The Band Structure of Solids

A quantitative description of this band structure can be obtained using what is known as a tight-binding model. 

In the following, we will describe the structure of solids in more qualitative terms. Let us consider a metal like aluminum to get a feeling for what happens. Aluminum has the electronic configuration ls 2s 2p 3s 3p, which sometimes is written as [Ne]3i 3p. This means that aluminum has three electrons circling a neonlike closed electronic shell. 

In catalysis, the interest is mainly focused on the transition metals. They distinguish themselves from other metals by having partially or completely filled i-shells. The orbitals of d electrons have specific shapes and very localized spatial extent. Hence, the overlap between d electronic states is much smaller than for s and p electronic states. When the interaction is weak, one still gets a continuum of states that form a band structure, but the band is much narrower. 

FIGURE 124 Schematics showing what happens in the limit of infinitely many overlapping 

The behavior of the electrons in A1 resembles the behavior of a gas of free electrons in three dimensions. To see this, we recall the energy solutions derived in basic quantum mechanics for a free electron gas  

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This is named the Hill determinant. After solving, the resulting secular determinant for the root of E (k) provides a more accurate method for calculating the band structure of solids, where n = 1 refers to the first band, n = 2 to the second, and so on. 

We collect here some simple treatments of arrays of atoms or particles in one dimension these will be quite useful in later analogies with the band structure of solids. In particular, the notions of Brillouin30 zone, of band edges, of... 

The band structure of solids has been studied theoretically by various research groups. In most cases it is rather complex as shown for Si and GaAs in Fig. 1.5. The band structure, E(kf is a function of the three-dimensional wave vector within the Brillouin zone. The latter depends on the crystal structure and corresponds to the unit cell of the reciprocal lattice. One example is the Brillouin zone of a diamond type of crystal structure (C, Si, Ge), as shown in Fig. 1.6. The diamond lattice can also be considered as two penetrating face-centered cubic (f.c.c.) lattices. In the case of silicon, all cell atoms are Si. The main crystal directions, F —> L ([111]), F X ([100]) and F K ([110]), where Tis the center, are indicated in the Brillouin zone by the dashed lines in Fig. 1.6. Crystals of zincblende structure, such as GaAs, can be described in the same way. Here one sublattice consists of Ga atoms and the other of As atoms. The band structure, E(k), is usually plotted along particular directions within the Brillouin zone, for instance from the center Falong the [Hl] and the [HX)] directions as given in Fig. 1.5. 

The tight-binding model is an approach to the electronic band structure from the atomic borderline case. It describes the electronic states starting from the limit of an isolated atom. It is assumed that the Fourier transform of the Bloch function can be approximated by the linear combination of atomic orbitals (LCAO). Thus, the band structure of solids is investigated starting from the Hamiltonian of an isolated atom centered at each lattice site of the crystal lattice. 

The band structure of solids accounts for their electrical properties, in order to move through the solid, the electrons have to change from one quantum state to another. This can only occur if there are empty quantum states with the same energy, in general, if the valence band is full, electrons cannot change to new quantum states in the same band. For conduction to occur, the electrons have to be in an unfilled band - the conduc-... 

For the acid-base interaction in solutions, in 1963, Pearson proposed the hard-soft acid-base (HSAB) principle to describe some basic rules about the kinetics and equilibrium of the reaction. In this paper, we attempt to apply the HSAB principle to solid interactions with the aid of the frontier orbital method. We shall first describe the HSAB principle as it has been evolved in recent years " and then the band structures of solids. After we demonstrate the compatibility between the HSAB principle and the band structures in the solid state, we then illustrate with several examples of adhesion and tribointeractions between metals and... 

Solids can be classified as metals, semimetals, intrinsic semiconductors and insulators. The band structures of solids can be illustrated in Fig. 3. Monovalent metals, e.g., Na , have a partially filled valence band, the lower half of which is occupied. The Fermi level is in the valence band but at the top of the occupied orbitals. Furthermore, there is still an energy gap between the valence band and the conduction band (unoccupied MO). In some metals, such as the bivalent metals, the valence band is full but overlaps a higher unoccupied conduction band. In this case, the Fermi level is in the conduction band and the overlapped valence band. Thus, the electrons close to the Fermi level are still free to move as the extra bands supply the unoccupied states. In the latter case, there appears to be no minimum energy gap. Eg" y which is generally reported in the literature. However, it is not... 

Now let us examine again the band structures of solids in Fig. 3. In Fig. 3a, the structure of metals with zero or low Eg signifies that all metals are "soft." However, they do not appear to be as "soft" as the q values calculated from the A-I data (Table 2). Indeed, metals have been classified as amphoteric materials. Most of the metals are "soft" acids, and some of them "soft" bases. When two metals are brought together into close contact, one of them assumes to be an acid while the other a base. In a recent paper by Cain et al. the interaction at the Cu/Cr interface has been treated as an acid-base Interaction. In this case, a soft base (Cu) and a soft acid (Cr) reacts preferentially. In fact, Cr (q = 3.1 eV) is very slightly softer than Cu (q= 3.3 eV) (See Table 2). 

This is not only important to understand the chemistry of loosely bound complexes but also contributes to elucidating the molecular structure in the transition region between free molecules and solids. One example for this is the formation of clusters in supersonic alkali beams where molecules Na have been observed from x = 2 to x = 12 [10.16]. The spectroscopy of such multimers (clusters) yields dissociation energies, ionization energies, and vibrational structure as a function of the number x of atoms in the cluster. The comparison of these figures with the values in the solid allows the proof of theoretical models, which explain the transition from molecular orbitals to the band structure of solids. 

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