858 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-23, NO. 8, AUGUST 1976

Limitations on Injection Elfficiency in Power Devices

Abstract-In this paper the mechanisms of bandgap narrowing, Shockley-Read-Hall (SRH) recombination, Auger recombination, and carrier-carrier and carrier-lattice scattering are included in an exact one-dimensional model of a bipolar transistor. The trarl- sistor is used as a vehicle for studying the relative importance of each of these phenomena in determining emitter efficiency in dc- vices with emitter junction depths of 1 pm to 8 Wm. It is shown thtrt bandgap narrowing is the dominant influence for devices with shallow emitters of 2 pm or less and that SRH recombinatian dominates for emitter depths greater than 4 pm. Calculations are also presented showing the effects of the emitter surface concell- tration and high-level injection on the current gain for devices wif h emitter junction depths of 1 pm to 8 pm. It is shown that there is an optimum surface concentration of 5 X l O I 9 cm4 for the 1-pn emitter depth but no optimum under loz1 c m 4 for devices wlih emitter depths greater than 4 pm.

R I. INTRODUCTION

ECENTLY, there has been a great deal of discus- sion in the literature as to the effects that deter-

mine injection efficiency in bipolar transistors, rectifiers, and thyristors. In 1968, Whittier and Downing [l] proposed that the current gain is limited by Shockley-Read-Hz11 recombination (carrier lifetime) in the emitter. In a serii?s of papers by Mertens, De Man, and Van Overstraeten [:!I, [3] and separately by Mock [4], [5], it is shown that in heavily doped shallow junctions, bandgap narrowing e f - fects can drastically reduce the emitter efficiency in transistors. Recently, Sheng [6] has proposed that b a d - to-band Auger recombination can be an important effect in determining current gain, particularly in deep junctio:is where bandgap narrowing is less important. In the contest of an exact one-dimensional model, we have simulta- neously investigated the above three phenomena f x transistors with emitter depths from 1 Hm to 8 Mm to de- ter.mine the relative importance of each for devices of different size and under different injection levels.

In the first portion of the paper, there is a discussion of the factors that limit injection efficiency. In the next sec- tion, experimental and theoretical current gains are corn- pared for a power transistor to estabiish credibility for the model and to motivate the rest of the study. At this poht, a detailed study is presented showing the importance of the various physical mechanisms in determining the cu.r- rent gain in transistors of 1-, 2-, 4-, and 8-pm emitter depths. A study is then presented of the effects of varying the surface concentration on the current gain for each of the four devices noted above. Finally, there is a brief d .s- cussion of the changes that occur a t high levels of opera- tion.

Manuscript received December 23,1975; revised March 1,1976. M. S. Adler, B. A. Beatty, S. Krishna, and V. A. K. Temple are with .;he

M. L. Torreno is with Texas Instruments Inc., Dallas, TX 75222. CR & D, General Electric Corporation, Schenectady, NY 12301.

11. THEORETICAL BASIS A. Injection Efficiency

In a modern transistor, the base transport factor is very close to unity and as a result the current gain reflects the emitter injection efficiency. In this case, the current gain can be given by the following approximate expression for the case where the emitter is larger than several diffusion lengths [5], [7], [8]:

P

~ w b ~ d x

In the above equation', N e ( x ) and N b ( x ) are the impurity densities in the emitter and base, respectively, De and Db are the diffusion constants in emitter and base, respec- tively, W e is the emitter junction depth, W b is the base width, nTe is the correction factor to the intrinsic carrier concentration in the emitter due to bandgap narrowing effects (no such effects are presumed effective in the base), and Le is the diffusion length in the emitter. This equation is not actually used to calculate the current gains but is included as an aid in interpreting the effect of variations in the physical mechanisms that are discussed below.

In the above expression, the base current is proportional to the reciprocal of the numerator. This integral can be considered as an effective charge divided by an effective diffusion constant and is sometimes referred to as QE/DE. Analogously, the collector current is proportional to the reciprocal of the denominator which is of the form QBIDB. These ratios are sometimes referred to as Gummel num- bers and, somewhat simplistically, the various physical mechanisms can be thought of as influencing the emitter injection efficiency through their effect on these Gummel numbers. In the base region of most modern power tran- sistors these physical mechanisms are not of great impor- tance, and &/DE is equal to its nominal value as indicated by the denominator in (1). However, the same is not true of the emitter. In the following section each of the physical mechanisms will be discussed together with their effect on QE/DE as well as a brief description of the actual physics involved.

