Transition temperature of strong-coupled superconductors reanalyzed

A through analysis is made of the dependence of the superconducting transition temperature T_c on material properties (λ, μ^*, phonon spectrum) as contained in Eliashberg theory. The most striking new feature of the analysis is in the asymptotic regime of very large λ where T_c is found to equal 0.15 (λ<ω²>)¹² (assuming μ^*=0.1). This result implies the surprising conclusion that within Eliashberg theory T_c is not limited by the phonon frequencies, and also shows that McMillan's "λ=2 limit" is spurious. The McMillan equation (with a prefactor altered from Θ_D1.45 to ω1.2) is found to be highly accurate for all known materials with λ<1.5 but in error for large values of λ. Correction factors to McMillan's equation are found in terms of λ, μ^*, and one additional parameter, (<ω²>)¹²ω. The frequency ω is defined as exp <lnω> where the averages <lnω> and <ω²> are defined using (2λω)α²F(ω) as a weight factor. These conclusions are based on a combination of analytic and numerical solutions of the Eliashberg equations, and are supported by a comparison with tunneling data. Especially strong support comes from a new experimental result for amorphous Pb_0.45Bi_0.55 reported herein. This material has parameters λ=2.59 and T_cω=0.284, in serious disagreement with McMillan's formula but in good agreement when the correction factors are included. The McMillan-Hopfield parameter η [or N(0) <I²>] is extracted from tunneling measurements or from a combination of empirical values of λ and neutron-scattering measurements of phonon dispersion. It is proposed that η (which is now known not to be accurately constant) is the most significant single parameter in understanding the origin of high T_c and the limitation of T_c by colvalent instabilities.