A k.p treatment of edge states in narrow 2D topological insulators, with standard boundary conditions for the wave function and its derivative

TL;DR

This paper extends the k.p Hamiltonian approach to analyze edge states in narrow 2D topological insulators, clarifying boundary conditions and correcting spurious solutions, with implications for quantum well systems like HgTe/CdTe.

Contribution

It provides a detailed treatment of boundary conditions for edge states in narrow 2D topological insulators using the 4x4 k.p Hamiltonian, addressing spurious solutions and quantifying edge state gaps.

Findings

Edge state gaps decrease with increasing sample width.

Standard boundary conditions eliminate spurious solutions.

Edge states exhibit pure exponential decay under certain symmetry conditions.

Abstract

For 2D topological insulators with strong electron-hole hybridization, such as HgTe/CdTe quantum wells, the widely used 4 x 4 k.p Hamiltonian based on the first electron and heavy hole sub-bands yields an equal number of physical and spurious solutions, for both the bulk states and the edge states. For symmetric bands and zero wave vector parallel to the sample edge, the mid-gap bulk solutions are identical to the edge solutions. In all cases, the physical edge solution is exponentially localized to the boundary and has been shown previously to satisfy standard boundary conditions for the wave function and its derivative, even in the limit of an infinite wall potential. The same treatment is now extended to the case of narrow sample widths, where for each spin direction, a gap appears in the edge state dispersions. For widths greater than 200 nm, this gap is less than half of the value reported for open boundary conditions, which are called into question because they include a spurious wave function component. The gap in the edge state dispersions is also calculated for weakly hybridized quantum wells such as InAs/GaSb/AlSb. In contrast to the strongly hybridized case, the edge states at the zone center only have pure exponential character when the bands are symmetric and when the sample has certain characteristic width values.