Gap opening and large spin-orbit splitting in MX2 (M=Mo,W X=S,Se,Te)   from the interplay between crystal field and hybridizations: insights from   ab-initio theory
  By means of first-principles density functional calculations, we study the maximally localized Wannier functions for the 2D transition metal dichalcogenides MX2 (M=Mo,W X=S,Se,Te). We found a M^+4-like ionic charge and a single occupied d-band. The center of the d-like maximally localized Wannier function associated with this band is distributed among three M sites. Part of the energy gap is opened by the crystal field splitting induced by the X^-2-like atoms. We extract the hopping parameters for the Wannier functions and provide a perspective on tight-binding model. From the analysis of the tight binding model, we have found an inversion of the band character between the Gamma and the K points of the Brillouin zone due to the M-M hybridization. The consequence of this inversion is the closure of the gap. The M-X hybridization is the only one that tends to open the gap at every k-point, the change in the M-X and M-M hybridization is the main responsible for the difference in the gap between the different dichalcogenide materials. The inversion of the bands gives rise to different spin-orbit splitting at Gamma and K point in the valence band. The different character of the gap at Gamma and K point offers the chance to manipulate the semiconductive properties of these compounds. For a bilayer system, the hybridizations between the out of plane orbitals and the hybridizations between the in plane orbitals split the valence band respectively at the Gamma and K point. The splitting in the valence band is opened also without spin-orbit coupling and comes from the M-M and X-X hybridization between the two monolayers. 
Carmine Autieri, Adrien Bouhon and Biplab Sanyal
cond-mat.mtrl-sci