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Physical properties of the Si$_{1-x}$Ge$_{x}$ alloys ($x$ being the composition of Ge) can be understood and predicted from their electronic band structures. In this paper, electronic band structures of the Si$_{1-x}$Ge$_{x}$ alloys are calculated using the first-principles density functional theory. The supper cell approach employed in our calculations leads to folding of electronic bands into the smaller Brillouin zone of the supercell, especially at the $\Gamma$ point. This often leads to the misinterpretation that the materials have direct band gap. The problem can be resolved by an unfolding band technique which allows us to recover the primitive cell picture of band structure of Si$_{1-x}$Ge$_{x}$. As a result, unfolded electronic bands correctly show an indirect band gap with the valence band maximum (VBM) at the $\Gamma $ point and the conduction band minimum (CBM) shifted away from $\Gamma$. CBM is gradually shifted from a point along $\Gamma X$ symmetry line (associated with Si) to the L point (associated with Ge) with the increased Ge composition $x$ and the switching occurs at $x$ in the range of 0.6$\sim$0.8 which is in accordance with the calculation using \textbf{\textit{kp}} method. Moreover, the additional electron pockets appear and develop at $\Gamma$ and $L$. This provides more comprehensive understanding for our recent experimental observations on the shift of the absorption energy assigned to $E1$ direct transitions within $L$ and $\Gamma$ points in the Brillouin zone of Si$_{1-x}$Ge$_{x}$ alloy nanocrystals.

Unfolding Band and Absorption Energy Shift of \(\text{Si-Ge}\) Nano Crystals from First-principles Calculations