Microscopic hopping transport

The value of the last integral is readily shown to be kB T, so that 

The total probability, Ptot, that the particle will be excited to a total energy exceeding the value 17max s then given by 

If all the energy given to the particle were associated with the forward component of the momentum, then Ptot could be substituted for PiUaux) n ecln- (65). For a three-dimensional system, however, there are also transverse momentum components py and pz in addition to the forward component px, since 

The second aspect of this problem is that the area densities may be different on the two sides of a given barrier. This difference in the area densities would lead to a difference in the forward and reverse components of the hopping current, even if the height of the barrier were the same as viewed from the forward and reverse directions. For example, if the barrier heights are W(f) and W(r) as viewed from the two directions, respectively, and the area density for forward hopping is denoted by nif) and the area density for reverse hopping is denoted by n(r), then the net current density over the barrier would be given by 

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In principle, i can be treated as the sum of diffusion and drift components, but we need to allow for the fact that disorder in transport energy levels leads to dispersion in the microscopic hopping rates and to charge trapping, which between them lead to diffnsion constants and mobilities that are effectively charge-density-dependent. In addition, the Einstein relation between diffusion constant and mobility is nnlikely to hold (Roichman and Tessler, 2002). 

In the previous section, we put forth evidence for the claim that the presence of point defects can completely alter the observed properties of materials. However, this discussion was set forth without entering into the question of how materials are brought to a state in which there is a given distribution of point defects. Many processes in materials are mediated by diffusion in which mass is transported from one part of the material to another. When viewed from the atomic scale, these diffusive processes may be seen as a conspiratorial effect resulting from the repeated microscopic hopping of particles over time. Our aim in the present section is to examine several different perspectives on diffusion, and to illustrate how a fundamental solution may be constructed for diffusion problems in much the same way we constructed the fundamental solution for elasticity problems in section 2.5.2. 

Flowever, the charge carrier motions in many organic semiconductors are between these two limits. It is thus expected that more sophisticated microscopic charge transport theories need to be developed to unify the concepts of the band-like and hopping transport. Indeed, many work along this line have been performed." It is known, however, that most of those rigorous quantum approaehes are limited to tens of sites because of the numerical convergence problem and computer memory limitations. 

It was pointed out later by Gill [16] that, although the model was able to predict the correct field dependence, and even its correct magnitude, it could be objected that the use of a trapped controlled mechanism assumes a transport in delocalized bands (see section 5.1.4). Experimental values of the microscopic mobility, which range from 10 to 10 , do not agree with such a transport. However, this objection could be removed by the model developed by Jonscher and Ansari [17], which assumes a thermally stimulated hopping transport in an energy distribution of localized states. 

Trap-Controlled Hopping. In trap-controlled hopping, the scenario described for trap-controlled band mobility applies. However, the microscopic mobility is associated now with carriers hopping in a manifold of localized states. Overall temperature and field dependence reflects the complicated convolution of the temperature and field dependence of both the microscopic mobility and the trap kinetic processes. Glearly, the observed behavior can now range from nondispersive to anomalously dispersive behavior as before, depending on the energy distribution of transport-interactive traps. 

Trap-Controlled Hopping. Physical defects that are associated with intermolecular conformations that deviate in some specific way from the normal statistical distribution of molecular arrangements and extrinsic chemical species can generate additional localized states that lie outside of the distribution of bulk states. The bulk states control the primary transport channel, the microscopic mobility, whereas the additional extrinsic states... 

When physical motion is either nonexistent or much slower than electron hopping, charge propagation is fundamentally a percolation process, because the microscopic distribution of redox centers plays a critical role in dictating the rate of charge transport [27-29]. Any self-similarity of the molecular clusters between successive electron hops imparts a memory effect, making the exact adjacent-site cormec-tivity between the molecules important. 

Transport in conducting polyaniline / nylon blends is observed to be independent on the composition, dopant anion, host matrix and structure of the conducting phase. The microscopic transport in all salt networks remains unchanged by the dilution of the salt in the nylon matrix. The conductivity follows a temperature dependence characteristic of generalized variable r e hopping mechanism with % = -1/4. The slope of the temperature dependence increases with dilution, indicating increased disorder in the polyaniline salt. 

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