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Space-charge-limited currents in germanium Nichol, Dennis William 1958

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SPACE-CHARGE-LIMITED CURRENTS IN GERMANIUM by DENNIS WILLIAh NICHOL . B . A . U n i v e r s i t y of B r i t i s h Columbia, 1 9 5 6 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF. MASTER OF SCIENCE i n the Department of • Phys ics We accept t h i s t h e s i s as conforming to the requi red standard The U n i v e r s i t y of B r i t i s h Columbia September 1 9 5 8 V ABSTRACT A study has been made of space-charge- l imited hole flow i n germanium by i n v e s t i g a t i n g the current-voltage-temperature c h a r a c t e r i s t i c s of se lec ted p-n-p t r a n s i s t o r s used as diodes with the base o p e n - c i r c u i t e d . These t r a n s i s t o r s were se lec ted so as to minimize the e f fec t of avalanche m u l t i p l i c a t i o n . 'These diodes pass hole current through the base a f t e r a voltage designated as the punchthrough voltage has been app l ied to deplete the n type base of e l e c t r o n s . The r e s u l t i n g space-charge- l imited current above punchthrough has been c l o s e l y s tudied and a l so i t s temperature dependence. To expla in the form of these c h a r a c t e r i s t i c s , publ i shed data have been used fo r the r e l a t i o n s h i p between e l e c t r i c f i e l d and c a r r i e r d r i f t v e l o c i t y f o r holes i n germanium i n order to consider the hole flow through the high f i e l d reg ion of the base. It was fur ther found necessary to consider the v a r i a t i o n of e f f e c t i v e emi t t ing area as the app l i ed voltage i s increased past punchthrough. For high appl ied vol tages and hence high a p p l i e d f i e l d s i n the base, a constant d i f f e r e n t i a l re s i s tance i s obtained of magnitude about equal to that expected t h e o r e t i c a l l y f o r a constant d r i f t v e l o c i t y of holes i n the base. The temperature dependence of t h i s current can be s a t i s f a c t o r i l y explained by the temperature v a r i a t i o n of the v i base-generated current and of the punchthrough vol tage i t s e l f . A s a t i s f a c t o r y model of the l a t t e r v a r i a t i o n has been made by cons ider ing the temperature v a r i a t i o n of the p o t e n t i a l "barrier at the emitter j u n c t i o n . I f the observed c h a r a c t e r i s t i c s are corrected for these v a r i a t i o n s , there i s found to be n e g l i g i b l e v a r i a t i o n of the space-charge- l imited hole -flow. Capacitance measurements were made on both junct ions f o r these d iodes . From these measurements the assumption of a step j u n c t i o n and of uniform impuri ty d i s t r i b u t i o n i n the base were j u s t i f i e d . No abrupt change of capacitance was observed as the voltage was increased through the punch-through voltage contrary to the f ind ings of Barker . The o r i g i n a l theor ie s of Shockley-Prim-Dacey were extended to inc lude the e f fec t s of ( i ) m o b i l i t y v a r i a t i o n over a wide range o f 1 - f ie lds ( i i ) non-planar geometry and ( i i i ) the p o t e n t i a l d i s t r i b u t i o n near the emitter j u n c t i o n . In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representative. It i s understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Physics  The University of British Columbia, Vancouver 8, Canada. Date September 29, 1958. i i TABLE OF COKTENTS CHAPTER PAGE I . INTRODUCTION . 1 1.1 Problems a r i s i n g i n the study of space-charge- l imi ted currents 1 1.2 Review of previous work . . . . . . . . . 4 I I . EXPERIMENTAL INVESTIGATION 10 2 .1 S e l e c t i o n of t r a n s i s t o r s f o r study . . . . 10 2.2 Measurement of punchthrough voltage and i t s temperature dependence 14 2.5 D . C . c h a r a c t e r i s t i c s of space-charge- l imited currents i n Germanium 14 2 .4 Capacitance Measurements 15 2.5 Temperature dependence of space-charge-l i m i t e d current 16 2 .6 Trans ient response of the space-charge-l imited. diode 16 I I I . THEORETICAL INVESTIGATION 17 5.1 R e l a t i v e importance of d r i f t and d i f f u s i o n components of hole current 17 5.2 Influence of f ie ld-dependent m o b i l i t y . . . 19 5.5 A model of m o b i l i t y - f i e l d dependence . . . 25 5.4 Temperature v a r i a t i o n of V p 25 5.5 Trans ient response of p-n-p diode 28 5 .6 Non-planar geometry fo r the constant v e l o c i t y case 29 5.7 High frequency impedance of a p-n-p diode past punchthrough • 50 i i i Table of Contents (Cont 'd) CHAPTER PAGE IV. DISCUSSION 3 2 4.1 I .V c h a r a c t e r i s t i c s and n o n - l i n e a r m o b i l i t y 32 4.2 Temperature dependence of space-charge-l i m i t e d c h a r a c t e r i s t i c s f o r the p-n-p diode * 3 4 4 . 3 Trans ient response of a space-charge-l i m i t e d diode 3 5 4 . 4 Capacitance of reverse biased junct ions . 3 6 V . CONCLUSIONS 3 7 5.1 Areas of agreement of theory and experiment 3 7 5 . 2 Outstanding problems 3 8 REFERENCES \ . 3 9 i v LIST OF TABLES AND FIGURES TABLE Facing Page 1 T r a n s i s t o r dsta 7 FIGURE 1 C i r c u i t l o r measurement of Schenkel-Statz punchthrough voltage 3 2 Temperature dependence of the punchthrough voltage 4 3 S tructure of GE 2N137 t r a n s i s t o r 5 4 Current-vol tage c h a r a c t e r i s t i c of p-n-p diode 14 5 Temperature dependence of p-n-p diode c h a r a c t e r i s t i c 15 6 C i r c u i t for measurement of j u n c t i o n capacitance 16 7 E l e c t r i c f i e l d dependence of d r i f t v e l o c i t y 19 8 E l e c t r o s t a t i c p o t e n t i a l across the base of a p-n-p diode 27 v i i ACKNOWLEDGEMENTS I wish to thank Professor R . E . Burgess f o r h i s help and guidance throughout t h i s i n v e s t i g a t i o n , and f o r h i s many valuable suggestions dur ing the prepara t ion of t h i s t h e s i s . Support from a Defence Research Board grant dur ing the summers of 1957 and 1958 i s g r a t e f u l l y acknowledged. 1 CHAPTER I INTRODUCTION 1.1 Problems a r i s i n g i n the study of space charge l i m i t e d cur ren t s . C e r t a i n se lec ted p-n-p j u n c t i o n t r a n s i s t o r s wi th su i tab le base widths and base impuri ty concentrat ions can be, used to study space-charge- l imited currents i n germanium, and i n p a r t i c u l a r , the e l e c t r i c f i e l d dependence of c a r r i e r d r i f t v e l o c i t y . C r i t e r i a f o r s e l e c t i o n of these t r a n s i s t o r s are e s t ab l i shed i n t h i s t h e s i s . These t r a n s i s t o r s are chosen so as to avoid s i g n i f i c a n t avalanche m u l t i p l i c a t i o n r e s u l t i n g from i o n i z a t i o n of l a t t i c e atoms by f a s t moving c a r r i e r s . I f one of these t r a n s i s t o r s i s used as a p-n-p diode wi th the base o p e n - c i r c u i t e d , then as a f i r s t approximation i t may be assumed that no current w i l l f low u n t i l a p a r t i c u l a r voltage ( c a l l e d the punch-through vol tage) i s reached. At t h i s voltage, the base l a y e r i s whol ly depleted of major i ty c a r r i e r s and the c o l l e c t o r space charge d e p l e t i o n reg ion makes contact with the emi t te r . Copious emission of holes then takes place from the emit ter in to the space charge reg ion of the base where a high e l e c t r i c f i e l d i s present . 