The emitter-base junction is forward biased and collector-base junction is reverse biased.Ignoring the reverse saturation current of the collector-base diode, the transistor can be represented as:

A small change in the base current would appear in an amplified form at the collector current. In other words, the BJT exhibits current gain and is called its forward current gain.

(Henceforth the subscript F will be dropped. Unless otherwise stated, the mode of operation by default is the forward active mode).
Let us look at current gain in more detail:

The base current supplies the holes needed to replenish those that are lost from the base
due to injection/recombination. It has four distinct components :

(i) this supplies the holes lost due to recombination in the neutral base region .
(ii) this supplies the holes that are injected into the emitter.
(iii) this supplies the holes that recombine within the EB space charge region.
(iv) this represents the holes that flow from the CB space charge region into the base as part of the reverse generation current.

The current gain can therefore be written as:

Eq. (5) shows that the component of current gain, which is the least, will determine the overall current gain. Let us try to estimate each one of the current gain components:

(i) The base current due to recombination within the neutral portion of the base can be written as

This requires determination of electron density within the base. It was shown earlier that if base recombination is neglected, then electron density varies linearly as shown below:

• It may seem strange that we are trying to estimate recombination within the base and using neglect of recombination within the base to find the electron density!
• However, it must be remembered that an assumption is good or bad depending on the context.
• It turns out that the profile of electron density in the base with recombination taken into account differs very slightly from the case where it is assumed to be linear, so we are justified in using this assumption.

The base current due to recombination within the base can then be expressed as :

• Current gain is proportional to the recombination lifetime in the base and inversely proportional to the base width. The current gain is also independent of collector current.
  

There is a parameter called the base transport factor, which is closely related to the component of current gain just calculated. The base transport factor is defined as the fraction of electron current, injected by the emitter, that reaches the collector:

The fraction of electrons that do not reach the collector junction are those that get lost due to recombination so that

For a uniformly doped emitter, the hole current can be assumed to be a diffusion current because low-level injection condition is always valid in emitter due to its heavy doping level:

The emitter region is as thin or even thinner than the base so that despite a lower recombination lifetime, the number of holes lost due to recombination within the emitter can be assumed to be negligible.

is the intrinsic carrier concentration in the emitter. The intrinsic carrier concentration depends on the band gap:

There are a variety of empirical models for this bandgap narrowing effect, the simplest of which is:

The base current due to hole injection into the emitter can then be expressed as:

Using Eq. (12) and Eq. (22) and the expressions for , derived earlier, the current gain can be expressed as:

Eq. (27) can be re-written as :

Using the expression for derived earlier, the emitter injection efficiency can be written as,

As noted earlier, the overall current gain is determined primarily by the base current resulting from injection of holes into the emitter. As a result, the current gain can be expressed in terms of emitter injection efficiency :

A high emitter injection efficiency is therefore necessary to obtain a high current gain.

The component of base current due to recombination within the emitter-base space charge region can be expressed using Eq. (89) in the chapter on PN junctions as:



It is assumed here that the space charge region lies entirely on the base side because the base doping is much smaller than the emitter doping. Using the expression for collector current given by Eq. (1), we obtain :

This component of current gain is sensitive to recombination lifetime and varies with collector current density.

Example 2.6: Determine the current gain component due to SCR recombination for current densities of
Take , base doping of , effective base thickness of and effective width of SCR recombination

Solution : Using Eq. (32) we obtain

• A quick calculation shows that this base current component is much smaller than the rest and can be ignored.

• There are however, situations where this current becomes the most important base current component. For example, there are devices called phototransistors, which are
  basically BJTs whose excitation is optical instead of electrical.

As a result of optical excitation, electron-hole pairs are generated, mostly in the collector-base depletion region. The phototransistor is operated with its base floating so that

The optically excited holes supply the rest of the base current components so that they are non-zero. As we noted earlier, the second component will be the largest so that

The optically excited current gets multiplied by the current gain and appears in an amplified form at the collector.

Overall current Gain

Taking all the components of the base current into account, we can draw the following conclusions:

1. at very low collector current densities, the current gain is determined primarily by recombination within the emitter-base space charge region. In this regime the current gain increases with the collector-current density and is sensitive to the recombination lifetime in the base.
   
   
2. At higher collector current densities, the current gain is determined primarily by injection of holes into the emitter or in other words emitter injection efficiency.
   
   
3. The current gain limited by emitter injection efficiency is independent of collector current as long as low level injection condition prevails in the base. As we shall see later, at very high collector current densities, the current gain begins to decrease.

The variation of current gain with collector current density can be represented by the following Figure:

(b) At what collector current density, does the current gain fall below unity?

(c) If recombination lifetime reduces to determine the collector current density beyond which the current gain will be practically constant.

Solution : (a) The different current gain components are
so that the current gain is primarily determined by emitter injection efficiency .

( b) This requires that the SCR limited current gain reduce almost to unity so that This limit is also inversely proportional to and thus is very sensitive to material purity.

(c) If recombination lifetime is reduced to then for a collector current density of , the different components become

Most of the time, the BJT is operated in region II where the current gain is constant. In this region, the current gain is given by Eq. (26) (reproduced here again):

• As mentioned earlier, the emitter doping is often kept at the maximum allowable limit so as to obtain the highest possible gain.

• A thin, lightly doped base region is needed to obtain a high current gain.
  
  
• Since emitter is much more heavily doped as compared to the base, the band gap narrowing is stronger in emitter so thatEq. (38) then shows that the current gain will decrease with decrease in temperature due to the exponential term involving band gap-narrowing terms.

