Abstract

We study localization phenomena in an SNS junction with a disordered metallic wire of length $L$ as its normal part. Standard description of the Josephson effect in such systems is based on the Usadel equation. However, the classical approach remains valid only while the junction is shorter than the localization length $\xi_\text{loc}$. In the opposite limit, quantum effects become important and the Usadel description is no longer valid.

We develop a general theory of the Josephson effect taking into account all localization (quantum) contributions. Our theory is based on the nonlinear sigma model and can be applied even in the limit of long junctions $L > \xi_\text{loc}$ when a fully quantum description is required. Applied to the Josephson effect in this limit, the theory predicts three qualitatively different regimes depending on relation between $L$, $\xi_\text{loc}$ and superconducting coherence length $\xi$. In all these regimes, we found analytical expressions for the current-phase relation. Motivated by experiments [1], we have also found limits of validity for the Ambegaokar-Baratoff relation and showed that it remains valid even in the presence of localization.


[1] A. Frydman and Z. Ovadyahu, Phys. Rev. B 55, 9047 (1997).