Impurity-induced polar Kerr effect in a chiral $p$-wave superconductor

We discuss the polar Kerr effect in a chiral $p$-wave ($p_x+i{p}_y$-wave) 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 (2006)].

Figure 1

Leading diagrams of ${\sum }_{ij}{ϵ}_{ij}{\pi }_{ij}\left(i{\omega }_n\right)$. The lines with the arrows $->-$, $-<->-$, and $->-<-$ denote quasiparticle Green’s functions ${\mathcal{G}}_{\mathbf{p}}\left(i{ϵ}_m\right)$, ${\mathcal{F}}_{\mathbf{p}}\left(i{ϵ}_m\right)$, and ${\mathcal{F}}_{\mathbf{p}}^{\ast }\left(i{ϵ}_m\right)$, respectively, in $G_{\mathbf{p}}\left(i{ϵ}_m\right)$. The lower lines have frequency ${ϵ}_{m+n}={ϵ}_m+{\omega }_n$, while the upper lines have ${ϵ}_m$. The dashed lines denote the impurity coupling ${\nu }_{\text{i}}$. The scattering at the black dots yields the factor $\text{Tr}\left({\tau }_3{\mathbf{g}}_{\mathbf{p}}\cdot \mathit{\tau }{\mathbf{g}}_{\mathbf{k}}\cdot \mathit{\tau }\right)$ in Eq. (3).

Figure 2

$\omega$ dependence of ${\theta }_K$ in Eq. (2) at $T/T_c=0.8$, obtained using parameters $d=6.8\text{Å}$, $l_i=10000\text{Å}$, ${\xi }_0=660\text{Å}$, ${\omega }_p=1.3\text{eV}$, ${ϵ}_F=0.14\text{eV}$, and ${\tau }_0^{-1}=6.6\times {10}^{-5}\text{eV}$ (Ref. 3). It is assumed that the cut-off energy of the pairing interaction ${\omega }_D>10{\omega }_p$.