Polaron models

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 perturbation theory used by Holstein in his small-polaron model confines its validity to an upper limit for J of around hto0, which corresponds to a non-adiabatic process. The adiabatic process, for which J > has been studied less extensively. In the high temperature limit, Emin and Holstein [46] arrive at the result that... 

Figure 35. Total conductivity, a, and oxygen stoichiometry, 3-d, at 1000 °C of Lao.9Sro.iMn03-(5, from measurements by Kuo et al. The model calculations are based on a large polaron model with equilibrium constants as given in ref 216. Thick line calculated stoichiometry, thin line calculated conductivity. (Reprinted with permission from ref 216. Copyright 2000 Elsevier.)...

In the 2-level limit a perturbative approach has been used in two famous problems the Marcus model in chemistry and the small polaron model in physics. Both models describe hopping of an electron that drags the polarization cloud that it is formed because of its electrostatic coupling to the enviromnent. This enviromnent is the solvent in the Marcus model and the crystal vibrations (phonons) in the small polaron problem. The details of the coupling and of the polarization are different in these problems, but the Hamiltonian formulation is very similar. ... 

There are other types of semiconducting polymers as well, some of the more important of which are listed in Table 6.10. The condnction mechanism in most of these polymers is the polaron model described above. Applications for these polymers are growing and inclnde batteries, electromagnetic screening materials, and electronic devices. 

Kenkre and Dunlap (1992) and Dunlap and Kenkre (1993) derived a polaron model based on the assumptions that the polarons are in the diabatic limit appropriate to small intersite transfer energies and that disorder plays a key role in the observed features. The crux of the explanation is the competition between energetic and spatial disorder. The method is based on the variable-rangehopping technique introduced by Apsley and Hughes (1974) and Triberis (1987). 

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. 

Besides the mentioned aperiodicity problem the treatment of correlation in the ground state of a polymer presents the most formidable problem. If one has a polymer with completely filled valence and conduction bands, one can Fourier transform the delocalized Bloch orbitals into localized Wannier functions and use these (instead of the MO-s of the polymer units) for a quantum chemical treatment of the short range correlation in a subunit taking only excitations in the subunit or between the reference unit and a few neighbouring units. With the aid of the Wannier functions then one can perform a Moeller-Plesset perturbation theory (PX), or for instance, a coupled electron pair approximation (CEPA) (1 ), or a coupled cluster expansion (19) calculation. The long range correlation then can be approximated with the help of the already mentioned electronic polaron model (11). 

Fig. 7 According to bi-polaron model of [115], the isotopic shift shows the largest signal only for intermediate values of hopping, t, which were carefully chosen to mimic the microstrain. It is for e-ph couplings. A, above A = 12 where the isotopic shift takes negative values, and this is possibly connected to the relevant properties of HTS (Upper panel). Conversely, the isotopic change of the polaron correlation shows a maximum for a given value of A (Lower panel), [116]...

While experimental evidence for polaronic relaxation is extensive, other experiments render the polaron models problematic (i) the use of the Arrhenius relation to describe the temperature dependence of the mobility (see above) leads to pre-factor mobilities well in excess of unity, and (ii) the polaron models cannot account for the dispersive transport observed at low temperatures. In high fields the electrons moving along the fully conjugated segments of PPV may reach drift velocities well above the sound velocity in PPV.124 In this case, the lattice relaxation cannot follow the carriers, and they move as bare particles, not carrying a lattice polarization cloud with them. In the other limit, creation of an orderly system free of structural defects, like that proposed by recently developed self-assembly techniques, may lead to polaron destabilization and inorganic semiconductor-type transport of the h+,s and e s in the HOMO and LUMO bands, respectively. 

The fundamental difference between disorder and polaron models is related to the difference in energy of hopping sites due to disorder and the change in molecular conformation upon addition or removal of a charge at a given site. In the disorder... 

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