Polaron representation

To conclude, after the canonical transformation we have two equivalent models (1) the initial model (145) with the eigenstates (160) and (2) the fictional free-particle model (154) with the eigenstates (158). We shall call this second model polaron representation. The relation between the models is established by (155)-(157). It is also clear from the Hamiltonian (148), that the operators < , d. ad. and a describe the initial electrons and vibrons in the fictional model. 

When the system is weakly coupled to the leads, the polaron representation (154), (162) is a convenient starting point. Here we consider how the sequential tunneling is modified by vibrons. 

Fig. 10 Two schematic representations of a polaron-like species in DNA. In the top drawing, the base pairs of DNA are represented by the horizontal lines the sugar diphosphate backbone is represented by the vertical lines. The polaronic distortion is enclosed in the box and extends over some number of base pairs. This is shown schematically by drawing the base-pair lines closer together. In the lower figure, a specific potential po-laron is identified, AAGGAA, and the radical cation is presented as being delocalized over this sequence. Movement of the polaron from one AAGGAA sequence to the next requires thermal activation...

Figure 3.69 Schematic representations of the nature and energetics of (a) polaron and (b) bipolaron formation (see text for details).

Figure 9.5 Schematic representation of a bound magnetic polaron (a) one bound magnetic polaron located on a charged defect and (b) overlapping bound magnetic polarons leading to ferromagnetic alignment of magnetic ions.

Fig. 2 Schematic representation of the doping of poly(p-phenylene ethynylene) (PPE) under formation of polarons (radical cations, radical anions) and bipolarons (dications, dianions). Note that multiple carriers can coexist on each macromolecule...

Despite the success of the disorder model concerning the interpretation of data on the temperature and field dependence of the mobility, one has to recognize that the temperature regime available for data analysis is quite restricted. Therefore it is often difficult to decide if a In p vs or rather a In p vs representation is more appropriate. This ambiguity is an inherent conceptual problem because in organic semiconductors there is, inevitably, a superposition of disorder and polaron effects whose mutual contributions depend on the kind of material. A few representative studies may suffice to illustrate the intricacies involved when analyzing experimental results. They deal with polyfluorene copolymers, arylamine-containing polyfluorene copolymers, and c-bonded polysilanes. 

Some conjugated polymers, such as polythiophene and polyaniline were synthesized already in the last century [8a,b], It is not surprising that, for example, polyaniline has played a major role in research directed toward synthetic metals because it possesses a relatively stable conducting state and it can be easily prepared by oxidation of aniline, even in laboratories without pronounced synthetic expertise (see section 2.6). It is often overlooked, however, that a representation of, for example, polypyrrole or polyaniline by the idealized structures 1 and 2 does not adequately describe reality, since various structural defects can occur (chart 1). Further, there is not just one polypyrrole, instead each sample made by electrochemical oxidation must be considered as a unique sample, the character of which depends intimately on the conditions of the experiment, such as the nature of the counterion or the current density applied (see section 2.5). Therefore, one would not at all argue against a practical synthesis, if the emphasis is on the active physical function and the commercial value of a material, even if this synthesis is quick and dirty . Care must be exercised, however, to reliably define the molecular structure before one proceeds to develop structure-property relationships and to define characteristic electronic features, such as effective conjugation length or polaron width. 

The first numerical results from a strictly quantum mechanical calculation were given a few years ago [89]. In particular, P. Kornilovitch formulated a path integral representation of a three-dimensional JT polaron. Applying a QMC algorithm, he calculated the energy of the ground state, the DOS and the effective mass of a single... 

Let us start with the E e JT polaron in the weak-coupling region (or for the case of small gEigie), in which the perturbation approach in momentum representation is useful. The thermal one-electron Green s function Gkya(ico ) with co the fermion Matsubara frequency is defined at temperature T by... 

FIGURE 2.2.5 The figure is a schematic representation of the formation of a polaron when a positive charge is placed on a molecule in a conjugated organic solid. The hexagons symbolize the core of the nuclei, while the circles represent the delocalized Jt-electrons. 

