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

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.

Proximity-induced gap in nanowires with a thin superconducting shell

“…Our results in the absence of disorder are in qualitative agreement with Ref. [18], which studies a onedimensional wire coupled to a thin superconductor. In Ref.…”

“…In Ref. 18, the coupling to the superconductor is described by a tunneling energy scale γ; our present approach features a continuum model, for which the coupling is described by the interface transmission probability |t ⊥ | 2 at perpendicular incidence. For weak coupling the two quantities are related by γ ∼ |t ⊥ | 2 v N /D N , and we find that our prediction for the suppression of the induced gap in Eq.…”

“…While early theoretical studies focused on nanowiresuperconductor heterostructures for which finite-size ef-fects can be neglected [13][14][15][16][17], more recent studies have considered the implications of a finite thickness of the superconductor. For a one-dimensional wire proximitized by thin two-or three dimensional superconducting coats, Reeg et al suggested that finite-size effects can be detrimental to the induced gap [18,19]. Other works considered the effects of spatially-varying electrostatic potentials.…”

Supercurrent Detection of Topologically Trivial Zero-Energy States in Nanowire Junctions

“…Further, we use L x = 520a, L NW = 1000a, and keep the N-junction 2a long, to reach realistic sizes with the outer ends of the NWs well within the SCs. The width of the SC, L y , is varied in order to tune the influence of the SC on the NW [39][40][41][42]. We have verified that our results remain qualitatively unchanged for ∆ sc and Γ both being smaller (or even larger), as well as when ∆ sc (i) is calculated self-consistently [50][51][52][53][54].…”

Entangling spins in double quantum dots and Majorana bound states

“…The two levels of the MBS qubit are |L = f † L |0 TNW and |R = f † R |0 TNW , with |0 TNW being the vacuum state of the TNW. We assume that the ST 0 qubit is coherently coupled to the MBS qubit and decoupled from states above the gap | 0 ± E Z j /2| ∆, with ∆ being the proximity induced gap of the topological superconductor, which should be close to the superconducting gap of the parent s-wave superconductor in the strong coupling regime [92][93][94][95][96][97][98] . Also we assume that the DQD has a well defined number of confined charge carriers -that it is protected from leakage to states with a different number of electrons in the DQD, t j /| 0 ± E Z j /2| 1.…”