Dynamical study of phase fluctuations and their critical slowing down in amorphous superconducting films

We report a comprehensive study of the complex ac conductance of amorphous superconducting InO${}_x$ thin films. We measure the explicit frequency dependency of the complex conductance and the phase stiffness over a range from 0.21 to 15 GHz at temperatures down to 350 mK using a novel broadband microwave Corbino spectrometer. The dynamic ac measurements are sensitive to the temporal correlations of the superconducting order parameter in the fluctuation range above $T_c$. Among other aspects, we explicitly demonstrate the critical slowing down of the characteristic fluctuation rate on the approach to the superconducting state and show that its behavior is consistent with vortex-like phase fluctuations and a phase-ordering scenario of the transition.

Figure 1

(a) and (b): frequency dependence of the real and imaginary conductances in the ranges $\omega /2\pi =0.21$–27 GHz and $T=0.35$–4 K. A color scale representing different temperatures is displayed in (a). The black curves are the conductances at $T_c$. The features in (a) at about 22 GHz are residual features imperfectly removed during calibration. (c) and (d): temperature dependence of the real and imaginary conductances in the frequency range $\omega /2\pi =0.660$–15 GHz. A color scale representing different frequencies is displayed in (d). The dashed black lines mark $T_c=2.36$ K.

Figure 2

Temperature dependence of phase stiffness at $\omega /2\pi =0.21$–15 GHz plotted against the vertical axis on the left. A color scale representing different frequencies is displayed as well. The black curve shows resistance per square of the same sample plotted with the vertical axis on the right. The dashed pink line is the KTB prediction, $4T_{\text{KTB}}=T_{\theta }$, for the universal jump in stiffness. The dark purple $▵$ marker is the temperature dependence of $T_{\theta }^0$ obtained via the scaling analysis described in the text. $T_c$ is marked by the black dashed line.

Figure 3

(a) Phase of $S(\omega /\Omega )$ normalized by $\pi /2$ as a function of reduced frequency $\omega /\Omega$. (b) Magnitude of $S(\omega /\Omega )$ as a function of reduced frequency. A color scale representing different temperatures for both plots is displayed in (a). Each plot is comprised of data measured at temperatures from 2.398 to 3 K and frequencies from 0.46 to 11 GHz.

Figure 4

(a) Fluctuation frequency as a function of temperature from 2.398 to 2.993 K. In (b) and (c), we plot $\Omega (T)/2\pi$ versus $1/\sqrt{T-T_c}$ and $T-T_c$, respectively, along with the fitting (black curves). (d) Excitation energy in units of degrees Kelvin as a function of $T-T_c$.