Josephson effect in strongly disordered metallic wires

We study localization effects in Josephson junctions with two superconductors connected by a strongly disordered metallic wire of length LL . The conventional description of the Josephson effect in such systems, based on the quasiclassical Usadel equation, neglects electron interference and is only applicable when LL is shorter than the localization length \xiξ 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 \xiL≫ξ , three qualitatively different regimes of the Josephson current arise depending on the ratio of the superconducting order parameter \DeltaΔ and the mean level spacing in the localization volume \Delta_\xiΔξ . 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 \Delta_\xi \gg \DeltaΔξ≫Δ and \xi \ll L \ll (\xi/\pi^2) \ln^2(\Delta_\xi/\Delta)ξ≪L≪(ξ/π2)ln2(Δξ/Δ) .