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PROFESSOR’S NOTES
12.1 THE BIPOLAR–JUNCTION TRANSISTOR: BASIC PRINCIPLES
The bipolar–junction transistor is one of the earliest types of semiconductor transistors. It likely would be the type
that your mother used when she was a young college student. The BJT is made up of two pn junctions back–to–
back, formed from three layers of semiconductor. It is therefore either of the form pnp or npn. The BJTs, like your
forebears, were rugged and strong, and this endurance and strength are important for many categories of contemporary circuits.
Most BJTs are formed by a planar technology in which an “epitaxial” layer is laid down on a substrate. This epilayer is of opposite gender to that of the substrate, e.g. n–type on a p–type substrate. The word epitaxial is one of those
words coined by the semiconductor industry, meaning that an “epi–”, or surface layer is grown on top of the crystalline substrate, acquiring the same crystalline “–taxial” structure as the substrate.
By means of implants, encirclement p–type diffusions are applied to the epilayer to create little “islands” of n–material in a p–type sea. If we had started with an n–type substrate and created a p–type epilayer, the islands would
have been p–type islands in an n–type sea. Each island is a place where a BJT may be constructed in isolated splendor. This structure with an implanted npn BJT is represented by Figure 12.1–1.
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Figure 12.1–1 Fabrication of the BJT. An npn transistor in cross–section, surrounded by p–type within
its little ’island’, is represented. The substrate is p–type and the epilayer is n–type.
The figure shows that the three layers of the BJT are three successively shallower layers, each counter–doped from
the previous layer. The epilayer is the outermost of these three layers. If we take a slice of npn (or pnp, if the
epilayer is p–type), across this structure, we have a basic 3–layer BJT, as represented by the inset to figure 12.1–1.
Note that the uppermost layer becomes the emitter of the BJT. This is an advantage, inasmuch as it helps to create a
BJT with large forward current gain βF.
Operation of the BJT in the forward–active mode is such that the base–emitter (BE) junction is forward–biased
while the base–collector (BC) junction is reverse–biased. Under this condition, carriers flow through the npn slice
in the manner represented by Figure 12.1–2. Because the junctions are very close to one another, the carriers injected across the BE junction by the forward bias find themselves in the vicinity of the strong E–field of the reverse–
biased BC junction. This strong E–field is from the collector toward the emitter, which is just exactly the direction
120
of field that excites the mobile (–) charge carriers, and sends them racing from the base region into the collector
region. Some of the carriers, of course, don’t make it across the base region, due to recombination. These casualties are carried off by the base terminal.
Note that current flows in the opposite direction to the electron flow. This flow is indicated by Figure 12.1–2.
#
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%
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Figure 12.1–2: Flow of charge carriers and current in the BJT. An npn transistor is represented
From this figure it is pretty clear that
'
IE
&
IC
IB
IC
&)( FI E
(12.1–1)
and that
where (
F
&
(12.1–2)
the forward current factor, which for a good quality device, is almost equal to 1. These two equations,
*
(12.1–1) and (12.1–2), are sufficient to define the forward gain current gain, I C I B
( F
+ F &
which is large provided that (
F
-
1
( F
&,+ F.
(12.1–3)
is close to 1. The more ideal transistors are able to control a large amount of collec-
tor current I C with only a very small amount of base current I B . Equation (12.2.3) may also be expressed in the
form:
1
+ F
1
&
( F
-
1
(12.1–4)
As we will see in section 12.6, equation (12.1–4) is handy for determining the effects of our fabrications on the
transistor current gain.
The BJT is, in principle, bilaterally symmetric. An np – pn junction structure exists, whether it is approached from
left–to–right or from right–to–left. The transistor may be inverted, with the BC junction forward–biased and the
BE junction reverse–biased, in which case we will have
I C ./&)(
I
R E.
and
121
(12.1–5)
1
R
1
R
(12.1–6)
1
where I C and I E are the ‘equivalent’ emitter and collector currents for the transistor in the inverted mode. The
principle of operation is the same. But if we compare “reverse” current gain R with “forward” current gain,
we would find that it is weaker by about two orders of magnitude. Typical values are:
F
100 whereas
R
F
,
0.5
The inverted mode of the BJT also is weak in another respect. The BC junction, due to its relatively light doping, is
able withstand a reverse bias of approximately 25 volts. However, the BE junction, since it is more heavily–doped,
can withstand a reverse bias of only about 7 volts. Therefore a BJT biased in an inverted mode, in which the BE
junction is reverse–biased, is in serious risk of junction breakdown if voltage rails of more than about 7 volts are
used.
12.2 EMITTER EFFICIENCY AND EMITTER DEFECT
Since the emitter–base junction is usually placed in forward bias, we must pay close attention to the features of this
junction. For any junction in forward bias, the n–type carriers injected into the p–type region will be of magnitude
np
n p0(e V Vt
1)
and the current due to these carriers will be diffusion current
Jn
qD n
dn p
dx
qD n
n (e V VT
L n p0
1)
Figure 12.2–1 represents the BE junction in forward bias. The forward emitter current will be n–type carriers injected into the p–type base, of current density
J nE
qD nB
dn p
dx
qD nB
n (e VBE VT
W B pB
1)
(12.2–1)
Figure 12.2–1: Excess minority carrier levels injected into the base region by BE junction forward bias.
An npn transistor is represented.
