Holstein model

The Holstein model is a model for structural reorganization just as the Marcus model. The Holstein model applies to a polaron localized in a lattice. The polaron is an electron moving with the lattice deformation caused by the electron. The polaron moves through the lattice with a certain effective mass. There are some similarities to the Marcus model, but also some differences. The Marcus model is, in principle, a many-electron model, since it assumes the existence of parabolic total energy surfaces. The Holstein model is a one-electron model that includes 

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

In Fig. 2.3 the dots represent the atoms in the Holstein model, and (a) shows the situation where an electron sets up a bond between two atoms, pulling them together. A physical example is a hole in the valence band of a solid rare gas (e.g. Xe), forming a molecule Xe . 

The Holstein model was preceded by the Pekar model [34] and in chemistry by the Marcus model [6]. In chemistry donor-acceptor systems are more frequent objects of study than conducting wires but the coupling between electronic and nuclear motion of similar nature. For example if the coupling is large a small nuclear displacement is sufficient to change the wave function much and in a way which corresponds to ET or EET. We use the effective, many-electron Hamiltonian H of eq.(4) and assume that it is solved for donor and acceptor, giving the energies Haa and Hdd, respectively. We use the new nuclear coordinates ... 

Equation (18) for explicitiy expresses the characteristic feature of the off-diagonal electron-phonon coupling, in contrast to the the diagonal electron-phonon coupling in the (gi a Holstein model [27], in which is described as... 

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... 

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]. 

Let us start with ht A a Holstein model. In the strong-coupling region, it is usually the case to employ the Lang-Firsov transformation [80], defined as... 

Fig. 13 Schematic ground-state phase diagram for the spin-1/2 XXZ model in the magnetic field in one [115] and two dimensions [119]. The line of A = 0 corresponds to the half-filling in the Holstein model. Antiferromagnetic, ferromagnetic, and XY phases correspond, respectively, to charge density wave, band insulating, and superconducting states in the Holstein model...

When gi = g2 and a>i = C02, this model is reduced to the iJ (g) e IT system. For simplicity, we assume that g /a>i 2/ 2 and treat the g2 term within second-order perturbation. By adopting a similar method in treating the Holstein model, we can map he E (bi + >2) model into the effective spin model as... 

When the molecule is rigid for the applied bias, approximations of harmonic motions for nuclei and linear couplings with electrons will be sufficient. Then we can adopt the (nonlocal) Holstein model [2], and the total Hamiltonian is expressed by... 

In the Holstein model the molecular crystal is described as a regular one-dimensional (ID) array of diatomic molecules. The Hamiltonian of the system is a sum of three terms Hi, H, and H t. The lattice... 

The results derived from the Holstein model are based on the solution to the time-dependent... 

Even though the resulting expressions from the Holstein and the Miller-Abrahams theories are very similar, they are conceptually different in that the Holstein model includes the concept of localization energy in terms of the polaronic effect but no static disorder, and the Miller-Abrahams theory does not involve the localization energy Instead, it is assumed that the localization occurs as a result of static disorder. In the Miller-Abrahams formula this is given an explicit form, which is related to the inverse localization length of the electronic wavefiinction associated with donor and acceptor states. 

For single-frequency phonons in ID, see Datta, S., Das, A., andYarlagadda, S., Many-polaron effects in the Holstein model, Phys. Rev. B, 71, 235118, 2005. 

Furthermore, if electron holes exist in narrow bands, there is often polarization around the hole. In that case, the band model cannot be used at all, in principle. An effective mass can still be defined, as in the Holstein model (see below). In the trapped case, the effective mass is very large. 

FIGURE 16.9 Holstein model with a linear chain of diatomic molecules (N ) with one added electron. 

Contrary to the Marcus model, the Holstein model is expressed in k space. In principle, the correction terms in the Born-Oppenheimer approximation are calculated. The effective mass is proportional to J" where J is the electronic coupling between the sites. 

Figures 10.16 and 17.5 are typical, fundamental Marcus models for electron pair transfer and the connected structural reorganization. The Holstein model can possibly be extended to include Hubbard U. This would not be an easy task, however. The two important obstacles are the inclusion of structural rearrangements more accurately than in the present Holstein models, and dealing with the correlation problem.

Before a proper evaluation of the matrix elements was available and before the new experimental results on ultrapure pentacene and rubrene were realized, Kenkre et al. [130] were able to fit the classical results of Karl [131] on the temperature dependence of the anisotropic mobility of pentacene with a three dimensional Holstein model. It now seems clear that the fitted parameters are not compatible with the computations (the hopping integral is about two orders of magnitude smaller than the typical value) and that the Holstein Hamiltonian is insufficient to capture the physics of organic semiconductors. 

From the combined consideration of [83] and [172] it emerges that the inclusion of the nonlocal electron phonon coupling considerably modifies the Holstein picture. Contrary to what is expected in a pure Holstein model, there is no abrupt change in transport mechanism between delocalized transport and activated transport. The polaronic band mechanism, appropriate at least until 100 K, gives way to the DLTD mechanism until 300 K without approaching the hopping limit. Since both mechanisms result in a similar temperature dependence of the mobility (fj. aT ) they can both be extended beyond their correct limit of applicability, making it relatively easy to interpolate between them (or extrapolate each of them). 

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