Page 140 - Electrical Properties of Materials

122 The band theory of solids

Exercises

7.1. Classify metals, semiconductors, and insulators on the 7.6. The reciprocal mass tensor for Bi close to the bottom of
basis of their band structure. Find out what you can about the conduction band is of the form
materials on the border lines of these classifications. ⎛ ⎞
a xx 0 0
⎠ .
⎝ 0 a yy a yz
7.2. X-ray measurements show that electrons in the conduc-
0 a yz a zz
tion band of lithium have energies up to 4.2 eV. Take this as the
highest filled energy level in the band. If you further identified
this energy level with the Fermi level, what average effective (i) Find the components of the effective mass tensor.
mass will give you the same result from free-electron the- (ii) Find the function E(k x , k y , k z ).
ory? Assume one free electron per atom. The atomic weight (iii) Show that the constant energy surfaces are ellipsoids.
3
of lithium is 6.94, and its density is 530 kg m .
7.7. Using the potential energy distribution of Fig. 7.2, de-
7.3. Show, using the Feynman model, that the effective mass termine, with the aid of the Ziman model, the width of the
at the bottom of the band is inversely proportional to the width first forbidden band. Take w = a/2.
of the band.
7.8. Assume, as in the Kronig–Penney model of Section 7.2,
that w and V 0 in Fig. 7.2 tend to zero and infinity, respectively,
7.4. Show from eqns (7.3) and (7.5) that the group velocity of but their product V 0 w is kept constant. Determine with the aid
the electron is zero at k = nπ/a. of the Ziman model the widths of the nth allowed band and
the nth forbidden band. Can you conclude that the higher the
7.5. In general, the reciprocal of the effective mass is a tensor band the wider it is?
whose components are given by the mixed derivatives. How
would the classical equation of motion 7.9. Thelowestenergybands in asolid areverynarrowbe-
cause there is hardly any overlap of the wave functions. In
dν general, the higher up the band is, the wider it becomes. As-
m = F suming that the collision time is about the same for the valence
dt
and conduction bands, which would you expect to have higher
be modified? mobility, an electron or a hole?