Supercurrents through proximity layers. II. Numerical solution of superconducting-normal-superconducting and superconducting-superconducting-superconducting weak links

There exist two distinctly different types of symmetric weak links. One contains a superconductor $S_1$, and the other contains a normal metal $N$ between the outside superconducting $S$ regions, where the states $S_1$ and $N$ refer to the properties of the isolated metals. Both types of weak links have multiple solutions of the pair potential for a fixed value of the dc current density. This multiplicity is theoretically elucidated. When the lengths of the $S_1$ and $N$ regions, $2d_n$, are comparable to $\left|{\xi }_n\right|$ or smaller, both types of weak links behave similarly. When $2d_n\gg \left|{\xi }_n\right|$, a superconducting-normal-superconducting (SNS) weak link behaves very much like a Josephson junction, but a superconducting-superconducting-superconducting (S-${\mathrm{S}}_1$-S) weak links behaves like a long superconducting wire with some of the Josephson-like properties remaining. Exact solutions of S-${\mathrm{S}}_1$-S and S-N-S weak links are calculated for different boundary conditions, lengths of the $S_1$ and $N$ regions, and current densities. The latter are compared to previous analytical approximations.