Band structure and end states in InAs/GaSb core-shell-shell nanowires

Abstract

Quantum wells in InAs/GaSb heterostructures can be tuned to a topological\nregime associated with the quantum spin Hall effect, which arises due to an\ninverted band gap and hybridized electron and hole states. Here, we investigate\nelectron-hole hybridization and the fate of the quantum spin Hall effect in a\nquasi one-dimensional geometry, realized in a core-shell-shell nanowire with an\ninsulator core and InAs and GaSb shells. We calculate the band structure for an\ninfinitely long nanowire using $\\mathbf{k \\cdot p}$ theory within the Kane\nmodel and the envelope function approximation, then map the result onto a BHZ\nmodel which is used to investigate finite-length wires. Clearly, quantum spin\nHall edge states cannot appear in the core-shell-shell nanowires which lack\none-dimensional edges, but in the inverted band-gap regime we find that the\nfinite-length wires instead host localized states at the wire ends. These end\nstates are not topologically protected, they are four-fold degenerate and split\ninto two Kramers pairs in the presence of potential disorder along the axial\ndirection. However, there is some remnant of the topological protection of the\nquantum spin Hall edge states in the sense that the end states are fully robust\nto (time-reversal preserving) angular disorder, as long as the bulk band gap is\nnot closed.\n