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Clearly, whatever the symmetry the expansion set of plane waves will always have the periodicity of the system, so just as before the scalar product of a reciprocal lattice vector, q = G' G, with a Bravais lattice vector, R, will be equal to an integral multiple of 2T, and hence e iq.R = 1. If there are N of these new generalised bases in the total volume of the crystal, then:
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Now the total volume of the crystal, O, divided by the number of the new general Bravais lattice points, N, is equal to the volume occupied by one of the bases, Ob (say), and therefore:
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The diamond, zinc blende and wurtzite crystals discussed so far all have a two-atom basis, and hence Oc (the volume of the primitive cell) has been the volume occupied by two atoms. In this more general basis with Na atoms (say), the volume occupied by its primitive cell would be:
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Therefore, the final expression for the crystal potential follows as:
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and the full expression for the Hamiltonian matrix elements is:
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At this point, the final equation appears to be merely a re-expression of the Hamiltonian matrix elements of the elemental and compound bulk semiconductors, and it can indeed be shown to reproduce those expressions. For example, consider a (Na=) 2-atom basis, with a cation at T = A0/8 (i^ + j^ + k^) and an anion at +T; then the potential term in equation (11.77) gives:
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which recalling that q = G' G, is equivalent to equation (11.54). However, the above is much more than just a generalised form for bulk semiconductor calculations; by thoughtful choice of the atomic basis and the primitive cell, equation (11.77) can be used to calculate the electronic structure of heterostructures of all dimensions, i.e. quantum wells, wires and dots. Such calculations are often referred to as large-cell calculations, however, the computational method of summing
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over atoms in a more extensive basis suggests the term 'large-basis' calculations to be more appropriate. The promise of this generalisation will be explored fully in subsequent chapters, however, for now, it is worthwhile pursuing these ideas for bulk systems.
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Table 11.6 The form factors (in eV) of a selection of III-V compound semiconductor, converted from the original values of Cohen and Bergstresser [236], where the symmetric form factor at q = 2 and the anti-symmetric at q = \/8 have been deduced by linear interpolation from the two adjacent values
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GaAs InAs GaP InP
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In particular, the brief digress into atomic form factors was in anticipation of this formalism this large-basis method relies on pseudopotential form factors which are associated with individual atoms, and hence in order to perform calculations of bulk compound semiconductors, the symmetric and anti-symmetric form factors listed in Table 11.4 need to be decomposed (as in Section 11.8), into their atomic components. In addition, however, the 'structure factor' associated with these atomic potentials is now of the form e iq.t , which is never zero, and hence the atomic form factors in this method need to be specified at all values of q that can arise from the basis set of reciprocal lattice vectors G. For a compound semiconductor, this implies that it is necessary to know the atomic form factors at q = 2 and \/8. For the example III-V materials given earlier in Table 11.4, this can be achieved as a zeroth-order approximation simply by linearly interpolating between the existing symmetric and anti-symmetric potentials (as in Table 11.6), and then decomposing the resulting form factors to give the data presented in Table 11.7. Fig. 11.11 shows the band structures of GaP and InP, calculated by using this largebasis method with a 65-element plane-wave basis set and the atomic form factors of Table 11.7. The validity of this approach, which turns out to be a small generalisation for the bulk case, is substantiated by the close agreement obtained with the original calculations of Cohen and Bergstresser [236].