76 C H A P T E R 3 Introduction to the Quantum Theory of Solids
forces are related to the viscosity of the liquids. Because of the difference in motion
of the marble in these two cases, the mass of the marble would appear to be different
in water than in oil. (As with any analogy, we must be careful not to be too literal.)
We can also relate the effective mass of an electron in a crystal to the E versus k
curves, such as is shown in Figure 3.11. In a semiconductor material, we will be deal-
ing with allowed energy bands that are almost empty of electrons and other energy
bands that are almost full of electrons.
To begin, consider the case of a free electron whose E versus k curve is shown
in Figure 3.7. Recalling Equation (3.28), the energy and momentum are related by
E ϭ p2͞2m ϭ ប2k2͞2m, where m is the mass of the electron. The momentum and
wave number k are related by p ϭ បk. If we take the derivative of Equation (3.28)
with respect to k, we obtain
_dE ϭ _ប2k ϭ _បp (3.38)
dk m m
Relating momentum to velocity, Equation (3.38) can be written as
_1 _dE ϭ _p ϭ v (3.39)
ប dk m
where v is the velocity of the particle. The first derivative of E with respect to k is
related to the velocity of the particle.
If we now take the second derivative of E with respect to k, we have
_d 2E = _ប2 (3.40)
dk2 m
We may rewrite Equation (3.40) as
_1 d_2E ϭ _1 (3.41)
ប2 dk2 m
The second derivative of E with respect to k is inversely proportional to the mass
of the particle. For the case of a free electron, the mass is a constant (nonrelativis-
tic effect), so the second derivative function is a constant. We may also note from
Figure 3.7 that d2E͞dk2 is a positive quantity, which implies that the mass of the
electron is also a positive quantity.
If we apply an electric field to the free electron and use Newton’s classical equa-
tion of motion, we can write
F ϭ ma ϭ ϪeE (3.42)
where a is the acceleration, E is the applied electric field, and e is the magnitude of
the electronic charge. Solving for the acceleration, we have
a ϭ _ϪeE (3.43)
m
The motion of the free electron is in the opposite direction to the applied electric field
because of the negative charge.
We may now apply the results to the electron in the bottom of an allowed energy
band. Consider the allowed energy band in Figure 3.16a. The energy near the bottom