Understanding topological phase transition in monolayer transition metal dichalcogenides

Despite considerable interest in layered transition metal dichalcogenides (TMDs), such as $M{X}_2$ with $M=(\mathrm{Mo},\mathrm{W})$ and $X=(\mathrm{S},\mathrm{Se},\mathrm{Te})$, the physical origin of their topological nature is still poorly understood. In the conventional view of topological phase transition (TPT), the nontrivial topology of electron bands in TMDs is caused by the band inversion between metal $d$- and chalcogen $p$-orbital bands where the former is pulled down below the latter. Here, we show that, in TMDs, the TPT is entirely different from the conventional speculation. In particular, $M{\mathrm{S}}_2$ and $M\mathrm{S}{\mathrm{e}}_2$ exhibits the opposite behavior of TPT such that the chalcogen $p$-orbital band moves down below the metal $d$-orbital band. More interestingly, in $M\mathrm{T}{\mathrm{e}}_2$, the band inversion occurs between the metal $d$-orbital bands. Our findings cast doubts on the common view of TPT and provide clear guidelines for understanding the topological nature in new topological materials to be discovered.

Figure 1

(a) A schematic for the topological phase transition in TIs. The valence band rises above the conduction band, and an interband hybridization opens the nontrivial energy gap. Based on this principle, the band inversion mechanism can be directly inferred from the band structure. (b) The structural transformation from the 1T to the 1T' phase in TMDs. The primitive cells are denoted as black solid lines, whereas red solid lines indicate the rectangular supercell for the 1T phase. (c) The band structure of monolayer 1T'-$\mathrm{Mo}{\mathrm{S}}_2$ near the Γ point strongly resembles the schematic in (a), exhibiting a suggestive inverted gap of $\mathrm{\Delta }=0.55\mathrm{eV}$ and thus misleading the band inversion mechanism in 1T'-TMDs.

Figure 2

Comparison of the orbital decomposed band structure with the OP decomposed band structure in 1T'-$\mathrm{Mo}{\mathrm{S}}_2$. Red, blue, and green dots denote the $\mathrm{Mo}-d_{yz}, \mathrm{S}-p_y$, and $\mathrm{Mo}-d_{z^2}$ orbital states, respectively. In the OP decomposed band structure, both the orbital character and the effective parity of the wave function are clearly visualized.

Figure 3

The evolution of the band structures during the structural transformation from 1T to 1T' in $M{X}_2$. (a) The OP decomposed band structures of (a) $\mathrm{Mo}{\mathrm{S}}_2$ and (b) $\mathrm{MoT}{\mathrm{e}}_2$. Purple and yellow dots indicate the $|\mathrm{S}-p_x^{-}\rangle$ and $|\mathrm{Mo}-d_{xz}^{+}\rangle$ states, respectively. Red, blue, and green dots indicate the $|M-d_{yz}^{+}\rangle , |X-p_y^{-}\rangle$, and $|M-d_{z^2}^{-}\rangle$ states, respectively. The rightmost panels show the band structures in the 1T' phase with including the spin-orbit coupling.

Figure 4

The OP decomposed band structures of group-VI TMDs in the 1T and 1T' phases without SOC. Purple and yellow dots denote the $|X-p_x^{-}\rangle$ and $|M-d_{xz}^{+}\rangle$ states, respectively, and their corresponding parities are labeled. For all the TMDs, the $|X-p_x^{-}\rangle$ state lies above the $|M-d_{xz}^{+}\rangle$ state, indicating that these states are not involved in the band inversion process.

Figure 5

Schematics for the (a) $p-d$ and (b) $d-d$ band inversion processes in $M{X}_2$. (c) The GGA (left panel) and HSE06 (right panel) calculations for the band ordering in 1T-$M{X}_2$.