7.2 Carrier Concentration
181
(a)
(b)
Fig. 7.1 Fermi integral ˆ
F 1/2 = (
√ π/2)F 1/2 with approximations in three regions of the argument: A 1 (x) =
(
√
π/2) exp(x) for x < 2, A 2 (x) = (
√ π/2)(1/4 + exp(−x)) −1 for −2 < x < 2, A 3 (x) = 2/3x 3/2 for x > 2.
a linear, b semilogarithmic plot
Table 7.1 Band gap, intrinsic carrier concentration, conduction band and valence-band edge density of states at T =
300 K for various semiconductors
E g (eV)
n i (cm −3 )
N C (cm −3 )
N V (cm −3 )
InSb
0.18
1.6 × 10 16
InAs
0.36
8.6 × 10 14
Ge
0.67
2.4 × 10 13
1.04 × 10 19
6.0 × 10 18
Si
1.124
1.0 × 10 10
7.28 × 10 19
1.05 × 10 19
GaAs
1.43
1.8 × 10 6
4.35 × 10 17
5.33 × 10 18
GaP
2.26
2.7 × 10 0
GaN
3.3
1
with
N C = 2
m e kT
2π 2
3/2
(7.8)
N V = 2
m h kT
2π 2
3/2
,
(7.9)
where N C (N V ) is called the conduction-band (valence-band) edge density of states. The masses in
(7.8) and (7.9) are the density of states masses given in (6.72) and (6.73). Values of N C,V for Si, Ge
and GaAs are given in Table 7.1.
Now, we assume that the Boltzmann approximation (E.23) can be used, i.e. the probability that a
band state is populated is 1. Then, the integral (7.1) can be executed analytically and the concentration
n of electrons in the conduction band is given as
n = 2
m e kT
2π 2
3/2
exp
E F − E C
kT
= N C exp
E F − E C
kT
.
(7.10)
For the Boltzmann approximation and a parabolic valence band, the density of holes is given by
p = 2
m h kT
2π 2
3/2
exp
−
E F − E V
kT
= N V exp
−
E F − E V
kT
.
(7.11)