Chapter 5: Bipolar Junction Transistors 
The ideal transistor model is based on the ideal pn diode model and provides a firstorder calculation of the dc parameters of a bipolar junction transistor. To further simplify this model, we will assume that all quasineutral regions in the device are much smaller than the minoritycarrier diffusion lengths in these regions, so that the "short" diode expressions apply. The use of the ideal pn diode model implies that no recombination within the depletion regions is taken into account. Such recombination current will be discussed in section 5.4.2. 
The discussion of the ideal transistor starts with a discussion of the forward active mode of operation, followed by a general description of the four different bias modes, the corresponding EbersMoll model and a calculation of the collectoremitter voltage when the device is biased in saturation. 
5.3.1. Forward active mode of operation 
The forward active mode is obtained by forwardbiasing the baseemitter junction. In addition we eliminate the basecollector junction current by setting V_{BC} = 0. The minoritycarrier distribution in the quasineutral regions of the bipolar transistor, as shown in Figure 5.3.1, is used to analyze this situation in more detail. 
Figure 5.3.1. :  Minoritycarrier distribution in the quasineutral regions of a bipolar transistor (a) Forward active bias mode. (b) Saturation mode. 
The values of the minority carrier densities at the edges of the depletion regions are indicated on the Figure 5.3.1. The carrier densities vary linearly between the boundary values as expected when using the assumption that no significant recombination takes place in the quasineutral regions. The minority carrier densities on both sides of the basecollector depletion region equal the thermal equilibrium values since V_{BC} was set to zero. While this boundary condition is mathematically equivalent to that of an ideal contact, there is an important difference. The minority carriers arriving at x = w_{B}  x_{p,BC} do not recombine. Instead, they drift through the basecollector depletion region and end up as majority carriers in the collector region. 
The emitter current due to electrons and holes are obtained using the "short" diode expressions derived in section 4.4.2.5, yielding: 
(5.3.1) 
and 
(5.3.2) 
It is convenient to rewrite the emitter current due to electrons, I_{E,n}, as a function of the total excess minority charge in the base, DQ_{n,B}. This charge is proportional to the triangular area in the quasineutral base as shown in Figure 5.3.1 a) and is calculated from: 
(5.3.3) 
which for a "short" diode becomes: 
(5.3.4) 
And the emitter current due to electrons, I_{E,n}, simplifies to: 
(5.3.5) 
where t_{r} is the average time the minority carriers spend in the base layer, i.e. the transit time. The emitter current therefore equals the excess minority carrier charge present in the base region, divided by the time this charge spends in the base. This and other similar relations will be used to construct the charge control model of the bipolar junction transistor in section 5.6.2. 
A combination of equations (5.3.1), (5.3.4) and (5.3.5) yields the transit time as a function of the quasineutral layer width, w_{B}^{'}, and the electron diffusion constant in the base, D_{n,B}. 
(5.3.6) 
We now turn our attention to the recombination current in the quasineutral base and obtain it from the continuity equation (2.9.3): 
(5.3.7) 
By applying it to the quasineutral base region and assuming steady state conditions: 
(5.3.8) 
which in turn can be written as a function of the excess minority carrier charge, DQ_{n,B}, using equation (5.3.3). 
(5.3.9) 
Next, we need to find the emitter efficiency and base transport factor. The emitter efficiency defined by equation (5.2.17), becomes: 
(5.3.10) 
It is typically the emitter efficiency, which limits the current gain in transistors made of silicon or germanium. The long minoritycarrier lifetime and the long diffusion lengths in those materials justify the exclusion of recombination in the base or the depletion layer. The resulting current gain, under such conditions, is: 
(5.3.11) 
From this equation, we conclude that the current gain can be larger than one if the emitter doping is much larger than the base doping. A typical current gain for a silicon bipolar transistor is 50  150. 
The base transport factor, as defined in equation (5.2.18), equals: 
(5.3.12) 
This expression is only valid if the base transport factor is very close to one, since it was derived using the “shortdiode” carrier distribution. This base transport factor can also be expressed in function of the diffusion length in the base: 
(5.3.13) 
Example 5.2  Consider a pnp bipolar transistor with emitter doping of 10^{18} cm^{3} and base doping of 10^{17} cm^{3}. The quasineutral region width in the emitter is 1 mm and 0.2 mm in the base. Use m_{n} = 1000 cm^{2}/Vs and mp = 300 cm^{2}/Vs . The minority carrier lifetime in the base is 10 ns. Calculate the emitter efficiency, the base transport factor, and the current gain of the transistor biased in the forward active mode. Assume there is no recombination in the depletion region. 
Solution  The emitter efficiency is obtained from: The base transport factor equals: The current gain then becomes: where the transport factor, a, was calculated as the product of the emitter efficiency and the base transport factor:

5.3.2. General bias modes of a bipolar transistor 
While the forward active mode of operation is the most useful bias mode when using a bipolar junction transistor as an amplifier, one cannot ignore the other bias modes especially when using the device as a digital switch. All possible bias modes are illustrated with Figure 5.3.2. They are the forward active mode of operation, the reverse active mode of operation, the saturation mode and the cutoff mode. 
