Supplementary material from Thomas–Fermi theory of out-of-plane charge screening in graphene

This paper provides a variational treatment of the effect of external charges on the free charges in an infinite free-standing graphene sheet within the Thomas–Fermi theory. We establish existence, uniqueness and regularity of the energy minimizers corresponding to the free charge densities that screen the effect of an external electrostatic potential at the neutrality point. For the potential due to one or several off-layer point charges, we also prove positivity and a precise universal asymptotic decay rate for the screening charge density, as well as an exact charge cancellation by the graphene sheet. We also treat a simpler case of the non-zero background charge density and establish similar results in that case.