Designing in-plane heterostructures of quantum spin Hall insulators from first principles: $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$ with adsorbates

Interfaces between normal and topological insulators are bound to host metallic states that are protected by time-reversal symmetry and are therefore robust against disorder and interface reconstruction. Two-dimensional topological insulators (quantum spin Hall insulators) offer a unique opportunity to change the local topology by adsorption of atoms or molecules and thus comprise an ideal platform for designing topological heterostructures. Here we apply first-principles calculations to show that the quantum spin Hall insulator $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$ exhibits a phase transition to a trivial insulator upon adsorption of various atoms. It is then demonstrated that one-dimensional metallic states indeed arise at the boundary of regions with and without adsorbed oxygen and that these boundary states generically constitute simple linear connections between valence and conduction bands in reciprocal space. This is in sharp contrast to topological edge states, which typically exhibit strong dispersion that are sensitive to a particular edge termination. The heterostructure is also suggestive of a simple design of one-dimensional metallic networks in sheets of $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$.

Figure 1

In-plane heterostructure of a quantum spin Hall insulator (QSHI) and a normal insulator (NI). The boundary regions host one-dimensional spin-polarized metallic states. In addition to the boundary state shown here there will be a counterpropagating state of opposite spin.

Figure 2

Left: Band structure of pristine $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$. Right: Top and side view of the $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$ structure. Unit cell indicated by black dashed lines and different edge terminations indicated by the three cuts.

Figure 3

Spectral functions of $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$ projected onto a single edge. The three different figures corresponds to the three edge terminations indicated in Fig. 2.

Figure 4

Left: Band structure at different snapshots along the configuration path where a single O atom adsorbs on the face of $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$. Here $d$ denotes the distance in $\AA$ to the equilibrium adsorption position. Right: Charge centers of hybrid Wannier functions localized along the $x$ direction as a function of the orthogonal crystal momentum $k_y$. The red color indicates oxygen character of the bands and the blue color indicates $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$ character.

Figure 5

Oxygen diffusion used for the nudged elastic band calculation. The transition barrier for the hopping is 2 eV.

Figure 6

Left: Spectral function of $1{\mathrm{T}}^{\prime }-{\mathrm{MoS}}_2$ heterostructures with a semi-infinite adsorbate layer of oxygen. The three different figures have different terminations of the adsorbate layer and correspond roughly to the three figures in Fig. 3.

Figure 7

Band structure of 2H-${\mathrm{MoS}}_2$. The colors indicate the spin character $S_{nk}=\langle nk|{\sigma }_z|nk\rangle$, with blue being spin down ($S_{nk}=-1$) and red being spin up ($S_{nk}=1)$, The bands without spin-orbit coupling are indicated by dashed gray lines.

Figure 8

Band structure of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ with (blue lines) and without (dashed gray lines) spin-orbit coupling.

Figure 9

Division of regions used for the iterative Green's function calculation of the interface spectral function.