Polaron and Hopping Models

More recently, a comprehensive model has been developed by Vissenberg and Matters [120] to account for these data. The model is based on a variable rangehopping system with an exponential distribution of localized states (Eq. (14.71)). 

The following expression is obtained for the temperature- and gate voltage-dependent mobility 

a is an effective overlap parameter that characterizes the tunneling of charges from one site to the other (it has the same meaning as a in Eq. (14.60)). To is the characteristic temperature of the exponential distribution and co and are adjustable parameters connected to the percolation theory. Be is the critical number of bonds reached at percolation onset. For a three-dimensional amorphous system, Be 2.8. Note that the model predicts a power law dependence of the mobility with gate voltage. 

The polaron model was also used by Waragai and Hotta [121] to analyze temperature-dependent data on methyl-substituted oligothiophenes. While their data did not cover a sufficient temperature range to discern a clear temperature dependence of the mobility, they noticed that [j. represented a drain voltage dependence, which they attributed to the dependence given by Eq. (14.69). Accordingly, the temperature-dependent mobility could be fitted to a thermally activated law of the form where Etot = fc/2 - ys/F, y being 

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