Josephson current in an anisotropicd-wave model

There should be no Josephson current between a ${\mathrm{CuO}}_2$ plane with a gap of ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ symmetry and a conventional s-wave superconductor. Nevertheless such a current is observed to exist between ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ (YBCO) and Pb although with a reduced magnitude. Penetration depth measurements have revealed a large anisotropy between a and b directions presumably due to the existence of the CuO chains. Using a simple anisotropic tight-binding model in qualitative agreement with the measured penetration depths and a standard model for the spin susceptibility, we obtain, on solution of the BCS gap equations, a gap function with mainly ${\mathit{d}}_{\mathit{x}}^2$-${\mathit{y}}^2$ symmetry but with a minor extended s-wave component. The resultant Josephson current for YBCO-Pb junctions is in good agreement with experiment.