A variational free-energy functional approach to the Schr\"{o}dinger-Poisson theory

In the numerical simulation of model electronic device systems, where electrons are typically under confinement, a key obstacle is the need to iteratively solve the coupled Schr\"{o}dinger-Poisson equation in order to obtain the electronic potential. We show that it is possible to bypass this obstacle by adopting a variational approach and obtaining the solution of the SP equation by minimizing a functional. We construct the required functional and establish some of its properties. We apply this formulation to the case of narrow channel quantum wells where the local density approximation yields accurate results.