Transport Mechanism of Polarons

Because polarons are localized species, their natural transport mechanism is hopping. We shall now briefly describe the small polaron model, as developed by Holstein and Emin [26, 29, 46]. 

The medium considered by Holstein is a one-dimensional crystal that contains a single excess electron. The Hamiltonian of the system is composed of three 

is the displacement of the nth molecule from its equilibrium position and M the reduced mass of each molecular site. Second, the electron is described within the frame of the tight-binding approximation, where it is assumed that the effect of the potential at a given site of the one-dimensional chain is limited to its nearest neighbors. In that case, the energy dispersion of the electron is given by 

Notice that the term ec /h has the dimension of a mobility, with a magnitude ranging between 1 and 5 cm V s in most molecular crystals. 

At very low temperatures, Holstein predicted that the small polaron would move in delocalized levels, the so-called small polaron band. In that case, mobility is expected to increase when temperature decreases. The transition between the hopping and band regimes would occur at a critical temperature Tt 0.40. We note, however, that the polaron bandwidth is predicted to be very narrow ( 10 or 10 eV for a typical phonon frequency of 1000 cm ), It is 

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