10.5 Electrons in Semiconductors
555
or
N ce
V
≈
2
h 3
e
−β
E g −μ+
1
2 p 2 /m ∗
ce
dp
=
2
h 3 4π e
−β(E g −μ)
∞
0
p
2 e
−β
1
2 p 2 /m ∗
ce dp
=
2
h 3 4π e
−β(E g −μ) (2m
∗
ce k B T )
3
2
∞
0
x
2 e
−x 2 dx .
Upon recognizing that the integral in the final line of this sequence has the value
√
π/4, we obtain for N ce the expression
N ce = 2
2m ∗
ce πk B T
h 2
3
2
V e
−β(E g −μ) .
(10.5.13)
Similarly, for N h we obtain the expression
N h
V
2
h 3
dp
e β[μ−ε (p )] + 1
=
2
h 3
dp
e
β
μ+
1
2 p 2 /m ∗
h
+ 1
2
h 3
e
−β
μ+
1
2 p 2 /m ∗
h
dp
or
N h 2
2πm ∗
h k B T
h 2
3
2
V e
−βμ .
(10.5.14)
Equality of N ce and N h allows us to solve for e βμ as
(m
∗
ce )
3
2 e
−β(E g −μ)
= (m
∗
h )
3
2 e
−βμ ,
or
e
2βμ
=
m ∗
h
m ∗
ce
3
2
e
βE g ,
from which we find that
e
βμ
=
m ∗
h
m ∗
ce
3
4
e
1
2 βE g .
(10.5.15)