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A theoretical and experimental investigation of metal-semiconductor contacts Horita, Robert Eiji 1962

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A T H E O R E T I C A L AND E X P E R I M E N T A L OF M E T A L - S E M I C O N D U C T O R  INVESTIGATION CONTACTS  by ROBERT. E I J I 3.A.Sc.,  University  A THESIS  of  HORITA  British  Columbia,  S U B M I T T E D I N P A R T I A L F U L F I L M E N T OF  THE R E Q U I R E M E N T S FOR THE D E G R E E OF MASTER OF A P P L I E D  in  the  SCIENCE  Department of  PHYSICS  We a c o e p t to  i960  the  this  required  THE UNIVERSITY  thesis  as  conforming  standard  OF B R I T I S H  September,  1962  COLUMBIA  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make i t freely available for reference and study.  I further agree that permission  for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  Department of  PHYSICS  The University of British Columbia, Vancouver 8, Canada. Date  September  25. 1962  ABSTRACT  Metal-semiconductor c o n t a c t s cally  have been known e m p i r i -  to obey a modified diode equation  (exp qV/akT  1 = 1  - 1)  a  where the parameter ' a ' o f t e n However, p r e v i o u s t h e o r i e s , thesis,  took values g r e a t e r than two.  which are b r i e f l y summarized i n  c o u l d not simply account f o r the anomaly where  g r e a t e r than two.  Previous t h e o r i e s  'a'  the is  are extended by c o n s i d e r i n g  one-dimensional b i p o l a r flow of c a r r i e r s and n e g l e c t i n g recombin a t i o n i n a homogeneous.semiconductor and an ohmic contact  at o p p o s i t e  c u r r e n t theory by Borneman e t .  ends.  al.  f i l a m e n t with a r e c t i f y i n g The z e r o - e l e c t r o n -  (1955) I  extended  s  to high  i n j e c t i o n l e v e l s by u s i n g the j u n c t i o n r e l a t i o n s of Misawa (1955)* Then the n o n - z e r o - e l e c t r o n - c u r r e n t theory i s developed. theory shows that  ' a ' i s u n i t y f o r low I n j e c t i o n Into  semiconductors and that  a = (3b-M)/(b-M)  semiconductors, where  extrinsic  for arbitrary  Into i n t r i n s i c semiconductors and, f o r high i n j e c t i o n extrinsic  This  injection  Into  'M' Is the e l e c t r o n to  hole  c u r r e n t r a t i o and ' b ' i s the e l e c t r o n to hole m o b i l i t y r a t i o . Thus,  ' a ' can take any value depending on the magnitude of M/b. To check the n o n - z e r o - e l e c t r o n - c u r r e n t t h e o r y ,  ments were performed only on n-type germanium. metal-semiconductor c o n t a c t s  experi-  Rectifying  were made by e l e c t r o p l a t i n g copper  and rhodium on germanium and ohmic c o n t a c t s  were made by  a l l o y i n g antimony-doped g o l d wire In a n i t r o g e n atmosphere. A side-arm probe adjacent to the p l a t e d - c o n t a c t voltage  across the  contact.  measured the  ill  For adjacent  to  the  the  rhodium-contact  rhodium contact  germanium f i l a m e n t resistance  as  measurement extraction from  were used  a function of  demonstrated at  the  moderate  longitudinal  to  current  two sides  measure  transverse  the  occurrence  and  i n the  that  the  of  between  of  the a-c  current.  This  i n j e c t i o n and  level  v i c i n i t y of  density  side-arms  on o p p o s i t e  l o n g i t u d i n a l d-c  the  contact high  to  specimen,  the  about  of  •  injection  contact  for  1 0 mA/cm  and  1  was  1 0 0 0 mA/cra .  Comparison with  the  of  the  I-V  measurements  non-zero-electron-current  intermediate  level  of  Injection  theory  occurred  of  the  indicated at  the  contacts that  rhodium  an contact  2 l o n g i t u d i n a l c u r r e n t d e n s i t y b e t w e e n a b o u t 0.1 mA/cm and 2 1 0 0 mA/cm a n d t h a t h i g h i n j e c t i o n o c c u r r e d , w i t h M = 4 . 9 4 , a t 2 t h e c o p p e r c o n t a c t f o r c u r r e n t d e n s i t y b e t w e e n a b o u t 1 mA/cm 2 for  and by  5 0 mA/cm . comparison of  Thus, the  zero-electron-current  the  level  of  experimental theory.  Injection value  of  c a n be  'a'  with  calculated the  non-  Iv  CONTENTS page CHAPTER  1.  INTRODUCTION  1  1.1  Historical  Review  1  1.2  Purpose  this  2  CHAPTER  2.  of  Thesis  T H E O R I E S FOR M E T A L - S E M I C O N D U C T O R C O N T A C T S  4  2.1  Unipolar Theories  2.2  Bipolar Theories  10  2.3  Z e r o - E l e c t r o n - C u r r e n t Theory  14  2.4  Extension of to  2.5  CHAPTER  the  N o n - Z e r o - E l e c t r o n - C u r r e n t Theory  22  2.51  E x t r i n s i c Semiconductors  22  2.52  I n t r i n s l o Semiconductors  29  H'  EXPERIMENTAL TECHNIQUES  3.2  Preparation of  3.3  Circuits for  of  32  Specimens  32  Specimens  I-V  33  Measurements  36  3.31  D-C M e a s u r e m e n t s  j6  3.32  Pulse  37  Transverse  4.  Measurements A-C R e s i s t a n c e  Measurement  38  E X P E R I M E N T A L R E S U L T S AND I N T E R P R E T A T I O N  4.1  Transverse  4.2  Current-Voltage Contact  4.3  Theory 18  Geometry  CHAPTER  Zero-Electron-Current  Arbitrary Injection Levels  3.1  3.4  4  A-C R e s i s t a n c e  Characteristic  of  .  Specimen  39  Rhodium-  Specimen  Current-Voltage Contact  Measurement  39  44 Characteristic  of  Copper46  V  4.4 CHAPTER  Comparison o f Experimental 5,.  BIBLIOGRAPHY  )  CONCLUSIONS  R e s u l t s w i t h Theory  47 49 51  vi  ILLUSTRATIONS Figure  Facing  Page  i  2.1  The M o t t B a r r i e r  2.2  Formation of  2.3  Equilibrium to  Schottky  Barrier  electron-energy  semiconductor  2.4  Surface-State  2.5  Equilibrium  6  diagrams  of  metal  contacts  8  Barrier  10  electron-energy  diagram  junction  P-N 2.6  5  of  a  >  Electronic-energy  diagram  of  11  a metal  - n  +  -  n  contact  14  2.7  Model f o r  2.8  The  dependence  current  2.10  for  The  dependence  of  Jsi'  on M f o r  the  The  dependence  an  'a'  on the  d i v i d e d by  ratio  voltages  3.1  of  ratio  mobility 2.9  metal-semiconductor  when  the the  the  contacts  15  electron  to  hole  electron  to  hole  i n t r i n s i c case  29  saturation current  density,  i n t r i n s i c case  o n M/b o f total  the  current  30  signs J is  of  currents  positive  and  in  i n t r i n s i c semiconductor  31  Equipotentlal configuration distorted  by  side-  arm p r o b e  33  3.2  Geometry  of  3.3  Circuits  for  3.4  Typical  3.5  Transverse tivity  Specimens I-V  Pulse a-c  3^  measurements  36  Forms resistance  modulation  37 measurement  of  conduc38  Transverse specimen  a-c  for  resistance  three  decades  of of  rhodium-contact longitudinal  current Transverse specimen  a-c  as  D-C r e v e r s e  resistance  a function of  of  rhodium-contact  longitudinal  current  characteristic  of  rhodium-contact  forward c h a r a c t e r i s t i c  of  rhodium-contact  specimen D-C  specimen Graphical  analysis  rhodium-contact Pulse  reverse  of  characteristic  for  specimen  characteristic  of  copper-contact  forward characteristic  of  copper-contact  specimen Pulse  specimen Graphical contact  analysis  specimen  of  characteristic  for  copper  ACKNOWLEDGMENT  I wish to supervised for  helpful The  Research the  the  thank  research  Professor  R.  reported  in this  c r i t i c a l discussions research  was  C o u n c i l In the  United States  and  financed form of  E. Burgess,  by  thesis,  suggestions. the  National  a Bursary  A i r Force Grant  w  and  by  AFOSR 6 5 - 0 2 4 0 .  1  CHAPTER  1  INTRODUCTION Historical  1.1  Review  Metal-semiconductor much  theoretical  vation  of their  classic  only  theories  barrier  many  Davydov  (1938)  showed  Interest  i n tunneling  (1958)  19^2)  sidered  usually  further  to thin  Next,  developed  surmounting  they  depended  work  functions  critically  the differences  to current  the metal-  this  potential  by W i l s o n  of current  (although  Holm  models  showed  were  proposed  recti-  recognized  i n  barrier  by  and the  thin,  con-  thermal  unsatisfactory between  until  and Schottky  model which  no o b v i o u s  functions  But  1951)  o f I t s importance (1938)  (1932)  transport  i t was  layers—e.g.  Mott  flow.  subsequently  and the semiconductor  i n work  at  through  and  the p o t e n t i a l  and germanium  theories,  the direction of  on the differences  of the metal  on s i l i c o n  rise  a theoretical  But the foregoing  (1949)  theories,  proposed  means  disappeared  of  obser-  Shockley's  to exist  observed  demonstration  junctions.  the  for rectification.  as t h e primary  almost  recent  electrons  excitation.  between  barrier  t u n n e l i n g gave  t u n n e l i n g was r e l e v a n t  (1939,  giving  Electron tunneling  to that  the contact  heavily-doped  ments  electrons  that  since  the e x i s t i n g  as t h e mechanism  (1932)  opposite  Esaki's  Before  the object  i n common w i t h m o d e r n  a potential  junction.  fication  that  features  ever  a t t h e J u n c t i o n was i n d e p e n d e n t l y  Nordheim  across  behaviour.  conduction  have been  study  of p-n junctions,  assumed  semiconductor  and  non-ohmic  containing  considered These  and experimental  treatment  although  contacts  because  the thermionic  whereas  experi-  correlation rectifying  properties this  which led  carrier  metal-semiconductor  have been  Stickler  to  the  a  by  Henlsch  Purpose  growth  of  of  this  of  properties  of  the  contact  current  of  always is  passed  theories.  understanding attempted possible  to and  techniques. assume  as  the  one-dlmenslonal  at  tunneling,  filament  opposite  the  followed  extensions  relevant  and a  theories  (Harrlck  to  this  Schwarz, detailed  c a n be  of  the  found  by  their  in  is  work  of  with  the It  The  of  complexity  of  the  presented  In  this  contact  will  in  through under  neglect  effects,  this  the  the the  when  cona  unsatis-  problem  In  thesis  s i m p l i f y i n g assumptions experimental  the  understanding  their  carriers  image  lack In  presented  flow  on  properties  reflected  simplest  theory  surface  Inherent  to  Furthermore,  their  them.  the  valid  'the  recombination,  hampered  1959).  changing  contacts  ends.  is  semiconductor  through  many  provide  Hence,  semiconductor contact  to  of  Because  contacts, make  given  work  of  article  and n o r i - r e p r o d u c i b i l l t y due  stable,  metal-semiconduotor  factory  discovery  and Borneman,  other  contacts  preparation nature  large  the  Those  state  Thesis  the  not  (1955) to  surface  of  (1957).  of  are  have  A study  the  definitive  rectification  importance  tacts  to  article.  References  Understanding uncertainty  (1948)  works  by JMlsawa  of  contacts.  propose  (1949)  Shockley's  (1955).  of  1.2  to  theoretical  given  review text  Shockley's  Subsequent  modifications  thesis  Bardeen  him and B r a t t a i n  Injection.  shortly.  and  various  non-correlation led  theory  and  of  as  conditions thesis  a  study  and  will  homogeneous and an ohmic  effects  of  field-dependent  trapping, mobility,  3  avalanche  breakdown,  field  emission,  conditions,  and p h o t o - e l e c t r i c  theoretical  assumptions,  studied  the  and  temperature Is  to  and, tal  o i l bath.  present  theories,  to  specimens  a brief develop  i n an attempt data  to  only  effects. area  were  summary o f  this  to  in a light-tight purpose  one  of  extension,  contacts.  