Quasiparticle band structures of $\beta$-HgS, HgSe, and HgTe

The electronic structures of mercury chalcogenides in the zinc-blende strucrure have been calculated by the LDA, $GW$ (one-shot, $G_{0}W_{0}$) and quasi-particle self-consistent $GW$ ($QSGW$) approximations including spin-orbit coupling (SO). The slight tendency to overestimation of the band gaps by $QSGW$ is avoided by using a $hybrid$ scheme (20$\%$ LDA and 80 $\%$ $QSGW$. The results of $G_{0}W_{0}$ depend strongly starting wave functions and are thus quite different from those from $QSGW$. Within $QSGW$ HgS is found to be a semiconductor, with a $\Gamma_{6}$ s-like conduction-band minimum state above the valence-band top $\Gamma_{7}$ and $\Gamma_{8}$ (``negative'' SO splitting). HgSe and HgTe have ``negative'' gaps (inverted band structure). In HgTe the $\Gamma_{7}$ state is below $\Gamma_{6}$ due to the large Te SO splitting, in contrast HgSe where $\Gamma_{6}$ is below $\Gamma_{7}$.