We measure and analyze the resistivity of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ crystals for different doping $\delta$. We obtain the fraction of carriers $\eta (T,\delta )=n_g∕n_{\mathrm{tot}}$ that do not participate in the $c$-axis conductivity. All the curves $\eta (T,\delta )$ collapse onto a universal curve when plotted against a reduced temperature $x=[T-ϴ\left(\delta \right)]∕{\Delta }^{*}\left(\delta \right)$. We find that at the superconducting transition $n_g$ is doping independent. We also show that a magnetic field up to 14 T does not affect the degree of localization in the $(a,b)$ planes but widens the temperature range of the $x$ scaling by suppressing the superconducting phase coherence.

• Received 15 July 2004

DOI:https://doi.org/10.1103/PhysRevB.70.214530