Critical dynamics of superconducting ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ films

We report on a systematic investigation of the critical fluctuations in the complex conductivity of epitaxially grown ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ films for $T\lesssim T_c$ using a two-coil inductive technique at zero applied field. We observe the static three-dimensional (3D) $\mathrm{XY}$ critical exponent in the superfluid density near $T_c.$ Linear scaling analysis close to the critical temperature yields a dynamic critical exponent of $z\approx 2.0$ for small drive currents, but nonlinear effects are seen to be important. At $T_c,$ a nonlinear scaling analysis also yields a 3D dynamic exponent of $z=2.0\mathrm{}\pm \mathrm{0.1.}$