UBC Theses and Dissertations

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UBC Theses and Dissertations

High field current fluctuations in n-type germanium Hart, Laurence Gilbert 1966

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THE UNIVERSITY-.OF BRITISH COLUMBIA FACULTY OF GRADUATE STUDIES. PROGEAMME OF THE FINAL ORAL EXAMINATION FOR THE DEGREE OF DOCTOR OF PHILOSOPHY  of  LAURENCE GILBERT HART  B.Sc,  U n i v e r s i t y of A l b e r t a  M.Sc,  U n i v e r s i t y of A l b e r t a  FRIDAY, MARCH 31, 1967 AT 2:30 P.M. IN ROOM 304, HENNINGS BUILDING  Chairman: R. B a r r i e J . W. B i c h a r d E. V. Bohn  I . McT. Cowan R. E . Burges F. W. D a l b y D. L . L i v e s e  E x t e r n a l Examiner: M. Lax B e l l Telephone L a b o r a t o r i e s Murray H i l l N.J., U.S.A.  Research S u p e r v i s o r :  R. E. Burges  HIGH FIELD CURRENT FLUCTUATIONS IN N-TYPE GERMANIUM •  ABSTRACT The work r e p o r t e d here theoretical  i s an e x p e r i m e n t a l  i n v e s t i g a t i o n of h i g h - f r e q u e n c y  c a l n o i s e generated  in extrinsic  type germanium a t h i g h e l e c t r i c f i e l d was  During  electri-  single-crystal fields.  p u l s e d so t h a t the l a t t i c e  remained near 77°K„  and  The  n-  electric  temperature  the p u l s e , the e l e c t r o n s  q u i c k l y r e a c h a n o n e q u i l i b r i u m s t e a d y - s t a t e due t h e i r g a i n i n g energy from the e l e c t r i c brought to a s t e a d y - s t a t e by means of w i t h the l a t t i c e v i b r a t i o n s . concerned angles  field  to  and  collisions  P r e v i o u s work has  been  w i t h n o i s e measurements made at r i g h t  to the e l e c t r i c  t r o p i c behaviour  was  field  d i r e c t i o n , where a n i s o -  observed„  The  present measure-  ments, made i n the d i r e c t i o n of the e l e c t r i c  field,  a l s o show a h i g h degree of anisotropy„" T h e . e l e c t r i c a l n o i s e generated the n o i s e temperature, Nyquist  formula  ment of T 30 Mc/s,  T , n  i s d e s c r i b e d by  o b t a i n e d by a d a p t i n g  to the n o n - e q u i l i b r i u m case.,  the  Measure-  performed a t f r e q u e n c i e s of 70 Mc/s  n J  i n d i c a t e d , a .uniform,^oise  spectrum i n t h i s  frequency  range f o r a l l the samples used.  t r o p y of T  n  suggested  that T  b a s i s of the many-valley  n  was  and  The  aniso-  e x p l a i n a b l e on  model of the  the  conduction  band of germanium, e s t a b l i s h e d by p r e v i o u s e x p e r i mental i n v e s t i g a t i o n s of the h i g h - f i e l d m o b i l i t y anisotropy. A f e a t u r e of the many-valley electrons  model i s t h a t  i n d i f f e r e n t v a l l e y s of the  conduction  band, w i l l  In general", e x h i b i t " d i f f e r e n t  behaviour and as a r e s u l t , t r a n s i t i o n s valleys w i l l as  between these  r e s u l t i n a n o i s e phenomenon  " i n t e r v a l l e y noise".  of'T^  i n the  the " i n t e r v a l l e y  a l l o w i n g a d i r e c t measure of the  e l e c t r o n " h e a t i n g " due to the e l e c t r i c electron noise".  directions,  In the  f i e l d , the  <Cm)>  and  <^1.1CT> "  both i n t e r v a l l e y and hot e l e c t r o n n o i s e  are.expected„  Both c o n t r i b u t i o n s to T  by means of B a r r i e ' s e x t e n s i o n  n  are e v a l u a t e d  to the case of many-  v a l l e y germanium of S t r a t t o n ' s h i g h - f i e l d theory„  described  However, f o r measurements  <^10c£> d i r e c t i o n ,  noise" will, vanish,  "hot  transport  transport  GRADUATE STUDIES Physics Nuclear Physics E l e c t r o m a g n e t i c Theory S o l i d State Physics N o i s e i n P h y s i c s Systems Theory, of Measurements  Related  J . B. Warren R. B a r r i e J . B. Gunn R. E. Burgess A. Crooker  Studies  Analogue Computers Communication Theory Network Theory  _,E. V. Bohn A. D. Moore A. D. Moore  HIGH  FIELD  CURRENT  FLUCTUATIONS  ,  IN  n-TYPE  GERMANIUM  by  LAURENCE  GILBERT  HART  B . S c ,  UNIVERSITY  OF  ALBERTA,  M . S c ,  UNIVERSITY  OF  ALBERTA,  A  THESIS . THE  SUBMITTED  IN  REQUIREMENTS DOCTOR  i n  OF  the  1954 1955  .  PARTIAL  FULFILMENT  FOR  DEGREE  THE  OF  OF  PHILOSOPHY  Department of  >  We  P h y s i c s  accept  t h i s  t h e s i s  required-  THE  UNIVERSITY  •  as  .  c o n f o r m i n g  t o  s t a n d a r d  OF  August  BRITISH  1966  •  COLUMBIA  the  In p r e s e n t i n g for  this  thesis  f u l f i l m e n t of the  an advanced degree at the U n i v e r s i t y of B r i t i s h  tiiat  tlu-j  study thesis  Library shall  for scholarly  without my w r i t t e n  Department  of  freely available  permission  representatives. thesis  9\JrfSf  (j>  Columbia  ) tC'] e  requirements  Columbia, for  I  It  agree  r e f e r e n c e and  f o r e x t e n s i v e copying of  this  by the Head of my  is understood that  for financial  permission.  The University of B r i t i s h Vancouver 8 , Canada  Apg-fL-  it  purposes may be granted  p u b l i c a t i o n of t h i s  Date  make  I f u r t h e r agree that  Department o r by h i s or  in p a r t i a l  gain s h a l l  copying  not be allowed  i i  ABSTRACT The  work  t h e o r e t i c a l n o i s e  r e p o r t e d  i n v e s t i g a t i o n  generated  germanium  at  p u l s e d  so  t h a t  D u r i n g  the  of  h i g h  e l e c t r i c  been  a n g l e s  the  the  temperature  remained  near  due  q u i c k l y  t o  t h e i r  of  the  n o i s e f i e l d The  a  r e a c h  d i r e c t i o n , present  77°K.  energy  from  by  means  P r e v i o u s made  where  a l s o  work:  at  r i g h t  a n i s o t r o p i c  measurements,  f i e l d ,  was  n o n -  s t e a d y - s t a t e  measurements  e l e c t r i c  a  g a i n i n g  v i b r a t i o n s .  d i r e c t i o n  show  made  a  i n  h i g h  a n i s o t r o p y .  e l e c t r i c a l  n o i s e  temperature, to  the  n o i s e  the  samples  T  was  T„  ,  generated o b t a i n e d  n o n - e q u i l i b r i u m  a  the  f i e l d  l a t t i c e  a't  of  e l e c t r i c  the  performed  M  The  w i t h  observed.  u n i f o r m  n - t y p e  t o  was  f o r m u l a  s i n g l e - c r y s t a l  brought  b e h a v i o u r  n o i s e  e l e c t r i c a l  and  w i t h  and  h i g h - f r e q u e n c y  e l e c t r o n s  e l e c t r i c  The  e x p e r i m e n t a l  f i e l d  concerned  of*  an  f i e l d s .  l a t t i c e  the  degree  of  e l e c t r i c  to  the  i s  e x t r i n s i c  s t e a d y - s t a t e  c o l l i s i o n s  has  i n  p u l s e ,  e q u i l i b r i u m t h e  here  f r e q u e n c i e s spectrum  u s e d .  The  e x p l a i n a b l e c o n d u c t i o n  e x p e r i m e n t a l  band  of  70Mc/s  i n  t h i s  the of  by  b a s i s  of  the  30Mc/s,  f r e q u e n c y of  T  of  the  the  by  the N y q u i s t  Measurements  and  g e r m a n i u m /  i n v e s t i g a t i o n s  d e s c r i b e d  a d a p t i n g  case.  a n i s o t r o p y  on  i s  of  range  f o r  t h a t  e s t a b l i s h e d  model  by  p r e v i o u s  m o b i l i t y  a n i s o t r o p y . A i n  f e a t u r e  d i f f e r e n t  of  the  v a l l e y s  m a n y - v a l l e y of  the  model  c o n d u c t i o n  i s  t h a t  band,  ,  a l l  m a n y - v a l l e y  h i g h - f i e l d  n  i n d i c a t e d  suggested  n  T  e l e c t r o n s  w i l l  i n  i i i  , g e n e r a l ,  e x h i b i t  d i f f e r e n t  t r a n s p o r t  r e s u l t ,  t r a n s i t i o n s  between  a  phenomenon  d e s c r i b e d  n o i s e  However, the  b o t h  measurements  " i n t e r v a l l e y  measure the  f o r  of  "hot  the  n o i s e "  e l e c t r o n  e l e c t r o n  i n t e r v a l l e y  c o n t r i b u t i o n s e x t e n s i o n h i g h - f i e l d  to  to the  n o i s e rt  \  T  n  i n  T  are  e v a l u a t e d  of  due  r e s u l t  t o  i n  d i r e c t i o n ,  (^00^  the  a  n o i s e " .  a l l o w i n g  n o i s e by  m a n y - v a l l e y  theory.  w i l l  as  a  d i r e c t  e l e c t r i c  f i e l d ,  the  e l e c t r o n  n  the  v a n i s h ,  " h e a t i n g " In  and  " i n t e r v a l l e y  hot  t r a n s p o r t  \  w i l l  as  v a l l e y s  and  case  v  of  these  behaviour  are  means  e x p e c t e d . of  germanium  B o t h  B a r r i e ' s of  S t r a t t o n ' s  iv  TABLE  OF  CONTENTS  Chapter  Page  1  INTRODUCTION  1  1.1  Review  of  1.2:  Transport V a l l e y High  1.4  B a r r i e ' s  1.5  Work  P r o p e r t i e s  1  of  a  Many-  7  Semiconductor  1.3  H i g h  P r e v i o u s  F i e l d  Transport  E x t e n s i o n  F i e l d  P r e v i o u s  Theory of  '  S t r a t t o n ' s  T r a n s p o r t  11 ' 14  Theory  D i s c u s s i o n s  of  High  F i e l d  16  A  18  N o i s e  •  .2  •' T H E O R Y -  3  MANY  •  OF  CURRENT  VALLEY  IN  SEMICONDUCTOR  EXPERIMENTAL 3.1  FLUCTUATIONS  APPARATUS  . Apparatus  D e s i g n  3.2.  Sample  H o l d e r  3.3  Measurements  3.4  N o i s e  AND  TECHNIQUE  25  C o n s i d e r a t i o n s  26  Design of  28  F i l t e r  Temperature  Constants  Measurement  30 32.  P r o c e d u r e \ 3 . 5  Measurement  3.6  D e t e r m i n a t i o n '.,  ,  of  P u l s e of  Sample  Sample  P r e p a r a t i o n  3.8  Sample  Temperature  3.9  34  E l e c t r i c  35  ^  36  F i e l d  3.7  •  V o l t a g e s  J o u l e  R i s e  H e a t i n g  I n f l u e n c e  of  D i f f e r e n t i a l  Due  to  37  Sample  39  i S k i n  E f f e c t  Conductance  on  TABLE OF CONTENTS Page  EXPERIMENTAL APPARATUS AND 3.10  Investigations of  3.11  TECHNIQUE(cont'd)  of S p u r i o u s  Sources  40  Noise  Sample N o i s e  Due t o H o l e G e n e r a t i o n  at the Current  41  Electrodes  EXPERIMENTAL RESULTS  44  COMPARISON OP EXPERIMENTAL RESULTS  . 45  WITH THEORY 5.1  D e t e r m i n a t i o n o f Group D r i f t Velocities  5.2  from B a r r i e  47  Theory  C a l c u l a t i o n o f Sample D i f f e r e n t i a l  51  Conductance 5.3  Determination of C a r r i e r Density  51  and E l e c t r o n P o p u l a t i o n 5.4  The H o t E l e c t r o n C o n t r i b u t i o n t o t h e Noise  52:  Temperature  CONCLUSIONS- AND- SUGGESTIONS FOR FUTURE  54  , WORK BIBLIOGRAPHY • \  57  APPENDICES F l u c t u a t i o n s i n E l e c t r o n P o p u l a t i o n of  58  Sample C a l c u l a t i o n o f Group P o p u l a t i o n S p e c t r a l Densities  65  T A B L E OF  CONTENTS  APPENDICES C a l c u l a t i o n the  of  I n t e r v a l l e y  C a l c u l a t i o n A r b i t r a r y  of  V a l l e y  T r a n s i t i o n V e l o c i t y  D i r e c t i o n  M a x w e l l i a n I n t e r v a l l e y  P o p u l a t i o n s  of  and  P r o b a b i l i t i e s  V a r i a n c e  i n  D i s p l a c e d  D i s t r i b u t i o n T r a n s i t i o n  A c o u s t o e l e c t r i c  . ' Rate  and  the  E f f e c t  J  LIST  F i g u r e  OF  FIGURES  T i t l e  1.1  E l l i p s o i d a l of  T y p i c a l  3.2  Sample  3.3  Band  Four-Probe  Holder  • Apparatus  Apparatus  Energy  E l e c t r o n s  U n i t  i n  Page 10  TT-space  31  Sample  D e t e r m i n a t i o n  and  Surfaces  Sample  Inside  f o r  Temperature  3.3a  Constant  Conduction  5.1  F o i l o w i n g  D i f f e r e n t i a l  Bath  of  ,  Noise  31  31  Conductance  T i t l e s  31  4.1  , N o i s e  Temperature  v s .  F i e l d  i n  <(lQ0^  45  4 . 2  Noise  Temperature  v s .  F i e l d  i n  <(ll0 >  45  4.3  Noise  Temperature  v s .  F i e l d  i n  ( i l l )  45  4.4-  4.5'  4.6  .  ,  D i f f e r e n t i a l  Conductance  v s .  F i e l d  i n ( l 0 0 )  ' D i f f e r e n t i a l  Conductance  v s .  F i e l d  i n ^ l l d )  D i f f e r e n t i a l  Conductance  v s .  F i e l d  i n  \  •  '  45  .  .  45  45  •  Acknowledgments The  author  s u g g e s t i n g Burgess  o f  t o  t h e o r i g i n a l  f o r a s s i s t a n c e  F i n a n c i a l Board  wishes  i s  P r o f e s s o r  problem w i t h  a s s i s t a n c e  Canada  thank  and P r o f e s s o r  t h e  from  g r a t e f u l l y  J . B . G u n n  f o r  R . E .  t h e o r y .  t h e Defence  Research  acknowledged.  CHAPTER  1.1  1  -  INTRODUCTION  REVIEW U n t i l  c a r r y i n g w i t h  OF  PREVIOUS  r e c e n t l y , s i n g l e  behaviour  where  dominant.  due  Recent  however,  have  the  f i e l d s .  T h i s  n o i s e  thermal  " h e a t e d " the  f i e l d An  t o  by  be  the  of  which  between  s t a t e s  w i t h  s h a l l  t r a n s p o r t  i s  a  of  under  the  to  f i r s t  a p p l i e d  and  the  t o  the  of  E l e c t r i c a l appearance and  n o i s e  a c r o s s i s  the  an  the  e l e c t r i c  e l e c t r o n s t o  b e i n g  e x p l a i n  m o b i l i t y .  p r e d i c t e d  the  t h e o r e t i c a l l y  e x p e r i m e n t a l s o - c a l l e d  e l e c t r o n  n o i s e  germanium  enhancement  e a r l i e r  e f f e c t i v e  " i n t e r v a l l e y  t r a n s i t i o n s  mass.  work  where  f l u c t u a t i o n s . n - t y p e  •  as  to  used  i s  done the  o f . b o t h  the  w i l l the  on  main  Then,  germanium  c a l c u l a t i o n  " i n t e r v a l l e y  due  r e s i s t o r s ,  p o p u l a t i o n  Gunn,1962)  h i g h  h i g h - c u r r e n t  the  are  t o  c o n d u c t i o n  r e v i e w  f l u c t u a t i o n s  s u b j e c t e d  n o i s e ,  d i f f e r e n t  f i e l d s ,  n - t y p e  e l e c t r o n  of  and  i n  n o i s e ,  the  concerned  n o i s e  i n t e r p r e t e d  experiment,  due  of  are  of  and  c u r r e n t -  been  f r e q u e n c i e s  concept  source  behaviour  and  source  been  semiconductor  c a r r i e r  v o l t a g e  has  t h i s  n o i s e " ,  a  f i e l d ,  measurable  c a r r y i n g  p o p u l a t i o n  Nyquist  a d d i t i o n a l  We  to  c a r r i e r  i n  have  low  dependence  c o n d i t i o n s  semiconductors  e l e c t r o n s  or  n o i s e  r e l a t i v e l y  r e v e a l e d  when  the  of  e x p e r i m e n t s ( E r l b a c h  present  of  i n v e s t i g a t i o n s  c r y s t a l at  e f f e c t s  WORK  be  "hot.  c u r r e n t concern  h i g h - f i e l d d i s c u s s e d e l e c t r o n "  noise'*. i n  any  t w o - t e r m i n a l  t e r m i n a l s  d e s c r i b e d  of  a  network  random  q u a n t i t a t i v e l y  by  i s  the  f l u c t u a t i n g means  i s  of  the  Thevenin  and Norton  d e s c r i p t i o n In  a  c i r c u i t  i s  as  i s  an  frequency  -f  c o n v e n t i o n , b a r  would  what  .  A?  most  . a  range  ,  A{  t h e  e q u i v a l e n t  of  over  on  a i f  pure  of  t h e  o f  view.  Ko  eCt)  at  b y v a l u e , ,  ett)  corresponds  bandwidth  d e f i n e d  g e n e r a t i n g K°  showed  g e n e r a t o r ,  thermal  a b s o l u t e  and  by  as  network  i n  c o n s i d e r e d He  much  o s c i l l o s c o p e  i s  an  a t  i n t e r v a l  r a y  r e s i s t a n c e  v o l t a g e  t o  i s ,  o s c i l l o s c o p e  n o i s e  was f i r s t  p o i n t  ett)  v o l t a g e  t h e  d e n s i t y  a  v o l t a g e  impedance  time  cathode  surroundings  r e l a t e d  a  f l u c t u a t i n g  case  of  c i r c u i t  a  root-mean-square  t w o - t e r m i n a l  i t s  i d e a l  i s  The  t h a t  value  Thevenin  w i t h  g e n e r a t o r  i t s  t e r m i n a l s ,  t h e o r e t i c a l  mean-square  The  T  s e r i e s  t h e  average  observed  b a s i c  T h i s  i s  b y  s p e c t r a l  w i t h  i n  v o l t a g e  an  ' .  t h e  i s  e q u i l i b r i u m  from  be  The  c o n s i d e r  To  (A?)  a c r o s s  The t o  t h e  i n d i c a t i n g  t o  frequency  2(0  r e p r e s e n t e d  than  were  where  and  g r e a t e r  p l a c e d  a t  i m p e d a n c e 2(^0  eCt) ,  c i r c u i t s .  f o l l o w s :  bandwidth  generator  the  e q u i v a l e n t  temperature Nyquist  t h a t i n T  a  ,  frequency by  N y q u i s t ' s  theorem: c? where  K  « 4 - K T . R . \ > C 0 A f i s  B o l t z m a n n ' s  c o n s t a n t , and  J>(4)  i s  t h e  P l a n c k  f a c t o r  h  b e i n g  P l a n c k ' s  c o n s t a n t .  