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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  •  Physics  We a c c e p t t h i s t h e s i s as required  conforming to  standard  The U n i v e r s i t y of B r i t i s h September 1 9 5 8  Columbia  the  V  ABSTRACT A study has been made of s p a c e - c h a r g e - l i m i t e d h o l e f l o w i n germanium by i n v e s t i g a t i n g the characteristics  current-voltage-temperature  of s e l e c t e d p - n - p t r a n s i s t o r s  w i t h the base o p e n - c i r c u i t e d . so as t o minimize the e f f e c t  used as  These t r a n s i s t o r s  diodes  were  selected  of avalanche m u l t i p l i c a t i o n .  'These d i o d e s pass h o l e c u r r e n t t h r o u g h the base a voltage designated  after  as the punchthrough v o l t a g e has been  a p p l i e d to d e p l e t e the n type base of e l e c t r o n s .  The  r e s u l t i n g s p a c e - c h a r g e - l i m i t e d c u r r e n t above punchthrough has been c l o s e l y s t u d i e d and a l s o i t s  temperature  dependence.  To e x p l a i n the form o f these c h a r a c t e r i s t i c s ,  published  d a t a have been used f o r the r e l a t i o n s h i p between e l e c t r i c  field  and c a r r i e r d r i f t v e l o c i t y f o r h o l e s i n germanium i n o r d e r to c o n s i d e r the h o l e flow t h r o u g h the h i g h f i e l d base.  I t was f u r t h e r found n e c e s s a r y  v a r i a t i o n of e f f e c t i v e is  r e g i o n o f the  to c o n s i d e r the  e m i t t i n g a r e a as the a p p l i e d  voltage  i n c r e a s e d p a s t punchthrough. For h i g h a p p l i e d v o l t a g e s  i n the b a s e ,  and hence h i g h a p p l i e d  a constant d i f f e r e n t i a l r e s i s t a n c e  is  fields  o b t a i n e d of  magnitude about equal t o t h a t expected t h e o r e t i c a l l y f o r a c o n s t a n t d r i f t v e l o c i t y o f h o l e s i n the b a s e . The temperature dependence of t h i s c u r r e n t can be satisfactorily  e x p l a i n e d by the temperature v a r i a t i o n o f the  vi  b a s e - g e n e r a t e d c u r r e n t and of the punchthrough v o l t a g e A satisfactory  itself.  model of the l a t t e r v a r i a t i o n has been made by  c o n s i d e r i n g the temperature v a r i a t i o n o f the p o t e n t i a l " b a r r i e r at  the e m i t t e r j u n c t i o n .  If  the observed c h a r a c t e r i s t i c s  c o r r e c t e d f o r these v a r i a t i o n s ,  there i s  are  found to be  n e g l i g i b l e v a r i a t i o n of the s p a c e - c h a r g e - l i m i t e d h o l e -flow. C a p a c i t a n c e measurements f o r these d i o d e s .  were made on b o t h j u n c t i o n s  From t h e s e measurements  the assumption o f  a step j u n c t i o n and o f u n i f o r m i m p u r i t y 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 c a p a c i t a n c e  observed as the v o l t a g e through v o l t a g e  was i n c r e a s e d t h r o u g h the p u n c h -  c o n t r a r y to the f i n d i n g s o f B a r k e r .  The o r i g i n a l t h e o r i e s  of S h o c k l e y - P r i m - D a c e y were  extended t o i n c l u d e the e f f e c t s over a wide range o f 1 - f i e l d s (iii)  was  (ii)  of  (i)  mobility variation  n o n - p l a n a r geometry and  the p o t e n t i a l d i s t r i b u t i o n near the e m i t t e r j u n c t i o n .  In presenting  t h i s thesis i n p a r t i a l fulfilment of  the requirements f o r an advanced degree at the  University  of B r i t i s h Columbia, I agree that the Library s h a l l make i t f r e e l y available for reference  and study.  I further  agree that permission for extensive copying of t h i s thesis for scholarly purposes may  be granted by the Head of  Department or by his representative.  my  It i s understood  that copying or publication of t h i s thesis f o r f i n a n c i a l gain s h a l l not be allowed without my written permission.  Department of  Physics  The University of B r i t i s h Columbia, Vancouver 8, Canada. Date  September 29,  1958.  ii TABLE OF COKTENTS CHAPTER I.  PAGE  INTRODUCTION 1.1  .  Problems a r i s i n g i n the study o f  space-  charge-limited currents 1.2 II.  1  Review of p r e v i o u s work  . . . . . . . . .  4  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  .  2.2  Measurement of punchthrough v o l t a g e  and  temperature 2.5  .  .  .  of  14 space-charge-limited  c u r r e n t s i n Germanium  14  2.4  C a p a c i t a n c e Measurements  15  2.5  Temperature dependence of  space-charge-  l i m i t e d current 2.6  T r a n s i e n t response o f the  16 space-charge-  limited. diode III.  10  its  dependence  D.C. characteristics  1  16  THEORETICAL INVESTIGATION 5.1  R e l a t i v e importance of d r i f t  17 and d i f f u s i o n  components o f h o l e c u r r e n t  17  5.2  I n f l u e n c e of f i e l d - d e p e n d e n t 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  T r a n s i e n t response of p-n-p d i o d e  28  5.6  N o n - p l a n a r geometry f o r the  . . .  constant  v e l o c i t y case 5.7  29  High f r e q u e n c y impedance o f a p - n - p d i o d e p a s t punchthrough  •  50  iii Table of Contents  (Cont'd)  CHAPTER IV.  PAGE DISCUSSION 4.1  3 2  I.V characteristics  and n o n - l i n e a r  mobility 4.2  32  Temperature dependence of limited characteristics diode  4.3  space-charge-  f o r the p-n-p  *  3 4  T r a n s i e n t response of a s p a c e - c h a r g e l i m i t e d diode  4.4 V.  Capacitance  35  of r e v e r s e b i a s e d  junctions  .  CONCLUSIONS 5.1  5.2  Areas of agreement  3 6 3 7  o f t h e o r y and  experiment  3 7  O u t s t a n d i n g problems  3 8  REFERENCES  \.  3 9  iv LIST OF TABLES AND FIGURES  TABLE  F a c i n g Page T r a n s i s t o r dsta  1  7  FIGURE 1  Circuit  l o r measurement o f  Schenkel-Statz  punchthrough v o l t a g e 2  3  Temperature dependence of the punchthrough voltage  4  2N137  3  S t r u c t u r e o f GE  transistor  4  Current-voltage characteristic  o f p-n-p  diode 5  14  Temperature dependence o f p-n-p d i o d e characteristic  6  5  Circuit  15  f o r measurement o f j u n c t i o n  capacitance field  16  7  Electric  8  E l e c t r o s t a t i c p o t e n t i a l a c r o s s the base o f a p-n-p diode  dependence of d r i f t v e l o c i t y  19  27  vii  ACKNOWLEDGEMENTS  I w i s h t o thank P r o f e s s o r and  R . E . Burgess f o r h i s h e l p  guidance throughout t h i s i n v e s t i g a t i o n ,  many v a l u a b l e  and f o r  s u g g e s t i o n s d u r i n g the p r e p a r a t i o n of  his this  thesis. Support from a Defence Research Board grant d u r i n g the summers o f 1957 and 1958 i s  gratefully  acknowledged.  1  CHAPTER I INTRODUCTION 1.1  Problems a r i s i n g i n the study o f space charge l i m i t e d currents. C e r t a i n selected p-n-p j u n c t i o n t r a n s i s t o r s  suitable  with  base w i d t h s and base i m p u r i t y c o n c e n t r a t i o n s can be,  used to study s p a c e - c h a r g e - l i m i t e d in particular,  the e l e c t r i c f i e l d  c u r r e n t s i n germanium, and dependence of c a r r i e r  drift  velocity. C r i t e r i a f o r s e l e c t i o n of t h e s e t r a n s i s t o r s established  i n this  are  thesis.  These t r a n s i s t o r s  are chosen so as t o a v o i d  significant  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 o f atoms by f a s t moving c a r r i e r s .  If  one o f these  lattice  transistors  i s used as  a p - n - p d i o d e w i t h the base o p e n - c i r c u i t e d , t h e n  as a f i r s t  a p p r o x i m a t i o n i t may be assumed t h a t no c u r r e n t  w i l l flow u n t i l a p a r t i c u l a r voltage through v o l t a g e )  is  reached.  ( c a l l e d the punch-  A t t h i s v o l t a g e , the base  l a y e r i s w h o l l y d e p l e t e d of m a j o r i t y c a r r i e r s collector  and t h e  space charge d e p l e t i o n r e g i o n makes c o n t a c t  the e m i t t e r .  Copious e m i s s i o n of h o l e s t h e n t a k e s  with  place  from the e m i t t e r i n t o the space charge r e g i o n o f the base where a h i g h e l e c t r i c f i e l d  is  present.  2  T r a n s i s t o r structures of t h i s type have been analyzed by Shockley and Prim diodes.  They consider n - i - n and  (1953).  p-n-p  However they do not take into account the e l e c t r i c  f i e l d dependence of m o b i l i t y which i s important at the high f i e l d s obtained near the c o l l e c t o r i n narrow base junction transistors. For the high f i e l d s i t u a t i o n , models of d r i f t  velocity  - e l e c t r i c f i e l d dependence may be made, and approximate integrations c a r r i e d out, leading to space-cnarge-limited diode c h a r a c t e r i s t i c s .  One  of the main problems to be  resolved i s a s a t i s f a c t o r y model of diode c h a r a c t e r i s t i c s in this  way. Further complications i n actual punchthrough  phenomena may  occur however.  are of d i f f e r e n t area. geometry may  Emitter and c o l l e c t o r contacts  Hence models assuming  non-planar  sometimes be necessary; although i t can be  shown that a saturated d r i f t v e l o c i t y (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 p a r t i c u l a r c y l i n d r i c a l geometry. At high temperatures,  the t r a n s i s t o r base region  becomes i n t r i n s i c and the usual e x t r i n s i c base a n a l y s i s must be f u r t h e r extended.  Also at these temperatures  the  saturation d i f f u s i o n current of the c o l l e c t o r j u n c t i o n ( I c o ) becomes comparatively l a r g e .  A l l these e f f e c t s lead to  modification of the diode c h a r a c t e r i s t i c s .  Figure 1 CIRCUIT FOR MEASUREMEEiT OF SCFENKEL-3TATZ FUSCHTHRQUGH VOLTAGE  p EMITTER  N B A S E  P COLLECTOR  COLLECTOR-EMITTER VOLTAGE CHARACTERISTIC FOR PNP PUNCHTHROUGH TRANSISTOR  3 The  technique used to determine  the punchthrough  voltage i s  due t o Schenkel and S t a t z (1954).  circuit  shown i n f i g u r e  is  1.  As the c o l l e c t o r v o l t a g e i s negative,  a v o l t a g e Vp  is  The necessary-  made  increasingly  reached at which the  emitter  v o l t a g e becomes l o c k e d t o t h a t of the c o l l e c t o r V  e  = V  - Vp .  c  The e m i t t e r t h e n a c t s as  connected t o the c o l l e c t o r  if  through a b a t t e r y  such t h a t  it  were  of e . m . f . - V  The p h y s i c a l mechanism e x p l a i n i n g t h i s phenomena that of c o l l e c t o r depletion l a y e r widening. reverse b i a s i s (which i s  increased,  depleted  As the  j u n c t i o n to i n c l u d e more f i x e d c h a r g e .  base r e g i o n , emitter  through which i t  j u n c t i o n at  It  is  spreads u n t i l  it  collector  touches  the punchthrough v o l t a g e .  assumed t h a t t h i s  layer,  resistivity  voltage i s  as the d i o d e v o l t a g e c o r r e s p o n d i n g t o g r e a t l y  the  This voltage  i s v e r y w e l l d e f i n e d by the Schenkel S t a t z type measurement.  is  T h i s space charge  d e p l e t i o n r e g i o n i s m a i n l y c o n f i n e d t o the h i g h  of the same  increased  c u r r e n t flow due to the onset of h o l e i n j e c t i o n from the emitter.  This latter  is  not so w e l l d e f i n e d ,  possibly  to the f a c t  t h a t the space charge zone i s not  delineated,  but has a boundary smeared out over  Debye l e n g t h s  (L0), L  D  due  sharply several  where = /0 € k T A T T q *  N  .  collector  the donor i o n space charge  of e l e c t r o n s ) spreads from the  p  (1.1.1)  Figure  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 c o n s t a n t k = Boltzmann's constant,  <[ = e l e c t r o n i c charge and N =  i m p u r i t y d e n s i t y i n the b a s e . 14-  20 ohm-cm. (N = 10  of germanium,  A t room t e m p e r a t u r e ,  in  - 3  cm.  )  germanium L^a?  1.3 x 10  cm.  A f u r t h e r problem w i t h r e g a r d t o the S c h e n k e l S t a t z punchthrough v o l t a g e effect  is  its  temperature  dependence.  This  cannot be e x p l a i n e d by t h e r m a l expansion of the b a s e ,  o r by v a r i a t i o n o f € w i t h t e m p e r a t u r e , which must be c o n s i d e r e d i s  but a f u r t h e r  factor  the temperature dependence of  the p o t e n t i a l b a r r i e r o f the e m i t t e r which i s due to space charge of h o l e s i n the base i n f r o n t of the  the  emitter.  The r e l a t i o n s h i p of V P t o t h i s p o t e n t i a l i s d e r i v e d from a double i n t e g r a t i o n of P o i s s o n ' s  e q u a t i o n a c r o s s the base  w i d t h , w i t h the c o n d i t i o n . " t h a t no c u r r e n t i s C r i t e r i a for transistor results,  are p r e s e n t e d  s e l e c t i o n and e x p e r i m e n t a l  i n Chapter I I  Theoretical investigations  flowing.  of t h i s  thesis.  are p r e s e n t e d i n C h a p t e r  and a d i s c u s s i o n of t h e i r success i n e x p l a i n i n g the results 1.2  is  III, observed  presented i n Chapter IV.  Review o f p r e v i o u s work. A major paper on space charge l i m i t e d e m i s s i o n i n  s e m i - c o n d u c t o r s i s by S h o c k l e y and Prim (1953). c o n s i d e r the s i t u a t i o n i n t r a n s i s t o r space charge  They  p h y s i c s analogous  l i m i t e d e m i s s i o n i n vacuum t u b e s .  The  to  first  Figure 3 STRUCTURE OF GE 2R137 TRANSISTOR  EMITTER  ^0 COLLECTOR  BASE  structure  they analyze  is  the n - i - n d i o d e ,  f o r which t h e y  o b t a i n the f o l l o w i n g r e l a t i o n between c u r r e n t d e n s i t y and a p p l i e d v o l t a g e analogous t o C h i l d ' s law i n vacuum tube electronics J  -  9£>U-V*  ,  1.2.1  where J = c u r r e n t d e n s i t y 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 .  This mobility i s  independent o f the e l e c t r i c A major premise that  the c u r r e n t i s  diffusion, than ~  c h i e f l y c a r r i e d by d r i f t  which i s a* v a l i d assumption  electronic  T is  constant,  field.  i n d e r i v i n g the.above  (25 m i l l i v o l t s at  constant,  assumed t o be a  relation rather  at v o l t a g e s  2 9 0 ° K ) where k i s  the a b s o l u t e t e m p e r a t u r e ,  than greater  Boltzmann's  and 9 i s  the  charge.  S h o c k l e y and Prim a l s o a n a l y z e p - n - p d i o d e s paper.  is  In i n t e g r a t i n g  base i n a p - n - p d i o d e ,  in their  P o i s s o n ' s e q u a t i o n a c r o s s the n type one must c o n s i d e r the f i x e d  positive  charge d e n s i t y 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 emitted h o l e s . charge  the f u r t h e r p o s i t i v e  For these diodes,  charge due  c o n d i t i o n s of  to  space  l i m i t e d e m i s s i o n o c c u r when the d e p l e t i o n l a y e r  t o i o n i z e d donors p e n e t r a t e s the b a s e .  This voltage  d e r i v e d from a double i n t e g r a t i o n of P o i s s o n ' s a c r o s s the base w i t h the c o n d i t i o n  J = 0.  due  is  equation  This leads  to  6 1.2.2  if  we n e g l e c t  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  emitter junction. through v o l t a g e .  This voltage i s  Por the case of c o n s t a n t m o b i l i t y ,  i n terms of the t r a n s i t  f o r any c u r r e n t  time of i n j e c t e d  A voltage-current  density  carriers  derived  constant  F o r l a r g e v a l u e s of the c u r r e n t  their derived characteristic  density  approaches a s y m p t o t i c a l l y  C h i l d ' s law a n a l o g f o r an i n t r i n s i c base diode as injected  carriers  i n the base  density  across  relation is  i n t h i s manner by S h o c k l e y and Prim f o r the mobility case.  the  c a l l e d the p u n c h -  P o i s s o n ' s e q u a t i o n may be i n t e g r a t e d  the b a s e .  