A Ternary Map of Ni–Mn–Ga Heusler Alloys from Ab Initio Calculations

Abstract

In the search for new magnetic functional materials, non-stoichiometric compounds remain a relatively unexplored territory. While experimentalists create new compositions looking for improved functional properties, their work is not guided by systematic theoretical predictions. Being designed for perfect periodic crystals, the majority of first-principles approaches struggle with the concept of a non-stoichiometric system. In this work, we attempt a systematic computational study of magnetic and structural properties of Ni–Mn–Ga, mapped onto ternary composition diagrams. Compositional stability was examined using the convex hull analysis. We show that the cubic austenite has its stability region close to the stoichiometric Ni${}_2$MnGa, in agreement with experimental data, while the tetragonal martensite spreads its stability over a wider range of Mn and Ni contents. The unstable compositions in both austenite and martensite states are located in the Ga-rich corner of the ternary diagram. We note that simultaneous stability of the austenite and martensite should be considered for potentially stable compounds suitable for synthesis. The majority of compounds are predicted to be ferrimagnetically ordered in both austenitic and martensitic states. The methodology used in this work is computationally tractable, yet it delivers some predictive power. For experimentalists who plan to synthesize stable Ni–Mn–Ga compounds with ferromagnetic order, we narrow the target compositional range substantially.

1. Introduction

During the last decades, shape memory Ni–Mn-based Heusler alloys have received notable attention as promising materials for a wide variety of engineering applications such as a magnetic actuator, controller, sensor, and damping technologies. Ni${}_2$MnGa is probably the most known Heusler alloy. Its distinguishing feature is the martensitic transformation in the ferromagnetic (FM) state between the high-temperature cubic austenite and the low-temperature modulated martensite with twin boundary structure occurring at about $T_m\approx 200$ K [1]. In 1996, Ulakko et al. [2] suggested the idea of magnetically induced reorientation of the twin variants structure in Ni${}_2$MnGa. These authors were the first to demonstrate 0.2% field-induced strains along the (001) direction in a Ni${}_2$MnGa single crystal with a magnetic field of 0.8 T applied at 265 K. That success was followed by extensive research of non-stoichiometric Ni–Mn–Ga compositions, covering various aspects such as the crystal structure of the austenite and martensite [3,4,5,6,7], magnetic and magnetoresistance properties [8,9], thermally and magnetically induced deformation [2,10,11], magnetocaloric properties [12,13], heat treatment processes [3,14], and phase diagrams [15,16,17,18,19,20,21,22].

At present, the most studied off-stoichiometric compounds are concentrated near Ni${}_2$MnGa. However, it is clear that expanding that region can lead to new functional phenomena. For example, we know that the martensitic and magnetic transition temperatures are sensitive to the valence electron concentration $e/a$ (electrons per atom). Ni-rich Ni${}_{2+x}$Mn${}_{1-x}$Ga ($0\le x<0.4$ and $7.5\le e/a<7.8$) [18] and Mn-rich Ni${}_2$Mn${}_{1+x}$Ga${}_{1-x}$ ($0\le x<0.6$ and $7.5\le e/a<8.0$) [22] provide an example. Some of these compositions reveal a strong magnetostructural coupling leading to an appearance of the first-order magnetostructural phase transition between the FM or antiferromagnetic (AFM) (ferrimagnetic (FIM)) martensite and the FM or paramagnetic (PM) austenite.

Compositions close to stoichiometric Mn${}_2$NiGa and Ga${}_2$MnNi are not investigated so deeply. According to experimental findings [23,24,25], Mn${}_2$NiGa austenite possesses an inverse Heusler structure ($F\overline{4}3m$, space group no. 216) and undergoes a martensitic transformation in the FIM state at $\approx 270$ K. In addition, a large thermal hysteresis of 50 K and a high Curie temperature $T_C=588$ K are reported. Ab initio studies References [26,27,28] confirm the inverse atomic arrangement. Ga${}_2$MnNi is found to be an FM martensite at room temperature with $T_C$ of about 330 K. The martensitic transformation in the PM state takes place at ≈780 K, and this temperature is the highest reported so far in the Ni–Mn–Ga family [29]. Experimental and theoretical study [29] revealed the regular L2${}_1$ structure of the austenitic phase, in which Ga atoms occupy equivalent positions.

