Zero-point motion and direct-indirect band-gap crossover in layered transition-metal dichalcogenides

Two-dimensional transition-metal dichalcogenides MX2 (MoS2, WS2, MoSe2,...) are among the most promising materials for band-gap engineering. Widely studied in these compounds, by means of ab initio techniques, is the possibility of tuning the direct-indirect gap character by means of in-plane strain. In such kind of calculations however the lattice degrees of freedom are assumed to be classical and frozen. In this paper we investigate in details the dependence of the band-gap character (direct vs indirect) on the out-of-plane distance h between the two chalcogen planes in each MX2 unit. Using DFT calculations, we show that the band-gap character is indeed highly sensitive on the parameter h, in monolayer as well as in bilayer and bulk compounds, permitting for instance the switching from indirect to direct gap and from indirect to direct gap in monolayer systems. This scenario is furthermore analyzed in the presence of quantum lattice fluctuation induced by the zero-point motion. On the basis of a quantum analysis, we argue that the direct-indirect band-gap transitions induced by the out-of-plane strain as well as by the in-plane strain can be regarded more as continuous crossovers rather than as real sharp transitions. The consequences on the physical observables are discussed.