CHAPTER 6
BIPOLAR JUNCTION TRANSISTORS
© Nezih Pala npala@fiu.edu
EEE 6397 – Semiconductor Device Theory
1
Introduction
We begin this chapter with a qualitative discussion to establish a sound physical
understanding of BJT operation.
Then we shall investigate carefully the charge distributions in the transistor and relate the
three terminal currents to the physical characteristics of the device.
We shall also discuss the properties of the transistor with proper biasing for amplification and
then consider the effects of more general biasing, as encountered in switching circuits.
We shall use the p-n-p transistor for most illustrations. The main advantage of the p-n-p for
discussing transistor action is that hole flow and current are in the same direction. It is simple
to relate them to the more widely used transistor, the n-p-n.
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2
Fundamentals of BJT Operation -1
Consider a reverse-biased p-n junction diode. According to the theory of p-n junction, the
reverse saturation current through this diode depends on the rate at which minority carriers
are generated in the neighborhood of the junction.
The reverse current due to holes being swept from n to p is essentially independent of the
size of the junction  field and hence is independent of the reverse bias. The reason given
was that the hole current depends on how often minority holes are generated by EHP
creation within a diffusion length of the junction—not upon how fast a particular hole is
swept across the depletion layer by the field. As a result, it is possible to increase the reverse
current through the diode by increasing the rate of EHP generation.
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3
Fundamentals of BJT Operation -2
We could control the junction reverse current simply by varying the rate of minority carrier
injection by a hypothetical hole injection device.
The current from n to p will depend on the hole injection rate and will be essentially
independent of the bias voltage. There are several obvious advantages to such external
control of a current; for example, the current through the reverse-biased junction would vary
very little if the load resistor RL were changed, since the magnitude of the junction voltage is
relatively unimportant. Therefore, such an arrangement should be a good approximation to a
controllable constant current source.
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Fundamentals of BJT Operation -3
A convenient hole injection device is a forward-biased p+-n junction. the current in such a
junction is due primarily to holes injected from the p+ region into the n material. If we make
the n side of the forward-biased junction the same as the n side of the reverse-biased
junction, the p+-n-p structure results.
With this configuration, injection of holes from the p+-n junction into the center n region
supplies the minority carrier holes to participate in the reverse current through the n-p
junction. It is important that the injected holes do not recombine in the n region before they
can diffuse to the depletion layer of the reverse biased junction. Thus we must make the n
region narrow compared with a hole diffusion length.
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5
Fundamentals of BJT Operation -4
The structure we have described is a p-n-p bipolar junction transistor.
The forward-biased junction which injects holes into the center n region is called the emitter
junction, and the reverse-biased junction which collects the injected holes is called the
collector junction. The p+ region, which serves as the source of injected holes, is called the
emitter, and the p region into which the holes are swept by the reverse-biased junction is
called the collector. The, center n region is called the base, for reasons which will become
clear later.
The biasing arrangement in the figure is called the common base configuration, since the base
electrode B is common to the emitter and collector circuits.
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Fundamentals of BJT Operation -5
To have a good p-n-p transistor, the first requirement:
 Almost all the holes injected by the emitter into the base should be collected.
 Thus the n-type base region Wb should be narrow, and the hole lifetime P should be
long. This requirement is summed up by specifying Wb << Lp, where Wb is the length of
the neutral n material of the base (measured between the depletion regions of the
emitter and collector junctions), and Lp is the diffusion length for holes in the base
(Dpp)1/2.
With this requirement satisfied, an average hole injected at the emitter junction will
diffuse to the depletion region of the collector junction without recombination in the
base.
Second requirement is that the current IE crossing the emitter junction should be
composed almost entirely of holes injected into the base, rather than electrons crossing
from base to emitter. This requirement is satisfied by doping the base region lightly
compared with the emitter.
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7
Fundamentals of BJT Operation -6
It is clear that current IE flows into the emitter of a properly biased p-n-p transistor and that IC
flows out at the collector, since the direction of hole flow is from emitter to collector.
However, the base current IB
requires a bit more thought. In a
good transistor the base current
will be very small since IE is
essentially hole current, and the
collected hole current IC is almost
equal to IE .There must be some
base current, however, due to
electron flow into the n-type base
region.
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EEE 6397 – Semiconductor Device Theory
8
Fundamentals of BJT Operation -7
We can account for IB physically by three dominant mechanisms:
(a) There must be some recombination of injected holes with electrons in the base,
even with Wb << Lp. The electrons lost to recombination must be resupplied through the
base contact.
(b) Some electrons will be injected from n to p in the forward-biased emitter junction,
even if the emitter is heavily doped compared to the base. These electrons must also be
supplied by IB.
(c) Some electrons are swept into the base at the reverse-biased collector junction due
to thermal generation in the collector. This small current reduces IB by supplying
electrons to the base.
The dominant sources of base current are
(a) recombination in the base, and
(b) injection into the emitter region.
Both of these effects can be greatly reduced by device design. In a well-designed transistor, IB
will be a very small fraction (perhaps one-hundredth) of IE.
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9
Amplification with BJTs -1
Basically, the transistor is useful in amplifiers because the currents at the emitter and
collector are controllable by the relatively small base current.
We shall use total current (d-c plus a-c) in this discussion, with the understanding that the
simple analysis applies only to d-c and to small-signal a-c at low frequencies. We can relate
the terminal currents of the transistor iE, iB, and iC by several important factors.
Neglecting the saturation current at the collector (component 3 in the figure above) and
recombination in the transition regions; the collector current is made up entirely of those
holes injected at the emitter which are not lost to recombination in the base.
Thus iC is proportional to the hole component of the emitter current iEp:
iC  BiEp
(7.1)
The proportionality factor B is simply the fraction of injected holes which
make it across the base to the collector; B is called the base transport factor.
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EEE 6397 – Semiconductor Device Theory
10
Amplification with BJTs -2
The total emitter current iE is made up of the hole component iEp and the electron
component iEn) due to electrons injected from base to emitter (component 5 in the figure
above). The emitter injection efficiency 

