The transition from an indirect to a direct gap semiconductor in unstrained as well as compressively and tensile strained ${\mathrm{Sn}}_x{\mathrm{Ge}}_{1-x}$ alloys is investigated as a function of the Sn content 0 ≤ $x$ ≤ 1 by means of both a very accurate supercell approach and the more approximate virtual crystal approximation (VCA). In the local density approximation we calculate the bowing parameter of the lattice constant of unstrained ${\mathrm{Sn}}_x{\mathrm{Ge}}_{1-x}$ alloys. Provided that pseudopotentials suitable for the VCA are used, the random supercell and VCA approaches yield consistent bowing parameters for the lattice constant of −0.21 and −0.28 Å, respectively, in the entire Sn concentration range. The band structures and energy gaps are calculated using the modified Becke-Johnson potential, which, for Ge, yields a one-electron band gap in very good agreement with experimental data. The crossover from an indirect to a direct gap semiconducting alloy is determined at about 4.5% Sn in unstrained ${\mathrm{Sn}}_x{\mathrm{Ge}}_{1-x}$. When ${\mathrm{Sn}}_x{\mathrm{Ge}}_{1-x}$ is grown commensurately and thus strained on Ge(100), a transition to a direct gap is also observed but at Sn concentrations of about 10%. We finally predict the direct and indirect band gaps as a function of the in-plane lattice constant and Sn concentration for ${\mathrm{Sn}}_x{\mathrm{Ge}}_{1-x}$ alloys grown on (100) substrates.

• Received 20 December 2013
• Revised 6 March 2014

DOI:https://doi.org/10.1103/PhysRevB.89.165201