bei48482_FM

The Solid State 375

Although there must be an energy gap between successive Brillouin zones in any

given direction, the various gaps may overlap permitted energies in other directions

so that there is no forbidden band in the crystal as a whole. Figure 10.47 contains

graphs of Eversus kfor three directions (a) in a crystal that has a forbidden band and

(b) in a crystal whose allowed bands overlap sufficiently to avoid having a forbidden

band.

As we know, the electrical behavior of a solid depends on the degree of occupancy

of its energy bands as well as on its band structure. Figure 10.48ashows the first and

second Brillouin zones of a hypothetical two-dimensional insulator. The first zone is

filled with electrons, and the energy gap between this zone and the second is much

wider than kT. This corresponds to the situation shown in Fig. 10.24 where the insu-

lator is diamond. In Fig. 10.48bthe zones are the same, but the first zone is only half

filled. This corresponds to the situation shown in Fig. 10.22, and the material is anal-

ogous to a metal such as sodium whose atoms have one valence electron each. In

Fig. 10.48cthe energies in the second zone overlap those in the first zone, so the

valence electrons partly occupy both zones. This corresponds to the situation shown

in Fig. 10.47b, and the material is analogous to a metal such as magnesium which has

two valence electrons per atom.

(a)

Forbidden band

k

k 1 k 2 k 3

E

k

(b)

E

k

No forbidden band

k 1 k 2 k 3

Figure 10.47Eversus kcurves for three directions in two crystals. In (a) there is a forbidden band,

in (b) the allowed energy bands overlap and there is no forbidden band.

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