Figure 5.3.2.:  Possible bias modes of operation of a bipolar junction transistor. 
The forward active mode is the one where we forward bias the baseemitter junction, V_{BE} > 0 and reverse bias the basecollector junction, V_{BC} < 0. This mode, as discussed in section 5.3.1, is the one used in bipolar transistor amplifiers. In bipolar transistor logic circuits, one frequently switches the transistor from the “off” state to the low resistance “on” state. This “off” state is the cutoff mode and the “on” state is the saturation mode. In the cutoff mode, both junctions are reversed biased, V_{BE} < 0 and V_{BC} < 0, so that very little current goes through the device. This corresponds to the “off” state of the device. In the saturation mode, both junctions are forward biased, V_{BE} > 0 and V_{CB} > 0. This corresponds to the low resistance “on” state of the transistor. 
Finally, there is the reverse active mode of operation. In the reverse active mode, we reverse the function of the emitter and the collector. We reverse bias the baseemitter junction and forward bias the basecollector junction, or V_{BE} < 0 and V_{BC} > 0. In this mode, the transistor has an emitter efficiency and base transport factor as described by equations ((5.3.10) and (5.3.12), where we replace the emitter parameters by the collector parameters. Most transistors, however, have poor emitter efficiency under reverse active bias since the collector doping density is typically much less than the base doping density to ensure high basecollector breakdown voltages. In addition, the collectorbase area is typically larger than the emitterbase area, so that even fewer electrons make it from the collector into the emitter. 
Having described the forward active mode of operation, there remains the saturation mode, which needs further discussion. Cutoff requires little further analysis, while the reverse active mode of operation is analogous to the forward active mode with the added complication that the areas of the baseemitter and basecollector junction, A_{E} and A_{C}, differ. The EbersMoll model describes all of these bias modes. 
5.3.3. The EbersMoll model 
The EbersMoll model is an ideal model for a bipolar transistor, which can be used, in the forward active mode of operation, in the reverse active mode, in saturation and in cutoff. This model is the predecessor of today's computer simulation models and contains only the “ideal” diode currents. 
The model contains two diodes and two current sources as shown in Figure 5.3.3. The two diodes represent the baseemitter and basecollector diodes. The current sources quantify the transport of minority carriers through the base region. These current sources depend on the current through each diode. The parameters I_{E,s}, I_{C,s}, a_{F} and a_{R} are the saturation currents of the baseemitter and base collector diode and the forward and reverse transport factors. 
Figure 5.3.3 :  Equivalent circuit for the EbersMoll model of an npn bipolar junction transistor 
Using the parameters identified in Figure 5.3.3, we can relate the emitter, base and collector current to the forward and reverse currents and transport factors, yielding: 
(5.3.14) 
(5.3.15) 
(5.3.16) 
The EbersMoll parameters are related by the following equation: 
(5.3.17) 
This relation ship is also referred as the reciprocity relation and can be derived by examining the minority carrier current through the base. For the specific case where the baseemitter and basecollector voltage are the same and the base doping is uniform, there can be no minority carrier diffusion in the base so that: 
(5.3.18) 
from which the reciprocity relation is obtained. 
The forward and reversebias transport factors are obtained by measuring the current gain in the forward active and reverse active mode of operation. The saturation currents I_{E,s} and I_{C,s} are obtained by measuring the baseemitter (basecollector) diode saturation current while shorting the basecollector (baseemitter) diode. 
5.3.4. Saturation 
In the low resistance “on” state of a bipolar transistor, one finds that the voltage between the collector and emitter is less than the forward bias voltage of the baseemitter junction. Typically the “on” state voltage of a silicon BJT is 100 mV and the forward bias voltage is 700 mV. Therefore, the basecollector junction is also forward biased. Using the EbersMoll model, we can calculate the “on” voltage from: 
(5.3.19) 
and using equations (5.3.15), (5.3.16) and the reciprocity relation (5.3.17), one obtains: 
(5.3.20) 
Saturation also implies that a large amount of minority carrier charge is accumulated in the base region. As a transistor is switched from saturation to cutoff, this charge initially remains in the base and a collector current will remain until this charge is removed by recombination. This causes an additional delay before the transistor is turned off. Since the carrier lifetime can be significantly longer than the base transit time, the turnoff delay causes a large and undesirable asymmetry between turnon and turnoff time. Saturation is therefore avoided in highspeed bipolar logic circuits. Two techniques are used to reduce the turnoff delay: 1) adding a Schottky diode in parallel to the basecollector junction and 2) using an emittercoupled circuit configuration. Both approaches avoid biasing the transistor in the saturation mode. The Schottky diode clamps the basecollector voltage at a value, which is slightly lower than the turnon voltage of the basecollector diode. An emittercoupled circuit is biased with a current source, which can be designed such that the collector voltage cannot be less than the base voltage. 
Example 5.3  Calculate the saturation voltage of a bipolar transistor biased with a base current of 1 mA and a collector current of 10 mA. Use a_{R} = 0.993 and a_{F} = 0.2. 
Solution  The saturation voltage equals:

Boulder, December 2004 