2 Transistor structures of this type have been analyzed by Shockley and Prim ( 1 9 5 3 ) . They consider n-i-n and p-n-p diodes. However they do not take into account the e l e c t r i c f i e l d dependence of mobility which i s important at the high f i e l d s obtained near the collector i n narrow base junction transistors. For the high f i e l d situation, models of d r i f t velocity - electric f i e l d dependence may be made, and approximate integrations carried out, leading to space-cnarge-limited diode characteristics. One of the main problems to be resolved i s a satisfactory model of diode characteristics i n this way. Further complications in actual punchthrough phenomena may occur however. Emitter and collector contacts are of different area. Hence models assuming non-planar geometry may sometimes be necessary; although i t can be shown that a saturated d r i f t velocity (which tends tto occur for large applied voltages) leads to a constant d i f f e r e n t i a l resistance independent of the particular c y l i n d r i c a l geometry. At high temperatures, the transistor base region becomes i n t r i n s i c and the usual extrinsic base analysis must be further extended. Also at these temperatures the saturation diffusion current of the collector junction ( I c o ) becomes comparatively large. A l l these effects lead to modification of the diode characteristics. F i g u r e 1 CIRCUIT FOR MEASUREMEEiT OF SCFENKEL-3TATZ FUSCHTHRQUGH VOLTAGE p N P E M I T T E R B A S E COLLECTOR COLLECTOR-EMITTER VOLTAGE CHARACTERISTIC FOR PNP PUNCHTHROUGH TRANSISTOR 3 The technique used to determine the punchthrough voltage i s due to Schenkel and Statz (1954). The necessary-c i r c u i t i s shown i n f i g u r e 1. As the c o l l e c t o r voltage i s made i n c r e a s i n g l y negat ive , a voltage Vp i s reached at which the emit ter voltage becomes locked to that of the c o l l e c t o r such that V e = V c - Vp . The emit ter then acts as i f i t were connected to the c o l l e c t o r through a ba t te ry of e .m. f . - V p . The p h y s i c a l mechanism exp la in ing t h i s phenomena i s that o f c o l l e c t o r d e p l e t i o n l a y e r widening. As the c o l l e c t o r reverse b i a s i s increased , the donor i o n space charge l a y e r , (which i s depleted of e l ec t rons ) spreads from the c o l l e c t o r j u n c t i o n to inc lude more f i x e d charge. Thi s space charge d e p l e t i o n reg ion i s mainly confined to the h igh r e s i s t i v i t y base r e g i o n , through which i t spreads u n t i l i t touches the emitter j u n c t i o n at the punchthrough vo l t age . T h i s voltage i s very w e l l def ined by the Schenkel Statz type of measurement. I t i s assumed that t h i s voltage i s the same as the diode voltage corresponding to g r e a t l y increased current flow due to the onset of hole i n j e c t i o n from the emit ter . Th i s l a t t e r i s not so w e l l de f ined , p o s s i b l y due to the fac t that the space charge zone i s not sharply de l inea ted , but has a boundary smeared out over severa l Debye lengths ( L 0 ) , where L D = /0 € k T A T T q * N (1.1.1) F i g u r e 2 TEMPERATURE DEPENDENCE OF THE PUNCHTHROUGH VOLTAGE In t h i s r e l a t i o n 6 = d i e l e c t r i c constant of germanium, k = Boltzmann's constant , <[ = e l e c t r o n i c charge and N = impuri ty dens i ty i n the base. At room temperature, i n 14- - 3 20 ohm-cm. (N = 10 cm. ) germanium L^a? 1.3 x 10 cm. A fur ther problem wi th regard to the Schenkel Statz punchthrough voltage i s i t s temperature dependence. Th i s e f fect cannot be explained by thermal expansion of the base, or by v a r i a t i o n of € with temperature, but a fu r ther f a c t o r which must be considered i s the temperature dependence of the p o t e n t i a l b a r r i e r of the emitter which i s due to the space charge of holes i n the base i n f ront of the emi t te r . The r e l a t i o n s h i p of V P to t h i s p o t e n t i a l i s der ived from a double i n t e g r a t i o n of Po i s son ' s equation across the base width , with the condit ion."that no current i s f l o w i n g . C r i t e r i a f o r t r a n s i s t o r s e l e c t i o n and experimental r e s u l t s , are presented i n Chapter I I of t h i s t h e s i s . T h e o r e t i c a l i n v e s t i g a t i o n s are presented i n Chapter I I I , and a d i s c u s s i o n of t h e i r success i n exp la in ing the observed r e s u l t s i s presented i n Chapter IV. 1.2 Review of previous work. A major paper on space charge l i m i t e d emiss ion i n semi-conductors i s by Shockley and Prim (1953). They consider the s i t u a t i o n i n t r a n s i s t o r phys ics analogous to space charge l i m i t e d emission i n vacuum tubes. The f i r s t Figure 3 STRUCTURE OF GE 2R137 TRANSISTOR EMITTER ^0 COLLECTOR BASE s t ructure they analyze i s the n - i - n d iode , f o r which they obta in the f o l l o w i n g r e l a t i o n between current d e n s i t y and app l ied vol tage analogous to C h i l d ' s law i n vacuum tube e l e c t r o n i c s J - 9£>U-V* , 1.2.1 where J = current dens i ty and M- = e l e c t r o n m o b i l i t y i n the base r e g i o n . Thi s m o b i l i t y i s assumed to be a constant , independent of the e l e c t r i c f i e l d . A major premise i n d e r i v i n g the.above r e l a t i o n i s that the current i s c h i e f l y c a r r i e d by d r i f t r a t h e r than d i f f u s i o n , which i s a* v a l i d assumption at vol tages greater than ~ (25 m i l l i v o l t s at 290 °K) where k i s Boltzmann's constant , T i s the absolute temperature, and 9 i s the e l e c t r o n i c charge. Shockley and Prim a l so analyze p-n-p diodes i n t h e i r paper. In i n t e g r a t i n g Po i s son ' s equation across the n type base i n a p-n-p d iode , one must cons ider the f i x e d p o s i t i v e charge dens i ty cjN i n the base due to i o n i z e d donor i m p u r i t i e s as w e l l as the fur ther p o s i t i v e charge due to emitted h o l e s . For these d iodes , condi t ions of space charge l i m i t e d emission occur when the dep le t ion l a y e r due to i on ized donors penetrates the base. Th i s vol tage i s der ived from a double i n t e g r a t i o n of Po i s son ' s equation across the base wi th the c o n d i t i o n J = 0 . Th i s leads to 6 1.2.2 i f we neglect the i n t e r n a l b a r r i e r p o t e n t i a l of the emit ter j u n c t i o n . Thi s vol tage i s c a l l e d the punch-through vo l t age . Por the case of constant m o b i l i t y , Po i s son ' s equation may be in tegra ted f o r any current dens i ty i n terms of the t r a n s i t time of i n j e c t e d c a r r i e r s across the base. A vo l tage-current dens i ty r e l a t i o n i s de r ived i n t h i s manner by Shockley and Prim f o r the constant m o b i l i t y case. For l a rge values of the current d e n s i t y t h e i r der ived c h a r a c t e r i s t i c approaches a symptot i ca l ly the C h i l d ' s law analog f o r an i n t r i n s i c base diode as the i n j e c t e d c a r r i e r s then dominate over the bound space charge i n the base reg ion . Shockley and Prim a l so inc lude i n t h e i r a n a l y s i s the e f fec t s of d i f f u s i o n currents i n p - i - p and n - i - n s t r u c t u r e s . At large d i s tances from the p o t e n t i a l maximum, the f i e l d and p o t e n t i a l i n the ba'se approach the C h i l d ' s law analog . The p o s i t i o n of the zero f i e l d point, i n the base i s s h i f t e d s l i g h t l y from the corresponding point of the C h i l d ' s law analog. The e f fec t of the d i f f u s i o n current i s not important however, because except near the p o t e n t i a l maximum the current i s c h i e f l y c a r r i e d by d r i f t . . Shockley-Prim theory has been extended by Dacey to the high f i e l d case by assuming the v e l o c i t y - f i e l d r e l a t i o n TABLE I BASIC PARAMETERS OF SELECTED TRANSISTORS (GE TYPE 2N137) TRANSISTOR NO 19 21 37 EMITTER AREA CM* 0.47x10" X ,.72xl0""X COLLECTOR AREA CM 3 0.88x10"* l . l x i o " * Ac I i 1.9 1.5 EASE WIDTH CM 1. 