Critical temperature and giant isotope effect in presence of paramagnons

TL;DR

This paper develops an analytical formula for the critical temperature in phonon-mediated superconductors considering spin fluctuations, revealing a significant isotope effect enhancement when spin fluctuation and phonon frequencies are comparable.

Contribution

It introduces a controllable approximation to accurately describe $T_{c}$ considering paramagnons, improving upon previous formulas and providing new insights into isotope effects.

Findings

Analytical formula for $T_{c}$ agrees with numerical solutions.

Strong isotope effect enhancement when spin fluctuation and phonon frequencies match.

Implications for near-magnetic superconductors like MgCNi$_{3}$.

Abstract

We reconsider the long-standing problem of the effect of spin fluctuations on the critical temperature and isotope effect in a phonon-mediated superconductor. Although the general physics of the interplay between phonons and paramagnons had been rather well understood, the existing 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. Based on 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. This effect may have important consequences for near-magnetic superconductors such as MgCNi$_{3}$