Electrical Conduction in Nonlinear Composites

Composite systems constitute a large class of naturally occurring or artificially synthesized disordered systems [1]. The systems are microscopically inhomogeneous and disordered but look homogeneous on the macroscopic scale. From the tunnelling electron micrographs (TEM) of such a composite material it can be seen that the typical dimension (ξ) of metallic islands embedded in the insulating matrix are much greater than the atomic size (a) but obviously much smaller than the macroscopic scale length (L): a ≪ ξ ≪ L. The effective conductivity of such a system depends upon the conductivities of the individual phases. For a low volume fraction (p) of the conducting phase, the system as a whole behaves like an insulator since the conducting regions do not form a continuous path through the sample. As p is increased, the conducting regions will in general tend to grow and eventually at a critical volume fraction (pc, called the percolation threshold) the conducting phase percolates through the sample. This may be considered as a classical insulator-to-metal transition or more popularly as a percolation transition. For all p > pc, the system is metallic, and if the conducting phase is Ohmic, so is the whole macroscopic system. Clearly this class of systems may be well described by the geometrical percolation theory. Now if an external voltage is applied across such composite systems (examples include dispersed metallic systems, carbon-black-polymer composites, sulphonated (doped) polyaniline networks etc., which are usually highly structured and give rise to some sort of universal behaviour.) a wide variety of interesting features associated with a nonlinear response emerge. Usually these composites exhibit an unusually low percolation threshold. Qualitatively identical nonlinear I − V (as well as dI/dV (≡ G)against V ) response have been reported [2, 3] both below and above the threshold for many of the composites although the nonlinearity exponent is found to be grossly different in the two regimes. Power-law growth of excess conductance for small V is another general feature of the class of composite systems where noninteger power-law has been observed. This in turn implies a power-law in the I − V relationship for small applied voltage (V ). The G − V curves are seen to saturate for an appropriately high enough voltage below the Joule-heating regime. The typical curve then looks like a nonlinear sigmoidal type function interpolating two linear regimes. Recent experiments on carbon-wax systems [2] as well as many earlier ones on disordered/ amorphous systems [4], find a non-integer power-law behaviour and a saturation in the DC-response as mentioned above. Composite systems show very interesting temperature-dependent conduction properties particularly in the low