Non-linear excitations

Various exact solutions to this problem are known. The most prominent one is the kink (soliton) for the case of polyacetylene being, 

Having gained some insight into the non-linear mechanisms giving rise to localized electronic (intragap) states, how dopants, mentioned in the previous Chapter, might influence these states. Two alternatives are feasible  

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In real tran -polyacetylene, the structure is dimerized with two carbon atoms in the repeat unit. Thus the tt band is divided into occupied tt and unoccupied n bands. The bond-alternated structure of polyacetylene is characterishc of conjugated polymers. Consequently, since there are no partially filled bands, conjugated polymers are expected to be semiconductors, as pointed out earlier. However, for conducting polymers the interconnection of chemical and electronic structure is much more complex because of the relevance of non-linear excitations such as solitons (Heeger, 2001). 

The production of acetylene as the major gaseous product in the reaction of butadiyne (equation 9) occurs in part via C2H radicals when short wavelength radiation is used , but with 254 nm radiation it is suggested that a molecular process occurs in a non-linear excited state. The evidence for this additional process is that no C2HD is formed when perdeuteriopropyne is present as a source of deuterium to trap the radicals. 

Besides some uncertainty, inherent in extrapolating results for finite-size systems to infinite ones, these calculations did not consider the effect on the optical properties of possible chain distortions, nor did they include the coupling of electrons to inter- and intra-molecular phonons. More recent efforts [9] have been focusing on half-filled systems, motivated by the current interest on linear and non-linear excitations in conducting polymers. 

Krinichnyi, V. I., 1996. The nature and dynamics of non-linear excitations in conducting polymers. Polyacetylene, Uspechi chimo (Russ.), 65, 84-97 (1996). 

Non-Linear Excitations and the Electronic Structure of Conjugated Polymers... 

At this stage we postpone the non-linear excitations of this model to a subsequent Chapter, we only discuss some of the symmetries of this model which are of relevance also for these excitations. First we note that the form of the electronic part of the Hamiltonian (Eq. 1) has exactly the form of a (relativistic invariant) Dirac operator in one dimension. This correspondence has been exploited [7] successfully in obtaining solutions of this model from results known in models of elementary particles. We remark here that similar analogies can be made for specific forms of the Fermi surface also in higher dimensions. Thus the connection between solid state physics and quantum field theory models can be used for a better understanding of both. 

In the first method we employ the first Born approximation for the impurity self-energy [11], This enables us to formulate equations of motion for the full space-dependent Green functions and thus consider the influence of disorder on the non-linear excitations as well [12] (see next Chapter). Furthermore, the replacement of Eq. 3 and Eq. 4 by random (Gaussian) fields... 

Summing up this Chapter we note that the non-linear excitations play a dominant role in various physical applications of conjugated polymers. But a hill understanding of the interplay of various mechanisms giving rise to these localized states has not been reached yet. 

In addition, from an application point of view, these polymers appear to be similar to conventional semiconductors. The recent proposal [21] that the non-linear excitations actually modify the conventional picture, e. g. at the interface between a polymer and a metal (or conventional semiconductor) space charge regions (depletion layers) differ from an inorganic semiconductor, finds renewed interest because transistors made from conjugated polymers are feasible and of interest for the integration of optical and electronic components. 

We can expect that in these highly conducting polymers electrons are much less localized and that they will reflect certain features of inorganic metals. But the chain-like structure of the conjugated polymers will give rise to a very anisotropic behaviour and some aspects of one-dimensionality will show up, such as the metal-insulator transition (Peierls-transition / /). The most exciting speculation is probably that on the existence of mobile conjugation defects, which can be described as non-linear excitations and are often referred to as solitons /5/ (Fig. U). These defects share many properties with non-linear excitations in other fields of physics and offer an interdisciplinary connection all the way from chemistry to elementary particle physics and field theory /6/. 

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