Linking dynamic and thermodynamic properties of cuprates: An angle-resolved photoemission study of (Ca$_{x}$La$_{1-x}$)(Ba$_{1.75-x}$La$_{0.25+x}$)Cu$_3$O$_y$ ($x$=0.1 and 0.4

We report angle-resolved photoemission spectroscopy (ARPES) on two families of high temperature superconductors (Ca$_{x}$La$_{1-x}$)(Ba$_{1.75-x}$ La $ _{0.25+x}$)Cu$_{3}$O$_{y}$ with $x=0.1$ ($T_{c}^{max}=56$ K) and $x=0.4$ ($ T_{c}^{max}=82$ K). The Fermi surface (FS) is found to be independent of $x$ or $y$, and its size indicates extreme sample-surface overdoping. This universal FS allowes the comparison of dynamical properties between superconductors of similar structure and identical doping, but different $ T_{c}^{max}$. We find that the high-energy ($\left\vert E\right\vert >50$ meV) nodal velocity in the $x=0.4$ family is higher than in the $x=0.1$ family. The implied correlation between $T_{c}^{max}$ and the hopping rate $ t $ supports the notion of kinetic energy driven superconductivity in the cuprates. We also find that the antinodal gap is higher for the $x=0.4$ family.