Cylindrical coordinate representation for multiband Hamiltonians

TL;DR

This paper develops a cylindrical coordinate formalism for multiband Hamiltonians in semiconductor heterostructures, enabling efficient modeling of electron states with cylindrical symmetry, and validates it through comparison with real quantum wire states.

Contribution

It introduces a complete formalism using cylindrical coordinates for multiband Hamiltonians, including boundary conditions, for the first time.

Findings

The formalism accurately models valence-band states in cylindrical semiconductor structures.

Cylindrical symmetry approximation is effective for quantum wires with hexagonal cross-sections.

The approach simplifies the analysis of low-dimensional semiconductor systems.

Abstract

Rotationally invariant combinations of the Brillouin zone-center Bloch functions are used as basis function to express in cylindrical coordinates the valence-band and Kane envelope-function Hamiltonians for wurtzite and zinc-blende semiconductor heterostructures. For cylindrically symmetric systems, this basis allows to treat the envelope functions as eigenstates of the operator of projection of total angular momentum on the symmetry axis, with the operator's eigenvalue conventionally entering the Hamiltonians as a parameter. Complementing the Hamiltonians with boundary conditions for the envelope functions on the symmetry axis, we present for the first time a complete formalism for efficient modeling and description of multiband electron states in low-dimensional semiconductor structures with cylindrical symmetry. To demonstrate the potency of the cylindrical symmetry approximation and establish a criterion of its applicability for actual structures, we map the ground and several excited valence-band states in an isolated wurtzite GaN quantum wire of a hexagonal cross-section to the states in an equivalent quantum wire of a circular cross-section.