Superconductivity and phonon softening: II. Lead alloys

The electron-phonon coupling parameter λ is investigated theoretically for alloys of Pb with Tl and Bi. Experiments reported in a preceding paper show that McMillan's suggestion λ∝〈ω²〉^-1 fails for these metals, indicating that the electronic factors N(0)〈I²〉 are playing an important role. On general grounds it is expected that the electronic coupling strength N(0)〈I²〉 is related to the amount of phonon renormalization ΩQ²-ωQ², where ΩQ is a bare phonon frequency. Thus, in spite of the failure of McMillan's suggestion, we conjecture that a formula exists which relates λ only to phonon frequencies; for nontransition metals we argue that this formula has the form λ=(1/N)f̄ΣQ(ΩQ²-ωQ²)ωQ². Microscopic theory indicates that the bare frequency ΩO 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 ΩQ² and f̄ for lead alloys. The actual phonon frequencies ωQ are known from inelastic neutron scattering and Born-von Karman extrapolation. Values of λ 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 λ and measured ωQ bypasses the need for detailed knowledge of the electronic nature of the material and all that information is disguised in the difference ΩQ²-ωQ². The extension of this work, however, to "nonpseudopotential" materials such as Nb or Nb₃Sn will require confronting the problem of identifying and calculating the "bare" frequencies ΩQ.