of  the  the  t h e n dominate over the bound space charge  region.  S h o c k l e y and Prim a l s o i n c l u d e i n t h e i r a n a l y s i s effects  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 l a r g e d i s t a n c e s from the p o t e n t i a l potential  zero f i e l d point, i n the base i s  s l i g h t l y from the c o r r e s p o n d i n g analog.  maximum, the f i e l d and  i n the ba'se approach the C h i l d ' s law a n a l o g .  p o s i t i o n of the  The e f f e c t  current i s  important however, because except near the  shifted  not  potential  c h i e f l y c a r r i e d by d r i f t . .  S h o c k l e y - P r i m t h e o r y has been extended by Dacey t o the h i g h f i e l d case by assuming the relation  The  p o i n t of the C h i l d ' s law  of the d i f f u s i o n  maximum the c u r r e n t i s  the  velocity-field  TABLE I BASIC PARAMETERS OF SELECTED TRANSISTORS (GE TYPE 2N137)  TRANSISTOR NO  21  19  37  EMITTER AREA CM*  0.47x10"  COLLECTOR AREA CM 3  0.88x10"*  Ac Ii  1.9  EASE WIDTH CM  1. 3 x l d "  IMPURITY DENSITY CM"3  8.7x10  MEASURED V P VOLTS RESISTIVITY OHM- CM E(COLLECTOR) AT V P KILOVOLTS/CM  3  3  1.3xl0  - 3  5.9xlo'  3  X  ,.72xl0""  l.lxio"*  1.5  1.3x10"  4.4xlo'  5.8  4.4  17  24  28  10  8.9  6.9  8.6  3  3  X  7 v = M (E  0  E)  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  at which t h i s half-power law comes into e f f e c t .  field  T h i s power  law i s j u s t i f i e d by v e l o c i t y - e l e c t r i c f i e l d measurements by Ryder (1953) who has used pulse techniques to obtain h i s data and by Gibson and G r a n v i l l e (1956) who have used a microwave absorption technique.  I t appears that t h i s law  becomes v a l i d f o r holes i n germanium from a f i e l d of about 1000 volts/cm. up to 10,000 volts/cm. For voltages w e l l beyond Vp, the f i x e d charge i s swamped by i n j e c t e d holes, and i f the approximation can be .x.  made that v = M ( E E) 0  throughout the entire base region,  then a new C h i l d ' s law analog i s obtained. J  =  1.43eMJst j |  1.2.4.  As i n the Shockley-Prim case of constant m o b i l i t y , the fixed charge i n the base cannot be neglected f o r intermediate voltages.  In a manner analogous to Shockley  and Prim a voltage-current density r e l a t i o n i s derived f o r t h i s case. Emeis and Herlet (1958) have made several experimental s i l i c o n a l l o y t r a n s i s t o r s and have f i t t e d Prim theory to t h e i r measured c h a r a c t e r i s t i c s . n-p-n  Shockley-  Two s e r i e s of  s i l i c o n a l l o y t r a n s i s t o r s were made, one with base .,  thicknesses of 40-45 microns and the other 55-60 microns.  8 V a r i o u s 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 transistors,  punchthrough t r a n s i s t o r s ,  showing b o t h these a t t r i b u t e s . negative resistance  In f i t t i n g it  Turnover c o n d i t i o n s  d i o d e s have been c o n s i d e r e d  (1957).  by B a r k e r  data,  transistors  The l a t t e r can have  characteristics.  f o r these n e g a t i v e r e s i s t a n c e  and  S h o c k l e y Prim t h e o r y t o e x p e r i m e n t a l  i s n e c e s s a r y to assume an e f f e c t i v e  r e l a t e current to current d e n s i t y .  area  to  T h i s a r e a can be  determined from c a p a c i t a n c e measurements on the e m i t t e r 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; o r from the c a p a c i t a n c e at punchthrough i f the base w i d t h i s  known.  Emeis and H e r l e t a p p l y S h o c k l e y P r i m t h e o r y t o e x p e r i m e n t a l d a t a by c h o o s i n g two parameters Vp  equivalent to  and the e m i t t i n g a r e a t o g i v e a b e s t f i t .  S c h e n k e l - S t a t z ounchthrough v o l t a g e s Besides determination of a r e a ,  their  Apparently  were not measured.  capacitance bias  measurement  on a j u n c t i o n can be used t o determine whether o r n o t the j u n c t i o n has a step i n i m p u r i t y c o n c e n t r a t i o n .  Por a step  j u n c t i o n , w i t h the a c c e p t o r i m o u r i t y d e n s i t y Np i n the p type m a t e r i a l v e r y much g r e a t e r t h a n the donor i m p u r i t y d e n s i t y Nn i n the n type m a t e r i a l , and i f Nn i s  constant,  C where-C i s  1.2.5.  the d i f f e r e n t i a l j u n c t i o n c a p a c i t a n c e  (as  measured w i t h a s m a l l - a m p l i t u d e , low f r e q u e n c y a . c .  signal)  and y i s the i n t e r n a l p o t e n t i a l d i f f e r e n c e of the c o l l e c t o r o  junction.  Hence a l o g C - log V p l o t of t h i s r e l a t i o n gives  a straight l i n e of slope  f o r Y>>%  and (fi can be estimated  from the departure from l i n e a r i t y at low V.  A p l o t of t h i s  sort can be used as a check on the degree to which a junction i s a step j u n c t i o n . Barker has reported that f o r some t r a n s i s t o r s studied these p l o t s show a marked step up i n the capacitance of the junction at the punchthrough voltage, which may be explained by assuming that the space charge d e p l e t i o n region has touched the emitter, switching the emitter capacitance i n t o the c i r c u i t . present  study.  This r e s u l t has not been confirmed  i n the  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 The p - n - p s t r u c t u r e s  for  study.  i n v e s t i g a t e d i n t h i s study were  chosen so as t o a v o i d avalanche breakdown due to c a r r i e r multiplication.  Fields greater  than 1000volts/cm  o b t a i n e d i n the base of t r a n s i s t o r s  are  always  s t u d i e d under space-  charge l i m i t e d c o n d i t i o n s , thus c a u s i n g a r e d u c t i o n o f mobility; present  this  thesis  effe'ct  is  one of those under study i n the  i n c o n s i d e r i n g m o d i f i c a t i o n s of the S h o c k l e y -  Prim theory. The v a r i o u s parameters i s presented i n table (a)  of the t r a n s i s t o r s  under study  I.  A c r i t e r i o n can be drawn up f o r the s e l e c t i o n o f  transistors  f o r punchthrough measurements m i n i m i z i n g  behaviour.  Miller  avalanche  (1955) has shown t h a t t h i s can be a n a l y z e d  i n terms of the avalanche m u l t i p l i c a t i o n f a c t o r M. to a p-n j u n c t i o n and i s  M pertains  the r a t i o o f the t o t a l c u r r e n t  through the j u n c t i o n t o t h a t  flowing  expected i f no a d d i t i o n a l  c a r r i e r s were b e i n g g e n e r a t e d due t o c o l l i s i o n i o n i z a t i o n by e n e r g e t i c  carriers.  E x p e r i m e n t a l l y he found t h a t  11 f o r n type G e . , where V g i s and V the r e v e r s e v o l t a g e a p-n-p d i o d e ,  the avalanche breakdown  a p p l i e d to the p - n j u n c t i o n .  i f we w i s h t o a v o i d avalanche T  necessary that  the c u r r e n t t h r o u g h the d i o d e .  I v  a s « »  0.98.  effects,  For it  is  — I  ( M - l ) < ^ ( l - ° 0 where ex. =  l e s s than 1.001  voltage,  c  o  and I  is  Hence we choose M to be  By 2 . 1 . 1  we then must  have  4 0.1.  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 Ge. p - n j u n c t i o n s was t h a t between j u n c t i o n breakdown  for voltage  and the n type i m p u r i t y c o n c e n t r a t i o n V where V g i s  = 1.0x10  g  2.1.2  N  _3 and N xn cm  in volts  Two t r a n s i s t o r  parameters which can be used  to  determine the base w i d t h and the base i m p u r i t y d e n s i t y  are  the punchthrough v o l t a g e  at  which the c u r r e n t g a i n i s is  and the frequency of <* c u t o f f reduced by 3 d b .  