#
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

- Heusler, F.; Starck, W.; Haupt, E. Magnetisch-chemische Studien. Verh. Dtsch. Phys. Ges.
**1903**, 5, 219–232. [Google Scholar] - Webster, P. Heusler Alloys. Contemp. Phys.
**1969**, 10, 559–577. [Google Scholar] [CrossRef] - Graf, T.; Felser, C.; Parkin, S.S.P. Simple rules for the understanding of Heusler compounds. Prog. Sol. State Chem.
**2011**, 39, 1–50. [Google Scholar] [CrossRef] - Picozzi, S.; Continenza, A.; Freeman, A. Co
_{2}MnX (X = Si, Ge, Sn) Heusler compounds: An ab initio study of their structural, electronic, and magnetic properties at zero and elevated pressure. Phys. Rev. B**2002**, 66, 094421. [Google Scholar] [CrossRef] - Webster, P. Magnetic and chemical order in Heusler alloys containing cobalt and manganese. J. Phys. Chem. Sol.
**1971**, 32, 1221. [Google Scholar] [CrossRef] - Kübler, J.; Williams, A.; Sommers, C. Formation and coupling of magnetic-moments in Heusler alloys. Phys. Rev. B
**1983**, 28, 1745–1755. [Google Scholar] [CrossRef] - Galanakis, I.; Dederichs, P.; Papanikolaou, N. Slater-Pauling behavior and origin of the half-metallicity of the full-Heusler alloys. Phys. Rev. B
**2002**, 66, 174429. [Google Scholar] [CrossRef] - Miura, Y.; Nagao, K.; Shirai, M. Atomic disorder effects on half-metallicity of the full-Heusler alloys Co
_{2}(Cr_{1-x}Fe_{x})Al: A first-principles study. Phys. Rev. B**2004**, 69, 144413. [Google Scholar] [CrossRef] - Galanakis, I.; Dederichs, P.; Papanikolaou, N. Origin and properties of the gap in the half-ferromagnetic Heusler alloys. Phys. Rev. B
**2002**, 66, 134428. [Google Scholar] [CrossRef] - Kandpal, H.C.; Fecher, G.H.; Felser, C. Calculated electronic and magnetic properties of the half-metallic, transition metal based Heusler compounds. J. Phys. D Appl. Phys.
**2007**, 40, 1507–1523. [Google Scholar] [CrossRef][Green Version] - Galanakis, I.; Mavropoulos, P.; Dederichs, P. Electronic structure and Slater-Pauling behaviour in half-metallic Heusler alloys calculated from first principles. J. Phys. D Appl. Phys.
**2006**, 39, 765–775. [Google Scholar] [CrossRef] - Picozzi, S.; Continenza, A.; Freeman, A. Role of structural defects on the half-metallic character of Co
_{2}MnGe and Co_{2}MnSi Heusler alloys. Phys. Rev. B**2004**, 69, 094423. [Google Scholar] [CrossRef] - Buschow, K.; Van Engen, P. Magnetic and magneto-optical properties of Heusler alloys based on aluminum and gallium. J. Mag. Mag. Mat.
**1981**, 25, 90–96. [Google Scholar] [CrossRef] - Nishino, Y.; Kato, M.; Asano, S.; Soda, K.; Hayasaki, M.; Mizutani, U. Semiconductor-like behavior of electrical resistivity in Heusler-type Fe
_{2}VAl compound. Phys. Rev. Lett.**1997**, 79, 1909–1912. [Google Scholar] [CrossRef] - Sakurada, S.; Shutoh, N. Effect of Ti substitution on the thermoelectric properties of (Zr,Hf)NiSn half-Heusler compounds. Appl. Phys. Lett.
**2005**, 86, 082105. [Google Scholar] [CrossRef] - Shen, Q.; Chen, L.; Goto, T.; Hirai, T.; Yang, J.; Meisner, G.; Uher, C. Effects of partial substitution of Ni by Pd on the thermoelectric properties of ZrNiSn-based half-Heusler compounds. Appl. Phys. Lett.
