The Jones Zone Gap

We turn now to the effects of a finite pseudopotential. Let us think specifically of silicon and begin with a free-elcctron gas of four electrons per ion, using the extended-zone representation, with energy equal to h k /2m for all k, rather than 

Nearest-neighbor LCAO bands for the homopolar semiconductors, found by using interatomic parameters predicted as in Fig. 18-1 (and listed in Table 2-1) and term values from the Solid Stale Table. Energies arc in electron volts. Notice that the vertical scale is reduced for the carbon bands. [After Froyen and Harrison, 1979.] 

Part (a) shows a frce-clectron Fermi sphere for silicon cut by various Bragg reflection planes, reducing the area of free Fermi surface. Part (b) shows the Jones Zone, made up of (220) Bragg planes, into which all of the silicon Fermi surface has disappeared. The view is along a [llO] direction in both parts. 

The Bragg planes are specified in Fig. 18-3 in terms of the lattice wave number giving rise to them. For example, there are lattice wave numbers [220]27c/fl, with a the cube edge of F ig. 3-1. The plane bisecting this vector is called the (220) plane. The (111) planes, which were shown in Fig. 16-9, are omitted here for clarity. The diagonal lines in part (a) are actually edges made by the intersections of planes of the (202) or (022) type with various combinations of signs of components. 

We know that when the pseudopotential is at full strength, all of the Fermi surface must disappear, since none is present in the semiconductor. We can, in fact, see from the figure that what must happen is that it disappears into the slanted and vertical planes of Fig. 18-3,a the horizontal planes as well as the omitted (111) planes are noncssential. Indeed, the vertical and slanted planes are among the twelve (220) Bragg planes that make up the Jones Zone (Mott and 

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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. 

We found earlier in this ciiapter that this is the dominant matrix element that opens up the gap at the Jones Zone, eliminates tlie Fermi surface, and dominates the bonding properties. As we slowly displace the second sublattice, as in... 

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