Crossover from 2D-XY to 3D-XY superconducting fluctuations with hole-doping in dynamical conductivity of La$_{2-x}$Sr$_x$CuO$_4$ thin films by broadnband technique

We report the systematic study of dynamical complex ac conductivity, $\sigma(\omega)=\sigma_1(\omega)+i\sigma_2(\omega)$, of high-quality La$_{2-x}$Sr$_x$CuO$_4$ (LSCO) ($x$=0.07 to 0.24) thin films ate temperatures ($T$s) just above the superconducting transition temperature, $T_c$, by using a broadband microwave technique, where both components of the complex conductivity can be obtained as a detailed function of frequency by sweeping the microwave frequency continuously (0.1 to 12~GHz). For all superconducting films, we observed a definite contribution of the superconducting fluctuation to $\sigma(\omega, T)$, which could not be described by the conventional Aslamazov- Larkin term. Detailed analyses using a dynamic scaling theory clearly indicated that the fluctuation can be well described by the 2D-$XY$ critical behavior for the underdoped LSCO samples, whereas it changes into the 3D-XY critical behavior in the optimally doped samples. We are now fixing the fluctuation behavior in the overdoped samples, which is very important to judge whether or not the quantum-criticality picture proposed in some theories is appropriate for the description of the electronic state of the high-$T_c$ cuprate supercondutors.