Role of anisotropic impurity scattering in anisotropic superconductors.

A theory of nonmagnetic impurities in an anisotropic superconductor including the effect of anisotropic (momentum-dependent) impurity scattering is given. It is shown that for a strongly anisotropic scattering the reduction of the pair-breaking effect of the impurities is large. For a significant overlap between the anisotropy functions of the scattering potential and that of the pair potential and for a large amount of anisotropic scattering rate in impurity potential the superconductivity becomes robust vis a vis impurity concentration. The implications of our result for YBCO high-temperature superconductor are discussed. The experimental data of electron irradiationinduced Tc suppression (Phys. Rev. B50, 15967 (1994)) is understood quantitatively and a good qualitative agreement with the ion (Ne) damage and Pr substitution-induced Tc decrease data (Phys. Rev. B50, 3266 (1994)) is obtained. 74.20.-z, 74.62.-c, 74.62.Dh Typeset using REVTEX 1 There now exists a considerable experimental evidence supporting the d-wave superconductivity in the cuprates (for review see Refs. 1-4). Nevertheless, this scenario still faces some theoretical difficulties. One of these is the predicted extreme suppression of the critical temperature Tc by nonmagnetic impurities. 5–10 Experimentally, however, the observed suppression of Tc by impurities or radiation damage in YBCO is much more gradual. 11,12 This issue was critically examined by Radtke et al. who considered isotropic impurity scattering within the second Born approximation by applying the Eliashberg formalism. Their predictions in both weakand strong-coupling theory gave a Tc suppression which was close to the Abrikosov-Gorkov scaling function with an effective impurity scattering rate. This led to an approximate universal dependence of Tc on the planar residual resistivity ρ0, which did not depend on the details of the microscopic pairing. In order to verify the results of Radtke et al. systematic electron irradiation experiments on YBCO were carried out by Giapintzakis et al. The measured initial slope of impurity induced Tc suppression was dTc/dρ ∼ −0.30K/μΩcm, 12 whereas the predicted value was in the range from ∼ −0.74K/μΩcm to −1.2K/μΩcm. While discussing the experimental results in Ref. 12 the authors invoked the issue of the anisotropic impurity scattering. They understood their data within a model of Millis et al. assuming a value of 0.5 for a dimensionless parameter gI which describes the anisotropy of the scattering potential and modifies the bare isotropic impurity scattering rate 1/τ according to 1/τ ⋆ = (1− gI) /τ , where 1/τ ⋆ is the effective scattering rate. Thus the analysis by Giapintzakis et al. brings out the significant role of the anisotropic scattering in understanding the impurity effect on d-wave superconductivity and calls for more detailed theoretical studies. In this paper we consider in detail the problem of nonmagnetic impurities in an anisotropic superconductor for the case of anisotropic (momentum-dependent) impurity scattering by applying weak-coupling approximation. We find a remarkable change in the Tc suppression which becomes more gradual when the anisotropy function defining anisotropy of the impurity potential overlaps with the anisotropy function of the order parameter. Al2 though our formalism is general and valid for any superconducting order parameter described by a one dimensional (1D) irreducible representation of the crystal point group we discuss the results for a d-wave superconductor in the context of high temperature superconductivity. In a certain limit, the effective scattering rate in our model is identical to that of Millis et al. We compute Tc as a function of planar residual resistivity. Within a certain range of scattering potential parameters values we find a quantitative agreement of our results with the electron irradiation data. Also for an appropriate choice of the impurity potential coefficients a good qualitative fit to the Pr substitution and Ne-irradiation data is obtained. We take h̄ = kB = 1 throughout the paper. We consider randomly distributed nonmagnetic impurities in an anisotropic superconductor. Treating the electron-impurity scattering within second Born approximation and neglecting the impurity-impurity interaction, the normal and anomalous temperature Green’s functions averaged over the impurity positions read G (ω,k) = − iω̃ + ξk ω̃ + ξk 2 + |∆̃ (k) | (1) F (ω,k) = ∆̃ (k) ω̃ + ξk 2 + |∆̃ (k) | (2) where the renormalized Matsubara frequency ω̃ (k) and the renormalized order parameter ∆̃ (k) are given by ω̃ (k) = ω + ini ∫ |w (k − k) |G (ω,k) dk (2π) (3) ∆̃ (k) = ∆ (k) + ni ∫ |w (k − k) |F (ω,k) dk (2π) (4) In above ω = πT (2n+1) (T is temperature and n is an integer number), ξk is the quasiparticle energy, ni is impurity (defect) concentration, w (k − k ) is momentum dependent impurity potential and ∆ (k) is the orbital part of a singlet superconducting order parameter defined as 3 ∆(k) = ∆e(k) (5) where e (k) is a real basis function of a 1D irreducible representation of an appropriate point group, which seems to be good approximation for high Tc superconductors. 1 We normalize e(k) by taking 〈e〉 = 1, where < ... >= ∫ FS dSkn (k) (...) denotes the average value over the Fermi surface (FS), n (k) is the angle resolved FS density of states normalized to unity, i.e. ∫ FS dSkn (k) = 1, and ∫ FS dSk stands for the integration over the Fermi surface. The impurity scattering potential is assumed to be separable and given by |w (k − k) | = |w0| 2 + |w1| f (k) f (k) (6) where |w0| (|w1|) is isotropic (anisotropic) scattering amplitude and f (k) is the momentumdependent anisotropy function. We assume that the overall scattering rate is determined by the isotropic component and impose the constraints |w1| 2 ≤ |w0| , 〈f〉 = 0, 〈