One-dimensional energy bands

Consider an electron in a one-dimensional energy band given by E(k) = — K2 cos ka in a Brillouin Zone, —n/a < k < nia. At time t = 0, with the electron having wave number fc = 0, apply an electric field [Pg.58]

Surface analytical techniques such as Auger electron spectroscopy (27), X-ray photoelectron spectroscopy (28), and secondary-ion mass spectrometry (29) have been used to study LB films. Synchrotron radiation is a particularly powerful probe enabling X-ray near-edge structure and extended X-ray absorption fine structure to be measured. Angle-resolved photoemission studies (30) confirmed the existence of a one-dimensional energy band along the (CH2)jc chain in a fatty acid salt film. 

Figure 5.2 Dispersion of the one-dimensional energy band formed by overlap of the s or ptt orbitals of an infinite chain of atoms...

A linear polymer consists of the repetition along the direction of the polymer chain of a small monomeric subunit. As such, it can be viewed as a one dimensional lattice. Accordingly, the electron energy level can be obtained by using Bloch functions, which results in one dimensional energy bands along the direction of the polymer chain. 

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 discussion above shows that TCV can be reasonably interpreted in the framework of known electronic states. The direct determination of interface states from TCV seems difficult because the dependence of (7° with the tip potential [78, 81] finds no simple explanation within a simple one-dimensional energy diagram. Tip-induced local modifications of the band diagram of the semiconductor may exist (see Sec. 4.2.3) which complicates the determination of energy levels. The experimental dependence of [7° on the pre-history of the electrode [78, 81] stem probably from... 

SchlUter et al. [247] used polarization selection rule effects with s-and p-polarized photons, together with energy distribution measurements, to show a good level of agreement between experiment and calculation for the one-fold site. This result was confirmed by Pandey et al. [249] and extended by Larsen et al. [250] to include the derivation of the two-dimensional energy bands from measurements of the energy positions of... 

Some recent studies of the electronic properties of poly silanes have suggested that the first UV transition is not a band-to-band transition, but is instead excitonic in nature [13,16,17,18].These predict that the onset of band-to-band transitions must therefore be at higher energies, with an exciton binding energy of an eV or more, as is common for Frenkel excitons in molecular systems. Since the vacuum ultraviolet (VUV) specmim has not been measured until now, the question of whetha- a one-dimensional semiconductor band gap model or a molecular orbital model is appropriate for the polysilanes has been unresolved. 

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... 

In this paper, we review progress in the experimental detection and theoretical modeling of the normal modes of vibration of carbon nanotubes. Insofar as the theoretical calculations are concerned, a carbon nanotube is assumed to be an infinitely long cylinder with a mono-layer of hexagonally ordered carbon atoms in the tube wall. A carbon nanotube is, therefore, a one-dimensional system in which the cyclic boundary condition around the tube wall, as well as the periodic structure along the tube axis, determine the degeneracies and symmetry classes of the one-dimensional vibrational branches [1-3] and the electronic energy bands[4-12]. 

Antiferromagnetic one-dimensional Kronig-Penney potentials, 747 Antiferromagnetic single particle potential, 747 energy band for, 747 Antilinear operator, 687 adjoint of, 688 antihermitian, 688... 

The second procedure, several aspects of which are reviewed in this paper, consists of directly computing the asymptotic value by employing newly-developed polymeric techniques which take advantage of the one-dimensional periodicity of these systems. Since the polarizability is either the linear response of the dipole moment to the field or the negative of the second-order term in the perturbation expansion of the energy as a power series in the field, several schemes can be proposed for its evaluation. Section 3 points out that several of these schemes are inconsistent with band theory summarized in Section 2. In Section 4, we present the main points of the polymeric polarization propagator approaches we have developed, and in Section 5, we describe some of their characteristics in applications to prototype systems. 

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