## 4 Results and Discussion
### Structural Transformations in \(\beta\)-FeSi\({}_{2}\) and \(\alpha\)-FeSi\({}_{2}\)
The decomposition temperature of the stoichiometric a-phase is seen to be lower than the temperature T\({}_{\rm C}\) of the \(\beta\)-FeSi\({}_{2}\)-to-\(\alpha\)-FeSi\({}_{2}\) transformation, indicating that the latter phase transition is unlikely. Now let us analyze two possible paths for the \(\beta\rightarrow\alpha\) transformation: the diffusion path and the deformation path. For this purpose, we considered the 6-atom fluorite cell (cf. Figure 5b). There are two cases: when the Fe atom in the cell center is moved towards the facet in the z-direction (diffusion path), and when the cell is under monoclinic (100)[001] strain. Both paths lead to the formation of the \(\alpha\)-phase, and the energetics of these two paths is shown in Figure 15. It is seen that the deformation path is more energetically favorable than the diffusion one. The \(\mathrm{E_{T}}(\varepsilon)\) dependence has a local minimum at \(\varepsilon=0.5\). All the strained structures have C2/m symmetry. The possibility of a direct \(\beta\rightarrow\alpha\) transition under monoclinic strain of the \(\beta\)-phase is confirmed by the results presented in Figure 16, where the atomic configurations of the strained \(\beta\)-structures are shown. The shear (001)[100] strain, \(\varepsilon=1.3\), transforms \(\beta\)-FeSi\({}_{2}\) into a distorted \(\alpha\)-FeSi\({}_{2}\), and after relaxation, into the perfect \(\alpha\)-phase. On the other hand, the \(\beta\)-FeSi\({}_{2}\) structure can be formed from \(\alpha\)-FeSi\({}_{2}\) under the corresponding monoclinic strain (not shown here). Finally, attempts were made to promote the transformation of \(\beta\)-FeSi\({}_{2}\) into \(\alpha\)-FeSi\({}_{2}\) by using FPMD simulations. At different temperatures, a strongly distorted version of the \(\beta\)-phase was revealed. However, at 1400 K, a structural transformation was detected. The total energy and lattice parameters of \(\beta\)-FeSi\({}_{2}\) are shown in Figure 17 as a function of simulation time at 1400 K. After about 1500 fs, the system reaches a stable state. This state after relaxation was identified as a 12-atom C2/m structure. A similar six-atom C2/m structure was fixed during simulations at 1800 K and 2000 K. The structures revealed at 1400 K and 1800 K are listed in Tables 4 and 8 as \(\beta\)-1400 and \(\beta\)-1800. Here, we note one characteristic feature: the structural transformations were accompanied by monoclinic-like strains of the initial unit cells, as in the case of \(\beta\)-FeSi\({}_{2}\) (see Section 4.6). For \(\gamma\)-FeSi\({}_{2}\), the driving force for the transformations was the softening of the \(\mathrm{\Gamma_{5+}}\) and \(\mathrm{L_{3}}\) phonon modes, whereas for \(\beta\)-FeSi\({}_{2}\), the transformations occurred due to the anisotropy of the elastic properties of this orthorhombic phase (see Section 4.4). At T \(>\) 2000 K, the system has P1 symmetry and resembles a melt. It is hard to estimate the melting point in our approach; nevertheless, a temperature of about 2400 K can be estimated, which is much higher than the experimental value of 1453 K. However, this does not come as a surprise since the real structures contain a number of defects that strongly affect (reduce) the melting point. So, it is fair to conclude that we failed in reproducing the \(\beta\)-to-\(\alpha\)-phase transition during FPMD simulations at different temperatures; instead, the final structure leads to monoclinic structures. There are two reasons for this. First, the system could reach a local minimum, as shown in Figure 15b, and, second, the indirect transition (see above) \(\beta\)-FeSi\({}_{2}\rightarrow\alpha\)-Fe\({}_{y}\)Si\({}_{2}+\varepsilon\)-FeSi would be most favorable. Also, we could not transform the \(\alpha\)-phase into the \(\beta\)-phase at low temperatures. However, there  
Figure 13: Difference in the Gibbs free energies (\(\Delta\)G) of the stoichiometric \(\alpha\)- and \(\beta\)-phases of FeSi\({}_{2}\) as a function of temperature.