We n o r m a l l y  t h e  c o n s i d e r  t h e  3 much  The  proof  of  t h i s  a c c o m p l i s h e d second which of  a  law  by  of  g i v e s  l e s s  c l a s s i c a l  p u r e l y  i t  i t s i n  a  were  to  d e r i v e  the In  and  very  simple  the  f o r m u l a  been  A l s o ,  constant  by  o b t a i n e d  agreement  determined  means  by  Another  of  very  l i m i t e d ,  e m i t t i n g  t r a v e l diode the  d i r e c t l y c u r r e n t  g i v e n  by  p r o c e s s  the at  cathode  at  h i g h  a l l  of  from  ah  and  are  e x a m i n a t i o n  of  H e l l e r ,  measurements  per  N y q u i s t the  by  Boltzmann.'s  cent  the  the  cathode  diode,  i t s  the  of  the  and  v a l u e s  at  by  n o i s e  i s  emitted Under  an  by  t h e s e  P o i s s o n .  t a k e s we  of  anode.  v o l t a g e s  a The  i n  the  c h a r a c t e r -  t e m p e r a t u r e -  f r e q u e n c i e s  i f  i n  c o n s i s t s  c u r r e n t - v o l t a g e  t r a n s i t - t i m e theorem,  shot  w h i c h  anode  c u r r e n t  anode.  i s  i s  surrounded  e l e c t r o n s  S c h o t t k y  the  case  H e l l e r ( 1 9 3 9 )  e x p e r i m e n t a l l y  p o s i t i v e  of  f l u c t u a t i o n s  r e c i p r o c a l  motion  and  Bakker  source  vacuum  r e g i o n  t o  one  n o i s e  c o n d i t i o n  because  the  methods.  e l e c t r o n  t h i s  of  v e r i f i e d  w i t h i n  hot  In  law,  f o r  d e r i v a t i o n  n o i s e  t e m p e r a t u r e - l i m i t e d  c u r r e n t - s a t u r a t e d  f o r m u l a  thermal  b a s i c  operated  Bakker  determined  the  i s  e l e c t r o n s  M o u l l i n  to  o t h e r  However,  i s  u s i n g . t h e  e q u i p a r t i t i o n  thermodynamic  d e r i v a t i o n  has  theorem  m o t i o n s .  m i c r o s c o p i c  Johnson(1928).  i s t i c .  N y q u i s t  a d d i t i o n  N y q u i s t  f r e e  theorem  arguments,  the  model,  e l e c t r o n  the  d e v i c e  the  the  and  N y q u i s t ' s  N y q u i s t ' s  v a l i d i t y .  m i c r o s c o p i c t o  of  thermodynamic  w h i c h  by  Then  l i m i t  g e n e r a l  d e s c r i b e d  of  u n i t y .  thermodynamics  c o n d u c t o r  able  than  the  cathode  c o n d i t i o n s , much on  assume  a  l e s s s i m p l e  t h a t  the  the than form e m i s s i o n  The  S c h o t t k y  where  S  i s  i s  x  i s  Xi  then  the  the  theorem  enhances  microwave  below  t h a t  s i n g l e emf.  i s  by  d i r e c t a  p a r a l l e l i s  F o r not  by  p r a c t i c a l  lower  w h i c h  the  N y q u i s t  theorem.  o r i e n t a t i o n  r e s u l t s of  and  the the  i n  s u r f a c e  Most  e x p e r i m e n t a l  and.  w i t h  the  frequency  s t r o n g l y  which  may  be  n o i s e  even  orders  f o r  c h a r a c t e r i z e  semiconductor o b t a i n e d  by  by  u s i n g  a  the  of  the  i n  n o i s e  work  of  t h i s  f l u c t u a t i o n  environment passage w h i c h  frequency, of  the  has low  g r e a t e r  n o n - e q u i l i b r i u m  N y q u i s t ' s  f l i c k e r  t o p i c  I t s  the  v a l u e s  of  of in  concerned  the  c u r r e n t  n o i s e  the  n o i s e  f o r m u l a  f o r  the  g e n e r a l  r e s i s t o r .  f r e q u e n c y than  i s  c r y s t a l  been  ,  n  at  c u r r e n t - c a r r y i n g  n o i s e ,  dependent  T  the  n o i s e  l o w e r  a  environment  parameter,  the  f r e q u e n c i e s  i t s  c u r r e n t ,  s m a l l  i n  d i o d e s ,  main  However,  magnitude  r e l a t i v e l y  at  the  w i t h  t h e o r e t i c a l  of  by  theorem.  a d d i t i o n a l  b i a s i n g  s m a l l  observed  r e s i s t a n c e .  e q u i l i b r i u m  c u r r e n t  n o i s e  e f f e c t  case  thermal  given  f r e q u e n c y very  and  and  c i r c u i t  f r e q u e n c i e s ,  i s  i n  f u n c t i o n  We  audio n o i s e  semiconductor  a  the  S c h o t t k y  s o u r c e ,  g e n e r a t o r  n o r m a l l y  e x p l a i n  spectrum  e q u i v a l e n t  w i t h  t r a n s i t - t i m e the  c u r r e n t  The  current  n o n - e q u i l i b r i u m  c r y s t a l  when  g i v e n  the  a  observed  r e g i o n  n o i s e  c u r r e n t .  which  the  the  g i v e n  Another t h e s i s ,  i n  does At  f l u c t u a t i n g  as  range.  f r e q u e n c i e s .  the  diode  theorem  f r e q u e n c y  e f f e c t  diode  admittance  S c h o t t k y a l l  s t a t e s ,  d e s c r i b e d  S c h o t t k y  r a d i o  the  mean  best  dependent  i s  theorem  of  n o i s e , thermal d e n s i t y .  the  temperature., n o n - e q u i l i b r i u m  5 case  and  l e t t i n g  temperature. v o l t a g e  where the  By  R'  i s  where  G  f l o w s ,  the  d e f i n i t i o n  the  d i f f e r e n t i a l  -  i s  the  d i f f e r e n t i a l  n o i s e  theorem.  s i x  Ziel,1954)  the  c u r r e n t  the  d i r e c t  c u r r e n t . t h e r m a l  f r e q u e n c y  r e c o m b i n a t i o n  van< d e r  i n  two  temperature,  c o n s t a n t s i s f i r s t The  i s  a  t h a t  the  l a s t  on  the  term, as  r e m a i n i n g  term  used,  r i g h t - h a n d of  i s  I  independent  because  i n t e r p r e t e d  The  sample  c o n s t a n t  term  n - t y p e  frequency-dependent  so  f o r  of  a s s o c i a t e s  of  n o i s e .  1  and  range  e x i s t e n c e  The  noise *.  thermodynamic  frequency  Z i e l  c u r r e n t  n e a r - i n t r i n s i c  and  dependence,  the  no  of  are  d i r e c t  over  the  When  c r y s t a l  n o i s e  and  t o  a  the  the  due  by  on  r e v e a l e d  Ai  i s :  s i m i l a r l y  conductance.  N y q u i s t ' s  M c / s  Here  mean-square  r e s i s t a n c e ;  equals  from  i n  the  r e p r e s e n t a t i o n  temperature  K c / s  components  ,  n  thermodynamic  r e p r e s e n t a t i o n :  performed  der  T  the  4 - K T n G ^ f  V-  t o  of  f o r  s e r i e s  Measurements  germanium  i s  the  g e n e r a t o r  temperature,  (van  s u b s t i t u t e  n  g e n e r a t o r . i n  c u r r e n t  one  T  of s i d e  i t s  " g e n e r a t i o n i s  c a l l e d  " e x c e s s  n o i s e " . \ g e n e r a t i o n - r e c o m b i n a t i o n  The i n  the  e l e c t r o n  e l e c t r o n  and  e l e c t r o n  p a i r  n e g l e c t i n g  and  hole  hole  g e n e r a t i o n  p o p u l a t i o n  c r e a t i o n  t r a p p i n g  and  • n o i s e  and  i s  r a t e s .  f l u c t u a t i o n s  h o l e - e l e c t r o n  t a k i n g  the  due  to  f l u c t u a t i o n s  Assuming o c c u r  due  t h a t t o  r e c o m b i n a t i o n ,  t r a n s i t - t i m e  much  h o l e -  greater than the l i f e t i m e , this  process  T«  given  t e m p e r a t u r e due t o .  by  lF Qu«.-»>vjV,, ? c  =  Here  is  the n o i s e  a,  is  F  the e l e c t r i c  s t r e n g t h , M-*. iP-*  field  e l e c t r o n and h o l e m o b i l i t y , r e s p e c t i v e l y and e l e c t r o n or hole l i f e t i m e . are given C\-  where  "the  fc, i s  The c o n d u c t i v i t i e s  the and o^,  0«.  -  by n^.Arv  |?  and  0"V = IP ^  Y\  M  r-  a r e t h e h o l e and e l e c t r o n  densities,  respectively. Talcing  T  f r o m t h e measured  6  value  of  -f,  ,  could —  t h e n be d e t e r m i n e d f r o m t h e  experimental value  of  g-r  IM  .  Ob  From  *  •  resistivity  and t h e measured of  of  i n t r i n s i c germanium was  t o be i n good a g r e e m e n t The e x c e s s  value  noise  has  G~p H - 0 T V  c a l c u l a t e d and  w i t h the accepted been o b s e r v e d  , the found  value.  t o have  the  following  behaviour: 1) The f r e q u e n c y d e p e n d e n c e from the order 2)  It  is  -f  to  c y c l e s per  second.  to  it  Its  "j?  existence results  240°K,  t o the s u r f a c e  on t h e d i r e c t c u r r e n t  is  condition  is  quadratic.  s p e c t r u m may be o b t a i n e d i n a p u r e l y  m a t h e m a t i c a l way i f used such t h a t  4°K  environment.  dependence  The  iO  over a range  constant.  extremely s e n s i t i v e  and a m b i e n t 4)  one Mc/s  roughly  Over a t e m p e r a t u r e range from relatively  3)  of  is  a distribution  f)Ctc)  is  9C^c) o f l i f e t i m e s  proportional to  is  t c ' . The  o f b o t h ."-'slow t r a p p i n g " a n a " f a s t  trapping"  i n a wide range of l i f e t i m e s necessary  to  1  obtain  7  the  spectrum  above.  "slow  The  "slow  over  s u r f a c e  i s  assuming  t h r o u g h  a  r e s u l t s  i n  f a r ,  c a r r y i n g h i g h  wide  been  1.2:  as o n l y  complex  e l e c t r i c  e l e c t r i c energy l a t t i c e  a s s o c i a t e d  the  oxide  w i t h  l a y e r  the  on  spectrum  the  l i f e t i m e s the  b a r r i e r  i f  s u r f a c e  of  we  the  t r a p s  v a r i a b l e  r e q u i r e d low  by  w i l l  by  w i d t h .  the t u n n e l l i n g  T h i s  a l s o  l i f e t i m e  experiment.  frequency, has  without  assume  temperature-independent  t r a n s p o r t  low  been  be  f i e l d  n o i s e  d i s c u s s e d .  d i s c u s s e d  p r o p e r t i e s  and  then  thev  l o s e  the  f i e l d , a c t s  v i b r a t i o n s  energy  depends  behaviour  momentum  g a i n  The  PROPERTIES  f i e l d  p r o p e r t i e s  the  i n  i n d i c a t e d  The  l a t e r ,  of  i n  c u r r e n t -  h i g h - f i e l d ,  a f t e r  germanium  the  have  examined.  Transport  i n  of  reach  n o i s e  TRANSPORT  the  been  e x p l a i n  semiconductors  frequency  r a t h e r  t o  p o t e n t i a l  d i s t r i b u t i o n , So  has  f r e q u e n c i e s  s u r f a c e .  e l e c t r o n s  a  of  e x i s t i n g  d i s t r i b u t i o n  c o n d u c t i o n  band  t r a p p i n g "  p o s s i b l e a  wide  states"'  semiconductor It  the  and to  on  the  and  the  l o s e  from  between  f i e l d  s o l u t i o n  e f f e c t i v e  d i r e c t i o n . of  energy  momentum  i n t e r a c t i o n  and  the  to  the  between  l a t t i c e .  i n c r e a s e  i n t e r a c t  f i e l d  The  t h e i r  w i t h  momentum.  s t e a d y - s t a t e  the  The  i n  which  at  the  t r a n s p o r t  the  same  e l e c t r o n s r a t e  v i b r a t i o n s . the  mass The  and  e l e c t r o n s  a  SEMICONDUCTOR  i n t e r a c t i o n s  e l e c t r o n s  d e s c r i b e  l a t t i c e  the  MANY-VALLEY  e l e c t r o n s ,  t-he  on  i t  A  i n v o l v e s  the  to  OF  of  f i e l d the  e f f e c t i v e  t h e ' S c h r o e d i n g e r  and  a  s i n g l e  e l e c t r o n mass  e q u a t i o n  i s f o r  f o r found the  e l e c t r o n motion from e l e c t r o n  in  the p e r i o d i c  values  EC^O  field  of the l a t t i c e .  f°r "the c o n d u c t i o n  functions  o f t h e wave v e c t o r  restricted  to vectors having  where  j*3  jp,  ?  the p r i m i t i v e lattice.  r  The c r y s t a l  which i n t u r n are  the values  L  k,,^,*^  3  are  v e c t o r s of the r e c i p r o c a l  i s assumed t o be a r e c t a n g u l a r  having  S| a*, , S a t  edges  2  one minimum o f t h e c o n d u c t i o n  which i s d i f f e r e n t  eigen-  band s t a t e s a r e  i n t e g e r s and  e  translation  parallelepiped Let  a  -k  The e n e r g y  ;  S^o^  band o c c u r ~x y, 2  from zero. I f the  •  }  at  -8?„  co-ordinate  s y s t e m i s c h o s e n so t h a t t h e o f f - d i a g o n a l components o f the  effective  expansion for  of  liquid  Levinger ^ = W  where  }  YY)0  the T a y l o r  ^4  X >  series !?  0  is  hl^ , k i ^ ;  e l e c t r o n mass f o r m o t i o n i n ) i s g i v e n by  from c y c l o t r o n resonance data at  temperatures.  The most r e c e n t  and F r a n k l ( 1 9 6 1 ) , wi = t  s  then  e n e r g y minimum o f germanium, t h e v a l u e s o f  obtained  helium  =  ij£-JL)  [l-t^,^  i-direction,  For the lowest  of  of  WU , t h e e f f e c t i v e  are  are zero,  a b o u t t h e minimum e n e r g y s t a t e  small values  where the  mass t e n s o r  =  quoted a s :  (o.08ISZ * . o o o o g ) rYlo  (U588 * .oo«r) mo  i s the free  e l e c t r o n mass.  data  are those  A  complete  germanium of  h i s  has  own  c y c l o t r o n  1)  been  p u b l i s h e d  resonance band,  the  work  band by  s t r u c t u r e  Herman  and  data  of  (1955)  o b t a i n e d  measurements.  C o n s i d e r i n g  i s  t h e r e  i t  found  t h a t  as  are  a  r e s u l t  from o n l y  two  the  c l a s s e s  v a l l e y : F o u r  i n  edge.  the  The  ( i l l )  g i v e n  the  z - d i r e c t i o n  One  minimum  s u r f a c e s The  any  at  The  t h i s  of  of  ( i l l ) .  about  w h i c h  o n l y of  t h i s  e l e c t r o n s  s c a t t e r i n g )  v a l l e y s ( i n t e r v a l l e y  accordance  w i t h  the  w i t h  ,  c o n s t a n t  energy •  above  the  temperatures  v a l e n c e  i s  there  minimum.  between and  s t a t e s  between  s c a t t e r i n g )  s e l e c t i o n  of  mass  0.8^te.v. h i g h  zone  e l l i p s o i d s  the  e f f e c t i v e  at  the  above,  the  i s  at  are  1.2.1  w i t h  occupancy  v a l l e y ( i n t r a v a l l e y d i f f e r e n t  i n  space  X  s u r f a c e s  minimum  that  s c a t t e r i n g  of  e q u a t i o n  k-Q  so  degree  by  s p h e r i c a l  of  4°K  energy  b e i n g  at  are  energy  band  d i r e c t i o n  c o n s t a n t  r e v o l u t i o n  2)  of  t h e o r e t i c a l  c o n d u c t i o n of  p i c t u r e  i n  the  s t a t e s  same  of  o c c u r s  i n  r u l e s :  S' = t±t where  M  v e c t o r  and  i n  t r a n s i t i o n  the  i s  the  f i n a l  i s  C o n s e r v a t i o n  the ( +  of  and wave  f o r  the  Jk  v e c t o r  i n i t i a l of  a b s o r p t i o n  energy  of  the  the and  system  e l e c t r o n  phonon —  wave  i n v o l v e d  f o r  e m i s s i o n ) .  r e q u i r e s  that  E(i')-E(£)±t.al^ w i t h the  the  +.  v e l o c i t y  f u n c t i o n  of  and of \  —  sound .  The  meaning i n  the  the  same  c r y s t a l ,  p r o b a b i l i t y  Pij  as  w h i c h  above may of  a  and be  U.  i s  a  t r a n s i t i o n  i n  u n i t  v a l l e y  time j  from  a  s t a t e  i n  v a l l e y  i  t o  a  s t a t e  i n  a t  t h e  s t a t e  t o  i t  i s  i s  'Ri-HirKE') where  i s  which  t h e  t h e  e l e c t r o n v a r i e s  The  m a t r i x  d e n s i t y i s  s t a t e s  s c a t t e r e d .  l i t t l e  element  of  along  Here  t h e  E  H t j =[n(^ j]' 'Dij >  a b s o r p t i o n  f o r  where  i s  i n i t i a l l y  i n  independent F o r  of  the of  mode  t h e  i n  of  t h e  t h e i r  t r a n s i t i o n  i n v o l v e d ,  l o c a t i o n s  i n  —^rn^ e*  w  F o r  b e i n g  t h e  a  n e a r l y  i n i t i a l Hence,  t r a n s i t i o n  v a l l e y  j  s t a t e s .  and f o r t o  f i n a l a b s o r p t i o n , a  s t a t e  i s  -1  w , K T t  W  f o r  i s  D L J  i s  Di}  of  present  t h e phonon  v a l l e y s .  p r o b a b i l i t y t*i  wo.  o f  e m i s s i o n  phonons and  s c a t t e r i n g  r e s p e c t i v e  energy  of  o c c u p a t i o n  i n t e r v a l l e y  independent s t a t e s  t h e  t h e number  that  s u r f a c e . f o r  i  assumed  constant  e m i s s i o n ,  U ^ U  U E - W )  f o r aE>U  \  f o r AE^I^U)  =*0  The  d e n s i t y  o f  the  s q u a r e - r o o t  s t a t e s o f  t h e  i n  both  f i n a l  cases s t a t e  i s  p r o p o r t i o n a l  energy.  t o  Fig.1.1  A ooi  Ellipsoidal  constant-energy  e l e c t r o n s of germanium minima a r e  of conduction  band  i n K - s p a c e . O n l y two o f t h e f o u r  shown.  BC and AC r e p r e s e n t , intervalley  surfaces  respectively, intravalley  transitions.  and  1.3  HIGH Many  h i g h of  f i e l d  the  d e a l t  changes  e l e c t r o n  at  used  i n  the  present  of  the  f i e l d  v e l o c i t i e s e l e c t r i c  t o  the  and of  the  approach  the and of  must  the the  r e c e n t '  m o b i l i t y Other  form  of  v e l o c i t y  the  e f f e c t  of  e l e c t r o n -  d i s t r i b u t i o n .  c u r r e n t  a  Any  f l u c t u a t i o n s  the  v e l o c i t y as  most  f i e l d .  determine  v a l l e y s  of  w i t h  the  e l e c t r o n  theory  of  the  observed  i s  under  found  to  v a r i a n c e and  the  f u n c t i o n  d r i f t  of  the  work;  the  f u n c t i o n  by  s o l v i n g  the  of  the  energy  o n l y  f o r  approach  R i s k e n ( 1 9 6 2 ) ,  s e m i -  e l e c t r o n  high be  who  energy  e l e c t r i c  f i e l d s .  M a x w e l l i a n ,  even  c o l l i s i o n s .  