Tunneling-percolation model

Experimental data relating to the conductivity of composite films with M/SC nanoparticles are described by the classical percolation model in terms of tunnel processes. Chemisorption of chemical compounds on the surface of M/SC nanoparticles in films and the subsequent reactions with participation of chemisorbed molecules change the concentration of conducting electrons and/or barriers for their tunnel transfer between the nanoparticles with the result of strong influence on the film conductivity. Such films are used as conductometric sensors for detecting various substances in an atmosphere. 

Charge transport in the accumulation channel is described by the percolation model [24] based on thermally activated tunneling of holes between localized states in an exponential density of states, described in Section 13.2.2. In the accumulation regime this Variable Range Hopping (VRH) model yields a gate-voltage dependent field-effect mobility of the form ... 

PEDOT PSS, 20-24, 20-45-20-46, 20-49 substituted PEDOTs, 20-25 PEDOT/PSS nanowires, 16-5 Peierls instability, 17-3 Pentacene, 2-2, 2-15, 2-16, 2-17, 2-18 Percolation models, 16-2 Percolation transition, 15-12 Pemigraniline, 7-25 Perturbation, 19-3, 19-11, 19-14 Phase diagram, 17-11-17-12, 17-15, 17-24-17-27 Phenyl-fused EDOT, 13-15-13-16 Phonon scattering, 15-13, 15-30 Phonon-assisted tunneling, 16-4 Phonon-induced delocalization, 15-8, 15-12—15-13, 15-41, 15-51, 15-69 Phosphonic add, 9-18, 9-19... 

Here, is an effective overlap parameter that characterizes the tunneling of chaiges from one site to the other (it has the same meaning as a in Eq. (14.60)). T0 is the characteristic temperature of the exponential distribution and a0 and Be are adjustable parameters connected to the percolation theory. Bc is the critical number of bonds reached at percolation onset. For a three-dimensional amorphous system, Bc rs 2.8. Note that the model predicts a power law dependence of the mobility with gate voltage. 

The percolation theory [5, 20-23] is the most adequate for the description of an abstract model of the CPCM. As the majority of polymers are typical insulators, the probability of transfer of current carriers between two conductive points isolated from each other by an interlayer of the polymer decreases exponentially with the growth of gap lg (the tunnel effect) and is other than zero only for lg < 100 A. For this reason, the transfer of current through macroscopic (compared to the sample size) distances can be effected via the contacting-particles chains. Calculation of the probability of the formation of such chains is the subject of the percolation theory. It should be noted that the concept of contact is not just for the particles in direct contact with each other but, apparently, implies convergence of the particles to distances at which the probability of transfer of current carriers between them becomes other than zero. 

According to this model the tunnel current arises due to formation of the infinite percolation cluster of contacting external spheres with i d. The... 

Fig. 10.6. Percolation cluster model of tunnel current in composite film containing M/SC nanoparticles (a) two-sphere model of spherical M/SC nanoparticle of radius Rq surrounded by outer sphere (radius Rd) that is defined by a degree of electron delocalization extending the nanoparticle and characterizes electron tunneling (see text) (b) the distribution of conductivity G(r) over the two-sphere particle (c) two-dimensional pattern of cluster from overlapping two-sphere particles (overlapping areas of outer spheres are shown).

One can then propose that, as for polyaniline, a heterogeneous model for conduction [27, 28] can also describe transport in PANI/SWNT below percolation. In contrast, Kaiser model describes transport above percolation with the system s metallic character reflected in a linear temperature dependence while lacking the exponential term associated with tunneling through conduction barriers [25], The... 

The temperature conductivity data of iodine doped PMQ3 (1.14 I per repeat from element analysis and weight gain.)(Figure 14) can be fitted not only with Equation 4, but also to a model of fluctuation-induced carrier tunneling (Equation 5). Either of them can be expected only when the metallic domain concentration increases to above the percolation threshold from broken paths. ... 

Most probably, the best known approach to the conduction by tunneling in random systems is the one used in the theories of hopping [6, 30]. However, as discussed below, this approach does not predict a percolation-like behavior as given by Eq. (5.6) with which we were concerned above. One starts the consideration of the hopping model by recalling the exponential decrease of the interparticle conductance g with the distance r, so that... 

As Balberg notes in a review The electrical data were explained for many years within the framework of interparticle tunneling conduction and/or the framework of classical percolation theory. However, these two basic ingredients for the understanding of the system are not compatible with each other conceptually, and their simple combination does not provide an explanation for the diversity of experimental results [17]. He proposes a model to explain the apparent dependence of percolation threshold critical resistivity exponent on structure of various carbon black composites. This model is testable against predictions of electrical noise spectra for various formulations of CB in polymers and gives a satisfactory fit [16]. 

CNTs in polymer-CNT composites are efficiently debundled and isotropically dispersed in polymer matrices, the efficient interaction between CNT and polymer provides good dispersion and a low percolation threshold, but only relatively low conductivity near and above percolation, frequently around 10 s cm is achieved at close to 2wt% CNT loading [70, 71], The polymer layer in the intemanotube connections is supposed to be the highest resistance section in the electrical pathway. This polymer layer is a barrier to efficient carrier transport between CNTs, and models for conductivity based on fluctuation-induced tunneling have been proposed [72]. A power law related to percolation theory can be used to model conductivity in the following form ... 

Clerc J P, Giraud G, Laugier J M and Luck J M (1990) The electrical conductivity of binary disordered systems, percolation clusters, fractals and related models, Adv Phys 39 191-209. Balberg I (1987) Tunneling and nonuniversal conductivity in composite materials, Phys Rev Lett 59 1305-1308. 

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