Figure 526 curves of equal energy inserted into the

Figure 5.26. Curves of equal energy inserted into the first Brillouin zone for a two-
dimensional square lattice.
Figure 5.27. A particular surface of equal energy (Fermi surface, see Section 6.1) and the
first Brillouin zone for copper. Adapted from A.B. Pippard, Phil. Trans. Roy. Soc. London ,
A 250 , 325 (1957).
60 I. Fundamentals of Electron Theory

Problems
1. What is the energy difference between the points L
2
0
and L
1
(upper) in the band diagram for
copper?
2. How large is the “gap energy” for silicon? ( Hint : Consult the band diagram for silicon.)
3. Calculate how much the kinetic energy of a free electron at the corner of the first Brillouin
zone of a simple cubic lattice (three dimensions!) is larger than that of an electron at the
midpoint of the face.
4. Construct the first four Brillouin zones for a simple cubic lattice in two dimensions.
5. Calculate the shape of the free electron bands for the cubic primitive crystal structure for
n ¼ 1 and n ¼ ´ 2 (see Fig. 5.6 ).
6. Calculate the free energy bands for a bcc structure in the k
x
-direction having the following
values for h
1
/ h
2
/ h
3
: (a) 1 11; (b) 001; and (c) 010. Plot the bands in k -space. Compare with
Fig. 5.18 .
7. Calculate the main lattice vectors in reciprocal space of an fcc crystal.
8. Calculate the bands for the bcc structure in the 110 [ G ´ N ] direction for: (a) (000);
(b) ð 0
±
10 Þ ; and (c) 1 11.
9. If b
1
· t
1
¼ 1 is given (see equation ( 5.14 )), does this mean that b
1
is parallel to t
1
?
5. Energy Bands in Crystals 61

.

CHAPTER 6
Electrons in a Crystal
In the preceding chapters we considered essentially only one electron, which
was confined to the field of the atoms of a solid. This electron was in most
cases an outer, i.e., a valence, electron. However, in a solid of one cubic
centimeter at least 10
22
valence electrons can be found. In this section we
shall describe how these electrons are distributed among the available
energy levels. It is impossible to calculate the exact place and the kinetic
energy of each individual electron. We will see, however, that probability
statements nevertheless give meaningful results.
6.1. Fermi Energy and Fermi Surface
The Fermi energy, E
F
, is an important part of an electron band diagram.
Many of the electronic properties of materials, such as optical, electrical, or
magnetic properties, are related to the location of E
F
within a band.
The Fermi energy is often defined as the “highest energy that the elec-
trons assume at T ¼ 0 K”. This can be compared to a vessel, like a cup, (the
electron band) into which a certain amount of water (electrons) is poured.
The top surface of the water contained in this vessel can be compared to the
Fermi energy. The more electrons are “poured” into the vessel, the higher
the Fermi energy. The Fermi energies for aluminum and copper are shown
in Figs. 5.21 and 5.22 . Numerical values for the Fermi energies for some
materials are given in Appendix 4. They range typically from 2 to 12 eV.