Journal of Superconductivity: Incorporating Novel Magnetism, Vol. 13, No. 1, 2000 Impurities in d-Wave Superconductors: Effect of Singular Density of States I. Grosu 1 Received 5 April 1999 We analyzed the effect of nonmagnetic impurities in d-wave superconductors with an energy dependent density of states. Two singular forms for the density of states have been considered. The strength of singularity affect the degradation of the critical temperature and reduces the pair-breaking effect. KEY WORDS: d-wave superconductivity; nonmagnetic impurities; density of states effects. 1. INTRODUCTION The effect of a small amount of nonmagnetic impurities produces a severe pair-breaking in d-wave superconductors, a result in contrast to the effect in isotropic s-wave superconductors [1]. The effect of nonmagnetic impurities is not only to reduce the su- perconducting transition temperature T c , but also leads to the appearance of a finite single particle density of states at the Fermi energy [2,3]. Taking this effect into consideration, Annett et al. [4], and Hirschfeld and Goldenfeld [5] explained the T 2 de- pendence of the superfluid density  s (T ) in YBCO films and single crystals, working in the unitary limit. Fehrenbacher and Norman [6] compared the effect of nonmagnetic impurity scattering on the properties of superconductors having a d x 2 -y 2 -order parameter, with an anisotropic s-wave superconductor with the same nodal structure. In both cases the T c suppression is given by an Abrikosov-Gorkov (AG) result, but the suppression is faster in the d x 2 -y 2 superconductor. While some experimental results show a good agreement with the AG theory, others observe a slower than predicted degradation of the critical tem- perature with the increase of the impurities concen- tration. Zhitomirsky and Walker [7] explained these results by the effect of Friedel oscilations that reduces the pair-breaking. The validity of the results based 1 Department of Theoretical Physics, University of Cluj, 3400 Cluj, Romania. 141 0896-1107/00/0200-0141$18.00/0  2000 Plenum Publishing Corporation on AG model were questioned by Franz et al. [8]. According to them, the AG theory is valid as long as the coherence length (T ) exceeds the mean spacing between impurities. This condition lead to a bounding temperature T b , and we require that T c should exceed this temperature. Tallon et al. [9] showed that in the overdoped region the AG theory is valid, and its validity increases with overdoping. Crisan et al. [10] reconsidered this problem and showed that the validity of the AG theory in d-wave superconductors is sensitive to the gap-to-T c ratio. In this paper we will recalculate the critical temperature degradation due to nonmagnetic impurities in d-wave superconductors, taking into consideration the effect of an energy-dependent density of states (singular in the limit   0). The results will be discussed in Section 3. 2. CRITICAL TEMPERATURE DEGRADATION The critical temperature degradation, due to a small amount of nonmagnetic impurities (such as Zn), in high-T c copper oxides have been studied (see Refs. [2,3,8,9]) to explain the substantially reduced superconducting transition temperature T c , and other properties of the high T c materials. Here we will give a model based on AG theory, taking into consideration the d-wave pairing mechanism and an energy-depen- dent density of states. For d-wave superconductors