Effect of Anisotropic Impurity Scattering in D-Wave Superconductors
Ze-Long Wang1, Rui-Ying Mao1, Da Wang1,2*, and Qiang-Hua Wang1,2*
1National Laboratory of Solid State Microstructures & School of Physics, Nanjing University, Nanjing 210093, China 2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
Abstract：In $d_{x^2-y^2}$-wave superconductors, the effect of s-wave point disorder has been extensively studied in literature. We study the anisotropic disorder in the form of $V_{kk'}^{\rm imp}=V_{\rm i}f_{k}f_{k'}$ with $f_k=\cos(2\theta)$ ($\theta$ 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 $\varGamma_k=\varGamma_{\rm i}f_k^2$. One consequence of the disorder effect is that the gap-to-$T_{\rm c}$ ratio $2\varDelta(0)/T_{\rm c}$ 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 $\varGamma_\theta=\varGamma_\alpha|\theta-\theta_0|^\alpha$ around each nodal direction $\theta_0$. We find the density of states $\rho(\omega)-\rho(0)\propto|\omega|$ ($\omega^2$) for all $\alpha\ge1$ ($\alpha < 1$) in the limit of $\omega\to0$. As a result, the superfluid density $\rho_{\rm s}$ exhibits two and only two possible scaling behaviors: $\rho_{\rm s}(0)-\rho_{\rm s}(T)\propto T$ ($T^2$) for $\alpha\ge1$ ($\alpha < 1$) in the low temperature limit.
2023, Vol. 40 
