S–I–S Josephson junction with a correlated insulator below its S–I transition

• We model an S–I–S junction, whose insulating layer is actually a superconductor below its S–I transition.
• We demonstrate that such a structure indeed behaves like a single Josephson junction.
• The coherence length diverges as the insulating layer approaches its S–I transition
• The model may describe so-called Cooper pair insulators studied in some experiments.

We consider a Josephson junction composed of two superconducting (S) regions separated by an insulating (I) region, but with the special property that the S and the I regions are superconducting films respectively above and below the superconducting–insulating (S–I) transition. To calculate the properties of this junction, we describe the system using an inhomogeneous quantum rotor Hamiltonian with a coupling energy J and spatially varying charging energy U  . The ratio J/UJ/U is chosen so that it is above the critical value for an S–I transition in the two superconducting regions, but below it in the insulating regime. Using both mean-field theory and perturbation theory, we show that the phase order parameter is finite in the S region and decays exponentially into the I region. Thus, the order parameter, which would be zero in the I region in isolation, is instead rendered nonzero by the adjacent S region, because of a proximity effect. As a result, there is a nonzero coupling energy between the two S regions. We show, using both mean-field theory and a quantum Monte Carlo calculation, that the phase stiffness constant, or helicity modulus, of this junction is nonzero, and falls off exponentially with separation of the two superconductors. We also analytically estimate the dependence of the coupling energy on the properties of the S and I regions, and suggest an analogy with conventional S–N–S junctions. Our results support the conclusion that this S–I–S sandwich structure, with a correlated insulating region, can be viewed as a single effective Josephson junction.