Miller-Abrahams hopping

In hopping transport the probability of a carrier moving from site i to site j is given by several expressions, the most commonly used one of which is the Miller-Abraham hopping rate (Eq. 5.1). 

We note a temperature dependence of the zero field mobility as exp[—( F()/F)2], a behavior which is indeed encountered in real organic semiconductors, and differs from both Millers-Abrahams fixed range and Moll s variable range hopping models. 

Fig. 20 Charge carrier mobility in P3HT as a function of the charge carrier concentration. Squares refer to an experiment performed on a field effect transistor while circles refer to experiments done on an electrochemically doped sample. In the latter case the mobility is inferred from the steady state current at a given doping level. Solid and dashed lines have been fitted using the theory of [101]. The fit parameters are the site separation a, the prefactor Vq in the Miller-Abrahams-type hopping rate, the inverse wavefunction decay parameter y and the dielectric constant e. From [101] with permission. Copyright (2005) by the American Institute of Physics...

In the Miller-Abrahams formalism, hopping between sites / and j is determined by the product of a prefactor, an electronic wavefunction overlap... 

In the Miller-Abrahams expression, the hopping rates are a product of a prefactor v0, an electron overlap quantity Wy = Wji9 and the term exp[-(ey - zj)/kT when y > e, and unity for the case where ey < e7. For a pair of sites separated Ry in an isotropic medium, Wy = exp(-2yft7y). where y is an inverse wavefunction decay constant. To incorporate positional or off-diagonal disorder, Pautmeier et al. (1988, 1990) assumed that sites / and j make independent contributions as... 

Majias and Casado (1995) analyzed the dependence of the carrier drift velocity on the degree of energetic disorder by a simulation technique (Casado and Majias, 1994). The simulations are based on jump rates derived from the Miller-Abrahams formalism. At low fields, the velocity always decreases with increasing disorder. At high fields, however, the presence of small amounts of disorder induces nonmonotonic behavior, showing a maximum of the velocity at nonzero disorder. The results were discussed by an argument based on the distribution of hopping rates in the three spatial directions. 

Shortly after Holstein published his work on polarons in molecular crystals. Miller and Abrahams introduced a very useful description of hopping conduction in terms of a phonon-assisted electron tunneling process [38]. Miller-Abrahams theory does not include the polaronic effect. Nevertheless it... 

Figm 5 Representation of the charge carrier transport in different models (a) OUW (b) OVB (c) Miller-Abrahams and (d) polaronic hopping. 

A celebrated derivation of the temperature dependence of the mobility within the hopping model was made by Miller and Abrahams 22. They first evaluated the hopping rate y,y, that is the probability that an electron at site i jumps to site j. Their evaluation was made in the case of a lightly doped semiconductor at a very low temperature. The localized states are shallow impurity levels their energy stands in a narrow range, so that even at low temperatures, an electron at one site can easily find a phonon to jump to the nearest site. The hopping rate is given by... 

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