Poole-Frenkel model

The release of the trapped carriers may be stimulated by the electric field. In the Pool-Frenkel model the decrease of the potential barrier of the trapped electron by the electric field (AEFp) obeys the formula... 

The main experimental results of charge carrier generation in PVC was explained in the frame of the Pool-Frenkel model [28-30]. The dependence of the recombination time on electric field was due to the change of the mobility in the electric field. Germinate recombination of the electron-hole pairs was investigated by means of luminescence decay characteristics [31]. 

Santos-Lemus and Yartsev (1995) analyzed earlier data on NIPC doped PC by Santos-Lemus (1983) and Santos-Lemus and Hirsch (1986). The temperature dependence was analyzed by a Poole-Frenkel model. From plots of the activation energy versus E /2, zero-field energies were determined for different NIPC concentrations. The values were in the range of 0.60 to 0.73 eV. From a plot of the zero-field activation energy versus p K a value of 0.98 eV was derived for the potential barrier of an isolated NIPC molecule. 

Summarizing the results obtained for charge transport in conjugated polymers, the Poole-Frenkel model seems to be a good description [126,127]. This model of transport assumes the existence of two types of carrier movement, one in conductive states (positive polarons) and the other via traps (negative polarons). 

Figure 50 Quenching efficiency (<5) as a function of dc electric field applied to the electrophosphorescent (EPH) and phosphorescent system. The curves are fits to the Poole—Frenkel (see lower inset) and Onsager (see upper inset) models for charge pair dissociation in external electric fields. The quenching efficiency is defined as a relative difference between the emission efficiency at a given field F[0(F)] and at a field F0[4>(F0)] where a decrease in the EPH efficiency becomes observed (<) (F0) (7 1)]/

Figure 101 Calculated Poole-Frenkel plots according to the correlated disorder model for different values of o jkT (from top curve downward). The calculations according to the Gaussian disorder model with

A limitation of the model of Abkowitz and coworkers is the assumption that the thermal generation process is field independent. A depletion model that includes a field-dependent thermal generation process has been proposed by Kasap et al. (1987). The model is based on the thermal generation of carriers from localized states via a Poole-Frenkel lowering of a Coulomb potential. The model assumes that the distribution of localized states is exponential in energy. According to Kasap et al., the field dependence of the thermal generation process is... 

Poole-Frenkel coefficient. A basic prediction of the model is the dependence of the discharge time tj on the initial potential, V0. At low fields, when VQ < Vpp (Vpp = EppL), log tj versus log VQ has a slope of Up, in agreement with Eq. (18). When VQ increases beyond Vpp, tj = VQU2p At high fields, when VQ Vpp, tj becomes constant. The basic difference in the models of Abkowitz and coworkers and Kasap et al. is the field dependence of the discharge time tj. 

Geminate recombination is the recombination of an electron with its parent cation. Geminate recombination models are premised on the assumption that the formation of a free electron-hole pair involves the dissociation of an intermediate charge-transfer state. Early models were based on the Poole-Frenkel effect. Most recent models have been based on theories due to Onsager. 

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. 

There have been many attempts to modify the Poole-Frenkel effect to make it more realistic. Three-dimensional models have been described by Jonscher... 

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. 

Santos-Lemus and Hirsch (1986) measured hole mobilities of NIPC doped PC. Over a range of concentrations, fields, and temperatures, the transport was nondispersive. The field and temperature dependencies followed logn / El/2 and -(T0IT)2 relationships. For concentrations of less than 40%, a power-law concentration dependence was reported. The concentration dependence was described by a wavefunction decay constant of 1.6 A. To explain a mobility that shows features expected for trap-free transport with a field dependence predicted from the Poole-Frenkel effect, the authors proposed a model based on field-enhanced polaron tunneling. The model is based on an earlier argument of Mott (1971). 

In order to predict absolute dielectric strengths we need to have more detailed information than is yet available about electronic states and mobilities in polymers. For the present we can only conclude that there is satisfactory agreement between the form of the theoretical results, based on a rather general electronic model, and the best experimental results. To the extent that the model is a very reasonable one, we can say that we can understand intrinsic breakdown behaviour. Measurement of pre-breakdown currents, especially with pointed electrodes which impose regions of very high field strength at their tips when embedded in the material, suggests that electronic carrier production either by injection from the electrodes (Schottky emission) or from impurities (Poole-Frenkel effect) may play a part in the breakdown process. More work is required, however, before this can be fully understood. 

SCLC given by Equation (8.50). The analysis of Many and Rakavy (1962) shows that the peak current is approximately 1.2 times the SCLC and that the maximum occurs at approximately 0.8 times the carrier transit time. Goldie (1999) has incorporated the experimentally observed Poole-Frenkel field dependence into this model and finds a range of possible numerical factors for the current maximum of from 1-1.2 times the SCLC and 0.7-0.8 times the transit time. Experimental data come close to these model profiles, see Abkowitz et al. (1994), Goldie (1999). 

A more searching analysis of the Poole-Frenkel mechanism performed for polysiloxane on the basis of the Hill model ( ) showed that charge carrier emission should proceed from the isolated Coulomb centre and should take place in the hemisphere related to that centre. The depth of the centres, determined form the activation dependence of the temperature, was =... 

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