One-dimensional spin-orbit coupled Dirac system with extended $s$-wave superconductivity: Majorana modes and Josephson effects

TL;DR

This paper investigates a one-dimensional spin-orbit coupled Dirac system with extended s-wave superconductivity, revealing Majorana modes, topological phase transitions, and unique Josephson effects, with potential experimental realizations.

Contribution

It introduces a novel 1D Dirac system with extended s-wave pairing, demonstrating Majorana zero modes, topological transitions, and unconventional Josephson phenomena.

Findings

Majorana zero modes appear at system ends with extended pairing.

Topological transition depends on Schrödinger and Dirac term ratio.

Unique Josephson effects with fractional Shapiro plateaus observed.

Abstract

Motivated by the spin-momentum locking of electrons at the boundaries of topological insulators, we study a one-dimensional system of spin-orbit coupled massless Dirac electrons with $s$-wave superconducting pairing. As a result of the spin-orbit coupling, our model has only two kinds of linearly dispersing modes, which we take to be right-moving spin-up and left-moving spin-down. Both lattice and continuum models are studied. In the lattice model, we find that a single Majorana zero energy mode appears at each end of a finite system provided that the $s$-wave pairing has an extended form, with the nearest-neighbor pairing being larger than the on-site pairing. We confirm this both numerically and analytically by calculating the winding number. Next we study a lattice version of a model with both Schr\"odinger and Dirac-like terms and find that the model hosts a topological transition between topologically trivial and non-trivial phases depending on the relative strength of the Schr\"odinger and Dirac terms. We then study a continuum system consisting of two $s$-wave superconductors with different phases of the pairing. Remarkably, we find that the system has a {\it single} Andreev bound state which is localized at the junction. When the pairing phase difference crosses a multiple of $2 \pi$, an Andreev bound state touches the top of the superconducting gap and disappears, and a different state appears from the bottom of the gap. We also study the AC Josephson effect in such a junction with a voltage bias that has both a constant $V_0$ and a term which oscillates with a frequency $\omega$. We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequency $\omega_J= 2eV_0/\hbar$ is a rational fraction of $\omega$. We discuss experiments which can realize such junctions.