3x ld " 3 1.3x l0 - 3 1.3x10"3 IMPURITY DENSITY C M " 3 8.7x10 3 5.9xlo ' 3 4.4x lo ' 3 MEASURED V P VOLTS 8.6 5.8 4.4 RESISTIVITY OHM- CM 17 24 28 E(COLLECTOR) AT V P KILOVOLTS/CM 10 8.9 6.9 v = M ( E 0 E) 7 1.2.3 where Al i s the low f i e l d mobility and E© the c r i t i c a l f i e l d at which this half-power law comes into effect. This power law i s j u s t i f i e d by velocity-electric f i e l d measurements by Ryder (1953) who has used pulse techniques to obtain his data and by Gibson and Granville (1956) who have used a microwave absorption technique. It appears that this law becomes v a l i d for holes i n germanium from a f i e l d of about 1000 volts/cm. up to 10,000 volts/cm. For voltages well beyond Vp, the fixed charge i s swamped by injected holes, and i f the approximation can be .x. made that v = M ( E 0 E) throughout the entire base region, then a new Child's law analog i s obtained. J = 1.43eMJst j | 1.2.4. As i n the Shockley-Prim case of constant mobility, the fixed charge in the base cannot be neglected for intermediate voltages. In a manner analogous to Shockley and Prim a voltage-current density relation i s derived for this case. Emeis and Herlet (1958) have made several experimental s i l i c o n alloy transistors and have f i t t e d Shockley-Prim theory to their measured characteristics. Two series of n-p-n s i l i c o n alloy transistors were made, one with base ., thicknesses of 40-45 microns and the other 55-60 microns. 8 Various c h a r a c t e r i s t i c s were p l o t t e d f o r avalanche breakdown t r a n s i s t o r s , punchthrough t r a n s i s t o r s , and t r a n s i s t o r s showing both these a t t r i b u t e s . The l a t t e r can have negative re s i s tance c h a r a c t e r i s t i c s . Turnover cond i t ions for these negative r e s i s t ance diodes have been considered by Barker (1957). data , i t i s necessary to assume an e f f e c t i v e area to r e l a t e current to current d e n s i t y . T h i s area can be determined from capacitance measurements on the emit ter j u n c t i o n i f base r e s i s t i v i t y i s known; or from the capacitance at punchthrough i f the base width i s known. Emeis and Her le t apply Shockley Prim theory to t h e i r experimental data by choosing two parameters equiva lent to Vp and the emit t ing area to give a best f i t . Apparent ly Schenkel-Statz ounchthrough voltages were not measured. Besides determinat ion of area , capacitance b ia s measurement on a j u n c t i o n can be used to determine whether or not the j u n c t i o n has a step i n impur i ty concentra t ion . Por a step j u n c t i o n , wi th the acceptor imouri ty dens i ty Np i n the p type mater i a l very much greater than the donor impur i ty dens i ty Nn i n the n type m a t e r i a l , and i f Nn i s constant , In f i t t i n g Shockley Prim theory to experimental C 1.2.5. where-C i s the d i f f e r e n t i a l j u n c t i o n capacitance (as measured wi th a smal l-ampli tude, low frequency a . c . s i gna l ) and yo i s the internal potential difference of the collector junction. Hence a log C - log V plot of this relation gives a straight line of slope for Y>>% and (fi can be estimated from the departure from l i n e a r i t y at low V. A plot of this sort can be used as a check on the degree to which a junction i s a step junction. Barker has reported that for some transistors studied these plots show a marked step up in the capacitance of the junction at the punchthrough voltage, which may be explained by assuming that the space charge depletion region has touched the emitter, switching the emitter capacitance into the c i r c u i t . This result has not been confirmed i n the present study. 10 CHAPTER II EXPERIMENTAL INVESTIGATION 2.1 S e l e c t i o n of t r a n s i s t o r s fo r study. The p-n-p s t ructures inve s t i ga ted i n t h i s study were chosen so as to avoid avalanche breakdown due to c a r r i e r m u l t i p l i c a t i o n . F i e l d s greater than 1000volts/cm are always obtained i n the base of t r a n s i s t o r s s tudied under space-charge l i m i t e d c o n d i t i o n s , thus causing a reduct ion of m o b i l i t y ; t h i s effe'ct i s one of those under study i n the present the s i s i n cons ider ing modi f ica t ions of the Shockley-Prim theory. The var ious parameters of the t r a n s i s t o r s under study i s presented i n table I . (a) A c r i t e r i o n can be drawn up fo r the s e l e c t i o n of t r a n s i s t o r s f o r punchthrough measurements minimizing avalanche behaviour. M i l l e r (1955) has shown that t h i s can be analyzed i n terms of the avalanche m u l t i p l i c a t i o n f ac tor M. M per ta ins to a p-n j u n c t i o n and i s the r a t i o of the t o t a l current f lowing through the j u n c t i o n to that expected i f no a d d i t i o n a l c a r r i e r s were being generated due to c o l l i s i o n i o n i z a t i o n by energet ic c a r r i e r s . Exper imenta l ly he found that 11 f o r n type G e . , where V g i s the avalanche breakdown vo l t age , and V the reverse voltage appl ied to the p-n j u n c t i o n . For a p-n-p d iode , i f we wish to avoid avalanche e f f e c t s , i t i s T — I necessary that (M-l)<^ ( l - ° 0 where ex. = c o and I i s the current through the d iode . Hence we choose M to be l e s s than 1.001 a s « » 0 .98. By 2.1 .1 we then must have I 4 0 . 1 . v 8 Another e m p i r i c a l r e l a t i o n observed by M i l l e r for Ge. p-n junct ions was that between j u n c t i o n breakdown voltage and the n type impuri ty concentra t ion V g = 1.0x10 N 2 .1 .2 _3 where V g i s i n v o l t s and N xn cm Two t r a n s i s t o r parameters which can be used to determine the base width and the base impuri ty dens i ty are the punchthrough voltage and the frequency of <* cu to f f at which the current ga in i s reduced by 3 db. Th i s frequency i s given by r 0.4 Dp U = ^ 2 .1 .3 vr Making use of the above r e l a t i o n s and the express ion for Vpo i t i s poss ib le to der ive the fo l lowing c r i t e r i o n fo r the s e l e c t i o n of t r a n s i s t o r punchthrough experiments such that V P . ^ 0.1 V . . 7 X 5 I.-7X5 ' ' f« Vp* < 4x10 where ~f«, i s expressed i n cyc le s / sec and Vpo i n v o l t s . 