Conductivity near the percolation threshold

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 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 ${\displaystyle p}$ 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.

Conductor-insulator mixture

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

${\displaystyle \sigma _{DC}(p)\propto \sigma _{m}(p-p_{c})^{t}}$

for ${\displaystyle p>p_{c}}$

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 ${\displaystyle \sigma _{m}=\infty }$. This case is useful for the description below the percolation threshold:

${\displaystyle \sigma _{DC}(p)\propto \sigma _{d}(p_{c}-p)^{-s}}$

for ${\displaystyle p<p_{c}}$

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]

${\displaystyle \sigma (p)\propto \sigma _{m}|\Delta p|^{t}\Phi _{\pm }\left(h|\Delta p|^{-s-t}\right)}$

with ${\displaystyle \Delta p\equiv p-p_{c}}$ and ${\displaystyle h\equiv {\frac {\sigma _{d}}{\sigma _{m}}}}$

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

${\displaystyle \sigma _{DC}(p_{c})\propto \sigma _{m}\left({\frac {\sigma _{d}}{\sigma _{m}}}\right)^{u}}$

with ${\displaystyle u={\frac {t}{t+s}}}$

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]

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

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:

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

References

1. Efros, A. L.; Shklovskii, B. I. (1976). "Critical Behaviour of Conductivity and Dielectric Constant near the Metal-Non-Metal Transition Threshold". Phys. Status Solidi B. 76 (2): 475–485. doi:10.1002/pssb.2220760205.
2. Clerc, J. P.; Giraud, G.; Laugier, J. M.; Luck, J. M. (1990). "The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models". Adv. Phys. 39 (3): 191–309. doi:10.1080/00018739000101501.
3. Bergman, D. J.; Stroud, D. (1992). "Physical Properties of Macroscopically Inhomogeneous Media". In H. Ehrenreich und D. Turnbull (ed.). Solid State Physics. 46. Academic Press inc. doi:10.1016/S0081-1947(08)60398-7.
4. Kumar, Ankush; Vidhyadhiraja, N. S.; Kulkarni, G. U . (2017). "Current distribution in conducting nanowire networks". Journal of Applied Physics. 122 (4): 045101. doi:10.1063/1.4985792.

See also

• Percolation theory