7.5 Shallow Defects
191
(a)
(b)
Fig. 7.9 a Position of the Fermi level in Si:P (N D = 10 15 cm −3 , E b
D = 45 meV, no acceptors) as a function of temperature
without consideration of intrinsic carriers. Zero energy refers to the (temperature-dependent, Table 6.4) conduction-band
edge E C with approximative solutions for low (dashed line, (7.31)) and high (dash-dotted line, (7.32)) temperatures. b
Corresponding density of conduction-band electrons as a function of temperature
Fig. 7.10 Electron
concentration as a function
of temperature for a Ge:As
sample with
N D ≈ 10 13 cm −3 . Solid
line is fit to the data with a
donor binding energy of
12.7 meV. Adapted
from [594]
10
10
10
10
14
13
12
11
3
0
1000/T (1/K)
20
n i
60
80
100
300 78
20.4
33.3
14.3
10
Ge:As
40
be considered. The neutrality condition (still in the absence of any acceptors) is
− n + p + N
+
D = 0 .
(7.35)
Using (7.10) and p = n
2
i /n, the equation reads:
N C exp
E F − E C
kT
−
n
2
i
N C exp(
E F −E C
kT
)
−
N D
1 + ˆ
g D exp(
E F −E D
kT
)
= 0 .
(7.36)
The solution can be given analytically but is more complicated
4 . The temperature-dependent position
of the Fermi level is shown in Fig. 7.11.
The three important regimes are the intrinsic conduction at high temperatures when n i N D , the
exhaustion at intermediate temperatures when n i N D and kT > E
b
D , and finally the freeze-out
regime for kT E
b
D at low temperatures when the electrons condense back into the donors. The three
4 It is given in the third edition of this book.