Advancements of the Frenkel-Poole model

The original model published by Frenkel 1938 [46] was one dimensional. In 1967 Jonscher [80] extended the model to three dimensions, by introducing an appropriate lowering of the Coulombic potential in other directions of the forward hemisphere (around the

11The term in the exponent for the field-dependent barrier lowering including the effect of polarisation in the material isβp

E_int=r _q

πǫ0ǫ^(dyn)r · ^Eext

ǫ^(stat)r

. Hence, ignoring the relative static permittivityǫ^(stat)r could be misinterpreted as too large dynamic permittivityǫ^(dyn)r , which istheusual deviation using the Frenkel-Poole model [163].

applied field) and integrating over the escape probabilities of the complete forward hemisphere. This was only one of several aspects of the publication of Jonscher. Maybe that is why one sometimes finds authors giving Hartke, who performed exclusively that exactly same derivation in 1968 [60], as source of the three-dimensional extension of the Frenkel-Poole model. Also all subsequent models by Hill [64], Ieda et al. [72], Connellet al.[28], Martin at al. [125], Potemski and Wilamowski [151] and Dallacasa and Paracchini [32] are three dimensional.

Besides the obviously more realistic description of the situation in the material, this advancement also addressed one of two major deviations of the original Frenkel-Poole model from the experimentally observed behaviour:the deviation at small fields. Until the works of Hill and Iedaet al.the conduction process at low fields had always been described apart from the mechanism at high fields [64, 72]. Experimental data, on the other hand, show almost identical activation energies for high and low fields [72], indicating that the low-field process should be connected with the thermally activated escape from the same defects. Unlike expected from the original model, the conductivity at low fields converges against a constant, higher value as predicted [151]. Every new variant of the Frenkel-Poole model shown here has lead to an improvement of the behaviour for low applied fields as compared to the initial theory (not necessarily each other), i. e. brings the theory closer to the experimentally observed behaviour around zero applied field[32].

After the introduction of three dimensionality, the next major extension concerned the charge exiting in directions of the backward hemisphere. Originally it has been as-sumed that the charge cannot escape in backward direction [46]. According to Hill [64], the three dimensional extensions by Jonscher [80] and Hartke [60] implicitly assumed that the emission probability in reverse direction remains constant with a probability exp(−_k^E_BTⁱ ), whereE_iis the ionisation energy of the trap. Re-examining both works, it is found that Hills assessment is only correct for the work of Hartke. Jonscher, like Frenkel, ignores emissions in backward direction (though, since the model is three-dimensional, this time the complete backward hemisphere is ignored). Hill [64] assumed that the barrier in the reverse direction increased in the same amount that it is lowered at the forward site. There is, however, a distinct difference between the forward and the back-ward site. At the forback-ward site, there is a maximal barrier height to overcome, while in the reverse direction the barrier is continuously increasing with distance. As noted by Connellet al.[28] neither assuming no change in barrier height in the reverse hemi-sphere nor considering the same amount of increase as decrease in the corresponding forward direction is appropriate. They propose to calculate the escape probability in reverse direction by integrating over the continuously decreasing probabilities of escape in reverse direction. Connellet al.could show, that approximating the backward-emission probability with its low-field approximation will always give an error below 6%. Ieda et al.[72] proposed a distance from the ionised centre r_δ= _4πǫ^e2

0ǫrδ at which a particle should be considered free. The energyδ should be sufficiently small, that a phonon would likely raise an electron only that small energy below the conductive band in the delocalised state, hence, in the order of the thermal energyk_BT. The barrier height (i. e.

the energetic distance to the ground state) at the projection of the distances_±r_δin the

corresponding hemispheres is then used to calculate the emission probabilities. Pai [144]

not only criticises the arbitrary choice of the cut-off energyδ, but also the possibility to manipulate Frenkel-Poole barrier-lowering coefficient freely with this constant, since it introduces a free offset to the exponent. Not exploiting this energy cut-off, the solution gives approximately the same characteristic as the extension of the Frenkel-Poole model derived by Connellet al.[144].

The alternative construction by Potemski and Wilamowski [151] in 1985 takes a com-pletely different path than the extensions of the Frenkel-Poole model above. They ex-tended the original model by Frenkel to include the influence of local fields caused by statistically distributed charged centres in the material. Using vector sums of the different fields (hence, allowing for arbitrary angles) acting on the potential around the trap, this is also a three-dimensional extension. Like the other extensions to three dimensions, it leads to Ohmic behaviour and increased zero-field conductivity as compared to the expected value from the original model if the applied field is much smaller than the local field. For fields significantly larger than the internal field, this extended model shows the conventional Frenkel-Poole behaviour. A remaining challenge is the unknown local-field distribution. In the absence of a better alternative, they assume the local field to be equally probable in every direction and a direction-independent mean value of the field strength.