the  on germanium  of  the to  were  constant  this  metal-semiconductor  an e x t e n s i o n  on metal-semiconductor  the  non-Isothermal  To c o n f o r m w i t h  contacts  placed  Therefore,  verify  degeneracy,  thesis contact  present obtain  theories, experimen*  2  CHAPTER THEORIES FOR 2.1  Unipolar Theories Before  section  charges  discovery  of c a r r i e r  sign only.  of both  approximations  titanium  dioxide,  and  compounds o f  table  exhibit  Important  (1932),  like  s e l e n i u m have g i v e n no like  injection.  o f the u n i p o l a r  (1939),  and  Schottky  and  are  cases  valid  such  as  relative  cuprous  oxide,  evidence of germanium, o f the  carrier  silicon,  periodic  T h e r e f o r e , the by  t h e o r i e s proposed  (1939,  considers  means o b s o l e t e  o r a t low  Semiconductors  pronounced c a r r i e r  no  in special  t h e g r o u p I I I and V e l e m e n t s  features  Mott  a r e by  semiconductors  i n j e c t i o n whereas o t h e r s e m i c o n d u c t o r s and  t h e phenomenon o f  semiconductors  f o r other semiconductors  of operation.  In  c o n t a c t s c o n s i d e r e d mobile  theories  for non-injection  heavily-doped e x t r i n s i c  temperatures  (defined  A l t h o u g h much o f t h e work now  signs, unipolar  they a r e v a l i d  Injection  to e x p l a i n  at metal-semiconductor  o f one  carriers for  Che  2.2), theories developed  rectification  in  METAL-SEMICONDUCTOR CONTACTS  19^2)  be  will  Wilson briefly  described.  Wilson considered a p o t e n t i a l between  obtained  e x p r e s s i o n s f o r t h e number o f e l e c t r o n s  with  their velocities  barrier cation the  unit  and  lying  and  hump f o r e l e c t r o n s  existing  conductor h i t t i n g  the metal  energy  a r e a o f the p o t e n t i a l  then  I n the  barrier  w i t h i n an e l e m e n t a l v e l o c i t y  hump and  He  range.  semi-  per  i n the d i r e c t i o n p e r p e n d i c u l a r to  o f t h e s e e x p r e s s i o n s by  potential  the s e m i c o n d u c t o r .  second the  Multipli-  the t r a n s m i s s i o n c o e f f i c i e n t  I n t e g r a t i o n over  the v e l o c i t y  ranges  of gave  electron energy  conduction  band  donors  SEMICONDUCTOR  / / / alence  / / / band  distance  Figure  2.1  The  Mott B a r r i e r — A n a p p l i e d forward v o l t a g e V r a i s e s the c o n d u c t i o n and v a l e n c e b a n d by qV, but the b a r r i e r thickness r e m a i n s t h e same. The e l e c t r i c field a c r o s s t h e b a r r i e r ( g i v e n by s l o p e o f l i n e ) i s uniform i n the b a r r i e r .  5  the  number o f  conductor passing  into  of  electron  ever,  i t  usually  inverse  across  at  observed  after  a barrier  high  and the b a r r i e r  function 0  by 0  Infinite  the  system  field  (1938) p r o v e d  -  E  p  where  case  surfaces  Figure before  where  approach  2.2a s h o w s contact  the  between exists  which  is  i s uniform i n the Independent  of  To e x p l a i n  the nature  of the  contact  Note  m  E  Q  is  a metal each  between w i t h an  work  p  2.2 t h a t  function are  i s the Fermi  energy  a n d come  where  into  profile  electron  at  level.  and semiconductor  other  of  electron  from F i g u r e  the e q u i l i b r i u m energy  has o c c u r r e d  a metal  the e l e c t r o n  the c o n d u c t i o n band and E an Ideal  that  however,  and' t h e s e m i c o n d u c t o r c  to  barrier,  0 0 « <  i n  there  levels  semiconductor s  How-  The S c h o t t k y  an i d e a l  function  that  i s a constant  thickness.  and an n-type  - X = E  g  plane  contact.  m  the  (1939) a n d S c h o t t k y  of donor  2.1).  consider  affinity  bottom of  Now c o n s i d e r  Importance  f o r the contact  the e l e c t r i c  (Figure  and work  electron  related the  barrier,  %  contact.  was shown t o g i v e  Mott proposed  thickness  gave  diodes.  Mott  theories  and the  flows  when Davydov  theory  depleted  so t h a t  a voltage-dependent  the  Wilson's  and a semiconductor.  has  affinity  secondary  contacts  simple  applied voltage  Schottky  electron  i n metal-semiconductor  the  work  moving  direction of r e c t i f i c a t i o n ,  sufficiently  semi-  electrons  the d i r e c t i o n of r e c t i f i c a t i o n opposite  the contact  barrier  the  The number o f  the metal-semiconductor  metal-semiconductor  (1939, 19^2) d e v e l o p e d metal  from  d i r e c t i o n was s i m i l a r l y o b t a i n e d  t h e two o p p o s i t e l y flow  gave  passing  the metal per second.  Shortly  a  area  t u n n e l i n g was shown t o b e o f  ordinary that  per unit  i n the reverse  difference net  electrons  with close of  energy  Is  Gap  Gap  0m  0  ffl  -e- -e- -e-  V////// (a)  (b)  M e t a l and s e m i c o n d u c t o r In equilibrium separated before contact  Figure  2.2  Equilibrium after close approach  Formation  of  Schottky  Barrier  (c)  Equilibrium i n close contact  plotted Fermi  vertically  levels  equilibrium  are (C.  and a p o s i t i o n c o - o r d i n a t e  aligned  since  the  that  situation  i n which  some  authors the  vacuum  levels  coincident.  Because  this  has  no  feature  different  important  the  before the  a Schottky  are  aligned  plane  sufficient  two  i n f i n i t e plane  approach  each  the of  donors  will  some  surfaces  other,  the  become  the  gap  Thus,  diminishes  conductor limiting  to  value (Figure  of  0  Fermi  -X  2.2c).  In' the  be  made  the  metal  acquires  donors  In  a region  a space-charge  Fermi and  which Is to  the  the  metal  and  is  the  n e c e s s a r y •> As  the  semiconductor  amount  surface  charge  positively.  semiconductor, extends  layer  is lt  the  well  formed. becomes  charge  of  the  into As  trans-  i n the  approaches  height  p-type  of  incorrect.  which  2.2b)  two  is  a negative  space  Figure  the  levels.  distances,  the  metal  gap between  This  the  state  (difference  between  up an e q u a l  Contacts  and  demonstrating  i n the  contact,  coincident  over  ,  levels  potential  physical  and 0 (see ffl  band diagrams  surface.  Meanwhile,  increases  arbitrary  the  lnter-atomlc  electrons.  semiconductor  between  must  of  and  established  exists  of  metal very  be  a non-equlllbrlum  also  charges  Ionized  the  thermal  should  authors  the  metal  low d e n s i t y  semiconductor.  to  the  It  Some  other  condition for  the  barrier  that  contact  semiconductor  Because  parent  barrier.  e q u i l i b r i u m , and n o t  and  while  i n energy  functions)  at  of  in  The  from o t h e r n o n - e q u i l i b r i u m  and a c o n t a c t  faces,  semiconductor  Thermal  so  1949).  situation is  e q u i l i b r i u m c a n be  t h e r m i o n i c work  infinite the  irrelevant  semiconductor  materials the  is  formation of  that and  lt  are  a r b i t r a r i l y consider  are  situations,  two m a t e r i a l s  H e r r i n g and M. H. N i c h o l s  mentioned  horizontally..  semi-  the contact  semiconductors  can  also  be  analyzed  In a  similar  From P o i s s o n ' s the  donor  thickness square  equation  levels  are  of  Schottky  root  the of  the  manner.  Ionized  In  and  the  barrier,  difference  the  assumption  space-charge x ,  Is  a l l  region,  found  to  vary  functions  of  the  layer,  x ,  Q  I n work  that  the as  the  metal  and  semiconductor:  Note on 0  that - 0  m  energy which  Is  qV,  rather  q  is  the  reverse  i n the  the  voltage diode  relations  theory.  barrier  layer  scattering (Bethe  of  19^2)  considers theories  two give  d e r i v a t i o n of the  be  the  Mott  electrons  I-V  is  the  potential  space-charge an  the  layer  externally  drop across  the  thickness  of  semiconductor  biased  by  on a p - t y p e  by  theory  Schottky  barrier the is  negatively  with  lattice  barriers,  current-  the  diffusion or  the  assumes  the  thickness  the  the  mean f r e e  vibrations.  collisions within  relation  semiconductor.)  either  compared w i t h  thermionic emitters the  lt  and the  be o b t a i n e d  large  neglects  Q  an n - t y p e  if  is  If  x .  charge,  a contact  The d i f f u s i o n to  the  potential  if  lt  depends  0  Thus  for  may  because  traversing  and d e c r e a s e s  With both  space-charge  - X  increases  true  ~ *s  electronic  metal.  Is  m  changes  It  positively to  the  t h a n on 0  V,  changes.  respect  of  an e l e c t r o n  involved  where  biased  of  voltage,  barrier  (The  S  thickness  change  applied by  the  \]K  o  x  facing  the  each  path  The d i o d e  barrier other.  of  for theory  region These  and  P-type Semiconductor  —  •—  •—  .  • Ep  N-type Semiconductor  Metal  .  Metal  777777 (a)  0  0 > m  (b)  S  0  <  m  E  7  0  s  t  Semiconductor Metal  (c) ' v  0  *m  =0 v  s  P-type Semiconductor —E  V,.  /  N-type Semiconductor  Metal  (d) ' s  0  v  m  Figure  r  <  0  2.3  (e)  0  m  >  E q u i l i b r i u m electron-energy diagrams metal to semiconductor contacts  0  s o  of  Ep  8  I where  I is  the  current,  the' metal  is  biased  conductor  or  when t h e  a  p-type  T the  Involving  the It  of  higher  metal  s h o u l d be contact  w i t h which  contact  lt  is  of  2.3e).  For  the  case  semiconductor  are  equal  the  neutral  (Figure  2.3c).  from  and  electrons  be  fixed  carrier  re-establish  concentration and h o l e s  at  and independent concentration  conductor  under  i n the the of  current  flow.  of  are  layer  are  unchanged  an  the in  treated  metal  already is  high  and  matched  formed  thermal  assumed  gener-  in  the  disturbed equilibrium  semiconductor,  current  of  the  the  metal-semiconductor  the  remains  any  semi-  semiconductor  functions  infinitely  metal  function of  f u n c t i o n can be  levels  arises  Different  than that  A p-type  carriers  function  with a  a p-type  work  greater  Fermi  of  of  function.  .  Boltzmann  barrier  semiconductor  work  if  which are  to  Schottky  space-rcharge  Furthermore,  k the  to  s semiconductor.  when t h e  or  semi-  a complicated  contact  work  the  metal  of  to  when t h e  c o n d i t i o n , no  able  the  work  result  and r e c o m b i n a t i o n r a t e s  carrier  the  lower  ation  of  that  an n-type  higher  (Figure  in  metal  when  with respect  charge,  and  the  positive  an n-type  negatively  electronic  makes c o n t a c t  with a metal  similarly  biased  to  of  equal  (2.1.1)  and V are  and I  profiles  semiconductor  (I  temperature,  f u n c t i o n or  energy  is the  of  of  1)  with respect  stressed  with a metal  equilibrium  q  parameters  work  conductor  n-type  metal  qV/kT -  voltage  positively  absolute  from the  (exp  8  V the  semiconductor),  constant,  either  = I  interface  flowing. throughout  Such a c o n t a c t  is  concentration will  Thus,  the  the  semi-  called  an  ohmic  9 contact  (see  section  2.2).  When c h e n - t y p e that  of  the  electrons  flow  a negative levels this  of  limited varies  of  i n contact  from  space the  type  emitting  metal  metal  the  In ordinary  conductor, the  but  applied  contacts  it  and c o n d u c t i v i t y field  strength  high,  sample  strong  the at  between  these  Mott  of the  the  are  Fermi  2.3d).  With  In  current  the  current  while  act  in  the  conductivity of  as  the  two  the  where high  good  semiis  the  semiconductor  length  and  ohmic  on the  low,  opposite  b a r r i e r models  properties  the  the  plane-parallel  sufficiently  contact  cathode  space-charge-  situations  contacts  and  the  space-charge-limited  and S c h o t t k y  metal  the  the contacts.  