122
Note that we need to use a region–dependent syntax. D nB represents the diffusion coefficient for n–type carriers
injected into the p–type base, and n pB represents the equilibrium minority–carrier level of n–type carriers in the
p–type base. J nE represents emitter current density of n–type carriers injected into the p–type base from the emitter
region. In this case, for convenience, a linear approximation has been made for n(x), as represented by the plot of
n p (x) vs depth within the base region. Assuming the linear approximation for n(x), the slope dn p dx is approximately n 1 W B , where
(12.2–2)
n1
n pB(e VBE VT 1)
n pBe VBE VT
where V BE
V T is assumed. WB is the thickness of the base region, also called the “base–width”. If we wish to
be more particular, the slope dnp /dx is actually a hyperbolic function that is matched to the boundary condition n =
n1 at the emitter end of the base region to n = n2 at the collector end of the channel. The carrier density n2 at the
collector side is:
n2
n pBe VBC VT
and therefore is practically zero since, in the forward active mode, VBC is negative. These hyperbolic functions do
not provide much in the way of added insight, so we will stick to equation (12.2–1).
We can avoid the hyperbolic functions as long as W B
L B , where LB is the recombination distance
LB
BD nB , also called the diffusion length.
B is the recombination lifetime of the n–type carriers in the p–
type base region. The recombination distance represents the characteristic distance for the level of remaining carriers after penetrating deep into the p–type territory.
Note that equation (12.2–1) may also be written in terms of the base doping NB , as
J nE
where we will usually assume ni = 1.5
qD nB n 2i V
e BE
WB NB
VT
(12.2–3)
1010, for silicon at room temperature (T = 300K).
Current is also injected from base to emitter by the forward bias, and will be of form similar to equation (12.2–3)
J pE
qD pE n 2i V
(e BE
WE NE
VT
(12.2–4)
1)
The use of emitter width WE in (12.2–4) is only valid if W E
L E , where L E
ED pE = the recombination
distance for the p–type carriers in the n–type emitter region. Otherwise we must either resort to hyperbolic functions or, if the emitter is of infinite thickness, we can use W E
L E , so that
J pE
qD pE n 2i V
e BE
LE NE
(12.2–5)
VT
As far as the operation of the transistor is concerned, only that portion of the emitter current that is injected into the
base, JnE, will be able to “link” to the collector. Therefore the emitter will have an efficiency of
J nE
JE
J nE
E
J nE
J pE
1
1
J pE J nE
Using equations (12.2–3) and (12.2–5), the ratio JpE / JnE , which is also called the emitter defect, will be:
E
J pE
J nE
D pE W B N B
D nB L E N E
so that the emitter efficiency is of the form
123
(12.2–6)
E
1
1
DE WB NB
DB LE NE
1
1
(12.2–7)
E
Note that we have dropped the subscripts relating to type of carriers, i.e. DpE
DE and DnB
DB . This choice is
made, both to reduce clutter and to recognize that exactly same generic form will exist for both npn and pnp transistor types. Unless we like to challenge ourselves with excess notation, which is very easy to do in dealing with BJT
transistors, we will ALWAYS assume that terms like DE, DB, µE, and µB represent diffusion coefficients and mobilities for the minority carriers, for the emitter and base regions, respectively, in this case.
If we had used the more exact hyperbolic functions, the emitter efficiency γE would have been of the form[12.2–1]
1
E
W
D EN B L B
tanh B
LB
D BN E L E
1
(12.2–8)
For WB /LB
1, tanh(x)
x and equation (12.2–8) collapses to equation (12.2–7). γE will be close to 100%
efficiency if NE > > NB . This requirement is consistent with the construction shown by Figure 12.1.1.
One of the reasons for bypassing the more exact hyperbolic analysis is that equations (12.2–7) and (12.2–8) assume
that the base and emitter layers are uniformly–doped. This condition is usually NOT the case. Because of the fabrication process, both the emitter and the base dopings vary considerably NB = NB (x) and NE = NE (x). The emitter
defect and efficiency should then be defined in terms of Gummel numbers GB and GE, where
GB
WB
0
N B(x)
dx
D B(x)
(12.2–9)
A similar equation may be identified for the emitter Gummel number, GE. The emitter defect will then be
E
GB
GE
(12.2–10)
and the emitter efficiency will be
E
1
GB GE
1
(12.2–11)
Note that, in like manner, we could define a defect factor for the collector,
c
GB
GC
Dc WB NB
DB Lc N c
(12.2–12)
For very high impurity doping of the emitter, which is fairly common for high–current devices, the bandgap is narrowed by a small but significant amount ∆EG . This reduces the level of injected current, and therefore increases the
emitter defect so that
E
e
E
EG kT
(12.2–13)
Since ∆EG may be on the order of .08eV at NE = 10 19#/ cm 3, the bandgap narrowing can increase the emitter defect
by a factor on the order of 25 greater than that indicated by equation (12.2–6).
124
Due to the bandgap narrowing and other effects, higher power BJT devices usually do not have as high a value of βF
as do signal–processing types of BJTs.
12.3 BASE TRANSPORT FACTOR AND BASE RECOMBINATION DEFECT
The carriers which are injected across the emitter junction into the base region fall into the collector as result of the
strong E–field in their favor at the collector junction. This strong field “sweeps” the excess charge–carriers into the
collector region and reduces the excess carrier level to nearly zero at the collector boundary. The behavior is represented by Figure 12.3–1.
All of the carriers do not make it across the base region because they are passing through territory where they are
subject to heavy attrition (due to recombination). For the linear approximation to n(x), represented by Figure
12.3–1, the carriers lost to recombination in traversing the base region are
WB
n(x)dx
1n W
2 1 B
0
Figure 12.3–1
Excess carrier levels in the base region of the npn transistor.
since the injected carrier density across the base region is approximately triangular. These carriers represent a recombination current density of
J nR
WB
1q
n(x)dx
B
qn 1W B
2 B
0
(12.3–1)
The fraction of carriers that succeed in making it across the base region are then
J nC
J nE
J nE
J nR
J nE
1
J nR
J nE
(12.3–2)
where, using equations (12.3–1), (12.2.1), and (12.2.2) the fraction of carriers lost to recombination is
J nR
J nE
qn 1W B
2 B
WB
qD Bn 1
2 WD 125
2
B
B
B
B
where, as before, we have dropped the subscript nB and replaced it by B, to make the formula generic and less
cluttered. The small factor δB is called the base recombination defect (or base defect). The fraction of carriers that
survive the transit of the base region, also known as the base–transport factor, is given by
J nC
J nE
1
T
2 WD
2
B
B
(12.3–3)
B
Since τB DB = L 2B , where LB is the recombination length in the base, then the base–transport factor can also be
expressed as
W 2B
(12.3–4)
1
1
T
B
2L 2B
If we had developed the base–transport factor in terms of the hyperbolic functions we would have obtained
sech
T
WB
LB
(12.3–5)
In this case, the hyperbolic form might be slightly simpler than the linear approximation. However, the canonical
simplicity of equation (12.3–4) will, in general, be of more value to our being able to interpret the physical effects of
our BJT device.
