Tunneling and Nonuniversality in Continuum Percolation Systems

The values obtained experimentally for the conductivity critical exponent in numerous percolation systems, in which the interparticle conduction is by tunneling, were found to be in the range of $t_0$ and about $t_0+10$, where $t_0$ is the universal conductivity exponent. These latter values are, however, considerably smaller than those predicted by the available “one-dimensional”-like theory of tunneling percolation. In this Letter, we show that this long-standing discrepancy can be resolved by considering the more realistic “three-dimensional” model and the limited proximity to the percolation threshold in all the many available experimental studies.

Figure 1

Left panel: Our calculated EMA conductance $G$ as a function of $p-p_c$ for dot particles ($b=0$) with a nearest-neighbor distribution that is given by Eq. (7). The different cases refer to different values of the parameter $\alpha =1-d/(a-b)$. Right panel: The same as for the left panel but for hard-core spheres with diameter $b>0$. The cases shown have the same $\alpha$ values as those in the left panel.

Figure 2

The local transport exponent $t(p)$, Eq. (9), as derived from the calculated data shown in Fig. 1.