Overall, we see that properly chosen deviations in stoichiometry can tune the magneto-structural coupling in Ni–Mn–Ga; however, the data are still sparse. Experimental studies of non-stoichiometric systems are associated with the complexity of synthesis as well as challenges of mechanical and segregational instabilities. Meaningful and systematic predictions from first-principles computations could allow for more efficient strategies of exploring wide ranges of non-stoichiometric compositions. Earlier, ab initio methods were used to explore structural, magnetic, vibrational, and thermodynamic properties of Ni${}_{2+x}$Mn${}_{1-x}$Ga and Ni${}_2$Mn${}_{1+x}$Ga${}_{1-x}$ alloys (for instance, see References [30,31,32,33,34,35,36]).

It was demonstrated that doping of Ni–Mn–Ga significantly influences lattice constants, bulk moduli, magnetic moments, and Curie temperatures of the austenite and provided qualitative information about the phase stability in terms of the formation energy [37,38,39].

Along with the conventional first-principles methods, chemical disorder in Ni${}_x$Mn${}_y$Ga${}_z$ was simulated using the coherent potential approximation (CPA) [37,38,40]. While this method treats atomic disorder very efficiently, it lacks the ability for complete structural relaxations, leaving some uncertainty that relies on the average description of the calculated properties of a crystal with ideal atomic positions. In order to verify our CPA results, we utilized simulations of the atomic disorder in Ni–Mn–Ga within 16-atom supercells, in [39], using the projector augmented wave (PAW) method [41,42]. Both approaches revealed similar compositional trends, which makes sense, since we did not allow the PAW calculations to fully relax atomic positions, keeping all atoms in the high-symmetry sites. Additional studies [37,38,39] showed similar compositional trends.

Thus, one significant drawback of our previous studies was the lack or internal relaxations of atomic positions, i.e., we assumed that in the high-symmetry cubic phase, atoms would naturally stick to the high-symmetry sites. Moreover, our previous work was focused on the stability of the austenitic phase only, assuming that martensite would have the same stability range. However, further analysis shows that the prediction of stable FM compositions requires simultaneous stability of the austenite and martensite. The challenge with this approach is that martensite requires relaxation of all degrees of freedom, allowing atoms to leave the high-symmetry sites. For Ni–Mn–Ga, local relaxations are essential, leading to such phenomena as structural modulations, which influences lattice parameters, structural stability, and magnetic order. Optimizing all those degrees of freedom for all possible compositions is a formidable task; therefore, we have to develop a simplified approach, which would be computationally manageable, and make a step in the right direction toward reliable ab initio predictions of compositional trends.

In this work, we analyze compositional trends in austenitic and martensitic phases of Ni${}_x$Mn${}_y$Ga${}_z$ alloys over the full ternary composition ranges. Our analysis of the chemical stability utilizes the convex hull methodology, which takes into account most possible reactions, providing a direct measure of the phase stability from the first-principles study [43].

2. Computational Methodology

Magnetic and structural properties were calculated using the PAW method implemented in VASP [41,42]. The exchange correlation effects were treated with the Perdew–Burke–Ernzerhof functional of the generalized gradient approximation [44]. The PAW pseudopotentials with the following electronic configurations were used: 3$p^6$3$d^8$4$s^2$ for Ni, 3$p^6$3$d^6$4$s^1$ for Mn, and 3$d^{10}$4$s^2$4$p^1$ for Ga. A uniform Monkhorst-Pack mesh of $8\times 8\times 8$ k-points was used to sample the Brillouin zone with the first-order Methfessel–Paxton method ($\sigma =0.1$ eV). The plane-wave basis kinetic-energy cutoff of 400 eV was applied, whereas the kinetic-energy cutoff for the augmented charge was chosen as 800 eV. Full crystal structure optimization, including atomic coordinates and lattice parameters, was performed using the conjugate-gradient algorithm. The convergence criterion for the residual forces is ${10}^{-5}$ eV/Å, while the self-consistent calculations are converged with an accuracy of ${10}^{-6}$ eV.

