#
An Ab Initio Study of Thermodynamic and Mechanical Stability of Heusler-Based Fe_{2}AlCo Polymorphs

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## Abstract

**:**

_{2}; AlCoFe

_{2}; Fe

_{2}CoAl; AlFe

_{2}Co; Heusler; disorder; ab initio; stability; elasticity

## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Schematic visualizations of 16-atom supercells of Fe${}_{2}$AlCo polymorphs used in our quantum-mechanical calculations (some atoms are shown with their periodic images). (

**a**) Full Heusler structure. (

**b**) Inverse Heusler structure. (

**c**) Specific distribution of Co (and Fe) atoms regarding the atoms in the 2NN shell (see the text). (

**d**) Special arrangement of atoms of all three chemical species. Selected atoms discussed in the text are marked by a star and/or a prime symbol.

**Figure 2.**Visualizations of single-crystal elastic properties of Fe${}_{2}$AlCo polymorphs in the form of directional dependences of Young’s modulus for (

**a**) the full-Heusler-structure variant, (

**b**) inverse-Heusler polymorph, (

**c**) specific distribution of Co (and Fe) atoms regarding the atoms in the 2NN shell (see the text), and (

**d**) special arrangement of atoms of all three chemical species. The directional dependences were computed from the single-crystal elastic stiffnesses of the studied polymorphs and visualized by the SC-EMA on-line tool [36,37] (http://scema.mpie.de).

**Figure 3.**Schematic visualizations of local magnetic moments of atoms in the four studied Fe${}_{2}$AlCo polymorphs in the case of (

**a**) the full-Heusler-structure variant, (

**b**) inverse-Heusler polymorph, (

**c**) specific distribution of Co (and Fe atoms) regarding the atoms in the 2NN shell (see the text), and (

**d**) special arrangement of atoms of all three chemical species. Magnetic moments of Al atoms are antiparallel to those of Fe and Co atoms but their magnitudes are so small, less than 0.05 ${\mu}_{\mathrm{B}}$, so that they can be considered as non-magnetic and are not shown in this figure. The numbers within the spheres indicate rounded values of local magnetic moments.

**Figure 4.**Estimated probabilities of occurrence of higher-energy polymorphs employing Boltzmann statistics: (

**a**) for the inverse Heusler phase and (

**b**) for the polymorph depicted in Figure 1d.

**Figure 5.**Relations between calculated properties of the four studied Fe${}_{2}$AlCo polymorphs, (

**a**) anti-correlation of the formation energy ${E}_{\mathrm{f}}$ and the bulk modulus B and (

**b**) correlation between the equilibrium volume and the magnetic moment of 16-atom supercells visualized in Figure 1. The dashed lines are linear fits obtained by the least-square method.

**Figure 6.**Computed energies and magnetic moments of the full Heusler polymorph of Fe${}_{2}$AlCo transformed along the Bain’s path: (

**a**) energies close to the undeformed state ($c/a$ = 1) with a shallow minimum, (

**b**) energies for a broader range of $c/a$ values with a deeper minimum, (

**c**) total magnetic moment per formula unit for states close to the undeformed state, (

**d**) total magnetic moment and local magnetic moments of Fe and Co atoms for a broader range of $c/a$ values. The energies in subfigures (

**a,b**) are depicted with respect to the energy of undeformed state with $c/a$ = 1 (indicated by vertical dashed lines and horizontal dash-dot lines). Mind the interruption of the vertical axes in part (

**d**).

**Table 1.**Calculated thermodynamic, structural, magnetic and elastic parameters of the studied Fe${}_{2}$AlCo polymorphs. Listed are values of the formation energy ${E}_{\mathrm{f}}$, equilibrium volume per atom ${V}_{\mathrm{eq}}$, equilibrium lattice parameter ${a}_{\mathrm{eq}}$ of 16-atom supercells shown in Figure 1, magnetic moments ${\mu}^{\mathrm{TOT}}$ per formula unit (f.u.), and single-crystal elastic stiffnesses ${C}_{11}$, ${C}_{12}$ and ${C}_{44}$ together with the bulk moduli B derived from these elastic stiffnesses (B = (${C}_{11}$ + 2${C}_{12}$)/3).

Polymorph | E${}_{\mathbf{f}}$ | V${}_{\mathbf{eq}}$ | a${}_{\mathbf{eq}}$ | ${\mathit{\mu}}^{\mathbf{TOT}}$ | C${}_{11}$ | C${}_{12}$ | C${}_{44}$ | B |
---|---|---|---|---|---|---|---|---|

(eV/atom) | (Å${}^{3}$/atom) | (Å) | (${\mathit{\mu}}_{\mathbf{B}}$/f.u.) | (GPa) | (GPa) | (GPa) | (GPa) | |

full Heusler | −0.145 | 11.740 | 5.727 | 5.70 | 148 | 158 | 115 | 155 |

inverse Heusler | −0.260 | 11.556 | 5.697 | 4.99(8) | 245 | 157 | 139 | 186 |

Co & Fe with mixed 2NN | −0.269 | 11.562 | 5.698 | 5.02 | 235 | 155 | 139 | 182 |

Co, Fe & Al with mixed 2NN | −0.232 | 11.617 | 5.707 | 5.08 | 233 | 145 | 125 | 174 |

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**MDPI and ACS Style**

Friák, M.; Oweisová, S.; Pavlů, J.; Holec, D.; Šob, M.
An *Ab Initio* Study of Thermodynamic and Mechanical Stability of Heusler-Based Fe_{2}AlCo Polymorphs. *Materials* **2018**, *11*, 1543.
https://doi.org/10.3390/ma11091543

**AMA Style**

Friák M, Oweisová S, Pavlů J, Holec D, Šob M.
An *Ab Initio* Study of Thermodynamic and Mechanical Stability of Heusler-Based Fe_{2}AlCo Polymorphs. *Materials*. 2018; 11(9):1543.
https://doi.org/10.3390/ma11091543

**Chicago/Turabian Style**

Friák, Martin, Sabina Oweisová, Jana Pavlů, David Holec, and Mojmír Šob.
2018. "An *Ab Initio* Study of Thermodynamic and Mechanical Stability of Heusler-Based Fe_{2}AlCo Polymorphs" *Materials* 11, no. 9: 1543.
https://doi.org/10.3390/ma11091543