2k a

2k a

Figure 2.13. Sketch of (a) unit cell in two-dimensional x,y coordinate space and (b) corresponding Brillouin zone in reciprocal space kx, ky for a rectangular Bravais lattice.

2.2.3. Energy Bands and Gaps of Semiconductors

The electrical, optical, and other properties of semiconductors depend strongly on how the energy of the delocalized electrons involves the wavevector k in reciprocal or k space, with the electron momentum p given by p = mv — Jik, as explained above. We will consider three-dimensional crystals, and in particular we are interested in the properties of the ni-V and the II-VI semiconducting compounds, which have a cubic structure, so their three lattice constants are the same: a = b = c. The electron motion expressed in the coordinates kx, ky, k2 of reciprocal space takes place in the Brillouin zone, and the shape of this zone for these cubic compounds is shown in Fig. 2.14. Points of high symmetry in the Brillouin zone are designated by capital Greek or Roman letters, as indicated.

The energy bands depend on the position in the Brillouin zone, and Fig. 2.15 presents these bands for the intrinsic (i.e., undoped) UI-V compound GaAs. The

Figure 2.14. Brillouin zone of the gallium arsenide and zinc blende semiconductors showing the high-symmetry points r, K, L, U, W, and X and the high-symmetry lines A, A, 2, Q, S, and Z. (From G. Burns, Solid State Physics, Academic Press, Boston, 1985, p. 302.)
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