ac Hopping Conduction at Extreme Disorder Takes Place on the Percolating Cluster

Simulations of the random barrier model show that ac currents at extreme disorder are carried almost entirely by the percolating cluster slightly above threshold; thus contributions from isolated low-activation-energy clusters are negligible. The effective medium approximation in conjunction with the Alexander-Orbach conjecture lead to an excellent analytical fit to the universal ac conductivity with no nontrivial fitting parameters.

Figure 1

Results for the ac conductivity at $\beta \equiv E_0/k_{B}T=320$ in rationalized units [8]; in these units the frequency marking onset of ac conduction, ${\omega }_m$, is of order ${10}^{-38}$. Ten independent $96\times 96\times 96$ samples were simulated. (a) Real part of $\sigma (\omega )$ with two cutoff’s: $k=3.2$ and $k=12.8$, averaged over the ten samples. The dashed line is the prediction of Eq. (2) scaled to the $k=12.8$ data. (b) Relative deviation from $k=12.8$ as a function of frequency plotted for each of the ten independent samples.

Figure 2

Contribution from isolated clusters for the real part of the ac conductivity, ${\sigma }_{\mathrm{IC}}^{\prime }(\omega )$, relative to ${\sigma }^{\prime }(\omega )$ as a function of the scaled real part of the conductivity, ${\tilde{\sigma }}^{\prime }(\omega )\equiv {\sigma }^{\prime }(\omega )/\sigma (0)$ (cutoff: $k=6.4$). The two dashed lines mark the relative masses of isolated clusters, i.e., low-activation-energy links outside the fat percolation cluster. Their contribution to the conductivity is much smaller than their relative mass, showing that the dominant part of the ac conduction takes place on the fat percolation cluster.

Figure 3

Testing Eqs. (2, 3). Data represent averaging over 100 independent $96\times 96\times 96$ samples ($\beta =320$, cutoff: $k=6.4$) and 100 independent $64\times 64\times 64$ samples ($\beta =160$, cutoff: $k=6.4$). For both temperatures the frequency was empirically scaled such that $\tilde{\sigma }=1+\tilde{\omega }$ in the low-frequency limit, where $\tilde{\sigma }\equiv \sigma (\omega )/\sigma (0)$, $\tilde{\omega }\equiv \omega \Delta ϵ{ϵ}_0/\sigma (0)$ [8]. (a) $|\ln \tilde{\sigma }|$ as a function of $|\tilde{\omega }/\tilde{\sigma }|$ in a log-log plot. (b) The apparent exponent $d\ln (|\ln \tilde{\sigma }|)/d\ln |\tilde{\omega }/\tilde{\sigma }|$ as a function of scaled frequency. A crossover from fractal behavior (exponent $2/3$) to homogeneous behavior (exponent one) is clearly visible. (c) The real and imaginary parts of the scaled conductivity compared to Eq. (3) (full lines).