iEp
iEn  iEp
(7.2)
For an efficient transistor we would like B and  to be very near unity; that is, the emitter
current should be due mostly to holes (  1), and most of the injected holes should
eventually participate in the collector current (B  1). The relation between the collector
and emitter currents is
Bi Ep
iC

 B  
iE iEn  iEp
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EEE 6397 – Semiconductor Device Theory
(7.3)
11
Amplification with BJTs -3
Bi Ep
iC

 B  
iE iEn  iEp
The product B is defined as the factor a, called the current transfer ratio, (common base
current gain) which represents the emitter-to-collector current amplification.
There is no real amplification between these currents, since  is smaller than unity.
On the other hand, the relation between iC and iB is more promising for amplification.
Consider the rates at which electrons are lost from the base by injection across the emitter
junction (iEn) and the rate of electron recombination with holes in the base.
In each case, the lost electrons must be resupplied through the base current iB. If the
fraction of injected holes making it across the base without recombination is B, then it
follows that (1 - B) is the fraction recombining in the base.
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EEE 6397 – Semiconductor Device Theory
12
Amplification with BJTs -4
Thus, neglecting the collector saturation current, the base current is
iB  iEn  (1  B)iEp
(7.4)
The relation between the collector and base currents is found from Eqs. (7.1) and (7.4):


