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

Abstract

Physical properties of the Si1-xGex 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 Si1-xGex alloys are calculated using the first-principles density functional theory. The supper cell approach employed in our calculations leads to the folding of electronic bands into the smaller Brillouin zone of the supercell, especially at the Γ point. This often leads to the misinterpretation that the materials have direct band gap. The problem can be resolved by the band unfolding technique which allows one to recover the primitive cell picture of band structure of Si1-xGex. As a result, unfolded electronic bands correctly show an indirect band gap with the valence band maximum (VBM) at the Γ point and the conduction band minimum (CBM) shifted away from Γ. The CBM is gradually shifted from a point along ΓX path (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 0.6~0.8 which is in accordance with the calculation using kp method. Moreover, the additional electron pockets appear and develop at Γ 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 Γ points in the Brillouin zone of Si1-xGex alloy nanocrystals.