Page 152 - Electrical Properties of Materials

134 Semiconductors

leading to a radius,

Ze 2
r s = . (8.41)
6π k B T
Next, we argue that the scattering power of the ion may be represented at the
radius by a scattering cross-section,

1 Ze 2 2
2
S s = r π = . (8.42)
s
π 6 k B T
Finally, assuming that the mean free path is inversely proportional to the
scattering cross-section, we obtain the relationship
–1 –1/2
τ ionized impurity = lT –1/2 ∼ S T ∼ T 3/2 . (8.43)
s
When both types of scattering are present the resultant collision time may
be obtained from the equation

If a high value of τ is required, 1 1 1
then one should use a very pure = + . (8.44)
τ τ thermal τ ionized impurity
material and work at low temper-
atures. A high value of τ means high mobility and hence high average velocity for the
electrons.
We do not know yet whether high electron velocities in crystals will have
∗ ∗
Two present applications are the Gunn many useful applications, but since fast electrons in vacuum give rise to in-
effect and very high frequency transist- teresting phenomena, it might be worthwhile making an effort to obtain high
ors, both to be discussed in Chapter 9.
carrier velocities in semiconductors.
How would mobility vary as a function of impurity density? It is bound
to decline. Instead of a mathematical model that is quite complicated, we are
going to give here actual measured curves for electron and hole mobilities at
T = 300 K for Ge and Si (see Fig. 8.7).
Note that electron mobilities are When both electrons and holes are present the conductivities add
higher than hole mobilities due to
2
2
the fact that in both materials the e τ e N e e τ h N h
σ = + . (8.45)
effective mass of electrons is smal- m ∗ m ∗
e h
ler than that of holes.
8.5 A relationship between electron and hole densities
Let us now return to eqns (8.17) and (8.20). These were derived originally
for intrinsic semiconductors, but they are valid for extrinsic semiconductors as
well. Multiplying them together, we get

3
2πk B T ∗ 3/2 E g
∗
N e N h =4 (m m ) exp – . (8.46)
h
e
h 2 k B T
It is interesting to note that the Fermi level has dropped out, and only the
‘constants’ of the semiconductor are contained in this equation. Thus, for a
∗
∗
given semiconductor (i.e. for known values of m , m , and E g ) and temperature
e
h