## 2 Crystallographic Examination and Relationships  
As was said before, there are two polytypes of bulk crystalline iron disilicide (FeSi\({}_{2}\)): the semiconducting orthorhombic \(\beta\)-phase (o548, space group Cmca, #64), stable below T\({}_{\rm C}\)\(\sim\)1275.5 K (1002.5 \({}^{\circ}\)C), and the metallic tetragonal \(\alpha\)-phase (tP3, space group P4/mmm, #123; \(\alpha\)-Fe\({}_{y}\)Si\({}_{2}\), (y \(<\) 1 - \(\
u\)), with vacancy composition, \(\
u\), on the Fe sublattice, stable above 1233 K (960 \({}^{\circ}\)C) and up to 1484 K (1211 \({}^{\circ}\)C), in addition to a nearby equiatomic-phase \(\varepsilon\)-FeSi (cP8, space group P2\({}_{1}\)3, #198), stable up to 1682 K (1409 \({}^{\circ}\)C). The thermodynamic reactions that take place in the Si-rich part of the phase diagram are:  
eutectoid: \(\alpha\)-FeSi\({}_{2}\rightarrow\beta\)-FeSi\({}_{2}\) + Si at 1233 K (960 \({}^{\circ}\)C)  
peritectoid: \(\varepsilon\)-FeSi + \(\alpha\)-FeSi\({}_{2}\rightarrow\beta\)-FeSi\({}_{2}\) at 1275.5 K (1002.5 \({}^{\circ}\)C)  
eutectic: liquid \(\rightarrow\alpha\)-FeSi\({}_{2}\) + FeSi at 1479 K (1206 \({}^{\circ}\)C)  
eutectic: liquid \(\rightarrow\alpha\)-FeSi\({}_{2}\) + Si at 1482 K (1209 \({}^{\circ}\)C)  
congruent melting: \(\alpha\)-FeSi\({}_{2}\rightarrow\) liquid at 1484 K (1211 \({}^{\circ}\)C)  
The orthorhombic \(\beta\)-phase has 48 atoms in the unit cell with the following lattice parameters: a = 9.863, b = 7.791, and c = 7.833, in A [30,31], described in Table 1 with 16 Fe sites and two types of Fe sites (Fe1 and Fe2), and with 32 Si sites with two types of Si sites (Si1 and Si2). This complex structure can be viewed in a first approximation as a stacking of Si-occupied cubes of parameter a\({}^{\prime}\) = a/4 with an alternate occupation of the cube centers by Fe and vacancy, \(\mathcal{V}\). A detailed description of this structure can be found in Ref. [30]. \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Site** & & **x\({}_{\rm C}\)** & **y\({}_{\rm C}\)** & **z\({}_{\rm C}\)** & **x\({}_{\rm A}\)** & **y\({}_{\rm A}\)** & **z\({}_{\rm A}\)** \\ \hline \multirow{3}{*}{Fe, 8d} & Fe1 & 0.2146 & 0 & 0 & 1/4 & 0 & 0 \\  & Fe1 & 0.7854 & 0 & 0 & 3/4 & 0 & 0 \\  & Fe1 & 0.7146 & 0 & 1/2 & 3/4 & 0 & 1/2 \\  & Fe1 & 0.2854 & 0 & 1/2 & 1/4 & 0 & 1/2 \\ \hline \multirow{3}{*}{Fe, 8f} & Fe2 & 1/2 & 0.3086 & 0.1851 & 1/2 & 1/4 & 1/4 \\  & Fe2 & 0 & 0.3086 & 0.3149 & 0 & 1/4 & 1/4 \\  & Fe2 & 1/2 & 0.1914 & 0.6851 & 1/2 & 1/4 & 1/4 \\  & Fe2 & 0 & 0.1914 & 0.8149 & 0 & 1/4 & 3/4 \\ \hline \multirow{3}{*}{Si, 16g} & Si1 & 0.1282 & 0.2746 & 0.0512 & 1/8 & 1/4 & 0 \\  & Si1 & 0.8718 & 0.2746 & 0.0512 & 7/8 & 1/4 & 0 \\ \cline{1-1}  & Si1 & 0.6282 & 0.2746 & 0.4488 & 5/8 & 1/4 & 1/2 \\ \cline{1-1}  & Si1 & 0.3718 & 0.2746 & 0.4488 & 3/8 & 1/4 & 1/2 \\ \cline{1-1}  & Si1 & 0.1282 & 0.2254 & 0.0512 & 1/8 & 1/4 & 1/2 \\ \cline{1-1}  & Si1 & 0.8718 & 0.2254 & 0.0512 & 7/8 & 1/4 & 1/2 \\ \cline{1-1}  & Si1 & 0.6282 & 0.2254 & 0.9488 & 5/8 & 1/4 & 1 \\ \cline{1-1}  & Si1 & 0.3718 & 0.2254 & 0.9488 & 3/8 & 1/4 & 1 \\ \hline \end{tabular}
\end{table} 
Table 1: Description of the \(\beta\)-FeSi\({}_{2}\) structure with cell content of (0 0 0) + (1/2 1/2 0) and the coordinates of the current (C) structure in units of the lattice parameters a, b, and c of the 24-atom unit cell, and those of the approximant (A) one within the unit cell. The tetragonal \(\alpha\)-phase has two atoms in the unit cell with the following lattice parameters: a = 2.7047, b = 2.7047, and c = 5.1430, in A [31; 32; 33], described in Table 2 with one Fe site and one Si site. The \(\alpha\)-FeSi\({}_{2}\) structure can be viewed as a defected CsCl-type order with alternate layers of Cs (Fe) in the c direction omitted. As a result, the axial ratio c/2a is slightly less than unity, and the Si layers are displaced towards each other. Thus in the [001] direction, the structure is made up of a 4\({}^{4}\) Fe layer surrounded on either side by 4\({}^{4}\) Si layers displaced (a + b)/2 relative to the Fe layer, giving three-layer units.