This  frequency  g i v e n by  r U  =  0 . 4 Dp ^  2.1.3  vr Making use of the f o r Vpo  it  is  possible  above r e l a t i o n s  P  I.-7X5  f« Vp* is  punchthrough experiments  0.1 V . . 7 X 5  where ~f«,  expression  to d e r i v e the f o l l o w i n g c r i t e r i o n  f o r the s e l e c t i o n of t r a n s i s t o r such t h a t V . ^  and the  '  <  '  4x10  expressed i n c y c l e s / s e c  and V p o i n  volts.  12 F o r the t r a n s i s t o r s left  I the e x p r e s s i o n on the  hand, s i d e of t h i s i n e q u a l i t y i s  and hence t h e s e t r a n s i s t o r s (b) is  i n table  a p p r o x i m a t e l y 1.2x10  comply w i t h the  criterion.  In o b t a i n i n g p - n - p d i o d e c h a r a c t e r i s t i c s  the  transistor  kept i n a c o n s t a n t temperature b a t h f o r a p a r t i c u l a r r u n .  However at h i g h c u r r e n t s and v o l t a g e s  the power d i s s i p a t e d  may become l a r g e enough t o r a i s e the temperature o f the t r a n s i s t o r w e l l above the ambient t e m p e r a t u r e .  An e s t i m a t e  of the temperature r i s e  can be made from the m a n u f a c t u r e r ' s  dissipation coefficient  expressed  i n temperature r i s e  above  ambient temperature p e r u n i t power d i s s i p a t e d . For i n s t a n c e f o r t r a n s i s t o r s c u r r e n t at 2Vp i s  w i t h Vp  =6  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 0.6 C/m.w. which would l e a d t o 100 C r i s e  transistor  the  t y p i c a l l y 12 ma c o r r e s p o n d i n g t o a power  d i s s i p a t i o n o f about 150 m i l l i w a t t s .  To reduce t h i s  volts  in air  of temperature!  s e r i o u s h e a t i n g i n the p r e s e n t s t u d y ,  l e a d s were f i t t e d w i t h l a r g e  whole assembly immersed i n o i l .  area f i n s  Furthermore the  the  and the characteristics  at h i g h c u r r e n t s were o b t a i n e d by p u l s i n g the t r a n s i s t o r pulses  of 10 microseconds d u r a t i o n at r e p e t i t i o n r a t e s o f  60-300/sec.  Good c o n s i s t e n c y between steady c u r r e n t  measurements and p u l s e d measurements were o b t a i n e d i n the r e g i o n of overlap of these techniques.  is  with  13 (c)  F o r symmetry of the c u r r e n t v o l t a g e  characteristics,  the v a r i a t i o n of c u r r e n t d e n s i t y a c r o s s the base s h o u l d be s m a l l and thus one s h o u l d 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 e m i t t e r areas (as capacitance  measurements) as n e a r l y e q u a l as  A l s o one s h o u l d choose resistivity. measured  determined by possible.  t r a n s i s t o r s w i t h a u n i f o r m base  T h i s can be d i s c l o s e d by e q u a l i t y of V p  from the two j u n c t i o n s and by the c a p a c i t a n c e  each j u n c t i o n b e i n g i n v e r s e l y p r o p o r t i o n a l to the  of  square  r o o t of the a p p l i e d b i a s  (d)  As body c o n d u c t i o n i s b e i n g s t u d i e d w i t h t h e s e  transistors,  they must be chosen so as  t o be f r e e  from s u r f a c e  c o n d u c t i o n channels between the e m i t t e r and c o l l e c t o r . may be determined by measurement p o t e n t i a l Ve, for a negative ~ < & | V C | < V p , which should V,  =  emitter-base  collector potential  such  give  will  lead to emitter  potentials  value.  T h i s phenomenon of s u r f a c e c o n d u c t i o n has been by Brown ( 1 9 5 3 ) .  that  ] p ' In ( l - « )  Any channels present many times t h i s  of f l o a t i n g  This  studied  F i g u r e 4CURRENT-VOLTAGE CHARACTERISTIC OF p-n-p  DIODE GE  2N137  100 10  CONSTANT  s o  DIFFERENTIAL RESISTANCE  10  60 s o  "  z or  OLID ( j  4-0 PULSE DAT* DC. ORTR  30 20  10  Vp  ^  VOLTAGE  *  io ix ABOVE PUNCH THROUGH G  9  14 2.2  Measurement of punchthrough v o l t a g e and i t s  temperature  dependence. The c i r c u i t p r e s e n t e d i n f i g u r e 1 was f i r s t  used by  Schenkel and S t a t z f o r the measurement o f punchthrough v o l t a g e s V« i s measured w i t h a v e r y h i g h impedance v o l t m e t e r . temperatures,  extremely h i g h impedances o c c u r f o r the e m i t t e r  junction diode. electrostatic  A t low  P r e c a u t i o n s must be taken to a v o i d  pickup o f 60 c p s .  voltages  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 s p l a c e the  potentials;  good s h i e l d i n g and the use o f a by-pass c a p a c i t o r minimize this  effect. The punchthrough v o l t a g e was measured at  from - 6 0 * C t o +80*C.  temperatures  T h i s v o l t a g e has a w e l l d e f i n e d  temperature dependence. F o r t r a n s i s t o r #37  ( t y p e GE2N157) i t was found  that  .o  V> was e s s e n t i a l l y c o n s t a n t below Q C and i n c r e a s e d at r a t e of 1.3 2.3  x 10  a  v o l t s / d e g above 0 C .  D.C. characteristics  o f space charge l i m i t e d ' c u r r e n t s  i n Ge. D.C.  2N137  characteristics  transistors  of p - n - p d i o d e s were t a k e n u s i n g  w i t h the base f l o a t i n g .  Runs were made at  o t e n degree temperature i n t e r v a l s up to 9 0 C . temperatures the r e v e r s e biased  s a t u r a t i o n c u r r e n t o f the  j u n c t i o n becomes l a r g e ,  are r e a c h e d .  At t h e s e h i g h reverse  and l i m i t i n g power c o n d i t i o n s  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 d e v i c e s by s t e a d y c u r r e n t - v o l t a g e measurements up t o power dissipations voltages,  o f 150 m i l l i w a t t s .  F o r h i g h e r c u r r e n t s and  the t r a n s i s t o r was p u l s e d and the v o l t a g e  pulses  a c r o s s the d i o d e and a c r o s s a known s e r i e s r e s i s t o r 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 voltage r e l a t i o n . p a s t the i n i t i a l  were  to o b t a i n a c u r r e n t -  The p o r t i o n of the p u l s e measured was transient current spike.  c o n s i d e r a t i o n s of c h a r a c t e r i s t i c s  As t h e o r e t i c a l  are concerned w i t h the  c u r r e n t a r i s i n g at p u n c h t h r o u g h , I p ( T ) , was s u b t r a c t e d these p u l s e d c h a r a c t e r i s t i c s . presented i n f i g u r e s 2.4  These c h a r a c t e r i s t i c s  from  are  4,'5-  C a p a c i t a n c e measurements were made on b o t h C o l l e c t o r  and e m i t t e r j u n c t i o n s at v a r i o u s r e v e r s e b i a s e s w i t h the b r i d g e arrangement d e p i c t e d i n f i g u r e 6 .  A G e n e r a l Radio  Twin T b r i d g e , type 821-A was used at f r e q u e n c i e s k i l o c y c l e s to 5 megacycles.  These c a p a c i t a n c e  from 100  measurements  were used t o determine the c o l l e c t o r and e m i t t e r a r e a s ,  as  the base w i d t h and the punchthrough v o l t a g e were known.  Cpt Cpt voltage, If slope of  is  = € ^  the c a p a c i t a n c e  and A i s  of a j u n c t i o n a t  2.4.1  the punchthrough  the c r o s s s e c t i o n a l a r e a o f the j u n c t i o n .  the c a p a c i t a n c e is  /?  expected.  is  plotted vs.  reverse bias  B a r k e r has r e p o r t e d t h a t  j u n c t i o n c a p a c i t a n c e measurements,  the c a p a c i t a n c e  a  for of  the  j u n c t i o n shows a marked s t e p up at the punchthrough v o l t a g e  Figure 6 CIRCUIT FOR MEASUREMENT OF JUNCTION CAPAC1S&KCE  N BIAS  4-  SUPPLY IMPEDANCE BRIDG-E  SIRS VOLTAGE  16 and the - s l o p e then c o n t i n u e s to he effect  He a t t r i b u t e d t h i s  to the e x t r a c a p a c i t a n c e o f the f l o a t i n g j u n c t i o n  becoming added t o t h a t of the j u n c t i o n b e i n g measured. r e s u l t was n o t confirmed i n the p r e s e n t plot is  shown i n f i g u r e 6 .  study.  This  A typical  Results for t y p i c a l junction  areas are g i v e n i n t a b l e I . 