We have measured the near-normal reflectance of Tl2Ba2CaCu2O8 (Tl2212) for energies from 0.1 to 4.0 eV at room temperature and used a Kramers-Kronig analysis to find the complex, frequency dependent dielectric function, from which the optical conductivity was determined. Using Thermal-Difference-Reflectance (TDR) Spectroscopy the reflectance of the sample in the normal state just above the superconducting transition, and in the superconducting state were then obtained. From these data we determined the ratio of the superconducting- to normal-state optical conductivities. Mattis and Bardeen had calculated this function within the BCS theory, where the gap is a fixed energy-independent quantity. Taking into account the retarded nature of the electron-phonon coupling results in a complex, energy dependent gap causing deviations from the Mattis-Bardeen plot at energies where the phonon coupling function is large. We find a typical deviation near the phonon energies in Tl2212, and in addition, at 1.2 and 1.7eV. The phonon, and these electronic terms can each be described by a coupling constant. None of which by itself gives rise to a high transition temperature, but the combination does. Using Resonant Inelastic X-Ray Scattering (RIXS) we find that the d-to-d excitations of the cuprate ion in Tl2212 fall at the same energies as the dips in the Mattis-Bardeen plot. We conclude that the high superconducting transition temperature of the cuprates is due to the sum of the phonon interaction, and interactions with the Cu-ion d-shell.
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