TY - JOUR

T1 - One-dimensional spin-orbit coupled Dirac system with extended s-wave superconductivity

T2 - Majorana modes and Josephson effects

AU - Udupa, Adithi

AU - Banerjee, Abhishek

AU - Sengupta, K.

AU - Sen, Diptiman

PY - 2021/4/7

Y1 - 2021/4/7

N2 - 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 Schrodinger 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 Schrodinger and Dirac terms. We then study a continuum system consisting of two s-wave superconductors with different phases of the pairing, with a delta-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 pi, 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 omega. We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequency omega(J) = 2eV(0)/PLANCK CONSTANT OVER TWO PI is a rational fraction of omega. We discuss experiments which can realize such junctions.

AB - 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 Schrodinger 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 Schrodinger and Dirac terms. We then study a continuum system consisting of two s-wave superconductors with different phases of the pairing, with a delta-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 pi, 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 omega. We find that, in contrast to standard Josephson junctions, Shapiro plateaus appear when the Josephson frequency omega(J) = 2eV(0)/PLANCK CONSTANT OVER TWO PI is a rational fraction of omega. We discuss experiments which can realize such junctions.

KW - topological superconductors

KW - Majorana fermions

KW - Josephson effects

U2 - 10.1088/1361-648X/abdd63

DO - 10.1088/1361-648X/abdd63

M3 - Journal article

C2 - 33470985

VL - 33

JO - Journal of Physics: Condensed Matter

JF - Journal of Physics: Condensed Matter

SN - 0953-8984

IS - 14

M1 - 145301

ER -