7.5 Shallow Defects
197
(a)
(b)
Fig. 7.17 a Position of Fermi level in partially compensated Si:P,B (N D = 10 15 cm −3 , E b
D = 45 meV, E b
A = 45 meV,
solid line: N A = 10 13 cm −3 , dashed line: N A = 0, dash-dotted line: N A = 10 12 cm −3 , short-dashed line: N A =
10 14 cm −3 , dash-double dotted line: N A = 5 × 10 14 cm −3 ) as a function of temperature. b Corresponding electron
concentration for N A = 10 13 cm −3 as a function of temperature (neglecting intrinsic carriers), dashed line for N A = 0
according to (7.34), dash-dotted line approximation for n N A as in (7.49)
with
α = 1 + ˆ
g D
N A
N C
exp
E
b
D
kT
= 1 +
N A
β
(7.46a)
β =
N C
ˆ
g D
exp
−
E
b
D
kT
.
(7.46b)
The carrier density is best obtained from (7.43),
2 n =
(N A − β) 2 + 4 N D β − (N A + β) .
(7.47)
For N A = 0 we have α = 1 and (7.30) is reproduced, as expected. For T = 0 (and N A = 0) the Fermi
energy lies at E F = E D since the donor level is partially filled (N
0
D = N D − N A ). For low temperatures
the Fermi level is approximated by
E F ∼ = E C − E
b
D + kT ln
N D /N A − 1
ˆ
g D
.
(7.48)
The corresponding carrier density at low temperatures is
n =
N C
ˆ
g D
exp
−
E
b
D
kT
N D
N A
− 1
.
(7.49)
For higher temperatures (7.34) holds approximately for n > N A ; the slope is now given by E
b
D /2 as in
the uncompensated case (Fig. 7.17b). For sufficiently high temperatures in the exhaustion regime (but
still n i < n) the electron density is given by
n ∼ = N D − N A .
(7.50)
At even higher temperatures the electron density will be determined by the intrinsic carrier concentration. Only in this case p = 0!