One-dimensional bands

Some researchers use molecule computations to estimate the band gap from the HOMO-LUMO energy separation. This energy separation becomes smaller as the molecule grows larger. Thus, it is possible to perform quantum mechanical calculations on several molecules of increasing size and then extrapolate the energy gap to predict a band gap for the inhnite system. This can be useful for polymers, which are often not crystalline. One-dimensional band structures are... 

Figure 14 Schematic diagram of simplified one-dimensional band structure for an nHg (top) and Kr (bottom) spacer layer film. (From Ref 232.)...

Below, the one-dimensional band structure of simple linear conjugated polymers is outlined, followed by a brief description of the use of some model 7t-conjugated molecules for conjugated polymers. Then, to close out the chapter, the description of the properties of conjugated polymers is extended to issues of optical absorption and photo-luminescence. 

One-dimensional band structure of linear conjugated polymers... 

The elementary excitations of a conjugated polymer chain can be described within the mono-electronic approach as electron and hole quasiparticles [74] in a one-dimensional band structure, possibly weakly bound into extended Wannier-type excitons [71,75]. Within this framework, electron-phonon interactions lead to a peculiar family of exotic excitations including solitons, polarons, polaron pairs and bipolarons. In many cases, however, disorder is so significant that the polymer films are better described as an ensemble of relatively short conjugated segments [76], essentially behaving... 

A one-dimensional band with band filling defined by p (0.5 for the TM2 salts forgetting about the structural dimerization) and a molecular volume V would lead to... 

The PPV spectra of Fig. 16 show all the signatures of exciton absorption and emission, such as in typical molecular crystals. The existence of well-defined structure in the absorption spectrum is not so easily accounted for in a band-to-band absorption model. In semiconductor theory, the main source of structure is in the joint density of states, and none is predicted in one-dimensional band structure calculations (see below). However, CPs have high-energy phonons (molecular vibrations) which are known (see, e.g., RRS spectra) to be coupled to the electron states. The influence of these vibrations has not been included in previous theories of band-to-band transition spectra in the case of such wide bands [176,183]. For excitons, the vibronic structure is washed out in the case of very intense transitions, corresponding to very wide exciton bands, the strong-coupling case [168,170]. Does a similar effect occur for one-electron bands Further theoretical work would be useful. 

If one assumes, as in Fig. 15 a, a one-dimensional band diagram and that tunneling electrons are collected from the sample within a disk of diameter 100 A, which is already larger than the radius of the tip, the current /jh is several orders of magnitude smaller than typical tunnel currents in the nanoamp range. To reach a thermionic current of 1 nA locally, the surface 9 must be a disk of diameter in the micrometer range. An explanation in terms of surface diffusion of carriers seems insufficient. 

Analysis of the NMR parameters and the dynamic processes depends on the exchange rates. The basic one-dimensional band-shape analysis is best suited to intermediate rates (10—10 per s). Slow exchange rates (—0.1-10 per s) are most accurately measured by using magnetization transfer experiments and Ti-relaxation times. Fast dynamic processes (> 10 per s) can be elucidated by investigating the spin-spin relaxation times. 

Peierls (15) showed that a partially filled one-dimensional band can always lower its energy by splitting into filled and empty bands. If the compound under study is in fact one-dimensional, the failure to observe the transition is somewhat puzzling. Hg2.8eAsF6 seems to fulfill the necessary conditions for a one-dimensional metallic system. These conditions... 

Depending on the crystal structure of the one-dimensional stacks and on whether a Peierls transition occurs or not (more on this subject wiU be given in Sect. 9.3), the states in the one-dimensional bands are wholly or partiaUy fUled. The CT crystals can therefore be semiconductors or metalHc conductors. If at high temperature metallic conductivity is present and at a lower temperature Tp a Peierls phase transition occurs, the metal becomes a semiconductor at T< Tp, or an insulator. 

Fig. 2.30 Schematic drawing of a one-dimensional band structure and density-of-states for free electrons (simple metals) according to Hartree-Fock theory.

The evolution of bands from N atoms in three dimensions is analogous, but with interesting differences. The energy width of the bands in a cubic lattice (in which only nearest neighbor interactions are considered) is 3 x AH, rather than 4//bb- In contrast to the density of states for a one-dimensional band, which peaks at the edges, the density of states in three dimensions peaks at the center of the band. ... 

For leucoemeraldine, the absorption spectrum (Figure 4) consists of a single asymmetric absorption with maximum at 3.7-3.8 dV, and witii a shape consistent with that expected for the joint density of states for an intrachain transition in a quasi-one dimensional band structure. This is the rc-rc transition of the (lA)n conjugated polymer. For the emeraldine base structure there are two transitions, as inferred from the data shown in Figure 4. The close agreement of the energies of the ultraviolet absorption peaks for the two materials, (1 A)n and [(1 A)(2A)]n, implies that they are of the same origin. 

Quasi-One-Dimensional Band Dispersion Along Polymer Chains... 

Experimental results on the band dispersion in o-bond polymers are very limited due to difficulty in preparing thin films with oriented chains [20, 31, 32, 62]. Here, we introduce the band dispersion of quasi-one-dimensional polymer polyethylene. Early work on the band structure study was carried out on systems with alkyl chains and was aimed at understanding the electronic structure of polyethylene, in particular, the possible existence of one-dimensional band structure in thin films where molecular chains assemble via weak interchain interactions. There is renewed interest in the band dispersions as they determine carrier transport properties in nanoscale molecular electronics [63]. 

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