Bi Ep
B iEp /(iEn  iEp )
iC


iB iEn  (1  B)iEp 1  B iEp /(iEn  iEp )

iC
B




iB 1  B 1  

(7.5)
(7.6)
The factor  relating the collector current to the base current is the base-to collector
current amplification factor (or more commonly common-emitter current gain). Since  is
near unity, it is clear that  can be large for a good transistor, and the collector current is
large compared with the base current.
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EEE 6397 – Semiconductor Device Theory
13
Amplification with BJTs -5
It remains to be shown that the collector current iC can be controlled by variations in the small
current iB. We can show from space charge neutrality arguments that iB can indeed be used
to determine the magnitude of iC.
Consider the transistor in the figure
where iB is determined by a biasing
circuit.
For simplicity, assume unity emitter
injection efficiency and negligible
collector saturation current.
Since the n-type base region is
electrostatically neutral between the
two transition regions, the presence
of excess holes in transit from emitter
to collector calls for compensating
excess electrons from the base
contact.
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14
Amplification with BJTs -6
There is an important difference in the times which electrons and holes spend in the base.
The average excess hole spends a time t defined as the transit time from emitter to
collector. Since the base width Wb is made small compared with Lp, this transit time is much
less than the average hole lifetime p in the base.
On the other hand, an average excess electron supplied from the base contact spends p
seconds in the base supplying space charge neutrality during the lifetime of an average
excess hole. While the average electron waits p seconds for recombination, many individual
holes can enter and leave the base region, each with an average transit time t . In particular,
for each electron entering from the base contact, p /t holes can pass from emitter to
collector while maintaining space charge neutrality.
Thus the ratio of collector current to base current is simply
p
iC
 
iB
t
(7.7)
for  = 1 and negligible collector saturation current.
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EEE 6397 – Semiconductor Device Theory
15
Amplification with BJTs -7
If the electron supply to the base (iB) is restricted, the traffic of holes from emitter to base is
correspondingly reduced. This can be argued simply by supposing that the hole injection
does continue despite the restriction on electrons from the base contact. The result would
be a net buildup of positive charge in the base and a loss of forward bias (and therefore a
loss of hole injection) at the emitter junction.
Clearly, the supply of electrons through iB can be used to raise or lower the hole flow from
emitter to collector.
Common-emitter circuit
The emitter junction is forward biased by
the battery in the base circuit. The voltage
drop in the forward-biased emitter
junction is small, so that almost all of the
voltage from collector to emitter appears
across the reverse-biased collector
junction.
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EEE 6397 – Semiconductor Device Theory
16
Example 7.1
(a) Show that the equation
p
iC
 
