Band structure of monolayer transition-metal dichalcogenides and topological properties of their nanoribbons: Next-nearest-neighbor hopping

Abstract

Tight-binding (TB) models exploited in calculating band structure of monolayer transition-metal dichalcogenides (TMDCs), namely ${MX}_{2}$ ($M=$ Mo and W; $X=$ S, Se and Te), can be divided into two groups: one is based on group theory and the other uses Slater-Koster (SK) method. The former in general is lack of flexibility to be extended to confined finite systems with lower symmetry, e.g., nanoribbons (NRs) and quantum dots. Unlike ubiquitous TB models, here we present an improved scheme of the flexible SK TB method in which the second-nearest-neighbor $M\text{\ensuremath{-}}M$ and $X\text{\ensuremath{-}}X$ hopping terms are included. Its improvement, being of comparable accuracy to first-principles calculations, is clearly elucidated through a comprehensive comparison between our results and those produced by widely accepted TB models in literature for monolayer ${\mathrm{MoS}}_{2}$. Besides, its high flexibility allows us to successfully extend our TB model from monolayer TMDCs to the $\mathrm{Mo}{X}_{2}$ ($X=$ S, Se, and Te) and $\mathrm{W}{X}_{2}$ ($X=$ S and Se) NRs of both zigzag and armchair types. We find that the zigzag NR could be either metallic or semiconducting, depending on the spin-orbit strength and band gap of its parent two-dimensional bulk TMDC, which is in contradiction to the usual consensus concerning TMDC NRs which exhibit the metallic behavior only. For a certain Fermi level, remarkably, we discovered that the ${\mathrm{MoS}}_{2}$ NRs demonstrate quantum valley Hall effect, while others only present topological insulator phase.