The Bassler Model

The hopping processes parallel to the electric field F take place either to neighbouring molecules with lower energies, or to neighbours with higher energies (cf Eqns. (8.78) and (8.79)). 

Numerous models have been proposed for hopping transport (see e.g. [Ml], [M2]). Conceptionally the simplest and physically most well-founded is the model of Bassler [47], which we will outline in the next section. In the sections thereafter, we will present typical experimental results for the temperature and electric-field dependencies of the mobility and for the temperature, field and thickness dependence of the dark current I(V) as a function of the applied voltage in disordered organic semiconductors. 

The Bassler model [47] for hopping transport in disordered organic solids is based on a few plausible hypotheses  

All these states are localised. The origin of the energy scale lies at the centre of the DOS. ff is the width of the distribution. The distribution itself is based on the stochastic variation of the polarisation energies (cf Sect. 8.3). This energy distribution is also termed diagonal disorder. 

The hopping rate vy between two sites i and j is assumed to be the product of a prefactor vq, a factor which takes the overlap of the wavefunctions into account, and a Boltzmann factor [48]  

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Fig. 8.45 Schematic of hopping transport in a disordered organic semiconductor. The energy distribution of the states (DOS) is assumed in the Bassler model to be a Gaussian distribution function C(E) with a width cr (compare Eq. (8.77)).

The overlap parameter 2yAf y is likewise not sharp, but rather statistically distributed. The distribution is assumed to be a Gauss function with a width S. The distribution of the overlap parameters is termed the non-diagonal disorder, a and S are the two important materials parameters in the Bassler model for hopping transport in disordered semiconductors. 

One result of the Bassler model is the relaxation of the excess charge carriers towards thermal equiUbrium after their production by photoexcitation (Fig. 8.46). The equilibrium energy (Eoo) of the charge carriers which were generated with... 

However, the Bassler model likewise gives a exp(VF) dependence. The reason for this is the field dependence of the hopping rate (Eq. (8.78)). The overall field and temperature dependence according to this model for the hopping conductivity in disordered materials at high fields (F> 10 V/cm) is given by ... 

S thus increases Hnearly as This dependence can be tested directly by experiment and used as an indicator for the applicabihty of the Bassler model. Figure 8.48 shows S(d ) for MPMP. Together with the temperature dependence with F = const and the electric-field dependence with T = const, the values quoted above for the two physically well-defined materials constants a and as well as for the empirical constant C were obtained. A further example of the Vf dependence will be treated in Sect 8.6.3. 

The interpretation of these experimental data for the TOF transients in PVK within the Bassler model [53] explains the temperature and time dependence of the hole transport quantitatively in terms of an intrinsic DOS of width a = 0.080 eV and additional traps with a molar concentration of 0.1% and a depth of 0.4eV. 

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-... 

The role of disorder in the photophysics of conjugated polymers has been extensively described by the work carried out in Marburg by H. Bassler and coworkers. Based on ultrafast photoluminescence (PL) (15], field-induced luminescence quenching [16J and site-selective PL excitation [17], a model for excited state thermalizalion was proposed, which considers interchain exciton migration within the inhomogenously broadened density of states. We will base part of the interpretation of our results in m-LPPP on this model, which will be discussed in some detail in Sections 8.4 and 8.6. 

In addition to three-dimensional models, there have been several onedimensional models based on the Onsager theory (Holroyd et al., 1972 Haberkom and Michel-Beyerle, 1973 Smetjek et al., 1973 Blossey, 1974 Singh and Bassler, 1975 Charle and Willig, 1978 Hong and Noolandi, 1978b Siddiqui, 1983, 1984). For the one-dimensional case, a field-independent slope-to-intercept ratio cannot be defined. One-dimensional models have been seldom used for organic materials. 

Yuh and Pai argued that the role of the polymer was related to the activation energy. Borsenberger and Bassler explained their results on a model based on dipolar disorder. According to the model, a is determined by the dipole moment of both the dopant molecule and the polymer repeat unit. The effect of the polymer host is then related to the difference in dipole moments of the dopant molecule and the polymer repeat unit as well as the dopant concentration. Most recent studies have been described by dipolar disorder arguments. 

An alternative framework for interpreting the temperature dependence of drift mobility is provided by the model suggested by Bassler and co-... 

A simple model, the Gaussian Disorder Model of Bassler and co-workers, has been very useful in rationalizing charge transport data on many amorphous molecular solids [59]. Its present version consists of the following assumptions. 

Reversible energy transfer between monomeric and dimeric forms of rhoda-mine 6G in ethylene glycol has been observed" and the concentration dependence of the overall fluorescence quantum yield has been modelled by Monte-Carlo simulations. Triplet energy transfer in disordered polymers has been analyzed on the basis of Bassler s model in which the trap energies have a Gaussian distribution." Energy transfer has also been observed in mono-layers and for photoswitchable molecular triads." The structural requirements for efficient energy transfer from a carotenoid to chlorophyll have been... 

The model cannot be solved analytically in closed form. Bassler et al. therefore constracted a model sample with 70 x 70 x 70 sites and, using a Monte-Garlo method, simulated the charge transport The simulation was thus an idealised experiment. 

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