Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

qec

Figure 6.12: Equivalent circuit for intrinsic bipolar transistor using the charge-control model.

Eqs. (6.73-6.75) constitute the charge-control model.

We are now able to draw an equivalent circuit for the above chargecontrol model. The term qF / Tp in ic expression is modeled as the current generators in the equivalent circuit because qF is not controlled by the VBC, the voltage drop across the circuit branch where the component is connected. The similar argument applies to qR/TR as well. This equivalent circuit is a large-signal circuit for there is no restriction as to how large the applied bias can be. It should be pointed out that the de components and their connection in this equivalent circuit are identical to those in the Ebers-Moll model [6.4].

6.5.2Small-Signal Equivalent Circuit

We next derive a small-signal equivalent circuit for transistors. Like any nonlinear system, the behavior of a bipolar transistor in response to an infinitesimally small variation in the bias can always be represented by a linear circuit. For a transistor biased in the active region, its

where Vi = kTIq, the thermal voltage, and the latter equations is from Eq. (6.27). The approximate form of 9m results from assuming that VBC ~ 0 and that IBc can be neglected. Thus we have

where 0 = TFITBF is called the defect factor [6.4] and is approximately equal to IBlIc = f3F1 • Using the same argument,

Approximation has been made to obtain the above equation because the base-width modulation due to the change of VBC is neglected, which is implied in the expression of Is (Eq. (6.24)) through the WB dependence on VBC. Using definitions of GdF, CdR, GBE, and GBC in Eqs. (6.63- 6.66), one obtains

An equivalent circuit corresponding to the above equations can be drawn as in Figure 6.13. Again we notice that terms 9mVBE and 9mRVBC are modeled as the current generators between nodes C and E because the

E

Figure 6.13: ac small-signal equivalent circuit for intrinsic bipolar transistor.

currents through these two-terminal elements are not controlled by their node voltages.

The above equivalent circuit can greatly be simplified in the active operation region where VBe < 0 for npn transistors. The basic assumption is thus eVBC /vt <t:: 1, and one obtains

and

(6.89)

The equivalent circuit of the transistor in the active operation region thus becomes much simpler, as shown in Figure 6.14, where GD = GdF+

GBE.

6.5.3ac Modeling of Junction Capacitances

The junction capacitances as defined in Eqs. (6.63-6.66) and used in deriving expressions for ac small-signal terminal currents are of quasisteady nature. That is the capacitance which is computed from the derivative of the (voltage-dependent) stored charges with respect to the applied voltage, which implies that for each change of bias, device has reached steady state before the stored charge is used for computation.

Cae

Figure 6.14: ac small-signal equivalent circuit for intrinsic bipolar transistor in active region.

It is certainly not the case when the frequency of the small signal is high and we expect to see that the transition processes will playa role in determining the device response. One might think that the quasisteady approach is at least accurate for low frequencies because the response of the device would be fast enough to follow the change of the external signal. It turns out, however, for those capacitances which are caused by the minority carriers (e.g. the diffusion capacitance), the real ac capacitance is always smaller than that from the quasisteady state analysis. On the other hand, those capacitances which are charged by the majority carriers (e.g. the depletion capacitances) have the same values as derived from the quasi-steady analysis. Lindmayer and Wrigley have an excellent discussion in their book [6.3]. In the following, we briefly introduce their analysis.

The rigorous definition of components for ac small signal circuits should come from the ac analysis, where the device is excited by either a voltage or current source. For the quasi-neutral region surrounded by junctions, such as the neutral base, it is easy to use voltage source. Then the ac terminal currents corresponding to the excitation can be found by solving the continuity equation for ac part of carrier concentration. The ratio of the ac current and the exciting voltage is usually a complex number and can be modeled as a susceptance. Its real part, if it is frequency independent, can naturally be modeled as a resistance, while the imaginary part, i.e., the admittance, if proportional to the frequency, can be modeled as a capacitance. On the other hand, once the ac carrier distribution is known, the stored charge can readily be obtained. Thus one can also use the derivative approach to compute the equivalent capacitance. The computation shows that the difference between the "true" ac capacitance and the one obtained using quasi-steady analysis

is not negligible even as the frequency approaches zero, and is dependent on the device structure. For the long-base device where the injected minority carriers vanish before reaching the end of the device, this ratio is one half, and for the short-base device and the receiving end is ac short-circuited, the ratio is one third. Interested readers can refer to Chapter 2 in [6.3]. The explanation to this discrepancy is that not all the stored charge can be reclaimed by the exciting source which provides the injected carriers. The situation is quite different from the plate capacitance or depletion capacitance, in which the the charge coming from one direction is always fully reclaimable, i.e., drawn out completely, by the source. In the diffusion capacitance case, once the signal applied to the injection end is withdrawn, the stored charge can go either way - through the injection end which constitute the normal capacitance term or through the receiving end which does not contribute to the displacement current hence the capacitive component.

