Impact of the transport supercurrent on the Josephson effect

We study the weak link between current-carrying superconductors, both conventional and d-wave. The state of the system is controlled by two parameters: the order parameter phase difference $\phi$ and the superfluid velocity $v_{s}$, which parameterizes the parallel to the boundary transport supercurrent which is injected externally. The low-temperature current-phase relations are derived. We consider two models of weak links: a constriction between two conventional superconductors and a plane boundary between two differently orientated d-wave superconductors. We show that for some relation between $\phi $ and $v_{s}$ quasiparticles create the current along the boundary which flows in the direction opposite to the transport supercurrent.