Nature of Transistor Quantities and Parameters

4.1. D-C QUANTITIES AND PARAMETERS

4

I CBO is the collector current when the collector is biased in the reverse (high resistance) direction with respect to the base, and the emitter is open-circuited. This current is made up of two components, one temperature-dependent and one voltage-dependent. The temperature-dependent component (Is in Fig. 4.1) is called the saturation current and results from thermal generation of electron-hole pairs, while the voltage-dependent component (h) results mainly from surface leakage through the collector-base junction.

I CBO is of primary concern in transistor biasing. Because of its extreme tempera-ture dependence it can become an appreciable part of the base current in low-level applications, and it can cause self-heating and thermal runaway in large-signal applications.

I CBO is generally measured at two voltages, at room temperature and at some elevated temperature. One measurement is made at a voltage low enough so that avalanche multiplication effects are negligible; at this voltage the elevated tempera-ture measurement is also usually made. The temperatempera-ture is set high enough to ensure that the saturation current is large compared to the leakage current. This allows the designer to use the known temperature dependence of the saturation

--Figure 4.1

29

current to determine the behavior of leBo at high temperature. Another measure-ment is made at or near the maximum voltage rating of the collector-base diode, usually at room temperature only.

Variation of leBO with Junction Temperature. As stated, leBo is made up of a saturation component (temperature-dependent) and a leakage component (voltage-dependent), expressed as leBo = Is

+

h This relationship is shown graphically in Fig. 4.2. At low temperatures, leBO is mainly the leakage component; at high temperatures, the saturation component becomes dominant. The temperature dependence of Is derived from Fermi-Dirac statistics has the form!

(1) where A and N are dependent on the physical properties of the semiconductor material and T is absolute temperature in degrees Kelvin.

dIs

=

AT3e^-NIT

l(3 ⁺ N) ⁼

^Is

(3 ⁺ N)

⁽²⁾

dT T T T T

Rearranging Eq. (2),

dIs

= (3 ⁺ N)

^dT

Is T T (3)

It is common to express the Is temperature dependence in terms of the number of Kelvin degrees (or centigrade degrees) temperature rise that it takes to double Is.

If we set dIs/Is

=

1 in Eq. (3), we arrive at llT

=

^T2

3T+N (4)

This doubling rate is also temperature-dependent; for a transistor that has a large leakage component, however, it has been observed that some constant doubling rate can be used as a conservative approximation over the useful temperature range.

For silicon transistors a llT of 10 Co has commonly been used, and for germanium transistors a llT of 14 Co. However, use of this approximation for transistors that have low leakages (such as planar transistors) can cause the designer trouble, especially at low temperatures. From Eq. (4) we find that llT = 10 CO at 130°C

Temperature, °C Figure 4.2

Nature of Transistor Quantities and Parameters 31

for silicon transistors (N

=

l4,OOooK for silicon). I::J.T

=

10 Co is a conservative approximation above l30°C for silicon transistors with low leakage currents, but at 25°C (room temperature), we find that I::J.T

=

5.97 CO. It is apparent that applying the I::J.T

=

10 Co rule of thumb to a value of I CBO measured at room tem-perature will yield far too optimistic values of I CBO at higher temperatures.

Variation of I CBO with Applied Voltage. At low voltages, the leakage com-ponent of I CBO varies almost linearly with applied voltage. At higher voltages, the very strong electric field in the narrow collector-base depletion layer causes a large increase in the kinetic energy of current carriers (holes and electrons) passing through this region; when the carriers collide with atoms of the crystal structure, enough energy is released to generate other electron-hole pairs, which in turn are accelerated by the strong electric field and may collide with other atoms, generating still more electron-hole pairs. This process is called avalanche multiplication, and it results in a rapid increase in collector current with collector voltage.

I cEo-The collector current when the collector is biased in the reverse (high resistance) direction with respect to the emitter and the base is d-c open-circuited.

I CE~ The collector current when the collector is biased in the reverse (high resistance) direction with respect to the emitter, and the base is shorted to the emitter.

I CE~ The collector current when the collector is reverse-biased with respect to the emitter and the base is returned to the emitter through an external resistance.

