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A Quantum-Mechanical Study of Antiphase Boundaries in Ferromagnetic B2-Phase Fe_{2}CoAl Alloy

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## Abstract

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## 1. Introduction

## 2. Methods

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**A schematic visualization of the computational supercells employed to model Fe${}_{2}$CoAl (a small deviation from the exact stoichiometry is discussed in the text). (

**a**,

**b**) show 54-atom supercells, which we refer to as variants 1 and 2, respectively, as special quasi-random models of B2-phase Fe${}_{2}$CoAl. (

**c**,

**d**) are doubles of these 54-atom supercells along the [001] direction, respectively. In order to model antiphase boundaries (APBs), we applied APB-related 〈111〉 shift marked by red arrows in (

**c**,

**d**) and obtained supercells shown in (

**e**,

**f**). To preserve the stoichiometry of the supercells when applying the APB shift to the upper half of 108-atom supercells in (

**c**,

**d**), one atomic plane was cyclically relocated to the top of each supercell; see red curved arrow in (

**c**,

**d**).

**Figure 2.**Calculated directional dependencies of Young’s modulus of supercells modeling Fe${}_{2}$CoAl, in particular variant 1 (

**a**) without APBs (

**b**) and with APBs. For variant 1 without APBs, we also show a directional dependence of the minimum and maximum value of Poisson’s ratio (

**c**) and their behavior within the (x,z) plane (

**d**), see Ref. [114] for details, with examples of negative values indicated by red arrows. The figures were produced using ELATE software [114] (open-access at http://progs.coudert.name/elate, accessed date 29 September 2021).

**Figure 3.**Computed local magnetic moments of atoms in supercells modeling Fe${}_{2}$CoAl, in particular variants 1 and 2 without ABPs (

**a**,

**b**), respectively, and variants 1 and 2 with APBs (

**c**,

**d**), respectively. The magnitude of local moments are indicated by the diameter of the spheres representing atoms—examples of the values for a few Fe and Co atoms (in Bohr magnetons) are listed in part (

**a**).

**Figure 4.**Calculated local magnetic moments of atoms (in Bohr magnetons, ${\mu}_{\mathrm{B}}$) as a function of the number of selected atoms in their first nearest neighbor shell (1NN) for supercells modeling Fe${}_{2}$CoAl with and without APBs. In particular, for supercells without APBs, we show magnetic moment of Fe atoms from the (Fe,Co) sublattice as a function of the number of Al atoms in the 1NN of Fe atoms (

**a**); magnetic moment of Co atoms from the (Fe,Co) sublattice as a function of the number of Al atoms in the 1NN of Co atoms (

**b**); and magnetic moment of Fe atoms from the (Fe,Al) sublattice as a function of the number of Co atoms in the 1NN of Fe atoms (

**c**). For supercells with APBs, we show magnetic moment of Fe atoms from the (Fe,Co) sublattice as a function of the number of Al atoms in the 1NN of Fe atoms (

**d**); magnetic moment of Co atoms from the (Fe,Co) sublattice as a function of the number of Al atoms in the 1NN of Co atoms (

**e**); and magnetic moment of Fe atoms from the (Fe,Al) sublattice as a function of the number of Co atoms in the 1NN of Fe atoms (

**f**).

**Table 1.**Calculated formation energy ${E}_{\mathrm{f}}$, volume per atom V, a two-atom B2 lattice parameter ${a}^{\mathrm{B}2}$, tetragonality ratio $c/a$ (the lattice parameter c is perpendicular to the APB interfaces), magnetic moment $\mu $ per atom, and the averaged APB interface energy $\langle {\gamma}^{\mathrm{APB}}\rangle $ for supercells with APBs.

Structure | ${\mathit{E}}_{\mathbf{f}}$ | V | ${\mathit{a}}^{\mathbf{B}2}$ | $\mathit{c}/\mathit{a}$ | $\mathit{\mu}$ | $\langle {\mathit{\gamma}}^{\mathbf{APB}}\rangle $ |
---|---|---|---|---|---|---|

eV/atom | Å${}^{3}$/atom | Å | ${\mathit{\mu}}_{\mathbf{B}}$/atom | mJ/m${}^{2}$ | ||

