Extended gapless regions in disordered d x 2 − y 2 wave superconductors

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 twodimensional dx22y 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 dx22y 2 wave superconductor. The associated enhancement of the residual density of states may be responsible for the rapid decrease of Tc and the increase of the London penetration depth with hole doping observed in overdoped cuprate superconductors. @S0163-1829~97!01633-0# The pair-breaking role of impurities in d-wave superconductors in the weak-coupling limit is well known. Recently the evidence that such weak-coupling descriptions apply in the overdoped region of the high-Tc cuprates has been growing. 1 However, the experiments that track the behavior of the gap function have come to divergent conclusions. 2 This raises the question of the evolution of the gap function in a d-wave superconductor as a function of the pairing interaction strength and the impurity concentration. In this paper we examine this question, allowing for angulardependent impurity scattering potentials. Our conclusion is that the evolution of the gap function depends on the character of the impurity scattering potential. In particular, we find that extended gapless regions grow as the critical point is reached when the impurity scattering potential is weighted towards forward scattering. In this case the scattering processes near the gap nodes, which connect regions of opposite sign pairing amplitude, are of increasing importance and reduce the gap in this region. This contrasts with the regions around the gap maxima, where the impurity scattering is less effective. The resulting evolution of the gap function is sketched in Fig. 1. In conventional Abrikosov-Gorkov ~AG! theory, the interaction of electrons with nonmagnetic impurities is governed by a momentum-independent scattering potential. 3 Here we generalize this approach to the momentumdependent case, with a scattering potential uk,k8 . The contribution to the self-energy of electrons scattering from this static potential is given by the self-consistent T-matrix equation,