The forward current ratio αF = JC / JE , is then a matter of taking ratios of currents, for which
JC
JE
F
J nC
J nE
J nE
JE
T E
(12.3–6)
If we have uniformly–doped base and emitter regions, which is at least our first approximation, αT and αE are represented by equations (12.3–4) and (12.2.6), respectively. More sophisticated devices and device analyses might
have need of equations (12.2.9) and (12.3–5).
**********************************************************************************
EXAMPLE 12.3–1: Determine αF and βF for an npn transistor with (uniform) doping levels of NE = 5
10 17, NB = 5 10 16, and NC = 5 10 15. Assume base width WB = 0.1LB and recombination time
constants τE = τB = 50ns. The mobilities within these regions are given by table E12.3–1.
Table E12.3–1: Representative mobilities for an npn transistor
mobility
NE
5
p
n
NB
1017
5
NC
1016
5
units
1015
#/cm3
cm2/V–s
cm2/V–s
Note that the ratio DB /DE = µB /µE, since the mobilty is proportional to the diffusion coefficient, i.e µn =
Dn /VT .
Also note that LB /LE
and
E
B
B
E
950 190 5 in this case, since L
B
.
Then
126
DB
B
, LE
DE E ,
E
1
5 1 1
10 1
5
0.1 1
1.0045 0.9955
The base transport factor αT is
T
1 W 2B
2L 2B αF = 0.9955 which gives
0.1 2
2 1 0.995
0.995 = 0.9906, and βF = 105.
Even though it is not a necessary part of these calculations it should be noted that
WB = 0.1LB = 0.1 BV T B 950 .02585 5 10 8 1.1 m.
*********************************************************************************
12.4 LINKING CURRENT AND THE EARLY EFFECT
For the npn transistor which we have used so far to tell us the story, the n–type carriers injected into the base from
the emitter, JnE are coupled directly to the collector current JnC by the diffusions and fields within the base region.
Since this is a semiconductor layer, current includes both drift and diffusion terms,
nE qD nB dn
dx
(12.4–1)
pE qD pB
dp
dx
(12.4–2)
Jn q
nB
0 q
pB
and
Jp Jp = 0 since none of the current across the base is due to Jp . It is all a great rampage of n–type carriers flowing across
the base. But the equation for Jp defines a relationship between E, the electric field, and the gradient of p(x). We
can apply this relationship to equation (12.4–1),
Jn q
n nB
D pB dp
pBp dx qD nB dn
dx
Now according to Einstein’s law, for which D = µVT,
nB
pB
D pB D nB
so that
Jn dp
D
q pnB n
dx
D
q pnB d (np)
dx
p dn dx (12.4–3)
This equation is called the linking–current relationship, since it defines Jn (x) anywhere within the channel. Solving
this differential equation and evaluating at x = WB , for which Jn = JnC ,
WB
pdx J nC W B) qD nB[np(x 0
2
nB i
qD n e
V BC V T
e
np(x VBE VT
127
0)]
(12.4–4)
Therefore the collector current IC is of the form
IC
q 2AD nBn 2i V
[e BE
Qb
AJ nC
VT
e VBC
VT
]
(12.4–5)
where A is the cross–sectional area of the junction. As indicated by the sign, the current flows from collector to
emitter, since it is due to n–type charge carriers (electrons flowing form emitter to collector).
Equation (12.4–5) neglects recombination. It primary virtue is that it recognizes that JnC is inversely proportional
to the somewhat malleable “base–charge” per area,
QB
WB
q
WB
q
pdx
0
N B(x)dx
(12.4–6)
0
NB = NB (x) need not necessarily be uniform. QB is voltage dependent, since it represents the majority carrier charge
in the zone that is between two depletion zones, each of which is voltage dependent. This situation is represented by
Figure 12.4–1. For the usual case, in which the transistor is in a forward active mode, variation of the collector
junction VBC has a small but noticeable effect and the emitter junction does not change.
We can evaluate increments by examining the derivative behavior of IC . Due to the charge dependence of the (reverse–biased) BC junction, IC will change slightly with respect to VCB as:
dI C
dV CB
I C dQ B
Q B dV CE
(12.4–7)
We have let dVCB = dVCE since, in forward bias, VBE is approximately constant. This change is handy, since dIC /
dVCE represents the small slope go , as shown by Figure 12.4–1, and defines the Early voltage VA :
dI C
dV CE
g0
IC
VA
(12.4–8)
dQB /dVCB is negative, since an increase in reverse bias VCB on the BC junction increases the reach of the junction
depletion charge QBC into the base, thereby decreasing QB by the same extent. This effect is illustrated by Figure
12.4–1
%)( * ,+.-%/( '
!"$#%&!"'
Figure 12.4–1:
Base–width modulation and the Early effect.
