Interatomic matrix elements

The two Ti curves do not covary in the quasigap. If the bonding was only between different atoms then the curves should covary. This indicates bonding between similar atoms. As noted above, the dd interatomic matrix elements for the Tii-Tii interaction are about 80 % larger than that in A3, supporting the idea of direct dd bonding. 

TABLE 5.1. Interatomic Matrix Elements for the Transition Metal Perovskite Oxides... 

IN THIS PART of the book, we shall attempt to describe solids in the simplest meaningful framework. Chapter 1 contains a simple, brief statement of the quantum-mechanical framework needed for all subsequent discussions. Prior knowledge of quantum mechanics is desirable. However, for review, the premises upon which we will proceed are outlined here. This is followed by a brief description of electronic structure and bonding in atoms and small molecules, which includes only those aspects that will be directly relevant to discussions of solids. Chapter 2 treats the electronic structure of solids by extending the framework established in Chapter 1. At the end of Chapter 2, values for the interatomic matrix elements and term values are introduced. These appear also in a Solid State Table of the Elements at the back of the book. These will be used extensively to calculate properties of covalent and ionic solids. 

To make the discussion of the electronic structure of diatomic molecules quantitative, it is necessary to have values for the various matrix elements. It will be found that for solids, a reasonably good approximation of the interatomic matrix elements can be obtained from the formula where d is the... 

Dimensionless cocdicicnts in Eq. (2-13) determining approximate interatomic matrix elements. 

The variation of matrix elements over the band is more subtle. The values of Xi(0) arc very large in covalent solids as compared to atoms, a fact that will be discussed in the next section, and their size would suggest that the interatomic matrix elements, such as those in Eq. (4-14), are dominant. The simplest approximation is the neglect of any dependence of the matrix elements upon initial and final states one should notice that this will give different answers depending upon whether one assumes that matrix elements of djdx are constant or that the i- arc constant. As usual, the assumption of equal probability on the basis of lack of information is not unique. A common approximation is that the matrix elements of X (or 1/m times the matrix elements of d/dx) are constant, which from Eq. (4-9) gives a X2( ) value directly proportional to the joint density of states ... 

The last form is obtained by using the expressions for the interatomic matrix elements from the Solid State Table. For polar semiconductors the splitting becomes 2(Vl+ with the polar energy- m contrast to the hybrid polar... 

Accurate energy bands obtained from first principles by computer calculation are available for most covalent solids. A display of the bands obtained by the Empirical Pseudopotential Method for Si, Ge, and Sn and for the compounds of groups 3-5 and 2-6 that are isoclec-tronic with Ge and Sn shows the principal trends with mctallicity and polarity. The interpretation of trends is refined and extended on the basis of the LCAO fitting of the bands, which provides bands of almost equal accuracy in the form of analytic formulae. This fitting is the basis of the parameters of the Solid State Table, and a plot of the values provides the test of the d dependence of interatomic matrix elements. 

The energy bands of tetrahedral solids have been studied in terms of LCAO s for many years the first study was that of Hall (1952), who used a Bond Orbital Approximation, keeping only nearest-neighbor interbond matrix elements in order to obtain analytic expressions for the bands over the entire Brillouin Zone. The recent study by Chadi and Cohen (1975), which did not use either of Hall s approximations, is the source of the interatomic matrix elements between. v and p orbitals, which appear in the Solid Stale Table. Pantelides and Harrison (1975) used the Bond Orbital Approximation but not the nearest-neighbor approximation and found that accurate valence bands could be obtained by adjusting a few matrix elements at the same time very clear interpretations of many features of the bands were achieved. The main features of the Pantiledes-Harrison interpretation will be presented here. 

To establish values for the parameters and we must decompose the bond orbitals into hybrids, as will be illustrated in Fig. 6-4, taking the coefficients of the two hybrids from Eq. (3-13) and using the polarities obtainable from Table 4-1. Interatomic matrix elements will also be included in Fig. 6-4. However, wc look first at the contributions arising only from the matrix elements P, and V i between hybrids on the same atoms. We call them V,-only bands. In terms of these. 

Furthermore, the contribution of the interatomic matrix element to the nearest-neighbor antibond matrix element is changed in sign, as can be seen in Fig. 6-7. Then Eq. (6-15) becomes... 

The value F, does not vary greatly with row in the periodic table, but V, an interatomic matrix element, does decrease with increasing lattice spacing. Evaluation of Eq. (6-35) shows this trend for diamond. Si, Gc, and Sn ( — 8.5, —0.2, 1.3, and 1.4 cV, respectively), while values obtained from band calculations arc -5.8, —0.5, 2.5, and 2.7, respectively -the value for diamond is from Herman ct al., 1967, and the others arc from Herman et al., 1967, 1968). Thus in diamond and silicon, another band has dropped below the simple conduction band shown in Fig. 6-9. As we noted in Section 6-C, the reordering of levels at F is a special feature of materials of low metallicity and low polarity. 

The calculation of vibration spectra in terms of force constants is similar to the calculation of energy bands in terms of interatomic matrix elements. Force constants based upon elasticity lead to optical modes, as well as acoustical modes, in reasonable accord with experiment, the principal error being in transverse acoustical modes. The depression of these frequencies can be understood in terms of long-range electronic forces, which were omitted in calculations tising the valence force field. The calculation of specific heat in terms of the vibration spectrum can be greatly simplified by making a natural Einstein approximation. 

Poole, Liesegang, Leckey, and Jenkin (1975) have reviewed published band calculations for the alkali halides and tabulated the corresponding parameters obtained by various methods. Pantclidcs (1975c) has used an empirical LCAO method that is similar to that described for cesium chloride in Chapter 2 (see Fig. 2-2), to obtain a universal one-parameter form for the upper valence bands in the rocksalt structure. This study did not assume only one important interatomic matrix clement, as we did in Chapter 2, but assumed that all interatomic matrix elements scale as d with universal parameters. Thus it follows that all systems would have bands of exactly the same form but of varying scale. That form is shown in Fig. 14-2. Rocksalt and zincblende have the same Brillouin Zone and symmetry lines, which were shown in Fig. 3.6. The total band width was given by... 

We may estimate the ion softening in perturbation theory as in Section 14-B, using the expansion of the wave function to first order in the interatomic matrix elements, Eq. (14-4), which we rewrite... 

In Chapter 2, it was mentioned that there is a strong resemblance between bands obtained from nearest-neighbor interactions in the LCAO approximation and the bands obtained from nearly-free-electron theory. In fact, formulae for the interatomic matrix elements based upon that similarity were used to estimate properties of covalent and ionic solids in the chapters that followed, Now that a... 

Eqs. (6-1) through (6-6), in terms of interatomic matrix elements defined in Eq. (3-26). We shall utilize the bands at points of highest symmetry, These are the s-like levels at F with energy a,+ 4K5 , the p-like levels at F with energy fip (4Fpp 4- 8Kpp )/3, and the two valence-band levels at X. We do not use the conduction-band levels at X, which are in poor agreement with the true bands. (This was shown in Fig. 3-8.) When we set the first six levels mentioned equal to their nearly-free-elcctron counterparts, the six equations may be solved to obtain the four interatomic matrix elements and the term values e, and fip. 

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 covalent solids, the Jones Zone gap should be identified with the principal optical absorption peak previously identified with LCAO interatomic matrix elements. Thus it allows a direct relation between the parameters associated with the LCAO and with the pseudopotential theories. It is best, however, to simplify the pseudopotential analysis still further before making that identification. 

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