Conductivity near the percolation threshold
Conductivity near the percolation threshold in physics, occurs in a mixture between a dielectric and a metallic component. The conductivity formula_1 and the dielectric constant formula_2 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. 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 formula_3 or not occupied with probability formula_4. There exists a critical value formula_5. For occupation probabilities formula_6 an infinite cluster of the occupied bonds is formed. This value formula_5 is called the percolation threshold. The region near to this percolation threshold  can be described by the two critical exponents formula_8 and formula_9 (see Percolation critical exponents). With these critical exponents we have the "correlation length", formula_10 formula_11 and the "percolation probability", P: formula_12 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 formula_13. And the dielectric component with conductivity formula_14 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 formula_15. This case describes the behaviour, if the percolation threshold is approached from above: formula_16 for formula_6 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 formula_18. This case is useful for the description below the percolation threshold: formula_19 for formula_20 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: formula_21 with formula_22 and formula_23 At the percolation threshold, the conductivity reaches the value: formula_24 with formula_25 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: formula_26 The R-C model. Within the R -C model, the bonds in the percolation model are represented by pure resistors with conductivity formula_27 for the occupied bonds and by perfect capacitors with conductivity formula_28 (where formula_29 represents the angular frequency) for the non-occupied bonds. Now the scaling law takes the form: formula_30 This scaling law contains a purely imaginary scaling variable and a critical time scale formula_31 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 (formula_32) can be written in terms of edge (wire) density (formula_33), resistivity (formula_34), width (formula_35) and thickness (formula_36) of edges (wires) as: formula_37