Semiconductor, topological semimetal, indirect semimetal, and topological Dirac semimetal phases of Ge${}_{1-x}$Sn${}_x$ alloys

Electronic structures of Ge${}_{1-x}$Sn${}_x$ alloys (0 $\le$ $x$ $\le$ 1) are theoretically studied by nonlocal empirical pseudopotential method. For relaxed Ge${}_{1-x}$Sn${}_x$, a topological semimetal is found for $\mathit{\text{x}}$ $>$ 41$\mathrm{\%}$ with gapless and band inversion at $\mathrm{\Gamma }$ point, while there is an indirect-direct bandgap transition at $x$ = 8.5$\mathrm{\%}$. For strained Ge${}_{1-x}$Sn${}_x$ on a Ge substrate, semimetals with a negative indirect bandgap appear for $x$ $>$ 43$\mathrm{\%}$, and the strained Ge${}_{1-x}$Sn${}_x$ on Ge is always an indirect bandgap semiconductor for $x$ $<$ 43$\mathrm{\%}$. With appropriate biaxial compressive strains, a topological Dirac semimetal is found with band inversion at $\mathrm{\Gamma }$ and one pair of Dirac cones along the [001] direction.