Книги2 / 1993 Dutton , Yu -Technology CAD_Computer Simulation

6.6.2High-Level Injection

We have analyzed the high-level injection phenomenon in pn junctions in Chapter 4, which has two direct consequences. One, the rate of increase of the injected minority carrier population at the junction space-charge region edge is reduced with the increased forward bias. That is, the dependence of exp(qV/kT) becomes exp(qV/2kT). The other is that the electric field in the quasi-neutral region becomes a function of bias and is no longer merely determined by the doping profile. This latter effect can be seen clearly for the uniformly doped region, which has no built-in electric field at the low-injection levels, as demonstrated in Chapter 4. At the high-level injection, however, the electric field arises and can often be determined by assuming that the drift and diffusion components of the majority carrier current is roughly equal, i.e., the net majority carrier current is small compared to its two competing (in sign) components.

We need now to expand our understanding of the high-level injection from the pn junction to the transistor. First let us look at the base quasi-neutral region under the high-level injection condition. For an npn transistor, the injected electron carrier concentration at the baseemitter junction edge of the base quasi-neutral region can be expressed

which is obtained by assuming the quasi-equilibrium through the baseemitter junction. At low or moderate injection levels, 4nr exp(VBE/vt) ~ N1(0), this expression is reduced to Eq. (6.19). At the high-level injection, the above expression becomes nCO) ~ ni exp(VBE/2vt), which actually implies that p(O) ~ nCO). Thus, there exists a concentration gradient towards the collector for majority carriers even in the uniformly doped base region. To prevent holes from flowing into the collector, there must be a countering electric field in the base region. The direction of this field is against the diffusion of the holes from the emitter side to the collector side of the base. This same field, however, will aid the flow of the injected minority carriers through the base and hence reduce the base transit time which is to be defined shortly. The quantitative discussion of this base electric field arising from the high-level injection will be delayed to the next section. The voltage dependence of the injected minority carriers at the base-emitter junction end of the

base quasi-neutral region directly determines the dependence of the collector current. For the forward-active region, the collector current is proportional to n(O), the injected electron concentration from the emitter. Hence if n(O) becomes proportional to exp(qVBE/2Vt), so does Ie.

We further investigate what happens in the collector region of the transistor under the high-level injection. Before we explain the physical nature of the effect, we need first describe the device structure under study. The impurity profile in the collector of the vertical transistor in integrated circuits usually consists of two parts - a lightly doped epilayer and a heavily doped buried layer, and they form a high-low (n+-n for npn transistor) junction in the collector region (see Appendix C for an example). The epi-Iayer is in a direct contact with the base region. When the base-collector junction is applied with a reverse bias, at low or moderate injection levels this bias is sustained by the voltage drop across the space-charge region located in the base-collector metallurgical junction. Due to the velocity saturation effect, when the electric field in the space-charge region exceeds a certain value (the critical field), carriers in this part of the space-charge region all have virtually the same velocity - the saturation velocity. Thus, for the finite collector current value there must be a finite density of electrons in the space-charge region. In other words, the depleted space-charge region is not really fully depleted. When the collector current further increases as a result of the increased forward base-emitter bias, the electron concentration in the epi-Iayer of the base-collector space-charge region will first reach and then exceed the background doping density because this is the most lightly doped part in the space-charge region. At the point where the mobile electron concentration in the space-charge epi-Iayer reaches the doping level, the potential barrier to the holes in the base region does not exist anymore and the holes flood into the once-depleted base side of the base-collector space-charge region. As a consequence, the space charge region in the collector starts to shift from the base-collector metallurgical junction to the high-low, buried - epi layer junction. The further increase of the collector current will bring more electrons into the epi-Iayer, eventually overwhelming the epi-Iayer doping. Hence, the polarity of the space-charge density in the space-charge epi-Iayer is reversed. The collector-base bias is now sustained by the voltage drop across the space-charge region in the high-low junction with epi-side of negative charge due to overcrowded electron population and the buriedlayer side of positive charge due to the depletion. We term the onset of

the high-level injection as the condition where the injected carrier concentration reaches the doping level at the epi-layer in the space-charge region. Specifically, we define the knee collector current, JK, as

