Energetic and Positional Disorder

Scher (1984, 1988), Silinsh and Jurgis (1985), Ries and Bassler (1987), Berlin et al. (1990), Rackovsky (1991), Arkhipov and Nikitenko (1993), and Scher (1993). Thus far, these models have not been widely used. 

Brophy, J. J., and Buttrey. J. W.. in Organic Semiconductors, MacMillan, New York, 1962. 

in Photoconductivity in Solids, John Wiley, New York, 1960. 

in Photoelectronic Properties of Semiconductors, Cambridge University Press, Cambridge, 1992. 

and DeMaeyer, L., in Progress in Reaction Kinetics (G. Porter, ed.), MacMillan, New York, 1964, p. 294. 

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Fishchuk II, Kadashchuk A, Bassler H, Abkowitz M (2004) Low-field charge-carrier hopping transport in energetically and positionally disordered organic materials. Phys Rev B 70 245212... 

Figure 9 The field dependencies of the mobility when both energetic and positional disorder occur.

Most polaron models consider only electron transfer steps parallel and antiparallel to the applied field. Van der Auweraer et al. (1994) derived an expression for the mobility that takes into account isotropic hopping in three dimensions. The treatment is based on the Marcus theoiy (Marcus, 1964, 1968, 1984 Kester et al., 1974 Jortner, 1976 Sumi and Marcus, 1986 Jortner and Bixon, 1988) and assumes that energetic and positional disorder can be neglected. 

The electronic structure of conjugated polymer semiconductors reflects the complex interplay between intrinsic rr-electron delocalization along the polymer backbone and strong electron-phonon coupling, and the existence of energetic and positional disorder in solution-processed thin films. In a hypothetical, infinitely straight polymer chain, the highest occupied molecular orbital (HOMO) and lowest unoccupied... 

Borsenberger, P. M., Richert, R., and Bassler, H., Dispersive and non-dispersive charge transport in a molecularly doped polymer with superimposed energetic and positional disorder, Phys. Rev. B, 47, 4289, 1993. 

Here, Co is an empirical constant, which is dependent on the chosen lattice, e.g., cubic, and its spacing, and /r-o is the prefactor mobility, which can be viewed as a theoretical upper limit to the carrier mobility at zero field and infinite temperature. The energetic and positional disorder parameters are a and E as previously defined. It shows an extremely strong thermal activation of the mobility tx sometimes termed super-Arrhenius. We also note that Eq. 5.2 follows the experimentally obtained general Poole-Frenkel like form for carrier mobility, at a given temperature, found in polymer semiconductors given in Eq. 5.3. 

Po is a prefactor mobility (zero held, inhnite temperature), C is an empirical constant of 2.9 X 10 (cm/V), a and S express the energetic (diagonal) and positional disorder (off-diagonal), respectively. Other approaches also exist, one of them being based on the Marcus theory of charge transfer [247, 248]. 

Conceptually, crystals are infinite, three-dimensional, and periodic entities constructed from atoms, ions, and molecules. Periodicity implies that every object contained in its smallest repeating unit, the unit cell, is systematically and infinitely repeated in three dimensions. However, real crystals are prone to lattice and other defects and parts of molecules or even entire molecules often exist in multiple crystallographically independent (or energetically similar) orientations. Moreover, the crystal structure derived from the diffraction pattern is the spatial average over the whole crystal and any deviation from the ideal three-dimensional regularity of the crystal is presented as disorder. Generally, there are two categories of disorder substitutional and positional disorder. 

To analyze the negative field dependence of the mobihty in EHO-OPPE within the Gaussian disorder transport formahsm and to determine the diagonal (energetic) disorder parameter a and the off-diagonal (positional) disorder parameter d, the following relation between the charge mobihty p and the disorder parameters was employed [75] ... 

In an extended version of the hopping concept, positional ( off-diagonal ) disorder in addition to energetic ( diagonal ) disorder has been introduced [54,63]. The simplest ansatz was to incorporate this by allowing the electronic overlap parameter 2ya to vary statistically. Operationally, one splits this parameter into two site contributions, each taken from a Gaussian probability density, and defines a positional disorder parameter I, in addition to the energetic disorder parameter cr. 

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