Josephson effect in strongly disordered metallic wires

Abstract

We study localization effects in Josephson junctions with two superconductors connected by a strongly disordered metallic wire of length $L$. The conventional description of the Josephson effect in such systems, based on the quasiclassical Usadel equation, neglects electron interference and is only applicable when $L$ is shorter than the localization length $\hat{I}¾$ in the wire. We develop a more general theory for the Josephson effect using the non-linear sigma model that fully accounts for electron interference, and hence localization. We show that for $L\gg \hat{I}¾$, three qualitatively different regimes of the Josephson current arise depending on the ratio of the superconducting order parameter $\hat{I}”$ and the mean level spacing in the localization volume $\hat{I}{”}_{\hat{I}}¾$. We derive the average supercurrent as a function of the phase difference for all three regimes. Quite unexpectedly, we observe that the Ambegaokar-Baratoff relation between the average critical current and the normal-state conductance still holds in the strongly localized state when $\hat{I}{”}_{\hat{I}}¾\gg \hat{I}”$ and $\hat{I}¾\ll L\ll (\hat{I}¾/\ddot{I}{€}^2){\ln }^2(\hat{I}{”}_{\hat{I}}¾/\hat{I}”)$.

11 pages, 5 figures