Exchange enhancement of the electron-phonon coupling in metals

It is well known that ferromagnetic tendency is destructive to superconductivity through the spin-fluctuation coupling effect. Experimentally, however, the relation between superconductivity and magnetism is not that simple. Recently, we pointed out that by softening phonons the exchange interaction between electrons enhances the electron-phonon coupling constant ${\lambda }_p$, and, accordingly, superconductivity. If the effect of the exchange enhancement of ${\lambda }_p$ overrides the destructive effect of the spin-fluctuation coupling, resultantly, superconductivity can be helped by magnetism. In order to substantiate the above idea, in the present paper we carry out a quantitative estimation of ${\lambda }_p$ for a model in which deviations from the jellium model are conveniently included by a parameter $\xi$. The size of the exchange enhancement of ${\lambda }_p$ is found to depend very sensitively on $\xi$. Thus, assuming different values for $\xi$, we can explain why in Pd the exchange enhancement of ${\lambda }_p$ is small, resulting in the absence of superconductivity, whereas it is large in ${\mathrm{V}}_3$Si, producing high superconducting transition temperature.