R-Matrix Theory for Transport in Interacting Semiconductor Quantum Devices: Local Density Approximation and Weak Formulation

In a series of publications, a scattering theory of quantum transport in mesoscopic semiconductor devices was developed in the R-matrix formalism. Recently, the R-matrix theory was cast in the weak formulation for non-interacting systems. In this paper we extend this theory to take into account the Coulomb interaction within the local density approximation of the density functional theory by Hohenberg, Kohn, and Sham. As a major advantage, the weak formulation formalism allows us to introduce a wide class of approximations by choosing appropriate basis functions. Here we implement the spectral element method for numerical purposes, which is a version of the finite element method that uses high-degree piecewise polynomials (Lagrange elements) as basis functions. Important for the density functional theory we derive an expansion of the carrier density in Wigner–Eisenbud functions in which the energy integration becomes space-independent. Finally we cast the Poisson equation in weak formulation as well to obtain a self-consistency loop for iteratively solving the local density approximation.