Critical temperature and the giant isotope effect in the presence of paramagnons

We reconsider the long-standing problem of the effect of spin fluctuations on the critical temperature and the isotope effect in a phonon-mediated superconductor. We discuss the general physics of the interplay between phonons and paramagnons, and show that the previously used approximate formulas fail to describe the correct behavior of T-c for general phonon and paramagnon spectra. Using a controllable approximation, we derive an analytical formula for T-c which agrees well with exact numerical solutions of the Eliashberg equations for a broad range of parameters. On the basis of both numerical and analytical results, we predict a strong enhancement of the isotope effect when the frequencies of spin fluctuation and phonons are of the same order. We discuss application to near-magnetic superconductors such as MgCNi3.