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 Of ILLINOIS 
 
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Digitized by the Internet Archive 
 
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 no. 149 DIGITAL COMPUTER LABORATORY 
 
 C ° P ' * UNIVERSITY OF ILLINOIS 
 
 URBANA, ILLINOIS 
 
 REPORT NO. 1^9 
 
 TRANSISTOR CURRENT --VOLTAGE RELATIONSHIPS 
 
 A consideration of high voltage dc operation 
 of junction transistors with special reference 
 to avalanche multiplication and related phenomena 
 
 by 
 
 L. Van Biljon 
 August 6, 1963 
 
 (This work was supported in part by the Office of Naval Research under 
 contract Nonr-l83M'l5 )• ) 
 
S'O 
 
 br 
 
 no. h9 
 
 Ce> p ■ J- 
 
 TABLE OF CONTENTS 
 
 Page 
 
 Introduction . . 
 
 2. Voltage Dependent Transistor Parameters 3 
 
 2.1 Influence of Operating Region . . 3 
 
 2.2 PN Junction Multiplication ............... 3 
 
 2.3 Multiplication in Practical Diodes . 6 
 
 2.4 Multiplication in Transistors ..... .... 7 
 
 3. Breakdown of Problem 13 
 
 3.1 Saturation Current . 13 
 
 3.2 Local Temperature Effects .... ..... 17 
 
 3.3 Long Time Constant Effects 19 
 
 3.4 Electric Fields Due to Current Flow 21 
 
 3.5 Base Width Modulation 27 
 
 3.6 General 31 
 
 4. Experimental Procedures 32 
 
 4.1 Preliminary Considerations 32 
 
 4.2 Measuring Circuit . 33 
 
 4.3 Measuring Results 34 
 
 4.3.1 Case of Constant p 4o 
 
 4.4 Influence of Resistivity Gradient 4l 
 
 5. Conclusion .................. ... 42 
 
 -11- 
 
SUMMARY 
 
 PN junction V-I curves in the higher-than-normal-voltage region are 
 discussed. The importance of effects other than avalanche multiplication, in 
 shaping these curves, is stressed. Experimental results of pulse measurements 
 on production type transistors are presented. It is concluded that the micro- 
 plasmic nature of the avalanche process and the apparent aporadic occurrance 
 of this breakdown, imparts an inherent "soft" characteristic to junction break- 
 down at reasonable current levels (a few micro-amp). If the formation of 
 microplasmas could be accurately controlled it would seem feasible to use true 
 avalanche breakdown in fractional nanosecond switching. 
 
 -111- 
 
1. Introduction 
 
 In Fig. 1 is shown a family of I - V curves, with I_ as parameter. 
 
 Conventional operation of transistors is limited to the region where 
 the equation 
 
 J C = aI E + J Q0 
 
 (1) 
 
 is approximately true, i.e., V < V in Fig. 1. 
 
 o 
 
 
 
 
 
 
 1 
 1 
 1 
 
 
 
 
 hk 
 
 
 
 v 
 
 
 
 
 i 
 
 E3 
 
 
 
 
 
 
 
 
 
 
 
 
 Z E2 
 
 
 
 i/ 
 
 
 
 
 
 
 
 
 
 Z E1 
 
 
 
 
 
 
 
 
 
 
 i 
 
 V 
 
 CB- 
 
 V. 
 
 M 
 
 Fig-ore 1. Conventional I - V__, Curves for Junction Transistors 
 
 .In equation (l), a is usually defined as 
 
 3l c 
 
 AV 
 
 CB 
 
 0, V__. small 
 
 L.J3 
 
 while I is the current in the base -collector circuit when I„ 
 l/J E 
 
 is relatively small. 
 
 = 0, and again V 
 
 CB 
 
 -1- 
 
These rather loose definitions of the quantities in eq. (l) are 
 perfectly in order for many conventional transistor applications. However, as will 
 be shown, more stringent requirements apply when defining these quantities for 
 higher voltage operation. 
 
 At larger V„._, it is seen in Fig. 1 how the curves indicate a departure 
 L/C 
 
 from low voltage conditions. Viewed externally, the collector current, and to some 
 lesser extent the emitter and/or base current start increasing sharply with voltage, 
 depending on the configuration of the external circuit. 
 
 \3v-J' 
 
 This sharp increase in\ ^7— j, commonly referred to as a 'breakdown' 
 
 CB' 
 
 1-30 
 
 phenomena, has teen thoroughly investigated for P-N junction diodes by many authors, 
 
 with fairly good agreement between theory and experiment. Depending on the relative 
 
 perfection of the diodes in question the theoretical treatment also varies in the 
 
 3^ 
 
 success of describing the ruling conditions. 
 
 For transistors, many studies of this region of operation have been 
 
 18 2° 37 
 made f ' but due to a larger number of complicating factors, the success of these 
 
 investigations has teen rather less than in the case of diodes. 
 
 Investigation of this special region of operation is normally prompted 
 by some especially useful properties. The significance of the breakdown or 'avalanche' 
 effect becomes apparent when it is realized that in a self-supporting breakdown, 
 currents may increase a thousand fold, or more, in a fraction of a nanosecond, 
 pointing the way to obvious importance in pulse circuits; a theoretical risetime 
 limit of 10 x ' ' seconds for this process has been predicted. Apart from actual 
 breakdown, the multiplication of carriers at below breakdown voltages may also be 
 of importance when current loss in some stage of a network is to be made up; the 
 ease with which an alpha of unity may be obtained illustrates this point. 
 
 Furthermore, since the breakdown process depends upon ionization phenomena, 
 a P-N function, stressed electrically to the verge of its breakdown field strength, 
 
 may augment effects due t;o an outside ionizing agent like particles from some 
 
 ,. • 22,36 
 
 nuclear fission process. J 
 
 -2- 
 
This report deals with some aspects of this breakdown phenomenon as 
 observed during an investigation of transistor operation under higher than normal 
 voltage conditions. 
 
 2. Voltage Dependent Transistor Parameters 
 
 2.1 Influence of Operating Region 
 
 Most transistor equivalent circuit parameters are both voltage and 
 current dependent. From the dc point of view though, the most important single 
 factor is alpha, and then to some lesser extent, also I m . 
 
