Transition temperature of strong-coupled superconductors reanalyzed

A through analysis is made of the dependence of the superconducting transition temperature Tc 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 Tc is found to equal 0.15 (λ〈ω²〉)1/2 (assuming μ*=0.1). This result implies the surprising conclusion that within Eliashberg theory Tc 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 ΘD1.45 to ωlog1.2) 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〉)1/2ωlog. The frequency ωlog 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 Pb0.45{Bi}0.55$ reported herein. This material has parameters ${λ}=2.59$ and ${{T}c}{{{ω}}log}=0.284$, in serious disagreement with McMillan's formula but in good agreement when the correction factors are included. The McMillan-Hopfield parameter ${η}$ [or $N(0) 〈{I}²〉$] 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.