c a r r i e d  out  c o l l i s i o n s  d i s t r i b u t i o n s .  a c o u s t i c a l ,  n e a r e s t  high  to  the  assuming  m a n y - v a l l e y  very  d i s t r i b u t i o n  has  energy  a  of  f o r  The  and  d e t e r m i n a t i o n  f i e l d s ,  i m p u r i t y ,  R e i k  r e s t r i c t i o n  a p p l y  h i g h  and  e q u a t i o n  e l e c t r o n - e l e c t r o n  average  i s  s c a t t e r i n g .  e l e c t r o n - e l e c t r o n  energy  problem  e q u a t i o n . f o r  c o l l i s i o n s  would the  the  d i s t r i b u t i o n  Boltzmann  w h i c h  the  of  v a r i o u s  3 t r a t t o n ( 1 9 5 7 ) of  f i e l d s  t h e ' f o r m  i n t e r v a l l e y  i d e a l  i g n o r i n g  a n i s o t r o p y  i n c l u d e  the  the  germanium,  p o p u l a t i o n  experiment  t r a n s p o r t  c o n d u c t o r  They  on  e x p l a i n  v e l o c i t y  o p t i c a l • a n d  o n l y ,  the  n o n - z e r o  e l e c t r o n - e l e c t r o n  s o l v e d  n - t y p e  on  f i e l d . i d e a l  t h i s  p u b l i s h e d  v a l l e y  component  of  Boltzmann  t o  i n w i t h  c o l l i s i o n s  t h e o r y  The  i n  THEORY  been  c o n s i d e r a t i o n s  d i s t r i b u t i o n  the  have  t r a n s p o r t  have  important  of  TRANSPORT  papers  w h i c h  and  FIELD  H i s  i n  order  exchanged  i n  an  a n a l y s i s  d e t e r m i n i n g of  of  the  the  magnitude  r o l e  v e l o c i t y  e s t i m a t e s  e l e c t r o n - e l e c t r o n  and  12 a c o u s t i c a l mode s c a t t e r i n g l e a d h i m t o a s s e r t critical  exists,  electron density,  where  T  u  is  the e l e c t r o n energy. energy  neighboring  on  .In  the value  at a l a t t i c e In  assumption for this  experiment,  than  order that  displaced  the  greater than  not v a l i d ,  momentum  analysis  for  then,  of Yamashita  and W a t a n a b e ( 1 9 5 4 ) ,  much  of S t r a t t o n ' s  analysis  may i n f l u e n c e t h e shape  with  is  those  the '  Maxwellian collisions  of the v e l o c i t y d i s t r i b u t i o n ,  necessary  the  solution,  electron-electron  the i n t e r p r e t a t i o n of the present it  Their  d i s t r i b u t i o n , d i f f e r s from the Although  found  who f o u n d t h e f o r m o f  of e l e c t r o n - e l e c t r o n c o l l i s i o n s .  zero f i e l d .  then  electron-electron  v e l o c i t y d i s t r i b u t i o n a t h i g h f i e l d s by i g n o r i n g  concerned,  tne  distribution  electron energies  the r o l e of  the r e s u l t s  f a r as  about  i n d e t e r m i n i n g t h e d i s t r i b u t i o n may be  the P i s a r e n k o  high  than  t h e c a r r i e r d e n s i t y was  by comparing  except at  for  . What f o r m t h e d i s t r i b u t i o n t a k e s  i s n o t known. H o w e v e r ,  effects  , the  i n the case of  o f a d i s p l a c e d M a x w e l l i a n momentum is  YV  77°K.  . According to S t r a t t o n ' s  case  collisions  E  a Maxwellian  must be g r e a t e r  in  temperature of  the present o x i o cw  of  and  by  greater  tend towards E  a  temperature  For densities  d i s t r i b u t i o n be M a x w e l l i a n , fields,  , given  the l a t t i c e  distribution will  energies  Ho,  that  experiment  t o know o n l y t h e e x t e n t  is to  as  w h i c h s u c h c o l l i s i o n s a f f e c t t h e mean v a l u e and t h e meansquare  v a l u e o f t h e v e l o c i t y d i s t r i b u t i o n . T h i s was  b y c a l c u l a t i n g t h e mean and m e a n - s q u a r e  v e l o c i t y at  a r b i t r a r y f i e l d o f 87 v o l t s p e r cm. f o r b o t h d i s p l a c e d M a x w e l l i a n and t h e P i s a r e n k o these  done an  the  distributions.  In  c a l c u l a t i o n s , a c o u s t i c a l mode s c a t t e r i n g and a  effective  mass were u s e d . The r e s u l t s  were t h a t  t h e mean v a l u e s  b y 1 1 $ . H e n c e i t was  mass as  is  calculations  d i f f e r b y 5% and t h e  mean-square  concluded t h a t the d i s p l a c e d M a x w e l l -  ian d i s t r i b u t i o n provides and s e c o n d , moments,  of the  scalar  a r e a s o n a b l e v a l u e of the  first  even f o r the case of a t e n s o r e f f e c t i v e  used i n B a r r i e ' s  extension of S t r a t t o n ' s  theory.  The t h e o r y o f B a r r i e was u s e d t o p r o v i d e a l l f i r s t . a n d . s e c o n d moments  of the v a r i o u s  valley velocity distributions  needed i n the t h e o r e t i c a l e x p r e s s i o n s fluctuations.  As w e l l ,  f o r the  current-  i t provided the v a l l e y  electron  t e m p e r a t u r e s , which were used t o c a l c u l a t e the h i g h valley populations.  The B a r r i e t h e o r y has  i m e n t a l l y as f a r as  the dependence of d r i f t  field(Barrie  a c t u a l values^ of d r i f t with the  v e l o c i t y are i n only f a i r  correctly the  agreement  theory.  is  that of Paige(1960).  field  transport  He assumed a M a x w e l l i a n  e n e r g y d i s t r i b u t i o n i n e a c h v a l l e y and b y scalar  exper-  v e l o c i t y on  at 77*K, but  Another approach i n i n t e r p r e t i n g high behaviour  been t e s t e d  and 3 u r g e s s , 1 9 6 2 ) . The t h e o r y p r e d i c t s  the o c c u r r e n c e of m o b i l i t y a n i s o t r o p y  field  e f f e c t i v e mass t h e o r y i n h i s  employing  expressions  for mobility .  was a b l e t o d e d u c e t h e v a l l e y e l e c t r o n t e m p e r a t u r e s  and  14  v a l l e y m o b i l i t i e s f o r the the  ^111^  current  current  d i r e c t i o n . Published  and  f i e l d , was  d i r e c t i o n close  d a t a on  the  to  angle between  used i n h i s a n a l y s i s . For  the  p r i n c i p a l d i r e c t i o n s t h i s method o f a n a l y s i s i s o f value  s i n c e the  1.4  current  BARBIE'S'EXTENSION OF In t h i s theory  in  Section  is  assumed low  1.2  o c c u p i e d and and 1)  and  field  i s assumed. A l s o , the only the  c o n t r i b u t e t o the  mean e l e c t r o n  current.  Other  energy  are  assumptions  following:  electrons  A-space i s M a x w e l l i a n at a l l f i e l d s about a f i e l d - d e p e n d e n t  discussed  f o u r ( i l l ) minima  the  •fUO o f  distribution.  same.  STRATTQN'S HIGH F I E L D THEORY  approximations used are The  d i r e c t i o n s ^ ' a r e the  t h e band scheme o f g e r m a n i u m  so t h a t  no  part  i n one  valley in  and  displaced  is  J^o . , t h a t i s ;  _E$-£O.)/KT  + U)* Here the in 2)  the  The  e  o r i g i n i n -K-space i s t a k e n a t t h e  (ill) direction.  steady-state  JKo  zone edge  and  T  equations from which the  parameters  factors \ <  are found c o n t a i n the  L  t  and  c  /BJF(JC)\ V  "3t~"/o>»  > "  f u n c t i o n due  tiie  r a _ t e  °f  change of  t o a c o u s t i c a l and  respectively.  The  sums i n v o l v i n g t h e  transition  ^  t r a n s i t i o n s by  absorption  a state  k  distribution  o p t i c a l mode s c a t t e r i n g ,  a b o v e f a c t o r s may  and  and  the  be  as  probabilities  > which represent o f a phonon  t r a n s i t i o n s by  expressed  emission  respectively, ^,  to a  state  of a phonon  ^.  t<  O r d i n a r y p e r t u r b a t i o n t h e o r y i s used t o c a l c u l a t e (k_ (X , £ +  ^ (j£ 4-^.-, JCj  and  £  Born approximation transition  where  . T h i s i m p l i e s use o f t h e  and t h e r e s u l t a n t  expression f o r the  probability,  E'  E  and  are the f i n a l  and i n i t i a l  energies,  respectively. BG^)  i s the square of the absolute value of the matrix  component o f t h e p e r t u r b i n g p o t e n t i a l , V t h e two s t a t e s ,  4>£. and  and  and i s d e f i n e d by  and  b e i n g t h e wave f u n c t i o n s o f t h e i n i t i a l  final  ft(^)  is  , connecting  states. t h e a v e r a g e number o f p h o n o n s i n t h e <£-mode,  i s given by:  is Since  assumed t o be i s o t r o p i c i n  intervalley scattering  ^,-space.  i s ignored i n the deter-  m i n a t i o n o f t h e v a l l e y m o b i l i t i e s , we n e e d c o n s i d e r the  i n t r a v a l l e y o p t i c a l and a c o u s t i c a l  According  t o .Herring(1955),  i n t h e case  only  scattering. only of i n t r a -  v a l l e y a c o u s t i c a l ^ s c a t t e r i n g may B(^3 be a n i s o t r o p i c - d u e to the p o s s i b i l i t y  of shear  strains, producing  deformation  potentials. The B a r r i e original  equations  equations  o f S e c t i o n 5.1 r e d u c e t o t h e  of Stratton(1958)  i f vtt-t  and W\i a r e  .equal. Since  the o r i g i n a l  S t r a t t o n t h e o r y p r e d i c t s an  isotropic m o b i l i t y at a l l f i e l d s , for  i t i s quite inappropriate  the i n t e r p r e t a t i o n of the present  1.5  experiment.  PREVIOUS DISCUSSIONS OF HIGH F I E L D NOISE P.J.Price(1959) f i r s t discussed the extension of the  Nyquist  theorem t o t h e case of a conductor  strong e l e c t r i c  fields  such t h a t t h e r e l a t i o n between t h e  c u r r e n t d e n s i t y and e l e c t r i c He c o n s i d e r e d  biased at  field  i s no l o n g e r  linear.  the f l u c t u a t i o n of v e l o c i t y of a s i n g l e  e l e c t r o n , n e g l e c t i n g any e l e c t r o n - e l e c t r o n i n t e r a c t i o n and  was a b l e t o r e l a t e t h e s p e c t r a l d e n s i t y o f t h e f i e l d  component o f e l e c t r o n v e l o c i t y f l u c t u a t i o n t o a. component o f t h e d i f f e r e n t i a l d i f f u s i o n d y a d i c . He a p p l i e s h i s r e s u l t t o two h i g h f i e l d  cases:  one i n w h i c h t h e c o n s t a n t  "if* e n e r g y s u r f a c e s i n Jk-space  a r e s p h e r i c a l and s c a t t e r i n g  o n l y b y a c o u s t i c a l mode i n t e r a c t i o n s ;  the other, applicable  o n l y i n t h e case of very h i g h f i e l d s ,  where o n l y  ing  by e m i s s i o n  cases  o f o p t i c a l phonons i s i m p o r t a n t .  I n both  t h e n o i s e t e m p e r a t u r e i s o f t h e same o r d e r a s  where  i s t h e mean e l e c t r o n e n e r g y . The  dependence o f  T  n  about  10  ,  frequency  i s u n i f o r m up t o f r e q u e n c i e s  of the r e c i p r o c a l of the mean-free-time, to  scatter-  the order  corresponding  c y c l e s p e r second.  Gurevich(196£} d i s c u s s e d t h e c u r r e n t f l u c t u a t i o n s i n t h e non.-equilip.rium. s t a t e o f a s e m i c o n d u c t o r a t h i g h field.  He assumed s p h e r i c a l e n e r g y s u r f a c e s i n A - s p a c e ,  a c o u s t i c a l mode s c a t t e r i n g a n d no e l e c t r o n - e l e c t r o n i n t e r a c t i o n . H i s e x p r e s s i o n f o r Si^)  i  n  "the d i r e c t i o n o f t h e  applied  field,  current  f l u c t u a t i o n s due t o t h e e l e c t r o n t h e r m a l  contains in  where  a parameter  S (w)  S (io)  i  x  "Ts  motions,  , which r e s u l t s i n d i s p e r s i o n  f o r frequencies  x  the s p e c t r a l density of  s  of the order  tV*  , where  -to  is  of the order  than  ^s'  [0  seconds. F o r frequencies  , h i s theory p r e d i c t s the current  much l e s s spectrum  t o be f r e q u e n c y i n d e p e n d e n t and t o be p r o p o r t i o n a l t o the  square-root of the applied Another source of noise  to high  frequencies  discussed uation  field.  also predicted  t o extend out  i s the i n t e r v a l l e y noise,  by P r i c e ( 1 9 6 0 ) .  This  source of current  i s due t o t h e random r a t e  between t h e v a r i o u s  first fluct-  of e l e c t r o n t r a n s i t i o n s  v a l l e y s o f t h e c o n d u c t i o n band a n d i s  p r e s e n t o n l y when t h e v a l l e y s a r e " q u a s i - i s o l a t e d " , i n t h e s e n s e t h a t a n e l e c t r o n l o c a l i z e d i n one v a l l e y t e n d s t o e x e c u t e many c o l l i s i o n s w i t h i n one v a l l e y b e f o r e  making  a t r a n s i t i o n t o another v a l l e y . A necessary c o n d i t i o n f o r for  the existence  of t h i s  e f f e c t i s the v a r i a t i o n of  e l e c t r o n m o b i l i t y between t h e v a l l e y s . -Using the experimental values of the i n t e r v a l l e y t r a n s i t i o n rate obtained at  b y W e i n r e i c h , S a n d e r s , and W h i t e ( 1 9 5 9 )  77°K, P r i c e u r e d i c t e d  tyooy d i r e c t i o n w i t h  and  field  that  valley  with  measured i n t h e i n the ^Lio) d i r e c t -  a joule heating  due t o t h e  o f t h e o r d e r o f 100 w a t t s p e r c u b i c  the spectral density  frequencies,  the noise  the d i r e c t current  i o n s h o u l d be d e t e c t a b l e applied  that  centimeter  s h o u l d drop o f f a t microwave  s i n c e t h e mean l i f e t i m e o f a n e l e c t r o n i n a  i s o f t h e o r d e r o f 10  seconds.  CHAPTER 2 - THEORY OF CURRENT F L U C T U A T I O N S I N A M A N Y - V A L L E Y SEMICONDUCTOR  Let the L  us s u p p o s e t h e s e m i c o n d u c t o r s a m p l e t o be i n  f o r m o f a homogeneous  s i n g l e - c r y s t a l b a r of l e n g t h  and o f u n i f o r m c r o s s - s e c t i o n w i t h  attached  t o i t s two e n d s . E l e c t r o n s  majority  carrier  and t h e h o l e  current  are. assumed t o be t h e  population  n e g l i g i b l e . We assume a u n i f o r m e l e c t r i c in  leads  t a k e n t o be . field  -F  exists  t h e sample r e s u l t i n g i n a s t a t i o n a r y f l u c t u a t i n g  c u r r e n t , 1 ^0  , whose  s p e c t r a l d e n s i t y , S^fa)  we w i s h  3  to c a l c u l a t e . The c u r r e n t electrons  wnere and  I(t)  i s due t o t h e m o t i o n o f t h e f r e e  o f t h e s a m p l e and i s g i v e n  by '  N(t) i s t h e f r e e e l e c t r o n p o p u l a t i o n  at time  t -  U j ( t ) i s t h e v e l o c i t y component o f t h e j t h e l e c t r o n  i n .the d i r e c t i o n o f t h e l o n g i t u d i n a l s a m p l e a x i s . L e t u s now assume t h a t t h e t o t a l composed o f two s e m i - i s o l a t e d N^Ct) w i t h  that  N(t) i s  groups o f p o p u l a t i o n  each group h a v i n g d r i f t  w h i c h are,, i n g e n e r a l ,  population  velocities  N*(t) and  i n the f i e l d  d i f f e r e n t - . B y s e m i - i s o l a t e d we mean  e l e c t r o n t r a n s i t i o n s . o c c u r between t h e groups b u t n o t  at a r a t e s u f f i c i e n t  t o a f f e c t t h e i n d i v i d u a l group  or v e l o c i t y d i s t r i b u t i o n s . N a t ) and  will  energy  fluctuate  due t o r a n d o m t r a n s i t i o n s o c c u r r i n g b e t w e e n t h e g r o u p s . will  also f l u c t u a t e , but i t s variance  i s sufficiently  small The  (Appendix l j ' t h a t  current  MCt) has been taken  i n t h e sample a t time  rt.it)  jwt)  j i  A=t  3  where  vrjL-t)  and  aneous a x i a l  w^tt)  t  constant,  i s then:  are, r e s p e c t i v e l y , the instant-  components o f v e l o c i t y  of the j t h e l e c t r o n  i n g r o u p 1 a n d o f t h e k t h e l e c t r o n i n g r o u p 2. The  c u r r e n t may be e x p r e s s e d  i n terms of t h e v a r i a b l e s  Nl.lt), Wa.fcfc) , AU^.Ct) , and AU3^(t), t h e l a t t e r the arithmetic-mean  velocity fluctuations,  i-»  two b e i n g  d e f i n e d by  N (.t)4_ z  where  and VTj U ) -  A  U j I t )- T r  A  iS^Ct) =  uT^Ct)  - U?  Then l ( t )  =  .•^-[N,Ct)(vr + A v t t t ) ) + N ( t ) ( w + AW-*C-t| z  c o v L ^ ^ ^ u r ^ C t +*)] eouals  We assume t h a t that  there  over"  i s assumed no a p p r e c i a b l e v e l o c i t y  f r o m one v a l l e y t o a n o t h e r a n d seems  since the matrix nearly final  zero. This  element f o r i n t e r v a l l e y  means  "carry-  reasonable  scattering i s  i n d e p e n d e n t o f the' l o c a t i o n s o f t h e i n i t i a l a n d states'* i n t h e i r  r e s p e c t i v e v a l l e y s ( H e r r i n g , 1955 ).  