2.5  Temperature dependence o f space charge l i m i t e d c u r r e n t . F o r the p - n - p d i o d e s #21 as  specified i n table I,  the temperature dependence o f the I-V c h a r a c t e r i s t i c i s not marked except f o r an a d d i t i o n a l c o n s t a n t c u r r e n t which flows, at Vp .  The temperature dependence o f the c h a r a c t e r i s t i c  v o l t a g e s p a s t punchthrough i s  at  s l i g h t l y a f f e c t e d by the  temperature dependence o f Vp . 2.6  T r a n s i e n t response  of the space charge l i m i t e d d i o d e .  T r a n s i s t o r #21 was p u l s e d w i t h i t s punchthrough v o l t a g e o f 6 v o l t s and an estimate made o f the t o t a l swept out o f the b a s e .  T h i s was e s t i m a t e d as 8 x 10  coulombs at room t e m p e r a t u r e .  J  charge  17  CHAPTER I I I THEORETICAL INVESTIGATION 3.1  Relative importance of d r i f t and d i f f u s i o n components of hole current. (i)  For constant m o b i l i t y , the r a t i o of hole d i f f u s i o n  current to hole d r i f t current i s :  D  §1  as  3.1.1  The resultant d e r i v a t i o n being due to the E i n s t e i n d i f f u s i o n r e l a t i o n qD  p  =  kTAc  p  where D  P  i s the  d i f f u s i o n constant f o r 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 m o b i l i t y . At the punchthrough voltage, the f i e l d increases 2V»  l i n e a r l y across the base to -ff- at the c o l l e c t o r j u n c t i o n . A f t e r the punchthrough voltage, when holes are i n j e c t e d 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 <  0  3.1.2.  18  So we may state that 2Vx  3.1.3  Also v/e know c. V X  sine e —  —  >  3.Ly-  0 we may make the approximation  dx  3.1.5  x  Hence the r a t i o of d i f f u s i o n to d r i f t current becomes  W  T ^ 7 ~  3  ,  1  ,  6  kT So i f V » - — the current i s c h i e f l y c a r r i e d by d r i f t except near the emitter.  But near the emitter the  d i f f u s i o n current becomes r e l a t i v e l y greater, and of course i s the sole component at the p o t e n t i a l e.g.  f o r T = 290°K, V = 5 v o l t s , Jdiff <  (ii)  max (E » 0 ) .  f o r x>2£  I f 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 v e l o c i t y saturates.  For t h i s case the r a t i o of  d i f f u s i o n to d r i f t current i s  0  pv as — = qp =» constant. i s c a r r i e d by d r i f t .  3.1.7  So f o r t h i s case a l l the current  Figure 7 ELECTRIC FIELD DEPENDENCE OF DRIFT VELOCITY •  m  —e  •  \  R o HOLES I N FT Y P E G E . G I B S O N ft I ID G R A N V I L L E A HOLES I N N T Y P E G E . R  Y  D  E  s  \ /O  too  IOOO  ELECTRIC  FIELD  10,000  1, | | | | ||  VOLT5/CM  t  (iii)  A f u r t h e r d i f f u s i o n c u r r e n t t o be c o n s i d e r e d  the r e v e r s e s a t u r a t i o n  is  c u r r e n t ( I ( r 0 ) a r i s i n g from the  t h e r m a l l y g e n e r a t e d h o l e s i n the base o f a p - n - p transistor. current  The c o n t r i b u t i o n o f t h i s c u r r e n t t o  I o f the p - n - p d i o d e may be a n a l y z e d i n terms T  of  the c u r r e n t a m p l i f i c a t i o n f a c t o r *  For  the p - n - p d i o d e I  c  = I  T x  3.2  limited  Poisson's  e  »-»m  revert*saiufx+iots'S  = •c  "  I  Z—~  Icq 1  -  .  TL  "I  Q  P . x . o  of.  o f f i e l d - d e p e n d e n t m o b i l i t y upon space  charge  characteristics.  p-n-p diode c h a r a c t e r i s t i c  can be a n a l y z e d u s i n g  e q u a t i o n and the e q u a t i o n f o r c u r r e n t  I f we assume p l a n a r geometry, c u r r e n t s are  —  , and hence I —  (f  Influence  The  the  and i f  density.  diffusion  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 e m i s s i o n o c c u r s at the punchthrough v o l t a g e when the d e p l e t i o n r e g i o n has  V As  V is  =  touched the  flgfl  emitter.  • '  i n c r e a s e d above V p , h o l e s are  3.2.2 i n j e c t e d by  the e m i t t e r i n t o the base r e g i o n , and P o i s s o n ' s -  equation  must be i n t e g r a t e d w i t h a p a r t i c u l a r v e l o c i t y -  electric  field  dependence.  A u s e f u l model f o r t h i s dependence i s : v =M  3.2.3  ( E » )  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 t h i s power law becomes v a l i d and K a constant designating f i e l d dependence. From experimental data due to Gibson, G r a n v i l l e and Ryder on holes i n germanium, i t i s observed that K v a r i e s 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 f i x e d value can be reasonably "ascribed to K. The following s p e c i f i c values of K are p a r t i c u l a r l y important as they have experimental j u s t i f i c a t i o n over c e r t a i n ranges of f i e l d . (i)  K = 1  f o r low f i e l d s  This i s the case of constant m o b i l i t y (v = uE) has been treated by Shockley and Prim.  and  Poisson's  equation may be integrated i n terms of the t r a n s i t time (t.) f o r minority c a r r i e r s across the base.  The  r e s u l t i n g equations f o r V and J may be expressed i n terms of the parameter S = exp 3.2.4-  J  32EA„  3.2.5  The c u r r e n t v o l t a g e approaches which  r e l a t i o n at  large voltages  a s y m p t o t i c a l l y t h a t f o r the i n t r i n s i c b a s e ,  is:  9£uv sir3  T  . , p.c.b  l  =  P  K » yk  (ii)  Prom the hole d r i f t v e l o c i t y measurements above,  listed  t h i s power exponent appears t o be a good  a p p r o x i m a t i o n over a c o n s i d e r a b l e field.  fi  (1000 v o l t s / c m . t o 10,000  Dacey (1953) has i n v e s t i g a t e d  range of  electric  volts/cm.). t h i s power lav/.  He  makes the a p p r o x i m a t i o n t h a t the power law v =  (EE)  h o l d s throughout the e n t i r e base and a r r i v e s  the  at  C h i l d ' s law a n a l o g  S t i l l making the above a p p r o x i m a t i o n , a current density r e l a t i o n i s  voltage-  a r r i v e d at i n a manner  analogous  to S h o c k l e y and Prim by i n t r o d u c i n g the  variable  r =  \  ^  The f o l l o w i n g r e l a t i o n i s V -  Y  "  (  then o b t a i n e d .  - 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 i g h a p p l i e d v o l t a g e s  and l a r g e  currents,,  f i x e d charge i n the "base i s dominated by the h o l e s and the above s o l u t i o n approaches  the  injected  asymptotically  eqn. 3.2.7. (iii)  K  =  This i s  0  the case o f a s a t u r a t i o n v e l o c i t y , which t a k e s  p l a c e at v e r y h i g h f i e l d s E  ( f o r h o l e s i n germanium  = 10,000 v o l t s / c m . ) . P o i s s o n ' s e q u a t i o n becomes  €'g  - H  • i -  9  5.2.10  I n t e g r a t i n g and c h o o s i n g the o r i g i n i n the base the p o i n t of zero  E Integrating  =  V  or  I =  at  field  ± [ qN +  /x  3.2.11  again  K  qH  +  Sr)£(V-V ) pi  i» + ^£  Y  5.2.12 3.2.13  Hence f o r the c o n s t a n t v e l o c i t y case we have a c o n s t a n t differential  resistance  p a s t punchthrough o f magnitude  23 3.3  A model o f m o b i l i t y - f i e l d dependence As the f i e l d i n the base o f a p - n - p t d i o d e  from zero at collector  the e m i t t e r j u n c t i o n to a h i g h v a l u e  varies  at  the  j u n c t i o n , i t may not always be a good a p p r o x i m a t i o n  t o 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 a c r o s s the e n t i r e b a s e . approaches  a s y m p t o t i c a l l y b o t h the c o n s t a n t  f o r low f i e l d s , T h i s model  The f o l l o w i n g model  and the c o n s t a n t  m o b i l i t y case  v e l o c i t y case f o r h i g h  fields.  is  3.3.1 where v i s v0  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  the s a t u r a t i o n  drift  velocity.  