128
In fact, since both QB and QBC are proportional to the density NB of the base doping, then dQB = – dQBC . If we make
this substitution, equation (12.2.7) can be rewritten as
dI C
dV CE
I C dQ BC
Q B dV CE
IC
C
Q B JC
(12.4–9)
For convenience, we have put the charge effects in terms of CJC , the depletion capacitance of the BC junction (capacitance due to the uncovered charge QBC ) since CJC = dQBC /dVCE . This equation tells us that the base charge QB
forms a relationship between the Early voltage VA and CJC :
VA
(12.4–10)
Q B C JC
If we had biased the transistor in the reverse mode, with VBC forward–biased and VBE reverse biased, the analysis
would have been completely the same, except in terms of the BE junction. Therefore an analogous relationship for
the emitter junction exists of the form:
VB
(12.4–11)
Q B C JE
where VB is the reverse Early voltage. VA and VB also may have aliases such as as VAF and VAR.
The model used by SPICE, the Gummel–Poon model, makes use of the Early effect relationships with the depletion
charges of the BE junction and BC junctions, for voltage modulation of the base–charge QB .
12.5 BASE TRANSIT TIME
The current linking the collector and emitter is due to the charges that were injected into the base, moving with
velocity v, as given by
IC
AJ nC
qAn pv
(12.5–1)
qAn p dx
dt
If we evaluate this differential equation, we find that it defines a base–transit time, τF, and a charge QF of forward
transport carriers, according to
F
1
IC
dt
F
0
qAn pdx
0
Qf
.
Ic
(12.5–2)
For the forward active mode, np = np (x) is approximately by the linear distribution n(x) = n1 ( 1 – x/WB ). Applying
this approximation to equation (12.5–2),
F
1 q
J nC
WB
n(x)dx
0
1 1 qn W
J nC 2 1 B
where we have used JnC = IC /A. If we neglect the recombination in the base region, assuming WB is small, then
equation we can use equation (12.2.1) to define JnC ,
J nC
J nE
qD B
n1
WB
129
for which
F
1 1 qn W
J nC 2 1 B
W 2B
2D nB
(12.5–3)
Does this look familiar? It should! In section 12.2.3 we showed that the ratio of recombination current JnR to injected current JnE was
B
W 2B
2 BD B
J nR
J nE
Therefore, tracing this definition through to its application to the base transport factor αT,, and using equation
(12.5–3), we can write the base transport factor, in alternative form, as
F
1
T
(12.5–4)
B
12.6 APPROXIMATIONS
Because the equations for αT and γE are relatively simple, we can make some informative approximations for
evaluating one of the principal parameters that we need for circuit evaluation, βF. In order to make these approximations it is handy to use equation (12.1.4), which was
1 1 Keep in mind that
αF
(12.6–1)
1
F
F
depends on the relationships for αT and γE ,
T
1
1 1
E
As a first approximation, suppose αT
1, then
1 1 F
F
D EN BW B
D BN EL E
W 2B
2L 2B
D EN BW B
D BN EL E
, and
E
1
D EN BW B
D BN EL E
E
(12.6–2)
where δE is the emitter defect, defined by equation (12.2.6). This approximation tells us that βF is proportional,
among other factors, to NE /NB . More explicitly, if we take E to be the more general form given by equation
(12.2.10), βF = GE /GB , where GB and GE are the Gummel numbers.
As an alternate approximation, we might assume
1 F
1
1
W 2B 2L 2B
Therefore, the forward current gain would be
1
1
W 2B
2L 2B
F
, so that
1, which makes
E
2
B
2
B
2L
W
130
1
B
1
F
W 2B
2L 2B
T
B
(12.6–3)
L W .
where δB is the base defect. In equation (12.6–3) we made the approximation 1/(1 – x)
(12.6–3) tells us that βF is proportional to
2
B
1 + x for x << 1. Equation
2
B
We can combine the emitter and base defects to give the approximation
1 E
(12.6–4)
B
F
Note that we can use equation (12.5.4), for αT, which tells us that, in approximation of high emitter efficiency,
B
F
(12.6–5)
F
This approximation is particularly useful for analysis of charge–control effects and time–domain response. As an
approximation, both emitter and base defects can be combined under equation (12.6–5) by incorporating the emitter defect into either τB or τF.
**********************************************************************************
EXAMPLE 12.6–1: Using the values of example 12.3.1, determine the base defect δB and the forward
and reverse emitter defects, δE and δC = δErev, (for the emitter and collector junctions respectively). From
these values find βF and βR using equation (12.6–4). Assume all recombination times, τE , τB , and τC = 50
ns, and that the emitter and collector regions to be large relative to their recombination lengths.
In example 12.3.1 the doping levels were:
NE = 5
10 17, NB = 5
10 16, and NC = 5
10 15.
and the base width WB = 0.1 LB . The mobilities were given as:
mobility
NE
5
p
n
NB
1017
5
NC
1016
units
1015
5
#/cm3
cm2/V–s
cm2/V–s
Since DE / DB = µE / µB and DC / DB = µC / µB , and since L B L E
result of all of the recombination times being equal, then
B
E
and L B L C
From the table, the mobility ratios µE /µB = 190/950 = 1/5 and µC /µB = 450/950 = 9/19.
The ratios of the doping levels are NB / NE = (5
NB / NC = (5 1016 ) / (5 1015 ) = 10
1016 ) / (5
This gives emitter defect
E
DE
DB
NB
NE
LB
LE
WB
LB
15 1
10
5
1017 ) = 1/ 10 and
0.1
0.4472
10 2
and collector defect, which is the same as the reverse emitter defect.
C
Erev
DC
DB
NB
NC
LB
LC
WB
LB
9 19 131
10
19 9 0.1
0.6882
B
C
as
The base defect is
W 2B
2L 2B
which tell us that
for which
Similarly
1 0.1 2
2
(0.4472
F
F
0.5
0.5)
10 10 2
2
0.9472
10 2
= 105.5. This result is almost the same as the result found under example 12.3.1.
