Current conservation in the self-consistent theory of Josephson junctions
Phys. Rev. B 112, 144508 – Published 10 October, 2025
DOI: https://doi.org/10.1103/1xbv-bvch
Abstract
Conventional treatments of Josephson junctions (JJs) are typically not current conserving. In the mean-field Bardeen-Cooper-Schrieffer 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 Bogoliubov–de Gennes equations with current conservation for quasi-one-dimensional 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, noncurrent 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.
