Theory of Electronic Raman Scattering in Disordered d−wave Superconductors

A theory for the effect of impurity scattering on electronic Raman scattering in unconventional superconductors is presented. The impurity dependence of the the spectra can be used to to distinguish between conventional (anisotropic s−wave) or unconventional (d-wave) energy gaps, and excellent agreement with the data on three cuprate superconductors is shown for dx2−y2 pairing. The full frequency dependence of the spectra can be reproduced by including inelastic scattering due to spin fluctuations and/or Coulomb scattering on a partially nested Fermi surface. PACS numbers: 78.30.-j, 74.72.-h, 74.25.Gz, 74.20.Mn Typeset using REVTEX 1 By now a more coherent picture of the pair state symmetry in cuprate superconductors has emerged, and it has become increasingly evident that a gap with predominantly dx2−y2 pairing symmetry can provide an adequate description of many experiments [1]. However, the question remains or not whether the order parameter changes sign around the Fermi surface. The available data on Josephson junction and point-contact tunneling measurements seems to in favor of the former, although examples exist to the contrary [2,1]. In addition, it has been argued that the impurity dependence of certain correlation functions can be used to determine whether the gap changes sign. In particular, a detailed analysis has been given for the case of the electromagnetic penetration depth [3], and the crossover from T to T 2 at a temperature T ∗ which grows with impurity concentration has be taken to support d−wave pairing. Similar considerations can be made for the Knight shift, NMR rates, etc. [1]. However, the evidence gained from the penetration depth or other correlation functions is incomplete in that only the topology of the energy gap nodes can be identified from the power-law behavior at low temperatures, which leaves the actual gap symmetry unspecified. For instance, in tetragonal superconductors, there are 5 pure representations, (dx2−y2 , dxy, d3r2−1, dxz and dyz), which yield the same low temperature power-laws and impurity dependences for, e.g., the penetration depth, NMR rates, density of states, etc. It has recently been shown that both the electronic and phonon contributions to Raman scattering in superconductors can be used to identify the order parameter symmetry and can distinguish between the various representations for d−wave pairing [4]. Raman scattering measures effective intracell density fluctuations ρ̃q, ρ̃q = ∑ k,σ γ(k)c†k−q/2,σck+q/2,σ, (1) where c†k,σ(ck,σ) is the creation (annihilation) operator for an electron with wavevector k and spin σ. For the case of non-resonant scattering [5], the Raman vertex γ is directly related to the curvature of the band dispersion ǫ(k) and incident e and scattered e polarization light vectors via