5. The electron-phonon spectral function a²F(w) is the phonon spectrum F(w) weighed by the interaction with electrons. A peak in the a²F(w) gives rise to a peak in the derivative of the tunneling conductance (see figure). In the framework of the Eliashberg theory it is possible to extract the electron-phonon spectral function by starting from the conductance curves of a tunnel junction and inverting the Eliashberg equations. We have discussed the limits and the correct utilization of the standard program for such an inversion in the case of high-T_c superconductors. In particular, we have introduced the results of the inversion procedure, applied to our tunneling data in Bi₂Sr₂CaCu₂O_8+x single-crystal break junctions with T_c = 93 K, in a direct program for the solution of Eliashberg equations. Then, we have compared the result to the original tunneling curve. Most of the observed differences at energy greater than the gap can be explained by a smooth energy dependence of the normal density of states that we introduced in the direct solution of the Eliashberg equations. Finally we show that the effects of the energy-dependent normal density of states can be simulated by an efficient electron-phonon spectral function but, also, by a negative, nonphysical, Coulomb pseudopotential.

R.S. Gonnelli, G.A. Ummarino and V.A. Stepanov
“Determination of the tunneling electron-phonon spectral function in high-T_c superconductors with energy dependence of the normal density of states”, Physica C 275, 162-171 (1997).