Impact of Disorder on Properties of Vacancies: A Case Study of B2 and A2 Polymorphs of Non-Stoichiometric Fe2CoAl

We have performed an ab initio study of vacancy-induced changes in thermodynamic, structural and magnetic properties of single-phase ferromagnetic Fe2CoAl with a chemically disordered (i) two-sublattice B2 phase or (ii) single-sublattice A2 phase. The two polymorphs of slightly non-stoichiometric Fe2CoAl (Fe27Co14Al13) were modeled by two different 54-atom supercells with atoms distributed according to the special quasi-random structure (SQS) concept. Both the lower-energy B2 phase and a higher-energy A2 phase possess elastic constants that correspond to an auxetic material that is mechanically stable. The properties of vacancies were computed by systematically removing different atoms (one at a time) from the supercells and quite wide ranges of values of vacancy-related characteristics were obtained. The increase in the level of disorder (when changing from the B2 to the A2 phase) results in an increase in the scatter of calculated values. The Fe and Co vacancies have lower vacancy formation energies than the Al ones. The total magnetic moment of the supercell decreases when introducing Fe and Co vacancies but it increases due to Al ones. The latter findings can be partly explained by an increase of the local magnetic moment of Fe atoms when the number of Al atoms in the first neighbor shell of Fe atoms is reduced, such as due to Al vacancies.

In our current study, we focus on the properties of vacancies in the disordered Fe 2 CoAl with the impact of disorder being illustrated by comparing two structural models with different level of disorder. First, we use a structural model that is based on the experimental work of Grover et al. [82] where a single-phase Fe 2 CoAl has effectively a chemically disordered B2 lattice with two sublattices: one containing equal amounts of Fe and Co atoms and the second exhibiting equal amounts of Fe and Al.
We further compare vacancy-related characteristics in the B2 phase with those obtained for a single-sublattice more-disordered A2 phase of Fe 2 CoAl. We use quantum-mechanical calculations to determine vacancy-induced changes in (i) thermodynamic properties by evaluating the vacancy formation energy, (ii) total magnetic moment, and (iii) structural properties by analyzing the volumetric changes.

Methods
When modeling a partially disordered B2 phase (with two compositionally different sublattices) and a single-sublattice disordered A2 phase of Fe 2 CoAl, we have utilized two different 54-atom supercells, see Figure 1a,b, respectively. The atoms on the two B2-phase sublattices and one A2-phase sublattice were distributed according to the special quasi-random structure (SQS) concept [83] as generated by the USPEX software [84][85][86]. The actual stoichiometry of our supercells, Fe 27 Co 14 Al 13 or Fe 2 Co 1.037 Al 0.963 , slightly deviates from the exact Fe 2 CoAl stoichiometry as 54 is not divisible by 4. The cube-shaped 54-atom supercells, 3 × 3 × 3 multiples of a 2-atom B2 cell, are convenient for modeling of elasticity of disordered systems. Single-crystal elastic constants were determined using the stress-strain method [87]. It should be mentioned that our computational supercells, that are used as models for partly-disordered B2 and disordered A2 phase of Fe 2 CoAl, do not have their anisotropic elastic properties equal along crystallographic directions that would be equivalent in the case of cubic-symmetry systems, such as along the [100], [010] and [001] directions. The computed differences are small (a few percent), nevertheless, we have used a proper mathematical approach [88] to determine the closest cubic-symmetry elastic tensor and its C 11 , C 12 and C 44 components are reported below. Regarding our quantum-mechanical calculations, we have used the Vienna Ab initio Simulation Package (VASP) [89,90] based on the density functional theory [91,92]. The Projector-Augmented-Wave (PAW) pseudopotentials [93,94] and the Generalized Gradient Approximation (GGA) in the parametrization by Perdew and Wang [95] (PW91) with the Vosko-Wilk-Nusair correction [96] were utilized for the exchange and correlation energy. The plane-wave energy cut-off was equal to 400 eV and a 6 × 6 × 6 Monkhorst-Pack [97] k-point mesh was used in the case of 54-atom supercells in Figure 1 (or their 53-atom variants with a vacancy). We have fully relaxed all studied supercells, i.e., the energy and forces were minimized with respect to atomic positions, cell shape and the volume (forces acting upon atoms were reduced under 0.01 eV/Å). All local magnetic moments were initially set up as parallel.