predicted  work  and  function  because  1  contacts of  the  dependence  difference V  low,  of  ( C h i l d ' s Law),  separation  of  Is  up a  normal d r i f t  and the  that  introducing  vacuum c a s e ,  the  in practical  the  (Figure  quadratic  Hence,  Because a  the  the  voltage is  seen  exceeds  made w i t h a t h e r m i o n i c  In  the  be  coincidence  setting  situations to  can  semiconductor,  may b e d o m i n a n t w h e n  field  small.  is  i960). of  it  work.function  semiconductor  semiconductor  negligible  the  a vacuum,  power  geometry.  i t ,  establish  the  into  in a  Is  to  an analogy  dependence  current  to  (Heljne  3/2  with  metal  and  contact  electrons  as  the  charge,  current  semiconductor's  the  on germanium were  work  function of  presence the  of  surface  semiconductor  conductivity, semiconductor.  the  found  the  metal  states  surface  to  used,  surface  properties Bardeen  and p r o d u c e s  states  independent  (194?) p o s t u l a t e d  which immobilizes the  inversion layer,  Such  have  a layer  of  electrons depleted  Just below  the  surface  w h i c h may  lie  within  of the  the  (b)  Figure  2.4  E q u i l i b r i u m ; the surface i s e q u a l and o p p o s i t e t o space charge  Surface-state  Barrier  charge the  normally of  the  due  f o r b i d d e n energy  crystal  to  adsorbed  filled  up t o  charge.  must  to  the  up  state  the  semiconductors  with  level of  the  the  the from  theory  surface the  or,  until  of  states  a net  with  the  surface surface  under  c o n d u c t i o n band  the  Fermi  present  partly,  surface  absence  the  highest level  at  the  (Figure  free  of  Bardeen  led  performed  to  the  to  will  filled 2.4b).  surfaces  germanium and s i l i c o n Independent  E x p e r i m e n t s w h i c h were  state  discontinuity  bulk material  bulk  may b e  the  associated  states  w i t h the  to  shows  in  Q  electrons  layers as  E  due  at  2.4a  that  surface  such  contact.  surface  Fermi  coincides  space-charge  metal  level  conditions,  f i l l  surface  Figure  energy  coincide  equilibrium  Thus,  the  may b e  periodicity  atoms.  Now s i n c e  states  tend  lattice  gap  of  a  confirm  discovery  of  of  the  hole  injection.  2.2  Bipolar  Theories  With Brattain be  the it  1948)  developed.  discovery became  with a  or  exhausted  completely  conductor  next  was p r e s u m e d bulk  of  charge  the  sidered are  to  the  be  of  to  the  theory  a simple  separated  by  for  and  layer  the  contributes,  the  p-n  partly in  along  with  the  A classic  J u n c t i o n (1949, 1950). model where  transition region  assumed  the to  to  depleted  the  semi-  semiconductor  theories,  rectification effect. the  theories  carriers of  and  associated  which i s  bulk  (Bardeen  contact  theories  In bulk contact  one-dimensional a  bulk  mobile charge  metal  ohmic.  for  injection  one-carrier  space-charge  semiconductor  layer,  Shockley's  to  carrier  possible  The p r e v i o u s  rectification  of  part  the  spaceexample  is  He c o n -  p and n be  of  regions  so n a r r o w  that  Figure  2.5  Equilibrium electron-energy of a P-N J u n c t i o n  diagram  11  recombination that  the  in  it  injected  is  negligible  carrier  (Figure  density  is  Assuming  2.5).  s m a l l he o b t a i n e d  the  expression  P<°) where the  p(0)  is  the  density  transition region  equilibrium applied with  density  voltage,  respect  Junction,  to  J , is  of  and  of  holes  J  is  g  the  where  n  0  is  p  re*glon, the  and  and D  electron  and L  n  the  are  the  s  are  and  lifetime  the p  (2.2.1) boundary  Q  is  n  the  p-reglon  The  current  are  current  thermal  and V i s biased  density  density  L  the  hole in  +  n  the positively  across  the  (2.2.2)  given  D  to  the  p-region  hole length  by  op n/ n> L  dlffuslvity  the  the  diffusion  related  the  is  thermal equilibrium electron n  between  ( e x p q V / k T - 1)  s  0  respectively  lengths  at  <1 ( P n V p  -  dlffuslvity  electron  qV/kT  by  saturation  J  P  In the n - r e g i o n ,  when  J = J where  x  n-reglon,  n-reglon.  given  e  holes  the  positive the  = Pon  density  In the  lifetime  the for  in  p-reglon. holes  the  n-region  respectively,  diffusion length in  in  and the  The  i n the  p-  and L  p  n-region diffusion  n-region, f  , P  the  for  electrons  L  L Equation predicts  the  d-c  2  P  (2.2.2) i s  p-region,  * ^ i n  by  = D ~f  p P  2 n  l n the  = D "C n Si of  current-voltage  great  Importance  characteristic  because  l n terms  lt of  basic  12 measurable to  semiconductor  metal-semiconductor  many  cases  the  contacts  contacts  metal-semiconductor  parameters.  are  The  since  lt  equation is  relevant  believed  effectively  p-n  have been  found  contacts  is  that  Junctions. to  in  Actually,  follow  the  I-V  relation j  I » I where  'a*  has  ten.  Previous  considering contact in  is  are  theories  the  that  2.5  assuming  to  have  contact  to  attempted  to  patchy,  that  be  to  f r o m one explain  point.  the  But  it  to  obtain  any  a patchy  contact,  if  plausible  as  this  is,  possible  to  high  as  anomaly  by  nature  of  the  will  be  shown  value  of  *a',  hypotheses  made.  is  necessary  theories. direction  The of  proceeding  to  define  any  easy  current  flow  current  can  contact  the  conductor biased  flow  and the  material, metal and  the  that  positively  for  p-type  material  with respect  in minority carrier  known as  injection  the  an i n j e c t i n g  concentration  to  density  contact  contact.  with reverse  the  current  occurs  whereas  current  flow  to the  semiconductor.  A decrease called  the  the  semi-  metal A  excess  is  large is  carriers  in minority extractlon.  contacts occurs  with forward current the  that  l i t t l e For  when  supplying  Is  to  high.  with respect  it  two-carrier  i n which is  theories,  refers  contact  forward d i r e c t i o n of  increase  known as  the  to  flow  resistance  biased  and  related  direction  is  negatively  with bipolar  current  through  to  when  terms  forward d i r e c t i o n of  direction refers  on n - t y p e  further  various  reverse  is  (2.2.3)  have v a l u e s  from p o i n t lt  - 1)  Is  Before lt  s  observed  different  Section  without  been  (exp qV/akT  carrier Similar  13  increases with as  and d e c r e a s e s  the  current  in minority carrier  directions  reversed  a c c u m u l a t i o n and e x c l u s i o n  contact  is  linear  defined  but  has  as  n  equilibrium potential  barrier  at  2.3c).  Therefore,  =  special  direction other  case of  of  current  d i r e c t i o n of  Ohmic easily  made  tivity  ones.  donor  if  to  an  used  to  in  is  the  "acceptor"  at  an  metal.  (1956) I n f e r r e d t h a t  one  and hole  implies that  the  the  the  for so  been  there  surface Is  the  greater  is  found  that lt  often  is be  that  to  the  a contact  Therefore contact  of  that  material  in  in  one  the  to  be  high  it  give  to  that rise  n-type.  to A  semiconductor  Dlemer, always  of  contacts  with a metal  make  more  resis-  ohmic  a p-type  Kroger,  merely  transparent  i n t r i n s i c semiconductor,  at  is  indi-  thickness  becomes  is,  so  (Figure  resistance  than  found  that  no  contact.  that  made  )  and  does n o t  in practice  it  4  recombina-  contact  than  the  may b e  small  can  where  2  infinite  linear  materials  this  time,  is  * *  2  mobilities  that  contact  through  semiconductor,  arises  only  (  A rectifying  always  flow  same  dope  the  situation  and  extrinsic  semiconductors  levels  similar  At  is  have  reason  barrier  flow.  n-type  would,  strongly  not  (2.3.1)) i n t h e s e m i c o n d u c t o r  current  contacts  The  potential  carrier to  to  equation  non-ohmic flow  An o h m i c  +  necessarily.  the  is  known  by  a n I-V r e l a t i o n w h i c h  an ohmic c o n t a c t  but  are  1955)•  I-V r e l a t i o n  electron  carriers  (Low  above,  >V>o>  o  contact  0 (equation  = Ap  the  of  n  the  This  the  rate  t h a t An  a  are  Q  densities.  tion-generation  cate  p  Q  whose  given  ^ n w h e r e j u , ,Up a n d n ,  those  respectively  a contact  a conductivity  from  concentration,  and  Klasens  diffuses  into  Figure  2.6  E l e c t r o n i c - e n e r g y diagram of metal - n - n contact +  a  14  the  semiconductor  layer  w o u l d be  through  making i t  formed  strongly  (Figure  the  thin  space-charge  essentially  like  that  n  -  in  n - metal  at  the  contact,  the  good  appears  increased area. or  by  the by  ohmic  This  is  conductor to  (a)  Only holes  current  based  (Figure  the  following  (b)  The  lifetime  (c)  There  is  J  diffused  the  barrier  Infinite  exist  generation  junction.  d i f f u s i o n of the  or  In  may  practise  be  i n the  contact  the  contact  material  semiconductor  surface  before  the  Philco  Theory and S t i c k l e r  ( 1 9 5 5 )  relations  on a model h a v i n g  2.7a).  To d e r i v e  a  these  for  o  f  metal-semi-  surface  barrier-  I-V r e l a t i o n s ,  they  assumptions.  can cross  density,  by  is  fabricated  recombination centres  current-voltage  contacts  made  often  no p o t e n t i a l  at  Schwarz,  derived  electrons  alloyed  tunnel  metal.  Borneman, Corporation  contact  -  required.  carriers  of  Zero-Electron-Current  2.3  are  a n n+  would  and r e c o m b i n a t i o n r a t e s  treatment  the  are  the  Hence,  require  accomplished  of  that  2.3d.  Thus,  carriers  contacts  addition of  mechanical  application  of  generation  the  so  condition that  and r e c o m b i n a t i o n r a t e s it  layer  ohmic c o n t a c t s  Besides  The  metal contacts  +  where  2.6).  Figure  and p - p -  +  devices  of  extrinsic.  , is  n  of  no b u l k  the  the  metal  contact  so  that  the  electron  zero. holes, ¥  space charge  A n = n - n  0  ,  Is  infinite.  so  = p - p  that  0  s A p  (2.3.1)  Spacecharge 'egion  (a)  Rectifying contact  Ohmic contact  (b) Figure  2.7  Model f o r (a)  (b)  metal-semiconductor  contacts.  Diagram of e l e c t r o n b a r r i e r at r e c t i f y i n g c o n t a c t on an n - t y p e semiconductor w i t h an ohmic c o n t a c t Semiconductor filament—one-dimensional flow of c a r r i e r s (magnitude of x i s exaggerated) 0  15  where n and p are the e l e c t r o n and hole d e n s i t i e s r e s p e c t i v e l y and n  Q  and p  are t h e i r thermal  0  equilibrium values.  (d)  The hole d e n s i t y a t the o p p o s i t e c o n t a c t i s p .  (e)  In the bulk,  and  Q  the e l e c t r o n and hole c u r r e n t d e n s i t i e s , J  n  Jp, are g i v e n by  where j j and p p  n  J  n  =  ^ n  J  p  =  ^HpPE ~  n  E +  k T  k  ^n T  g  a  S  u p  a r e the constant  r  d  (2.3.2)  n  r a d  (2.3.3)  P  m o b i l i t i e s o f the holes and  e l e c t r o n s r e s p e c t i v e l y , q i s the e l e c t r o n i c charge,  kis  Boltzmann's c o n s t a n t , T i s the absolute temperature, and E i s the e l e c t r i c (f)  field.  The v o l t a g e across  v  where p(0)  c  the b a r r i e r , V , i s g i v e n by c  - kT/q  l n P<0> Po  (2.3.4)  i s the hole d e n s i t y at the boundary o f the space-  charge r e g i o n i n the semiconductor. From the above assumptions, Borneman, Schwarz, and S t i c k l e r have d e r i v e d I-V c h a r a c t e r i s t i c s i n g e n e r a l i z e d o r t h o gonal  co-ordinates.  