12 For the t r a n s i s t o r s i n tab le I the expression on the l e f t hand, s ide of t h i s i n e q u a l i t y i s approximately 1.2x10 and hence these t r a n s i s t o r s comply wi th the c r i t e r i o n . (b) In obta in ing p-n-p diode c h a r a c t e r i s t i c s the t r a n s i s t o r i s kept i n a constant temperature bath fo r a p a r t i c u l a r run . However at high currents and voltages the power d i s s i p a t e d may become large enough to r a i s e the temperature of the t r a n s i s t o r w e l l above the ambient temperature. An estimate of the temperature r i s e can be made from the manufacturer 's d i s s i p a t i o n c o e f f i c i e n t expressed i n temperature r i s e above ambient temperature per u n i t power d i s s i p a t e d . For instance fo r t r a n s i s t o r s wi th Vp =6 v o l t s the current at 2Vp i s t y p i c a l l y 12 ma corresponding to a power d i s s i p a t i o n of about 150 m i l l i w a t t s . A t y p i c a l value of the d i s s i p a t i o n c o e f f i c i e n t f o r these t r a n s i s t o r s i n a i r i s 0.6 C/m.w. which would lead to 100 C r i s e of temperature! To reduce t h i s serious heat ing i n the present s tudy, the t r a n s i s t o r leads were f i t t e d with large area f i n s and the whole assembly immersed i n o i l . Furthermore the c h a r a c t e r i s t i c s at high currents were obtained by p u l s i n g the t r a n s i s t o r wi th pulses of 10 microseconds durat ion at r e p e t i t i o n rates of 60-300/sec. Good cons i s tency between steady current measurements and pulsed measurements were obtained i n the reg ion of overlap of these techniques . 13 (c) For symmetry of the current vol tage c h a r a c t e r i s t i c s , the v a r i a t i o n of current dens i ty across the base should be small and thus one should choose t r a n s i s t o r s which have t h e i r c o l l e c t o r and emit ter areas (as determined by capacitance measurements) as near ly equal as p o s s i b l e . Al so one should choose t r a n s i s t o r s wi th a uniform base r e s i s t i v i t y . Thi s can be d i s c l o s e d by e q u a l i t y of V p measured from the two junct ions and by the capacitance of each j u n c t i o n being i n v e r s e l y p r o p o r t i o n a l to the square root of the app l i ed b ia s (d) As body conduction i s being s tudied with these t r a n s i s t o r s , they must be chosen so as to be free from surface conduction channels between the emit ter and c o l l e c t o r . Th i s may be determined by measurement of f l o a t i n g emitter-base p o t e n t i a l Ve , f o r a negative c o l l e c t o r p o t e n t i a l such that ~<&|V C | < Vp , which should give V, = ] p ' In ( l - « ) Any channels present w i l l lead to emitter p o t e n t i a l s many times t h i s va lue . Th i s phenomenon of surface conduction has been studied by Brown (1953). Figure 4-CURRENT-VOLTAGE CHARACTERISTIC OF p-n-p DIODE GE 2N137 100 10 s o 10 60 s o " 4-0 30 z or OL-ID ( j 20 10 CONSTANT DIFFERENTIAL RESISTANCE PULSE DAT* DC. ORTR Vp ^ * G 9 io ix V O L T A G E ABOVE PUNCH THROUGH 14 2.2 Measurement of punchthrough voltage and i t s temperature dependence. The c i r c u i t presented i n f i gure 1 was f i r s t used by Schenkel and Statz f o r the measurement of punchthrough voltages V« i s measured with a very high impedance vol tmeter . At low temperatures, extremely h igh impedances occur fo r the emitter junc t ion diode. Precautions must be taken to avoid e l e c t r o s t a t i c pickup of 60 cps. vol tages which become r e c t i f i e d by the t r a n s i s t o r and d i sp l ace the p o t e n t i a l s ; good s h i e l d i n g and the use of a by-pass capac i tor minimize t h i s e f f e c t . The punchthrough voltage was measured at temperatures from -60*C to +80*C. Th i s voltage has a we l l def ined temperature dependence. For t r a n s i s t o r #37 (type GE2N157) i t was found that . o V> was e s s e n t i a l l y constant below Q C and increased at a ra te of 1.3 x 10 v o l t s / d e g above 0 C. 2.3 D . C . c h a r a c t e r i s t i c s of space charge l imi ted ' currents i n Ge. D . C . c h a r a c t e r i s t i c s of p-n-p diodes were taken us ing 2N137 t r a n s i s t o r s with the base f l o a t i n g . Runs were made at o ten degree temperature i n t e r v a l s up to 90 C . At these high temperatures the reverse sa tura t ion current of the reverse biased j u n c t i o n becomes l a r g e , and l i m i t i n g power condi t ions are reached. D . C . c h a r a c t e r i s t i c s were measured f o r these VOLTAGE VOLTS 15 devices by steady current-vol tage measurements up to power d i s s i p a t i o n s of 150 m i l l i w a t t s . For higher currents and vo l tages , the t r a n s i s t o r was pulsed and the vol tage pulses across the diode and across a known se r i e s r e s i s t o r were measured on a c a l i b r a t e d o s c i l l o s c o p e to obta in a c u r r e n t -voltage r e l a t i o n . The p o r t i o n of the pulse measured was past the i n i t i a l t r a n s i e n t current sp ike . As t h e o r e t i c a l cons iderat ions of c h a r a c t e r i s t i c s are concerned with the current a r i s i n g at punchthrough, I p ( T ) , was subtracted from these pulsed c h a r a c t e r i s t i c s . These c h a r a c t e r i s t i c s are presented i n f i gures 4 , ' 5 -2.4 Capacitance measurements were made on both C o l l e c t o r and emitter junct ions at var ious reverse biases wi th the bridge arrangement depic ted i n f igure 6. A General Radio Twin T b r i d g e , type 821-A was used at frequencies from 100 k i l o c y c l e s to 5 megacycles. These capacitance measurements were used to determine the c o l l e c t o r and emitter areas , as the base width and the punchthrough vol tage were known. Cpt = € ^ /? 2 .4 .1 Cpt i s the capacitance of a j u n c t i o n at the punchthrough vo l tage , and A i s the cross s e c t i o n a l area of the j u n c t i o n . I f the capacitance i s p l o t t e d v s . reverse b ia s a slope of i s expected. Barker has reported that f o r j u n c t i o n capacitance measurements, the capacitance of the j u n c t i o n shows a marked step up at the punchthrough voltage Figure 6 CIRCUIT FOR MEASUREMENT OF JUNCTION CAPAC1S&KCE N BIAS 4-SUPPLY IMPEDANCE BRIDG-E SIRS VOLTAGE 16 and the - slope then continues to he He a t t r i b u t e d t h i s e f fec t to the extra capacitance of the f l o a t i n g j u n c t i o n becoming added to that of the j u n c t i o n being measured. T h i s r e s u l t was not confirmed i n the present study. A t y p i c a l p l o t i s shown i n f i gure 6. Results f o r t y p i c a l j u n c t i o n areas are g iven i n tab le I. 2.5 Temperature dependence of space charge l i m i t e d cur rent . For the p-n-p diodes #21 as s p e c i f i e d i n t ab le I, the temperature dependence of the I-V c h a r a c t e r i s t i c i s not marked except fo r an a d d i t i o n a l constant current which flows, at Vp . The temperature dependence of the c h a r a c t e r i s t i c at vol tages past punchthrough i s s l i g h t l y a f fected by the temperature dependence of Vp . 