Temperature effects in topological insulators of transition metal dichalcogenide monolayers

We investigate the role of temperature on the topological insulating state of metal dichalcogenide monolayers, 1T$^\prime$-MX$_2$ (M=W, Mo and X=S, Se). Using first principles calculations based on density functional theory, we consider three temperature-related contributions to the topological band gap: electrons coupling with short-wavelength phonons, with long-wavelength phonons \textit{via} Fr\"ohlich coupling, and thermal expansion. We find that electron-phonon coupling promotes the topology of the electronic structures of all 1T$^\prime$-MX$_2$ monolayers, while thermal expansion acts as a counteracting effect. Additionally, we derive the band renormalization from Fr\"ohlich coupling in the two-dimensional context and observe its relatively modest contribution to 1T$^\prime$-MX$_2$ monolayers. Finally, we present a simplified physical picture to understand the "inverse Varshni" effect driven by band inversion in topological insulators. Our work reveals that, among the four 1T$^\prime$-MX$_2$ studied monolayers, MoSe$_2$ is a promising candidate for room temperature applications as its band gap displays remarkable resilience against thermal expansion, while the topological order of WS$_2$ can be tuned under the combined influence of strain and temperature. Both materials represent novel examples of temperature promoted topological insulators.