The Poole-Frenkel Effect

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. 

The Poole-Frenkel effect (Poole, 1916, 1917 Frenkel, 1938, 1938a) is the field-induced reduction in the ionization energy required to separate an electron from a fixed positive charge. Although largely discounted in recent years, arguments 

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The dependence of the drift mobility p on the electric field is represented by formula p (p-E1/2/kTcf) which corresponds to the Pool-Frenkel effect. The good correspondence between experimental and theoretical quantity for Pool-Frenkel coefficient 3 was obtained. But in spite of this the interpretation of the drift mobility in the frame of the Coulombic traps may be wrong. The origin of the equal density of the positive and negative traps is not clear. The relative contribution of the intrinsic traps defined by the sample morphology is also not clear [17,18]. This is very important in the case of dispersive transport. A detailed analysis of the polymer polarity morphology and nature of the dopant molecules on mobility was made by many authors [55-58]. 

We treat here the case of shallow Gaussian traps. The equations reduce to those applicable to single level traps by making the standard deviation at = 0. If the Poole-Frenkel effect (PEE) is included the J-V relation for a device containing shallow traps is given... 

Here is the hole trap density, g is the degeneracy factor (taken as unity in the calculations), Etp is the ionization energy of the hole traps, Ev is the valence band edge (i.e. HOMO), Ny is the effective density of states for holes, at is the standard deviation of the Gaussian distribution of traps, and the factor expOS /F/fcr) arises due to the Poole-Frenkel effect. Analytical solution of (3.58) and (3.60) can not be obtained. Numerically computed results are shown in Fig. 3.29. 

Fig. 3.31. Schematic illustration of the lowering of trap ionization energy by the Poole-Frenkel effect.

The electric field at the interface of the oxide and the active layer is large. As discussed earlier, the field assists the ionization of the traps due to the Poole-Frenkel Effect. The trap depth is reduced by an amount /9 /F and the number of trapped carriers is reduced. Following Horowitz and Delannoy [157] we consider the traps at a single level located near the band edge [158] in an n-type polymer. The treatment is quite general and can be extended to p-type polymers quite easily. When PFE is included, Eq. (6.3) changes to,... 

The conductivity and j-U characteristic given by Eq. (187) is similar to the result following the Poole-Frenkel effect on SCL currents [376,377]. Such a situation is clearly not expected in the case of the standard solution for trapping by a discrete set of separated microtraps expressed by Eqs. (167) and (168). This is the case if one extrapolates Eq. (185) to l 0, with = (Neff/N0) exp(—Et/kT). The physical meaning of this extrapolation is that we deal with one discrete trap level (Et), the trap potential being the infinitely sharp point well for which the barrier lowering can be neglected. 

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 Fig. 8.17, the idea behind the Poole-Frenkel effect is illustrated. The basic assumption is that charged trapping states exist which are neutralised immediately... 

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. 

In special cases, inelastic processes contribute too, for example, by the Poole-Frenkel effect [61]. In photoelectrochemistry, finally, the photocurrent is initiated by vertical electron/hole pair generation with following charge separation. 

Schottky devices have recently been fabricated by thermal evaporation of indium on polyaniline, poly(o-anisidine) and poly(aniline-co-or oanisidine), respectively [103]. The values of the rectification ratio, the ideality factor and the barrier height of an indium/poly(o-anisidine) have been experimentally determined as 300,4.41 and 0.4972, respectively. The observed deviation from the Schottky behaviour for these devices seen at higher voltages has been explained in terms of either the Poole-Frenkel effect or due to the presence of a large number of defects containing the trapped charges existing at the indium/poly (aniline-co-or oanisidine) interface. 

By comparing the field dependence of jUj) with that of a, one can distinguish the Poole-Frenkel effect from the mechanisms which affect only jUj). Marshall and Owen find o (F) to rise somewhat less rapidly with field than )LId- DeWit and Crevecoeur (1971), on the other hand, find ao exp (F/Fq) and a considerably smaller field dependence than that observed by Marshall and Owen. 

The field dependence of Eq. (5.67) is commonly attributed to the Poole-Frenkel effect (Frenkel (1938)) and derived in the following manner. Let us assume the concentration of mobile carriers is governed by the ionization of donors. The Coulomb potential of each donor is modified by the field F in the x-direction as follows... 

The Poole-Frenkel effect is the high-field assisted thermally-activated escape from traps. The derivation of an expression for the rate of escape from traps is similar to the derivation of the Schottky equation. Because the charge on the trap is fixed in position, the attractive force is proportional to l/x instead of to l/(2x). This gives an apparent difference of 2 in the slope of the plot of log / vs. E as compared with the Schottky mechanism. 

The usual expression for the Poole-Frenkel effect is thus the rate of escape from traps. This gives the current in the situation where only a small fraction of traps are ionized at any given moment, and where the field in the electrically neutral oxide controls the current. In a thick enough film the ionized traps near the interface will provide space charge which will accumulate until the field across the interface is increased to the value necessary to keep the rate of injection at the required level. With a thin enough film... 

In addition to the Poole-Frenkel effect and the field-induced tunneling from traps to conduction band states, the Zener effect (field-induced transitions from valence band to conduction band) and various forms of avalanche breakdown effects, can give a bulk conductivity rising sharply with field. These effects are difficult to assess in the present systems, because little is known about the electronic states in amorphous oxides, the electronic transport process, or the lattice vibration spectrum. 

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