A generalization of the Abrikosov-Gorkov equations for nonmagnetic impurities in unconventional superconductors is proposed, including higher harmonics in the expansion of the momentum-dependent gap function and a momentum-dependent impurity scattering potential. This model is treated within a self-consistent calculation to obtain the electronic density of states, the optical conductivity, and the gap function in a two-dimensional $d_{x^2-y^2}$ wave superconductor. It is argued that momentum-dependent scattering from the impurities may lead to extended gapless regions in the gap function centered around the nodes of the pure $d_{x^2-y^2}$ wave superconductor. The associated enhancement of the residual density of states may be responsible for the rapid decrease of $T_c$ and the increase of the London penetration depth with hole doping observed in overdoped cuprate superconductors.

• Received 27 February 1997

DOI:https://doi.org/10.1103/PhysRevB.56.5108