We study the doping and temperature dependence of the single-particle coherent weight, $z_A$, for high- $T_c$ superconductors ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+x}$ using angle-resolved photoemission. We find that at low temperatures the coherent weight $z_A$ at $\left(\pi ,0\mathrm{}\right)$ is proportional to the carrier concentration $x$ and that the temperature dependence of $z_A$ is similar to that of the $c$-axis superfluid density. We show that, for a wide range of carrier concentration, the superconducting transition temperature scales with the product of the low-temperature coherent weight and the maximum superconducting gap.

• Received 13 October 2000

DOI:https://doi.org/10.1103/PhysRevLett.87.227001