Exploring the conventional and anomalous Josephson effects at arbitrary disorder strength in systems with spin-dependent fields

TL;DR

This paper develops a comprehensive theory for Josephson currents in SNS junctions with spin-dependent fields, accounting for arbitrary disorder, and explores effects like SOC, anomalous currents, and disorder-induced transitions.

Contribution

It derives a compact expression for the Josephson current applicable to various disorder regimes and investigates the robustness of anomalous effects and $0$-$ ext{pi}$ transitions under disorder.

Findings

Josephson critical current varies with magnetic field and SOC, enabling SOC probing.

The anomalous Josephson ($ ext{$oldsymbol{ extphi}_0$}$) effect persists and can be enhanced by moderate disorder.

Disorder suppresses the $0$-$ ext{ extpi}$ transition in systems with altermagnets.

Abstract

We present a theory of the Josephson current in superconductor-normal metal-superconductor (SNS) junctions in the presence of generic spin-dependent fields, such as spin-orbit coupling (SOC), Zeeman fields, and altermagnetism. We consider systems with arbitrary disorder strength, going beyond the usual diffusive and ballistic approximations. Using the linearized quasiclassical Eilenberger equation, we derive a compact expression for the Josephson current, which is then applied to various situations of experimental interest. First, we investigate the evolution of the Josephson critical current in an applied magnetic field in the presence of Rashba and Dresselhaus SOC, and discuss how this dependence can be used to probe SOC in the junction. We then study the anomalous Josephson ($\varphi_0$) effect in systems with Rashba SOC and show that it remains robust over a wide range of disorder strength, and can even be enhanced by moderate disorder in sufficiently long junctions. Finally, we investigate the Josephson current in disordered junctions with altermagnets, and show how the $0$-$\pi$ transition in such systems is suppressed by disorder. Our results may be useful for describing experimental setups with high-mobility samples, which nevertheless always contain some amount of disorder, and where neither purely ballistic nor diffusive approximations are adequate.