We used 16-atom supercells to model 105 compositions covering uniformly the ternary Ni–Mn–Ga diagram, as shown in Figure 1. In order to facilitate discussions, we split the diagram schematically into the three zones depending on Mn content y and denoted them as Area I ($y\le 25$ at.%), II ($25<y<50$ at.%), and III ($y\ge 50$ at.%).

The choice of these areas based on Mn content is justified by similarities in magnetic configurations for each area, which are attributed to Mn atoms sitting in different sub-lattices. We examine both regular ($Fm\overline{3}m$, space group #225) and inverse ($F\overline{4}3m$, space group no. 216) Heusler structures for all compositions in the austenitic phase and find that the regular structure is preferable for Ni- and Ga-rich compositions (areas I and II). In this case, the alloys assume the #225 and $\#{225}^{\prime }$ structures, as shown in Figure 1. The inverse Heusler structure turns out to be more favorable for Mn-rich compositions (area III). One can find more information about crystal structures in the Supplementary Material (SM) (Table S1 and Figures S1 and S2).

The magnetic ordering of Mn-excess compositions is rather complex due to the magnetic moments of Mn atoms that occupy the Ga and Ni sites and can interact antiferromagnetically with those on the Mn sublattice. In view of this fact, we considered several types of magnetic configurations, which can be grouped as follows: (i) area I: FM only; (ii) area II: FM, FIM-1, and FIM-2 (Mn excess atoms have a reversed magnetic moment to that of Mn atoms on the Mn sublattice); (iii) area III: FM one and seven FIM configurations. All magnetic configurations are listed in Table S2 and visualized in Figures S3 and S4 of the SM.

To analyze the phase stability of the compounds, we used a new three-step procedure, with an improvement over our previous studies [37,38,39], where only the formation energy ($E_{form}$) was used as the stability criterion. At first, we evaluate the stability of austenitic (martensitic) phases in the context of the preservation of cubic (tetragonal/orthorhombic) symmetry after the full relaxation for each composition. In the second step, the $E_{form}$ of the compounds in austenitic and martensitic phases is analyzed. It is assumed that a compound is stable against the decomposition into its pure components if $E_{form}$ has a negative sign in accordance with the equation $E_{form}=E_0^{{\mathrm{Ni}}_x{\mathrm{Mn}}_y{\mathrm{Ga}}_z}-(x{E}_0^{\mathrm{Ni}}+y{E}_0^{\mathrm{Mn}}+z{E}_0^{\mathrm{Ga}})$. Here $E_0^{{\mathrm{Ni}}_x{\mathrm{Mn}}_y{\mathrm{Ga}}_z}$ is the ground-state energy of Ni${}_x$Mn${}_y$Ga${}_z$, while $E_0^{\mathrm{Ni}}$, $E_0^{\mathrm{Mn}}$, and $E_0^{\mathrm{Ga}}$ are the ground-state energies of corresponding components in their preferable structures. Compositions with a positive $E_{form}$ are eliminated from the consideration. In the third step, the phase stability is determined using the convex hull analysis in conjunction with the evaluation of decomposition into pure elements and stable binary compounds. To plot the convex hull, 13 pivot points were used as follows: Ga, Ni, Mn, Ga${}_3$Ni${}_2$, Ga${}_3$Ni${}_5$, Ga${}_5$Ni, Ga${}_7$Ni${}_3$, Ga${}_9$Ni${}_{13}$, GaNi${}_3$, MnGa, MnGa${}_4$, MnNi${}_3$, and Ni${}_2$MnGa. Structural parameters for the pivot points are listed in SM (Table S3). To investigate the decomposition trends, all possible combinations of the reaction products (pivot points) were generated for each compound. We took into account the decomposition of two and three reaction products, each of these might be one of 13 pivot points. The stoichiometric coefficients for reactions are evaluated by solving linear equations, only considering equations with the positive stoichiometric coefficients. For each composition, a fraction of stable reactions is normalized on its own number of possible reactions depending on Ni, Mn, and Ga content. By an example of Ni${}_9$Mn${}_4$Ga${}_3$, 27 possible decomposition reactions are listed in Table S4 of the SM. This compound is stable against decomposition in the case of 22 reactions, which gives us 81% of stable reactions.