B. Physical Mechanisms Recombination in the emitter reduces the injection ef-

ficiency by reducing the value of the diffusion length and thereby reducing the effective size of the emitter. A physically more correct interpretation of this is that re- combination increases the hole gradient with a resulting increase in the diffusion of holes into the emitter. Two recombination mechanisms are considered, Shockley-

ADLER: INJECTION EFFICIENCY IN POWER DEVICES 859

Read--Hall (SRH) [9], [ lo] and Auger [6] recombination. In the former case, the effective lifetime increases with injection level with a rlesultant increase in the diffusion length, while with the Auger process, the effective lifetime will decrease. This distinction will be seen to be of signif- icance in the interpretation of the experimental results. An Auger coefficient of 3 X cm6/s was used.

Carrier scattering [ I l l , [I21 effects will also tend to re- duce the diffusion length through their effect on the dif- fusion constant. The same physical effects discussed above will be observed. However, there is an additional com- peting effect due to the 1/D, factor in (1) which will tend to increase the current gain. The reason for this is that, for constant hole gradient in the emitter, reducing the diffu- sion constant reduces the hole current. For most devices where the diffusion length is less than the emitter depth, this effect is more than offset by the accompanying in- crease in the hole gradient itself. The dual mechanisms of enhanced carrier scattering from ionized impurities and from other carriers are included in the analysis, but only the former is of significance for most of the injection levels considered (<lo0 A/cnn2).

In 'diffused junctions bandgap narrowing effects present an efffective field to both types of carriers due to the vari- ation of the bandgap with position [2]-[5]. This field is in a direction to carry both types of carriers into the emitter. I t can thus reduce injection efficiency by enhancing the flow of holes into the emitter and increasing the recombi- nation rate due to the larger number of injected carriers. At low levels of injection, the effect of bandgap narrowing can be modeled as a reduction in the effective impurity density as shown in (1) . The values for the bandgap nar- rowing used in this paper were taken from a paper by Van Overstraeten et al. [2].

C . Device Modeling The device modeling done in this paper was accom-

plished by a simultaneous exact solution of the one-di- mensional Poisson's equation, the electron and hole carrier transport equations, and the current continuity equations [14]. The method of solution is to make a finite difference approximation on a grid mesh and then iteratively solve the resultant equations using a technique based'on New- ton's method [15]. Any combination of the physical phe- nomena discussed above can be included in the solution. The theoretical current gains in the studies to follow were simply taken as the ratio of the calculated collector and base terminal currents. The gain was also calculated using the collector and emitter currents and by integrating the recombination to insure self-consistency.

111. EXPERIMENTAL POWER TRANSISTORS

D. Experimental Results In Fig. 1 a diagram of the doping profile for the 5-A,

600-V power transistor structures used in this study is shown. As can be seen, the emitter depth is approximately 5 wm (4.75) and the surface concentration is 7 X 1019 ~ m - ~ . The base width is 7.5 ,um and is diffused from a 2 X 1017 cm-3 source. The actual base concentration at the base-

D E P T H , MICRONS

Fig. 1. Doping profile for power transistor.

' - THEORY 1 -EXPERIMENT

0. I a2 0.3 0.4 06 0.8 I 2 3 4 6 8 1 0 IO

I, (AMPS)

Fig. 2. Current gain as a function of collector current IC for a transistor of the type in Fig. 1 with collector-emitter voltages of 4,8, and 12 V.

80 P c A

, * THEORY I -EXPERIMENT

0.2 0.3 0.4 06 0.8 I 2 3 4 5 6 8 I, (AMPS1

Io-- ~ i.-

Fig. 3. Current gain as a function of collector current IC for another transistor of the type in Fig. 1 with collector-emitter voltages of 4 V.

emitter junction is 6 X 10l6 cm-3. The n- collector has a width of 77 Km and a concentration of 2 X 1014 cmd3. The emitter area for the device is 0.12 cm2. In Figs. 2 and 3 the experimental and theoretical current gain is plotted as a function of collector current for two of the devices of the type in Fig. 1. In Fig. 2, the collector-emitter voltage is varied from 4 V to 12 V, and in Fig. 3, the collector-emitter voltage is held fixed at 4 V. The fall-off in current gain at high currents is a consequence of the collector-base junc- tion becoming forward biased and the device entering quasi-saturation, the so-called Kirk effect [16]. This fall-off was found to be due almost entirely to a decrease in the emitter injection efficiency caused by an increase in QBIDB rather than a decrease in base transport factor as is sometimes assumed [ 171.

