Fluctuations, dissipation, and nonuniversal superfluid jumps in two-dimensional superconductors

We report a comprehensive study of the complex ac conductivity of thin effectively two-dimensional amorphous superconducting $\mathrm{In}{\mathrm{O}}_x$ films at zero applied field. Below a temperature scale $T_{c0}$ where the superconducting order parameter amplitude becomes well defined, there is a temperature where both the generalized superfluid stiffness acquires a frequency dependence and the dc magnetoresistance becomes linear in field. We associate this with a transition of the Kosterlitz-Thouless-Berezinskii (KTB) type. At our measurement frequencies the superfluid stiffness at $T_{\mathit{KTB}}$ is found to be larger than the universal value. Although this may be understood with a vortex dielectric constant of ${ϵ}_v\approx 1.9$ within the usual KTB theory, this is a relatively large value and indicates that such a system may be out of the domain of applicability of the low-fugacity (low-vortex-density) KTB treatment. This opens up the possibility that at least some of the discrepancy from a nonuniversal magnitude is intrinsic. Our finite-frequency measurements allow us access to a number of other phenomena concerning the charge dynamics in superconducting thin films, including an enhanced conductivity near the amplitude fluctuation temperature $T_{c0}$ and a finite dissipation at low temperature which appears to be a universal aspect of highly disordered superconducting films.

Figure 1

(Color online) dc sheet resistance. Temperature dependence showing the broad resistive transition resulting from fluctuations. The normal-state resistance curves are generated using the procedure described in the text. $T_{c0}$ is believed to be $2.28\mathrm{K}$.

Figure 2

(Color online) Temperature dependence of the superfluid stiffness $T_{\theta }$. The prediction of the dirty BCS model is shown as a dashed line which goes to zero at $T_{\mu }$. The lower-frequency data show a significant deviation from the mean-field behavior. The temperature where $T_{\theta }$ acquires a frequency dependence can be identified with $T_{\mathit{KTB}}$.

Figure 3

(Color online) The dc magnetoresistance isotherms showing a power-law dependence on the applied field for temperature $T=0.236$, 0.499, 0.749, 0.999, 1.250, and $1.497\mathrm{K}$. The dashed line is a linear fit.

Figure 4

(Color online) Temperature dependence of ${\sigma }_1$ at 9, 11, 22, and $106\mathrm{GHz}$. Note the shift in the peak to higher temperatures and the decrease in its height with increasing frequency.

Figure 5

(Color online) $\tau$ vs temperature at various given frequencies extracted via the simple relaxation model in Eq. (7). Also shown is the Ginzburg-Landau result using $T_{c0}=2.28\mathrm{K}$. The dashed horizontal lines mark the time scales set by the various probing frequencies, $1∕\omega$. Notice how the $\tau$ levels off at lower temperatures, indicating the freezing out of amplitude fluctuations.