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Bipolar Transistor
CHAPTER OBJECTIVES This chapter introduces the bipolar junction transistor (BJT) operation and then presents the theory of the bipolar transistor I-V characteristics, current gain, and output conductance. High-level injection and heavy doping induced band narrowing are introduced. SiGe transistor, transit time, and cutoff frequency are explained. Several bipolar transistor models are introduced, i.e., Ebers–Moll model, small-signal model, and charge control model. Each model has its own areas of applications.
he bipolar junction transistor or BJT was invented in 1948 at Bell Telephone Laboratories, New Jersey, USA. It was the first mass produced transistor, ahead of the MOS field-effect transistor (MOSFET) by a decade. After the introduction of metal-oxide-semiconductor (MOS) ICs around 1968, the high- density and low-power advantages of the MOS technology steadily eroded the BJT’s early dominance. BJTs are still preferred in some high-frequency and analog applications because of their high speed, low noise, and high output power advantages such as in some cell phone amplifier circuits. When they are used, a small number of BJTs are integrated into a high-density complementary MOS (CMOS) chip. Integration of BJT and CMOS is known as the BiCMOS technology. The term bipolar refers to the fact that both electrons and holes are involved in the operation of a BJT. In fact, minority carrier diffusion plays the leading role just as in the PN junction diode. The word junction refers to the fact that PN junc- tions are critical to the operation of the BJT. BJTs are also simply known as bipolar transistors.
8.1 INTRODUCTION TO THE BJT
A BJT is made of a heavily doped emitter (see Fig. 8–1a), a P-type base , and an N-type collector. This device is an NPN BJT. (A PNP BJT would have a P+^ emitter, N-type base, and P-type collector.) NPN transistors exhibit higher transconductance and
● (^) ●
T
292 Chapter 8 ● Bipolar Transistor
speed than PNP transistors because the electron mobility is larger than the hole mobility. BJTs are almost exclusively of the NPN type since high performance is BJTs’ competitive edge over MOSFETs. Figure 8–1b shows that when the base–emitter junction is forward biased, electrons are injected into the more lightly doped base. They diffuse across the base to the reverse-biased base–collector junction (edge of the depletion layer) and get swept into the collector. This produces a collector current , I C_. I_ C is independent of V CB as long as V CB is a reverse bias (or a small forward bias, as explained in Section 8.6). Rather, I C is determined by the rate of electron injection from the emitter into the base, i.e., determined by V BE. You may recall from the PN diode theory that the rate of injection is proportional to e qV^. These facts are obvious in Fig. 8–1c. Figure 8–2a shows that the emitter is often connected to ground. (The emitter and collector are the equivalents of source and drain of a MOSFET. The base is the equivalent of the gate.) Therefore, the I C curve is usually plotted against V CE as shown in Fig. 8–2b. For V CE higher than about 0.3 V, Fig. 8–2b is identical to Fig. 8–1c but with a shift to the right because V CE = V CB + V BE. Below V CE ≈ 0.3 V,
FIGURE 8–1 (a) Schematic NPN BJT and normal voltage polarities; (b) electron injection from emitter into base produces and determines I C ; and (c) I C is basically determined by V BE and is insensitive to V CB.
N^ N Emitter Collector
0
B
E C
V BE V CB
E c
I c
E Fn V BE
V BE
V CB
E Fp
E v V CB E Fn
Base
P
(a)
(b)
(c)
BE ⁄ kT
294 Chapter 8 ● Bipolar Transistor
τB and D B are the recombination lifetime and the minority carrier (electron) diffusion constant in the base, respectively. The boundary conditions are [Eq. (4.6.3)]
(8.2.3)
(8.2.4)
where n B0 = n i^2 / N B , and N B is the base doping concentration. V BE is normally a forward bias (positive value) and V BC is a reverse bias (negative value). The solution of Eq. (8.2.1) is
Equation (8.2.5) is plotted in Fig. 8–4. Modern BJTs have base widths of about 0.1 μm. This is much smaller than the typical diffusion length of tens of microns (see Example 4–4 in Section 4.8). In the case of W B << L B , Eq. (8.2.5) reduces to a straight line as shown in Fig. 8–4.
n iB is the intrinsic carrier concentration of the base material. The subscript, B, is added to n i because the base may be made of a different semiconductor (such as SiGe alloy, which has a smaller band gap and therefore a larger n i than the emitter and collector material).
