Percolation tunneling

Equation (8-11) can be simplified and the tunnel feature directly disclosed if 1) only the shortest tunneling percolation path through nearest-neighbour interactions is important and 2) all the intermediate group exchange interactions and energy gaps take the same values, p and As, respectively, i.e. = p and s - Sk oi - k = As. Eqn. (8-11) then reduces to... 

Rubin, Z. et al. (1999) Critical behavior of the electrical transport properties in a tunneling-percolation system. Phys. Rev., B59 (19), 12196-12199. 

Vionnet-Menot S, Grimaldi C, Maeder T, Strassler S, Ryser P (2005) Tunneling-percolation... 

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. 

The electron transport properties described earlier markedly differ when the particles are organized on the substrate. When particles are isolated on the substrate, the well-known Coulomb blockade behavior is observed. When particles are arranged in a close-packed hexagonal network, the electron tunneling transport between two adjacent particles competes with that of particle-substrate. This is enhanced when the number of layers made of particles increases and they form a FCC structure. Then ohmic behavior dominates, with the number of neighbor particles increasing. In the FCC structure, a direct electron tunneling process from the tip to the substrate occurs via an electrical percolation process. Hence a micro-crystal made of nanoparticles acts as a metal. 

In some of the metal-insulator transitions discussed here the use of classical percolation theory has been used to describe the results. This will be valid if the carrier cannot tunnel through the potential barriers produced by the random internal field. This may be so for very heavy particles, such as dielectric or spin polarons. A review of percolation theory is given by Kirkpatrick (1973). One expects a conductivity behaving like... 

When a film is very thin, it may not be continuous, and conduction is subject to the percolation effect, whereby charge migrates by hopping or tunneling between island sites [50,51]. Such a process is activation controlled, and such thin films do not obey Ohm s law. The activation energy can be decreased by the presence of an applied electric field, making development of a rigorous theory difficult. The resistivity can be expressed by the relationship [5]... 

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).

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. 

A simple consideration of granular metals in the framework of the classic percolation theory when granules are treated as metal balls, embedded into insulating material, appears to be very limited. Taking into account quantum effects and, first of all, possibility of the tunnel transitions between nanogranules leads to the change in parameters of the percolation theory and even to diminishing of the percolation threshold [15,38,39]. Even in... 

Attempts to take into account both localization and percolation or, in other words, to allow for quantum effects in percolation go back to Khmel-nitskii s pioneer paper [68]. The experimental attempts to study quantum effects in conductivity close to the percolation threshold have been undertaken in Refs. [69-71]. The physical sense of these results is stated in Ref. [71] and could be described as follows. The percolation cluster is non-uniform it includes both big conductive regions ( lakes ) and small regions (weak links or bottlenecks) which connect lakes to each other. On approaching the percolation threshold from the metallic side of the transition, these weak links become thinner and longer, and at x = xc the cluster breaks or tears into pieces just in such areas. As a result, exactly these conditions start to be sufficient for the electron localization. Thus, a percolation provokes an Anderson localization in bottlenecks of the percolation cluster. Sheng and collaborators [36,37,72] tried to take into account the influence of tunneling on conductivity for systems in the vicinity of the percolation transition. Similar attempts have been made in papers [38,56]. The obtained results prove that the possibility of tunneling shifts the percolation threshold toward smaller x values and affects material properties in its vicinity. 

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... 

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 ... 

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. ... 

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