Bond polarization matrix elements

Notice that the gap vanishes for a homopolar semiconductor, which is true also for the exact bands, and if V were equal to V , it would simply be equal to times the predicted band width, p4K,. Thus, qualitatively, the gap is in very simple correspondence with the polarity of the system. The observed splittings are from 30 percent to 45 percent lower than those predicted in the K,-only theory by Eq. (6-12). The value is not modified as we add additional matrix elements within the Bond Orbital Approximation (Pantelides and Harrison, 1975). P rom Eqs. (6-3) and (6-4) we see that the situation is greatly complicated if the Bond Orbital Approximation is not u.scd (that is, bonding antibonding matrix elements are added), though of course the predicted gaps do go to zero as the polarity goes to zero in any case. 

The bond polarity parameter jam affects remarkably (in the first order) only the diagonal density matrix elements the off-diagonal ones acquire the corrections of the second order in /im. 

Bond orbitals are constructed ft om s/r hybrids for the simple covalent tetrahedral structure energies are written in terms of a eovalent energy V2 and a polar energy K3. There are matrix elements between bond orbitals that broaden the electron levels into bands. In a preliminary study of the bands for perfect crystals, the energies for all bands at k = 0 arc written in terms of matrix elements from the Solid State Tabic. For calculation of other properties, a Bond Orbital Approximation eliminates the need to find the bands themselves and permits the description of bonds in imperfect and noncrystalline solids. Errors in the Bond Orbital Approximation can be corrected by using perturbation theory to construct extended bond orbitals. Two major trends in covalent bonds over the periodic table, polarity and metallicity, arc both defined in terms of parameters from the Solid State Table. This representation of the electronic structure extends to covalent planar and filamentary structures. 

Eq, (4-12), as applied to hybrid states, is used in the last step it shows also that = 0. The remarkable result is that the matrix elements are not expected to vary with polarity in an isoelectronic scries. This feature also applies to the intra-atomic terms in the matrix element between bonding and antibonding orbitals on adjacent sites. A similar treatment of the matrix elements ofx indicates that they vary inversely with covalcncy 1 isoelectronic... 

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. 

Use these matrix elements to obtain the shift in bond energy to second order in 0. Use this to correct the value of C, obtained in Problem 8-1 for errors arising from the Bond Orbital Approximation. (Notice that = 20.) The finding that there arc only corrections for polar semiconductors carries over to the tetrahedral case. 

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