  We present systematic ab initio calculations of the charge carrier
correlations between adjacent layers of two-dimensional materials in the
presence of both charged impurity and strain disorder potentials using the
examples of monolayer and bilayer graphene. For the first time, our analysis
yields unambiguous first-principles quantum corrections to the Thomas--Fermi
densities for interacting two-dimensional systems described by orbital-free
density functional theory. Specifically, using density-potential functional
theory, we find that quantum corrections to the quasi-classical Thomas-Fermi
approximation have to be taken into account even for heterostructures of
mesoscopic size. In order for the disorder-induced puddles of electrons and
holes to be anti-correlated at zero average carrier density for both layers,
the strength of the strain potential has to exceed that of the impurity
potential by at least a factor of ten, with this number increasing for smaller
impurity densities. Furthermore, our results show that quantum corrections have
a larger impact on puddle correlations than exchange does, and they are
necessary for properly predicting the experimentally observed Gaussian energy
distribution at charge neutrality.