Figure I3.Z6. Schematic representation of successive (a) p-doping, and (b) n-doping in a band model. From left to right undoped state, polaron states (here symmetric) for lightly doped anT, bipolaron states (above, here symmetric) or polaron bands (below) for intermediate to strongly doped anT, bipolaron bands for strongly doped anT. The polaron and bipolaron states originate from the valence and conduction band near edgestates of the undoped material. The dashed areas mark occupied bands.

FIGURE 1.13 Schematic representation of the one-electron energy diagram for a polaron (a) localized on a single conjugated chain and (b) delocalized over two cofacial chains. The symmetry of the orbitals and the relevant electronic excitations are indicated. 

FIGURE 22.3 Energy levels and associated optical transitions of positive polaron and bipolaron excitations. The full and dashed arrows represent allowed and forbidden optical transitions, respectively. H, S, and L are HOMO, SUMO, and LUMO levels, respectively, and u and g are odd and even parity representations, respectively. 2too(P) and 2o)o(BP) are assigned. (From Vardeny, Z.V. and Wei, X., Handbook of conducting polymers, 2nd ed., eds. T.A. Skotheim, R.L. Elsenbaumer, and ]. Reynolds, Marcel Dekker, New York, 1998. With permission.)... 

Figure 4 Upper panel basic photophysical scenario for an isolated molecule. Dotted (dashed) arrows show vibronic absorption (fluorescence) shaded large arrows internal conversion (1C), vibrational relaxation (VR), and intersystem crossing (ISC) processes. Solid black arrow shows photoinduced absorption transitions of triplet excitons taking place in the microsecond time domain. Dot-dashed arrow indicates phosphorescence. Lower panel schematic representation of negative polaron levels and their spin population within the monoelectronic scheme. Black arrows show photoinduced absorption transitions (Pi and P2).

Fig. 174. Schematic representations of the band structures of (a) soliton lattice, (b) bipolaron lattice, and (c) polaron lattice forms.

Fig. 15. Schematic representations of (a) a defect in trans- A (a mirror plane can be dawn through the defect itself), (b) a soliton in cis-PA, (c) a bipolaron in cis-PA and (d) a polaron in cis-PA.

Fig. 38. Schematic representation of the band structure obtained Irom the electronic polaron model (Liu 1987, 1988).

The absence of any electron transition between the HOMO level and the upper polaronic level is in fact directly related to the selection rules imposed by the symmetry of the oligomers. Taking account of Civ symmetry, for instance, we find this transition to be strongly limited because the two levels that are involved belong to the same a irreducible representation this therefore leads to an excitation that is polarized in a direction transverse to the chain axis (the symmetry constraints are even more drastic when dealing with C2h symmetry. 

The simplest spectroelectrochemical measurement which yields information on electrochromic properties of CPs is the UV-Vis-NIR spectroelectrochemical curve, an in-situ or sometimes ex-situ measurement of the transmission-mode UV-Vis-NIR spectrum of the CP at various applied potentials. Such a measurement, which we shall hereinafter abbreviate as a SPEL curve (or just SPEL), is depicted in Fig. 3-1 this figure is a re-representation of the optical spectra of poly(pyrrole) (P(Py)) discussed in Chapter 2, with an abscissa in terms of wavelength, and represents a particularly well-behaved CP system. To recap again here, the single, prominent valence conduction (tt tt ) band transition in the pristine polymer (at ca. 388 nm) is accompanied by three additional polaron based transitions at low doping level (ca. 590 nm, 885 nm, 1,771 nm), which finally evolve into two bipolaron based bands (ca. 459 nm, 1,240 nm). 

The P(Py) system depicted in Fig. 3-1 represents a particularly well behaved and more "ordered system, where the evolution from the polaron to the bipolaron bands is clearly visible. Fig. 3-2 shows another such SPEL of an experiment P(Py) system, showing more clearly the disappearance of the polaron absorption (ca. 520 nm) with increased doping, to yield the broader bipolaron absorptions (beyond 700 nm). Figs. 3-3. 3-4. and 3 5 show more representative CP systems Fig. 3-5 also shows the alternative, %-Transmission representation. 

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