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

## TL;DR

This study explores how out-of-plane atomic distances influence the bandgap nature in layered transition-metal dichalcogenides, revealing strain-induced transitions and the role of quantum zero-point motion in these electronic properties.

## Contribution

It provides a detailed quantum analysis of how out-of-plane lattice fluctuations affect the direct-indirect bandgap crossover in layered $MX_2$ materials, extending beyond classical lattice assumptions.

## Key findings

- Bandgap character is highly sensitive to out-of-plane distance $h$.
- Strain can induce continuous crossover between direct and indirect gaps.
- Quantum zero-point motion smooths the transition, preventing sharp phase changes.

## Abstract

Two-dimensional transition-metal dichalcogendes $MX_2$ (es. MoS$_2$, WS$_2$, MoSe$_2$, \ldots) are among the most promising materials for bandgap 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 bandgap character (direct vs. indirect) on the out-of-plane distance $h$ between the two chalcogen planes in each $MX_2$ unit. Using DFT calculations, we show that the bandgap 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 bandgap transitions induced by the out-of-plane strain as well 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.