Band structure calculations of Si-Ge-Sn alloys: Achieving direct band gap materials

Alloys of silicon (Si), germanium (Ge) and tin (Sn) are continuously attracting research attention as possible direct band gap semiconductors with prospective applications in optoelectronics. The direct gap property may be brought about by the alloy composition alone or combined with the influence of strain, when an alloy layer is grown on a virtual substrate of different compositions. In search for direct gap materials, the electronic structure of relaxed or strained Ge_1-xSn_x and Si_1-xSn _x alloys, and of strained Ge grown on relaxed Ge _1-x-ySi_xSn_y, was calculated by the self-consistent pseudo-potential plane wave method, within the mixed-atom supercell model of alloys, which was found to offer a much better accuracy than the virtual crystal approximation. Expressions are given for the direct and indirect band gaps in relaxed Ge_1-xSn_x, strained Ge grown on relaxed Si_xGe_1-x-ySn_y and strained Ge _1-xSn_x grown on a relaxed Ge_1-ySn_y substrate, and these constitute the criteria for achieving a (finite) direct band gap semiconductor. Roughly speaking, good-size (up to ∼0.5 eV) direct gap materials are achievable by subjecting Ge or Ge_1-xSn_x alloy layers to an intermediately large tensile strain, but not excessive because this would result in a small or zero direct gap (detailed criteria are given in the text). Unstrained Ge_1-xSn_x bulk becomes a direct gap material for Sn content of >17%, but offers only smaller values of the direct gap, typically ≤0.2 eV. On the other hand, relaxed Sn _xSi_1-x alloys do not show a finite direct band gap.