Polaron small

The theory of the small polaron was first introduced by Yamashita and Kurosawa [18] to account for the very small mobility found in transition metal oxides. It has been extensively developed later by Holstein [19]. A review of the small polaron 

7 is the electron-lattice coupling constant, the calculation of which is both crucial and complex. In the continuum approximation, Yamashita and Kurosawa [18] arrive at a coupling constant 

The condition of validity of Eqs (19) were specified by Holstein. First, the existence of the small polaron requires that J E, /2. Moreover, the use of perturbation theory restricts the formula to J Hujq. This upper limit is in fact that for a non-adiabatic process. The adiabatic process, for which J fiwo, has been studied by Emin and Holstein [22]. The high temperature mobility in that case is given by 

It is stressed that, since the overlap integral J is treated as a perturbation, the condition J E ll should still hold. 

The low temperature regime has been extensively dealt with by Holstein [19]. He arrives at the result that, under given circumstances, a band regime can occur when T is lower than about 0.40. The mobility is now given by 

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Such renormalization can be obtained in the framework of the small polaron theory [3]. Scoq is the energy gain of exciton localization. Let us note that the condition (20) and, therefore, Eq.(26) is correct for S 5/wo and arbitrary B/ujq for the lowest energy of the exciton polaron. So Eq.(26) can be used to evaluate the energy of a self-trapped exciton when the energy of the vibrational or lattice relaxation is much larger then the exciton bandwidth. 

In Fig. 1 the absorption spectra for a number of values of excitonic bandwidth B are depicted. The phonon energy Uq is chosen as energy unit there. The presented pictures correspond to three cases of relation between values of phonon and excitonic bandwidths - B < ujq, B = u)o, B > ujq- The first picture [B = 0.3) corresponds to the antiadiabatic limit B -C ljq), which can be handled with the small polaron theories [3]. The last picture(B = 10) represents the adiabatic limit (B wo), that fitted for the use of variation approaches [2]. The intermediate cases B=0.8 and B=1 can t be treated with these techniques. The overall behavior of spectra seems to be reasonable and... 

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 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... 

Emin stresses the point that, as in the non-adiabatic process, Eq. (14.66) is only valid for J < Eu/2 to preserve the small polaron character of the charge carrier. 

Importantly, deep oxidation of polyaniline leads to a material that becomes insulating and spinless. This phenomenon was demonstrated in case of poly(fV-methylaniline) by monitoring ESR signal and electric conductivity of the sample (Wei et al. 2007). Deep oxidation results in the formation of the so-called polaron pairs that are evidenced by optical spectra. Because the hopping probability of two polarons on a single chain is too small, polaron pairs do not contribute to electric conductivity and ESR signal. 

Schein LB, Glatz D, Scott JC (1990) Observation of the transition from adiabatic to nonadiabatic small polaron hopping in a molecularly doped polymer. Phys Rev Lett 65 472... 

With mixed-valence compounds, charge transfer does not require creation of a polar state, and a criterion for localized versus itinerant electrons depends not on the intraatomic energy defined by U , but on the ability of the structure to trap a mobile charge carrier with a local lattice deformation. The two limiting descriptions for mobile charge carriers in mixed-valence compounds are therefore small-polaron theory and itinerant-electron theory. We shall find below that we must also distinguish mobile charge carriers of intennediate character. 

Small polarons are mobile electrons (or holes) of a mixed-valence configuration that either tunnel or hop firom site to site (e.g. Fe + Fe " Fe " + Fe ) in a time T, > cor ... 

Fig. 4a-c. Models for electron hopping correlated by electrostatic electron-electron interactions plus strong electron-phonon interactions for a valence ratio Fe /Fe = 1 (a) small pola-rons, (b) diatomic polarons, (c) small polaron coupled to slower (only one phase shown) dimerization... 

The Seebeck coefficient a becomes temperature-independent only above a temperature Tt, where T, — 300 K for x < 0.1 In magnetite, T, can be identified with the onset of strong electron-phonon coupling. The temperature-independent a shows a continuous evolution from the value for P = 1 described by Eq. (16) at x = 0.1 to that by Eq. (15) for x > 0.8. Although small-polaron formation is observed for x > 0.2, regions apparently persist where multielectron jumps can occur. 

At lower temperatures (T < T,), a exhibits a temperature dependence characteristic of a small activation energy (= 0.03 eV) for excitation of charge carriers from stationary trap sites It is reasonable to suspect that small polarons tend to be trapped at impurity centers at low temperature. 

As X increases through x = 1, the Fermi energy drops from the bottom of the Fe3+ 2+ 3 (j6 bands to the top of the Mn " 3 d band, and the activation energy of the conductivity increases abruptly from 0.05 eV for small-polaron Fe ions to 0.3 eV for small-polaron Mn holes . The Mn ion is a strong Jahn-Teller ion (configuration Eg), so the holes in the Mn 3d band form more stable small polarons. 

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