Pairing fluctuation theory of high $T_c$ superconductivity in the   presence of nonmagnetic impurities
  The pseudogap phenomena in the cuprate superconductors requires a theory beyond the mean field BCS level. A natural candidate is to include strong pairing fluctuations, and treat the two-particle and single particle Green's functions self-consistently. At the same time, impurities are present in even the cleanest samples of the cuprates. Some impurity effects can help reveal whether the pseudogap has a superconducting origin and thus test various theories. Here we extend the pairing fluctuation theory for a clean system [Phys. Rev. Lett. 81, 4708 (1998)] to the case with nonmagnetic impurities. Both the pairing and the impurity $T$ matrices are included and treated self-consistently. We obtain a set of three equations for the chemical potential $\mu$, $T_c$, the excitation gap $\Delta(T_c)$ at $T_c$, or $\mu$, the order parameter $\Delta_{sc}$, and the pseudogap $\Delta_{pg}$ at temperature $T<T_c$, and study the effects of impurity scattering on the density of states, $T_c$ and the order parameter, and the pseudogap. Both $T_c$ and the order parameter as well as the total excitation gap are suppressed, whereas the pseudogap is not for given $T\le T_c$. Born scatterers are about twice as effective as unitary scatterers in suppressing $T_c$ and the gap. In the strong pseudogap regime, pair excitations contribute a new $T^{3/2}$ term to the low $T$ superfluid density. The initial rapid drop of the zero $T$ superfluid density in the unitary limit as a function of impurity concentration $n_i$ also agrees with experiment. 
Qijin Chen and J. R. Schrieffer