iB
t
is valid from arguments of the steady state replacement of stored charge. Assume that n = p .
(b) What is the steady state charge Qn = Qp due to excess electrons and holes in the neutral
base region for the transistor in the previous slide?
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EEE 6397 – Semiconductor Device Theory
17
BJT Fabrication -1
The first transistor invented by Bardeen
and Brattain in 1947 was the point
contact transistor. In this device two
sharp metal wires, or "cat's whiskers,”
formed an "emitter" of carriers and a
"collector" of carriers. These wires were
simply pressed onto a slab of Ge which
provided a "base" or mechanical support,
through which the injected carriers
flowed.
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EEE 6397 – Semiconductor Device Theory
18
BJT Fabrication -2
The p-n junctions in BJTs can be formed in a variety of ways using thermal diffusion, but
modern devices are generally made using ion implantation
Let us review a simplified version of how to make a double polysilicon, self-aligned n-p-n Si
BJT. This is the most commonly used, state-of-the-art technique for making BJTs for use in an
IC. Use of n-p-n transistors is more popular than p-n-p devices because of the higher
mobility of electrons compared to holes.
The process steps are shown in cross-sectional view in the following figures:
STEP 1: A p-type Si substrate is oxidized,
windows are defined using photolithography
and etched in the oxide. Using the
photoresist and oxide as an implant mask, a
donor with very small diffusivity in Si, such as
As or Sb, is implanted into the open window
to form a highly conductive n+ layer.
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EEE 6397 – Semiconductor Device Theory
19
BJT Fabrication -3
STEP 2: The photoresist and the oxide are
removed, and a lightly doped n-type epitaxial
layer is grown. During this high temperature
growth, the implanted n+ layer diffuses only
slightly towards the surface and becomes a
conductive buried collector (also called a subcollector). After a B channel stop implant, LOCOS
isolation layer is grown to ensure that there is no
electrical cross-talk between adjacent BJTs.
STEP 3: A polysilicon layer is deposited by
LPCVD, and doped heavily p+ with B either
during deposition or subsequently by ion
implantation. An oxide layer is deposited next
by LPCVD. Using photolithography with the
base/emitter mask, a window is etched in the
polysilicon/oxide stack by RIE.
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EEE 6397 – Semiconductor Device Theory
20
BJT Fabrication -4
STEP 4: A heavily doped "extrinsic" p+ base is
formed by diffusion of B from the doped
polysilicon layer into the substrate in order to
provide a low-resistance, high-speed base
ohmic contact. An oxide layer is then
deposited by LPCVD, which has the effect
of closing up the base window that was
etched previously, and B is implanted
into this window.
STEP 5: Then, another LPCVD oxide layer is
deposited to close up the base window further,
and the oxide is etched all the way to the Si
substrate by RIE, leaving oxide spacers on the
sidewalls. Heavily n+ doped (typically with As)
polysilicon is then deposited on the substrate,
patterned and etched.
Finally, an oxide layer is deposited by CVD, windows are etched in it corresponding to the
emitter (E), base (B), and collector (C) contacts, and a suitable contact metal such as Al is sputter
deposited to form the ohmic contacts.
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EEE 6397 – Semiconductor Device Theory
21
Minority Carriers and Terminal Currents -1
We will examine the operation of a BJT in more detail. We will use analysis used to analyze
the problem of hole injection into a narrow n-type base region.
Basically, we assume holes are injected into the base at the forward-biased emitter, and
these holes diffuse to the collector junction.
The first step is to solve for the excess hole distribution in the base,
and the second step is to evaluate the emitter and collector currents (IE , IC ) from the
gradient of the hole distribution on each side of the base.
Finally, the base current (IB) can be found from a current summation or from a charge control
analysis of recombination in the base.
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EEE 6397 – Semiconductor Device Theory
22
Minority Carriers and Terminal Currents -2
We shall at first simplify the calculations by making several assumptions:
1. Holes diffuse from emitter to collector; drift is negligible in the base region.
2. The emitter current is made up entirely of holes; the emitter injection efficiency is  = 1.
3. The collector saturation current is negligible.
4. The active part of the base and the two junctions are of uniform cross-sectional area A;
current flow in the base is essentially one-dimensional from emitter to collector.
5. All currents and voltages are steady state.
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EEE 6397 – Semiconductor Device Theory
23
Solution of the Diffusion Equation in the Base Region -1
Since the injected holes are assumed to flow from emitter to collector by diffusion, we can
evaluate the currents crossing the two junctions by techniques used in p-n junction analysis.
Neglecting recombination in the two depletion regions, the hole current entering the base at
the emitter junction is the current IE , and the hole current leaving the base at the collector is
IC. If we can solve for the distribution of excess holes in the base region, it is simple to
evaluate the gradient of the distribution at the two ends of the base to find the currents.
Consider the case in the figure: In equilibrium, the Fermi level is flat, and the band diagram
corresponds to that for two back-to-back p-n junctions.
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EEE 6397 – Semiconductor Device Theory
24
Solution of the Diffusion Equation in the Base Region -2
For a forward-biased emitter and a reverse-biased collector
(normal active mode), the Fermi level splits up into quasiFermi levels.
The barrier at the emitter-base junction is reduced by the
forward bias, and that at the collector-base junction is
increased by the reverse bias.
The excess hole concentration at the edge of the emitter
depletion region pE and the corresponding concentration
on the collector side of the base pc are
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pE  pn (e qVEB / kT  1)  pn e qVEB / kT
(7.8)
pC  pn (e
(7.9)
qVCB / kT
EEE 6397 – Semiconductor Device Theory
 1)   pn
25
Solution of the Diffusion Equation in the Base Region -3
We can solve for the excess hole concentration as a function of distance
in the base p(xn) by using the proper boundary conditions in the diffusion
equation:
d p( xn ) p( xn )

