Hard wall edge confinement in 2D topological insulators and the energy of the Dirac Point

TL;DR

This paper investigates the effects of boundary conditions on edge state confinement in 2D topological insulators, revealing that explicit treatment of the wall and passivation layers significantly influence the Dirac point energy and edge state behavior.

Contribution

It introduces a boundary condition approach that explicitly models the wall and passivation layers, improving physical accuracy over traditional methods.

Findings

Weak confinement leads to unphysical gap solutions.

Passivation layers can align the Dirac point near mid-gap.

Interface band mixing shifts the dispersion, especially at the Dirac point.

Abstract

In 2D topological insulators (TIs) based on semiconductor quantum wells such as HgTe/CdTe or InAs/GaSb/AlSb, spin polarized edge states have been predicted with a massless Dirac like dispersion. In a hard wall treatment based on the 4 x 4 BHZ Hamiltonian and open boundary conditions (OBCs), the wave function is weakly confined near the edge, with which it makes no contact. In contrast, standard boundary conditions for the wave function and its derivative (SBCs) lead to strong confinement with a peak amplitude at the edge. Unfortunately, weak confinement exhibits unphysical behavior related to a spurious gap solution that is included in the OBC wave function. This is confirmed by the gap solutions of the parent multiband Hamiltonian from which the smaller Hamiltonian is derived, which exhibit physical behavior and do not satisfy OBCs. Unlike OBCs or other approaches based on phenomenological boundary conditions, SBCs treat the wall explicitly. Using a basis of empty crystal free electron states for the vacuum with the same symmetry as the TI states, it is shown that a large wall band gap overlapping that of the TI can only be achieved by including a thin passivation layer. For passivation materials such as silicon dioxide where the mid gap energy is nearly degenerate with that of the TI, the Dirac point is very close to mid gap and virtually independent of the TI band asymmetry. The treatment also demonstrates that a significant shift of the dispersion may be introduced by interface band mixing. The shift is largest at the Dirac point and decreases monotonically with edge state wave vector, vanishing when the edge states merge with the bulk band edges.