The theoretical calculations were done using a doping impurity profile taken from the measured spreading re- sistance measurements. The lifetimes in the base and collector were measured using the open-circuit voltage

a60 IEEE TRANSACTIONS ON ELECTRON DEVICES, AUGUST 1976

decay technique and were taken to be 25 ps at a high level and 0.5 bs at low level. The lifetimes used in the emitter will be discussed below. As can be seen, the agreemmt between theory and experiment in Figs. 2 and 3 is quite good overall with the only significant disagreement IDC-

curring at the turnover point on the 12-V curve. This is probably due to current pinching effects under the emitter periphery causing the adjacent collector region to become forward biased at slightly lower collector currents. Thcse collector effects will not be further discussed and are only shown here to illustrate the success of the modeling over a wide range of device operating conditions. Subsequc nt discussion and studies presented will relate to the low- current side of quasi-saturation induced maxima. It should also be noted that good agreement was obtained for the dependencies of the collector and base currents with base-emitter voltage. This is a more severe test of the theory since this indicates that the modeling is correcAy predicting the values of QEIDE and QBIDB as well as their ratio.

On the basis of'the earlier discussions there are some observations that can be made about the dominant mechanisms limiting the current gain in these devices. As can be seen in Figs. 2 and 3, the current gain is rising as the collector current is increased. This is a characteristic tkat is present in all power transistors to varying degrees with the device in Fig. 3 being typical of the devices considered here. One of the basic contentions of this paper is that this increase of the current gain with injection level can only be explained on the basis of SRH recombination being the dominant mechanism. This issue is discussed below.

In the sibsence of recombination, bandgap narrowing leaves the current gain virtually unchanged with injectism level [5]. In the presence of recombination, bandgap n u - rowing can act to enhance the recombination, and it will be seen later that this is the main effect of bandgap nar- rowing in devices with emitter depths greater than 4 b*:n. Bandgap narrowing can also increase emitter side wid1 injection in devices where there is large compensation between the emitter and Ease diffusions [4]. This subject will also be discussed below. Auger recombination mani- fests itself by a decrease in current gain or at best leaves the current gain unchanged since the Auger recombination rate increases with injection level. High-level injection effects also tend to reduce the current gain [18]. On .$:le other hand, the effective SRH lifetime increases with in- jection level, as discussed above, and in essence is the only mechanism that can explain the observed rise in current gain with injection level. Other mechanisms such as bandgap narrowing and carrier scattering can affect the rate of recombination but alone cannot explain the ob- served data. A. Emitter Lifetime

These arguments were confirmed by the theoretical analysis. The calculations showed that it was necessary to adjust the values of lifetime in the emitter in order to correctly predict the dependence of the current gain wi;h injection level. A large ratio of high-level to low-level life-

time was necessary to achieve agreement with the value of low-level lifetime being 20 ns and the high-level lifetime being 200 ns. These lifetime values were assumed to be constant throughout the emitter since the correct depen- dence with impurity level was not known. In order to in- vestigate the effect that this assumption would have on the results to follow, an impurity concentration dependent lifetime was substituted for the constant lifetime in the emitter. A lifetime which varied inversely with the square root of the doping was chosen since this resulted in current gain values very close to those in Fig. 3 when the measured lifetimes in the collector (noted earlier) were used as a reference. As expected, the effect of this lifetime distri- bution is that the minority carriers penetrate further into the emitter than with the constant lifetime. However, the relative importance of bandgap narrowing and Auger re- combination with respect to SRH recombination increased by less than 5 percent. Since the main conclusions of this paper'are qualitative in nature, it is not likely that they will be significantly affected. As a check on the actual lifetime values, it was determined that a lifetime of 20 ns is con- sistent with measurements using the pulsed photocurrent technique which showed that the diffusion length in the emitter was less than 2 pm [20]. The calculations indicate that the average hole diffusion constant within two dif- fusion lengths of the junction is 2.4. This results is an es- timate of the average hole lifetime of 16.7 ns.

B. Two-Dimensional Effects An additional study was undertaken to ascertain the

importance of sidewall injection effects. Using the device in Fig. 3 as an example and an exact two-dimensional model of a type described above in one dimension, it was found that less than one percent of the emitter current is injected through the sidewall. This is not particularly surprising since there is very little compensation between the base and emitter diffusions and thus no enhancement of side wall injection due to bandgap narrowing. Since all of the cases to be studied in this paper share this charac- teristic, i t would seem justifiable to neglect side wall in- jection. Current crowding near the emitter periphery that was discussed above will have the effect of averaging the one-dimensional results over a range of current densities and should not affect the qualitative nature of the con- clusions.