2.6 Trans ient response of the space charge l i m i t e d d iode . T r a n s i s t o r #21 was pulsed with i t s punchthrough voltage of 6 v o l t s and an estimate made of the t o t a l charge swept out of the base. Th i s was estimated as 8 x 10 coulombs at room temperature. J 17 CHAPTER III THEORETICAL INVESTIGATION 3.1 Relative importance of d r i f t and diffusion components of hole current. (i) For constant mobility, the ratio of hole diffusion current to hole d r i f t current i s : The resultant derivation being due to the Einstein diffusion relation qDp = kTAcp where DP i s the diffusion constant for holes; p i s the hole density, which i s a function of distance across the base; k i s Boltzmann's constant; T i s the absolute temperature; and/fpis the hole mobility. At the punchthrough voltage, the f i e l d increases 2V» linearly across the base to -ff- at the collector junction. After the punchthrough voltage, when holes are injected into the base, the f i e l d i s increased throughout the base above the value obtaining at punchthrough. As the hole density decreases across the base D §1 a s 3.1.1 < 0 3.1.2. 18 So we may state that 2Vx 3.1.3 Also v/e know c. V X 3.Ly-sine e — — > 0 we may make the approximation dx x 3.1.5 Hence the ratio of diffusion to d r i f t current becomes T ^ 7 ~ W 3 , 1 , 6 kT So i f V»-— the current i s chiefly carried by d r i f t except near the emitter. But near the emitter the diffusion current becomes re l a t i v e l y greater, and of course i s the sole component at the potential max (E » 0 ) . e.g. for T = 290°K, V = 5 volts, ( i i ) If the f i e l d i n the base i s s u f f i c i e n t l y high, the d r i f t velocity saturates. For this case the ratio of diffusion to d r i f t current i s Jdiff < for x>2£ 0 3.1.7 pv as — = qp =» constant. So for this case a l l the current i s carried by d r i f t . Figure 7 ELECTRIC FIELD DEPENDENCE OF DRIFT VELOCITY m • • \ —e R Y D E o HOLES IN F G I B S O N ft Is A HOLES I N N R T Y P E G E . ID GRANVILLE T Y P E G E . \ 1 , | | | | | t| /O too IOOO 10,000 E L E C T R I C F I E L D V O L T 5 / C M ( i i i ) A fur ther d i f f u s i o n current to be considered i s the reverse s a tura t ion current (I ( r 0 ) a r i s i n g from the thermal ly generated holes i n the base of a p-n-p t r a n s i s t o r . The c o n t r i b u t i o n of t h i s current to the current I of the p-n-p diode may be analyzed i n terms T — I of the current a m p l i f i c a t i o n f a c t o r * = • c Z—~ . For the p-n-p diode I c = I e , and hence I T — Icq TL "I Q x ( f »-»m revert*saiufx+iots'S " 1 - of. P . x . o 3.2 Influence of f ie ld-dependent m o b i l i t y upon space charge l i m i t e d c h a r a c t e r i s t i c s . The p-n-p diode c h a r a c t e r i s t i c can be analyzed us ing Po i s son ' s equation and the equation f o r current d e n s i t y . I f we assume p lanar geometry, and i f d i f f u s i o n currents are neglected where .v(E) i s the f i e l d dependent v e l o c i t y . Space charge l i m i t e d emission occurs at the punchthrough vol tage when the d e p l e t i o n reg ion has touched the emi t ter . V = flgfl • ' 3.2.2 As V i s increased above Vp, holes are i n j e c t e d by the emit ter i n t o the base r e g i o n , and Poisson's- equation must be in tegra ted with a p a r t i c u l a r v e l o c i t y - e l e c t r i c f i e l d dependence. A useful model for this dependence i s : v = M ( E » ) 3.2.3 where M i s the low f i e l d mobility, E.the c r i t i c a l f i e l d at which this power law becomes valid and K a constant designating f i e l d dependence. From experimental data due to Gibson, Granville and Ryder on holes i n germanium, i t i s observed that K varies from one to zero as E increases. Nevertheless under some conditions the range of E throughout most.Qf the base i s such that a fixed value can be reasonably "ascribed to K. The following specific values of K are particularly important as they have experimental j u s t i f i c a t i o n over certain ranges of f i e l d . (i) K = 1 for low f i e l d s This i s the case of constant mobility (v = uE) and has been treated by Shockley and Prim. Poisson's equation may be integrated i n terms of the transit time (t.) for minority carriers across the base. The resulting equations for V and J may be expressed i n terms of the parameter S = exp 3.2.4-J 3 2 E A „ 3.2.5 The current vol tage r e l a t i o n at large vol tages approaches a sympto t i ca l ly that fo r the i n t r i n s i c base, which i s : T 9 £ u v l . , P fi = sir3 p . c . b ( i i ) K » yk Prom the hole d r i f t v e l o c i t y measurements l i s t e d above, t h i s power exponent appears to be a good approximation over a considerable range of e l e c t r i c f i e l d . (1000 v o l t s / c m . to 10,000 v o l t s / c m . ) . Dacey (1953) has inve s t i ga ted t h i s power lav/. He makes the approximation that the power law v = ( E E ) holds throughout the e n t i r e base and a r r i v e s at the C h i l d ' s law analog S t i l l making the above approximation, a v o l t a g e -current dens i ty r e l a t i o n i s a r r i v e d at i n a manner analogous to Shockley and Prim by in t roduc ing the va r i ab l e r = \ ^ The fo l lowing r e l a t i o n i s then obta ined. V - Y " ( - 1 6 B * +36B-4-8B* +61nB+25) $.2.6 3(B-4B*+lnB+3) : l J = 2 . 846^ES V£, (B-4B* +lnB+3) * 3-2 .9 22 where B = exp ^qNr^ Por h igh appl ied vol tages and large currents, , the f i x e d charge i n the "base i s dominated by the i n j e c t e d holes and the above s o l u t i o n approaches a symptot i ca l ly eqn. 3.2.7. ( i i i ) K = 0 Thi s i s the case of a s a tura t ion v e l o c i t y , which takes place at very high f i e l d s ( for holes i n germanium E = 10,000 v o l t s / c m . ) . Po i s son ' s equation becomes €'g - 9H • i - 5.2.10 Integra t ing and choosing the o r i g i n i n the base at the point of zero f i e l d E - ± [ qN + / x 3.2.11 Integrat ing again V= KqH + Sr)£- Yi» + ^ £ 5.2.12 or I = (V-V pi) 3.2.13 Hence f o r the constant v e l o c i t y case we have a constant d i f f e r e n t i a l re s i s tance past punchthrough of magnitude 3.3 A model of m o b i l i t y - f i e l d dependence 23 As the f i e l d i n the base of a p-n-ptdiode va r i e s from zero at the emitter junc t ion to a high value at the c o l l e c t o r j u n c t i o n , i t may not always be a good approximation to use a p a r t i c u l a r power law f o r the v e l o c i t y - f i e l d dependence across the en t i r e base. The fo l lowing model approaches a symptot ica l ly both the constant m o b i l i t y case f o r low f i e l d s , and the constant v e l o c i t y case f o r h igh f i e l d s . Thi s model i s where v i s the d r i f t v e l o c i t y , / * t h e low f i e l d m o b i l i t y and base r e g i o n , i t i s convenient to introduce the fo l lowing dimensionless parameters f o r f i e l d s trength and dis tance across the base. 3.3.1 v 0 the sa tura t ion d r i f t v e l o c i t y . In i n t e g r a t i n g Po i s son ' s equation throughout the F = (qNv, + J)M. E Jv„ 3.3.2 and y = CqHvo + J)M x v * J€ 3.3-3 then i n t e g r a t i n g Po i s son ' s equation with the boundary c o n d i t i o n F = 0 when x = 0, we have y = F - l n ( F + 1) 3.3.4 For the low f i e l d , and therefore constant m o b i l i t y case we have F « 1 and hence F 2 7 - ^ - 3 . 3 . 5 For the high f i e l d case F 1 i . e . E >>v— we have a saturated d r i f t v e l o c i t y and y = F 3 . 3 . 