Energy bands in solids

The extension of the fashionable energy band theory to other crystals than adaman-toid semiconductors is frequently justified with the theorem that the one-electron 

Finally, we derived the BLOCH FUNCTION to show that these energy bands, in reciprocal space, do have some validity in quantum mechanics. It also gives Insight as to the nature of the Fermi level. We also illustrated band models in 5.3.9. What is Important to realize that the valence band there is drawn in two dimensions. Actually, it follows the Brlllouin Zone or k-space of the crystal lattice in three-dimensions. The Fermi level surface is also affected by both k-space and temperature. It is constrained by reciprocal space, just as the BrUlouin Zone is. We use the band model to illustrate certain aspects of each unique crystal. Otherwise, the required model would be quite complex, particularly those crystals with low symmetry. We usually illustrate some specific defect and the band model immediately adjoining it. For a phosphor, this would be the activator (impurity) center. Since we have already (Chapter 2) examined various point defects, let us now illustrate them within the periodic lattice as a function of the energy bands and the Band Model. 

The effect of a periodic lattice upon the wavefunctions of an electron is illustrated in the following diagram, given as 5.3.68. on the next page. Note that the positions of the individual atoms within the lattice are shown, i.e.- lattice sites . 

At the left side, we have shown the energy levels of a single electron as the energy, E, increases. When lattice sites are imposed, we get a series of bands (shown horizontedly) as E increases. Note also the unoccupied zones, indicated by the hatched area. Occupation of the allowed bands is a function of the types and numbers of electrons present. At the bottom is the wavefunction associated, as a function of atom-position in the lattice. 

Effect of a Periodic Lattice on the Wave Function of an Electron 

Let us now construct a band model for our semi-conductor, Ge. We illustrate both p-lype and n- e defects, as follows  

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Transitions between Electron Energy Bands in Solids... 

Cornwell, J.F. Group theory and electronic energy bands in solids. In Wolhlfarth, E.P. (ed.) Selected topics in solid state physics, vol.X, p. 52, North Holland, London (1969)... 

J.F. Cornwell Group Theory and Electronic Energy Bands In Solids (North-Holland, Amsterdam 1969) ... 

Figure Al.3.6. An isolated square well (top). A periodic array of square wells (bottom). This model is used in the Kronig-Peimey description of energy bands in solids.

We will steirt by examining the methods of calculation of energy bands in solids. 

As stated above, the basic equations used to describe electron energy bands in solids are called the BLOCH FUNCTIONS. These are a part of quantum mechanics, so we begin by borrowing one of its more famous concepts, that of "The Particle in a Box . 

An alternative approach to extend the molecular orbital method is offered by the work of Hubbard for the study of narrow energy bands in solids with the aim to study magnetism. The main idea of this work is to analyze the many-electron problem for the case of separated atoms, which means the limit of zero bandwidth. 

FIGURE 30.2 Schematic of electron energy bands in solids. The valence band (VB) and conduction band (CB) are indicated. 

Duffy, J.A. (1990) Bonding, Energy Levels and Bands in Inorganic Solids, Longman Scientific and Technical, Harlow, Essex, UK. Straightforward description of energy bands in solids. 

Energy bands in solids 189 3.1.6. Other alkali-metal compounds 212... 

The introduction to Chap. Two on energy bands in solids and their association to electrostatic adhesion should be referred to, since static electrification has the same bases as electrostatic adhesion forces. Krupp (1971) has given a convenient diagram showing the interactions between static electrification and solid-state physics. See Fig. 6-1. 

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