Structure of the quantum spin Hall states in HgTe/CdTe and InAs/GaSb/AlSb quantum wells

Abstract

A solution of the $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ model is presented for bulk and quantum spin hall (QSH) edge states in semiconductor topological insulator (TI) quantum wells (QWs), bounded at the edge by an infinite wall potential. The edge states are exponentially localized, with a nonzero amplitude at the QW edge, and obey standard boundary conditions for the wave function and its derivative. Single helical edge states with spin locked to the direction of motion are found in the TI band gap $({E}_{\mathrm{TI}})$ of QWs with both strong (HgTe/CdTe) and weak (InAs/GaSb/AlSb) $s\text{\ensuremath{-}}p$ hybridization, but in the second case only below a small critical band gap, ${E}_{\mathrm{crit}}\ensuremath{\sim}1.6\phantom{\rule{0.16em}{0ex}}\mathrm{meV}$. For ${E}_{\mathrm{TI}}>{E}_{\mathrm{crit}}$, there appear to be two degenerate states for each spin direction. It is suggested that ${Z}_{2}$-like topological properties can still be maintained if one of these states is spurious or suppressed by disorder. The effect of interface band mixing, and band mixing due to structural inversion asymmetry and bulk inversion asymmetry is also considered. Simple model Hamiltonians are developed for the bulk and edge states which are calibrated against a bulk eight-band $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ calculation close to the TI transition. At the transition, the zero gap bulk states exhibit a spin splitting, essentially changing the Dirac point to a circle. In the TI phase, there is a small change in the dispersion of the QSH edge states. These results confirm the robustness of the QSH edge states to spatial symmetry breaking interactions.