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

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## Abstract

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## 1. Introduction

## 2. Computational Methodology

## 3. Results and Discussion

## 4. Conclusions

## Supplementary Materials

**2017**, 152, 125–132.). The points map the most studied compositions (about 900) reported in the literature; Table S1: Positions of the Ni, Mn and Ga atoms considered in the regular (space group #225) and inverse (space group #216) Heusler structures; Table S2: The considered orientation of spin magnetic moment of Mn atoms located at different Wyckoff positions in the regular (#225) and inverse (#216) Heusler structures. Here the reversed spin orientation is marked in a red color to enhance visibility. Notice that the Ni spin magnetic moment is aligned along the direction ⇑; Table S3: Space group, lattice parameter a (in Å), $c/a$ ratio, and total energy ${E}_{tot}$ (in eV/atom) per atom for the pivot points of the ternary Ni–Mn–Ga convex hull; Table S4: Possible decomposition reactions and the decomposition energy ${E}_{dec}$ (in eV/atom) between the investigated alloy and reaction products for Ni${}_{9}$Mn${}_{4}$Ga${}_{3}$. The positive value indicates the phase instability against the decomposition process and vice versa; Table S5: Optimized lattice constants a, b, c (in Å) and their ratios, the total energy ${E}_{tot}$ and the formation energy ${E}_{form}$ (in eV/atom) as well as the preferable structure for compounds in the austenite phase from the area I; Table S6: Optimized lattice constants a, b, c (in Å) and their ratios, the total energy ${E}_{tot}$ and the formation energy ${E}_{form}$ (in eV/atom) as well as the preferable structure for compounds in the austenite phase from the area II; Table S7: Optimized lattice constants a, b, c (in Å) and their ratios, the total energy ${E}_{tot}$ and the formation energy ${E}_{form}$ (in eV/atom) as well as the preferable structure for compounds in the austenite phase from the area III; Table S8: Optimized lattice constants a, b, c (in Å) and their ratios, the total energy ${E}_{tot}$ and the formation energy ${E}_{form}$ (in eV/atom) for compounds in the martensite phase from the area I; Table S9: Optimized lattice constants a, b, c (in Å) and their ratios, the total energy ${E}_{tot}$ and the formation energy ${E}_{form}$ (in eV/atom) for compounds in the martensite phase from the area II; Table S10: Optimized lattice constants a, b, c (in Å) and their ratios, the total energy ${E}_{tot}$ and the formation energy ${E}_{form}$ (in eV/atom) for compounds in the martensite phase from the area III; Table S11: Element resolved magnetic moments (in ${}_{\mathrm{B}}$), total magnetic moments (in ${}_{\mathrm{B}}$/f.u.) and favorable magnetic reference state for compounds in the austenite and martensite phase from the area I; Table S12: Element resolved magnetic moments (in ${}_{\mathrm{B}}$), total magnetic moments (in ${}_{\mathrm{B}}$/f.u.) and favorable magnetic reference state for compounds in the austenite phase from the area II. Here ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ga}\right)}}$ and ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ni}\right)}}$ are the magnetic moments of Mn atoms placed at the Ga- and Ni sublattice, respectively; Table S13: Element resolved magnetic moments (in ${}_{\mathrm{B}}$), total magnetic moments (in ${}_{\mathrm{B}}$/f.u.) and favorable magnetic reference state for compounds in the martensite phase from the area II. Here ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ga}\right)}}$ and ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ni}\right)}}$ are the magnetic moments of Mn atoms placed at the Ga- and Ni sublattice, respectively; Table S14: Element resolved magnetic moments (in ${}_{\mathrm{B}}$), total magnetic moments (in ${}_{\mathrm{B}}$/f.u.) and favorable magnetic reference state for compounds in the austenite phase from the area III. Here ${\mu}_{{\mathrm{Mn}}^{\left(4a\right)}}$ and ${\mu}_{{\mathrm{Mn}}^{\left(4b\right)}}$ are the magnetic moments of Mn atoms, which occupy 4a and 4b Wyckoff sites while ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ga}\right)}}$ and ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ni}\right)}}$ are the Mn magnetic moments at the Ga- and Ni sites, respectively; Table S15: Element resolved magnetic moments (in ${}_{\mathrm{B}}$), total magnetic moments (in ${}_{\mathrm{B}}$/f.u.) and favorable magnetic reference state for compounds in the martensite phase from the area III. Here ${\mu}_{{\mathrm{Mn}}^{\left(4a\right)}}$ and ${\mu}_{{\mathrm{Mn}}^{\left(4b\right)}}$ are the magnetic moments of Mn atoms, which occupy 4a and 4b Wyckoff sites while ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ga}\right)}}$ and ${\mu}_{{\mathrm{Mn}}^{\left(\mathrm{Ni}\right)}}$ are the Mn magnetic moments at the Ga- and Ni sites, respectively; Table S16: Martensitic transition temperature ${T}_{m}$ (in K) for compositions from the areas I, II, and III; Table S17: Experimental value of the lattice constants a (in Å) for austenitic phase; Table S18: Experimental value of the total magnetic moments (in ${}_{\mathrm{B}}$/f.u.) for compounds in the martensite phase; Table S19: Experimental value of the martensitic transition temperature ${T}_{m}$ (in K).

