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 ($\lambda$, ${\mu }^{*}$, phonon spectrum) as contained in Eliashberg theory. The most striking new feature of the analysis is in the asymptotic regime of very large $\lambda$ where $T_c$ is found to equal $0.15 {\left(\lambda 〈{\omega }^2〉\right)}^{\frac{1}{2}}$ (assuming ${\mu }^{*}=0.1\mathrm{}$). 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 "$\lambda =2\mathrm{}$ limit" is spurious. The McMillan equation (with a prefactor altered from $\frac{{\Theta }_D}{1.45}$ to $\frac{{\omega }_{log}}{1.2}$) is found to be highly accurate for all known materials with $\lambda <1.5\mathrm{}$ but in error for large values of $\lambda$. Correction factors to McMillan's equation are found in terms of $\lambda$, ${\mu }^{*}$, and one additional parameter, $\frac{{\left(〈{\omega }^2〉\right)}^{\frac{1}{2}}}{{\omega }_{log}}$. The frequency ${\omega }_{log}$ is defined as $\exp 〈\ln \omega 〉$ where the averages $〈\ln \omega 〉$ and $〈{\omega }^2〉$ are defined using $\left(\frac{2}{\lambda \omega }\right){\alpha }^{2}F\left(\omega \right)$ 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 $\lambda =2.59\mathrm{}$ and $\frac{T_c}{{\omega }_{log}}=0.284\mathrm{}$, in serious disagreement with McMillan's formula but in good agreement when the correction factors are included. The McMillan-Hopfield parameter $\eta$ [or $N\left(0\right)\mathrm{} 〈I^2〉$] is extracted from tunneling measurements or from a combination of empirical values of $\lambda$ and neutron-scattering measurements of phonon dispersion. It is proposed that $\eta$ (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.