Percolation Behavior When the Interparticle Conduction Is by Tunneling

Percolation Behavior When the Interparticle Conduction Is by Tunneling 

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 

We define then the average critical local conductance by gc=g(fc) in Eq. (5.12). For simplicity, let us consider the case that is commonly discussed in the literature ]6], that is, the b/rc 0 limit. One also assumes then that tc so that, practically, for all the interparticle distances r rc, g = 0 (blocked conductance), while for all r tc distances one assumes that g=oo (shorts). One can show, as expected, that the 

We have then that the hopping model, while frequently called the percolation model [6], is essentially a percolation threshold model that is not associated a priori with the critical behavior of the conductivity as in percolation theory (Eq. (5.6)). In passing we further note that in the limiting case of a high particle density (i.e., Tc —2b-C2i)) but still assuming a uniform N (see Section 5.3), the combination of Eqs. (5.12) and (5.13) yields that log(cr) varies as rather than the above log(cr) oc result of the dilute Nlimit [11] that is given by Eq. (5.14). This yields that the conductance of the system, as a function of x, will be given by [ 11 ] 

While a more accurate expression has been given for t — tyii [36], we will consider here this more simple transparent equation in what follows. The fact that both Eqs. (5.14) and (5.16) depend on the interpartide distances suggests that when a varies with N, the dominant effect in the case of tunneling, when there is an appropriate interpartide distance distribution, will be the variation of the local conductances rather than the connectivity of the system. 

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