1 (0.6882
R
.005)
0.6932
for which βR = 1.44. Note that almost all of the current defect for the inverted mode of the transistor is due
to the collector defect. (= reverse emitter defect)
*********************************************************************************
12.7 CHARGE–CONTROL ANALYSIS
In section 12.5, we identified the base transit time τF as a relationship between collector current and a forward
carrier charge in the base, QF. This charge, which, for the npn transistor, is associated with injected n–type carriers,
n(x), forms an approximately triangular distribution within the base layer, as shown by figure 12.7–1. This charge–
current relationship, for steady–state, is:
IC
QF
F
! "$#
"&%
Figure 12.7–1
Carrier charge QF in the base under forward–active bias.
The figure also shows that IB plays a role in control of the forward carrier charge. The change of QF with respect to
time, dQF /dt, should increase as result of base current and decrease as result of the recombination of carriers. This
can be represented by
dQ F
dt
IB
Q
' F
B
132
(12.7–1)
Note that for steady–state, dQ F
0 and Q F
Then
.
B
I C IB
F
B
(12.7–2)
F
Note that this result is just like equation (12.6.4), where we were analyzing charge in terms of recombination using
the base–transport factor.
If the collector junction were forward–biased and the emitter junction reverse–biased, a triangular distribution of
carrier charge Q R would result, of a reverse orientation. Just like the analysis above, we could identify, in a similar
way.
I C IB R
B
(12.7–3)
R
where R
the reverse transit time. SPICE uses R
TR. This equation is a little deceptive, in that βR is low
primarily because of a poor junction efficiency, NOT because of transit times. But, assuming that we can equivalently “embed” the junction effects γE and γC into the transit–times, then equations (12.7–2) and (12.7–3) are perfectly alright.
If both junctions are forward biased, these triangular distributions could be superimposed, as shown in figure
12.7–2, to form a trapezoidal distribution. This situation corresponds to the saturation mode. This distribution
creates something of a problem for the carriers, since only the upper triangular part of the trapezoid has a slope
which can define a collector current. This upper triangular distribution, which we have labeled as QF ’, is identical
in behavior to the triangular distribution that we considered in figure 12.7–1. It therefore can be represented by
time constant F , so that saturation current
I CS
Q F F.
Figure 12.7–2: When the BJT is in saturation, carrier charge QF and carrier charge QR overlap to form a
trapezoidal distribution.
The charge distribution below the upper triangular distribution, QF ’ represents a charge which is not swept across
the base, since there is no slope dn/dx that can be associated with it. This part of the distribution therefore represents
a stored charge of carriers, QS , in the base. The only means by which this charge may be diminished is by recombination, or by the feed currents IB and IBS , as represented by figure 12.7–2(b). Following the model represented by
this figure, the behavior of charge QS is define by a charge–control equation, similar to (12.7–1), of the form
133
dQ S
dt
(I B
QS
I BS)
(12.7–4)
B
where IBS represents the base current needed to supply the saturation
I CS F. Other than the relatively slow
recombination process, QS can only be diminished by reversal of IB , such that it drains the charge out of the base.
This process is indicated by the logic–switch example shown by figure 12.7–3.
Figure 12.7–3: Saturated logic: The switch switches IB from IBF, which is large, and sufficient to put the
transistor in saturation, to – IBR , which acts as a “base–drain” to remove the stored charge QS .
This case is fairly important. For t < 0, QS is assumed to be in steady–state, for which dQS / dt = 0. Now, using
IB (t < 0 ) = IBF, equation (12.7–4) gives an initial boundary condition on QS ,
Q S(t
0)
I BF
B
I BS
(12.7–5)
At t = 0, the switch is toggled, and IB –> – IBR . The differential equation that results, using (12.7–4), is then
dQ S
dt
(
I BR
I BS)
Q S(t)
(12.7–6)
b
This equation is subject to the boundary condition on QS given by equation (12.7–5). Solution is straightforward,
since equation (12.7–6) is just another one of those standard decay–time equations that we always encounter with
circuits. Its solution is
Q S(t)
(I BR I BF)e t B (I BR I BS)
(12.7–7)
B
Note that for t
0, QS
IBF – IBS , which is the boundary condition (12.7–5).
This equation tells us that after some time, t then
ts , the stored charge QS
(I BR
I BF)e t S B
(I BR
0. If we let QS
0 in equation (12.7–7)
I BS)
for which we can get the storage time ts ,
tS
1n
B
(I BR
(I BR
I BF)
I BS) !
(12.7–8)
The storage time is an important aspect of BJT logic circuits where the transistor is allowed to go into saturation. It
represents a finite time delay on the order of several ns. Even though it is a well–recognized problem, for which a
number of good solutions have been applied, it is a circuit design aspect that must always be addressed for any of the
134
BJT logic circuit families, including such mixed technologies as BICMOS.
12.8 THE SMALL–SIGNAL PARAMETERS
The small–signal model is readily developed from the charge–control analysis. It is necessary that the transistor be
in the forward active mode, for which collector current IC and the forward carrier charge QF, are related according
to
QF
(12.8–1)
IC
F
where, according to the definition of QF given by equation (12.5.2) and the definition for n1 given by (12.2.2),
QF
VT
Q F0e VBE
(12.8–2)
where QF0 is a constant, = 1/ 2 qAnpB WB if we happen to have a uniformly doped base.
The small–signal change of IC , from equation (12.8–1), is then
dI C
dQ F
Q FOe
V BE V T
F
F
dV BE
VT
IC
dV
V T BE
Since small–signals i C
transducer (VCT) form
dI C and
BE
dV BE, this equation represents the small–signal voltage–to–current
iC
gm
gm
IC
VT
BE
where
(12.8–3)
Equation (12.8–2) also defines a capacitance due to the excess charge carriers injected into the base, of the form
CD
dQ F
dV BE
d ( I )
dV BE F C
F
dI C
dV BE
g
F m
(12.8–4)
Since QF is a diffusion charge, CD is usually called the diffusion capacitance of the emitter junction.
