The Band Model

The band model derives directly from quantum mechanics. It has proven highly successful to explain satisfactorily both the high conductivity of metals, and the properties of high mobility conventional inorganic semiconductors. Although it is far less appropriate to the case of organic compounds, it is worth giving here a concise overview of this basic model. 

The concept of energy bands in a solid can be physically understood by considering a one dimensional crystal, with a lattice constant a, and a nearly free electron [5], for which the bands are treated as a weak perturbation. The energy and wavefunc-tion of a free electron are respectively of the form 

The model above describes a free electron in a one dimensional crystal at the BZ boundaries. In the more general case, the solution of the time independent SchrSdinger equation (5) in a three dimensional crystal can be obtained with the help of the Bloch theorem, which states that if the potential energy V(r) is periodic, the solutions ti k(r) of 

Calculated band structures show a series of allowed and forbidden bands. The highest energy occupied allowed band is called the valence band (VB), and the lowest energy unoccupied one the conduction band (CB). Between these bands lies a forbidden band, the width of which defines the energy gap of the solid. Note that in a molecular solid, a one to one correspondence can be made between the VB and the CB on the one hand, and the Highest Occupied Molecular Orbital (HOMO) and Lowest Unoccupied Molecular Orbital (LUMO) of the isolated molecule on the other hand. 

Another important concept in band theory is that of effective mass. In a semiconductor, most of the charges reside at the edge of the conduction or valence band. Band edges can be approximated to parabolic bands, by analogy with the free electron dispersion law, Eq. (3) 

As already mentioned the possibility of delocalization of electrons within a macroscopic aggregate is a characteristic of the crystalline solid state. The outer electrons of metals, in particular, behave as a confined electron gas. This is readily recognized by considering an arrangement of 4 Na atoms. Such an ensemble represents a markedly electron-deficient state (one valence electron per Na) the energetically relevant mesomeric structures are  

For a further description, we use the single-electron approximation and confine a single electron for simplicity in a one-dimensional box of length L. We set the potential energy within the box to zero, while it is infinite at the walls of the box. 

As a result of this the Hamiltonian operator reduces to the operator for the kinetic energy that is —h (5/9x) /(87r m). 

The Schrodinger equation for the wave function k) leads to a hnear homogeneous differential equation of the form ( / ) k) oc — k) for which sine and cosine functions are relevant solutions. At the two ends of our box (k k) = 0 and, hence, k) = 0 must hold. The disappearance of the function at x = 0 only permits the sine function  

The parameter k designates the wave vector which is reduced here to a component in the x direction A is the associated wavelength (A = 27r/k). The relationship between k (and hence also A) and the energy eigenvalues e is obtained by substitution in the Schrodinger equation (—h /87r m)(5/dx) k) = e k) as 

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Fig. 8. (a) Energy levels for the band model of silver haUde crystals. The band bending at the surface (-) is exaggerated. The extent of bending is at... 

In this diagram, we show the band model structure at the juncture of two metals, each of which has its own Fermi Level. The Fermi Level is the energy level of the electrons contained in the metal. That is- when metal atoms (each... 

The electronic conductivity of metal oxides varies from values typical for insulators up to those for semiconductors and metals. Simple classification of solid electronic conductors is possible in terms of the band model, i.e. according to the relative positions of the Fermi level and the conduction/valence bands (see Section 2.4.1). 

Although the band model explains well various electronic properties of metal oxides, there are also systems where it fails, presumably because of neglecting electronic correlations within the solid. Therefore, J. B. Good-enough presented alternative criteria derived from the crystal structure, symmetry of orbitals and type of chemical bonding between metal and oxygen. This semiempirical model elucidates and predicts electrical properties of simple oxides and also of more complicated oxidic materials, such as bronzes, spinels, perowskites, etc. 

It has also to be remembered that the band model is a theory of the bulk properties of the metal (magnetism, electrical conductivity, specific heat, etc.), whereas chemisorption and catalysis depend upon the formation of bonds between surface metal atoms and the adsorbed species. Hence, modern theories of chemisorption have tended to concentrate on the formation of bonds with localized orbitals on surface metal atoms. Recently, the directional properties of the orbitals emerging at the surface, as discussed by Dowden (102) and Bond (103) on the basis of the Good-enough model, have been used to interpret the chemisorption behavior of different crystal faces (104, 105). A more elaborate theoretical treatment of the chemisorption process by Grimley (106) envisages the formation of a surface compound with localized metal orbitals, and in this case a weak interaction is allowed with the electrons in the metal. 

