Maximum 2Δ0/kBTcfor electron-phonon superconductors

For an electron-phonon superconductor the question of a possible maximum value for the size of the ratio 2${\Delta }_0$/$k_B$ $T_c$ is addressed. Here ${\Delta }_0$ is the gap edge and $T_c$ the critical temperature. We were able to obtain well-converged solutions of the Eliashberg equations for a δ-function electron-phonon spectral density centered at ${\Omega }_E$ with ${\Omega }_E$ as low as 0.75 meV, in which case 2${\Delta }_0$/$k_B$ $T_c$ is still rising and equal to ∼9. In the limit ${\Omega }_E$→0, we can prove analytically that both ${\Delta }_0$ and $T_c$ scale like (${\Omega }_E$ ${)}^{1/2}$ so that their ratio remains finite. A rough estimate gives a value of about 7 for the ratio in this limit. In the course of the discussion a new upper bound on ${\Delta }_0$ is established and tested.