# 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.[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 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 $p < p_c$

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:[2]

$\sigma(p) \propto \sigma_m |\Delta p|^t \Phi_{\pm} \left(h|\Delta p|^{-s-t}\right)$

with $\Delta p \equiv p - p_c$ and $h \equiv \frac{\sigma_d}{\sigma_m}$

At the percolation threshold, the conductivity reaches the value:[1]

$\sigma_{DC}(p_c) \propto \sigma_m \left(\frac{\sigma_d}{\sigma_m}\right)^u$

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.[1] Clerc et al.[2] Bergman et al.[3]
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:[1]

$\epsilon_1(\omega=0,p) = \frac{\epsilon_d}{|p-p_c|^s}$

## 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 $\sigma_d = i C \omega$ (where $\omega$ represents the angular frequency) for the non-occupied bonds. Now the scaling law takes the form:[2]

$\sigma(p, \omega) \propto \frac{1}{R} |\Delta p|^t \Phi_{\pm} \left(\frac{ i \omega}{\omega_0}|\Delta p|^{-(s+t)}\right)$

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.[2]

## References

1. ^ a b c d A. L. Efros and B. I. Shklovskii, Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non-Metal Transition Threshold, Phys. Status Solidi B 76, 475 (1976)
2. ^ a b c d J. P. Clerc, G. Giraud, J. M. Laugier and J. M. Luck, The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models, Adv. Phys. 39, 191 (1990)
3. ^ D. J. Bergman and D. Stroud, Physical Properties of Macroscopically Inhomogeneous Media, hg. von H. Ehrenreich und D. Turnbull, Bd. 46, Solid State Physics (Academic Press inc., 1992)