Effect of Anisotropic Impurity Scattering in D-Wave Superconductors

In d_x²–y²-wave superconductors, the effect of s-wave point disorder has been extensively studied in literature. We study the anisotropic disorder in the form of $V_{k{k}^{\prime }}^{\mathrm{imp}}=V_{\mathrm{i}}{f}_k{f}_{k^{\prime }}$ with f_k = cos(2θ) (θ the azimuthal angle of k), as proposed to be caused by apical oxygen vacancies in overdoped La-based cuprate films, under the Born approximation. The disorder self-energy and d-wave pairing affect each other and must be solved simultaneously self-consistently. We find that the self-energy is reduced at low frequencies and thus weakens the pair-breaking effect. This frequency dependence vanishes in the dirty limit for which the disorder is well described by a scattering rate ${\Gamma }_k={\Gamma }_{\mathrm{i}}{f}_k^2$. One consequence of the disorder effect is that the gap-to-T_c ratio 2Δ(0)/T_c is greatly enhanced by the d-wave disorder, much larger than the s-wave disorder and the clean BCS value 4.28. Lastly, we generalize the d-wave scattering rate to a general form ${\Gamma }_{\theta }={\Gamma }_{\alpha }|\theta -{\theta }_0{|}^{\alpha }$ around each nodal direction θ₀. We find the density of states ρ(ω)–ρ(0)∝|ω| (ω²) for all α ≥ 1 (α < 1) in the limit of ω → 0. As a result, the superfluid density ρ_s exhibits two and only two possible scaling behaviors: ρ_s(0)–ρ_s(T)∝ T (T²) for α ≥ 1 (α < 1) in the low temperature limit.