Finite-size effects in a nanowire strongly coupled to a thin superconducting shell

TL;DR

This paper investigates how the finite size of a superconducting shell affects the proximity effect in a nanowire, revealing that size-induced level spacing can hinder the realization of topological phases relevant for Majorana fermions.

Contribution

It provides a theoretical analysis of finite-size effects in superconductor-nanowire systems, highlighting conditions for achieving a hard gap and topological phases without fine-tuning.

Findings

Finite-size effects significantly impact the proximity-induced gap.

Large tunneling energy can shift the chemical potential and alter topological phase boundaries.

Achieving a hard gap requires tunneling energy to exceed superconductor level spacing.

Abstract

We study the proximity effect in a one-dimensional nanowire strongly coupled to a finite superconductor with a characteristic size which is much shorter than its coherence length. Such geometries have become increasingly relevant in recent years in the experimental search for Majorana fermions with the development of thin epitaxial Al shells which form a very strong contact with either InAs or InSb nanowires. So far, however, no theoretical treatment of the proximity effect in these systems has accounted for the finite size of the superconducting film. We show that the finite-size effects become very detrimental when the level spacing of the superconductor greatly exceeds its energy gap. Without any fine-tuning of the size of the superconductor (on the scale of the Fermi wavelength), the tunneling energy scale must be larger than the level spacing in order to reach the hard gap regime which is seen ubiquitously in the experiments. However, in this regime, the large tunneling energy scale induces a large shift in the effective chemical potential of the nanowire and pushes the topological phase transition to magnetic field strengths which exceed the critical field of Al.