Effect of thermal phonons on the superconducting transition temperature

There is no consensus in the literature on whether or not thermal phonons depress the superconducting transition temperature $T_c$. In this paper it is shown by accurate numerical solution of the real-frequency Eliashberg equations for the pairing self-energy $\phi$ and renormalization function $Z$ that thermal phonons in the kernel for $\phi$ raise $T_c$ but those in $Z$ lower it by a larger amount so that the net effect is to depress $T_c$. (A previous calculation which ignored the effect of thermal phonons in $\phi$ overestimated the suppression of $T_c$ by at least a factor of 3.) It is shown how to switch off the thermal phonons in the imaginary-frequency Eliashberg equations, exactly for $Z$ and approximately for $\phi$. The real-frequency and approximate imaginary-frequency results for the depression of $T_c$ by thermal phonons are in very satisfactory agreement. Thermal phonons are found to depress the transition temperature of ${\mathrm{Nb}}_3$Sn by only 2%. It is estimated that the suppression of $T_c$ by thermal phonons saturates at about 50% in the limit of very strong electron-phonon coupling.