The hole c u r r e n t c r o s s i n g the metal contact  is Ip = C k T ^ where  i p = p(0)  -  where V = V  - p  "o+Po ,2  Q  + Vg  p  2<$p - ( n - p ) I n ( 1 + £ p / n ) Q  0  (2.3-5)  0  (2.3.6)  0  -1 +  4ppn 1  +  (n  0 + P o  0  )^  < P ^/M ex  -  1)  i s the t o t a l v o l t a g e a c r o s s the specimen, Vg  16  is  the voltage across  depending only contact  on t h e geometry o f t h e d i o d e .  of radius  hole  flow  then  C = A/(L-x )  is  r e g i o n , and C i s a c o n s t a n t  the bulk  r  1  1  on a s e m i - i n f i n i t e  i s one-dimensional  For a circular C = kr.  slab,  a l o n g p o s i t i o n c o - o r d i n a t e 'x',  where A i s t h e a r e a o f t h e c o n t a c t  Q  the l e n g t h o f t h e bulk  I f the  region  ( F i g u r e 2.7b).  From  and L - x  Q  equations  (2.3-5) a n d (2.3.6)  < ( n +Po)  - 1  l + 4 n p ( n + p ) " ( e x p qV/kT -1) J  - 1  2  0  no+Po  1 +  - 1) J  jl + 4n p (n +p r (exp  G  0  0  o  qV/kT  2  o  o  o  0  (2.3.7)If (a)  V i s not too l a r g e , the f o l l o w i n g l i m i t i n g n /p » Q  0  1  n /p 0  = 1  0  n /p « 0  Q  1  = CkTju p  p  p  0  ( e x p qV/kT  0  ( e x p qV/2kT  (2.3.8)  - 1)  (intrinsic) I  (c)  result. ,  (n-type) I  (b)  cases  p  = 2CkTu p p  - 1)  (2.3-9)  (p-type) I  Equation  p  = Cqu  p P o  (2.3.10)  V  (2.3.8) has t h e f o r m o f t h e u s u a l  diode  equation.  I t c o u l d a l s o have b e e n d e r i v e d by a s s u m i n g t h a t t h e  hole d r i f t  current i s negligible  current  i n equation  (Einstein relation). Intrinsic f o r n-type  compared  t o the hole  diffusion  (2.3.3) when we n o t e t h a t kTu = qD P P Equation  (2.3.9) p r e d i c t s t h a t f o r  m a t e r i a l s t h e s a t u r a t i o n c u r r e n t w o u l d be d o u b l e semiconductors  and t h a t t h e r e c t i f i c a t i o n  ratio  that  17  w o u l d be d e c r e a s e d . negligible equation  compared  (2.3.10)  the  strict  the  conductivity  type I-V  curves  type  to the hole  as  of  Therefore,  It  i n form from  through  is (2.3.3)1  relation in  I - V r e l a t i o n and because for highly  (2.3.7) that  varies  intrinsic  i n equation  i s a n ohmic  value  equation  diffusion current  current  the l i n e a r  the r e s i s t i v i t y  materials  drift  has the c o r r e c t  should vary  lines  the hole  c a n be o b t a i n e d .  sense because  materials.  ohmic  By a s s u m i n g  extrinsic  predicts  that  of an i d e a l  from h i g h l y  to h i g h l y  pthe  diode  to  extrinsic  extrinsic  n-  p-type  materials.  Borneman, qualitatively behaved  very  Schwarz,  and q u a n t i t a t i v e l y , much a s p r e d i c t e d  trie  rectification ratio  and  that,  relative  i n the n-type  expected  theoretically.  included  i n the theory  a manner  mental some  as  data.  current  Investigation current  l e d to  theory.  investigation, to the  arbitrary  samples They  were noted  slightly that  the d i s c r e p a n c i e s  levels.  of  the course  of  this  predicted the  than  the of  were  would change  strongly  the  ln  experi-  suggested  that  theoretical  non-zero-electronthis  theory  form  chapter.  than  that  samples,  with  that  curves  i f recombination  and the ensuing  These  found  except  larger  curves  the zero-electron-current  remaining sections  less  i n the p-type  the development  injection  (2.3.7)  the discrepancies  during  have  experimental  somewhat  the t h e o r e t i c a l  existed  Also,  (1955)  by e q u a t i o n  was u s u a l l y  to reduce  However,  electron  their  to the currents  currents  such  and S t i c k l e r  theoretical was  extended  the substance  of  18  Extension of  2.4  Injection The only  for  which of  low  is  the  where  to by  n(0)  x = x ,  Mlsawa  by  arbitrary  is  £p_ A r b i t r a r y  is  used  theory. by  valid  the  can  i  exp  q C ^ - ^ J / k T  p(0)  =  n  i  exp  q(f  relation  -^)/kT  p  electron  and h o l e  densities  at  intrinsic carrier  density,  ^  and ^  (imrefs).  be  Shockley,  n  n  (2.3.4),  derivation  theory  using  =  the  is  equation  i n the  This  n(0)  and  levels  relation,  levels  Prom  theory  are  the  is  the  electro-  electron  and  hole  By m u l t i p l y i n g t h e  above  equations,  obtained  assuming  space  p(0)  » n  charge  x  2  exp  (2.4.1)  qV /kT Q  neutrality  he  obtained  the  sym-  forms  p(0)  n  0  + p(0)  -  p  0  = n i  n(0)  p  0  + n(0)  -  n  0  = x^  remarkable-that  tremendous the  i n that  injection  are  the  Shockley's injection  Mlsawa (1955).  potential,  metrical  It  low  presented  n(0)  and  since  for  a n d p(0)  quasl-Fermi  zero-electron-current  injection  n^ i s  0  static  previous  v a l i d only  extended  Z e r o - E l e c t r o n - C u r r e n t Theory  Levels  formulae  derived  the  theoretical  definitions  derivation  of  of  these  the  these  2  2  exp  qV /kT  (2.4.2)  exp  qV / k T  (2.4.3)  Mlsawa r e l a t i o n s  significance  are  so  quasl-Ferml levels.  equations,  low  c  which are  easily  derived  Previous  i n j e c t i o n was  of  to  from  the  Invariably  19  assumed  at the Junctions.  But with  the development  of  (2.4.1), (2.4.2), a n d (2.4.3), i n j e c t i o n a t a r b i t r a r y could  be t r e a t e d For  reduces  l o w i n j e c t i o n when  to Shockley's  equivalent  relation  equation  i n <<p  when  densities  high  p(0) a n d n(0)  semiconductor  - n  » n  0  obtain  V = V  B  f i  exp we h a v e  from  I  where used  - P  (2.2.1) a n d (2.3.4)) (2.4.4)  c  r e l a t i o n f o r low i n j e c t i o n  W  = n(0) - n  = op)  Q  exp qV /kT  (2.4.5)  c  0  that  I s , when t h e c a r r i e r  we h a v e  f o r both  types  of  p(0) » n  e x p qV /2kT  1  (2.4.6)  c  (2.4.2) c a n b e s o l v e d  for V  which  c  is  then  to  V  Now  Q  (2.4.2)  material  Equation  to  equation  >  exp qV /kT  levels,  n(0)  added  0  (note  injection  0  (see equation  = p  by  n(0) At  o p « n  to Shockley's  i s given  0  levels  theoretically.  p(0)  An  equations  p  - JE + V  Q  which  qV/kT -  satisfies  = CkT.Mp  0  .  2^p  the equation  (1 + < * p / n ) 0  the previous  £p = p(0) - p to solve  c  d x « ( k T / q l n (1 + i p / n )  -  2  (1 +  h/p ) 0  section  that  (n -p )  l n (1 + ( $ p / n )  Equations  0  0  0  (2.4.7)  (2.3.5)  (2.4.7) a n d (2.3.5) c a n now b e  f o r the d-c I-V r e l a t i o n .  The i n t r i n s i c case c a n  20  be  s o l v e d e x a c t l y to y i e l d an I-V  injection  levels.  However, the e x t r i n s i c  s o l v e d e x a c t l y and (a)  HQ/PO  >  >  approximations  (n-type  l  i)  Low  p  Q  The  cases  are;  0  p  (exp qV/kT  0  High i n j e c t i o n  Ip = C k T u n  must be made.  be  (op«n )  = CkTu p  p  cases cannot  material)  injection I  11)  r e l a t i o n h o l d i n g at a l l  2(pb/n )  -  (2.4.8)  1)  (ip»n ) Q  exp qV/3kT  l / 3  0  - qV/3kT -  1/3  In p / n 0  0  (2.4.9) (b)  n /p 0  = 1  Q  (Intrinsic  material)  Ip = 2 C k T j u p p  (c)  n  o/Po ^  1  1)  (p-type  0  (exp qV/3kT  -  1)  (2.4.10)  material)  Low I n j e c t i o n  (<Sn«p ) 0  (2.4.11)  Ip = |Cqu p V p  ii)  High i n j e c t i o n  Ip = Ck J)i Po f  p  2(n /p ) / 2  Q  0  3  0  (<£n»p ) 0  exp qV/3kT  + qV/3kT + 1/3  In p / n Q  0  (2.4.12)  These r e s u l t s  are s i g n i f i c a n t  because the value of  'a' i n the e x p o n e n t i a l term (see equation high i n j e c t i o n  (2.2.3)) i s 3 f o r  l n a l l types of semiconductors.  'a' has been observed Injection conditions.  f r e q u e n t l y i n p-n The  T h i s value of  J u n c t i o n s under high  high i n j e c t i o n  formulae  f o r the c u r r e n t  21  in  extrinsic  Independent level  of  tially diode  on  the  applies  but  that  intrinsic current be  V  (2.3.4)  .  equation  In  i s less  The  referring  i n the  However,  Mlsawa e q u a t i o n  reduces to  Injection  l e v e l approximation.  from  the  cal  form  the  exact  Is to  For  injection  (2.4.10)  (2.3.10) is  identical  p-type  i n the  and  zero-electron-  in current  (2.4.1). theory  although  it is  the  can  Notice by  equation  Mlsawa  f o r n-type m a t e r i a l ,  levels  when  o n « P  0  ,  for  the  symmetrical  form o f  Shockley's  This  symmetrical  form  equation  i n t r i n s i c m a t e r i a l no The  is  t e r m s n e e d be  effect  of  the  use  dropped of  the  symmetri-  of S h o c k l e y ' s a p p r o x i m a t i o n f o r p-type m a t e r i a l Mlsawa e q u a t i o n  Shockley's  low  for i n t r i n s i c material  for a given level  p(0),  f o r a given  zero-electron-current  (2.4.10)  the  approximation.  i s to y i e l d ,  t h a n l n the equations  across  I-V  the  Mlsawa e q u a t i o n .  Increase,  extension  low  and  Mlsawa e q u a t i o n  (2.3.4)  usual  a linear  reduction  extension,  reduces to equation  .  voltage  exponen-  equation  zero-electron-current  l n the  low  and  exact  than p r e d i c t e d  to the  any  the  f o r n-type m a t e r i a l  reason f o r t h i s  at  case,  When t h e  the  and  depends o n l y  injection  and  f o r a given  at  (2.4.5)  low  ln V  current  (2.3.8), (2.3.9)  equations  p-type m a t e r i a l  low  the  (2.4.11)  and  current  i s given  Q  i n i n t r i n s i c material  material  by  f o r m u l a f o r the  diode equation  theory.  found  that  the  the  terms l i n e a r  f o r p-type m a t e r i a l s .  (2.4.8)  that  contain  a p p l i e s to n-type m a t e r i a l s  compared w i t h  found  the  voltage.  equation  equations  also  of V w h i l e  Injection  relation  are  materials  and  (2.4.11).  value Thus,  V,  of the  a smaller  theory.  This  V  l n the c  and extension  obtained  by  effect  on  the  value  of  current  Is  seen l n  22  The above low I n j e c t i o n reverse d i r e c t i o n .  However,  equations are v a l i d i n  the high i n j e c t i o n  the  equations are  not v a l i d i n the r e v e r s e d i r e c t i o n s i n c e there i s a l i m i t number of m i n o r i t y c a r r i e r s that can be e x t r a c t e d . occurs  p  saturation:  = -CkTp  2p  p  + (n -p ) 0  0  Hence, the magnitude of the n-type m a t e r i a l i f  In (1 -  Q  p /no)  saturation current is greater  the amount of doping i s  l e s s and,  saturation current for i n t r i n s i c material i s  the  h i g h l y doped n-type m a t e r i a l .  for  p-type  material since i t  furthermore,  double  No s a t u r a t i o n c u r r e n t  satisfies  in  that  for  exists  a linear I - V relation.  N o n - Z e r o - E l e c t r o n - C u r r e n t Theory Extrinsic  2.51  Semiconductors  The previous  two s e c t i o n s of t h i s chapter  s i d e r e d the case of zero e l e c t r o n some e l e c t r o n characteristic  current only.  c u r r e n t always e x i s t s ,  its  w i l l now be d e r i v e d f o r  with no r e c o m b i n a t i o n . are  (2.4.13)  0  the  2.5  material  Equation (2.3.5) then g i v e s the c u r r e n t i n n -  type m a t e r i a l at I  the  This l i m i t  i n n-type m a t e r i a l when p(0) = 0, and i n p-type  when n(0) = 0.  to  have c o n -  But  influence  since  on the I - V  the one-dimensional  For convenience the r e l e v a n t  case  equations  listed. J. n  qu nE + k T u  n  grad n  (2.51.1)  J  q u p E - kT;j  p  grad p  (2.51.2)  P  n  p  J =  P  (2.51.3)  23  'dn/ot =  -(n-n )/^ 0  •ap/at « - ( p - p ) / f 0  p(0)  1  - q"  1  p  p  0  p = p(0) p = p three equations  (2.51.4)  div J  p  (2.51.5) (2.51.6) (2.51.7)  2  = p - p  0  n  c  = n^  Q  divJ  exp (qV /kT)  2  An s n - n  first  + q"  n ( 0 ) = nr n  The  n  Q  at x = x  s Ap  (2.51.8)  Q  (2.51.9) (2.51.10)  at x = L  Q  are the expressions f o r the electron  c u r r e n t d e n s i t y , t h e h o l e c u r r e n t d e n s i t y , and t h e t o t a l c u r r e n t . Equations  (2.51.4)  and ( 2 . 5 1 . 