We have already shown, from equation (12.4.8) that
g0
dI C
dV CE
IC VA
(12.8–5)
where VA is the forward Early voltage.
A base–collector diffusion capacitance CA can also be defined, as follows:
CA
dQ F
dV CB
dQ F
dV CE
135
F
dI C
dV CE
g
F 0
(12.8–6)
Since go is approximately 4 orders of magnitude smaller than gm , then CA is likewise very much less than CD , being
on the order of fF. Junction depletion capacitances CJE and CJC are larger than CA by several orders of magnitude,
typically on the order of pF.
Other small–signal terms relate to the base current, IB, = IC / βF. In this respect there are two base conductances,
g and g given by:
dI B
dV BE
1 dVdI g
dI B
dV CE
1 dVdI g
g
F
C
m
BE
F
(12.8–7)
and
g
F
C
0
CE
F
(12.8–8)
g represents such a small conductance that it is usually omitted from the small–signal model. Even the relatively
small conductance go is often neglected, for simplicity. The complete small–signal BJT model, also called the
hybrid–pi model, is shown by figure 12.8–1.
Figure 12.8–1:
The hybrid–pi small–signal model of the BJT
Both depletion capacitances and diffusion capacitances contribute to C and C . C
junction,
C
C D C JE
is associated with the BE
(12.8–9)
whereas Cµ is associated with the BC junction,
C
CA
C JC
(12.8–10)
C JC
Since we have well–defined capacitances and conductances, the hybrid–pi small–signal model of figure 12.8–1 has
a well–defined high–frequency corner. If we look at the case under which we get maximum output current, in
which case the output is shorted, the hybrid–pi circuit has a pole at
!#" g
C
C
Since the current gain is βF, the gain–bandwidth product is
! T
C
g
F
C
gm
C
136
C
(12.8–11)
is usually called the unity gain–bandwidth product, and is often a specification of the transistor upper
fT
T 2
frequency limit at some test current IC. Note that if we use equations (12.8–9) and (12.8–10), and neglect CA , then
g
m
C JE
T
C JC
g
(12.8–12)
(12.8–13)
m F
or, we can determine τF from gm , CJE , and CJC , as follows:
1 F
1
g m (C JE
T
C JC)
Our favorite circuit simulation software, SPICE, requires input values for F (TF), CJE (CJE), and CJC (CJC) to
be included in the .MODEL statement in order to make a reasonable assessment of circuit frequency response.
We have to be a little wary of this equation, because as the current I C
0, g m
case we could end up with a nonsense value for τF. In the limit,
gm
gm
T
C
C JE C JC
C
0, and C D
0,
in which
representing a cutoff ”floor” for the frequency response of a transistor.
**********************************************************************************
EXAMPLE 12.8–1: A BJT has test specifications fT = 318 MHz at IC = 1 mA. The collector capacitance
C = 1 pF, and CJE = 2 CJC . (a) Find the SPICE parameters and (b) find the small signal capacitance C
and fT when IC = 0.1 mA.
SOLUTION: (a) Assume that VT = .025 V. Then gm0 = transconductance under test conditions = 40
mA/V. Since CA << CJC , then CJC
C = 1 pF, and CJE = 2
CJC = 2 pF.
Also we find that
T
=2
318 MHz = 2 Gr/s
Then, using equation (12.8–13), we find
1
1
(2pF
F
2Gr s 40mA V
1pF)
0.425ns
Units were included in this case to indicate that our result should be always be in ns, since we are dealing
with pF and Gr/s.
The SPICE parameters are then:
TF = 0.2NS, CJE = 2PF and CJC = 1 PF.
(b) For current IC = 0.1 mA, we will have gm = 40
we find that
CD
gM
F
4ma V
0.1 = 4 mA/V. Then from equation (12.8–4),
0.425ns
1.7pF
for which we will have C = 1.7 pF + 2 pF = 3.7 pF
C
gm
T
C
and
4
(3.7
1)
0.851Gr s
corresponding to fT = 135 MHz.
**********************************************************************************
137
12.9 THE GUMMEL–POON MODEL AND SPICE PARAMETERS
In simulation of BJT circuits, the BJT functions as a current–to–current transducer (CCT). The emitter and base
defects may be included with βF. The parameter βR is sufficient to define the BJT in the inverted mode. The complete flow of currents in the forward–active mode of the BJT is shown by figure 12.9–1.
Figure 12.9–1
Current flow in a one–dimensional npn transistor, showing emitter and base defects.
At the level of analysis where hand calculations are the norm, we assume that IC / IB = constant = βF. Over a wide
range of currents, which is what we expect with real circuits, and which is the way in which we desire the circuit
simulation software to work, this assumption is not particularly good. There are two current–driven effects that
degrade the ratio IC / IB , one at low levels of current, and one at high–levels of current. These effects are indicated
by figure 12.9–2.
Note that at low–levels, the ratio IC / IB , which we also call “βF ”, rolls off, because IB has a smaller slope than IC . At
high–levels, IC / IB rolls off because the slope on IC is smaller than that for IB . It appears that our favorite assump
tion, I C I B constant, is valid only within some median range.
,+ ! #
"
$ #"
%&
')( * %
Figure 12.9–2
Logarithmic plots of IC and IB vs VBE , showing low–level and high–level “knees”.
But not to worry. We merely include these effects under the base current IB and the base charge QB that we identified under the linking–current analysis in section 12.4. This approach is called the Gummel–Poon model [12.9–1],
138
developed at Bell labs in the early 1970’s.
Other than the efficiencies γΕ and γC , we have seen that the operating characteristics of the BJT depend almost
exclusively on the electrical and physical profile of the base region and on the injection currents into and across this
region. The linking–current analysis, developed in section 12.4, tie these together in the form
IC
1 e 1 ]
VT
I S [ e VBE
where
VBC V T
n 2i
QB
(12.9–2)
p(x)dx
(12.9–3)
q 2AD B
IS
and
WB
q
QB
(12.9–1)
0
Note that, in the forward active mode, where V BE
0, V BE
IC
0, I C is of the form of an ideal diode current, i.e.
