Influence of magnetic imperfections on the low-temperature properties ofd-wave superconductors

We consider the influence of planar ‘‘magnetic’’ imperfections which destroy the local magnetic order, such as Zn impurities or ${\mathrm{Cu}}^{2+}$ vacancies, on the low-temperature properties of the cuprate superconductors. In the unitary limit, at low temperatures, for a ${\mathit{d}}_{{\mathit{x}}^2\mathrm{-}{\mathit{y}}^2}$ pairing state such imperfections produce low-energy quasiparticles with an anistropic spectrum in the vicinity of the nodes. We find that for the ${\mathrm{La}}_{2\mathrm{-}\mathit{x}}$ ${\mathrm{Sr}}_{\mathit{x}}$ ${\mathrm{CuO}}_4$ system, one is in the quasi-one-dimensional regime of quasiparticle scattering, discussed recently by Altshuler, Balatsky, and Rosengren, for impurity concentrations in excess of ∼0.16% whereas Ya-Ba-Cu-${\mathrm{O}}_7$ appears likely to be in the true two-dimensional scattering regime for Zn concentrations less than 1.6%. We show the neutron-scattering results of Mason et al. [Phys. Rev. Lett. 71, 919 (1993)] on ${\mathrm{La}}_{1.85}$ ${\mathrm{Sr}}_{0.14}$ ${\mathrm{CuO}}_4$ provide strong evidence for ‘‘dirty d-wave’’ superconductivity in their samples. We obtain simple expressions for the dynamic spin susceptibility and ${}_{}{}^{63}{}_{}{}^{}\mathrm{Cu}$ spin-lattice relaxation time, ${}_{}{}^{63}{}_{}{}^{}$ ${\mathit{T}}_1$, in the superconducting state.