@article{WOS:000616591800001,
 abstract = {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.},
 article-number = {145301},
 author = {Udupa, Adithi and Banerjee, Abhishek and Sengupta, K. and Sen, Diptiman},
 doi = {10.1088/1361-648X/abdd63},
 eissn = {1361-648X},
 issn = {0953-8984},
 journal = {JOURNAL OF PHYSICS-CONDENSED MATTER},
 month = {APR 7},
 number = {14},
 title = {One-dimensional spin-orbit coupled Dirac system with extended s-wave
superconductivity: Majorana modes and Josephson effects},
 unique-id = {WOS:000616591800001},
 volume = {33},
 year = {2021}
}