as the system stabilises by fill-ing in the gap, that is by re-distributing the states intothe lower energy states. The delocalisation and stronghybridisation of the MnI and Fe minority states in L10structure is the outcome of this. The quenching of theFe moment in the Xa structure can also be understoodfrom the electronic structure. The nearly full Fe statesin both spin channels reduces the exchange splitting as-sociated with Fe. When the system tetragonally distortstowards the L10 structure, the Fe states fill out signifi-

cant part of the occupied region in the minority band,while the spectral weight associated with the Fe statesin the majority band shifts towards the unoccupied re-gion, thus recovering the exchange splitting observed inthe Cu3Au and DO19 phases. The MnII states in themajority bands associated with the Cu3Au and DO19

structures are distributed in a nearly identical way, pro-viding an explanation to the almost equal MnII momentsin these two phases. The proximity of the MnI momentsin these two phases also emerge due to the near identi-cal minority spin densities of states of MnI which impliesnearly same exchange splitting associated with the MnIatoms. The Fe moments are also nearly equal in thesephases which can be explained in a similar way as Fehybridises with both Mn in relevant spin channels. Themoments associated with MnII in these two phases areless than that of MnI. The relatively less exchange split-ting associated with the MnII atoms can be understoodas the effect of stronger hybridisation with the Fe states.In the Xa and L10 states, the exchange splitting of MnIIis more than that of MnI. This happens due to appear-ance of more localised MnII and Fe states in the majorityspin band, thus reducing hybridisations.

The densities of states imply that Cu3Au, DO19 andL10 phases are stable as was observed experimentally.The strong hybridisations between Mn and Fe states ineither spin channel are found to be responsible behind it.The reduced hybridisations can be correlated with the in-stability in the Xa phase as implied by the large densitiesof states at the Fermi level. In order to understand theorigin of this instability and to predict whether this phasecan ever be synthesised, we first check the mechanicalstability by computing the elastic constants. The elasticconstants for all four phases are given in Table III.

TABLE III: The calculated elastic constants (in GPa) ofMn2FeGa in Cu3Au, Xa, L10 and DO19 phases.

Structure B C11 C12 C13 C33 C44 C66 C′

Cu3Au 131.3 141.6 126.1 - - 33.9 - 7.8Xa 147.2 166.8 137.4 - - 127.3 - 14.7L10 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

The calculated elastic constants satisfy all the stabil-ity criteria for Cu3Au, Xa, DO19 and L10 phases [33, 38].Among all, the shear modulus(C′ = (C11−C12)/2) givesan insight to the stability of all the structures with re-spect to the shear(for hexagonal, C66 = C ′). For all thestructures, C ′ is positive and indicate the relative me-chanical stability of the structure against shear. The C ′

value of tetragonal structure is higher than other struc-tures which might indicate that it is the most stablestructure against shear instability. For the Xa structure,however, the value of C′ is sufficiently small, in compar-ison to C44. The elastic anisotropy ratio A

(= C44/C ′

)is a measure of the stability of a crystal with cubic sym-metry against stress along (110) planes [39]. In the Xa

8

phase of Mn2FeGa, this ratio is 8.66, indicating that thesystem can be unstable against anisotropic stress. In or-der to gain further insight into the possible consequencesof it, and thus understand the mechanism of the Xa toL10 transformation in this system, we have computed thephonon dispersion relations in both Xa and L10 phases(Figs. 8 and 9, supplementary material). The disper-sion relations indicate dynamical stability in both phases.None of the acoustic modes in the Xa phase were imagi-nary, or produced any pronounced softening. These indi-cate that the transformation from Xa to L10 phase is notphonon driven as was the case for Mn2NiGa [20]. Thus,the phase transition from Xa to L10 can be considereddue to the well known Jahn-Teller effect.

