Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

Figure 6.9: Analytic example of computing minority stored charge factors (a) piecewise doping profile approximation, (b) inner integral based on Eqs. (6.30-31), (c) outer integral to obtain QSF and QSR.

6.4Generalized Charge Storage Model

At this point it is useful to realize that excess minority charge storage occurs not only in the base but also in the collector and emitter. Before proceeding the extension, we first make clear the physical meaning to terms Q sFb(O) and Q sRb(WB). These terms actually represent those parts of the stored excess carriers which are controlled by their respective junction biases. Take Q sFb(O) as an example. For a uniformly doped base region, Eqs. (6.34) and (6.35) are reduced to the same form of IsW~/2Dn, and further, using the expression of Is in Eq. (6.23), one

Is[b(O)-b(WB)]

Be

b(WB)

Figure 6.10: Bipolar transistor model reformulated in terms of the relative excess concentrations at the BE and Be junctions.

This is exactly the stored charge of the excess electrons in the base when the distribution of electrons is assumed to be linear and the excess electron concentration at the emitter boundary of the base quasi-neutral region is set to b(O)nUNA while that at the collector boundary of the region is set to zero. We designate this portion of the excess electrons as injected from the emitter. For the same reason, we can say that Q sRb(WB) is the storage of excess electrons injected from (or extracted to for negative b(WB» the collector.

Excess minority charge in the collector and emitter can be defined in terms of their respective relative excesses as

where QEF and QCR are determined solely by the doping profiles in the quasi-neutral regions of the emitter and collector, respectively, in a manner similar to Eqs. (6.34-6.35). In fact, the effects due to QEF and QSF are additive as are those due to QCR and QSR. What this means is that to consider all minority charge storage effects we can replace the elements shown in Figure 6.10 with those of values

262 CHAPTER 6. BIPOLAR TRANSISTORS

Clearly, this means that the figure no longer represents simply effects in the base. However, we have come to a point where we can refine our terminal device model shown in Figure 1.6. Specifically, we can now include QF and QR as elements in parallel with the diodes IBF and

Eo--,...---r---t.-I----r---...--o C

B

(a) Model including both de transport and minority charge storage effects.

(b) Model including both de space charge and minority charge storage effects as well as de transport components.

Figure 6.11: Different level of npn transistor models

IBR. This configuration is shown in Figure 1.11 (a). Note that qF and

(6.42)

An important feature of the pn junctions, which has been neglected so far, is the charge storage within the space-charge regions. The expressions for lEE and lEG are de equations and take no account of transient or ae effects in the space-charge regions. To complete the transistor model we must include these charge storage effects. We can do so easily by using the junction capacitance given in Chapter 4. The definition of incremental charge, q(Va), is a convenient representation for the spacecharge storage effect since it tells how much charge must be added to or subtracted from the equilibrium condition Q(O) in order to achieve a given junction voltage Va (and associated space charge Q(Va». Thus,

(6.43)

where q(Va) is a positive charge added to the p-side of the pn junction when Va has a polarity to forward bias the junction. For the abruptly doped junction, Eqs. (4.78) and (4.75-76) give

(6.44)

where

(6.45)

and

(6.46)

For the linearly graded junction one can easily show that the depletion layer width is proportional to ("po - Va )1/3, where "po has a different expression from Eq. (6.45). To determine the space charge Q(Va), note that since the space-charge density varies linearly with the distance away from the metallurgical junction, the total charge amounts to the area of a triangle. The base of this triangle is the depletion layer width, I, and the height is al where a is the slope of the doping profile. Thus,

where the exact parameter dependences of Qo and "po should easily be derived. Using the relationships given by Eqs. (6.44) and (6.48) in the above, one obtains

for the linearly graded junction where q(Va) is positive for a forwardbias condition. Again it should be emphasized that this is a positive charge added to the p-side of the junction for forward bias.

It is useful to add subscripts to the above space-charge expressions to reflect which junctions are referred to. Thus

(6.51)

and

(6.52)

where n = 2 for an abrupt junction and n = 3 for a linearly graded junction, and the subscripts indicate the base-emitter and base-collector space-charge regions, respectively. The inclusion of these space-charge effects in the model leads to the modifications shown in Figure 6.11 (b).