3. Results and Discussion

To gain information about the structural stability of compounds under study, we first discuss the arrangement of atoms within a crystalline cubic structure for each composition. Figure 2 illustrates the ternary diagram of the cubic structures distribution with the preferable atomic arrangement for Ni${}_x$Mn${}_y$Ga${}_z$.

The full geometric optimization of structures with the cubic supercell suggested that 54 out of the 105 compositions preserve a cubic symmetry, as opposed to Reference [39] where all 105 compositions were considered with the cubic structure. Another distinctive feature of the full geometric optimization (as opposed to the relaxation without change in atomic positions) is the prediction of slightly different atomic configurations. Thus, the favorable configurations of atoms are turned out to be as follows: (i) the Mn-rich compounds ($y\ge 50$ at.%) have the inverse structure (#216); (ii) the compounds of $x>45$ at.%Ni, $25<y<50$ at.%Mn, and $z<30$ at.%Ga possess the regular structure ($\#225$) in which Ni atoms occupy 8c sites; (iii) the compounds of $x<45$ at.%Ni and $y<25$ at.%Mn tend to favor the $\#{225}^{\prime }$ structure, in which Ga atoms are located at 8c sites. However, neglecting of the atomic relaxations [39] does not provide the #225${}^{\mathrm{mix}}$ atomic ordering for compositions with $30<x<45$ at.%Ni, $6<y<25$ at.%Mn, and $40<z<70$ at.%Ga and the #216 one for Mn${}_{50}$Ni${}_{25}$Ga${}_{25}$. In the latter case, the inverse Heusler structure (#216) has been observed experimentally [26,27]. For the remaining 51 compounds, the initial cubic structure is transformed into the tetragonal and orthorhombic configurations. Among those are 27 compounds mainly with Ga- and Ni-rich content from area I; 15 compounds of area II with variable content of the all constituent elements; and 9 Mn-rich compositions with $z<15$ at.%Ga from the area III.

At the same time, a full geometric optimization of structures with a tetragonal supercell reveals 13 compositions, in which the tetragonal structures are higher in energy than the cubic one. These are 11 Ga-rich and near-equiatomic Ni and Ga compositions from the area I and two compositions with nearly equal Mn and Ga content from the area II. The list of compositions with stable austenite and martensite phases is presented in Tables S5–S10, see the SM.

Figure 3 shows formation energies for compounds in the austenitic and martensitic phases mapped onto the ternary diagram. We have eliminated here the unfavorable compositions in the corresponding phases in accordance with Figure 2.

As can be seen from Figure 3a,b, the formation energy is negative for the majority of compounds. The maps of $E_{form}$ for favorable austenitic and martensite structures are similar. For both cases, the most stable compounds are close to Ni${}_{50}$Mn${}_{25}$Ga${}_{25}$ in concentration ranges of $45<x<65$ at.%Ni, $6<y<30$ at.%Mn, and $18<z<45$ at.%Ga.

In general, the supercell calculations with structural relaxations reproduce the corresponding static relaxation results for Ni–Mn–Ga in the austenitic phase [37,38,39]. However, comparative analysis of the two approaches shows that the Ga- and Mn-rich compounds located at the left and top corners of the ternary map turn out to be unstable ($E_{form}$ > 0) after the full geometric optimization as compared to the static relaxation procedure, which predict the Ga-rich compositions to be unstable only in the austenitic phase [37,39]. In the martensitic phase, among 92 compositions, only two (Ni${}_{6.25}$Mn${}_{12.5}$Ga${}_{81.25}$ and Ni${}_{6.25}$Mn${}_{18.75}$Ga${}_{75}$) orthorhombic structures have positive formation energies and decompose into the elements involved. We have excluded these compositions from further consideration.