2
dxn
L2p
2
(7.10)
The solution of this equation is
p( xn )  C1e
xn / L p
 C2e
 xn / L p
(7.11)
where Lp is the diffusion length of holes in the base region. Unlike the simple problem of
injection into a long n region, we cannot eliminate one of the constants by assuming the
excess holes disappear for large xn. In fact, since Wb << Lp in a properly designed transistor,
most of the injected holes reach the collector at Wb.
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EEE 6397 – Semiconductor Device Theory
26
Solution of the Diffusion Equation in the Base Region -4
The solution is very similar to that of the narrow base diode problem. In this case the
appropriate boundary conditions are
p( xn  0)  C1  C2  pE
p( xn  Wb )  C1e
Wb / L p
 C2 e
Wb / L p
 pC
(7.12)
Solving for the parameters C1 and C2 we obtain:
C1 
pC  pE e
Wb / L p
e
e
Wb / L p
Wb / L p
C2 
,
pE e
Wb / L p
Wb / L p
e
e
 pC
(7.13)
Wb / L p
Inserting into Eqn. (7.11) and assuming collector junction is strongly reverse biased and the
equilibrium hole concentration pn is negligible compared with the injected concentration pE,
the excess hole distribution is given by:
p( xn )  pE
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Wb / L p
e
e
 xn / L p
Wb / L p
e
e
e
Wb / L p
e
xn / L p
Wb / L p
EEE 6397 – Semiconductor Device Theory
(for pC  0)
(7.14)
27
Solution of the Diffusion Equation in the Base Region -5
Some observations:
p(xn) varies almost linearly between the emitter and collector junction depletion regions.
The excess electron concentration in the p+ emitter decays exponentially to zero,
corresponding to a long diode. This is because, at high emitter doping levels, the minority
carrier electron diffusion length is often shorter than the thin emitter region.
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28
Evaluation of the Terminal Currents -1
Knowing the excess hole distribution in the base region, we can evaluate the emitter and
collector currents from the gradient of the hole concentration at each depletion region edge.
dp( xn )
I p ( xn )  qAD p
dxn
(7.15)

d
xn / L p
 xn / L p
 qAD p
C1e
 C2 e
dxn

This expression evaluated at xn = 0 gives the hole component of the emitter current,
I Ep  I p ( xn  0)  qA
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Dp
Lp
(C2  C1 )
EEE 6397 – Semiconductor Device Theory
(7.16)
29
Evaluation of the Terminal Currents -2
Similarly, if we neglect the electrons crossing from collector to base in the collector reverse
saturation current, IC is made up entirely of holes entering the collector depletion region
from the base. Evaluating Eq. (7.15) at xn = Wb we have the collector current
I C  I p ( xn  Wb )  qA
Dp
Lp
(C2e
Wb / L p
 C1e
Wb / L p
)
(7.17)
When the parameters C1 and C2 are substituted from Eqs. (7-13), the emitter and collector
currents take a form that is most easily written in terms of hyperbolic functions:
I Ep
D p  pE (eWb / L p  e Wb / L p )  2pC 
 qA


Wb / L p
Wb / L p
L p 
e
e

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EEE 6397 – Semiconductor Device Theory
coth x = (ex + e-x)/(ex - e-x)
csch x = 2/(ex - e-x)
30
Evaluation of the Terminal Currents -3
I Ep
Dp 
Wb
Wb 
 qA
 pC csch 
pE ctnh
L p 
Lp
L p 
Dp 
Wb
Wb 
I C  qA
 pC ctnh 
pE csch
L p 
Lp
L p 
(7.18)
Now we can obtain the value of IB by a current summation, noting that the sum of the base
and collector currents leaving the device must equal the emitter current entering. If IE  IEp
for   1,


