Resonant states and order-parameter suppression near pointlike impurities in d-wave superconductors

We examine the role of order-parameter suppression in the development of low-energy peaks (i.e., resonances) in the tunneling density of states near a nonmagnetic impurity in a d-wave superconductor. Without order-parameter suppression, the zero-energy resonance appears only in the unitary (i.e., strong impurity) limit. However, suppression makes the resonance appear even when the impurity is much weaker. To model this situation, we make the physical hypothesis that the order parameter is reduced whenever one electron of a Cooper pair encounters the impurity, a hypothesis that retains the exact solvability of the problem. In this way, we determine that suppression of the order parameter drives the effective strength of the impurity towards the unitary limit. We determine the order-parameter reduction variationally, and show that the ratios between the main energy scales—the bandwidth and superconducting gap—strongly affect this reduction and, in consequence, the position and width of the resonance.