In i n t e g r a t i n g P o i s s o n ' s base r e g i o n , i t dimensionless a c r o s s the  is  e q u a t i o n throughout  the  c o n v e n i e n t to i n t r o d u c e the f o l l o w i n g  parameters f o r f i e l d s t r e n g t h and  distance  base. F = (qNv, + J)M. E Jv„  3.3.2  y = CqHvo + J)M v * J€  3.3-3  and  then i n t e g r a t i n g P o i s s o n ' s  x  e q u a t i o n w i t h 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  F o r the low f i e l d , and t h e r e f o r e case we have F «  constant m o b i l i t y  1 and hence F  2  7 - ^-  3.3.5  F o r the h i g h f i e l d case F  1 i.e.  E >>v— we have  a s a t u r a t e d d r i f t v e l o c i t y and y = F This is  3.3.6  s i m p l y the case o f a constant  differential  r e s i s t a n c e , which was encountered e a r l i e r i n t r e a t i n g diode  characteristics. For intermediate f i e l d s  approximate  (F~l)  we can make the  inversion F = y + Cy*  where the c o n s t a n t  C is  3.3.7  chosen from a g r a p h i c a l  w i t h the i n v e r t e d r e l a t i o n y = F -  must i n t e g r a t e  comparison  ln(F+l).  To o b t a i n a v o l t a g e - c u r r e n t  is  p-n-p  d e n s i t y r e l a t i o n we  the above f i e l d r e l a t i o n a c r o s s the b a s e .  c o n v e n i e n t t o i n t r o d u c e another d i m e n s i o n l e s s  c o r r e s p o n d i n g t o the  U = \ Fdy J  quantity  voltage.  =  (qNvp + j f JLX V X  € v03  o  3.3.8  J*  E q u a t i o n 3 . 3 . 7 then becomes  w i t h U m 0 at y • 0 .  y + cy*  3.3.9  It  S o l v i n g , and e x p r e s s i n g t h e r e s u l t  i n terms o f t h e  u s u a l p a r a m e t e r s , we h a v e -L  3  The constant  first  term  differential  a correction  i s the previous expression f o rthe  resistance,  f a c t o r a t lower  a n d t h e s e c o n d t e r m becomes  currents.  For low f i e l d s C approaches J 2 . 1 . 3 g i v e s a f i t o f 3.3*4 t o 3-3.7 lC%.over a f i e l d The  range  A value of C of  w i t h an e r r o r o f l e s s  than  o f 500 - 14,000 v o l t s / c m .  correction  term i n t h i s  model i s o f g r e a t  importance.  Fortypical  v a l u e s o f the parameters  o f 2N137  transistors,  t h e two t e r m s do n o t become e q u a l u n t i l  I  = 30 ma,  x The  correction  term corresponds t o  and i s a n a l o g o u s t o t h e C h i l d ' s Law a n a l o g u e C<vJ2,  t h e c o r r e c t i o n term  except that  ( ~YJP^  I  = ^ At  I  = 9A£AV2.  Ag  V  As  i s i d e n t i c a l w i t h t h e analog."  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 The  variation  punchthrough  o f Vp voltage  i snormally given by the  aNW* e x p r e s s i o n Vpo- g g » 1  variation  °ne obvious p o s s i b i l i t y  o f temperature  i s t h e t h e r m a l e x p a n s i o n o f t h e base w i d t h and t h e  temperature dependence o f £ . temperature^  (N i s assumed i n d e p e n d e n t o f  as the donors a r e a l l i o n i z e d  a t a temperature  w e l l below any c o n s i d e r e d i n t h e p r e s e n t  study).  C o n s i d e r i n g t h i s form o f temperature dependence  Vp*(2a-b)  where a = ^ (Conwell  $pjt  and b = ^  1952) and  §§'  a  n  3.4.1  a  s  v  a  l  u  e  b i s o f the order o f 2 x  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 A n t o n c i k . Hence f ^ x / -  2 x 10-%  ^.1 x 10~"^/°C  10"V°C  according  (Antonelk  by these a g e n c i e s .  1956). 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 ~ v o l t s / d e g . 5  3.4.2  A f u r t h e r mechanism o f temperature v a r i a t i o n o f Vp must be c o n s i d e r e d however. In a p-n-p t r a n s i s t o r i n the s p a c e - c h a r g e - l i m i t e d condition,  t h e r e w i l l be a p o t e n t i a l r i s e t o a maximum  immediately i n f r o n t o f t h e e m i t t e r . Schenkel-Statz difference  In determining the  punchthrough v o l t a g e , one measures t h e  i n p o t e n t i a l between c o l l e c t o r and e m i t t e r .  As  the i n t e r n a l c o n t a c t p o t e n t i a l s a r e n e a r l y the same f o r b o t h the e m i t t e r and c o l l e c t o r j u n c t i o n s , observed  t h e i r e f f e c t upon the  punchthrough v o l t a g e i s n e g l i g i b l e as t h e y  each o t h e r .  cancel  However i n measuring t h i s d i f f e r e n c e , no account  i s taken o f t h e p o t e n t i a l r i s e immediately i n f r o n t o f t h e emitter.  Figure  8  ELECTROSTATIC POTENTIAL ACROSS THE BASE OF p-n-p  DIODE  In i n t e g r a t i n g Poisson's equation across the base, it  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. and  However, t h e l o c a t i o n  m a g n i t u d e 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 d e p e n d e n t  on c u r r e n t , and hence o n 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 punchthrough  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  voltage i s  derived by i n t e g r a t i n g Poisson's equation, the o r i g i n f o r the 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 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 b e t a k e n a s x = - x of t h e c o l l e c t o r as x =  t o be a t  c  and t h a t  W-x . 0  Integrating Poisson's equation w i t h these  boundary  c o n d i t i o n s we h a v e  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  w h e n c e we c a n w r i t e V where  p  V P  p e  - 2(V  -.  V  =  x qNW . 2£  p  o  p o  V )* e  -* '  3.4.5  28  Neglecting the r e l a t i v e l y i n s i g n i f i c a n t of W and £ with temperature dV  e  dT  ~  =  =  we then have  qNW  dx<>  W  dVo  E  -  variation  W  ,  j,  ,  As Shockley and Prim have shown that the hole d i s t r i b u t i o n near the p o t e n t i a l maximum i s s t i l l  approximately  Maxwellian, we can assume I , the^current flowing through the p  p-n-p  diode at V , i s of the form P  I  =  P  1  exp  0  (- g  7  3.4.7  ?-)  corresponding to hole flow across the b a r r i e r of height V © .  Since ^ transistors 40 v o l t "  1  f f " i s t y p i c a l l y 0.09 d e g " f o r the 1  r  examined, V  i s a few tenths of a v o l t , and ^ J T ^  p  we can estimate that  \^\ >  Therefore to a good approximation dV dT  _ ~  0  whence  3.5  .  kT  ql  p  ^  dl£ dT  «  3.4.9  I f a p-n-p t r a n s i s t o r 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 c a r r i e r s i n the n type base.  From the area of t h i s  capacitance  spike,  it  quantity  of  charge  swept  compared  to  the  equal  out  estimated  of  the  total  to  base.  charge  determine This can  i n the  base  the  be which  is  to  Q where A  3.6  s h o u l d be p o s s i b l e  m  is  the  =  mean c r o s s  N o n - p l a n a r geometry For  may b e  for  i n general  and depends  the  the  area  on the  of  type  From Gauss  q  N  the  of  A  velocity  Poisson's  case.  equation  co-ordinates  3.6.I  equipotential surface  curvilinear co-ordinate  =  at  r  system.  3.6.2  qNAdr  )<• \  A  T  T the  base.  I  +  T p . . -T(W) =  For  the  theorem  €d(EA)  and  of  constant  curvilinear  =  dr" is  sectional area  any n o n - p l a n a r geometry,  expressed  where A ( r )  3.5.1  A J W  case  of  A =  d  r  J.6.4  ?  I  dr  a -  constant  •  const.  3.6.5  .30which i s  the case of a c o n s t a n t  saturation  drift velocity.  case of a c o n s t a n t differential curvilinear  differential  From 3.6.5  resistance for  we see  d r i f t v e l o c i t y we have a  that  the  constant  r e s i s t a n c e i n d e p e n d e n t l y o f the  particular  geometry.  High frequency impedance of a p - n - p diode p a s t punchthrough.  3.7  If  i n the i n t e g r a t i o n of P o i s s o n ' s  equqtion across  the case of a p-n-p d i o d e one assumes a s m a l l v a r i a t i o n of the a p p l i e d v o l t a g e V about there w i l l  a l s o be s m a l l v a r i a t i o n s  about t h e i r r e s p e c t i v e valves  whereM  is  a d.c.  integrated,  of E ( x ) ,  values.  If  where  -  "  V,  -  v>  " " TS  =  £—1  = 27Tf and f i s  N(x),  v(x)  J(x),  oscillatory  and E,  then  3.7.1 - -  =  w  If  the a . c .  