Two Bounds on the Maximum Phonon-Mediated Superconducting Transition Temperature

Two simple bounds on the $T_c$ of conventional, phonon-mediated superconductors are derived within the framework of Eliashberg theory in the strong coupling regime. The first bound is set by the total electron-phonon coupling available within a material given the hypothetical ability to arbitrarily dope the material. This bound is studied by deriving a generalization of the McMillan-Hopfield parameter, $\widetilde{\eta}(E)$, which measures the strength of electron-phonon coupling including anisotropy effects and rigid-band doping of the Fermi level to $E$. The second bound is set by the softening of phonons to instability due to strong electron-phonon coupling with electrons at the Fermi level. We apply these bounds to some covalent superconductors including MgB$_2$, where $T_c$ reaches the first bound, and boron-doped diamond, which is far from its bounds.