The Band Model of Electrons in Solids

A simple description of electrons in a solid is the model of a free electron gas in the lattice of the ions as developed for the description of metals and metal clusters. The interaction of electrons and ions is restricted to Coulomb forces. This model is called a jellium model. Despite its simplicity, the model explains qualitatively several phenomena observed in the bulk and on the surface of metals. For a further development of the description of electrons in solids, the free electron gas can be treated by the rules of quantum mechanics. This treatment leads to the band model. Despite the complexity of the band model, Hoffmann presented a simple description of bands in solids based on the molecular orbital theory of organic molecules that will also be discussed below. 

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The situation described here is based on a simple one-electron model which can hardly be expected to predict the behaviour of complex many-electron systems in quantitative detail. There can be no doubt however, that the qualitative picture is convincing and probably that the broad principles of electronic behaviour in solids have been identified. The most significant feature of the model is the band structure that makes no sense except in terms of the electron as a wave. Important, but largely unexplored aspects of solid-state reactions and heterogeneous catalysis must also relate to the nearly-free models of electrons in solids. 

Optical properties of metal nanoparticles embedded in dielectric media can be derived from the electrodynamic calculations within solid state theory. A simple model of electrons in metals, based on the gas kinetic theory, was presented by Drude in 1900 [9]. It assumes independent and free electrons with a common relaxation time. The theory was further corrected by Sommerfeld [10], who incorporated corrections originating from the Pauli exclusion principle (Fermi-Dirac velocity distribution). This so-called free-electron model was later modified to include minor corrections from the band structure of matter (effective mass) and termed quasi-free-electron model. Within this simple model electrons in metals are described as... 

A second fundamental aspect to be considered in relation to optical transitions in solids, is that electron states, other than definite energy assignments, are also characterized by a distribution in the momentum space, related to the movement (i.e. to the kinetic energy) of electrons in the soUd. For the sake of pictorial simplicity, bidimensional models of crystals, conceived as a square well potential, are usually employed in this respect, portraying the parabolic valley dependence (in one direction) of energy from the momentum vector k, as schematized in Figure 2.4A. The significance of the downward curvature of the valence band is that if electrons could have a net motion in such a band (i.e. if it were not completely filled), they would be accelerated in the opposite direction with respect to those in the conduction band. 

It is important to emphasize that the band model of solids, while extremely successful, is simply one approach among several that can be used in describing the properties of solids. It is an approach that is elegant, powerful, and amenable to quantification. However, the same conclusions can be deduced by starting from other assumptions. For instance, the band gap can be viewed simply as the energy required to break the covalent bond in a covalently bonded solid, or to ionize the anions in an ionic solid. At absolute zero, there are no atomic vibrations, the electrons are trapped, and... 

So far, in this chapter, the band model of the solid has been ignored. This is because magnetism is associated with the d and f orbitals. These orbitals are not broadened greatly by interactions with the surroundings and even in a solid remain rather narrow. The resulting situation is quite well described in terms of localised electrons placed in d or f orbitals on a particular atom. However, some aspects of the magnetic properties of solids can be explained only by band theory concepts. 

Formulating the extended perturbation model through infinite ordered quantum well in crystal Kronig-Penney model) so generating the energetic multiple gaps in spectra of electrons in solids - the electronic stmcture of bands in crystals ... 

A pure transition metal is best described by the band theory of solids, as introduced in Chapter 10. In this model, the valence s and d electrons form extended bands of orbitals that are delocalized over the entire network of metal atoms. These valence electrons are easily removed, so most elements In the d block react readily to form compounds oxides such as Fc2 O3, sulfides such as ZnS, and mineral salts such as zircon, ZrSi O4. ... 

We have shown the least complicated one which turns out to be the simple cubic lattice. Such bands are called "Brilluoin" zones and, as we have said, are the allowed energy bands of electrons in any given crystalline latttice. A number of metals and simple compounds have heen studied and their Brilluoin structure determined. However, when one gives a representation of the energy bands in a solid, a "band-model is usually presented. The following diagram shows three band models ... 

The electronic conductivity of metal oxides varies from values typical for insulators up to those for semiconductors and metals. Simple classification of solid electronic conductors is possible in terms of the band model, i.e. according to the relative positions of the Fermi level and the conduction/valence bands (see Section 2.4.1). 

Chapter 2 introduces the band theory of solids. The main approach is via the tight binding model, seen as an extension of the molecular orbital theory familiar to chemists. Physicists more often develop the band model via the free electron theory, which is included here for completeness. This chapter also discusses electronic condnctivity in solids and in particular properties and applications of semiconductors. 

Electron correlation plays an important role in determining the electronic structures of many solids. Hubbard (1963) treated the correlation problem in terms of the parameter, U. Figure 6.2 shows how U varies with the band-width W, resulting in the overlap of the upper and lower Hubbard states (or in the disappearance of the band gap). In NiO, there is a splitting between the upper and lower Hubbard bands since IV relative values of U and W determine the electronic structure of transition-metal compounds. Unfortunately, it is difficult to obtain reliable values of U. The Hubbard model takes into account only the d orbitals of the transition metal (single band model). One has to include the mixing of the oxygen p and metal d orbitals in a more realistic treatment. It would also be necessary to take into account the presence of mixed-valence of a metal (e.g. Cu ", Cu ). 

Landau (26) proposed that an additive electron in a dielectric can be trapped by polarization of the dielectric medium induced by the electron itself. Applying the model to electrons in the conduction band of an ionic crystal is rather complicated since the translational symmetry of the solid must be considered and the interaction of the excess electron with the lattice vibrations must be treated properly (I, 13, 14). 

We must emphasize that the processes described in the language of the band model (electron excitation, photogeneration of electron hole pairs, migration and capture of charge carriers) are essentially processes whereby chemical bonds in the solid and at the surface of the solid are excited, broken and formed again. 

The electron gas model adequately describes the conduction of electrons in metals however, it has a problem, that is, the electrons with energy near the Fermi level have wavelength values comparable to the lattice parameters of the crystal. Consequently, strong diffraction effects must be present (see below the diffraction condition (Equation 1.47). A more realistic description of the state of the electrons inside solids is necessary. This more accurate description is carried out with the help of the Bloch and Wilson band model [18],... 

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