Maximal symetrization and reduction of fields: application to wavefunctions in solid state nanostructures

TL;DR

The paper introduces a comprehensive formalism called MSRF for symmetry-based reduction of fields, enhancing analysis and computation of wavefunctions in solid state nanostructures with high symmetry.

Contribution

It presents a new general formalism for maximal symmetry reduction applicable to various coupled PDEs, with a systematic spatial domain reduction and optimal basis construction, improving analysis and computational efficiency.

Findings

Enables sharper symmetry analysis of eigenstates

Reduces computational domain size for simulations

Applicable to a wide range of coupled PDE systems

Abstract

A novel general formalism for the maximal symetrization and reduction of fields (MSRF) is proposed and applied to wavefunctions in solid state nanostructures. Its primary target is to provide an essential tool for the study and analysis of the electronic and optical properties of semiconductor quantum heterostructures with relatively high point-group symmetry, and studied with the $k\cdot p$ formalism. Nevertheless the approach is valid in a much larger framework than $k\cdot p$ theory, it is applicable to arbitrary systems of coupled partial differential equations (e.g. strain equations or Maxwell equations). For spinless problems (scalar equations), one can use a systematic Spatial Domain Reduction (SDR) technique which allows, for every irreducible representation, to reduce the set of equations on a minimal domain with automatic incorporation of the boundary conditions at the border, which are shown to be non-trivial in general. For a vectorial or spinorial set of functions, the SDR technique must be completed by the use of an optimal basis in vectorial or spinorial space (in a crystal we call it the Optimal Bloch function Basis - OBB). The advantages are numerous: sharper insights on the symmetry properties of every eigenstate, minimal coupling schemes, analytically and computationally exploitable at the component function level, minimal computing domains. The formalism can be applied also as a postprocessing operation, offering all subsequent analytical and computationnal advantages of symmetrization. The specific case of a quantum wire (QWRs) with $C_{3v}$ point group symmetry is used as a concrete illustration of the application of MSRF.