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 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 MgB2, where T-c reaches the first bound, and boron-doped diamond, which is far from its bounds.