Transcendental momentum quantization in semiconducting Rashba nanowires and zero energy states in their normal and superconducting phase

TL;DR

This paper investigates the unique momentum quantization and zero energy states in semiconducting Rashba nanowires, both normal and superconducting, revealing complex eigenstate structures and conditions for zero energy states relevant to topological phases.

Contribution

It introduces a transcendental quantization condition for finite Rashba nanowires and derives criteria for zero energy states in both trivial and topological phases.

Findings

Eigenstates involve multiple momentum components due to spin-orbit coupling

Quantization condition is transcendental, not quantum box-like

Zero energy states can exist in both trivial and topological phases

Abstract

We study finite system properties of the canonical low energy model for a semiconducting nanowire with Rashba spin-orbit coupling. The case of an isolated wire as well as of one proximitized by an s-wave superconductor are considered. Already for the normal wire, the presence of spin-orbit coupling leads to eigenstates of the finite system composed of more than two momentum eigenstates. The quantization condition for the wavevectors is not that of a quantum box, but given instead by a transcendental equation linking the involved wavevectors. For the wire with superconducting pairing, the presence of electron and hole channels complicates the composition of the eigenstates. In this case we derive an approximate quantization condition close to the phase boundary, and a condition for the appearance of exact zero energy states. It can be satisfied both in the topological and in the trivial phase. Both the trivial and topological zero energy states contribute to the linear transport through Andreev reflection and direct transmission processes, with their relative importance depending on the degree of the states' localization at the boundary.