391
Bipolar Junction Transistor Compact Models
where:
A E  = emitter area
n p is the injected electron concentration at the edge of EB-junction
depletion region as shown in Figure 11.17
W B is the width of the neutral base region
Then the minority carrier transit time across the base is given by
τ B
B
CC
Q
I
=
(11.42)
Since the built-in electric field within base is assumed to be negligible, from the Fick’s first law of diffusion, we can show that the
electron diffusion current (Equation 2.40) is
I
qA D
dn
dx
CC
E n
p
=
(11.43)
where:
D n is the average electron diffusivity in the p-type base region of
an npn-BJT
Assuming the equilibrium electron concentration, n p0  << n p , we can
express Equation 11.43 as
I
qA D
n
W
CC
E n
p
B
≅
(11.44)
Now, substituting for Q B and I CC from Equations 11.41 and 11.44,
respectively, in Equation 11.42, we get the expression for base transit
time as
τ B
B
n
W
D
=
2
2
(11.45)
To understand the importance of τ B in determining the speed of BJTs,
let us consider a vertical npn-BJT with W B  = 1 μm and lightly doped
base so that D n  ≅ 38 cm 2  sec −1 . Then from Equation 11.45, we find that
the value of τ B  ≅ 132 psec for a uniformly doped base region. In reality, the base doping is graded, and therefore, an aiding electric field
speeds up the carrier transit through the base. As a result, τ B is further reduced. Also, in order to maintain the charge neutrality under
high-level injection, the hole concentration in the base has a gradient
similar to the electron gradient. This sets up an electron field, which
also speeds up the electron transit through the base. Thus, τ B is not
the dominant frequency limitation in advanced IC BJTs.
Similarly, we can derive the expression for the reverse diffusion
capacitance C DC and transit time τ Rdc with reference to Figure 11.18.