Quantum Square Wells with Capacitive Walls: A Toy Model for Quantum Capacitors

This research aims to determine the energy quantization in a one-dimensional infinite square well modified by capacitive walls. The electric field inside the wall produces a linear potential. The solution to the Schrödinger equation is the Airy function for an infinite square well. Furthermore, the Wentzel–Kramers–Brillouin (WKB) approach is applied to finite wells, and the energy quantization for both cases based on this modified potential has been derived. In this paper, we also examine the quantum capacitance of the system, which is determined from the density of states and depends on dimensionality. The results obtained show that there is compatibility between the simple model that we work on and several systems, for example, graphene-base system.