5 )  arerespectively  e q u a t i o n s f o r e l e c t r o n s and h o l e s . Mlsawa r e l a t i o n , Equation  (2.51.7)  (2.51.6), the  was d e r i v e d a t t h e b e g i n n i n g o f S e c t i o n 2 . 4 . s t a t e s that the product  brium c a r r i e r d e n s i t i e s i s assumed when e q u a t i o n last  Equation  the continuity  I s a constant. (2.51.8)  of the thermal  Space c h a r g e  i sapplied.  equili-  neutrality  Finally, the  two r e l a t i o n s a r e t h e b o u n d a r y c o n d i t i o n s i n t h e s e m i c o n d u c t o r , Now i f t h e l i f e t i m e s o f h o l e s a n d e l e c t r o n s , X  p  X , n  a r e assumed i n f i n i t e ,  then f o r t h e steady J J  From e q u a t i o n s  and  state  - constant  n  P  = constant  (2.5I.I) t o (2.51.3) grad p = g  a  P  P J ( 1 V P - 1 - N/p) kTu (2p+N)(l+M) p  (2.51.ID  24  where  M= J / J , n  b = J%/)Jp, and N = n  p  a p p l y i n g the boundary c o n d i t i o n s j  kTu (M+l) (L-x )(M-b)  =  • ^P 2  0  Note that when j  = s  p(0)  = 0,  +  J = - J  2p  I n t e g r a t i n g and  Q  / po + b N ( b - M ) " l  ^n(l+M) (L-x )(b-M)  - p .  we o b t a i n  M+b  c  n  0  (p(0)  n  + bN(b-M)-  so that  s  b+M  If  «  (b-M)p /bN Q  = kTu p (l+M)(L-x )  s  p  expression  0  f o r the  equals p-type  (2.51.12), - J  .  s a t u r a t i o n c u r r e n t can be  and n o t i n g that  An a l t e r n a t i v e  material is  discussed  and  g  -1  0  d e r i v e d f o r p-type m a t e r i a l by s e t t i n g equation  (2.51.13)  1 J  An e q u i v a l e n t  material  (b-M)pp bN  1 +  c  (2.5L12)  A  f o r n-type  Q  and  1  n(0)  = 0 i n the c u r r e n t  the c u r r e n t at  saturation  procedure f o r d e r i v i n g  J  s  to apply the symmetry t r a n s f o r m a t i o n s ,  later  in this  section,  From  E = - dV/dx,  the boundary c o n d i t i o n s ,  to equation  equations  for to be  (2.5I.I3).  (2.51.2),  (2.5I.H),  the v o l t a g e across  the b u l k , Vg,  is  found to be w V  Equation  l  (2.51.6)  v  in  _ kT M+b , B " q M-b  c  g i v e s the v o l t a g e across - kT/q  which, of c o u r s e ,  equation  n  ,-1 \ Pp + bN(b-M)' p(0) + bN(b-M) - 1  (2.51.8).  the  (2.51.14)  barrier  In (1 + < * p / p ) ( l + 5n/n )  Sp = p(0)  - p  Q  = n(0)  Equations ( 2 . 5 1 . 1 2 ) ,  cannot be a p p l i e d e x a c t l y  (2.51.15)  Q  0  - n  0  = Sn from  (2.51.14),  and ( 2 . 5 1 . 1 5 )  to g i v e the I - V c h a r a c t e r i s t i c  25  explicitly. the  exact  Note  that  equations  (2.51.12). and (2.51.14)  give  relation  Now,  as i n the p r e v i o u s  must  be  (a)  Low I n j e c t i o n  section,  ,low a n d h i g h  injection  cases  considered,  1)  n  / p  0  If equations  » l  0  (n-type  jop/n |«l  material) (b-M)p(0)/bn, « 1 , *o  and  0  (2.51.14) and (2.51.15)  B * f  V  and  approximate  V  n  (2.51.16)  (2.51.17)  ( j + iE)  Q Now f r o m e q u a t i o n s  to  ip  s i i i  c  then  (2.51.18)  PQ  to (2.51.18)  we o b t a i n  q u n (M+l) n  °  J  (L-Xo)(iltb)  V  B  (2.51.19)  kTp_p-(M+l) J  ?L-x )—  =  {  e  x  0  which bulk  shows  a linear  and the usual 11)  n /p 0  If equations  and also  diode 0  « 1 0  equation  « 1  B  V  k  T  -  (2.5L20)  U  I-V r e l a t i o n f o r the  f o r the space-charge  region.  material)  and  (2.51.14) and (2.51.15)  V  ^ c/  an ohmic  (p-type  4n/p  p  |(M-b)n(0)/Mp approximate  * - f | ^ i »  o  « 1 ,  then  to  (3.51.21,  26  and  V c  q  In (1 + *&) n  (2.51.22)  0  A p p l y i n g equation (2.51.16) to the f o r e g o i n g equations  j  _  J  W Q C H - D  V  (L-x )(M+b)  V  0  and  kTu n (M+l) = tr \u "" n  J  B  gives  '  (  2  U  ,  5  1  5  #  i  2  ,  3  )  O  J  rt  (  e  x  P 1 cAT v  "  1)  (2.51.24)  \ ti— XQ y n  which shows a l i n e a r and a l s o an ohmic I - V r e l a t i o n f o r bulk and usual diode equation f o r the space-charge  region.  Note should be taken of the symmetry of the equations fact,  f o r low i n j e c t i o n i n t o n -  a l l the equations  the  above  and p-type m a t e r i a l .  In  f o r the low i n j e c t i o n of c a r r i e r s i n t o  p-type m a t e r i a l can be obtained from those f o r the n-type by simply making the symmetry  transformations n p  n  C o r o l l a r i e s of these t r a n s f o r m a t i o n s are of  Thus,  f o r example,  (2.51.21), Into for  b  >-l/b  M  1/M  equation (2.51.17) transforms i n t o  (2.51.18) i n t o  why t h i s  equation  (2.51.22), and (2.51.19) and (2.51.20)  (2.51.23) and (2.51.24). the d r i f t  course  Other examples are the  expressions  and d i f f u s i o n c u r r e n t s i n S e c t i o n 2.52. The reason  symmetry d i d not occur i n the  zero-electron-current  27 theory  was b e c a u s e  asymmetry  into  High  <Wn  and  rules  should  Immediately  B u t when  Introduced  M ^ 0, t h e n  the  an  symmetry  hold.  Injection For  and  M= 0  the analysis.  transformation (b)  setting  0  sufficiently so  >>1,  that  high  injection  (b-M)p(0)/bN  (2.51.15) a p p r o x i m a t e  such  <£P/P » 1  that  0  equations  »1,  (2.5L14)  to (2.51.25)  In  kT(M+b)/(M-b)q  1 + (b-M)p /bN 0  (b-M)p(0)/bN  - In  u and so  V that  V = Vg + V  V =  Finally, high  p  «• (2kT/qj l n  c  i s given  3b-M  injection  J(L-Xp)(b-M) kT^ (M+l)  by  2  "b+FT B  k  0  2  H  (b+M)/(3b-M)  p T n ( e  qN b M +  kT  3b-M u _ ~  where  H  high injection  v  b  +  2  N  x  p  aV_to=M» k T 3b-M (  N  + (b-M)p (b-M)  the  is  }  (2.5L28)  3b-M  0  n i  (2.2.3)) g i v e n b y  a  semiconductors  b+M  equation predicts  a  (2.51.27)  (2.51.25), a n d (2.51.27),  (2.51.16),  n  This  1 + (b-M)p /bN (b-M)ni/bN  T  I-V relation for extrinsic  „  (2.51.26)  1  j . ——- 1l n +  V  from equations  (p(0)/n )  _ 3b - M  b - M  an ' a '  value  (see  equation  28  which reduces to  a = 3  when  M = 0.  equation reduces to equations  (2.4.9) and (2.4.12) when  and the type of semiconductor i s  taken i n t o account.  a s ' t h e l e v e l of i n j e c t i o n i n c r e a s e s the value of the value of  1  ficance next  is  Note  that  materials  from u n i t y to a value determined by  H'.  is  Thus,  it  seen that the value of  significant  1  M  plays  1  the I - V c h a r a c t e r i s t i c .  I t can be seen that n  in extrinsic  M= 0  'a' increases  a major r o l e i n the form of  M = J /Jp,  T h i s high i n j e c t i o n  the e l e c t r o n to hole c u r r e n t r a t i o ,  i n the above e q u a t i o n s ,  and i t s  signi-  can be seen more c l e a r l y i n the i n t r i n s i c case i n the  section.  'M' Is r e l a t e d to the i n j e c t i o n r a t i o , Y,  the  r a t i o of hole c u r r e n t to the t o t a l c u r r e n t by V * Its  significance  J =  P _ j ~  i F T T  (2.51.29)  can a l s o be seen l n the f o l l o w i n g  analysis.  Examining  J  n =  J  P =  (<Vdrlft (  Vdrlft  where  +  +  ( n>dlff  <p>dlff J  -1 < M < 0 , l t  J  = 1F  ( p) d r i f t  = ^pP  J  ( J  n diff  ( J  p^diff  }  =  k T  Mn 6  = ~ ^p § k T  i s found that J  d i f f u s i o n c u r r e n t dominates  n  (2.51.30)  = /(W D  < n>drift J  If  = JM/(M+1)  J  (2.51.3D  +  n E  n  E  r a d  r  a  n  d  P  < 0 and that  the e l e c t r o n d r i f t  the  electron  c u r r e n t while  F i g u r e 2.8 The d e p e n d e n c e o f ' a ' on the e l e c t r o n to hole current r a t i o d i v i d e d by t h e e l e c t r o n to hole mobility ratio for the i n t r i n s i c case (dashed l i n e s asymptotes)  are  29 Jp  > 0 and  current the  equal  the  semiconductor  is  true  strongly  is  = 0  n  and  diffusion current  if  the  and o p p o s i t e .  hole  drift  true  always  it  is  dominates  .strongly is  the  n-type.  to  the  intrinsic  the  electron  the  hole  n-type  equations  be  features  (2.51.30)  If  M > 0,  current is  the  diffusion  and  M = 0,  n-type  and  electron  current.  The  while  is  true  feature  reverse to  be  J = q(^ n n  0  noticed  the  the  drift  the  the  are  dominates  if  where  drift  reverse current  hole  drift  semiconductor if  is  it  is  M/b =  n /p 0  Q  + ;jpP ). 0  Semiconductors  transcental  interesting  p-type  diffusion current  A special  can  hole  and d i f f u s i o n c u r r e n t s  semiconductor For  the  weakly  p-type.  diffusion  ohmic case  case  or  The h o l e  The p r e v i o u s due  n-type  drift  Intrinsic  2.52  Is  dominates  electron  if  or weakly  the  it  p-type.  dominates  p-type  which  current  if  current Is  hole  reverse  then J  Is  if  the  extrinsic  case  nature  the  solved of  this  of  theory.  ( n)drift J  be  equations.  explicitly  (2.51.31)  and  could not  c a n be  = t  J  However,  and r e v e a l s  First,  it  solved  from shown  M+T  exactly the  many  the  two  current  that  (2.52.1)  M-b (  (  ,  Vdlff  "V  drift  ( p)dlff J  and  therefore  we o b t a i n  ' * J M+7  the  t  =  ratios  £  J  J  b(M+l)  b(i!l)  (2.52.2) (2.52.3) (2.52.4)  F i g u r e 2.9 The d e p e n d e n c e o f t h e saturation current density, J s i , o n M ( t a k i n g b=2) f o r t h e intrinsic case. Hence, the dependence of the I-V c h a r a c t e r i s t i c on M = J / J p n  (dashed  lines  are  asymptotes)  M  30  n'drlft  M+b M-b  _  * n>diff J  Now when the equations this intrinsic  ( A d r i f t  p'dlff  (2.52.5)  (2.51.1) to (2.51.15) are m o d i f i e d f o r  case, we o b t a i n  g  J  =  r  a  d  "  p  (2.52.6)  2kT^U+M)  ^M-bHL-x ) [  ° '  P  Q  p  (  0  )  (2.52.7)  (2.52.8)  2kT  „  kT Ib-M  J = J where  ln  a = (3b-M)/(b-M)  s  l  ,  (p(0)/ )  (2.52.9)  n i  , ,n\/  ( e x p qV/akT  \  (2.52.10)  - 1)  (2.52.11)  and 2kTu n (l+M) n  'si  =  Q  Thus, we have o b t a i n e d equation of  1  (b-M)(L-x ) (2.52.11) which p r e d i c t s values  'a' of any magnitude, depending on the value of M. ( F i g u r e 2.8) A n a l y s i s o f these f o r e g o i n g equations  the f a c t o r  M/b  i s important  I-V c h a r a c t e r i s t i c equation  i n d i c a t e s that  l n determing the nature o f the  ( F i g u r e 2.9). I t can be shown by examining  (2.52.11) t h a t the r e g i o n s o f i n t e r e s t where p h y s i c a l l y  r e a l i z a b l e I-V c h a r a c t e r i s t i c s are obtained are  Figure  2.10  The d e p e n d e n c e o n M/b o f t h e s i g n s o f c u r r e n t s a n d v o l t a g e s when t o t a l c u r r e n t J Is p o s i t i v e i n an i n t r i n s i c semiconductor  31  (a)  -l/b  (b)  M/b = 1  (c)  1 < M/b < 3  The  < M/b < 1  values  lead  to  (region  of  M/b  current  characteristic  J  the  ohmic  is  of  the  in region  where,  as  above,  above  that  the  going  from r e g i o n  = J  for  J  = J  the is  lt  c a n be  and d r i f t  diffusion  region  -  is,  the  1,  I-V  1)  (2.52.12)  the  1-2,  I-V  relation  is  Q  seen  Is  both n  form  (1 -  exp  (-qV//3kT))  of  both  region  the  that  zero. feature  J  > 0  Note  has  given  at  Figure  the one  and d r i f t for  diffusion  currents  2 the  of  2 is  Another and  curious -l/b  in  components  beginning  In region  the  2.10)  0 < M / b < 1.  region  from  reversed  and d i f f u s i o n  i n region  dominant  positive.  (See  2.  drift  (2.52.14)  positive.  rectification  diffusion  < 0 for  (2.52.13).  the  are  is  + ^p^jV  i  has  currents  A curious  1, J  n  into  1  current  dominant.  = q(>i n  direction  and e l e c t r o n  V and Vg are In  regions that  In region  qV/pkT  In region  and  0  - l / b < M / b < 0,  M / b = 0,  characteristics,  voltage.  (exp  0  2 lt  Now e x a m i n i n g  section,  I-V  these  one  finally  hole  outside  n  form  ^3 >0.  