Charge-transfer gap size and oxygen hole content as two mechanisms controlling $T_c$ in the Emery model

Investigating the drivers of superconducting critical temperature trends in cuprates is crucial for uncovering the mechanism of high-temperature superconductivity. Here we study this problem in the canonical model of the copper-oxygen plane, the Emery model, with cellular dynamical mean-field theory. Using the Zaanen-Sawatzky-Allen diagram as a guiding framework, we systematically quantify how the maximum superconducting critical temperature $T_c^{\rm max}$ depends on the copper-oxygen energy distance and on the local repulsion on the copper orbital. Unexpectedly, $T_c^{\rm max}$ is optimized not only near the charge-transfer insulator to metal boundary, consistent with previous findings, but also deep in the charge-transfer regime, revealing an unexplored mechanism. Then we link model parameters to physical observables, identifying the charge-transfer gap size and the oxygen hole content as two mechanisms controlling $T_c^{\rm max}$. $T_c^{\rm max}$ increases monotonically as the oxygen hole content increases and the charge gap size decreases. The oxygen hole content is the dominant variable in varying $T_c^{\rm max}$. Our work provides predictions for proposed realizations of the Emery model with ultracold atoms and a theoretical framework for understanding key experimental trends in hole-doped cuprates.