This leads t o the f o l l o w i n g expression:  4-  i? ur | c o v [ N , f t ) M C t + r ) | + c o v [ M x ( - t ) , M , ( t t t j ] J +  Nilt)Ni(t+*)  )  x  c o M ^ t t ) , u ^ C t + t ) ] * Wa.Ct)N (t+r) c o v [ o r a ) , u r j t + r j x  a  The  current spectral  Weiner-Khintchine  density,  $ (u5) , i s g i v e n by t h e x  theorem,  Ni(t)N»(t+t) COY [vr^oUt-rcjJ cos cordis +  + 1 wo  .0)  -v- 4  tixLt)$ lUr) x  cov[w&.Ci\uk(t+r)jcosior  The  group p o p u l a t i o n s p e c t r a are found i n Appendix  The  field  and c r y s t a l - o r i e n t a t i o n  velocities,  vr  and  appropriate transport calculation T h e s e may  of  M~  T  dependent  drift  a r e d e t e r m i n e d f r o m some  theory. There remains o n l y the  cov(ykCt) U"o. C-t ^ k  and  covjwktt), Wa.(-t-t^)j .  be o b t a i n e d d i r e c t l y f r o m t h e i r d e f i n i t i o n s ,  as f o l l o w s ,  f o r group  Multiplying  t h e s e ' two ^ e q u a t i o n s t o g e t h e r and  Nj(t) Nil (*+'*)  at time  "fc + t  \r(t) electron. The  averaging;  i s the f r a c t i o n of e l e c t r o n s p r e s e n t i n  groun 1  t  -  1:  C o V  where  time  2.  that  r e m a i n i n group 1  until  .  i s the a x i a l v e l o c i t y  component o f any g r o u p  1  ,  d e p a r t u r e o f e l e c t r o n s f r o m group 1  governed by the e q u a t i o n (Appendix 2),.  t o g r o u p 2. i s  P12.N1  where  transition  i s the p r o b a b i l i t y per unit  from  Hence  group  isions. result  Then,  i f *^\,  of a s i n g l e  i s t h e time  performed  Similar  Si  i s the angle  i  i s g i v e n by  between c o l l i s i o n s  over  of d e f l e c t i o n  indicated  ^a.  as a  (Wannier):  f o r electrons i n  by t h e a n g u l a r  a l l possible intravalley  i n which case  coll-  event,  f  time  assumption  involves only e l a s t i c  e x p r e s s i o n s may a l s o  electrons,  i f we make t h e  scattering  g r o u p 1. The a v e r a g e is  ,p,iY  scattering  where t h e r e l a x a t i o n  •Si  £(.t)= e" .  may be f o u n d  intravalley  of a  t o group 2.  has t h e f o r m  cov^vrCt),vr(t that  1  time  brackets  transitions.  be w r i t t e n f o r g r o u p and  Sa,  replace  2 t\  and  respectively.  Assuming t h a t  ^  i s much g r e a t e r t h a n  ^> , ia  dependence  o f t h e r i g h t - h a n d - s i d e of equation  determined  by  takes  c o v ^ t f ( t ) , v ( t 0] +-c  , the time 2.2 i s  so t h a t e q u a t i o n  2.2  the form  Similarly  i f " r ^ " i s much g r e a t e r t h a n  Pa,i  ,  N*tt) N»Ct -**) c o v ^ t ^ ^ c C t - K c ) ) = Na.vav.ure" The with  current spectral t h e above  d e n s i t y , as g i v e n b y . e q u a t i o n 2 . 1 ,  expressions  along with  the population  spectra  t a k e n f r o m A p p e n d i x 2, now t a k e s t h e  —  form:  r o  Carrying  oat the i n t e g r a t i o n s  The d i f f e r e n t i a l , c o n d u c t a n c e ,  gives.  G i t a ) , due t o t h e  electrons  of t h e i t h group i s , (Wannier) \ _  r  where  W \ i  the f i e l d  i s t h e e f f e c t i v e mass o f t h e i t h e l e c t r o n i n d i r e c t i o n and  relaxation We may  Zi  Xl  i s the i n t r a v a l l e y velocity-  time. c h e c k t h e s e f o r m u l a s by c o n s i d e r i n g  e q u i l i b r i u m zero f i e l d  c a s e . Then  V  , and  the  ur  are  z e r o and t h e i n t r a v a l l e y v e l o c i t y r e l a x a t i o n t i m e s a r e equal t o 'to If  , t h e same f o r a l l v a l l e y s .  we assume t h e t o t a l  conductance i s equal t o the  sum o f t h e - s e p a r a t e g r o u p c o n d u c t a n c e s ,  ignoring the  c o n t r i b u t i o n o f i n t e r - g r o u p t r a n s i t i o n s t o Gt<*0 , the'n, in  equilibrium  The  current spectral density i s , at equilibrium,  I n t h e above e x p r e s s i o n ,  a n  &  \'^\  a r e t a k e n as  zero, which i s c o n s i s t e n t with the e a r l i e r  that  assumption  G°(w) = G ° ^ ) 4 G° Cw) x  In the e q u i l i b r i u m case,  y&.r\r  yavur  and  obey t h e .  s i m p l e r e l a t i o n s g i v e n i n A p p e n d i x 3,  Wi varvr = The  K T  w  U  x  v u v u j =. K T  current s p e c t r a l density then  f o r m -given b y the- N y q u i s t  U  c o r r e c t l y assumes t h e  formula, KTu  =  4 GT(co) K T u  R e t u r n i n g t o t h e n o n - e q u i l i b r i u m c a s e , we make t h e following inequalities  i n accordance with.the  experimental  conditions:  The  c u r r e n t s p e c t r u m t h e n assumes i t s l o w - f r e q u e n c y  1  The  L*  fcu?  (fc* +  noise temperature,  T  n  , o f t h e sample under non-  e q u i l i b r i u m c o n d i t i o n s i s a convenient i t s behaviour.  I t i s found S  x  form,  = 4 K T n G  way o f d e s c r i b i n g  u s i n g the. N y q u i s t  formula  where • G  i s t h e sample  differential  g i v e n b y t h e sum o f t h e c o n t r i b u t i o n s of t h e e l e c t r o n  where  Xi.  electrons  V  From t h e f i e l d  from each  c a r r i e d by group  i s the applied  dependence o f  and  "uj  , Giy  conductance  temperature  " l?G> (^+K,?K  Thus we f i n d t h a t contribution the  and  then  and t h e n o i s e  k  i  voltage.  G^-x. may be f o u n d . The t o t a l d i f f e r e n t i a l is  group  population.  i s t h e mean c u r r e n t and  conductance,  KG,  r  the noise temperature contains  due t o t h e d i f f e r e n c e  i n drift  d i f f e r e n t v a l l e y s . This contribution,  a  v e l o c i t y of termed  i n t e r v a l l e y n o i s e b y P r i c e (1965) i s , due t o t h e (vr—ur)*' factor, The ion,  strongly  dependent  on t h e s a m p l e  second c o n t r i b u t i o n ,  represents the increase  increase  i n the variance  each v a l l e y .  orientation.  the "hot e l e c t r o n " i n sample  contribut-  n o i s e due t o t h e  of the v e l o c i t y d i s t r i b u t i o n i n  CHAPTER 3 - EXPERIMENTAL APPARATUS AND A series and  conductance,  ions of a p p l i e d e l e c t r i c crystal the  of noise t e m p e r a t u r e , T n >  of d e t e r m i n a t i o n s  differential  G  field  TECHNIQUE  , were made as on r e c t a n g u l a r ,  single  n - t y p e germanium s a m p l e s whose l e n g t h s  ( i l l ) , ^lio) >  samples  ^ o)  a n ( i  i n each case  temperatures frequencies  and of  ment t e c h n i q u e Basically, 1) A low  output  crystal  1Q  were c o o l e d  and  70 Mc/s  . The  The  nitrogen  measurements were p e r f o r m e d 30 Mc/s  were i n  directions.  to l i q u i d  funct-  at  same measure-  was; u s e d f o r a l l s a m p l e s . the  apparatus  consisted  of:  impedance v o l t a g e p u l s e  supplying lO^sec. sample.resistances  pulses  to  generator  for  samples u n d e r s t u d y .  were o f the  order  o f one  The  hundred  ohms. 2.) A v a r i a b l e f r e q u e n c y voltage  generator  multivibrator for triggering  and  the  the  delay multivibrator(see 4  below). 3)  A low-noise  gated  and  p r e a m p l i f i e r s and  70 Mc/s  i n g the centered 4)  One  30 Mc/s  sample n o i s e on t h e  d e l a y m u l t i v i b r a t o r and the  A temperature-limited noise reference  Mc/s  .meter f o r measur-  a 5>isec  interval  pulse.  square-wave v o l t a g e  gating pulse noise  along with'30  output  during  sample v o l t a g e  ator for supplying 5)  only  amplifier,  diode  t o the  gener-  amplifier.  which s e r v e d  as  a  source.  6) A s e t o f d e p o s i t e d  c a r b o n r e s i s t o r s whose  conductance  at  the  7) A  two  measuring frequencies  sample mount  and  designed  ments. The liquid the  3.1  mount h a d ' t o have p r o v i s i o n f o r be  compact  between the  so  as  measureholding  to  s a m p l e and  minimize the  but  of the p r e a m p l i f i e r .  apparatus  circuit  with  range,  of t h e  sample  least  required heating the  to determine  o f 10  sizes order  cm  used  results of 100  pulses The  was per  chosen, w i t h  next  field  per  210  the  per  the  noise  sample must be  i n mind a p u l s e  very  duration  r a t e of from  must be  measured  measurement  of  of  nine  frequency  of the  pulse be  This  t o 100  Mc/s.  A  o f 10 Mc/s  the  o n l y under-  should  range  Hence,  second.  the r e c i p r o c a l  i n d i c a t e s a value  joule  are used.  noise  in this  fields  c e n t i m e t e r • are.  so t h a t a number o f c y c l e s o f t h e  frequency  liquid  electric  short pulses  a repetition  c o n d i t i o n s , the  much g r e a t e r t h a n  impurity  experiment,  c o n s i d e r a t i o n i s t h e measurement  sample n o i s e . S i n c e pulsed  conditions  desired  i n the  ohms. S i n c e  very  requirements  second t o  the  equiv-  a m e a s u r a b l e amount o f n o i s e ,  i s extreme u n l e s s  small. With these  noise  in resistances in  a v e r a g e power d i s s i p a t e d by  lO^sec.  For  s e v e r a l hundred v o l t s  to produce  the  sample u n d e r h i g h f i e l d  sample h e a t i n g . is .3  order  n i t r o g e n of the at  served  of the  minimal  practical  of  input  APPARATUS DESIGN CONSIDERATIONS  alent  be  filters  f o r f o u r - t e r m i n a l sample n o i s e  lead inductance  The  of  c o n s i s t i n g of a p p r o p r i a t e  n i t r o g e n and, y e t  terminals  were known.  should  duration visible. choice  i s also desirable since  the  l/f  and t h e e x c e s s n o i s e  small,  as d i s c u s s e d  i n Chapter  frequency-dependent ment o f n o i s e uencies  frequencies ability higher  of commercial  sources  a t two w i d e l y  30 Mc/s a m p l i f i e r c o n s i s t e d by f i v e  o r was f o l l o w e d  by a two s t a g e  stage  as u s e d  a cathode  arbitrarily.  o f a low n o i s e  to achieve  only  non-zero. This frequency by  o f t h e same  maximum s e n s i t i v i t y  suppressor  grid  A gating pulse applied pulse so  a 6AK5 p e n t o d e ,  connected  that i t s  t o the cathode.  t o "'cut-off'" t h e 6AS6 was grid.  The d u r a t i o n that  any t r a n s i e n t v o l t a g e s  then  of the g a t i n g  o f t h e sample  pulse  from e n t e r i n g the  due t o t h e s w i t c h - o n and s w i t c h - o f f  current. .  was  the l a s t . r a d i o -  t o t h e 6AK5 e x c e p t  was s e t a t a b o u t o n e - h a l f  amplifier  output  on" t i m e was t h e g a i n  by-replacing  i s net i n t e r n a l l y sufficient  of the  the a m p l i f i e r gain  the "pulse  i s similar  to i t s suppressor  as t o a v o i d  sample  during  apparatus  stage.  of the a m p l i f i e r , normally  a 6AS6, w h i c h  detect-  t h e 50 Mc/s p r e a m p l i f i e r was  was a c h i e v e d  stage  30 Mc/s  video.amplifier with i t s  consisted  a t 30 Mc/s e x c e p t  so t h a t  tuned  cascode  d e t e c t i o n was u s e d . The  m e t e r t o t h e p u l s e ' sample n o i s e , gated  A  follower.  70 Mc/s a m p l i f e r  order  o f t h e two  frequency.  r e p l a c e d by a 70 Mc/s c a s c o d e and a m i x e r In  freq-  of the a v a i l -  synchronously  Vacuum d i o d e  measure-  different  a m p l i f i e r s at t h i s  pentode stages.  The  are n e g l i g i b l e ,  o f 70Mc/s was c h o s e n  followed  t o be  1. As a c h e c k t h a t t h e  was c h o s e n a t 30 Mc/s, b e c a u s e  preamplifier  last  be e x p e c t e d  r a n g e a r e d e s i r a b l e . The l o w e s t  frequency  The  noise  temperature  in this  terms would  of the  28 Such t r a n s i e n t s ,  i f p r e s e n t , w o u l d be v i s i b l e on t h e  monitoring o s c i l l o s c o p e . T h e i r p o s s i b l e presence a l s o checked by p l a c i n g a metal amplifier  resistor  was  across the  i n p u t t e r m i n a l s i n p l a c e o f t h e germanium  s a m p l e and p u l s i n g i t w i t h t h e same v o l t a g e and c u r r e n t m a g n i t u d e s . S i n c e no i n c r e a s e i n t h e a m p l i f i e r m e t e r r e a d i n g o c c u r r e d , i t was c o n c l u d e d  output  t h a t the system  was f r e e f r o m t r a n s i e n t e f f e c t s . Due t o t h e l o w r e p e t i t i o n r a t e and t h e s h o r t d u r a t i o n of t h e n o i s e p u l s e from t h e a m p l i f i e r , the noise.power of the pulsed Instead, a "peak-reading"  s a m p l e was p u t o f t h e q u e s t i o n .  v o l t m e t e r was u s e d f o r t h e o u t p u t  meter. S i n c e t h e measuring procedure  called  o f two n o i s e s o u r c e s , t h e p e a k - r e a d i n g to indicate the equivalence 3. 2.  f o r a comparison  v o l t m e t e r was u s e d  o f two n o i s e  sources.  SAMPLE HOLDER DESIGN The  first  styrofoam two  t h e measurement o f  experimental  s a m p l e h o l d e r t r i e d was a  bottle with stainless  s t e e l connections  t o the  c u r r e n t l e a d s o f t h e s a m p l e . The c u r r e n t l e a d s  served  a l s o a s t h e n o i s e s e n s i n g e l e c t r o d e s . The s a m p l e was c u t i n the form of a r e c t a n g u l a r c r o s s - s e c t i o n bar w i t h  gold  w i r e l e a d s a l l o y e d t o t h e two e n d s , s o t h a t t h e p u l s e c u r r e n t and t h e s a m p l e n o i s e were b o t h  conducted  b y t h e same  e l e c t r o d e s . T h i s arrangement 'is p e r f e c t l y s a t i s f a c t o r y i f no a d d i t i o n a l n o i s e i s i n t r o d u c e d a t t h e j u n c t i o n s b e t w e e n the semiconductor  and t h e w i r e l e a d . H o w e v e r , s i n c e p r e -  c a u t i o n s a g a i n s t j u n c t i o n n o i s e were t a k e n b y v a n d e r Z i e l and  associates,(van der Ziel,1954}  i n t h e i r low frequency  m e a s u r e m e n t s r e f e r r e d , t o i n C h a p t e r 1, j u n c t i o n n o i s e a l s o be p r e s e n t  a t t h e h i g h e r f r e q u e n c i e s as w e l l . 'Prelim-  inary two-terminal (ill) ent  n o i s e t e m p e r a t u r e measurements i n  s a m p l e s showed c o n s i s t e n c y b e t w e e n s a m p l e s ,  a l s o of the d i r e c t i o n of pulse  n o i s e and d e p e n d e n t on t h e d i r e c t i o n  preted  t h e s a m p l e was o b s e r v e d . as e v i d e n c e  independ-  c u r r e n t . I n t h e (iOO)  s a m p l e s , h o w e v e r , n o i s e much i n e x c e s s  through  may  of the ( i l l ) s a m p l e  of current flow .  This behaviour  was  inter-  o f j u n c t i o n n o i s e and t h e t w o - t e r m i n a l  m e a s u r e m e n t s were d i s c o n t i n u e d i n f a v o u r  of f o u r - t e r m i n a l  determinations. The  f o u r - t e r m i n a l m e a s u r e m e n t s were made on s a m p l e s  c u t a s shown i n F i g . 3 . 1 . The p u l s e c u r r e n t l e a d s a r e attached  t o t h e two ends and t h e n o i s e m e a s u r e m e n t s made  b e t w e e n t h e two s i d e c o n t a c t s . I n t h i s manner, a n y n o i s e generated  by t h e passage of c u r r e n t through  a  metal-semi-  conductor  j u n c t i o n does n o t a p p e a r d i r e c t l y  a c r o s s the'  i n p u t t e r m i n a l s ' o f t h e a m p l i f i e r . However, p a r t o f t h e noise generated  a t t h e c u r r e n t e l e c t r o d e s may s t i l l  the a m p l i f i e r unless  enter  e f f o r t s a r e made t o i s o l a t e t h e c u r r e n t  e l e c t r o d e s f r o m g r o u n d p o t e n t i a l . T h i s was done b y means of f i l t e r s , center  as described  i n S e c t i o n 3.3, t u n e d  o f t h e a m p l i f i e r pass band. I t i s then n e c e s s a r y  check t h a t any j u n c t i o n n o i s e small  to the to  entering the amplifier i s  compared t o t h e n o i s e a p p e a r i n g  across  the side  c o n t a c t s o f t h e s a m p l e . T h i s was done by u s i n g two v a l u e s of impedance f o r ' each f i l t e r .  A l l filters  had b a n d - w i d t h s  much l e s s t h a n t h e a m p l i f i e r p a s s b a n d and h e n c e t h e  a b s o l u t e v a l u e o f i m p e d a n c e , \Z\, p a s s band was d e t e r m i n e d  o f each f i l t e r  hy i t s c a p a c i t a n c e ,  i n d u c t a n c e , L , v a l u e s . The r e l a t i o n  1^1 « _uL  where  , and  Z is  of the f i l t e r .  uJo.  and  "  onant f r e q u e n c y  C  (si  i s the frequency  ^TT  for  over the  i s the res-  air  The f i l t e r  capacitance,  C o , i s composed o f an e x t e r n a l v a r i a b l e p a r t and t h e d i s t ributed  coil  capacitance. F i l t e r s having  different  values  of  Co  any  junction noise present at the current electrodes.  were u s e d t o v a r y t h e amount o f a t t e n u a t i o n o f  S i n c e no d e p e n d e n c e o f n o i s e t e m p e r a t u r e a nc e was f o u n d , successful  i t was c o n c l u d e d  on f i l t e r  that a l l f i l t e r s  capacitwere  i n a t t e n u a t i n g t h e j u n c t i o n n o i s e . w e l l below  the side e l e c t r o d e noise. 