Dp 
W
W
b
b

I B  I E  I C  qA
 csch
pE  pC  ctnh

L p 
L
L
p
p


Dp 
Wb 
 qA
pE  pC  tanh

L p 
2 L p 
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EEE 6397 – Semiconductor Device Theory
(7.19)
31
Example 7.2
a)
Find the expression for the current I for the transistor
connection shown if =1.
b)
How does the current I divide between the base lead
and the collector lead?
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EEE 6397 – Semiconductor Device Theory
32
Approximations of the Terminal Currents -1
The general equations of the previous section can be simplified for the case of normal
transistor biasing, and such simplification allows us to gain insight into the current flow.
For example, if the collector is reverse biased, pc = -pn from Eq. (7.9). Furthermore, if the
equilibrium hole concentration pn is small (see Fig), we can neglect the terms involving pc.
For   1, the terminal currents reduce to
I E  qA
Dp
Lp
pE ctnh
Wb
Lp
Dp
Wb
I C  qA
pE csch
Lp
Lp
Dp
Wb
I B  qA
pE tanh
Lp
2Lp
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EEE 6397 – Semiconductor Device Theory
33
Approximations of the Terminal Currents -2
Dp
1 y y3
ctnh y     ...
y 3 45
Dp
1 y 7 y3
csch y   
 ...
y 6 360
Dp
y3
tanh y  y   ...
3
Wb
I E  qA
pE ctnh
Lp
Lp
Wb
I C  qA
pE csch
Lp
Lp
Wb
I B  qA
pE tanh
Lp
2Lp
Using the expansions of the hyperbolic function, for small values of Wb/Lp we can neglect
terms above the first order of the argument. It is clear that IC is only slightly smaller than IE,
as expected. The first-order approximation of tanh y is simply y, so that the base current
is
I B  qA
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Dp
Lp
pE
Wb
qAWb pE

2Lp
2 p
EEE 6397 – Semiconductor Device Theory
(7.21)
34
Approximations of the Terminal Currents -3
The same approximate expression for the base current is found from the difference in the
first-order approximations to IE and IC:
I B  I E  IC
 1
Wb / L p   1
Wb / L p 


 qA
pE 


Lp
3   Wb / L p
6 
 Wb / L p
D pWb pE qAWb pE
 qA

2
2Lp
2 p
Dp
(7.22)
This expression for IB accounts for recombination in the base region.
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EEE 6397 – Semiconductor Device Theory
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Current Transfer Ratio -1
The value of IE calculated thus far in this section is more properly designated IEp, since we
have assumed that  = 1 (the emitter current due entirely to hole injection). Actually, there is
always some electron injection across the forward-biased emitter junction in a real transistor,
and this effect is important in calculating the current transfer ratio. The emitter injection
efficiency of a p-n-p transistor can be written in terms of the emitter and base properties:
1
 Ln
 Wb nn 
Wb 
  1 
tanh n   1  p
L p 
 L p 
 Ln p p 
n
p
p n n
p
n
n p p
p
n
n
p



1
(7.25)
In this equation we use superscripts to indicate which side of the emitter-base junction is
referred to. For example, Lpn is the hole diffusion length in the n-type base region and np is
the electron mobility in the p-type emitter region. In an n-p-n the superscripts and
subscripts would be changed along with the majority carrier symbols.
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EEE 6397 – Semiconductor Device Theory
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Current Transfer Ratio -2
Using Eq. (7.20) for IEp, and Eq. (7.20) for IC, the base transport factor B is
I C csch Wb / L p
Wb
B

 sech
I Ep ctnh Wb / L p
Lp
(7.26)
and the current transfer ratio  is the product of B and  as in Eq. (7.3).
 Wb nn 
  B  1  p
 Ln p p 
p
n
n
p
© Nezih Pala npala@fiu.edu
-1