The criterion of stability in terms of formation energy is not rigorous since some multicomponent compounds can decompose into stable binary and ternary prototypes together with elemental components. Therefore, it is necessary to consider the thermodynamic phase stability of Ni${}_x$Mn${}_y$Ga${}_z$ in terms of the energy convex hull analysis. The main idea is to compare $E_{form}$ of the ternary compound under study with a possible combination of stable phases (pivot points). The three-dimensional convex hull and its cross-sections into the formation energy-composition space of Ni–Mn–Ga with austenite and martensite phases are shown in Figure S5 of the SM. The majority of investigated compounds have negative formation energies but lie above or closely above the convex hull. This finding indicates that these compositions are in a thermodynamically metastable austenitic or martensitic phase and can decompose.

Figure 4a,b display contour maps of the decomposition reactions indicating the stable, metastable, and unstable Ni${}_x$Mn${}_y$Ga${}_z$ compositions in the austenite and martensite. One can see that for both phases, the Ga-rich compounds concentrated near the left corner of the ternary diagram are unstable and reveal a tendency for decomposing into stable sub-compounds as suggested by the convex hull. It happens because the Ga-rich compounds lie sufficiently above the convex hull. Compounds from the middle part of the diagram are found to be metastable with a fraction of decomposition reactions at about 50%. Finally, compounds of $z<30$ at.%Ga located at the right side of the ternary diagram are potentially stable or nearly stable (i.e., lying on or right above the convex hull).

In particular, the fraction of compounds with the stable tetragonal structure is sufficiently greater than those with the stable cubic structure. We note that the Ni–Mn–Ga compounds predicted to be stable in the austenite phase are concentrated near the stoichiometric Ni${}_2$MnGa. This finding reproduces the experimental ternary diagram well, revealing the most studied Ni–Mn–Ga compounds (see Figure S6 in the SM).

Figure 5a illustrates the contour map of the optimized lattice constant $a_0$ of the cubic austenite Ni${}_x$Mn${}_y$Ga${}_z$. It can be seen that equilibrium lattice parameters depend mostly on the Ga content. Compounds with Ga excess $z>55$ at.% exhibit the large $a_0>6$ Å due to the larger atomic radius of Ga as compared to those of Ni and Mn. A gradual decrease in Ga doping leads to a reduction in $a_0$. For compositions with $x<55$ at.%Ni and $25<z<55$ at.%Ga, $a_0$ takes values in the range between 5.85 and 5.95 Å, whereas the further reduction of Ga at the same Ni content decreases the lattice parameter to $a_0\approx$ 5.78 Å. A sufficiently smaller lattice constant (≈5.68 Å) is observed for Ni-rich compositions with $x>70$ at.%Ni.

In Figure 5b, we display the contour map of the tetragonal ratio $c/a$ of martensitic Ni${}_x$Mn${}_y$Ga${}_z$. For Mn- or Ni-rich compounds with the Ga content of $z<20$ at.%, the tetragonal structure is characterized by the ratios $1.3<c/a<1.4$. An increase in both Ga and Ni or Ga and Mn concentrations simultaneously leads to a reduction in the $c/a$ range to $1.2<c/a<1.3$. The quantity of these compounds is the largest and they are represented within the middle area of diagram with the exception of compositions with $25<x<57$ at.%Ni, $y<35$ at.%Mn, and $31<z<57$ at.%Ga (see the blue map in Figure 5b), which are stable in the cubic phase ($c/a=1$) only. The largest tetragonal ratios varied in the range of $1.4<c/a<1.5$ are found for tetragonally distorted Ga-rich compounds ($x<18$ at.%, $18<y<30$ at.%, and $55<z<70$ at.%). We would like to mention that for Ga-rich compounds ($x<18$ at.%, $y<18$ at.%, and $z>64$ at.%), the orthorhombic structure with $0.9<b/a<1$ and $1.3<c/a<1.4$ is favorable. The calculated lattice constants for both austenitic and martensitic structures are summarized in the SM (see Tables S5–S10).

