Conductivity near the percolation threshold in physics, occurs in a mixture between a dielectric and a metallic component. The conductivity ${\displaystyle \sigma }$ and the dielectric constant ${\displaystyle \epsilon }$ of this mixture show a critical behavior if the fraction of the metallic component reaches the percolation threshold.^[1]

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. This behavior 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 in electronic components, 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 ${\displaystyle p}$ 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:

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

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 ${\displaystyle \sigma _{m}}$. 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.

Dielectric constant

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

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

The R-C model

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]

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.^[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:

${\displaystyle R_{sn}={\frac {\pi }{2}}{\frac {\rho }{wt{\sqrt {N_{E}}}}}}$

See also

• Percolation theory

References