Semiconductor, topological semimetal, indirect semimetal, and topological Dirac semimetal phases of Ge1−xSnx alloys

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