Current Conservation in the Self-Consistent Josephson Junction

TL;DR

This paper introduces a numerical method for self-consistently conserving current in Josephson junction models, revealing significant effects on the current-phase relation, especially in regimes with non-negligible lead current density.

Contribution

It presents a new self-consistent numerical approach for solving BdG equations in Josephson junctions, emphasizing the importance of current conservation for accurate CPR predictions.

Findings

Current conservation affects Josephson harmonic content.

It can reverse the forward skewness of the CPR.

The method improves modeling accuracy in certain regimes.

Abstract

Conventional treatments of Josephson junctions (JJs) are typically not current-conserving. In the mean-field BCS theory, current conservation is only guaranteed if the superconducting order parameter is treated self-consistently. We show that this requirement has significant consequences for the current-phase relation (CPR) in certain regimes, where the current density in the superconducting leads is non-negligible. To this end, we introduce a numerical method for the self-consistent treatment of the BdG equations with current conservation for quasi-1D superconductor-normal (metal)-superconductor (SNS) JJs. Our model incorporates a phase gradient of the order parameter in the leads, which is set to match the Josephson current through the weak link. We compare our method to standard, non-current-conserving approaches by calculating the CPR for SNS JJs while varying lengths and gate voltages controlling the normal metal. We show that current conservation has significant implications for the Josephson harmonics and can weaken or even reverse forward skewness of the CPR.