J  the  n  impossible  J and  M = J /J )  with negative  > 0 and  Q  2)  (where  J where  1)  1-2)  (region  physically  positive  (region  of is  are  of this  dominant  equal  at  In region drift  that  feature  V  1-2  current Q  is  <  0  when  that  < M/b < 0. ( F i g u r e  2.10)  32  2i  CHAPTER EXPERIMENTAL Geometry  3•1  It across ohmic  the bulk  of Is  Specimens experimentally  contact of  under  the  contact  will  interact. tally  to  that  require  studied  obtain. by  the  best  configuration.  electrical contact the  the  studied,  With  be  is  Is  that  avoid  potential  probes  1949).  as  potential  probes.  contacts  on  inhomogeneltles into  Therefore,  where  the  the  was at  metal  Also the  specimen  at were  contacts  the impossible a  complete  the  then rectifying  point-  avoided  side  w h i c h may  side-arms  'ohmic'  it  exist  Therefore,  potential  Impurities  probes  which  is  probe  influence  upon,  'ohmic'  temperatures.  may y e t  decided  potentials  be  through  technique  alloyed  alloying  contact  contacts  since  measured.  floating  the  current  mutually  experimen-  it  contact  a  the  will  upon to  although  a such  with a  contact  gold-antimony  d i f f u s i o n of  decided  but  two  the  as  the  was  'ohmic'  the  semiconductor  pass  (Shockley  of  the  with  to  avoided  because  specimen  required  were  specimen  difficult  some  make  opposite  are  recognized,  of  to  contacts  'ohmic'  the  choose  contact  voltage  Including  contacts  of  the  negligible  an  a probe  to  is  for  properties  necessary  always  two  measured  It  measure  the  a long  necessity  circuit  electrical  contacts  to  'ohmic'  'ohmic'  to  One may  otherwise  ohmic b u l k  the  eliminate  without  ohmic b u l k  a good  Hence,  near  to  contact  study  the  because  Such good  separated  difficult  semiconductor.  s p e c i m e n ^so s h o r t specimen  TECHNIQUES  of  were  the  result the  due  high  decided  could  to  be  upon  alloyed  1  I  O  <  00  o o X  m  <  I ON O O X  <  1  1  i.  o o  o  O O  X  <  o  *  I  8 o  X  <  X  <  •  •  F i g u r e 3.1 Equlpotential configuration distorted by side-arm probe  7  • •  v>>  to  o  o  o  <  X"  TV  o  o  c5  o  X  <  <  <  o  o <  ON  c$  o  p o  x*  x  <  -o  o  X  00  o o X  vO  o q  X <  *  •  \  \. 1  \ . -. 1  I  75 u V 50 u V -  75 uV 50 UV  30 uV 20 UV -  30 UV 20 uv  10 u v -  10 uV  A l f o i l l e n g t h = 100 cm Probe l e n g t h = 20 cm Current = 500 mA  to  the  which  ends  of  the  side-arras.  equlpotential  was  probe,  a probe-analog  foil.  The  foil  projection ends  of  was  being  Since  there  sampled  in  experiment  cut  into  was  a long  simulating a potential  the  s t r i p were clamped passed  was  co p l o t  equlpotential  used  this  the  probe-analog  probes  measured  through  experiment  the  strip  it  the  equlpotential  on  with a the  A d-c  (Figure  on the  aluminum  The  constant VTVM  probe  From  3.1).  the  plane  the  centre.  foil.  that  to  perpendicular  and a  found  as  by  thin  plates  lines  was  specimen  near  i n brass  5 0 0 mA, w a s  the  some d o u b t  performed  probe  current,  was  side-arm  bisecting  the  side-arm.  3.2  Preparation of Both  a  boule  boule  of  the  on a w i r e  wafer  Probe  saw. and  and  then was  obtained  that  the of  the  is  photographic  plate-glass  dry  overnight.  surface  of  the  wafer  was  subsequent  Ultrasonic  It  1954).  were  adhering  ultrasonic  cut  i n rough  made from and  made  specimen  was  was  Eastman 9 1 0  Important the  cutting  G r i n d e r the  made  to  glass  the  fine  Point resis-  of  the  is  26.3  glued  to  Adhesive ensure plate  operation  wafer  the  rhodium-contact  Each wafer  to  to  from  Next,  Atomic Four  The r e s i s t i v i t y  from which the  was  were  was  abraded  corrections  w i t h Kodak  Impact  slice  on a B a i r d  2 3 . 5 ohm-cms.  to  Raytheon  measured  wafer  allowed  a  thesis  i n de-ionized water.  copper-contact  made  the  was  rinsed  (Valdes  s p e c i m e n was  during  in this  First,  slice  and s u i t a b l e  from which  ohm-cms  This  resistivity  value  reported  germanium.  slurry  Assembly  tivity wafer  specimens  n-type  carborundum  Specimens  may  on  and  the  entire  because the  either  become  T csi  - o.soo (a)  Copper-contact specimen ( t h i c k n e s s = O.865)  o 00 <*•  -M0-  8.0 90  o  4.881  (b)  a*  Rhodium-contact specimen  in  1/  4.890 —  _i  0.224  —5". 48 3  Figure  3.2  Geometry  of  Specimens  S c a l e : 20 mm = 1 mm ( 2 0 x a c t u a l size) ( t h e numbers a r e l n m i l l i m e t e r s )  Sideview  34  pitted  where  the  surface  was  may c r a c k .  The  specimens  on  saw  (Figure  the  wire  The by  They  were  water,  cleaned  etched  the  specimens  0.6%  antimony  the  ends  The  alloying  to  be  of  to  a  were  used  with  the  were  kept  with  acetone,  the  in  any  lt  etched,  alloyed  performed  by  placing  the  centre  just  bring  the  oxide  form  was  then  eutectic  by  the  germanium  coll  was  followed have  water.  The  was  kept  slowly by  formed  specimens  the  specimen  millimeter the  wire and  portion about  in  the  contact germanium  associated at  on  the the  cooled  a quick on  5  to  Micro-manipulators  all  melted  with  specimens.  temperature,  wire  cleaning  wire  the  of  C.  After  atmosphere  heating  Ge-Au-Sb.  that  As p r e s s u r e  heating  coil  150°  again.  portion of  (antimony-doped)  clean and  and  plate  de-ionized  gold  end of  the  glass  in  rinsed  paper,  'ohmic'  system  gold  w h i c h may  in de-lonlzed  the  Provided  procedure  lt  approximately  rinsed  and  a heating  length  phase  wire  the  of  above  the  or  final  from the  to  peroxide,  the  immediately.  remove  heating  to  scrupulously  This  separated  and  three  germanium,  rinse  was  germanium.  ture.  by  rinsed,  probes  glass  their  in a nitrogen  the to  then  w i t h hydrogen  and 3 m i l l i m e t e r  of  into  doping  was  C,  cut  the  filter  the  356°  then  to  had d r i e d on c l e a n  temperature  contact  were  adhesive  i n CP-4,  alloyed  diameter  the  were  adhering  3.2a,b).  specimens  decomposing  not  the  dip  point wire to  of towards  room  i n CP-4  specimen  were  tools  then  and  ready  temperato a for  electroplating.  Turner produced  (1959)  rectifying  found  contacts  all  plated  on n - t y p e  metals  germanium  except  antimony  and ohmic  contacts  on p-type  germanium.  rectifying  contacts  rhodium were specimens.  as  only  end.  gradually  that  the  the  The  it  15 m i n u t e s ,  then  plated  5 minutes.  thinner  of  contacts  c o u l d be  the  these  metallic  measured  Figure  were  to  as  scraped easily  mA f o r mA f o r  as  in a  small  end  into  the  through  contact  was  45 m i n u t e s .  at  off  fairly  lead  c o u l d not  be  mA f o r  markedly. easily  contacts scraped  specimen  on the  so  end  plated The  a  to  low v a l u e  at  of mA  0.1  rhodium mA  0.10  32-power  rhodium p l a t i n g  0.02  differed  plating  the  and at  examination under the  metallic  made c o n t a c t  45 m i n u t e s  where  by  rhodium p l a t i n g  f i r m l y and e v e n l y  copper  the  fluoborate  held  passing  Since  and  formed  from an i n i t i a l l y  resumed  contacts  appeared  2 hours. Whereas  w i t h the  copper  thumb  c o u l d be  wiped  off  by  even  The  nail  off),  using  a  point.  Finally, then  0.5  patches  rhodium contact  sharp  The  0.05  p l a t i n g was  not  copper  electroplated  a meniscus  density  But because  tenacity  (although  at  at  revealed  the  up u n t i l  would deposit  for  microscope  be  dipping Its  up s t a r t i n g  germanium f i l a m e n t .  for  by  current  built  contact  was  to  Each specimen,  the  contact  metals  opposite.  thesis,  Sigmund Cohn Mfg. Co.  electroplated  and d r a w i n g  was  ln this  the  w i t h H a r n s h a w C h e m i c a l Company c o p p e r  solution the  Just  and r h o d i u m c o n t a c t s  respeccively.  was  desired  the  The c o p p e r  s o l u t i o n and  solution clamp,  were  chosen  electroplating plating  Antimony behaved  the  on the  3.2a,b g i v e s  and c o p p e r - c o n t a c t  physical  Carl these  Zeiss  dimensions  the  specimens  Measuring Microscope  measurements  specimens.  of  for  the  (0 -  were 50).  rhodium-contact  (a)  D-C m e a s u r e m e n t  of  contact  characteristic  1  V  X  Dual beam 'scope (different i a l Inputs)  Pulse Generator  ^^yVW10012-  Vertical  —o  cru  Vertical  -o  (b)  P u l s e measurement  Figure  3.3  of  Circuits for  contact  I-V  characteristic  measurements  2  36  Circuits  3.3  for  D-C  3.31  The holder area the  Measurements specimens  prepared  under  pressure  contact,  study.  contact  and  particularly  scraped  off.  The  Soft the  the  lucite  copper  the  s p e c i m e n w o u l d be  paraffin  o i l  (white,  was  contained  temperature oil  bath  each the  bath  was  gold second  run  through  extended the  shielding  from the  10  rheostats  mA.  contact it  taken  specimen the  was  lead  plastic  from c o a x i a l  were  inadvertently  that  all  o i l bath  four  of  125/135).  d o w n by going  w i t h an  constant of  the  two  to  to  screws,  the  insulating  This  being shielding  Junction box.  soldered  the  stability  tubing before  cables.  This  Co.  For mechanical  fastened  the  metal  temperature  were c o a t e d  to  the  be  on  between  Instrument  C.  a copper  holder  leads  the  constant  run through  that  viscosity  23.4°  contact  placed  so  specimen  Within  A m p h e n o l UHF c o -  connectors.  The d - c with  was  leads  to  Bayley  The  fastening  so  designed  domestic,  specimen  copper  lead  J u n c t i o n box  axial  also  The  and each  was  at  was  would not  exposed  ( M o d e l 134).  the  foil  one,  in a lucite  pressure  contact  in a light-tight  from  screw  Junction box. film  light,  a r b i t r a r i l y set  lead  lead  holder  placed  bronze  area  of  oil  were  which provided a phosphor  contact  sides  Measurements  I-V  was  values  current  which  The c u r r e n t specimen found caused  permitted was  a n d 10  in practice an  was  not  s u p p l i e d from a b a t t e r y a range  allowed  mA i n t h e that  irreversible  to  of  current  exceed  change  to  1 jxA  place  in  of the  to  copper-  specimen  much i n e x c e s s take  series  1 mA l n t h e  rhodium-contact  currents  from  in  for  these contact.  (a)  (b)  Pulse  Pulse  forms  forms  Figure  3.4  —  --  forward d i r e c t i o n ( I = 1.4 mA, V = 250  reverse direction ( I =-8.7 JJA, V = -100  Typical  Pulse  Forms  mV)_rhodium-contact  mV)-rhodium-eontact  37 The  voltage  (Hewlett of  the  circuit.  obtained the  I-V  was  readings,  After  the  were  for  412A).  Figure  increase  increase  checked,as  the  the  the  for  the  the  pulse  operated  at  rheostats  was  changed.  forms  oscilloscope,  at  the  flat  technique  checked,  per the  portion of  current  the  pulse  ln  d-c  used  had  not  and  reverse  the  voltage  Increased  changed.  polnt-  current  the  readings,  points  to  ensure  that  procedure  and c u r r e n t  voltage This  directions.  was  measurements  display  was  the  on the  decreased, was  to  502  current  pulse  the  and  were  taken had  similar  taken  I-V curve  ensure  the  was  followed.  were  followed  dual-  transients a  the  width.  readings  I-V curve  readings  except  generator  was  were  which replaced  the  The p u l s e  points  procedure  measurements  T e k t r o n i x type  measurements  and the  pulse  form a f t e r  as  the  the  take  reduced,  and v o l t a g e  To o b t a i n  as  taken  w i t h 500 j j s e c  3.4).  