Symmetric Galerkin boundary element method for computing the quantum states of the electron in a piecewise-uniform mesoscopic system

TL;DR

This paper introduces a symmetric Galerkin boundary element method to accurately compute quantum states of electrons in piecewise-uniform mesoscopic systems, improving numerical efficiency and precision over previous approaches.

Contribution

It presents a novel symmetric discretization of Helmholtz boundary integral operators for better accuracy in quantum state calculations.

Findings

Achieved highly accurate energy spectra and wave functions.

Demonstrated the method's efficiency for two-region mesoscopic systems.

Provided compact boundary integral expressions.

Abstract

The quantum behavior of charge carriers in semiconductor structures is often described in terms of the effective mass Schr\"{o}dinger equation, neglecting the rapid fluctuations of the wave function on the scale of the atomic lattice. For systems with piecewise-constant mass and potential energy, this amounts to solving a set of Helmholtz equations with wavenumbers dictated by the physical parameters of each homogeneous subregion. Making use of the Green function method, the system of differential equations can be expressed in boundary integral form to enable efficient numerical solution. In the present study, this strategy is applied in combination with a Galerkin technique to compute the energy spectrum and the wave functions of the electron in a mesoscopic structure composed of two regions. The proposed formulation differs from those presented before for the same scenario in that it implements a symmetric discretization of the four Helmholtz boundary integral operators, which leads to compact expressions and very accurate results.