Impurity-induced polar Kerr effect in a chiralp-wave superconductor

Abstract

We discuss the polar Kerr effect in a chiral p-wave ͑p x + ip y -wave͒ superconductor. It is found that the off-diagonal component of a current-current correlation function is induced by impurity scattering in the chiral p-wave condensate, and a nonzero Hall conductivity is obtained using the Kubo formula. We estimate the Kerr rotation angle by using this impurity-induced Hall conductivity and compare it with experimental results ͓J. Xia et al., Phys. Rev. Lett. 97, 167002 ͑2006͔͒. Recently, the quasi-two-dimensional ͑quasi-2D͒ superconductor Sr 2 RuO 4 with T c = 1.5 K has attracted considerable attention and it has been investigated extensively. 1 It is plausible that the order parameter in Sr 2 RuO 4 has the spin triplet p x Ϯ ip y -wave symmetry. One of the fascinating properties of this state is the spontaneous breaking of parity in a twodimensional ͑2D͒ sense ͑p x → p x and p y → −p y ͒ and in a timereversal symmetry due to the presence of nonzero chirality characterized by l z = Ϯ 1, where l z is the z component of the relative orbital angular momentum of the Cooper pair.The polar Kerr effect ͑PKE͒, in which the direction of polarization of reflected linearly polarized light is rotated, has been known as an effective tool for understanding ferromagnetism. 2 Because of the analogy between ferromagnetic order-for instance, with s z = 1 and chiral pair condensation with l z = 1-it is naively expected that the PKE is induced in the chiral p-wave state at zero field. In fact, the PKE has been observed in the superconducting state of Sr 2 RuO 4 . 3 Up-to-date theoretical reports on the PKE in the chiral p-wave state are given in Refs. 4 and 5, in which interesting mechanisms have been proposed by the field theoretical approach; however, obtained results of the Kerr rotation angle are considerably smaller than the experimental results.Therefore, it is crucial to elucidate the fundamental nature of the PKE in the chiral p-wave superconductor. We will show that a Kerr rotation angle comparable to that obtained experimentally is obtained by taking into account nonmagnetic and short-ranged impurity scatterings of quasiparticles in a chiral p-wave condensate. It is also found that this impurity-induced PKE is suppressed or is zero for any superconducting state other than the chiral p-wave state. 6 This result is contrary to the naive analogy with a ferromagnet, since the effect is not proportional to chirality analogous to magnetization but suppressed in higher chirality states with l z = Ϯ 2, Ϯ 3,¯. The natural unit ប = c = k B = 1 is used throughout this Rapid Communication.We reviewed the phenomenology of the PKE in timereversal symmetry-breaking superconducting systems, which is an extension of the argument for itinerant ferromagnetic systems. 2,8 We also refer to the discussion of anyon superconductivity. 9 Let us suppose that z Ͼ 0 is empty, z Ͻ 0 is filled by the superconductor, and incident light is linearly polarized and propagating along the z direction perpendicular to the superconducting plane with a w...

Frequency and temperature dependence of the anomalous ac Hall conductivity in a chiralpx+ipysuperconductor with impurities

“…(5) of Ref. [48] for the same model in the same asymptotic limit. Given the causality requirement σ xy (ω) = σ * xy (−ω) [56], it appears that the factor of i = √ −1 was overlooked in the calculation of Ref.…”

“…The effect of disorder on the anomalous Hall conductivity in a p x + ip y superconductor was studied by Goryo [48]. He considered the skew-scattering processes, which involve triple scattering events on a given impurity.…”

“…Goryo [48] calculated the anomalous ac Hall conductivity σ xy (ω) for a p x + ip y superconductor in the limit of high frequencies ω ≫ ∆ 0 , where 2∆ 0 is the superconducting gap, and for T close to T c . This limit is relevant for the experiment [1], which utilized ω = 0.8 eV, whereas the gap at T = 0 can be estimated from the BCS relation as 2∆ 0 = 3.5 T c = 0.46 meV.…”

“…We find that the factor of i = √ −1 was overlooked in the calculation of Ref. [48]. Then we consider a more conventional model of Gaussian disorder and calculate the Feynman diagrams where electrons scatter on two different impurities.…”

“…In the case of a chiral p-wave superconductor, the anomalous Hall conductivity due to the skew-scattering diagrams of Fig. 3 was first calculated by Goryo [48] for high frequencies ω ≫ ∆ 0 and T close to T c . In Sec.…”