 When considering either the voltage or the current influence on transistor 
 
 parameters, very careful notice has to be taken of the carrier levels in the various 
 
 19 
 regions. Rittner has shown that currents up to 25 amperes may be accommodated 
 
 in the emitter and collector regions of low frequency, medium power transistors; 
 
 for the higher resistivity base however, he pointed out that carrier level effects 
 
 could only be neglected while the total current was still of the order of micro -amps, 
 
 20 21 kl 
 
 Fletcher, Webster and Misawa have, amongst others, treated current variation 
 
 of alpha very fully and when investigating voltage influences at normal operating 
 currents, effects due to carrier level variation should not be lost sight of. 
 
 In this preliminary investigation, experimental observations were limited 
 to the micro-amp region so as to minimize carrier level effects. 
 
 Of the various voltage -dependent processes in a transistor, avalanche 
 multiplication is certainly one of the most important. This phenomena has been 
 thoroughly investigated by several workers, notably Wolff, Shockley, McKay, ' 
 McAfee, 23 Miller, and Baraff. 35 
 
 2.2 P-N Junction Multiplication 
 
 The electric field in the base-collector depletion region of a transistor 
 accellerates motile charges which enter this region. Should the field be strong 
 enough, the accelleration may be sufficient to cause ionizing collisions between the 
 mobile carriers and lattice ions in the depletion region. 
 
 -3- 
 
By analogy with the Townsend gaseous "breakdown, a voltage dependent multiplication 
 factor, M, was originally defined by McKay and McAfee as 
 
 (1 - h = a. .w 
 
 M i 
 
 where a. = rate of ionization, i.e., number of hole -electron pairs produced by 
 one carrier per cm. travel in the direction of the field. 
 
 W = effective width of depletion region. 
 
 McKay at a later stage published more data on the above relationship writing it 
 more generally as: 
 
 i r w 
 
 which incorporates the idea that an "effective rate of ionization" may be a compli- 
 cated concept in a P-N junction. 
 
 I Depletion 
 Region 
 
 carriers 
 
 b-^^-l 
 
 Figure 2. Definition of M 
 
The important assumptions made in the above analysis were: 
 
 1. a. is a function of the field, E , only. 
 
 2. No carrier recombination or significant interaction takes 
 place in the depletion region. 
 
 The I-V relation of a junction will be essentially dependent upon the 
 
 law which connects the rate of ionization with the electric field and hence with the 
 
 applied voltage. 
 
 17 
 Wolff originally assumed that the energy loss per collision would be 
 
 much less than the energy gain between collisions for fields producing appreciable 
 
 multiplication. He postulated a quasi -Maxwellian energy distribution for the hot 
 
 electrons producing multiplication and arrived at an ionization coefficient, a. , 
 
 depending on the field, E , according to: 
 
 _ B 
 
 E £ 
 a.(E) = A • e (2) 
 
 E 2 
 
 3^ 
 A different point of departure was assumed by Shockley who stated that only 
 
 those carriers who do not suffer randomizing collisions, but are subject to fairly 
 
 uninterupted accelleration in the direction of the field, will produce ionization. 
 
 His relation between ionization rate and electric field may be written as: 
 
 -D 
 E 
 a (E) = G • e ~ (3) 
 
 Some controversy arose here since very careful experiments provided proof in support 
 of both the above laws, depending upon the sample under investigation. 
 
 -5- 
 
35 
 Recent work by Baraff seems to clear up most of the apparent discrepancies 
 
 observed. He proves that at low fields, ionization most probably takes place as 
 
 3^ 
 analyzed by Shockley where sustained acceleration only will produce pair 
 
 production. 
 
 At high fields, more random motion obtains as initially assumed by 
 
 17 
 Wolff and the process tends to follow the law proposed by him. At intermediate 
 
 fields a combination of these effects will be present, considerably complicating 
 
 the analysis. 
 
 The case to apply to an individual transistor will depend on such 
 parameters as resistivities, material, shape of junction, etc.; this latter data 
 can only be determined from a very complete knowledge of both electrical and 
 geometrical characteristics . 
 
 2.3 Multiplication in Practical Diodes 
 
 The high field strength ionization process mentioned above is rarely, if 
 
 ever, observed alone in practical P-N junctions . Technological limitations impose 
 
 rather severe restrictions on the degree of perfectness attainable in diode production. 
 
 o 
 However, Miller has been successful in obtaining data on junction diodes 
 
 providing some understanding of the multiplication in these units. 
 
 He found, amongst other things, that the previously mentioned multiplication 
 factor, M, may be expressed in terms of the applied voltage V, and the ' ultimate 
 breakdown voltage' of the junction, V , : 
 
 "<"> ' — T« <*) 
 
 ft) 
 
 where n is some constant depending upon the material and the resistivities, 
 
 
 -6- 
 
This empirical relation thus provided a link between the electric field 
 
 produced by the applied voltage, and the multiplication due to the ionization 
 
 35 
 process. The results of Baraf f ' s work should stress the point that this empirical 
 
 relation at most, is an approximate expression. 
 
 Apart from the depletion layer multiplication mentioned above, it was 
 
 found that most junctions exhibit a current ' leakage, ' probably along the free 
 
 2 
 surfaces of the structure. Garrett and Brattain proposed a theory for this 
 
 surface effect and found, as Miller had, that this leakage current most probably 
 
 also was multiplicative, although not necessarily obeying the empirical expression 
 
 of eq. (k) . The analytic description of diode I-V relations was thus found to 
 
 be formidable, and even as yet in many instances, not possible. 
 
 2.k Multiplication in Transistors 
 
 After publication of his work on multiplication in diodes, Miller applied 
 the idea of this kind of multiplication to transistors where multiplicative break- 
 downs had been observed. The basic equation 
 
 Z C = aI E + ^0 
 
 was modified by Miller so as to include high voltage effects and was written as: 
 
 I (V) = a MI E + MI CQ0 (5) 
 
 where 
 
 a. = a at very low voltage 
 
 I C00 = I G0 at very loW voltage 
 
 -7- 
 
The applicability of the empirical relation of eq. (k) to transistor phenomena 
 
 was tested by comparing the base -collector breakdown voltage, V_, with the emitter- 
 
 B 
 
 collector breakdown voltage, V (base open) . 
 
