10.5 Electrons in Semiconductors
559
Here, however, we shall prefer to write this result as
N ce = N c e
βμ
;
N c ≡ 2
2πm ∗
ce k B T
h 2
3
2
,
(10.5.27)
so that
e
βμ
=
N ce
N c
=
N ce
N c
.
(10.5.28)
We may now combine this last result with our result Eq. (10.5.24) to obtain
N de
N D − N de
= 2e
βE D e
βμ
= 2
N ce
N c
e
βE D ,
(10.5.29)
from which we obtain the relation
N ce (N D − N de )
N de
=
1
2 N c e
−βE D .
(10.5.30)
This expression may be interpreted in terms of a dissociation equilibrium for the
donors D via
D D
+
+ e
− .
(10.5.31)
If, for example, we recall the relationship between number density n A ≡ N A and
concentration [A] of a chemical species A as
n A = N 0 [A] ,
(10.5.32)
with N 0 being the Avogadro number, N 0 = 6.02214 × 10 23 mol
−1 , we have
n D
N 0
= [D] ,
1
N 0
(n D − n de ) = [D
+
] ,
n ce
N 0
= [e
−
] .
(10.5.33)
The equilibrium constant K eq for process (10.5.31) is defined as
K eq (T ) ≡
[D + ] [e − ]
[D]
,
(10.5.34)
with the temperature dependence of the equilibrium constant K eq (T ) given explicitly as
K eq (T ) =
1
2 n c e
−βE D .
(10.5.35)