Theory of hopping conduction

Shante s theory of hopping conduction in highly anisotropic disordered materials ... 

The theory of hopping conduction described above differs greatly in its fundamental structure from typical theories of conduction which start with electrons in... 

Theory of Hopping Conduction in Very Narrow Band Poiymers and in Disordered Poiymers with Appiications... 

Finally, we note that after determining the primary jump rates between different sites of a multicomponent disordered polypeptide, these data can be used as input in a stochastic (random walk) theory of hopping conductivity in random media. Since, however, this theory has not yet been applied to any real polymers, we do not describe this method here but refer to the appropriate papers. ... 

Shortly after Holstein published his work on polarons in molecular crystals. Miller and Abrahams introduced a very useful description of hopping conduction in terms of a phonon-assisted electron tunneling process [38]. Miller-Abrahams theory does not include the polaronic effect. Nevertheless it... 

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... 

Treatment of hopping conduction in terms of Marcus theory stresses the importance of the reorganization energy term as a major factor in determining the charge... 

The differentiation between whether delocalized (band theory) conductivity or diffusionlike hopping conductivity best explains experimental conductivity results is not always easy in practice but can be made by a comparison of the theoretical expressions for electrical conductivity and mobility of the charge carriers in a solid. 

This notion of occasional ion hops, apparently at random, forms the basis of random walk theory which is widely used to provide a semi-quantitative analysis or description of ionic conductivity (Goodenough, 1983 see Chapter 3 for a more detailed treatment of conduction). There is very little evidence in most solid electrolytes that the ions are instead able to move around without thermal activation in a true liquid-like motion. Nor is there much evidence of a free-ion state in which a particular ion can be activated to a state in which it is completely free to move, i.e. there appears to be no ionic equivalent of free or nearly free electron motion. 

Such a mechanism is not incompatible with a Haven ratio between 0.3 and 0.6 which is usually found for mineral glasses (Haven and Verkerk, 1965 Terai and Hayami, 1975 Lim and Day, 1978). The Haven ratio, that is the ratio of the tracer diffusion coefficient D determined by radioactive tracer methods to D, the diffusion coefficient obtained from conductivity via the Nernst-Einstein relationship (defined in Chapter 3) can be measured with great accuracy. The simultaneous measurement of D and D by analysis of the diffusion profile obtained under an electrical field (Kant, Kaps and Offermann, 1988) allows the Haven ratio to be determined with an accuracy better than 5%. From random walk theory of ion hopping the conductivity diffusion coefficient D = (e /isotropic medium. Hence for an indirect interstitial mechanism, the corresponding mobility is expressed by... 

Eq. (18) is usually attributed to the variable-range hopping conductivity in presence of the Coulomb gap [34]. However, the analysis [72,75,76] shows that it is unrealistic explanation for the case of nanocomposites, because to fit experimental value on the basis of this theory one has to assume that the length of a single hop is less than the size of granules D and the electron... 

Neither of the above theories takes the macrostructure of carbon black into account in any quantitative way, although in practice this is crucial in selecting a material for a particular application. Sheng (1980) was, however, able to develop the first of these theories and extend its range of applicability. He noted the well-established model for hopping conduction by tunnelling... 

The standard assumption of the theory of the AC resonance hopping conductivity that the energies, Ej- of the sites in isolated pairs are not... 

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