Thickness dependence of spin polarization and electronic structure of ultra-thin films of MoS$_2$ and related transition-metal dichalcogenides

Thickness dependence of electronic structures of transition-metal dichalcogenides (TMDs) MX$_2$ (M=Mo or W; X=S, Se or Te) is investigated using first-principles calculations. When spin-orbit coupling (SOC) is included in the computations, the electronic structure of monolayer MX$_2$ films exhibits significant band splittings due to the breaking of spatial inversion symmetry. In particular, spin-split states appear around the valence band maximum with nearly 100\% out-of-the-plane spin polarization with the spin oriented oppositely at the K and K$'$ symmetry points in the Brillouin zone. For bilayer films, the spin-polarization can be tuned by an out-of-the-plane electric field, and the spin-polarized states are weakly coupled between the layers with small $k_z$ dispersion. We confirm a transition from an indirect to a direct band gap as the thickness is reduced to a monolayer in MoX$_2$, in agreement with recent experimental findings. Our study provides insight into the thickness dependence of electronic structure and the degree of spin polarization of the valence bands in ultra-thin TMD films and their viability for spintronics applications. \\[4pt] [1] T.-R. Chang, H. Lin, H.-T. Jeng, and A. Bansil, Sci. Rep. {\bf 4}, 6270 (2014).