Holstein polaron

In fact, there is no qualitative difference between the IT polaron and the Holstein polaron in this region. Even quantitatively, they are exactly the same, as long as the coupling constants are normalized according to (24). 

Fig. 2 Inverse of the polaron mass enhancement factor, m/m, as a function of for the T (8) a (HP Holstein polaron) and the (8) e JT polaron. In the latter, the result in the infinite chain d = 1) is compared with that in the two-site system as weU as the analytic result in (26). The anti-adiabatic condition of (Uo// = 5 is assumed...

It is worth stressing that the SWAP and the Holstein Polaron can exist in any polymer, for example in polyethylene or nylon, as well as in conjugated polymers. 

It is useful to note that the reorganization energy. A, is directly related to the (Holstein) polaron binding energy ( poi = A/2) [9,10] in addition, for a self-exchange reaction, the driving force AG° is zero. 

The vibrational kinetic energy operator is shown here for completeness but is neglected in the derivation of the Holstein polaron (stationary solution). 

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... 

Meisel KD, Vocks H, Bobbert PA (2005) Polarons in semiconducting polymers study within an extended Holstein model. Phys Rev B 71 205206... 

Up to this point we have discussed the formation of polarons in ionic crystals. Polarons of another type can also form in elements and other systems, such as the valence bands of alkali and silver halides, where the polarizability is not the relevant factor. In fact Holstein s (1959) original discussion of the small polaron was of this form. This kind of polaron is sometimes called a molecular polaron, and is illustrated in Fig, 2.3(a), and in Fig. 2.3(b) in the activated configuration of the atoms when the electron can move from one site to another. There is nothing analogous to the large polaron in this case in three-dimensional systems either a small polaron is formed or there is little effect on the effective mass from interaction with phonons. 

There are two different temperature regimes of diffusive behavior they are analogous to those described by Holstein [1959] for polaron motion. At the lowest temperatures, coherent motion takes place in which the lattice oscillations are not excited transitions in which the phonon occupation numbers are not changed are dominant. The Frank-Condon factor is described by (2.51), and for the resonant case one has in the Debye model ... 

At the end of the 80s, the first reports of carriers with a polaronic nature were measured with photo-induced absorption, optical conductivity and infrared reflexivity in LaCu04+x and NdCu04 y [46,47]. The advocate idea was that with the assistance of absorbed light, polarons gain the ability to hop from one localized site to another. The origin of the mid-infra red peak oscillations that register on these experiments was linked to polarons. This picture was supported by related theoretical studies of polarons and bipolarons [48]. It is important to emphasize that these carriers were successfully modeled by Holstein-Hubbard and Holstein t — J... 

The Holstein and the Frohlich polarons are two of the most studied types of polarons, typically the former are used for short range interactions and the latter for long range lattice interactions in real space they are represented as ... 

In order to address the properties of a JT-polaron, we compare it with the two classes of polarons we have already presented. We use the phonon interaction of the molecular Holstein or Frohlich Hamiltonian, given by (7) and we compare it with a JT Hamiltonian,... 

By comparing the result of w /w for the infinite-site system obtained by VED [96] (see. Fig. 2), we are confident that the two-site calculation provides a reasonably good result for m /m in the whole range of g at least in the anti-adiabatic region of t/a>o. The relevance of the two-site calculation has also been seen in the Holstein model [78]. Thus we can expect that the same is true for the r (g) t JT polaron. In Fig. 3, we show the result of m/m for the T (g) r system solid curve) which is obtained in the anti-adiabatic region by implementing an... 

Fig. 3 Inverse of the mass enhancement factor, mim, as a function of g

Physically the polaron mass enhancement is brought about by the virtual excitation of phonons. In the H (g a Holstein model no restriction is imposed on exciting multiple phonons, implying that all the terms in Fig. lb for the vertex function contribute, while in the g e JT model, there is a severe restriction due to the existence of the conservation law intimately related to the 50(2) rotational symmetry in the pseudospin space. Actually, among the first- and second-order terms for the vertex function, only the term T2/ contributes, leading to the smaller polaron mass enhancement factor m jm than that in the Holstein model in which the correction r 1 is known to enhances m /m very much. In this way, the applicable range of the Migdal s approximation [48] becomes much wider in the g e JT system [63]. 

The formation of a bipolaron (or a bound pair of two polarons) is established, if the ground-state energy of the two-electron system is lower than twice the ground-state energy of a polaron. This issue has been studied rather intensively for the Holstein bipolaron [78], but it is not the case for the JT bipolaron. In [96], the electron-electron correlation function and the effective mass of an (gi e bipolaron was studied in one dimension in comparison with the corresponding results for the Holstein bipolaron [107]. In Fig. 5, we plot the phase diagram for the bipolaron formation, from which we find that the JT bipolaron is less stable than the Holstein one. 

Holstein, T. (1959) Studies of polaron motion Part II. The small polaron. Ann. Phys., 8, 343-389. 

At low temperatures, the small-polaron moves by Bloch-type band motion, while at elevated temperatures it moves by thermally activated hopping mechanism. Holstein (1959), Friedman and Holstein (1963), Friedman (1964) performed the theoretical calculations of small-polaron motion and showed that the temperature dependencies of the small-polaron mobility in the two regimes are different. In the high-temperature hopping regime, the electrical conductivity is thermally activated and it increases with increasing temperature. As shown by Naik and Tien (1978), its temperature dependence is characterized by the following equation... 

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