2.3  MEASUREMENT OF F I L T E R CONSTANTS It  was d e c i d e d  t o mount t h e f i l t e r s  along with the  sample i n t h e l i q u i d n i t r o g e n , s i n c e c o o l i n g t h e c o i l s increased the value of the p a r a l l e l and  hence d e c r e a s e d  filter ance  L  t o c o n s i s t of a p a r a l l e l , distributed  0  C\  capacitance L-o  and  were found for  the f i l t e r  Ci  value of  Cj  r e s i s t a n c e , R,  b a n d w i d t h . We assume e a c h combination  capacitance  C  , and e x t e r n a l  c  , w i t h shunt r e s i s t a n c e f o r each f i l t e r  of the induct-  R  4  i n liquid nitrogen  b y a d d i n g known v a l u e s o f f i x e d m i c a  and d e t e r m i n i n g t h e r e s o n a n t Ci  . The r e s o n a n t  ,  frequency  frequency was f o u n d  condensers f o r each using a  g r i d - d i p m e t e r whose f r e q u e n c y was compared a g a i n s t a Marconi  M e t e r as s t a n d a r d . Lo  slope of a p l o t  of  hy s u b t r a c t i n g  C\  for  each v a l u e of  found  (S  was  vs. C  0  , and  x  from C\  t h e n found; f r o m t h e was  obtained  the t o t a l  capacitance  «  . The f i n a l  v a l u e o f C<j  L  0  ,  0  was  by a v e r a g i n g e a c h o f t h e s e v a l u e s . r e s i s t a n c e , R,  The s h u n t  way. The v a r i a b l e m i c a parallel  w i t h each  , was f o u n d  condensers  coil,  the v a r i a b l e  coil  condenser  w e r e now p l a c e d i n  t h e c o m b i n a t i o n now  the form of the f i l t e r s used measurements. B o t h  i n the f o l l o w i n g  i n the l a t e r  constituting  noise  l o s s e s and d i e l e c t r i c will  contribute to  temperature  losses i n  Rj  The f i l t e r was mounted i n a s t y r o f o a m c o n t a i n e r h o l d ing  liquid nitrogen. Electrical Vs i n c h by /s 3  connections c o n s i s t i n g of  inch brass s t r i p s  were s e a l e d i n t o t h e  b a t h b o t t o m so t h a t t h e f i l t e r , c o o l e d t o l i q u i d n i t r o g e n t e m p e r a t u r e s . c o u l d be c o n n e c t e d  directly  a c r o s s t h e Q-  M e t e r - t e r m i n a l s . The c o n n e c t i o n s b e i n g , s h o r t and o f l o w inductance contributed l i t t l e measurement  procedure  with a high Q c o i l filter  to the c o i l  i m p e d a n c e . The  c o n s i s t e d o f r e s o n a t i n g t h e Q-Meter  at the f i l t e r resonant  was t h e n c o n n e c t e d  frequency.  a c r o s s the condenser  The  terminals  of  t h e Q-Meter and t h e Q-Meter c a p a c i t o r c h a n g e d s l i g h t l y ,  if  necessary,  to b r i n g i t i n t o resonance.  Then t h e d e c -  r e a s e o f t h e Q-Meter r e a d i n g , due t o t h e s h u n t of  t h e f i l t e r , a l l o w s t h e v a l u e of t h e shunt  to  be  found.  resistance  resistance  F i g . 3 . 1 - T y p i c a l f o u r probe sample  to  yrofoam  -Scale  17:1  amplifier liquid  container  nitrogen  filter coil  lucite  base voltage in  Fig.3.2.  Sample  holder  pulse  and n i t r o g e n b a t h  - Scale  1.66  ®  Apparatus used i n the t e m p e r a t u r e and and  determination  differential  corresponding  unit t i t l e s  of sample  noise  c o n d u c t a n c e . Numbers follow  in  Fig.5.5a.  ~ £ 1g ,<•/<J .m. ^o d.  TP  i •  \  Apparatus U n i t J I i t l e s Number  (refer to Fig.3.3) Description  1  Variable frequency  multi-  vibrator 2  lOjjLsec. r e c t a n g u l a r  pulse  generator 3  Pulse  amplifier  4  Fixed  resistor  5  Pulse  attenuator  6  Oscilloscope  7 2c 8  Filters  9  Germanium  10  Liquid nitrogen  • 11  sample  5722 N o i s e  12  30 Mc/s  bath  diode  o r 70 Mc/s  pre-  amplifier 13 14  100 Mc/s •  Peak, r e a d i n g  15 16  oscillator  30 Mc/s ^  gated  voltmeter. amplifier  4^360. rectangular  pulse  generator 17  ,•  3^csec. d e l a y i n g vibrator  multi-  32  3.4  NOISE TEMPERATURE MEASUREMENT PROCEDURE The  first  s t e p i n the' d e t e r m i n a t i o n o f t h e  noise temperature  c o n s i s t e d i n determining the  ential  sample conductance under the p u l s e d  field.  The  .Set  f o l l o w i n g procedure  the n o i s e diode  v a l u e and  across the  to  give a reading  we  d e s i g n a t e hy  OM.  r e a d s X.  accomplished  this.  Then, r e p l a c e field.  "til© p u l s e d f i e l d  amplifier  Adjust the a m p l i f i e r  on t h e a m p l i f i e r  p u l s e d by t h e e l e c t r i c  OM  electric  i n p u t t e r m i n a l s of the  X  differ-  c u r r e n t n e a r i t s maximum r a t e d  p l a c e a known c o n d u c t a n c e , G.  VJ© i n c r © 8 . s©  sample  G  output  meter,' w h i c h  by  sample,  the  N o t i n g the r e a d i n g of a p p l i e d t o the  I f the noise diode  c u r r e n t has  s a m p l e o r t h e known c o n d u c t a n c now  procedure The  been chosen  the  have been overwhelmed  i t i s only necessary  s h o u l d be  obtained  Once t h e s a m p l e d i f f e r e n t i a l  to repeat  i n both  p r o c e e d s i n e i t h e r o f two  c o n d u c t a n c e has  ways, d e n o t e d b y  (Used f o r n o i s e t e m p e r a t u r e room  check  by  the  above  current.  cases.  t h e measurement o f t h e s a m p l e n o i s e  Case A  i n the  sample  f o r a s m a l l e r value of the n o i s e diode  same f i e l d  found,  , then  until.  e q u a l t h e known c o n d u c t a n c e . To  that a l l other noise sources the noise diode,  G  OM,  sample  h i g h enough t o "swamp o u t " a l l o t h e r n o i s e s o u r c e s  conductance w i l l  gain  been  temperature cases  A and  g r e a t e r t h a n ..  temperature)  C o n s i d e r i n g the sample's Norton  noise  equivalent •i  circuit,  the mean-square n o i s e c u r r e n t g e n e r a t o r  i n the  3.  frequency  It  range  is  i s i n parallel  With  with  i | " and i s g i v e n b y  t h e conductance  t h e sample a c r o s s t h e a m p l i f i e r  m e t e r r e a d i n g i s some v a l u e , s a y  G.  input  X .The known c o n d u c t a n c e  along with i t s noise current generator, i s then  substituted  current varied two  current generators  cases  w i t h t h e same v a l u e  noise diode  current generator  in  ,  diode  X once a g a i n . T h e n , t h e  in parallel  where  i£  a r e t h e same, s i n c e t h e m e a s u r e d  n o i s e i s t h e same i n b o t h  If  denoted by  f o r t h e s a m p l e and t h e n o i s e  u n t i l OM r e a d s  are independent,  the.output  and b o t h  generators are  of conductance. Since the  and t h e s a m p l e n o i s e  generator  we w r i t e  = IF  +IF  i s t h e n o i s e diode< c u r r e n t  S u b s t i t u t i n g the formulas  generator.  f o r t h e above c u r r e n t  generators,  leads to  where The  i s t h e mean n o i s e d i o d e  noise temperature  Case B  current.  i s then  (Used f o r n o i s e t e m p e r a t u r e s  l e s s than  room  temperature) In  this  c a s e , t h e known c o n d u c t a n c e i s f i r s t  across the a m p l i f i e r OM  reads  connected  i n p u t and' t h e g a i n a d j u s t e d s o t h a t  some c o n v e n i e n t  value  X . The s a m p l e u n d e r  field,  i s then  connected  diode  current  ad/justed  current  and  until  -  - T l  U  and e r r o r a l o n e  B i s tried, sample  that  l i .  determine whether case  then  i f Tn  OM w i l l  temperature  room t e m p e r a t u r e ,  and c a s e  determine the noise  current  greater  drop  X when  and i t i s o b v i o u s  A p r o c e d u r e must be used t o  temperature.  each d e t e r m i n a t i o n I  of noise  and t h e v o l t a g e  across  a fixed  V  obtained  a 6.15 t o 1 p u l s e  temperature,  applied across  I  was f o u n d  118 ohm w i r e  resistor  sample b u t a t room t e m p e r a t u r e .  ments were  by a p p l y i n g  opened  equalled The uator  from the v o l t a g e i n series  attenuator  the attenuated resistors  used  were o f d e p o s i t e d  pulse  point>  through the switch'  V . . was v a r i e d u n t i l i t B  0  voltage.  i n t h e compensated p u l s e carbon  pulses  t o w h i c h a v a r i a b l e L.C.  t h e o s c i l l o s c o p e as a d e t e c t o r ,  and c l o s e d , w h i l e  with  measure-  t h e unknown v o l t a g e  c  a switch. Using  t h e sample  i t s current  A l l voltage  v o l t a g e , Vj>.. , was a p p l i e d a t an i n t e r n a l  was  than  and  of t h e sample i s g r e a t e r than .  t e r m i n a l s were a l s o f o u n d .  to  read  To  MEASUREMENT OF PULSE VOLTAGES For  the  A or 3 i s  i s greater than  i s applied to a m p l i f i e r input  the noise  3.5  . Equating  2.KG  o  a p p r o p r i a t e . F o r example,  the  X  temperature i s T  case  OM r e a d s  and t h e n o i s e  generators.gives,  the noise  Trial  to a m p l i f i e r input  atten-  c o n s t r u c t i o n and o f low  voltage 280  c o e f f i c i e n t . The a t t e n u a t i o n was c h e c k e d a t a  volt pulse  a m p l i t u d e and f o u n d t o be w i t h i n one p e r  cent of t h e low v o l t a g e 5.6  value.  DETERMINATION 0? SAMPLE ELECTRIC F I E L D • Experimentally  i t was f o u n d t h a t f o r t h e s a m p l e  g e o m e t r y u s e d , a s i d e e l e c t r o d e and t h e c e n t e r the  l i n e of  s i d e - a r m extended t h r o u g h t h e sample form an e c u i -  potential part  s u r f a c e . The e l e c t r i c f i e l d  o f t h e sample, t h a t  F  i n the "active"  i s , t h e p a r t between t h e s i d e  p r o b e s was t h e n t a k e n as  F Y  where L  x  and  i s the center  Empirically,  where and  V,  3  3  L a r e t h e s i d e e l e c t r o d e v o l t a g e s and.  l i n e separation of the side-arms.  i s the voltage  that  across the current  electrodes,  C^ i s c o n s t a n t f o r e a c h s a m p l e . T h i s means t h a t t h e  V, a l o n e .  length  current  may be d e t e r m i n e d f r o m a measurement o f  That i s , tr  due  A  i t was f o u n d  electric, field  The  V  V -V  «  =  C^L  V*-V3  ^  VL.  i s f o u n d t o be l e s s t h a n t h e -sample  electrode separation;  the difference i s l i k e l y  t o penetration of the current  electrodes  s a m p l e i n t e r i o r and t h e c o n s e q u e n t l o w e r i n g electrode  separation.  into the c f the current  3.7  SAMPLE PREPARATION' The  '  s i n g l e c r y s t a l f r o m w h i c h t h e samples were  p a r e d was g r o w n f r o m t h e -melt b y t h e C z o c h r a l s k i nique.  pre-  tech-  The a x i s o f t h e c r y s t a l was i n t h e ( l l O ^ d i r e c t i o n .  A value  o f i m p u r i t y d e n s i t y was c h o s e n s o a s t o p r o d u c e  a resistivity crystal.  o f about f i v e  T h i s was t h e v a l u e  most r e c e n t  experimental  ohm-cm a t t h e c e n t e r of r e s i s t i v i t y  ofthe  f o r which the  w o r k on a n i s o t r o p y  of hot e l e c t r o n  c o n d u c t i v i t y h a d b e e n done and s o f o r t h e s a k e o f c o n t i n u i t y i n t h e experiments t h i s choice.  The grown c r y s t a l  seemed t h e most  e x h i b i t e d s i x faces  reasonable along the  d i r e c t i o n o f g r o w t h . Laue x - r a y a n a l y s i s e s t a b l i s h e d of these f a c e s  four  a s ^ 1 1 1 ^ a n d two a s ^LOO^. The ( l l O ^ d i r e c t -  ion-was found as w e l l . N e x t , s l i c e s were c u t a t r i g h t  angles t o t h e growth  a x i s o f t h e c r y s t a l b y means o f a 0.003 i n c h tungsten  wire,  d r i v e n i n a r e c i p r o c a t i n g w i r e saw.  C a r b o r u n d u m was u s e d a s t h e c u t t i n g a b r a s i v e . were;then lapped  to a thickness  with a v a r i a t i o n i n thickness 5$.  using  11  The s l i c e s  o f a b o u t 0.4 m i l l i m e t e r  o v e r each s l i c e  F o u r t e r m i n a l samples' ( F i g . 5.1)  slices  diameter  o f about  were c u t f r o m t h e  a n u l t r a s o n i c c u t t i n g t o o l and c a r b o r u n d u m  as t h e c u t t i n g a b r a s i v e . The s a m p l e s were t h e n e t c h e d i n medium C?4 s o l u t i o n f o r a b o u t 45 s e c o n d s t o c l e a n a n d prepare the surface wire gold,  f o r t h e attachment o f t h e four  l e a d s . These l e a d s  c o n s i s t e d o f 0.005 i n c h  gold  diameter  doped w i t h 0.6% a n t i m o n y , • a c c o r d i n g t o t h e S i g m u n d  Cohn Co., t h e m a n u f a c t u r e r . When b r o u g h t . i n t o . c o n t a c t  with  ths etched  g e r m a n i u m i n a n i t r o g e n gas  a t e m p e r a t u r e above the erature, a l i q u i d i m o n y , and  gold  ed and  liquid  the  between the  atmosphere  g o l d - g e r m a n i u m e u t e c t i c temp-  phase c o n s i s t i n g of.germanium,  i s f o r m e d . The  temperature  phase s o l i d i f i e s  gold.wire  and  a  n^-n  j u n c t i o n , the  lower-  t o f o r m an a l l o y  t h e s a m p l e . Upon  sample s u r f a c e . T h i s  ant-  i s then  region,  solidification,  a germanium r e g r o w t h l a y e r , h i g h l y doped w i t h f o r m s on t h e  at  antimony  j u n c t i o n has  been termed  P r e f e r r i n g t o t h e h i g h l y doped  r e g r o w t h l a y e r , and  the  Electrically,  j u n c t i o n s p e r f o r m i n s u c h a manner  that they  these  n  to the  are h i g h l y impervious  sample c o n d u c t i v i t y .  to hole  current  a l l o w i n g e l e c t r o n current passage. Expressions d e r i v e d f o r the hole difficult  c u r r e n t but  possibility p l a c e and As  j u n c t i o n . In t h i s  3.8  W i t n no  surrounding  now  consider  p  sufficiently  TO  takes  i t s importance. for  five  JOULE SEATING  a p p l i e d t o the  e q u i l i b r i u m w i t h the  sample, i t w i l l  liquid  nitrogen  i t a t t h e a m b i e n t t e m p e r a t u r e , Go. . the  of v o l t a g e p u l s e s where  the  solution.  voltage pulse  remain i n thermal  We  injection  s a m p l e s were e t c h e d  SAMPLE TEMPERATURE R I S E DUE  bath  h i g h l y doped  i n d i r e c t , means a d o p t e d t o a s s e s s  s e c o n d s i n a medium CP4  the  experiment,  i s a l w a y s assumed t h a t h o l e  a f i n a l step, the  have b e e n  depend I n p a r t on  t o determine p r o p e r t i e s of the  r e g r o w t h l a y e r of the  while  i s the  s a m p l e t e m p e r a t u r e when a  of d u r a t i o n  . <*  and  succession  frequency  ,  o f f - t i m e between p u l s e s . A f t e r a  l o n g time the  sample w i l l  be  in' a  steady-state  when t h e h e a t g e n e r a t e d by a s i n g l e p u l s e  i s removed  from the  heat  sample d u r i n g the o f f t i m e .  c a n be hy gen  and  conduction  by  and  conduction  Vve assume t h a t  convection  throught  the thermal  the  cyclic  s a m p l e t h e n may act)  temoerature, i n time  In the  Qi  8—  For  a  steady-state  at the  end  start  start  of the  of  n c 0  of the next  get,  for  time  t  pulse. i n the  i s i t s s p e c i f i c heat,  pulse, and  "t  0  constant.  <x ^ t  interval  the  'to  b  s a m p l e mass. C\>  experiment,  is  pulse,  i s t h e p u l s e power d i s s i p a t e d d u r i n g t h e  i t s thermal  ,  to  to  was  a l l measurements,  Then, f o r  at the  o f c o o l i n g , we  oLt  By  liquid  ,  at  V\o i s t h e  is  i s small  c h a r a c t e r i z e d by  Gi.  at the  range from zero t o  In the  sample i s  to the  which, i n the  steady-state,  A s s u m i n g Newton's l a w  is  wires.  t.  t h e n decays back t o  P  the l e a d s  surfaces  be  v o l t a g e p u l s e , r i s e s t o 0-f  where  nitro-  temperature gradient  compared t o t h e l o s s e s f r o m t h e  uniform  sample l e a d  t h a t t h e h e a t c o n d u c t e d by  n i t r o g e n . The  removal  to the l i q u i d  c o n d u c t i v i t y of the  h i g h e n o u g h so t h a t t h e r a d i a l s m a l l and  This  f o u n d t o be  a b o u t 0.05  i s constant  much l e s s t h a n u n i t y ,  and  seconds. _5  e q u a l t o 10  sec.  Putting  i n extreme v a l u e s  of  P = -Xoo watt —-5  M  = 4-4-  x  0  / 0  C = 0-153  Jo"'e  fr  9t - 0 * ^  Then  0-4°  - a . So,  the  than  degrees,  of 3  Taking repetition  Then  and  the  rise  d u r i n g the  i s less  pulse  of  the  degrees. the  case  o f low  rate results  l-4-°  Q i - Q ^  3.. 