Unfolding the band structure of van der Waals heterostructures

Layer-by-layer stacking of two-dimensional materials results in van der Waals heterostructures (vdWH) of nanometer thickness and clean interfaces that often exhibit exceptional properties and present novel challenges. We perform first-principles calculations using density functional theory to examine the electronic properties of vdWH consisting of graphene (Gr) and semiconducting transition metal dichalcogenide (TMD) monolayers at several twist angles. We describe in detail our methodology for the creation of simulation cells which are almost free of strain due to lattice mismatch (less than 1%) and for unfolding the electronic bands of the vdWH in a way that allows for a straightforward comparison with the electronic structure of the constituent monolayers. The weak interlayer interactions in Gr/MoS2 and Gr/WS2 heterobilayers leave the Gr and TMD band structures almost unaffected but move the Fermi level closer to the TMD conduction band minimum. Careful examination of transitions from direct to indirect band gaps for some twist angles reveals that these are due to very small strain remaining in the simulation cells. In WS2/MoS2 and WSe2/MoSe2 heterobilayers, interlayer interactions do not affect the conduction band minimum at K but lead to eigenenergy splitting and eigenstate hybridization at the valence band around the I' point, which is very sensitive to interlayer distance and determines whether the valence band maximum is at the I' or K point. Our results unveil a small redshift of the intralayer electronic transitions of TMDs when interacting with either Gr or other TMD monolayers.