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We provide an extended mathematical description of the strong correlationbetween the experimentally observed softening of Raman modes associated within-plane oxygen motions and the corresponding superconducting criticaltemperature $T_c$, as a function of oxygen doping $x$, in YBa$_2$Cu$_3$O$_x$.The model provides a direct link between physical trends of soft optical $A_g$(in-plane) oxygen modes, the level of oxygen doping $x$, and thesuperconducting $T_c$. Different regimes observed in the trend of $T_c$ vsdoping can be mechanistically explained in terms of corresponding regimes ofoptical phonon softening in the Raman spectra. These results provide furtherevidence related to the physical origin of high-temperature superconductivityin rare-earth cuprate oxides and to the significant role of electron-phononcoupling therein.

Correlation between optical phonon softening and superconducting $T_c$ in YBa$_2$Cu$_3$O$_x$