6 T h i s i s simply the case of a constant d i f f e r e n t i a l r e s i s t ance , which was encountered e a r l i e r i n t r e a t i n g p-n-p diode c h a r a c t e r i s t i c s . For intermediate f i e l d s ( F ~ l ) we can make the approximate i n v e r s i o n F = y + Cy * 3 . 3 . 7 where the constant C i s chosen from a g raph ica l comparison with the inver ted r e l a t i o n y = F - l n ( F + l ) . To obta in a vo l tage-current dens i ty r e l a t i o n we must integrate the above f i e l d r e l a t i o n across the base. I t i s convenient to introduce another dimensionless quant i ty corresponding to the vo l t age . U = \ Fdy = (qNvp + j f JLXX V 3 . 3 . 8 J o € v 0 3 J * Equation 3 . 3 . 7 then becomes - y + cy* 3 . 3 . 9 with U m 0 at y • 0 . S o l v i n g , and e x p r e s s i n g the r e s u l t i n terms o f the u s u a l parameters, we have 3 - L The f i r s t term i s the p r e v i o u s e x p r e s s i o n f o r the const a n t d i f f e r e n t i a l r e s i s t a n c e , and the second term becomes a c o r r e c t i o n f a c t o r a t lower c u r r e n t s . F o r low f i e l d s C approaches J 2 . A v a l u e o f C o f 1 .3 g i v e s a f i t of 3.3*4 t o 3-3.7 w i t h an e r r o r o f l e s s than l C % . o v e r a f i e l d range o f 500 - 14,000 v o l t s / c m . The c o r r e c t i o n term i n t h i s model i s of g r e a t importance. F o r t y p i c a l v a l u e s o f the parameters o f 2N137 t r a n s i s t o r s , the two terms do not become equal u n t i l I = 30 ma, x The c o r r e c t i o n term corresponds t o I = ^ A g A t (V~YJP^ and i s analogous to the C h i l d ' s Law analogue I = 9A£AV 2. As C<vJ2 , the c o r r e c t i o n term i s i d e n t i c a l w i t h the analog." except t h a t c u r r e n t b e g i n s t o f l o w a t V = Vpo r a t h e r than 7 = 0. 3.4 Temperature v a r i a t i o n of Vp The punchthrough v o l t a g e i s n o r m a l l y g i v e n by the aNW* e x p r e s s i o n Vpo- g g 1» °ne obvious p o s s i b i l i t y of temperature v a r i a t i o n i s the thermal expansion o f the base w i d t h and the temperature dependence of £ . (N i s assumed independent o f temperature^ as the donors are a l l i o n i z e d a t a temperature well below any considered i n the present study). Considering t h i s form of temperature dependence Vp*(2a-b) 3.4.1 where a = ^  $pjt and b = ^  § § ' a n a s v a l u e ^.1 x 10~"^/°C (Conwell 1952) and b i s of the order of 2 x 10"V°C according to a t h e o r e t i c a l i n v e s t i g a t i o n by Antoncik. (Antonelk 1956). Hence f ^ x / - 2 x 1 0 - % by these agencies. For 2N137 t r a n s i s t o r s Vp =10 v o l t s and hence •j|E?/v- 2 x 1 0~ 5volts/deg. 3.4.2 A f u r t h e r mechanism of temperature v a r i a t i o n of Vp must be considered however. In a p-n-p t r a n s i s t o r i n the space-charge-limited condition, there w i l l be a p o t e n t i a l r i s e to a maximum immediately i n front of the emitter. In determining the Schenkel-Statz punchthrough voltage, one measures the difference i n p o t e n t i a l between c o l l e c t o r and emitter. As the i n t e r n a l contact p o t e n t i a l s are nearly the same f o r both the emitter and c o l l e c t o r junctions, t h e i r e f f e c t upon the observed punchthrough voltage i s n e g l i g i b l e as they cancel each other. However i n measuring t h i s d i f f e r e n c e , no account i s taken of the p o t e n t i a l r i s e immediately i n front of the emitter. F i g u r e 8 ELECTROSTATIC POTENTIAL ACROSS THE BASE OF p-n-p DIODE I n i n t e g r a t i n g P o i s s o n ' s e q u a t i o n a c r o s s t h e b a s e , i t i s u s u a l l y assumed t h a t t h e c h e m i c a l e m i t t e r p l a n e i s i d e n t i c a l w i t h t h e p o t e n t i a l maximum. However, t h e l o c a t i o n and magnitude o f t h i s i n t e r n a l p o t e n t i a l maximum i s dependent on c u r r e n t , and hence on t e m p e r a t u r e , t h u s p r o v i d i n g a model f o r t h e dependence o f p u n c h t h r o u g h v o l t a g e on t e m p e r a t u r e . When t h e e x p r e s s i o n f o r t h e p u n c h t h r o u g h v o l t a g e i s d e r i v e d by i n t e g r a t i n g P o i s s o n ' s e q u a t i o n , t h e o r i g i n f o r t h e p l a n a r gemoetry c o n s i d e r e d i s c o n v e n i e n t l y chosen t o be a t E = 0, t h a t i s , t h e p o t e n t i a l maximum. The p o s i t i o n o f t h e c h e m i c a l e m i t t e r p l a n e may t h e n be t a k e n a s x =-x c and t h a t o f t h e c o l l e c t o r as x = W-x0. I n t e g r a t i n g P o i s s o n ' s e q u a t i o n w i t h t h e s e boundary c o n d i t i o n s we have where V i s t a k e n t o be a n e g a t i v e v o l t a g e . The h e i g h t o f t h e p o t e n t i a l maximum f r o m t h e c h e m i c a l e m i t t e r p l a n e i s t h e n whence we c a n w r i t e V p - . V p o - 2 ( V p o V e ) * -* ' 3.4.5 x where V p e = qNW . P 2£ 2 8 Neglecting the relatively insignificant variation of W and £ with temperature we then have dVe qNW dx<> d T = ~ E W = - W dVo , j, , As Shockley and Prim have shown that the hole distribution near the potential maximum i s s t i l l approximately Maxwellian, we can assume I p , the^current flowing through the p-n-p diode at VP , i s of the form I P = 10 exp (- g 7?-) 3.4.7 corresponding to hole flow across the barrier of height V © . Since ^ r f f " i s typically 0.09 deg" 1 for the transistors examined, Vp i s a few tenths of a volt, and ^ J T ^ 40 v o l t " 1 we can estimate that \^\ > Therefore to a good approximation dV0 _ kT dl£ dT ~ qlp dT whence . ^ « 3 . 4 . 9 3.5 If a p-n-p transistor i s pulsed with a voltage equal to i t s punchthrough voltage, one expects a capacitance spike on the current pulse associated with the sweep out of majority carriers i n the n type base. From the area of this c a p a c i t a n c e s p i k e , i t s h o u l d b e p o s s i b l e t o d e t e r m i n e t h e q u a n t i t y o f c h a r g e s w e p t o u t o f t h e b a s e . T h i s c a n b e c o m p a r e d t o t h e e s t i m a t e d t o t a l c h a r g e i n t h e b a s e w h i c h i s e q u a l t o Q = A J W 3.5.1 w h e r e A m i s t h e m e a n c r o s s s e c t i o n a l a r e a o f t h e b a s e . 3.6 N o n - p l a n a r g e o m e t r y f o r t h e c o n s t a n t v e l o c i t y c a s e . F o r a n y n o n - p l a n a r g e o m e t r y , P o i s s o n ' s e q u a t i o n may b e e x p r e s s e d i n g e n e r a l c u r v i l i n e a r c o - o r d i n a t e s = q N A + I 3.6.I dr" w h e r e A ( r ) i s t h e a r e a o f t h e e q u i p o t e n t i a l s u r f a c e a t r a n d d e p e n d s o n t h e t y p e o f c u r v i l i n e a r c o - o r d i n a t e s y s t e m . F r o m G a u s s t h e o r e m € d ( E A ) = q N A d r 3.6.2 and T p . . -T(W) = )<• \ A d r ? J . 6 . 4 T I T d r a - • c o n s t . 3.6.5 F o r t h e c a s e o f A = c o n s t a n t .30-which i s the case of a constant d i f f e r e n t i a l re s i s tance f o r a s a tura t ion d r i f t v e l o c i t y . From 3.6.5 we see that f o r the case of a constant d r i f t v e l o c i t y we have a constant d i f f e r e n t i a l res i s tance independently of the p a r t i c u l a r c u r v i l i n e a r geometry. 