Supercurrents through superconducting-normal-superconducting proximity layers. I. Analytic solution

A general equation is derived for the maximum lossless current density which can be sustained through a thick normal layer which is part of a superconducting-normal-superconducting (S N S ) sandwich. This result is adapted to nonuniform current densities and compared with experiments on Pb-Cu-Pb by Clarke. Agreement is obtained when the ratio of the pair potentials at the S Nboundaries is made a constant, independent of temperature and thickness of the Cu layer. The pair potential in Cu is interpreted as an effective one which is induced in Cu by the proximity effect due to Pb. The superconducting phase difference is calculated across the $N$ metal as a function of current for various thicknesses of the $N$ layer. It reduces to the Josephson dc current relation when the $N$ metal becomes thick in relation to the coherence length in the $N$ region. We find that the value of the pair potential in the center of the $N$ metal is current dependent and finite (not zero) for all current densities including the critical.