 In the latter case a breakdown will obviously be initiated when the 
 loop gain becomes unity and one may write: 
 
 When Ig = 0: I C (V) = I E = (QM) • Ig 
 
 i.e., aM = 1 (6) 
 
 Good agreement between the behavior of the transistors tested by him, and the 
 
 o 
 
 proposed model for the breakdown process was reported by Miller. It was stated by 
 him that this method of determining M was used in order to "minimize purely 
 transistor effects." Considering however that interpretation of the data hinges on 
 eq. (6), transistor effects influencing 0C n must have been included. 
 
 By determining M, and n, in this way effects like base width modulation, 
 sweeping fields due to ohmic voltdrops in the base and leakage currents are con- 
 sidered to be negligible; this most probably is not the case for production-type 
 transistors, as will be shown. 
 
 It is thought that 'M' in eq. (k) should probably be written as 
 'aM' and the factor as a whole be considered. Recent publications on transistor 
 and diode multiplication stresses the fact that several different phenomena, and 
 not only avalanche impact ionization determine high voltage operation- -this will 
 again be referred to in par. k.3> 
 
 After publication of Miller's paper on determining M, Miller and Ebers 
 
 22 
 
 jointly published a paper on avalanche transistors, making use of earlier data 
 
 of M, as found by Miller. Equation (5) was used to incorporate voltage influences 
 on transistor currents. (Reference 22, eq. (7)0 
 
In this paper, experimental data was presented in the form of I - V 
 
 curves with I as parameter- -the curves presented for a PNP-transistor are re- 
 E 
 
 produced in Fig. 3 "below. 
 
 = OmA 
 
 16 20 2U 28 32 
 
 Collector Voltage V (volts' 
 
 Figure 3. I - V Curves for PNP -Trans is tor (Ebers and Miller, Ref. 22) 
 
 2k 25 
 
 As was pointed out by Heymann and later reported by Heymann and van Biljon ' 
 
 the curves of Fig. 3 deviate rather considerably from a set to be described by eq. (S). 
 
 Using this equation, Fig. *]- shows how the curves in Fig. 3 require M 
 and (aM) to vary with voltage, considering the L, = and L = 1 ma curves. (The 
 low voltage value of I is unfortunately not indicated in the set of curves 
 presented by Ebers and Miller, and a likely value of 5 microamp was assumed; however, 
 a value of even 20 microampere would not significantly change the discrepancy 
 observed in Fig. 4.) 
 
 .9. 
 
t 
 
 M(I E ) 
 
 12 
 
 10 
 
 1200 
 1100 
 1000 
 900 
 
 8< v. < 
 
 h'/co 
 
 6oo 
 
 r '.v )i . 
 i+co 
 
 - 00 
 I '0 
 
 L 
 
 M <W 
 
 16 20 2k 28 3? 
 
 Figure k. Multiplication Factors as Determined from Figure 3 
 
 Representative figures from the above curves show that M(l ) has a value of 5 at 
 V = 26 volts, while M(l ) is 3; at V = 28 volts, M(l ) has increased to 300, 
 
 O ill v/ L/>-/ 
 
 while M(l ) has only doubled to 6. Closer inspection of Fig. 3 also reveals that 
 
 according to these curves M(l_) would vary with I_ as shown in Fig. 5« 
 
 hi 
 
 In connection with Fig. h it must be pointed out that Miller had stressed 
 the point that due to complicating effects, M should nbt be determined from the 
 relation 
 
 I coo 
 
 as was done here. However, if eq. (5) is to apply, the M-f actors of Fig. h which 
 bear no relation whatsoever to each other, suggest that in conventional transistors, 
 the complicating effects may be more important than a cursory investigation would 
 indicate. in 
 
t 
 
 H 
 
 i — S — fe — * — I) 
 
 Ij(mA)' 
 
 Figure 5- M as a Function of I_, as in Figure 3 
 
 It is not clear why M(lp) should vary as shown above, with voltage, and 
 no indication is given in the text of Ref. 22. A probably explanation is that 
 thermal influences may have been present, thus yielding higher apparent multiplication 
 as the voltage V ( and thus I ) rises. For this to have been an adequate explanation 
 the slope of the curves in Fig. 5 should have been positively increasing with I_, 
 instead of decreasing as shown. 
 
 It is well known that practical transistors break down in widely different 
 
 ways, and although the theory may at times be a fair approximation to an idealized 
 
 3^ 
 structure, practical transistors require a more complete approach. 
 
This point is well illustrated in Fig. 6, where oscillographs of the 
 
 I - V-t-. ( I„=0 ) breakdown curves for three different alloyed junction transistors 
 C CB ill 
 
 are shown . 
 
 Steepest Curve - Type 2N58l 
 Middle Curve - Type 2N382 
 Bottom Curve - Type 2N^C4 
 
 Verticle Scale -•-: 2 imicroamp/div. 
 Horizontal Scale - 5 volts/div. 
 
 Temperature 
 
 - 313 K. 
 
 Figure 6. Different Types of Breakdown Curves --Alloyed Junction Transistors 
 
 Detailed information on some experimental procedures for obtaining 
 breakdown data will be given at a later stage but it is well to keep in mind, that 
 due to several factors to be listed, reliable data of this nature is difficult to 
 obtain from ordinary transistors. 
 
 -12- 
 
3. Breakdown of Problem 
 
 Having established that the phenomena observed are either in need of 
 further explanation, or not reliably represented by the experimental data presented, 
 it is appropriate at this stage to mention some of the probable sources of error 
 in the simplified approach to multiplication processes in transistors. 
 
 3.1 Saturation Current 
 
 Ik 
 
 Sah, Noyce and Shockley have shown that it is impossible for a PN- 
 
 junction 'saturation' current to saturate with voltage. 
 
 Taking account of the thermal pair formation within the confines of the 
 depletion region they have shown how, particularly in Si, and to a lesser extent 
 in Ge, the saturation current is bound to increase with voltage. 
 
 If depletion layer recombination may be neglected due to the strong 
 electric field present, the saturation current originating within the depletion 
 region may be comparable to the conventional body and surface contributions; 
 keeping in mind the depletion width variation with voltage this effect may be 
 important when M becomes appreciable. 
 
