# Transport in superconductor--normal metal--superconductor tunneling structures: Spinful p-wave and spin-orbit-coupled topological wires

## TL;DR

This paper theoretically analyzes transport in superconductor-normal metal-superconductor junctions with topological superconductors, focusing on Majorana zero modes, conductance peaks, and effects of Andreev bound states, providing insights relevant to experimental efforts in topological superconductivity.

## Contribution

It introduces a comprehensive theoretical framework for understanding tunneling conductance in topological superconductor junctions, highlighting the behavior of Majorana zero modes and conductance quantization.

## Key findings

- Majorana zero modes produce conductance peaks at specific voltages.
- Conductance peak values depend on junction transparency and lead properties.
- Finite-energy Andreev bound states shift conductance peaks away from gap voltages.

## Abstract

We theoretically study transport properties of voltage-biased one-dimensional superconductor--normal metal--superconductor tunnel junctions with arbitrary junction transparency where the superconductors can have trivial or nontrivial topology. Motivated by recent experimental efforts on Majorana properties of superconductor-semiconductor hybrid systems, we consider two explicit models for topological superconductors: (i) spinful p-wave, and (ii) spin-split spin-orbit-coupled s-wave. We provide a comprehensive analysis of the zero-temperature dc current $I$ and differential conductance $dI/dV$ of voltage-biased junctions with or without Majorana zero modes (MZMs). The presence of an MZM necessarily gives rise to two tunneling conductance peaks at voltages $eV = \pm \Delta_{\mathrm{lead}}$, i.e., the voltage at which the superconducting gap edge of the lead aligns with the MZM. We find that the MZM conductance peak probed by a superconducting lead $without$ a BCS singularity has a non-universal value which decreases with decreasing junction transparency. This is in contrast to the MZM tunneling conductance measured by a superconducting lead $with$ a BCS singularity, where the conductance peak in the tunneling limit takes the quantized value $G_M = (4-\pi)2e^2/h$ independent of the junction transparency. We also discuss the "subharmonic gap structure", a consequence of multiple Andreev reflections, in the presence and absence of MZMs. Finally, we show that for finite-energy Andreev bound states (ABSs), the conductance peaks shift away from the gap bias voltage $eV = \pm \Delta_{\mathrm{lead}}$ to a larger value set by the ABSs energy. Our work should have important implications for the extensive current experimental efforts toward creating topological superconductivity and MZMs in semiconductor nanowires proximity coupled to ordinary s-wave superconductors.