9  0  maximum " b a c k g r o u n d " t e m p e r a t u r e r i s e  0.5  order  3  electric  i n the  and  INFLUENCE OF  field  and  high  pulse  f o l l o w i n g values::  d $ - Q ± — \ - S °  SKIN EFFECT ON  SAMPLE DIFFERENTIAL  CONDUCTANCE The  skin  where jx  0  &  , i s given  i s the frequency, i s the  For of  depth,  the  cr  p e r m e a b i l i t y o f the  a cylindrical  £  0  the  c o n d u c t i v i t y , and  medium.  sample o f r a d i u s  r e s i s t a n c e at frequency  resistance,  by-  f  , i s approximated  T  t o the as  0  , the  ratio  zero-frequency  The  above a p p r o x i m a t i o n  greater than At  77° K,  holds  o n l y l o r s k i n d e p t h much  the r a d i u s .  s a m p l e r e s i s t i v i t y was  t h e mean c r o s s - s e c t i o n a l a r e a Then,  and  o  _  skin effect In the  effect  i s a p p r e c i a b l e , but  resistivity diameter  of the  0.1  ram .  g o l d w i r e l e a d s , we c a n be n e g l e c t e d  o f t h e g o l d w i r e was  a b o u t 10  o f 0.005 i n c h e s , t h e r a t i o  a b o u t 0.01  x  the  find  anyway.  c a n t h e r e f o r e be  INVESTIGATIONS OF  i s about  ible  o f one  SPURIOUS SOURCES OF  soldered  five.  sample  NOISE the  ampere were a p p l i e d t o t h e s a m p l e . The  i n the  a  neglected.  e x i s t e n c e o f t r a n s i e n t v o l t a g e s as w e l l as  generated  The  effective resistance  Hundred v o l t p u l s e s r e s u l t i n g i n c u r r e n t s of order  skin  ohm-cm. F o r  ohms, w h i c h i s s m a l l compared t o t h e  r e s i s t a n c e and 3.10  order  and  negligible.  F o r a t y p i c a l l e n g t h o f 3 mm,, is  ohm-cm.  7  i s completely  case of the f i n e  a b o u t 0.3  j u n c t i o n s and  poss-  noise  pressure  metal-to-  m e t a l ' c o n t a c t s e n t e r i n g t h e a m p l i f i e r must be i n v e s t i g a t e d . P o s i t i o n i n g the  ,!  on"' p u l s e t o t h e a m p l i f i e r a t  c e n t e r of t h e v o l t a g e p u l s e i z e s t h e t r a n s i e n t s . Any the  amplifier  i n g the output the course  should  be  s u p p l i e d t o the' s a m p l e m i n i m -  l a r g e t r a n s i e n t s p i c k e d up visible  on t h e  n o i s e of the a m p l i f i e r .  of the  the  experiments.  Small  by  o s c i l l o s c o p e monitorNone was  observed  transients as.well  in as  spurious  noise  pressure  contacts  A metal stituted The as  generated  resistor  a typical  inductive  germanium  No a d d i t i o n a l when t h e w i r e  and  wire  of l i q u i d  was r e s i s t i v e  was  sub-  nitrogen.  o f t h e same  s a m p l e and. h a d a s w e l l a  order  small  i n series.  noise  was o b s e r v e d  s a m p l e was p u l s e d .  from the a m p l i f i e r  Hence,  i t was  concluded  transient effects  as w e l l as s p u r i o u s  electrode  from the  noise  external  the sample. Another  generation the  possible  source  room i l l u m i n a t i o n .  bath  3.11  This  i s due t o  checked  optical cue t o  and f o u n d  sample n o i s e  was c h a n g e d  when a l l l i g h t s  covered  was  since the pulsed  when t h e i l l u m i n a t i o n zero,  of noise  o f h o l e - e l e c t r o n p a i r s i n t h e sample  non-observable  to  of a fine  i n the bath  t h e s y s t e m was f r e e f r o m  sample p u l s e to  i n the form  of the wire  component  joints  i n v e s t i g a t e d i n t h e f o l l o w i n g way.  f o r t h e sample  impedance  that  were  a t the. s o l d e r e d  d i d not  from- t h e n o r m a l  were turned  t o be vary  amount  o f f and t h e sample  w i t h a "txlack c l o t h . .  S A M P L E N O I S E DUE TO H O L E G E N E R A T I O N AT THE  CURRENT  ELECTRODES The hole can four  possible existence  injection only  at the positive  sample  be i n v e s t i g a t e d i n d i r e c t l y  t e r m i n a l samples.  ments a more d i r e c t The  of a d d i t i o n a l  first  noise  F o r the•two  noise  current  due t o electrode  i n the case  of the  terminal noise  measure-  c h e c k was p o s s i b l e . temperature  measurements, performed .  on  rectangular  bars  same a s o b t a i n e d eliminate inal  block to  filament  e n d was  block  holes  average  ing  the pulse  injection  a special  of the block  only  the block  duration.  were p o s i t i v e ,  during  holes.  wire  were  it  was c o n c l u d e d ,  direction In  with  temperature and  measurements  30 M c / s . A n y n o i s e  exhibit since  a strong  the hole  current  of  resistance injected  on t h i s  sample t h e same  samples.  i n the  samples,  was o b t a i n e d  o n t h e same  due t o h o l e  frequency  dar-  Hence  (ill")  injection.  terminal  information on hole -injection  length  and were  measured  on  electrode  free  two t e r m i n a l  any  drift  the high  temperature  the noise  of the four  would  pulse  be r e l a t i v e l y  was n o t due t o h o l e  the case  alloyed  so t h a t  i f the block  the voltage  the previous that  rect-  large  were  electrode  showed no d e p e n d e n c e o n p u l s e , p o l a r i t y obtained  term-  of a  a  leads  chosen  the block  Measurements of noise  as  with  electrode  Hence,  o f the sample would  consisting  To  and t o t h e end o f t h e f i l a m e n t .  dimensions  one-half  samples.  i n ,the t w o  direction gold  were about t h e  terminal  noise  sample  c u t . The u s u a l  iniected.into  direction  the' f o u r  iii the ( i l l )  electrode  the  part  with  possible hole  one c o r n e r  The  later  measurements,  angular  i n the ( i l l )  spectral  with  sample  over  density  noise  at both  injection  dependence  further-  should  this  should  70 M c / s  range, be o f t h e  form S  where  Ij>  r  (00) =  4 - 1 ^ ll-cosuai^  i s the hole  current  a n d • tj>  , the hole  transit  time  i s of  the  order  Here i t i s assumed ed  due".to t h e i r  was  determined  10"'*' s e c o n d s ,  than  the The  used  -7  seconds.  transit  for a typical  r e c o m b i n a t i o n may  be  time.  The  lifetime  sample  and  found  which i s s e v e r a l o r d e r s  absence  considered  10  that hole  brief  transit  then  of  of  neglectof  holes  t o be  about  magnitude  greater  time. of  frequency dependent n o i s e  i n d i c a t e s that hole small.  injection  noise  i n the could  sample be  CHAPTER 4 - EXPERIMENTAL RESULTS The Tn  experimental results  , and d i f f e r e n t i a l  consist  conductance,  30 M c / s a n d 70 M c / s , a s a f u n c t i o n for  t h e sample a x i s  ions. one  value of impurity  The are  , measured  of e l e c t r i c  at both  field,  o f n - t y p e 'germanium  density.  The l a t t i c e  E,  direct-  was o f  temperature  t h e r a n g e o f 77°£ t o 80°K.  most p r o m i n e n t  i t s apparent  range  G  temperature,  i n t h e ( i l l ) , <(llo) , a n d ^ 1 0 0 )  The sample m a t e r i a l - ,  was w i t h i n  of noise  lack  features  of the.noise temperature  o f f r e q u e n c y dependence  50 M c / s t o 70 M c / s a s w e l l  over the  as i t s h i g h degree o f  anisotropy. The events  lack  of frequency dependence  contributing 'X  mean d u r a t i o n  Certainly that  the carrier  average  MX  i s less  mean f r e e  time  an-  uency  i n t h e measured  fluctuations  arising  anisotropic  than  thai!  t o be f r e q u e n c y Also,  since the of the  seconds,(Appendix from  also-be independent  such  valley-  of freq-  range.  and Gunn(1962), by m e a s u r i n g at right  a  seconds.  Sy-io''"  i s less  i s o f t h e o r d e r o f £xl0  transitions\would  Srlbach  found  time  than  e l e c t r o n ' spends i n one v a l l e y  a n y m o d u l a t i o n o'f t h e - c u r r e n t  have  i s much l e s s  i n t h e range o f measurements.  c o n d u c t i o n band  valley  fluctuations  t h e r m a l n o i s e would be e x p e c t e d  independent  5)-  , such that  At 70-Mc/s t h e n , - X  unity.  so  t o the current  suggests that the  the current  angles to the e l e c t r i c  behaviour at a l a t t i c e  field  also  temperature of  77 K. However, t h e d e p e n d e n c e o i ' t h e n o i s e t e m p e r a t u r e , representing the current fluctuations field,  on f i e l d  was f o u n d  n  p e r p e n d i c u l a r t o zhe  t o be l i n e a r w i t h e l e c t r i c  o v e r most o f t h e r a n g e i n v e s t i g a t e d  \  field  by them, w h i c h was o f  t h e r e g i o n up t o one t h o u s a n d v o l t s p e r c e n t i m e t e r . F o r our e x p e r i m e n t a l log?  r e s u l t s , both  p l o t s w e r e made t o s e e i f a n y s i m p l e  existed  between  In  and  F  b u t none was  Most o f t h e e x p e r i m e n t a l was  due t o f l u c t u a t i o n s  error  and l o g i n v s . relationship found.  i n both ' T  n  and  t h e maximum e s t i m a t e d  a s i n g l e r e a d i n g b e i n g a b o u t 10$> f o r t h e n o i s e  u r e a n d a b o u t 7% f o r t h e error  G  i n t h e meter m o n i t o r i n g t h e out-  put n o i s e o f the a m p l i f i e r , in  logT* vs. F  differential  o f a b o u t 4yo was p r e s e n t  error temperat-  c o n d u c t a n c e . An  i n the values of f i e l d .  .  :  :  N o i s e T e m p e r a t u r e vs.' E l e c t r i c  400-  for of  s a m p l e a x i s i n ^00}  Ex£.*,l  Field  and l a t t i c e  temperature  77°K. — -.  experimental theoretical  a)  u  •p cd  a?  © co  o  >7»  25o  y  200 \  y  y  y  y  y y  y ISO) y  \OQY  y  So  y  y  y  y  y  y  y  y  too  I So  2.00  z$o  Electric  ooo  35b  Field,(y/cm)  4-co  Noise Temperature vs. El.ectric for of W  s a m p l e a x i s i n (Lid)  Field  and l a t t i c e t e m p e r a t u r e  77° K. experimental  7  theoretical  fn ZS -P co  <D (o • P. £•!  CO E-<  0  co  O c 5  / /  100  ZOO  260 Electric  3QQ 3SO Field,(v/cm)  400  \lo<-  ? j r-.A  Noise Temperature vs. E l e c t r i c for 1560.  of  sample  axis  i n( i l l )  and  . 7,  Field .  l a t t i c e temperature  77° K.  7  experimental theoretical SH  1300  in CD  0 EH CD CO •H  IIOO  O  <?00  intervalley & hot e l e c t r o n  / /  / /  loo  500  300  |O0  5o  ico  |$o  3oo  2oo  Electric  350  Field,(v/cm)  4-oo  \ \  \  \  \  Differential for of  \  s.  Conductance v s . E l e c t r i c  Field  s a m p l e a x i s i n <^.od]>. and l a t t i c e t e m p e r a t u r e 77°K. experimental theoretical  0  S o  joo  1^0  2.00  i  z$o  Electric  300  l 3£o  Field,(v/cm)  4<?o  SO  )00  jSO  XOo  250 E l e c t r i c  300  35o  F i e l d , ( v / c m )  400  CHAPTER  In with the  -  5  this  the  chapter,  formulas  The  and  owing 1)  for  2,  Chapter and  the  the  hot  evaluate  the  intervalley  G-  4  v/ITH  are  THEORY  compared  obtained  appropriate parameters  from  fieldare  sample n o i s e  shows two  found.  temperature,  independent  terms:  the  electron contributions. contribution,  the  foll-  i s required:  The  drift, velocity  of  the  electric  i n any  field  The  the  and  ptj  principal  crystal  field  jp^  m u s t be  populations This  and  i s done  1962)  any  field  directions. taken  5). For  used the  to  the  values  drift  of  Stratton's later,  from  non  the the  zero as  deduce  transition  strength in.the  For  case data  found  of  three  zero  of Yceinreich  fields,  the  intervalley  sample d i f f e r e n t i a l  principal  velocities.  , at  be  for  discussed  intergroup  The  From the  as  the  i n Appendix  of  three  T  e l e c t r o n temperatures,  theory,  the  of  may  a l , (Appendix  of  valley,  B a r r i e s extension  Burgess,  values  probabilities,  the  one  required drift  absolute  valley  e l e c t r o n s i n any  orientations.  supplies  et  Tn  the  theory(Barrie  3)  of  and  crystal  2)  of  2 after  expression  i n Chapter  intervalley  results  values  Chapter  RESULTS  EXPERIMENTAL  orientation-dependent  final  derived  To  of  OF  the  theoretical  dependent  as  COMPARISON  the' d i f f e r e n t  from the  Barrie  steady-state  valley  transition  probabilities.  2. conductance,  drift- velocity velocities  the  given  by  calculated the  mobilities  Barrie are  from theory.  found:  47  the  Q u a n t i t i e s f i n a l l y used f o r t h e d i f f e r e n t i a l  d u c t a n c e a r e A-C^ifO The  and <L 0^?-^)  n o i s e t e m p e r a t u r e c o n t r i b u t i o n due t o " h o t e l e c t r o n s "  requires the d i f f e r e n t i a l valley  groups; t h i s  as i n 3) a b o v e . the  con-  valley  conductance  o f each o f t h e  i s o b t a i n e d from t h e m o b i l i t y  As w e l l as t h e d i f f e r e n t i a l  results,'  conductance,  e l e c t r o n t e m p e r a t u r e s a r e needed;  these are .  g i v e n by t h e B a r r i e t h e o r y . 5.1  DETERMINATION OF GROUP DRIFT VELOCITY FROM THE BARRIE THEORY B a r r i e ' s extension of Stratton's high f i e l d  ( B a r r i e and B u r g e s s , 1 9 6 2 ) r e l a t i n g the d r i f t  c o n s i s t s o f two e q u a t i o n s  v e l o c i t y , vT*  single valley to the e l e c t r i c  of electrons i n a  v  field,  , where t h e  s t a r denotes v a l u e s i n t h e t r a n s f o r m e d &-space. The  theory  equations a r e :  4-  V-A?  K!  +  4^KT 7e L  Wo  H o •=  where  C and  ;  3  X  e / 0  are constants  Q  Ience, i f  the v a l l e y  vr**  rather field  _  ©a.  noted  situation  he  found  . 1 Wo  *  from the  ^ Jrtt  U  V  volume.'  01  IT  relations:  JTMI  A  that the B a r r i e equations  w e l l  i n . the. p r e s e n t  were o r i e n t a t e d  0  velocity,  easy t o a p p l y i f the d i r e c t i o n i s known, as  d  T  i s known, t h e d r i f t  . and  s h o u l d be  «  and V ' i s t h e c r y s t a l  e l e c t r o n s may  Wo  It  V  .  Tu  of the  are  electric"  as i t s m a g n i t u d e . T h i s i s t h e experiment,  along the p r i n c i p a l  i n 'which the  samples  c r y s t a l " d i r e c t i o n s -  because o f the symmetrical, arrangement of the v a l l e y s  Then, about  the l o n g i t u d i n a l in  the  sample a x i s ,  same d i r e c t i o n  The  give the  as w e l l field,  the  complete procedure  of the Barrie- t h e o r y will  as  principal 1) S o l v e  Tt  This  the  equations  f o r any  proceeding  value  and  r  field  may  and  r *  gives  T  be  found.  of each v a l l e y  onent's o f  in  the  by T  , in  inserting the•equations, and  upwards.  /V**  ulations  applied  to the...lattice temperature  , allows  From the dependence o f  knowing  of  vallejr  '  temperature,  2), K n o w i n g t h e a p p l i e d found.  solution .  sample c u r r e n t a r e  vr**  for  equations  , f o r each  crystal directions.  s t a r t i n g w i t h T equal  VT  group d r i f t The  and  axis.  the f i n a l  follows.  temperature,  the f i e l d  values of electron  V  i s as  as t h e d r i f t . v e l o c i t i e s provided  v:as known t o be-  longitudinal  in solving  then  electron  the f i e l d  be  F*  .,. and  hence t h e v a l l e y  From  AT*"*  nd ir**  the d r i f t  popvelocity  i s o b t a i n e d . Then, t a k i n g  velocities,  shown i n a t a b l e  on  to  are p a r a l l e l , - gives  i n the f i e l d  relationship  AT**  T  vf  r and  direction  gives the  W  b e t w e e n some o f t h e s e on t h e f o l l o w i n g  comp-  page.  quantities  is  :  The f o l l o w i n g the  table  group t r a n s p o r t  summarizes  cn  properties:  Samples. Axis \^'. Directions.  100  the i n f o r m a t i o n  F.*  c *  F  F  54°. 8  _vr \)-«"*  ---  P|  ur  ---  Pi  ' . 110 Gold Hot  Valleys  35°. 2 '90°  Valleys  111 Cold V a l l e y Hot  Valleys  K  ?i  energy  •©  =  0° 70°. 5  i s the angle ellipsoid  / i s s . ^  and  ?5  between  the major a x i s  the f i e l d  « . =  of a constant  direction.  (Bus.  +  B a  •  "  51  5.2  CALCULATION OF D I F F E R E N T I A L CONDUCTANCE The  low  frequency  a n c e f o r one  valley  v a l u e of t h e d i f f e r e n t i a l  i s given  b- —  conduct-  by  — —  u  mi L 2  This i s also eaual to  Mx  where  i s t h e group  c u r r e n t of group i applied Values  _ a Mi  i  oU^H  m o b i l i t y , It  e l e c t r o n s , and  ^ i F  50 v o l t p e r  were o b t a i n e d  cm.  intervals,  differentiated,  f o r m u l a . The estimated repeated  -the  s  from B a r r i e ' s theory.  