Hopping field dependent mobility

Besides its temperature dependence, hopping transport is also characterized by an electric field-dependent mobility. This dependence becomes measurable at high field (namely, for a field in excess of ca. 10d V/cm). Such a behavior was first reported in 1970 in polyvinylcarbazole (PVK) [48. The phenomenon was explained through a Poole-ITenkel mechanism [49], in which the Coulomb potential near a charged localized level is modified by the applied field in such a way that the tunnel transfer rale between sites increases. The general dependence of the mobility is then given by Eq. (14.69)... 

The second approach deals with charge transport macroscopically. An amorphous organic semiconductor can be treated as an ensemble of disordered hopping sites through which injected carriers drift xmder the influence of an external applied field. Historically, the Poole-Frenkel (PF) model was one of the first models to explain the electric field dependence of charge carrier drift mobilities [21,22], The field-dependent mobility can be written as... 

The second approach (e.g. [35]), which allows for modifications of the dimensionality of the hopping motion, is to perform computer simulations for fixed assumed values for all but one parameter and vary the parameter of interest, e.g., a. The resulting families of simulated mobility - electric field results are then compared to the experimentally determined values and the one closest to the experimental result yields the fitted value. In the CDM formalism by Novikov and co-workers correlations are introduced between neighbouring site energies. This was done in order to reproduce the experimentally observed field dependent mobility, at low fields which commonly occurs in semiconducting polymers. They also obtain a semi empirical expression for the full field and temperature dependence of the mobility, see Eq. 5.4, although their parameters differ slightly from the Bassler GDM. 

In the above consideration it has been tacitly assumed that the charge carrier mobility docs not depend on the electric field. This is a good approximation for molecular crystals yet not for disordered systems in which transport occurs via hopping. Abkowitz et al. [37] have solved that problem for a field dependence of ft of the form p-po (FIFU) and trap-free SCL conduction. Their treatment predicts... 

Figure 12-16. Field dependence of die charge carrier mobility in an undiluted hopping system al various values of the disorder parameter a = a/kT (Kef. [67]).

In discussing low temperature-dependent mobility, we should mention charge transport by polarons, an intermolecular phonon-assisted hopping process 24>25>. Polarons (charge carriers trapped in their polarization field) arise from a strong electron-phonon interaction where there is a weak overlap of wave functions of... 

The effective carrier mobilities and their dependence on concentration for benztriazole derivatives embedded in polycarbonate were explained by the percolative aspects in photoconductivity [296]. The observed field dependence of the mobility for polycarbonate films doped with diethynylaminobenzaldehy-de-diphenyl hydrazone cannot be accounted for by any known hopping model [297]. The influence of the nature of the polymer matrix on photogeneration and transport properties of the molecule doped polymers was investigated in some papers [57, 58, 298, 299],... 

Charge transport in the accumulation channel is described by the percolation model [24] based on thermally activated tunneling of holes between localized states in an exponential density of states, described in Section 13.2.2. In the accumulation regime this Variable Range Hopping (VRH) model yields a gate-voltage dependent field-effect mobility of the form ... 

A key result of the analysis of Van der Auweraer et al. is that the drift velocity increases with increasing field at low fields, then decreases with increasing field at high fields. This gives rise to a negative field dependence of the mobility at high fields. The results are illustrated in Fig. 30. For these calculations, the temperature was 298 K. the rale constant kQ 10-6 s-f and the hopping distance... 

Pai et al. (1983) measured hole mobilities of a series of bis(diethylamino)-substituted triphenylmethane derivatives doped into a PC and poly(styrene) (PS). The mobilities varied by four orders of magnitude, while the field dependencies varied from linear to quadratic. In all materials, the field dependencies decreased with increasing temperature. The temperature dependencies were described by an Arrhenius relationship with activation energies that decrease with increasing field. Pai et al. described the transport process as a field-driven chain of oxidation-reduction reactions in which the rate of electron transfer is controlled by the molecular substituents of the hopping sites. 

The first time-of-flight measurements of PVK were by Regensburger (1968). At 4.0 x K)5 V/cm, a mobility of 10-6 cm2/Vs was reported. The mobility was field dependent and thermally activated with a high-field activation energy of 0.7 eV. No evidence for deep trapping was observed. The field dependence was attributed to the ionization of shallow traps or field-assisted hopping. The study of Regensburger, and the earlier work of Vannikov (1967), were the first to clearly show that electronic transport occurs in polymers. Kiyszewski et al. (1968) and Szymanski and Labes (1969) later reported a... 

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