Disordered Josephson Junctions of d-Wave Superconductors

We study the Josephson effect between weakly coupled d-wave superconductors within the quasiclassical theory, in particular, the influence of interface roughness on the current-phase relation and the critical current of mirror junctions and $45^\circ$ asymmetric junctions. For mirror junctions the temperature dependence of the critical current is non-monotonic in the limit of low roughness, but monotonic for very rough interfaces. For $45^\circ$ asymmetric junctions with a linear dimension much larger than the superconducting coherence length we find a $\sin(2\phi)$-like current-phase relation, whereas for contacts on the scale of the coherence length or smaller the usual $\sin\phi$-like behavior is observed. Our results compare well with recent experimental observations.