FIGURE 8–4 When W B << L B, the excess minority carrier concentration in the base is approximately a linear function of x.
n ' 0( ) n B0 e qV BE ⁄ kT = ( – 1 )
n ' ( W B) n B0 e
qV BC ⁄ kT = ( – 1 ) ≈ – n (^) B0≈ 0
n ' ( x ) n B0 e
qV BE ⁄ kT ( – 1 )
W B – x L B
sinh^
sinh( W B ⁄ L B)
n ' ( x ) = n ' 0( ) ( 1 – x ⁄ W B)
n iB 2
N B
-------- e
qV BE ⁄ kT ( – 1 ) 1 x W B
0.0 0.2 0.4 0.6 0.8 1.
n' n' (0) n' = 1.0 W B^ ^ 0.01 L B W B 0.5 L B W B 0.9 L B
x/W B
n^2 i N B (e qV BE/ kT 1)
8.2 ● Collector Current 295
As explained in the PN diode analysis, the minority-carrier current is dominated by the diffusion current. The sign of I C is defined in Fig. 8–2a and is positive.
A E is the area of the BJT, specifically the emitter area. Notice the similarity between Eq. (8.2.7) and the PN diode IV relation [Eq. (4.9.4)]. Both are proportional to and to Dn i^2 / N. In fact, the only difference is that d n' /d x has produced the 1/ W B term in Eq. (8.2.7) due to the linear n' profile. Equation (8.2.7) can be condensed to
(8.2.8)
where I S is the saturation current. Equation (8.2.7) can be rewritten as
In the special case of Eq. (8.2.7)
where p is the majority carrier concentration in the base. It can be shown that Eq. (8.2.9) is valid even for nonuniform base and high-level injection condition if G b is generalized to [1]
G B has the unusual dimension of s/cm^4 and is known as the base Gummel number. In the special case of n iB = n i , D B is a constant, and p ( x ) = N B( x ) (low-level injection),
(8.2.12)
Equation (8.2.12) illustrates that the base Gummel number is basically proportional to the base dopant density per area. The higher the base dopant density is, the lower the I C will be for a given V BE as given in Eq. (8.2.9). The concept of a Gummel number simplifies the I C model because it [Eq. (8.2.11)] contains all the subtleties of transistor design that affect I C : changing base material through n iB ( x ), nonconstant D B , nonuniform base dopant concen- tration through p ( x ) = N B ( x ), and even the high-level injection condition (see Sec. 8.2.1), where p > N B. Although many factors affect G B , G B can be easily determined from the Gummel plot shown in Fig. 8–5. The (inverse) slope of the
I C A E qD B d n d x
------- A E qD B^ n ' 0
W B
A E q
D B
W B
n iB 2
N B
-------- e qV BE ⁄ kT = ( – 1 )
( e qV kT ⁄^ – 1 )
I C I S e
qV BE ⁄ kT = ( – 1 )
I C A E
qn i 2
G B
--------- e qV BE ⁄ kT = ( – 1 )
G B
n i 2
n iB
N B
D B
-------- W B
n i 2
n iB
p D B
= = -------- W B
G B
n i 2
n iB^2
-------- p D B
-------- d x 0
W B ≡ ∫
G B^1
D B
-------- N B ( x ) d x 0
W B = (^) ∫
D B
= -------- ×base dopant atoms per unit area
8.3 ● Base Current 297
8.3 BASE CURRENT
Whenever the base–emitter junction is forward biased, some holes are injected from the P-type base into the N+^ emitter. These holes are provided by the base current, I B.^1 I B is an undesirable but inevitable side effect of producing I C by forward biasing the BE junction. The analysis of I B, the base to emitter injection current, is a perfect parallel of the I C analysis. Figure 8–6b illustrates the mirror equivalence. At an ideal ohmic contact such as the contact of the emitter, the equilibrium condition holds and p' = 0 similar to Eq. (8.2.4). Analogous to Eq. (8.2.9), the base current can be expressed as
G E is the emitter Gummel number. As an exercise, please verify that in the special case of a uniform emitter, where n iE, N E (emitter doping concentration) and D E are not functions of x ,
2
(^1) In older transistors with VERY long bases, I B also supplies holes at a significant rate for recombination in the base. Recombination is negligible in the narrow base of a typical modern BJT.
FIGURE 8–6 (a) Schematic of electron and hole flow paths in BJT; (b) hole injection into emitter closely parallels electron injection into base. 2
(^2) A good metal–semiconductor ohmic contact (at the end of the emitter) is an excellent source and sink of carriers. Therefore, the excess carrier concentration is assumed to be zero.
