Impurity Effect on Frequency Dependent Superconductivity: Odd-Frequency Pairing and Even-Frequency Pairing

We study an effect of non-magnetic impurities on frequency dependent superconductivity such as the odd-frequency superconductivity to which the conventional Abrikosov-Gorkov (AG) formula cannot directly be applied. Using a realistic odd-frequency gap function, we investigate the behavior of the transition temperature against the impurity concentration. We find that the transition temperature decrease is more robust than that of the AG formula, and that the transition temperature exhibits a reentrant structure even though it does not appear in the system in the clean limit. For even-superconductivity, we show that the transition temperature suppression by the impurity becomes weaker than that of the AG formula if the gap Delta(omega(n)) has a peak around omega(n) = 0.