Band theory of semiconductors

Appendix 17 in M. Ladd (1994) Chemical Bonding in Solids and Fluids, Ellis Horwood, Chichester. 

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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. 

More extensive discussions of the band theory of semiconductors are given in standard texts such as Kittel (1996). Taylor and Heinonen (2002) is a more... 

Band theory of semiconductors is applicable and the theorem of effective mass is valid. 

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. 

For many years, during and after the development of the modem band theory of electronic conduction in crystalline solids, it was not considered that amorphous materials could behave as semiconductors. The occurrence of bands of allowed electronic energy states, separated by forbidden ranges of energy, to become firmly identified with the interaction of an electronic waveform with a periodic lattice. Thus, it proved difficult for physicists to contemplate the existence of similar features in materials lacking such long-range order. 

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. 

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. 

Illumination with light having a wavelength larger than the band gap of silicon generates a photocurrent under an anodic potential on an n-Si electrode but has essm-tially no effect on p-Si, as would be expected from the basic theories of semiconductor electrochemistry. However, the photocurrent may not be sustained because of the formation of an oxide film, which passivates the silicon surface to various degrees depending on the electrolyte composition. In solutions without fluoride species, the photocurrent is only a transient phenomenon before the formation of the oxide film. In fluoride solutions, in which oxide film is dissolved, a sustained photocurrent can be obtained. 

In the past, there have been two main types of interpretations of chemisorption and catalysis on chromia. One is more or less of the type given in this chapter (for example, 39, 56, 63a, 66). The other is based upon theories of semiconductors (for example, 68-70). Chromia is a semiconductor at high temperatures (20). No complete theory of chemisorption on chromia is possible at present one can only use approximate treatments. However, in our opinion, the first type of approximation (which is related to coordination chemistry and cry.stal field theory) is much more useful than the second type for reactions in reducing atmospheres at lower temperatures, say below 300°. Morin (70a) has given an analysis of transition metal oxides which indicates that the 3d band in a-Ci 203 is so narrow as to correspond to 3d charge carriers localized on the cations. 

Consequently, DFT is restricted to ground-state properties. For example, band gaps of semiconductors are notoriously underestimated [142] because they are related to the properties of excited states. Nonetheless, DFT-inspired techniques which also deal with excited states have been developed. These either go by the name of time-dependent density-functional theory (TD-DFT), often for molecular properties [147], or are performed in the context of many-body perturbation theory for solids such as Hedin s GW approximation [148]. 

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