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Motivated by the spin–momentum locking of electrons at the boundaries of certain 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, and we take these 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. We find that the continuum model also has zero energy end modes. Next we study a lattice version of a model with both Schrödinger 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 Schrödinger and Dirac terms. We then study a continuum system consisting of two  s -wave superconductors with different phases of the pairing, with a  δ -function potential barrier lying at the junction of the two superconductors. Remarkably, we find that the system has a  single  Andreev bound state (ABS) which is localized at the junction. When the pairing phase difference crosses a multiple of 2 π , an ABS 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  ω . We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequency  ω J  = 2 eV 0 / ℏ  is a rational fraction of  ω . We discuss experiments which can realize such junctions.

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