1
I S e VBE
IC
In the reverse–active mode, for which V BC
0, V BC
VT
IF
0, I C it is of similar and opposite form
I e 1
S
V BC V T
IR
where IS is saturation current for the base region. The linking–current analysis of section 12.4 shows that IS is the
exactly the same for either the forward or the reverse direction.
Low–level currents are dominated by recombination in the junction space–charge regions. In the forward–active
mode this current will be primarily that associated with the emitter junction. In the more comprehensive model we
also include the collector junction. Recombination current in the SCR does not link across the base region, and
therefore becomes a contribution to the base current IB , for each of the two junctions. These two exponential terms
must have emission factors nE and nC , respectively, which we expect to be of value approximately 2, consistent
with recombination in the SCR..
The more comprehensive form for IB , with terms including recombination in each SCR, is then
IB
I e 1 I e 1 I e 1 I e 1
S
VBE V T
SE
V BE n EVT
F
S
V BC V T
SC
VBC n CV T
R
where nE and nC are the emission factors for recombination in the emitter and collector junctions, respectively. ISE
and ISC are the saturation coefficients for these terms. All of these coefficients are SPICE parameters, as identified
by table 12.9–1.
High–level current effects and the Early effect are associated with base–charge QB . QB represents the components
of base charge which are not part of the drift–diffusion process for charge–control analysis between emitter and
collector junctions. It includes (1) the depletion charge for the two junctions and (2) excess charges associated with
high–level current effects. With these factors included, QB then takes the form
139
QB
Q BO
Q BE
Q BC
V BE
VB
Q BO 1
Q KF
V BC
VA
Q KR
IS V V
e BE T
I KF 1
IS V V
e BC T
I KR (12.9–5)
1 QB0 is the zero–bias value of QB , i.e. the value when VBE and VBC are = 0. QBE and QBC are the uncovered “depletion” charge regions of the BE and BC junctions, respectively, that encroach the base region. QKF and QKR are the
excess charges in the base as result of high–level injection.
The high–level excess charges QKF and QKR begin to become significant at levels of current defined by IKF and IKR .
These are the “knee current” levels, for forward and reverse modes of operation, respectively, at which the linking
current IC begins to roll–off.
It is also convenient to identify the zero–bias value of IS as IS0 . With this simplification equation (12.9–3) is of the
form
q 2AD Bn 2i
QB
IS
q 2D BAn 2i Q BO
Q BO Q B
I SO
qB
(12.9–6)
where
QB
Q BO
qB
(12.9–7)
qB is the “normalized” form of QB . Inasmuch as it contains the the complete electrical behavior of the contributions
to the base charge, it is self–sufficient as a relationship which will let us analyze the high–level collector current
effects as a function of bias:
qB
1
q1
V BE
VB
V BC
VA
1 I SO V V
q B I KF e BE T
1
I SO V V
e BC T
I KR 1
(12.9–8)
q2
qB
q1 represents the terms associated with only with the doping NB . Note that the uncovered (depletion) charges on the
base side of the BE and BC junctions are associated only with NB . These terms should dominate at low current
levels. The term q represents excess charges associated with the high–level injection, which therefore involves IS .
These are expected to dominate at high current levels. Equation (12.9–8) is quadratic in qB and can be written as
q 2B
q Bq 1
q2
(12.9–9)
0
which has a (positive) solution of
qB
q1
2
(q 1 2) 2
q2
(12.9–10)
Note that in the limits of low–level and high–level currents, equation (12.9–10) gives:
qB
qB
q1
if q 2
q 21 4
(low
q 21 4
(high
level injection)
(12.9–11)
q 2 if q 2
level injection)
In terms of the normalized use of qB , equation (12.9–1) is of the form
140
I SO V V
q B e BE T
IC
1
e VBC VT
1 (12.9–12)
We can see that equation (12.9–12), along with (12.9–11), gives a behavior for IC that is sufficient to describe a
roll–off at high–level injection. For example, suppose the BJT is in forward–active mode and at a state of high–lev
el current, for which q B
q2 .
Then equation (12.9–12) gives
IC
IS (e VBE VT
I S I KF(e
1)
VBE V T 1)
(12.9–13)
I KFI S e VBE 2VT
1
This is the appropriate IC behavior at high–level injection that we noted with figure 12.9–2.
The basic Gummel–Poon model for the BJT consists of equations (12.9–5), (12.9–8), and (12.9–12). The CCT
ratio IC / IB is a realistic representation of the device current gain response, illustrated by figure 12.9–3.
Figure 12.9–3 Plot of IC / IB vs VBE . The maximum value is βF, which occurs at approximately VBE =
0.65V in this case.
The Gummel–Poon model has a total of some 21 parameters associated with the charge–control analysis. These
include a few higher–order parameters needed to model variations in τF which we have not detailed. In addition to
the charge–control parameters there are also 5 parameters for built–in resistances, and 4 parameters which detail
specifics of the pn junction, such as grading coefficients and the built–in potentials. 6 parameters for a substrate
junction JS are also included, for those situations in which the substrate junction inadvertently may be forward
biased. In addition there are noise parameters and bandgap parameters, for a total parameter count of approximately 40.
A complete parameter list for the Gummel–Poon model is given by table 12.9–1. Note that some of these parameters may not have the same keywords from one version of SPICE to another.