Using these assumptions, the mean local field may be extracted. For their measurement of CdF₂:Y this leads to local field strengths around10³V cm⁻¹. The same idea of includ-ing the local fields by other randomly distributed charge centres was implemented by Dallacasa and Paracchini in 1987 [32]. Using an approximated result in the end and a slightly different derivation, this work arrives at local field strengths around10⁴V cm⁻¹ for CdF₂:Y crystals. The comparability of the samples of both publication cannot be assessed from the sparse information about the samples in the work of Potemski and Wilamowski [151]. Although both theories include important properties, that can be expected to play a significant role in the local potential environment around the trap while simultaneously making it more realistic, the theories (as well as the others) are unable to include the strong directional dependence of the bonds that are present in many investigated materials. Both works include, for a specific trap, only a local field in one direction superimposed to the applied field and,still, the Coulombic potential. Up to now, all models are therefore restricted to describe the situation in materials with either weakly oriented bonds or materials where the central part of the potential of strongly oriented bonds can be reduced to its role as origin of the binding energy. The two lat-ter models may then be suited especially in those situations mentioned above, where a high concentration of defects renders the recognition of contributions by neighbouring defects important. Tetrahedral amorphous carbon, an example which shows Frenkel-Poole behaviour, has covalent bonds which are strongly directional, whilst due to its amorphous characteristic many possible defect types with very different and complicated local potential environments around the trap are thinkable. Hence, extraction of more detailed information about the local structure around the defects might, due to the many possible very different individual local situations, not be possible. An exception would be, if the distance of the bound charge carrier is rather large from the centre, so that the complicated local fields can be approximated by a (screened) Coulombic field. A very

large volume of localisation of the charge bound to the trap would render comparable permittivities possible in the first place, as well. For an amorphous system this could also validate the assumption of isotropy. In this context, it may be important to note that Frenkel-Poole conduction is observed for various, very different, materials (e. g. Si₂N₄ [175], Ta₂O₅[129, 170, 60], SiO [170, 80, 60], Al₂O₃[60], layered crystalline solids [GaTe, CaSe] [82], ta-C [68] and many others) The asymmetric central potential of a covalent bond, and consequently also of highly localised structural defects that are expected to be a significant source of deep-level trap states in otherwise intrinsic materials, is very sensitive even to slight changes of the involved species [41, pp. 281-334]. From the fact that the Frenkel-Poole effect is observed almost universally for a wide variety of materials it may, hence, be deduced that the very different central potential parts of the traps do not dominate the response of this high-temperature process. A more thorough discussion of the local environment is given in section 5.6.2, which critically reviews the assumptions of the different models, and section 5.6.4, where the actual local potential is described and speculated why the Frenkel-Poole models leads to a good description despite its deficits.

In 1967, Simmons [170] is the first to analyse the energetic distribution and the oc-cupation statistics of traps as well as donor states in the band gap in the context of Frenkel-Poole conduction. One of his assumptions is that the traps which experience field-assisted barrier-lowering are not necessarily the donor states. Consequently, different concentrations of trap- versus donor-states and their energetic arrangement, to themselves as well as with respect to the Fermi level, are discussed. An important finding is that, de-pendent on energetic arrangement and concentration of the respective states, the barrier lowering and the initial barrier height are, within the scope of the usual Frenkel-Poole effect, divided by a factor of one or two. This is essential, because the Schottky-emission model (an electrode limited process) has the identical current-voltage relation as the Frenkel-Poole model (a bulk-limited phenomenon) except for a barrier-lowering coef-ficient of half the size of the initial Frenkel-Poole model. Hence, Simmons could show, that the value of the barrier-lowering coefficient cannot be used to interpret whether a bulk- or electrode-limited process is observed. Furthermore, this is a possible explanation for the typical observation of shallower slopes.

To decide whether the measured conductivity is electrode or bulk limited, Mead [129]

has suggested a method which does not require the knowledge of the film thickness.

However, due to the high density of states in the band gap, amorphous materials, for which Frenkel-Poole conductivity is often observed, rarely are electrode limited [83]. As confirmed also in this work, the difference between a chromium/gold top contact and an aluminium/titanium top contact on tetrahedral amorphous carbon were negligible.

The work by Connellet al.[28] picks up the occupation statistics of the traps and their energetic distribution from Simmons [170] while discussing it from the standpoint of compensation between acceptor and donor levels. Furthermore, they clear up with the common misconception that finding a single activation energy indicates one energetic distance of the traps to the conductive band. A single activation energy can also be found for a continuous distribution of donor levels, as it would be expected in an amorphous semiconductor. The degree of compensation, in such a more complicated system, then leads to a continuous factor m ∈ [1, 2]by which initial barrier and barrier-lowering