3.7 High frequency impedance of a p-n-p diode past punchthrough. I f i n the i n t e g r a t i o n of Po i s son ' s equqtion across the case of a p-n-p diode one assumes a small s u n i s o i d a l v a r i a t i o n of the appl ied voltage V about i t s d . c . v a l v e , then there w i l l a l so be small v a r i a t i o n s of E ( x ) , J ( x ) , N(x ) , v(x) about t h e i r respect ive d . c . va lues . I f the small o s c i l l a t o r y valves of e l e c t r i c f i e l d and v e l o c i t y are v, and E, then dv d"E = M* = 5 3.7.1 E, whereM i s a d i f f e r e n t i a l m o b i l i t y . - -Po i s son ' s equation may then be expressed i n two par t s , a d . c . and an a . c . components. I f the a . c . component i s in tegra ted , then the a . c . impedance Z may be determined 7 - v> - w * ~ 1 ) * n o Z " " " F " v V 2> / 3 * 7 ' 2 V, TSy = I~v ^ where 2 = £—1 + J & W, where ^ = 27Tf and f i s the a . c . frequency. For the high W f i e l d s i n the base past punchthrough, — « v f t h e t r a n s i t time i n » the base, and fo r the constant v e l o c i t y case M =0. 3 1 Hence f o r the high f i e l d constant / .velocity s i t u a t i o n 7 « W l , / 1 2 - 4-,1to!t- (tor)* ^ " n Z~-V*.\ 2 4 \ . J 3-7*5 Consequently R ^ g ] = tan -^p . where £ i s the phase angle . The reactance i s accordingly small compared to the res i s tance astv"r'<<l fo r frequencies up to hundreds of megacycles. For instance i n a t y p i c a l c a s e ( v « ? 1 0 * m / s e c W^IO m, hence c = — = 10 sec . Thi s small value of t i s a l so a j u s t i f i c a t i o n f o r the neglect of hole recombination or t rapping e f fec t s i n the base r e g i o n . A fur ther point to be considered however i s whether t i s so small that the number of c o l l i s i o n s a hole suffers i n cross ing the base has dropped to the point where conventional cons iderat ions of d i f f u s i o n and m o b i l i t y break down. Using the r e l a t i o n mP M- where "Eg i s the r e l a x a t i o n time fo r h o l e s , m^ the e f f e c t i v e mass of holes and M the mobi l i ty , the magnitude of may be est imated. From t h i s i t appears that the number of c o l l i s i o n s across the base would average about 300. 32 CHAPTER IV DISCUSSION OF RESULTS 4.1 I , V c h a r a c t e r i s t i c s and n o n - l i n e a r m o b i l i t y . For the t r a n s i s t o r s cons idered , the f i e l d i n 90% of the base at the punchthrough voltage i s i n the range f o r the E * v e l o c i t y — f i e l d law di scussed i n s e c t i o n 3.2. As the vol tage i s increased through punchthrough the f i e l d throughout the base r a p i d l y r i s e s near the c o l l e c t o r to that requ i red f o r a saturated d r i f t v e l o c i t y . In applying the theor ie s of m o b i l i t y - f i e l d dependence developed i n sect ions 3.2 and 3.3 however, an area must be chosen to r e l a t e c u r r e n t , d e n s i t y to c u r r e n t . I f t h i s area i s taken to be the area of the emi t ter , then the t h e o r e t i c a l c h a r a c t e r i s t i c s show an increase i n current with voltage which i s much f a s t e r than that observed exper imenta l ly . An explanat ion of t h i s phenomenon can be made i n terms of emi t t ing area . I t may be assumed that at the punchthrough vol tage the space charge column touches the emit ter and the emit t ing area increases wi th i n c r e a s i n g voltage u n t i l the e n t i r e emit ter i s covered. A model exp l a in ing t h i s e f fec t could be made by cons ider ing the t r a n s i s t o r as two or more t r a n s i s t o r s i n p a r a l l e l , one with the Schenkel-Statz punchthrough voltage and an area which i s a small f r a c t i o n of the emit ter area , and the others wi th the remainder of the emit ter area and higher punchthrough vo l tages . 3 3 At h igh voltages where the f i e l d i n the base becomes s u f f i c i e n t l y h igh the d r i f t v e l o c i t y saturates and leads to a constant d i f f e r e n t i a l r e s i s t a n c e . The value of t h i s r e s i s t ance c a l c u l a t e d fo r t r a n s i s t o r # 3 7 from equation ( 3 . 2 . 1 4 ) i s 14 ohms while a d i f f e r e n t i a l re s i s tance of about 12 ohms r e s u l t s from . the measured c h a r a c t e r i s t i c . I t i s observed from l o g I - log V c h a r a c t e r i s t i c s that J (V-Vp f over two decades of c u r r e n t . However i n s ec t ion 3 . 3 the model which takes i n t o account both low and high f i e l d m o b i l i t y - f i e l d dependence leads to T ~ . 9 A € x c (V-V D ) x 8 W f o r currents up to severa l mil l iamps before the other term i n ( 3 . 3 . 1 1 ) p e r t a i n i n g to the s a tura t ion d r i f t - v e l o c i t y becomes dominant. I f i t i s assumed that the space charge d e p l e t i o n reg ion touches the emit ter as a sphere touches a plane, then for,^a small increase of r (the radius of the sphere) above the value r 0 r equ i red to touch the plane at punchthrough,we have a covered area equal to A = 2 7T r e ( r - r 0 ) 4 .1 .1 Now f o r vol tages jus t past punchthrough i f r T „ V-Vp. = flSp-(r-r#) 4 .1 .2 and hence A - (V-V f . ) ~ p 34 Hence i f t h i s l i n e a r v a r i a t i o n of A with excess voltage i s taken in to account the low f i e l d component of (3.3.11) lead to a cubic dependence of current on vol tage above punchthrough exact ly as observed. The.above assumption of non-planar geometry however would have no e f fect on the ul t imate attainment of a constant d i f f e r e n t i a l r e s i s t a n c e , as t h i s i s independent o f any choice of c u r v i l i n e a r geometry with c y l i n d r i c a l symmetry as shown i n sec t ion 3.6 . 4.2 Temperature dependence of space-charge- l imited c h a r a c t e r i s t i c s f o r the p-n-p d iode . Two fac tor s must be taken i n t o account when cons ider ing the temperature dependence of space-charge- l imited c h a r a c t e r i s t i c s . They are the base-generated current Ip f lowing at Vp f o r the temperature at which the c h a r a c t e r i s t i c i s measured, and'the temperature dependence of the punchthrough vo l t age . The measurement of the l a t t e r i s discussed i n sec t ion 2.5 and. a t y p i c a l p l o t o f Vp v s . T f o r t r a n s i s t o r #37 i s g iven i n f i gure 2. A t h e o r e t i c a l explanat ion of t h i s temperature dependence i s made i n s e c t i o n 3 . 4 . The temperature c o e f f i c i e n t f o r t r a n s i s t o r #37 from 0*C to 30*0 was 1.3x10*"* v o l t s / d e g . I f a value of 0.2 i s assumed for where x 0 i s the dis tance from the chemical emitter j u n c t i o n to the p o t e n t i a l b a r r i e r , • t h e n equation 3 .4.9 leads to a theore t i ca l value of 1.1x10 v o l t s / d e g . f o r . The height of the p o t e n t i a l b a r r i e r V„ has the 3 5 value of 0.18 v o l t s which i s reasonable i n r e l a t i o n to the hole current t ransmitted across i t . I f the base-generated current Ip f lowing at Vp i s subtracted from the current observed at higher temperatures and the corrected current i s p l o t t e d against V-Vp where Vp i s the punchthrough vol tage at the temperature at which the c h a r a c t e r i s t i c i s taken, a l l the high temperature po in t s f a l l c l o s e l y on the room temperature curve i n f igure 2. There i s no fur ther change of the c h a r a c t e r i s t i c down to lower temperatures (-60 C) as Ip becomes very small and the punch-through vol tage remains e s s e n t i a l l y constant . Hence the corrected space charge l i m i t e d c h a r a c t e r i s t i c i s s u b s t a n t i a l l y © o independent of temperature from -60 C . t o 90 C. which i s i n complete accordance with the theory of space charge l i m i t e d cur rent s . '-It i s noted that the c o n d i t i o n of i n t r i n s i c conduction i n the base i s reached at the high temperatures used without any marked change of the c h a r a c t e r i s t i c . 