Free-energy functional of the electronic potential for Schr\"{o}dinger-Poisson theory

TL;DR

This paper develops a variational free-energy functional for the Schrödinger-Poisson system, enabling more efficient simulations of electronic potentials in nanostructures by bypassing iterative solutions.

Contribution

It introduces a true free-energy functional for the Schrödinger-Poisson theory applicable to different quantum well regimes, facilitating computational efficiency.

Findings

Functional accurately describes narrow quantum wells using local density approximation.

Functional applicable to wider channels with Thomas-Fermi approximation.

Reduces computational costs in electronic structure simulations.

Abstract

In the study 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 (SP) equation. It is possible to bypass this obstacle by adopting a variational approach and obtaining the solution of the SP equation by minimizing a functional. Further, using molecular dynamics methods that treat the electronic potential as a dynamical variable, the functional can be minimized on the fly in conjunction with the update of other dynamical degrees of freedom leading to considerable reduction in computational costs. But such approaches require access to a true free-energy functional, one that evaluates to the equilibrium free energy at its minimum. In this paper, we present a variational formulation of the Schr\"{o}dinger-Poisson (SP) theory with the needed free-energy functional of the electronic potential. We apply our formulation to semiconducting nanostructures and provide the expression of the free-energy functional for narrow channel quantum wells where the local density approximation yields accurate physics and for the case of wider channels where Thomas-Fermi approximation is valid.