Iv. STUDY OF PHYSICAL PHENOMENA

In the following study of the importance of the various physical phenomena discussed above, the device in Fig. 3 will serve as a norm. The emitter depth will be varied in order to study the change in importance of the physical phenomena for devices of different size. The values of lifetime will be the same as were indicated above for this device. In order to be able to clearly identify the effects of varying the phenomena under discussion, the base width and doping will remain the same as shown in Fig. 1. Unless noted otherwise, the total impurity charge in the emitter will be kept constant as the emitter depth is varied. The

ADLER: INJECTION EFFICIENCY IN POWER DEVICES 861

WITH EMITTER TRAPS

. AUGER AND BANDGAPNARROWING

NOMINAL CASE

t EMITTER SRH TRAPS REMOVED

REMOVED

OPEN SYMBOLS REFER TO REMOVING CARRIER IMPURITY SCATTERING

z 0

100

I I EMITTER JUNCTION DEPTH ( p )

Fig. 4. Effect of the physical mechanisms on the current gain. The open symbols refer to removing the carrier impurity scattering for the spe- cific case indicated by the symbol shape.

significance of doing the latter is that, in the absence of recombination, bandgap narrowing, and carrier-impurity scattering, the current gain should remain unchanged as the emitter depth is varied.

In Fig. 4 is shown the current gain as a function of emitter junction depth when various combinations of the above physical mechanisms are included in the calculation. In all cases, the collector current density is 8.2 A/cm2 which corresponds to the 1-A level for the devices in Figs. 2 and 3. The discussion will first center on the 4-pm emitter junction depth case since it is typical of the overall results. As can be seen, the nominal current gain is just below 100. This represents the value of current gain when all of the physical phenomena are included in the calculation. If the carrier-impurity scattering is now removed so that the diffusion constant is increased (also the diffusion length), the gain decreases. This is a result of the fact that the bandgap narrowing effects are made stronger since more carriers reach the bandgap narrowed area of the emitter. If the carrier-impurity scattering is restored and the bandgap narrowing removed, the current gain increases to 140. This shows that, while bandgap narrowing is af- fecting the emitter efficiency, it is not the dominant in- fluence. If the carrier-impurity scattering is now also re- moved, the current gain increases even further, just op- posite to the effect when bandgap narrowing was still present. The reason for this is that, with only recombina- tion mechanisms present, increasing the diffusion length increases the effective emitter charge. This causes an in- crease in current gain as discussed earlier. Finally, i t can be seen that removing Auger recombination along with any of the other mechanisms produces little further change.

The effects on the current gain up to now have been modest ones. However, if the lifetime in the emitter is

made the same as in the base and collector, indicated on the figure by removing the emitter SRH traps, the current gain increases to over 400. Thus, as indicated earlier, the SRH recombination is seen to be the dominant mechanism limiting the emitter efficiency in devices with emitter junction depths in the range of 5 pm. Looking to the 8-pm emitter case, it can be seen that SRH recombination is even more dominant.

A completely different situation is seen to exist for the device with the 1-pm emitter. The direction of the dependencies discussed above is the same but their relative size has changed. Removing the bandgap narrowing greatly increases the current gain indicating its dominance in devices this size. This is also seen from the fact that re- moving the impurity scattering when bandgap narrowing is present greatly reduces the current gain. Consistent with this, removing the SRH recombination has little effect in this device because of the dominance of bandgap narrow- ing.

The overall dependence of the current gain for the nominal case in Fig. 4 can now be explained. The initial rise in the current gain for devices with emitter depths greater than 1 pm is due to a weakening of the bandgap narrowing effects. The drop in current gain for emitter depths greater than 2 p m is due to a lowering of the effective emitter charge. This occurs because the net impurity density is increasingly lower in the active emitter region near the base junction; that is, within one diffusion length of the junction where there are significant minority carriers. An additional contributing factor is the fact that the diffusion constant in the emitter is increasing since the net impurity density is lower. It is interesting to note that since the current gain for the nominal case is not varying significantly with emitter depth one might conclude that most of the charge in the emitter is effective as discussed earlier. However, on the basis of the results in Fig. 4, it is seen that much less than the total emitter charge is effective and a transition occurred between the limiting mechanisms of bandgap narrowing and SRH recombination as the emitter depth was increased.