Our data allow for a systematic comparison of the calculated lattice parameters with available experimental data. The calculated equilibrium lattice constant $a_0$ of the stoichiometric compositions are ≈5.81 Å for Ni${}_2$MnGa, 5.84 Å for Mn${}_2$NiGa, and 5.95 Å for Ga${}_2$MnNi. These values are close to experimental ones: 5.82 Å for Ni${}_2$MnGa [1], 5.9 Å for Mn${}_2$NiGa [23], and 5.84 Å for Ga${}_2$MnNi [29]. Despite the slight difference in theoretical and experimental values for Mn${}_2$NiGa and Ga${}_2$MnNi, it should be pointed out that our results are found to be in excellent agreement with other ab initio studies. For instance, Barman et al. [29] reported $a_0=5.96$ Å for Ga${}_2$MnNi and Kundu et al. [28] obtained 5.84 Å for Mn${}_2$NiGa.

As for the off-stoichiometric compositions, there is a limited list of compounds studied experimentally, and most of them are concentrated near Ni${}_2$MnGa (see Figure S6 in the SM). Figure 5c,d show magnifications of the theoretical and experimental data in the vicinity of Ni${}_2$MnGa maps of $a_0$. The lattice constants of the most experimental compounds take values between 5.81 and 5.86 Å, except the Ni${}_{51.2}$Mn${}_{31.1}$Ga${}_{17.7}$ and Ni${}_{52}$Mn${}_{24.4}$Ga${}_{23.6}$ with the largest $a_0=5.91$ Å and lowest $a_0=5.76$ Å [45], respectively. The discrepancy between experimental and calculated values is ≈1%. Apart from the accuracy of ab initio simulations, the difference could arise from the fact that the experimental values of $a_0$ were obtained at finite temperatures (i.e., at temperatures above $T_m$). For instance, $T_m$ for Ni${}_{51.2}$Mn${}_{31.1}$Ga${}_{17.7}$ with the largest $a_0$ is about 446 K [45]. In contrast, ab initio methods assume zero temperatures and lattice constants have to be slightly less. As Figure 5d suggests, the lattice constant reduces and reaches the value of about 5.76 Å with increasing (decreasing) Ni (Mn) content at the fixed ≈25 at.%Ga. On the other hand, the successive replacement of Ga by Mn and vice versa at the fixed 50 at.%Ni does not change the lattice constant sufficiently, which is about 5.84 Å. The similar trends for Ni${}_{50\pm x}$Mn${}_{25\pm y}$Ga${}_{25}$ and Ni${}_{50}$Mn${}_{25\pm y}$Ga${}_{25\pm z}$ can be seen in the theoretical $a_0$ map, as shown in Figure 5c.

Total magnetic moments (${\mu }_{tot}$) for stable compositions in the austenite and martensite are shown in contour maps of Figure 6a,b, correspondingly, where we also marked areas with favorable magnetic configurations. One can see a similarity in the magnetic moments of the austenitic and martensitic phases. Namely, Ga- and Ni-rich compounds (area I) are ordered ferromagnetically. Ga-rich compositions have a small magnetic moment less than 1.5 ${}_{\mathrm{B}}$/f.u.; which is correlated with larger contents of non-magnetic Ga diluting the ferromagnetically ordered phase. Compositions of area II are predominantly ordered ferrimagnetically with FIM-1 type. At the same time, there are regions with FIM-2, FIM-1(2), and FM spin configurations on the right and left sides of the area II. We note that inverted spins are assigned to Mn-excess atoms located at Ni sites for FIM-1 and at Ga sites for FIM-2. This effect takes place for all Mn excess atoms in the case of the FIM-1(2) state. The largest magnetic moments of 5.5 ${}_{\mathrm{B}}$/f.u. and 4.52 ${}_{\mathrm{B}}$/f.u. are obtained for FM austenite Ni${}_{62.5}$Mn${}_{31.25}$Ga${}_{6.25}$ and FM martensite Ni${}_{68.75}$Mn${}_{25}$Ga${}_{6.25}$. In addition, the largest difference between ${\mu }_{tot}$ in austenite and martensite is about 2.5 ${}_{\mathrm{B}}$/f.u. for composition $50<x<75$ at.%Ni and $z<25$ at.%Ga.

