bei48482_FM

The Solid State 371

Brillouin Zones

The region in k-space (here an imaginary plane whose rectangular coordinates are kx

and ky) that low-kelectrons can occupy without being diffracted is called the first

Brillouin zone,shown in Fig. 10.41. The second Brillouin zone is also shown; it con-

tains electrons with kathat do not fit into the first zone yet which have suffi-

ciently small wave numbers to avoid diffraction by the diagonal sets of atomic planes

in Fig. 10.40. The second zone contains electrons with kvalues from ato 2a

for electrons moving in the xand ydirections, with the possible range of kval-

ues narrowing as the diagonal directions are approached. Further Brillouin zones can

be constructed in the same manner. The extension of this analysis to actual three-

dimensional structures leads to Brillouin zones such as those shown in Fig. 10.42.

The significance of the Brillouin zones becomes apparent when we look at the en-

ergies of the electrons in each zone.

The energy of a free electron is related to its momentum pby

E (10.21)

and hence to its wave number kby

E (10.22)

In the case of an electron in a crystal for which ka, there is practically no

interaction with the lattice, and Eq. (10.22) is valid. Since the energy of such an electron

^2 k^2



2 m

Energy and wave

number

p^2



2 m

Energy and

momentum

First Brillouin

zone

Second

Brillouin

zone

ky = +πa

kx = +πa

ky = –πa

kx = –πa

ky

kx

Figure 10.41The first and second Brillouin zones of a two-dimensional square lattice.

Figure 10.42First and second

Brillouin zones in a face-centered

crystal.

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