Reversible magnetization and critical fluctuations in systematically doped YBa${}_2$Cu${}_3$O${}_{7-\delta }$ single crystals

The temperature and field dependence of reversible magnetization have been measured on a YBa${}_2$Cu${}_3$O${}_{7-\delta }$ single crystal at six different doping concentrations. It is found that the data above 2 T can be described by the scaling law based on the GL-LLL (lowest Landau level approach based on Ginzburg-Landau theory) critical fluctuation theory yielding the values of the slope of upper critical field $-\mathrm{d}{H}_{\mathrm{c}2}(T)\mathrm{/}\mathrm{d}T$ near $T_{\mathrm{c}}$. This set of values is self-consistent with that obtained in doing the universal scaling for the six samples. Based on a simple Ginzburg-Landau approach, we determined the doping dependence of the coherence length $\xi$ which behaves in a similar way as that determined from $\xi= \hbar v_$\mathrm{F}$/E_$\mathrm{s}\mathrm{c}$$with$E_$\mathrm{s}\mathrm{c}$$ the superconducting energy scale. Our results may suggest a growing coherence length towards more underdoping.