 Consider the saturation current due to pair formation in the region 
 adjoining a junction and within one diffusion length from it. For this current 
 density, one may write taking account of both sides of the junction: 
 
 J = -2q • D • -r— (carrier level) 
 
 -2* £ ( 7 ) 
 
 C 
 
 where for the most general case: 
 
 n = carrier level on either side of junction 
 L = diffusion length 
 c = lifetime of carrier in depletion region. 
 
 -13- 
 
The contribution from the space charge region itself is determined by the rate 
 of pair formation within this region, modified by the appropriate life time 
 considerations. Making use of an idealized model and restricting attention to low 
 current levels, it has been shown that the contribution from this source may be 
 expressed by: 
 
 J 2 = - 4 • n. 4- (8) 
 
 2c 
 
 where 
 
 n. 5= electron or hole concentration in intrinsic material 
 W = width of depletion region. 
 
 The ratio of (7) and (8) now indicates the relative importance of the two main 
 sources of saturation current, viz: 
 
 'Depletion layer' current _ n t W # » 
 
 'Body' current " 5n. L 
 
 This ratio may vary from 0.1 for Ge to several thousand for Si structures and 
 noting that for a step junction for instance: 
 
 Wa n/V 
 
 the futility of defining a current like I mn becomes apparent. Equally, the term 
 'low voltage' loses much of its significance as there is no voltage below which I_ n 
 may be regarded as constant. 
 
 •Ik- 
 
Typical data for a Ge transistor will serve to illustrate this point: 
 
 n. = 2.k x 10 13 / cm 3 . W = «/pV x 10 cm. 
 
 n = 1.6 x 10 15 / cm 3 . L = 10~ 3 cm. 
 
 n ' 
 
 p = 1 ohm cm. 
 
 The ratio of eq. (9) tus yields: 
 
 Ratio = — -^ 4v x 10 
 
 n. L 
 
 l 
 
 - 0.24 ^V 
 
 If V = 1 volt, Ratio =0.2^. A saturation current of 1 microamp would thus have 
 
 0.20 microamp depletion layer contribution. If V = 10 volts, the ratio is O.76 
 
 and a much larger proportion than "before of the total saturation current would originate 
 
 in the depletion region. 
 
 Ik 
 
 It has been shown that this space charge contribution was subject to 
 
 the same multiplication as that charge coming from the base region and thus helps 
 determine the shape of M(v) for diodes and transistors, if it is endeavored to find 
 M from saturation current data. The above indicates that this procedure is highly 
 unreliable since excessive values of M will be indicated as is well illustrated in 
 Fig. h. It would seem that the more elegant methods used by Miller, McKay and 
 others are the only possible ways for obtaining information of this nature. 
 
 Apart from the saturation currents mentioned above, there also is a 
 
 2 
 
 contribution from surface currents. Garrett and Brat tain have observed that surface 
 
 currents also are subject to multiplication, as was also concluded by Miller; 
 quantitive analysis of surface effects in transistors is practically impossible at 
 the present stage. 
 
 ■15- 
 
The separation of true '"body 1 saturation current from surfa.ce leakage 
 current is a formidable problem and believed to be as yet unresolved. Recent detailed 
 measurements by Kuper - r were made in order to separate ..these components, ,but it 
 is believed that these results are not conclusive. They -are based on the assumption 
 that, in the circuit of" Fig„ J 9 I consists solely of "surface, space charge re- 
 combination and internal field emission current" due to the reverse bias V-^. 
 
 PNP 
 
 Figure 7» Circuit Used by Kuper 
 
 3^ 
 
 Measurements using this circuit have, however, shown that no matter how large V , 
 it never succeeds in ^drawing all 'body' currents to the emitter and contribution to 
 I from this source always remains. A typical curve is shown in Fig. 8. 
 
 O 
 
 H 
 
 w 
 
 M 
 
 I 
 
 V. 
 
 EB 
 Figure 8. Division of Saturation Current Between Emitter and Collector 
 
 -16- 
 
3.2 Local Temperature Effects 
 
 Since avalanche phenomena often are associated with higher than normal 
 
 voltages in transistors and not uncommonly with above normal currents, thermal 
 
 effects may seriously hamper experimental work. 
 
 26 
 Mortenson has considered the influence of dissipation on transistor 
 
 junction temperature under conditions of pulsed excitation and his results may be 
 
 significant since most curve tracing of transistor phenomena is done by repetative 
 
 signals, dc procedures would of course, thermally -wise, present the worse possible 
 
 case. 
 
 It was found by Mortenson that short time junction temperatures could 
 be 200$ higher than the average temperature even under relatively low voltage 
 pulsed operation in the milli -ampere current range. Noting that to a rough ap- 
 proximation the gross I of a Ge transistor doubles for every 10 K temperature 
 rise, this effect may present serious difficulty in determining actual isothermal 
 conditions inside a transistor; Fig. 9 indicates the order of magnitude of 
 environmental temperature influence. 
 
 Thermal runaway following on avalanche breakdown is of course a common 
 phenomenon which often leads to the destruction of a transistor. By severely cooling 
 the transistor however, junctions seem to recover even from excessive breakdowns, 
 the recovery time being of the order of seconds. 
 
 50 r- 
 
 V^C volts) 
 
 Figure 9« Influence of Temperature on Saturation Current 
 
 -17- 
 
The following results will, illustrate this point: 
 
 A 2N^-0U transistor was found to tend towards breakdown, under pulsed 
 conditions, at V = kO volts, when at 313 K. By cooling to 213 K, 60 volts 
 dc or more could be applied and after the initial severe breakdown, the junction 
 recovered within about one second. Although subject to subsequent sporadic break- 
 downs, the average current remained at the low level of a few micro -amperes. The 
 
 higher the applied voltage the more frequent the sporadic breakdowns but also the 
 
 c 
 smaller the current peaks, as had also been observed for diodes by McKay; a typical 
 
 curve is shown in Fig. 10. 
 
 In order to limit junction dissipation during experiments, an intermittent 
 type of dc measurement could be resorted to. However, due to long time constant 
 effects in junction V-I characteristics, serious difficulties may arise here, as 
 mentioned below in par. 3.3- The importance of thermal effects may however not be 
 overlooked; the curves of Ebers and Miller contain dissipations of up to 300 mw 
 instantaneous; whether this was a dc or pulsed measurement is however not known, but 
 this level of dissipation very likely did influence their results. 
 