Results
A primary aim of our study consists in properties of vacancies but we will report their characteristics with respect to those of vacancy-free states. Therefore, we start with summarizing thermodynamic, structural and magnetic properties of both the B2 and A2 phases without defects. As the properties of the B2 phase can be found in our previous study [98], we will focus on the A2 phase. First, we analyze the thermodynamic stability by evaluating formation energies. The formation energy E f per atom is defined as the difference between the energy of the studied phase, in our case, the energy E(Fe 27 Co 14 Al 13 ) of the 54-atom supercell, and the sum of energies of constituting atoms in their elemental phases, i.e., ferromagnetic (FM) body-centered cubic (bcc) Fe, E(Fe), ferromagnetic hexagonal close-packed (hcp) Co, E(Co), and nonmagnetic (NM) face-centered cubic (fcc) Al, E(Al), weighted by their amount, i.e., The formation energies of the B2 and A2 phases are equal to −0.243 eV/atom and −0.157 eV/atom, respectively. The disordered A2 phase has a significantly less negative energy and, therefore, a lower thermodynamic stability. It is an excited state with respect to the B2 phase that is supposed to be the ground state as proposed by Grover et al. [82]. The A2 phase has also a higher configurational entropy but the difference is quite small, 0.03 meV/(K·atom), see the Appendix A, and it is only of minor importance. We further check the mechanical stability of the studied phases by determining a set of single-crystal elastic constants (C 11 , C 12 , C 44 ) corresponding to a cubic-symmetry system. The A2 phase has them (203 GPa, 140 GPa, 123 GPa) different from those for the more ordered B2 phase (244 GPa, 141 GPa, 131 GPa) [98] but both systems are mechanically stable (fulfill the stability conditions [99]). We visualize the elastic properties of both phases in the form of a directional dependence of Young's modulus in Figure 2 using the MELASA software [100] (open access available online: https://melasa.cerit-sc.cz/ (accessed on 6 October 2021)). Further, we have also used the ELATE software [101] (open access at http://progs. coudert.name/elate, accessed on 6 October 2021) to determine both the minimum (−0.243) and the maximum (0.857) value of single-crystal Poisson ratio ν. Interestingly, as our analysis predicts that the Poisson ratio of the A2-phase Fe 2 CoAl is negative for certain directions of loading, it is an auxetic material (as was also the case of the B2 phase [98]). Thorough information is provided in Figure 3 that visualizes a directional dependence of both maximum and minimum value of Poisson ratio and the values in the x-z plane (for details, see ref. [101]). The negative values of ν min are marked by red color in Figure 3. Regarding the magnetic properties, the computed local magnetic moments in both the B2 and A2 phase are shown in Figure 4. The total magnetic moment in the case of the B2 phase amounts to 68.26 µ B per 54-atom supercell, while the A2 phase has the total magnetic moment by 18.7% higher, equal to 81.0 µ B per 54-atom supercell. Common magneto-volumetric correlations can help us to connect this difference in the total magnetic moment with the fact that the volume of the A2 supercell (641.6 Å 3 per 54-atom supercell) is higher than that of the B2 phase (624.3 Å 3 per 54-atom supercell). Figure 4 also neatly visualizes the differences between the local magnetic moments in the B2 and A2 phases as an illustration of the impact of the different level of order in the B2 and A2 polymorphs.
Next, we systematically remove each of the 54 atoms in each of the two phases to determine the properties of vacancies and compare the results for each atom type. The computed results in the case of Fe vacancies are shown in Figure 5. The vacancy formation energies (E(Fe (27−1) Co 14 Al 13 + E(Fe) − E(Fe 27 Co 14 Al 13 )) in Figure 5a,b are quite clearly different for the Fe atoms belonging to the two different sublattices in the B2 phase. The Fe vacancies at the (Fe, Al) sublattice have higher vacancy formation energies, see full circles in Figure 5a, than those at the (Fe, Co) sublattice, full triangles in Figure 5a. The Fe vacancy formation energies in the A2 phase cover a broader range of values, see Figure 5b, including some quite low ones. The difference of the total magnetic moments, µ(Fe (27−1) Co 14 Al 13 ) − µ(Fe 27 Co 14 Al 13 ), of supercells with and without a vacancy, respectively, is shown in Figure 5c,d. It is mostly negative (the total magnetic moment is lower when a Fe atom is removed). The reduction in the B2 (A2) phase is often smaller (bigger), respectively, than the magnitude of the magnetic moment of FM bcc Fe, 2.2 µ B , that is indicated by the horizontal dashed line. A volume difference V(Fe (27−1) Co 14 Al 13 ) − V(Fe 27 Co 14 Al 13 ) of the supercells with and without vacancy, respectively, is negative, see Figure 5e,f, i.e., the volume is reduced due to a missing Fe atom, and the reductions cover a wider range in the A2 phase.
The results for Co vacancies are summarized in Figure 6. There are only 14 Co atoms in the 54-atom supercell representing the B2 and A2 phase and, therefore, the number of data points in Figure 6 is about twice lower than in the case of 27 Fe atoms in Figure 5 discussed above. All the Co atoms are located only in one of the two sublattices in the B2 phase and their vacancy formation energies cover quite a narrow range of values, see Figure 6a. Not having this limitation in the case of the A2 phase, the vacancy formation energies are spread over a wider range, see Figure 6b. Regarding the vacancy-induced change of the total magnetic moment in Figure 6c,d, its value in the B2 (A2) phase is typically reduced by less (more) than the magnitude of the magnetic moment of one FM hcp Co atom, 1.5 µ B , see the horizontal dashed line in Figure 6c,d. Our findings related to the volumetric reduction of the supercells due to Co vacancies are very similar to those that we found for Fe vacancies (see above).
Rather different trends are found in the case of Al vacancies, see Figure 7. First, the corresponding Al-vacancy formation energies in Figure 7a are much higher in the B2 phase than those related to Fe and Co vacancies. The change of the ordering from the B2 to the A2 phase leads to a reduction of the vacancy formation energy, cf. Figure 7a,b. Regarding the change of the total magnetic moment due to Al vacancies, it increases very significantly in the B2 phase, see Figure 7c. This increase can be partly explained as an opposite to the reduction of local magnetic moment of Fe atoms when increasing the number of Al atoms in the first nearest neighbor (1NN) shell of Fe atoms. We have reported this trend in Fe-Al alloys [71] or Fe-Al-Ti alloys [102]. As an Al vacancy in the (Fe, Al) sublattice lowers the number of Al atoms in the 1NN shell of Fe atoms in the (Fe, Co) sublattice of the B2 phase, local magnetic moments of Fe atoms from the (Fe, Co) sublattice next to the Al vacancy increase and so does the total magnetic moment of the whole supercell. We previously found an increase of the total magnetic moment due to Al vacancies also in the Fe-Al alloys [77]. The above-discussed mechanism is polymorph-sensitive. In contrast to the results for the B2 phase, where the Al atoms are limited to only one half of all possible atomic positions (i.e., one sublattice of the two in the B2 polymorph), the Al atoms are statistically distributed over all lattice sites in the A2 phase. Consequently, the above-mentioned increase of the total magnetic moment due to Al vacancies in the B2 polymorph is less pronounced in the A2 phase and the changes of the total magnetic moment are both positive and negative covering a significantly narrower range close to the zero value, see Figure 7d. The computed changes in magnetic moments can be possibly explained in terms of charge transfer that can be analyzed using, e.g., molecular orbital calculations [103]. Regarding the vacancy-induced volumetric changes, see Figure 7e,f, they are qualitatively similar to those in the case of Fe and Co vacancies.