Wb
 sech
Lp

EEE 6397 – Semiconductor Device Theory
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Example 7.3
Extend Eq. (7.20a)
Dp
Wb
I E  qA
pE ctnh
Lp
Lp
to include the effects of nonunity emitter injection efficiency (< 1). Derive Eq. (7.25) for .
1
 Ln
 Wb nn 
Wb 
  1 
tanh n   1  p
L p 
 L p 
 Ln p p 
n
p
p n n
p
n
n p p
p
n
n
p



1
Assume that the emitter region is long compared with an electron diffusion length.
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EEE 6397 – Semiconductor Device Theory
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Example 7.3
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EEE 6397 – Semiconductor Device Theory
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Generalized Biasing -1
The expressions derived in the last section describe the terminal currents of the transistor, if
the device geometry and other factors are consistent with the assumptions. Real transistors
may deviate from these approximations. For example, if the roles of emitter and collector are
reversed, these equations predict that the behavior of the transistor is symmetrical. Real
transistors, on the other hand, are generally not symmetrical between emitter and collector.
This is a particularly important consideration when the transistor is not biased in the usual
way.
We have discussed normal biasing (sometimes called the normal active mode), in which the
emitter junction is forward biased and the collector is reverse biased. In some applications,
particularly in switching, this normal biasing rule is violated.
In this section we shall develop a generalized approach which accounts for transistor
operation in terms of a coupled-diode model, valid for all combinations of emitter and
collector bias. This model involves four measurable parameters that can be related to the
geometry and material properties of the device.
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EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -1
If the collector junction of a transistor is forward biased, we cannot neglect pc; instead,
we must use a more general hole distribution in the base region. The figure illustrates a
situation in which the emitter and collector junctions are both forward biased, so that pE
and pc are positive numbers.
We can handle this situation with Eqs. (7.18) and (7.19) for the symmetrical transistor. It is
interesting to note that these equations can be considered as linear superpositions of the
effects of injection by each junction.
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EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -2
The straight line hole distribution of Fig.a can be broken into the two
components of Figs.b and c. One component (Fig. b) accounts for the holes injected by the
emitter and collected by the collector. We can call the resulting currents (IEN and ICN) the
normal mode components, since they are due to injection from emitter to collector.
The component of the hole distribution illustrated by Fig. c results in currents IEI and ICI
which describe injection in the inverted mode of injection from collector to emitter. Of
course, these inverted components will be negative, since they account for hole flow
opposite to our original definitions of IE and IC.
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EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -3
For the symmetrical transistor, these various components are described
by Eqs. (7.18). Defining
a 
qAD p
Lp
 Wb 
ctnh  
L 
 p
and
b
qAD p
Lp
 Wb 
csch  
L 
 p
we have
I EN  apE
and
I EI  bpC
and
I CN  bpE
with
I CI  apC
with
pC  0
(7.27)
pE  0
The four components are combined by linear superposition in Eq. (7-18):
I E  I EN  I EI  apE  bpC  A(eqVEB / kT  1)  B(eqVCB / kT  1)
(7.28)
I C  I CN  I CI  bpE  apC  B(eqVEB / kT  1)  A(eqVCB / kT  1)
where A = apn and B = bpn.
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EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -4
These equations show that a linear superposition of the normal and inverted components
does give the result derived previously for the symmetrical transistor.
To be more general, we must relate the four components of current by factors which allow for
asymmetry in the two junctions. For example, the emitter current in the normal mode can be
written
I EN  I ES (eqVEB / kT  1),
pC  0
(7.29)
where IES is the magnitude of the emitter saturation current in the normal mode. Since we
specify pC = 0 in this mode, we imply that VCB = 0 in Eq. (7.8). Thus we shall consider IES to be
the magnitude of the emitter saturation current with the collector junction short circuited.
Similarly, the collector current in the inverted mode is
I CI   I CS (eqVCB / kT  1),
pE  0
(7.30)
where ICS is the magnitude of the collector saturation current with VEB = 0.
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EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -5
The corresponding collected currents for each mode of operation can
be written by defining a new  for each case:
I CN   N I EN   N I ES (eqVEB / kT  1)
I EI   I I CI   I I CS (eqVCB / kT  1)
(7.31)
where N and I at are the ratios of collected current to injected current in each mode. We
notice that in the inverted mode the injected current is ICI and the collected current is IEI.
The total current can again be obtained by superposition of the components:
I E  I EN  I EI  I ES (eqVEB / kT  1)   I I CS (eqVCB / kT  1)
(7.32)
I C  I CN  I CI   N I ES (eqVEB / kT  1)  I CS (eqVCB / kT  1)
These relations are referred to as the Ebers-Moll equations.
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The Coupled-Diode Model -6
Ebers-Moll equations is that IE and IC
are described by terms resembling
diode relations (IEN and ICI), plus
terms which provide coupling
between the properties of the
emitter and collector (IEI and ICN ).
This coupled-diode property is
illustrated by the equivalent circuit
in the figure. In this figure we take
advantage of Eq. (7.8) to write the
Ebers-Moll equations in the
following form:
I E  I ES
pC I ES
pE
pE   N pC 
  I I CS

