Electrons energy bands

Wang C S and Callaway J 1978 BNDPKG. A package of programs for the calculation of electronic energy bands by the LCGO method Comput. Phys. Commun. 14 327... 

Fig. 2. (a) A schematic diagram of a n—p junction, including the charge distribution around the junction, where 0 represents the donor ion 0, acceptor ion , electron °, hole, (b) A simplified electron energy band diagram for a n—p junction cell in the dark and in thermal equilibrium under short-circuit... 

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... 

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]. 

The reciprocal lattice is useful in defining some of the electronic properties of solids. That is, when we have a semi-conductor (or even a conductor like a metal), we find that the electrons are confined in a band, defined by the reciprocal lattice. This has important effects upon the conductivity of any solid and is known as the "band theory" of solids. It turns out that the reciprocal lattice is also the site of the Brillouin zones, i.e.- the "allowed" electron energy bands in the solid. How this originates is explciined as follows. 

Heaton, R.A., Harrison, J.G. and Lin, C.C. (1983) Self-interaction correction for density-functional theory of electronic energy bands of solids, Phys. Rev., B28, 5992-6007. 

Our model of positive atomic cores arranged in a periodic array with valence electrons is shown schematically in Fig. 14.1. The objective is to solve the Schrodinger equation to obtain the electronic wave function ( ) and the electronic energy band structure En( k ) where n labels the energy band and k the crystal wave vector which labels the electronic state. To explore the bonding properties discussed above, a calculation of the electronic charge density... 

Detailed electronic energy-band calculations have revealed the existence of appropriate surface states near the Fermi energy, indicative of an electronically driven surface instability. Angle-resolved photoemission studies, however, showed that the Fermi surface is very curved and the nesting is far from perfect. Recently Wang and Weber have calculated the surface phonon dispersion curve of the unreconstructed clean W(100) surface based on the first principles energy-band calculations of Mattheis and Hamann. ... 

Fig. 2-3. Formation of electron energy bands in constructing a solid crystal X from atoms of X ro = stable atom-atom distance in crystal BB = bonding band ABB = antibonding band e, = band gap.

Fig. 2-4. Lattice potential energy and electron energy bands in crystals IB s inner band FB s frontier band.

Fig. 2-12. Electron energy band formation of silicon crystals from atomic frontier orbitals number of silicon atoms in crystal r = distance between atoms rg = stable atom-atom distance in crystals, sp B8 = bonding band (valence band) of sp hybrid orbitals sp ABB = antibonding band (conduction band) of sp hybrid orbitals.

Fig. 2-21. Formation of electron energy bands in metal oxides from isolated metal ions and oxide ions.

Fig. 2-22. Fonnation of electron energy bands in aluminum oxide. [From Vyh, 1970.]...

Fig. 2-24. Electron energy bands in a-FejOs CB = conduction band of vacant 3d orbitals of Fe VBj = valence band comprising occupied 3d orbitals of Fe and antibonding 2p orbitals of O " VB = valence band of occiq>ied 2p orbitals of [From Anderman-Kennedy, 1988.]...

In addition to the electron energy bands and impurity levels in the semiconductor interior, which are three-dimensional, two-dimensional localized levels in the band gap exist on the semiconductor surface as shown in Fig. 2-28. Such electron levels associated with the surface are called surface states or interfacial states, e . The siuface states are classified according to their origin into the following two categories (a) the surface dangling state, and (b) the surface ion-induced state. 

Fig. 2-29. Formation of electron energy bands and surface danj ing states of silicon crystals DL-B = dangling level in bonding DL-AB = dangling level in antibonding.

SoUd ice forms a crystal of diamond structure, in which one water molecule is hydrogen-bonded with four adjacent water molecules. Most (85%) of the hydrogen bonds remain even after solid ice melts into liquid water. The structure of electron energy bands of liquid water (hydrogen oxide) is basically similar to that of metal oxides, 6dthough the band edges are indefinite due to its amorphous structure. 

Figure 2-34 shows the diagram of electron energy bands of liquid water. As in the case of metal oxides, the oxygen 2p orbital constitutes the valence band. 

Fig. 2-34. Electron energy bands of liquid water formed from atomic orbitals of hydrogen and oxygen atoms.

The electrode potential at which the electron energy band is flat in semiconductor electrodes is caUed the flat band potential, . The flat band potential is used as a characteristic potential of individual semiconductor electrodes in the same way as the potential of zero charge is used for metal electrodes. At the flat band potential the space charge, Ogc, is zero but the interfacial charge, + oh + o, is not zero. The electrode interface is composed of only the compact layer at the flat band potential if no diffuse layer exists on the solution side. 

Figure 8-1 shows the potential energy barrier for the transfer reaction of redox electrons across the interface of metal electrode. On the side of metal electrode, an allowed electron energy band is occupied by electrons up to the Fermi level and vacant for electrons above the Fermi level. On the side of hydrated redox particles, the reductant particle RED is occupied by electrons in its highest occupied molecular orbital (HOMO) and the oxidant particle OX, is vacant for electrons in its lowest imoccupied molecular orbital (LUMO). As is described in Sec. 2.10, the highest occupied electron level (HOMO) of reductants and the lowest unoccupied electron level (LUMO) of oxidants are formed by the Franck-Condon level sphtting of the same frontier oihital of the redox particles... 

We consider a simple redox electron transfer of hydrated redox particles (an outer-sphere electron transfer) of Eqn. -1 at semiconductor electrodes. The kinetics of electron transfer reactions is the same in principal at both metal and semiconductor electrodes but the rate of electron transfer at semiconductor electrodes differs considerably from that at metal electrodes because the electron occupation in the electron energy bands differs distinctly with metals and semiconductors. 

The behavior of the magnetic susceptibility as a function of temperature is in accord with what one should expect from an array of small isolated metal clusters, with splitting between the filled and the empty intra-cluster electronic energy bands of the order of 30 K. The magnetic moment per cluster of less than one electron spin still requires clarification. 

Figure 9.10 Electron energy bands in nonconductors and conductors. Filled bands are shown hatched.

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