Band energy landscapes in twisted homobilayers of transition metal dichalcogenides

Twistronic assembly of 2D materials employs the twist angle between adjacent layers as a tuning parameter for designing the electronic and optical properties of van der Waals heterostructures. Here, we study how interlayer hybridization, weak ferroelectric charge transfer between layers, and piezoelectric response to deformations set the valence and conduction band edges across the moir{\'e} supercell in twistronic homobilayers of MoS$_2$, MoSe$_2$, WS$_2$ and WSe$_2$. We show that, due to the lack of inversion symmetry in the monolayer crystals, bilayers with parallel (P) and anti-parallel (AP) unit cell orientations display contrasting behaviors. For P-bilayers at small twist angles we find band edges in the middle of triangular domains of preferential stacking. In AP-bilayers at marginal twist angles ($\theta_{AP} < 1^\circ$) the band edges are located in small regions around the intersections of domain walls, giving highly localized quantum dot states.