Example 2.8 : Determine the injection efficiency limited current gain for the following sets of temperatures:

The table below shows the variation of current gain with temperature predicted by Eq.(38)

• The current gain varies significantly with temperature and drops to less than unity at 77 K, which is the temperature of liquid nitrogen.
  
  
• Since liquid nitrogen is readily available at a very low cost, it is one of the most widely used methods of cooling a device. It is of interest to cool a BJT so as to remove excess heat from it. Circuits such as emitter coupled logic (ECL) consume lot of power and their scale of integration is limited by the ability to extract heat generated in the bipolar transistors. A ver simple method of overcoming this would have been to refrigerate the circuit in liquid nitrogen, but as we have seen, the current gain falls to less than unity, making the transistor useless.

• All the discussion regarding current gain so far has been for the dc current gain. As we shall see later, the small signal current gain is also of interest in many cases

The small signal gain is sometimes also called the ac gain because, the small signal is often a sinusoidal signal in many analysis.

The small signal current gain can be expressed in terms of the dc current gain using the following general expressions for collector and base currents:

• The collector and base currents have ideality factors respectively. The ideality factor may depart from unity at low or very high current densities.

Using Eq. (39-41), we obtain

• The small signal current gain will be different from the dc gain if either the base or the collector ideality factor is different from unity.

• For example, at very low collector current densities, the base current is dominated by SCR recombination current, which has an ideality factor of 2. Since the collector current
  has ideality factor of unity at low or moderate currents, the small signal current gain will be twice that of the dc gain.
• At very high collector current densities, we shall see later, collector current has ideality factor greater than unity, while the base current has ideality factor of unity. In this case
  the small signal gain will be smaller than the dc gain.
• For intermediate collector current densities, the dc gain equals the small signal current gain.

The collector current in active mode of operation for a uniformly doped base is given by
the expression:

It appears from this equation as if the collector current is independent of the collector- base voltage. However, this is not so because the effective base width depends on the
collector bias as illustrated by the Figure below:

• As the collector-base voltage increases, the depletion region associated with the collector base junction widens as illustrated by the Figure above. The base width narrows to As a result of the decrease in base width, the collector current would increase. This base width modulation effect described first by James Early, is known as the Early Effect.

The change in collector current as a result of change in collector-base voltage can be evaluated using Eq. (43)

Using Eq. (44), we obtain

Using the expression for the magnitude of collector-base depletion charge, we can obtain the following results:

Defining a parameter called the Early voltage:

• The expression for early voltage shows that its value depends on the collector-base bias but this dependence is weak so that for collector-base voltages not far from zero:

Representing we obtain

• Noting that and base emitter voltage in active mode being less than unity is often much smaller than the early voltage, we obtain

Eq. (54) shows that the collector current would slowly increase with the collector-emitter voltage as illustrated by the Figure below:

Through extrapolation of the characteristics in active mode, the early voltage can be measured!

Ignoring the small reverse current of the collector-junction, the base current can be written as

Eq. (22) for show that these currents have no dependence on the collector-base voltage so that

Use of Eq. (51) allows Eq. (56) to be re-written as:

Eq. (57) can be re-written in terms of the base transport factor

• Because of very small value of base transport factor, the voltageso that the base current is much less affected by the applied collector bias as compared to the
  collector current.

Example 2.9 Determine the Early voltage for an NPN transistor having the following characteristics at and -5 Volts.

Take, base doping of effective base thickness of collector doping of

Solution : The collector depletion width and junction capacitance/area can be calculated to be :

The early voltage then can be calculated to be:

the collector depletion width and junction capacitance become The new early voltage

• In all the analysis carried out so far, the doping in emitter, base and collector were all assumed to be uniform. However, in most devices, due to the nature of fabrication,
  doping in emitter and base is nonuniform, approximating gaussian or error function most of the time. We will next look at the model for collector current when base doping is
  nonuniform.
  

• As a result of nonuniform base doping, an electric field is present in the base so that the electron cannot be assumed to be simply diffusive. The magnitude of the electric field
  can be obtained through analysis of the hole current:

This hole current supplies the holes that are injected into the emitter or those that recombine in the base. If the hole current is assumed to be very small so that:

This expression for electric field can be used to write the expression for electron current or equivalently collector current as:

Eq. (62) can be re-written as

Integration of this equation across the base gives:

Since doping is nonuniform in the base, the mobility and therefore the diffusion constant would also vary in the base. This variation, however, is small and represents its average value in the base.

·The electron density at the edge of reverse biased collector-base junction will be negligible so that and the electron-hole product at the edge of emitter-base
junction can be written as :

Eq. (64) can now be simplified to

·Eq. (66) is known as the representation of the collector current. A model forBJT will be derived using this equation later on.

is known as the Gummel number for the base. Using quasi-neutrality condition for the base:

For low level injection: so that

For uniformly doped base :

Substitution of Eq. (70) in Eq. (66) gives the same expression for collector current as Eq. (1) that we have used throughout the previous analysis.

Example 2.10 (a) An NPN transistor has a nonuniform doping in the base described by the expression :

Assume that average diffusion constant in the base is and that effective base thickness is Determine the magnitude and direction of electric field in the base
(b) Obtain an expression for electron density profile within the base
(c) Determine the injection efficiency limited current gain at room temperature. Assume that

(a) Under low level injection conditions, so that

The negative sign indicates that the field is directed towards emitter. It therefore aids in drift of electrons towards the base.

(b) The electron current density within the base can be written as

Using the expression for electric field obtained earlier, we can re-write the above expression as

This equation can be expressed as

Integration of the above expression from x to WB with the boundary condition that electron density at the collector end of the base is negligible, we obtain

(c) The expression for current gain can be modified to

For calculating the band gap narrowing in the base, we find the average doping in the base and use it in the expression given in the text. With that we obtain a current gain of 116.