141
TABLE 12.9–1 Gummel–Poon Model Parameters
Symbol
Description
Keyword Alt keywordsDefault
Units
IS
BF
BR
NF
NR
ISE
JS
10–16
A
nF
nR
ISE
Forward carrier saturation current
Maximum forward current gain
Maximum reverse current gain
Forward emission coefficient
Reverse emission coefficient
BE junction saturation current
C2
100
1
1
1
0
A
ISC
BC junction saturation current
ISC
C4
IKF
IKR
nE
nC
VA
VB
RC
RE
RB
RBM
IRB
High–level injection forward knee current
High–level injection reverse knee current
BE junction emission coefficient
BC junction emission coefficient
Forward Early voltage
Reverse Early voltage
Collector resistance
Emitter resistance
Base resistance
Minimum high–current base resistance
50 pct drop point for RB
forward transit time
reverse transit time
Coefficient for bias dependence of F
Voltage coefficient for bias dependence of
JBF
JBR
NLE
NLC
VA/VAR
VB/VBR
F
IKF
IKR
NE
NC
VAF
VAR
RC
RE
RB
RBM
IRB
TF
TR
XTF
VTF
High–current coefficient for behavior of F
Excess phase factor at t = F
Zero–bias BE junction depletion capacitance
BE junction built–in potential
BE junction grading coefficient
Zero–bias BC junction depletion capacitance
BC junction built–in potential
BC junction grading coefficient
Zero–bias substrate depletion capacitance
Substrate junction built–in potential
Substrate junction exponent factor
Base–collector capacitance fraction
Coefficient for depletion capacitance
F and R temperature coefficient
Saturation current temperature coefficient
Bandgap energy
Flicker noise coefficient
Flicker noise exponent
ITF
PTF
CJE
VJE
MJE
CJC
VJC
MJC
CJS
VJS
MJS
XCJC
FC
XTB
XTI
EG
KF
AF
IS
F
R
F
R
XTF
VTF
ITF
PTF
CJE
E
mE
CJC
C
mC
CJS
S
mS
XCJC
FC
XTB
XTI
EG
KF
AF
142
JRB
PE
ME
PC
MC
TB
PT
0
∞
∞
1.5
2
∞
∞
0
0
0
0
∞
0
0
0
∞
∞
0
0
0.75
0.33
0
0.75
0.33
0
0.75
0
1
0.5
0
3
1.12
0
1
A
A
A
V
V
A
s
s
V
A
deg
F
V
F
V
F
V
eV
12.13 ANALYSIS OF THE BJT: EXERCISES
12–1. Find emitter defect δE , base recombination defect δB , and βF for the following BJT. Let τE = 10 ns, τΒ
= 20 ns, τC = 20 ns, and assume that both the emitter and base are shallow, with WE = 1.0 µm and WB = 1.2 µm.
Assume that the transistor is of type npn.
mobility
NE
1017
5
p
NB
NC
1016
1
1015
2
n
units
#/cm3
cm2/V–s
cm2/V–s
12–2. Evaluate the transistor of exercise 12–1, for δE, δB and βF, if it is of type pnp. Why does this transistor have
so much smaller βF than the npn version?
12–3. Evaluate the collector defect δC and the reverse–current gain factor, βR for both of the types (npn and pnp)
identified by exercises 12–1 and 12–2. Note that you can use the base defect, δB , as already determined.
12–4. The dopings and mobilities for a symmetrically–doped transistor are shown. The transistor has a shallow
emitter, of depth WE = 2 µm , and thickness WB = 2 µm of the base region. The recombination times are all equal, τE
= τB = τC = 20 ns. (a) Assuming an npn transistor, determine δE , δB , and δC. (b) Why is the collector defect
smaller than the emitter defect? (c) Determine βF and βR and comment on their relative magnitudes.
mobility
NE
1
p
1017
NB
1
NC
1016
1
1017
n
units
#/cm3
cm2/V–s
cm2/V–s
12–5. How is the (metallurgical) base width WB affected by the transistor biases? Assume the same parameters
(dopings, dimensions, mobilities) as given by exercise 12–1 and determine the “shrink” of the basewidth WB as
result of the encroachment of the junction depletion widths. Assume forward bias VBE = 0.6 V and a CE bias VCE =
4 V. Note that VCB = VCE – VBE ( = reverse–bias on BC junction). (a) Determine the built–in potential φBE and φBC
for each of the junctions and (b) the encroachment widths ∆WBC = NC /(NC + NB )WBC , where WBC is the depletion
width of the BC junction. A similar form for ∆WBE results for the BE junction. (c) What reduced basewidth WB
results?
12–6. Determine the base transit time τF for exercise 12–1, using the base–width and the diffusion coefficient for
the base as given. The charge–control model assumes that βF = τB /τF, and that βR = τB /τR . Let βF = 50, and determine τΒ . Note that it is nowhere near that of τB used in exercise 12–1. In fact, SPICE uses it as an idealized charge–
control parameter. Assume βR = 1.25 and determine τR . The parameters βF , βR , τF, and τR are four of the parameters that SPICE needs in order to use the transistor in circuit simulation.
12–7. When carrying a current IC = 0.5 mA an npn transistor with βF = 80, βR = 0.5, VA = 100V, Cµ = 1 pF has
unity–gain bandwidth of fT = 318 MHz. Assuming that the junctions have CJE 2CJC and that CJC >> CA ,
(a) determine SPICE parameters CJE , CJC , τF = TF and τR = TR.
(b) determine the small–signal capacitances C C and CA.
(c) What values do C and fT have if the current is reduced to 0.1 mA?
143
12–8. The transistor used in the circuit shown has SPICE parameters BF = 100, VA = 120, BR = 0.5, TF = 0.25 ns,
CJE = 2CJC = 1 pF.
(a) Determine the small–signal conductances gm , g , go , g .
(b) Determine the diffusion capacitances CD and CA
(c) From (a) and (b) determne the small–signal circuit capacitances C and C and cutoff frequency fT.
(d) If the current (IC ) is reduced by a factor of 2.5, what is the circuit capacitance C and what is fT ?
Figure P12.8
Exercise for relating SPICE BJT parameters to the small–signal parameters.
144
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