Band structure of representative solids

Electron transitions in transition-metal ions usually involve electron movement between the d orbitals (d-d transitions) and in lanthanides between the / orbitals (/-/ transitions). The band structure of the solid plays only a small part in the energy of these transitions, and, when these atoms are introduced into crystals, they can be represented as a set of levels within the wide band gap of the oxide (Fig. 9.15). 

The two 3-functions in Eq. (3.2.2.19), providing a Idnematical description of the photoemission process, represent a very stringent condition under which photoemission intensity can be observed, especially when many-body interactions are neglected, and + E - Ei = eB(kj) represents the single-particle band structure of the solid according to Koopman s theorem. The convenient picture of direct transitions arises [19] in a band structure plotted in the reduced zone scheme, the conservation of wave vectors up to a reciprocal lattice vector means that an electron is excited vertically, that is, at constant crystal momentum, from an initial-state band to a final-state band eB(k/) —> C (kj) with energy levels separated by the photon energy hv. In the free-electron final-state approximation, the upper level is... 

To determine the BEs (Eq. 1) of different electrons in the atom by XPS, one measures the KE of the ejected electrons, knowing the excitation energy, hv, and the work function, electronic structure of the solid, consisting of both localized core states (core line spectra) and delocalized valence states (valence band spectra) can be mapped. The information is element-specific, quantitative, and chemically sensitive. Core line spectra consist of discrete peaks representing orbital BE values, which depend on the chemical environment of a particular element, and whose intensity depends on the concentration of the element. Valence band spectra consist of electronic states associated with bonding interactions between the... 

Little theoretical work has been done on the electronic structure of a solid with a free surface. The main contributions are those of Tamm (4), Shockley (5), Goodwin 6), Artmann (7), and Kouteck (5), and the main conclusion is that, in certain circumstances, surface states may exist in the gaps between the normal bands of crystal states. In this section we investigate the problem in the simplest way. The solid is represented by a straight chain of similar atoms, and its two ends represent the free surfaces. This one-dimensional model exhibits the essential features of the problem, and the results are easily generalized to three dimensions. 

Fig. 7.5 The hybrid NFE-TB band structure of fee and bee iron in the nonmagnetic state. The solid circles represent the first principles energy levels of Wood (1962). (From Pettifor (1970e).)...

Semi-conductors have conducting properties between those of insulators and conductors. The band structures of conductors, intrinsic semiconductors and insulators are represented schematically in Figure 1.4. The widths and the separations of the bands are dependent upon the intemuclear spacings of the constituents, so that the band structure may be modified in the vicinity of the surfaces of the crystal, by the occurrence of surface reactions, or by interactions with gases, liquids or other solids. 

In Fig. 3 the stopping power is presented for a proton traveling parallel to the Cu (111) surface, with velocity v = 1 a.u., as a function of the distance from the proton to this surface. The results obtained with the model, which includes the band structure of the target are represented by a solid line. In the inner part of the solid the stopping power shows oscillations reflecting the layer structure of the target. In the outer part, the stopping power decreases as the distance to the top-most layer increases. 

Figure 1. Schematic of the electronic band structures of different types of solids. Electrons are represented by shaded areas.

Structure of the solid are fraught with difficulties. There is disparity in the literature over the values for activation energies, and this is compounded with the problem of identifying the relevant solid-state energies (see Chapter 5 of this volume). The correlations proposed on the basis of Table I depend crucially on the calculated thermal band gaps and first exciton levels. In the calculations, use has been made of the results of de Boer and van Geel [17] and Mott [6], who showed that the thermal energy is always less than that measured optically by an amount which represents lattice relaxation after the optical transition. The relationship between the two is... 

Detailed analysis of the band structure of SWCNTs [103,104,106,107] shows that the tubes are metallic or semiconducting depending on the chirality of the tube. Classification of SWCNTs based on their electronic structure is shown in Table 4.2. A schematic showing the relationship between the chirality and electronic properties of CNTs is shown in Figure 4.15. Solid circles denote the metallic CNTs while the semiconducting ones are represented by open circles. 

Fig. 9 Band structure of the HPC structure left-hand side) and a single isolated chain (right-hand side). The Fermi energy is represented by the solid dark line...

Figure 4.8. Band structure of four representative covalent solids Si, C, SiC, GaAs. The first and the last are semiconductors, the other two are insulators. The small diagram above the band structure indicates the Brillouin Zone for the FCC lattice, with the special k-points X, L, K, W, U identified and expressed in units of In fa, where a is the lattice constant F is the center of the BZ. The energy scale is in electronvolts and the zero is set at the Valence Band Maximum. (Based on calculations by LN. Remediakis.)...

Figure 4.11. Band structure of two representative metallic solids Al, a free-electron metal, and Ag, a d-electron metal. The zero of energy denotes the Fermi level. (Based on calculations by I.N. Remediakis.)...

The band structure of a representative three-dimensional solid (/eft) is parabolic, with a band gap between the lower-energy valence band and the higher-energy conduction band. The energy bands of 2D graphene right) are smooth-sided cones, which meet at the Dirac point... 

Figure 1.13 Valence band structure of ZnO near the F point. The open circles represent the calculation results using the ASA-LMTO method including spin-orbit coupling. The solid lines are fits to the Rashba-Sheka-Pikus effective Hamiltonian. (After Ref [60].)...

FIGURE 26 Experimentally determined band structure of copper along the E direction. [From Baalmann, A., et at. (1985). Solid State Commun. 54, 583]. The dashed lines represent theoretical results derived by Eckardt, H., Fritsche, L., and Noffke, J. (1983). J. Phys. F14, 97. 

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. 

Our discussion of electronic structure has been in terms of band filling only. Of course, there is a lot more to know about band structures. The density of states represents only a highly simplified representation of the actual electronic structure, which ignores the three-dimensional structure of electron states in the crystal lattice. Angle-dependent photoemission gives information on this property of the electrons. The interested reader is referred to standard books on solid state physics [9,10] and photoemission [16,17]. The interpretation of photoemission and X-ray absorption spectra of catalysis-oriented questions, however, is usually done in terms of the electron density of states only. 

The empirical approach [7] was by far the most fruitful first attempt. The idea was to fit a few Fourier coefficients or form factors of the potential. This approach assumed that the pseudopotential could be represented accurately with around three Fourier form factors for each element and that the potential contained both the electron-core and electron-electron interactions. The form factors were generally fit to optical properties. This approach, called the Empirical Pseudopotential Method (EPM), gave [7] extremely accurate energy band structures and wave functions, and applications were made to a large number of solids, especially semiconductors. [8] In fact, it is probably fair to say that the electronic band structure problem and optical properties in the visible and UV for the standard semiconductors was solved in the 1960s and 1970s by the EPM. Before the EPM, even the electronic structure of Si, which was and is the prototype semiconductor, was only partially known. 

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