component  impedance Z may be determined *  ~  F"vV  2>  1  )  * n o7 3  /  * '  2  I~v ^ +  J & W,  the a . c .  frequency.  i n the base p a s t punchthrough,  the b a s e ,  then  the s m a l l  components.  then the a . c .  2  valve,  e q u a t i o n may then be expressed i n two  y  fields  d.c.  mobility.  and an a . c .  7 Z  where ^  its  =E,5  = M*  a differential  Poisson's parts,  d.c.  sunisoidal  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, dv d"E  is  for  a  and f o r the c o n s t a n t  F o r the h i g h  W  —«vfthe  v e l o c i t y case M  transit  »  =0.  time i n  31  Hence f o r the h i g h f i e l d 7  « W  l  ,  /  ~-V*.\  Z  4-,1to!t- (tor)*  12 -  24  Consequently R ^ g ] phase a n g l e .  The r e a c t a n c e  the r e s i s t a n c e megacycles. W^IO  constant /.velocity s i t u a t i o n ^  "  \ . J  = tan  -^p.  where £ i s  For instance c  in a typical  suffers  the  = — = 10  case(v«?10*m/sec  sec. is  also a j u s t i f i c a t i o n  the n e g l e c t o f h o l e r e c o m b i n a t i o n o r t r a p p i n g e f f e c t s  is  5  i s a c c o r d i n g l y s m a l l compared t o  This small value of t  whether t  -*  astv"r'<<l f o r f r e q u e n c i e s up t o hundreds o f  m, hence  base r e g i o n .  n  3 7  for  i n the  A f u r t h e r p o i n t to be c o n s i d e r e d however i s so s m a l l t h a t the number of c o l l i s i o n s a h o l e  i n c r o s s i n g the base has dropped to the p o i n t where  conventional considerations  of d i f f u s i o n and m o b i l i t y break  down. U s i n g the r e l a t i o n time f o r h o l e s , m^ the e f f e c t i v e the magnitude of  m P M- where "Eg i s  the  relaxation  mass o f h o l e s and M the m o b i l i t y ,  may be e s t i m a t e d .  From t h i s i t appears  the number of c o l l i s i o n s a c r o s s the base would average about  that 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 . F o r the t r a n s i s t o r s c o n s i d e r e d ,  the base a t the punchthrough v o l t a g e i s  the f i e l d i n 90% o f i n the range f o r  E * v e l o c i t y — f i e l d law d i s c u s s e d i n s e c t i o n is  increased  3.2.  As t h e  through punchthrough the f i e l d throughout  r a p i d l y r i s e s n e a r the c o l l e c t o r t o t h a t r e q u i r e d f o r saturated d r i f t mobility-field however, current.  velocity.  I n a p p l y i n g the t h e o r i e s  dependence  this  of  t a k e n t o be the a r e a of the  c h a r a c t e r i s t i c s show an i n c r e a s e  c u r r e n t w i t h v o l t a g e which i s experimentally.  much f a s t e r  An e x p l a n a t i o n  i n terms of e m i t t i n g a r e a . punchthrough v o l t a g e the emitter  a  developed i n s e c t i o n s 3.2 and 3.3  area i s  then the t h e o r e t i c a l  It  of t h i s  than t h a t  the e n t i r e  space charge column touches  explaining this t r a n s i s t o r as  effect  is  in  covered.  the the  increasing A model  c o u l d be made by c o n s i d e r i n g  two o r more t r a n s i s t o r s i n p a r a l l e l ,  the S c h e n k e l - S t a t z  emitter,  observed  may be assumed t h a t at  emitter  to  phenomenon can be made  and the e m i t t i n g a r e a i n c r e a s e s w i t h  voltage u n t i l  voltage  the base  an a r e a must be chosen to r e l a t e c u r r e n t , d e n s i t y If  the  the one w i t h  punchthrough v o l t a g e and an a r e a which i s  s m a l l f r a c t i o n o f the e m i t t e r  area,  a  and the o t h e r s w i t h the  remainder o f the e m i t t e r a r e a and h i g h e r punchthrough v o l t a g e s .  33  At h i g h v o l t a g e s where the f i e l d i n the base becomes sufficiently constant  h i g h the d r i f t  differential  calculated  the measured  J  resistance.  The v a l u e  f o r t r a n s i s t o r # 3 7 from e q u a t i o n  while a d i f f e r e n t i a l  It  v e l o c i t y s a t u r a t e s and l e a d s to a  is  (V-Vp f  of t h i s (3.2.14)  resistance is  14 ohms  r e s i s t a n c e of about 12 ohms r e s u l t s from .  characteristic. observed from l o g I - l o g V c h a r a c t e r i s t i c s over two decades o f c u r r e n t .  However i n  that section  3 . 3 the model which t a k e s i n t o account b o t h low and h i g h f i e l d m o b i l i t y - f i e l d dependence T ~ .  9A€xc  leads  to  (V-VD  )x  8W  for currents (3.3.11) dominant.  up to s e v e r a l  m i l l i a m p s b e f o r e the o t h e r term i n  p e r t a i n i n g to the s a t u r a t i o n If  r e g i o n touches  it  is  assumed t h a t the space charge  the e m i t t e r as  a sphere touches  for,^a s m a l l i n c r e a s e of r (the r a d i u s value  r0  d r i f t - v e l o c i t y becomes  A  a plane, then  o f the sphere) above the  r e q u i r e d to t o u c h the p l a n e at  a covered a r e a equal  depletion  punchthrough,we  to  =  2 7T r e  (r-r0)  Now f o r v o l t a g e s j u s t p a s t punchthrough i f r T V-Vp. = and hence  A - (V-V f .)  have  flSp-(r-r#) ~ p  4.1.1 „ 4.1.2  34 Hence i f t h i s l i n e a r v a r i a t i o n o f A w i t h excess is  taken i n t o account the low f i e l d component of  t o a c u b i c dependence of c u r r e n t on v o l t a g e e x a c t l y as o b s e r v e d .  voltage  (3.3.11)  lead  above punchthrough  The.above assumption of n o n - p l a n a r  geometry however would have no e f f e c t attainment of a c o n s t a n t  on the  ultimate  differential resistance,  as  this  is  independent o f any c h o i c e o f c u r v i l i n e a r geometry w i t h cylindrical 4.2  symmetry as  shown i n s e c t i o n  Temperature dependence of  characteristics  space-charge-limited  f o r the p - n - p d i o d e .  Two f a c t o r s the temperature  3.6.  must be t a k e n i n t o account when c o n s i d e r i n g  dependence o f s p a c e - c h a r g e - l i m i t e d  characteristics.  They are the b a s e - g e n e r a t e d c u r r e n t Ip  f l o w i n g at Vp  temperature  at which the c h a r a c t e r i s t i c  i s measured,  temperature  dependence o f the punchthrough v o l t a g e .  measurement  of the l a t t e r  t y p i c a l p l o t o f Vp figure  2.  vs.  is  discussed  f o r the and'the  The  i n s e c t i o n 2.5  T for transistor  #37 i s  A t h e o r e t i c a l explanation of t h i s  and. a  given i n  temperature  dependence i s made i n s e c t i o n 3 . 4 . The temperature  coefficient  0*C to 30*0 was 1.3x10*"* v o l t s / d e g . assumed f o r  where x 0  is  for transistor If  the d i s t a n c e  a v a l u e o f 0.2  #37 from is  from the c h e m i c a l  e m i t t e r 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 e q u a t i o n 3 . 4 . 9 l e a d s to a t h e o r e t i c a l for  .  value o f 1.1x10  volts/deg.  The h e i g h t o f the p o t e n t i a l b a r r i e r V„ has  the  35  v a l u e of 0.18 v o l t s which i s current transmitted across If subtracted  reasonable  i n r e l a t i o n to the h o l e  it.  the b a s e - g e n e r a t e d c u r r e n t Ip from the c u r r e n t observed at  f l o w i n g at Vp higher  is  temperatures  and the c o r r e c t e d c u r r e n t i s p l o t t e d a g a i n s t V-Vp where Vp is  the punchthrough v o l t a g e  characteristic  is  at the temperature  t a k e n , a l l the h i g h temperature p o i n t s  c l o s e l y on the room temperature  curve i n f i g u r e 2.  no f u r t h e r change of the c h a r a c t e r i s t i c temperatures  (-60  through v o l t a g e  a t which the fall  There  is  down t o lower  C) as Ip becomes v e r y s m a l l and the punch-  remains e s s e n t i a l l y  constant.  c o r r e c t e d space charge l i m i t e d c h a r a c t e r i s t i c ©  independent o f temperature  Hence the is  substantially  o  from -60 C . t o 90 C . which i s  in  complete accordance w i t h the t h e o r y of space charge l i m i t e d currents.  '-It i s n o t e d t h a t the c o n d i t i o n of i n t r i n s i c  c o n d u c t i o n i n the base i s  reached at the h i g h  used w i t h o u t any marked change of the 4.3  temperatures  characteristic.  T r a n s i e n t response o f a s p a c e - c h a r g e - l i m i t e d The charge  diode.  swept out at punchthrough f o r the p - n - p  d i o d e #21 was observed i n terms of the t r a n s i e n t  s p i k e on  _H  sudden a p p l i c a t i o n " of Vp and found t o be 8x10 The v a l u e expected f o r charge considerations  swept out o f the base by simple  of base volume and base i m p u r i t y d e n s i t y i s by  e q u a t i o n 3 . 5 * 1 e q u a l t o qNAW = 8x10 discrepancy i s  coulombs.  at p r e s e n t  unexplained.  coulombs.  This  large  I t might be argued t h a t the charge content of the is  determined by c h a r g i n g the c a p a c i t a n c e  the a p p l i e d v o l t a g e ;  of the d i o d e up t o  b u t i t may r e a d i l y be seen t h a t t h i s  i d e n t i c a l w i t h the v a l u e c o n s i d e r e d . a b o v e . = 4.4  C a p a c i t a n c e of r e v e r s e In the present  on the t r a n s i s t o r s  biased  AqNW  junctions.  study capacitance  measurements  were made  mentioned i n T a b l e 1 on each j u n c t i o n f o r w i t h no  change of the smooth l i n e of S l o p e - ^ on a l o g - l o g p l o t .  This  i s d i f f i c u l t t o e x p l a i n t h e o r e t i c a l l y as the base s h o u l d  be c o m p l e t e l y d e p l e t e d o f m a j o r i t y c a r r i e r s capacitance However as  at V P and. the  one would t h i n k would t h e r e a f t e r  remain  the base wafer i s v e r y much l a r g e r  e m i t t e r and c o l l e c t o r a r e a s (see d e p l e t i o n r e g i o n may i n f a c t  figure  that  f o r one GE  junction capacitance i n c r e a s e d through Vp .  2N137  constant.  i n a r e a t h a n the  3) the space  transistor  showed an abrupt  Barker  s t u d i e d the  increase  as  e x p l a i n t h i s phenomenon.  appears to be n e c e s s a r y  floating  the  A f u r t h e r study of many t y p e s  punchthrough t r a n s i s t o r s  charge  c o n t i n u e to expand away from the  e m i t t e r and c o l l e c t o r i n t o the e n t i r e base l a y e r . reports  is  For:  r e v e r s e b i a s e s up t o double the punchthrough v o l t a g e  result  spike  to  voltage of  fully  37 ,  CHAPTER V CONCLUSIONS  5.1  Areas of agreement  V  o f t h e o r y and  The p - n - p d i o d e s  experiment.  s t u d i e d a l l have a w e l l ' d e f i n e d  Schenkel-Statz  punchthrough v o l t a g e  irregularities  i n the dependence  and show no marked  of f l o a t i n g e m i t t e r  on c o l l e c t o r p o t e n t i a l p a s t punchthrough. v a r i a t i o n of e m i t t i n g a r e a w i t h v o l t a g e  The  apparent  above punchthrough l e a d s  o n l y t o a r e a s o n a b l e e x p l a n a t i o n of the observed at  lower v o l t a g e s and the c o n s t a n t  differential  p r e d i c t e d by t h e o r y f o r the h i g h f i e l d c o n s t a n t is  also obtained, After  temperature  and i s  o f the magnitude  the diode c h a r a c t e r i s t i c s  as  dependence  r e q u i r e d by the t h e o r i e s  h o l e flow d e v e l o p e d i n t h i s The temperature is  resistance v e l o c i t y case  expected.  are  t h e y are found to be s u b s t a n t i a l l y  temperature  characteristics  corrected  for  dependence o f the b a s e - g e n e r a t e d c u r r e n t f l o w i n g a t  punchthrough and the temperature voltage,  potential  of  of the punchthrough independent  of  space-charge-limited  thesis.  dependence o f the punchthrough v o l t a g e  e x p l a i n e d i n b o t h s i g n and magnitude w i t h a model which  t a k e s 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 l e a d s to a v a l u e which i s current.  of p o t e n t i a l b a r r i e r  height  q u i t e r e a s o n a b l e i n r e l a t i o n to the t r a n s m i t t e d  hole  38 The c a p a c i t a n c e emitter junctions  measurements  justify  the assumptions  and uniform r e s i s t i v i t y throughout i n the t h e o r e t i c a l 5.2  made on the c o l l e c t o r and of s t e p  junctions  the "base which was made  discussion.  Outstanding problems. A major problem r e m a i n i n g i n the e x p l a n a t i o n o f p - n - p  diode c h a r a c t e r i s t i c s  is  that  of the e x t e n s i o n of the  charge d e p l e t i o n r e g i o n throughout the base to the Evidently after  the punchthrough v o l t a g e  measurements  it  appears t h a t  and from  emitter.  has, been reached  e m i t t e r j u n c t i o n a r e a covered by the space charge increases with applied voltage;  space  the  region  capacitance  the space charge c o n t i n u e s t o widen  i n t o the e n t i r e base l a y e r p a s t punchthrough. The charge the t r a n s i e n t  swept out o f the base l a y e r ' as  current pulse  at punchthrough i s much l a r g e r  than t h a t expected from c o n s i d e r a t i o n s base w i d t h , no ready  i n d i c a t e d by  and mean a r e a of j u n c t i o n s .  of impurity d e n s i t y , T h i s phenomenon has  explanation.  A f u r t h e r problem which may be i n v e s t i g a t e d temperature  dependence of punchthrough.  as p o s i t i o n i n the base and h e i g h t front  is  the  The parameters  used^such  o f the p o t e n t i a l b a r r i e r i n  of the e m i t t e r , m a y p o s s i b l y be determined d i r e c t l y by  suitable  experiments  temperature  and the v a l i d i t y o f the model f o r  dependence  r i g o u r o u s l y checked.  o f the punchthrough v o l t a g e  more  39 REFERENCES A n t o n c i k , E . , On the Theory o f the Temperature Dependence the R e f r a c t i v e Index of Homopolar C r y s t a l s , C z e c h . J .  6, p p . 204-216, 1956.  of Phya.,  B a r k e r , A . S . , A Study of Space-Charge and. Avalanche M u l t i p l i c a t i o n P r o c e s s e s i n Germanium, T h e s i s f o r Master of S c i e n c e d e g r e e , U n i v e r s i t y of B r i t i s h C o l u m b i a , September 1957* Brown, W . L . , N-Type S u r f a c e C o n d u c t i v i t y on P-Type Germanium, P h y s . R e v . , 91, p p . 518-527, August 1953. C o n w e l l , E . M . , P r o p e r t i e s o f S i l i c o n and Germanium;I,  IRE, 4 0 , p p . 1327-1337, 1952.  ]  Proc.  Dacey, G . C , Space-Charge L i m i t e d Hole C u r r e n t i n Germanium, Phys. R e v . , 90, p p . 759-763, June 1953> t  E m e i s , ; R . , H e r l e t , A . , The B l o c k i n g C a p a b i l i t y o f _ A l l o y e d S i l i c o n Power T r a n s i s t o r s , -^roc. I R E , 4 6 , pp. 1216-1220. June 1958. G i b s o n , A . F . , G r a n v i l l e , J . W . , The Measurement o f D r i f t M o b i l i t y i n Germanium at H i g h 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, p p . 259-266, November 1956. M i l l e r , S . L . , Avalanche Breakdown i n Germanium, P h y s . R e v . , p p . 1234-1241, August 1955. i Ryder, E . J . , Fields,  M o b i l i t y o f H o l e s and E l e c t r o n s i n H i g h Phys. R e v . , 90, p p . 766-769, June 195%  99i  Electric  S c h e n k e l , H., S t a t z , H . , V o l t a g e Punch-Through and Avalanche Breakdown and t h e i r E f f e c t on the JWaximum O p e r a t i n g V o l t a g e s f o r J u n c t i o n 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, p p . 614-625, 1954.—  S h o c k l e y , W . , The Theory of P-N J u n c t i o n s i n Semiconductors P-N J u n c t i o n T r a n s i s t o r s , B e l l S y s . T e c h . J r . 28, 435-4S9, J u l y ' 1949. p p <  and  S h o c k l e y , W . , and P r i m , B . C . , Space-Charge L i m i t e d E m i s s i o n Semiconductors, P h y s . Rev. 90, pp. 753-758, June.1953.  in  

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