Pseudopotential form factor

Thus, the screened pseudopotential form factor of aluminium normalized by the Fermi energy will approach q = 0 at —2/3 as observed in Fig. 5.12. 

This completes the specification of the pseudopotential as a perturbation in a perfect crystal. We have obtained all of the matrix elements between the plane-wave states, which arc the electronic states of zero order in the pseudopotcntial. We have found that they vanish unless the difference in wave number between the two coupled states is a lattice wave number, and in that case they are given by the pseudopotential form factor for that wave number difference by Eq. (16-7), assuming that there is only one ion per primitive cell, as in the face-centered and body-centered cubic structures. We discuss only cases with more than one ion per primitive cell when we apply pseudopotential theory to semiconductors in Chapter 18. Tlicn the matrix element will be given by a structure factor, Eq. (16-17),... 

The ratio wJEf) of pseudopotential form factor to frec-clcctron Fermi energy for silicon, showing that the direct Joncs-Zone diffraction [220J should be weak. 

This view runs into difficulties that have only recently been completely resolved. The principal one is that the pseudopotential form factor happens to be very small for this particular diffraction. In Fig. 18-4 is sketched the pseudopotenlial form factor for silicon obtained from the Solid Stale Table the form factor that gives the [220] diffraction is indicated. Because it lies so close to the crossing, it is small and the diffraction is not expected to be strong. Heine and Jones (1969) noted, however, that a second-order diffraction can take an electron across the Jones Zone this could be a virtual diffraction by a lattice wave number of [1 ll]27t/fl followed by a virtual diffraction by [I lT]27c/a. (Virtual diffraction is an expression used to describe terms in perturbation theory it can be helpful but is not essential to the analysis here.) This second-order diffraction would involve the large matrix elements associated with the [11 l]27t/a lattice wave number indicated in Fig. 18-4, and Heine and Jones correctly indicated that these are the dominant matrix elements. 

The identification of the covalent energy with a pseudopotential has relevance to the f/ -dependence that has recurred throughout our studies. The pseudopotential form factors, when divided by the Fermi energy and plotted against, as in Fig. 18-3, are almost a universal eurve, approaehing —2/3 at small q and crossing the axis near q/k = 1.6 or 1.7 in most systems. To the extent that this curve is universal, the value at q ky= 1.108, and therefore, IT, would be a universal constant times as we have found to be true. The case for such a... 

Cohen, M. L., and T. K. Bergstresser (1966). Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures. Phys. Rev. 141, 789-96. 

Pseudopolentials. See also Pseudopotential form factor Pseudopolential theory application to covalent bonding. 407-429 core radius. See Core radius in covalent solids, 41,407-429 defined,352 energy dependence, 545 formulation for metals, 360ff, 543-545 history of, 343... 

The pseudopotential form factor, hybridization term, and total form factor, obtained by Moriarty (1972) for copper. This includes a greatly improved treatment of exchange due to Lindgren (1971), in comparison to that used earlier by Harrison (1969, 1970) and Moriarty (1970). [After Moriarty, 1972b.]... 

An empirical pseudopotential method (EPM) calculation (23) was done to reproduce the gap and the reflectivity spectrum adjusting the pseudopotential form factors. This study led to a minimum direct gap at L and the lowest conduction state was obtained at Ff. A band stmcture calculation using a semi-ab initio approach (10) obtained an indirect gap (Hs Ff) of 2.0 eV and a comparable direct gap (Hs Tj) of approximately 2.0 eV. The minimum band gaps of BP have been reliably estimated from the experimental optical absorption. However, the direct band gaps and other excitation energies must be estimated from structure in the optical response versus frequency. The accuracy of the resulting experimental values depends on the correct identification of features in, e.g., the reflectivity with particular transitions between band states. Then the GW results may be more reliable estimates than the experimental direct band gaps. 

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