Miller-Abrahams jump rates

The starting point of all transport models is an expression of the probability Wy for a carrier to jump from site i to site j. Two types of expressions are usually considered for disordered materials with a weak electron/phonon coupling and at low temperatures, only disorder is considered and the polaronic effects are neglected. In this case the Miller-Abrahams jump rate is used... 

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. 

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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