The elements of the model can briefly be categorized and described as follows. The qBE and qBG terms represent the voltage-dependent space-charge storage. Their voltage dependence is as given in Eqs. (6.51- 6.52) and is of a "power-law" form. The qF and qR terms represent the injected minority charge storage. As shown, they include effects of the minority charge storage in both the base-emitter and base-collector, respectively. The voltage dependence of these terms is exponential with respect to the applied junction voltages as indicated by Eqs. (6.41-6.42). The IBE and IBG terms represent the base current components. They model both the collector and emitter "reverse injection" as well as parasitic recombination effects in the space-charge regions and elsewhere. Thus their voltage dependences can include terms of the form eqVa/kT and eQVa/2kT as given in Eqs. (6.25-6.26). In all but the most idealized situations one uses experimental data to give appropriate parameter fits.

Finally, the component associated with Is represents the transport of minority carriers through the base. As we have found, this term exhibits a "reciprocal" behavior as observed in terminal currents. Its form can be confirmed theoretically even for nonuniform base doping. This result is shown in Eq. (6.23).

With the aid of the equations mentioned above and Figure 6.11 (b) we are now in a position to consider current flow for other than de conditions. For time-variant conditions (ae or transient), the displacement current was defined for the space-charge capacitance by Eq. (4.80) in Chapter 4. Thus the base current needed to provide charge to alter the space-charge regions under the time-variant condition is

(6.53)

where the subscript Be designates "space charge."

The base region is p-type in an npn transistor and the current flows into the base from the base terminal since in order to increase VBE and VBC the space-charge regions must be narrowed and holes must be supplied from the base terminal. These two current components must leave the emitter and collector terminals, respectively (to maintain current continuity as governed by the Kirchoff's current law). Thus,

In a similar manner we can discuss the displacement current for the diffusion capacitance which is formed by the stored minority carriers in the quasi-neutral regions. Let us first consider the emitter component. To forward bias (or increase the forward bias on) the base-emitter junction, electrons must come from the emitter lead and be stored in qF. This means that the actual current is out of the emitter terminal. Thus,

(6.57)

where dif f stands for the diffusion capacitance. Extending the above argument to the collector displacement term, one has

Since all these stored electrons have to be compensated by the holes coming from the base lead to maintain approximate charge neutrality, the base displacement current for diffusion capacitances has the positive sign with expression of

The behavior of de components of the currents have already been summarized in Eqs. (6.27-6.29). Combining with the equations above we can now write the expressions of time-variant terminal currents for the npn configuration given in Figure 6.11 (b):

(6.62)

The bracketed terms before the dVBE/dt and dVBC/dt terms represent the parallel combination of two nonlinear capacitances. Namely, we can define the diffusion capacitances as

(6.63)

(6.64)

Similarly, the depletion layer capacitances are defined as

(6.66)

Note that all these capacitances are nonlinear and are taken as the slope of their respective q(Va) vs. Va curve at appropriate bias.

We now have the fundamental relationships which describe the terminal currents of an npn transistor as a function of base-emitter and base-collector voltages under transient conditions. Unfortunately, these equations are nonlinear. In the next section, we will develop a chargecontrol model for bipolar transistor operation, and based on this model both large-signal and small-signal equivalent circuits will be derived.

6.5Transistor Equivalent Circuits

The previous section has discussed the generalized charge-storage model in the bipolar device. Even though the time-variant components of the terminal currents can be expressed using the time derivatives of the stored charges, the de parts are still directly related to the junction voltages, which gives rise to the nonlinearity of the overall equations. In this section we will first justify the introduction of time constants which relate the stored charge to the de currents, and then develop a charge control model which enables the terminal currents to be expressed as linearly dependent upon the stored charge. Based on this model, both large-signal and small-signal equivalent circuits for the transistor operation will be derived.

6.5.1Charge Control Model

In the above development of the charge-storage model, we noticed that both the minority charge-storage and the de current components have the same dependence on the applied junction voltage. So we might consider to link the current directly to the stored charge using some (time) constants. We start to look at the possibility of this modeling approach called the charge control model. First, let us look at the expression for the de collector current (Eq. (6.27)), which can be regrouped as

(6.67)

The first term on the right-hand-side (RHS) of the above equation represents the current component controlled by the base-emitter bias, VBE.