was  again,  ( M o d e l B7B)  to  3.3b).  second  d-c  the  (Figure  point-by-point voltage  used  (Figure  20 p u l s e s  the  out  schematic  take  A similar  the  i n the  and f o r  w h i c h was  On  died  the  Measurements  beam o s c i l l o s c o p e voltage  VTVM  direction.  i d e n t i c a l with those  and  voltage,  had been  Rutherford Pulse Generator  battery  a d-c  ln plotting  voltage  The c o n d i t i o n s u n d e r w h i c h made w e r e  by  shows  3.3a  the  voltage  had not  reverse  measured  followed  maximum r e a d i n g s  Pulse  3.32  was  to  characteristic  followed  contact  The p r o c e d u r e  I-V curve  voltage  etc.  the  Packard Model  by-point and  across  that  In the  The as  the  were the  contact  forward  Impedance Bridge  R  a-c 1  Figure  3.5  pF  T r a n s v e r s e a-c r e s i s t a n c e measurement of c o n d u c t i v i t y modulation  38  3.4  Transverse  The attempt the  A-C  Resistance  t r a n s v e r s e a-c  to d e t e r m i n e  his infra-red  to ensure  'ohmic' c o n t a c t s on  The  r e s i s t a n c e never current  varied  in  F i g u r e 3.5  of  the  A General at The  one  Radio  kilocycle  100  ohm  Voltmeter at  was  about  as  3%  to 20 ^uA.  the r e g i o n u n d e r  (Model 403A)  the  a-c  measured  11.6  Then,  at  at  I-V  one kXL  However, the  measure-  where as  the  the  the c i r c u i t  shown  t r a n s v e r s e a-c r e s i s t a n c e the r h o d i u m  Hewlett t h e a-c  contact.  measured t h e a-c r e s i s t a n c e  l o n g i t u d i n a l d-c  and  occur  a linear  from  t o measure t h e  the  occurs  measurements  t r a n s v e r s e d-c  I65OA Impedance B r i d g e  resistor  5 JuA.  ^uA  extraction  d i d not  r e l a t i o n was  more t h a n  -80  used and  or e x t r a c t i o n  I-V  i n an  absorption technique.  side-arms,  varied  from  side-arms  the  measured  to H a r r l c k ' s (1956)  free c a r r i e r  that i n j e c t i o n  ments were made.  r e s i s t a n c e was  whether i n j e c t i o n or  area contact analogously  with  Measurement  c u r r e n t was Packard  varied.  Transistor  c u r r e n t w h i c h was  kept  39  CHAPTER  4  E X P E R I M E N T A L R E S U L T S AND I N T E R P R E T A T I O N  4.1  Transverse The  measurement occurrence at  the  Initial  on t h e of  was  into  to  at  the  to  determine  rhodium  the  the  contact.  experimental  for  which  in  Section  electron  to  saturation  bulk the  of  transverse  extraction  the  level  of  validity  of  assuming  i n j e c t i o n was  current  values  of  ratio,  'od'  of  determine  minority  injection  injection being  an e x p l i c i t  to  resistance  semiconductor.  of  analysis  a-c  s p e c i m e n was  level  gives  high  required  place in  injection  at  order the  provisionally  assumed  because  the  for I-V  is  extrinsic  for  only  case  semiconductors  r e l a t i o n whereby  c a n be  (defined  carriers  Another  taking  lt  the  calculated  ' M , the 1  from  the  in  Section  4.2),  and  relation  between  transverse  the  current.  Figure  shows  4.1  R _ »  through  rhodium contact.  a  the  tivity  of  the  occurs  when  respect  to  c  the  specimen  the the  l o n g i t u d i n a l d-c It  semiconductor  occurs  direction.  The  when t h e  same  shows  is  contact  empirically  that  and t h e r e f o r e biased  (forward  variables  current,  demonstrates  increases  rhodium contact  extraction  graph  and  the  resistance,  log-log  the  I-V c h a r a c t e r i s t i c s  hole  experimental  the  theoretical  2.51  of  i n j e c t i o n or  High  the  the  purpose  determine  contact,  Measurement  rhodium-contact  either  contact  purpose  A-C R e s i s t a n c e  of  is  that  the  that  biased 4.1  (Figure  flowing conduc-  injection  positively  direction),  Figure  I,  with  and  that  i n the  reverse  plotted  on  4.2)  a-c  a  20  Figure  -f-  4.2  T r a n s v e r s e a-c r e s i s t a n c e of r h o d i u m - c o n t a c t specimen as a f u n c t i o n of l o n g i t u d i n a l current  (a)  I  (|iA)  (b)  I  (juA)  fi a-c (kA)  10 8  o  o  "0 -~  H  a-c  I -324  =3?  x  10'  -0  k S L  0„  2-r  I 1  H-H-  6 Note:  8  10  20  40  c u r r e n t , s c a l e f o r c u r v e (a) ( s q u a r e s ) r a n g e s from 1 t o f r o m 1 0 0 t o 1 0 , 0 0 0 jiA f o r c u r v e ( b ) (circles)  60  80  1 0 0 pA  100 and  (|1A)  40  R  A a  _ = 11.6 k_Q_ "° -o v 37 I  t o be n o t e d  resistance,  c  H _ , a  equals  b e c a u s e no d - c c u r r e n t a  transverse  ICA.  H  here  f o r 100 J J A  Is that  a-c  <  I < 10 mA  d-c r e s i s t a n c e , (£<3_ )t , c  < d-o>t  =  flows  d-c c u r r e n t  (4.1.1)  t h e t r a n s v e r s e a-c  the transverse  H  uA ^  ^?4 ° "  U  A subtle point  for 1 ^ 1  (4.1.2)  B  i n the transverse  direction.  e x i s t e d , then H _ = dV^/dlj. ^ a  where V* a n d 1^. a r e t h e d-c v o l t a g e t  c  and c u r r e n t  If  (^d-c^t  i n the transverse  direction. An relation Several  attempt  between R _ a  was made t o f i n d  i s dominant, t h a t  I t was assumed  one-dlmenslonal v a r i a t i o n of hole  longitudinal  distance,  x, |and t h a t  i n an e x p o n e n t i a l  manner.  ApU)  where  P  1  L • which  diffuse  y  Q  with  i n t o the  (4.1.3)  p  = VjLj/qDp  (4.1.4)  = Ap(0) exp ( x - L ) / L  (4.1.5)  = Ap{L) exp - y / L  (4.1.6)  Q  Ap(x) = p ( x ) - p  same l e t t e r  density  Thus,  0  Ap(0)  A  the holes  = Ap(0) exp ( x - x ) / L  Ap(x)  that  t h e s i d e - a r m s do n o t a f f e c t t h e  longitudinal  side-arms  expected  and t h e l o n g i t u d i n a l d-c c u r r e n t , I .  c  a s s u m p t i o n s were made h e r e .  diffusion  the t h e o r e t i c a l l y  and  stood  Ap  y  = p  y  p  - p .  f o r the length  Q  Note t h a t t h e  of the diode  l n the  41  theory of Chapter 2 Is purposely set  equal to the  distance  between the plane b i s e c t i n g the side-arms and the edge of space-charge the v o l t a g e of 'y'  r e g i o n In the semiconductor.  This was done  is  edge of the specimen.  of the side-arm I s ,  the  c o n d u c t i v i t y so that the Invoking equation  a-c  (4.1.2),  / dy/^A'  R  .(4.1.7)  1 + CjJ  7 * i»  *o-s where R _ 0  tance,  l  g  sectional  s  1 + C j J exp  (-1 /L S  ) P' '  i s the e q u i l i b r i u m value of the side-arm a-c is  the length of the s i d e - a r m , A' i s  area of the s i d e - a r m , (f i s  semiconductor,  Letter  The d i f f u s i o n of m i n o r i t y c a r r i e r s i n t o  the side-arm can modulate i t s  ( a-c>s =  theory  at that p l a n e .  the t r a n s v e r s e c o - o r d i n a t e whose o r i g i n l i e s at  resistance  since  V (which corresponds to that at x = L i n the  Chapter 2) was measured e x p e r i m e n t a l l y  the  J is  the  resis-  cross-  the c o n d u c t i v i t y of  the  the l o n g i t u d i n a l d-c c u r r e n t d e n s i t y ,  and  ftp-sA'qupU+^yLp  By using the E i n s t e i n r e l a t i o n and n o t i n g that Vq(l+b)L  fi  _ A'/l  0  s  =  s  ^o"  1  D  Ci = A rough computation was made f o r the value of C Y=  taking  1, q/kT = 40 v o l t " , (TQ = 0.04 ohm" c m " , and L 1  since and  1  it  the hole l i f e t i m e  1  1  of the m a t e r i a l i s  was taken that Dp = 50 cm  2  sec" , 1  it  P  b = 2,  = 0.07 cm  approximately  lO'^sec  was found that  as  42  ranged from 0 . 1 mA to 10 mA, the f a c t o r C^J  the c u r r e n t , I ,  ranged from 3 to 300 since J = I / A  where A, the area of  i s approximately 7 . 5 x 10  rhodium c o n t a c t ,  J  cm .  the  Also l n t h i s  c u r r e n t range, the f a c t o r C^J exp ( - 1 / L )  ranged from 0 . 5 to  50.  exceeds 1 mA that  B  Hence, i t  p  Is only when the c u r r e n t , I ,  the two f a c t o r s above are l a r g e compared with u n i t y so  that  e q u a t i o n (4.1.7) can be approximated by " W s  ^  W  _  (exp  0  7  >V  S  l /L s  (  p  - 1) t 4  i"b)i  A rough a n a l y s i s of the a-c r e s i s t a n c e of the p o r t i o n of the specimen would assume that the electric other.  ~  f  J x  a  c  r W [ P  l  i s the t r a n s v e r s e a-c v o l t a g e across the c e n t r e p o r t i o n  the specimen, t  is  the specimen t h i c k n e s s ,  l  c  is  the  distance  the c e n t r e p o r t i o n from one side-arm to the o t h e r , and  Xj^ and x  2  are the l o n g i t u d i n a l d i s t a n c e s  to the s i d e s of the s i d e - p r o b e s . p,  (4.1.2),  1 „  tva-o  where V _  the  Xazc  - —  across  transverse  Then the a-c c u r r e n t i s given by, u s i n g e q u a t i o n a-c  8 >  centre  f i e l d i s uniform from one side of the specimen to  T  of  -V ,  the hole d e n s i t y ,  N= n  e q u i l i b r i u m values of defined before. i s g i v e n by  Thus,  0  - p  0  from the rhodium contact  The other symbols such as the d i f f e r e n c e of the thermal  the e l e c t r o n and hole d e n s i t i e s , the a-c r e s i s t a n c e  have been  of the c e n t r e p o r t i o n  43  where  R -c 0  i  s  t  h  e  equilibrium value  (4.1.9)  1/Ro -c  >J +  0  of the centre  a-c  resistance  and tqft(l+b)Lp  ° Now,  used  C  =0.55  2  In the analysis  taken with x  (Figure  /x  volt  B  s  that  0  2  obtained  current  the total  from equations  s  A rough compared exceeds be  calculation to that  where  inversely  lt  was o b s e r v e d  approximately  x  1-  resistance are-  i n the current J  2  range  1/150 o h m " "  1  transverse  +  2  (4.1.10) a-c resistance  a-c resistance,  is valid,  since  R  so t h a t H  Thus,  (4.1.8).  resistance  is  = 2(fi In this  „)„,  current  a-c  reason  root  for this  i s inversely  c a n be seen  range  However,  proportional to  o f the l o n g i t u d i n a l current.  discrepancy  can  resistance  proportional to the longitudinal current.  t h e cube  small  when t h e c u r r e n t , I ,  the transverse  the resistance  is  ^ a-c^s  the centre  of the side-arms  that  of  exceeds  I/C2J  ~  R  that  / o  range  ( a-c)c  =  shows  (4.1'. 8)  is  possible  a-c  from e q u a t i o n  equation  exp  (4.1.7) a n d (4.1.9)  1 mA, t h e t o t a l  obtained  -  when t h e n u m e r i c a l  2  of C  (Ra-c)c Therefore,  X-L  = 0.0593 c m , t = 0.057 c m , a n d Hence,  3.2b).  In this  cm  - 1  0.1 mA t o 10 m A , t h e m i n i m u m v a l u e l/ o-c  -  0  of the side-arm  = O.0369 c m , x  1  = 0 . 1 4 cm  c  — m „  =  approximately,  values  l  2  exp  A  by e x a m i n i n g  a I-  Figure  (a)  (b)  (c)  35-  3500,  7000  4.4  D-C f o r w a r d characteristic of rhodium-contact specimen (scales (a), ( b ) , and correspond to curves (a), ( b ) , and ( c ) )  (c) /  / 3 0 - - 3000, 6000  25  2500, 5000  20  2000,  4000  15  1500,  3000  1 0 — 1000,  2000  500,  /O  1000  25 250 500  / °/  (a) (b) (c)  equation so  that  the l i f e t i m e  the d i f f u s i o n  injection not  If  (4.1.8).  level  remain  obtained  varies,  inversely The  from  length  level  then  injection  dinal  x 10  =  current  it  c a n be seen  to  high.  4.2  the previous JC  1 3  Figures characteristic teristic  of  3  4.3  a n d 4.4  a high voltage' to  the presence  parallel  the diode = I of  the e f f e c t s  of  B  these  relation  (4.1.4),  that  we f i n d  f o r the  changes  that  longitu-  from 3 to from  300,  moderate  Specimen  current-voltage The  charac-  equation  -1)  (4.2.1)  resistance,  In order  the contact  of  values  specimen.  ( e x p <*V  R .  by a  (4.1.11)  the d-c  a low s e r i e s  resistance,  characteristic  subtract  show  current.  