Conclusions
We have performed a quantum-mechanical study of vacancy-induced changes in thermodynamic, structural and magnetic properties of single-phase ferromagnetic slightly non-stoichiometric Fe 2 CoAl with a chemically disordered either two-sublattice B2 or singlesublattice A2 phase. The two polymorphs of Fe 2 CoAl were modeled by two different 54atom supercells with atoms on either two B2 sublattices or a single A2 sublattice distributed according to the special quasi-random structure (SQS) concept. Both the lower-energy B2 and higher-energy A2 phases were found to possess elastic constants that correspond to auxetic and mechanically stable systems. Our systematic removal of different atoms (one at a time) from the supercells resulted in quite wide ranges of values of vacancy-related characteristics. The increase in the level of disorder (when changing from the B2 phase to the A2 one) results in a further increase in the scatter of calculated values. In general, the Fe and Co vacancies have lower vacancy formation energies than the Al ones. The change from the B2 phase to the A2 phase typically means that the vacancy formation energies cover a wider range of values, in particular, including some quite low energies in the case of the A2 phase. The total magnetic moment of the whole supercell decreases when introducing Fe and Co vacancies but it increases in the B2 phase when an Al vacancy is introduced. The latter finding can be explained by an increase of the local magnetic moment of Fe atoms when the number of Al atoms in the first neighbor shell of Fe atoms is reduced, such as here due to an Al vacancy. In general, it often is easier to form the vacancies in the A2 structure than in the more stable B2 one. In the A2 structure, vacancies cause a more significant decrease (or a much lower increase in case of Al vacancy) in the total magnetic moment of the supercells and a higher decrease in their volume.   Figures 1 and 4 were visualized using the VESTA [104].

Conflicts of Interest:
The authors declare no conflict of interest.

Appendix A
To evaluate the configurational entropy of the B2 and A2 phases, we use a generalized formula [105] derived for the sublattice model [106] S conf = −R ∑ j a j ∑ i f j i ln f j i where R is the universal gas constant, i runs over different chemical species, j is the index of different sublattices, a j is the number of lattice sites of a sublattice j divided by the total number of all lattice sites and f j i is the fraction of a chemical species i on a sublattice j. The configurational entropy is higher in the A2 phase than in the B2 phase by 0.03 meV/(K·atom). If the total energy difference of 86 meV per atom between the A2 and B2 phase should be compensated solely by the difference in the configurational entropy, it would happen at the temperature of 2880 K.