As mentioned in Sec. 1.3, the electrochemical potential of electrons in condensed phases corresponds to the Fermi level of electrons in the phases. There are two possible cases of electron ensembles in condensed phases one to which the band model is applicable (in the state of degenera< where the wave functions of electrons overlap), and the other to which the band model cannot apply (in the state of nondegeneracy where no overlap of electron wave functions occurs). In the former case electrons or holes are allowed to move in the bands, while in the latter case electrons are assumed to be individual particles rather than waves and move in accord with a thermal hopping mechanism between the a4jacent sites of localized electron levels. 

We now consider the relationship which connects the electrochemical potential of electrons in the hopping model with that in the band model. The total concentration, N, of electron sites for the hopping model may be replaced by the effective state density, JVc, for the band model. For the two models thereby we obtain from Eqn. 2-27 the following equation ... 

Thus, for the band model the activity coefficients, y. and Vh, of electrons and holes in the conduction and valence bands are, respectively, given in Eqn. 2-33 ... 

The passive film is composed of metal oxides which can be semiconductors or insulators. Then, the electron levels in the passive film are characterized by the conduction and valence bands. Here, we need to examine whether the band model can apply to a thin passive oxide film whose thickness is in the range of nanometers. The passive film has a two-dimensional periodic lattice structure on... 

Observations with microcrystals of semiconductor sihcon have shown that the transition from the model of localized electron levels quantum size) to the band model of delocalized electron levels (microscopic or macroscopic size) occurs at about 2 nm [Kanemitsu-Uto-Masumoto, 1993]. It appears, then, that the band model can apply to passive films thicker than 2 nm. Further, accoimting that the film interacts with the substrate metal, the band model may apply even to the range of thickness less than 2 nm. 

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. 

When sodium lignosulfonate or sulfur lignin are compounded, for instance, with iodine or bromine, complexes supposedly form (16-17). These systems are conductors with mixed ionic and electronic nature. Presumably they are charge transfer complexes, since the electronic conductivity predominates (18-19). These compounded materials form charge transfer structures (20). Water is supposed to introduce ionic conductivity to the system. Impurities affect conductivity, too (21). In any case, the main models of conductivity are probably based on the band model and/or the hopping model. 

It is known that within the framework of the band model variation of the potential with coordinate in a semiconductor is equivalent to the bending of energy bands. The bands near the surface are bent downward if sc > 0 and... 

The above discussion of the band model and its application to a description of equilibrium chemisorption has been brief and incomplete. It is not the principal purpose in this paper to discuss surface barrier effects in equilibrium chemisorption, but rather surface barrier effects in the irreversible region of chemisorption. However, before we begin the consideration of these latter effects, we will digress and examine the basic properties of zinc oxide, which material will be used both as motivation and as illustration in the following text. 

One can see the fruitfulness of the hopping and band model synthesis for heterogeneous polymer structures. In this case we may consider the charge transfer inside the conjugated section of the chain in the frame of the band model and transitions between these parts of the chain as jumps or as an activated surmounting of the barriers. 

On the terminology of the band model c- bonds form the completely filled low band, and rc-bonds make the partially filled band, which defines the electronic properties of the polyacetylene. 

For poly(phenylacetylene) doped with acceptor iodine molecules the long wavelength bands at 940 nm were ascribed to CT complexes [178, 179]. The band model with trap controlled conductivity was used to the photoconductive... 

This situation can be expressed in terms of the band model as shown in Fig. 1.24. Stoichiometric NiO is an intrinsic semiconductor, having an energy gap of Eq (=Eq—E ). Non-stoichiometric Nij O, which has metal vacancies or electronic defects, has an acceptor level A between the valence... 

From Figure 5, with Ep (SCE) defined as zero, and (as common with the band model) the potentials more positive toward the bottom of the figure, we find ... 

We must emphasize that the processes described in the language of the band model (electron excitation, photogeneration of electron hole pairs, migration and capture of charge carriers) are essentially processes whereby chemical bonds in the solid and at the surface of the solid are excited, broken and formed again. 

The promotion of an electron or defect electron to the conduction band or valence band, respectively, is only a part of the whole reaction. This excitation is identical with the destruction of a bond. In the band model only that part of the bond destruction is described which is connected with electron movement the shift of the cores from the energy valleys is not taken into account. The activation energies of the conductivity and of the chemical reaction are proportional but not identical. 

Because of these properties, ZnO has been considered to be an ideal material for testing the electronic theories of catalysis, specifically the band model (based on the collective properties of the solid) or the localized site... 

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