bei48482_FM

374 Chapter Ten

maxima at the lattice points. Since the charge density corresponding to an electron

wave function ise^2 , the charge density in the case of  1 is concentrated between

the positive ions, while in the case of  2 , it is concentrated atthe positive ions. The

potential energy of an electron in a lattice of positive ions is greatest midway between

each pair of ions and least at the ions themselves, so the electron energies E 1 and E 2

associated with the standing waves  1 and  2 are different. No other solutions are

possible when kaand accordingly no electron can have an energy between

E 1 and E 2.

Figure 10.46 shows the distribution of electron energies that corresponds to the

Brillouin zones pictured in Fig. 10.43. At low energies (in this hypothetical situation

for E2 eV) the curve is almost exactly the same as that of Fig. 9.11 based on the

free-electron theory. This is not surprising since at low energies kis small and the elec-

trons in a periodic lattice then dobehave like free electrons.

With increasing energy, however, the number of available energy states goes be-

yond that of the free-electron theory owing to the distortion of the energy contours

by the lattice. Hence there are more different kvalues for each energy. Then, when

ka, the energy contours reach the boundaries of the first zone and energies

higher than about 4 eV (in this particular model) are forbidden for electrons in the

kxand kydirections although permitted in other directions. As the energy goes far-

ther and farther beyond 4 eV, the available energy states become restricted more and

more to the corners of the zone, and n(E) falls. Finally, at approximately 6^12 eV, there

are no more states and n(E) 0. The lowest possible energy in the second zone is

somewhat less than 10 eV and another curve similar in shape to the first begins. Here

the gap between the possible energies in the two zones is about 3 eV, and so the

forbidden band is about 3 eV wide.

Forbidden

band

Second zone

n(E)

First zone

051015

E, eV

Figure 10.46The distributions of electron energies in the Brillouin zones of Fig. 10.43. The dashed

line is the distribution predicted by the free-electron theory.

Table 10.2 Effective Mass

Ratios m*mat the Fermi

Surface in Some Metals

Metal m*m

Lithium Li 1.2

Beryllium Be 1.6

Sodium Na 1.2

Aluminum Al 0.97

Cobalt Co 14

Nickel Ni 28

Copper Cu 1.01

Zinc Zn 0.85

Silver Ag 0.99

Platinum Pt 13

B

ecause an electron in a crystal interacts with the crystal lattice, its response to an external

electric field is not the same as that of a free electron. Remarkably enough, the most important

results of the free-electron theory of metals discussed in Secs. 9.9 and 9.10 can be incorporated

in the more realistic band theory merely by replacing the electron mass mby an average effec-

tive massm*. For example, Eq. (9.56) for the Fermi energy is equally valid in the band theory

when m* is used in place of m. Table 10.2 is a list of effective mass ratios m*mfor several

metals.

Effective Mass

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