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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 ofSn composition $x$ are identified, and the impact of epitaxial strain isincluded by constraining supercell lattice constants perpendicular to the [001]growth direction to the lattice constants of Ge, zinc telluride (ZnTe), orcadmium telluride (CdTe) substrates. It is found that application of 1% tensilestrain reduces the Sn composition required to bring the (positive) direct bandgap to zero by approximately 5% compared to a relaxed Ge$_{1-x}$Sn$_{x}$ alloyhaving the same gap at $\Gamma$. On the other hand, compressive strain hascomparatively less impact on the alloy band gap at $\Gamma$. Using DFTcalculated alloy lattice and elastic constants, the critical thickness forGe$_{1-x}$Sn$_{x}$ thin films as a function of $x$ and substrate latticeconstant is estimated, and validated against supercell DFT calculations. Theanalysis correctly predicts the Sn composition range at which it becomesenergetically favourable for Ge$_{1-x}$Sn$_{x}$/Ge to become amorphous. Theinfluence of stoichiometry and strain is examined in relation to reducing themagnitude of the inverted (``negative'') $\Gamma_{7}^{-}$-$\Gamma_{8}^{+}$ bandgap, which is characteristic of semimetallic alloy electronic structure. Basedon our findings, strategies for engineering the semimetal-to-semiconductortransition via strain and quantum confinement in Ge$_{1-x}$Sn$_{x}$nanostructures are proposed.

Impact of stoichiometry and strain on Ge$_{1-x}$Sn$_{x}$ alloys from first principles calculations