Breakdown of Eliashberg Theory for Two-Dimensional Superconductivity in the Presence of Disorder

We have performed tunneling measurements on ultrathin, homogeneous, strong-coupled superconducting Pb films in the regime where $T_c$ varies with sheet resistance. The energy gap $\Delta$ and the strong-coupling corrections to the density of states have been measured as functions of the sheet resistance $R_{\square }$. We conclude that the standard picture of enhanced Coulomb repulsion with increasing sheet resistance is too naive and that, in the presence of disorder, an analysis more sophisticated than Eliashberg theory is necessary.