Page 189 - Semiconductor For Micro- and Nanotechnology An Introduction For Engineers

Statistics
is a good approximation. This allows us to solve (5.37a) and (5.37b)
which gives
 µ – E C 
n = D exp  ---------------- (5.39a)
k T 
C
e
B
 E – µ 
V
n = D exp  ---------------- (5.39b)
k T 
h
V
B
with
/
1 2m k T 32
e B 

D = --- ------------------- (5.40a)
C 4  πh 
/
1 2m k T 32

h B 
D = --- -------------------- (5.40b)
V 4  πh 
(5.40a) and (5.40b) are called the effective density of states. As a rule of
thumb we say that the approximations made calculating the density are
only good if the densities are lower than the respective density of states.
Charge neutrality now requires
V
/
( m ) 32 exp  µ – E C ( m ) 32 exp  E – µ  (5.41)
/
---------------- =
----------------
e  k T  h  k T 
B B
For (5.41) to hold, it is
E + E V 3k T  m h 
B
C
µ = -------------------- + -------------ln ------ (5.42)
i  m 
2 4 e
where the index i stands for “intrinsic”. We see that for equal electron
and hole masses m = m the chemical potential lies exactly half way
e h
between the conduction band edge and the valence band edge. Further-
more it will not show any temperature dependence. The intrinsic carrier
concentration thus reads
/
E
G 
e
/
⋅
-------------
n = n = n = 4.9 10 15  m m h  34 T 32 exp   – --------- (5.43)
m 
i
e
h
B
0 k T 
186 Semiconductors for Micro and Nanosystem Technology