(6.103)

where Ne is the doping concentration in the epi-layer, assumed constant, and Vsat is the saturation velocity. The reason JK is called the knee current is that starting around this current value, the rate of collector current increase with the forward base-emitter bias decreases, i.e., the In Ie vs. VEE curve starts to knee over or level off. Further increase of the collector current results in a higher electron concentration in the epi space-charge region and a reduction of the width of the space-charge region in the epi-layer. In other words, there appears a quasi-neutral region at the collector side of the base-collector junction in the epi-layer. This quasi-neutral region concatenates with the neutral base region, resulting in the so-called base widening effect. This base-width widening is one of the causes for the collector current to level off when the collector current exceeds the knee current. The other reason is the decrease of the rate of increase for the injected minority carrier concentration to the base from the emitter.

In summary, we should realize that the first consequence or sign of the high-level injection in the transistor is the shift of the space-charge region in the collector from the base-collector metallurgical junction to the epiburied layer, high-low junction or towards the collector contact if there is no epi-layer in the structure. This is followed by the basewidth widening which, combined with the exp(qV/2Vt) dependence of n(O), results in the collector current level-off.

There is yet another subtle phenomenon which is revealed from the numerical device simulation. When the neutral region appears on the epi-side of the base-collector junction, the original space-charge region at the base-collector metallurgical junction is not entirely gone. There actually is a residue of the electric field with the same polarity as that at the low or moderate injection. This is probably due to the spill over of the majority carriers (holes) in the base region to the epi-side of the junction. However, for a heterojunction device, if the bandgap in the base region is smaller than that in the collector region, and especially if the valence band edge forms a barrier for holes to spill over into the collector, such as in a Sit-xGex base (where x is the Ge mole fraction

(a)

rc

Figure 6.15: Typical npn transistor device geometry, (a) showing area factors and key extrinsic device resistors, (b) circuit model with parasitics shown.

end to the receiving end is defined as the time for an injected minority carrier to traverse the entire region. The minority carrier current, either due to drift or diffusion or both, can be considered as the excess minority carriers moving in the current direction with the same directional velocity. Thus,

where n' is the excess electron density and In represents the magnitude of the current density. The reason for using the excess rather than the total minority carrier concentration in defining directional velocity is that the equilibrium carrier concentration does not contribute to the

-6

-9'

Figure 6.16: Semi-logarithmic plot of Ie and IB versus VBE with base and collector resistances as parameters.

net current. Then

(6.105)

where WB is the width of the base quasi-neutral region. The expression for vn(x) can be found in Eq. (6.104), leading to

if In is assumed constant in the base region. It may be noticed that the integral is actually the stored excess minority carrier charge in the base region. In general, one can say that the transit time over a region equals the stored charge divided by the current carried by this charge. In the forward-active bias region and including the high-level injection in the base, In can be written as

Comparing to Eq. (6.23), we see two changes. One is the replacement of NA(x) with p(x) to accommodate the high-level injection. This is

actually a resumption of the original form as in Eq. (6.7). The other change is an approximation of the position-dependent diffusivity, Dn( x), as a constant value, Dn, which is the average of Dn( x) in the base region. In this case, the base transit time becomes

(6.108)

Take pure diffusion as an example, and assume constant diffusivity,

This is the usual form of the base transit time for a uniformly doped base region under the low-level injection (n( x) ~ p(x» condition. For a nonuniformly doped base there will be a built-in electric field at all injection levels due to the doping gradient. Even for a uniformly doped base, at high-level injection there arises an electric field, as analyzed in the previous section. Both cases add a drift component to the carrier transport in the base region and will therefore affect the base transit time. In the case of the built-in electric field, the direction of the field depends on the specific doping profile. But for a double-diffused (emitter and base) transistor, the overall effect of this built-in field is to aid the traverse of the injected minority carriers, hence reducing the base transit time. For the latter case, the direction of the field arising from the high-level injection is always favorable to the transport of the minority carriers across the base. The TB reduction can be as much as twofold at

286 CHAPTER 6. BIPOLAR TRANSISTORS

the high-level injection [6.4]. The significance of the base transit time is that it constitutes a major portion of the transit time for a carrier from the emitter to reach the collector. In fact, the transit time through the collector space-charge region is typically one tenth the base transit time because the carriers travel in this region with saturated velocity, which is around 107 cmls in silicon. In the next section we will relate the cutoff frequency to this transit time.

