Transition from parabolic to ring-shaped valence band maximum in few-layer GaS, GaSe, and InSe

Abstract

By performing first-principles electronic structure calculations in frames of density functional theory we study the dependence of the valence band shape on the thickness of few-layer III-VI crystals (GaS, GaSe, and InSe). We estimate the critical thickness of transition from the bulklike parabolic to the ring-shaped valence band. Direct supercell calculations show that the ring-shaped extremum of the valence band appears in $\ensuremath{\beta}$-GaS and $\ensuremath{\beta}$-GaSe at a thickness below 6 tetralayers $(\ensuremath{\sim}4.6nm)$ and 8 tetralayers $(\ensuremath{\sim}6.4nm)$, respectively. Zone-folding calculations estimate the $\ensuremath{\beta}$-InSe critical thickness to be equal to 28 tetralayers $(\ensuremath{\sim}24.0nm)$. The origin of the ring-shaped valence band maximum can be understood in terms of $\text{k}\ifmmode\cdot\else\textperiodcentered\fi{}\text{p}$ theory, which provides a link between the curvature of the energy bands and the distance between them. We explain the dependence of the band shape on the thickness, as well as the transition between two types of extremes, by the $k$-dependent orbital composition of the topmost valence band. We show that in the vicinity of critical thickness the effective mass of holes in III-VI compounds depends strongly on the number of tetralayers.