Page 130 - Electrical Properties of Materials

112 The band theory of solids

Thus, in the three-dimensional case, the energy band extends from the
minimum energy

E min = E 1 –2(A x + A y + A z ) (7.35)

to the maximum energy
E max = E 1 +2(A x + A y + A z ). (7.36)

7.5 The effective mass

The mass of an electron in a crys- It has been known for a long time that an electron has a well-defined mass,
tal appears, in general, different and when accelerated by an electric field it obeys Newtonian mechanics. What
from the free-electron mass, and is happens when the electron to be accelerated happens to be inside a crystal?
usually referred to as the effective How will it react to an electric field? We have already given away the secret
mass. when talking about cyclotron resonance.
We shall obtain the answer by using a semi-classical picture, which, as the
name implies, is 50% classical and 50% quantum-mechanical. The quantum-
mechanical part describes the velocity of the electron in a one-dimensional
lattice by its group velocity,

1 ∂E
v g = , (7.37)
∂k
which depends on the actual E–k curve. The classical part expresses dE as the
work done by a classical particle travelling a distance, v g dt, under the influence
of a force eE yielding

dE = eE v g dt
1 ∂E
= eE dt. (7.38)
∂k
We may obtain the acceleration by differentiating eqn (7.37) as follows:

2
dv g 1 d ∂E 1 ∂ E dk
= = . (7.39)
2
dt dt ∂k ∂k dt
Expressing now dk/dt from eqn (7.38) and substituting it into eqn (7.39) we
get
2
dv g 1 ∂ E
= eE . (7.40)
2
dt ∂k 2
Comparing this formula with that for a free, classical particle
dv
m = eE , (7.41)
dt
we may define

∂ E
2 –1
2
∗
m = (7.42)
∂k 2