4 . 3 Trans ient response of a space-charge- l imited d iode . The charge swept out at punchthrough for the p-n-p diode #21 was observed i n terms of the t rans ient spike on _ H sudden app l i ca t ion" of Vp and found to be 8x10 coulombs. The value expected for charge swept out of the base by simple cons iderat ions of base volume and base impuri ty dens i ty i s by equation 3 . 5 * 1 equal to qNAW = 8x10 coulombs. Th i s large discrepancy i s at present unexplained. I t might be argued that the charge content of the spike i s determined by charging the capacitance of the diode up to the app l ied vo l t age ; but i t may r e a d i l y be seen that t h i s i s i d e n t i c a l with the value considered.above. F o r : = AqNW 4 .4 Capacitance of reverse b iased j u n c t i o n s . In the present study capacitance measurements were made on the t r a n s i s t o r s mentioned i n Table 1 on each j u n c t i o n f o r reverse b iases up to double the punchthrough voltage with no change of the smooth l i n e of Slope- ^ on a l o g - l o g p l o t . Th i s r e s u l t i s d i f f i c u l t to exp l a in t h e o r e t i c a l l y as the base should be completely depleted of major i ty c a r r i e r s at V P and. the capacitance one would th ink would therea f te r remain constant . However as the base wafer i s very much l a r g e r i n area than the emitter and c o l l e c t o r areas (see f i gure 3) the space charge d e p l e t i o n reg ion may i n fac t continue to expand away from the emit ter and c o l l e c t o r in to the en t i r e base l a y e r . Barker reports that fo r one GE 2N137 t r a n s i s t o r s tudied the f l o a t i n g junc t ion capacitance showed an abrupt increase as the voltage increased through Vp . A fur ther study of many types o f punchthrough t r a n s i s t o r s appears to be necessary to f u l l y exp la in t h i s phenomenon. 37 , CHAPTER V CONCLUSIONS V 5.1 Areas of agreement of theory and experiment. The p-n-p diodes s tud ied a l l have a wel l ' de f ined Schenkel-Statz punchthrough voltage and show no marked i r r e g u l a r i t i e s i n the dependence of f l o a t i n g emit ter p o t e n t i a l on c o l l e c t o r p o t e n t i a l past punchthrough. The apparent v a r i a t i o n of emit t ing area wi th voltage above punchthrough leads only to a reasonable explanat ion of the observed c h a r a c t e r i s t i c s at lower vol tages and the constant d i f f e r e n t i a l r e s i s t ance p r e d i c t e d by theory fo r the high f i e l d constant v e l o c i t y case i s a l so obta ined, and i s of the magnitude expected. A f t e r the diode c h a r a c t e r i s t i c s are corrected f o r temperature dependence of the base-generated current f lowing at punchthrough and the temperature dependence of the punchthrough vo l t age , they are found to be s u b s t a n t i a l l y independent of temperature as required by the theor ie s of space-charge- l imited hole flow developed i n t h i s t h e s i s . The temperature dependence of the punchthrough voltage i s explained i n both s ign and magnitude wi th a model which takes i n t o account the p o s i t i o n of the p o t e n t i a l b a r r i e r i n the base and leads to a value of p o t e n t i a l b a r r i e r height which i s qui te reasonable i n r e l a t i o n to the t ransmit ted hole c u r r e n t . 38 The capacitance measurements made on the c o l l e c t o r and emitter junct ions j u s t i f y the assumptions of step junct ions and uniform r e s i s t i v i t y throughout the "base which was made i n the t h e o r e t i c a l d i s c u s s i o n . 5.2 Outstanding problems. A major problem remaining i n the explanat ion of p-n-p diode c h a r a c t e r i s t i c s i s that of the extension of the space charge dep le t ion region throughout the base to the emi t ter . E v i d e n t l y a f t e r the punchthrough voltage has, been reached the emitter j u n c t i o n area covered by the space charge r eg ion increases wi th appl ied vo l t age ; and from capacitance measurements i t appears that the space charge continues to widen in to the en t i r e base l a y e r past punchthrough. The charge swept out of the base layer ' as i n d i c a t e d by the t rans ient current pulse at punchthrough i s much l a r g e r than that expected from cons iderat ions of impuri ty d e n s i t y , base width, and mean area of junc t ions . Thi s phenomenon has no ready explanat ion. A fu r ther problem which may be inves t i ga ted i s the temperature dependence of punchthrough. The parameters used^such as p o s i t i o n i n the base and height of the p o t e n t i a l b a r r i e r i n f ront of the emitter,may p o s s i b l y be determined d i r e c t l y by su i tab le experiments and the v a l i d i t y of the model fo r temperature dependence of the punchthrough voltage more r igourous ly checked. 39 REFERENCES Antonc ik , E . , On the Theory of the Temperature Dependence of the Ref rac t ive Index of Homopolar C r y s t a l s , Czech. J . P h y a . , 6, pp. 204-216, 1956. Barker , A . S . , A Study of Space-Charge and. Avalanche M u l t i p l i c a t i o n Processes i n Germanium, Thes i s f o r Master of Science degree, U n i v e r s i t y of B r i t i s h Columbia, September 1957* Brown, W . L . , N-Type Surface C o n d u c t i v i t y on P-Type Germanium, Phys. R e v . , 91, pp. 518-527, August 1953. Conwel l , E . M . , Proper t i e s of S i l i c o n and Germanium;I, P roc . IRE, 40, pp. 1327-1337, 1952. ] Dacey, G . C , Space-Charge L imi ted Hole Current i n Germanium, Phys. Rev . , 90, pp. 759-763, June 1953>t Emei s , ; R . , H e r l e t , A . , The Blocking C a p a b i l i t y o f_Al loyed S i l i c o n Power T r a n s i s t o r s , -^roc. IRE, 46, pp. 1216-1220. June 1958. Gibson, A . F . , G r a n v i l l e , J . W . , The Measurement of D r i f t M o b i l i t y i n Germanium at High E l e c t r i c F i e l d s , Journal of E l e c t r o n i c s , V o l . 2, pp. 259-266, November 1956. M i l l e r , S . L . , Avalanche Breakdown i n Germanium, Phys. R e v . , 99i pp. 1234-1241, August 1955. i Ryder, E . J . , M o b i l i t y o f Holes and E l e c t r o n s i n High E l e c t r i c  F i e l d s , Phys. Rev . , 90, pp. 766-769, June 195% Schenkel , H., S t a t z , H . , Voltage Punch-Through and Avalanche Breakdown and t h e i r E f f e c t on the JWaximum Operat ing Voltages  f o r Junct ion T r a n s i s t o r s , J?x>oc. flat''j. E l e c t r o n i c s C o n f . . ' TO, pp. 614-625, 1954.— Shockley, W., The Theory of P-N Junctions i n Semiconductors and  P-N Junct ion T r a n s i s t o r s , B e l l Sys . Tech. J r . 28, p p < 435-4S9, July' 1949. Shockley, W., and Pr im, B . C . , Space-Charge L imi ted Emiss ion i n  Semiconductors, Phys. Rev. 90, pp. 753-758, June.1953. 

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