of Rhodium-Contact  the rhodium-contact  does n o t obey  of  level  will  p  10 mA, J C ^ v a r i e s  Characteristic  I because  q(l+b)^  the i n j e c t i o n  Current-Voltage  and equation  numerical  mA t o  the  a-c r e s i s t a n c e  may b e e x a m i n e d  c m " . By n o t i n g  X  range,0.1 that  L , changes as  to the l o n g i t u d i n a l  f o r C,  Ap(0)  Ap(0) = 4.5  c a r r i e r s , c h a n g e s  the transverse  proportional of  the  for holes,  the expression  a n d by i n s e r t i n g  of  to obtain  proper,  associated  i t  is  R , and s  the  current  necessary  resistances  by the  relations  V, m  = V  T  I :fi  S  (4.2.2)  Figure  4.5  G r a p h i c a l analysis of c h a r a c t e r i s t i c f o r rhodium-contact specimen  45  I  m  = I - V/R  (4.2.3)  p  where I and V are the e x p e r i m e n t a l l y measured values of the c u r r e n t and  the v o l t a g e whereas I  v a l u e s of the c u r r e n t and procedure Ig_ »  immediately  are the c o r r e c t e d m  the v o l t a g e of the c o n t a c t .  This,  g i v e s the value of the s a t u r a t i o n c u r r e n t ,  In the r e v e r s e d i r e c t i o n .  r  and V m  &  Then f o r the values o f . t h e c u r -  r e n t , I , l n the r e v e r s e d i r e c t i o n , l o g ( I + I _ ) i s p l o t t e d ffl  m  a g a i n s t V. is o( .  ( F i g u r e 4.5).  The  initial  In order to o b t a i n  r  value of I  r  slope of the l i n e  f o r the forward d i r e c t i o n and  ffl  where, f o r v a l u e s of  ( F i g u r e 4.5).  The  and  f  Subsequently,  l o g (I + I _ ) s  the slope of the l i n e y i e l d s  f  o< . f  >> kT/q,  t  the  l i n e a r p o r t i o n i s extended  to i n t e r s e c t the v e r t i c a l a x i s and the i n t e r c e p t s  the  s  log I i s plotted against V  to I _ .  obtained  (4.2.1) f o r the forward d i r e c t i o n ( I _ 0 ,  of equation  curve becomes l i n e a r  s  corresponds  i s plotted against  V"  m  T h e r e f o r e , the c o r r e c t e d  c u r r e n t - v o l t a g e c h a r a c t e r i s t i c has been reduced to the f o u r parameters c< _, I_ „, o< , and I . The values of these I ' s-i» r' s-r parameters o b t a i n e d f o r the rhodium-contact  specimen by the  d-c  measurement are: o<  = 34.5  volt"  1  f  <=^  = 31.2  volt"  1  =  9.0  UA  =  8.8  r  Ig_ I  f  s-r  These values of o ( value of kT/q values of oi  /  uA  are l e s s than the u s u a l l y  i n the o r d i n a r y diode e q u a t i o n . l e s s than or equal to kT/q  expected  However,  have been  observed  Figure  4.7  Pulse forward characteristic copper-contact specimen ( v o l t a g e s c a l e s (a) and (b) c o r r e s p o n d t o c u r v e s (a) and  of  (b))  46  frequently  before  by  others  and  Is  not  s u r p r i s i n g here.  The  p  saturation  current  rhodium-contact mA/cm  0.95 Since  the  reverse o(  X  f  specimen  reported  2  by  saturation  c<  it  and n e g a t i v e  appears  this  experiment  the  parameters  I  (4.6) and  manner as  kT/q  so  was  4.2.  the  V^ i n the  that  the  copper-contact  done  The  of  closely  the  forward  equal  o( f o r  and  the with  (1955)» and »  since  satisfies both  equation  positive  for  only  is  that  of  of  the  specimen  the  I  measured  in  this  against  not V  by  i n exactly  rhodium-contact  d i r e c t i o n do  log  Specimen  current-voltage  analyzed  difference  forward  curve  Copper-Contact  (4.7) show  The c h a r a c t e r i s t i c  method.  of  and  Characteristic  pulse  values  from  rhodium contact  for  Section  agrees  for  and S t i c k l e r  approximately  characteristic  in  Schwarz,  observed  currents.  Figures  same  mA/cm  1.1  obtained  are  that  same  Current-Voltage  4.3  in  about  current  measurements  r >  of  Borneman,  I-V  with the  (4.2.1)  density  case  is  extend  does not  the specimen  that much  the  the past  become  m linear.  Therefore,  straight  line  values  of  the  of  values  log  (I  I  I _ s  + I _f)  were  f  against  s  parameters  I  of  obtained  are  o<  f  = 24.7  volt"  CX  r  = 10.0  VOlt"  s-f  =  s-r  =  M  9 0  1  6  ?  A  V  k  added V  1  1  m  was  to  I  until  obtained.  a  best The  10  3  V  20  (mV)—reverse  Figure  4.8  30 -200  Graphical analysis for copper-contact  40 -150  V  f f l  (mV)--forward  -100  of c h a r a c t e r i s t i c specimen  -50  0  4?  The c o p p e r appreciably values are  less  obtained  not  from  than  forward  values  than the  reverse  Similarly, the  has  q/kT  from the  equal.  obtained  contact  the  of  o< m u c h m o r e  rhodium contact  and f o r w a r d  values  of  and r e v e r s e  I-V  the  and  the  measurements  saturation  characteristics  current  are  differ-  2 ent.  The r e v e r s e  saturation current  density  of  15.8  mA/cm  and  2 •the  forward  saturation current  density  of  8.5  mA/cm  agree  in  order  2 of  magnitude  w i t h 9.2  Stickler  (1955).  contact,  the  contact 4.4  Comparison of  copper hole  the  same  by  used  ratio.  Borneman,  behaviour  of  diode  equation  values  with  obtained  to o b t a i n v a l u e s  Schwarz,  the  characteristics  Experimental Results  were  current  the  and r e v e r s e  experimental  contacts  reported  unlike  satisfy  The  to  But,  forward  do n o t  mA/cm  and  rhodium  of  the  copper  numerically.  Theory for  of  the  r h o d i u m and  M, the  The f o r w a r d p a r a m e t e r s  o<^,  electron were  Inserted  Into (4.4.1)  where-  a = q/o( kT.  taking  the  =  1900  values  indicated the  of  the  cmVvolt-sec.  (2.51.28) i s  for  The v a l u e  f  v a l i d only  also  current  assumption,  from  the  less  <£p/n  0  »  b = JJ /Mr) n  w  a  obtained  s  m o b i l i t i e s at  the  temperature  This  for  M =  for  relation  high  injection.  transverse  than 1,  of  1 mA, is  lt  a-c  r/ p J  the  f  However,  resistance  appears  Invalid for  J  that  the  by  of  r  o  m  as  the  equation was  measurement high  injection  rhodium contact  since  48  M  = -28.1 I s I n t h e r e g i o n where p h y s i c a l l y  relations value of  result.  Moreover,  oC^. = 34.5 v o l t  l y i n g b e t w e e n q/2kT and q/kT,  Injection  between 0.13 mA/cm  low  levels.  2  a n d 130 mA/cm  The  less  Injection  occurs  for  2  0< = 24.7 v o l t "  1  f  i s signifi-  indicates  that  b e t w e e n 10 pk and 500 ]xk ( J  high  between  2  1 mA/cm  and 50 mA/cm  possible  I-V r e l a t i o n s .  reveals  shows i t i s p o s s i b l e t o  M = 4.94 f u r t h e r I  this <  however, a p p e a r s t o s a t i s f y t h e  c r i t e r i o n since  t h a n q/kT.  for  level of injection.  copper contact,  injection  the l e v e l  ) I s between t h e h i g h and  t h e above p r o c e d u r e  determine the approximate  cantly  2  i sa  I s between 1 JAA a n d 1 mA  T h u s , a l t h o u g h M c a n n o t be o b t a i n e d  Intermediate case,  *  i t appears that  i n t h e e x p e r i m e n t where I  (J  high  since  i m p o s s i b l e I-V  that  diffusion  ) since  the electron  current.  this  value  From S e c t i o n drift  leads  to physically  2.51, t h i s  current  value  of M  dominates the e l e c t r o n  49  CHAPTER  ji  CONCLUSIONS The work  In this  metal-semiconductor  contacts.  zero-eleotron-current Section  theory.  The o u t s t a n d i n g  into in  into  on the  feature  of  E x p e r i m e n t s were non-zero-electron-current measurements verse  a-c  relation  to  on the  resistance,  However,  carrier density, reduce  data  the  snowed  the  then  (2.2.3)  of  a =  can  take  electron  moderate  with  to  injection  that  high  for  injection  (3b-M)/(b-M) any  value  current to  a-c  ratio.  check  gave  a  trans-  longitudinal current, the  expected  lifetime  is  I,  theoretical a function  relation  Injection  the  resistance  specimen  theoretical  high  is  i n an attempt  Rough c a l c u l a t i o n s to  of  existing  hole  Transverse  carrier the  theory  arbitrary  and f o r  value  versus  agree  the  section  rhodium-contact  discrepancy.  that  the  theory.  a  if  to  this  performed  R -c»  which d i d not  relation. the  made  of  M, the  2.4  the  non-zero-electron-current  semiconductors  equation  value  extended  extended  the  semiconductors  modified diode  depending  is  develops  intrinsic  extrinsic  the  2.5  has  In Section  theory  levels.  injection  thesis  may  made  change  from  occurred  of  at  this the  2 rhodium contact  for  current  The e x p e r i m e n t a l specimen contact was 50  true  Indicated for for  that  J between the  high  0.1  s p e c i m e n by  J between  parameters  and  for  10  the  100  occur  mA/cm , whereas  specimen  2  for  and  1000  mA/cm .  rhodium-contact  i n j e c t i o n d i d not  copper-contact  mA/cm . • T h e r e f o r e ,  contact  density  at  the  J between  lt  was  f o u n d M = 4.94  for  the  using  the  high injection  equation  that converse 1  and  copper  (2.51.28).  50  Thus,  metal-semiconductor  Injection  by  ratio,  from the  measured  Finally,  it  M,  between There  calculating  contacts  the  Is  theory  ranges.  related  to  increasing more  of  majority  that  carriers  and  Houghton  (1954)  the it  another  necessary  have  equilibrium  value  Is  whether  doubtful  In ductor which nation  the  can value  s h o u l d be  dependent  M.  presented  in this  rectifying  the  be  To  was  M at  considered  thesis  this  hole  since  must  local  that  the Is  be  assumed  the  the  to  contact.  in  X  n-type ln  theory  the and  assumptions,  its  thermal  side-arms,  but  lt  experiments.  of  metal-semiconthe  mechanism  Furthermore,  implies  stressed  of  Banbury  on  negligible  determining  an ohmic  concentration  contacts  equal  with  neutrality,  theoretical  In the  this  diminish  diminution  between  the  is  contribution  expected  assumed  exist  a l l  Increased.  understanding  the  (which  charge  the  may  throughout  space  current  reasons.  to  bisecting  by  hole  to  density  true  to  expected  current  satisfy  future  it  so  high  several  ratio,  discrepancy  approached  of  Finally,  one.  of  for  through point  plane  this  due  w h i c h was  the  conclusion,  contacts sets  at  total  current  results. to  is  for  a discrepancy  constant  injected,  observed  source  that  data  increases  Recombination,  is  noted  since,  the  have  forward  experimental Is  to  the  injection  are  also  majority  theory,  of  M remains  current,  carriers  germanium.  values  (2.51.29))  minority carriers  increased  electron  the  equation  the  with  of  and e x p e r i m e n t a l  forward  as  value  the  In fact,  M by  checked  s h o u l d be  no g u a r a n t e e  voltage  can be  that  a  recombi-  positionthe  contact  theory  opposite  the  51  BIBLIOGRAPHY Banbury,  P . C . and Houghton,  Bardeen,  J .  Bardeen,  J . and B r a t t a l n ,  J .  R e v . 21,  Phys.  Physlca  20, 1050 U954)  717-727 (1947)  W.H.  Phys.  R e v . 2k, 230-231 a n d  231-232 (1948) *Bethe,  H.A.  Borneman,  Massachusetts Report  E . H . , Schwarz,  Esaki,  B.  Tech.  L.  Phys.  Phys.  Radiation  andS t i c k l e r ,  (USSR)  Rev. 102,  J . J .  Lab.  J . Appl.  2, 87-95 (1938)  603-604 (1958)  102, H 7 3 - H 8 1  N.J.  Phys.  Rev.  Harrick,  N.J.  Phys.  Rev.H  L.  (1942)  26, 1021-1028 (1955)  Harrick,  Heijne,  of Tech.,  R.F.,  Phys. "Davydov,  Inst,  43/12  i,  (1956)  876-882 (1959)  Thesis,  U n i v e r s i t y o f A m s t e r d a m (i960); (Philips R e s e a r c h R e p t s . S u p p l . N o . 4, (I96D) R e c t i f y i n g Semiconductor Contacts. Oxford U n i v . P r e s s , L o n d o n (1957)  Henisch,  H.K.  Herring,  C. and N i c h o l s ,  M.H.  R e v . Mod. Phys.  21, 185-270  (1949) Holm,  R.  Kroger,  J . Appl. F.A.,  Phys.  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