Conductivity near the percolation threshold

For describing such a mixture of a dielectric and a metallic component we use the model of bond-percolation. On a regular lattice, the bond between two nearest neighbors can either be occupied with probability ${\displaystyle p}$ or not occupied with probability ${\displaystyle 1-p}$. There exists a critical value ${\displaystyle p_{c}}$. For occupation probabilities ${\displaystyle p>p_{c}}$ an infinite cluster of the occupied bonds is formed. This value ${\displaystyle p_{c}}$ is called the percolation threshold. The region near to this percolation threshold can be described by the two critical exponents ${\displaystyle \nu }$ and ${\displaystyle \beta }$ (see Percolation critical exponents).

With these critical exponents we have the correlation length, ${\displaystyle \xi }$

${\displaystyle \xi (p)\propto (p_{c}-p)^{-\nu }}$

and the percolation probability, P:

${\displaystyle P(p)\propto (p-p_{c})^{\beta }}$

For the description of the electrical percolation, we identify the occupied bonds of the bond-percolation model with the metallic component having a conductivity ${\displaystyle \sigma _{m}}$. And the dielectric component with conductivity ${\displaystyle \sigma _{d}}$ corresponds to non-occupied bonds. We consider the two following well-known cases of a conductor-insulator mixture and a superconductor–conductor mixture.

The dielectric constant also shows a critical behavior near the percolation threshold. For the real part of the dielectric constant we have:[1]

${\displaystyle \epsilon _{1}(\omega =0,p)={\frac {\epsilon _{d}}{|p-p_{c}|^{s}}}}$

Within the R-C model, the bonds in the percolation model are represented by pure resistors with conductivity ${\displaystyle \sigma _{m}=1/R}$ for the occupied bonds and by perfect capacitors with conductivity ${\displaystyle \sigma _{d}=iC\omega }$ (where ${\displaystyle \omega }$ represents the angular frequency) for the non-occupied bonds. Now the scaling law takes the form:[2]

${\displaystyle \sigma (p,\omega )\propto {\frac {1}{R}}|\Delta p|^{t}\Phi _{\pm }\left({\frac {i\omega }{\omega _{0}}}|\Delta p|^{-(s+t)}\right)}$

This scaling law contains a purely imaginary scaling variable and a critical time scale

${\displaystyle \tau ^{*}={\frac {1}{\omega _{0}}}|\Delta p|^{-(s+t)}}$

which diverges if the percolation threshold is approached from above as well as from below.[2]

For a dense network, the concepts of percolation are not directly applicable and the effective resistance is calculated in terms of geometrical properties of network.[4] Assuming, edge length << electrode spacing and edges to be uniformly distributed, the potential can be considered to drop uniformly from one electrode to another. Sheet resistance of such a random network (${\displaystyle R_{sn}}$) can be written in terms of edge (wire) density (${\displaystyle N_{E}}$), resistivity (${\displaystyle \rho }$), width (${\displaystyle w}$) and thickness (${\displaystyle t}$) of edges (wires) as:

${\displaystyle R_{sn}\,=\,{\frac {\pi }{2}}{\frac {\rho }{w\,t\,{\sqrt {N_{E}}}}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}$

• Percolation theory