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**(Color online) (

**a**) The set of 105 composition points formed by the 16-atom supercells and represented as a ternary diagram of the Ni–Mn–Ga system. Color circles denote the stoichiometric compositions. (

**b**) Regular ($\#225$ and $\#{225}^{\prime}$) and inverse ($\#216$) cubic Heusler structures with the corresponding Wyckoff positions for the stoichiometric compounds.

**Figure 2.**(Color online) Distribution of stable austenitic crystal structures mapped into the ternary diagram of Ni${}_{x}$Mn${}_{y}$Ga${}_{z}$. Open circles indicate the compositions with tetragonally and orthorhombically distorted structures afforded by the geometric optimization procedure of an initial cubic structure. Here #225${}^{\mathrm{mix}}$ indicates the regular structure with $8c$ sites simultaneously occupied by Ni and Ga.

**Figure 3.**(Color online) Formation energy mapped into the ternary diagram of Ni${}_{x}$Mn${}_{y}$Ga${}_{z}$ compounds with favorable (

**a**) cubic and (

**b**) tetragonal and orthorhombic crystal structures.

**Figure 4.**(Color online) The ternary maps of the stable reactions for Ni${}_{x}$Mn${}_{y}$Ga${}_{z}$ in the (

**a**) austenitic and (

**b**) martensitic phases against the decomposition into a mixture of stable compounds. Here, 0% and 100% correspond to the unstable and stable Ni${}_{x}$Mn${}_{y}$Ga${}_{z}$ compositions.

**Figure 5.**(Color online) Calculated (

**a**) equilibrium lattice parameter ${a}_{0}$ and (

**b**) tetragonal ratio $c/a$ mapped into the ternary diagram of Ni${}_{x}$Mn${}_{y}$Ga${}_{z}$ compounds in austenite and martensite. (

**c**) Theoretical and (

**d**) experimental large-scale ${a}_{0}$ maps. The list of experimental compositions is tabulated in Table S17, see the SM. Label O in (

**b**) denotes the region with a favorable orthorhombic structure. For the orthorhombic structure, the $b/a$ ratio is presented in Table S8, see the SM.

**Figure 6.**(Color online) Calculated total magnetic moments together with preferable spin alignment of Ni${}_{x}$Mn${}_{y}$Ga${}_{z}$ in the (

**a**) austenitic and (

**b**) martensitic phase. (

**c**) Theoretical and (

**d**) experimental large-scale maps of the total magnetic moment in the martensitic phase. The experimental compositions are listed in Table S18, see the Supplementary Materials.

**Figure 7.**(Color online) (

**a**,

**b**) Theoretical and (

**c**) experimental martensitic transition temperature of Ni${}_{x}$Mn${}_{y}$Ga${}_{z}$. In the case of (

**b**,

**c**), the large-scale contour maps in the vicinity of Ni${}_{50}$Mn${}_{25}$Ga${}_{25}$ are illustrated. The list of experimental compositions is presented in Table S19, see the SM.

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**MDPI and ACS Style**

Sokolovskaya, Y.; Miroshkina, O.; Baigutlin, D.; Sokolovskiy, V.; Zagrebin, M.; Buchelnikov, V.; Zayak, A.T.
A Ternary Map of Ni–Mn–Ga Heusler Alloys from Ab Initio Calculations. *Metals* **2021**, *11*, 973.
https://doi.org/10.3390/met11060973

**AMA Style**

Sokolovskaya Y, Miroshkina O, Baigutlin D, Sokolovskiy V, Zagrebin M, Buchelnikov V, Zayak AT.
A Ternary Map of Ni–Mn–Ga Heusler Alloys from Ab Initio Calculations. *Metals*. 2021; 11(6):973.
https://doi.org/10.3390/met11060973

**Chicago/Turabian Style**

Sokolovskaya, Yulia, Olga Miroshkina, Danil Baigutlin, Vladimir Sokolovskiy, Mikhail Zagrebin, Vasilly Buchelnikov, and Alexey T. Zayak.
2021. "A Ternary Map of Ni–Mn–Ga Heusler Alloys from Ab Initio Calculations" *Metals* 11, no. 6: 973.
https://doi.org/10.3390/met11060973