● (^) ●
I B A E
qn i^2 G E
--------- e
qV BE ⁄ kT = ( – 1 )
G E
n i 2
n iE^2
-------- n D E
-------- d x 0
W E = ∫
I B A E q
D E
W E
n iE 2
N E
-------- e
qV BE ⁄ kT = ( – 1 )
contact Emitter Base Collector
I E
Electron flow Hole flow
p E' n B'
(a)
(b)
contact
I C
I B
W E W B
298 Chapter 8 ● Bipolar Transistor
8.4 CURRENT GAIN
Perhaps the most important DC parameter of a BJT is its common-emitter current gain , βF.
Another current ratio, the common-base current gain, is defined by (8.4.2)
αF is typically very close to unity, such as 0.99, because βF is large. From Eq. (8.4.3), it can be shown that
(8.4.4)
I B is a load on the input signal source, an undesirable side effect of forward biasing the BE junction. I B should be minimized (i.e., βF should be maximized). Dividing Eq. (8.2.9) by Eq (8.3.1),
A typical good βF is 100. D and W in Eq. (8.4.5) cannot be changed very much. The most obvious way to achieve a high βF , according to Eq. (8.4.5), is to use a large N E and a small N B. A small N B , however, would introduce too large a base resistance, which degrades the BJT’s ability to operate at high current and high frequencies. Typically, N B is around 10^18 cm–3. An emitter is said to be efficient if the emitter current is mostly the useful electron current injected into the base with little useless hole current (the base current). The emitter efficiency is defined as
EXAMPLE 8–1 Current Gain A BJT has I C = 1 mA and I B = 10 μA. What are I E, βF, and αF?
● (^) ●
βF
I C
I B
I C= αF I E
αF
I C
I E
I C
I B + I C
I C ⁄ I B
1 + I C ⁄ I B
βF 1 + βF
βF
αF 1 – αF
βF
G E
G B
D B W E N E n iB^2
D E W B N B n iE
γE
I E – I B
I E
I C
I C + I B
1 + G B ⁄ G E
I E = I C + I B= 1mA + 10 μA=1.01mA
βF
I C
I B
----- 1mA 10 μA
αF
I C
I E
----- 1mA 1.01mA
300 Chapter 8 ● Bipolar Transistor
8.4.3 Poly-Silicon Emitter Whether the base material is SiGe or plain Si, a high-performance BJT would have a relatively thick (>100 nm) layer of As doped N +^ poly-Si film in the emitter (as shown in Fig. 8–7). Arsenic is thermally driven into the “base” by ~20 nm and converts that single-crystalline layer into a part of the N +^ emitter. This way, βF is larger due to the large W E , mostly made of the N+^ poly-Si. This is the poly-Silicon emitter technology. The simpler alternative, a deeper implanted or diffused N + emitter without the poly-Si film, is known to produce a higher density of crystal defects in the thin base (causing excessive emitters to collector leakage current or even shorts in a small number of the BJTs).
8.4.4 Gummel Plot and βF Fall-Off at High and Low I C High-speed circuits operate at high I C, and low-power circuits may operate at low I C. Current gain, β, drops at both high I C and at low I C. Let us examine the causes.
EXAMPLE 8–2 Emitter Band-Gap Narrowing and SiGe Base Assuming D B = 3 D E, W E = 3 W B, N B = 10 18 cm–3, and = What is βF for (a) N E = 10 19 cm–3^ , (b) N E = 10^20 cm–3, and (c) N E = 10^20 cm–3^ and the base is substituted with SiGe with a band narrowing of ∆ E gB = 60 meV? SOLUTION: a. At N E = 10^19 cm–3^ , ∆ E gE ≈ 50 meV
From Eq. (8.4.5),
b. At N E = 10^20 cm–3^ , ∆ E gE ≈ 95 meV
Increasing N E from 10 19 cm–3^ to 10^20 cm–3^ does not increase βF by anywhere near 10 × because of band-gap narrowing. βF can be raised of course by reducing N B at the expense of a higher base resistance, which is detrimental to device speed (see Eq. 8.9.6).
c.
n iB 2 n i 2 .
