Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

where the direction of the terminal currents is defined as in Figure 6.4 (Le. positive for current flowing into the device) and except for the emitter all regions are assumed to constitute "short bases". Note that the component @has specifically been neglected and will be discussed later. If the terms IBF, Is, and IBR as defined in the above equations, those equations can be simplified to give

It can be interpreted that Is represents the base transport current and IBF represents the "forward" component of the base current into the emitter region. This component is most important for the normal (or Forward) active mode and hence the "BF" notation. Note that this component corresponds directly to the "E" charge component. Finally, IER refers to the base current in the collector region which dominates for the "Reverse" active mode and thus the "BR" notation. Note also that this component corresponds directly to the "C" charge component.

The representation of these equations in a "model" is shown in Figure 6.4. The two diodes represent "loss elements" in the sense that they directly reflect current components across the collector and emitter junctions, which contribute only to the base current (see Figure 6.3 (c)). The dependent current source represents the base transport current and depends on both the EB and CB boundary conditions.

For the fBF and fBR recombination terms, only an assumed "ideal'; dependence is considered. The space-charge recombination terms,

Figure 6.4: pnp transistor model with diodes representing recombination currents and the current source represents the "ideal" base transport current.

which have an eqV/2kT dependence, have been neglected. For most practical situations these recombination terms are of equal importance to the "ideal" terms. To include these terms simply amounts to modifying the equations used in association with the diodes shown in Figure 6.4. For example, to include space-charge recombination, the following additions can be made:

where the subscript "sr" is used to indicate recombination in the spacecharge region. These terms can still be interpreted in terms of the diodes which are modified by additive current terms to account for the nonideal current components.

The development of the equations describing the bipolar junction transistor to this point have assumed a uniformly doped structure which consists of a pnp structure. These assumptions are appropriate for considering a lateral pnp structure very much like the MOSFET geometry discussed in Chapter 5. Moreover, in Figure 6.1 we see lateral surface devices - PNPiL and PNPwellL. We now consider the double-diffused bipolar structure, typically an npn type device, to optimize the high gain aspects of the BJT. Such a structure is shown in Figure 6.5(a), with dimensions and auxiliary features typical of a device used for bipolar integrated circuits. The surrounding p-region isolates the npn transistor from other components in the silicon. Since the collector contact

Figure 6.5: (a) Plane and section views for a vertical npn transistor, assuming junction isolation, (b) typical doping profiles (log concentration) versus depth for the double-diffused portion of the npn device, (c) the net acceptor profile (arbitrary units) across the base region.

is made on the upper surface, there can be substantial resistance between the contact and the base-collector junction (near the center of the structure). For the double-diffused structure a typical doping profile of impurities across this region is shown in Figure 6.5 (b). The doping profile in the emitter is rather steep, and high concentration diffusion effects are typically observed at the peak. In the base region, the boron compensates (changes doping polarity) the epitaxial doping. The emitter-base junction is defined by the n+ again compensating the boron doping. The net acceptor impurity concentration in the base region is shown in Figure 6.5 (c) as a function of position. To consider the transport of injected electrons across this nonuniformly doped region requires more than the simple diffusion equation used in considering the pnp device shown in Figures 6.3(a) and described by Eqs. (6.3).

6.3Transport Current Analysis

To begin the consideration of current transport across the base we must realize that the doping nonuniformity shown in Figure 6.5 (c) gives rise to an electric field across the base. That is, to maintain no net flow for majority carriers in the base, the following condition must apply:

and using the Einstein relationship with the quasi-neutrality approximation at the low injection level that p(x)::::: NA(x):

Turning our attention now to the minority carriers moving across this region, the expression for the electron current flow is

It is useful to realize that with no bias applied, and hence no injection, I n = O. That is, for equilibrium conditions,

(6.10)

254 CHAPTER 6. BIPOLAR TRANSISTORS

nonuniform base doping. To simplify the discussion we make the assumption that we can neglect recombination of the injected carriers as they are traversing the base region - hence I n is not a function of position. At this point we can use the same approach for the MOSFET in Chapter 5. That is, since we do not precisely know b(x) or 8b(x)/8x at all positions, let us integrate the defining current equation upto the boundaries where the conditions are nicely posed. Considering the current, In, instead of the current density, I n, (In = AJn), one has from Eq. (6.14)

Note that the spatial dependence of Dn has now been included. This dependence comes from the doping dependency of the mobility. The result of the integration gives

It is important to see how b(WB) and b(O) depend on the terminal parameters. The excess carrier concentration at each boundary, from our discussion of pn junction operation, is set by the equilibrium minority concentration and the applied voltage across the junction there. That is,

This is the result from our discussion of pn diodes in Chapter 4 that applied voltages appear only across the space-charge regions independent of base doping and that the quasi-Fermi energy levels remain flat in the space-charge region and are separated by an amount of qVa , where Va is the applied forward bias. Using the boundary values for b given in Eqs. (6.21-6.22), Eq. (6.18) becomes

The denominator of the expression for Is is often called the Gummel integral (Gummel number), CB.

Note that we are now defining the term Is in a more general way. It is useful and reassuring at this point to show that the expression for Is given in Eq. (6.23) reduces to the form of Eq. (6.3) for the case of uniform doping. Looking at Eq. (6.23) for constant NA and Dn ,

(6.24)

which is identical to Is in Eq. (6.3) except for the carrier type. Thus Eq. (6.23) gives the general base-transport equation for a nonuniform base doping, which reduces to our earlier result for the uniform doping case.

At this point we can easily incorporate this new equation into a model for the npn transistor. As with the pnp structure we can introduce diodes between base-emitter and base-collector terminals to represent reverse injection and space-charge recombination terms. You may have noticed that the BE and Be notation in Eq. (6.23) is different from that used for the pnp. This reflects the choice to have the order of the subscripts reflect the "+" to "-" connection required to give a forward biased junction. Figure 6.6 shows an npn model. The base-emitter and base-collector diodes have the following voltage dependences:

Figure 6.6: npn transistor model with base-current diodes and generalized base transport factor including the Gummel integral to account for the nonuniform base doping.

and IB == IBE + IBG. Compared to Eq. (6.5), the "ideal" component of the current (i.e., the component with exponent factor qVa / kT where Va is the applied bias across the junction) has a pre-exponential factor expressed in terms of Is and the so-called common-emitter current gains f3F and f3R for IBF and IBR, respectively. If we can neglect the spacecharge recombination terms, the maximum achievable f3 is determined by the ratio of the transport current divided by the respective IBE or IBG term. This point will be discussed in greater detail shortly.

The notation above emphasizes "forward" and "reverse" components. Notice that the direction (Le. the arrow) in the current source in Figure 6.6 has been reversed from that of Figure 6.4 and the minus sign for Eq. (6.4) has also been reversed to account for the change. The equations describing terminal currents are

This brings us to a result of major importance. Consider the following two experiments. First, ground the base and collector terminals

Figure 6.8: Experimental results corresponding to measurement configurations shown in Figure 6.7.

At this point it is useful to consider the total minority charge storage in the base and to see how this charge storage can be related to terminal parameters. The result will allow us to describe capacitive effects due to the base transport charge. To begin with we note that the excess charge in the base can be obtained as follows:

(6.30)