 V_ = 60 volts dc 
 GJ3 
 
 Vertical scale 
 Horizontal scale 
 
 h A/div. 
 1 sec/div. 
 
 Figure 10. Recovery of Base -Collector Junction When at 213 K 
 
 -18- 
 
3.3 Long Time Constant Effects 
 
 It is common knowledge that when a voltage is suddenly applied to a PN- 
 j unction, the initial current surge (apart from a capacitive one) seems to indicate 
 a '"breakdown' voltage considerably lower than when a dc voltage is applied by 
 starting at zero and steadily increasing it. 
 
 Some indication of the differences observed are shown in the following 
 
 oscillographs of base -collector behavior (emitter open) of a 2N650 transistor. 
 
 Figure 1.1(a) shows three cases for the same transistor; the highest curve results 
 
 from single shot V,_ excitation- -a single 100 microsecond pulse gave each point 
 
 OB 
 
 of the curve; one minute intervals were allowed between each point of the trace. 
 
 The middle curve represents 100 microsecond pulses applied every 5 
 milliseconds, with 10 seconds being allowed for ' settling' at each value of voltage; 
 this means that in contrast to the single shot, the dots of the middle curve 
 represent the I-V relation after at least 2000 pulses. The lowest curve represents 
 the dc case with 30 second ' settling' time being allowed at each voltage, before 
 the reading was taken. 
 
 It is felt that if multiplication effects are to be used in fast circuits, 
 data like that shown above is often more relevant than dc curves as the obvious 
 discrepancies illustrated will certainly affect circuit design. 
 
 A further illustration of the long time constant effect is given in 
 Fig. ll(b). The three closely spaced pulses are spaced 1 second in time after 
 the transistor had been allowed to 'rest' for 15 minutes. The trace fourth from 
 the bottom shows the equilibrium reached after long excitation at a frequency of 
 2 c/s; the top curve shows the shape of the applied V_. 
 
 The actual recovery of the base -collector junction used in the photographs 
 of Fig. II is shown in Fig. 12. Noting that the horizontal time scale is 1 sec/cm, 
 the order of time involved is seen to be considerable. 
 
 -19- 
 

 
 
 
 .j _ — — — — — — — 
 
 i 
 
 Tve- — ^m ?VT~~r~*'. — " " 
 
 .*■■ 
 
 (a) 
 
 Horizontal scale- 5 volts/cm 
 Vertical scale -k microamp/cm 
 
 CD) 
 
 Horizontal scale -2 millisec/cm 
 Vertical scale -1 microamp/cm 
 
 Figure 11. Long Time Constant Effects 
 
 -20- 
 
Horizontal scale - 1 sec/cm 
 Vertical scale - 15 microamp/cm 
 
 = k-0 volts dc 
 
 CB 
 
 Figure 12. Recovery of Junction- -Type 2N650 
 
 3.^ Electric Fields Due to Current Flow 
 
 The shape of the curves in Fig. 6 makes it obvious that apart from M, 
 
 other voltage dependent factors are indeed of importance as already stated in the 
 
 former sections. 
 
 8 15 
 
 It was mentioned by Miller and analyzed by Schenkel and Statz and 
 
 12 
 still later by Pell that currents flowing through the region adjoining a PN- junction 
 
 cause ohmic voltdrops representing an E-field such as to aid the flow of carriers 
 
 (see Fig. 13). 
 
 -21- 
 
Base 
 
 Collector 
 
 Electron drift current 
 
 Electron concentration "^ 
 gradient 
 
 Hole concentration ~"\ 
 gradient 
 
 Hole diffusion 
 current 
 
 Depletion 
 Region 
 
 Figure 13. Electric Field Near PN Step Junction 
 
 It is conceivable that in the presence of multiplication this field 
 aided effect could initiate an increase in junction current and thus cause external 
 measurements to he interpreted as a faster increase in M with voltage than is actually- 
 present. This tendency is reflected in base-collector V-I relationships in transistors, 
 compared with data of M alone, obtained by alpha bombardment and photon excitation 
 methods ' in diodes. 
 
 3.^-.l Preliminary Considerations 
 
 In a first order analysis, effects due to large current levels may be 
 neglected; since it is possible to adequately observe the breakdown phenomena in 
 
 question at currents of 5 or 10 microamps, (representing current densities much 
 
 19 N 
 
 below the limits set by Rittner ) this is a legitimate assumption. 
 
 -22- 
 
Starting with the two basic transport equations: 
 
 ^P 
 
 J = -qD • 3*; + qp [I E (10) 
 
 p * p Ox ^n^p~ v y 
 
 J = qD r§ + qn |i E (ll) 
 
 n n ox * n n~ 
 
 the field E, may be expressed in terms of the total current, (j + J ), as 
 
 (j +J ) + qD p. _ qD || 
 
 E - n P V^x n ^ (12) 
 
 *> qu p + qu n v ' 
 
 p n n n 
 
 Some speculation about the origin of the field, E, will be in order at this stage. 
 
 It is generally accepted that a field E is present to preclude diffusion 
 of electrons from emitter to collector even though there is an electron concentration 
 gradient suitably oriented; this gradient is set up by the hole concentration 
 gradient and the tendency towards space charge neutrality in the base region. This 
 field is known to aid the transport of holes and to effectively increase the diffusion 
 constant . 
 
 When multiplication is present in the base -collector depletion region, 
 the electrons produced here enter the base and have to be removed against the existing 
 concentration gradient, next to the base -collector depletion region. Since diffusive 
 flow is opposed by the gradient, this current will have to be a drift current and an 
 extra field component, aiding the field of the former paragraph, will be established. 
 
 The field, E, in eq. (12) will thus be caused by two complementary 
 processes as outlined. 
 
