London Penetration Depth and Pair Breaking

The London penetration depth is evaluated for isotropic materials for any transport and pair-breaking Born scattering rates. Besides known results, a number of new features are found. The slope $|d\rho/d\theta |$ of the normalized superfluid density $\rho=\lambda^2(0)/\lambda^2(\theta)$ at the transition $\theta=T/T_c=1$ has a minimum near the value of the pair-breaking parameter separating gapped and gapless states. The low-$T$ exponentially flat part of $\rho$ for the s-wave materials is suppressed by increasing pair breaking. For strong $T_c$ suppression by magnetic impurities the "Homes scaling" $\lambda^{-2}(0) \propto \sigma T_c$ with $\sigma$ being the normal conductivity gives way to $\lambda^{-2}(0) \propto \sigma T_c^2$. For the d-wave order parameter, the transport and spin-flip Born scattering rates enter the theory only as a sum, in particular, they affect the $T_c$ depression in the same manner. We confirm that the linear low temperature behavior of $\rho$ in a broad range of the combined scattering parameter turns to the $T^2$ behavior only when the critical temperature is suppressed at least by a factor of 3 relative to the clean limit $T_{c0}$. Moreover, in this range, $\rho (\theta)$ is only weakly dependent on the scattering parameter, i.e. it is nearly universal.