# Conductivity near the percolation threshold

In a mixture between a dielectric and a metallic component, the conductivity $\sigma$ and the dielectric constant $\epsilon$ of this mixture show a critical behavior if the fraction of the metallic component reaches the percolation threshold. The behavior of the conductivity near this percolation threshold will show a smooth change over from the conductivity of the dielectric component to the conductivity of the metallic component and can be described using two critical exponents s and t, whereas the dielectric constant will diverge if the threshold is approached from either side. To include the frequency dependent behavior, a resistor-capacitor model (R-C model) is used.

## Geometrical percolation

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 $p$ or not occupied with probability $1-p$ . There exists a critical value $p_{c}$ . For occupation probabilities $p>p_{c}$ an infinite cluster of the occupied bonds is formed. This value $p_{c}$ is called the percolation threshold. The region near to this percolation threshold can be described by the two critical exponents $\nu$ and $\beta$ (see Percolation critical exponents).

With these critical exponents we have the correlation length, $\xi$ $\xi (p)\propto (p_{c}-p)^{-\nu }$ and the percolation probability, P:

$P(p)\propto (p-p_{c})^{\beta }$ ## Electrical percolation

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

### Conductor-insulator mixture

In the case of a conductor-insulator mixture we have $\sigma _{d}=0$ . This case describes the behaviour, if the percolation threshold is approached from above:

$\sigma _{DC}(p)\propto \sigma _{m}(p-p_{c})^{t}$ for $p>p_{c}$ Below the percolation threshold we have no conductivity, because of the perfect insulator and just finite metallic clusters. The exponent t is one of the two critical exponents for electrical percolation.

### Superconductor–conductor mixture

In the other well-known case of a superconductor-conductor mixture we have $\sigma _{m}=\infty$ . This case is useful for the description below the percolation threshold:

$\sigma _{DC}(p)\propto \sigma _{d}(p_{c}-p)^{-s}$ for $p Now, above the percolation threshold the conductivity becomes infinite, because of the infinite superconducting clusters. And also we get the second critical exponent s for the electrical percolation.

### Conductivity near the percolation threshold

In the region around the percolation threshold, the conductivity assumes a scaling form:

$\sigma (p)\propto \sigma _{m}|\Delta p|^{t}\Phi _{\pm }\left(h|\Delta p|^{-s-t}\right)$ with $\Delta p\equiv p-p_{c}$ and $h\equiv {\frac {\sigma _{d}}{\sigma _{m}}}$ At the percolation threshold, the conductivity reaches the value:

$\sigma _{DC}(p_{c})\propto \sigma _{m}\left({\frac {\sigma _{d}}{\sigma _{m}}}\right)^{u}$ with $u={\frac {t}{t+s}}$ ### Values for the critical exponents

In different sources there exists some different values for the critical exponents s, t and u in 3 dimensions:

Values for the critical exponents in 3 dimensions
Efros et al. Clerc et al. Bergman et al.
t 1,60 1,90 2,00
s 1,00 0,73 0,76
u 0,62 0,72 0,72

## Dielectric constant

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

$\epsilon _{1}(\omega =0,p)={\frac {\epsilon _{d}}{|p-p_{c}|^{s}}}$ ## The R-C model

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

$\sigma (p,\omega )\propto {\frac {1}{R}}|\Delta p|^{t}\Phi _{\pm }\left({\frac {i\omega }{\omega _{0}}}|\Delta p|^{-(s+t)}\right)$ This scaling law contains a purely imaginary scaling variable and a critical time scale

$\tau ^{*}={\frac {1}{\omega _{0}}}|\Delta p|^{-(s+t)}$ which diverges if the percolation threshold is approached from above as well as from below.

## Conductivity for dense networks

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. 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 ($R_{sn}$ ) can be written in terms of edge (wire) density ($N_{E}$ ), resistivity ($\rho$ ), width ($w$ ) and thickness ($t$ ) of edges (wires) as:

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