Quantum spin Hall effect and topological phase transition in two-dimensional square transition-metal dichalcogenides

Two-dimensional (2D) topological insulators (TIs) hold promise for applications in spintronics based on the fact that the propagation direction of an edge electronic state of a 2D TI is locked to its spin orientation. Here, using first-principles calculations, we predict a family of robust 2D TIs in monolayer square transition-metal dichalcogenides $M{X}_2(M=\mathrm{Mo},\mathrm{W};X=\mathrm{S},\mathrm{Se},\mathrm{Te})$, which show sizeable intrinsic nontrivial band gaps ranged from 24 to 187 meV, thus ensuring the quantum spin Hall (QSH) effect at room temperature. Different from the most known 2D TIs with comparable band gaps, these sizeable energy gaps arise from the strong spin-orbit interaction related to $d$ electrons of the Mo/W atoms. A pair of topologically protected helical edge states emerges at the edge of these systems with a Dirac-type dispersion within the bulk band gap. The topologically nontrivial natures are confirmed by the nontrivial ${\mathrm{Z}}_2$-type topological invariant. More interestingly, with applied strain, a topological quantum phase transition between a QSH phase and a trivial insulating/metallic phase can be realized, and the corresponding topological phase diagram is well established.

Figure 1

Crystal structures of monolayer 1S-$M{X}_2$ from (a) top and (b) side views. (c) 2D Brillouin zone of monolayer 1S-$M{X}_2$ and one-dimensional (1D) projected Brillouin zone of the corresponding nanoribbon with high-symmetry points. (d) Band structures of monolayer 1S-$\mathrm{Mo}{\mathrm{S}}_2$ and 1S-$\mathrm{W}{\mathrm{S}}_2$ with and without SOC. The Fermi level is set to zero.

Figure 2

Phonon band dispersion of (a) 1S-$\mathrm{Mo}{\mathrm{S}}_2$, (b) 1S-$\mathrm{MoS}{\mathrm{e}}_2$, (c) 1S-$\mathrm{MoT}{\mathrm{e}}_2$, (d) 1S-$\mathrm{W}{\mathrm{S}}_2$, (e) 1S-$\mathrm{WS}{\mathrm{e}}_2$, and (f) 1S-$\mathrm{WT}{\mathrm{e}}_2$.

Figure 3

(a) The evolution of orbital-resolved band structures of 1S-$\mathrm{Mo}{\mathrm{S}}_2$ and 1S-$\mathrm{W}{\mathrm{S}}_2$ when taking into account SOC. (b) Band structure of 1S-$\mathrm{Mo}{\mathrm{S}}_2$ around the Fermi level in 2D $k$ space with energy as the third dimension. (c) The global band gap $\left(E_g\right)$ and the SOC-induced direct band gap at the $\mathrm{\Gamma }$ point $\left(E_{\mathrm{\Gamma }}\right)$ of 1S-$M{X}_2$. (d) The parities of HVB-1, HVB, LCB, and $\mathrm{LCB}+1$ at the $\mathrm{\Gamma }$ and $M$ points for 1S-$M{X}_2$; the products of the occupied bands at the $\mathrm{\Gamma }$ and $M$ points are also listed in the brackets. $\delta {\left(X\right)}^2\equiv 1$ has no effect on band topology, and therefore the results at the $X$ points are not shown here.

Figure 4

Atomic and band structures of (a) 1S-$\mathrm{Mo}{\mathrm{S}}_2$ and (b) 1S-$\mathrm{W}{\mathrm{S}}_2$ nanoribbons. The red lines denote electronic states localized on edge atoms, while blue ones denote nonlocalized states. The Fermi level is set to zero.

Figure 5

Topological phase diagram of 1S-$M{X}_2$ as a function of strain. The insert shows the value of the global band gap $\left(E_g\right)$ of 1S-$M{X}_2$ as a function of strain.