Conductivity near the percolation threshold

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 ν and β (see Percolation critical exponents).

With these critical exponents we have the correlation length, ξ

ξ ( p ) ∝ ( p c − p ) − ν

and the percolation probability, P:

P ( p ) ∝ ( p − p c ) β

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 σ m . And the dielectric component with conductivity σ 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 σ d = 0 . This case describes the behaviour, if the percolation threshold is approached from above:

σ D C ( p ) ∝ σ 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 σ m = ∞ . This case is useful for the description below the percolation threshold:

σ D C ( p ) ∝ σ 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:

σ ( p ) ∝ σ m | Δ p | t Φ ± ( h | Δ p | − s − t )

with Δ p ≡ p − p c and h ≡ σ d σ m

At the percolation threshold, the conductivity reaches the value:

σ D C ( p c ) ∝ σ m ( σ d σ m ) u

with u = 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:

Dielectric constant

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

ϵ 1 ( ω = 0 , p ) = ϵ 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 σ m = 1 / R for the occupied bonds and by perfect capacitors with conductivity σ d = i C ω (where ω represents the angular frequency) for the non-occupied bonds. Now the scaling law takes the form:

σ ( p , ω ) ∝ 1 R | Δ p | t Φ ± ( i ω ω 0 | Δ p | − ( s + t ) )

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

τ ∗ = 1 ω 0 | Δ p | − ( s + t )

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