The band theory of solids

Molecular orbital theory can be appHed to aggregates of a virtually infinite number of atoms, as in metals or diamond, for example. Whereas in the study of metal complexes one can restrict oneself to the m.o.s directly related to the metal—hgand interactions, now it is necessary to consider the whole sample of solid. Thus, if each atom of an aggregate of N atoms contributes with one valence orbital, s, there will be N orbitals which, in principle, extend to the whole aggregate. One could still call them molecular orbitals, which is equivalent to considering the whole sample as a giant molecule. 

Among such a huge number of molecular orbitals, one is maximally bonding having no nodes  

Between those two extremes (corresponding to extreme values of energy) there will lie A — 2 m.o.s with a varying number of nodes and various energies. The result is a band of m.o.s having a nearly continuous band of energy values. 

Problem 11.7 Interpret the following form for the Htickel secular determinant for a row of N atoms contributing just one s orbital each  

The extension to three dimensions requires a knowledge of the geometric arrangement of the atoms. For a square prism arrangement of 16 atoms (ref. 148) 

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The other place to read an authoritative histoi7 of the development of the quantum-mechanical theory of metals and the associated evolution of the band theory of solids is in Chapters 2 and 3 of the book. Out of the Crystal Maze, which is a kind of official history of solid-state physics (Hoddeson et al. 1992). 

Understanding alloys in terms of electron theory. The band theory of solids had no impact on the thinking of metallurgists until the early 1930s, and the link which was eventually made was entirely due to two remarkable men - William Hume-Rothery in Oxford and Harry Jones in Bristol, the first a chemist by education and the second a mathematical physicist. 

All these properties of metals are consistent with a bonding description that places the valence electrons in delocalized orbitals. This section describes the band theory of solids, an extension of the delocalized orbital ideas... 

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. ... 

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. 

The results of the electron theory as developed for semiconductors are fully applicable to dielectrics. They cannot, however, be automatically applied to metals. Contrary to the case of semiconductors, the application of the band theory of solids to metals cannot be considered as theoretically well justified as the present time. This is especially true for the transition metals and for chemical processes on metal surfaces. The theory of chemisorption and catalysis on metals (as well as the electron theory of metals in general) must be based essentially on the many-electron approach. However, these problems have not been treated in any detail as yet. 

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. 

The energy states of gaseous atoms split because of the overlap between electron clouds. Obviously, therefore, atoms must come much closer before the clouds of the core electrons begin to overlap compared with the distance at which the clouds of outer (or valence) electrons overlap (Fig. 6.119). Hence, at the equilibrium interatomic distances, the energy levels of the core electrons (in contrast to the valence electrons) do not show any band structure and therefore will be neglected in the following discussion. This simplified picture of the band theory of solids will now be used to explain the differences in conductivity of metals, semiconductors, and insulators. 

ELECTRON GAS. The term electron gas is used to denote a system of mobile electrons, as. for example, the electrons in a metal that are free to move. In the free electron theory of metals, these electrons move through the metal in the region of nearly uniform positive potential created by the ions of the crystal lattice. This theory when modified by the Pauli exclusion principle, serves to explain many properties of metals, especially the alkali metals. For metals with more complex electronic structure, and semiconductors, the band theory of solids gives a better picture. 

Conductivity phenomena should not be considered in isolation the interplay between these and excitonic processes underlying photoconduction and luminescence must be taken into account. The reader should, therefore, possess a prior knowledge of those aspects of solid-state science that deal with periodic structures. Adequate explanations are given in numerous recent reviews and texts dealing with the band theory of solids, in general, and excitonic behaviour... 

The theoretical prediction of the optical absorption profile of a solid using first-principles methods has produced results in reasonable agreement with experiment for a variety of systems [2-4], For example, several ionic crystals were studied extensively, generally using the Hartree-Fock one-electron approximation [5], through the extreme-ultraviolet. Lithium fluoride was the focus of a particularly detailed comparison [6-8], providing excellent confirmation of the applicability of the band theory of solids for optical absorption. 

The band theory of solids has been developed to account for the electronic properties of materials. Two distinct lines of approach will be described. 

All these methods, based on methods used for the band theory of solids, appear to have a most welcome and useful application to large molecules. At present they probably give, for transition-metal complexes, the most worthwhile method other than the ab initio approach. Even this latter qualification is disputed by some workers. 

The band theory of solids provides a clear set of criteria for distinguishing between conductors (metals), insulators and semiconductors. As we have seen, a conductor must posses an upper range of allowed levels that are only partially filled with valence electrons. These levels can be within a single band, or they can be the combination of two overlapping bands. A band structure of this type is known as a conduction band. 

The structure of semiconducting solids provides a convenient basis for understanding the important electronic properties of these materials. The important optical and electrical characteristics of semiconducting solids arise from the delocalized electronic properties of these materials. To understand the origin of this electronic delocalization, we must consider the nature of the bonding within semiconductor crystals. The basic model that has been successfully used to describe the electronic structure of semiconductors is derived from the Band Theory of solids. Our treatment of band theory will be qualitative, and the interested reader is encouraged to supplement our discussion with the excellent reviews by... 

The importance of metals and semiconductors to modem technology is difficult to overestimate. We consider in this section the band theory of solids, which can account for many of the characteristic properties of these materials. 

We will carry on here the study of the degradation of the energy of these ions, electrons and photons. The properties of the irradiated solid, that exert negligible influence on the absorption of high energy radiation, must be taken into account in this part devoted to the absorption of low energy radiation. Let us recall that we are exclusively interested in this paper with nonmetallic solids (semiconductors and insulators) frequently, however, we will use the band theory of solids. 

The band theory of solids filled and 2N vacant anti-bonding m.o.s (at 0 K) ... 

The band theory of solids explains the three broad classes of electronic conductivity seen in nature in terms of the number density of charge carriers available in classes of solids. 

Bloch wave function - A solution of the Schrodinger equation for an electron moving in a spatially periodic potential used in the band theory of solids. 

As mentioned, the band theory of solids leads to a clear distinction between metals and insulators metals are associated with systems with partially filled bands, whereas insulators are associated with systems with completely full and empty bands. However, for a system that, according to a delocalized description of electrons, would have partially filled bands, it may be advantageous to keep these electrons locaHzed because in that way electrostatic repulsion between them could be minimized. These kind of localized states are usually known as Mott-Huhhard... 

In the previous chapters, we discussed various models of bonding for covalent and polar covalent molecules (the VSEPR and LCP models, valence bond theory, and molecular orbital theory). We shall now turn our focus to a discussion of models describing metallic bonding. We begin with the free electron model, which assumes that the ionized electrons in a metallic solid have been completely removed from the influence of the atoms in the crystal and exist essentially as an electron gas. Freshman chemistry books typically describe this simplified version of metallic bonding as a sea of electrons that is delocalized over all the metal atoms in the crystalline solid. We shall then progress to the band theory of solids, which results from introducing the periodic potential of the crystalline lattice. 

In summary, the band theory of solids can be used to explain the structure, spectroscopy and electrical properties of one-dimensional solids, such as the linear chain tetracyanoplatinates. However, most crystalline solids have translational symmetry in more than one dimension. Can band theory be extended to two and even three dimensions The answer is, of course, yes. However, the shapes of the bands become significantly more complicated as the number of dimensions increases. Let us just consider two dimensions for the moment and an imaginary lattice composed of only H atoms. In two dimensions, the wavefunction for the Bloch orbitals is given by Equation (I 1.27) ... 

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