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 ($\ensuremath{\lambda}$, ${\ensuremath{\mu}}^{*}$, phonon spectrum) as contained in Eliashberg theory. The most striking new feature of the analysis is in the asymptotic regime of very large $\ensuremath{\lambda}$ where ${T}_{c}$ is found to equal $0.15 {(\ensuremath{\lambda}〈{\ensuremath{\omega}}^{2}〉)}^{\frac{1}{2}}$ (assuming ${\ensuremath{\mu}}^{*}=0.1$). 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 "$\ensuremath{\lambda}=2$ limit" is spurious. The McMillan equation (with a prefactor altered from $\frac{{\ensuremath{\Theta}}_{D}}{1.45}$ to $\frac{{\ensuremath{\omega}}_{log}}{1.2}$) is found to be highly accurate for all known materials with $\ensuremath{\lambda}<1.5$ but in error for large values of $\ensuremath{\lambda}$. Correction factors to McMillan's equation are found in terms of $\ensuremath{\lambda}$, ${\ensuremath{\mu}}^{*}$, and one additional parameter, $\frac{{(〈{\ensuremath{\omega}}^{2}〉)}^{\frac{1}{2}}}{{\ensuremath{\omega}}_{log}}$. The frequency ${\ensuremath{\omega}}_{log}$ is defined as $\mathrm{exp} 〈\mathrm{ln}\ensuremath{\omega}〉$ where the averages $〈\mathrm{ln}\ensuremath{\omega}〉$ and $〈{\ensuremath{\omega}}^{2}〉$ are defined using $(\frac{2}{\ensuremath{\lambda}\ensuremath{\omega}}){\ensuremath{\alpha}}^{2}F(\ensuremath{\omega})$ 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 ${\mathrm{Pb}}_{0.45}$${\mathrm{Bi}}_{0.55}$ reported herein. This material has parameters $\ensuremath{\lambda}=2.59$ and $\frac{{T}_{c}}{{\ensuremath{\omega}}_{log}}=0.284$, in serious disagreement with McMillan's formula but in good agreement when the correction factors are included. The McMillan-Hopfield parameter $\ensuremath{\eta}$ [or $N(0) 〈{I}^{2}〉$] is extracted from tunneling measurements or from a combination of empirical values of $\ensuremath{\lambda}$ and neutron-scattering measurements of phonon dispersion. It is proposed that $\ensuremath{\eta}$ (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.