Microscopic dielectric permittivities of graphene nanoribbons and graphene

We derive a microscopic Poisson equation using the density-density response function. This equation is valid for any realistic potential perturbation and permits the study of dielectric response in nanostructures, especially in one-dimensional nanostructures and quantum dots. We apply this equation to simulate a nanoscale parallel-plate capacitor (nanocapacitor) with graphene as dielectric and two nanocapacitors with a graphene nanoribbon (GNR) as dielectric. The density-density response function is calculated using first-order perturbation theory and empirical pseudopotentials. From the microscopic electric field of the graphene nanocapacitor, we calculate the out-of-plane microscopic dielectric constant of graphene and from the electric field of GNR nanocapacitors, we calculate the full microscopic dielectric tensor of several GNRs with different widths. We find that the out-of-plane microscopic dielectric constants of GNRs and graphene do not depend on their energy band gap. We also study the effect of a surrounding dielectric on the dielectric permittivity of graphene and we conclude that the surrounding dielectric barely affects the dielectric permittivity of graphene.