  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.