Let us consider the Mn-rich compounds mapped in the area III. Since Mn atoms can occupy four $fcc$ sublattices constituting the inverse Heusler structure depending on nominal composition, several FIM alignments can be realized. Among the seven considered spin configurations, FIM-3 and FIM-7 are predicted to be favorable. For FIM-3, Mn atoms at the 4a site have reversed spin alignment, while Mn atoms at both 4a and $4b$ have opposite spins in the case of FIM-7. The total magnetic moment is smaller than 1.5 ${}_{\mathrm{B}}$/f.u particularly in the tetragonal martensitic phase because of the complex magnetic order. The largest difference in total magnetic moments between the austenite and martensite is ≈2.5 ${}_{\mathrm{B}}$/f.u. for compounds with $10<x<35$ at.% and $z<25$ at.%. The partial magnetic moments for all compositions are listed in Tables S11–S15 of the SM.

In contrast to Reference [39], the relaxation of the atomic positions allows to reveal distinguishing features of the magnetic phase diagram, which are: (i) the FIM-1 phase with a larger magnetic moment (We would like to note that there is the misprint in Figure 3a of Ref. [39]. The unit for total magnetic moment plotted in the area II of diagram is ${}_{\mathrm{B}}$/atom instead ${}_{\mathrm{B}}$/f.u. As consequence, there is a difference between magnetic moments calculated within the full geometric and static relaxation procedures.) exists for compounds with $y<50$ at.%Mn and (ii) the FIM-3 phase occurs for compounds with $y\ge 50$ at.%Mn.

In Figure 6c,d, we illustrate the zoomed-in contour maps of ${\mu }_{tot}$ for a set of theoretical and experimental compounds in the martensitic phase for compositions near Ni${}_{50}$Mn${}_{25}$Ga${}_{25}$. Close inspection of the data shows that the theoretical ${\mu }_{tot}$ correlates with experimental ones. In both cases, the largest ${\mu }_{tot}\approx 4$${}_{\mathrm{B}}$/f.u. is found for Ni${}_{50}$Mn${}_{25}$Ga${}_{25}$, whereas deviations from the stoichiometry such as Ni${}_{50\pm x}$Mn${}_{25\mp x}$Ga${}_{25}$, Ni${}_{50}$Mn${}_{25\pm x}$Ga${}_{25\mp x}$, and Ni${}_{50\pm x}$Mn${}_{25}$Ga${}_{25\mp x}$ result in the magnetic moment reduction to 2 ${}_{\mathrm{B}}$/f.u. For experimental samples $46<x<57$ at.%Ni, $y<19$ at.%Mn, and $28<z<35$ at.%Ga, the magnetic moment is less than 1 ${}_{\mathrm{B}}$/f.u. Importantly, these calculations do not predict any compounds with a favorable tetragonal structure in the present concentration range. Such distinction can be caused by the presence of modulated martensitic structure in the experimental compounds, while modulations were not taken into account in calculations due to an increase in computational problems.

Having the total energies of the austenitic and martensitic phases, we can estimate the martensitic transformation temperature $T_m$ using the relation $\mathsf{\Delta }E\approx k_B{T}_m$, where $\mathsf{\Delta }E$ is the energy difference between the cubic and tetragonal structures and $k_B$ is the Boltzmann constant. Obtained $T_m$ for Ni${}_x$Mn${}_y$Ga${}_z$ are presented in Figure 7. One can see that the martensitic transformation is predicted for ≈40% of compounds under study, most of those are compositions with $z<30$ at.%Ga. The lowest $T_m$ is obtained for compositions with $25<z<60$ at.%Ni, $15<y<30$at.%Mn, and $25<z<55$ at.%Ga, while the largest ones are found for Mn-rich compounds with a small Ga content. In particular, we predict $T_m\approx 1160$ K for Ni${}_{43.75}$Mn${}_{50}$Ga${}_{6.25}$. This compound is close in composition to a binary NiMn, which demonstrates the B2-L1${}_0$ martensitic transformation at about 1000 K [46,47].

