Gap opening and large spin–orbit splitting in (M = Mo,W; X = S,Se,Te) from the interplay between crystal field and hybridisations: insights from ab-initio theory

By means of first-principles density functional calculations, we study the maximally localised Wannier functions for the 2D transition metal dichalcogenides (M = Mo,W; X = S,Se,Te). We have found that part of the energy gap is opened by the crystal field splitting induced by the -like atoms. The inversion of the band character between the and the K points of the Brillouin zone is due to the M–M hybridisation. The consequence of this inversion is the closure of the gap in absence of the M–X hybridisation. The M–X hybridisation is the only one that tends to open the gap at every k-point. It is found that the change in the M–X and M–M hybridisation 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 and K point in the valence band. The different character of the gap at and K point offers the chance to manipulate the semiconducting properties of these compounds. For a bilayer system, the hybridisation between the out-of-plane orbitals and the hybridisation between the in-plane orbitals split the valence band respectively at the and K point. The splitting in the valence band is opened also without spin–orbit coupling and occurs due to the M–M and X–X hybridisation between the two monolayers. The transition from direct to indirect band gap is governed by the hybridisation between out-of-plane orbitals of different layers and in-plane orbitals of different layers.