Effect of anisotropic impurity scattering in d-wave superconductors
In d$_{x^2-y^2}$-wave superconductors, the effect of s-wave point disorder has been extensively studied in the literature. In this work, we study the anisotropic disorder with a the form of $V_{kk'}^{\rm imp}=V_if_kf_{k'}$ with $f_k=\cos(2θ)$ (with $θ$ 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 have to be solved simultaneously self-consistently. We find 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 $Γ_k=Γ_if_k^2$. One consequence of the disorder effect is 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 $4.28$. At last, we generalize the d-wave scattering rate to a general form $Γ_θ=Γ_α|θ-θ_0|^α$ around each nodal direction $θ_0$. We find the density of states $ρ(ω)-ρ(0)\propto|ω|$ ($ω^2$) for all $α\ge1$ ($α<1$) in the limit of $ω\to0$. As a result, the superfluid density $ρ_s$ exhibits two and only two possible scaling behaviors: $ρ_s(0)-ρ_s(T)\propto T$ ($T^2$) for $α\ge1$ ($α<1$) in the low temperature limit.