The two gap transitions in Ge$_{1-x}$Sn$_x$: effect of non-substitutional complex defects

The existence of non-substitutional $\beta$-Sn defects in Ge$_{1-x}$Sn$_{x}$ was confirmed by emission channeling experiments [Decoster et al., Phys. Rev. B 81, 155204 (2010)], which established that although most Sn enters substitutionally ($\alpha$-Sn) in the Ge lattice, a second significant fraction corresponds to the Sn-vacancy defect complex in the split-vacancy configuration ( $\beta$-Sn ), in agreement with our previous theoretical study [Ventura et al., Phys. Rev. B 79, 155202 (2009)]. Here, we present our electronic structure calculation for Ge$_{1-x}$Sn$_{x}$, including substitutional $\alpha$-Sn as well as non-substitutional $\beta$-Sn defects. To include the presence of non-substitutional complex defects in the electronic structure calculation for this multi-orbital alloy problem, we extended the approach for the purely substitutional alloy by Jenkins and Dow [Jenkins and Dow, Phys. Rev. B 36, 7994 (1987)]. We employed an effective substitutional two-site cluster equivalent to the real non-substitutional $\beta$-Sn defect, which was determined by a Green's functions calculation. We then calculated the electronic structure of the effective alloy purely in terms of substitutional defects, embedding the effective substitutional clusters in the lattice. Our results describe the two transitions of the fundamental gap of Ge$_{1-x}$Sn$_{x}$ as a function of the total Sn-concentration: namely from an indirect to a direct gap, first, and the metallization transition at higher $x$. They also highlight the role of $\beta$-Sn in the reduction of the concentration range which corresponds to the direct-gap phase of this alloy, of interest for optoelectronics applications.