Low-energy hole subband dispersions in a cylindrical Ge nanowire: the effects of the nanowire growth direction

TL;DR

This study investigates the validity of the spherical approximation in calculating hole subband dispersions in cylindrical Ge nanowires, revealing growth direction dependencies and conditions where the approximation is accurate.

Contribution

It provides a detailed analysis of hole subband dispersions without the spherical approximation, highlighting growth direction effects and validating the approximation in specific cases.

Findings

Realistic subband dispersions show a double-well anticrossing structure.

Subband parameters depend on nanowire growth direction.

Spherical approximation accurately predicts dispersions in certain directions.

Abstract

We examine the validity of the spherical approximation $\gamma_{s}=(2\gamma_{2}+3\gamma_{3})/5$ in the Luttinger-Kohn Hamiltonian in calculating the subband dispersions of the hole gas. We calculate the realistic hole subband dispersions (without the spherical approximation) in a cylindrical Ge nanowire by using quasi-degenerate perturbation theory. The realistic low-energy hole subband dispersions have a double-well anticrossing structure, that consists with the spherical approximation prediction. However, the realistic subband dispersions are also nanowire growth direction dependent. When the nanowire growth direction is restricted in the (100) crystal plane, the detailed growth direction dependences of the subband parameters are given. We find the spherical approximation is good approximation, it can nicely reproduce the real result in some special growth directions.