Band structure and optical transitions in atomic layers of hexagonal gallium chalcogenides

We report density-functional-theory calculations of the electronic band structures and optical absorption spectra of two-dimensional crystals of Ga${}_{2}{X}_{2}$ ($X=$ S, Se, and Te). Our calculations show that all three two-dimensional materials are dynamically stable indirect-band-gap semiconductors with a sombrero dispersion of holes near the top of the valence band. We predict the existence of Lifshitz transitions---changes in the Fermi-surface topology of hole-doped Ga${}_{2}{X}_{2}$---at hole concentrations ${n}_{S}=7.96\ifmmode\times\else\texttimes\fi{}{10}^{13}$ cm${}^{\ensuremath{-}2}$, ${n}_{Se}=6.13\ifmmode\times\else\texttimes\fi{}{10}^{13}$ cm${}^{\ensuremath{-}2}$, and ${n}_{Te}=3.54\ifmmode\times\else\texttimes\fi{}{10}^{13}$ cm${}^{\ensuremath{-}2}$.