The LCAO Bands

Energy bands are accurately known for most tetrahedral semiconductors Chapter 6 contains a discussion of the details of these bands as obtained by more accurate methods than will be discu.ssed here. In this section we shall continue the construction of electron states in terms of the bond orbitals introduced in the 

The numbers within the ovals represent tlie numbering of bond orbitals in a zincblendc lattice. The small circles represent atoms the numbers within them represent the distance below the plane of the figure in units of fi/4. 

A wave number must again be associated with every state for each wave number we construct eight different Bloch sums, four for the bond orbital types, 

In contrast, in CsCl wc sufficiently simplified the matrix elements that we were able to use single Bloch sums, of the kind in Eq. (3-19), as eigenstates we cannot do that here. 

For any wave number in the Brillouin Zone we seek the lowest energy solution by minimizing i//k H jjk K k iAk with respect to in direct analogy to the minimization of energy for the system based upon two basis states, as taken up in 

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In earlier chapters, it was seen how a qualitative energy-level diagram for the smallest repeating chemical point group, or lattice point (known to crystallographers as the basis, or asymmetric unit), can be used to approximate the relative placement of the energy bands in a solid at the center of the BZ. This is so because the LCAO-MO theory is equivalent to the LCAO band scheme, minus consideration of the lattice periodicity. The present chapter will investigate how the orbital interactions vary for different values of the wave vector over the BZ. 

It is of interest also to compare the true bands with the frce-electron bands, as we did in CsCl frce-clectron bands arc shown in Fig. 3-8,c. It is remarkable how close the relation is between free-electron bands and LCAO bands at low energies. The resemblance between the free-electron bands and the true bands is good at high energies, and clearly, if we wished to study highly excited states, the free-electron basis would provide a much better starting point for this than the. LCAO bands. 

Energy bands for germanium. Part (a) shows the LCAO bands fitted by Chadi and Cohen (1975), based on nearest-neighbor matrix elements. Part (b) shows the nearly-frce-electron bands. Two energy difTerences are shown in part (a) which can be matched immediately with part (b) to obtain predictions for the corresponding interatomic matrix elements. [After Froyon and Harrison, 1979.]... 

In the central panel, part (b), are shown the LCAO bands for body-centered cubic chromium, with four parameters adju.stcd slightly to accord with the bands for chromium from Fig. 20-1, which arc reproduced to the right, in part (c). The corresponding adjusted values appear in the Solid State Table, except for which was taken to be twice the valtic obtained from the Solid State Table. (The matrix element for nearest neighbors docs not enter.) In part (a) arc shown the bands obtained if all. second-neighbor interactions arc dropped. [Part (c), after Mattheiss, 1964.]... 

The reason for this dependence recently became very clear in a study of the bands of covalent solids by Froyen and Harrison (1979). They took advantage of the similarity of the LCAO bands and free-electron bands, noted in Fig. 2-2. By equating selected energy differences obtained in the two limits, they derived formulae that had this dependence for all of the interatomic matrix elements. We may in fact see in detail how this occurs by considering Fig. 2-2. The lowest band, labelled s in Fig. 2-2,a, was given by Eq. (2-5). For k in an. x-direction, it becomes E k) = — 4V2 — 2V2 cos ka, varying by 4V2 from F (where /c = 0) to X (where... 

We begin by expanding the LCAO bands for small /c we can, for example, expand the expressions in Eqs. (6-11) and (6-12). Expansion of Eq. (6-11), with b taken from (6-16), gives the heavy-hole bands,... 

A plot comparing values for the optical gap, as observed (vertical scale) and as predicted from the LCAO bands by Eq. (6-19). The line corresponds to observed values 1/3 as great as predicted values. Data are taken from Table 6-3. 

The IR bands in a number of nickel complexes of triaryl formazans have been assigned by Arnold and Schiele.415 A similar assignment of the electronic bands has been carried out.414 LCAO-MO calculations correlate well with these assignments417 and have been extended to include both inner ligand transitions as well as charge transfer bands and d—d transitions.418 EPR spectra have been used to study the nature of bonding in copper complexes of heterocyclic-containing formazans.419 Metal formazan complexes have also been studied by electrochemistry.283,398 420-422... 

Band Theory of Metals, Three approaches predict the electronic band structure of metals. The first approach (Kronig-Penney), the periodic potential method, starts with free electrons and then considers nearly bound electrons. The second (Ziman) takes into account Bragg reflection as a strong disturbance in the propagation of electrons. The third approach (Feynman) starts with completely bound electrons to atoms and then considers a linear combination of atomic orbitals (LCAOs). 

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. The presumption is, of course, that said electronic structures are more appropriately described from the standpoint of an MO (or Bloch)-type approach, rather than the Heitler-London valence-bond approach. This chapter will start with the many-body Schrodinger equation and the independent-electron (Hartree-Fock) approximation. This is followed with Bloch s theorem for wave functions in a periodic potential and an introduction to reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the free-electron model and the LCAO method, both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal-nonmetal transition is discussed in Chapter 7. 

Figure 4.7. The LCAO (tight-binding) band structure for ReOs. The dashed line represents the Fermi energy. To the far right is the density-of-states (DOS) curve for states of one spin. The occupied states (up to the Fermi level) are shaded gray. Note that the valence band is completely filled while the conduction band is partially filled. Hence, ReOs should be metallic.

Both the LCAO and NFE methods are complementary approaches to one-electron band theory, in which electrons are allowed to move independently of one another, through an averaged potential generated by all the other electrons. The true Hamiltonian is a function of the position of all the electrons in the solid and contains terms for all the interactions between these electrons, that is, all of the electron-electron Coulombic repulsions. Electronic motion is correlated the electrons tend to stay away from one another because of Coulombic repulsion. 

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