  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.
