Phonon assisted hopping, tunneling

For g(x) to be proportional to 1/x, it requires the relaxation time to be an exponential function of some random variable such that x = xq exp ( ), where itself has a flat distribution. It means that ( ) = constant, and rt(x) = n ). (d /dx) oc x. If a, the polarisability, is also a function of then it can lead to a sub-linear frequency dependence of a (co). The functional form given for the variation of x can arise from two different relaxation mechanisms. The first is a classical barrier hopping, in which two energetically favourable sites like in a double well potential are separated by a barrier fV and = W/kT. The second mechanism is a phonon assisted quantum tunneling through a barrier, which separates two equilibrium positions, in which case = 2aR, where a is the localization length and R is the separation between the sites. In the first case, by treating JV as independent of R, it has been shown (Poliak and Pike, 1972) that... 

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... 

This suggests that the rate determining steps are associated with tunnel 1 ling across barriers (chain ends or to another chain) rather than phonon assisted hopping processes. 

At energies near the electrons are confined to channels and are thus not typical three dimensional free electrons. But, we have ignored quantum mechanics and in reality the electrons can tunnel from one channel to another and between various allowed regions. Furthermore as already indicated if we allow the electrons to hop there will be an additional component due to thermally assisted processes (or phonon... 

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