The relationship among I CEO , ICES, I CER, and I CBO can be found with the aid of the equivalent circuit of Fig. 4.3. The resistor R from base to emitter represents the general termination at this point. For any value of R, Ic

=

^{ICER (for R}

=

^00,

Ic = ICEO). The current generator Is is the saturation component of I CBO, the leakage component is accounted for by the resistance ^rCL, and ^rEL is the base spreading resistance. For most practical transistors, the current generator ^IE is obtained by recognizing that²

(5)

Figure 4.3

where aN, al = normal and inverse common-base current gains of the transistor

(aN~ -h],B)

lEBO = reverse saturation current of the inverted transistor CPE

=

emitter diode potential

q = electronic charge ob-tained from a data sheet, but it is useful for qualitative analysis. If we examine Eq. (10) with R set equal to zero and let rEL approach infinity and A approach zero simultaneously, we find that

leER

=

leBO

=

Is

+

VOE (1 - hpB)

=

^Is

⁺

^{h (11)}

rOL

Here our expression reduces to the sum of the thermal saturation current, Is, and a term representing the collector-base diode leakage current,

h.

Letting R ap-proach infinity in Eq. (10),

Nature of Transistor Quantities and Parameters 33

Equation (13) can be reduced to a more familiar expression by making the approximation

and

which is a good approximation for a practical transistor. With these assumptions, Eq. (l3) reduces to

1 VCE h*

ICER

=

Is I

+ -

(1 - FB)

- OiNOiI rCL

(14) In each case we have the sum of a saturation current and a leakage current, which is what we would have intuitively expected before any analysis.

Equation (10) is plotted in Fig. 4.4. This type of plot is sometimes included in data sheets. It should be pointed out that ICES assumes that the collector-to-emitter path is not punched through, i.e., that the collector depletion layer does not extend into the emitter.

ICEr-The collector current when the collector is reverse-biased with respect to the base and the base is forward- or reverse-biased with respect to the emitter. This quantity will be approximately equal to I CBO unless the base-emitter junction is reverse-biased by a voltage which exceeds the breakdown rating of this junction.

BVcBo-The breakdown voltage between the collector and base electrodes with the emitter open-circuited.

BVcEo-The breakdown voltage between the collector and emitter with the base open-circuited.

BVCEs-The breakdown voltage between the collector and emitter with the base short-circuited to the emitter.

BVOEg--The breakdown voltage between the collector and emitter with the base returned to the emitter through an external resistance.

BVOEr-The breakdown voltage between the collector and emitter with a volt-age applied between base and emitter.

BVEBO- The breakdown voltage between the emitter and base electrodes with the collector open-circuited.

log leER

leEO

logR Figure 4.4

The breakdown voltage BVOBO in most transistors is due to the avalanche multi-plication of lOBo discussed previously. BVOEO is less than BVOBO, and is quite often less than half of BVOBO. The collector current can be written as

(15) M is the multiplication factor that accounts for the rapid rise in leBo near BVOBO.

An expression for M has been given,³

M= I

1 - (VoE/BVOBO)n (16)

where n is an empirical constant dependent on physical properties of a semicon-ductor. From Eq. (15) we see that the value of M that will make lB = 0 is M = ljhFB. Substituting this result into Eq. (16),

BVOEO

=

BVOB0Ct

+\F~)lIn

⁽¹⁷⁾

Equation (17) predicts the voltage at which the total alpha of the transistor equals one. At this voltage, the common-emitter current gain is infinite and the collector current increases unchecked.

It is not possible in practice to measure the true values of these breakdown voltages, since the true value implies that they are measured at infinite collector (or emitter) currents. The values of BVOBO, BVOEO, BVOES, etc., given in a data sheet are measured by applying a constant current to the proper electrodes of the transistor and measuring the voltage between the electrodes. The magnitude of the constant current depends usually on whether the transistor is designed for small-signal, medium-power, or power applications. When the product of the measured current and the breakdown voltage is sufficient to cause heating of the junction, it is customary to make the breakdown voltage measurement with a low duty-cycle pulse (see Pulse Testing, Sec. 5.2). If the junction temperature increases, the measured breakdown voltage decreases because the saturation current increases.

This is illustrated in Fig. 4.5.

BVCE01 BVCE02

Figure 4.5

Constant measuring

current

Ic

Nature of Transistor Quantities and Parameters 35

Vrr-Punch-through voltage-the voltage between the collector and emitter electrodes at which the collector depletion layer extends into the emitter.

BVcEs-Breakdown voltage, usually limited by the avalanche breakdown effect previously described rather than by VPT.