the f u n c t i o n ^ i F  using a five-point  was  i n t h e f o l l o w i n g way.  The  differentiation  erence  found  greater than total  2.5%  was  conductance  c o n t r i b u t i o n s from both DETERMINATION OF  G  was  was  differentiation  u s i n g a t h r e e - p o i n t f o r m u l a , and  At  numer-  e r r o r involved i n the d i f f e r e n t i a t i o n  formula  5.3  ±  F-v/i_  compared w i t h the f i v e - p o i n t  The  i s t h e mean  field. of  ically  ^Ti  the  was  results  v a l u e s . No  diff-  b e t w e e n them. found  by a d d i n g  the  groups.,  CARRIER DENSITY AND  ELECTRON  POPULATION I n the  ( l i d ) d i r e c t i o n sample  found  t o be  o f 6%  due  2.46  ohm-cm a t 23°C, w i t h an  t o t h i c k n e s s and  estimated  as b e i n g due  c o n t r i b u t i o n of s c a t t e r i n g o n l y w i t h the  was error  width v a r i a t i o n s along  sample l e n g t h . Taking, the m o b i l i t y lattice  , the r e s i s t i v i t y  the  to  the  imp-  urity the  component  carrier  s m a l l , r e s u l t s i n a value  d e n s i t y . To  check  that  o f 6.5x10 m f o r  the i m p u r i t y s c a t t e r -  i n g i s s m a l l a t room t e m p e r a t u r e , t h e m o b i l i t y a s g i v e n by  the Debye-Conwell theory,  o f b o t h i m p u r i t y .and l a t t i c e and  which includes the e f f e c t s c a t t e r i n g , was c a l c u l a t e d  f o u n d t o be i n a g r e e m e n t w i t h t h e l a t t i c e  alone  mobility  t o w i t h i n l e s s t h a n one p e r c e n t .  I n t h e (lOO*) and ( i l l )  d i r e c t i o n s , t h e measured  r e s i s t i v i t i e s were 2.03 and 2.00 ohm-cm. r e s p e c t i v e l y , w i t h maximum  e r r o r s o f a b o u t &fo. The c a r r i e r d e n s i t i e s 710  are  essentially  In c a l c u l a t i n g the t o t a l the  noise  generating  the present  -3  t h e same a t 8.0x10 ra .  part  sample e l e c t r o n p o p u l a t i o n ,  o f t h e sample o f i n t e r e s t i n  i n v e s t i g a t i o n was c o n s i d e r e d  t o c o n s i s t of  the main body o f t h e s a m p l e ( F i g . 3 . 1 ) between t h e c e n t e r lines  of. t h e s i d e p r o b e s and e x c l u d i n g t h e s i d e  t h e m s e l v e s , w h i c h w o u l d be l a r g e l y The  values  <110>  N =  (l00>  N =  (ill)  f\ -  5.4  of t h e t o t a l p o p u l a t i o n  regions.  obtained  T '°' — • n ——  were:  i.fc&y.io"  S--I7 x i o "  THE HOT ELECTRON C O N T R I B U T I O N TO T H E 'NOISE As  W  low f i e l d  probes  given  i n C h a p t e r 2, t h i s  Ni WW w u -  c  \  , r  + la. ^  KM d_('ju.,F +  M*F)  contribution i s  v « r ur « L U x F ) di-  TEiviPERATURE  I n A p p e n d i x 3, i t i s shown t h a t f o r t h e d i s t r i b u t i o n i n A - s p a c e assumed b y B a r r i e , given  the v e l o c i t y  variances are  by  vY\ •  where  t  and  Wrt^,  are, respectively",  the group 1  and g r o u p 2 e f f e c t i v e m a s s e s i n t h e f i e l d T-  t  i s the electron The f i n a l  ion  T  and  temperature of the i t h group.  e x p r e s s i o n f o r t h e hot e l e c t r o n  contribut-  i s then  =  dF  ,  The v a l u e s o f ^ ' / N population ratio, direction,-  For  direction  and  ;  ^VH  a r e found from the v a l l e y  ^ V Y \ . F o r t h e sample a x i s %  i n the ( l l O )  n a.  the (ill') d i r e c t i o n ,  _Hv_ __ N  nX 3  +  a.  V a l u e s o f -711 .are. f o u n d u s i n g r e l a t i o n ' A£.l  o f Appendix 2  and r e q u i r e o n l y t h e v a l u e o f t h e g r o u p e l e c t r o n , t e m p e r a t res,  T, . and T ^ .  1  C H A P T E R 6 - C O N C L U S I O N S AND S U G G E S T I O N S FOR F U T U R E w'ORK  The first  work d e s c r i b e d  experimental  frequency  noise  i n the. p r e c e d i n g  investigation  i n the f i e l d  chapters  of high  direction  i s the  c u r r e n t , •'high-  i n - an  extrinsic  semiconductor. Two  independent  interpret noise.  sources  the r e s u l t s ,  intervalley  An a d d i t i o n a l s o u r c e ,  total  electron population,  to  negligible.  be  o f . n o i s e have,been used t o  I n ^two c r y s t a l intervalley  noise  electron noise, erature values imental poorly  results  and h o t e l e c t r o n  due t o f l u c t u a t i o n s i n t h e  has been shown  theoretically  o r i e n t a t i o n s , <^lll) and -  with  the r e s u l t i n g  ( i l l ) noise  temperature  agree w e l l w i t h  f o r the ( i l l )  to vanish  ion  appearing  hot  electron noise  direction  the exper-  and . r a t h e r . .  with  and o n l y  i n the n o i s e  i n the other  roughly  noise  the experimental  sample, a x i s w i t h  noise i s  t e m p e r a t u r e . The c a l c u l a t e d  two c r y s t a l  temperature.is  the i n t e r v a l l e y  o r i e n t a t i o n s . . T h i s agrees results.  of e r r o r i n the experimental due t o s l i g h t  misalignment  respect t o the principal' c r y s t a l  T h i s would be p a r t i c u l a r l y  -  t h e hot. e l e c t r o n c o n t r i b u t -  i s much s m a l l e r .than  • One p o s s i b l e s o u r c e of  temp-  f o r t h e (lio) .  predicted  the  o f the- ( l i d ) . . The ' c a l c u l a t e d ,  I n t h e (lOO^ d i r e c t i o n , , t h e i n t e r v a l l e y  noise  s'lio) ,  i s " c a l c u l a t e d t o dominate t h e hot  greater than'that of noise  noise  undesirable  values  of the direction  i n t h e "(lOO)  samples  where t h e  noise  the  o r i e n t a t i o n s . Any  other  result erent  t e m p e r a t u r e i s so much s m a l l e r such misalignment  tnan f o r would  i n d i f f e r e n t t r a n s p o r t p r o p e r t i e s of the v a l l e y s and  t h i s reason,  consequently,  diff-  i n t e r v a l l e y noise. For  i t i s somewhat s p e c u l a t i v e t o draw  conclusions  from the  discrepancies  Discrepancies t e m p e r a t u r e s may ed  i n the  theory.  1)  In the  ( i l l ) and  due  and  direction.  b e t w e e n a c t u a l and be  any  between t h e o r y  e x p e r i m e n t f o r measurements i n t h i s  predicted  noise  to several s i m p l i f i c a t i o n s  These  (lio)  current  d i r e c t i o n s , the  inter-  c a l c u l a t i o n s include t r a n s i t i o n s only  een  ( i l l ) minima of the  four  conduction  a t h i g h f i e l d , s t r e n g t h s • when t h e t h e s e . v a l l e y s are take  low  betw-  band'. H o w e v e r ,  e l e c t r o n temperatures (pOO)  h i g h , t r a n s i t i o n s to the  p l a c e . Because of the  e f f e c t i v e mass o f  would c o n t r i b u t e t o the  of  minimum 0.034  e l e c t r o n m a s s e s , e l e c t r o n s i n t h i s minimum s h o u l d h i g h m o b i l i t y and  contain  are:  v a l l e y noise the  .  possess  intervalley  noise. 2) The  i n t e r v a l l e y t r a n s i t i o n r a t e under e q u i l i b r i u m  c o n d i t i o n s was  found from r a t h e r scanty  et a l . Only f i v e v a l u e s i g a t e d and  values  of the  i t may  be  differential  d a t a was  higher  f i e l d s may ing  i n *the  values be  of f i e l d .  Weinreieh  uncertain.  experimental  c o n d u c t a n c e show q u i t e good  The  a t t r i b u t e d t o the  theory.  rather  added t h a t , t h e  agreement w i t h t h e p r e d i c t i o n s of the the  of  o f s a m p l e i m p u r i t y were i n v e s t -  interpolating their  In conclusion,  data  B a r r i e theory  p o o r agreement at neglect  at low  of i m p u r i t y  scatte  From t h e r e s u l t s been p r o v e n t h a t  of t h i s  investigation  i t has n o t  the a n i s o t r o p y of noise temperature  i s due t o t h e i n t e r v a l l e y  n o i s e component a l o n e .  M e a s u r e m e n t s made o v e r a much g r e a t e r reported here and'extending i n t o  frequency range  t h e microwave  than  region  w o u l d be e x p e c t e d t o show a f r e q u e n c y d e p e n d e n c e i f t h e major  s o u r c e o f n o i s e were due t o i n t e r v a l l e y  whereas t h e h o t e l e c t r o n the  microwave  transitions  n o i s e s h o u l d be u n i f o r m  beyond  spectrum, dropping o f f w i t h frequency a t 10/  the  o r d e r o f 10  c y c l e s per second.  Some a d d i t i o n a l  i n f o r m a t i o n might  a l s o be o b t a i n e d  f r o m n o i s e t e m p e r a t u r e d e t e r m i n a t i o n s made a t d i f f e r e n t lattice  temperatures. Increasing the l a t t i c e  f r o m 77° K d e c r e a s e s t h e e l e c t r o n  mobility  the  resulting  intervalley  transition  i n the intervalley  rate,  might  and i n c r e a s e s i n a reduction  noise.  Other semiconductors possessing, l e s s structure  temperature  a l s o be i n v e s t i g a t e d .  complex  In particular,  a n t i m o n i d e , w h i c h i s b e l i e v e d t o have s p h e r i c a l energy s u r f a c e s i n A-space  would  band indium  constant  show no i n t e r v a l l e y  noise.  N o i s e a n a l o g o u s t o i n t e r v a l l e y n o i s e , a n d due t o h o l e transitions  between t h e d e g e n e r a t e v a l e n c e bands o f p - t y p e  germanium m i g h t be o b s e r v a b l e w i t h t h e p r e s e n t e x p e r i m e n t a l arrangement. are  Since- t h e c o n s t a n t energy  spherical,  surfaces of the holes  t h e n o i s e t e m p e r a t u r e s h o u l d be  isotropic.  BIBLIOGRAPHY B a k k e r , C . J . and H e l l e r ,  G.  B a r r i e , R. and B u r g e s s , R.R. B u r g e s s , R.E.  G u r e v i c h , V.L.  (1963) S o v i e t P h y s i c s - J.E.T.P. 16,  ( 1 9 5 5 ) B . S . T . J . 34,  J o h n s o n , J.B. L e v i n g e r , B.W.  381.  (1962) S e m i c o n d u c t o r C o n f e r e n c e a t E x e t e r p.128.  ( 1 9 5 5 ) P r o c . I . R.E.  H e r r i n g , C.  263.  (1962.) C a n . J . P h y s . 4 0 , 1056.  (1965) R a d i o S c i e n c e J . 69D,  E r l b a c h , E. and Gunn, J.B.  Herman, F.  ( 1 9 3 9 ) P h y s i c a 6,  (1928) P h y s . Rev.  1252.  43,. 1 7 0 3 . 237. 32, 97.  and F r a n k l , D . R . ( 1 9 6 1 ) J . P h y s . C h e m . S o l i d s 20, 2 6 1 .  P a i g e , E.G.3. (1960) P r o c . P h y s . S o c . 7 5 , 174. P r i c e , ' P . J . ( 1 9 5 9 ) I.B.M. J o u r n a l 3, 1 9 1 . P r i c e , P . J . (1960) J . A p p l . P h y s . 3 1 , 949. P r i c e , P . J . and H a r t man,  R.L. ( 1 9 6 4 ) J . P h y s .'Chem. S o l i d s 25,  R e i k , H.G.  567.  and R i s k e n , H . ( 1 9 6 1 ) P h y s . R e v . 1 2 4 ,  R e i k , I-I.G. and R i s k e n , H. (1962) P h y s . R e v . 1 2 6 ,  777. 1757.  S t r a t t o n , R.  (1957) P r o c . R o y a l Soc.London A242, 355.  Stratton,  (1958) S o l i d  R.  State- P h y s i c s i n E l e c t r o n i c s  Telecommunications - B r u s s e l s , van der Z i e l ,  A.  (1954)  Noise  and  p343.  Prentice-Hall  W a n n i e r , G.H.  Elements of S o l i d S t a t e Theory - Cambridge University Press W e i n r e i c h , G. , S a n d e r s , T.M. , and W h i t e , H.G. ( 1 9 5 9 ) ' P h y s . R e v . 114, 3 3 . W i l s o n , A.H. ( 1 9 5 0 ) T h e o r y o f P e t a l s - C a m b r i d g e U n i v e r s i t y Press Y a m a s h i t a , J . and W a t a n a b e , M.  (1954) P r o g r . T h e o r . P h y s . 12 443. t  APPENDIX 1 - FLUCTUATIONS • ' .:• _  • OF 'SAMPLE  We w i s h . u l t i m a t e l y in' the  total  electron  .under h i g h e l e c t r i c first  I N ELECTRON' POPULATION '  to calculate  field  conditions.  fluctuations.  i s a -homogeneous  semiconductor c r y s t a l • c o n t a i n i n g  electrons  To do t h i s we must  population  The s y s t e m t o be c o n s i d e r e d  positive  fluctuations  p o p u l a t i o n o f a t y p i c a l sample  consider the e q u i l i b r i u m  stationary  the.  a f i x e d number  No  i o n s and a f l u c t u a t i n g number N  whose mean number  equals  No  .a p l a s m a i n . w h i c h t h e p o s i t i v e  ions instead  a r e , r i g i d l y bound t o f i x e d l o c a t i o n s .  The  of H&)  of  so a s t o  p r e s e r v e s p a c e - c h a r g e n e u t r a l i t y . Such' a s y s t e m  of  impurity  constitutes  of being  free  characteristics  s u c h a s y s t e m a r e d e t e r m i n e d by two i n t e r a c t i o n s : t h e  coulomb i n t e r a c t i o n w h i c h ' t e n d s . t o c r e a t e a u n i f o r m d i s t r i b u t i o n of electrons effects  amongst t h e f i x e d i o n s and t h e  of thermal a g i t a t i o n  to .disrupt  o f t h e e l e c t r o n s .-which t e n d s '.  t h e u n i f o r m d i s t r i b u t i o n . T h e s e two c o m p e t i n g  p r o c e s s e s r e s u l t "in e l e c t r o s t a t i c s h i e l d i n g and  .electrons.  I n t h e c a s e - w h e r e t h e mean  of both  electron-electron  c o u l o m b i n t e r a c t i o n i s s m a l l c o m p a r e d t o t h e mean kinetic the  effect  of s h i e l d i n g  shielding  results  length,  f r o m the. c h a r g e i s g i v e n by  and  i n the formula  4-rv£r  f o r the average e l e c t r o s t a t i c p o t e n t i a l f  electron  energy, the Debye-Elickel theory i s a p p l i c a b l e ,  "  ance  ions  -  \  V(?)  . The. l e n g t h . . .  at a d i s t D  , t h e Debye  where  C  •i s the p e r m i t t i v i t y  and 'Ho  t h e mean  e l e c t r o n d e n s i t y . Thus, e n c l o s i n g every f i x e d charge  i n t h e medium i s a - s p h e r e o f c h a r g e  whose r a d i u s i s o f t h e o r d e r Since the  neutrality•  o f t h e Debye l e n g t h .  t h e e l e c t r o n s a r e i n t h e r ma 3. e q u i l i b r i u m w i t h T  L a t t i c e at a temperature  .thermal motions- a p p r o p r i a t e the  or mobile  coulomb i n t e r a c t i o n .  }  t o ~T  t h e y p o s s e s s -random but c o n s t r a i n e d by  S u c h random m o t i o n s ' and t h e i r  associated current density 'fluctuations are described the f l u c t u a t i o n - d i s s i p a t i o n theorem, i z a t i o n of t h e N y q u i s t theorem fluctuations i n a dissipative  by  which i s a 'general-  of current  and v o l t a g e  network.  By a p p l y i n g t h e f l u c t u a t i o n - d i s s i p a t i o n . t h e o r e m t o the  l o n g i t u d i n a l modes o f p r o p a g a t i o n , . ,  found f o r the spectrum  The  of charge d e n s i t y ' f l u c t u a t i o n s :  F o u r i e r component o f wave v e c t o r ^  B u r g e s s (1965)  P  has t h e form  v  w h i c h , when i n t e g r a t e d o v e r t h e v o l u m e o f t h e s a m p l e y i e l d s an e x p r e s s i o n  for  f l u c t u a t i o n due t o t h e •^(AN^p  i s then found, a f t e r  , the oonulation Fourier  component.  averaging  o v e r an  ensemble,  50  ((<\NT)  •g^T£^  the phase f a c t o r  (\-COS^L)  ^  vf  having  (l-COS  P^B)  r A)(j-COS A  L  disappeared.  a r e t h e .dicaensions o f t h e p a r a l l e l e p i p e d  , A  , and B  sample i n r e s p -  e c t i v e l y t h e x,y,  and z d i r e c t i o n s . The e x p r e s s i o n  varN  hy i n t e g r a t i n g o v e r a l l modes o f V• ,-  i s obtained  t h e mode d e n s i t y b e i n g to  for  volume o f P  per unit  space,  yield: oo co  varN  KT£.  r  r  r  (i-cosr L) (\- co6r A)(i-c sP B )AR,,in,ip. N  s  x  upper, l i m i t  ignore This  z  n a1  An  P  to  varN  "3  i s f o u n d i m m e d i a t e l y ' i f we  t h e ' one t e r m i n t h e d e n o m i n a t o r o f t h e ' i n t e g r a n d .  c o r r e s p o n d s t o t h e case o f an i n f i n i t e  pebye  length,  o r the. a b s e n c e o f c o u l o m b i n t e r a c t i o n . . T h i s ' - l i m i t i s ' : -  Returning  t o the  general  of t h e components o f the  P  case,  t h e i n t e g r a t i o n o v e r one  " say'. Vx  i s performed  f o l l o w i n g r e l a t i o n , , which'' i s e s t a b l i s h e d  contour  integration:  CO  — CO  The  expression for  a> co VdrM =  KTL  varN  t h e n becomes  using  e a s i l y by  For i n t e g r a t i o n over following  w e - r e q u i r e -the s o l u t i o n o f t h e  integral:  0-cosn,A)af;  Letting  In.the  3  -00 HjA = X  A  ' results i n  much l e s s t h a n u n i t y , X|  limit' of  an u p p e r  »< = D^T^  and  limit',  • .aooroaches  -  •  00 o  Returning over  17  varN  .  t o the expression leads ''  f o r varN  t o an e x p r e s s i o n '' '' . 