Small polaron hopping transport

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]. 

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 T, 0.40. We note, however, that the polaron bandwidth is predicted to be very narrow ( IO Viojo, or lO 4 eV for a typical phonon frequency of 1000 cm-1). It is therefore expected that this band transport mechanism would be easily disturbed by crystal defects. 

The experiments just discussed made it clear that the motion of the hole on the series of As represents a different mechanism of transport than tunneling. Giese [13] and Bixon and Jortner [18] suggested that this mechanism is incoherent hopping of the hole between neighboring bases. This means that the hole wavefunction is Hmited to one base. The wavefunctions of the remaining electrons on that base would of course be distorted by the presence of the hole. Thus in this view of the transport process the base on which the hole sits could be called a molecular polaron, or a small polaron because it is limited to one site. 

In Ae small polaron picture, on the other hand, the electron-phonon interaction, compounded by the localization effects introduced by the disorder, leads to the formation of small polarons. The polaron binding energy is then the largest energy in the problem, and charge transport involves multiphonon-assisted hopping of small polarons (Emin, 1984). 

It was pointed out later by David Emin [11] that hopping transport could be divided into four categories, which result from two couples of jump regimes. First, a jump can be either strongly coupled (Emin uses the term small-polaronic ) or weakly coupled [12]. Last, a hop is either adiabatic or nonadiabatic [13]. 

An attractive model, specifically designed for the case of molecular semiconductors is that of the molecular nearly small polaron (MP) developed by SiUnsh [23]. Initially, the model was developed to resolve the contradiction between a power law temperature dependence of the mobiUty (Eq. 12) reported in naphthalene and pery-lene, which is typical of band theory, and a mean free path of the order of the lattice constant, which would lead to hopping transport. As no transport theory was able to give a satisfactory quantitative description of the experimental facts, a phenomenological approach was adopted. In the model, the carrier is considered as a polaron-type particle resulting of interactions of the carrier with vibrations of the lattice. 

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