C. Magnetic Exchange interactions

In order to understand the origin of the magnetic struc-tures in various phases as depicted in sub-section III-A,we now look at the results of various inter-atomic andintra-atomic exchange interactions for the Xa, L10 andDO19 phases. Since the L10 and DO19 phases are themost promising ones from applications point of view andthe Xa phase has strong connection with the L10 phaseas has been established in the previous sub-sections,we have investigated the exchange interactions for thesethree phases. The results for the Xa and the L10 phasesare presented in Fig. 7(a) and (b), respectively. Forboth phases, the nearest neighbour strong antiferromag-netic and ferromagnetic interactions between two differ-ent pairs of atoms dominate the scene. The higher neigh-bour interactions are too small to influence the magneticstructure, which would be collinear. In case of the tetrag-onal DO22 phase of Mn3Ga, neutron diffraction experi-ments [16] found a small canting of Mn spins occupyingthe 2b sites (the sites occupied by Fe in Fig. 2 (b)).The calculated exchange parameters established frustra-tion associated with that particular Mn site, which to-gether with the in-plane magnetic anisotropy at thatsite, was used to explain the spin canting at that site.In the present case, we do not see any such possibil-ity arising out of the qualitative and quantitative na-ture of the three major inter-atomic parameters associ-ated with MnI-MnII, Fe-MnI and Fe-MnII pairs. Forall three of them, at least the first two nearest neigh-bour exchange parameters are either ferromagnetic or an-tiferromagnetic, thus ruling out possibility of magneticfrustration. In Both Xa and L10 structures, the over-whelmingly dominating interaction is the antiferromag-netic MnI-MnII. Same was the case for DO22 Mn3Ga.The qualitative difference between Mn2FeGa and Mn3Gamagnetic interactions is observed in cases of the exchangeparameters involving Fe. In the Xa phase, the first neigh-bour antiferromagnetic Fe-MnI exchange parameter is ofthe same magnitude as that of the first neighbour ferro-magnetic Fe-MnII one although the former pair is at alarger distance. In the L10 phase, we observe a role rever-

0.4 0.6 0.8 1 1.2 1.4d (units of a)

-30

-20

-10

0

10

J ij (

meV

)

MnI - MnIIMnI - FeMnII - FeMnI - MnIMnII - MnIIFe - Fe

(a)

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2d (units of a)

-30

-20

-10

0

10

J ij (

meV

)

MnI - MnIIMnI - FeMnII - FeMnI - MnIMnII - MnIIFe - Fe

(b)

FIG. 7: Magnetic exchange interaction(Jij) of Mn2FeGa asa function of interatomic distance(d) in (a) Xa and (b) L10

phases.

sal of these two interactions: the Fe-MnI is now stronglyferromagnetic in the first neighbour, the interaction be-ing ferromagnetic even for higher neighbours, while theFe-MnII exchange is now weakly antiferromagnetic. Thisrole reversal can be explained from the structural defor-mation of Xa that leads to the L10. The contraction inthe basal plane now reduces the Fe-MnI distances alongwith a subsequent increase in the Fe-MnII distance dueto elongation in the [001] direction. This increase in theFe-MnII distances weaken the exchange interactions be-tween this pair and makes it weakly antiferromagnetic,while the opposite happens for the Fe-MnI pairs whichare now closer. The presence of strong ferromagneticand antiferromagnetic components in the first neighbourexchange interactions of Mn2FeGa in the Xa and L10phases along with weak higher neighbour interactions arethe reasons behind the collinear ferrimagnetic structurein these phases. Thus, the presence of Fe in the sys-tem inflicts a significant ferromagnetic component in themagnetic interactions, in comparison to Mn3Ga.

The collinear-like magnetic configuration obtained inthe DO19 phase of Mn2FeGa, as opposed to the non-collinear magnetic structure in Mn3Ga where the neigh-bouring atoms(both in-plane and out-of plane) alignthemselves in a triangular structure with an angle of1200 between them indicates that Fe plays a vital role.

9

a

b

FIG. 8: The nearest neighbour Mn-Mn and Mn-Fe bonds inDO19 structure of Mn2FeGa are shown(a view from c axis).Network with two adjacent basal planes(the unit is shown onthe top right corner with black border) are shown. Atomsmarked ‘i ’ and ‘o’ correspond to whether they are in planeor out of plane, respectively. The relatively darker circlesassociated with each colour are at the top kagome plane andthe relatively lighter circles are at the bottom kagome plane.The colour codes are the ones used in other figures.