VEB~ The d-c open-circuit voltage (floating potential) between the emitter and base electrodes with the collector reverse-biased. This measure-ment can determine VPT, for if punch-through occurs,

(18) VBW-The voltage between the base and emitter electrodes with the

base-emitter junction forward-biased and the collector-base junction reverse-biased. In Fig. 4.6, the voltage represented by the ideal characteristic has a negative temperature coefficient; i.e., it decreases with increasing temperature.⁴ However, that part of the actual input characteristic beyond the knee of the curve is due mainly to bulk resistance and has a positive temperature coefficient. It is apparent that variation of VBE with temperature depends on the bias current level, the coefficient being negative (approximately -2.5 mv/°C) at low emitter currents, becoming less negative as IE increases, and possibly going positive at high values of IE.

VBE(sat)-The voltage between the base and emitter electrodes with both the emitter-base and collector-base junctions forward-biased. This quantity is generally measured with the base current greater than the value needed to saturate the lowest hFE transistor of a given type. The previous discussion of VBE vs. temperature applies here also.

VCE(sat)-The voltage between the collector and emitter electrodes with both emitter-base and collector-base diodes forward-biased. VCE(sat) is of particular importance in switching applications. It is the minimum switch contact potential, and is usually measured under the same conditions as VBE(sat). It has a positive temperature coefficient since it is partly due to an ohmic drop across the collector bulk resistance.

F Ideal characteristic

Figure 4.6

Transistor D-C Parameters

hF~ The static value of the common-emitter short-circuit current gain.

The short-circuit gain is the most important of the transistor param-eters. In circuit analyses where reasonable approximations are made, all parameters can be neglected at one time or another save hFE (or hFB). Since the transistor is a current control device, we would expect the current gain parameter to be important.

hFE Variation with Emitter Current. Figure 4.7 shows the variation of hFE with collector current at several junction temperatures for a silicon double-diffused mesa transistor. This graph appears on the data sheet of the TI 2N697. As Ie increases from very small values, hFE increases to a maximum and then decreases at high values of Ie. This can be related to hFB (the static value of the common-base current transfer ratio) variation by recognizing that

-hFB- 1 hFE h

+

^FE

and IE

=.!.£

hFB

hFB can be expressed as the product of three components, ⁵ IhFBI

=

^yf3M

(19)

(20)

(21) where y = emitter efficiency, the fraction of the emitter current that is carried by minority carriers in the base side of the base-emitter transition region f3

=

transport factor, the fraction of injected minority carriers in the base

that arrive at the collector junction

M

=

collector multiplication factor, the number of current carriers collected per minority carrier presented at the base side of the collector junction

~

140~---~~---,---~

Pulse measurement

2% duty cycle _ _ _ TA = 150°C - + - - - 1

(300 j.lsec pulse width) VcE=10v

~ 20r--~~~~---l---1---l

Ic , collector current, ma Fig. 4.7. hFE vs. Ie characteristics of 2N697.

Nature of Transistor Quantities and Parameters 37

Equation (21) is illustrated by Fig. 4.8, which is the classic one-dimensional current-flow model of an NPN transistor. In the base of an NPN transistor, the majority carriers are holes and the minority carriers are electrons. Three processes have been described⁶ to account for the change in hFB with emitter current. Each process dominates at a particular current level. At very low emitter currents, the recom-bination of electrons and holes in the emitter depletion layer is high compared to the emitter current and lowers emitter efficiency y. As the emitter current increases, the recombination current in the depletion layer remains constant, causing y to rise.

For still higher emitter currents, an increasing electric field develops in the base region and accelerates the minority electrons toward the collector, increasing {3. As the emitter current is increased further, the high minority-carrier density causes an increase in base conductivity, lowering y. This is known as conductivity modulation.

Thus, IhFB

I

will pass through a maximum, then fall off at still higher currents.

The recombination centers in the emitter depletion layer are caused by crystalline defects, both in the bulk of the crystal and at the surface. The recombination current is composed of a volume recombination component and a surface recom-bination component.7 In planar transistors, the surface recomrecom-bination is negligible, owing to the oxide coating over the junction. Planar transistors maintain reason-ably high current gains at emitter currents on the order of 1 /La.

The aiding electric field which develops in the base region can be explained with the aid of Fig. 4.9. The charge concentration gradients shown in Fig. 4.9 are valid for step-junction transistors with uniform impurity density in the base.

In diffused transistors, a more complex situation exists; the majority impurity concentration (acceptors) in the base sets up an electric fieldS that retards the minority carriers (electrons) for a short distance from the emitter junction, then accelerates them the remainder of the way to the collector. The field set up by the current flow acts to increase the accelerating field similar to the action in step-junction transistors. Figure 4.9 is used as an illustration because of its simplicity.