n iE^2 n i^2 e ∆ E gE ⁄ kT = = n i^2 e 50 26 meV⁄^ = n i^2 e 1.92^ =6.8 n i^2
βF
D B W E
D E W B
N E n^ i 2
N B n iE
= ×---------------- 2
9 ⋅ 1019 ⋅ n i^2
10 18 6.8 n i 2 ⋅
n iE^2 n i^2 e
∆ E gE ⁄ kT = = n i^2 e 95 26 meV⁄^ = n i^2 e 3.65^ = 38 n i^2
βF
D B W E
D E W B
N E n i^2
N B n iE
× ---------------- 2
9 ⋅ 1020 ⋅ n i^2
10 18 38 n i 2 ⋅
n iB 2 n i 2 e
∆ E gB ⁄ kT n i 2 e 60 26 meV⁄ 10 n i 2 = = =
βF
D B W E
D E W B
N E n iB 2
N B n iE^2
× ----------------
20 10 n iB 2 ⋅ 10 18 ⋅ 39 n i^2
8.4 ● Current Gain 301
We have seen in Fig. 8–5 (Gummel plot) that I C flattens at high V BE due to the high-level injection effect in the base. That I C curve is replotted in Fig. 8–8. I B, arising from hole injection into the emitter, does not flatten due to this effect (Fig. 8–8) because the emitter is very heavily doped, and it is practically impossible to inject a higher density of holes than N E. Over a wide mid-range of I C in Fig. 8–8, I C and I B are parallel, indicating that the ratio of I C/ I B , i.e., βF , is a constant. This fact is obvious in Fig. 8–9. Above 1 mA, the slope of I c in Fig. 8–8 drops due to high-level injection. Consequently, the I c/ I B ratio or βF decreases rapidly as shown in Fig. 8–9. This fall-off of current gain unfortunately degrades the performance of BJTs at high current where the BJT’s speed is the highest (see Section 8.9). I B in Fig. 8–8 is the base–emitter junction forward-bias current. As shown in Fig. 4–22, forward-bias current slope decreases at low V BE or very low current due to the space-charge region (SCR) current (see Section 4.9.1). A similar slope change is sketched in Fig. 8–8. As a result, the I c/ I B ratio or βF decreases at very low I C. The weak V BC dependence of βF in Fig. 8–9 is explained in the next section.
FIGURE 8–7 Schematic illustration of a poly-Si emitter, a common feature of high- performance BJTs.
FIGURE 8–8 Gummel plot of I C and I B indicates that βF ( = I C / I B) decreases at high and low I C.
N-collector
P-base
SiO (^2)
Emitter
N-poly-Si
0.2 0.4 0.6 0.8 1.0 1.
10 ^12
10 ^10
10 ^8
10 ^6
10 ^4
10 ^2
I C
(A)
V BE
I B
I C
Excess base current
bF
injection in base
High level
8.5 ● Base-Width Modulation by Collector Voltage 303
FIGURE 8–10 BJT output conductance: (a) measured BJT characteristics. I B = 4, 8, 12, 16, and 20 μA. (From [3]); (b) schematic drawing illustrates the definition of Early voltage, V A.
FIGURE 8–11 As V C increases, the BC depletion layer width increases and W B decreases causing d n’/ d x and I C to increase. In reality, the depletion layer in the collector is usually much wider than that in the base.
I C
0
(a)
(b)
I B
I B
I B
V CE V A
0 1 2 3 4 5 V ce (V)
I c
(mA)
I B
N^ P N Emitter Base (^) Collector
E C
x
V BE B
W B W B2 V CE1 V CE2 V CE W B
V CE
n'
304 Chapter 8 ● Bipolar Transistor
[see Eq. (4.2.5)]. For the same reason, (c) would tend to move the depletion region into the collector and thus reduce the depletion region thickness on the base side, too. Both (a) and (b) would depress βF [see Eq. (8.4.5)]. (c) is the most acceptable course of action. It also reduces the base–collector junction capacitance, which is a good thing. Therefore, the collector doping is typically ten times lighter than base doping. In Fig. 8–10, the larger slopes at V CE > 3V are caused by impact ionization (Section 4.5.3). The rise of I c due to base-width modulation is known as the Early effect , after its discoverer.
8.6 EBERS–MOLL MODEL
So far, we have avoided examining the part of the I–V curves in Fig. 8–12 that is close to V CE = 0. This portion of the I–V curves is known as the saturation region because the base is saturated with minority carriers injected from both the emitter and the collector. (Unfortunately the MOSFET saturation region is named in exactly the opposite manner.) The rest of the BJT operation region is known as the active region or the linear region because that is where BJT operates in active cir- cuits such as the linear amplifiers.
