Hard-wall edge confinement in two-dimensional topological insulators and the energy of the Dirac point

Abstract

In 2D topological insulators (TIs) based on semiconductor quantum wells such as HgTe/CdTe, hard wall spin-polarized edge states are calculated from an $8\ifmmode\times\else\texttimes\fi{}8$ linear-$k$ multiband (LKMB) Hamiltonian based directly on $\mathbf{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbf{p}$ theory, and from a $4\ifmmode\times\else\texttimes\fi{}4$ BHZ Hamiltonian derived from it by the elimination of remote states using perturbation theory. Both approaches lead to similar results with standard boundary conditions obtained by integrating the eigenstate equation across an interface (SBCs). In contrast, open boundary conditions (OBCs) yield unphysical results and do not work for the LKMB Hamiltonian. Their failure is traced to a spurious solution introduced by the elimination process. In the BHZ treatment, SBCs are shown to be consistent with perturbation theory on both sides of the boundary, and a wall hybridization parameter is estimated for a vacuum using a basis of empty crystal free electron states. A Dirac point is not expected for a vacuum, but can exist when a thin passivation layer is used whose midgap energy is nearly degenerate with that of the TI. In the absence of interface band mixing (IBM), the Dirac point is then very close to midgap and virtually independent of the TI band asymmetry. IBM introduces a significant energy shift, which decreases monotonically with edge state wave vector. Using SBCs, the paradoxical extension of the BHZ edge states across the topological phase transition is also resolved.