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Two-dimensional transition-metal dichalcogendes $MX_2$ (es. MoS$_2$, WS$_2$,MoSe$_2$, \ldots) are among the most promising materials for bandgapengineering. Widely studied in these compounds, by means of ab-initiotechniques, is the possibility of tuning the direct-indirect gap character bymeans of in-plane strain. In such kind of calculations however the latticedegrees of freedom are assumed to be classical and frozen. In this paper weinvestigate in details the dependence of the bandgap character (direct vs.indirect) on the out-of-plane distance $h$ between the two chalcogen planes ineach $MX_2$ unit. Using DFT calculations, we show that the bandgap character isindeed highly sensitive on the parameter $h$, in monolayer as well as inbilayer and bulk compounds, permitting for instance the switching from indirectto direct gap and from indirect to direct gap in monolayer systems. Thisscenario is furthermore analyzed in the presence of quantum lattice fluctuationinduced by the zero-point motion. On the basis of a quantum analysis, we arguethat the direct-indirect bandgap transitions induced by the out-of-plane strainas well by the in-plane strain can be regarded more as continuous crossoversrather than as real sharp transitions. The consequences on the physicalobservables are discussed.

Zero point motion and direct/indirect bandgap crossover in layered transition-metal dichalcogenides