How can we relate the critical temperature and the superconducting gap amplitude in cuprate superconductors?

We explore the superconducting state of hole-doped cuprates by electronic Raman scattering as a function of both temperature and doping level. We observe a loss of coherent quasi-particles in the anti-nodal region and show that coherent Bogoliubov quasiparticles are confined around the nodes. This contrasts to conventional superconductors where superconductivity develops uniformly along the normal-state Fermi surface. We define the fraction of coherent Fermi surface, f$_{c}$ around the nodes for which quasi-particles are well defined and superconductivity sets in. We establish that T$_{c} \quad \propto $ f$_{c}\Delta _{max}$. $\Delta _{max}$ is the maximum amplitude of the d-wave superconducting gap. This new relation differs from the standard BCS theory and gives us some clues for increasing T$_{c}$ in the cuprates. S. Blanc et al. Phys. Rev. B \textbf{82}, 144516 (2010); S. Blanc et al. Phys. Rev. B \textbf{80}, 140502 (2009).