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Band inversion transition in HgTe nanowire grown along the [001] direction

Rui Li

TL;DR

This work analyzes gap closing and reopening in a cylindrical HgTe nanowire grown along [001] by building a six-band Kane-model-based low-energy Hamiltonian that incorporates both an anisotropic term $H''$ and bulk inversion asymmetry (BIA). The authors employ envelope-function theory with a hard-wall radial confinement and project $H'$ and $H''$ onto the lowest electron and top hole subbands, producing a $6\times6$ effective Hamiltonian with two $3\times3$ blocks. They show that $H''$ converts the $E_1$–$H_1$ crossing into an anticrossing at $k_z=0$, and that a gap-closing-and-reopening persists at finite $k_z$ for a critical radius $R\approx3.45$ nm with $k_zR\approx\pm0.24$, signaling a topological phase. Moreover, the BIA contribution vanishes in the low-energy subspace for the [001] cylinder, so spin splitting is absent in this geometry, and for $R>3.45$ nm the system resides in a quasi-1D topological insulator phase with end states.

Abstract

The low-energy effective Hamiltonian of a cylindrical HgTe nanowire grown along the [001] crystallographic direction is constructed by using the perturbation theory. Both the anisotropic term and the bulk inversion asymmetry term of the Kane model are taken into account. Although the anisotropic term has converted the crossing between the $E_{1}$ and $H_{1}$ subbands into an anticrossing at $k_{z}R\!=\!0$, the gap-closing-and-reopening transition in the subband structure can still occur at the wave vectors $k_{z}R\!\approx\!\pm0.24$ for critical nanowire radius $R\!\approx\!3.45$ nm. The bulk inversion asymmetry does not contribute to the low-energy effective Hamiltonian, i.e., there is no spin splitting in the $E_{1}$, $H_{1}$, and $H_{2}$ subbands for a [001] oriented cylindrical nanowire.

Band inversion transition in HgTe nanowire grown along the [001] direction

TL;DR

This work analyzes gap closing and reopening in a cylindrical HgTe nanowire grown along [001] by building a six-band Kane-model-based low-energy Hamiltonian that incorporates both an anisotropic term and bulk inversion asymmetry (BIA). The authors employ envelope-function theory with a hard-wall radial confinement and project and onto the lowest electron and top hole subbands, producing a effective Hamiltonian with two blocks. They show that converts the – crossing into an anticrossing at , and that a gap-closing-and-reopening persists at finite for a critical radius nm with , signaling a topological phase. Moreover, the BIA contribution vanishes in the low-energy subspace for the [001] cylinder, so spin splitting is absent in this geometry, and for nm the system resides in a quasi-1D topological insulator phase with end states.

Abstract

The low-energy effective Hamiltonian of a cylindrical HgTe nanowire grown along the [001] crystallographic direction is constructed by using the perturbation theory. Both the anisotropic term and the bulk inversion asymmetry term of the Kane model are taken into account. Although the anisotropic term has converted the crossing between the and subbands into an anticrossing at , the gap-closing-and-reopening transition in the subband structure can still occur at the wave vectors for critical nanowire radius nm. The bulk inversion asymmetry does not contribute to the low-energy effective Hamiltonian, i.e., there is no spin splitting in the , , and subbands for a [001] oriented cylindrical nanowire.

Paper Structure

This paper contains 6 sections, 15 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Description of the model
  3. Low-energy effective Hamiltonian
  4. Transformation of representation
  5. The effect of the bulk inversion asymmetry
  6. Summary

Figures (3)

  • Figure 1: The lowest electron and the two topmost hole subbands at $k_{z}=0$ as a function of the nanowire radius $R$. (a) Result without the anisotropic term $H"$. (b) Result with the anisotropic term $H"$. The anisotropic term $H"$ causes the crossing between the $E_{1}$ and $H_{1}$ subbands to become as an anticrossing. The anticrossing occurs at $R\approx3.4$ nm, and the energy splitting at the anticrossing is about $8.2$ meV.
  • Figure 2: The lowest electron subband dispersion $E_{1}$ and the two topmost hole subband dispersions $H_{1}$ and $H_{2}$ near $k_{z}R=0$. (a) Result for radius $R=3.10$ nm with normal band structure. (b) Result for radius $R=3.45$ nm with critical band structure, the subband gap closes at $k_{z}R\approx\pm0.24$. (c) Result for radius $R=3.80$ nm with inverted band structure.
  • Figure 3: The energies in the finite nanowire of radius $R=3.8$ nm and length 400 nm as a function of state number $N$ (a). There are two energy states (marked green) in the subband gap. The density probability distributions of these two energy states are given in (b) and (c), respectively.