Theory of Josephson effects in anisotropic superconductors

An analytical formula for the dc Josephson current in anisotropic singlet superconductor/insulator/anisotropic singlet superconductor junctions is presented. The formula is applicable for junctions with arbitrary insulating-potential height and thickness and with any symmetries including $d$-wave superconductors. In contrast to the formulas for conventional $s$-wave superconductors, the formula includes two additional effects. One is the intrinsic phase of the pair potential originating from the pairing symmetry in anisotropic superconductors. The other is the formation of localized states around the insulator. Using this formula, the Josephson current is calculated in $s$-wave superconductor/insulator/$d_{x^2-y^2}$-wave superconductor $\mathrm{}(s/I/d)\mathrm{}$ and $d_{x^2-y^2}$-wave superconductor/insulator/$d_{x^2-y^2}$-wave superconductor $\mathrm{}(d/I/d)\mathrm{}$ junction configurations. In the case of the $\mathrm{}(d/I/d)\mathrm{}$ junction, the anomalous temperature dependence of the maximum Josephson current is calculated. This behavior is completely different from that expected for $s$-wave superconductors. The validity of a phenomenological theory by Sigrist and Rice [J. Phys. Soc. Jpn. 61, 4283 (1992)] is also discussed.