Correlation between optical phonon softening and superconducting Tc in YBa2Cu3Ox within d-wave Eliashberg theory

We provide a mathematical description, based on d-wave Eliashberg theory, of the strong correlation between the experimentally observed softening of Raman modes associated with in-plane oxygen motions and the corresponding superconducting critical temperature Tc, as a function of oxygen doping x, in YBa₂Cu₃O_x. The theoretical model provides a direct link between physical trends of soft optical A_g (in-plane) oxygen modes, the level of oxygen doping x, and the superconducting Tc. Different regimes observed in the trend of Tc vs doping can be related to corresponding regimes of optical phonon softening in the Raman spectra. These results provide further evidence related to the physical origin of high-temperature superconductivity in rare-earth cuprate oxides and to the significant role of electron–phonon coupling therein.