The band theory

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. 

A book edited by Levinson (1981) treated grain-boundary phenomena in electroceramics in depth, including the band theory required to explain the effects. It includes a splendid overview of such phenomena in general by W.D. Kingery, whom we have already met in Chapter I, as well as an overview of varistor developments by the originator, Matsuoka. The book marks a major shift in concern by the community of ceramic researchers, away from topics like porcelain (which is discussed in Chapter 9) Kingery played a major role in bringing this about. 

The high electrical conductivity of metals as well as the high electron (and hole) mobility of inorganic covalently bound semiconductors have both been clarified by the band theory [I9, which slates that the discrele energy levels of individual atoms widen in the solid stale into alternatively allowed and forbidden bands. The... 

Slater, J. C., Revs. Modern Phys. 25, 199, "Ferromagnetism and the band theory."... 

It was pointed out in my 1949 paper (5) that resonance of electron-pair bonds among the bond positions gives energy bands similar to those obtained in the usual band theory by formation of Bloch functions of the atomic orbitals. There is no incompatibility between the two descriptions, which may be described as complementary. It is accordingly to be expected that the 0.72 metallic orbital per atom would make itself clearly visible in the band-theory calculations for the metals from Co to Ge, Rh to Sn, and Pt to Pb for example, the decrease in the number of bonding electrons from 4 for gray tin to 2.56 for white tin should result from these calculations. So far as I know, however, no such interpretation of the band-theory calculations has been reported. 

The SCF method for molecules has been extended into the Crystal Orbital (CO) method for systems with ID- or 3D- translational periodicityiMi). The CO method is in fact the band theory method of solid state theory applied in the spirit of molecular orbital methods. It is used to obtain the band structure as a means to explain the conductivity in these materials, and we have done so in our study of polyacetylene. There are however some difficulties associated with the use of the CO method to describe impurities or defects in polymers. The periodicity assumed in the CO formalism implies that impurities have the same periodicity. Thus the unit cell on which the translational periodicity is applied must be chosen carefully in such a way that the repeating impurities do not interact. In general this requirement implies that the unit cell be very large, a feature which results in extremely demanding computations and thus hinders the use of the CO method for the study of impurities. 

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. 

Diffusion and migration in solid crystalline electrolytes depend on the presence of defects in the crystal lattice (Fig. 2.16). Frenkel defects originate from some ions leaving the regular lattice positions and coming to interstitial positions. In this way empty sites (holes or vacancies) are formed, somewhat analogous to the holes appearing in the band theory of electronic conductors (see Section 2.4.1). 

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. 

According to the band theory, the motion of electrons under an external electric field is possible only for electrons in partially filled energy bands. The probability function. 

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 Band Theory of Crystalline Solids. Consider a crystalline solid. The atoms are arranged according to a three-dimensional pattern (or lattice) in which they have equilibrium interatomic distances. A thought experiment is now performed. The lattice is expanded, i.e., the interatomic distances are increased. 

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. 

A description of charge transport in molecular conductors has been adapted from the band theory of semiconductors (79MI11300). The conductivity is given by the product of the concentration of charge carriers, expressed in the format of an activation energy, and the carrier mobility which is inversely proportional to an exponent of the absolute temperature. Both expressions contain parameters specific to each sample and the general approach is of little use in the design and synthesis of new materials. 

The electrical properties of many solids have been satisfactorily explained in terms of the band theory . Briefly, the motion of an electron detached from its parent atom but free to move in a periodically varying potential field, such as that existing between atoms on a crystal lattice, is expressed in terms of a wave function (Boch Function). This particular... 

According to the band theory model, this process corresponds to thermal excitation across the forbidden gap and as such the law of mass action may be applied to the equilibrium between creation and recombination of holes and electrons. 

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. 

---