Kerr Effect from Diffractive Skew Scattering in Chiral $p_x\pm i{p}_y$ Superconductors

We calculate the temperature dependent anomalous ac Hall conductance ${\sigma }_H(\mathrm{\Omega },T)$ for a two-dimensional chiral $p$-wave superconductor. This quantity determines the polar Kerr effect, as it was observed in ${\mathrm{Sr}}_2{\mathrm{RuO}}_4$ [J. Xia et al., Phys. Rev. Lett. 97, 167002 (2006)]. We concentrate on a single band model with an arbitrary isotropic dispersion relation subjected to rare, weak impurities treated in the Born approximation. As we explicitly show by detailed computation, previously omitted contributions to the extrinsic part of an anomalous Hall response, physically originating from diffractive skew scattering on quantum impurity complexes, appear to the leading order in the impurity concentration. By direct comparison with published results from the literature we demonstrate the relevance of our findings for the interpretation of the Kerr effect measurements in superconductors.

Figure 1

Zero temperature Hall conductivity for a chiral $p_x\pm i{p}_y$ superconductor with weak impurities and, for concreteness, a quadratic dispersion relation (ac frequency $\mathrm{\Omega }$, superconducting pairing amplitude ${\mathrm{\Delta }}_0$, elastic scattering time $\tau$ and ${\sigma }_H(0)=\mp e^2/[105\pi ({\mathrm{\Delta }}_0\tau {)}^2\hslash ]$). The solid red (blue dashed) curve represents the imaginary (real) part of ${\sigma }_H$. Inset: real space illustration of quantum mechanical probabilities for processes contributing to ${\sigma }_H$ and corresponding to diagrams (2a)–(2h) from Fig. 2. While generally all of those diagrams contribute, in the specific case of a parabolic band, the response stems from the processes (2b)–(2d), only. These contributions rely on diffractive scattering from quantum impurity complexes (yellow ellipses) with spatial extension $|{\mathit{R}}_1-{\mathit{R}}_2|$ comparable to the Fermi wavelength ${\lambda }_F$.

Figure 2

Diagrams (2a)–(2o) represent ${\sigma }_H(\mathrm{\Omega })$ to second order in impurity concentration for the model defined by Eqs. (2). Diagrams (2a) and (2e)–(2h) were presented in Ref. [26]. Diagrams (2i)–(2o) are zero. Diagrams (2b)–(2d) are the diffractive contributions, which are the major focus in this work. Diagram $(1_3)$: Mercedes star diagram [26, 33] occurring for a model with non-Gaussian disorder.