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

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-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-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-wave superconductors with different phases of the pairing, with a-function potential barrier lying at the junction of the two superconductors. Remarkably, we find that the system has aAndreev bound state (ABS) which is localized at the junction. When the pairing phase difference crosses a multiple of 2, 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 constantand a term which oscillates with a frequency. We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequency= 2/is a rational fraction of. We discuss experiments which can realize such junctions.