There are several "figures of merit" for the operation of the
transistor. The first of these is called the emitter
injection efficiency, γ
γ. The emitter injection efficiency is just the ratio
of the electron current flowing in the emitter to the total
current across the emitter base junction:
γ=
I
e
I
Ee
+
I
Eh
γ
I
e
I
Ee
I
Eh
(1)
If you go back and look at the diode equation you will note that the
electron forward current across a junction is proportional to
N d
N d
the doping on the n-side of the
junction. Clearly the hole current will be proportional to
N
a
N
a
,
the acceptor doping on the p-side of the junction. Thus, at
least to first order
γ=
N
d
E
N
d
E
+
N
a
B
γ
N
d
E
N
d
E
N
a
B
(2)
(There are some other considerations which we are ignoring in
obtaining this expression, but to first order, and for most
"real" transistors, Equation 2 is a very good
approximation.)
The second "figure of merit" is the base transport factor,
α
T
α
T
.
The base transport factor tells us what fraction of the electron
current which is injected into the base actually makes it to
collector junction. This turns out to be given, to a very good
approximation, by the expression
α
T
=1−WB2Le2
α
T
1
WB
2Le
2
(3)
Where
W
B
W
B
is the physical width of the base region, and
L
e
L
e
is the electron diffusion length, defined in the electron diffusion length
equation.
L
e
=
D
e
τ
r
L
e
D
e
τ
r
(4)
Clearly, if the base is very narrow compared to the diffusion
length, and since the electron concentration is falling off like
e−x
L
e
x
L
e
the shorter the base is compared to
L
e
L
e
the greater the fraction of electrons who will actually make it
across. We saw before that a typical value for
L
e
L
e
might be on the order of 0.005 cm or 50 μm. In a
typical bipolar transistor, the base width,
W
B
W
B
is usually only a few μm and so
αα can be quite close to
unity as well.
Looking back at this
figure, it should be clear that, so long as the
collector-base junction remains reverse-biased, the collector
current
I
Ce
I
Ce
, will only depend on how much of
the total emitter current actually gets collected by the
reverse-biased base-collector junction. That is, the collector
current IC is just some fraction of the total emitter current
I
E
I
E
.
We introduce yet one more constant which reflects the ratio
between these two currents, and call it simply
"αα." Thus we say
I
C
=α
I
E
I
C
α
I
E
(5)
Since the electron current into the base
is just
γ
I
E
γ
I
E
and
α
T
α
T
of that current reaches the collector, we can write:
I
C
=α
I
E
=
α
T
γ
I
E
I
C
α
I
E
α
T
γ
I
E
(6)
Looking back at the structure of an npn bipolar transistor, we
can use Kirchoff's current law for the transistor and say:
I
C
+
I
B
=
I
E
I
C
I
B
I
E
(7)
or
I
B
=
I
E
−
I
C
=
I
C
α−
I
C
I
B
I
E
I
C
I
C
α
I
C
(8)
This can be re-written to express
I
C
I
C
in terms of
I
B
I
B
as:
I
C
=α1−α
I
B
≡β
I
B
I
C
α
1
α
I
B
β
I
B
(9)
This is the fundamental operational equation for the bipolar
equation. It says that the collector current is dependent only
on the base current. Note that if
αα is a number close to (but
still slightly less than) unity, then
ββ which is just given by
will be a fairly large number. Typical values for a will be on
the order of 0.99 or greater, which puts
ββ, the current gain, at
around 100 or more! This means that we can control, or amplify
the current going into the collector of the transistor with a
current 100 times smaller going into the base. This all occurs
because the ratio of the collector current to the base current
is fixed by the conditions across the emitter-base junction, and
the ratio of the two,
I
C
I
C
to
I
B
I
B
is always the same.
