7.5. HIGH-FREQUENCY BEHAVIOR OF A BJT 329
Inserting the above equation and equation 7.5.12 into equation 7.5.10 gives us forτBC 1
τBC 1 =CBCΔVBC
ΔIC
=
CBC(ΔVBE+ΔIC(RE+RC))
ΔIC
(7.5.14)
=
(
ΔVBE
ΔIC
)
CBC+(RE+RC)CBC
τBC 1 =(rE+RE+RC)CBC (7.5.15)In calculatingτC, we will assume that the electron velocity profile in the base-collector deple-
tion region does not necessarily need to remain constant. This would, for example, be the case if
the material composition in the collector was varied, such as in a double heterojunction bipolar
transistor structure, or in the case of non-stationary transport in short collectors. The increased
electron concentration in the collectorΔn(x)asafunctionofΔICis then given by
Δn(x)=ΔIC
AEeve(x)(7.5.16)
whereAEis the emitter area andve(x)is the electron velocity at a pointxin the collector.
figure 7.12 shows a schematic plot ofΔn(x)in the collector for an arbitrary velocity distribution
ve(x).
We first need to calculateΔQC. To do this, we find the induced charged(ΔQC)atx=0
caused by a sheet of charge−eΔn(x)dxat a pointx, and integrate fromx=0tox=wC,as
illustrated in figure 7.13a (note that we assumewd,BC wC, since the base and subcollector
are doped highly and the collector is typically fully depleted when the device is under bias).
The electric field induced in the depletion region by each sheet charge element is shown in
figure 7.13b. Using Gauss’ Law, we can related(ΔQC)andeΔn(x)dxtodE+(x)anddE−(x).
dE+(x)=d(ΔQC)
AE(7.5.17)
dE+(x)+dE−(x)=eΔn(x)dx
(7.5.18)
Also, since the change in voltage in the collector due to the induced charge must be zero, the
area underdE+(x)in figure 7.13b must equal the area abovedE−(x),or
x·dE+(x)=(wC−x)dE−(x) (7.5.19)Solving forΔE−(x)in this equation gives us
dE−(x)=(
x
wC−x)
dE+(x) (7.5.20)We can then substitute this result into equation 7.5.18 to get
dE+(x)+(
x
wC−x)
dE+(x)=eΔn(x)dx