**2001**, 79, 4165–4167. [Google Scholar] [CrossRef] - Chadov, S.; Qi, X.; Kuebler, J.; Fecher, G.H.; Felser, C.; Zhang, S.C. Tunable multifunctional topological insulators in ternary Heusler compounds. Nat. Mat.
**2010**, 9, 541–545. [Google Scholar] [CrossRef] [PubMed][Green Version] - Lin, H.; Wray, L.A.; Xia, Y.; Xu, S.; Jia, S.; Cava, R.J.; Bansil, A.; Hasan, M.Z. Half-Heusler ternary compounds as new multifunctional experimental platforms for topological quantum phenomena. Nat. Mater.
**2010**, 9, 546–549. [Google Scholar] [CrossRef] [PubMed] - Planes, A.; Manosa, L.; Acet, M. Magnetocaloric effect and its relation to shape-memory properties in ferromagnetic Heusler alloys. J. Phys. Cond. Matter
**2009**, 21, 233201. [Google Scholar] [CrossRef] [PubMed] - Entel, P.; Buchelnikov, V.; Khovailo, V.; Zayak, A.; Adeagbo, W.; Gruner, M.; Herper, H.; Wassermann, E. Modelling the phase diagram of magnetic shape memory Heusler alloys. J. Phys. D Appl. Phys.
**2006**, 39, 865–889. [Google Scholar] [CrossRef] - Kainuma, R.; Imano, Y.; Ito, W.; Morito, H.; Sutou, Y.; Oikawa, K.; Fujita, A.; Ishida, K.; Okamoto, S.; Kitakami, O. Metamagnetic shape memory effect in a Heusler-type Ni
_{43}Co_{7}Mn_{39}Sn_{11}polycrystalline alloy. Appl. Phys. Lett.**2006**, 88, 192513. [Google Scholar] [CrossRef] - Gilleßen, M.; Dronskowski, R. A combinatorial study of full Heusler alloys by first-principles computational methods. J. Comput. Chem.
**2009**, 30, 1290–1299. [Google Scholar] [CrossRef] [PubMed] - Gilleßen, M.; Dronskowski, R. A combinatorial study of inverse Heusler alloys by first-principles computational methods. J. Comput. Chem.
**2010**, 31, 612–619. [Google Scholar] [CrossRef] [PubMed] - Grover, A.K.; Pillay, R.G.; Nagarajan, V.; Tandon, P.N. Site preference and local environment effects in ferromagnetic ternary alloys. J. Magn. Magn. Mater.
**1980**, 15, 699–700. [Google Scholar] [CrossRef] - Kresse, G.; Hafner, J. Ab initio molecular dynamics for liquid metals. Phys. Rev. B
**1993**, 47, 558–561. [Google Scholar] [CrossRef] - Kresse, G.; Furthmüller, J. Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set. Phys. Rev. B
**1996**, 54, 11169–11186. [Google Scholar] [CrossRef] - Hohenberg, P.; Kohn, W. Inhomogeneous electron gas. Phys. Rev. B
**1964**, 136, B864–B871. [Google Scholar] [CrossRef] - Kohn, W.; Sham, L.J. Self-consistent equations including exchange and correlation effects. Phys. Rev. A
**1965**, 140, A1133–A1138. [Google Scholar] [CrossRef] - Blöchl, P.E. Projector augmented-wave method. Phys. Rev. B
**1994**, 50, 17953–17979. [Google Scholar] [CrossRef][Green Version] - Kresse, G.; Joubert, D. From ultrasoft pseudopotentials to the projector augmented-wave method. Phys. Rev. B
**1999**, 59, 1758–1775. [Google Scholar] [CrossRef] - Perdew, J.P.; Wang, Y. Accurate and simple analytic representation of the electron-gas correlation energy. Phys. Rev. B
**1992**, 45, 13244–13249. [Google Scholar] [CrossRef] - Vosko, S.H.; Wilk, L.; Nusair, M. Accurate spin-dependent electron liquid correlation energies for local spin density calculations: a critical analysis. Can. J. Phys.
