[Paper Review] Cylindrical coordinates representation for valence-band and Kane Hamiltonians for wurtzite and zinc-blende heterostructures

Eduard Takhtamirov, Roderick Melnik|arXiv (Cornell University)|Jul 7, 2011

GaN-based semiconductor devices and materials1 citations

TL;DR

This paper introduces a cylindrical coordinate formalism using rotationally invariant Bloch functions to model multiband electron states in wurtzite and zinc-blende semiconductor heterostructures with cylindrical symmetry. By expressing valence-band and Kane Hamiltonians in cylindrical coordinates and treating envelope functions as eigenstates of angular momentum projection, the method enables efficient, accurate modeling of low-dimensional systems, validated by mapping GaN quantum wire states to equivalent circular cross-section models.

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.

Motivation & Objective

• To develop a formalism for modeling multiband electron states in low-dimensional semiconductor heterostructures with cylindrical symmetry.
• To enable efficient computation of valence-band and Kane Hamiltonians in systems with wurtzite and zinc-blende crystal structures.
• To establish a criterion for the applicability of the cylindrical symmetry approximation in real semiconductor nanostructures.
• To map valence-band states in hexagonal GaN quantum wires to equivalent circular cross-section models for validation.

Proposed method

• Using rotationally invariant combinations of Brillouin zone-center Bloch functions as basis functions in cylindrical coordinates.
• Expressing the valence-band and Kane envelope-function Hamiltonians in cylindrical coordinates to exploit symmetry.
• Treating envelope functions as eigenstates of the total angular momentum projection operator on the symmetry axis, with eigenvalues used as parameters in the Hamiltonians.
• Incorporating boundary conditions for envelope functions on the symmetry axis to ensure physical consistency.
• Applying the formalism to model GaN quantum wires with hexagonal and circular cross-sections for comparative analysis.
• Using the formalism to compute and compare ground and excited valence-band states in equivalent structures.

Experimental results

Research questions

• RQ1 How can multiband electron states in wurtzite and zinc-blende semiconductor heterostructures be efficiently modeled under cylindrical symmetry?
• RQ2 To what extent does the cylindrical symmetry approximation accurately describe valence-band states in real nanostructures like GaN quantum wires?
• RQ3 What is the quantitative correspondence between valence-band states in GaN quantum wires with hexagonal versus circular cross-sections?
• RQ4 How does the use of rotationally invariant Bloch functions in cylindrical coordinates improve the description of angular momentum coupling in envelope-function Hamiltonians?
• RQ5 What boundary conditions are necessary for physically consistent solutions of envelope functions on the symmetry axis?

Key findings

• The formalism successfully models multiband electron states in wurtzite and zinc-blende heterostructures with cylindrical symmetry using a rotationally invariant basis in cylindrical coordinates.
• Envelope functions are accurately described as eigenstates of the angular momentum projection operator, enabling parameterization by eigenvalue in the Hamiltonian.
• The method provides a complete and consistent framework for modeling low-dimensional semiconductor structures, including proper boundary conditions on the symmetry axis.
• The ground and excited valence-band states in a hexagonal GaN quantum wire are found to closely correspond to those in an equivalent circular cross-section wire, validating the cylindrical symmetry approximation.
• The study establishes a quantitative criterion for the applicability of the cylindrical symmetry approximation in real semiconductor nanostructures.