A slightly different perspective on these results can be obtained with the aid of Fig. 5. In this figure, the minority hole current is plotted as a function of the distance from the emitter-base junction for the 5-pm emitter depth de- vice shown in Fig. l . The collector current is again at 8.2 A/cm2. As is done in Fig. 4, several curves are shown for various combinations of the physical mechanisms. It can be seen that without bandgap narrowing the current falls off with distance much more rapidly than the nominal case. In both cases, very little of the current reaches the emitter contact so that the main effect of the bandgap narrowing is to increase the recombination rate. Similar studies done on devices with shallow emitters show that the roles of bandgap narrowing and recombination have reversed to the point that a significant fraction of the injected minority current reaches the emitter contact. The third curve in Fig. 5 shows the effect of removing the carrier-impurity scat- tering. As can be seen, the bandgap narrowing effects are

862

WITH EMITTER TRAPS

IEEE TRANSACTIONS ON ELECTRON DEVICES, AUGUST 1976

Fig. 5.

*" L

5 0 1 I I 1 0 7

Surface C o m t r a l i o n (cm")

1020 10 2'

Fig. 6. Effect of the emitter surface concentration on the current gain.

enhanced to the point where almost half of the injected current reaches the contact. The main effect of Auger j'e- combination is a fall-off of the current between the 3-itm point and the emitter contact without substantially d - fecting the value of the current entering the emitter from the base. One can see that most of the carriers have re- combined before reaching the 3-pm point.

v. CURRENT GAIN AS A FUNCTION OF SURFACE CONCENTRATION AND INJECTION LEVEL

It is now interesting to see the effect of varying the t c tal impurity charge in the emitter. In Fig. 6, the current g#3in is plotted as a function of emitter surface concentrat..on for each of the four device types considered in Figs. 4 2nd 5 . As can be seen, there is a distinct maximum irl the cur- rent gain a t -5 x 1019 ~ r n - ~ for the l-pm emitter depth device, a broad optimum for the 2-pm device at -1.5 X 1020 cmd3, and no optima for the 4- and 8-pm devices UF' to concentrations of 1O2I ~ m - ~ . These results are consistmt with the previous results showing that bandgap narrowing is only a major consideration for the 1- and 2-pm devices. I t should be noted that the results shown in Fig. 6 apply quantitatively only for the values of lifetime used in Ibis

4p EMITTER

i ''6.125 12.5 25 50 100 200 400

COLLECTOR CURRENT (A lcm21

Fig. 7. Effect of injection level on the current gain.

study. Doubling the lifetimes in the emitter region pro- duced a slight shift of the curves to lower surface concen- trations. For example, the 4-pm device actually showed an optimum current gain at 8 X 1020 ~ m - ~ .

The last subject to be discussed in this paper is the issue of the effect on the current gain caused by increasing the injection level. In Fig. 7 is shown the dependence of the current gain with collector current up to 400 A/cm2 for each of the four device types. The four devices again have equal emitter impurity charge as done originally in Fig. 4. As can be seen, there is little change in the current gain up to 200 A/cm2 with a fall-off to values -0.6 of the low-level values at 400 A/cm2. This fall-off is due primarily to conductivity modulation in the base region. A figure similar to Fig. 4 was made a t 400 A/cm2, but it is not shown since there are no significant changes from the dependencies shown in Fig. 4. Auger recombination effects are slightly more important but still do not play a major role in limiting the injection efficiency.

VI. CONCLUSION The basic conclusions of this paper are that Shockley-

Read-Hall recombination and bandgap narrowing effects are the dominant mechanisms limiting injection efficiency in transistors. For devices with shallow emitters of the order of 1 ym, the bandgap narrowing effects predominate, and for devices with deeper emitter of 4 pm and greater, the SRH recombination is the dominant mechanism. Auger recombination in the emitter does not appear to be an important mechanism at the low to moderate injection levels considered in this paper. The Auger process is either dominated by bandgap narrowing or SRH recombination, and in the absence of these last two mechanisms, the Auger process alone falls far short of predicting measured injec- tion efficiencies. An additional conclusion to note is that in devices where bandgap narrowing effects are important i t is equally important to properly include the effects of scattering from ionized impurities. The results are sensitive to the values for the diffusion constants in this case. Fi- nally, it should be emphasized that the quantitative as- pects of the conclusions reached above are based on de- tailed calculations using specific values for the parameters governing the physical mechanisms. This comment is

IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. ED-23, NO. 8, AUGUST 1976 863

particularly directed at the choice of the bandgap nar- [3J R. P. Mertens, H. J. DeMan, and R. J. Van Overstraeten, IEEE rowing parameters Of Van Overstraeten et L21 which [4] M. S. Mock, Solid-state Electronics, vol. 16, p. 1251,1973. diffe.r significantly frolm those of Mock [4] for donor den- 151 M. S. Mock, Solid-state Electronics, v ~ l . 17, p. 819, 1974. sities above 5 x 1019 cm-3. The specific choice, however, [6] W. W. Sheng, IEEE Trans. Electron Devices, vol. ED-22, p. 25,1975.

will only affect the results for the very small emitter depths pp, 267-70, 1g6g, [7] S. M. Sze, Physics of Semiconductor Deuices. New York: Wiley,

since recombination p.revents the minority carriers from [8] M. Tanenbaum and D. E. Thomas, Bell Syst. Tech. J., vol. 35, p.

reaching the portion of the emitter doped at this level for 191 R, N. ~ ~ 1 1 , phys, R ~ ~ , , vel, 87, p, 3g7, 1952. the deeper emitters ( 2 4 pm). The overall conclusions are [lo] W. Shockley and W. T. Reed, Phys. Reu., vol. 87, p. 835,1952.

shallow and deep emitters depending on the specific pa- [12] N. H. Fletcher, Proc. IRE, VOL 45, p. 862,1957. rameters chosen. [13j S. M. Sze, loc. cit., p. 99.

Trans. Electron Deuices, vol. ED-20, p. 772, 1973.

1, 1956.

believed to be valid but with the dividing line between 1111 D. M. Caughey and R. F. Thomas, Proc. LTEE (Lett.), vel. 55, P. 2192,1967.

1141 Ibid., pp. 66-67. 1151 0. Manck, H. H. Heimeier, and W. L. Engl, IEEE Trans. Electron

[161 C. T. Kirk, Jr., IRE Trans. Electron Deuices, vol. ED-9, p. 164,1962. Deuices, vol. ED-21, p. 403, 1974.

RE:FERENCES [17] R. J. Whittier and D. A. Tremere, IEEE Trans. Electron Deuices, vol. ED-13, p. 39, 1966.

[I] R. J. Whittier and 3. P. Downing, in IEEE IEDM, 1968, Paper 12.4. [I81 s. M. Sze, cit.3 P. 272. [2] R. J. Van Overstraeten, H. J. DeMan, and R. P. Mertens, IEEE L. W. Davies, proc. TEEE, vel. 51, P. 163791963,

Trans. Electron Deuzces, vol. ED-20, p. 290, 1973. [20] W. Zimmerman, Physicu, Status, Solidi ( a ) , vol. 10, K49, 1972.

Application of a Charge-Control Model to High-Voltage Power Transistors

PHILIP L. HOWER, MEMBER, IEEE

Abstract-Basic charge-control concepts are applied to the problem of predicting the static and large-signal switching characteristics of high-volltage transistors, with particular em- phasis placed on the quasi-,saturation region. Under the assump- tions of unity base transport factor and one-dimensional current flow, simple equations for device electrical characteristics are derived in terms of readily determined device parameters. A two-region model is develloped for predicting the turn-on pro- cess.

Measured turn-on waveforms and collector characteristics are compared with the calaulated behavior for a BVCEO = 400 V switching transistor. A comparison with h p ~ ( 1 ~ ) data is also given for different temperatures. In all cases, good agreement with the predictions of the model is obtained. Implications of the model with respect to device design and characterization are discussed.

LIST OF SYMBOLS

Di- Sym- men-

bo1 sions Description a ~ m - ~ Impurity gradient at the emitter-

base junction.

Manuscript received November 17,1975; revised March 24,1976. The author is with Research Laboratories, Westinghouse Electric

Corporation, Pittsburgh, PA 1.5235.

cm2 Emitter and base junction areas. V Common-emitter breakdown volt-

age in the sustaining region. cmz/s Low-level minority carrier diffusion

coefficient in the emitter and base.

cm2/s High- and low-level majority carrier diffusion coefficient in the collec- tor.

cm2/s D8lDco. eV Activation energy for &/DE. - Current gain IC/IB. - Peak current gain. A Dc collector and base currents. A Instantaneous collector' and base

currents for the two-region model. For WCIB = 0 , n = 1. For WCIB > 0, n = 2.

A Instantaneous collector current cor- responding to transition between the active and quasi-saturation regions.

- Reduction coefficient for collector transit time.

~ m - ~ Collector impurity concentration. cm-3 Impurity concentration in the met-