# Band Theory of Solids

## Band theory of solids

A quantum-mechanical theory of the motion of electrons in solids which predicts certain restricted ranges, or bands, for the electron energies.

If the atoms of a solid are separated from each other to such a distance that they do not interact, the energy levels of the electrons will then be those characteristic of the individual free atoms, and thus many electrons will have the same energy. As the distance between atoms is decreased, the electrons in the outer shells begin to interact, thus altering their energy and broadening the sharp energy level out into a range of possible energy levels called a band. One would expect the process of band formation to be well advanced for the outer, or valence, electrons at the observed interatomic distances in solids. Once the atomic levels have spread into bands, the valence electrons are not confined to individual atoms, but may jump from atom to atom with an ease that increases with the increasing width of the band.

Although energy bands exist in all solids, the term energy band is usually used in reference only to ordered substances, that is, those having well-defined crystal lattices. In such a case, an electron energy state can be classified according to its crystal momentum p or its electron wave vector k = p /&planck; (where &planck; is Planck's constant h divided by 2π). If the electrons were free, the energy of an electron whose wave vector is k would be as shown in the equation below, where E0 is the energy of

the lowest state of a valence electron and m0 is the electron mass. In a crystal, however, the electrons are not free because of the effect of the crystal binding and the forces exerted on them by the atoms; consequently, the relation E( k ) between energy and wave vector is more complicated. The statement of this relationship constitutes the description of an energy band.

The bands of possible electron energy levels in a solid are called allowed energy bands. There are also bands of energy levels which it is impossible for an electron to have in a given crystal. Such bands are called forbidden bands, or gaps. The allowed energy bands sometimes overlap and sometimes are separated by forbidden bands. The presence of a forbidden band immediately above the occupied allowed states (such as the region A to B in the illustration) is the principal difference in the electronic structures of a semiconductor or insulator and a metal. In the first two substances there is a gap between the valence band or normally occupied states and the conduction band, which is normally unoccupied. In a metal there is no gap between occupied and unoccupied states. The presence of a gap means that the electrons cannot easily be accelerated into higher energy states by an applied electric field. Thus, the substance cannot carry a current unless electrons are excited across the gap by thermal or optical means.

Electron energy E versus wave vector k for a monatomic linear lattice of lattice constant a

Under external influences, such as irradiation, electrons can make transitions between states in the same band or in different bands. The interaction between the electrons and the vibrations of the crystal lattice can scatter the electrons in a given band with a substantial change in the electron momentum, but only a slight change in energy. This scattering is one of the principal causes of the electrical resistivity of metals. See Electrical resistivity

An external electromagnetic field (for example, visible light) can cause transitions between different bands. Here momentum must be conserved. Because the momentum of a photon hv/c (where v is the frequency of the light and c its velocity) is quite small, the momentum of the electron before and after collision is nearly the same. Such a transition is called vertical in reference to an energy band diagram. Conservation of energy must also hold in the transition, so absorption of light is possible only if there is an unoccupied state of energy hv available at the same k as the initial state. These transitions are responsible for much of the absorption of visible and near-infrared light by semiconductors.

The results of energy-band calculations for ordinary metals usually predict Fermi surfaces and other properties that agree rather well with experiment. In addition, cohesive energies and values of the lattice constant in equilibrium can be obtained with reasonable accuracy, and, in the case of ferromagnetic metals, calculated magnetic moments agree with experiment. However, there are significant discrepancies between theoretical calculations and experiments for certain types of systems, including heavy-fermion systems (certain metallic compounds containing rare-earth or actinide elements), superconductors, and Mott-Hubbard insulator (compounds of 3d transition elements, for which band calculations predict metallic behavior). Also, band calculations for semiconductors such as silicon, germanium, and gallium arsenide (GaAs) predict values for the energy gap between valence and conduction bands in the range one-half to two-thirds of the measured values. In all these cases, the failures of band theory are attributed to an inadequate treatment of strong electron-electron interactions.

McGraw-Hill Concise Encyclopedia of Physics. © 2002 by The McGraw-Hill Companies, Inc.
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Band Theory of Solids

a branch of quantum mechanics that examines the motion of electrons in solids. Free electrons may have any energy—their energy spectrum is continuous. Electrons that belong to isolated atoms have definite discrete energy values. In solids the energy spectrum of electrons is quite different and consists of separate allowed bands separated by bands of forbidden energies.

Band theory is the basis of the modern theory of solids. It has made it possible to understand the nature of and explain the most important properties of metals, semiconductors, and dielectrics.

## band theory of solids

[′band ‚thē·ə·rē əv ¦säl·ədz]
(solid-state physics)
A quantum-mechanical theory of the motion of electrons in solids that predicts certain restricted ranges or bands for the energies of these electrons. Also known as energy-band theory of solids.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Working from lectures, Swanson starts with the foundations of quantum physics, including Fourier transforms, the postulatory basis, operators, and properties, then moves to the Schrodinger equation in one dimension and three dimensions, total angular momentum, approximation methods, atomic spectroscopy, quantum statistics, band theory of solids, emission, absorption, lasers, and scattering theory.

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