Layer-Dependent Energy of Two Parallel Charged Nano-Layers

We consider a bilayer nanosystem consisting of two parallel charged nano-layers. Such a system represents the typical building block of a multi-layered nanostructure formed by alternatively depositing thin metal-insulator-metal layers. We calculate exactly in closed form the electrostatic interaction energy between two parallel charged nano-layers where each of them is modeled as a finite two-dimensional square-shaped system with constant surface charge density. We apply suitable mathematical transformations that allow us to derive an exact expression for the interaction energy between nano-layers as a function of both size and inter-layer separation distance. We also derive an approximate expression for the interaction energy linearized with respect to the inter-layer separation that works well for small inter-layer separation. We show that the interaction energy of this configuration of nano-layers depends on the ratio of the inter-layer separation distance relative to the characteristic length of the structure.