  We calculate the electronic structure of germanium-tin (Ge$_{1-x}$Sn$_{x}$)
binary alloys for $0 \leq x \leq 1$ using density functional theory (DFT).
Relaxed alloys with semiconducting or semimetallic behaviour as a function of
Sn composition $x$ are identified, and the impact of epitaxial strain is
included by constraining supercell lattice constants perpendicular to the [001]
growth direction to the lattice constants of Ge, zinc telluride (ZnTe), or
cadmium telluride (CdTe) substrates. It is found that application of 1% tensile
strain reduces the Sn composition required to bring the (positive) direct band
gap to zero by approximately 5% compared to a relaxed Ge$_{1-x}$Sn$_{x}$ alloy
having the same gap at $\Gamma$. On the other hand, compressive strain has
comparatively less impact on the alloy band gap at $\Gamma$. Using DFT
calculated alloy lattice and elastic constants, the critical thickness for
Ge$_{1-x}$Sn$_{x}$ thin films as a function of $x$ and substrate lattice
constant is estimated, and validated against supercell DFT calculations. The
analysis correctly predicts the Sn composition range at which it becomes
energetically favourable for Ge$_{1-x}$Sn$_{x}$/Ge to become amorphous. The
influence of stoichiometry and strain is examined in relation to reducing the
magnitude of the inverted (``negative'') $\Gamma_{7}^{-}$-$\Gamma_{8}^{+}$ band
gap, which is characteristic of semimetallic alloy electronic structure. Based
on our findings, strategies for engineering the semimetal-to-semiconductor
transition via strain and quantum confinement in Ge$_{1-x}$Sn$_{x}$
nanostructures are proposed.
