Dielectric screening at TMD:hBN interfaces: Monolayer-to-bulk transition, local-field effect, and spatial dependence

The dielectric effects of a substrate have been shown to be important in modulating the electronic properties of an adsorbate, especially in van der Waals heterostructures. Here, using the first-principles dielectric embedding $GW$ approach within the framework of many-body perturbation theory, we perform a case study on the dielectric screening effects of hexagonal boron nitride (hBN) on various transition-metal dichalcogenides (TMDs). We consider three systems: monolayer MoS${}_2$, bilayer MoS${}_2$, and mixed WS${}_2$/MoS${}_2$ bilayer adsorbed on hBN, and examine three aspects of the substrate dielectric screening: (i) thickness dependence and the monolayer-to-bulk transition, where we consider the effects of one-, two-, three-, and four-layer hBN; (ii) local-field effect, where we numerically assess a common approximation of neglecting the in-plane local-field components of the substrate polarizability; and (iii) spatial dependence, where we consider mixed WS${}_2$/MoS${}_2$ bilayer adsorbed on hBN with either side facing the substrate. Our results provide quantitative insight into how the substrate screening effects can be leveraged for band structure engineering.