**1980**, 58, 1200. [Google Scholar] [CrossRef] - Zhou, L.; Holec, D.; Mayrhofer, P.H. First-principles study of elastic properties of cubic Cr
_{1-x}Al_{x}N alloys. J. Appl. Phys.**2013**, 113, 043511. [Google Scholar] [CrossRef] - Mouhat, F.; Coudert, F.X. Necessary and sufficient elastic stability conditions in various crystal systems. Phys. Rev. B
**2014**, 90, 224104. [Google Scholar] [CrossRef] - Jain, V.; Nehra, J.; Sudheesh, V.D.; Lakshmi, N.; Venugopalan, K. Comparative study of the structural and magnetic properties of bulk and nano-sized Fe
_{2}CoAl. AIP Conf. Proc.**2013**, 1536, 935–936. [Google Scholar] [CrossRef] - Titrian, H.; Aydin, U.; Friák, M.; Ma, D.; Raabe, D.; Neugebauer, J. Self-consistent Scale-bridging Approach to Compute the Elasticity of Multi-phase Polycrystalline Materials. Mater. Res. Soc. Symp. Proc.
**2013**, 1524. [Google Scholar] [CrossRef] - Friák, M.; Counts, W.A.; Ma, D.; Sander, B.; Holec, D.; Raabe, D.; Neugebauer, J. Theory-guided materials design of multi-phase Ti-Nb alloys with bone-matching elastic properties. Materials
**2012**, 5, 1853–1872. [Google Scholar] [CrossRef] - Raghavan, V. Ternary and aluminum phase higher order diagram updates. J. Phase Equi. Diff.
**2005**, 26, 623. [Google Scholar] - Raghavan, V. Ternary and higher order aluminum phase diagram updates. J. Phase Equi. Diff.
**2005**, 26, 348. [Google Scholar] [CrossRef] - Ducher, R.; Kainuma, R.; Ohnuma, I.; Ishida, K. Phase equilibria and stability of B2 and L2
_{1}ordered phases in the Co-Fe-Ga Heusler alloy system. J. Alloys Compd.**2007**, 437, 93–101. [Google Scholar] [CrossRef] - Kumar, A.; Srivastava, P.C. Synthesis and characterization of Co
_{2}FeAl Heusler alloy nanoparticles. Mater. Sci. Pol.**2013**, 31, 501–505. [Google Scholar] [CrossRef] - Zhu, L.F.; Friák, M.; Dick, A.; Grabowski, B.; Hickel, T.; Liot, F.; Holec, D.; Schlieter, A.; Kuehn, U.; Eckert, J.; Ebrahimi, Z.; Emmerich, H.; Neugebauer, J. First-principles study of the thermodynamic and elastic properties of eutectic Fe-Ti alloys. Acta Mater.
**2012**, 60, 1594–1602. [Google Scholar] [CrossRef] - Hemzalová, P.; Friák, M.; Šob, M.; Ma, D.; Udyansky, A.; Raabe, D.; Neugebauer, J. Ab initio study of thermodynamic, electronic, magnetic, structural, and elastic properties of Ni
_{4}N allotropes. Phys. Rev. B**2013**, 88, 174103. [Google Scholar] [CrossRef] - Maisel, S.B.; Hoefler, M.; Mueller, S. A canonical stability-elasticity relationship verified for one million face-centred-cubic structures. Nature
**2012**, 491, 740. [Google Scholar] [CrossRef] [PubMed] - Friák, M.; Všianská, M.; Holec, D.; Zelený, M.; Šob, M. Tensorial elastic properties and stability of interface states associated with Σ 5(210) grain boundaries in Ni
_{3}(Al,Si). Sci. Technol. Adv. Mater.**2017**, 18, 273–282. [Google Scholar] [CrossRef] [PubMed] - Craievich, P.J.; Weinert, M.; Sanchez, J.M.; Watson, R.E. Local stability of nonequilibrium phases. Phys. Rev. Lett.
