Three-band tight-binding model for monolayers of group-VIB transition metal dichalcogenides

Abstract

We present a three-band tight-binding (TB) model for describing the low-energy physics in monolayers of group-VIB transition metal dichalcogenides $M{X}_{2}$ ($M=\text{Mo}$, W; $X=\text{S}$, Se, Te). As the conduction- and valence-band edges are predominantly contributed by the ${d}_{{z}^{2}}$, ${d}_{xy}$, and ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ orbitals of $M$ atoms, the TB model is constructed using these three orbitals based on the symmetries of the monolayers. Parameters of the TB model are fitted from the first-principles energy bands for all $M{X}_{2}$ monolayers. The TB model involving only the nearest-neighbor $M$-$M$ hoppings is sufficient to capture the band-edge properties in the $\ifmmode\pm\else\textpm\fi{}K$ valleys, including the energy dispersions as well as the Berry curvatures. The TB model involving up to the third-nearest-neighbor $M$-$M$ hoppings can well reproduce the energy bands in the entire Brillouin zone. Spin-orbit coupling in valence bands is well accounted for by including the on-site spin-orbit interactions of $M$ atoms. The conduction band also exhibits a small valley-dependent spin splitting which has an overall sign difference between Mo${X}_{2}$ and W${X}_{2}$. We discuss the origins of these corrections to the three-band model. The three-band TB model developed here is efficient to account for low-energy physics in $M{X}_{2}$ monolayers, and its simplicity can be particularly useful in the study of many-body physics and physics of edge states.