pn
pn
pn
I B  1   N I ES
I C   N I ES
© Nezih Pala npala@fiu.edu
pC
pE
 1   I I CS
pn
pn
pC I CS
pE
 I pE  pC 
 I CS

pn
pn
pn
EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -7
It is often useful to relate the terminal currents to each other as well as
to the saturation currents. We can eliminate the saturation current from the coupling term in
each part of Eq. (7.32). For example, by multiplying Eq. (7.32a) by N and subtracting the
resulting expression from Eq. (7-32b), we have
I C   N I E  1   N I I CS (eqVCB / kT  1)
(7.35)
Similarly, the emitter current can be written in terms of the collector current:
I E   I I C  1   N I I ES (eqVEB / kT  1)
(7.36)
The terms (1 - NI)ICS and (1 - NI)IES can be abbreviated as ICO and IEO, respectively,
where ICO is the magnitude of the collector saturation current with the emitter junction
open (IE = 0), and IEO is the magnitude of the emitter saturation current with the collector
open.
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EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -8
The Ebers-Moll equations then become
I E   I I C  I EO (e qVEB / kT  1)
(7.37)
I C   N I E  I CO (e qVCB / kT  1)
and the equivalent circuit is shown in figure.
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EEE 6397 – Semiconductor Device Theory
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The Coupled-Diode Model -9
In this form the equations describe both the emitter and collector currents in terms of a
simple diode characteristic plus a current generator proportional to the other current.
For example, under normal biasing the equivalent circuit reduces to the form shown in the
figure. The collector current is N times the emitter current plus the collector saturation
current, as expected. The resulting collector characteristics of the transistor appear as a
series of reverse-biased diode curves, displaced by increments proportional to the emitter
current in the figure.
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EEE 6397 – Semiconductor Device Theory
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Example 7.4
A symmetrical p+-n-p+ bipolar transistor has the following properties:
Emitter
Base
A=10-4 cm2
Na=1017
Nd=1015 cm-3
Wb=1 m
n = 0.1 s
p = 10 s
p = 200
n = 1300 cm2/Vs
n = 700
p = 450
(a) Calculate the saturation current IES = ICS
(b) With VEB = 0.3 V and VCB = -40 V, calculate the base current IB, assuming perfect emitter
injection efficiency,
(c) Calculate the base transport factor B, emitter injection efficiency 
and amplification factor , assuming that the emitter region is long
compared with Ln.
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EEE 6397 – Semiconductor Device Theory
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Example 7.4
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EEE 6397 – Semiconductor Device Theory
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Example 7.4
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