00 Au  and-integrating  f o r the upper  l i m i t of  -- - • ,  1  L7I  % +  -OO  + J£MJ&LS\  The  i n t e g r a t i o n over  except  t h e parameter  an u p p e r  limit  expression ion  that  0 -  cos  n ^ U r *  Q  i s similar t o that f o r P  °<  i s now e q u a l to. u n i t y . . F i n d i n g  to the  f o r t h e upper  y  ,  \  .integration leads,to-the. f i n a l limit'of  a l l sample d i m e n s i o n s  Debye l e n g t h : .  \L-  A  varN  , w i t h t h e assumpt  a r e much g r e a t e r  than the  P u t t i n g i n the for  a typical  f o l l o w i n g v a l u e s ," which' a r e . a p p r o p r i a t e sample i n the p r e s e n t  experiment under  equilibrium conditions:.. 8x10  Y\ = 0  and  We  *\  •  1=  using the r e s u l t  see  from t h i s the  determining  .  \Uo  for  A^3*/o  varN . r e s u l t s  in  m  ....  i m p o r t a n c e o f t h e Debye l e n g t h : i n  o f t h e p l a s m a . Of p a r t i c u l a r , i m p o r t a n c e , i s t h e dimension.of  V  the p o p u l a t i o n f l u c t u a t i o n s i n a volume  V  i n cases  minimum  of samples shaped l i k e  s l a b s or l o n g r e c t a n g u l a r b a r s ,  as  thin  i n the present  "exper-  iment . In ium  a p p l y i n g these  s a m p l e , we  s i n c e we to  r e s u l t s t o an  are o v e r e s t i m a t i n g the  fluctuations,  h a v e assumed a l l s u r f a c e s o f V-  t h e p a s s a g e of, l o n g i t u d i n a l  whereas.only the  ohmic end  f l o w , t o and' f r o m t h e In  e q u i l i b r i u m german-  due  fluctuations,  '••.•'"  ,  t h i s a n a l y s i s to the, n o n - e q u i l -  c a s e , we  of e l e c t r o n d r i f t ,  permeable  c o n t a c t s a l l o w unimpeded . e l e c t r o n  sample.  order, t o extend  ibrium high f i e l d  density  ; t o be  must f i r s t  t o the  evaluate the  applied f i e l d ,  on  .  .  effect  the  \ \  •  p o p u l a t i o n f l u c t u a t i o n s . T h i s may  be  .  done by  considering  the p o p u l a t i o n f l u c t u a t i o n s " i n a volume i d e n t i c a l sample volume, which; i s moving at the d r i f t in,  f o r convenience,  to. the  v e l o c i t y - U. •  the x - d i r e c t i o n of a s e t of axes a t  rest with respect  t o t h e e q u i l i b r i u m p l a s m a . As i n t h e , the V -spectrum i n  e q u i l i b r i u m c a s e , we need. ^ ) the moving c o o r d i n a t e  system.  A p l a n e wave o f c i r c u l a r  frequency  vector  P  appear  i n the moving c o o r d i n a t e s  vector  r  In  as o b s e r v e d i n t h e f i x e d  but c i r c u l a r  showed t h a t t h e T  and wave  coordinates  will  t o h a v e t h e same wave  frequency  the stationary coordinate  60  CO = CO — n i x  system,  Burgess(1965)  -spectrum of the l o n g i t u d i n a l  space-  c h a r g e d e n s i t y modes i s g i v e n b y  Since  where  i n t h e moving system only a f r e q u e n c y s h i f t  CO - G O u Q . . 1  -  I n t e g r a t i n g (^rJ) moving  occurs,  bi'  over  gives  (^f*)  f ° "the r  system.  co  00  «"  U"^.  ur, The  first  the  second,due  Since  t e r m i s i d e n t i c a l t o t h e e q u i l i b r i u m v a l u e , and t o d r i f t , w e now  ^ ^_~£(p m  (J)\  i s  a  n  e  v  e  consider. n  f  unc  "ti°n o f t o  , the  64  drift  t e r m i s odd i n T%  term i n . t h e varN  ^  ; hence, i n c l u d i n g the d r i f t  -space/integral  r e s u l t ' s i n no c h a n g e f r o m t h e e q u i l i b r i u m  . . A second  i s i n .the p o s s i b l e  the  thermal v e l o c i t i e s  the  d r i f t . just considered. we  value.  enhancement o f  of the e l e c t r o n s as d i s t i n c t  from  T h i s may b e . t a k e n i n t o , account.' .  adopt t h e concept o f . a n e l e c t r o n t e m p e r a t u r e t o  describe  the non-equilibrium  i n the .Barrie theory, length  of  e f f e c t o f t h e h i g h ' f i e l d on f l u c t u a t i o n s ••in .'  the'total population  if  i n the evaluation  velocity distribution,  and assume a n o n - e q u i l i b r i u m  as Debye  e x i s t s which i s p r o p o r t i o n a l to' the .square-root  of t h e e l e c t r o n temperature.'  i  ..  APPENDIX 2 - CALCULATION OF GROUP POPULATION . SPECTRAL D E N S I T I E S Let  us c o n s i d e r t h e j t h group, s t i l l  restricting  o u r s e l v e s t o t h e c a s e of- two g r o u p s i n a l l . Then, f o r s m a l l we may w r i t e the  fluctuations  the following  k ANjCt) _ b  jJk  AN3  ::  A Nj (t) + v|». (t)  (A 2.1)  J  j-taKj^  where  population  two L a n g e v i n e q u a t i o n s f o r  population fluctuations,  1 A M j It) « tit  i n t h e group  i  "  s  tils  a t r a n s i t i o n from group j N^Gt) i s a s t o c h a s t i c  probability per unit  time of  t o g r o u p k.  term r e p r e s e n t i n g  the fluctuations  i n eLANljfr). Let  us denote t h e F o u r i e r  function  xCt) o f t i m e b y 2(u))2  Then, F o u r i e r two  t r a n s f o r m o f any s u i t a b l e  (xttJe  i M t  at  t r a n s f o r m i n g e q u a t i o n A2.1 y i e l d s t h e  equations:  with  the solution  where  .2.  The  group p o p u l a t i o n s p e c t r a l d e n s i t i e s a r e d e f i n e d by  and  T-^co  T  F r o m t h e s o l u t i o n o f e q u a t i o n A2.1 f o l l o w i n g e x p r e s s i o n f o r SN^W)  The  spectrum  uting  if  of  M^i  we a r r i v e a t t h e  :  i s r e l a t e d t o the events  constit-  by:  the events a r e mutually independent.  Here  l>  i s the  mean r a t e o f o c c u r r e n c e o f e v e n t s , a t y p i c a l  event  the form  equals zero.  v(-fc)  , and commencing a t t i m e ' t  Two p r o c e s s e s c o n t r i b u t e t o V> between groups, e n t r y and e x i t passage  : electron  having  transitions  o c c u r r i n g a t t h e r a t e ft^Nv -V- J? \\  x  and  f r o m t h e s a m p l e b o u n d a r i e s due t o t h e  of current, occurring at the rate  L = Hi where  tj  i s the ••transit-time.  Since  i s much s m a l l e r t h a n  ributor to  V  rfct) i s a d e l t a  j  3  ^  , t h e major cont-  i s the intergroup t r a n s i t i o n function,  ± £ ( £ - t o )  where'  time of t r a n s i t i o n , w i t h the p o s i t i v e  "to  rate. i s the  s i g n . f o r e n t r y and  the n e g a t i v e  S«¥ ^  sign for exit.  The s p e c t r u m  ignore f l u c t u a t i o n s  i n the t o t a l  i s then:  ^YiIt) + ^xtt)  i s found by c o n s i d e r i n g  i(  S^;  . I f we  e l e c t r o n p o p u l a t i o n , as  d i s c u s s e d , i n A p p e n d i x . 1, d t  d-t  Then, a d d i n g  equations  The s p e c t r u m o f ty| + 4^.  Since  S^*.^  A2.1  i s related  to  and 9 f , A  ,  i s zero,  The c o e f f i c i e n t s i n e q u a t i o n  Equation  gives  A2.2 then  A2.2. h a v e t h e v a l u e s :  y i e l d s f o r the population  spectrum  of the j t h group:  For frequencies  60  , SN^W)  approaches  SMJ  ,  68  where  Since  and  Hence  \ \  APPENDIX 3 - CALCULATION  OF V A L L E Y POPULATIONS AND  I N T E R V A L L E Y TRANSITION As s t a t e d electron of  PROBABILITIES  i n Chapter 1, the p r o b a b i l i t y  t r a n s i t i o n between v a l l e y s phonon o f e n e r g y ft to  a lattice  \ I _. \ / (AE +\QS?  where  THE  o f an  due t o a b s o r p t i o n is:  X  i s the i n i t i a l  e n e r g y o f the  electron  above t h e v a l l e y minimum.. For  e m i s s i o n o f a phonon o f e n e r g y  fto)  , the t r a n s i t i o n  probability is  f o r AE^*w  We " W t A E - U f -O  AE<^  Assuming that i n a, s i n g l e erature  the energy d i s t r i b u t i o n of the  valley  i s Iviaxwellian  -f(Ae) = a ( T ) ( ^ e ^ o.tT)is r e l a t e d  to the s i n g l e  e  /  K  electrons  a t an e l e c t r o n  T, t h e e n e r g y d i s t r i b u t i o n i s g i v e n  W  temp-  by  T  valley  population  density,  n,CT)by  C o n s i d e r i n g now in  one v a l l e y ,  probability  the t r a n s i t i o n s by a b s o r p t i o n ,  per unit  we  f r o m an e n e r g y find  that  time i s p r o p o r t i o n a l  shelleCCde)  the t r a n s i t i o n to  e The  c o r r e s p o n d i n g  Hence, a  the  v a l l e y  of-  —I  t o t a l  at  e x p r e s s i o n ^ f o r  t r a n s i t i o n  e l e c t r o n  a r b i t r a r y  e m i s s i o n  p r o b a b i l i t y ,  temperature  e l e c t r o n  T  temperature,  to i s  i s  BICT), another  above  B e s s e l  are  r e a d i l y  e v a l u a t e d  f u n c t i o n  r(V) \  Then,  0,Cr)rf  If.ggf)  ^ T ) f  e E x p r e s s i n g  aCT)  t o  +  i n t e g r a l s  '  v a l l e y  p r o p o r t i o n a l  ACT),  The  from  i  n  terms  of  -I vitCT)  + ,  |-e  i n  terms  of  the  71  6,CT) * n , ( T ) T  K, k j e  _e  +  T h e n t h e t r a n s i t i o n p r o b a b i l i t y p e r e l e c t r o n , d e n o t e d by  to ' 1  S u p p o s e two v a l l e y s h a v i n g T  x  e l e c t r o n temperatures  a r e i n a s t e a d y - s t a t e o f p o p u l a t i o n and t h a t a l l f o u r  v a l l e y s a r e c h a r a c t e r i z e d by e i t h e r the  T, a n d  T  4  or  T . z  Then  c o n d i t i o n 8 \ C T 0 ~ ®»^T».') a l l o w s t h e  steady-state  valley population ratio  ~Y\  t o be f o u n d .  X  The r e l a t i o n i s  where  e and  T  equal  for T  Since TL.  I— e  has t h e bounds:  for and  '  T  u  (  9C*T,.)=cscL  much g r e a t e r t h a n i i u ) •> 3 ~  co+k, ^ T - ^  the v a l l e y population at zero f i e l d , , i s one-quarter  absolute  values  of  the t o t a l  rtt  and  T  equals  electron population, the  Y\%. may now be f o u n d .  The i n t e r g r o u p t r a n s i t i o n p r o b a b i l i t i e s introduced  when  i n Chapter 2 a r e found  f o r the v a l l e y concerned, together  |>,  x  and j ^ i  from the values with the value  o f ^it" ") 1  of ^ C I T K )  and  the a b s o l u t e  are  evaluated  For  example,  of  i n Appendix for  temperature ures  value  the  case  and  at  77°K, w h i c h  at  electron  5. of  one v a l l e y  Tj and t h e o t h e r  T^, we h a v e ,  ^  denoting  three  the  at  single  electron valley  temperat-  by- s u b -  script 1 :  where per  e l e c t r o n at  valley valleys 5,  and  V£L  Vai  are the t r a n s i t i o n for  the  at e l e c t r o n temperature  T,  at  zero f i e l d  electron temperature;T^.  V°ia. e q u a l s  \  \  3 ^  .  !  probabilities  special  case  and t h e o t h e r As  shown i n  of  one  three Appendix  APPENDIX 4 - CALCULATION OF VELOCITY VARIANCE I N ARBITRARY DIRECTION OF DISPLACED MAXWELLIAN DISTRIBUTION L e t i , i , I t be t h e c o m p o n e n t s o f a v e c t o r x  taking  a  the o r i g i n of the c o o r d i n a t e system at the center  o f one o f t h e c o n s t a n t directed  e n e r g y e l l i p s o i d s . The 4ta a x i s i s  a l o n g the major a x i s  Taking  an a r b i t r a r y ,°lx , ^3  cosines  laced Maxwellian /oi e l e c t r o n <**. v<i, v-a->  of the e l l i p s o i d .  direction  o(  with  direction  we r e q u i r e t h e v a r i a n c e o f t h e d i s p distribution i n this  i n state  related  }  A  to M  direction.  has v e l o c i t y components by : **A.  Wit  Denoting hy  in'"-space,  i t s component o f v e l o c i t y  i n t h e d i r e c t i o n <?C  we h a v e  Averaging  over  a l l electrons i n the v a l l e y ,  denoting  t h e a v e r a g e by A T ^ :  The m e a n - s q u a r e v a l u e i s  \  where. 3 ^ ^ Since  i s  t  n  e  •••  electron  velocity  distribution.  74  U s i n g i a n  the  e x p r e s s i o n  f o r  of  the  d i s p l a c e d  M a x w e l l -  d i s t r i b u t i o n ,  * t T )  c  w i t h  and  Then,  by  where the f o r  d i r e c t  ©<* VT^  c a l c u l a t i o n ,  i s  the  a x i s ,  and  a c c e l e r a t i o n  angle'between i s  W\  s  i n  the  the c<  the  d i r e c t i o n  i n e r t i a l  d i r e c t i o n ,  e f f e c t i v e g i v e n  1  l A j j o i n . Qcl + W * C o S  dot  «<  and mass  b y ( W i l s o n )  APPENDIX 5 ' -  INTERVALLEY TRANSITION RATE AND  THE  AGOUSTOELECTRIC EFFECT  The of  introduction  n-type  an  acoustical  germanium atoms w i l l  valleys  of  lattice  the  the  of  the  minima have t h e  occupation  will  change and  as  a  will  redistribute  acoustical  redistribution greater  t h a n the  ribution  will  ribution  as  of  gaining .  The  allows The  relevant  i s the  be  of the  complete, rate,  to  be  and  as  i s the  of  intervalley  Kia.  from the  (n,  _a  i s the  group 2 from group  net 1.  enough,  about  a d.c.  redist-  the  redist-  a direct  G.  rate.  et  the  voltage  means  a l . 1958),  redistribution. in  the  a travelling acoustical  transitions  transition  the  no  intervalley transtion  are  q u a n t i t y determined from the  o  populations  or  frequency afford  appearance of  intervalley  valleys'  i f i t i s much  little  determined  that  transitions.  effect(Weinreich,  of p r o p a g a t i o n  authors define  where  valley  or  undeformed  i n the  o c c u r . Hence, measurements o f  information  far  states  intervalley  intervalley  a function  lattice  f o u r minima  wave f r e q u e n c y i s h i g h  acoustoelectric  direction  effect  to  not  ciata a b o u t  effect  As  will  of  c o n s e q u e n c e the  due  a  same e n e r g y . T h i s means  of  the  the  c o n d u c t i o n hand, whereas i n t h e  probability  If  shift  wave i n t o  wave.  concerned,  the  '  acoustoelectric time,t,„  , which  the  equation,  - n j  rate  Here,  of  e n t r y of  e a c h of  the  electrons two  groups  into consists  o f two v a l l e y s , and  Y\i  u n i t volume i n g r o u p  i s t h e number o f e l e c t r o n s  per  i .  The  above r e l a t i o n a l l o w s  the  p r o b a b i l i t y p e r u n i t t i m e o f an e l e c t r o n m a k i n g a  t r a n s i t i o n from group and  j  each c o n s i s t  c a s e when  t o group  ,  j , where g r o u p s i  o f two v a l l e y s , f o r t h e  \>*j  equilibrium  >  i n equilibrium,  = J? *  then  Y\ =-*\x t  a  R.a = \L (.n,-"*")  Hence Equating  Eor  i  of  =  I n terms of  Since,  the determination  R>%  to Weinreich's expression  gives  t h e c a s e o f one v a l l e y i n one g r o u p and t h r e e  i n the  .0  other,  we may  \>°?  a l s o determine the  for this  case t o avoid  f^'s  , d e n o t e d now  confusion  with  by  the previous  c a s e o f two v a l l e y s i n e a c h g r o u p . In t h i s case, the gross r a t e of entry second group of t h r e e group  i s ,at  where n  of,electrons  v a l l e y s from the one-valley  to the first  equilibrium  i s the t o t a l f r e e  electron  concentration.  E q u a t i n g t h i s t o t h e g r o s s r a t e f r o m group 2 t o group  ft - s Considering containing  1:  Kl  a s i n g l e v a l l e y i n t h e c a s e o f two  groups,each  two v a l l e y s , a t e q u i l i b r i u m t h e g r o s s r a t e  from  the  one v a l l e y  since  *I  t h e r a t e  s u b s t i t u t i n g  gross  i n  r a t e  i n  terms  o f  terms  %v  t h e two v a l l e y s  of  t h e  equal  o f ,  have  been  r e p o r t e d  b y ' P r i c e  the  donor  f o r  t h e : l a t t i c e 100* K .  experiment  f o r  g i v e s ,  t h e case  obtained  by  o f  Rjun«-  t h e donor  was, t h e o r d e r  T  t o  antimony  u  ,  10  T h e i r  i n  t h e range  cm .  as t h e  v a l u e  i s :  which  r a t e ,  and T e l l ,  R  d e n s i t y ,  of  t r a n s i t i o n  ^ e i n r e i c h  a n d H a r t m a n (196-4).  temperature,  i s  ,  t h e i n t e r v a l l e y  c o n t r i b u t i o n ,  N t  \ ^  f o r  - i -  t o  i m p u r i t y  of  obtained  ,  Measured.values  to  t o  i s  f o r t h e e x p r e s s i o n s  \>°'  R,  v a l l e y s  group.  Hence,  and,  a l l t h e other  i s  A- '  second  the  t o  i n  Hence,  o f  from 2 0 " K  t h e f o r  and  present t h i s  experiment Rjto*»»r =  From 77*K,  W e i n r e i c h read  from  et  A'0  al(1959),  h i s graphs  &  $l<? t h e phonon i s  c o n t r i b u t i o n  a t  Since  the  decrease  donor at  temperatures  c o n t r i b u t i o n  h i g h  f i e l d s  higher  when  than  the  i s the  c o n t r i b u t i o n  was  f i e l d  value  i n t e r v a l l e y  phonon  the  c o n t r i b u t i o n  n e g l e c t e d  a l o n e .  and  v a l l e y s  l a t t i c e  donor  of  s m a l l  i s are  at  then  i s  to  e l e c t r o n  temperature,  a l t o g e t h e r .  r a t e  expected  The t a k e n  the zero as  the  

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