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Table 11.1 The first 65 reciprocal lattice vectors of a face-centred cubic crystal (in units of 2TT/A0)
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[000] _ _ [111] [Til] [111] [111] [Til] [1T1] [Til] [TTT] [111] [TTT] [200] [020] [002] [220] [202] [022] [220] [202] [022] [311] [131] [113] [311] [131] [113] [311] [131] [IT3] [113] [311] [131] [111] [222] [222] [222] [222] [222] [400] [040] [004]
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[111] [ill] [ITT] [111] [TTl] [111]
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[200] [220] [220] [311] [113] [311] [3TT] [311] [222] [020] [202] [202] [131] [311] [131] [131] [222]
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[ 0 0 2 ] 2 [002] 8 [022] [022] [022] 11 [113] [131] [113] [113] [113] [222] [222] 12
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the 1s22s22p6 electrons. The remaining four valence electrons, which in an isolated Si atom are found in the 3s and 3p orbitals, are the subject of the investigation; it is their energy levels and charge distributions which determine the electronic properties of the crystal and thus provide the motivation for this theoretical derivation. The empirical nature of the pseudopotential method is incorporated by adjusting the values of Vf (q) in order to achieve the closest agreement of the calculated energy levels with those measured by experimental methods. Note, therefore, that Vf (q} summarizes many of the microscopic electrostatic properties of the crystal. For example, it accounts for the nuclear charge, the inner-shell electrons, the screening provided by these electrons, and under the auspices of the independent electron approximation described earlier, it also accounts for the electron electron interaction experienced between the valence electrons. As mentioned above, q (the difference between two reciprocal vectors), is also a reciprocal lattice vector, and in a bulk crystal, such as a face-centred cubic, it takes discrete values as deduced in 1 (see Table 11.1). The pseudopotential form factor Vf (q} is, therefore, also a discrete function, only having non-zero values for particular q. Cohen and Bergstresser [236] found that the experimentally determined band structure features of Si could be reproduced by using the values of Vf (q) given in Table 11.2. It should be noted that as Vf (q} is truncated for q > and as Vf (0) only has the effect of shifting the energies up or down then Vf (q) has only three non-zero values, which occur for q = 3 8 and 11. Note also that for q = 2, the structure factor
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Table 11.2 The form factors (in eV) of the common group-IV semiconductor elements, converted from the original values of Cohen and Bergstresser [236]
Material Si Ge
A0 (A) 5.43 5.66
Vf ^ +0.27 +0.07
-1.43 -1.57
+0.54 +0.41
for the face-centred cubic crystals of interest here, which is given by equation (11.38) becomes:
With q = (2TT/Ao)2i, for example, this then gives:
which is zero, and hence the value of Vf(q) is irrelevant. Reverting back to the mathematics, then:
which when substituted back into equation (11.24) finally gives the complete form for the matrix elements as
Fig. 11.2 displays the results of calculations of the bulk band structure of Si by using the form factors of Table 11.2, together with the 65-element plane wave basis set in Table 11.1. The graph shows the three highest-energy valence-band levels, which (in this simple approach, ignoring spin orbit coupling) are all degenerate at k = 0, and two of which remain degenerate across all k. In addition, the two lowest-energy conduction-band states can be clearly seen. In this first calculation, the energy levels are plotted along one of the (100) directions. The continuous energy band nature of the solutions is visible, as expected, which is in contrast to the solutions of the Schrodinger equation in heterostructures, the focus of the majority of this work so far. Furthermore, Fig. 11.2 demonstrates the periodic nature of the band structure, with equal energy levels separated by an electron wave vector equal to [200], i.e. the smallest reciprocal lattice vector along the direction of interest (see Table 11.1). Such periodicity is, of course, merely reflecting the Brillouin zone symmetry structure of the crystal [1] and the edge of the first Brillouin is at exactly half of the reciprocal lattice vector in question, i.e. k = 2TT/A0. Note that there is a slight discrepancy in