Interaction between charged particles in a 2D slab (and the Rytova-Keldysh approach)

Charged particles in a three dimensional medium with uniform dielectric constant interact via Coulomb potential. This potential solves Poisson equation for a point charge in such a situation. However, in 2D materials and thin films/layers/slabs, the dielectric constant in the system depends on the direction perpendicular to the slab, so that the common Coulomb interaction is no longer suitable. Here, I discuss how to solve Poisson equation in this situation and show the approximations needed to reach the so called Rytova-Keldysh potential, which has been widely used for interactions in 2D materials lately. References: For the non-hydrogenic Rydberg series of exciton energies in a monolayer: https://doi.org/10.1103/PhysRevLett.1... Natalia Rytova's paper: https://arxiv.org/abs/1806.00976 Keldysh's paper: http://jetpletters.ru/ps/1458/article... Cudazzo et al. approach to obtain the Rytova-Keldysh potential: https://doi.org/10.1103/PhysRevB.84.0... Further reading: https://doi.org/10.1103/PhysRevB.97.1... and the Supplemental Material of https://doi.org/10.1103/PhysRevB.103....