Energy bands, relation

The most extensive calculations of the electronic structure of fullerenes so far have been done for Ceo- Representative results for the energy levels of the free Ceo molecule are shown in Fig. 5(a) [60]. Because of the molecular nature of solid C o, the electronic structure for the solid phase is expected to be closely related to that of the free molecule [61]. An LDA calculation for the crystalline phase is shown in Fig. 5(b) for the energy bands derived from the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for Cgo, and the band gap between the LUMO and HOMO-derived energy bands is shown on the figure. The LDA calculations are one-electron treatments which tend to underestimate the actual bandgap. Nevertheless, such calculations are widely used in the fullerene literature to provide physical insights about many of the physical properties. 

Fig. 18. One-dimensional energy dispersion relations for (a) armchair (5,5) nanotubes, (b) zigzag (9,0) nanotubes, and (c) zigzag (10,0) nano tubes. The energy bands with a symmetry arc non-degenerate, while the e-bands are doubly degenerate at a general wave vector k [169,175,176].

The ID electronic energy bands for carbon nanotubes [170, 171, 172, 173, 174] are related to bands calculated for the 2D graphene honeycomb sheet used to form the nanotube. These calculations show that about 1/3 of the nanotubes are metallic and 2/3 are semiconducting, depending on the nanotube diameter di and chiral angle 6. It can be shown that metallic conduction in a (n, m) carbon nanotube is achieved when... 

Because the ID unit cells for the symmorphic groups are relatively small in area, the number of phonon branches or the number of electronic energy bands associated with the ID dispersion relations is relatively small. Of course, for the chiral tubules the ID unit cells are very large, so that the number of phonon branches and electronic energy bands is also large. Using the transformation properties of the atoms within the unit cell transformation... 

Figure 5 Schematic representation of absorbance of porphyrin compounds in relation to tissue transmittance at various wavelengths (see text). The lowest energy band (Band I) is shown in each case, apart from the porphyrin spectrum (etio type shown) on the left. The transmittance curve refers to a fold of human scrotal sac 0.7 cm thick (Wan, S. Parrish, J. A. Anderson, R. R. Madden, M. Photochem. Photobiol. 1981, 34, 679-681). The broad feature at ca. 500-600 nm is ascribed to haemoglobin (reproduced by permission of the Royal Society of Chemistry from Chem. Soc. Rev. 1995, 24, 19-33).

First, we describe the various system parameters, primarily adapted from Newns (1969). From the energy dispersion relation (2.32), the bulk states are distributed through a band, centered at a, and with width Wb = 4 / . The Fermi level Ef is taken to be at the center of this band, and is chosen to be the energy zero (so that Ef = a = 0, for all systems). The position of /, relative to the vacuum level, is given by the work function (j>, whence the isolated H adatom level, relative to Ef is... 

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