As electrons are injected into the base, they move toward the collector (by diffusion) and establish a concentration gradient (ne). In order to maintain space-charge neutrality⁹ in the base, an equal hole distribution is established (np). It must be remembered that, even though there was an original hole concentration (no) due

~)IE (M-l)-yfHE

Fig. 4.8. One-dimensional transistor model.

to the impurity doping, the crystal was electrically neutral, and electrons injected into the emitter cause an unbalance that must be neutralized by an equal number of holes. (This must not confuse the reader into thinking that the base and emitter currents are equal, for they differ considerably.)

Once the electron and hole distributions are set up in the base, electrons are jerked into the collector region (by the high reverse bias at the collector junction) at a much higher rate than holes can be injected into the emitter by the forward bias on the base-emitter diode. The electrons that are jerked into the collector are balanced by electrons injected into the base from the emitter, while holes that are injected into the emitter from the base are supplied by the generation of electron-hole pairs by the externally applied field at the ohmic (nonrectifying) base contact.

The large hole density in the base tends to induce a flow of holes in the same direc-tion as electron flow (toward the collector). This happens until an electric field is set up that prevents further hole movement. An electric field that prevents hole flow in one direction will accelerate electrons in that direction; thus, this field accelerates the minority of electrons toward the collector.

The conductivity modulation referred to previously is caused by the increased number of charge carriers in the base. The conductivity of the base region (O"b) is given by the formula

O"b

=

^qp,p(Na

+

ne) where q

=

electron charge (1.6019 X 10-¹⁹ coulomb)

f.Lp

=

hole mobility

Na

=

acceptor density in the base

ne

=

density of emitted electrons in the base

(22)

At low currents the electron density in the P base (for NPN transistors) is negligible compared to the hole density, and the conductivity is relatively independ-ent of emitter currindepend-ent. Increasing emitter currindepend-ents will evindepend-entually cause the electron density to be an appreciable part of the original hole density (no). Thus, the increased hole density caused by the equal increased electron density (space-charge

Emitter Base Collector

~

I O~---~~~---~

~ Original doping level

~ nor---~----~=---~~

x

~---w---~~

Fig. 4.9. Charge distribution in base.

Nature of Transistor Quantities and Parameters 39

neutrality requirement) will cause a rapid increase in base conductivity. This increased conductivity will cause the emitter efficiency to decrease according to

y

=

¹

1

+

aBW/a~PE (23)

where W

=

base width

aE

=

emitter conductivity

LpE

=

diffusion length for holes in the emitter

In simple terms, the increased hole and electron densities near the emitter junction increase the probability of electrons and holes recombining. This recombination current causes the decrease in y.

hFB Variation with Junction Temperature. It is difficult to arrive at a general-ized expression for the temperature dependence of hFB. The three factors (y, {3, M) in hFB are all complex functions of absolute temperature; transistors can be designed so that hFB has almost any desired temperature dependence. As an example, the emitter efficiency, y, depends¹ on carrier mobility, carrier concentration, and carrier lifetime (among other things). All these quantities are functions of temperature, 10

and the functions depend on the type of material (silicon or germanium), the type of doping, the doping density, and even the type of construction (alloy, mesa, grown junction, etc.).

All that can be said generally is that the temperature dependence of hFB is not usually a prime consideration in transistor design. It is usual for hFB to fall off at temperatures below room temperature and to increase above room temperature (see Fig. 4.7 for an example).

hI~ The static value of the short-circuit common-emitter input impedance.

In Fig. 4.10, hIE is just the slope of the line drawn through the origin and the point of measurement P.

4.2. A-C PARAMETERS

CTc-Collector transition capacitance.

CT~Emitter transition capacitance.

A transition capacitance is formed by the diffusion mechanism of carriers in an unbiased semiconductor junction. Diffusion creates a region about the junction

v

p

Figure 4.10 I

40 Fundamental Considerations

rbb,

b'

^Cb'c

b /

^C

Vb'e

~

^{Cb'e rb'e}

ree

e

Fig. 4.11. Transcurrent small-signal common-emitter equivalent circuit.

which is depleted of carriers, and an electrostatic potential across this depletion region. If an externally applied voltage forces a change in this junction potential, the charge which is thus added or removed corresponds precisely to that from a capacitor with plates having an area and a separation corresponding to the cross section and thickness of the depletion region and having a dielectric with permittivity equal to that of the semiconductor material.

There are actually two types of capacitance at a semiconductor junction: a transition capacitance, as defined above, which is primarily dependent on junction

There are actually two types of capacitance at a semiconductor junction: a transition capacitance, as defined above, which is primarily dependent on junction

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