696
24 Transistors
where γ is the emitter efficiency, α T the base transport factor and M the collector multiplication factor.
Since the collector is normally operated below the threshold for avalanche multiplication, M = 1.
The current amplification in the common emitter circuit is
β 0 = h FE =
∂ I C
∂ I B
.
(24.14)
Using (24.12), we find
β 0 =
∂ I E
∂ I B
− 1 =
∂ I E
∂ I C
∂ I C
∂ I B
− 1 =
1
α 0
β 0 − 1 =
α 0
1 − α 0
.
(24.15)
Since α 0 is close to 1 for a well-designed transistor, β 0 is a large number, e.g. β 0 = 99 for α 0 = 0.99.
The emitter efficiency is (A denotes the device area)
γ =
A j Ep
I E
=
1 +
n E
p B
D E
D B
L B
L E
tanh
w
L B
−1
.
(24.16)
The base transport factor, i.e. the ratio of minority carriers reaching the collector and the total number
of injected minority carriers, is (for reverse bias |βU CB | | kT )
α T =
j Cp
j Ep
=
exp(βU EB ) − 1 + cosh w/L B
1 + (exp(βU EB ) − 1) cosh w/L B
≈
1
cosh (w/L B )
≈ 1 −
w
2
2L
2
B
.
(24.17)
The first approximation is for βU EB 1 (emitter diode injecting in forward direction), the second
approximation is for w L B . If the base length is a tenth of the diffusion length, the base transport
factor is α T > 0.995. M is also very close to 1; for reverse bias U CB and w L B we find
M ≈ 1 +
w
L C
D C
D B
δn C
δ p C − δ p E
≈ 1 +
w
L C
D C
D B
n C
p B
exp(−βU EB ) .
(24.18)
Thus for w L B , α 0 is dominated by γ and given as (approximating (24.16))
α 0 ≈ γ ≈ 1 −
w
L E
n E
p B
D E
D B
.
(24.19)
The gain β 0 is then also determined by γ:
β 0 = h FE ≈
γ
1 − γ
≈
1
1 − γ
∝
N E
N B
L E
w
,
(24.20)
N E and N B being the doping levels in the emitter and base, respectively. The base and collector
current are shown in Fig. 24.6 as a function of the emitter–base voltage, i.e. the voltage at the injection
diode. The collector current is close to the emitter–base diode current and displays a dependence
∝ exp(eV EB /kT ). The base current shows a similar slope but is orders of magnitude smaller in
amplitude. For small forward voltages of the emitter–base diode, the current is typically dominated by
a nonradiative recombination current that flows through the base contact and has an ideality factor (m
in Fig. 24.6) close to 2.
24 Transistors
where γ is the emitter efficiency, α T the base transport factor and M the collector multiplication factor.
Since the collector is normally operated below the threshold for avalanche multiplication, M = 1.
The current amplification in the common emitter circuit is
β 0 = h FE =
∂ I C
∂ I B
.
(24.14)
Using (24.12), we find
β 0 =
∂ I E
∂ I B
− 1 =
∂ I E
∂ I C
∂ I C
∂ I B
− 1 =
1
α 0
β 0 − 1 =
α 0
1 − α 0
.
(24.15)
Since α 0 is close to 1 for a well-designed transistor, β 0 is a large number, e.g. β 0 = 99 for α 0 = 0.99.
The emitter efficiency is (A denotes the device area)
γ =
A j Ep
I E
=
1 +
n E
p B
D E
D B
L B
L E
tanh
w
L B
−1
.
(24.16)
The base transport factor, i.e. the ratio of minority carriers reaching the collector and the total number
of injected minority carriers, is (for reverse bias |βU CB | | kT )
α T =
j Cp
j Ep
=
exp(βU EB ) − 1 + cosh w/L B
1 + (exp(βU EB ) − 1) cosh w/L B
≈
1
cosh (w/L B )
≈ 1 −
w
2
2L
2
B
.
(24.17)
The first approximation is for βU EB 1 (emitter diode injecting in forward direction), the second
approximation is for w L B . If the base length is a tenth of the diffusion length, the base transport
factor is α T > 0.995. M is also very close to 1; for reverse bias U CB and w L B we find
M ≈ 1 +
w
L C
D C
D B
δn C
δ p C − δ p E
≈ 1 +
w
L C
D C
D B
n C
p B
exp(−βU EB ) .
(24.18)
Thus for w L B , α 0 is dominated by γ and given as (approximating (24.16))
α 0 ≈ γ ≈ 1 −
w
L E
n E
p B
D E
D B
.
(24.19)
The gain β 0 is then also determined by γ:
β 0 = h FE ≈
γ
1 − γ
≈
1
1 − γ
∝
N E
N B
L E
w
,
(24.20)
N E and N B being the doping levels in the emitter and base, respectively. The base and collector
current are shown in Fig. 24.6 as a function of the emitter–base voltage, i.e. the voltage at the injection
diode. The collector current is close to the emitter–base diode current and displays a dependence
∝ exp(eV EB /kT ). The base current shows a similar slope but is orders of magnitude smaller in
amplitude. For small forward voltages of the emitter–base diode, the current is typically dominated by
a nonradiative recombination current that flows through the base contact and has an ideality factor (m
in Fig. 24.6) close to 2.