One should realize from the above discussion that even though the exact values used for some capacitive components, namely diffusion capacitances, have to be modified according to more rigorous analysis, the circuit configuration as demonstrated in this section does not need to be changed. Hence, we should be able, from this point on, to compute the device characteristics and parameters based on the above equivalent circuits.

One of the applications of the ac equivalent circuit is to predict the high-frequency behavior which is usually characterized by the cutoff frequency. We will define this frequency later on and discuss the way to compute in both analytical way and by simulation. Before we discuss the high-frequency behavior, however, let us first look at the second order effects. By the second order effects we mean those not covered by the formulation in the previous sections and yet affect the transistor characteristics in one way or other.

6.6Second Order Effects

The discussion about transistors so far is based on the so-called rigidboundary approximation, i.e., the edges of the depletion (space-charge) regions in the base-emitter and base-collector junctions do not change appreciably as the bias is changed. Furthermore, the built-in electric field in the quasi-neutral region in the base is also assumed unaltered by

the applied bias. There are, however, certain circumstances where the above assumptions are not valid. The device behaves under the influence of several second-order effects, and they will be discussed in this section. First, we will discuss the effect on the device characteristics of change of the base-collector space-charge region boundary due to VBe change. We then consider the situation where the population of the injected minority carriers in the base region is comparable to the background doping concentration (majority carrier population). The effect will be twofold - first it will alter or result in the electric field in the quasineutral base region, the so-called Webster effect; second the space charge region in the base-collector metallurgical junction will be overwhelmed by the injected carriers from the emitter and the compensating base majority carriers, resulting in a widening ofthe quasi-neutral base region and the shift of the junction space charge region towards the collector. There are a number of phenomena associated with this shift and they are all associated with the high-level injection. Finally because the series resistance and the spread resistance also play an important role in determining both de and ae characteristics of the device, we will briefly discuss their impact.

6.6.1Base-Width Modulation Due to Base-Collector Bias - Early Effect

When the base-collector bias changes, not only qR and qBC are changed, but also the boundaries of the base-collector space-charge region changes. In fact, the change of qBC is due to the movement of the boundaries. The effect of the boundary displacement at the base side of the junction on the transistor I - V characteristics is especially obvious because it changes the width of the base quasi-neutral region, WB, and the base transport current critically depends on this width. We can see this dependency through the following analysis.

For VBe ~ 0, we have

from Eqs. (6.27) and (6.24). Note that NA in Eqs. (6.23-6.24) has been replaced by the majority carrier concentration in the base, p, because of the consideration that one of the second-order effects soon to be discussed is the high-level injection. This change has also been discussed

in Chapter 4 where it is pointed out that the built-in field in the base quasi-neutral region is determined by the majority carriers rather than by the net doping distribution. For the moment, we neglect the bandgap narrowing effect, which mainly plays a critical role in the heaVily-doped emitter region and hence mostly affects the magnitude of the base current. We now must consider WB, the base width, as a function of VBC. The collector current of a transistor, when its base-collector junction is reverse biased, is affected by the base-collector bias mainly through this base width modulation. By differentiating the above approximate expression of Ic with WB dependency explicitly included, one obtains

This seemingly rather complicated formula can greatly be simplified by introducing a new parameter with units of voltage, VA, such that

the shrinking base-collector space-charge region, VA thus defined is a negative quantity for npn transistors.

The change of the width of the base quasi-neutral region due to variation of the applied bias on the transistor is conventionally named the base-width modulation, and in particular that due to the basecollector bias change is called the Early effect, named after James Early, who first analyzed this phenomenon [6.1]. Because VA as defined in Eq. (6.92) is a good measure ofthe Early effect, it is called the Early

We may also define the total charge of the base majority carriers at the thermal equilibrium as

This change in the base majority charge can be related to the basecollector depletion capacitance in the following way:

(6.99)

where the derivative of qBC with respect to WB has been evaluated as the base doping density at the edge of the base-collector space-charge region, NA(WB), multiplied by the area of the base-collector junction, Ac, and q. Combining Eqs. (6.97) and (6.99) in Eq. (6.96), one obtains

Therefore the Early voltage can be expressed as the ratio of the base majority charge to the base-collector junction capacitance, scaled by the ratio of the base-collector to base-emitter junction areas. In a typical IC bipolar transistor, Ac is substantially larger than AE. It is desired that the influence of the base-collector voltage on the collector current be minimized, i.e., VA should be large in magnitude. The above equation gives the right direction one should take in order to improve the Early voltage. The ultimate goal is to reduce the change of the base-collector boundary in the base region.

The introduction of the Early effect also has a consequence on the equivalent circuit of the transistor. Considering only the forward-active

where the parameter", = Vt/VA can be interpreted as the ratio of the relative strength of VBC'S controllability over Ic versus that of VBE.