 ■23- 
 
If a multiplication of M is present in the depletion region, and a hole 
 
 current J enters from the base, M * J will leave so that the electron current 
 
 P P 
 
 density produced will be J , where 
 
 J = (M-l) • J 
 n p 
 
 If the resistivity of the base region is taken to be P, the field set up by this 
 current flow will be: 
 
 E = (M-l) • J • P (13) 
 
 r<j P 
 
 By replacing E in eq. (12) by (13) above, the hole current density in terms of 
 M may be found to be: 
 
 * P it k- ¥) 
 
 j = J (!4) 
 
 P (M-l) n [qp fl+«£V ^- N ,1 " M 
 v y p ^pL^n \ up J * up dJ 
 
 Equation (l^-) contains the simplifying assumptions that approximate space charge 
 neutrality obtains in the base and that the hole and electron concentration gradients 
 are just about equal, i.e.: 
 
 n = p + N, 
 n . n d 
 
 and 
 
 If):© 
 
 -2k- 
 
It is seen in eq. (l4) that when M tends to unity, no drift field will he present 
 and the total current will be the sum of two diffusion components, viz: 
 
 M 
 
 •J = -«) p + qD |S ( 15 ) 
 
 v v ox n ox x ' 
 
 Only the field due to the electron current is thus taken account of here, 
 and not that originally present before multiplication set in. 
 
 In eq. (l^) the expression 
 
 (< 
 
 '».-jS 
 
 represents the hole saturation current J which would flow under quiescent 
 conditions so that the rest of this expression represents the influence of 
 multiplication via M. 
 
 By making use of the appropriate constants listed below, eq. (l^-) 
 can be approximated to eq. (l6): 
 
 -1Q 2 
 
 q = 1.6 x 10 y coul. D = 100 cm /sec. 
 
 n ' 
 
 2 2 
 
 l_i = 1900 cm /volt sec. D = ^9 cm /sec. 
 
 H n = 3900 cm 2 /volt sec. N = 1.5 x 10 15 /cm 3 
 
 11 3 
 p = 1 ohm cm. p = 3.6 x 10 /cm 
 
 n ' 
 
 Hence : 
 
 J p = F(M) ' J g (16) 
 
 -25- 
 
F(M) as a function of M, is shown graphically in Fig. Ik below. 
 
 Figure ik. F(M) vs. M 
 
 The significance of F(M) is as follows: 
 
 If M = 1, and say J = 0.1 amp/cm , a certainly) will exist. 
 
 and hence: 
 
 If M = 5, F(M) = 0.84 
 
 J = 0.84 J 
 P s 
 
 so that for the same o 
 
 } ( ■**- j has tc increase by a factor i / \ 
 
 Attention to the 
 
 signs of eq. (10) shows that an increase in \^-J means it must becomes less negative; 
 
 the gradient will thus be less by this factor. For the same gradient therefore, the 
 current will have increased by this factor. 
 
 Thus, if J =0.1 a/ cm when M = 1, the current density will change with 
 M as shown in Fig. 15 below. 
 
 -26- 
 
1 
 
 OJ 
 
 Ph 
 
 0.2 
 
 0.1 
 
 
 5 10 
 
 M ► 
 
 15 
 
 Figure 15. Current Density Increase Due to F(m) 
 
 Quantitive evaluation of this effect in a particular transistor is not readily 
 accomplished, "but Fig. 15 does illustrate that the field aiding of current due to 
 the products of multiplication, can account for a certain amount of the current 
 increase with voltage, as distinct from the multiplication increase itself. 
 
 3-5 Base Width Modulation 
 
 Early analyzed the influence of base width modulation on most transistor 
 parameters, and considering the importance of his results one could expect that the 
 effect he analyzed could also he of significance in the operation at present being 
 considered. 
 
 From the one -dimensional geometry which has been considered so far in this 
 
 report, it seems obvious that the factor most likely to be influenced by variation 
 
 dp 
 of the effective base width will be! -v 2 - . It thus remains to express this term in 
 
 eq. (l^) as a function of the base width, and thence as a function of the applied 
 
 voltage . 
 
 -27- 
 
Proceeding along conventional lines, assuming simple recombination 
 characterized by life time i, the hole concentration in the base may be written as 
 
 2 2 
 
 p = A ± e + A 2 e + A (17) 
 
 where 
 
 qE 
 e = — + 
 kl 
 
 q £\ 2 4 
 
 . 
 
 nW ' v 
 
 + 
 
 In the model considered here, p is essentially a decreasing function of x, and 
 since in practice 
 
 + 1 1 • 
 e > > - e 
 
 A is essentially zero. 
 
 f ~b \ 
 From (17)1 t*-J may be found and noting that we are interested in currents 
 
 at the collector where x = W, there follows from (17) and (lh): 
 
 qDl ^(i -ja 
 
 e - 1 
 
 -28- 
 
In (l8), € is a function of E , which in turn is a function of J , so that a 
 graphical solution only may he possible. 
 
 It may be noted though, that while E is small, the last term in (l8) 
 
 approximates to 
 
 W 
 
 so that variation of this term with voltage is, at low voltages, 
 
 due mostly to variation of W, the "base width. In Ge, the depletion width of an 
 ahrupt junction varies approximately according to the relation: 
 
 Width - VpV x 10 cm. 
 
 With a typical p of 1 ohm. cm the last term of (l8), denoted hy F(E), is 
 shown as a function of voltage in Fig. l6. 
 
 on 
 
 i 
 o 
 
 H 
 
 X 
 
 K 
 
 10 20 30 40 50 60 
 V( volts) ► 
 
 Figure l6. F(E) as a Function of V 
 
 If quant it ive calculations are to he made, it will be necessary that the value of M 
 to be used is that value which applies for an isolated junction- -this must be done 
 to separate the F(e) and the M increase in J . 
 
 M(V) may be found from published data--McKay and McAfee, Ref. 23, Fig. 9. 
 
 -29- 
 
The current density may thus be written as 
 
 _ F(E) , , 
 
 J p ~ M • K Q {19) 
 
 where 
 
 pajj. N, 
 
 n d 
 
 K = 
 
 * p 2 V |ip / 
 
 Although no quant it ive results on a practical transistor have been calculated, 
 Fig. l6 does indicate that the base width variation effect on( 5*-) may be important. 
 
 The reason why eq. 18 was employed to show this effect and not simply the 
 obvious expression: 
 
 \ ^-) = G • (base width - C„ vV) where C. and C are constants 
 
 is that at large E , \^J will also be influenced by the field. Also, when con- 
 sidering the above, the following should be borne in mind: 
 
 If in eq. 18, F(E) of Fig. l6 is used, while a constant p is assumed, 
 the result will not be true for many practical cases. If it were, it would lead 
 
 to tne fallacy that infinite^ :r— / > an ^ hence infinite current, could be derived 
 from a finite rate of pair formation in the base. 
 
