Quantum spin Hall effect in monolayer and bilayer ${\mathrm{TaIrTe}}_4$

Generally, stacking two quantum spin Hall insulators gives rise to a trivial insulator. Here, based on first-principles electronic structure calculations, we confirm that monolayer ${\mathrm{TaIrTe}}_4$ is a quantum spin Hall insulator and remarkably find that bilayer ${\mathrm{TaIrTe}}_4$ is still a quantum spin Hall insulator. Theoretical analysis indicates that the covalentlike interlayer interaction in combination with the small band gap at the time-reversal invariant $\mathrm{\Gamma }$ point results in new band inversion in bilayer ${\mathrm{TaIrTe}}_4$, namely, the emergence of quantum spin Hall phase. Meanwhile, a topological phase transition can be observed by increasing the interlayer distance in bilayer ${\mathrm{TaIrTe}}_4$. Considering that bulk ${\mathrm{TaIrTe}}_4$ is a type-II Weyl semimetal, layered ${\mathrm{TaIrTe}}_4$ thus provides an ideal platform to realize different topological phases at different dimensions.

Figure 1

(a) Crystal structure of bulk ${\mathrm{TaIrTe}}_4$. The red, green, and blue balls denote Ta, Ir, and Te atoms, respectively. The dashed rectangle highlights the layered structure. (b) Top view of monolayer ${\mathrm{TaIrTe}}_4$, where Te atoms are hidden for clarity. Several typical interatomic distances are labeled. The dashed lines with labels “left” and “right” denote two different terminations of thin-layer ${\mathrm{TaIrTe}}_4$, respectively. (c) Surface Brillouin zone (BZ) of monolayer ${\mathrm{TaIrTe}}_4$. The red dots represent the high-symmetry $k$ points.

Figure 2

Evolution of the total energy of bulk ${\mathrm{TaIrTe}}_4$ with interlayer distance $d$. Here $d_0$ denotes the equilibrium interlayer distance.

Figure 3

Electronic structure of monolayer ${\mathrm{TaIrTe}}_4$: (a) density of states; band structures calculated (b) without and (c) with spin-orbit coupling.

Figure 4

Spectral function of monolayer ${\mathrm{TaIrTe}}_4$ with a semi-infinite termination edge. The corresponding edge states are highlighted by red arrows. The white-black color scale represents the spectral density with the unit of states per eV.

Figure 5

Band structures of bilayer ${\mathrm{TaIrTe}}_4$ calculated (a) without and (b) with SOC. Inset shows the enlarged band structure around the $\mathrm{\Gamma }$ point.

Figure 6

Spectral function of bilayer ${\mathrm{TaIrTe}}_4$ left (a) and right (b) terminations. Refer to Fig. 1 for the definition of left and right terminations. The corresponding edge states are highlighted by red arrows. The white-black color scale represents the spectral density with the unit of states per eV.

Figure 7

Topological invariant $Z_2$ with increasing interlayer distance for bilayer ${\mathrm{TaIrTe}}_4$. Differential charge densities for (a) equilibrium structure ($d=d_0$) and (b) enlarged interlayer distance ($d=d_0+2$ Å), respectively. Here only positive differential charge densities are illustrated.