## 4 Results and Discussion
### Electronic Structural Properties  
For the FeSi\({}_{2}\) phases, the symmetry and the structural and energetic parameters are summarized in Table 4, and the \(\beta\)-structures that were identified during FPMD simulations at 1400 K (\(\beta\)-1400) and 1800-200 K (\(\beta\)-1800) are also included.  
Figure 7 shows the densities of states (DOS) of different phases of iron disilicide. We compared the calculated DOSs with those reported in other theoretical investigations [21; 29] for the stoichiometric \(\alpha\)-, \(\beta\)-, and \(\gamma\)-phases, and found that they are in good agreement (not shown here) with those reported here. Our results show that the \(\beta\)-phase is semiconducting, while the \(\gamma\)-phase should display metal conductivity. For \(\beta\)-FeSi\({}_{2}\), the band gap narrows with temperature before its disappearance at T = 1800 K. Both \(\alpha\)-FeSi\({}_{2}\) and \(\alpha\)-Fe\({}_{0.8}\)Si\({}_{2}\) exhibit the properties inherent to semimetals.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Structure** & **N\({}_{\mathbf{a}}\)** & **Symmetry** & **a (Å)** & **b (Å)** & **c (Å)** & **V (Å\({}^{3}\)/atom)** &
\begin{tabular}{c} \(\Delta\)E\({}_{\mathrm{T}}\)** \\ **(eV/atom)** \\ \end{tabular} \\ \hline \(\beta\)-FeSi\({}_{2}\) & 24 & & & 9.874 & 7.769 & 7.819 & 12.495 & \\  & & & (9.869) \({}^{\mathrm{a}}\) & (7.791) \({}^{\mathrm{a}}\) & (7.833) \({}^{\mathrm{a}}\) & (12.540) \({}^{\mathrm{a}}\) & \\  & & & (9.869) \({}^{\mathrm{b}}\) & (7.774) \({}^{\mathrm{b}}\) & (7.846) \({}^{\mathrm{b}}\) & (12.540) \({}^{\mathrm{b}}\) & \\ \hline \(\beta\)-1400 & 12 & C2/m, \#12 & 9.612 & 7.669 & 4.068 & 12.247 & 0.0357 \\ \(\beta\)-1800 & 6 & C2/m, \#12 & 6.889 & 3.834 & 6.520 & 12.584 & 0.1064 \\  & & & 2.705 & 2.705 & 5.143 & 12.541 & \\ \(\alpha\)-FeSi\({}_{2}\) & 3 & P4/mmm, & (2.690) \({}^{\mathrm{c}}\) & (2.690) \({}^{\mathrm{c}}\) & (5.130) \({}^{\mathrm{c}}\) & 12.374 & 0.0564 \\  & & & \#123 & (2.710) \({}^{\mathrm{d}}\) & (2.710) \({}^{\mathrm{d}}\) & (5.140) \({}^{\mathrm{d}}\) & 12.583 & \\ \(\alpha\)-Fe\({}_{0.8}\)Si\({}_{2}\) & 45 & P1, \#1 & 2.680 & 2.680 & 5.129 & 12.280 \({}^{\mathrm{*}}\) & - \\ \(\gamma\)-FeSi\({}_{2}\) & 3 & Fm3m, \#225 & 5.400 & 5.400 & 5.400 & 13.122 & 0.1743 \\  & & & (5.400) \({}^{\mathrm{d}}\)Based on these DFT results, total energies versus atomic volume are displayed for \(\beta\)-FeSi\({}_{2}\), \(\alpha\)-FeSi\({}_{2}\), and \(\gamma\)-FeSi\({}_{2}\), and their bcc- and fcc-based approximant counterparts in Figure 8. As expected, the stability of FeSi\({}_{2}\) increases from \(\Theta\)-C16 to \(\alpha\), and to \(\beta\), which is the ground state at 0 K. Then, the next class of stable structures is bcc based, followed by the class associated with fcc-based structures. This is a posteriori expected, since intrinsically, the two constitutive elements, Fe and Si, have the signature of the bcc phase (and not the fcc one). This result justifies the ground-state analysis of the Ising model discussed in Section 4.1 on the simple cubic lattice, since the other simple cubic lattice is fully occupied by Si atoms and therefore can be ignored in the search for ground-state configurations made of Fe atoms and vacancies.  
As shown in Table 4, the calculated lattice parameters for \(\alpha\)-Fe\({}_{0.8}\)Si\({}_{2}\) are closer to those obtained experimentally for the \(\alpha\)-phase than the calculated values for \(\alpha\)-FeSi\({}_{2}\), which is explained by the fact that the high-temperature \(\alpha\)-phase has vacancies primarily located on the Fe sublattice [18; 57; 58]. The most stable phase among those listed in Table 4 is \(\beta\)-FeSi\({}_{2}\). However, there is an unexpected result: the total energy of the intermediate phase, \(\beta\)-1400, is lower compared to that of the a-phase that was revealed in the experiment at T \(>\) 1240 K [59]. This fact will be discussed below.