Low-energy in-gap states of vortices in superconductor-semiconductor heterostructures

TL;DR

This paper develops an analytical perturbation theory approach to predict low-energy vortex states in superconductor-semiconductor heterostructures, aiding the understanding of Majorana modes for quantum computing.

Contribution

It introduces a new perturbation theory method to analytically calculate low-energy vortex states, including magnetic and gap profiles, for both topological insulators and electron gases.

Findings

Analytical formulas for low-energy states are derived.

Results cover a wide range of vortex and chemical potential parameters.

The method is validated against numerical simulations.

Abstract

The recent interest in the low-energy states in vortices of semiconductor-superconductor heterostructures are mainly fueled by the prospects of using Majorana zero modes for quantum computation. The knowledge of low-lying states in the vortex core is essential as they pose a limitation on the topological computation with these states. Recently, the low-energy spectra of clean heterostructures, for superconducting-pairing profiles that vary slowly on the scale of the Fermi wavelength of the semiconductor, have been analytically calculated. In this work, we formulate an alternative method based on perturbation theory to obtain concise analytical formulas to predict the low-energy states including explicit magnetic-field and gap profiles. We provide results for both a topological insulator (with a linear spectrum) as well as for a conventional electron gas (with a quadratic spectrum). We discuss the spectra for a wide range of parameters, including both the size of the vortex and the chemical potential of the semiconductor, and thereby provide a tool to guide future experimental efforts. We compare these findings to numerical results.