Renormalization of the electron-phonon interaction by strong electronic correlations in high-${\mathit{T}}_{\mathit{c}}$ superconductors

The renormalization of the electron-phonon interaction by strong electronic correlations is studied using a one-band Hubbard model with infinite repulsion and nearest (t model) and nearest and second-nearest (tt′ model) neighbor hopping terms and an on-site electron-phonon coupling. Using Hubbard’s X operators and an extension from 2 to N degrees of freedom for the electrons the leading contributions for the electron self-energy and the vertex function in 1/N and the electron-phonon coupling constant are given and numerically evaluated for a square lattice. We find that the momentum dependence of the vertex function depends strongly on doping: For large dopings it is rather weak, with decreasing doping it becomes more and more pronounced leading to a strong reduction of the vertex at larger momentum transfers until, for very small dopings, the vertex essentially consists of a forward scattering peak with a width proportional to the doping. This behavior occurs both in the t and the tt′ models and also in one dimension where analytic expressions are derived. Correlation effects also change ${\mathrm{\alpha }}^2$F in general: The full-symmetric component ${\mathrm{\alpha }}^2$${\mathit{F}}_1$ is in the t model somewhat, in the tt′ model rather strongly suppressed, especially near half-filling; the other symmetry components ${\mathrm{\alpha }}^2$${\mathit{F}}_{\mathit{i}}$ with i=2,...,5 increase strongly with decreasing doping and are of similar magnitude as ${\mathrm{\alpha }}^2$${\mathit{F}}_1$ near half-filling.

Including also direct Coulomb repulsion nontrivial symmetries such as d wave become more stable than the s-wave order parameter below a critical value for the doping even for the considered phonon-mediated superconductivity. Most dramatic, however, is the quenching of the resistivity due to electron-phonon scattering both in the t and the tt′ models at intermediate and small dopings. This result may explain the absence of phonon features in the experimental transport coefficients of high-${\mathit{T}}_{\mathit{c}}$ compounds. © 1996 The American Physical Society.