Figure 7b,c compare the theoretical and experimental zoomed-in contour maps of martensitic for compositions concentrated near Ni${}_{50}$Mn${}_{25}$Ga${}_{25}$. One can see a noticeable correlation between our results and experimental results. For the Ni${}_{50}$Mn${}_{25\pm x}$Ga${}_{25\mp x}$ compound, the subsequent replacement of Ga by Mn atoms leads to an increase in $T_m$. The theoretically obtained change in $T_m$ of about 150 K between Ni${}_{50}$Mn${}_{25}$Ga${}_{25}$ and Ni${}_{50}$Mn${}_{31.25}$Ga${}_{18.75}$ is in agreement with an experimental one between Ni${}_{50}$Mn${}_{25}$Ga${}_{25}$ and Ni${}_{50}$Mn${}_{28.9}$Ga${}_{21.1}$ [22]. A similar trend can be observed for $T_m$ of Ni${}_{50+x}$Mn${}_{25}$Ga${}_{25-x}$ and Ni${}_{50+x}$Mn${}_{25-x}$Ga${}_{25}$ but with a smaller change of $T_m$ with x. On the other hand, an increase in Ga content leads to $T_m$ reduction as it follows from theoretical and experimental ternary maps. Since substitution of Mn or Ni for Ga provides a change in the $e/a$ ratio, it results in an increasing $T_m$. Figure 7a,b show that the majority of considered compounds demonstrate this tendency and reproduce the experimental $T_m(e/a)$ behavior [17,20,22] in accordance with the Hume–Rothery mechanism. A correlation between Figure 5b and Figure 7 allows us to conclude that a high martensitic transition temperature is observed for compositions with a large tetragonality ($c/a$ ratio) of the martensite. A detailed inspection of Figure 7b,c suggests that the theoretical values of $T_m$ are calculated to be less than the experimental ones. This disagreement can be minimized by an extension of zero-temperature ab initio calculations to finite temperatures in terms of the balance of free energies in the different magnetic, premartensitic, and martensitic phases as proposed by Dutta et al. [35].

4. Conclusions

In summary, we carried out a systematic investigation of Ni–Mn–Ga austenite and martensite with respect to their magnetic and structural properties. A mesh of 105 Ni–Mn–Ga compositions was studied using the first-principles supercell calculations. The ground state atomic and magnetic configurations for cubic austenitic and tetragonal martensitic phases were simulated and considered based on their chemical and structural stability. It is shown that 51 compositions located mainly on the left ($z>45$ at.%Ga) and right ($y>45$at.%Mn) sides of triangle diagrams are unstable in the cubic austenite. In the case of the martensitic phase, the majority of compositions prefer the tetragonal structure with the $c/a$ ratio varied from 1.15 to 1.5. The exception is the compounds $25<x<57$ at.%Ni, $y<35$ at.%Mn, and $31<z<57$ at.%Ga, which prefer the cubic structure. Ga-rich compounds have an orthorhombic structure. The phase stability analysis conducted in terms of the convex hull showed that (i) Ga-rich compositions with $z>60$ at.%Ga decompose with utmost probability into stable pure components and binary phases that lie on the convex hull; (ii) compounds with $35<z<60$ at.%Ga tend to decompose with a probability of about 50%; (iii) alloys with $z<35$ at.%Ga are stable against decomposition, and their phase stability rises with decreasing Ga content.

The magnetic phase diagrams are found to be similar for austenite and martensite, revealing the regions of FM and FIM spin alignment. The ferromagnetically ordered austenitic and martensitic phases are available predominantly in compounds with Mn content $y<30$ at.%. In contrast, the complex FIM order with various types of Mn spin alignment takes place for the compositions with $y>30$ at.%Mn. Alloys with $10<x<35$ at.%Ni and $z<25$ at.%Ga reveal the largest magnetization difference between the austenitic and martensitic phases, which turned out to be ≈2.5 ${}_{\mathrm{B}}$/f.u.

Overall, this work demonstrates the predictive power of first-principles calculations, which can be used for the heuristic search of functional magnetic materials and their characterization and for the prediction of compounds with novel functionalities. Despite limitations in ab initio methods, results obtained in this work demonstrate our improved understanding of what it takes to predict stable martensitic off-stoichiometric magnetic compounds. With the increasing expertise in this field, one could undertake a full-feature study with larger supercells, allowing for various types of structural degrees of freedom, like the modulations in the case of Ni–Mn–Ga.