Effect of anisotropic impurity scattering in d-wave superconductors

Abstract

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 $V_{k{k}^{\prime }}^{\mathrm{imp}}=V_i{f}_k{f}_{k^{\prime }}$ with $f_k=\cos (2\hat{I}¸)$ (with $\hat{I}¸$ 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 $\hat{I}{“}_k=\hat{I}{“}_i{f}_k^2$. One consequence of the disorder effect is the gap-to-$T_c$ ratio $2\hat{I}”(0)/T_c$ 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 $\hat{I}{“}_{\hat{I}}¸=\hat{I}{“}_{\hat{I}}\pm |\hat{I}¸-\hat{I}{¸}_0{|}^{\hat{I}}\pm$ around each nodal direction $\hat{I}{¸}_0$. We find the density of states $\ddot{I}(\ddot{I}‰)-\ddot{I}(0)\propto |\ddot{I}‰|$ ($\ddot{I}{‰}^2$) for all $\hat{I}\pm \ge 1$ ($\hat{I}\pm <1$) in the limit of $\ddot{I}‰\to 0$. As a result, the superfluid density $\ddot{I}{}_s$ exhibits two and only two possible scaling behaviors: $\ddot{I}{}_s(0)-\ddot{I}{}_s(T)\propto T$ ($T^2$) for $\hat{I}\pm \ge 1$ ($\hat{I}\pm <1$) in the low temperature limit.

5 pages, 6 figures