Composite Majorana Fermion Wavefunctions in Nanowires

TL;DR

This paper analytically investigates Majorana fermion wavefunctions in nanowires with normal and superconducting segments, revealing their composite nature, oscillatory behavior, and delocalization in the topological phase.

Contribution

It provides explicit analytic solutions for Majorana wavefunctions in different spin orbit regimes and uncovers their composite structure and interference effects.

Findings

Majorana wavefunctions are superpositions of states with different localization lengths.

Interference causes oscillations in the Majorana probability density.

Majorana modes delocalize into the normal section in the topological phase.

Abstract

We consider Majorana fermions (MFs) in quasi-one-dimensional nanowire systems containing normal and superconducting sections where the topological phase based on Rashba spin orbit interaction can be tuned by magnetic fields. We derive explicit analytic solutions of the MF wavefunction in the weak and strong spin orbit interaction regimes. We find that the wavefunction for one single MF is a composite object formed by superpositions of different MF wavefunctions which have nearly disjoint supports in momentum space. These contributions are coming from the extrema of the spectrum, one centered around zero momentum and the other around the two Fermi points. As a result, the various MF wavefunctions have different localization lengths in real space and interference among them leads to pronounced oscillations of the MF probability density. For a transparent normal-superconducting junction we find that in the topological phase the MF leaks out from the superconducting into the normal section of the wire and is delocalized over the entire normal section, in agreement with recent numerical results by Chevallier et al. (arXiv:1203.2643).