**1994**, 72, 3076–3079. [Google Scholar] [CrossRef] [PubMed] - Šob, M.; Wang, L.G.; Vitek, V. Local stability of higher-energy phases in metallic materials and its relation to the structure of extended defects. Comput. Mater. Sci.
**1997**, 8, 100–106. [Google Scholar] [CrossRef] - Wang, L.G.; Šob, M.; Zhang, Z. Instability of higher-energy phases in simple and transition metals. J. Phys. Chem. Solids
**2003**, 64, 863–872. [Google Scholar] [CrossRef][Green Version] - Friák, M.; Šob, M.; Vitek, V. Ab initio calculation of phase boundaries in iron along the bcc-fcc transformation path and magnetism of iron overlayers. Phys. Rev. B
**2001**, 63, 052405. [Google Scholar] [CrossRef] - Qiu, S.L.; Marcus, P.M.; Ma, H. Tetragonal equilibrium states of Mn and Fe. J. Appl. Phys.
**2000**, 87, 5932–5934. [Google Scholar] [CrossRef] - Spišák, D.; Hafner, J. Complex reconstruction of γ-iron multilayers on Cu(100): Ab initio local-spin-density investigations. Phys. Rev. B
**2000**, 61, 16129–16136. [Google Scholar] [CrossRef] - Friák, M.; Hickel, T.; Körmann, F.; Udyansky, A.; Dick, A.; von Pezold, J.; Ma, D.; Kim, O.; Counts, W.A.; Šob, M.; et al. Determining the elasticity of materials employing quantum-mechanical approaches: From the electronic ground state to the limits of materials stability. Steel Res. Int.
**2011**, 82, 86–100. [Google Scholar] [CrossRef] - Friák, M.; Šob, M.; Vitek, V. Ab initio calculation of tensile strength in iron. Phil. Mag.
**2003**, 83, 3529–3537. [Google Scholar] [CrossRef] - Legut, D.; Friák, M.; Šob, M. Phase stability, elasticity, and theoretical strength of polonium from first principles. Phys. Rev. B
**2010**, 81, 214118. [Google Scholar] [CrossRef] - Legut, D.; Friák, M.; Šob, M. Why is polonium simple cubic and so highly anisotropic? Phys. Rev. Lett.
**2007**, 99, 016402. [Google Scholar] [CrossRef] [PubMed] - Šob, M.; Friák, M.; Legut, D.; Vitek, V. Theoretical strength, magnetism and stability of metals and intermetallics. In Complex Inorganic Solids; Turchi, P., Gonis, A., Rajan, K., Meike, A., Eds.; Springer: New York, NY, USA, 2005; pp. 307–325. [Google Scholar]
- Šob, M.; Legut, D.; Friák, M.; Fiala, J. Magnetism of Ni
_{3}Al and Fe_{3}Al under extreme pressure and shape deformation: An ab initio study. J. Mag. Mag. Mat.**2004**, 272, E205. [Google Scholar] [CrossRef] - Friák, M.; Šob, M. Ab initio study of the bcc-hcp transformation in iron. Phys. Rev. B
**2008**, 77, 174117. [Google Scholar] [CrossRef] - Zelený, M.; Friák, M.; Šob, M. Ab initio study of energetics and magnetism of Fe, Co, and Ni along the trigonal deformation path. Phys. Rev. B
**2011**, 83, 184424. [Google Scholar] [CrossRef] - Momma, K.; Izumi, F. VESTA: A three-dimensional visualization system for electronic and structural analysis. J. Appl. Crystallogr.
**2008**, 41, 653–658. [Google Scholar] [CrossRef] - Momma, K.; Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data. J. Appl. Crystallogr.
**2011**, 44, 1272–1276. [Google Scholar] [CrossRef]

**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