Band theory of solids

This approximation has the benefit of allowing us to introduce the Born-von Karman boundary conditions given by Equation (11.20), where N is the number of atoms in the ring  

In 1928, the Swedish physicist Felbc Bloch proved that the solutions to the Schrodinger equation for a periodic potential take the form shown in Equation (11.21), where e is the equation for a plane wave and u(x) is simply a function having the same periodicity as that of the lattice. 

Setting Equation (11.21) equal to Equation (3.35), as shown in Equation (11.22), we obtain the result given in Equation (11.23), which is equivalent to that obtained in Equation (11.14). Thus, the quantum number k is, in one sense, a wavenumber. 

Representation of (a) a linear chain of atoms and (b) the approximation of a linear chain of atoms as the enlarged area from a ring of atoms having a very large radius. 

Molecular orbital diagram for the fictional ring structures of Li metal having n atoms. [Reproduced from Hoffmann, R. Solids and Surfaces A Chemists View of Bonding in Extended Structures, Wiley-VCH New York, 1989. This material is reproduced with permission of John Wiley Sons, Inc. ] 

The next three subsections address the not-so-transparent concept of how and why bands form in solids. Three approaches are discussed. The first is a simple qualitative model. The second is slightly more quantitative and sheds some light on the relationship between the properties of the atoms making up a solid and its band gap. The last model is included because it is physically the most tangible and because it relates the formation of bands to the total internal reflection of electrons by the periodically arranged atoms. 

In the same way as the interaction between two hydrogen atoms gave rise to two orbitals, the interaction or overlap of the wave functions of atoms in a solid gives rise to energy bands. To illustrate, consider 10 atoms of Si in their ground state (Fig. 2.1 a). The band model is constructed as follows  

Assign four localized tetrahedral sp hybrid orbitals to each Si atom, for a total of 4 X 10 hybrid orbitals (Fig. 2. b). 

The overlap of each of two neighboring sp lobes forms one bonding and one antibonding orbital, as shown in Fig. 2.1 IJ. 

The two electrons associated with these two lobes are accommodated in the bonding orbitals (Fig. l.Wd). 

The properties of semiconductor electrodes and their differences compared with metal electrodes can be understood by examining their electronic structure. Semiconductors are unique in their electronic properties due to their band structure. The origin of the energy bands is generally discussed using band or zone theory, where the motion of a single electron in the crystal lattice is considered. It is assumed that there is no interaction between individual electrons or between the electrons and lattice points (the potential energy is zero). Therefore, the model is sometimes referred to as the nearly free electron model. 

The time-dependent Schrodinger equation for a free electron is written as 

For semiconductors (and insulators), the completely filled lower band is called the valence band (VB) and the higher energy band immediately above it is called the conduction band (CB). The average energy gap between these two bands is called the band gap (fig). Based on equation (9.2.8), the density of energy states above the CB edge is given by 

Similarly, the density of states below the VB edge f is given by 

Electron-hole pairs can also be introduced by substitution of acceptor and donor atoms by a process called doping. These doped semiconductors are called extrinsic semiconductors. 

---

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. 

S. L. Altmann, Band Theory of Solids. An Introduction from the Point of View of Symmetry. Clarendon, 1991. 

I. Solids—Spectra. 2. Energy-band theory of solids. 3. Solid state chemistry. 

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. 

Our results will be based on the one electron energy band theory of solids (13) that forms the basis for the present-day understanding of metal and semiconductor physics. It is the counterpart of the chemist s molecular orbital theory, and we shall try to relate our results back to the underlying atomic structure. 

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. 

We note that the notions optical and electrical gap are here used in the context of the classical band theory of solids and can be confusing in application to molecular (van der Waals bonded) solids, where they have the opposite meaning the optical gap reflects the energy of excitonic (localized) states, while the electrical gap stands for the lowest energy between free carrier states. 

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. 

R. Altmann, Band Theory of Solids An Introduction from the Point of View of Symmetry , Oxford University Press, Oxford, 1991 C. lung and E. Canadell, Description Orbitalaire de la Structure Electronique des Solides , Ediscience International, Paris, 1997. 

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. 

---