● Early on Early Voltage ● Anecdote contributed by Dr. James Early, November 10, 1990
“In January, 1952, on my way to a Murray Hill Bell Labs internal meeting, I started to think about how to model the collector current as a function of the collector voltage. Bored during the meeting, I put down the expression for collector current I C = βF I B. Differentiating with respect to V C while I B was held constant gave:
How can βF change with V C? Of course! The collector depletion layer thickens as collector voltage is raised. The base gets thinner and current gain rises. Obvious! And necessarily true. Why wasn’t this found sooner? Of those who had thought about it at all before, none was educated in engineering analysis of electron devices, used to setting up new models, and bored at a meeting.”
∂ I C
∂ V C
---------- I B
∂ βF ∂ V C
FIGURE 8–12 In the saturation region, I C drops because the collector–base junction is significantly forward biased.
● (^) ●
Saturation region
Active region
V CE
I C
0
I B
306 Chapter 8 ● Bipolar Transistor
What causes I C to decrease at low V CE_? In this region, both the BE and BC junctions are forward biased. (For example: V_ BE = 0.8 V, V BC = 0.6 V, thus V CE = 0.2 V.) A forward- biased V BC causes the n' at x = W B to rise in Fig. 8–4. This depresses d n'/ d x and therefore I C.
8.7 TRANSIT TIME AND CHARGE STORAGE
Static IV characteristics are only one part of the BJT story. Another part is its dynamic behavior or its speed. When the BE junction is forward biased, excess holes are stored in the emitter, the base, and even the depletion layers. We call the sum of all the excess hole charges everywhere Q F. Q F is the stored excess carrier charge. If Q F = 1 pC (pico coulomb), there is +1 pC of excess hole charge and −1 pC of excess electron charge
stored in the BJT.^5 The ratio of Q F to I C is called the forward transit time, τF.
Equation (8.7.1) states the simple but important fact that I C and Q F are related by a constant ratio, τF. Some people find it more intuitive to think of τF as the storage time. In general, Q F and therefore τF are very difficult to predict accurately for a complex device structure. However, τF can be measured experimentally (see Sec. 8.9) and once τF is determined for a given BJT, Eq. (8.7.1) becomes a powerful concep- tual and mathematical tool giving Q F as a function of I C , and vice versa. τF sets a high-frequency limit of BJT operation.
8.7.1 Base Charge Storage and Base Transit Time To get a sense of how device design affects the transit time, let us analyze the excess hole charge in the base, Q FB, from which we will obtain the base transit time, τFB. Q FB is qA E times the area under the line in Fig. 8–15.
FIGURE 8–14 Ebers–Moll model (line) agrees with the measured data (symbols) in both the saturation and linear regions. I B = 4.3, 11, 17, 28, and 43 μA. High-speed SiGe-base BJT. A E = 0.25 × 5.75 μm 2. (From [3].)
(^5) This results from Eq. (2.6.2), n' = p'.
V CE (V)
I C
I C
(A)
0.5 1.0 1.
● (^) ●
τF
Q F
I C
8.7 ● Transit Time and Charge Storage 307
Dividing Q FB by I C and using Eq. (8.2.7),
To reduce τFB (i.e., to make a faster BJT), it is important to reduce W B.
8.7.2 Drift Transistor−Built-In Base Field
The base transit time can be further reduced by building into the base a drift field that aids the flow of electrons from the emitter to the collector. There are two ways of accomplishing this. The classical method is to use graded base doping, i.e., a large N B near the EB junction, which gradually decreases toward the CB junction as shown in Fig. 8–16a. Such a doping gradient is automatically achieved if the base is produced by dopant diffusion. The changing N B creates a d E (^) v / d x and a d E c / d x. This means that there is a drift field [Eq. (2.4.2)]. Any electrons injected into the base would drift toward the collector with a base transit time shorter than the diffusion transit time ,
Figure 8–16b shows a more effective technique. In a SiGe BJT , P-type epi- taxial Si (^) 1- ηGe η is grown over the Si collector with a constant N B and η linearly varying from about 0.2 at the collector end to 0 at the emitter end [4]. A large
FIGURE 8–15 Excess hole and electron concentrations in the base. They are equal due to charge neutrality [Eq. (2.6.2)].