6.7.2Cutoff Frequency

The cutoff frequency, IT, of a transistor is defined as the signal frequency applied to it with which the value of the common-emitter, short-circuit current gain becomes unity, that is,

where TEO is the transit time from the emitter contact to the collector contact for minority carriers. Physically, TEO can be divided into five parts as injected carriers traverse from the emitter contact to the collector contact.

Those time constants on the right hand side of the equation correspond to, respectively, the emitter transit time, TE, emitter-base depletionlayer charging time, TEB, base transit time, TB, collector-base depletionlayer charging time, TBO, and the collector transit time, TO. For detailed discussion, readers are referred to [A. 7]-[6.7].

From the simulation point of view, iT can be evaluated in three ways - using frequency-dependent ae analysis to find the signal frequency at which the ac f3 becomes unity (Eqs. (6.112-6.113», to find TEO and

then using Eq. (6.114) to compute Jr, to evaluate fT based on the ac small-signal equivalent circuit (Figure 6.14). The first two methods are discussed in Appendices A and C, respectively, and we will consider here only the equivalent circuit approach. Based on Figure 6.14, if the collector is ac grounded in a common emitter configuration, one can evaluate the current gain as

simplification in order to obtain a concise formulation for fT. It is observed that the frequency independent part of the imaginary trem for the denominator is at least f3F times bigger than that for the numerator for CBC+CD is always larger than CBC. For the frequency at which the magnitude of ac f3 becomes unity the imaginary term in the numerator remains much smaller than one, so this term can be neglected for f ,...., fT. Thus the above equation is simplified to

which is a rather simple expression. Moreover, all parameters needed to calculate fT can readily be obtained from the device simulation. We now deliberate this observation. It is noticed that CD = CdF + CBE is the overall base-emitter junction capacitance including both the diffusion and depletion parts, and for the base-collector junction the diffusion part is any way negligible due to the reverse bias condition, so CBC can also be represented by the overall junction capacitance. All these junction capacitance can be obtained during the low-frequency ac analysis in the simulation. gm is nothing but the transconductance which is also available from the low-frequency analysis in the device simulation. All these means that mere low frequency analysis can provide enough information leading to the value of fT. In Appendix C, we will give

comparison in JT evaluation using Eq. (6.118) and low frequency analysis to the actual (frequency-dependent) ac analysis in finding the exact frequency where the ac current gain becomes unity. It turns out that the results from both methods is very close before HLI occurs. The computation effort, however, is much less using the former method.

6.8Application of Simulation Tools to Bipolar Device Design

The bipolar transistor, while conceptually straightforward in terms of base transport analysis as given by Eq. (6.23), exhibits a variety of technological dependences which require careful analysis. In this section the use of device simulators in modeling the technological complexities of the bipolar device is briefly discussed. Especially the general problem of parameter calibration for bipolar devices is discussed. This includes the adjustment of carrier physical parameters such as the carrier lifetime and bandgap narrowing. The general methodology of optimizing the simulation results to fit the experimental data is illustrated using SEDAN as an example.

Consider again the bipolar device structure shown in Figure 6.15 (a) Figure 6.17 shows the Gummel plot between VBE = 0.1 and 1.0 V with no extrinsic resistance. The solid curve represents a TO value of hole and electron lifetimes of 0.1 millisecond and the dashed curve corresponds to a value of 1DJ.t s. While there is little change in the collector current, the low-bias base current changes dramatically. This increased base current also implies a reduction in low-level current gain. Hence, it is important to have an accurate calibration of parameters such as lifetime before extensive design activities are leveraged based on simulation.

In order to calibrate the models to match the experiment, the schematic approach shown in Figure 6.18 is recommended [6.5]. Based on comparison of simulation and measurements in the low and medium bias range (see Figure 6.17), the carrier lifetime parameters are adjusted to match the base current. This could include doping dependence data (refer to the formulation for T in Figure 4.3 (b)) if available. Next, in the intermediate bias range the bandgap narrowing and ni terms are varied to match both IB and Ie. Finally, over the entire bias range other parameters such as Auger recombination rate are adjusted. Although the various parameters do interact, by starting from low bias and the