Theory of $SNS$ sandwiches with nonmagnetic impurities of arbitrary concentration for near-critical temperatures

A microscopic theory of an $SN$ contact and an $SNS$ sandwich is developed for nearcritical temperatures and arbitrary impurity concentrations. A boundary condition for the Ginzburg–Landau equation at the boundary of the superconductor with the normal metal is established. The current states in an $SNS$ sandwich with large thickness $d$ of the normal layer are calculated; it is shown that the effective length $\xi$ over which the current decreases by $e$ times as $d$ is increased is $1/\xi=(1/\xi_0+1/l)f(l/\xi_0)$, where $\xi_0$ is the coherence length in the pure superconductor. $l$ is the mean free path, and $f(l/\xi_0)$ is a root of a transcendental equation. The function $f$ is such that as $l$ varies from infinity to $l\ll\xi_0$ there is a smooth transition from effective length $\xi_0$ to $(\xi_0l/3)^{1/2}$.