EXAMPLE 8–3 Base Transit Time What is τFB if W B = 70 nm and D B = 10 cm^2 /s? SOLUTION:
2.5 ps is a very short time. Since light speed is 3 × 108 m/s, light travels less than 1 mm in 2.5 ps.
0
x W B
p n n (0)
Area equals stored charge per unit of A E
n^2 iB N B(e
q V BE/ kT (^) 1)
Q FB = qA E n ' 0( ) W B ⁄ 2
Q FB
I C
-----------≡ τFB
W B
2
2 D B
τFB
W B^2
2 D B
2
2 × 10 cm 2 ⁄s
W B
2 ⁄ 2 D B.
8.7 ● Transit Time and Charge Storage 309
When I C is small, ρ = qN C as expected from the PN junction analysis (see Section 4.3), and the electric field in the depletion layer is shown in Fig. 8–18a. The shaded area is the potential across the junction, V CB + φbi. The N +^ collector is always present to reduce the series resistance (see Fig. 8–22). No depletion layer is
FIGURE 8–17 Transit time vs. I C / A E. From top to bottom: V CE = 0.5, 0.8, 1.5, and 3 V. The rise at high I C is due to base widening (Kirk effect). (From [3].)
FIGURE 8–18 Electric field ( x ), location of the depletion layer, and base width at (a) low I C such as 0.1 mA/μm^2 in Fig. 8–17; (b) larger I C ; (c) even larger I C (such as 1 mA/μm^2 ) and base widening is evident; and (d) very large I C with severe base widening.
J C (mA/m 2 )
t f
(ps)
0 0
5
10
15
20
25
30
0.5 1 1.5 2 2.5 3
(a) (b)
x
Base
Base width
Depletion layer
N Collector
N Collector
(c)
x
Base
Base width
Depletion layer
N Collector
N Collector
(d)
x
Base
Base width
Depletion layer
N Collector
N Collector
x
Base
Base width
Depletion layer
N Collector
N Collector
310 Chapter 8 ● Bipolar Transistor
shown in the base for simplicity because the base is much more heavily doped than the collector. As I C increases, ρ decreases [Eq. (8.7.5)] and d / d x decreases as shown in Fig. 8–18b. The electric field drops to zero in the very heavily doped N + collector as expected. Note that the shaded area under the ( x ) line is basically equal to the shaded area in the Fig. 8–18a because V CB is kept constant. In Fig. 8–18c, I C is even larger such that ρ in Eq. (8.7.5) and therefore d / d x has changed sign. The size of the shaded areas again remains unchanged. In this case, the high- field region has moved to the right-hand side of the N collector. As a result, the base is effectively widened. In Fig. 8–18d, I C is yet larger and the base become yet wider. Because of the base widening , τF increases as a consequence [see Eq. (8.7.3)]. This is called the Kirk effect. Base widening can be reduced by increasing N C and V CE. The Kirk effect limits the peak BJT operating speed (see Fig. 8–21).
8.8 SMALL-SIGNAL MODEL
Figure 8–19 is an equivalent circuit for the behavior of a BJT in response to a small input signal, e.g., a 10 mV sinusoidal signal, superimposed on the DC bias. BJTs are often operated in this manner in analog circuits. If V BE is not close to zero, the “1” in Eq. (8.2.8) is negligible; in that case
(8.8.1)
When a signal v BE is applied to the BE junction, a collector current g (^) m v BE is produced. g (^) m , the transconductance , is
At room temperature, for example, g (^) m = I C /26 mV. The transconductance is determined by the collector bias current, I C. The input node, the base, appears to the input drive circuit as a parallel RC circuit as shown in Fig. 8–19.
(8.8.4)
Q F in Eq. (8.7.1) is the excess carrier charge stored in the BJT. If Q F = 1 pC, there is +1 pC of excess holes and −1 pC of excess electrons in the BJT. All the excess hole charge, Q F, is supplied by the base current, I B. Therefore, the base presents this capacitance to the input drive circuit:
(8.8.6)
● (^) ●
I C I S e qV BE ⁄ kT =
g (^) m
d I C d V BE
≡ --------------- d d V BE
--------------- I S e
qV BE ⁄ kT = ( )
q kT
= ------- I S e
qV BE ⁄ kT I C^ kT q
g (^) m I C kT q
r π
d I B d V BE
βF
d I C d V BE
g (^) m βF
r π =βF ⁄ g (^) m
C π
d Q F d V BE
--------------- d d V BE
= = --------------- τF I C= τF g (^) m