var. 1 no APBs | −0.243 | 11.56 | 2.849 | 1.000 | 1.266 | — |

var. 1 with APBs | −0.226 | 11.65 | — | 1.011 | 1.344 | 199 |

exp. A2-phase [112] | — | 11.77 | 2.866 | 1.000 | 1.18–1.23 | — |

var. 2 no APBs | −0.244 | 11.58 | 2.850 | 1.000 | 1.274 | — |

var. 2 with APBs | −0.218 | 11.72 | — | 1.006 | 1.380 | 310 |

**Table 2.**Calculated single-crystal elastic constants of variants 1 and 2 as models of Fe${}_{2}$CoAl with and without APBs. The values of ${C}_{11}$, ${C}_{33}$, ${C}_{12}$, ${C}_{13}$, ${C}_{44}$, ${C}_{66}$, Young’s moduli ${Y}_{\mathrm{min}}$, ${Y}_{\mathrm{max}}$, and shear moduli ${G}_{\mathrm{min}}$ and ${G}_{\mathrm{max}}$ are given in GPa, and we expect their error bar to be 1–2 GPa. The minimum and maximum values of Y, G, and $\nu $ were obtained using ELATE software [114].

Single-Crystal | ${\mathit{C}}_{11}$ | ${\mathit{C}}_{33}$ | ${\mathit{C}}_{12}$ | ${\mathit{C}}_{13}$ | ${\mathit{C}}_{44}$ | ${\mathit{C}}_{66}$ |

Elastic Constants | (GPa) | (GPa) | (GPa) | (GPa) | (GPa) | (GPa) |

var. 1 without APBs | 244 | 244 | 141 | 141 | 131 | 131 |

var. 1 with APBs | 232 | 228 | 138 | 143 | 134 | 132 |

var. 2 without APBs | 247 | 247 | 141 | 141 | 132 | 132 |

var. 2 with APBs | 234 | 227 | 136 | 141 | 133 | 132 |

${\mathit{Y}}_{\mathbf{min}}$ | ${\mathit{Y}}_{\mathbf{max}}$ | ${\mathit{G}}_{\mathbf{min}}$ | ${\mathit{G}}_{\mathbf{max}}$ | ${\mathit{\nu}}_{\mathbf{min}}$ | ${\mathit{\nu}}_{\mathbf{max}}$ | |

(GPa) | (GPa) | (GPa) | (GPa) | |||

var. 1 without APBs | 141 | 315 | 52 | 131 | −0.083 | 0.626 |

var. 1 with APBs | 117 | 318 | 43 | 134 | −0.139 | 0.728 |

var. 2 without APBs | 145 | 317 | 53 | 132 | −0.075 | 0.614 |

var. 2 with APBs | 120 | 316 | 45 | 133 | −0.125 | 0.715 |

**Table 3.**Polycrystalline bulk modulus B, Young’s modulus Y, shear modulus G, and Poisson’s ratio $\nu $ computed according to Voigt, Reuss, and Hill homogenization methods in the of variants 1 and 2 of Fe${}_{2}$CoAl with and without APBs as obtained using the ELATE software [114] (open-access at http://progs.coudert.name/elate, accessed date 29 September 2021). The expected error bar of the values of the three moduli (B, Y and G ) is 1–2 GPa.

Polycrystal | B | Y | G | $\mathit{\nu}$ |
---|---|---|---|---|

Elasticity | Voigt/Reuss/Hill Values in GPa | Voigt/Reuss/Hill | ||

APB-free var. 1 | 175/175/175 | 250/211/231 | 99/81/90 | 0.262/0.300/0281 |

var. 1 with APBs | 171/171/171 | 247/194/221 | 98/74/86 | 0.260/0.311/0.285 |

APB-free var. 2 | 176/176/176 | 253/215/234 | 100/83/92 | 0.261/0.297/0.279 |

var. 2 with APBs | 170/170/170 | 247/198/223 | 98/76/87 | 0.258/0.306/0.282 |

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**MDPI and ACS Style**

Friák, M.; Gracias, J.; Pavlů, J.; Šob, M.
A Quantum-Mechanical Study of Antiphase Boundaries in Ferromagnetic B2-Phase Fe_{2}CoAl Alloy. *Magnetochemistry* **2021**, *7*, 137.
https://doi.org/10.3390/magnetochemistry7100137

**AMA Style**

Friák M, Gracias J, Pavlů J, Šob M.
A Quantum-Mechanical Study of Antiphase Boundaries in Ferromagnetic B2-Phase Fe_{2}CoAl Alloy. *Magnetochemistry*. 2021; 7(10):137.
https://doi.org/10.3390/magnetochemistry7100137

**Chicago/Turabian Style**

Friák, Martin, Josef Gracias, Jana Pavlů, and Mojmír Šob.
2021. "A Quantum-Mechanical Study of Antiphase Boundaries in Ferromagnetic B2-Phase Fe_{2}CoAl Alloy" *Magnetochemistry* 7, no. 10: 137.
https://doi.org/10.3390/magnetochemistry7100137