Bond alternated chains, energy bands

Figure 5. Energy bands (—), dipole transition moment (- -), and density of states (—) for bond alternated chain.

Fig. 10.2. One-dimensional semiconductor model of interacting 2/>z-electrons in polymer chain from C-C- bonds of alternating length (it-electron system of polyacetylene) [9] (a)-the configurations of chain with repeat union 2a (b)-energy band scheme for 2/>z-electrons G the gap between valent and conduction bands.

Alternation of the CC bond lengths along the chain and the existence of a large energy gap are well-established facts in PA (see Chapter 12, Section II.C.2). However, since each carbon atom contributes one tt electron, there is at first sight no obvious reason why CC bonds should not be equivalent. If they were, and taking into account the electron spin, the tt electrons should generate a half-filled band such a material is a metal. If there is bond alternation, the one-dimensional unit cell is doubled and a gap opens at the Brillouin zone boundary the material is a semiconductor. 

All carbon-carbon bonds in the skeleton have 50% double bond character. This fact was later confirmed by X-ray diffraction studies. A simple free-electron model calculation shows that there is no energy gap between the valence and conduction bands and that the limit of the first UV-visible transition for an infinite chain is zero. Thus a simple free-electron model correctly reproduces the first UV transition with a metallic extrapolation for the infinite system. Conversely, in the polyene series, CH2=CH-(CH=CH) -CH=CH2, he had to disturb the constant potential using a sinusoidal potential in order to cover the experimental trends. The role of the sinusoidal potential is to take into account the structural bond alternation between bond lengths of single- and double-bond character. When applied to the infinite system, in this type of disturbed free-electron model or Hiickel-type theory, a non-zero energy gap is obtained (about 1.90 eV in Kuhn s calculation), as illustrated in Fig. 36.9. 

The ideal chain-structures employed in theoretical models bear little resemblance to real materials. Disorder has been modelled by either a distribution of conjugation lengths or electron-lattice interactions. Bond length and substitutional disorder has been shown to lead to the disappearance of bond alternation and closure of the band gap, though with a low density of states at the Fermi energy. A simple model has been developed for the influence of defects on the electrical conductivity of doped samples. Pinning of solitons by defects has been discussed, but it is clear that the role of disorder requires further study. ... 

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