Superconductivity and phonon softening: II. Lead alloys

The electron-phonon coupling parameter $\lambda$ is investigated theoretically for alloys of Pb with Tl and Bi. Experiments reported in a preceding paper show that McMillan's suggestion $\lambda \propto {〈{\omega }^2〉}^{-1\mathrm{}}$ fails for these metals, indicating that the electronic factors $N\left(0\right)\mathrm{}〈{\mathcal{I}}^2〉$ are playing an important role. On general grounds it is expected that the electronic coupling strength $N\left(0\right)\mathrm{}〈{\mathcal{I}}^2〉$ is related to the amount of phonon renormalization ${\Omega }_Q^2-{\omega }_Q^2$, where ${\Omega }_Q$ is a bare phonon frequency. Thus, in spite of the failure of McMillan's suggestion, we conjecture that a formula exists which relates $\lambda$ only to phonon frequencies; for nontransition metals we argue that this formula has the form $\lambda =\left(\frac{1}{N}\right)\overline{f}{\Sigma }_Q\frac{({\Omega }_Q^2-{\omega }_Q^2)}{{\omega }_Q^2}$. Microscopic theory indicates that the bare frequency ${\Omega }_O$ must include not only the Coulombic point-ion repulsion, but also the restoring force on a displaced point-ion due to the rigid-crystal charge density. Pseudopotential theory is used to estimate ${\Omega }_Q^2$ and $\overline{f}$ for lead alloys. The actual phonon frequencies ${\omega }_Q$ are known from inelastic neutron scattering and Born-von Karman extrapolation. Values of $\lambda$ are calculated and compared with the tunneling results reported in a preceding paper. It is found that excellent agreement compared with the tunneling results reported in the preceding paper. It is found that excellent agreement is obtained throughout the series if $f$ is chosen to be 0.023, independent of material. It is hoped that this relation between $\lambda$ and measured ${\omega }_Q$ bypasses the need for detailed knowledge of the electronic nature of the material and all that information is disguised in the difference ${\Omega }_Q^2-{\omega }_Q^2$. The extension of this work, however, to "nonpseudopotential" materials such as Nb or ${\mathrm{Nb}}_3$Sn will require confronting the problem of identifying and calculating the "bare" frequencies ${\Omega }_Q$.