Josephson junctions of multiple superconducting wires

TL;DR

This paper develops a scattering matrix formalism to analyze Andreev bound states and Josephson currents in multi-wire superconducting junctions, revealing quantized transconductance, fractional Shapiro steps, and spin-dependent effects.

Contribution

It introduces a novel formalism for multi-wire superconducting junctions and provides analytical results for Andreev states and Josephson effects, including quantization and spin-dependent phenomena.

Findings

Quantized transconductance at zero temperature.

Fractional Shapiro plateaus in voltage response.

Spin-dependent Andreev bound states and magnetic moments.

Abstract

We study the spectrum of Andreev bound states and Josephson currents across a junction of $N$ superconducting wires which may have $s$- or $p$-wave pairing symmetries and develop a scattering matrix based formalism which allows us to address transport across such junctions. For $N \ge 3$, it is well known that Berry curvature terms contribute to the Josephson currents; we chart out situations where such terms can have relatively large effects. For a system of three $s$- or three $p$-wave superconductors, we provide analytic expressions for the Andreev bound state energies and study the Josephson currents in response to a constant voltage applied across one of the wires; we find that the integrated transconductance at zero temperature is quantized to integer multiples of $4e^2/h$, where $e$ is the electron charge and $h = 2\pi \hbar$ is Planck's constant. For a sinusoidal current with frequency $\omega$ applied across one of the wires in the junction, we find that Shapiro plateaus appear in the time-averaged voltage $\langle V_1 \rangle$ across that wire for any rational fractional multiple (in contrast to only integer multiples in junctions of two wires) of $2e \langle V_1 \rangle/(\hbar \omega)$. We also use our formalism to study junctions of two $p$- and one $s$-wave wires. We find that the corresponding Andreev bound state energies depend on the spin of the Bogoliubov quasiparticles; this produces a net magnetic moment in such junctions. The time variation of these magnetic moments may be controlled by an external applied voltage across the junction. We discuss experiments which may test our theory.