 -30- 
 
However, if a transistor is fed from a voltage source, as for instance 
 when measuring the I - V characteristic while maintaining V constant, the 
 condition of constant p will he approached and a very fast increase of current with 
 voltage may be expected, as distinct from the case where I is kept constant. A 
 practical curve illustrating this influence, is shown in Fig. 26, par. U.3. 
 
 Another point worth noting at this stage is that since the field effect 
 could he of importance in the base and the 'soft' breakdown effect at least partly 
 attributed to it, an opposing field in this region may help establish a sharper 
 knee in the breakdown characteristic. 
 
 Such a field could be established by retrograding a junction and by 
 sufficiently increasing the doping level as the junction is approached; the field 
 mentioned above may be neutralized or even reversed. Retrograded structures have 
 received some attention recently and more data about their breakdown characteristics 
 should become available. 
 
 In order to test this effect, a diffused base, planar PKP Ge transistor, 
 Type 2N6^3 was taken and its B-G and B-E breakdown characteristics compared at 
 constant V_,. The B-C junction will be a normal forward graded junction while the 
 B-E junction will be retrograded; however, if out-diffusion had been applied to the 
 B-E junction during manufacture so as to improve its breakdown voltage, the effect 
 sought will have been partly destroyed. The curves obtained are shown in Fig. 27, 
 par. k.3. 
 
 3.6 General 
 
 In the preceding paragraphs it has been shown that apart from the increase 
 in current due to avalanche multiplication in a transistor, several subsidiary effects 
 aid in the increase of current observed in practice. 
 
 There are many more influences which play a part in determining transistor 
 V-I relations such as the forming of microplasms due to statistical variation of 
 impurity density in the mother crystal or variation of diffusion depth in diffused 
 *ase struct OT es> 
 
 •31- 
 
Also, no mention has been made of positive feedback due to circuit 
 elements when a transistor is actually connected in a circuit. 
 
 However, it should be clear that interpretation of experimental data 
 of the nature discussed here will be very difficult indeed except in the isolated 
 cases where special units are made under closely controlled circumstances. 
 
 k. Experimental Procedures 
 
 k.l Preliminary Considerations 
 
 In preceding paragraphs it was pointed out that both thermal and long time 
 constant effects made practical determination of avalanche phenomena very difficult 
 indeed. 
 
 Also, due to the extreme sensitivity of the current to voltage in the near 
 avalanche region, accurate and above all, reproduceable results are very hard to 
 obtain. 
 
 Keeping in mind the complications mentioned, it was decided to make an 
 essentially dc measurement but of an intermittent nature in order to limit dissipation. 
 This would also yield results akin to those resulting from pulsed application of 
 transistors. Experience had shown furthermore, that only a null -balancing bridge 
 method of determining cur-rent would produce the desired accuracy while the same 
 applied to the voltage measurement. 
 
 In particular the following was decided upon: 
 
 1. To minimize thermal effects, the average dissipation would 
 be kept below 10 microwatts. 
 
 2. The environmental temperature of the transistor would be very carefully 
 controlled. 
 
 3. Voltages would be applied intermittently to allow fair sized 
 currents to be passed while satisfying dissipation requirements; 
 a 50 microsecond pulse every 5 milliseconds would be used. 
 Capacitive transients in transistors with a cut-off frequency f 
 of 5 mo/s or more would thus not be of importance, if circuit 
 impedances were kept low. 
 
 -32- 
 

 h. Current would be read on a null -balancing bridge. It was 
 found that voltages could be read satisfactorily on a 
 digital voltmeter which also enabled remarkable resetability 
 of any specific voltage. 
 
 k.2 Measuring Circuit 
 
 The first circuit constructed was that shown in Fig. 18, non-inductive 
 resistors being used. G is a standard moving coil galvanometer of 3500 ohm resistance 
 and 0.006 microamp/mm deflection sensitivity. 
 
 The pulse generator supplying the V^-, and V^ pulses had an internal 
 resistance of 50 ohms, and the pulse overshoot and droop was less than U$> in 
 all cases. During balancing of the bridge, the voltmeter was disconnected from 
 the circuit. 
 
 r 
 
 To Digit 
 Voltmeter 
 
 v. 
 
 Figure 18. Circuit for Measuring I 
 
 V Curves with L, as Parameter 
 
 -33- 
 
k.3 Measuring Results 
 
 In Fig. 19 are shown some results obtained with a 2N1305 transistor. The 
 full lines are those obtained with the circuit of Fig. 18 while the dotted curves 
 are the ones found with a Tektronix 575 transistor curve tracer. 
 
 30 40 
 
 Vcb (VOLTS) — ► 
 
 Figure 19. I - V__, Characteristics --2N1 305 
 
 -3k- 
 
The curve tracer results display the normal type of characteristics expected but 
 the other set of curves indicate that the 'breakdown' observed is not a multiplicative 
 one but either a 'punch-thru' or unidentified surface breakdown; this is concluded 
 from the even, straight -lined characteristics observed practically right up to the 
 sharp increase; also after the apparent breakdown, a constant resistance of about 
 60 kilohm is maintained in contrast to the practically zero resistance shown by 
 the curve tracer. It is thought that thermal effects may be the cause of this 
 latter sharp increase of current with voltage; Fig. 20 compares the voltage vs. 
 time characteristics of the two sweeps employed. 
 
 t 
 
 50 usee pulse every 5 millisec 
 Curve 
 
 2^0 c/s half sine wave 
 
 6 7 8 9 10 
 
 Figure 20. Comparison of Sweep Dissipations 
 
 Further experimental results obtained with the circuit of Fig. 18 and 
 a 2N404 transistor are shown in Fig. 21. In these curves the I curve has been 
 subtracted from the measured values of I so that these curves in effect represent 
 the terms (aM)l of eq. (5), par. 2.4. 
 
