Poole-Frenkel formalism

The computer simulation model [30] mentioned above predicts that exp (EIEq), which has been observed in several cases [32-34]. In other cases, however, /a was found to be proportional to E, which is consistent with the Poole-Frenkel formalism of field-lowering of hopping barriers [14, 35]. In many cases, however, the graphic representation cannot differentiate between the models owing to difficulties in assessment of the linearity or nonlinearity of the field dependence over a narrow field range. 

The carrier mobility p is temperature- and field-dependent. Many theories have been developed to explain the temperature dependence, but no comprehensive model is yet available. It is still not clear whedier the charge carrier mobility follows a simple Arrhenius relationship (log p 1/7) as predicted by Gill [30] or if the more complex relationship log p 1/ 7 proposed by Busier et al. [35] is valid. The relationship between the mobility p and the electrical field strength E is equally unclear. Here Gill s model predicts a log pi E dependence which is consistent with a Pool-Frenkel formalism, whereas Bassler s calculations lead to a log pi E dependence. A detailed description of the different models and results obtained by fitting experimental mobility data to those models is beyond the scope of this chapter. It shall only be pointed out here that the main difficulty is the limited range of temperature and electric field in which carrier mobilities can be measured [36]. Additional experi-... 

Unlike the Poole-Frenkel effect, the dipole trap argument does not require high concentrations of charged traps. Further, the problem of small distances between the hopping sites relative to the position of the potential energy maxima, which is a major limitation of Poole-Frenkel arguments, is avoided. The model predicts field and temperature dependencies that are similar to the disorder formalism. The dipole trap model and the disorder formalism both lead to activation energies that are temperature dependent. 

Hole mobilities of p-diethylaminobenzaldehyde diphenylhydrazone (DEH) doped PC were measured by Schein et al. (1986). The field and temperature dependencies were described as logjU PE1/2 and -(T0/T)2. While the field dependencies could not be described by any existing theory, the temperature dependencies were consistent with the disorder formalism. The field dependencies were further investigated by Schein et al. (1989). The measurements were made over an extended range of fields, 8.0 x 103 to 2.0 x 106 V/cm. The results were compared to predictions of models proposed by Bagley (1970), Seki (1974), Facci and Stolka (1986), and a modified Poole-Frenkel argument due to Hill (1967). The only model that agreed with the results was based on the Poole-Frenkel effect. The authors discounted this explanation for reasons cited in Chapter 7. 

Formally, it is possible to derive Eq. (8.85) from the Poole-Frenkel effect To be sure, there are considerations that speak against this model conception. For example, the distance of the potential maximum from the trapping state at typical values of the electric field is about an order of magnitude greater than the usual hopping distance between two molecules of ca. 0.5-1 run (x = 6 run for F = 10 V/cm and e = 4.0). Therefore, the Vf behaviour in the exponent of the mobility cannot simply be explained by the Poole-Frenkel effect. 

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