The substantial magnetisation obtained in the measure-ments [27] on the DO19 phase of Mn2FeGa was connectedwith the possibility of creation of Fe-Mn ferromagneticmatrix in the system. The magnetic exchange interac-tions calculated for this phase of Mn2FeGa and presentedin Fig. 9 helps understanding the origin of the collinearlike magnetic structure obtained from the total energycalculations. The network of the nearest neighbour mag-netic atoms in the two neighbouring planes in the DO19

structure is shown in Fig. 8. It is clear from the struc-ture that there are two types of triangular networks inthe basal planes-one with the shared Ga neighbours andone without. Consequently, the strengths of the inter-actions between the same pair of specie at the same dis-tance would be different in two networks. This was foundout to be the case for Mn3Ga where the in-plane interac-tion between Mn atoms with shared Ga were compara-ble to the interaction without shared Ga, which ensuredthe 1200 orientation between the Mn neighbours belong-ing to different networks [19]. In the present case ofMn2FeGa, we find that the in plane exchange parametersfor the three pairs of magnetic atoms are heterogeneous.While the MnI-MnII and MnI-Fe first neighbour interac-tions without shared Ga are strongly antiferromagnetic,the MnII-Fe interactions are strongly ferromagnetic withmagnitudes of the three interactions comparable. Thesame qualitative feature are observed for the interactionswith networks having Ga atoms in between, althoughthe strength of the interactions are reduced, yet non-negligible. These ensure that the frustration in the sys-tem is destroyed and we obtain a collinear like magneticstructure. The out-of-planes interactions too fail to in-

0.4 0.5 0.6 0.7 0.8 0.9 1 1.1d (units of a)

-30

-20

-10

0

10

20

J ij (

meV

)

MnI - MnII (in plane)

MnI - MnII (out of plane)

MnI - Fe (in plane)

MnI - Fe (out of plane)

MnII - Fe (in plane)

MnI - MnI (out of plane)

MnII - MnII (out of plane)

Ga

Ga

Ga

FIG. 9: Magnetic exchange interaction(Jij) of Mn2FeGa as afunction of inter atomic distance(d) in the DO19 phase.

duce any frustration as the dominating second neighbourout-of-plane ferromagnetic interactions are about 3 timessmaller than the first neighbour out of plane antiferro-magnetic interaction. Thus, the presence of Fe in thesystem indeed introduces substantial Fe-Mn ferromag-netic interactions in the system getting rid of the possiblefrustration as was conjectured [27].

IV. CONCLUSIONS

The presented investigation of the structural andmagnetic properties of four different structural phasesof Mn2FeGa in the framework of same first-principlesmethod enables us to understand the microscopic physicsassociated with different physical aspects as well as thesimilarities and differences with the prototype Mn3Gafrom which the system under present investigation can bederived, and which is having similar promising applica-tions in the area of spintronics. We find striking similar-ity between the two systems with regard to relative sta-bilities of the structural phases. Particularly importantwas to investigate the importance of the Xa phase whichhas not been synthesised experimentally. We find thatthe Xa phase provides an important connection betweenthe hexagonal DO19 and the tetragonal L10 phase. How-ever, we find that the phase is mechanically unstable toanisotropic stress. The origin of the electronic instabilityassociated with this phase appears to be Jahn-Teller ef-fect. The analysis of the electronic structures reveal thatthe significant hybridisations between Mn and Fe atomsare responsible for the stabilities and atomic magnetismof the other three phases investigated. While Mn2FeGahas substantial similarities with Mn3Ga in these aspects,there is significant qualitative differences in the magneticstructures. Unlike the tetragonal DO22 phase of Mn3Ga,where a canted spin structure associated with one ofthe Mn sites was observed, the tetragonal L10 phase ofMn2FeGa has a collinear magnetic structure. The resultsof magnetic exchange interactions clearly showed that the

10

equally strong ferromagnetic Fe-Mn interactions and an-tiferromagnetic Mn-Mn interactions are responsible forthe collinear ferrimagnetic structure in this phase as wellas in the Xa phase. The magnetic structure in the hexag-onal DO19 phase is where we observe substantial impactof the presence of Fe in the system. The non-collinearmagnetic structure in Mn3Ga, which was an artefactof the geometric frustrations and magnetic frustrationsdue to Mn-Mn exchange interactions now gives way to acollinear like magnetic structure with spins confined tobe a − b plane, which is energetically almost degenerateto a collinear configuration with z-axis as the easy axis.

The origin of this is found to be strong ferromagneticFe-Mn components in an antiferromagnetic host as wassuggested in the experiment [27].

ACKNOWLEDGMENT

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

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