 -35- 
 
8o 
 70 
 6o 
 50 
 ko 
 30 
 
 20 
 10 
 
 t 
 
 x c - *so YS v cb 
 
 2NUo4 
 
 h = 10 ^ " ^0 
 
 Figure 21. I_- I_. vs. V_ for 2NUo4 Transistor 
 
 For each of these curves another curve of (aM) vs. V,„ may be plotted: the I_ = 10 
 microamp curve yields the graph shown in Fig. 22. Assuming an ultimate breakdown 
 for this transistor of 56 volts, Miller's M as given by eq. (k) is also plotted for 
 comparison, showing good agreement; n was taken as 3> as suggested by Miller for Ge, 
 
 •36- 
 
6 
 
 2N404 
 
 aM 
 M(I C0 ) 
 
 I 
 
 \% =56v. 
 
 K> 
 
 20 
 
 30 40 
 
 V CB (VOLTS)— -* 
 
 50 
 
 60 
 
 Figure 22. (aM), ^ and M(l ) vs. V CB for 2N^04 
 
 Further experimental results obtained on alloyed junction transistors GE-1 and 
 2Np8l are shown in Figs. 23 and 2h, showing varying degrees of correspondence 
 between theory and experiment. 
 
 The increase of I c with V CB as discussed in pars. 3.k and 3.5 will not be 
 very noticeable here due to the current source in the emitter tightly controlling 
 the available current. 
 
 \ 
 
 -37- 
 
12 
 
 10 
 
 8 
 
 6 . 
 
 10 
 
 20 
 
 30 
 
 1+0 
 
 50 
 
 V CT ( volts) 
 
 Figure 23. Multiplication Data for GE-1 Transistor 
 
 —J 
 60 
 
 
 -38- 
 
12 
 
 t 
 
 10 
 
 o 
 P 8 
 
 
 2N581 
 
 
 1.0 
 
 
 / ><\ / 
 
 ■0.8 ¥ 
 
 <= 1 
 
 '0.6 w 
 
 
 (0M) / 
 
 '0.4 
 
 
 ^^^ CO 
 
 •0.2 . 
 
 
 
 10 
 
 20 
 
 30 
 V CT ( volts) 
 
 40 
 
 Figure 24. Multiplication Data for 2N58l Transistor 
 
 Figure 24, giving results for a typical 2N58l PNP alloyed junction Ge 
 transistor (one from a test batch of ten) shows clearly that multiplication data 
 can most certainly not be obtained for this type of transistor by means of the method 
 outlined. Of the ten transistors tested, all showed the tendency at higher voltages 
 to have 
 
 M 
 
 ^oo > (aM 
 
 I E + M 
 
 W 
 
 This could be the result of an influence by the current on the breakdown characteristic, 
 but the cause of the behavior is not known. 
 
 -39- 
 
U.3.-1 Case of Constant p 
 
 n 
 
 To illustrate the effect of constant p , the circuit of Fie. 18 was 
 
 n 7 to 
 
 modified to that shown below: 
 
 ion 
 
 V 
 
 IN 
 
 Figure 25. Circuit for Maintaining Constant V. 
 
 EB 
 
 Results obtained with a 2N^04 transistor are shown in Fig. 26. The increase in I 
 will of course also partly be the result of regenerative feedback due to the series 
 
 base resistance r. 
 
 bb' * 
 
 -40- 
 
140 
 S20 
 100 
 80 
 60 
 40 
 20 
 
 
 
 t 
 
 2N404 
 300 °K 
 
 .tfl* 
 
 >4*£ 
 
 10 
 
 15 20 25 
 
 V CB (VOLTS) 
 
 30 
 
 35 
 
 Figure 26. I_- V_^ Curves with V_ = Constant 
 
 k.k influence of Resistivity Gradient 
 
 As mentioned in par. 3»5 it is expected that the sign of the impurity 
 gradient next to the junction will influence the 'sharpness' of the breakdown 
 characteristic . 
 
 Making use of the circuit of Fig. 25, curves obtained on the B-E and B-C 
 junctions of a diffused base planar PNP Ge transistor are shown in Fig. 27- Some 
 indication of a sharper breakdown is present although not enough units were available 
 for test to yield conclusive results. 
 
 41- 
 
The variation of depletion width and hence of the electric field with 
 applied voltage in the two cases mentioned above may also play a role in determining 
 the shape of the breakdown curve; this has been well illustrated in the publication 
 of Nathanson and Jordan. 
 
 300 r 
 
 200 
 
 100 
 
 t 
 
 313°K 
 
 V CB ( volts) 
 
 ■*> 
 
 10 
 
 20 
 
 30 
 
 50 
 
 Figure 27. Emitter-Base and Collector-Base Breakdown Curve for Diffused Base 
 Transistor 
 
 5. Conclusion 
 
 Extensive measurements on several types of transistors indicate that 
 
 multiplication factors of about 10 or 20 may be reproduceably obtained in any one unit, 
 
 Higher values tend to occur in a region of behavior where production type transistors 
 are subject to sporadic breakdowns making results unreliable. 
 
 Thermal effects play a role in determining I-V relations to the extent 
 that it may not be possible to obtain isothermal multiplication data from measurements 
 on conventional transistors. 
 
 -h2- 
 
 
It is believed that the multiplication factor determined from transistor 
 measurements should "be regarded as (ctM.) as distinct from the value of M alone, as 
 
 found in diodes. 
 
 Long time constant effects seriously complicate intermittent measurement 
 of transistor I-V relations. 
 
 Both thermal and long time constant influences may be minimized by using 
 short pulses of supply voltage spaced suitably in time; this method however requires 
 extreme accuracy of measurement due to the very! low level of effective dc currents 
 and voltages present. 
 
 Electric fields set up by currents produced by multiplication, together 
 with base width modulation, may be the cause of transistor experimental data being 
 interpreted as a faster rise in multiplication than is actually present. It would 
 seem as if the micro-plasmic nature of the junction avalanche breakdown and the 
 apparent sporadic occurrance of the process, imparts an inherent "soft" character- 
 istic to junction breakdown at reasonable current levels (a few micro-amp). If 
 the extreme speed of avalanche build-up is to be effectively utilized in fractional 
 
 nanosecond switching, either perfect control over micro-plasma formation, and 
 
 . -Q . 
 
 probably very low-level operating currents (10 amp) will be required. 
 
 -h3- 
 
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 -k5-