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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. B81, 155204 (2010)], which established that although most Sn enterssubstitutionally ($\alpha$-Sn) in the Ge lattice, a second significant fractioncorresponds to the Sn-vacancy defect complex in the split-vacancy configuration( $\beta$-Sn ), in agreement with our previous theoretical study [Ventura etal., Phys. Rev. B 79, 155202 (2009)]. Here, we present our electronic structurecalculation for Ge$_{1-x}$Sn$_{x}$, including substitutional $\alpha$-Sn aswell as non-substitutional $\beta$-Sn defects. To include the presence ofnon-substitutional complex defects in the electronic structure calculation forthis multi-orbital alloy problem, we extended the approach for the purelysubstitutional alloy by Jenkins and Dow [Jenkins and Dow, Phys. Rev. B 36, 7994(1987)]. We employed an effective substitutional two-site cluster equivalent tothe real non-substitutional $\beta$-Sn defect, which was determined by aGreen's functions calculation. We then calculated the electronic structure ofthe effective alloy purely in terms of substitutional defects, embedding theeffective substitutional clusters in the lattice. Our results describe the twotransitions of the fundamental gap of Ge$_{1-x}$Sn$_{x}$ as a function of thetotal Sn-concentration: namely from an indirect to a direct gap, first, and themetallization transition at higher $x$. They also highlight the role of$\beta$-Sn in the reduction of the concentration range which corresponds to thedirect-gap phase of this alloy, of interest for optoelectronics applications.

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